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11institutetext: J. Schütz 22institutetext: Faculty of Sciences, Universiteit
Hasselt,
Agoralaan Gebouw D, BE-3590 Diepenbeek, Belgium
22email<EMAIL_ADDRESS>33institutetext: D. C.
Seal44institutetext: United States Naval Academy, Department of Mathematics,
572C Holloway Road, Annapolis, MD 21402, USA
44email<EMAIL_ADDRESS>55institutetext: J. Zeifang 66institutetext: Institute
of Aerodynamics and Gas Dynamics, University of Stuttgart,
Pfaffenwaldring 21, DE-70569 Stuttgart, Germany
66email<EMAIL_ADDRESS>
# Parallel-in-time high-order multiderivative IMEX solvers
Jochen Schütz David C. Seal Jonas Zeifang
(Received: date / Accepted: date)
###### Abstract
In this work, we present a novel class of parallelizable high-order time
integration schemes for the approximate solution of additive ODEs. The methods
achieve high order through a combination of a suitable quadrature formula
involving multiple derivatives of the ODE’s right-hand side and a predictor-
corrector ansatz. The latter approach is designed in such a way that
parallelism in time is made possible. We present thorough analysis as well as
numerical results that showcase scaling opportunities of methods from this
class of solvers.
###### Keywords:
Multiderivative IMEX parallelism in time time integration
## 1 Introduction
In this work, we consider numerical approximations of nonlinear differential
equations of the form
$\displaystyle w^{\prime}(t)$
$\displaystyle=\Phi(w)\equiv\Phi_{\text{I}}(w)+\Phi_{\text{E}}(w),\quad
t\in(0,T_{end}),$ (1) $\displaystyle w(0)$ $\displaystyle=w_{0}.$
It is assumed that the right-hand side $\Phi$ is split into the contributions
$\Phi_{\text{I}}$ (‘stiff’) and $\Phi_{\text{E}}$ (‘non-stiff’), to account
for different scales in the solution. The rationale is that the term
$\Phi_{\text{I}}$ will be treated implicitly to obtain a stable method, while
the term $\Phi_{\text{E}}$ will be treated explicitly, typically for
efficiency reasons by reducing the complexity of the algebraic solves required
to address the implicit terms per time step.
Differential equations that can be put into this form arise frequently, often
resulting from the discretization of singularly perturbed partial differential
equations. Some examples include problems in meteorology giraldo2010semi ,
geophysics BispenIMEXSWE , aerodynamics BoQiuRussoXiong19 ; HaJiLi12 ;
RSIMEXFullEuler , molecular dynamics filbet2010class and many more. The list
is by no means exhaustive. Here, we assume a splitting has already been
performed. Our goal is to develop a high-order implicit-explicit (IMEX) method
that lends itself to time-parallelism.
At the core of this work is the novel discretization method of Eq. 1 developed
in SealSchuetz19 . To the best of the authors’ knowledge, this method is,
together with the very recent publication AbdiHojjati2020 , the first one to
combine the IMEX (implicit/explicit, see, e.g., Ascher1997 ; Ascher1995 ;
Bos09 ; pareschi2000implicit ; christopher2001additive ) and the
multiderivative paradigms AiOk2019 ; TC10 ; Seal2015b ; KastlungerWanner1972 ;
TurTur2017 ; Seal13 . In this context, the term _multiderivative_ refers to
the fact that higher derivatives of the unknown solution $w$ are used in the
numerical method. While more established methods (Runge-Kutta, Adams methods,
backward differentiation formulas (BDF) and the like) rely on only evaluating
$\Phi_{\text{E}}$ and $\Phi_{\text{I}}$, multiderivative methods also take the
temporal derivatives into account. To be more precise, in this work, we use
two derivatives, i.e., also the quantities111The dot here ($\cdot$) refers to
the time derivative $d/dt$, whereas the prime (′) refers to the Jacobian of
the vector valued $\Phi_{\text{I}}$ and $\Phi_{\text{E}}$.
$\displaystyle\overset{\boldsymbol{.}}{\Phi}_{\text{I}}(w):=\Phi_{\text{I}}^{\prime}(w)\Phi(w),\qquad\text{and}\qquad\overset{\boldsymbol{.}}{\Phi}_{\text{E}}(w):=\Phi_{\text{E}}^{\prime}(w)\Phi(w)$
are taken into account. Note that there holds
$w^{\prime\prime}(t)=\overset{\boldsymbol{.}}{\Phi}(w)\equiv\overset{\boldsymbol{.}}{\Phi}_{\text{I}}(w)+\overset{\boldsymbol{.}}{\Phi}_{\text{E}}(w)$.
This has of course the potential to increase the order of consistency without
adding more stages or steps, respectively. The limitation to two derivatives
is only for the ease of presentation. The method developed in SealSchuetz19
has already been extended to cope with higher derivatives alex2020highorder ;
and also in the context of this present work, this is easily possible.
There are three unique features to consider for the class of methods presented
here:
* •
The methods we present use a predictor-corrector strategy for increasing the
overall order of accuracy. This means that in each timestep, a predicted value
(we will indicate values corresponding to the predictor with $[0]$), as well
as several corrector steps (indicated with $[k]$, $1\leq k\leq k_{\max}$) are
computed. Here, $k_{\max}$ is a user-defined parameter that defines the
maximum number of corrections in a single time step. Similiar to deferred
correction, or DC-type methods (cf. CausleySeal19 ; DuttGreenRokh00 ;
OngSpiteri and the references therein), the corrected steps pick up an order
of accuracy each time, until some maximum accuracy is reached based on the
underlying quadrature rule.
* •
The methods we construct are derived from two-derivative Runge-Kutta methods,
however, they are modified in such a way that they can be applied to any
implicit-explicit (IMEX) splitting
$\Phi(w)=\Phi_{\text{I}}(w)+\Phi_{\text{E}}(w)$ of the ODE. This ultimately
leads to the fact that the methods are not one-step, but two-step methods,
with a very mild dependency on the previous step. As there are multiple stages
involved, this leads to general linear methods with associated convergence
theory Butcher1966 , but we have higher derivatives of the unknown involved in
our analysis.
* •
The methods we produce can all leverage parallel-in-time implementations to
leverage distributed or multicore computer architectures.
The ability to parallelize an ODE solver in time has become more and more
important in recent years with the number of processors available growing
constantly. Because of the causality principle, parallelization in time poses
different challenges then parallelization in space. For an overview of
temporally parallel methods, we refer to Gander’s work Gander2015 and the
very recent overview article by Ong and Schroder OngSchroder2020 . Also the
website PintURL is a beautiful source of information on the topic. Temporally
parallel methods can be cast into different classes. The scheme we present in
this work uses the concept of pipelining, i.e., the information and data flow
is directed in such a way that the correction iterations can be put on
different processors. We have been inspired by the RIDC (revisionist integral
deferred correction) schemes ChriOng11 ; ChriOng10 ; ChriOng2015 , however,
similar ideas have been around for a long time, see MirankerLiniger1967 . The
essence of pipelining is that, given one has enough processors at one’s
disposal, the higher order achieved through the corrections can be obtained in
roughly the same wallclock-time that it takes to compute the lower-order
predictor. In our case, the predictor’s convergence order is two, while the
corrections can increase this order up to the order of the underlying Runge-
Kutta quadrature. In our examples, this is four, six and eight, respectively,
but conceptually, the order is not limited to this.
The paper is organized as follows: In Sec. 2, we describe the underlying
algorithm in detail and introduce the necessary notation. Thereafter, in Sec.
3, we analyze stability and convergence of the method through writing it in a
one-step fashion. The method is inherently time-parallel, which is laid out in
Sec. 4. Also, two alternative approaches are shown and later compared. In Sec.
5 we show numerical results for well-known test cases. We find some
improvements that one can make to the numerical algorithm described earlier;
these are shown in Sec. 6. Subsequently, scaling results are presented. As
usual, the last section, Sec. 7 offers conclusion and outlook.
## 2 Numerical algorithm
We start with setting the notation that is used in this work. For expository
purposes, we work with a fixed timestep $\Delta t$, and therefore with a fixed
total number of timesteps, denoted by $N$, we must have
$\displaystyle\Delta t:=\frac{T_{end}}{N}.$
The discrete time levels are defined as
$\displaystyle t^{n}:=n\Delta t,\qquad 0\leq n\leq N.$
Note that the uniform timestep we assume here is not necessary in practical
computations, but it simplifies the analysis. Throughout the whole
publication, we assume that $\Phi$ and the split functions $\Phi_{\text{I}}$,
$\Phi_{\text{E}}$ and the solution $w$ are sufficiently smooth to warrant
picking up high-order accuracy from the numerical method. Formally, we make
the following assumption:
###### Assumption 1
All occuring functions $\Phi$, $\Phi_{\text{I}}$, $\Phi_{\text{E}}$ and their
temporal derivatives
$\overset{\boldsymbol{.}}{\Phi},\overset{\boldsymbol{.}}{\Phi}_{\text{I}}$,
$\overset{\boldsymbol{.}}{\Phi}_{\text{E}}$ are assumed to be smooth and
Lipschitz continuous.
In this work, we are interested in multiderivative Runge-Kutta methods. These
are a relatively straightforward extension of well-known Runge-Kutta methods
that include extra derivatives of the right hand side function (Seal13, ,
Definition 1)). To set notation, we include a full definition of a _two_
-derivative Runge-Kutta method. Higher derivatives can be included by
including more terms in the expansion for each of the stages.
###### Definition 1
Let $w^{n}$ denote an approximate solution to $w$ at point $t^{n}$, and let
$w^{n,l}$, $1\leq l\leq s$ denote the stage values to be computed. A _two-
derivative Runge-Kutta method_ is any method that can be cast as
$\displaystyle w^{n,l}:=w^{n}+\Delta
t\sum_{j=1}^{s}B_{lj}^{(1)}\Phi(w^{n,j})+\Delta
t^{2}\sum_{j=1}^{s}B_{lj}^{(2)}\overset{\boldsymbol{.}}{\Phi}(w^{n,j}),\qquad
1\leq l\leq s,$
and update given by
$w^{n+1}=w^{n}+\Delta t\sum_{j=1}^{s}b_{j}^{(1)}\Phi(w^{n,j})+\Delta
t^{2}\sum_{j=1}^{s}b^{(2)}_{j}\overset{\boldsymbol{.}}{\Phi}(w^{n,j}),$
where the $B^{(1)}$, $B^{(2)}$, $b^{(1)}$, and $b^{(2)}$ are the Butcher
tableaux that define the scheme. We say the method is _globally stiffly
accurate_ if $b^{(1)}_{j}=B^{(1)}_{sj}$ and $b^{(2)}_{j}=B^{(2)}_{sj}$, i.e.,
the last stage defines the update:
$\displaystyle w^{n+1}:=w^{n,s}.$
Note that this is a fully coupled system of (potentially nonlinear) equations
whenever the tableaux are not lower triangular. In this form, the Runge-Kutta
method is hence typically not suited for high-dimensional ODEs, nonetheless,
they can serve as excellent background methods for creating more efficient
solvers.
Of particular note are the two-derivative Runge-Kutta _collocation_ methods.
These methods are derived by fitting a Hermite-Birkhoff polynomial interpolant
through the time instances given by $c\Delta t$ and integrating the result. In
this work, we use equidistant collocation points, which implies:
* •
the method is of order $q$, with $q=2s$ being twice the number of stages
(hence, number of elements in $c$);
* •
the stage order is also $q$;
* •
the last stage is equal to the update step, which corresponds to the globally
stiffly accurate or first-same-as-last property already known in standard
Runge-Kutta methods BOSCARINO201760 ; HaiWan1 ; HaiWan and will have an
impact on asymptotic properties of the method SKN15 .
###### Example 1 (Two-derivative Hermite-Birkhoff collocation methods)
In this work, we use the following two-derivative Runge-Kutta methods:
* •
A fourth-order method ($q=4$) with two stages ($s=2$, one being fully
explicit), which exactly corresponds to the method used in SealSchuetz19 :
$\displaystyle c=\begin{pmatrix}0\\\ 1\end{pmatrix},\quad
B^{(1)}=\begin{pmatrix}0&0\\\\[5.0pt]
\frac{1}{2}&\frac{1}{2}\end{pmatrix},\quad
B^{(2)}=\begin{pmatrix}0&0\\\\[5.0pt]
\frac{1}{12}&\frac{-1}{12}\end{pmatrix}.$ (2)
* •
A sixth-order method ($q=6$) with three stages ($s=3$, one being fully
explicit), as also used in SSJ2017 :
$\displaystyle c=\begin{pmatrix}0\\\ \frac{1}{2}\\\ 1\end{pmatrix},\quad
B^{(1)}=\begin{pmatrix}0&0&0\\\\[5.0pt]
\frac{101}{480}&\frac{8}{30}&\frac{55}{2400}\\\\[5.0pt]
\frac{7}{30}&\frac{16}{30}&\frac{7}{30}\\\ \end{pmatrix},\quad
B^{(2)}=\begin{pmatrix}0&0&0\\\\[5.0pt]
\frac{65}{4800}&-\frac{25}{600}&-\frac{25}{8000}\\\\[5.0pt]
\frac{5}{300}&0&-\frac{5}{300}\end{pmatrix}.$ (3)
* •
An eigth-order method ($q=8$) with four stages ($s=4$, one being fully
explicit):
$\displaystyle c=\begin{pmatrix}0\\\\[5.0pt] \frac{1}{3}\\\\[5.0pt]
\frac{2}{3}\\\\[5.0pt] 1\end{pmatrix},\quad B^{(1)}$
$\displaystyle=\begin{pmatrix}0&0&0&0\\\\[5.0pt]
\frac{6893}{54432}&\frac{313}{2016}&\frac{89}{2016}&\frac{397}{54432}\\\\[5.0pt]
\frac{223}{1701}&\frac{20}{63}&\frac{13}{63}&\frac{20}{1701}\\\\[5.0pt]
\frac{31}{224}&\frac{81}{224}&\frac{81}{224}&\frac{31}{224}\end{pmatrix},$ (4)
$\displaystyle B^{(2)}$ $\displaystyle=\begin{pmatrix}0&0&0&0\\\\[5.0pt]
\frac{1283}{272160}&-\frac{851}{30240}&-\frac{269}{30240}&-\frac{163}{272160}\\\\[5.0pt]
\frac{43}{8505}&-\frac{16}{945}&-\frac{19}{945}&-\frac{8}{8505}\\\\[5.0pt]
\frac{19}{3360}&-\frac{9}{1120}&\frac{9}{1120}&-\frac{19}{3360}\end{pmatrix}.$
(5)
We choose to use the above-mentioned class of methods with equispaced abscissa
for the sake of a clearer presentation of results. Other multiderivative
Runge-Kutta methods could also be used, of course with the necessary
modifications. Higher derivatives can also be incorporated as an alternative
option to further increase the overall order.
With each Runge-Kutta method, there is an underlying quadrature rule connected
to the solver. That is, given the quantities $\phi^{j}$, $1\leq j\leq s$, we
define the “quadrature rule” as the multiderivative Runge-Kutta flux update,
given by
$\displaystyle\mathcal{I}_{l}(\phi^{1},\ldots\phi^{s}):=\Delta
t\sum_{j=1}^{s}B^{(1)}_{lj}\phi^{j}+\Delta
t^{2}\sum_{j=1}^{s}B^{(2)}_{lj}{\overset{\boldsymbol{.}}{\phi}}^{j}.$
These terms are precisely the terms required to produce each stage in the
Runge-Kutta method. They may or may not be high order quantities, unless
further assumptions are made about the solver. In the case of the Hermite-
Birkhoff-type approach, they are of higher order:
###### Lemma 1
For each Hermite-Birkhoff Runge-Kutta method, there holds,
$\displaystyle\int_{t^{n}}^{t^{n+1}}\Phi(w(t))\mathrm{d}t=\mathcal{I}_{l}\left(\Phi(w(t^{n}+c_{1}\Delta
t)),\ldots,\Phi(w(t^{n}+c_{s}\Delta t))\right)+\mathcal{O}(\Delta t^{q+1}).$
###### Proof
This is due to the construction: The Hermite-Birkhoff polynomial is of order
$2s-1=q-1$, which gives an integral approximation of order $q+1$. (Note that
the length of the integration area is $\Delta t$.)
∎
In the sequel, we exploit precisely this quadrature to extend the method shown
in SealSchuetz19 to higher orders and parallelism in time. Before we actually
define the algorithm, let us clarify the notation used. The algorithm to be
presented relies on the approximated quantities:
$\displaystyle w^{n,[k],{l}}\approx w(t^{n}+c_{l}\Delta t),\quad 0\leq n\leq
N,\quad 0\leq k\leq k_{\max},\quad 1\leq l\leq s.$ (6)
Here, $n$ is the usual discrete time level, and the $l$ refers to the stages
that are present within the Runge-Kutta method. The index $k$ is a parameter
that is associated to ‘correction’ steps (to be explained below). The final
index $k_{\max}$ is a fixed parameter chosen by the user to describe the total
number of iterations sought. For non-stiff problems, it is typically dictated
by the maximum order that can be reached, while for stiff problems, it can be
advantageous to use a larger $k_{\max}$ to overcome convergence issues
SealSchuetz19 , or even allow the Algorithm to choose this adaptively.
Finally, we make the abbreviations
$\displaystyle\Phi^{n,[k],{l}}:=\Phi\left(w^{n,[k],{l}}\right),\quad\Phi_{\text{I}}^{n,[k],{l}}:=\Phi_{\text{I}}\left(w^{n,[k],{l}}\right),\quad\Phi_{\text{E}}^{n,[k],{l}}:=\Phi_{\text{E}}\left(w^{n,[k],{l}}\right),$
to simplify the notation.
We are now ready to formulate the following algorithm. Given the Butcher
tableaux $B^{(1)}$ and $B^{(2)}$ and the time instances $c$, (e.g., any of the
methods from Example 1), the Hermite-Birkhoff Predictor Corrector method is as
follows:
###### Algorithm 1 ($\text{HBPC}(q,k_{\max},$))
To advance the solution to Eq. (1) from time level $t^{n}$ to time level
$t^{n+1}$, fill the values $w^{n,[0],{l}}$ using a second-order IMEX-Taylor
method:
1. 1.
Predict. Solve the following expression for $w^{n,[0],{l}}$ and $1\leq l\leq
s$:
$\displaystyle\begin{split}w^{n,[0],{l}}:={\color[rgb]{1,0.19,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.19,1}\pgfsys@color@cmyk@stroke{0}{0.81}{0}{0}\pgfsys@color@cmyk@fill{0}{0.81}{0}{0}w^{n-1,[0],{s}}}&+c_{l}\Delta
t\left(\Phi_{\text{I}}^{n,[0],{l}}+{\color[rgb]{1,0.19,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.19,1}\pgfsys@color@cmyk@stroke{0}{0.81}{0}{0}\pgfsys@color@cmyk@fill{0}{0.81}{0}{0}\Phi_{\text{E}}^{n-1,[0],{s}}}\right)\\\
&+\frac{(c_{l}\Delta
t)^{2}}{2}\left({\color[rgb]{1,0.19,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.19,1}\pgfsys@color@cmyk@stroke{0}{0.81}{0}{0}\pgfsys@color@cmyk@fill{0}{0.81}{0}{0}\overset{\boldsymbol{.}}{\Phi}_{\text{E}}^{n-1,[0],{s}}}-\overset{\boldsymbol{.}}{\Phi}_{\text{I}}^{n,[0],{l}}\right).\end{split}$
(7)
Subsequently:
2. 2.
Correct. Solve the following for $w^{n,[k+1],{l}}$, for each $2\leq l\leq s$
and each $0\leq k<k_{\max}$:
$\displaystyle\begin{split}w^{n,[k+1],{1}}&:={{\color[rgb]{1,0.19,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.19,1}\pgfsys@color@cmyk@stroke{0}{0.81}{0}{0}\pgfsys@color@cmyk@fill{0}{0.81}{0}{0}w^{n-1,[k+2],{s}}}},\\\
w^{n,[k+1],{l}}&:={\color[rgb]{1,0.19,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.19,1}\pgfsys@color@cmyk@stroke{0}{0.81}{0}{0}\pgfsys@color@cmyk@fill{0}{0.81}{0}{0}w^{n-1,[k+2],{s}}}\\\
&+{\Delta
t}\left(\Phi_{\text{I}}^{n,[k+1],{l}}-\Phi_{\text{I}}^{n,[k],{l}}\right)-\frac{\Delta
t^{2}}{2}\left(\overset{\boldsymbol{.}}{\Phi}_{\text{I}}^{n,[k+1],{l}}-\overset{\boldsymbol{.}}{\Phi}_{\text{I}}^{n,[k],{l}}\right)\\\
&+\mathcal{I}_{l}(\Phi^{n,[k],{0}},\Phi^{n,[k],{1}},\ldots,\Phi^{n,[k],{s}}).\end{split}$
(8)
In order to close the recursion, if $k=k_{\max}-1$, then replace each of the
$k+2$ with $k_{\max}$.
3. 3.
Update. In order to preserve the first-same-as-last property, we put
$\displaystyle w^{n+1}:=w^{n,[k_{\max}],{s}}.$
Finally, to seed initial conditions for this solver, we define
$\displaystyle w^{-1,[k],{s}}:=w_{0},\qquad 0\leq k\leq k_{\max}.$
Note that the starting value of $k=0$ denotes a straightforward second-order
IMEX-Taylor scheme; each $k>0$ represents a corrected version thereof, taking
into account the quadrature rule obtained from the Runge-Kutta scheme. The
reasoning is that with each additional correction, we see one extra order of
convergence until the maximal order of the Runge-Kutta method, $q$, has been
found.
###### Remark 1
The difference between this algorithm and our previous one found in
SealSchuetz19 lies in the highlighted red terms:
$\left\\{{w^{n-1,[0],{s}}},{\Phi_{\text{E}}^{n-1,[0],{s}}},{\overset{\boldsymbol{.}}{\Phi}_{\text{E}}^{n-1,[0],{s}}},{w^{n-1,[k+2],{s}}}\right\\}.$
In our original work each of these would have been evaluated at $w^{n}$. Here,
we work with different iterate numbers. This potent modification makes it
possible to parallelize the method in time without sacrificing the order of
accuracy. The idea that will be layed out in this publication is very similar
to – and in fact inspired by – the strategy proposed in ChriOng11 ; ChriOng10
.
###### Remark 2
Please note that Alg. 1 is not a Runge-Kutta method.
###### Remark 3
There is no implicitness in the Runge-Kutta quadrature $\mathcal{I}_{l}$. The
only implicitness is in the difference of the $\Phi_{\text{I}}$ and
$\overset{\boldsymbol{.}}{\Phi}_{\text{I}}$. This term is used to introduce
the necessary stability for stiff problems.
In the next three sections we present a thorough analysis of this solver
followed by a discussion of a parallization strategy and then numerical
results. In Section 6, we suggest an alternative formulation that provides a
low-storage alternative and even better scaling results.
## 3 Convergence analysis
Due to the changes made in the algorithm in comparison to the one proposed in
SealSchuetz19 , this is no longer a one-step multiderivative time integrator,
but a general linear method. It has both a multistage as well as a multistep
flavour, and we need to follow the convergence analysis outlined in the
seminal work by Butcher Butcher1966 to conduct our analysis. To ease the
presentation, we will in the sequel assume that we are treating a scalar
differential equation, i.e., we postulate that
$\displaystyle\Phi:\operatorname{\mathbb{R}}\rightarrow\operatorname{\mathbb{R}}.$
We start by rewriting Alg. 1 to make it more accessible to a convergence
analysis. Define the vector
$Y^{n+1}\in\operatorname{\mathbb{R}}^{s\cdot(k_{\max}+1)}$ consisting of all
stages and all correction steps at time instance $t^{n}=n\Delta t$, $n>0$,
i.e.,
$\displaystyle Y^{n+1}:=\left(\begin{matrix}w^{n,[0],{1}}\\\ \colon\\\
w^{n,[0],{s}}\\\ w^{n,[1],{1}}\\\ \colon\\\ w^{n,[1],{s}}\\\ \colon\\\
w^{n,[k_{\max}],{s}}\end{matrix}\right)\in\operatorname{\mathbb{R}}^{s\cdot(k_{\max}+1)}.$
Note that we have made a shift in the $n$ for a clearer exposition. For $n=0$,
we define $Y^{0}$ to be the vector with entries consisting of the solution at
time $t=0$:
$\displaystyle Y^{0}:=\left(\begin{matrix}w_{0}\\\ \colon\\\
w_{0}\end{matrix}\right)\in\operatorname{\mathbb{R}}^{s\cdot(k_{\max}+1)}.$
Then, we can write the time integrator from Alg. 1 in a “one-step-fashion” as
$\displaystyle Y^{n+1}=AY^{n}$ $\displaystyle+\Delta
t\left(B_{E}\Phi_{\text{E}}(Y^{n+1})+B_{I}\Phi_{\text{I}}(Y^{n+1})\right)$
$\displaystyle+\frac{\Delta
t^{2}}{2}\left(\tilde{B}_{E}{\overset{\boldsymbol{.}}{\Phi}_{\text{E}}}(Y^{n+1})+\tilde{B}_{I}{\overset{\boldsymbol{.}}{\Phi}_{\text{I}}}(Y^{n+1})\right)$
for matrices $A$, $B_{I}$, $B_{E}$, $\tilde{B}_{E}$ and $\tilde{B}_{I}$ in
$\operatorname{\mathbb{R}}^{s\cdot(k_{\max}+1)\times s\cdot(k_{\max}+1)}$.
In the sequel, we use the well-known inf-norm $\|\cdot\|_{\infty}$ for both
matrices and vectors. For the sake of readability, we write $\|\cdot\|$
instead of $\|\cdot\|_{\infty}$. That is, for
$v\in\operatorname{\mathbb{R}}^{\kappa}$, we define $\|v\|:=\max_{1\leq
i\leq\kappa}|v_{i}|$; for matrices
$C\in\operatorname{\mathbb{R}}^{\kappa\times\kappa}$, there holds
$\displaystyle\|C\|:=\sup_{v\in\operatorname{\mathbb{R}}^{\kappa}}\frac{\|Cv\|}{\|v\|}.$
###### Lemma 2
The matrix $A$ is power-bound by one, i.e., for each
$n\in\operatorname{\mathbb{N}}$, there holds
$\displaystyle\|A^{n}\|\leq 1.$
###### Proof
Define the matrix
$\operatorname{\mathfrak{E}}\in\operatorname{\mathbb{R}}^{s\times s}$ that is
zero except for the last column, which is filled with ones, i.e.,
$\displaystyle\operatorname{\mathfrak{E}}=\left(\begin{matrix}0&0&\cdots&0&1\\\
0&0&\cdots&0&1\\\ \vdots&\vdots&&\vdots&\vdots\\\
0&0&\cdots&0&1\end{matrix}\right).$
Then, $A$ is given as the block matrix
$\displaystyle
A=\left(\begin{matrix}\operatorname{\mathfrak{E}}&0&0&0&\cdots&0&0\\\
0&0&\operatorname{\mathfrak{E}}&0&\cdots&0&0\\\
0&0&0&\operatorname{\mathfrak{E}}&\cdots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&0&\cdots&\operatorname{\mathfrak{E}}&0\\\
0&0&0&0&\cdots&0&\operatorname{\mathfrak{E}}\\\
0&0&0&0&\cdots&0&\operatorname{\mathfrak{E}}\end{matrix}\right).$ (9)
There holds $\|Av\|\leq\|v\|$, which can be seen due to the fact that for each
$1\leq i\leq s\cdot(k_{\max}+1)$, there is a $1\leq j\leq s\cdot(k_{\max}+1)$
such that $(Av)_{i}=v_{j}$. This concludes the proof.
∎
###### Corollary 1
There exists a constant $C\in\operatorname{\mathbb{R}}$ such that for each
$j\in\operatorname{\mathbb{N}}^{\geq 0}$, there holds
$\displaystyle\|A^{j}B_{E}\|\leq C,\quad\|A^{j}B_{I}\|\leq
C,\quad\|A^{j}\tilde{B}_{E}\|\leq C,\quad\|A^{j}\tilde{B}_{I}\|\leq C.$
Similarly to Butcher1966 , we define the vector $W^{n}$ as the vector of the
exact solutions at time instant $t^{n}$, i.e.,
$\displaystyle W^{n+1}:=\left(\begin{matrix}w(t^{n}+c_{1}\Delta t)\\\
\colon\\\ w(t^{n}+c_{s}\Delta t)\\\ w(t^{n}+c_{1}\Delta t)\\\ \colon\\\
w(t^{n}+c_{s}\Delta
t)\end{matrix}\right)\in\operatorname{\mathbb{R}}^{s\cdot(k_{\max}+1)},\qquad
n\geq 0.$
We define $W^{0}:=Y^{0}$. For a convergence analysis, we are interested in the
error, or the differnce between the approximate and the exact solution:
$Z^{n}:=Y^{n}-W^{n}$. This quantity fulfills the following time-discrete
equation:
$\displaystyle\begin{split}Z^{n}&\equiv Y^{n}-W^{n}\\\ =&AZ^{n-1}+\Delta
tB_{E}\left(\Phi_{\text{E}}(Y^{n})-\Phi_{\text{E}}(W^{n})\right)+\Delta
tB_{I}\left(\Phi_{\text{I}}(Y^{n})-\Phi_{\text{I}}(W^{n})\right)\\\
&+\frac{\Delta
t^{2}}{2}\tilde{B}_{E}\left(\overset{\boldsymbol{.}}{\Phi}_{\text{E}}(Y^{n})-\overset{\boldsymbol{.}}{\Phi}_{\text{E}}(W^{n})\right)+\frac{\Delta
t^{2}}{2}\tilde{B}_{I}\left(\overset{\boldsymbol{.}}{\Phi}_{\text{I}}(Y^{n})-\overset{\boldsymbol{.}}{\Phi}_{\text{I}}(W^{n})\right)\\\
&+E_{n},\end{split}$ (10)
with $E^{n}$ the local consistency error
$\displaystyle E^{n}:=AW^{n-1}$ $\displaystyle+\Delta
t\left(B_{E}\Phi_{\text{E}}(W^{n})+B_{I}\Phi_{\text{I}}(W^{n})\right)$
$\displaystyle+\frac{\Delta
t^{2}}{2}\left(\tilde{B}_{E}\overset{\boldsymbol{.}}{\Phi}_{\text{E}}(W^{n})+\tilde{B}_{I}\overset{\boldsymbol{.}}{\Phi}_{\text{I}}(W^{n})\right)-W^{n}.$
As in SealSchuetz19 , we can prove that the local consistency error at
correction level $k$ is given by $\mathcal{O}\left(\Delta
t^{\min\\{2+k,q\\}+1}\right)$. For $k=0$, this is straightforward – this is
second-order IMEX-Taylor. For $k>0$, techniques as in SealSchuetz19 (and long
known to the spectral deferred correction (SDC) community, see, e.g.,
DuttGreenRokh00 ) can be employed.
###### Lemma 3
Subdivide $E^{n}$ into $k_{\max}+1$ blocks of size $s$ each; call the
individual blocks $E^{n,[k]}$, $0\leq k\leq k_{\max}$. Then, there holds
$\displaystyle E^{n,[k]}=O(\Delta t^{\min\\{2+k,q\\}+1}).$
###### Proof
The proof goes along the same lines as in SealSchuetz19 and is hence omitted.
∎
The following lemma, although straightforward to prove, is of utmost
importance to the convergence analysis.
###### Lemma 4
The recursion for $Z^{n}$ given in (10) can be explicitly written as:
$\displaystyle\begin{split}Z^{n}=&\Delta
t\sum_{j=0}^{n-1}A^{j}B_{E}\left(\Phi_{\text{E}}(Y^{n-j})-\Phi_{\text{E}}(W^{n-j})\right)\\\
&+\Delta
t\sum_{j=0}^{n-1}A^{j}B_{I}\left(\Phi_{\text{I}}(Y^{n-j})-\Phi_{\text{I}}(W^{n-j})\right)\\\
&+\frac{\Delta
t^{2}}{2}\sum_{j=0}^{n-1}A^{j}\tilde{B}_{E}\left(\overset{\boldsymbol{.}}{\Phi}_{\text{E}}(Y^{n-j})-\overset{\boldsymbol{.}}{\Phi}_{\text{E}}(W^{n-j})\right)\\\
&+\frac{\Delta
t^{2}}{2}\sum_{j=0}^{n-1}A^{j}\tilde{B}_{I}\left(\overset{\boldsymbol{.}}{\Phi}_{\text{I}}(Y^{n-j})-\overset{\boldsymbol{.}}{\Phi}_{\text{I}}(W^{n-j})\right)\\\
&+\sum_{j=0}^{n-1}A^{j}E^{n-j}.\end{split}$ (11)
###### Proof
This can be easily proved using (10) and an induction over $n$. Note that one
has to take into account that $Z^{0}=0$ by definition.
∎
In order to show convergence, we need the following result on the analytical
solution to a linear recursion equation:
###### Lemma 5
Consider the following linear recursion:
$\displaystyle x_{n}=\alpha(x_{0}+\ldots+x_{n-1})+\beta_{n},\qquad n\geq 1,$
(12)
with $\alpha,\beta_{n}\in\operatorname{\mathbb{R}}$ given for all
$n\in\operatorname{\mathbb{N}}$. (As indicated, $\beta_{n}$ is allowed to
depend on $n$, while $\alpha$ is not.) Also $x_{0}$ is supposed to be known.
The analytical solution to this recursion is given by
$\displaystyle
x_{n}=\alpha(\alpha+1)^{n-1}x_{0}+\sum_{j=1}^{n-1}\beta_{j}\alpha(1+\alpha)^{n-j-1}+\beta_{n}.$
(13)
###### Proof
We proceed by induction. For $n=1$, we have $x_{1}=\alpha x_{0}+\beta_{1}$,
which agrees with formula (13). Now assume that (13) is correct for all $n<N$.
Then, there holds:
$\displaystyle x_{N}$ $\displaystyle=\alpha\sum_{n=0}^{N-1}x_{n}$
$\displaystyle=\alpha
x_{0}+\alpha\sum_{n=1}^{N-1}\left(\alpha(\alpha+1)^{n-1}x_{0}+\sum_{j=1}^{n-1}\beta_{j}\alpha(1+\alpha)^{n-j-1}+\beta_{n}\right)+\beta_{N}$
$\displaystyle=\underbrace{\alpha
x_{0}+\alpha\sum_{n=1}^{N-1}\alpha(\alpha+1)^{n-1}x_{0}}_{=:I}+\underbrace{\alpha\sum_{n=1}^{N-1}\sum_{j=1}^{n-1}\beta_{j}\alpha(1+\alpha)^{n-j-1}+\alpha\sum_{n=1}^{N-1}\beta_{n}}_{=:II}+\beta_{N}.$
We compute the terms separately:
$\displaystyle I$ $\displaystyle=\alpha
x_{0}\left(1+\sum_{n=0}^{N-2}\alpha(\alpha+1)^{n}\right)=\alpha
x_{0}\left(1+\alpha\frac{(\alpha+1)^{N-1}-1}{\alpha}\right)$
$\displaystyle=\alpha x_{0}(\alpha+1)^{N-1}.$
Now for $II$, there holds via a switching of the summation that
$\displaystyle II$
$\displaystyle=\alpha\sum_{n=1}^{N-1}\sum_{j=1}^{n-1}\beta_{j}\alpha(1+\alpha)^{n-j-1}+\alpha\sum_{n=1}^{N-1}\beta_{n}$
$\displaystyle=\alpha\sum_{j=1}^{N-2}\beta_{j}\sum_{n=j+1}^{N-1}\alpha(1+\alpha)^{n-j-1}+\alpha\sum_{n=1}^{N-1}\beta_{n}$
$\displaystyle=\alpha\sum_{j=1}^{N-2}\beta_{j}\left({(1+\alpha)^{N-j-1}}-1\right)+\alpha\sum_{n=1}^{N-1}\beta_{n}=\alpha\sum_{j=1}^{N-1}\beta_{j}(1+\alpha)^{N-j-1}.$
All together, this shows that also $x_{N}$ can be formed according to (13).
The statement is hence true.
∎
Using the above results, we can immediately show that the method is second-
order convergent:
###### Theorem 3.1
Given that $\Delta t$ is sufficiently small, and given the assumptions on the
functions $\Phi$, $\Phi_{\text{I}}$ and $\Phi_{\text{E}}$ as done in
Assumption 1, there is a constant $C$ such that
$\displaystyle Z^{n}\leq C\Delta t^{2},$
and hence, the method is second-order convergent.
###### Proof
Note that due to Corollary 1, we can estimate terms like $\|A^{j}B_{E}\|$
against fixed constants. Using the expression (11), we can hence compute
$\displaystyle\|Z^{n}\|\leq$ $\displaystyle\ \Delta
tL_{\Phi_{\text{E}}}\sum_{j=0}^{n-1}\|A^{j}B_{E}\|\|Z^{n-j}\|+\Delta
tL_{\Phi_{\text{I}}}\sum_{j=0}^{n-1}\|A^{j}B_{I}\|\|Z^{n-j}\|$
$\displaystyle+\frac{\Delta
t^{2}}{2}L_{\overset{\boldsymbol{.}}{\Phi}_{\text{E}}}\sum_{j=0}^{n-1}\|A^{j}\tilde{B}_{E}\|\|Z^{n-j}\|+\frac{\Delta
t^{2}}{2}L_{\overset{\boldsymbol{.}}{\Phi}_{\text{I}}}\sum_{j=0}^{n-1}\|A^{j}\tilde{B}_{I}\|\|Z^{n-j}\|$
$\displaystyle+\sum_{j=0}^{n-1}\|A^{j}E^{n-j}\|$ $\displaystyle\leq\ $
$\displaystyle C\Delta
t\sum_{j=0}^{n-1}\|Z^{n-j}\|+\sum_{j=0}^{n-1}\|E^{n-j}\|.$
Given that $C\Delta t$ is smaller than one, we get the inequality
$\displaystyle\|Z^{n}\|\leq\ $ $\displaystyle C\Delta
t\sum_{j=1}^{n-1}\|Z^{n-j}\|+\sum_{j=0}^{n-1}\|E^{n-j}\|,$
with another constant $C$ that does not depend on $\Delta t$ or $N$. Note that
summation of the first term begins at $j=1$. Now there holds that
$\|Z^{n}\|\leq\varepsilon_{n}$, with $\varepsilon_{0}=0$ and
$\varepsilon_{n}$, $n>0$, defined as
$\displaystyle\varepsilon_{n}=C\Delta
t(\varepsilon_{0}+\ldots+\varepsilon_{n-1})+\sum_{j=0}^{n-1}\|E^{n-j}\|.$
This is exactly of form (12), and has hence the analytical solution
$\displaystyle\varepsilon_{n}=\sum_{j=1}^{n-1}C\Delta
t\left(\sum_{k=0}^{j-1}\|E^{n-j}\|\right)(1+C\Delta
t)^{n-j-1}+\sum_{j=0}^{n-1}\|E^{n-j}\|.$
We can estimate $(1+C\Delta t)^{n-j-1}$ against $e^{CT_{end}}$, and because of
La. 3, there then holds that
$\displaystyle\varepsilon_{n}\leq C\Delta t^{2}.$
Note that the property $\sum_{k=0}^{j-1}\|E^{n-j}\|$ can be estimated against
$C\Delta t^{2}$. Note also that constants are subject to change. They do
however never depend on $N$ or $\Delta t$.
∎
Second order convergence is of course not completely what we expect from Alg.
1. In the following, we explore the nature of the higher orders in some more
detail.
###### Remark 4
The structure of the matrix $A$ as given in (9) is very interesting: it
“pushes elements in vectors downwards.” To facilitate the understanding of the
following analysis, we show different exponents of $A$. Obviously,
$A^{0}=\operatorname{Id}$, the identiy matrix, and $A^{1}$ is given in (9).
There holds, always given that $k_{\max}$ is large enough:
$\displaystyle
A^{2}=\left(\begin{matrix}\operatorname{\mathfrak{E}}&0&0&0&0&\cdots&0&0&0\\\
0&0&0&\operatorname{\mathfrak{E}}&0&\cdots&0&0&0\\\
0&0&0&0&\operatorname{\mathfrak{E}}&\cdots&0&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
0&0&0&0&0&\cdots&\operatorname{\mathfrak{E}}&0&0\\\
0&0&0&0&0&\cdots&0&\operatorname{\mathfrak{E}}&0\\\
0&0&0&0&0&\cdots&0&0&\operatorname{\mathfrak{E}}\\\
0&0&0&0&0&\cdots&0&0&\operatorname{\mathfrak{E}}\\\
0&0&0&0&0&\cdots&0&0&\operatorname{\mathfrak{E}}\end{matrix}\right),\qquad
A^{3}=\left(\begin{matrix}\operatorname{\mathfrak{E}}&0&0&0&0&\cdots&0&0&0\\\
0&0&0&0&\operatorname{\mathfrak{E}}&\cdots&0&0&0\\\ 0&0&0&0&0&\cdots&0&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&0&0&\cdots&\operatorname{\mathfrak{E}}&0&0\\\
0&0&0&0&0&\cdots&0&\operatorname{\mathfrak{E}}&0\\\
0&0&0&0&0&\cdots&0&0&\operatorname{\mathfrak{E}}\\\
0&0&0&0&0&\cdots&0&0&\operatorname{\mathfrak{E}}\\\
0&0&0&0&0&\cdots&0&0&\operatorname{\mathfrak{E}}\\\
0&0&0&0&0&\cdots&0&0&\operatorname{\mathfrak{E}}\end{matrix}\right),\qquad\ldots$
Eventually, there is a limit matrix $A^{\infty}$, given as
$\displaystyle
A^{\infty}=\left(\begin{matrix}\operatorname{\mathfrak{E}}&0&0&0&0&\cdots&0&0&0\\\
0&0&0&0&0&\cdots&0&0&\operatorname{\mathfrak{E}}\\\
0&0&0&0&0&\cdots&0&0&\operatorname{\mathfrak{E}}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&0&0&\cdots&0&0&\operatorname{\mathfrak{E}}\\\
0&0&0&0&0&\cdots&0&0&\operatorname{\mathfrak{E}}\\\
0&0&0&0&0&\cdots&0&0&\operatorname{\mathfrak{E}}\\\
0&0&0&0&0&\cdots&0&0&\operatorname{\mathfrak{E}}\\\
0&0&0&0&0&\cdots&0&0&\operatorname{\mathfrak{E}}\\\
0&0&0&0&0&\cdots&0&0&\operatorname{\mathfrak{E}}\end{matrix}\right).$
More formally, we have $A^{j}=A^{\infty}$ for all $j\geq k_{\max}-1$. This
structure is important, because it significantly simplifies the convergence
analysis. Essentially, it states that most contributions to the overall error
come from higher corrections. Intuitively, this is desirable, because higher
corrections are supposed to have smaller error contributions, at least
asymptotically.
###### Lemma 6
Subdivide $Z^{n}$ into $k_{\max}+1$ blocks of size $s$ each, similarly to La.
3. Define $\varepsilon_{k}^{n}$, $0\leq k\leq k_{\max}$ to be the infinity-
norm of the $k-$th block. For $n=0$, there is no error, and hence
$\displaystyle\varepsilon_{k}^{0}=0,\qquad 0\leq k\leq k_{\max}.$
Furthermore, for $k=0$, the underlying consistency analysis is trivial, and
hence $\varepsilon_{0}^{n}$ can be bound by a positive constant $C$ times
$\Delta t^{2}$, i.e.,
$\displaystyle\varepsilon_{0}^{n}\leq C\Delta t^{2},\qquad 1\leq n\leq N.$
The constant $C$ depends of course on the analytical solution, the fluxes
$\Phi$, the initial conditions $w_{0}$ and the like, but it does not depend on
$\Delta t$ or $n$. Furthermore, there is a constant $C$ such that there holds
for $1\leq k<k_{\max}$ and $1\leq n\leq N$:
$\displaystyle\varepsilon^{n}_{k}$ $\displaystyle\leq C\Delta
t\left(\varepsilon_{k}^{n}+\varepsilon_{k-1}^{n}\right)+C\Delta
t\sum_{j=1}^{n-1}\sum_{h=k}^{k_{\max}}\varepsilon_{h}^{n-j}+C\Delta
t^{\min\\{2+k,q\\}},$ (14) $\displaystyle\varepsilon^{n}_{k_{\max}}$
$\displaystyle\leq C\Delta
t\sum_{j=0}^{n-1}\left(\varepsilon_{k_{\max}-1}^{n-j}+\varepsilon_{k_{\max}}^{n-j}\right)+C\Delta
t^{\min\\{2+k_{\max},q\\}}.$ (15)
###### Remark 5
Before proving this lemma, we point out a couple of salient features:
* •
Note that $\varepsilon^{n}_{k_{\max}}$ has a global (i.e., over _all_ $n$)
dependence on $\varepsilon^{n}_{k_{\max}-1}$, while all the other
$\varepsilon^{n}_{k}$ do not have this kind of dependence on
$\varepsilon^{n}_{k-1}$.
* •
We have chosen to take the same constant $C$ for both the expression of
$\varepsilon_{0}^{n}$ as well as for all the terms in the inequality for
$\varepsilon_{k}^{n}$. This can be done without loss of generality, as we take
the largest of all these constants.
###### Proof
To prove (14), we consider the analytical expression of $Z^{n}$ given in (11).
Let us first treat the last term in (11), $\sum_{j=0}^{n-1}A^{j}E^{n-j}$. Due
to La. 3, we know that there is a constant $C$ such that $E^{n-j,[k]}\leq
C\Delta t^{\min\\{2+k,q\\}+1}$. There holds
$\displaystyle\left(\sum_{j=0}^{n-1}A^{j}E^{n-j}\right)^{[k]}\leq NC\Delta
t^{\min\\{2+k,q\\}+1}\leq T_{end}C\Delta t^{\min\\{2+k,q\\}}.$
By $(\cdot)^{[k]}$, we denote the $k-$the block of a vector, again, $0\leq
k\leq k_{\max}$. We have used the fact that $\Delta t=\frac{T_{end}}{N}$ and
that $A^{j}$ “pushes elements downwards,” so there will be no $\Delta t$
contributions of lower orders than the ones given here. This explains why we
carefully chose the red terms in Alg. 1 the way we did.
In a similar way, we treat the other terms. They can all be handled alike, so
we only consider the term
$\displaystyle\Delta
t\sum_{j=0}^{n-1}A^{j}B_{I}\left(\Phi_{\text{I}}(Y^{n-j})-\Phi_{\text{I}}(W^{n-j)}\right)$
$\displaystyle=$ $\displaystyle\Delta
tB_{I}\left(\Phi_{\text{I}}(Y^{n})-\Phi_{\text{I}}(W^{n)}\right)+\Delta
t\sum_{j=1}^{n-1}A^{j}B_{I}\left(\Phi_{\text{I}}(Y^{n-j})-\Phi_{\text{I}}(W^{n-j)}\right).$
Note that the first term couples errors at correction level $k-1$ to those of
$k$. Estimated, this gives, independent of whether $k=k_{\max}$ or not, a
contribution of
$\displaystyle C\Delta
t\left(\varepsilon_{k}^{n}+\varepsilon_{k-1}^{n}\right).$
For $k<k_{\max}$, due to the peculiar structure of $A^{j}$, $A^{j}B_{I}$
remains a block-matrix; on the $k-$th block-row, only two blocks are
filled.222Let us clarify what we mean by block-rows and block-columns: The
matrices we are dealing with here are of size $s\cdot(k_{\max}+1)\times
s\cdot(k_{\max}+1)$. They can hence be subdivided into $(k_{\max}+1)^{2}$
blocks of size $s\times s$. According to the notation in Alg. 1, we begin
counting by $k=0$. Block-row $k$ means hence in the big matrix the rows from
$k(s+1)+1$ to $(k+1)(s+1)$. The block-columns thereof are $j$ and $j+1$, with
$j\geq k$. A generous estimate yields the contribution
$\displaystyle C\Delta
t\sum_{j=1}^{n-1}\sum_{h=k}^{k_{\max}}\varepsilon_{h}^{n-j}.$
(In fact, for $j\geq k_{\max}-1$, the block columns are $k_{\max}-1$ and
$k_{\max}$, so sharper estimates could be used.) For $k=k_{\max}$, the non-
zero block-columns are $k_{\max}-1$ and $k_{\max}$. This then gives the
contribution
$\displaystyle C\Delta
t\sum_{j=1}^{n-1}\left(\varepsilon_{k_{\max}-1}^{n-j}+\varepsilon_{k_{\max}}^{n-j}\right).$
This yields the desired result.
∎
We do not have a proof for the following statement, which is why we formulate
it as a proposition.
###### Proposition 1
From the results in La. 6, we conjecture that the method is convergent with
the $k-$dependent orders. That is, we postulate that each of the correction
steps satisfy
$\displaystyle e^{n}_{k}$ $\displaystyle=\mathcal{O}\left(\Delta
t^{\min\\{2+k,q\\}}\right),$ $\displaystyle\quad k$ $\displaystyle<k_{\max},$
and that the final correction step satisfies
$\displaystyle e^{n}_{k_{\max}}$ $\displaystyle=\mathcal{O}\left(\Delta
t^{\min\\{1+k_{\max},q\\}}\right),$ $\displaystyle\quad k$
$\displaystyle=k_{\max}.$
###### Remark 6
Please note that the last correction step does not increase the order. This is
different to the behavior of the algorithm in SealSchuetz19 ; it is a
consequence of the fact that we have modified the values at time level $t^{n}$
to make the algorithm ready for parallelism.
###### Remark 7
Although we only present numerical results that support Proposition 1 – and we
only know from Thm. 3.1 that the method is at-least second order convergent –
we can very well see the underlying structure from Eqs. (14)-(15): For
$k<k_{\max}$, $\varepsilon_{k}^{n}$ essentially possesses the contributions
$\Delta t\varepsilon_{k-1}^{n}$, $\Delta
t\sum_{j=1}^{n-1}\sum_{h=k}^{k_{\max}}\varepsilon_{h}^{n-j}$ and $\Delta
t^{\min\\{2+k,q\\}}$. (We have neglected the constants here.) If we assume
that for increasing $k$, the order in $\Delta t$ of $\varepsilon_{k}^{n}$
would also increase or at least not decrease, the contribution $\Delta
t\sum_{j=1}^{n-1}\sum_{h=k}^{k_{\max}}\varepsilon_{h}^{n-j}$ scales as
$\mathcal{O}(\varepsilon_{k}^{n})$. Furthermore, if the order between
$\varepsilon_{k}^{n}$ and $\varepsilon_{k-1}^{n}$ differs by one, then $\Delta
t\varepsilon_{k-1}^{n}$ scales again as $\varepsilon_{k}^{n}$. The final
constant term $\Delta t^{\min\\{2+k,q\\}}$ then gives the overall order.
## 4 Parallel implementation
The method presented in Alg. 1 has been carefully designed in such a way that
it can be implemented in a parallel setting. The ideas we use are similar to
the ones shown in ChriOng10 – and in fact, the much older ideas presented in
MirankerLiniger1967 – and are roughly based on the fact that dependencies are
chosen in such a way that pipelining becomes possible. The following
observations are key to being able to parallelize this solver:
* •
The predictor, i.e., the values $w^{n,[0],{l}}$ that correspond to zeroth
iteration number $k=0$, do _not_ depend on the values of any other correction
step. They only depend on the final value $w^{n-1,[0],{s}}$ obtained by the
_predictor_ in the previous time step.
* •
For $1\leq k<k_{\max}$, the $(k)-$th corrector step, i.e., the values
$w^{n,[k],{l}}$, depend on the $(k-1)$-st corrector step at time level $n$, as
well as on the next correction at the previous time step, $w^{n-1,[k+1],{s}}$.
* •
The $(k_{\max})$-th corrector step, $w^{n,[k_{\max}],{l}}$ depends only on the
$(k_{\max}-1)$-st corrector step at time level $n$ as well as the final
correction at the previous time level, $w^{n-1,[k_{\max}],{s}}$.
These dependencies are illustrated in more detail on the left hand side of
Fig. 1. On the $x-$axis, the time instances $n$, $n+1,\ldots$, are indicated,
while on the $y-$axis, the correction iterates $k$ and the predictor ($k=0$)
are sketched. The circles at position $(n,k)$ correspond to the computation of
$w^{n,[k],{l}}$ for all $1\leq l\leq s$. Circles with the same number on it
can be computed in parallel, while those with a higher number have to wait for
those with a lower number to finish. The arrows indicate direct dependencies,
required for the calculation of $w^{n,[k],{l}}$. Given the dependencies, it
also makes sense to always group the correction iterates $2k$ and $2k+1$ on
the same process, since one would otherwise create idle processor time. In
addition, in order to use computational resources as efficiently as possible,
we also do this for the predictor, so the predictor $(k=0)$ is grouped
together with the first correction iterate $k=1$, which would strictly
speaking not be necessary. Red arrows then imply communication over processes.
Note that communication is always unidirectional. For comparison, the
dependencies of the original serial algorithm from SealSchuetz19 are
visualized in the center of Fig. 1. One can clearly see that without the
modifications proposed in Alg. 1, the predictor and all correction steps
depend on the $(k_{\max})$-th iteration at time level $n-1$.
HO-Parallel | Original | LO-Parallel
---|---|---
Figure 1: Detailed view on the dependencies of the parallelization flavors.
Here we include an illustrative example with a total of four time steps and
$k_{\max}=5$ total iterations. On the $x-$axis, time instances $n$,
$n+1,\ldots$ are indicated, while on the $y-$axis, the correction iterates $k$
and the predictor ($k=0$) are sketched. The circles at position $(n,k)$
correspond to the computation of $w^{n,[k],{l}}$ for all $1\leq l\leq s$.
Circles with the same number on it can be computed in parallel, while those
with a higher number have to wait for those with a lower number to finish. The
arrows indicate direct dependencies required for the calculation of
$w^{n,[k],{l}}$, with red arrows indicating that communication between
processors is needed. Left: Higher order parallel-in-time MD scheme. Middle:
Original serial algorithm proposed in SealSchuetz19 . Right: Low order
parallel-in-time MD scheme, enabling the use of more processors.
The High-Order Parallel (HO-Parallel) version is designed to deliver parallel
speedup while retaining high-order accuracy. This is the method described in
Alg. 1, but here we describe one such grouping of processors that yields a
parallel solver. Theoretically, this speedup can be estimated as follows:
Denote the number of timesteps again by $N$. The unparallelized algorithm has
to perform $N\cdot(k_{\max}+1)$ operation blocks (the computation of
$w^{n,[k],{l}}$ for given $n$, $k$ and for all $1\leq l\leq s$). Each process
in the parallelized version has to perform $2N$ operations, and the last
process has to wait $k_{\max}-1$ cycles before it can start. This gives,
neglecting all other sources of imbalance and overhead such as communication
and the like, a maximum theoretical speedup of
$\frac{N\cdot(k_{\max}+1)}{2N+k_{\max}-1}\rightarrow\frac{k_{\max}+1}{2},\qquad
N\rightarrow\infty.$ (16)
Please note that this is the speedup in comparison to letting the algorithm
run in its unparallelized version, it is not the theoretical speedup in
comparison to the underlying Runge-Kutta method, as is done, e.g., in
EmmettMinion12 . This speedup is harder to determine, in particular in the
IMEX setting that we follow here, because it depends on the convergence of the
correction iterates as well as on the (nonlinear) algebraic solves, which can
differ tremendously from the fully implicit Runge-Kutta method and the semi-
implicit correction iterates used here.
Grouping the processors has the obvious disadvantage that it reduces the
parallelization capabilities, and therefore if attaining parallel speed is of
chief concern, we suggest the Low-Order Parallel (LO-Parallel) method
described in the right frame of Fig. 1. In short, the basic idea is to
dedicate an entire process to each correction step and let each correction run
completely independent from the other corrections insofar as they need to wait
for the previous correction to finish in a queue like fashion. That is, the
LO-Parallel version changes the red terms in Alg. 1 from $w^{n-1,[k+2],{s}}$
to $w^{n-1,[k+1],{s}}$:
$\displaystyle\begin{split}w^{n,[k+1],{1}}&:={w^{n-1,[k+1],{s}}},\\\
w^{n,[k+1],{l}}&:={w^{n-1,[k+1],{s}}}\\\ &+{\Delta
t}\left(\Phi_{\text{I}}^{n,[k+1],{l}}-\Phi_{\text{I}}^{n,[k],{l}}\right)-\frac{\Delta
t^{2}}{2}\left(\overset{\boldsymbol{.}}{\Phi}_{\text{I}}^{n,[k+1],{l}}-\overset{\boldsymbol{.}}{\Phi}_{\text{I}}^{n,[k],{l}}\right)\\\
&+\mathcal{I}_{l}(\Phi^{n,[k],{0}},\Phi^{n,[k],{1}},\ldots,\Phi^{n,[k],{s}}).\end{split}$
(17)
This means that on correction iterate $k+1$, we only take the previous
$(k)-$th correction iterate into account, and not as in the novel algorithm,
the $(k+2)$-nd iterate. Provided we lag each of the corrections sufficiently
through a startup procedure, then each of the correctors can follow up and
clean up the previous iterate at the same cost of the predictor. In comparison
to the serial evaluation of Alg. 1, the speedup would be
$\frac{N\cdot(k_{\max}+1)}{N+(k_{\max}+1)}\to k_{\max}+1,\quad N\to\infty$
(18)
under ideal circumstances. Unfortunately, this modfification reduces the
overall method to a second-order method, and therefore we do not recommend
this LO-Parallel strategy unless parallelization is of utmost importance.
## 5 Numerical results
In this section, we first experimentally validate the theoretical findings of
Sec. 3 with sample scalar ODE test cases. Here, we show that the desired
orders of accuracy are reached after the predictor and each of the correction
steps. We then increase the complexity of our test cases to systems and stiff
problems. There we study the influence of choosing one of the algorithms
presented in Fig. 1. It is shown that the modifications of the original
algorithm from SealSchuetz19 described in Alg. 1 only have a small influence
on the obtained solution. Finally, it is shown that the proposed method
converges to the limiting Runge-Kutta method given in Def. 1 even for very
stiff problems.
### 5.1 ODE for Convergence Testing
We start by considering a scalar model problem
$\displaystyle w^{\prime}(t)=-w^{-\frac{5}{2}},\qquad w_{0}=1,$ (19)
to validate the order of convergence of the introduced methods. The analytical
solution of this ODE is $w(t)=\left(-\frac{7}{2}t+w_{0}^{7/2}\right)^{2/7}$;
the error is evaluated at $T_{end}=0.25$. For this test problem we introduce
an artificial IMEX splitting via
$\displaystyle\Phi_{\text{E}}(w)=\alpha
w^{\prime}(t),\qquad\Phi_{\text{I}}(w)=(1-\alpha)w^{\prime}(t),$
for $\alpha=0.2$.
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Figure 2: Error of the iterates of the parallel IMEX-MD schemes for the simple
ODE problem $w^{\prime}=-w^{-\frac{5}{2}}$, see Eq. (19). The error is
computed at $T_{end}=0.25$. Different Runge-Kutta tables and $k_{\max}=3$
(left) and $k_{\max}=9$ (right) are used.
Fig. 2 shows the convergence of the fourth-, sixth- and eighth-order schemes
after all correction steps. The figure shows that the predictor step is second
order accurate and the correction steps increase the order of accuracy
successively. With $k_{\max}=3$ fourth order of accuracy is achievable and one
can note that the last two correction steps are always of the same order.
Increasing $k_{\max}$ shows that with each correction step the scheme picks up
one order of accuracy - until the maximum order defined by the quadrature rule
is reached. Hence, the maximum achievable order is
$\min({k_{\max}+1},{\text{accuracy of quadrature rule}})$, see also Prop. 1.
### 5.2 Pareschi-Russo Problem
Next, we consider the model problem introduced by Pareschi and Russo (PR)
pareschi2000implicit to additionally consider the ability of treating stiff
ODEs with the novel method. The system of ODEs is given by
$\displaystyle w_{1}^{\prime}(t)$ $\displaystyle=-w_{2},\qquad
w_{2}^{\prime}(t)=w_{1}+\frac{\sin(w_{1})-w_{2}}{\varepsilon},\qquad
w_{0}=\left(\frac{\pi}{2},1\right),$ (20)
and is computed until $T_{\text{end}}=5$. To account for the stiff behavior,
similar as it is done in pareschi2000implicit , we split the problem into a
non-stiff, explicitly treated part $\Phi_{\text{E}}$ and a stiff, implicitly
treated part $\Phi_{\text{I}}$
$\displaystyle\Phi_{\text{E}}=\begin{pmatrix}-w_{2}\\\
w_{1}\end{pmatrix},\qquad\Phi_{\text{I}}=\frac{1}{\varepsilon}\begin{pmatrix}0\\\
\sin(w_{1})-w_{2}\end{pmatrix}.$
We use this test problem to investigate the modifications of Alg. 1 compared
to the original one in SealSchuetz19 and the straight-forward, but low-order
parallelization variant. For an overview on the three different algorithms see
Fig. 1.
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Figure 3: Error for Pareschi-Russo IMEX problem pareschi2000implicit , see Eq.
(20), at $T_{end}=5$ with $4^{th}$ (top), $6^{th}$ (middle) and $8^{th}$
(bottom) order IMEX-MD scheme with $k_{\max}=9$ with different stiffness
parameters $\varepsilon\in\\{10^{0},10^{-2},10^{-3}\\}$.
In order to investigate the different behaviors for stiff and non-stiff
problems, $\varepsilon$ in Eq. (20) is varied from $\varepsilon=1$ to
$\varepsilon=10^{-3}$. The comparison is done for all three quadrature rules
described in Eq. (2)–(4). A fixed $k_{\max}=9$ is used. In Fig. 3 the results
of the calculations are summarized. One can clearly see that the parallel low-
order scheme is always second order accurate, regardless of the equipped
quadrature rule and stiffness of the problem (orange lines). Additionally, the
error of the parallel-low order scheme is always higher than with the other
schemes. Considering the high-order parallel and the original serial scheme,
the figure shows that the modifications introduced in Alg. 1 compared to the
original algorithm only slightly influence the obtained error and have no
effect on the achieved order (compare red and green lines). Nevertheless, some
differences between Alg. 1 and the original algorithm are present, mostly
showing up for large timesteps and non-stiff problems. Here, the original
algorithm gives slightly better results. This is most probably due to the very
few timesteps used for large $\Delta t$ (the coarsest resolution uses only
four timesteps). Differently to the original algorithm, the information of the
last corrector step has not yet reached the first corrector step in those
settings, see Fig. 1 for illustration. This reduces the accuracy of the
algorithm if only a few timesteps are performed. Moreover, in the original
algorithm a better $w^{n}$ is used for all steps $k\neq k_{\max}$ what
naturally has a favorable influence on the solution.
One can see that while for the fourth order scheme no order reduction can be
observed for stiff problems, $\text{HBPC}(6,k_{\max})$ and
$\text{HBPC}(8,k_{\max})$ show order reduction for an increasing stiffness. In
SealSchuetz19 it has been shown, that increasing the number of correction
steps can cure this problem. Therefore, we perform a convergence study with
respect to the number of correction steps $k_{\max}$ for the stiffest setting
$\varepsilon=10^{-3}$ with $\text{HBPC}(6,k_{\max})$ and
$\text{HBPC}(8,k_{\max})$. Fig. 4 shows how the increase of corrector steps
improves the accuracy of the obtained results. It is shown numerically that
for an increasing $k_{\max}$, the predictor-corrector scheme described in Alg.
1 converges towards the limit method, i.e. the fully coupled two-derivative
Runge-Kutta method, see Def. 1.
Summarizing, the presented simulations of the PR problem show that the
modifications of the predictor-corrector scheme from SealSchuetz19 described
in Alg. 1 have only a slight influence on the solution quality. Using the
straightforward possibility to parallelize the scheme (Fig. 1 right) is
clearly inferior to the original method and Alg. 1 even for stiff problems.
Moreover, for the high order methods, it gets obvious that for stiff problems
it can be beneficial to increase the number of corrector steps $k_{\max}$.
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Figure 4: Influence of $k_{\max}$ iterations on error of PR-IMEX problem with
$\varepsilon=10^{-3}$ for Alg. 1 with $6^{th}$ order (left) and $8^{th}$ order
(right) quadrature formula. Limit denotes the corresponding sixth- or eigth-
order Runge-Kutta method as given in Def. 1.
### 5.3 Van-der-Pol Equation
The van-der-Pol equation (vdP) allows us to study the convergence properties
for very stiff problems in more detail. It is given by
$\displaystyle w_{1}^{\prime}(t)$ $\displaystyle=w_{2},\qquad
w_{2}^{\prime}(t)=\frac{(1-w_{1}^{2})w_{2}-w_{1}}{\varepsilon},\qquad
w_{0}=\left(2,\frac{-2}{3}+\frac{10}{81}\varepsilon\right),$ (21)
with $T_{\text{end}}=0.5$. Again, we split the equation into a non-stiff
explicitly treated part $\Phi_{\text{E}}$ and a stiff, implicitly treated part
$\Phi_{\text{I}}$ via
$\displaystyle\Phi_{\text{E}}=\begin{pmatrix}w_{2}\\\
0\end{pmatrix},\qquad\Phi_{\text{I}}=\frac{1}{\varepsilon}\begin{pmatrix}0\\\
(1-w_{1}^{2})w_{2}-w_{1}\end{pmatrix}.$
We use this test problem to study the convergence properties with a typical
number of correction steps one would use for practical applications for stiff
and non-stiff problems. The calculations are performed on one Intel skylake
node with $36$ cores. We compare calculations with using all processors on the
node ($k_{\max}=71$), to half the number of the processors on the node
($k_{\max}=35$). We choose the minimum number of iterations to obtain the
desired order for non-stiff problems, ($k_{\max}=3$, $k_{\max}=5$ and
$k_{\max}=7$, respectively for the $4^{th}$, $6^{th}$ and $8^{th}$ order
method). Similar to the previously performed evaluations with the PR test
problem, we compare the obtained results with $k_{\max}\rightarrow\infty$ and
the limiting method from Def. 1.
In order to compute the limiting method as $k_{\max}\rightarrow\infty$, we
compare the numerical errors w.r.t. the exact solution of simulations with
$k_{\max}$ and $k_{\max}/2$. If the relative difference between those two
errors differs by more than $1\%$, $k_{\max}$ is increased by a factor of two
and the calculation is repeated. That means, repeat the simulation with
$2k_{\max}$ if
$\displaystyle\frac{\left|\left(\|w-w_{h}\|_{2}\right)_{k_{\max}}-\left(\|w-w_{h}\|_{2}|\right)_{k_{\max}/2}\right|}{\left(\|w-w_{h}\|_{2}\right)_{k_{\max}}}>0.01,$
else use the solution with corresponding $k_{\max}$ as the “converged”
solution. With this investigation we can experimentally show that our method
converges towards the limiting Runge-Kutta method, i.e. the algorithm does not
diverge.
###### Remark 8
Note that a similar strategy can be pursued to obtain an adaptation criterion
for the IMEX-MD scheme. Since the exact solution is not known for general
problems, one could think about evaluating the change of the numerical
solution after each iteration. If the iterates do not differ more than a user-
defined threshold, no further correction steps are performed. Also timestep
adaptation could be taken into account. We leave the detailed analysis of an
adaptation of Alg. 1 in this direction to future investigations.
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Figure 5: Error for van-der-Pol equation at $T_{end}=0.5$ of simulations with
$\text{HBPC}(6,k_{\max})$. Orders 4 and 6 are omitted for brevity, but show
similar behavior. In each image, we increase the level of stiffness of the
problem by shrinking $\varepsilon$, and in each frame we consider the impact
of increasing the total number of correction steps $k_{\max}$. The last
picture (second on the second line) shows the results of solving the fully
coupled limiting Runge-Kutta method, (cf. Def. 1). The first image on the
second line shows the numerically determined limit if
$k_{\max}\rightarrow\infty$. This is in excellent agreement with the
underlying Runge-Kutta method in the right frame, as we expect to be the case
when the iterates converge.
In Fig. 5 the results of the simulation are summarized for the $6^{th}$-order
method. For an increasing stiffness, order reduction can be observed. This
can, at least partially, be cured by increasing the number of correction
steps; perfect convergence cannot be observed. This is consistent with the
limiting method and one can see that with $k_{\max}\rightarrow\infty$ the
limiting method is reached. The results also extend to the $4^{th}$\- and
$8^{th}$-order scheme (not shown here for the sake of brevity): small values
of $k_{\max}$ produce order reduction for some regimes. The effect can be
mitigated by increasing the number of iterates with $k_{\max}\to\infty$. For
the $4^{th}$-order limiting-scheme, order reduction is completely ruled out,
which is not the case for $6^{th}$ and $8^{th}$-order schemes.
## 6 Improved algorithm and scaling results
Given the flexibility of the solver, there are several opportunities to
enhance the proposed Alg. 1. In this subsection, we present one such variation
that improves the accuracy and scalability of the solver by making a total of
two modifications:
* •
Improved predictor. The predictor in Alg. 1 is completely independent of the
correction steps, as the used initial condition is given by the previously
computed predictor results. A minor modification would be to use the more
accurate result from the first (or higher) correction from the previous time
step. This is advantageous whenever the main computational load lies on the
predictor. This could happen, e.g., if the number of Newton iterations for the
predictor is higher than for the corrector, which is often observed in
practice. Moreover, the predictor could, in principle, be leveraged as the
cornerstone for adaptive timestepping, which is crucial for many large-scale
applications. Based on our observations, we find we can accomplish this and
have a predictor that is overall third-order accurete in time, which reduces
the total number of corrections required.
* •
Improved quadrature. The $l-$th stage of the $(k)$-th corrector with $l>1$
uses a quadrature rule $\mathcal{I}_{l}$ in Alg. 1 that is still fully
computed with values from correction step $k$, although for $\iota<l$, the
values $w^{n,[k+1],{\iota}}$ are readily available. If we consider a Gauss-
Seidel type approach, similar to the one found in Crockatt2018 , we change
this so that in $\mathcal{I}_{l}$, we replace $w^{n,[k],{\iota}}$ by
$w^{n,[k+1],{\iota}}$ for $\iota<l$. Here, we show that that this not only
leads to a more accurate scheme, but it is also lower storage.
We formalize the proposed changes to Alg. 1 as its own algorithm.
###### Algorithm 2 ($\text{HBPC*}(q,k_{\max},$))
To advance the solution to Eq. (1) in time, we compute values $w^{n,[k],{l}}$.
The meaning of the parameters $n$, $k$ and $l$ is explained in Eq. (6) and
thereafter. To account for the initial conditions, define
$\displaystyle w^{-1,[k],{s}}:=w_{0}.$
First, the values $w^{n,[0],{l}}$ are filled using a straightforward second-
order IMEX-Taylor method with an improved initial condition.
1. 1.
Predict. Solve the following expression for $w^{n,[0],{l}}$ and $1\leq l\leq
s$:
$\displaystyle\begin{split}w^{n,[0],{l}}&:={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}w^{n-1,[1],{s}}}\\\
&+c_{l}\Delta
t\left(\Phi_{\text{I}}^{n,[0],{l}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Phi_{\text{E}}^{n-1,[1],{s}}}\right)+\frac{(c_{l}\Delta
t)^{2}}{2}\left({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\overset{\boldsymbol{.}}{\Phi}_{\text{E}}^{n-1,[1],{s}}}-\overset{\boldsymbol{.}}{\Phi}_{\text{I}}^{n,[0],{l}}\right).\end{split}$
(22)
The modified terms (${w^{n-1,[1],{s}}},{\Phi_{\text{E}}^{n-1,[1],{s}}}$, and
${\overset{\boldsymbol{.}}{\Phi}_{\text{E}}^{n-1,[1],{s}}}$, marked in blue)
simply use one higher iterate than the one found in Alg. 1. This makes the
predictor third-order accurate due to its improved local truncation error.
2. 2.
Correct. Next, the correction steps take place to fill the values of
$w^{n,[k],{l}}$ for $1\leq k\leq k_{\max}$. Solve the following for
$w^{n,[k+1],{l}}$, for each $2\leq l\leq s$ and each $0\leq k<k_{\max}$:
$\displaystyle\begin{split}w^{n,[k+1],{1}}&:={\color[rgb]{1,0.19,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.19,1}\pgfsys@color@cmyk@stroke{0}{0.81}{0}{0}\pgfsys@color@cmyk@fill{0}{0.81}{0}{0}w^{n-1,[k+2],{s}}},\\\
w^{n,[k+1],{l}}&:={\color[rgb]{1,0.19,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.19,1}\pgfsys@color@cmyk@stroke{0}{0.81}{0}{0}\pgfsys@color@cmyk@fill{0}{0.81}{0}{0}w^{n-1,[k+2],{s}}}\\\
&+{\Delta
t}\left(\Phi_{\text{I}}^{n,[k+1],{l}}-\Phi_{\text{I}}^{n,[k],{l}}\right)-\frac{\Delta
t^{2}}{2}\left(\overset{\boldsymbol{.}}{\Phi}_{\text{I}}^{n,[k+1],{l}}-\overset{\boldsymbol{.}}{\Phi}_{\text{I}}^{n,[k],{l}}\right)\\\
&+\;\mathcal{I}_{l}(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Phi^{n,[k+1],{0}},\ldots,\Phi^{n,[k+1],{l-1}},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Phi^{n,[k],{l}},\ldots,\Phi^{n,[k],{s}})\end{split}$
(23)
The red term(s) ($w^{n-1,[k+2],{s}}$) are the same modified values we
implemented in Alg. 1. The blue terms
$\Phi^{n,[k+1],{0}},\ldots,\Phi^{n,[k+1],{l-1}}$ are evaluated at iterate
$k+1$ instead of $k$. This produces smaller errors and requires less storage.
Again, if $k=k_{\max}-1$, then the $k+2$ in the red terms are replaced by
$k_{\max}$ in order to close the recursion.
3. 3.
Update. In order to retain a first-same-as-last property, we update the
solution with
$\displaystyle w^{n+1}:=w^{n,[k_{\max}],{s}}.$
The parallelization strategy suggested in Fig. 1 remains largely unaltered due
to the suggested grouping. Given that the first thread operates on both the
$k=0$ and the $k=1$ iterates, the predictor simply draws information from the
$k=1$ iterate instead of the $k=0$ iterate. The corrector also remains
unaltered - in fact there is slightly looser coupling given that the
quadrature rule uses information that is being updated.
In the following, we explore the impact of the further modifications to the
algorithm numerically. We find that in most cases the modified algorithm
behaves better than the standard one. Scaling results are therefore given on
the basis of Alg. 2 only.
### 6.1 Influence of the modifications
We study the influence of the two modifications with previously considered
Pareschi-Russo problem (Eq. (20)) with $\varepsilon=1$. In Fig. 6 the results
with Alg. 1 and the modified Alg. 2 are compared side-by-side. Although the
modification in the predictor and the corrector are presented simultaneously,
we carry out a separated analysis of both changes in order to highlight the
individual influence.
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Figure 6: Error for Pareschi-Russo IMEX problem pareschi2000implicit with
$\varepsilon=1$ at $T_{end}=5$ with our IMEX-MD schemes. We use a fixed
$k_{\max}=9$ for all the simulations. We display the impact of increasing the
iterate number $k$ and the fact that each additional correction improves the
overall order of accuracy by one, save the final one. We compare side-by-side
the analytically investigated Alg. 1 (Left column) and the modified Alg. 2
(Right column). The modifications made for Alg. 2 increase the order of the
predictor and therefore the following correction steps by one until the
maximum order of convergence is reached. Moreover, the errors especially for
large timesteps are significantly reduced with Alg. 2.
We make the following observations concerning the additional modifications:
* •
Modification of the predictor. The modification of the predictor has the
drawback that the predictor loses its independence from the correction steps
as it now uses the solution of the first corrector step as the solution at
$t^{n}$. In Fig. 6 we see the influence of modifying the predictor step: the
predictor’s order of accuracy is increased from second to third order. This
results in an additional order of the following correction steps, of course
only until the maximum achievable order is reached. For the $6^{th}$\- and
$8^{th}$-order method we additionally observe improvements in the error found
after the last iteration.
* •
Modification of the corrector. The primary advantage in the modification in
the corrector is that less overall storage is required for the solver. Simply
put, as the stage values of the previous iteration are replaced with the stage
values from the current iteration $k+1$, no additional storage array for the
old stage values is required. Moreover, potentially better values are used for
the quadrature rule. We call this modification “Gauß-Seidel style” as “better”
information is used as soon as it is available Saad2003 . Note that similar
ideas have been pursued for an integral deferred correction method in
Crockatt2018 .
In Fig. 6 we can also observe the influence of using the Gauß-Seidel style
corrector by comparing the iterates to each other. The improved quadrature
significantly decreased the error. This effect is especially visible for large
timesteps since each correction step improves the solution with
$\mathcal{O}(\Delta t)$ (of course only until the maximum order of convergence
is reached). Although not shown here, this effect is even more pronounced for
stiff problems. Moreover, the required amount of Newton iterations is
significantly reduced for such cases that require large timesteps. This could
certainly be beneficial for multiscale problems. Hence, using values of the
predictor that are “one correction step better” as soon as they are available
facilitates the solution of the non-linear problem and at the same time allows
for more accurate results.
### 6.2 Parallel performance
We use the improved Alg. 2 to test the parallel performance of the proposed
method. In order to do this, we run Alg. 2 in both serial and parallel
settings. The method is implemented in MATLAB using the parallel computing
toolbox ParallelToolbox . Calculations are done on one Intel skylake node with
2 Xeon Gold 6140<EMAIL_ADDRESS>GHz, with 18 cores each, provided by the Vlaams
Supercomputing Centrum (VSC). The algorithm necessitates the solution of a
non-linear system of equations
$\displaystyle F(\tilde{w}):=f(\tilde{w})-\text{rhs}=0,$
with
$\displaystyle\tilde{w}$ $\displaystyle=w^{n,[0],{l}},\quad
f(\tilde{w})=w^{n,[0],{l}}-c_{l}\Delta
t\left(\Phi_{\text{I}}^{n,[0],{l}}\right)+\frac{(c_{l}\Delta
t)^{2}}{2}\left(\overset{\boldsymbol{.}}{\Phi}_{\text{I}}^{n,[0],{l}}\right),$
rhs $\displaystyle=w^{n-1,[1],{s}}+c_{l}\Delta
t\left(\Phi_{\text{E}}^{n-1,[1],{s}}\right)+\frac{(c_{l}\Delta
t)^{2}}{2}\left(\overset{\boldsymbol{.}}{\Phi}_{\text{E}}^{n-1,[1],{s}}\right),$
for the predictor and
$\displaystyle\tilde{w}$ $\displaystyle=w^{n,[k+1],{l}},\quad
f(\tilde{w})=w^{n,[k+1],{l}}-\Delta
t\left(\Phi_{\text{I}}^{n,[k+1],{l}}\right)+\frac{\Delta
t^{2}}{2}\left(\overset{\boldsymbol{.}}{\Phi}_{\text{I}}^{n,[k+1],{l}}\right),$
rhs $\displaystyle=w^{n-1,[k+2],{s}}-\Delta
t\left(\Phi_{\text{I}}^{n,[k],{l}}\right)+\frac{\Delta
t^{2}}{2}\left(\overset{\boldsymbol{.}}{\Phi}_{\text{I}}^{n,[k],{l}}\right)+\mathcal{I}_{l},$
for the corrector. We choose to solve this using a damped Newton’s method with
starting point $\tilde{w}_{0}=w^{n-1,[1],{s}}$ for the predictor and
$\tilde{w}_{0}=w^{n-1,[k+2],{s}}$ for the corrector. We define a relative
convergence criterion of
$\displaystyle\frac{\|F(\tilde{w})\|_{2}}{\|F(\tilde{w}_{0})\|_{2}}\leq\varepsilon_{\text{Newton}}=10^{-6}$
and an absolute convergence criterion of
$\|F(\tilde{w})\|_{2}\leq\varepsilon^{\prime}_{\text{Newton}}=10^{-14}$. A
maximum of $1000$ iterations is allowed. Starting with a damping factor of
$1$, the damping factor is halved if one Newton step’s residual exceeds $0.9$
times the previous Newton step’s residual. If none of the convergence criteria
is met after $1000$ iterations, we mark the solver as converged. Typically,
when this happens it is then due to machine accuracy, and not to a lack of
convergence for Newton’s method.
The hierarchical structure of the method allows us to improve the iterative
scheme by replacing the initial $\tilde{w}$ with $w^{n,[k],{l}}$ for the
correction steps. This improves the starting point of Newton’s method and can
reduce the amount of Newton iterations. We could also think about replacing
$\tilde{w}_{0}$ with $w^{n,[k],{l}}$. This would have an influence on the
results presented below. Nevertheless, we observe that these changes do not
significantly alter the reported results. Hence, we can only conclude that the
proposed method is quite robust against such algorithmic variants.
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Figure 7: Speed-up over number of timesteps for different $k_{\text{max}}$
with non-stiff Pareschi-Russo problem (solid lines) and stiff van-der-Pol
equation (dashed lines) for $\text{HBPC*}(4,k_{\max})$ (red),
$\text{HBPC*}(6,k_{\max})$ (orange) and $\text{HBPC*}(8,k_{\max})$ (green)
with the more efficient Alg. 2. Black line indicates theoretically achievable
speed-up $\frac{N\cdot(k_{\max}+1)}{2N+k_{\max}-1}$.
The results regarding speed-up are reported in Fig. 7 for an increasing number
of correction steps $k_{\max}$ and hence also an increasing number of
processors ranging from $\\#\text{procs}=2$ ($k_{\max}=3$) to
$\\#\text{procs}=36$ ($k_{\max}=71$). We choose a non-stiff (PR-IMEX with
$\varepsilon=1$) and a stiff (vdP-IMEX with $\varepsilon=10^{-3}$) problem to
see if this has an influence on the parallel performance. All calculations are
done with the $4^{th}$, $6^{th}$ and $8^{th}$ order method. The figure shows
that the achieved speed-ups are quite similar for both considered problems,
but differently for the three different quadrature rules: with
$\text{HBPC*}(8,k_{\max})$ the highest speed-up can be achieved. This is most
probably caused by the more ‘processor-local’ work due to the more stages
compared to the other considered schemes. As the communication introduces some
overhead, the scaling properties of the algorithm are increased by performing
more processor-local operations for an almost similar amount of communication.
This is a typical observation made when parallelizing numerical methods. It
has, e.g., also been observed for high-order discontinuous Galerkin methods
where a better parallel performance can be achieved if higher order ansatz
polynomials are chosen as they increase the amount of processor-local work
MunzDG12 . As the parallel performance results are almost the same for the
stiff and non-stiff problem, we anticipate that the obtained performance gain
is transferable to other problems.
One can see that if too few timesteps are used, no speed-up is achieved and a
deceleration of the simulation is obtained. This can be caused by the overhead
introduced by establishing the communication. Nevertheless, if more timesteps
are used, a significant speed-up can be achieved with the proposed
parallelization strategy. Considering for example the case with $36$
processors ($k_{\max}=71$), one can achieve a speed-up of up to a factor of
$\approx 13$ for $\text{HBPC*}(8,k_{\max})$, $\approx 11$ for
$\text{HBPC*}(6,k_{\max})$ and up to $\approx 8$ for
$\text{HBPC*}(4,k_{\max})$. These values are in the same range as reported in
the review paper by Ong and Schroder OngSchroder2020 . In terms of their
paper, our method belongs to the class of direct time-parallel methods.
The maximum theoretical speed-up is not achieved for all cases. There can be
two reasons for this: First, the overhead introduced by the communication
slows down the computations. Second, the processor-local work is distributed
unevenly. The first point is strongly influenced by the current implementation
and architecture and is therefore beyond the scope of this work. To gain more
insight into the amount of processor-local work we consider the amount of
Newton iterations performed by each processor.
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Figure 8: Required amount of Newton iterations for different correction steps
(grouped by the processor on which they are performed) and two selected test
setups (left: PR-IMEX with $\varepsilon=1$; right: vdP-IMEX with
$\varepsilon=10^{-3}$) over the number of timesteps.
In Fig. 8 the Newton iterations on the first $6$ and the last processor are
shown for the $\text{HBPC*}(8,k_{\max})$ scheme for the non-stiff PR-IMEX
problem and the stiff vdP-IMEX equation. It is shown that the required amount
of iterations decreases with an increasing index of the correction step, i.e.
with the index of the processor. Especially the first processor which is
responsible for the predictor and the first corrector step requires
significantly more iterations than the other processors. This trend is even
enhanced for the considered stiff problem. To cure this imbalance one can
either think about a different distribution of the processors than presented
in Fig. 1, or a termination criterion for Newton’s method depending on the
current index of the correction step.
### 6.3 Arenstorf Orbit
Finally, we use the Arenstorf orbit problem HaiWan1 ; OngSpiteri for further
illustration of the proposed method’s capabilities. The Arenstorf orbit
problem describes a three body problem, where the movement of a light object
is influenced by two heavy objects. This can, for example, be a satellite
being influenced by two planets. The problem is actually a second-order
differential equation, here formulated as a system of first-order ODEs:
$\displaystyle w^{\prime}=\begin{pmatrix}w_{3}\\\ w_{4}\\\
w_{1}+2w_{4}-\mu^{\prime}\frac{w_{1}+\mu}{D_{1}}-\mu\frac{w_{1}-\mu^{\prime}}{D_{2}}\\\
w_{2}-2w_{3}-\mu^{\prime}\frac{w_{2}}{D_{1}}-\mu\frac{w_{2}}{D_{2}}\end{pmatrix}.$
Here, we have defined
$\displaystyle D_{1}$
$\displaystyle:=\left((w_{1}+\mu)^{2}+w_{2}^{2}\right)^{\frac{3}{2}},\quad
D_{2}:=\left((w_{1}-\mu^{\prime})^{2}+w_{2}^{2}\right)^{\frac{3}{2}},$
$\displaystyle\mu$ $\displaystyle:=0.012277471,\quad\mu^{\prime}:=1-\mu,$
and the initial conditions
$\displaystyle w_{0}=\left(0.994,~{}0,~{}0,~{}-2.001585106379\right)^{T}.$
Although there is no multi-scale character of the problem, we artificially
split the equation into an explicit and an implicit part. For that purpose,
all parts of the equation which are divided by $D_{1}$ or $D_{2}$ are simply
treated implicitly, the remaining parts are treated explicitly.
The solution is a closed orbit with a period of $17.065216560159$. Similar to
OngSpiteri , we choose $10^{5}$ equidistant timesteps to simulate one period
of the problem. We choose the $\text{HBPC*}(8,k_{\max})$ method with Alg. 2
for our calculations, which results in an error after one period of
$\|w-w_{0}\|_{2}=1.7818\cdot 10^{-9}$ for $k_{\max}=71$.
Figure 9: Solution of the Arenstorf orbit problem for the two spatial
variables. $10^{5}$ timesteps of the $8^{th}$ order IMEX-MD scheme described
in Alg. 2 are used to simulate one period.
In Fig. 9 the solution of $w_{1}$ and $w_{2}$ for the Arenstorf orbit problem
is shown. One can see that after one period, the initial condition is reached
with good accuracy and a periodic orbit is obtained. With $k_{\max}=71$, and
hence choosing $36$ processors, the speed-up is $\approx 14$.
To obtain a comparison in a “real-world” setting we consider the scenario of
having a serial code and one performs $7$ correction steps, what is a
reasonable $k_{\max}$ for the $8^{th}$ order method. Having now the
possibility to use a parallel scheme on one compute node with $36$ processors,
one might use $k_{\max}=71$. Comparing those two computing times, one still
obtains a speed-up of $\approx 1.5$ and at the same time a potentially better
solution.
As a very last example, we consider the Arenstorf orbit problem with
relatively few timesteps, $N=5{,}000$. (Note that in this work, we only use
timesteps that are uniformly spaced. The Arenstorf orbit is an example of a
difficult problem that asks for adaptive timestepping, which is beyond the
scope of this work.) We use the $8^{th}$ order scheme and $k_{\max}=7$. In
Fig. 10 we plot the orbits for the predictor and the last correction step,
once for Alg. 1 (left) and once for Alg. 2 (right). All other parameters are
exactly the same. It is clear that at least the predictor of Alg. 1 is rather
useless here, and also the last correction value is not a closed orbit. On the
other hand, the modifications done in Alg. 2 lead to a better solution, where
both predictor and the last correction step are, in the ‘eyeball-norm’, closed
orbits. This once again shows the superiority of Alg. 2 over its more
straightforward counterpart.
Figure 10: Solution of the Arenstorf orbit problem for the two spatial
variables. A total of $N=5{,}000$ timesteps of the $8^{th}$ order IMEX-MD
scheme described in Alg. 1 (left) and in Alg. 2 (right) are used to simulate
until final time $T=17.065216560159$, which should correspond to one period.
The improved algorithm produces something that is close to what is expected to
be a closed orbit, whereas the first algorithm is far from the exact solution,
with the predictor being wildly off.
## 7 Conclusions and outlook
In this work, we have developed a novel IMEX solvers for the numerical
treatment of ODEs that operate on multiple derivatives of the ODE’s flux
function. The algorithm has been specifically designed to be
* •
of high-order (we showed results until order eight, higher orders are easily
achievable by increasing the number of collocation points or the number of
derivatives used),
* •
and parallelizable in time.
There are two flavors of the method, one (Alg. 1) is a softer modification to
our previous result SealSchuetz19 that is more ammenable to analysis, and the
second one (Alg. 2) is more storage efficient and has better scaling results.
We have shown numerical results demonstrating the behavior of the algorithms.
In the light of OngSchroder2020 , the scaling results seem to be very much in
line with other state-of-the-art methods.
There are multiple important open areas that need to be addressed with future
work. The most accessible problem to consider would be to increase the total
number of derivatives used, or work on different integration points $c$. Of
notable interest would be to explore collocation solvers described in Ex. 1
constructed from the so-called Gauss-Turan-type StroudStancu1965 quadrature
rules. These solvers would ideally offer low-storage alternatives to the
present solvers. Next, it would be interesting to explore versions of this
solver that make use of arbitrary multiderivative Runge-Kutta methods, not of
the collocation variety. In addition, it is imperative that we further explore
implementations for PDEs. To date, multiderivative methods have largely been
sidelined, even though they can outperform lauded and highly optimized MATLAB
routines such as the builtin ode15s (cf. AbdiConte2020 for one such case
study). One reason for this lack of broader interest is arguably the difficult
computation of the second (and third, …) derivatives in practical
applications. For some ODEs stemming from a semi-discrete PDE, this can be
done rather elegantly MultiDerHDG2015 , but for many, it is a tedious task,
which requires leveraging Lax-Wendroff type time discretizations. We therefore
suggest exploring the possibilities of using finite-difference approximations,
as done in BAEZA201887 ; ZorioEtAl in the context of explicit Taylor methods.
It is not yet clear how this interacts with the stability properties of the
overall method, in particular for highly stiff problems, as these type of
discretizations require fully discrete stability analyses, but is certainly
worth looking into. Furthermore, the extension and testing of this time-
stepping option to the classical IMEX application areas remains to be
explored.
###### Acknowledgements.
This study was initiated during a research stay of D.C. Seal at the University
of Hasselt, which was supported by the Special Research Fund (BOF) of Hasselt
University. Additional funding came from the Office of Naval Research, grant
number N0001419WX01523 and N0001420WX00219. J. Zeifang was partially supported
by the German Research Foundation (DFG) through the project GRK 2160/1
“Droplet Interaction Technologies”. The HPC-resources and services used in
this work were provided by the VSC (Flemish Supercomputer Center), funded by
the Research Foundation - Flanders (FWO) and the Flemish Government.
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|
††thanks: Corresponding author
# Investigation of $P_{cs}(4459)^{0}$ pentaquark via its strong decay to
$\Lambda J/\psi$
K. Azizi<EMAIL_ADDRESS>Department of Physics, University of Tehran,
North Karegar Avenue, Tehran 14395-547, Iran Department of Physics, Doğuş
University, Acıbadem-Kadıköy, 34722 Istanbul, Turkey School of Particles and
Accelerators, Institute for Research in Fundamental Sciences (IPM) P.O. Box
19395-5531, Tehran, Iran Y. Sarac<EMAIL_ADDRESS>Electrical and
Electronics Engineering Department, Atilim University, 06836 Ankara, Turkey
H. Sundu<EMAIL_ADDRESS>Department of Physics, Kocaeli
University, 41380 Izmit, Turkey
###### Abstract
Recently the observation of a new pentaquark state, the hidden-charmed strange
$P_{cs}(4459)^{0}$, was reported by the LHCb Collaboration. The spin-parity
quantum numbers of this state were not determined as a result of insufficient
statistics. To shed light on its quantum numbers, we investigate its decay,
$P_{cs}(4459)^{0}\rightarrow J/\psi\Lambda$, the mode that this state has been
observed, within the QCD sum rule framework. We obtain the width of this decay
assigning the spin-parity quantum numbers of $P_{cs}(4459)^{0}$ state as
$J^{P}=\frac{1}{2}^{-}$ and its substructure as diquark-diquark-antiquark. To
this end, we first calculate the strong coupling constants defining the
considered decay and then use them in the width calculations. The obtained
width is consistent with the experimental observation, confirming the quantum
numbers $J^{P}=\frac{1}{2}^{-}$ and compact pentaquark nature for
$P_{cs}(4459)^{0}$ state.
## I Introduction
In the past two decades, starting with the observation of the $X(3872)$
Choi:2003ue , we witnessed the observations of many exotic hadrons candidates
for tetraquarks Zyla:2020zbs and pentaquarks Aaij:2015tga ; Aaij:2016ymb ;
Aaij:2019vzc . The first observation of pentaquark states was announced in
2015 by the LHCb collaboration Aaij:2015tga and two pentaquark states in
$J/\psi p$ invariant mass spectrum of the $\Lambda_{b}^{0}\rightarrow J/\psi
pK^{-}$ decays were reported with the following resonance parameters
Aaij:2015tga : $m_{P_{c}(4380)^{+}}=4380\pm 8\pm 29~{}\mathrm{MeV}$,
$\Gamma_{P_{c}(4380)^{+}}=205\pm 18\pm 86~{}\mathrm{MeV}$ and
$m_{P_{c}(4450)^{+}}=4449.8\pm 1.7\pm 2.5~{}\mathrm{MeV}$,
$\Gamma_{P_{c}(4450)^{+}}=39\pm 5\pm 19~{}\mathrm{MeV}$. The LHCb
Collaboration supported this observation later, in 2016, with a full amplitude
analysis for $\Lambda_{b}^{0}\rightarrow J/\psi p\pi^{-}$ decays Aaij:2016ymb
. In 2019, a new pentaquark resonance, $P_{c}(4312)^{+}$, was reported by the
LHCb Collaboration with the following mass and width Aaij:2019vzc :
$m_{P_{c}(4312)^{+}}=4311.9\pm 0.7^{+6.8}_{-0.6}~{}\mathrm{MeV}$ and
$\Gamma_{P_{c}(4312)^{+}}=9.8\pm 2.7^{+3.7}_{-4.5}~{}\mathrm{MeV}$. Together
with the $P_{c}(4312)^{+}$ state, the LHCb also announced the split of the
peak corresponding to $P_{c}(4450)^{-}$ into two peaks which have the
following masses and widths: $m_{P_{c}(4440)^{+}}=4440.3\pm
1.3^{+4.1}_{-4.7}~{}\mathrm{MeV}$, $\Gamma_{P_{c}(4440)^{+}}=20.6\pm
4.9^{+8.7}_{-10.1}~{}\mathrm{MeV}$ and $m_{P_{c}(4457)^{+}}=4457.3\pm
0.6^{+4.1}_{-1.7}~{}\mathrm{MeV}$, $\Gamma_{P_{c}(4457)^{+}}=6.4\pm
2.0^{+5.7}_{-1.9}~{}\mathrm{MeV}$ Aaij:2019vzc . These observations and the
advances in experimental facilities and techniques indicate the possibility to
observe more exotic states in the future.
On the other hand, there is still uncertainties in the sub-structures and
quantum numbers of these observed pentaquark states. In that matter, there are
different proposals and theoretical works about these resonances in the
literature analyzing their parameters and giving consistent predictions with
their observed properties. It is obvious that deeper investigations are
required not only to differentiate these proposals but also to help better
identify the nature of these states. Understanding the inner structures and
properties of these exotic states may also support their future
investigations. Besides, they may provide improvements in understanding the
dynamics of the quantum chromodynamics (QCD) in its nonperturbative domain.
With their non-conventional quark substructures that are different from the
conventional baryons composed of three quarks/antiquarks or mesons composed of
a quark and an antiquark, they provide an attractive ground for the
understanding of the nonperturbative nature of strong interaction. Although
the investigations of such exotic states extend before their observations,
with their observations the pentaquarks have become a hot topic in all these
respects. With these motivations and the excitement brought by their
observations, their various properties were investigated widely with different
approaches to shed light on their nonspecific sub-structures and quantum
numbers. Based on their close masses to meson-baryon threshold, they were
assigned as meson baryon molecular states in Refs. Chen:2015loa ; Chen:2015moa
; He:2015cea ; Meissner:2015mza ; Roca:2015dva ; Azizi:2016dhy ; Azizi:2018bdv
; Azizi:2020ogm ; Chen:2020opr . They were interpreted with diquark-diquark-
antiquark Lebed:2015tna ; Li:2015gta ; Maiani:2015vwa ; Anisovich:2015cia ;
Wang:2015ava ; Wang:2015epa ; Wang:2015ixb ; Ghosh:2015ksa ; Wang:2015wsa ;
Zhang:2017mmw ; Wang:2019got ; Wang:2020rdh ; Ali:2020vee ; Wang:2016dzu and
diquark-triquark Wang:2016dzu ; Zhu:2015bba models. To investigate their
properties in Ref. Liu:2017xzo a variant of the D4-D8 model and in Ref.
Scoccola:2015nia the topological soliton model were used. They were also
explained as kinematical effects Guo1 ; Guo2 ; Mikhasenko:2015vca ; Liu1 ;
Bayar:2016ftu . Besides the observed ones, the possible other candidate
pentaquark states were also considered in the literature with different quark
contents Liu:2020cmw ; Chen:2015sxa ; Feijoo:2015kts ; Lu:2016roh ;
Irie:2017qai ; Chen:2016ryt ; Zhang:2020cdi ; Paryev:2020jkp ; Gutsche:2019mkg
; Azizi:2017bgs ; Cao:2019gqo ; Azizi:2018dva ; Zhang:2020vpz ; Wang:2020bjt ;
Xie:2020ckr .
Recently, in a talk, implications of LHCb measurements and future prospects,
the evidence for a pentaquark including a strange quark in its quark content
was first announced by the LHCb Collaboration Wang111 and later it was
reported in the Ref. Aaij:2020gdg . The $P_{cs}(4459)^{0}$ was observed in
$\Xi_{b}^{-}\rightarrow J/\psi K^{-}\Lambda$ decays with the following
measured mass and width Aaij:2020gdg :
$\displaystyle M=4458.8\pm
2.9^{+4.7}_{-1.1}~{}\mathrm{MeV},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Gamma=17.3\pm
6.5^{+8.0}_{-5.7}~{}\mathrm{MeV},$ (1)
with statistical significance exceeding $3\sigma$ and there is no
determination for its spin parity quantum numbers, yet. With a mass just below
$\Sigma_{c}\bar{D}^{*}$ threshold, the $P_{cs}(4459)^{0}$ was interpreted as
$\bar{D}^{*}\Sigma_{c}$ hadronic molecular state in Ref. Chen:2020uif . The
analyses were conducted using QCD sum rule method and the results supported
its possibility to be $\bar{D}^{*}\Sigma_{c}$ molecular state with either
$J^{P}=\frac{1}{2}^{-}$ or $J^{P}=\frac{3}{2}^{-}$ giving mass values
consistent with the experimentally reported one Chen:2020uif . Molecular
explanation for the $P_{cs}(4459)^{0}$ was also discussed in Ref. Peng:2020hql
using effective field formalism and the masses were predicted considering the
$\bar{D}^{*}\Xi_{c}$ molecular pictures with $J^{P}=\frac{1}{2}$ and
$J=\frac{3}{2}$ as $4469$ MeV and $4453-4463$ MeV, respectively. With these
results the spin of the $P_{cs}(4459)^{0}$ state was suggested to possibly be
$J=\frac{3}{2}$. In Ref. Chen:2020kco molecular interpretation was taken into
account using one-boson-exchange model and $P_{cs}(4459)^{0}$ was interpreted
as a coupled
$\Xi_{c}\bar{D}^{*}/\Xi_{c}^{*}\bar{D}/\Xi_{c}^{{}^{\prime}}\bar{D}^{*}/\Xi_{c}^{*}\bar{D}^{*}$
bound state that has $I(J^{P})=0(\frac{3}{2}^{-})$. In Ref. Wang:2020eep the
mass analysis was made via QCD sum rule approach for a pentaquark state
containing strange quark with an interpolating current in the scalar-diquark-
scalar-diquark-antiquark form. Based on the mass value obtained for the state
as $M=4.47\pm 0.11$ MeV, which was consistent with the experimentally observed
one, $P_{cs}(4459)^{0}$ was assigned to have the quantum numbers
$J^{P}=\frac{1}{2}^{-}$.
As is seen, the quantum numbers for the $P_{cs}(4459)^{0}$ state were not
determined by the experiment, and from different studies there are different
assumptions for its quantum numbers and sub-structure, indicating the
necessity for further investigations of the properties of this state. Inspired
by this, we investigate the $P_{cs}(4459)^{0}$ state through its strong decay
via QCD sum rule method Shifman:1978bx ; Shifman:1978by ; Ioffe81 . This
method has a wide range of applications in the literature, which resulted in
successful predictions consistent with the experimental observations. To
analyze the pentaquark states within the QCD sum rule approach a proper choice
of the interpolating current is necessary. So far, it has been observed that,
in various QCD sum rules analyses for observed pentaquark states, the
different choices of the interpolating fields, either in the molecular form or
in the diquark-diquark antiquark form, were applied. These analyses have
resulted in mass predictions that are consistent with the experimental
observations. In Refs. Wang:2021itn ; Wang:2020rdh , it was pointed out that
the hadronic dressing mechanism, which also works for $X$, $Y$, $Z$ states
Wang:2019iaa , may compromise the interpretation of these states as a molecule
or diquark-diquark-antiquark states considering the result of the QCD sum
rules. This may be attributed to the possibility that the pentaquark states
may have both the diquark-diquark-antiquark and meson-baryon type Fock
components. The pentaquark states may have a typical size of the conventional
baryon with a diquark-diquark-antiquark type kernel, and the strong couplings
to the meson-baryon pairs may cause spending a considerable part of their life
time as meson-baryon molecules. The local interpolating current in the
diquark-diquark-antiquark form can be formed from special superposition of the
meson-baryon type interpolating currents and the opposite is also possible,
that is a meson baryon current can also be written as a special superposition
of diquark-diquark-antiquark type currents that carries the net effect, (see
for instance Ref. Wang:2019got ). To interpolate the pentaquark state, one can
choose either diquark-diquark-antiquark or molecular type currents. Any
current type with the same quark structure and quantum numbers of the
pentaquark’s Fock states may couple to the pentaquark. The main component of
the Fock states may give the sub-structure of the pentaquark. For more
details, we refer to the Refs. Wang:2021itn ; Wang:2020rdh ; Wang:2018waa ;
Wang:2019iaa ; Wang:2019hyc . Therefore, besides mass predictions, the
investigations of their decay mechanisms, using different choices for
interpolating currents, may help to distinguish their substructure with
comparisons of the results to experimental results. In this work, to provide
the width, we first calculate the strong coupling constants defining the decay
$P_{cs}(4459)^{0}\rightarrow J/\psi\Lambda$ using three-point QCD sum rule
approach with an interpolating current in the scalar-diquark-scalar-diquark-
antiquark form of $J^{P}=\frac{1}{2}^{-}$. Then, the obtained results for the
strong coupling constants are used to determine the corresponding width value.
We compare the obtained result with the experimental observation to shed light
on the quantum numbers and quark sub-structure of the considered state.
The organization of the paper is as follows: In next section we give the
details of the QCD sum rule calculations for the strong coupling constants
defining the $P_{cs}(4459)^{0}\rightarrow J/\psi\Lambda$ decay. The numerical
analyses of the obtained sum rules as well as the width of the considered
decay are also presented in Sec. II. Last section is devoted to a summary and
comparison of the obtained result for the width to that of the experiment.
## II The strong decay $P_{cs}(4459)^{0}\rightarrow J/\psi\Lambda$
In this section the details of the calculations for the strong coupling
constants and the width of the strong decay $P_{cs}(4459)^{0}\rightarrow
J/\psi\Lambda$ and their numerical analyses are given. The correlation
function required for the calculations has the following form:
$\Pi_{\mu}(p,q)=i^{2}\int d^{4}xe^{-ip\cdot x}\int d^{4}ye^{ip^{\prime}\cdot
y}\langle
0|\mathcal{T}\\{\eta^{\Lambda}(y)\eta_{\mu}^{J/\psi}(0)\bar{\eta}^{P_{cs}}(x)\\}|0\rangle,$
(2)
where the $\eta^{P_{cs}}$, $\eta^{\Lambda}$ and $\eta^{J/\psi}$ are the
interpolating currents of the considered states which have the same quantum
numbers with these states and $\mathcal{T}$ is used to represent the time
ordering operator. The interpolating currents are given as:
$\displaystyle\eta^{P_{cs}}$ $\displaystyle=$
$\displaystyle\epsilon^{ila}\epsilon^{ijk}\epsilon^{lmn}u^{T}_{j}C\gamma_{5}d_{k}s^{T}_{m}C\gamma_{5}c_{n}C\bar{c}^{T}_{a},$
$\displaystyle\eta^{\Lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\epsilon^{lmn}\sum_{i=1}^{2}\Big{[}2(u^{T}_{l}CA_{1}^{i}d_{m})A_{2}^{i}s_{n}+(u^{T}_{l}CA_{1}^{i}s_{m})A_{2}^{i}d_{n}+(d^{T}_{n}CA_{1}^{i}s_{m})A_{2}^{i}u_{l}\Big{]},$
$\displaystyle\eta_{\mu}^{J/\psi}$ $\displaystyle=$
$\displaystyle\bar{c}_{l}i\gamma_{\mu}c_{l},$ (3)
where the sub-indices, $a,~{}i,~{}j,~{}k,~{}l,~{}m,~{}n$ represent the color
indices $u,~{}d,~{}s,~{}c$ are the quark fields, $C$ is charge conjugation
operator; and $A_{1}^{1}=I$, $A_{1}^{2}=A_{2}^{1}=\gamma_{5}$ and
$A_{2}^{2}=\beta$ is an arbitrary parameter to be determined from the
analyses. The above correlation function is calculated in two representations
which are called hadronic and QCD representations. The QCD sum rules for the
physical quantities are obtained from the matches of the coefficients of the
same Lorentz structures attained on both sides.
In the hadronic representation of the correlation function, the interpolating
currents are treated as operators creating or annihilating the hadronic
states. To proceed in the calculation of this side, complete sets of related
hadronic states that have the same quantum numbers with the given
interpolating currents are inserted inside the correlator. After taking four
integrals the results turn into
$\displaystyle\Pi_{\mu}^{\mathrm{Had}}(p,q)=\frac{\langle
0|\eta^{\Lambda}|\Lambda(p^{\prime},s^{\prime})\rangle\langle
0|\eta_{\mu}^{J/\psi}|J/\psi(q)\rangle\langle
J/\psi(q)\Lambda(p^{\prime},s^{\prime})|P_{cs}(p,s)\rangle\langle
P_{cs}(p,s)|\eta^{P_{cs}}|0\rangle}{(m_{\Lambda}^{2}-p^{\prime
2})(m_{J/\psi}^{2}-q^{2})(m_{P_{cs}}^{2}-p^{2})}+\cdots,$ (4)
where $\cdots$ is used to represent the contributions of higher states and
continuum, the $p$, $p^{\prime}$ and $q$ are the momenta of the $P_{cs}$ and
$\Lambda$ and $J/\psi$ states, respectively. The matrix elements in this
result are defined in terms of the masses and current coupling constants, and
they have the following forms:
$\displaystyle\langle 0|\eta^{P_{cs}}|P_{cs}(p,s)\rangle$ $\displaystyle=$
$\displaystyle\lambda_{P_{cs}}u_{P_{cs}}(p,s).$ $\displaystyle\langle
0|\eta^{\Lambda}|\Lambda(p^{\prime},s^{\prime})\rangle$ $\displaystyle=$
$\displaystyle\lambda_{\Lambda}u_{\Lambda}(p^{\prime},s^{\prime}),$
$\displaystyle\langle 0|\eta_{\mu}^{J/\psi}|J/\psi(q)\rangle$ $\displaystyle=$
$\displaystyle f_{J/\psi}m_{J/\psi}\varepsilon_{\mu},$ (5)
where $\varepsilon_{\mu}$ is the polarization vector and $f_{J/\psi}$ is the
decay constant of the $J/\psi$ state, $\lambda_{P_{cs}}$, $\lambda_{\Lambda}$
are the current coupling constants of the $P_{cs}$ and $\Lambda$ states,
$u_{P_{cs}}$ and $u_{\Lambda}$ are the corresponding spinors, respectively.
$|P_{cs}(p,s)\rangle$ is used to represent one-particle pentaquark state with
negative parity. The matrix element $\langle
J/\psi(q)\Lambda(p^{\prime},s^{\prime})|P_{cs}(p,s)\rangle$ is given in terms
of the coupling constants, $g_{1}$ and $g_{2}$ as
$\displaystyle\langle
J/\psi(q)\Lambda(p^{\prime},s^{\prime})|P_{cs}(p,s)\rangle=\epsilon^{*\nu}\bar{u}_{\Lambda}(p^{\prime},s^{\prime})\big{[}g_{1}\gamma_{\nu}-\frac{i\sigma_{\nu\alpha}}{m_{\Lambda}+m_{P_{cs}}}q^{\alpha}g_{2}\big{]}\gamma_{5}u_{P_{cs}}(p,s).$
(6)
In the next step, the matrix elements given in Eqs. (5) and (6) are placed in
the Eq. (4) and following summations over spins of spinors and polarization
vectors are applied
$\displaystyle\sum_{s}u_{P_{cs}}(p,s)\bar{u}_{P_{cs}}(p,s)$ $\displaystyle=$
$\displaystyle({\not{p}}+m_{P_{cs}}),$
$\displaystyle\sum_{s^{\prime}}u_{\Lambda}(p^{\prime},s^{\prime})\bar{u}_{\Lambda}(p^{\prime},s^{\prime})$
$\displaystyle=$ $\displaystyle({\not{p}^{\prime}}+m_{\Lambda}),$
$\displaystyle\varepsilon_{\alpha}\varepsilon^{*}_{\beta}$ $\displaystyle=$
$\displaystyle-g_{\alpha\beta}+\frac{q_{\alpha}q_{\beta}}{m_{J/\psi}^{2}}.$
(7)
And finally, after the Borel transformation, which is applied to suppress the
contributions coming from higher states and continuum, the result of physical
side is obtained as
$\displaystyle\tilde{\Pi}_{\mu}^{\mathrm{Had}}(p,q)$ $\displaystyle=$
$\displaystyle
e^{-\frac{m_{P_{cs}^{2}}}{M^{2}}}e^{-\frac{m_{\Lambda}^{2}}{M^{\prime
2}}}\frac{f_{J/\psi}\lambda_{\Lambda}\lambda_{P_{cs}}m_{\Lambda}}{m_{J/\psi}(m_{\Lambda}+m_{P_{cs}})(m_{J/\psi}^{2}+Q^{2})}\big{[}-g_{1}(m_{\Lambda}+m_{P_{cs}})^{2}+g_{2}m_{J/\psi}^{2}\big{]}\not\\!pp_{\mu}\gamma_{5}$
(8) $\displaystyle+$ $\displaystyle
e^{-\frac{m_{P_{cs}^{2}}}{M^{2}}}e^{-\frac{m_{\Lambda}^{2}}{M^{\prime
2}}}\frac{f_{J/\psi}\lambda_{\Lambda}\lambda_{P_{cs}}m_{J/\psi}m_{\Lambda}}{(m_{\Lambda}+m_{P_{cs}})(m_{J/\psi}^{2}+Q^{2})}\big{[}g_{1}(m_{\Lambda}+m_{P_{cs}})+g_{2}(m_{\Lambda}-m_{P_{cs}})\big{]}\not\\!p\gamma_{\mu}\gamma_{5}$
$\displaystyle+$ $\displaystyle\mathrm{other~{}structures}+\cdots$
where $M^{2}$ and $M^{\prime 2}$ are the Borel parameters to be determined
from the analyses imposing some necessary criteria and $Q^{2}=-q^{2}$. The
result contains more Lorentz structures than the ones given explicitly in Eq.
(8). However, in the last equation, we present only the ones that are used
directly in the analyses, and the others and the contribution of the excited
states and continuum are represented as $\mathrm{other~{}structures}+\cdots$.
The second representation of the correlation function, the QCD side, is
obtained using the interpolating currents explicitly in the correlation
function. To this end, the possible contractions between the quark fields are
attained using Wick’s theorem that renders the result to the one given in
terms of heavy and light quark propagators as:
$\displaystyle\Pi_{\mu}^{\mathrm{QCD}}(p,p^{\prime},q)$ $\displaystyle=$
$\displaystyle i^{2}\int d^{4}xe^{-ip\cdot x}\int d^{4}ye^{ip^{\prime}\cdot
y}\epsilon^{klm}\epsilon^{i^{\prime}l^{\prime}a^{\prime}}\epsilon^{i^{\prime}j^{\prime}k^{\prime}}\epsilon^{l^{\prime}m^{\prime}n^{\prime}}\frac{1}{\sqrt{6}}\sum_{i=1}^{2}\bigg{\\{}-2Tr[\gamma_{5}CS_{u}^{Tkj^{\prime}}(y-x)CA_{1}^{i}S_{d}^{lk^{\prime}}(y-x)]$
(9) $\displaystyle\times$ $\displaystyle
A_{2}^{i}S_{s}^{mm^{\prime}}(y-x)\gamma_{5}CS_{c}^{Tnn^{\prime}}(-x)C\gamma_{\mu}CS_{c}^{Ta^{\prime}n}(x)C+A_{2}^{i}S_{d}^{mk^{\prime}}(y-x)\gamma^{5}CS_{u}^{Tkj^{\prime}}(y-x)CA_{1}^{i}S_{s}^{lm^{\prime}}(y-x)$
$\displaystyle\times$
$\displaystyle\gamma_{5}CS_{c}^{Tnn^{\prime}}(-x)C\gamma_{\mu}CS_{c}^{Ta^{\prime}n}(x)C+A_{2}^{i}S_{u}^{kj^{\prime}}(y-x)\gamma^{5}CS_{d}^{mk^{\prime}}(y-x)CA_{1}^{i}S_{s}^{lm^{\prime}}(y-x)\gamma_{5}C$
$\displaystyle\times$ $\displaystyle
S_{c}^{Tnn^{\prime}}(-x)C\gamma_{\mu}CS_{c}^{Ta^{\prime}n}(x)C\bigg{\\}},$
where $S_{q}^{ab}(x)=S_{u,d,s}^{ab}(x)$ and $S_{c}^{ab}(x)$ are the light and
heavy quark propagators with the following explicit expressions:
$\displaystyle S_{q}^{ab}(x)=$ $\displaystyle
i\frac{x\\!\\!\\!/}{2\pi^{2}x^{4}}\delta_{ab}-\frac{m_{q}}{4\pi^{2}x^{2}}\delta_{ab}-\frac{\langle\overline{q}q\rangle}{12}\Big{(}1-i\frac{m_{q}}{4}x\\!\\!\\!/\Big{)}\delta_{ab}-\frac{x^{2}}{192}m_{0}^{2}\langle\overline{q}q\rangle\Big{(}1-i\frac{m_{q}}{6}x\\!\\!\\!/\Big{)}\delta_{ab}$
$\displaystyle-\frac{ig_{s}G_{ab}^{\theta\eta}}{32\pi^{2}x^{2}}\Big{[}x\\!\\!\\!/\sigma_{\theta\eta}+\sigma_{\theta\eta}x\\!\\!\\!/\Big{]}-\frac{x\\!\\!\\!/x^{2}g_{s}^{2}}{7776}\langle\overline{q}q\rangle^{2}\delta_{ab}+\cdots,$
and
$\displaystyle S_{c}^{ab}(x)=$ $\displaystyle\frac{i}{(2\pi)^{4}}\int
d^{4}ke^{-ik.x}\Big{\\{}\frac{\delta_{ab}}{k\\!\\!\\!/-m_{c}}-\frac{g_{s}G_{ab}^{\alpha\beta}}{4}\frac{\sigma_{\alpha\beta}(k\\!\\!\\!/+m_{c})+(k\\!\\!\\!/+m_{c})\sigma_{\alpha\beta}}{(k^{2}-m_{c}^{2})^{2}}$
(11)
$\displaystyle+\frac{\pi^{2}}{3}\Big{\langle}\frac{\alpha_{s}GG}{\pi}\Big{\rangle}\delta_{ab}m_{c}\frac{k^{2}+m_{c}k\\!\\!\\!/}{(k^{2}-m_{c}^{2})^{4}}+\cdots\Big{\\}}.$
The same Lorentz structures obtained in the hadronic side are also present in
this side, and the ones used in our analyses are $\not\\!pp_{\mu}\gamma_{5}$
and $\not\\!p\gamma_{\mu}\gamma_{5}$, whose contributions are represented in
the below equation explicitly, and the contributions of the others are
represented with the last term stated as $\mathrm{other\,\,\,structures}$.
$\displaystyle\Pi_{\mu}^{QCD}(p,q)$ $\displaystyle=$
$\displaystyle\Pi_{1}\,\not\\!pp_{\mu}\gamma_{5}+\Pi_{2}\,\not\\!p\gamma_{\mu}\gamma_{5}+\mathrm{other\,\,\,structures}.$
(12)
To obtain the coefficients, $\Pi_{i}$, of these Lorentz structures, we use the
propagators explicitly in Eq. (9) and transform the results into momentum
space. After computation of the four integrals the spectral densities,
$\rho_{i}$ are obtained from the imaginary part of the results,
$\rho_{i}(s,s^{\prime},q^{2})=\frac{1}{\pi}Im[\Pi_{i}]$. These spectral
densities are used in the following dispersion relation:
$\displaystyle\Pi_{i}=\int ds\int
ds^{\prime}\frac{\rho_{i}^{\mathrm{pert}}(s,s^{\prime},q^{2})+\rho_{i}^{\mathrm{non-
pert}}(s,s^{\prime},q^{2})}{(s-p^{2})(s^{\prime}-p^{\prime 2})},$ (13)
giving us the final results of the QCD representation of the correlation
function. In the last equation $i=1,2,..,12$ and
$\rho_{i}^{\mathrm{pert}}(s,s^{\prime},q^{2})$ and $\rho_{i}^{\mathrm{non-
pert}}(s,s^{\prime},q^{2})$ represent the perturbative and non-perturbative
parts of the spectral densities, respectively. The results of the spectral
densities that are used in the analyses ($i=1,2$) are:
$\displaystyle\rho_{1}^{\mathrm{pert}}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\int_{0}^{1-x}dy\frac{1}{1024\sqrt{6}\pi^{6}\chi^{3}\chi^{\prime
6}}(1+5\beta)m_{s}xy(Q^{2}xy+s^{\prime}\chi\chi^{\prime}+m_{c}^{2}\chi^{\prime
2})^{2}\Theta[L(s,s^{\prime},Q^{2},x,y)],$
$\displaystyle\rho_{1}^{\mathrm{non-pert}}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\int_{0}^{1-x}dy\Bigg{\\{}-\frac{1}{128\sqrt{6}\pi^{4}\chi^{2}\chi^{\prime
5}}\big{[}\big{(}(\beta-1)\langle\bar{d}d\rangle+\langle\bar{s}s\rangle(1+5\beta)+(\beta-1)\langle\bar{u}u\rangle\big{)}xy\big{(}Q^{2}xy+s^{\prime}\chi\chi^{\prime}+m_{c}^{2}\chi^{\prime
2}\big{)}\big{]}$ (14) $\displaystyle-$
$\displaystyle\frac{\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle}{36864\sqrt{6}\pi^{4}\chi^{4}\chi^{\prime
5}}(1+5\beta)m_{c}x^{4}y^{2}+\frac{\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle}{(9216\sqrt{6}\pi^{4}\chi^{3}\chi^{\prime
4})}(1+5\beta)m_{s}xy\big{[}9x^{2}+9(y-1)^{2}+x(19y-18)\big{]}$
$\displaystyle+$ $\displaystyle\frac{1}{256\sqrt{6}\pi^{4}\chi\chi^{\prime
4}}\big{[}m_{0}^{2}\big{(}(\beta-1)\langle\bar{d}d\rangle+\langle\bar{s}s\rangle(1+5\beta)+(\beta-1)\langle\bar{u}u\rangle\big{)}xy\big{]}\Bigg{\\}}\Theta[L(s,s^{\prime},q^{2},x,y)],$
and
$\displaystyle\rho_{2}^{\mathrm{pert}}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\int_{0}^{1-x}dy\frac{1}{2048\sqrt{6}\pi^{6}\chi^{3}\chi^{\prime
4}}\Big{[}-(1+5\beta)m_{c}m_{s}\big{(}Q^{2}xy+m_{c}^{2}\chi^{\prime
2}+s^{\prime}(x^{2}+(y-1)y+x(2y-1))\big{)}^{2}\Big{]}$ $\displaystyle\times$
$\displaystyle\Theta[L(s,s^{\prime},Q^{2},x,y)]$
$\displaystyle\rho_{2}^{\mathrm{non-pert}}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\int_{0}^{1-x}dy\Bigg{\\{}\frac{1}{512\sqrt{6}\pi^{4}\chi^{2}\chi^{\prime
4}}\Big{[}m_{s}\big{(}\langle\bar{s}s\rangle(1+5\beta)-2(\beta-1)\langle\bar{u}u\rangle\big{)}\chi\big{(}2Q^{2}xy+s^{\prime}\chi\chi^{\prime}\big{)}-2m_{c}\big{(}\langle\bar{s}s\rangle(1+5\beta)$
(15) $\displaystyle+$
$\displaystyle(\beta-1)\langle\bar{u}u\rangle\big{)}\chi^{\prime}\big{(}Q^{2}xy+s^{\prime}\chi\chi^{\prime}+m_{c}^{2}\chi^{\prime
2}\big{)}-2(\beta-1)\langle\bar{d}d\rangle\big{(}m_{s}\chi(2Q^{2}xy+s^{\prime}\chi\chi^{\prime})+m_{c}\chi^{\prime}(Q^{2}xy+s^{\prime}\chi\chi^{\prime}$
$\displaystyle+$ $\displaystyle m_{c}^{2}\chi^{\prime
2})\big{)}\big{]}+\frac{\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle}{147456\sqrt{6}\pi^{4}\chi^{4}\chi^{\prime
4}}(1+5\beta)m_{c}\big{[}m_{c}xy\chi^{\prime}(4x^{2}-3xy+4y^{2})-8m_{s}\chi(2x^{4}-x^{3}y-xy^{3}+2y^{4})\big{]}$
$\displaystyle-$
$\displaystyle\frac{\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle}{36864\sqrt{6}\pi^{4}\chi^{3}\chi^{\prime
4}}(1+5\beta)\big{[}-(Q^{2}+s-s^{\prime})x^{2}y\chi+2m_{c}m_{s}\chi^{\prime
2}\big{(}9x^{2}+9(y-1)^{2}+x(17y-18)\big{)}\Big{]}$ $\displaystyle-$
$\displaystyle\frac{m_{0}^{2}}{1536\sqrt{6}\pi^{4}\chi\chi^{\prime
4}}\big{[}2m_{s}\big{(}\langle\bar{s}s\rangle(1+5\beta)-3(\beta-1)\langle\bar{u}u\rangle\big{)}xy+3m_{c}\big{(}\langle\bar{s}s\rangle(1+5\beta)+(\beta-1)\langle\bar{u}u\rangle\big{)}\chi^{\prime
2}$ $\displaystyle+$ $\displaystyle
3(\beta-1)\langle\bar{d}d\rangle\big{(}-2m_{s}xy+m_{c}\chi^{\prime
2}\big{)}\big{]}+\frac{1}{5184\sqrt{6}\pi^{4}\chi\chi^{\prime
4}}(1+5\beta)g_{s}^{2}(\langle\bar{d}d\rangle^{2}+\langle\bar{s}s\rangle^{2}+\langle\bar{u}u\rangle^{2})xy$
$\displaystyle+$ $\displaystyle\frac{1}{48\sqrt{6}\pi^{2}\chi\chi^{\prime
4}}\big{[}(\beta-1)\langle\bar{s}s\rangle\langle\bar{u}u\rangle+\langle\bar{d}d\rangle\big{(}(\beta-1)\langle\bar{s}s\rangle+\langle\bar{u}u\rangle+5\beta\langle\bar{u}u\rangle\big{)}\big{]}xy\Bigg{\\}}\Theta[L(s,s^{\prime},Q^{2},x,y)]$
where
$\displaystyle\chi$ $\displaystyle=$ $\displaystyle(x+y-1),$
$\displaystyle\chi^{\prime}$ $\displaystyle=$ $\displaystyle(x+y),$
$\displaystyle L(s,s^{\prime},Q^{2},x,y)$ $\displaystyle=$
$\displaystyle\frac{-Q^{2}xy-s^{\prime}(x+y-1)(x+y)-m_{c}^{2}(x+y)^{2}}{(x+y)^{2}},$
(16)
with $\Theta[\ldots]$ being the unit-step function.
Completing the calculations of both representations, we match the results
considering the coefficients of the same Lorentz structures. This step gives
two results both containing $g_{1}$ and $g_{2}$. Solving these two coupled
expressions together, we obtain the sum rules giving the considered coupling
constants, $g_{1}$ and $g_{2}$, analytically as
$\displaystyle g_{1}$ $\displaystyle=$ $\displaystyle
e^{\frac{m_{P_{cs}}^{2}}{M^{2}}}e^{\frac{m_{\Lambda}^{2}}{M^{\prime
2}}}\frac{m_{J/\psi}(m_{J/\psi}^{2}+Q^{2})\left[(m_{P_{cs}}-m_{\Lambda})\tilde{\Pi}_{1}+\tilde{\Pi}_{2}\right]}{f_{J/\psi}\lambda_{\Lambda}\lambda_{P_{cs}}m_{\Lambda}(m_{\Lambda}^{2}+m_{J/\psi}^{2}-m_{P_{cs}}^{2})}$
$\displaystyle g_{2}$ $\displaystyle=$ $\displaystyle
e^{\frac{m_{P_{cs}}^{2}}{M^{2}}}e^{\frac{m_{\Lambda}^{2}}{M^{\prime
2}}}\frac{(m_{P_{cs}}+m_{\Lambda})(m_{J/\psi}^{2}+Q^{2})\left[m_{J/\psi}^{2}\tilde{\Pi}_{1}+(m_{P_{cs}}+m_{\Lambda})\tilde{\Pi}_{2}\right]}{f_{J/\psi}\lambda_{\Lambda}\lambda_{P_{cs}}m_{\Lambda}m_{J/\psi}(m_{\Lambda}^{2}+m_{J/\psi}^{2}-m_{P_{cs}}^{2})},$
(17)
where $\tilde{\Pi}_{i}$ is the Borel transformed form of the $\Pi_{i}$
function. As is seen from the results for computations of these coupling
constants, we are in need of some input parameters. These parameters are
collected in Table 1. In the calculations, the masses of light $u$ and $d$
quarks are taken as zero.
Parameters | Values
---|---
$m_{s}$ | $93^{+11}_{-5}~{}\mathrm{MeV}$ Zyla:2020zbs
$m_{c}$ | $(1.27\pm 0.02)~{}\mathrm{GeV}$ Zyla:2020zbs
$m_{P_{cs}}$ | $(4.47\pm 0.11)~{}\mathrm{GeV}$ Wang:2020eep
$m_{J/\psi}$ | $(3096.900\pm 0.006)~{}\mathrm{MeV}$ Zyla:2020zbs
$m_{\Lambda}$ | $(1115.683\pm 0.006)~{}\mathrm{MeV}$ Zyla:2020zbs
$\lambda_{P_{cs}}$ | $(1.86\pm 0.31)\times 10^{-3}~{}\mathrm{GeV}^{6}$ Wang:2020eep
$\lambda_{\Lambda}$ | $(0.013\pm 0.02)~{}\mathrm{GeV}^{3}$ Aliev:2002ra
$f_{J/\psi}$ | $(481\pm 36)~{}\mathrm{MeV}$ Veliev:2011kq
$\langle\bar{q}q\rangle$ | $(-0.24\pm 0.01)^{3}$ $\mathrm{GeV}^{3}$ Belyaev:1982sa
$\langle\bar{s}s\rangle$ | $0.8\langle\bar{q}q\rangle$ Belyaev:1982sa
$m_{0}^{2}$ | $(0.8\pm 0.1)$ $\mathrm{GeV}^{2}$ Belyaev:1982sa
$\langle\overline{q}g_{s}\sigma Gq\rangle$ | $m_{0}^{2}\langle\bar{q}q\rangle$
$\langle g_{s}^{2}G^{2}\rangle$ | $4\pi^{2}(0.012\pm 0.004)$ $~{}\mathrm{GeV}^{4}$Belyaev:1982cd
Table 1: Some input parameters entering the calculations.
Besides the given parameters in Table 1, there are five additional parameters
which are the threshold parameters $s_{0}$ and $s^{\prime}_{0}$, Borel
parameters, $M^{2}$ and $M^{\prime 2}$ and the mixing parameter $\beta$ which
is coming from the interpolating current of the $\Lambda$ baryon. These
parameters are determined from the analyses of the results imposing the
standard criteria of the method such as weak dependence of the results on the
auxiliary parameters, pole dominance and convergence of operator product
expansion (OPE). Considering these conditions, the threshold parameters are
fixed as follows:
$\displaystyle 21.0\,\,\mathrm{GeV}^{2}$ $\displaystyle\leq$ $\displaystyle
s_{0}\leq 23.0\,\,\mathrm{GeV}^{2},$ $\displaystyle 1.7\,\,\mathrm{GeV}^{2}$
$\displaystyle\leq$ $\displaystyle s^{\prime}_{0}\leq
2.3\,\,\mathrm{GeV}^{2}.$ (18)
The upper limits of Borel parameters are determined by imposing the condition
of pole dominance for the selected working regions of continuum thresholds. To
this end, we consider the following ratio using the continuum subtracted and
Borel transformed invariant amplitude
$\Pi_{i}(s_{0},s^{\prime}_{0},M^{2},M^{\prime 2},\beta)$ obtained from the QCD
side:
$\displaystyle PC=\frac{\Pi_{i}(s_{0},s^{\prime}_{0},M^{2},M^{\prime
2},\beta)}{\Pi_{i}(\infty,\infty,M^{2},M^{\prime 2},\beta)},$ (19)
where $PC$ denotes the pole contribution and $i$ stands for the selected
structures. To fix the upper limit of the Borel parameters we impose this
ratio to be larger or at least equal to $20\%$, which is typical in the
analyses of the exotic states. For the calculations of their lower limits, the
convergence of the series of OPE is considered: the dominance of perturbative
part over the nonperturbative ones and ”the higher the dimension of the
nonperturbative operator, the lower its contribution”. To extract the lower
limit, using this criteria, we fix the ratio of the higher dimensional term,
that is the term having dimension 6 in the QCD side, to the whole result as
follows:
$\displaystyle R(M^{2},M^{\prime
2})=\frac{\Pi_{i}^{6}(s_{0},s^{\prime}_{0},M^{2},M^{\prime
2},\beta)}{\Pi_{i}(s_{0},s^{\prime}_{0},M^{2},M^{\prime 2},\beta)},$ (20)
and keep this ratio as $R(M_{min}^{2},M^{\prime}{}^{2}_{min})=0.02$ to certify
the convergence of the OPE. With these conditions, the Borel parameters are
fixed as
$\displaystyle 5.0\ \mathrm{GeV}^{2}\leq M^{2}$ $\displaystyle\leq$
$\displaystyle 7.0\ \mathrm{GeV}^{2},$ $\displaystyle 1.4\
\mathrm{GeV}^{2}\leq M^{\prime 2}$ $\displaystyle\leq$ $\displaystyle 2.6\
\mathrm{GeV}^{2}.$ (21)
As the final parameter, we determine the working intervals of $\beta$ from the
analyses by considering a parametric plot of the results as functions of
$\cos\theta$ where $\beta=\tan\theta$. We select the regions that show least
variations with respect to the changes in $\cos\theta$, which read
$\displaystyle-1\leq\cos\theta\leq-0.5~{}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}~{}~{}0.5\leq\cos\theta\leq
1.$ (22)
Our analyses show that the physical quantities show weak dependence on the
auxiliary parameters in the above windows for $s_{0}$ and $s^{\prime}_{0}$,
$M^{2}$, $M^{\prime 2}$ and $\cos\theta$. To depict the dependence of the
results of coupling constants on the auxiliary parameters, we plot the Figures
1 and 2 for $g_{1}$ and $g_{2}$ at $Q^{2}=0$. From these figures and the
numerical values we see a good stability of the results with respect to the
Borel parameters in their working window. However, the results show some weak
dependencies on the continuum thresholds in their working intervals, which
remain inside the limits allowed by the method. The variations with respect to
the auxiliary parameters appear as the main sources of the uncertainties in
the numerical results.
Figure 1: Left: The variation of the strong coupling constant $g_{1}(Q^{2}=0)$
as a function of threshold parameters $s_{0}$ and $s_{0}^{\prime}$ at the
central values of the Borel parameters $M^{2}$ and $M^{\prime 2}$ and the
parameter $\beta$. Right: The variation of the strong coupling constant
$g_{1}(Q^{2}=0)$ as a function of Borel parameters $M^{2}$ and $M^{\prime 2}$
at the central values of the threshold parameters $s_{0}$ and $s_{0}^{\prime}$
and the parameter $\beta$.
Figure 2: Left: The variation of the strong coupling constant $g_{2}(Q^{2}=0)$
as a function of threshold parameters $s_{0}$ and $s_{0}^{\prime}$ at the
central values of the Borel parameters $M^{2}$ and $M^{\prime 2}$ and the
parameter $\beta$. Right: The variation of the strong coupling constant
$g_{2}(Q^{2}=0)$ as a function of Borel parameters $M^{2}$ and $M^{\prime 2}$
at the central values of the threshold parameters $s_{0}$ and $s_{0}^{\prime}$
and the parameter $\beta$.
Using the given input parameters in Table 1 and the determined windows for
auxiliary parameters, we calculate the strong coupling constants for the
considered decay channel. The following fit functions represent the
$Q^{2}$-behavior of the strong coupling form factors:
$\displaystyle g_{i}(Q^{2})$ $\displaystyle=$ $\displaystyle
g_{0}e^{c_{1}\frac{Q^{2}}{m_{P_{cs}}^{2}}+c_{2}(\frac{Q^{2}}{m_{P_{cs}}^{2}})^{2}}.$
(23)
with $g_{0}$, $c_{1}$ and $c_{2}$ being the fit parameters that take the
values given in Table 2.
Coupling Constant | $g_{0}$ | $c_{1}$ | $c_{2}$
---|---|---|---
$g_{1}$ | $4.22\pm 0.51$ | $1.54$ | $1.16$
$g_{2}$ | $10.54\pm 1.26$ | $1.54$ | $1.16$
Table 2: : Parameters of the fit functions for coupling constants, $g_{1}$ and
$g_{2}$.
We, then, use the fit functions to determine the coupling constants at
$Q^{2}=-m_{J/\psi}^{2}$ as
$\displaystyle g_{1}=2.63\pm
0.31~{}~{}~{}~{}~{}~{}~{}~{}\mathrm{and}~{}~{}~{}~{}~{}~{}~{}~{}g_{2}=5.25\pm
0.63,$ (24)
where the errors are due to the uncertainties present in the input parameters
entering the calculation and in the determinations of the auxiliary
parameters, as well.
Having determined the strong coupling constants, the next task is to compute
the corresponding width for $P_{cs}\rightarrow J/\psi\Lambda$ decay channel in
terms of the strong coupling constants and other related parameters. The
standard calculations lead to the width formula as
$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{f(m_{P_{cs}},m_{J/\psi},m_{\Lambda})}{16\pi
m_{P_{cs}}^{2}}\Bigg{[}-\frac{2(m_{J/\psi}^{2}-(m_{\Lambda}+m_{P_{cs}})^{2})}{m_{J/\psi}^{2}(m_{\Lambda}+m_{P_{cs}})^{2}}\Big{(}g_{2}^{2}m_{J/\psi}^{2}(m_{J/\psi}^{2}+2(m_{\Lambda}-m_{P_{cs}})^{2})$
(25) $\displaystyle+$ $\displaystyle
6g_{1}g_{2}m_{J/\psi}^{2}(m_{\Lambda}-m_{P_{cs}})(m_{\Lambda}+m_{P_{cs}})+g_{1}^{2}(2m_{J/\psi}^{2}+(m_{\Lambda}-m_{P_{cs}})^{2})(m_{\Lambda}+m_{P_{cs}})^{2}\Big{)}\Bigg{]},$
where
$\displaystyle f(x,y,z)$ $\displaystyle=$
$\displaystyle\frac{1}{2x}\sqrt{x^{4}+y^{4}+z^{4}-2x^{2}y^{2}-2x^{2}z^{2}-2y^{2}z^{2}}.$
(26)
Using the values of the strong coupling constants, we compute the width for
the considered channel to be
$\displaystyle\Gamma(P_{cs}\rightarrow J/\psi\Lambda)$ $\displaystyle=$
$\displaystyle\left(15.87\pm 3.11\right)~{}\mathrm{MeV}.$ (27)
## III Summary and conclusion
The recently observed pentaquark state, the hidden-charmed strange
$P_{cs}(4459)^{0}$, added a new member to the pentaquark family. Its
experimentally observed mass and width were reported as $M=4458.8\pm
2.9^{+4.7}_{-1.1}~{}\mathrm{MeV}$ and $\Gamma=17.3\pm
6.5^{+8.0}_{-5.7}~{}\mathrm{MeV}$, respectively Aaij:2020gdg . However, its
quantum numbers, $J^{P}$, could not be determined as a result of insufficient
statistics in the experiment Aaij:2020gdg . Using the QCD sum rule method,
$P_{cs}(4459)^{0}$ state was studied both in the molecular form assigning its
quantum numbers as $J^{P}=\frac{1}{2}^{-}$ or $\frac{3}{2}^{-}$ Chen:2020uif
and in the diquark-diquark-antiquark form with quantum numbers
$\frac{1}{2}^{-}$ Wang:2020eep . Its mass was obtained in these studies and
compared to experimental data to shed light on its nature. Both of these
interpretations resulted in mass predictions consistent with the experimental
data creating a need for further investigations of this state, for instance
its width.
In this study, we investigated the strong $P_{cs}\rightarrow J/\psi\Lambda$
decay and obtained the strong coupling constants representing the amplitude of
this decay using the QCD sum rule method. To this end, we adopted an
interpolating current in the diquark-diquark-antiquark form for the
substructure of this particle. In the analysis, we considered the quantum
numbers of $P_{cs}(4459)^{0}$ state as $J^{P}=\frac{1}{2}^{-}$. The obtained
strong coupling constants were used in the determination of the corresponding
width, which is obtained as $\Gamma(P_{cs}\rightarrow
J/\psi\Lambda)=\left(15.87\pm 3.11\right)~{}\mathrm{MeV}$. Compared to the
experimental value, the obtained width is in good consistency with
experimental data, which favors the quantum numbers $J^{P}=\frac{1}{2}^{-}$
and compact pentaquark nature of diquark-diquark-antiquark form for
$P_{cs}(4459)^{0}$ state.
In Ref. Wang:2021itn , also, the authors have considered the molecular
interpretations for this state and concluded that it is either
$\bar{D}\Xi^{\prime}_{c}$ with $J^{P}=\frac{1}{2}^{-}$ and $I=0$ or
$\bar{D}\Xi^{*}_{c}$ with $J^{P}=\frac{3}{2}^{-}$ and $I=0$. In Ref.
Zhu:2021lhd $P_{cs}(4459)$ state was interpreted as $\Xi_{c}\bar{D}^{*}$ with
$J^{P}=\frac{3}{2}^{-}$ without excluding the possibility of its being two-
pole structure $\Xi_{c}\bar{D}^{*}$ states with $J^{P}=\frac{1}{2}^{-}$ and
$J^{P}=\frac{3}{2}^{-}$. Another molecular interpretation was given in Ref.
Chen:2021tip in which its two-body strong decay behaviors supported its being
$\Xi_{c}\bar{D}^{*}$ state with $I(J^{P})=0(\frac{3}{2}^{-})$.
All of the above-mentioned investigations indicate that the quantum numbers
and nature of the $P_{cs}(4459)^{0}$ state are still ambiguous and need
clarification not only from further theoretical studies of its various
properties but also from future experiments. Comparison of the theoretical
results on various parameters of this state with future experimental data will
shed light on the nature, quark-gluon organization and quantum numbers of this
state.
## ACKNOWLEDGEMENTS
K. Azizi is thankful to Iran Science Elites Federation (Saramadan) for the
partial financial support provided under the grant number ISEF/M/99171.
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|
# Learning Abstract Task Representations
Mikhail M. Meskhi,1 Adriano Rivolli,2 Rafael G. Mantovani,3 Ricardo Vilalta,1
###### Abstract
A proper form of data characterization can guide the process of learning-
algorithm selection and model-performance estimation. The field of meta-
learning has provided a rich body of work describing effective forms of data
characterization using different families of meta-features (statistical,
model-based, information-theoretic, topological, etc.). In this paper, we
start with the abundant set of existing meta-features and propose a method to
induce new abstract meta-features as latent variables in a deep neural
network. We discuss the pitfalls of using traditional meta-features directly
and argue for the importance of learning high-level task properties. We
demonstrate our methodology using a deep neural network as a feature
extractor. We demonstrate that 1) induced meta-models mapping abstract meta-
features to generalization metrics outperform other methods by $\sim 18\%$ on
average, and 2) abstract meta-features attain high feature-relevance scores.
## Introduction
Meta-learning (MtL) allows rational agents to improve on their learning
abilities through a process known as learning to learn (Hospedales et al.
2020; Schmidhuber 1987; Vanschoren 2018; Vilalta and Drissi 2002). One major
goal is to build self-adaptive systems that adjust their learning mechanism
automatically with new tasks. Automatic adaptation can be described in a
plethora of ways; it can be as simple as tuning hyper-parameters, selecting a
different family of learning algorithms, or simply warm-starting a model.
Meta-learning relies on past experience stored in the form of meta-knowledge.
One type of meta-knowledge encompasses families of meta-features used as a
form of data (or task) characterization. Meta-features capture various types
of data properties such as statistical, e.g., number of numerical attributes;
degree of class separation, e.g., Fisher’s Linear Discriminant (Ho and Basu
2002); or level of concept complexity, e.g., concept variation (Vilalta 1999;
Pérez and Rendell 1996). The proper identification of data properties is
essential to map tasks to learning mechanisms.
Several approaches in meta-learning use families of meta-features as input to
quantify task similarity. It is common to compute task similarity as the
Euclidean distance between two meta-feature vectors. While this approach has
shown to be effective in simple scenarios (Vanschoren 2018), it exhibits clear
limitations. First, selecting a subset of relevant meta-features is a non-
trivial task. What criteria should we invoke to select or discard a family of
meta-features? For example, statistical meta-features are not always intuitive
and lack expressiveness. Previous work has shown how different datasets may
share identical statistical properties but markedly different data
distributions (Matejka and Fitzmaurice 2017). Second, computing certain types
of meta-features on large datasets is computationally costly. For example,
topological meta-features perform multiple passes over the training dataset to
compute a single figure of merit (Ho and Basu 2002; Lorena et al. 2018).
Ultimately, the selection of meta-features is an ad hoc process based on
domain knowledge.
In this paper, we propose an approach that learns abstract meta-features from
families of traditional meta-features using a deep neural network. We argue
that traditional meta-features are not always capable of capturing crucial
task characteristics. This can be attributed to inherent limitations such as
being hand-crafted, and not being tuned to specific tasks (lacking in general
applicability). Extracting meta-features on large datasets can quickly scale
up computational costs and execution times. For example, complexity meta-
features involve geometrical computations resulting in long computing times.
The main contribution of this paper is a novel method of inducing abstract
meta-features as latent variables learned by a deep neural network;
experimental results demonstrate the efficacy of the learned abstract meta-
features in improving generalization performance estimation.
## Related Work
Data characterization consists of extracting meaningful task properties.
Simple, statistical, and information-theoretic meta-features can be
straightforwardly extracted from datasets by capturing information concerning
data dimensionality, distribution, and the amount of information present in
the data. Model-based and landmarking meta-features characterize datasets
indirectly by using the induced learning models; these meta-features comprise
model properties and model performance (Rivolli et al. 2018).
Another family of meta-features is based on capturing the complexity of a
learning task (Ho and Basu 2002), and has been successfully used in different
scenarios (Lorena et al. 2018). Most complexity measures are computationally
expensive; one approach to reduce the associate computational cost is to train
a metaregressor using traditional meta-features as input (Garcia et al. 2020).
The metaregressor estimates complexity values for any dataset at low cost; the
predicted values are called simulated meta-features.
Meta-features can be transformed to summarize the data, e.g., by reducing data
dimensionality. Principal Component Analysis PCA (Hotelling 1933) is the most
straightforward and generic approach, even though it ignores the metatarget.
For example, after running PCA, Bilalli, Abelló, and Aluja-Banet (2017) select
the most relevant components (according to the cumulative total variance);
next, a filter capturing the correlation with the metatarget identifies the
most discriminating variables. Muñoz et al. (2018) propose a new
dimensionality reduction technique suited for data visualization; the most
statistically significant traditional meta-features representing the hardness
of the datasets are first identified, after which the metadataset is
transformed into a 2-dimensional space to search for an optimal projection
through an iterative optimization process; the new 2D space can project
different model-performance footprints to investigate their strengths and
weaknesses.
A different approach that has achieved popularity in recent years invokes Deep
Neural Networks (DNNs). A strength behind DNNs is the capacity to learn data
characteristics from a diverse and large amount of data. DNNs have had a
strong impact in application areas such as speech recognition and image
understanding (Deng and Yu 2014). However, its use in meta-learning is still
incipient and requires further investigation. A few studies explore deep
learning for feature generation, representing different tasks and datasets in
terms of embeddings generated by pre-trained DNNs (Achille et al. 2019).
Gosztolya et al. (2017) solve different automatic speech recognition tasks
through a two-step learning process: performing classification with DNNs,
followed by the extraction of intrinsic features from the DNN output. In the
second phase, features were used to improve model predictions. The same
strategy is explored by Notley and Magdon-Ismail (2018), using DNNs to extract
features from both images and numeric data.
Our hypothesis is that DNNs provide the means to extract intrinsic features
from data. Our approach lies between transformation and deep learning methods
since a DNN is induced from traditional meta-features and the knowledge
captured by the hidden layers are used to generate abstract meta-features. In
the process, traditional meta-features are transformed into latent variables
used by the deep learning model to make predictions. Once extracted, abstract
meta-features can be used by any meta-learning algorithm.
Our approach differs from traditional meta-learning settings. While model-
based and landmarking induce a learning model for each dataset (metainstances)
and extract features from this model, in the current approach, a model is
induced using the metabase (all datasets). This is akin to the simulated meta-
feature approach; however, instead of using the model’s predicted values, we
use the abstract representation of the DNN model to extract meta-features.
## Problem Statement
Given a classification task on a dataset $\mathcal{D}$ with $n$ instances, our
goal is to compute a meta-feature $f$ on $\mathcal{D}$. A meta-feature is
usually a hand-crafted characterization function capturing a specific property
of interest on a given task. Meta-features are regarded as a form of meta-
knowledge collected over a distribution of tasks to learn _how to learn_. Not
all meta-features are informative, and some of them are very task specific.
Learning relevant meta-features can prove useful in identifying hidden
relationships across tasks, and is necessary to build accurate meta-learners.
Knowledge extracted across tasks, a.k.a. meta-knowledge, is key to the success
of meta-learning by obviating learning from scratch on new tasks. By
exploiting meta-knowledge, the metalearner can effectively construct an
optimal solution based on past experience (Hospedales et al. 2020; Vanschoren
2018). For example, a meta-learner can identify that a new task is similar to
previous tasks and warm-start a similar model with near optimal
hyperparameters. This avoids the –sometimes painstakingly– slow processes of
error and trial in building a new model. Meta-knowledge can be understood as
meta-features, model hyperparameters, performance measures, etc. In our work,
meta-knowledge consists of meta-features and performance measures gathered
from previous tasks.
We formally define the process of meta-feature extraction as a function $f$
that receives as input a dataset $\mathcal{D}$, and returns as output a set of
$k$ values characterizing the dataset (Rivolli et al. 2018):
$f(\mathcal{D})=\sigma(m(\mathcal{D})),$ (1)
where $m$ is a characterization metric mapping
$\mathcal{D}\rightarrow\mathbb{R}^{k^{\prime}}$, $k^{\prime}$ is the original
number of meta-features, and $\sigma$ is a summarization function mapping
$\mathbb{R}^{k^{\prime}}\rightarrow\mathbb{R}^{k}$. The purpose of the
summarization function is to reduce the size of the output to a fixed size. A
dataset $\mathcal{D}$ is characterized by a meta-feature space $\mathcal{F}$.
Our goal is to find a subset of meta-features, $F\subset\mathcal{F}$,
capturing relevant task information.
## Abstract Meta-features
We propose a novel approach to learning new abstract meta-features by
constructing new representations from traditional meta-features using a deep
neural network. A neural network –parameterized by $\mathbf{w}$– is a
universal function estimator. The goal is to learn a function $g$ to predict
the true target $y$ through an approximation $\hat{y}$, where
$g(\mathbf{x})=\hat{y}$. Training is achieved by computing gradients of the
loss function with respect to the weights $\nabla_{\mathbf{w}}J(\mathbf{w})$
and then back-propagating the errors through the network to update the
weights. The architecture of a neural network is described by the number of
neurons per layer and by the number of hidden layers $\ell_{h}^{n}$, where $h$
is the index for the hidden layer and $n$ is the total number of neurons in
that layer. Forward propagating input $\mathbf{x}$ through the network leads
to a sequence of non-linear transformations; non-linearity is achieved via an
activation function $\phi$. Increasing the number of hidden layers and neurons
allows the neural network to approximate highly non-linear functions. Each
layer contains a learnt latent representation of the input data. The last
hidden layer comprises the final learnt latent variables, $\\{z_{i}\\}$, where
each latent variable $z_{i}$ is a representation of the original input in an
abstract space. The number of latent variables is user-defined by controlling
the number of neurons in that layer. By training a deep neural network on the
traditional meta-feature space $\mathcal{F}_{t}$, we can learn a new latent
representation $\mathcal{F}_{a}$ (abstract meta-features). The resulting deep
neural network serves as a feature extractor where the learnt latent space $Z$
is extracted from the last hidden layer. This process is highlighted in Figure
1.
Hyper-parameter | Value
---|---
Learning rate | 0.005
Hidden layers | 5
Latent variables | 16
Criterion | $\textnormal{Smooth}_{L_{1}}$
Optimizer | Adam
Activation | ReLU
Table 1: AbstractNet hyper-parameters.
### AbstractNet
Our methodology constructs a deep neural network (DNN) to act as a feature
extractor on each pair of input and output meta-instances. After providing
traditional meta-features as input and a performance measure of three
different algorithms as target, we train our AbstractNet to learn a meaningful
abstract representation of the meta-dataset. AbstractNet consists of $5$ fully
connected layers with sixty four neurons in each of the first four layers
($\ell^{64}_{h},1\leq h\leq 4$), and a final latent layer with sixteen neurons
($\ell^{16}_{5}$). Our target is a three dimensional output consisting of a
performance estimation for each of three learning algorithms applied to a
given task (represented using meta-features). Non-linearity between layers is
achieved via the ReLU activation function,
$\phi(q)=\max(0,q),$ (2)
where $q=\mathbf{wx}+\mathbf{b}$ is the linear transformation of the input.
Variance is controlled via dropout; regularization is applied between layers
two and four with probabilities $p=(0.1,0.05)$ respectively. We selected the
smooth $L_{1}$ loss function (Girshick 2015) as our criterion function:
$J(\hat{y},y)=\frac{1}{n}\sum_{i}\nu(\hat{y}_{i}-y_{i}),$ (3)
where the summation goes over all training examples, $\hat{y}$ and $y$ are the
estimated and true response values respectively, and $\nu$ is defined as
$\nu(u)=\left\\{\begin{array}[]{ll}(0.5*u^{2})/\lambda&\textnormal{if}\quad|u|<\lambda\\\
|u|-0.5*\lambda&\textnormal{otherwise.}\end{array}\right.$ (4)
Parameter $\lambda$ specifies the threshold that defines the step function.
The smooth $L_{1}$ loss is less sensitive to outliers, has a smoother
landscape, and prevents exploding gradients. The list of full neural network
hyper-parameters is available in Table 1. Once the deep neural network is
trained, we forward propagate the meta-dataset validation partition to extract
the latent variables $\\{z_{i}\\}$ from the last hidden layer to induce our
meta-models.
Figure 1: A deep neural network for learning abstract meta-features. The last
hidden layer is used to extract the learnt latent variables $\\{z_{i}\\}$ to
produce abstract meta-features.
## Experiments
In this section we describe the experimental design to induce abstract meta-
features. Figure 2 provides an overview of the entire process, with three
different phases corresponding to data characterization, meta-database
construction, and induced meta-model evaluation. We explain these steps in
detail next.
---
Figure 2: Sequence of steps to generate abstract meta-features and to assess
meta-model performance.
### Datasets and Performance Evaluation
Phase 1 of our experiments includes dataset selection, performance evaluation,
and meta-feature extraction. We collected a total of $517$ classification
datasets from OpenML (Vanschoren et al. 2014), a free scientific platform for
standardization of experiments and sharing of empirical results. These
datasets were selected following some criteria: the number of features does
not exceed $500$; there are no missing values; there are at least $2$ classes;
and the minority class must have at least $10$ examples.
Next, we evaluated three common learning algorithms: Support Vector Machine
(SVM), Random Forest (RF), and Multilayer Perceptron (MLP) on each dataset,
recording their generalization performance in terms of the Area Under the ROC
Curve (AUC) (Hand and Till 2001).
In parallel, we extracted the traditional family of meta-features using the
PyMFE library (Alcobaça et al. 2020) for these categories: general,
statistical, info-theoretical, concept, model-based and landmarking. The
functions _max, min_ and _mean_ were used to summarize multi-valued results.
### Meta-database
In phase 2, we combined meta-features and performance values per dataset to
construct a meta-database, $\mathcal{D}_{meta}$. A pre-processing step was
also performed at the meta-database by removing meta-instances with more than
$100$ missing values, meta-features with more than $70\%$ of missing values,
constant meta-features, and highly correlated meta-features. The remaining
missing values were imputed using a $k$-NN with $k=10$. Lastly, we generated a
meta-database with three different targets, one for each performance
prediction problem (SVM, RF, MLP). The final meta-database has a total of
$517$ meta-examples (instances) and $265$ meta-features (characteristics); it
was used to train the deep neural network to induce the latent abstract meta-
features.
### Evaluation
In phase 3, we evaluated our approach under four different settings:
* •
Abstract: this strategy explores our approach alone, by constructing abstract
meta-features through the AbstractNet and extracting a latent representation
from the last hidden layer.
* •
Traditional: we induce meta-models on traditional meta-features only.
* •
Hybrid: we induce meta-models using a combination of traditional and abstract
meta-features.
* •
PCA: as a baseline for comparison, PCA (Hotelling 1933) is invoked to
transform traditional meta-features through linear combinations. PCA also
generates latent features keeping the components with 95% of the cumulative
variance.
The hybrid approach is instrumental to assess feature relevance; it can show
the value of abstract meta-features in predicting model performance over
traditional meta-features.
We performed $10\times 10$-fold cross-validation; in each iteration, nine
folds are used to obtain the abstract meta-features (AbstractNet) and induce
the different meta-models; and the remaining fold is used to validate the
meta-models. We employed three learning algorithms: Decision Trees, Random
Forest, and Support Vector Machines as meta-inducers. Finally, we evaluated
the performance of our models on the validation sets and report the Root Mean
Squared Error (RMSE) and the coefficient of determination ($R^{2}$). We also
repeated the experiments 10 times with different seeds to perform statistical
validations using the Hierarchical Bayesian correlated t-test (Benavoli et al.
2017). Here, we compared the performance of the meta-inducers using distinct
subsets of meta-features. The test evaluates in pairs, resulting in
probabilities concerning which approach is better (left and right) for a
particular evaluation measure. It also defines a region of equivalence (rope)
that indicates the probability that the difference in performance is
insignificant. The complete experimental methodology is shown in Table 2.
Element | Feature | Value
---|---|---
Base level | Datasets | 517
Target Algorithms | SVM, RF, MLP
Performance | AUC
Meta-features | Abstract | 16
Traditional | 154
Hybrid | 170
PCA | 60 (95% of var)
Meta-level | Tasks | 3 (base level)
Regressor | RF, DT, SVM
Resampling | 10 x 10-CV
Performance | RMSE, $R^{2}$
Statistical Validation | Test | Hier. Bayesian
Table 2: Experimental settings across base and meta-levels.
## Results and Discussion
Experimental results are shown in Table 3. Results in bold stand for best
results. As we can see, abstract meta-features achieve best RMSE and $R^{2}$
scores across all four settings. For instance, abstract meta-features perform
vastly better than traditional meta-features with Decision Trees; the
dimensionality of abstract meta-features is sixteen, while that of traditional
meta-features is one hundred and fifty four. The reduced size of the space of
abstract meta-features leads to high-generalization meta-models. Similar
results are seen when different learning algorithms are used to induce meta-
models. PCA transformation ranked lowest in terms of performance across the
four settings; we hypothesize that PCA’s inherent linear components are not
expressive enough. AbstractNet is capable of learning highly abstract
representations to capture complex relationships between meta-features and the
target variable.
We can see that abstract meta-features closely follow the true AUC values
across datasets, while traditional and PCA meta-features exhibit instability
and poor performance. Variance along our performance metrics per learning
algorithm is shown in Figures 4, 4. The hybrid approach allows us to combine
traditional and abstract meta-features. By ranking the top fifteen most
important features using Gini index from the Random Forest meta-model, seven
out of sixteen learnt abstract meta-features ranked in the top fifteen
features.
Table 4 shows probabilities obtained with the Bayesian Hierarchical $t$-test
over different meta-databases and performance values. Abstract meta-features
improved the use of traditional and PCA meta-features significantly,
confirming the generalization ability of our approach. Unlike PCA, our deep
neural network learnt a non-linear abstract transformation of traditional
meta-features while increasing their predictive power.
Inducer | Meta-features | $R^{2}$ Score | | RMSE
---|---|---|---|---
SVM AUC | RF AUC | MLP AUC | | SVM AUC | RF AUC | MLP AUC
DT | Abstract | 0.906 (0.072) | 0.839 (0.17) | 0.864 (0.129) | | 0.043 (0.014) | 0.057 (0.032) | 0.048 (0.022)
| Traditional | 0.554 (0.132) | 0.510 (0.155) | 0.561 (0.14) | | 0.101 (0.019) | 0.114 (0.024) | 0.097 (0.018)
| Hybrid | 0.877 (0.078) | 0.803 (0.163) | 0.841 (0.128) | | 0.050 (0.015) | 0.067 (0.031) | 0.054 (0.022)
| PCA | 0.424 (0.136) | 0.384 (0.132) | 0.418 (0.12) | | 0.121 (0.018) | 0.131 (0.022) | 0.117 (0.016)
RF | Abstract | 0.925 (0.059) | 0.856 (0.17) | 0.882 (0.125) | | 0.038 (0.012) | 0.052 (0.033) | 0.044 (0.023)
| Traditional | 0.761 (0.076) | 0.715 (0.094) | 0.752 (0.079) | | 0.071 (0.012) | 0.081 (0.018) | 0.070 (0.011)
| Hybrid | 0.928 (0.055) | 0.851 (0.167) | 0.880 (0.123) | | 0.037 (0.012) | 0.054 (0.032) | 0.044 (0.023)
| PCA | 0.688 (0.083) | 0.641 (0.085) | 0.685 (0.075) | | 0.081 (0.012) | 0.092 (0.016) | 0.080 (0.011)
SVM | Abstract | 0.889 (0.050) | 0.862 (0.109) | 0.873 (0.083) | | 0.062 (0.008) | 0.076 (0.017) | 0.068 (0.011)
| Hybrid | 0.829 (0.049) | 0.774 (0.100) | 0.805 (0.073) | | 0.070 (0.007) | 0.083 (0.016) | 0.074 (0.010)
| Traditional | 0.685 (0.082) | 0.603 (0.108) | 0.644 (0.081) | | 0.088 (0.010) | 0.102 (0.016) | 0.092 (0.009)
| PCA | 0.715 (0.067) | 0.640 (0.105) | 0.704 (0.075) | | 0.082 (0.009) | 0.096 (0.016) | 0.085 (0.009)
Table 3: Generalization performance of meta-models using different meta-databases containing all families of meta-features. Meta-databases | Measure | left | rope | right
---|---|---|---|---
Traditional $\times$ Abstract | RMSE | 0.001 | 0.000 | 0.999
$R^{2}$ Score | 0.000 | 0.000 | 1.000
PCA $\times$ Abstract | RMSE | 0.000 | 0.000 | 1.000
$R^{2}$ Score | 0.001 | 0.000 | 0.999
Traditional $\times$ PCA | RMSE | 0.292 | 0.696 | 0.012
$R^{2}$ Score | 0.911 | 0.000 | 0.089
Table 4: Hierarchical Bayesian statistical probabilities by comparing pairs of
meta-databases. The columns _left_ and _right_ indicate each meta-database’s
probability outperforming the other. The _rope_ column indicates the
probability of them being similar.
## Conclusions
Data-driven meta-learning (MtL) requires new forms of data characterization.
Given the difficulty inherent to the size of the meta-feature space, this
paper explores a promising direction to solve the meta-feature selection
problem by learning abstract meta-features via a deep neural network. We
introduce and discuss data characterization as meta-knowledge. In order to
optimally meta-learn over a distribution of tasks, the right form of meta-
knowledge is required _a priori_. By defining our meta-objective as
generalization performance, we construct a deep neural network to learn an
abstract representation of traditional meta-features, i.e., we generate
abstract meta-features as meta-knowledge to be used by our meta-models. We
contend that abstract meta-features are more expressive and effectively
capture hidden variable relationships.
Our experimental results demonstrate the efficacy of abstract meta-features as
strong predictors of generalization performance, while reducing the size of
the meta-feature space. Feature importance values computed on hybrid meta-
features show that abstract meta-features frequently achieve top results. Our
results show that PCA linear transformations are not as expressive as the non-
linear transformations learnt by our deep neural network.
### Limitations & Future work
There is no clear subset of traditional meta-features capable of capturing
task properties well over highly diversified tasks. Just as deep learning
usually outperforms handcrafted features, learning meta-features with DNN can
offer great insight into improving the data characterization process. Future
research directions involve exploring abstract meta-features in a cost-
effective fashion, and decomposing complex task characteristics as functions
of simpler building properties. The goal is to identify fundamental
characteristics that cover a broad spectrum of complex tasks.
Figure 3: Violin plot of RMSE scores of meta-models on three approaches:
abstract, traditional, and PCA.
Figure 4: Violin plot of $R^{2}$ scores of meta-models on three approaches:
abstract, traditional, and PCA.
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|
# Rapid Convergence: The Outcomes of Making PPE during a Healthcare Crisis
Kelly Mack Paul G. Allen School of Computer Science, University of
WashingtonSeattleWAUSA , Megan Hofmann Human Computer Interaction Institute,
Carnegie Mellon UniversityPittsburghPAUSA , Udaya Lakshmi School of
Interactive Computing, Georgia Institute of Technology,AtlantaGAUSA , Jerry
Cao Paul G. Allen School of Computer Science, University of
WashingtonSeattleWAUSA , Nayha Auradkar Paul G. Allen School of Computer
Science, University of WashingtonSeattleWAUSA , Rosa I. Arriaga School of
Interactive Computing, Georgia Institute of Technology,AtlantaGAUSA , Scott
E. Hudson Human Computer Interaction Institute, Carnegie Mellon
UniversityPittsburghPAUSA and Jennifer Mankoff Paul G. Allen School of
Computer Science, University of WashingtonSeattleWAUSA
###### Abstract.
The NIH 3D Print Exchange is a public and open source repository for primarily
3D printable medical device designs with contributions from expert-amateur
makers, engineers from industry and academia, and clinicians. In response to
the COVID-19 pandemic, a collection was formed to foster submissions of low-
cost, local manufacture of personal protective equipment ( Personal Protective
Equipment (PPE)). We systematically evaluated the 623 submissions in this
collection to understand: what makers contributed, how they were made, who
made them, and key characteristics of their designs. Our analysis reveals an
immediate design convergence to derivatives of a few initial designs
affiliated with NIH partners (e.g., universities, the Veteran’s Health
Administration, America Makes) and major for-profit groups (e.g., Prusa). The
NIH worked to review safe and effective designs but was quickly overloaded by
derivative works. We found that the vast majority were never reviewed (81.3%)
while 10.4% of those reviewed were deemed safe for clinical (5.6%) or
community use (4.8%). Our work contributes insights into: the outcomes of
distributed, community-based, medical making; features the community accepted
as “safe” making; and how platforms can support regulated maker activities in
high-risk domains (e.g., healthcare).
personal protective equipment, COVID-19, makers, making, survey
## 1\. Introduction
Medical making is emerging alongside maker efforts (e.g., hobbyists,
engineers, designers, digital fabrication enthusiasts) to apply crafting and
digital fabrication to invent, manufacture, and repair medical devices.
Research on maker practices across domains has developed rich insights into
material practices of collaboration in shared repositories (Cheliotis et al.,
2014; Buehler et al., 2015; Alcock et al., 2016) and social norms (Fox et al.,
2015; Vyas, 2019; Lindtner et al., 2015). Unlike many other domains of making,
medical making raises vital concerns about safety and efficacy because medical
devices can pose significant risks to life and limb. Prior to the COVID-19
pandemic, the National Institute of Health (NIH) 3D Print Exchange served as
an “open, comprehensive, and interactive website for searching, browsing,
downloading, and sharing biomedical 3D print files, modeling tutorials, and
educational material” (National Institute of Health, 2014)—an open repository
for medical making. However, as the demand for PPE in the pandemic overwhelmed
global supply chains, the Food and Drug Administration (FDA) and NIH began
sourcing and reviewing alternative, open source designs created by a variety
of institutions and hobbyist makers. To ensure that designs are safe to use,
the NIH, in partnership with the Veteran’s Health Administration (VHA), FDA,
and Center for Disease Control (CDC), began reviewing these submissions. By
analysing this collection, we contribute a better understanding of the effect
of a critical, safety-review process on what maker’s create, reuse, and share.
To understand trends from this extraordinary occurrence of medical making, we
present a mixed-methods analysis of this NIH 3D Print Exchange’s COVID-19
Collection. We use a combination of qualitative data from a thematic analysis
and quantitative data from web scrapped details of the 623 submissions. We
reviewed every submission to the COVID special collection between its start
date, March 20th 2020, and January 1st, 2021. Our quantitative analysis shows
that: submissions rapidly dropped off in April after the initial surge of
designs and that most (83.5%) designs are variations of a few key classes of
devices reviewed in this early period. Our qualitative analysis further
demonstrates designs within these classes (masks, face shields, and straps)
converged to a few common forms. We also find that makers not affiliated with
large institutions struggled to fully document or test their designs. This
insufficient documentation of many designs led to wasted NIH reviewer time and
demonstrates a lack of clarity in maker perceptions of review criteria.
Our results reveal that NIH’s goal of collecting diverse and innovative
designs from makers was not met. Instead of generating a diverse array of
designs, the submission requirements and rating designations led to a rapid
convergence of the design space. Even though the NIH did not request any
particular types of designs, the majority of designs fell into three narrow
categories of PPE: face shields, masks, and straps (i.e., tension-relief
bands, ear-savers). Within these categories, diversity in the designs was low,
particularly among face shields where the main defining design feature was the
presence or absence of a visor to provide protection from above.
Open maker repositories with no formal review process tend to included a wide
range of unique designs (Buehler et al., 2015; Kuznetsov and Paulos, 2010).
Contrary to these observations, we observed only a few design archetypes and
numerous derivatives which made small changes to the manufacturing method
(e.g., 3D printer, slicing settings, bed arrangement) or scale (e.g., fit) of
the designs. We discuss possible factors for this shortened idea generation
phase resulting in a quick design convergence. We further discuss the groups
who made these designs, how designers interacted on the forum, and factors
that contributed better review outcomes. In short, teams of expert makers
outperformed expert-amateur makers in review outcomes. Finally, we discuss how
the current review process led to confusion of requirements among makers; this
confusion wasted scarce reviewer time.
Based on our findings, we recommend different ways that maker repositories
with review processes can support alternative interactions with the community
and yield greater design diversity while maintaining safety. First, make the
reviewing process more transparent and effective by 1) ensuring that all key
information required for a review is marked mandatory, and 2) providing
feedback about why designs received their review rating. Second, introduce a
required field to explain updates made to a design in remixes. Small updates
can be more rapidly reviewed than more involved changes. Third, pose clear
requests to the community. These communications can help ensure that designs
diverge rather than converge on what is already positively reviewed. Finally,
support and motivate innovation by denoting “work-in-progress” submissions and
explicitly encouraging designs that diverge from the norm. These features
together save reviewers time and position innovation and creativity as values
to the community in addition to safety.
Our work lies at the intersection of CSCW themes on material practices
summarized in Rosner’s literature review (Rosner, 2012). CSCW scholars have
explored material practices in collaborative communities engaged in craft
(Rosner, 2012), music (Cheliotis et al., 2014), and sites of peer-production
(Yamauchi et al., 2000; Kriplean et al., 2008). Our work informs and draws
from three related themes. The first has developed an understanding of
materials and tools as non-human actors in collaborative practices. The second
discovers emergent, temporal interactions while the third has evolved around
designers’ notions of affordances in digital-physical materials. Our inquiry
specifically extends Rosner’s observations on the temporal and heterogeneous
facets of collaboration with materials (Rosner, 2012) aligned with other
studies on making (Oehlberg et al., 2015). Medical makers balanced safety,
urgency, and uncertainty in the digital-physical fabrication process during a
pandemic. Their open source contributions required greater articulation work
that was critical for design remix and reuse designs in a flexible manner. The
repository as a material actor is designed for innovation, not speed, among
other factors. We discuss how fabrication work is embedded in material
practices of medical making. We provide novel implications for remix, reuse,
and coordination to build open source infrastructure for medical making. We
reflect on social relations in a community marked by norms of safety,
reliability, and regulation. Our findings on rapid design convergence in the
NIH 3D Print Exchange challenge expectations of novelty, variety, and wider
reach of open source activity.
## 2\. Related Work
### 2.1. Digital Fabrication and Peer Production in Medical Making
Maker activities are characterized by their community’s norms and material
practices. Tanenbaum et al. describe hobbyist makers tend towards a
hedonistic preference of maker-technologies (i.e., 3D printers) that offer
speed, replicability, and collective skill to democratize material-driven
innovation (Tanenbaum et al., 2013). In contrast to this
technocentric/utilitarian view, others call attention to an ecosystem of
sociopolitical actors (Lindtner et al., 2015), community structures (Fox et
al., 2015)), and future opportunities (Lindtner et al., 2014) to critique
notions of empowerment within material constraints at sites of making. Studies
on digital fabrication in healthcare communities reveal how care for
recipients of Assistive Technology devices (Buehler et al., 2015; Parry-Hill
et al., 2017) and motivations in DIY Health (O’Kane et al., 2016; Slegers et
al., 2020) impact peer production. Relatively less is known about a similar
trend in digital fabrication practices applied to medical practice in HCI.
Medical applications for digital fabrication are on the rise with advances in
3D printing (Mukhopadhyay and Poojary, 2018; Ventola, 2014; Kalaskar, 2017).
Most studies track clinician experiments with the novel use of fabrication
technologies in bio-printing (Vijayavenkataraman et al., 2016), surgical
guides (Malik et al., 2015), dentistry (Dawood et al., 2015), implants
(Curodeau et al., 2000), prosthetics (Gretsch et al., 2016), and orthotics
(Chimento et al., 2011). This trend correlates to a history of crafting
practices and device improvisations (Gomez-Marquez and Young, 2016) and open
source infrastructures. However, recent HCI studies indicate a wider variety
of ”medical makers” (Lakshmi et al., 2019) engage in medical device
development and deployment in care delivery roles. They adapt the fabrication
process to suit specialized practice (Hofmann et al., 2019) and generalized
care norms (Lakshmi et al., 2019). Hofmann et al. found that occupational
therapists limit material iterations to integrate digital fabrication into
their standard practices, packed schedules, and keep costs to a minimum
(Hofmann et al., 2019; Slegers et al., 2020). Lakshmi et al. discuss how
clinician-makers hesitate to distribute prototype designs without due
regulatory approval or licensing extending from an ethos of safety and a risk
aversion to personal medical liability (Lakshmi et al., 2019). While 3D
printing advocates in medicine proved prescient in the COVID-19 PPE crisis
(Novak and Loy, 2020; Lakshmi et al., 2021; Hofmann et al., 2021), regulatory
and policy infrastructure in the medical space is underdeveloped. Similar to
the open source software development communities (Yamauchi et al., 2000),
flexible and ad hoc coordination is key for efficient medical maker response.
Medical makers already defer to their professional norms to uphold safety and
reliability with risk-averse approaches. In uncertain times, these factors may
conflict with expectations of novelty and variety with 3D printing overtly
recognized as a tool for innovation within the medical research community
(e.g., VA’s Innovator’s Network (Razak, 2018)). It is unclear how these social
and material constraints influence peer production mechanisms for medical
makers engaged in digital fabrication.
### 2.2. Barriers to Reuse and Remix in Digital Fabrication
Reuse and adaptation of shared designs is a perceived benefit in maker
communities (Cheliotis et al., 2014). These activities, described as remixing,
are motivated by collective learning among makers by contributing to peer
production activity on repositories (Kuznetsov and Paulos, 2010; Buehler et
al., 2016). Makers, medical makers included, expect to adapt designs and re-
share them with an articulation of their efforts for future reuse. However,
these expectations are constrained by factors specific to the digital-physical
material process for both adaptation and articulation work acting as barriers
to collaboration.
Unlike physical artifacts, novelty of an adapted digital artifact can be
attributed to the extent of variation from the original as Cheliotis et al.
note in their study on a musician community (Cheliotis et al., 2014). On
Thingiverse, Kim et al. describe how popular contributions of preferred
digital file types rely on real world constraints around printer filaments and
reliable outcomes (Kim et al., 2017). To support collaboration between users,
it is better to share the source files generated on modeling tools (e.g.,
OpenCAD) to retain the original geometry of the model and make editing easier
(Hudson et al., 2016). However, Alock et al. reports an overwhelming
preference for STLs (84%) over OpenSCAD files (3.7%) on Thingiverse (Alcock et
al., 2016) possibly because it signals a convenience to download-and-print the
model. Regardless of their popularity, STLs are inflexible file types lacking
metadata, forcing makers to rebuild designs from scratch.
When makers share editable models, they fail to articulate key details such as
the design’s purpose and manufacturing details (e.g., slicing instructions).
Even more experienced users can struggle with inferring the details of a print
when there is insufficient documentation on materials, print settings, and/or
assembly (Ludwig et al., 2014; Oehlberg et al., 2015). One part of this
challenge is that makers rarely document these aspects of their designs as
they go, and they avoid the work when sharing online (Ashbrook et al., 2016).
Further, many novices in 3D-modeling struggle to understand the intricacies of
models such as dealing with print uncertainties (Kim et al., 2017) and
figuring how 3D-models interact with real-world geometries (Ashbrook et al.,
2016; Hofmann et al., 2018; Chen et al., 2015). This makes it difficult for
them to explicitly document how their design works. One solution may be to
integrate the documentation process directly into 3D modeling processes
(Hofmann et al., 2018), however no widely adopted standard tools support this
workflow. In the rare case that all relevant information included, variations
in printer and filament can still cause prints to fail (Kim et al., 2017). On
sharing platforms, insufficient documentation is partly addressed on user
forums by the community’s discussion on specific 3D-models.This reactive
process is not sustainable over time as users continue to remix the model.
Documentation can be lost with each iteration leaving gaps for the successive
author might not understand everything about the model and be unable to answer
questions (Flath et al., 2017; Alcock et al., 2016). Moreover, the process
increases the burden of articulating their designs on the authors.
Articulation work is embedded in complex cooperative arrangements around the
artifact itself (Kriplean et al., 2008; Morgan et al., 2014). For example, on
Wikipedia, Kriplean et al’s case study analyses how moderators’ contribution
from core editing shifts to “meta-work activities” that ultimately build the
collective reputation by overseeing participation, support, and quality of
outcome. Morgan et al. in their analysis of alternate WikiProjects found open
collaborations persist when they maintain low barriers for participation and
community-adapted social structures (Morgan et al., 2014). Most maker
communities favor a flexible, informal structure (Khanapour et al., 2017) to
maximize participation especially from volunteers (Yamauchi et al., 2000) over
defined roles for critical meta-work to ensure quality. Eventually, this leads
to inconsistent information on core properties, evaluation methods, or use
cases, leaving most digital fabrication repositories riddled with insufficient
documentation of design files. It is not surprising that time constrained
medical makers avoid adopting open source designs (Lakshmi et al., 2019).
Working within institutional infrastructure, their efforts to make medical
devices are further subject to available technical expertise, uncertainties
around physical materials, and licensing or regulatory mandates to ensure safe
use. Yet, repositories like the NIH 3D Print Exchange and the limited
distribution of digital files on hospital sites indicate that medical makers
publish their designs for use, reuse, and distribution. We examine the
emergent practices around the recent push to make and design PPE (Lakshmi et
al., 2021; Hofmann et al., 2021) on the NIH 3D Print Exchange. Novak and Loy
undertake a wider analysis of COVID-19 response efforts in early 2020 (Novak
and Loy, 2020). Our study takes a deep dive into the medical maker community
on a single platform.
## 3\. Background: The NIH 3D Print Exchange and COVID-19
The NIH 3D Print Exchange is a 3D model repository hosted by the US government
with an exclusive focus on collecting “bioscientific” models. Before the
COVID-19 pandemic, the collection was an open library of bio-medical models
(e.g., molecules, organs), a small collection of open source prosthetic-like
devices from e-NABLE, and simple 3D printable lab equipment (e.g., test tube
holders). The goal of the project was to be the authoritative source for
medical makers’ designs. It included the extensive documentation needed to
reliably make such models wherever physical equipment was available
nationwide. By early 2020, there were efforts to add an expert review process
to the exchange that would enable makers to receive feedback from VHA and FDA
experts. The program roll out was hastened to completion in response to the
surging COVID-19 pandemic in March.
Due to the pandemic, traditional PPE and medical device manufactures could not
keep up with demand and there were global shortages of respirators, face
shields, masks, and other critical supplies. The NIH 3D Print Exchange
released its new review process early through the new COVID-collection. The
collections’ goal was specifically: “to inform decision-making on PPE and
medical device production, without stifling innovation…by filtering designs
through a systematic review process.” The collection was intended to connect
innovative makers and manufacturers to produce products to fill in supply
gaps. Anyone could submit their designs to the collection, queuing them for
review by medical and engineering experts within the NIH and other government
affiliates.
Makers submit their designs through an extensive, publicly available form
(National Institute of Health, 2020b). They could provide: a textual
description, manufacturing details (e.g., 3D printer model; materials; design
files, pre-processing, assembly, cleaning, and use instructions), licensing
information, and documentation (e.g., images, testing procedures and data);
though, few of these fields were mandatory. All submissions are marked as
“prototypes” before they are reviewed. Submissions are reviewed based on a
priority determined by (1) demand (i.e., the design meets an unmet need), (2)
feasibility (i.e., it seems reasonable that the design works as described),
and (3) detail (i.e., the submission includes enough information to make
review possible) (National Institute of Health, 2020a). Reviewed designs are
independently produced and tested with actual materials by reviewers to
determine what classification, if any, is appropriate. More detailed criteria
for review is listed on the collection’s FAQ and in a document detailing the
different types of masks (general use face masks, surgical face masks, and N95
respirators). Specific criteria for other types of PPE are not present.
Besides the default prototype status, submissions’ review status can be:
“reviewed for clinical use”, “reviewed for community use”, and “warning”. Note
that none of these terms include the word “approved”; this is a purposeful
decision to remove confusion between the exchange’s review process and FDA
approval processes. Positively reviewed designs on the NIH 3D Print Exchange
still do no have FDA approval. Submissions reviewed for clinical use are
deemed to be the safest and most effective submissions. These are appropriate
to use in a high-risk clinical environment. Community use denotes a lower
standard where the device itself is expected to be safe but its efficacy
cannot be guaranteed; it will not hurt the user, but it might not protect them
either. The warning category was used in rare cases where the design itself is
not safe. Usually this was received for high risk designs like ventilator
parts. If a design did not meet the clinical or community standards but was
not so risky to merit a warning, the reviewers would privately provide
feedback and leave the design marked as a prototype. Occasionally, reviewers
left public comments before deciding the design’s final status. We cannot
determine if reviewers left comments in all cases or only in the absence of a
private email response.
This review process was quickly overloaded with a surge of new designs. On
July 24th, the Exchange stopped considering common face shield and ear-saver
designs for review “due to the volume of submissions, unless the face shield
is a novel design adapted for a specific use”. They turned their review
efforts exclusively to nasal swabs for COVID-19 tests which make up only seven
of the 623designs.
The NIH 3D Print Exchange presents an unique opportunity for researchers to
study what medical makers do when collectively tasked to address one global
problem (i.e., PPE production in a pandemic). Unlike open maker repositories,
the NIH 3D Print Exchange includes an explicit review process and heightened
community standards that are in line with the standards clinicians strive to
uphold. However, similar to other traditional repositories, makers
contributing to the exchange likely still face challenges in learning how to
make safely and documenting the quality and safety of their designs so that
others can reproduce and build on them.
## 4\. Methods
To understand NIH 3D Print Exchange PPE submissions, we collected fields from
each submission for quantitative analysis. We further qualitatively coded 520
submissions made before January 1st, 2021. We coded three types of PPE which
made up the majority of the submissions (83.5%): face shields, masks, and ear-
savers. Each submission was reviewed manually to determine if it was a face
shield, mask, ear-saver, or another device. Our final sample of masks, face
shields, and ear-savers was 520 of the 623 total submissions made prior to
January 1st, 2021.
Based on the submission form structure, for each submission we programatically
collected, where ever appropriate, the:
* •
Entry name
* •
Submission date
* •
Remixing attribution and the original design
* •
Manufacturing method (e.g., 3D printing, laser cutting)
* •
3D printer model, if applicable
* •
3D modeling software
* •
Slicing software
* •
3D printer materials
* •
Review status
* •
External documentation (e.g.,images, videos, PDFs, website links)
* •
Pre- and post-processing instructions
* •
Licenses
* •
Comment counts
Each of these pieces of information was either scraped from a well-formatted
field on the design submission page or found by searching the text associated
with each entry for relevant keywords.
We performed an additional layer of processing on this scraped data to gain
insights into makers’ reuse of other submissions in their designs (i.e,
remixing). The form did not require makers to declare changes made in remixes,
though many makers noted it in text. To capture differences across remixes we
compared fields between original and derivative designs and logged differences
in key fields (e.g., manufacturing method, materials, modeling software,
printer-used). Additionally, we searched all text associated with a model for
a list of qualifying words that we saw repeatedly in our qualitative coding
(e.g., more, less, faster, slower, thicker, thinner, safer).
In addition to this automatically collected data, four authors deductively
coded each entry. We derived our codes by inductively coding 50 entries
selected through stratified random sampling across the three design
categories. Additional codes were developed based on a review of the
literature and current media coverage of makers’ response to the pandemic. We
applied these codes in a top down fashion to all 520 face shield, mask, and
ear-saver entries. We all coded in batches of 50 stratified random samples
until saturation across the coders was reached, updating and removing codes
based on group consensus. We reached saturation with an average inter-rater
reliability of 0.87 (range=0.64-1.00) across all accepted codes. Three of
these authors went on to individually review the remainder of the data set. We
met weekly to update each other and discuss any uncertainties that arose.
Based on a thematic analysis of these codes, we present themes on the
community’s values, how trade-offs between values were made in designs, and
how remixing behaviors supported convergence of the design space. We developed
a shared understanding of the data through weekly meetings where PPE and codes
were examined.
## 5\. Results
In March 2020, the NIH opened the 3D Print Exchange as a place for people to
post design ideas and generate discussion and feedback. In our dataset of the
623 submissions between March 20th and January 1st, the designs fell into
three main categories: face shields (N=263/623, 42.2%), face masks (N=177/623,
28.4%), and ear-savers (N=80/623, 12.8%) (Figure 1). The remaining submissions
(N=103/623, 16.5%) included mask cases, ventilator parts, or door-openers. In
this section, we characterize the dataset of face shield, mask, and ear-savers
that we qualitatively coded (N=520). First, we discuss key properties; how
they were designed, manufactured, and by whom. Then, we narrow our focus to
key properties for medical making: replicability and safety.
(a) Face shield; 42.2% of submissions (3dpx-013359 pictured)
(b) Mask; 28.4% of submissions (3dpx-013677 pictured)
(c) Ear savers; 12.8% of submissions (3dpx-013860 pictured)
Figure 1. Examples of the three types of PPE we use in our analysis. (a) shows
a face shield (42.2%), (b) shows a mask (28.4% of submissions), and (c) shows
a tension relief band (ear-savers comprised 12.8% of submissions).
The figure shows the most popular three types of maker made PPE. (A) is a face
shield. It has a 3D printed frame that goes around the face, touching the face
in the forehead. There is a small part that covers the forehead from above
contaminants. Attached to the front of the frame is a clear plastic sheet that
protects the face from contaminants from the front. (B) is a a mask. Like a
surgical mask, it covers the mouth and nose of the person. This mask is 3D
printed and has a pear-shape. There is a black part attached to the snout
(front) of the mask that contains the filter material. It is attached to the
face with elastic straps that wrap around the back of the head. (C) is a
tension relief band. It is a small 3D printed bad that is about the size of a
bookmark. It has notches at both skinny ends which allow rubber bands/straps
to wrap around them. These means that the plastic holds the tension of these
straps, rather than the ears.
Figure 2. The number of submissions per type of PPE were most popular at the
end of March and early April. Face shields are denoted by blue bars, masks by
orange, and ear-saver by grey.
A a bar graph that shows the number of submissions of each type of entry -
face shield, mask, and ear-saver - as a function of time. The x-axis is split
up into time intervals by week, starting with March 29 and ending December 31.
In each time interval, the left most bar represents the number of face shields
submitted, the middle bar represents the number of masks submitted, and the
right most bar represents the number of ear-savers submitted. We see an
increase in number of submissions of all types from March to April and then an
exponential decrease from April on wards.
### 5.1. Temporal Trends
Submissions surged right after the collection was created in immediate
response to the pandemic in the United States. The total number of submissions
steadily increased until and peaked in the first week of April (Figure 2) then
the submission rate dramatically decreased. It increased slightly with the
resurgence of the virus in the United States in May.
Makers tended to submit designs with greater perceived importance or
complexity. The first submissions (prior to March 29th) were two ventilator
valves and one face shield, which are simpler to model and manufacture than a
face mask. The media had also expressed that these were more important for
saving lives than ear-savers which only increase mask comfort. This is an
early example of a repeated pattern in our data. Makers tended to focus on
designing what attracts attention or was deemed important, especially by news
sources, rather than what could be reliably produced to fulfill PPE needs.
### 5.2. Material Trade-offs
Due to resource scarcity induced by the pandemic, makers made careful trade-
offs when selecting manufacturing methods and materials. Makers had to balance
between competing goals of broadening participation, using available
materials, rapid manufacturing, and the safety of a design. We present three
examples below that highlight these tensions and the trade-offs that were made
that were perceivable in the designs.
#### 5.2.1. Material Selection, Safety, and Participation
Many submissions could be made by expert-amateur makers. The most common
filaments used were all widely available to consumers: PLA (N=223/520, 42.9%),
PET (N=34/520, 6.5%), PETG (N=140/520, 26.9%), and ABS (N=64/520, 12.3%). PLA,
PET, and PETG are common and easy to print with. ABS, however, requires more
advanced setups due to toxic off-gassing. Similarly, for designs that
specified a particular 3D printer, the majority (279/312, 89.4%) used printers
available to consumers for less than $10,000. Many submissions listed multiple
filament options (N=147/520, 28.3%) (e.g., printing a face shield in PLA or
PETG). Notably, the most commonly remixed face shields (3DPX-013306,
3DPX-013359) could be made with several variations of PLA or PETG, and could
be manufactured on a consumer printer. The prevalence of easy-to-use materials
afforded opportunities for hobbyists and broadened participation.
On the other hand, more complex materials or printers could improve safety at
the expense of participation. 16% of designs used materials that require
special equipment or additional expertise to work with (e.g., TPU, Nylon, PC,
ASA). The most commonly remixed mask was printed with nylon that requires an
industrial printer. Nylon was chosen because it can be sanitized, unlike PLA
or PETG. Therefore, substitutions other filaments could be unsafe. Similarly,
some designs combined multiple filaments to meet particular design goals at
the expense of easy manufacturability. For example, the “Helmet-Compatible
Community Face Mask” (3DPX-013354) used a rigid material (e.g, PLA, ABS, PETG)
for the snout to ensure the filter was held away from the nose and mouth. It
used a flexible material (e.g., TPU) where the mask touches the face to
improve comfort and air-seal. Choices by some makers to trade off
manufacturability for other goals shows that they believed advanced methods
were required at the cost of supporting more makers.
#### 5.2.2. Powerful Tools that Limit Participation
A design’s manufacturing method determines who can make a design and how much
work is required. Unsurprisingly, 3D printing was by far the most popular
method (N=482/520, 92.7%) followed by laser cutting (N=49/520, 9.4%) and
injection molding (N=22/520, 4.2%). The fact that most designs supported 3D
printing by hobbyist makers broadened who could manufacture PPE.
Several entries listed more than one manufacturing method (N=173/520, 33.3%),
such as the “Georgia Tech Face Shield for Injection Molding, 3D Printing,
Waterjet, Laser Cutting” (3DPX-013314). Often these gave makers choices. For
example, the “NAVAIR - TDP for 3DVerkstan Protective Face Shield”
(3DPX-014090) lists that the submission can either be “printed on non-
industrial 3D printers or laser cut.” While 3D printers are relatively slow
and require post processing, they are widely available. Injection moldering,
on the other hand, is fast but inaccessible to most for hobbyists. Makers
designed for multiple manufacturing methods to both support makers and
increase manufacturing efficiency.
Other designs utilized multiple manufacturing techniques for the same design.
For example, the “Southern Tier Face Shield” with model ID 3DPX-014082 was one
of several face shield designs that required a 3D printed frame that goes
across the wearer’s forehead and a laser cut PC barrier to prevent droplets
from reaching the face. They chose materials like PC because they can be
quickly and automatically cut. Alternatively makers may increase post
processing requirements to avoid using additional manufacturing machines. For
example, regular, office hole-punchers could be easily used with transparent,
plastic, 3-ring binder sheets to create the clear face shield without laser
cut plastic (N=63/520, 12.1%). The “Livingston Shield v2.2” (3DPX-014416)
instructs users to use a hole-puncher to create 4 holes in a transparency
sheet to attach to the 3D printed face shield frame. Though the materials were
common and unlikely to run out in the pandemic, this design requires more
manual post processing to punch and attach the sheets to the 3D printed frame
than laser-cut alternatives. Makers traded-off increases in production speed
through advanced manufacturing with slow manual process that increased
participation.
Aside from manufacturing tools, makers’ choices of software can effect
participation. The most used modeling and slicing tools are listed in Table 1.
Most modeling tools were oriented towards professionals (e.g., Solidworks) or
expert-amateurs (e.g., Autodesk Fusion 360). These require more experience to
use than novice-oriented tools (e.g., TinkerCAD). 70% (N=364/520) and 25.6%
(N=133/520) of submissions provided the modeling and slicing software used,
respectively. The ability to edit another user’s model depends on the software
used (e.g., it is difficult to edit a model made in Blender in Solidworks).
Table 1. Submission Counts and data set percentages for reported manufacturing methods, 3D printer filaments, CAD tools, and Slicing Tools. Manufacturing Method | Submission Count | Portion of All Submission
---|---|---
3D Printing | 482 | 92.7%
Laser Cutter | 49 | 9.4%
Injection Mold | 22 | 4.2%
CNC | 14 | 2.7%
None Reported | 319 | 61.3%
3D Printer Filament | |
PLA | 223 | 42.9%
PET | 34 | 6.5%
PETG | 140 | 26.9%
ABS | 64 | 12.3%
TPU | 47 | 9%
None Reported | 332 | 63.8%
CAD Tool | |
Fusion 360 | 126 | 24.2%
SolidWorks | 102 | 19.6%
Autodesk Inventor | 13 | 2.5%
Rhino | 25 | 4.8%
TinkerCAD | 22 | 4.2%
None Reported | 161 | 31%
Slicing Tool | |
Cura | 59 | 11.3%
Simplify3D | 28 | 5.4%
None Reported | 391 | 75.2%
### 5.3. Community Members and Interactions
Prior work positions maker communities as mainly hobbyists working on
independent projects in a shared space (Khanapour et al., 2017). But the NIH
3D Print Exchange’s community additionally included what we call affiliated
makers who were affiliated with a university, healthcare facility, and/or for-
profit company. Affiliated makers usually represented larger teams of experts.
Interaction between community members on the exchange was uncommon, making it
difficult for makers to seek support through the repository.
#### 5.3.1. Individuals and Affiliated Teams
The NIH 3D Print Exchange was built to support the open exchange of designs
and foster collaboration across stakeholders (e.g., healthcare professionals,
universities, companies, entrepreneurs, hobbyist makers). 448 unique authors
submitted designs. The median number of designs submitted per person was 1 and
the range was 1-9. Our qualitative review revealed that most (N=344/520,
66.2%) authors listed no affiliation with their submission. We suspect this
indicates a lone maker who is not affiliated with a relevant organization.
Those submissions with listed affiliations had team members from industry
(N=84/520, 16.2%), academia (N=67/520, 12.9%), and the healthcare industry
(N=59/520, 11.3%).
Numerous designs were the result of collaborations within and across
institutions. As shown in Figure 3, 30 projects involved people with different
affiliations. The most common type of collaboration was between universities
and health care facilities (N=19/30, 63.3%). The “Stopgap Surgical Facemask”
(3DPX-013429) lists 59 team members from for-profit institutions,
universities, hospitals, the FDA, and the VHA. While affiliated makers often
worked in teams, unaffiliated makers rarely collaborated.
#### 5.3.2. Community Interactions
The NIH 3D Print Exchange supports lightweight interaction between makers
through submission comments, but commenting was rare. 78.5% (N=408/520) of
designs had zero comments and 10.8% (N=56/520) had only one comment. Based on
studies on other studies of COVID-19 medical making (Hofmann et al., 2021;
Lakshmi et al., 2021), we expect that makers were primarily communicating in
other platforms. On the exchange itself, feedback between makers was not the
norm. For instance, the “Helmet-Compatible Community Face Mask” (3DPX-013354)
designer stated in the submission “I welcome all feedback in the comments
section to further iterate and optimize”, and the “USCSW modified 2 part build
with Pencil Popper” (3DPX-013404) mask designer after receiving a 1/5 star
community-rating on the NIH 3D Print Exchange commented “Please contact me to
explain the one star - I’d be happy to modify anything you didn’t like”.
Neither designer received a response. Among the 10.8% of designs that had at
least two comments, constituting a conversation, the median number of comments
was 3. The only outlier was the, “Stopgap Surgical Face Mask (SFM)”
(3DPX-013429), the first revision of a mask that later received clinical usage
rating (see 3DPX-014168), that had 111 comments. We found no evidence of the
NIH 3D Print Exchange broadly being used to collaborate or directly
communicate between makers and/or other stakeholders (e.g., NIH, clinicians).
Figure 3. The 176 mask, face shield, and ear-savers that were designed by an
affiliated person divided up according to affiliation of the members. Though
most designs were carried out by a single type of organization, we see 30
designs with multiple types of contributors.
A Venn diagram showing the number of designs created by universities, for
profit companies, and health care workers. For the designs that only had one
type of affiliation, 43 were university, 73 were for-profit companies, and 30
were healthcare. 1 design was by for-profit and university, 6 designs were
for-profit and healthcare, 19 designs were university and healthcare. 4
designs had all three of these affiliations.
### 5.4. Replicability and Documentation
For the NIH 3D Print Exchange to be useful, makers need to be able replicate
submissions. We found evidence of remixing behavior (179/520, 34.4%), but only
found 61 (11.7%) entries that comments reported as successfully replicated.
Thus, remixing was prevalent, but its unclear if they were manufacturing
others’ designs. We have no way of measuring the number of people who made a
design and chose not to share that on this site. Thus, we examine other
factors which may influence replicability (e.g., documentation, ease of
manufacturing, and licensing). We expect that submissions with more complete
documentation, that are easy to make, and have open licences (e.g., public
domain) would be more readily adopted. Other factors, such as media attention
or affiliation with famous groups (e.g., Prusa, e-NABLE) are also likely
contributors beyond the scope of this study.
#### 5.4.1. Prototype Remixes
The NIH 3D Print Exchange facilitated collaboration and iteration for
“remixing”, similar to other popular maker forums like Thingiverse and
Instructables (Cheliotis et al., 2014; Oehlberg et al., 2015). 131 out of 520
(25.2%) of entries were listed as remixes or “other versions” of models on the
NIH 3D Print Exchange. Figure 4 presents the remixing network (186 designs),
omitting submissions that are neither a remix or remixed. Many nodes (56,
30.1%) were only remixed once. There were notable outliers: one design, the
“3DVerkstan 3D printed face shield head band” (3DPX-013306), was remixed 12
times and 4 additional designs were derivative of those remixes. Another, the
“DtM-v3.1 Face Shield PPE” (3DPX-013359), was remixed 16 times with 6
additional derivatives. Both of these designs were made by affiliated makers.
“3DVerkstan 3D printed face shield head band” is made by 3DVerkstan, a
European 3D printing company, and the “DtM-v3.1 Face Shield PPE” involved team
members from Microsoft, three universities, and three hospitals. The mask and
ear-saver that had the highest number of remixes were the “Stopgap Surgical
Face Mask” (3DPX-013429) (5 remixes), which was made by an expansive team
crossing companies, universities, and hospitals, and the “Surgical Mask
Tension Release Band for Ear Comfort & Extended Use” (3DPX-013410) (6
remixes), which was designed by a VHA employee. It is important to note that
three of these four designs were rated for clinical use, and that no designs
in our remixing graph were given a warning usage rating. Makers did not
iterate to remix designs flagged with warnings to fix those flaws; they
remixed successful designs to work under their local manufacturing
constraints. Overall, we see that safety and affiliation of designs influenced
remixing behavior. This implies that safety was a community norm and
affiliated makers were trusted sources of designs.
Some remixing behavior is not captured by explicit links between submissions.
For example, many designs shared a similar shape to the popular “Montana Mask”
(3DPX-013443) which was spotlighted on Good Morning America on April 12th
(Good Morning America, 2020). Further, not all makers attributed credit. For
example, the maker of the “3 Hole Punch Minimal Face Shield” (3DPX-013501)
found that someone had remixed their design by putting two copies of the
original design in their printing file without attributing. They commented:
“at least credit the creator”.
Our qualitative analysis showed that remixes were primarily incremental
changes to support alternative manufacturing techniques. Few changes were
intended to significantly influence use or efficacy. 54 of the remixes listed
a change in materials, of which 20 added new materials not mentioned in the
original design. 25 designs strictly limited the number of materials
options/materials used in a design. However, the majority (N=17) of these
designs only removed complex filament to use (e.g., TPU, Nylon, ABS). 21
remixes used different modeling software than the original submission, which
may make it easier for makers to replicate the design in the CAD tool of their
choice. 13 remixes used different 3D printers models enabling more people to
manufacture the design and 6 tailored a design suited for “many” printers for
an individual printer model. For example, the “FDM Printable version of Stop
Gap Mask” (3DPX-013771) remixed the popular “Stopgap Surgical Mask” to make it
“allow printing on hobby style FDM printers (PLA, PET-g etc)”. The original
design required an industrial Powder Bed Fusion Nylon printer. Note that this
change effects the mask’s porosity, making it harder to to disinfect. Other
common reasons for remixing designs included adjusting designs to fit
different size print beds, take less time to manufacture, require less
material, or to print more than one design at a time. Occasionally, designs
affected comfort or ease or use in small ways (e.g., “[This change] makes it a
bit more comfortable for different head sizes” (3DPX-013659)) While some of
these changes may effect safety, none constitute divergence from the original
design. On the NIH 3D Print Exchange, remixing behavior was almost exclusively
tweaking designs to support new makers.
There are a few examples of substantial feature changes, often motivated by
local user feedback. One face shield design, “Anvil Verkstan Visor”
(3DPX-014089), significantly modified the popular “3DVerkstan V3 - Face
Shield” based on community feedback: “The entire visor has been redesigned and
model[ed] from scratch so there will be variances in widths, curves, length,
etc. when compared to the original. We re-made this model to better support
our local community in our efforts to help the workers on the front lines.”
Another design, the “Surgical Mask Tension Release Band with Hair Stabilizer”
(3DPX-013819), iterated on a clinically reviewed design to improve it based on
issues experienced by clinician users: “They requested a way to keep the band
from moving around/flying off while attempting to put on or take off their
masks. I incorporated a section of hair pick so that the part can be inserted
into the hair, where it will stay on it’s own, allowing both hands to be used
for putting on or taking off the mask.” We observed few remixes like these,
which implies that makers either created designs from scratch when addressing
more significant design requirements (e.g., clinical usage, fit) or that more
makers were interested in tweaking designs to support manufacturing under
their resource constraints.
Figure 4. A network showing remixing relationships. An arrow starts at the
original design and points to the remix of that design. Colors represent usage
rating, with blue nodes as unreviewed, yellow nodes as rated for community
usage, and green nodes as rated for clinical usage. The two grey nodes are
designs that linked to pages that no longer exist.
A network graph consisting of colored circular nodes and arrows pointing to
other nodes. An overwhelming majority of the 186 nodes are blue, meaning
unreviewed, and only have one or two edges. There are a few very dense parts
of the graphs where there are a lot of edges. Notably, one cluster with 17
edges are all coming from one clinically reviewed design (the DTM face
shield). There is another cluster of 12 edges coming from another clinically
reviewed design (the Verkestan face shield). There are two smaller clusters
for the most popularly remixed strap and mask, each with around 5 edges.
#### 5.4.2. Documentation
The NIH 3D Print Exchange was created for sharing PPE materials for
collaboration, education, and practical use. Thus, it is crucial for entries
to be documented to foster communication between makers, reviewers,
manufacturers, and PPE users. Documentation was often presented as static
documents (N=183/520, 35.2%) (e.g., PDFs), and video links (N=31/520, 6%).
Images were also a popular form of documentation. All entries included at
least one thumbnail image, by default a view of the 3D model, and the majority
included additional photographs or diagrams (N=429/520, 82.5%). A majority of
entries included at least one web link (N=315/520, 60.6%), often to a
portfolio or alternate repository (e.g., Thingiverse, GitHub). External
website content is dynamic, but the NIH required static documentation to be
included on the exchange itself. We stopped reviewing links because we found
several broken links during our qualitative analysis. Finally, a majority of
entries (N=290/520, 55.8%) also included pre/post processing information, such
as printer settings, cleaning instructions, and material recommendations,
which are critical to ensure proper manufacturing and safe use. Overall,
makers tended to provide documentation that required the least additional work
from them, preferring easy to update websites over creating static documents,
or adding easy to capture images instead of videos. Documentation did not
appear to be makers’ top priority.
In the medical domain, reproducible testing procedures and results are
critical. Test results are needed to quantify the level of protection a design
provides. Their importance to reviewers is supported by the correlation
between presence of testing and community or clinical approval
($\chi^{2}=4.1,p<.05$). Only 44 (8.5%) designs documented rigorous testing
results: 6 face shields and 46 mask. A $\chi^{2}$ test reveals that
affiliation with healthcare facilities or universities correlated with the
presence of testing results (healthcare: $\chi^{2}=22.3,p<.00001$ ;
university: $\chi^{2}=21.4,p<.00001$). This is likely because testing requires
specialized equipment that consumers cannot easily access. The community’s
importance of testing advantage affiliated makers over unaffiliated makers.
#### 5.4.3. Licensing
All submissions included a licenses that set permissions for sharing,
adapting, selling, or remixing submissions. Several designs offered little or
no restrictions to use and adaptations: N=67/520, 12.9% were public domain and
N=330/520, 63.5% were CC BY or CC BY-SA; both of these do not restrict usage,
but require attribution to the original. Other designs limited themselves to
non-commercial use only (N=96/520, 18.5% were CC BY-NC or CC BY-NC-SA), while
still others used the strictest of licenses which do not allow for
modifications to be made to the design (N=14/520, 2.7% were CC BY-NC-ND or CC
BY-ND). Overall, the tendency for authors to use less restrictive licences is
aligns with prior work that shows that maker communities tend to value
openness (Kuznetsov and Paulos, 2010).
### 5.5. Convergence of Designs
Our dataset was characterized by rapid convergence of design ideas; there was
little exploration of new forms of PPE. The COVID Collection was broad in it’s
call for design, stating that it was created to “inform decision-making on PPE
and medical device production, without stifling innovation”. Interestingly,
the community who submitted to this collection narrowed its focus to the
production of three types of PPE: masks, face shields, and ear-savers; 520 of
the 623 total submissions (83.5%) fell into these three categories. The 103
“other” submissions focused on meeting a range of needs (e.g., ventilator
parts, shoe covers, gowns, hand-less door openers, nasal swabs).
We further saw convergence of designs within these three overarching PPE
categories. Consider face shields. In our preliminary analysis of a random
sample of face shields, the only common difference between the designs was
protection from liquid droplets from above (Figure 5a and b). Besides this
feature, face shields almost exclusively consisted of a 3D printed frame that
braces against the forehead and a clear plastic sheet that attaches to the
front of the frame to protect the face. The two most commonly remixed face
shields (“3DVerkstan 3D printed face shield head band” and “DtM-v3.1 Face
Shield PPE”) followed this archetype. The convergence to only a few archetypes
over a period of about a month is unusual. Generally, makers are espoused for
their creativity and presentation of novel, innovative, even wild ideas. But
those ideas were largely absent from the NIH 3D Print Exchange.
(a) A face shield without protection from above (3DPX-013343)
(b) A face shield with protection from above (3DPX-013325)
(c) The scuba mask/face shield design (3DPX-013396)
Figure 5. Examples of three types of face shields. The first two examples show
the most common arcehtypes we found, those providing coverage from above (b)
and those that do not (a). The third image (c) is an example of the
“scuba/snorkel” designs that relied on a consumer face mask or snorkel mask
that covers the whole face and air is breathed through the snorkel pipe.
Three types of face shield. The first (sub-figure a) has a white 3d printed
frame. It looks like a thin headband that wraps around the head, pressing
against the wearer’s forehead. this band has two hooks protruding from the
front (over the forehead) on which a plastic, transparent sheet hangs to
prevent liquid from reaching the face. Notable, there is a large gap that you
can see between the 3D user’s face and the transparency sheet, where droplets
of liquid could fall. The second (sub-figure b) is the same as figure a except
there is a small visor that starts at where the face shield touches the
transparency sheet and slants up towards the top of the wearer’s head. This
visor prevents liquid from entering the space between the transparency sheet
and the person’s face. The third (sub-figure c) shows a person wearing a scuba
mask. It has a tight seal around the user’s entire face (chin to forehead) and
is clear so the user can see out of it. There is a small tube that comes out
of the sealed mask and points up towards the sky to allow air to reach the
user under the sealed mask.
There was one notable design for a combined face shield-mask that starkly
deviated from this norm: the “Five-minute zero-print full-face snorkel mask
with filter” (3DPX-013396), shown in Figure 5c. It required no 3D printing and
only attachment of filtering material over the spout of a full-face, sealed,
snorkel mask. There were 13 other scuba-mask-based designs that all used the
same concept but used a 3D printed adapter to attach the filter material.
Across our entire qualitative review, this was the only archetype that varied
significantly from a design that was reviewed for clinical use before the
rapid drop off in submissions in April. It is the exception that proves the
rule.
### 5.6. Safety
The review system is the core component that distinguishes the NIH 3D Print
Exchange from any other maker repository. The process enforces clinical norms
of safety and quality. Overall, the risks associated with different types of
PPE was the primary determinant in review status. More subtle details that
contribute to safety quality were difficult to analyse because, to date, 81.3%
of designs have not been reviewed. However, some traits that we expect
contributed to a design’s safety-level could be found across the whole
dataset. Though we are not experts in the safety of PPE, we identified three
relevant safety traits through our analysis: coverage, fit, and the presence
of cleaning instructions. The safety criteria for masks and face shields
differ, and so we discuss them separately below. Ear-savers, on the other hand
pose little risk as an accessory to improve comfort, so we do not discuss
their safety features. There are no examples of ear-savers with a “warning”
usage rating status.
#### 5.6.1. Usage Ratings and Safety Results
The NIH 3D Print Exchange created four different usage ratings to classify
entries based on the prototype’s level of safety (Table 2). The vast majority
of entries (N=464/520, 81.3%) had a “prototype” status, which is the default
rating of submissions uploaded, indicating that the submission has not been
fully reviewed. Though not an official rating, we did note that 8.8%
(N=41/464) of these submissions had received notes from the reviewer which
indicates that these submissions were not acceptable given the level of
documentation included in the submission. There were three statuses for
designs that completed review, “Clinical Use”, “Community Use”, and “Warning”.
These categories dictate the level of trust reviewers had in the designs’
safety and efficacy. Design affiliation with healthcare correlated with both
likelihood of receiving reviewer attention ($\chi^{2}=11.4,p<.001$) and
community or clinical ratings ($\chi^{2}=11.2,p<.001$). This may indicates
that affiliated makers were sought out for review and were better suited to
submit designs that reviewers viewed favorably (i.e., considered safe).
Table 2. The usage rating given to PPE across the three main categories of face shield, mask, and ear-saver. The majority of designs received an unreviewed “prototype” status. Rating | Face Shield | Mask | Ear-Saver
---|---|---|---
Clinical use | 16 | 2 | 11
Community use | 2 | 22 | 1
Warning | 0 | 2 | 0
Checked but no rating given | 7 | 28 | 6
Prototype (not checked) | 238 | 123 | 62
29 entries (5.6%) received clinical usage ratings, meaning the entry had been
evaluated in a clinical setting and reviewers deemed appropriate for
healthcare workers in contact with COVID-19 patients—their highest mark of
safety. For example, the “Stopgap Surgical Face Mask (SFM) Revision B”
(3DPX-014168) was evaluated in a clinical setting and was given a clinical
usage rating. Others (N=25/520, 4.8%) received a community usage rating,
meaning that the entry is suitable for workers in retail stores, law
enforcement, and other community activities. 2 entries (0.4%), both of which
were masks, received a “warning” rating, indicating that the entry needed FDA
approval or has design flaws that make it unsafe to use. Outside of our
dataset of masks, face shields, and ear-savers, 5.5% of all submissions
(N=34/623) had a warning rating. A majority of these entries with warning
status (N=15/34, 44.1%) were ventilator parts. Many of these entries had notes
from the author saying the entry had not been tested; for example, the author
of the “Ventilator Circuit Splitters - reinforced & thicker walls”
(3DPX-013347) stated that they “make no representations as to the safety of
this device.” Other entries with the warning status included parts for other
respiration devices and mask sanitizers. As we expected, classes of devices
that pose more risk (e.g., masks, ventilator parts) received the more
scrutiny, and devices that pose less risks (e.g., face shields, ear savers)
received less scrutiny.
There were 41 submissions that were not yet reviewed but had reviewer notes.
The reviewer notes in a majority of these submissions (N=31/41, 72.1%)
requested documentation, specifically best printing parameters or use
instructions. Some reviewer notes (N=8/41, 18.9%) pointed out that the
printing instructions and instructions for use were on external links and this
documentation needed to be statically embedded in the submission to prevent
modifications after review. Other reviewer notes (N=4/41, 9.3%) requested that
submissions be renamed so as not to imply incorrect usage and protection
properties. For example, reviewers asked for “respirator” to be taken out of
the title of the submission “3D Printed Respirator Mask, 4 sizes, XSM, SM, M,
L” (3DPX-013948) because the term “respirator” is a medical term that implies
a specific level of protection that this mask did not meet (National Institute
of Health, 2020a). Two reviewer notes on masks requested testing information.
For example, the “The Unity Mask PRO”(3DPX-014364) listed that the mask met or
exceeded National Institute for Occupational Safety and Health (NIOSH) N95
filtration criteria, but did not provide the test results. Based on these
reviewer comments that makers’ and reviewers’ value of documentation were
misaligned.
#### 5.6.2. Characteristics of Safe Designs
Beyond the designs that were reviewed, we could only identify three
characteristics of designs that we have a high confidence influence whether or
not their reproduction is safe for clinical use: (1) mask sizing/fit, (2) face
shield coverage, and (3) the presence of cleaning/disinfecting instructions).
The main safety trait that varied across masks was the inclusion of different
size 3D models. For masks, one size does not fit all; sizing is a key factor
that influences fit and fit ensures safety. Facial features do not scale
uniformly, so scaling model sizes is not a solution. Most masks will not
create a secure air seal on a diverse set of human faces. In these cases,
contaminated air could enter through the gaps between the mask and face,
rather than through the filter. Different sizes are needed to ensure that
people of different ages and genders are protected. Only 43 (25%) of the masks
offered at least two sizes. In practice, users may find that different designs
fit them better, but most wearers do not have the opportunity to print a range
of masks and pick the best fit. The lack of sizing features in this data set
shows that most makers were not considering this key safety feature in their
design process.
The main safety trait that varied across face shields was forehead coverage.
Face shields at a minimum need to protect the front of the face (eyes, nose,
and mouth) from liquid droplets, and almost all designs did so; 2 did not.
However, several designs (192/263, 36.9% of face shields) also protected the
wearer from liquid droplets from above by covering the forehead. Most designs
either created a “visor” like piece to connect to the top of the mask or a
closed gap so that there is no open space between where the frame touches the
forehead and the clear sheet (see Figure 5). Forehead coverage may not be as
critical as mask sizing for safety, but the additional feature demonstrates
that many designers were considering increase safety when designing face
shields.
A final piece of information that was critical to safe use of reusable PPE in
a pandemic was cleaning instructions. Cleaning instructions are necessary to
ensure proper disinfection and safe reuse. Only 84 (16.2%) and were included
in instances of all three types of prototypes. Affiliation and usage rating
were both correlated with presence of cleaning instructions. A $\chi^{2}$ test
reveals that affiliation with a health care organization or a university
correlated with the presence of cleaning instructions (healthcare:
$\chi^{2}=5.0,p<.05$ ; university: $\chi^{2}=11.8,p<.001$), and the presence
of cleaning instructions was correlated with a community or clinical usage
rating ($\chi^{2}=33.3,p<.00000001)$. Many cleaning protocols are based on
common protocols for medical devices that are already in clinical use. Perhaps
affiliated makers were more readily aware of these practices than unaffiliated
makers. Notably, while cleaning instructions effected the review process,
there was no submission field for including them explicitly.
Overall, there were only a few features that we could demonstrate impacted the
safety rating of submissions. This may be because of how small the sample of
reviewed designs is, making it difficult to identify common flaws in makers
designs or characteristics of high-quality designs.
## 6\. Discussion
The COVID-19 pandemic spurred one of the most widespread efforts of medical
making to date. Makers sustained efforts to make PPE with existing technical
and human infrastructure (Novak and Loy, 2020). The NIH 3D Print Exchange
COVID collection was meant to “inform decision-making on PPE and medical
device production, without stifling innovation” (National Institute of Health,
2020a). Decision-making was supported by the formal review process, but
reviewers were overwhelmed by the surge of designs. Further, the platform’s
goal of fostering innovation was undermined by regulated, inflexible
structures which have been shown to create environments that can be less
conducive for maker contributions (Khanapour et al., 2017). The decision to
regulate the repository stems from the medical making community norm of safety
and reliability. These prerogatives led medical makers to limit remixed
designs, causing a rapid convergence of ideas. Hobbyist makers’ contributions
were subject to social and institutional structures of medical maker groups.
In the context of medical making during a health crisis, we discuss how these
material practices (Rosner, 2012) inform future design of online repositories
that support supporting innovation through peer production in high-risk
domains.
### 6.1. The Effects of Novel, Scarce Review
NIH reviewers were inundated with hundreds of designs, and few designs were
reviewed. Reviewer time is a scarce resource. Though the NIH provided some
guidance for safely designing one type of PPE, masks, it was not widely
adopted by makers; few masks were approved and many lacked documentation.
Further, the NIH provided makers little details on what the review process
would consider. We suggest that this unclear reviewing process advantaged
makers affiliated with healthcare institutions over lone-makers.
#### 6.1.1. The Benefits of Affiliation
Makers who were affiliated with a university, the healthcare facility, or for-
profit company were reviewed more positively because they tended to more
successfully adhere to the NIH’s value of safety. The designs affiliated
makers contributed received more reviewer attention and higher usage ratings.
While the NIH 3D Print Exchange aspired to support collaboration, the
collaborations that were positively reviewed tended to rely on institutional
systems for accountability, reliability, and perceived value of contribution.
Prior literature shows that maker communities are driven by social norms (Fox
et al., 2015; Toombs, 2015). We contribute to this literature by demonstrating
that social norms derived from clinical institutions advantage makers working
in the healthcare domain.
Affiliated makers had better access to resources pools that aligned with the
NIH’s clinical expectations of safety. For instance, interviews with makers
working in close proximity to healthcare workers helped affiliated makers
evaluate PPE usability (Lakshmi et al., 2021). Additionally, they could access
rare medical expertise (e.g., infectious disease teams). Another study of
COVID-19 maker communities showed that they struggle to curate and analyze
scientific information in an evolving crisis plagued with widespread
misinformation (Hofmann et al., 2021). This challenge further advantages
affiliated makers with experience reading scientific literature (e.g.,
healthcare workers, university workers). In terms of material resources,
universities and for-profit companies often had resources like 3D printers,
filament, and CAD and fabrication experts. Further, many healthcare and
university workers had access to testing facilities, which explains the
statistical correlations between these affiliations and presence of testing
results . Finally, affiliated makers had access to teams of specialized
experts when attempting faster iteration. Access to such resources is
demonstrative of an institutional culture that supports and demands thorough
documentation in order to decrease safety risks. It is to be expected that
affiliated makers would bring these practices and the values they represent
with them to the NIH 3D Print Exchange.
#### 6.1.2. Reviewing Expert-Amateurs
Individual or hobbyist makers may be unfamiliar with formal review processes,
while affiliated makers are often accustomed to peer-review. Review is not
common on maker repositories (e.g., Thingiverse), which support relatively
unrestricted sharing of designs. Therefore, makers likely were unclear about
expectations going in. In our study, we saw that most submissions had the
level of documentation we would expect to see on hobbyist repositories.
However, the NIH reviewers needed documentation in order to replicate designs
to review them. We see three plausible explanations for the discrepancy.
First, makers may be defaulting to their usual practices and simply following
the norms established on other repositories. Second, makers may lack an
understanding of what details reviewers need, particularly early on when few
accepted designs could be used as exemplars. Alcock et al. demonstrated with
survey of Thingiverse that makers rarely provide enough documentation to
support other makers in remixing their work (Alcock et al., 2016). It follows
that they would continue to struggle to provide documentation on the NIH 3D
Print Exchange, but with greater consequence since it halts the review
process. Finally, makers may not value the review process as it excludes their
contribution to the collaborative in clear terms (Morgan et al., 2014).
Instead, we expect that many makers are using the NIH 3D Print Exchange as a
repository for COVID specific designs. The lack of documentation poses unique
challenges for regulated maker repositories. How can makers be encouraged to
document their work for review without discouraging makers with less resources
or experience?
Lone makers seemed to struggle to create safe, documented designs, outside of
making minor remixes of affiliated designs. Based on the designs they
provided, unaffiliated makers appear to be unaware of the necessity of
cleaning instructions or manufacturing instructions in the review process;
understandably so, since cleaning designs is “invisible” (Morgan et al., 2014)
to makers outside of healthcare settings. However, the correlation between
documentation and usage ratings indicates that this work is valued by
reviewers. Therefore, novice medical makers can submit more effective designs
if they are alerted to reviewers’ values by encountering mandatory form fields
asking for this safety critical information.
One could argue that affiliated makers appear better prepared than lone makers
to support healthcare settings—i.e., leave it to the professionals. But, this
view erases the contributions of non-affiliated medical makers who provided
critical support in the COVID-19 crisis (Hofmann et al., 2021). It also
undermines the potential in harnessing low-skill craft alternatives in
supplementing collective action beyond established practices (Lakshmi et al.,
2021). Instead, we propose that the NIH 3D Print Exchange naively positioned
lone makers at the same level as large, professional, collaborative
institutions. This forces makers to perform along standards set by larger
institutions without those institutional resources. To do this, most makers
tweaked accepted designs, rather than innovating in diverse ways.
### 6.2. Safety’s Impacts on Design Diversity
There were only a few distinct types of designs on the NIH 3D Print Exchange
and remixes with small incremental changes were frequent. However, the NIH 3D
Print Exchange’s goal was not to “stifle innovation”; they wanted makers to
contribute unique and novel solutions. We hypothesize that the norm of safety
and the uncertainty of the review process encouraged individual makers to only
make small changes when remixing.
We observed that when makers adapted others’ designs, they limited their edits
compared to the remixing behavior in other creative communities; few makers
made large, structural changes to designs. Affiliated makers contributed the
designs that were remixed the most. These had also received community or
clinical usage ratings. This data reveals a tendency of the community to
follow norms set by perceived authorities including, these professionally
affiliated groups and the NIH itself. However, as described, lone makers face
greater challenges than institutions to document and test their designs.
Instead, most makers likely work with scarce resources (e.g., one 3D printer
model, few types of filament) and need to adapt designs to their emergent
needs.
Lone makers had to two main strategies to meet the NIH’s value of safety with
their limited resources. One option was submitting a more innovative design
without full documentation, and, therefore, receiving less reviewer attention
or lower usage ratings. Otherwise, they posted derivative works of clinically
reviewed designs. However small, there is inherent value in makers’
incremental modifications of existing designs. Many modifications consisted of
adjusting a design to allow faster manufacturing or printing with a different
materials or devices. Unlike a comparable domain of open source software,
Hudson et al. has shown that adapting designs to new fabrication techniques is
no trivial feat (Hudson et al., 2016). In some cases, small changes (e.g.,
changing materials) may have unintended safety consequences, but, for the most
part, these changes can help more people make a design.
The remix behavior on the NIH 3D Print Exchange emphasizes the affordances of
physical materials and how they shape the designing and manufacturing process.
That many remixes adjusted the materials or tools highlights an intention to
interact with design materials as “heterogeneous enactments” over time rather
than “fixed forms” (Rosner, 2012). We question measures of novelty as a degree
of divergence from the original (e.g, (Cheliotis et al., 2014)). The small
remixes of designs modified material-determined affordances and expanded who
could make well-rated designs. This helps more makers to aid in supplying
PPEduring the pandemic. While we recognize these small and critical
innovations, they are not inline with the NIH’s expectations of makers. In a
pandemic characterized by shortages in PPE, using a range of materials or
manufacturing methods and prioritizing speed along with safety are all
valuable design goals. Therefore, seeking more unique design archetypes is a
valuable goal.
Finally, makers may submit more divergent designs if they better understood
higher level healthcare needs; one cannot innovate without first understanding
stakeholder practices, values, and needs. The large number of masks, face
shields, and ear-savers was linked to the media coverage of need for these
types of PPE. As specific designs got more positive coverage from their NIH
reviews, makers were inclined to copy them. One solution to encourage novelty
is to make the repository’s high-level goals explicit. For example, while not
directly connected to the NIH, the Department of Defense’s America Makes
initiative’s design innovation contests were successful at producing novel PPE
that was submitted to the NIH exchange. Without clear calls to innovate,
makers likely directed their efforts at the meeting the review process’s
afforded value of safety. Rather than “re-imagining what a face shield looks
like”, they tweaked accepted designs.
## 7\. Design Recommendations
The COVID-19 collection within the NIH 3D Print Exchange has been an
experiment in online sourcing of community medical device designs. In many
ways, this tool is follows Lakshmi et al’s recommendations to use “partially-
open repositories” to collect, review, and regulate medical makers’ designs
(Lakshmi et al., 2019). We understand, from the repository’s statement, that
it is intended to create an environment for purpose-driven collaboration with
amateurs and experts (Kuznetsov and Paulos, 2010). Based on our findings, we
conclude there are gaps to bridge in realizing this goal. We found that this
iteration of the exchange was not able to review designs fast enough, and
makers tended to submit risk averse designs rather than proposing novel and
unique designs. While this trade-off may be inevitable, we expect that a
balance between novelty and review could be struck with interface variations
that support and reinforce community values like safety and innovation. We
offer three suggestions.
### 7.1. Clarify Reviewing Criteria
A reviewing process is novel for maker repositories, and makers need
clarifications to use it effectively. Confusion could be clarified by
including reviewer comments in accepted designs. These comments are needed on
incomplete and accepted designs so that makers can differentiate between the
two. Makers would benefit from information about what made a design safe or
effective. An additional way clarify requirements is to mark critical fields
as mandatory for submission. However, the NIH 3D Print Exchange may have
limited requirements in an effort to not overwhelm makers. As a compromise, we
propose that fields be marked as “recommended for review”. This option would
still highlight safety and documentation requirements without disenfranchising
makers who prioritize sharing a design over achieving a usage rating. Reviewer
time would be conserved since they could easily prioritize designs that have
the details needed to replicate a design.
### 7.2. Identify Remixes
In it’s current form, the NIH 3D Print Exchange does not include a structure
for annotating how a design was remixed. While this reduces burden of
documenting changes, it also obfuscates “material” (Rosner, 2012) distinctions
between designs. Without a field to describe changes, it is up to makers to
add this information in unrelated fields or to omit it entirely. Moreover,
requiring these details could conserve reviewers’ efforts by reviewing just
the highlighted changes rather that re-reviewing derivative work.
### 7.3. Motivate Innovation
Designs on the NIH 3D Print Exchange rarely diverged from a few common forms.
Instead, it seems that safety was valued at the expense of innovation. To an
extent we agree with how priorities were set; while innovation and creativity
are important, safety is nonnegotiable. However, we expect two strategies can
encourage innovation while ensuring safety. First, more diverse designs can be
encouraged through explicit calls (e.g., “build a better face shield”).
Second, innovation can be rewarded along with safety, clarifying to makers
that the two do not have to conflict. Safety was rewarded with clinical usage
ratings. Similarly, we recommend that commendable innovation and creativity be
noted in the review process. A “uniqueness votes” or “tags” system could
encourage makers to explore new ideas. Reviewers could prioritize reviewing
highly innovative designs over ones similar to reviewed designs. Since makers
tend to modify designs with positive reviews this could have a snow ball
effect where more makers remix designs that are increasingly divergent.
### 7.4. Support Innovation
To support the sharing of more creative, novel designs, collaboration
interfaces for medical making must provide a structure to indicate the
progress and/or intent of a design. All designs submitted to the NIH 3D Print
Exchange automatically received the “prototype” status, which put it in the
queue for review. There was no way to designate a design as an “seeking
feedback”, “not intended for production”, or “ready for review.” Consequently,
we suspect this lack of affordance limited the scope of submitted designs to
those that were close to current PPE designs or reviewed designs on the NIH 3D
Print Exchange. Indeed, it is hard to determine the safety of a creative,
novel design that is unlike previously reviewed designs. Posting such a design
without indicating it is not ready to be manufactured can be unsafe,
especially since new makers to the community may mistakenly view a designs
affiliation with the NIH’s website as a sign of authority or approval.
Introducing an “in progress” label to designs will encourage the sharing of
more diverse ideas and seeking out feedback without risk of others adopting a
design not ready for production.
## 8\. Limitations
We recognize that our work was affected by our own experiences. Two of the
authors were deeply involved in making PPE this spring and contributed to
designs submitted to the NIH 3D Print Exchange. Though we have established
relationships with the creators of the NIH 3D Print Exchange, in this paper,
we only draw from publicly available evidence. As researchers in computer
science fields our recommendations focus on the design of tools, but we
recognize that public policy determines what designs can be created and when
and how they can be used. While we engage in wider discussions of policy, they
are out of the scope of this paper. In particular, we have avoided making
judgements about what makes designs safe or who should be doing this work. We
leave such questions up to medical makers and suggest tools that could bolster
these critical conversations.
## 9\. Conclusion
The NIH 3D Print Exchange houses 623 makers’ designs for PPE, the results of
one of the most expansive efforts of medical making yet recorded. The forum
was created to strike a balance between providing guidance through a formal
review process and not stifling creativity. Our analysis of these 623 reveals
makers’ misconceptions about the review process and criteria which lead to a
rapid convergence of the design space. A few key designs created by
university, for-profit company, and clinically affiliated makers received
clinical usage ratings. Following submissions, particularly those made by
unaffiliated makers, were derivatives of these designs. Often these
submissions made small changes to optimize or increase flexibility of
manufacturing. Overall, few designs were reviewed, and several of the designs
that received reviewer attention were missing key pieces of information that
prevented full review for clinical use.
In sum, our results suggest that affiliated makers received more positive
ratings and more reviewer time than non-affiliated makers due to the knowledge
and practices they bring from their clinical work. At the same time, many
makers, particularly unaffiliated makers, often left out key pieces of
information from their design submissions, leading to wasted review cycles. To
make a more efficient and understandable review process without stifling maker
creativity, we make three recommendations. First, prioritize unique designs
for review to provide more examples of divergent and safe designs. Second,
pose explicit requests to the community calling for diverse ideas and allow
for makers to denote a design as “seeking feedback” so as to be clear that the
mask is not ready for mass-manufacturing. Finally, establish clear metrics of
safety. These changes aim to bridge the gap between the NIH’s goals and
unaffiliated makers’ understanding of the review process and the values it
implies.
## References
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|
# The Binary INformation from Open Clusters Using SEDs (BINOCS) Project:
Reliable Photometric Mass Determinations of Binary Star Systems in Clusters
Benjamin Thompson Department of Physics & Astronomy, Texas Christian
University, TCU Box 298840,
Fort Worth, TX 76129, USA (p.frinchaboy, t.spoo<EMAIL_ADDRESS> GitHub,
Inc. Peter M. Frinchaboy Department of Physics & Astronomy, Texas Christian
University, TCU Box 298840,
Fort Worth, TX 76129, USA (p.frinchaboy, t.spoo<EMAIL_ADDRESS> Taylor Spoo
Department of Physics & Astronomy, Texas Christian University, TCU Box 298840,
Fort Worth, TX 76129, USA (p.frinchaboy, t.spoo<EMAIL_ADDRESS> John Donor
Department of Physics & Astronomy, Texas Christian University, TCU Box 298840,
Fort Worth, TX 76129, USA (p.frinchaboy, t.spoo<EMAIL_ADDRESS>
(Received October 3, 2019; Revised May 31, 2020)
###### Abstract
We introduce a new binary detection technique, Binary INformation from Open
Clusters using SEDs (binocs), which we show is able to determine reliable
stellar multiplicity and masses over a much larger mass range than current
approaches. This new technique determines accurate component masses of binary
and single systems of the open clusters main sequence by comparing observed
magnitudes from multiple photometric filters to synthetic star spectral energy
distributions (SEDs) allowing systematically probing the binary population for
low mass stars in clusters for 8 well-studied open clusters. We provide new
deep, infrared photometric catalogs ($1.2-8.0$ microns) for the key open
clusters NGC 1960 (M36), NGC 2099 (M37), NGC 2420, and NGC2682 (M67), using
observation from NOAO/NEWFIRM and Spitzer/IRAC. Using these deep mulit-
wavelength catalogs, the binocs method is applied to these clusters to
determine accurate component masses for unresolved cluster binaries. We
explore binary fractions as a function of cluster age, Galactic location and
metallicity.
open clusters: general — open clusters: individual (NGC 2099 (catalog M 37),
NGC 2682 (catalog M 67))
††journal: AJ
## 1 Introduction
Binary stars have long been recognized to have an effect on stellar evolution,
allowing the formation of non-standard stars, such as blue stragglers. While
these stars may be formed in the field, their abundance in star clusters leads
us to explore the effects of environment on the frequency of these stars.
Additionally, it has long been recognized that the frequency of binaries will
also affect the long term dynamics and stellar distributions within a cluster.
In order to more fully understand these effects on stellar and cluster
evolution, the number of binaries within a cluster must be accurately
determined, as well as other properties about the binary population. Having
this data will allow correlations to be made between properties of the binary
sample and dynamical traits of the cluster, as well as the frequency of
anomalous stars. Correctly determining the number and composition of binary
systems will allow deeper understanding of the physical processes behind
stellar and cluster evolution.
Internal process can result in stars being ejected from the cluster due to
gravitational interaction with other member stars. When a less-massive star
gravitationally interacts with a more massive one, it may pick up enough
energy to be accelerated beyond the escape velocity of the cluster. Binary
systems may amplify this process by contributing part of their orbital energy
to interactions, which is usually greater than the kinetic energy of the
system moving through the cluster, which is fairly easy in poorly bound, low-
mass, low-dispersion systems like open clusters.
Due to the vast timescale over which clusters evolve, stellar ejection cannot
be studied observationally. Cluster ejection is usually studied via detailed
N-Body simulations (Hurley et al., 2001, 2005), which can give a detailed
description of what stars were ejected, when they were ejected, and how fast
they were moving at the time of ejection. Each of these parameters dictate how
the field population of the galaxy may have been built up by open cluster
dissolution.
N-Body body simulations have already been run to analyze the binary
population’s effect on escaping stars, discovering only a slight difference
when varying the cluster binary percentage from 0 to 70% (Moyano Loyola &
Hurley, 2013). These studies have made assumptions about the primordial binary
population, however, such as an even mass-ratio (ratio of larger to smaller
star mass) distribution, which may not be the case in reality.
Measuring the cluster binary fraction and mass-ratio distribution (as a
function of primary mass) for real clusters will go a long way in calibrating
these N-Body simulations. Analyzing open clusters with various ages will also
allow N-Body simulations to check intermediate steps against these
“benchmarks,” further improving their accuracy and predicting power.
It has also long been established that members of star clusters experience
mass segregation. Binary systems, on average are more massive than single
stars, have generally been thought to become more centrally concentrated than
singles due to the same mechanism. A decreasing binary fraction with radius,
indicative of mass segregation, has been observationally confirmed for several
open and globular clusters (e.g., Geller & Mathieu, 2012; Milone et al.,
2012). Similar analyses have been conducted on the young ($15-30$ Myr),
massive cluster NGC 1818, located in the Large Magellanic Cloud (LMC),
producing conflicting results, which having more detailed clusters with binary
characterization will allow us to fully explore.
While the advent of of high precision photometry, from Kepler and TESS, has
opened up new studies for analyzing eclipsing binary systems, systematic
methods for reliably and efficiently probing cluster binary populations for
large numbers of clusters has remained elusive.
## 2 Binary Detection
### 2.1 Previous Large-Scale Methods in Clusters
Currently, binary systems within open clusters are detected using one of two
methods. The first is _two-band detection_ , leveraged by both Elson et al.
(1998) and de Grijs et al. (2013), which uses a cluster color-magnitude
diagram (CMD) to quickly separate stars into singles and binaries. Stars lying
far enough from the cluster single-star main sequence are classified as
binaries, while all others are deemed single stars. While this method is quick
and easy (only requiring imaging in two filters), it is plagued by
degeneracies when attempting to determine accurate masses. In their paper, de
Grijs et al. admit that this method will only work inside a small mass range
of NGC 1818. Outside of this region “…the CMD is too steep to easily
disentangle single from binary stars and blends. In addition, toward fainter
magnitudes, photometric errors start to dominate any potential physical
differences…” To explore radial distributions of binary systems with a wide
range of masses, two-band detection is not feasible.
The ideal method for this work is using radial velocity (RV) measurements to
determine stellar multiplicity, as this approach will yield the most
information about each system. There are, however, significant drawbacks: due
to the number of stars in each cluster, and the velocity precision necessary
to detect most binary systems, RV surveys can take hundreds of nights over
many years, if not decades, to complete. Additionally, RV surveys cannot
accurately measure cluster stars fainter than $V\sim 16$ without significant
observing time on large telescopes, which removes a majority of cluster stars
from the RV studies conducted so far.
With the growth of large-scale, space and ground-based photometric surveys
(e.g., Pan-STAARS, ESA Gaia, LSST, 2MASS, UKIDSS, VVV, Roman Space Telescope,
Spitzer/GLIMPSE, WISE; Chambers et al., 2016; Gaia Collaboration et al., 2016;
Ivezić et al., 2019; Cutri et al., 2003; Lucas et al., 2008; Minniti et al.,
2010; Spergel et al., 2013; Churchwell et al., 2009; Wright et al., 2010),
another photometry method would be ideal that is not cost-prohibitive in terms
of telescope time, but will also yield accurate and detailed binary
information over a wide range of stellar masses.
### 2.2 BINOCS
We introduce the new Binary INformation from Open Clusters using Seds (binocs)
binary detection method, which determines accurate component masses of binary
and single systems within open clusters by comparing observed magnitudes from
multiple photometric filters to synthetic star spectral energy distributions
(SEDs). An example of this method is shown in Figure 1.
Figure 1: SED fitting of observed star in NGC 2682. _(Left)_ Comparison of
observed fluxes (grey dots) to best-fit single star model SEDs. _(Right)_
Comparison of observed fluxes to best-fit single model. For each model, the
fit’s $\Phi$ value, defined in §2.3, is shown.
A star in NGC 2682 was observed in 11 bands ($ugriJHK_{S}$
[3.6][4.5][5.8][8.0])111IRAC [5.8] and [8.0] magnitudes provide little
strength to the fit since they are on the Rayleigh- Jeans tail of the SED, and
they are significantly more shallow than the [3.6] and [4.5] magnitudes, thus
while they were used in testing the technique, we choose not to use them in
the analysis presented here.. When the observed magnitudes are compared
against all single star model SEDs, two close-fitting models are detected,
however neither fit the entire spectrum well (left panel of Figure 1). While
one model (with a mass of 1.062 M⊙) fits accurately in the optical and $J$
band, it diverges for IR fluxes. The 1.024 M⊙ model fits oppositely:
overestimating optical fluxes, while accurately mapping the IR.
Next, the star is compared to binary model SEDs, where the best-fit is much
more accurate (right panel of Figure 1). A binary star in NGC 2682 with a
primary mass of 1.009 M⊙ and secondary mass of 0.647 M⊙ fits within the
observational uncertainties for 10 of the 11 observed bands. This star, while
classified as a ’single’ in a previous RV study (Mathieu & Latham, 1986), is
matched much more closely as a binary system using the binocs approach.
Possible reasons for this mismatch will be further discussed in §7.
A similar, but different Bayesian-based analysis has also recently been
introduced by Cohen et al. (2020).
### 2.3 BINOCS Code
This binocs detection method is implemented through a publicly available
code222http://github.com/bathompso/binocs. The steps implemented by this code
are described below.
First, the binocs code creates a library of synthetic cluster star SEDs using
an isochrone, which lists stellar parameters ($T_{\text{eff}}$, $\log g$) and
absolute magnitudes for a model star, given a cluster age, metallicity,
reddening333The method works well over a range of reddening, but would be
significantly adversely affected by differential reddening, which we plan to
improve in future versions of the binocs code., and stellar mass. Isochrone
sets often come in coarse mass grids, which hampers the binocs code’s ability
to compute accurate mass estimates. Stellar parameters are interpolated
cubically in mass, generating new isochrone points in steps of 0.01 M⊙. This
interpolation only works along the main sequence, however, where mass
increases monotonically. For evolved stars, at the turn-off or RGB, the
original isochrone points are used.
Using the new mass-interpolated isochrone model, SEDs are created by computing
up to 15 filter magnitudes ($UBVRIugrizJHK_{S}$[3.6][4.5]) for every possible
combination of single synthetic stars in an isochrone. The isochrone absolute
magnitudes are then adjusted to observed apparent magnitudes using the
cluster’s distance and reddening.
Next, each star in the cluster is compared to every possible model (binary and
single) using:
$\Phi=\sum_{\text{filters}}\frac{1}{\lvert
m_{\text{star}}-m_{\text{model}}\rvert+\eta_{soft}}$ (1)
Here, $m_{\text{star}}$ is the observed magnitude of the star in a particular
band, while $m_{\text{model}}$ is the apparent magnitude of the synthetic
model star. $\eta_{soft}$ is a single global softening parameter for which we
use $\eta_{soft}=0.01$ (mag), the typical uncertainty of the photometry.
If any of the sum elements is below a threshold value, e.g., $\frac{1}{\lvert
m_{\text{star}}-m_{\text{model}}\rvert+\eta_{soft}}<10$ (meaning the absolute
difference in magnitudes is $>0.09$), that element is declared to be “distant”
and is not added to the sum. The selection of this threshold values is
explored in §6.3. Only models with 3 good optical magnitudes ($UBVRIugriz$), 3
good near-IR magnitudes ($JHK_{S}$) and 2 good mid-IR magnitudes ([3.6][4.5])
are considered. Figure 1 illustrates why such a requirement is necessary:
binary SEDs differ from those of single stars only when compared across a
large wavelength range. In a small region, the differences between a binary
and single SED are negligible. Requiring a minimum number of good filter
magnitudes across the entire wavelength range will ensure that all models
accurately fit the entire SED, not just a single portion of it.
After discarding those models with too many “distant” magnitudes, all $\Phi$
values are normalized by the number of “close” magnitudes used in the sum. The
model with the highest $\Phi$ is chosen as the best fit. If no models have
enough “close” magnitudes, the star is marked as a non-member of the cluster.
After comparing each star to the full model library, it is also compared to
only single models as a secondary check, using a much less stringent “close”
magnitudes cut — each sum element must be $>1$. The purpose of this comparison
is two-fold: first is to be able to compare best-fitting single and binary
models for illustrative purposes, as shown in Figure 1. Secondly, some stars,
while classified as binaries through the binocs method, are better classified
as singles (these cuts will be explained in §6.4). If a star is forced to be
classified as single, the parameters from this stage of fitting will be used
to estimate its mass.
This fitting process is iterated 300 times, with each run randomly sampling
magnitudes from a Gaussian distribution, with $\sigma$ equal to the
photometric uncertainty. After all 300 resamples, the binocs code determines
whether the star is a member or not. If a majority of the chosen best-fits
denote that the star is a non-member, then that star is declared to be a non-
member. Of the stars which are members, primary and secondary masses are
determined by the median of all the best-fits. Uncertainties in the mass
estimates are computed using the standard deviation of all best-fit masses.
Similarly for the single-only fitting runs, the best-fit mass and uncertainty
are the median and standard deviation of all results, respectively.
By using multiple filters over a large wavelength range (0.3 - 4.5 $\mu$m),
individual photometric errors become less important than in the two-band
detection method. This means that the binocs method can determine mass
information for stars outside of the small mass window available for two-band
detection. Additionally, only a small amount of telescope time, relative to RV
surveys, is needed to capture cluster photometry across the optical to mid-IR
(assuming access to the correct observing facilities). This allows for the
detection of binaries in many clusters using a minimum of resources.
Additionally, we initially tested this technique using only (0.3 - 2.2
$\mu$m), and found that the addition of the [3.6][4.5] significantly improved
the resultant fitting. Without utilizing the mid-IR filters resulting in
significantly larger fitting errors and greater uncertainty in distinguishing
binarity.
§4 will cover the photometric data used by the binocs routine in this work,
while §5 will discuss the underlying isochrone models used. §6 will explore
the assumptions in the method description above (number of good filters
necessary, number of resamples, magnitude threshold level). §7 will compare
binocs results to that of a previous radial velocity (RV) studies of NGC 2168
and 2682.
## 3 Cluster Sample
In total, 8 clusters were targeted for use in this work. The distribution of
cluster parameters for our targeted sample is shown in Table 1.
Figure 2: Distribution of the eight targeted clusters in age and [Fe/H] (Dias et al., 2002). The ’X’ represents the cluster NGC 1960 which does not have any published metallicity information, so here we assume solar metallicity. Table 1: Adopted Cluster parameters for all clusters in dataset (Dias et al., 2002). Cluster | Age (Gyr) | [Fe/H] | Dist(pc) | E(B-V)
---|---|---|---|---
NGC 188 | 6.30 | -0.02 | 1820 | 0.06
NGC 1960 (M36) | 0.03 | $\cdots$ | 1320 | 0.22
NGC 2099 (M37) | 0.35 | +0.08 | 1390 | 0.30
NGC 2158 | 1.10 | -0.23 | 5080 | 0.36
NGC 2168 (M35) | 0.13 | -0.21 | 870 | 0.20
NGC 2420 | 2.00 | -0.23 | 2500 | 0.03
NGC 2682 (M67) | 3.50 | +0.01 | 860 | 0.04
NGC 6791 | 8.00 | +0.38 | 4170 | 0.15
The cluster sample covers a large area of the parameter space: ages range from
25 Myr to 9 Gyr while [Fe/H] varies from $-0.23$ to $+0.38$ — 40% to 200% the
Iron content of the Sun. Exploiting this parameter range is critical in
answering the posed science questions. In reference to science question 1,
there are three clusters with ages $<500$ Myr. Using binocs results from these
three clusters, an understanding of the primordial cluster binary population
can be conceived.
## 4 Photometry Data
The binocs method requires photometric data over a wide range of the spectrum
(optical to mid-IR) to detect binaries effectively. Photometric data over this
range was compiled from a variety of sources listed in Table 2.
Table 2 summarizes the available data for use in this project, from the
sources listed above, as well as from literature. 2MASS, WISE, and IRAC data
are available for all clusters and are therefore not listed in Table 2. Data
sources in italics are not yet reduced, and not currently available for
analysis.
Table 2: Photometry data fopr Clusters analyzed in this study Cluster | Visual Data | Near-IR Data | Membership Data
---|---|---|---
NGC 188 | von Hippel & Sarajedini (1998) | 2MASS | Geller et al. (2008)
Stetson et al. (2004) | Platais et al. (2003)
NGC 1960 (M36) | MOSAIC | NEWFIRM | Sanner et al. (2000)
NGC 2099 (M37) | Hartman et al. (2008) | NEWFIRM |
NGC 2158 | MOSAIC | 2MASS |
NGC 2168 (M35) | MOSAIC | NEWFIRM | Geller et al. (2010)
NGC 2420 | An et al. (2009) | NEWFIRM |
NGC 2682 (M67) | An et al. (2009) | NEWFIRM | Mathieu et al. (1997)
Yadav et al. (2008) | Yadav et al. (2008)
NGC 6791 | An et al. (2009) | Carney et al. (2005) |
Each of the cluster datasets in Table 2 have different levels of completeness,
which will dictate which analysis projects the cluster can be included in.
Clusters with complete photometry, although some may only have shallow 2MASS
near-IR magnitudes, can have bulk binary population parameters determined,
while complete deep photometry is necessary for the more detailed radial
distribution analysis.
### 4.1 Optical Photometry
Many open clusters have been studied exhaustively using optical filters,
including NGC 2099 and NGC 2682, and thus optical photometry for these
clusters come from previously published sources.
NGC 2099 optical photometry is pulled from Hartman et al. (2008), which used
both short- and long-exposure images to provide $gri$ magnitudes for $10\leq
r\leq 23$.
NGC 2682 falls within the Sloan Digital Sky Survey (SDSS; York et al., 2000)
imaging region. The aperture photometry employed by SDSS caused problems for
cluster photometry due to crowding in cluster core regions. An et al. (2009)
reanalyzed the $ugriz$ SDSS images, extracting magnitudes using point-spread
function (PSF) photometry, which can handle the dense cluster cores.
Photometry in An et al. (2009) only covers two SDSS imaging regions near the
core of NGC 2682, but does not touch further out regions. In these sparse
outer regions, the original SDSS data release 7 (DR7; Abazajian et al., 2009)
aperture photometry is accurate enough to be used.
Due to the length of exposure (54s) and telescope size (2.5-m), stars in the
SDSS images begin to saturate above $r\sim 13$. Unfortunately, this removes
almost all stars above the turn-off in NGC 2682. To fill in brighter stars
that are not included in the SDSS catalog, $BVI$ photometry from Yadav et al.
(2008) is used as a supplement. The combination of these two photometry
sources provides nearly complete coverage of the cluster in the optical, from
$V\sim 10$ to $g\sim 23$.
#### 4.1.1 MOSAIC:
The MOSAIC instrument (Sawyer et al., 2010), outfitted with $UBVRI$ filters,
contains an array of eight 2048-by-4096 pixel CCD chips to create a single
8192-by-8192 pixel image. While it has been attached previously to the 4-m
telescope at Kitt Peak National Observatory (KPNO), the data used in this
project is from the WIYN 0.9-m telescope at KPNO. With roughly a square degree
field of view, the MOSAIC images will allow us to analyze the entire spatial
extent of any cluster observed.
Images of several open clusters were obtained with MOSAIC over several nights
in Feb 2000 (Sarajedini & Kinemuchi, _private communication_). $UBVI$
photometry was obtained on three clusters in the same set: M35, M36, M37. For
all clusters, both short and long sequences of images were taken. Short images
had exposure lengths of 25s, 8s, 5s, 5s in $UBVI$, respectively. Four images
of the same exposure length were taken in each filter. Long sequence images,
also four per filter, had 10 times the exposure length of the short set: 250s,
80s, 50s, 50s. Using both sequences together allows for photometry of the
brightest and faintest stars within the cluster.
Figure 3: Reduced MOSAIC 50s $I$-band image for NGC 1960.
Two of the clusters have already been analyzed here: NGC 2168 in Thompson et
al. (2014) and NGC 1960. For our analysis, the four images in each filter were
combined to form a higher signal-to-noise master image, and to provide a
complete covering of the cluster. Note the wide gap between chips in the
individual MOSAIC images, shown in Figure 3. Each of the four images per
filter were _dithered_ (slightly offset) such that the combined image had no
gaps in coverage.
These master images were then split into the 8 individual chips on the MOSAIC
image. This splitting was done to accommodate the DAOPHOT PSF photometry
package, which has limits on image size. The individual 2k$\times$4k chips
were the largest DAOPHOT could handle. In each chip (and for each master
image), the process was the same. First, 400 candidate template stars were
chosen to create a PSF. Next, the trimming process described in Thompson et
al. (2014) was run, trimming the candidate list down to 250-300 template
stars. Using this cleaned list, PSF parameters were determined, and then
applied through ALLSTAR.
Photometric quality plots for the short and long sets are shown in Figure 4.
For reference, high quality photometry has uncertainties less than 0.05. The
MOSAIC images provide this high quality data for $11\leq V\leq 20$, covering
nearly all of the stars within these clusters.
Figure 4: MOSAIC photometric quality plots for NGC 1960 in $UBVI$. _Left:_
Short set of exposures. _Right:_ Long set of exposures.
The ALLSTAR-derived magnitudes must be transformed to a standard system, in
order to be comparable to other results. For calibration, photometry from the
individual chips were re-combined to produce single photometry files for each
master image, then matched to previously published “standard” photometry. For
NGC 1960, the previously published $UBVI$ photometry from Sharma et al. (2006)
was used to transform the instrumental MOSAIC magnitudes to the standard
system.
Sources detected in the MOSAIC images were matched to the published photometry
for each cluster, producing between $500-600$ matches for each filter. Using
these common stars, the instrumental ALLSTAR magnitudes were transformed via
the following equations:
$u=U+a_{U}+b_{U}\times(U-B)$ (2) $b=B+a_{B}+b_{B}\times(B-V)$ (3)
$v=V+a_{V}+b_{V}\times(B-V)$ (4) $i=I+a_{I}+b_{I}\times(V-I)$ (5)
Table 3: Transformation coefficients for MOSAIC photometry. Cluster | Filter | Length | a | b
---|---|---|---|---
NGC 1960 | $U$ | Short | $1.843\pm 0.009$ | $0.008\pm 0.011$
Long | $-0.650\pm 0.010$ | $-0.053\pm 0.008$
$B$ | Short | $1.191\pm 0.004$ | $-0.105\pm 0.005$
Long | $-1.305\pm 0.005$ | $-0.127\pm 0.006$
$V$ | Short | $1.536\pm 0.003$ | $0.048\pm 0.004$
Long | $-0.928\pm 0.005$ | $0.034\pm 0.006$
$I$ | Short | $1.993\pm 0.004$ | $0.002\pm 0.004$
Long | $-0.562\pm 0.011$ | $-0.000\pm 0.011$
Here, lowercase filter letters indicate instrumental (ALLSTAR-derived)
magnitudes, while uppercase filters are those of the standard photometry. The
transformation coefficients for each cluster and filter are located in Table
3. Transformations were done separately for the short and long exposure
sequences. Residuals for these transformations are shown in Figure 5. Once the
instrumental magnitude were calibrated to the standard system, all photometry
was combined into a single master catalog.
Figure 5: Residuals from transformation to standard system for NGC 1960 MOSAIC
photometry.
### 4.2 $JHK_{S}$ Near-IR Photometry
While Two-Micron All Sky Survey (2MASS; Skrutskie et al., 2006) $JHK_{S}$
photometry is available for all open clusters, it is too shallow ($J\sim 16$)
to provide photometry for low-mass members of the cluster. New $JHK_{S}$ near-
IR photometry was obtained, by us, using the NEWFIRM instrument (Hoffman et
al., 2004) on the Kitt Peak 4-m telescope. Images were taken on two nights in
Feb 2008. Observation and reduction processes are the same as used in Thompson
et al. (2014) from which we published the NGC 2168 data.
Observations were taken in “4Q” mode, which offsets the telescope in a pattern
to align the center of the cluster on each of the four NEWFIRM detectors. This
allows for larger area coverage than a single NEWFIRM field of view. To
minimize errors in flat-fielding and negate cosmetic defects within the chips,
the telescope was dithered between exposures for both clusters. Clusters have
effective exposure times of 2400s in $J$ and $H$, and a total of 3600s in
$K_{S}$.
All images were reduced (dark correction, flat fielding, sky subtraction)
through the NEWFIRM pipeline (Swaters et al., 2009). The reduced images were
stacked into master frames for each filter. Photometry was carried out using
the DAOPHOT II and ALLSTAR programs (Stetson, 1987), using a detection
threshold of 3$\sigma$ in all filters. Initially, 2000 stars were chosen to
determine the PSF for each stacked image. This list was then trimmed to remove
stars which degraded the fit. First, stars near bad or saturated pixels were
removed, so as to avoid PSF distortion by these outliers. Next, stars that
were less than 4 full-width at half-maximum (FWHM) from another source were
removed from the PSF list, ensuring that the PSF would not be contaminated by
crowding. Finally, stars whose PSF $\chi^{2}$ fit values were 2$\sigma$ or
more above the mean were removed. After trimming, approximately 500 and 800
stars remained for determining the PSF in NGC 2682 and NGC 2099, respectively.
The DAOPHOT-derived magnitudes were tied to the standard system by matching to
2MASS photometry. Only 2MASS sources with the highest photometric quality
(‘AAA’) were used in the standard catalog. NGC 2682 frames matched
approximately 700 stars between the NEWFIRM and 2MASS datasets, while NGC 2099
had almost 2000 overlapping sources. Using these matches, transformations were
determined to the standard system for each cluster:
$j=J+a_{j}(J-K_{S})+b_{j}$ (6) $h=H+a_{h}(H-K_{S})+b_{h}$ (7)
$k=K_{S}+a_{k}(J-K_{S})+b_{k}$ (8)
In equations (6)–(8), lowercase letters denote DAOPHOT magnitudes, while
uppercase letters denote 2MASS standard magnitudes. Transformation
coefficients for each of the clusters is listed in table 4.
Table 4: NEWFIRM Transformation Coefficients Cluster | $J$ | $H$ | $K_{S}$
---|---|---|---
NGC 1960 | $a_{j}=-0.056\pm 0.006$ | $a_{h}=-0.177\pm 0.018$ | $a_{k}=+0.042\pm 0.006$
$b_{j}=+2.441\pm 0.004$ | $b_{h}=+2.620\pm 0.003$ | $b_{k}=+3.063\pm 0.004$
NGC 2099 | $a_{j}=-0.121\pm 0.008$ | $a_{h}=-0.354\pm 0.016$ | $a_{k}=+0.112\pm 0.009$
$b_{j}=+2.434\pm 0.004$ | $b_{h}=+2.318\pm 0.003$ | $b_{k}=+3.020\pm 0.005$
NGC 2168 | $a_{j}=-0.099\pm 0.005$ | $a_{h}=-0.296\pm 0.012$ | $a_{k}=+0.093\pm 0.007$
$b_{j}=+2.397\pm 0.003$ | $b_{h}=+2.297\pm 0.002$ | $b_{k}=+3.030\pm 0.005$
NGC 2420 | $a_{j}=-0.036\pm 0.008$ | $a_{h}=-0.234\pm 0.020$ | $a_{k}=+0.130\pm 0.011$
$b_{j}=+2.752\pm 0.005$ | $b_{h}=+2.739\pm 0.003$ | $b_{k}=+3.179\pm 0.006$
NGC 2682 | $a_{j}=-0.100\pm 0.010$ | $a_{h}=-0.250\pm 0.021$ | $a_{k}=+0.113\pm 0.014$
$b_{j}=+2.444\pm 0.007$ | $b_{h}=+2.277\pm 0.004$ | $b_{k}=+2.956\pm 0.010$
A plot of transformation residuals is shown in Figure 6, along with
measurement uncertainty as a function of magnitude. For almost all stars in
the NEWFIRM images, magnitude uncertainties are $<0.1$.
The NOAO/NEWFIRM photometry reaches a depth of approximately ($J$,$H$,$K_{S}$
= 18.6, 18.1, 17.8).
Figure 6: _Left:_ NEWFIRM magnitude vs uncertainty for for NGC 2099 (top) &
NGC 2682 (bottom). _Right:_ Transformation residuals between NEWFIRM
magnitudes and 2MASS.
### 4.3 Mid-IR Photometry
Deep Mid-IR photometry ([3.6][4.5][5.8][8.0]) for NGC 2099 and NGC 2682 was
gathered using the Infrared Array Camera (IRAC; Fazio et al., 2004) on the
Spitzer Space Telescope. NGC 2682 data was obtained as part of cycle 2
proposal 20710 (PI Skrutskie), and NGC 2099 data was obtained from cycle 3
proposal 30800 (PI Frinchaboy). The data were taken in High Dynamic Range
(HDR) mode, allowing measurement of the brightest and faintest stars in the
cluster simultaneously. The data were reduced and photometered using the
GLIMPSE (Galactic Legacy Infrared Mid-Plane Survey Extraordinare; Benjamin et
al., 2003) pipeline, which was modified to handle the HDR data. The
Spitzer/IRAC photometry reaches a depth of approximately ([3.6][4.5][5.8][8.0]
= 18.0, 16.5, 14.6, 13.8 ). Although within this work using the binocs method
we use only the [3.6] and [4.5] bands, for completeness purposes, we provide
this new photometry for all four Spitzer bands to the community in Table 5.
Supplemental Mid-IR photometry ($3-4.6\mu$m) is available from the Wide-field
Infrared Survey Explorer (WISE; Wright et al., 2010) for the entire sky. WISE
photometry was pulled for a 1∘ radius around the cluster, extending the
spatial coverage of the mid-IR data. Unfortunately, WISE and IRAC use slightly
different filters, and a correction must be applied to the WISE photometry in
order to merge it with the deeper IRAC data.
WISE and IRAC photometry of NGC 2099 were compared, and transformation
equations were found using more than 800 common sources. Transformations were
limited to [3.4]${}_{\text{WISE}}<14$ and [4.6]${}_{\text{WISE}}<13.5$, beyond
which the transformations become problematic. Residuals between IRAC and WISE
photometry shows no correlation with color, and only small magnitude offsets.
Figure 7: Residuals from transformation between WISE and IRAC magnitudes for
sources near NGC 2099.
[3.4]WISE and [3.6]IRAC are interchangeable in the specified region, while
[4.6]${}_{WISE}=$ [4.5]${}_{IRAC}+0.03$. A plot of residuals for this
transformation are shown in Figure 7. Using these simple transformation
equations, WISE photometry was merged with the IRAC sources.
### 4.4 Merged Dataset
Optical, near- and mid-IR photometry sets for each cluster are merged into a
final dataset. Final cluster color-magnitude diagrams (CMDs) and spatial
distributions are shown in Figures 8 and 9.
Figure 8: Cluster CMDs for merged datasets. NGC 2682 includes supplementary
$BVI$ photometry to include stars above the turn-off. CMDs are shown only for
sources within 20′ of the cluster centers. Figure 9: Cluster spatial diagrams
for merged datasets. _Solid lines:_ $gri$ photometry datasets: Hartman et al.
(2008) for NGC 2099, An et al. (2009) for NGC 2682. _Dotted lines:_ NEWFIRM
$JHK_{S}$ photometry. _Dashed lines:_ IRAC mid-IR photometry. _Dot-dash
lines:_ Supplemental $BVI$ photometry from Yadav et al. (2008) for NGC 2682.
2MASS near-IR and WISE mid-IR photometry is available for all-sky. NGC 2682
SDSS DR7 photometry is available over the entire plotted area.
The IRAC coverage area in NGC 2682 is only a thin stripe in declination,
designed to overlap the 2MASS 6x calibration area. WISE photometry is
necessary to implement the binocs detection method across the entire cluster
area.
Table 5: New NEWFIRM and IRAC photometry.
Cluster | RA | Dec | $J$ | $H$ | $K_{S}$ | [3.6] | [4.5] | [5.8] | [8.0]
---|---|---|---|---|---|---|---|---|---
(NGC) | (2000.0) | (2000.0) | (mag) | (mag) | (mag) | (mag) | (mag) | (mag) | (mag)
2682 | 132.73465 | 11.77622 | 14.189$\pm$0 | 13.613$\pm$0.013 | 13.445$\pm$0 | 13.410$\pm$0.024 | 13.451$\pm$0 | 13.415$\pm$0.040 | 13.402$\pm$0 | | |
2682 | 132.73955 | 11.58258 | 13.921$\pm$0 | 13.410$\pm$0.011 | 13.297$\pm$0 | 13.208$\pm$0.020 | 13.247$\pm$0 | 13.181$\pm$0.035 | 13.205$\pm$0 | | |
2682 | 132.74037 | 12.01129 | 14.141$\pm$0 | 13.702$\pm$0.011 | 13.671$\pm$0 | 13.557$\pm$0.023 | 13.618$\pm$0 | 13.536$\pm$0.036 | 13.608$\pm$0 | | |
2682 | 132.74144 | 12.08310 | 14.283$\pm$0 | 13.744$\pm$0.012 | 13.624$\pm$0 | 13.566$\pm$0.022 | 13.551$\pm$0 | 13.488$\pm$0.037 | 13.573$\pm$0 | | |
2682 | 132.74359 | 11.80183 | 14.073$\pm$0 | 13.635$\pm$0.011 | 13.556$\pm$0 | 13.486$\pm$0.023 | 13.524$\pm$0 | 13.489$\pm$0.033 | 13.367$\pm$0 | | |
2682 | 132.74379 | 11.81218 | 13.993$\pm$0 | 13.571$\pm$0.009 | 13.512$\pm$0 | 13.438$\pm$0.021 | 13.443$\pm$0 | 13.416$\pm$0.030 | 13.422$\pm$0 | | |
2682 | 132.74596 | 11.80808 | 14.299$\pm$0 | 13.778$\pm$0.010 | 13.713$\pm$0 | 13.586$\pm$0.018 | 13.512$\pm$0 | 13.536$\pm$0.033 | 13.527$\pm$0 | | |
2682 | 132.74651 | 11.97064 | 14.615$\pm$0 | 13.908$\pm$0.014 | 13.887$\pm$0 | 13.670$\pm$0.019 | 13.732$\pm$0 | 13.689$\pm$0.038 | 13.613$\pm$0 | | |
2682 | 132.74748 | 12.16521 | 14.022$\pm$0 | 13.567$\pm$0.021 | 13.569$\pm$0 | 13.283$\pm$0.030 | 13.393$\pm$0 | 13.362$\pm$0.033 | 13.278$\pm$0 | | |
2682 | 132.74765 | 11.91457 | 14.125$\pm$0 | 13.673$\pm$0.018 | 13.573$\pm$0 | 13.480$\pm$0.022 | 13.504$\pm$0 | 13.451$\pm$0.033 | 13.329$\pm$0 | | |
$\cdots$ | | |
Note. — This table is available in its entirety in machine-readable form in
the online journal. A portion is shown here for guidance regarding its form
and content.
## 5 Stellar Isochrone Models
The binocs code uses synthetic SEDs from isochrones to determine best-fit
masses for each star in the clusters. Therefore, the mass determination from
the binocs code is only as accurate as the underlying isochrones themselves.
Modern stellar models are still affected by non-negligible discrepancies due
to variation in input physics (Valle et al., 2013). This is apparent in the
comparison of isochrone tracks computed by different sets of authors. In
Figure 10, two popular isochrone sets are over-plotted on NGC 2682 and NGC
2099 CMDs: Dartmouth (Dotter et al., 2007) and Padova (Girardi et al., 2002)
or PARSEC (Bressan et al., 2012). For NGC 2682, 3.5 Gyr isochrones with [Fe/H]
= +0.01, E($B-V$) = 0.04 and a distance of 855 pc were used. For NGC 2099, 355
Myr isochrones were used, with [Fe/H] = +0.08, E($B-V$) = 0.3 and a distance
of 1386 pc. Dartmouth isochrones can only be interpolated in [Fe/H] for ages
$>1$ Gyr. Because no 355 Myr Dartmouth isochrones with [Fe/H] = +0.08 can be
generated, they are not shown in Figure 10.
Figure 10: Comparison of popular isochrone sets to combined cluster photometry
in several CMDs. _Dotted Line:_ Dartmouth (Dotter et al., 2007, NGC 2682
only). _Solid Line:_ Padova (Girardi et al., 2002). _Dashed Line:_ PARSEC
(Bressan et al., 2012). NGC 2682 isochrones: 3.5 Gyr, [Fe/H] = +0.01, E($B-V$)
= 0.04, 1386 pc. NGC 2099 isochrones: 355 Myr, [Fe/H] = +0.08, E($B-V$) = 0.3,
1386 pc.
It is clear from Figure 10 that all the isochrone sets deviate from the
observed main sequence, especially for low-mass stars. To quantify the
deviation between the models and observations, residuals between a by-eye
empirical ridgeline and isochrones in various filters are shown in Figure 11.
The NGC 2682 $ugriz$ empirical ridgelines are pulled from the same source as
the data, An et al. (2009). For stars with magnitudes above or below the
ridgeline area, colors are shifted such that the adjusted ridgeline is
continuous.
Isochrone models vary anywhere from $\sim 0.1$ to over 0.3 in color, depending
on filter. This large discrepancy in color may significantly affect results
from the binocs fitting. Before being used to compute synthetic SEDs, these
isochrones will have to be altered.
Figure 11: Residuals between empirical ridgelines and isochrones for various
filters. Same isochrones as shown in Figure 10. _Dotted Line:_ Dartmouth
(Dotter et al., 2007). _Solid Line:_ Padova (Girardi et al., 2002). _Dashed
Line:_ PARSEC (Bressan et al., 2012).
To correct the isochrones so that they more closely match the data, isochrones
are adjusted to align with the hand-drawn ridgelines. This is done by assuming
the bolometric correction to the $r$ filter (and hence the $r$ magnitude
itself) is correct, and adjusting all colors accordingly to match the
empirical ridgelines. This updated isochrone is then fed into the binocs code
to create synthetic SEDs.
For all binocs runs, PARSEC isochrones were used. While the PARSEC isochrones
showed the most deviation from the empirical ridgelines in Figure 11, this
error is corrected out by the empirical transformation. Of the isochrone sets
considered, PARSEC provides the largest mass range of synthetic stars, and is
therefore the most advantageous for this approach.
## 6 binocs Testing
When the binocs code was introduced in §2.3, several parameters were assumed:
the number of iterations of the fitting, the number of “good” fitting filters,
and the threshold to consider a magnitude “good.” Each of these parameters
were tested, and the results are shown below.
### 6.1 Number of Iterations
The binocs fitting is iterated a number of times to produce best-fit masses
and uncertainties. While the binocs code has random elements (sampling of
Gaussian error distribution), if the process is iterated enough times, the
final results will not vary greatly. Running excess iterations beyond this
will use more computing time, but not enhance the results in any meaningful
way. To determine the minimum number of iterations required, the combined NGC
2682 dataset was run through the binocs code with varying numbers of
iterations: 3, 10, 30, 90, 150, 200, 300, 400, 500, 600, 700, 1200.
For each number of iterations, the binocs code was run five times. Using these
five runs, a “% uncertainty”, $\Sigma$, was computed for each star. $\Sigma$
is defined as the standard deviation of all five resulting masses divided by
the average of the resulting masses for which the star is classified as a
member. $\Sigma$’s for primary and secondary mass determinations are computed
independently. Stars that were classified as non-member stars in all five runs
(and hence not given any best-fit masses) were removed from the set.
After computing $\Sigma$’s for all stars in the NGC 2682 dataset, median and
95${}^{\text{th}}$ percentile $\Sigma$’s were computed for each iteration
value. The results are shown in Figure 12.
Figure 12: Results of the number of iterations test. Circles correspond to
$\Sigma$’s for primary masses, while squares correspond to secondary mass
$\Sigma$’s. Solid lines show median $\Sigma$, dashed lines show
95${}^{\text{th}}$ percentile. Grey line denotes average 10% uncertainty
between runs.
Median $\Sigma$ values are overall quite low; both primary and secondary
median $\Sigma$’s are equal to zero for any number of iterations $\geq 150$.
In order to ensure that a majority of stellar mass determinations are roughly
constant between runs, we require the 95${}^{\text{th}}$ percentile $\Sigma$
to be less than 0.1: on average, there will be a less than a 10% difference in
derived stellar masses between runs for 95% of stars in the dataset. Using 300
iterations of the binocs fitting will satisfy this requirement (as seen in
Figure 12), and is chosen as the ideal number of iterations in the final
computations.
### 6.2 Number of Good Filters
While a comprehensive sampling of the SED over all 10 filters ($UBVRI$ or
$ugriz$, $JHK_{S}$, [3.6][4.5]) is ideal, it is often impractical to obtain
quality photometry in this number of bands for every cluster we wish to study.
In practice, the binocs code will have to produce accurate results using a
less-than-ideal number of filters.
The library of synthetic SED models generated from the best-fit isochrone for
NGC 2682 ([Fe/H] = +0.01, age = 3.55 Gyr) was used as an input into the binocs
code. Using the combined NGC 2682 dataset, average photometric uncertainties
were computed for all bands in bins of 0.5 mag. Every magnitude in the input
library was randomized using a gaussian with $\sigma=$ 2 $\times$ the average
photometric uncertainty in the corresponding bin.
The binocs code was run on the input library for various combinations of
usable filters. For each run, only certain filter magnitudes in the randomized
library were transferred to the final input file, listed in the first column
of table 6.
Each filter combination was run 5 times, and each time the % error in the
primary and secondary mass determination was recorded. After all 5 runs, stars
were binned into steps of 0.1 in mass ratio, and average % error $+$ 1
standard deviation was computed for all stars in the bin. This 1$\sigma$ %
error is shown for each bin and filter combination in table 6.
Table 6: 1$\sigma$ % errors in mass estimates for various combinations of filters. | Mass Ratio |
---|---|---
Filters | | 0.0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
101: $.g......$[3.6]. | | 6.6 | 0.0 | 0.1 | 1.1 | 1.2 | 2.5 | 4.1 | 4.9 | 5.1 | 6.2 | 9.6 | 1$\sigma$ % Error in $M_{\text{pri}}$
111: $.g...J..$[3.6]. | | 6.0 | 0.0 | 0.1 | 0.9 | 1.0 | 1.8 | 3.1 | 3.7 | 3.9 | 4.7 | 9.1
202: $.gr.....$[3.6][4.5] | | 2.3 | 0.0 | 0.0 | 0.7 | 0.8 | 1.5 | 3.3 | 3.6 | 3.7 | 4.5 | 7.6
211: $.gr..J..$[3.6]. | | 2.7 | 0.0 | 0.0 | 0.7 | 0.7 | 1.8 | 3.0 | 3.6 | 3.8 | 4.5 | 7.9
212: $.gr..J..$[3.6][4.5] | | 2.5 | 0.0 | 0.0 | 0.7 | 0.7 | 1.3 | 2.9 | 3.4 | 3.4 | 4.1 | 7.7
222: $.gr..J.K_{S}$[3.6][4.5] | | 2.6 | 0.0 | 0.0 | 0.6 | 0.7 | 1.2 | 2.2 | 3.0 | 3.3 | 3.9 | 6.9
322: $.gri.J.K_{S}$[3.6][4.5] | | 1.5 | 0.0 | 0.0 | 0.6 | 0.7 | 0.9 | 1.9 | 2.4 | 2.7 | 3.3 | 6.1
332: $.gri.JHK_{S}$[3.6][4.5] | | 1.4 | 0.0 | 0.0 | 0.5 | 0.6 | 0.9 | 1.9 | 2.5 | 2.8 | 3.3 | 6.2
532: $ugrizJHK_{S}$[3.6][4.5] | | 0.6 | 0.0 | 0.0 | 0.2 | 0.2 | 0.5 | 0.8 | 1.4 | 1.8 | 1.9 | 4.4
101: $.g......$[3.6]. | | … | 0.0 | 66.5 | 37.3 | 26.1 | 18.7 | 17.1 | 17.3 | 14.5 | 12.1 | 13.0 | 1$\sigma$ % Error in $M_{\text{sec}}$
111: $.g...J..$[3.6]. | | … | 0.0 | 43.7 | 29.4 | 20.7 | 16.1 | 11.3 | 10.4 | 9.9 | 8.8 | 11.2
202: $.gr.....$[3.6][4.5] | | … | 0.0 | 51.6 | 24.4 | 15.7 | 11.9 | 10.5 | 9.0 | 7.5 | 7.2 | 10.0
211: $.gr..J..$[3.6]. | | … | 0.0 | 39.9 | 32.8 | 18.5 | 12.6 | 9.7 | 10.5 | 8.0 | 7.3 | 10.3
212: $.gr..J..$[3.6][4.5] | | … | 0.0 | 44.9 | 23.5 | 15.8 | 11.4 | 8.9 | 8.3 | 6.9 | 6.4 | 10.1
222: $.gr..J.K_{S}$[3.6][4.5] | | … | 0.0 | 27.6 | 16.6 | 13.4 | 9.1 | 7.2 | 7.3 | 6.5 | 6.2 | 8.6
322: $.gri.J.K_{S}$[3.6][4.5] | | … | 0.0 | 38.1 | 18.8 | 11.1 | 7.1 | 5.7 | 5.8 | 5.0 | 4.8 | 7.7
332: $.gri.JHK_{S}$[3.6][4.5] | | … | 0.0 | 19.7 | 15.0 | 10.6 | 8.3 | 5.5 | 5.6 | 5.3 | 4.8 | 7.8
532: $ugrizJHK_{S}$[3.6][4.5] | | … | 0.0 | 37.7 | 13.5 | 10.4 | 6.0 | 3.3 | 3.8 | 3.7 | 2.9 | 5.3
The ‘332’ ($griJHK_{S}$[3.6][4.5]) filter combination is chosen as the
preferred option for this work. Only requiring 3 optical filters increases the
number of usable clusters while only marginally increasing the uncertainty in
the final binocs results. For primary mass estimates the ’332’ combination
produces the second-lowest 1$\sigma$ % errors of all scenarios, and produces
secondary mass estimates good to within 20%. The only better scenario is the
full photometry set (532), which is difficult to obtain for many open
clusters, especially depth in the $u$ filter.
Using these results, there are 1500 stars in NGC 2682 and 3500 stars in NGC
2099 which have the necessary number of filters for a good binocs fitting.
### 6.3 “Close” Filter Threshold
In addition to generating accurate mass estimates for cluster stars, the
binocs method can mark stars as non-members if they do not have the required
number of “close” filters. Therefore, the threshold ($\frac{1}{\lvert
m_{\text{star}}-m_{\text{model}}\rvert+\eta_{soft}}<X$) which defines whether
a filter is “close” will adjust the level of field star contamination within
the sample. Conversely, if the threshold is too stringent, many legitimate
member stars may be discarded from the sample.
To test for the optimal threshold level, an input catalog was created
similarly to that used in §6.2. The input file to the binocs code contained
three copies of the synthetic library created in §6.2: one at the same
distance as NGC 2682, one shifted a distance modulus of 0.8 nearer, and one
shifted a distance modulus of 0.8 further than NGC 2682. As the magnitude
difference between the single-star main sequence and equal-mass binary
sequence is 0.753, there should be no degeneracies between the three copies of
the input library.
The binocs code was run on the input file for various values of the threshold.
After the run was complete, two numbers were computed: the percentage of
member stars (from the copy of the library at NGC 2682’s distance) that were
classified as non-members, and the percentage of non-member stars (from the
other two copies of the library) that were classified as members. The best-fit
“close” threshold value is chosen such that the sum of these two values is at
a minimum.
Figure 13: Results of the threshold test for NGC 2682. Best threshold chosen
to minimize % Total.
Figure 13 shows the results of the threshold testing. A minimum in % Total is
found at a threshold value of 10. Contamination from foreground and background
stars quickly increases when threshold values are larger than 10, while the
percentage of missed member stars only decreases gradually.
A formal $\chi^{2}$ form of (1) was also explored in addition to the fixed-
width threshold form used:
$\Theta=\sum_{\text{filters}}\frac{(m_{\text{star}}-m_{\text{model}})^{2}}{\sigma_{\text{star}}^{2}}$
(9)
While the $\chi^{2}$ form produced less theoretical contamination in this
test, it also classified more than 90% of the stars in the NGC 2682 dataset as
non-members! As this is unrealistic for a cluster so far from the galactic
plane, the fixed-width threshold form of (1) was used in the final binocs
code.
### 6.4 Minimum Mass Ratio
It is often impossible to tell the difference between a single star and a low
mass-ratio binary, even when using a minimum of 8 filters. A minimum mass
ratio, as a function of primary mass, is determined to be the maximum of three
values:
* •
_Lowest mass-ratio model:_ For the PARSEC isochrones being used, the lowest-
mass model has a mass of 0.13 M⊙, defining a minimum model mass ratio for each
primary mass.
* •
_Synthetic best-fit mass ratio:_ After each run of the binocs code, a test
similar to that in §6.2 is run. Minimum mass ratios are defined as the
resulting best-fit of each synthetic single star.
* •
_Constant threshold:_ After a detailed comparison to clusters that have RV
comparisons available, we find that binocs results prove unreliable,
single/binary indistinguishable, for stars with $q<0.3$ (see §7). Even if the
synthetic tests estimate a value less than this, the minimum threshold for
identifying a binary is a mass ratio $q\geq 0.3$.
Figure 14: Minimum mass ratios for NGC 2099, as a function of primary mass.
Stars are defined to be singles if they have best-fit mass ratios below the
specified value. Minimum mass ratios for NGC 2099 are shown in Figure 14. For
systems with primary masses below 0.5 M⊙, minimum mass ratios are dominated by
the minimum model mass of 0.13 M⊙. Above a primary mass of 2.5 M⊙, minimum
mass ratios are dominated by degeneracies at the turn-off. As shown in Figure
14, the binocs technique works well bewteen 2.5–0.5 M⊙ for NGC 2099.
In their analysis, de Grijs et al. stated their analysis was only sensitive to
binaries with $q\geq 0.55$. The binocs method shows an improvement in mass
sensitivity, with minimum mass ratios closer to 0.35 for a large mass range.
## 7 Results
The binocs results for NGC 2682 were compared to a published RV study of 104
cluster members (26 binaries) by Mathieu & Latham (1986). A comparison of
multiplicity determinations is shown in table 7. The comparison was limited to
stars with $14.5\leq g\leq 16.5$, avoiding the degeneracies at the turn-off,
and ending at the lower magnitude limit for the RV study.
Figure 15: CMD comparison of RV and binocs SED-fitting results for NGC 2682.
Stars considered in comparison are those within dashed line limits. Color of
dot indicates which cell of Table 7 star belongs to. Black circles indicate RV
singles which were classified as binocs best-fit binaries with mass ratios
$<0.3$.
Using the updated minimum mass ratio calculation in §6.4, there is good
agreement between binocs and RV results, with 60% of RV singles being
confirmed as single stars by the binocs routine, and 56% of RV binaries being
confirmed.
Table 7: Comparison of binocs and Mathieu & Latham (1986) RV multiplicity results for NGC 2682. as shown in Figure 15. | binocs
---|---
| Single | Binary | Non-member
RV Single | 81 | 29 | 25
RV Binary | 4 | 24 | 15
Each method has its own limitations and biases, and exact agreement is not
expected. RV surveys cannot detect long-period binaries, or those with high
inclination, while binocs is insensitive to these parameters. These types of
systems may account for many of the 29 RV-single stars that the binocs routine
fit as binaries. The 4 RV-binary stars that binocs determined to be singles
may be systems with small secondaries. RV shifts for small companions may
still be appreciable, while the amount of contributed light to the SED is not.
The RV and binocs methods are complementary techniques, but still show a large
amount of overlap in the results. Unfortunately neither NGC 2682 nor NGC 2099
have published double-line spectroscopic binaries with masses determined.
The binocs results for NGC 2168 were compared to a published RV study of the
cluster in Geller et al. (2010). A comparison of multiplicity determinations
is shown in Table 7. To avoid complications from the turn-off, and poor faint
data in the RV studies, the comparison is limited to a specific magnitude
range. For NGC 2168, only stars with $14.2\leq V\leq 16.5$ are considered.
or stars classified as binaries by binocs , many are also classified as
binaries by RV detection methods, with a 69% overlap in NGC 2168. The binocs
code shows a lower match when classifying RV singles444The term “RV singles”
is used to denote a system which does not show an appreciable velocity shift.,
with only 59% of RV singles being classified as singles by binocs in NGC 2168.
To ensure reasonable agreement between binocs and Geller et al. (2010), a
floor of 0.3 was set in the minimum mass ratio calculation.
Figure 16: CMD comparison of RV and binocs SED-fitting results for NGC 2168. Stars considered in comparison are those within dashed line limits. Color of dot indicates which cell of table 8 star belongs to. Black circles indicate RV singles which were classified as binocs best-fit binaries with mass ratios $<0.3$. Table 8: Comparison of binocs and Geller et al. (2010) RV multiplicity results, as shown in Figure 16. | binocs
---|---
| Single | Binary | Non-member
Single | 113 (45%) | 98 (39%) | 40 (16%)
Binary | 8 (18%) | 31 (69%) | 6 (13%)
Using the updated minimum mass ratio calculation in §6.4, there is good
agreement between binocs and RV results, with 60% of RV singles being
confirmed as single stars by the binocs routine, and 56% of RV binaries being
confirmed.
### 7.1 Mass Determination
While not a part of the analysis set due to the lack of deep near-IR
photometry, the clusters NGC 188 and NGC 6819 have also been the subject of
detailed RV studies (Geller et al., 2008; Hole et al., 2009, respectively). A
comparison to the RV studies can be completed in the region where 2MASS
photometry is available.
Of the 1046 stars studied in NGC 188, 13 were _double-lined_ binaries. Further
follow-up on these stars, published in Geller et al. (2009), characterized the
orbits of these double-lined binaries, and produced accurate binary mass
ratios. Similarly, NGC 6819 stars were followed up in Milliman et al. (2014),
and 15 double-lined binaries were detected. The RV-determined mass ratios are
compared to those from binocs in Figures 17 and 18.
Figure 17: _Left:_ NGC 188 CMD in $B-V$. Solid line is isochrone used to
generate models for binocs fitting. Dashed line is equal-mass binary sequence.
Black circles are double-lined binaries. _Right:_ Comparison of RV mass ratios
black) from Geller et al. (2009) to binocs (grey) for NGC 188 double-lined
binaries. Stars outlined in red are those complicated by degeneracies close to
the turn-off. Figure 18: Same as Figure 17 for NGC 2168. RV data from Leiner
et al. (2015). The stars outlined in red is below the $q\geq 0.3$ threshold
level.
There are several highly discrepant mass ratio determinations in NGC 188 and
NGC 6819. Many of these double-lined systems lie near the turn-off of each
cluster, where the single star main sequence and equal mass binary sequence
overlap (as seen in the left hand panels of Figures 17. In these regions,
there are natural degeneracies, and the binocs code cannot accurately
determine parameters. Stars marked by red circles in Figures 17 lie extremely
close to these degeneracies and therefore exhibit large errors with respect to
the RV results.
Ignoring those stars very close to the crossing of single star main sequence
and equal mass binary sequence, there is close agreement between RV and binocs
mass ratios. Including the quoted uncertainties in mass from binocs
(uncertainties from the RV surveys are negligible), mass ratios largely agree
to within 10%.
Combining this 10% mass ratio accuracy with the previous conclusion that
binocs results are largely agreeing with RV multiplicity determinations, it is
clear that the binocs code is producing accurate results that can be
extrapolated to lower-mass stars.
### 7.2 Membership Comparison to Gaia
One additional effect of the binocs analysis is that stars are classified as
single or binary members and non-members. The ability to reject SEDs that
cannot fit for stars of the correct distance, reddening, and luminosity class
can be a powerful tool to exploring faint membership and binarity of simple
populations.
To test the effectiveness to photometric “cleaning” of non-member stars from
the cluster CMD for main sequence stars, we have made a comparison to the Gaia
DR2 (Gaia Collaboration et al., 2016, 2018) proper motion-based membership
probability using the method from Donor et al. (2018) for two of our clusters
NGC 2099 and NGC 2682. While the binocs method will not be as effective as
Gaia, the simplicity of using only photometry, allows probing much deeper than
Gaia.
Figure 19: 2MASS CMD of cluster NGC 2682 stars analyzed by binocs with Gaia proper motion-based membership data. Grey diamonds represent stars where both methods agree that they are non-members. Cyan triangles represent stars that are considered members with both the Gaia method and the binocs method. Orange squares represent stars that are considered members with Gaia and non-members with binocs. Magenta circles represent stars that are considered non-members with Gaia and members with binocs. Star counts in each category can be found in Table 9. For $H\geq 12$ where the binocs method is effective, the two membership methods agree for 74% of the stars. Figure 20: Same as Figure 19 for the cluster NGC 2099. These two methods agree for 75% of the stars. Table 9: Comparison of binocs membership to Gaia membership | | binocs | binocs
---|---|---|---
| | member | Non-member
| NGC 2682 ($H\geq 12$) |
Gaia Member | | 354 | 60
Gaia non-member | | 138 | 229
| NGC 2099 (all) |
Gaia Member | | 886 | 82
Gaia non-member | | 558 | 1152
We cross-matched the Gaia to the binocs analyzed stars, where we then compared
two methods to see how reliable they are in identifying clusters members: 1)
the binocs membership method and 2) the Gaia proper motion-based membership
method. We found for the cluster NGC 2682 (Figure 19 and Table 9) that for
stars fainter than $H\sim 12$, main sequence stars, the binocs method agrees
fairly well with Gaia with binocs finding members for ($\sim$86%) of the Gaia
members with significant overlap in members along the main sequence, which
cuts-off at magnitudes $H\sim 16$ due to the limit Gaia photometry and proper
motions. However we do find that binocs method is not quite as discriminating,
as it finds 543 members compared to 414 with Gaia.
We preformed the same comparison for the cluster NGC 2099 (Figure 20 and Table
9). The binocs method members again overlap with many of the Gaia members in
main sequence portion of the CMD and cut-offs at magnitudes around 16, due to
the same limitations. In this plot, far more stars in the binocs sample is
considered to be members of NGC 2099, as compared to NGC 2682, since this
younger cluster has a longer main sequence where the binocs works well. For
this cluster, binocs categorizes $\sim$92% of the Gaia members correctly as
members, but again includes more Gaia non-member stars as members.
As a comparison to Gaia proper motion membership, one of the best membership
methods available, we find binocs agrees with Gaia membership $>75$% of the
time on the main sequence, but is more inclusive on non-members. However,
binocs can be used at large distances, unlike Gaia, such as to explore
membership and binarity in simple stellar populations in other galaxies, given
sufficient photometric depth (e.g., Hubble + JWST).
### 7.3 Binary Fractions
After validating the binocs code, it can begin to be applied to the clusters
in the analysis set with the requisite photometry. The binocs code was run on
each of the 8 clusters available for this analysis (see Table 1), and the
overall binary fraction was recorded. A list of clusters, their parameters,
and the associated overall binary percentage is shown in Table 10.
Table 10: Overall binary fractions for the 8 clusters considered in this analysis. | Binary | Age | | Rgc | Number of | Mass
---|---|---|---|---|---|---
Cluster | Fraction | (Gyr) | [Fe/H] | (kpc) | Members | Range (M⊙)
NGC 188 | 0.44 | 6.30 | $-$0.02 | 9.54 | 405 | 0.80 – 1.14
NGC 1960 (M36) | 0.66 | 0.03 | $\cdots$ | 9.81 | 941 | 0.65 – 6.46
NGC 2099 (M37) | 0.48 | 0.35 | $+$0.08 | 9.88 | 1632 | 0.32 – 3.21
NGC 2158 | 0.49 | 1.10 | $-$0.23 | 13.56 | 266 | 1.00 – 1.98
NGC 2168 (M35) | 0.61 | 0.13 | $-$0.21 | 9.37 | 2258 | 0.55 – 3.19
NGC 2420 | 0.41 | 2.00 | $-$0.23 | 10.81 | 748 | 0.35 – 1.63
NGC 2682 (M67) | 0.44 | 3.50 | $+$0.01 | 9.11 | 642 | 0.19 – 1.40
NGC 6791 | 0.39 | 8.00 | $+$0.38 | 8.11 | 524 | 0.89 – 1.16
#### 7.3.1 Binary Fraction Versus Age
One of the main science questions of this work is how the binary population of
a cluster evolves over time. The trend of overall binary fraction with cluster
age is shown in Figure 21. Overall, there seems to be a decreasing trend with
age. Gravitational interactions between stars can easily disrupt some binary
systems, while creating binaries from two single stars is much less common. It
appears the majority of binary disruption occurs quickly during the first 200
Myr of a cluster’s lifetime, after which the binary fraction becomes fairly
constant.
Figure 21: (Top) Overall cluster binary fraction, as a function of cluster
age. (Middle) verall cluster binary fraction, as a function of cluster [Fe/H].
(Bottom) Overall cluster binary fraction, as a function of cluster Rgc.
After about 200 Myr, the binary fraction stabilizes to around 0.42, which is
slightly higher than the measured binary percentage of 0.33 for field stars
(Raghavan & McAlister, 2009). This small difference may be attributable to the
fact that during the strong gravitational interaction which could eject a
cluster binary system into the field population, the binary system may also be
disrupted. Without a better understanding of the ejection processes of binary
systems, the overall binary fraction of cluster and field stars cannot be
easily compared.
Completing an analysis such as the one in Figure 21 using only RV surveys
could take decades to build up enough analysis clusters to produce any useful
insights. Two-band analysis, though fast, is dominated by degeneracies, and is
limited to small magnitude ranges across the main sequence. With new, deep
photometric surveys becoming available (UKIDSS, VVV, ESA Gaia, LSST), more
clusters could be added to the list with minimal effort using the binocs code.
Generating the plot in Figure 21 using hundreds of open clusters would yield
significant insights into the true distribution of binary fractions.
#### 7.3.2 Binary Fraction Versus Metallicity
It is not well-understood how differences in metallicity of a pre-cluster
cloud may affect the formation of binary systems. The distribution of binary
fractions as a function of metallicity is shown in Figure 21. There are only 7
clusters shown in Figure 21 due to the fact that M36 does not have a published
metallicity value.
It is clear from Figure 21 that any observed trend will be dominated by the
contribution from NGC 6791, at [Fe/H] $=+0.38$. Without this metallicity
outlier, there is hardly any trend in binary fraction. The absence of a trend
is still significant: the metallicity of a primordial cluster may have no
effect on the binary population, at least on the aggregate level. This insight
could be important for initial conditions of numerical simulations.
Similarly to the distribution with age, more data points can be added to this
plot with minimal effort when new deep photometry becomes available. Filling
in the remainder of the metallicity range will give more insight into whether
a trend exists or not. Additionally, with a much larger number of clusters,
binary fraction can be modeled as a function of both metallicity _and_ age.
#### 7.3.3 Binary Fraction Versus Galactocentric Radius
The above two comparisons have linked binary fraction to intrinsic cluster
parameters, but clusters are not isolated systems, and the galactic
environment plays a large part in cluster evolution. Clusters that are born
near the center of the Galaxy experience a higher rate of tidal stripping
events and other interactions which may alter the binary population. Figure 21
shows the overall binary fraction of clusters as a function of galactocentric
radius (Rgc; the distance the cluster is from the center of the galaxy).
In Figure 21, any observed trend is dominated by the two very young clusters,
and thus high binary fraction, in the sample. Removing these two data points,
a slight increasing trend with radius is observed. This would indicate that
the gravitational shocking experienced at lower Galactic radii cause more
binaries to be destroyed or ejected. However, NGC 2158, with a Rgc of 13.5, is
a high leverage point; its removal would result in there being no trend in
Rgc. Additionally, the most central cluster is NGC 6791, with an age of 8 Gyr,
while NGC 2158 has an age of 1.1 Gyr, an age difference which may explain the
trend without Rgc.
As with the metallicity comparison, more clusters are needed to fill in the
gaps in Rgc, disentangle correlations with age, and determine whether a trend
truly exists. A more complete Figure 21 would allow the exploration of
cluster-environment interactions, and would inform cluster simulations on the
correct treatment of tidal stripping events and other gravitational
collisions.
## 8 Conclusions
Understanding main sequence low-mass binary populations is essential for fully
characterizing the masses and evolution of stellar clusters. The
characteristics of binary populations, such as the mass function and radial
distribution are important for understanding the underlying physics of cluster
evolution. It is well established that cluster stars, as well as high mass
binary systems, undergo mass segregation over time, but the extent that this
affects the low mass binary population has not been fully explored. In this
work:
* •
We present new deep Near-IR and Mid-IR photometry for the open clusters NGC
2099 (M37) and NGC 2682 (M67). The NOAO/NEWFIRM photometry reaches a depth of
($J$,$H$,$K_{S}$ = 18.6, 18.1, 17.8) for NGC 2099 and ($J$,$H$,$K_{S}$ = 18.8,
19.0, 18.0) for NGC 2682. The Spitzer/IRAC photometry reaches a depth of
([3.6][4.5][5.8][8.0] = 18.0, 16.5, 14.6, 13.8 ) for NGC 2099 and
([3.6][4.5][5.8][8.0] = 18.5, 17.4, 15.0, 14.0) for NGC 2682.
* •
We introduce the Binary INformation from Open Clusters using SEDs (binocs) a
purely photometric method for determination of unresolved binaries and
determination of the masses of both stars, for main sequence stars with
primary masses below the turnoff to 0.5 M⊙ (2.5–0.5 M⊙ for NGC 2099). We
showed that the binocs method is a significant improvement over current binary
detection techniques; requiring an order of magnitude less time, generating
mass estimates on an order of magnitude more stars, and enabling quantitative
exploration of faint binary systems, which are unreachable by RV studies. The
binocs method allows for robust, quick binary classification that will become
especially powerful as new all-sky surveys are released.
* •
We tested the binocs code to ensure it produced reasonable results for binary
detection and mass determination, when compared to previously-published
studies based on multi-decade RV work. Overall binary fractions can be
computed quickly using binocs for clusters with sufficient photometry.
* •
The results for NGC 188 are consistent with the result of Cohen et al. (2020),
which compared to the binocs results as preliminarily presented in Frinchaboy
& Thompson (2015).
* •
We find a clear decrease in binary fraction with respect to cluster age, due
likely to disruption of wide-binary systems in the cluster environment.
The authors would like to thanks the referee for their patience and
comments/suggestions that improved this paper. The authors would like to
acknowledge graduate thesis travel support from NOAO as well as financial
support from the Texas Space Grant Consortium and NSF-AST grant 1311835 and
AST-1715662. Spitzer Cycle 3 observations were funded under NASA/JPL sub-award
grant GO-30800. The authors would especially like to thank the GLIMPSE team,
specifically Brian Babbler & Marilyn Meade, for their contribution in
modifying the GLIMPSE pipeline to handle our HDR IRAC data. The authors would
also like to thank the Max-Planck-Institut für Astronomie (MPIA Heidelberg)
for hosting PMF and JD during the completion of this work. This research uses
services or data provided by the NOAO Science Archive. NOAO is operated by the
Association of Universities for Research in Astronomy (AURA), Inc. under a
cooperative agreement with the National Science Foundation. This work has also
made use of the NASA/IPAC Infrared Science Archive, which is operated by the
Jet Propulsion Laboratory, California Institute of Technology, under contract
with the National Aeronautics and Space Administration.
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|
2.55em section[1.5em]2.8em1pc subsection[3.7em]3.4em1pc
subsubsection[7.6em]3.2em1pc paragraph[10.3em]3.2em1pc
# Bi-traceable graphs, the intersection of three longest paths and Hippchen’s
conjecture
Juan Gutierrez1 1 Departamento de Ciencia de la Computación Universidad de
Ingeniería y Tecnología (UTEC<EMAIL_ADDRESS>and Christian
Valqui2,3 2Pontificia Universidad Católica del Perú, Sección Matemáticas,
PUCP, Av. Universitaria 1801, San Miguel, Lima 32, Perú. 3Instituto de
Matemática y Ciencias Afines (IMCA) Calle Los Biólogos 245. Urb San César. La
Molina, Lima 12, Perú<EMAIL_ADDRESS>
###### Abstract.
Let $P,Q$ be longest paths in a simple graph. We analyze the possible
connections between the components of $P\cup Q\setminus(V(P)\cap V(Q))$ and
introduce the notion of a bi-traceable graph. We use the results for all the
possible configurations of the intersection points when $\\#V(P)\cap V(Q)\leq
5$ in order to prove that if the intersection of three longest paths $P,Q,R$
is empty, then $\\#(V(P)\cap V(Q))\geq 6$. We also prove Hippchen’s conjecture
for $k\leq 6$: If a graph $G$ is $k$-connected for $k\leq 6$, and $P$ and $Q$
are longest paths in $G$, then $\\#(V(P)\cap V(Q))\geq 6$.
###### Key words and phrases:
Hippchen’s conjecture, three longest paths, traceable graph, intersection of
longest paths
###### 2020 Mathematics Subject Classification:
primary 05C38; secondary 05C45
Christian Valqui was supported by PUCP-DGI-CAP-2020-818.
###### Contents
1. 1 Preliminaries and an illustrative example
2. 2 Bi-traceable graphs
3. 3 Bi-traceable graphs with $\\#(V(P)\cap V(Q))\leq 4$
4. 4 Bi-traceable graphs with $\\#(V(P)\cap V(Q))=5$
5. 5 In the three exceptional cases $V(P)\cap V(Q)$ is a separator
6. 6 Three longest paths in the exceptional cases
## Introduction
Different configurations of the intersection points of longest paths in simple
graphs have been analyzed by various authors with different purposes.
Axenovich in [A] used two configurations $Q_{1}$ and $Q_{2}$ that described
forbidden connections in $G\setminus V(P)\cap V(Q)$, in order to prove that in
outerplanar graphs every three longest paths share a common point. Hence, for
outerplanar graphs the following conjecture is true.
Conjecture 1: For every connected graph, any three of its longest paths have a
common vertex.
In [dR] this conjecture is proven for a broader class of graphs and in [FFNO]
an approach to this conjecture using certain distance functions is given. It
is known that two longest paths in a connected graph have necessarily a common
vertex, but there are connected graphs, where seven longest paths do not have
a common vertex (See for example [S]). The equivalent question for 3,4,5 and 6
longest paths is still open. See [SZZ] for a survey on this and similar
problems.
On the other hand Hippchen in [H] analyzed forbidden connections in
$G\setminus V(P)\cap V(Q)$ when $\\#(V(P)\cap V(Q))=2$ in order to prove that
in $3$-connected graphs the intersection of two longest path contains at least
3 vertices. He then stated the following conjecture:
Conjecture 2: The intersection of two longest paths in a $k$-connected simple
graph has cardinality at least $k$.
This conjecture was an adaptation of a similar conjecture for longest cycles
instead of paths, which appeared first in [Gr]*p.188 and was attributed to
Scott Smith, and where some partial results are known (See for example [G],
[ST] and [Ch]).
In [G] the first author proved Hippchen’s conjecture for $k=4$, and in [CCP]
Hippchen’s conjecture is proven for $k=5$. In both cases the possible
configurations and forbidden connections of the graph induced by $V(P)\cap
V(Q)$ is analyzed.
The configuration of the intersection vertices of longest cycles (instead of
paths) has been considered for example in [B79], [Gr] and [ST]. Babai
[B79]*Lemma 2 uses a forbidden connection in order to prove that in a
3-connected graph, any two longest cycles have at least three points in
common. Groetschel in [Gr] proved that the complement of the intersection of
two longest cycles is not connected when there are at most 5 intersection
points, Steward and al. used a computer program in [ST] to analyze the
different configurations of the intersection points in order to show that this
remains true for $k=6,7$ (for $k=8$ there is a counterexample).
In the present paper we will make a complete analysis of the possible
configurations of the intersection points of two longest paths $P$ and $Q$,
when there are $\ell\leq 5$ intersection points. We will analyze the different
possibilities for the sequential order of the intersection vertices in each of
the two longest paths $P$ and $Q$, which is given by a permutation of the
$\ell$ intersection vertices. Two permutations give the same configuration if
they can be transformed into each other by exchanging $P$ and $Q$, or by
changing the direction in which we travel through the points in each path.
If the intersection is a single point or has two points, there is only one
configuration. If the intersection has three points, there are two
configurations, already described in [G]. If the intersection has 4 points,
then the 24 different permutations give 7 different configurations, which
correspond to 7 different cases described in [CCP]. When the intersection has
5 points, we find 23 different cases arising from the 120 different
permutations.
For each configuration we analyze in detail the possible connections between
the components of the graph $(P\cup Q)\setminus(V(P)\cap V(Q))$. We find a
large class of graphs $P\cup Q$, such that in each exterior swap unit (see
Definition 2.9) $P\setminus(V(P)\cap V(Q))$ cannot be connected with
$Q\setminus(V(P)\cap V(Q))$ outside $V(P)\cap V(Q)$ (See Proposition 2.14 and
Remark 2.15). In these graphs, $V(P)\cap V(Q)$ is a separator, if we
additionally assume that $V(P)\neq V(Q)$. For graphs not in this class we
analyze in detail whether two components of $(P\cup Q)\setminus(V(P)\cap
V(Q))$ can be connected such that $P$ and $Q$ are still longest paths.
We will use our results in order to approach three different problems on
longest paths, the intersection of three longest paths, Hippchen’s conjecture
and a path version of Groetschel’s result on the connectedness of the
complement of the intersection of two longest paths. One of our main results
is the following result on a possible counterexample of Conjecture 1 in
Theorem 6.5:
If the intersection of three longest paths $P,Q,R$ is empty, then
$\\#(V(P)\cap V(Q))\geq 6$.
By symmetry, this result implies that in this case we also have $\\#(V(R)\cap
V(Q))\geq 6$ and $\\#(V(P)\cap V(R))\geq 6$. Hence we can rephrase our result
in the following way:
Let $P,Q,R$ be three longest paths in a graph. If one of $V(P)\cap V(Q)$,
$V(P)\cap V(R)$ or $V(Q)\cap V(R)$ has less than 6 points, then the three
paths have a common vertex.
On the other hand, if Conjecture 1 is false and there exist three longest
paths in a graph with empty intersection, then with our methods one can carry
out a systematic search for a counterexample.
As second main result of the present paper we prove Hippchen’s conjecture for
$k=6$ (Corollary 5.3) and give a new proof for $k=5$ (Corollary 3.5). The same
methods should be useful in order to prove Hippchen’s conjecture for higher
$k$, or to prove it completely.
The third problem to which we apply our methods is the path-version of
[Gr]*Theorem 1.2(a) (see also [ST]). We arrive at the following results
(Theorem 5.7 and Corollary 5.8):
$[Gr(\ell\leq 5)]$ Assume that $P$ and $Q$ are two longest paths in a simple
graph $G$. If $V(Q)\neq V(P)$ and $V(P)\cap V(Q)$ has cardinality $\ell\leq
5$, then it is a separator (called an articulation set in [Gr]), which means
that the complement is not connected.
$[Gr(n\leq 7)]$ If $V(Q)\neq V(P)$ and $n=|V(G)|\leq 7$ then $V(Q)\cap V(P)$
is a separator.
Open question: Which are the maximal $\ell$ and $n$ such that the above
results remain true? In an hypotraceable graph we can choose two longest paths
$P$ and $Q$ that leave out two connected vertices, and so $V(Q)\cap V(P)$ is a
not separator in that case. Since there exists a hypotraceable graph on 34
vertices (see [Th]), we know automatically that $n_{max}\leq 33$ and
$\ell_{max}\leq 31$.
Now consider the following simple graph with 11 vertices, that has two longest
path $P$ and $Q$ of length 9, which satisfy $V(Q)\neq V(P)$ and moreover, the
complement of $V(Q)\cap V(P)$ is connected. Since $\\#(V(P)\cap V(Q))=9$ we
have $n=11$ and $\ell=9$.
Simple graph, $n=11$ verticesTwo longest paths, $\ell=\\#(V(P)\cap V(Q))=9$
Thus $n_{max}\leq 10$ and $\ell_{max}\leq 8$. Moreover, we have verified all
cases up to $n=10$ and will write down the lengthy computations in a
systematic way in a forthcoming article. This implies that $n_{max}=10$ and by
the above result we already know that $5\leq\ell_{max}\leq 8$.
The article is organized as follows. In the first section we prove Theorem
1.2, which illustrates our method in a simple case. The theorem says that if
the intersections vertices of two longest path $P$ and $Q$ have the same
sequential order in $P$ and in $Q$, then $V(P)\cap V(Q)$ is a separator, thus
$P$ and $Q$ cannot be a counterexample to the Hippchen conjecture.
In the second section we introduce our main new tools: bi-traceable (BT)
graphs, the standard representations $BT(P,Q)$ of these graphs and what we
call exterior swap units (ESU) (see Definition 2.1). We also introduce in
Definitions 2.2, 2.12 and 2.17 three different conditions on how you can
connect, or rather on how you cannot connect, two components of $(P\cup
Q)\setminus(V(P)\cap V(Q))$. They can be not directly connectable (NDC), non
connectable (NC) or locally non connectable (LNC). We also prove that if
$V(P)\neq V(Q)$ and if all ESU’s in $BT(P,Q)$ are NC, then $V(P)\cap V(Q)$ is
a separator. In the third section we prove that for every graph $BT(P,Q)$ with
$\ell\leq 4$ all ESU’s are NC, which reproves the Hippchen conjecture for
$k=5$. In section 4 we analyze the case $\\#V(P)\cap V(Q))=5$ and find that in
all but three exceptional cases all ESU’s are NC. In section 5 we prove that
in these three cases $V(P)\cap V(Q)$ is a separator, proving the Hippchen
conjecture for $k=6$. In the last section we prove that in none of the three
cases there can be a third longest path $R$ with $P\cap Q\cap R=\emptyset$,
which proves that if the intersection of three longest paths $P,Q,R$ is empty,
then $\\#(V(P)\cap V(Q))\geq 6$.
If one wants to continue our analysis for the case where the intersection
consists of 6 points, it will be necessary to make a computer assisted
analysis of the 720 permutations and the approximately 100 configurations
resulting from them, and find the exceptional configurations. The methods
developed in the present paper will be useful in order to treat exceptional
cases that will arise for $\ell\geq 6$. The complete tables of possible
configurations can be useful for other problems involving intersections of
longest paths.
## 1 Preliminaries and an illustrative example
Along the present paper $G$ is a simple graph, and $P,Q$ are longest paths. By
definition, a path cannot pass twice through the same vertex, and the length
of a path is the number of its edges. We will construct a large class of
graphs $P\cup Q$, such that in each exterior swap unit (see Definition 2.9)
$P\setminus(V(P)\cap V(Q))$ cannot be connected with $Q\setminus(V(P)\cap
V(Q))$ outside $V(P)\cap V(Q)$ (See Proposition 2.14 and Remark 2.15). As
usual, removing the vertices also removes the incident edges. In these graphs,
$V(P)\cap V(Q)$ is a separator, if we additionally assume that $V(P)\neq
V(Q)$.
We define an operation ${}^{\prime\prime}+^{\prime\prime}$ on two paths $P$,
$Q$, which is defined only if $P$ and $Q$ share exactly one endpoint, and
$P+Q$ is simply the union of both paths. If we write $P+Q+R$, then $P$ and $Q$
share one endpoint, and the other endpoint of $Q$ is also an endpoint of $R$,
and $P+Q+R$ is the union of the three paths.
###### Notation 1.1.
Along this work we assume that $\\#V(P)\cap V(Q)=\ell$ for some $\ell\geq 1$.
If we write $P=P_{0}+P_{1}+\dots+P_{\ell}$ such that $P_{i}\cap
P_{i-1}=\\{a_{i}\\}$, and $\\{a_{1},\dots,a_{\ell}\\}=V(P)\cap V(Q)$, and we
write $Q=Q_{0}+Q_{1}+\dots+Q_{\ell}$ such that $Q_{i}\cap
Q_{i-1}=\\{b_{i}\\}$, and $\\{b_{1},\dots,b_{\ell}\\}=V(P)\cap V(Q)$, then the
two paths $P,Q$ determine a permutation $\sigma$ on $\\{1,2,\dots,\ell\\}$
given by $b_{j}=a_{\sigma(j)}$.
Write $Q_{i}^{\prime}:=Q_{i}\setminus(V(P)\cap V(Q))$ and
$P_{i}^{\prime}:=P_{i}\setminus(V(P)\cap V(Q))$. These are the components of
$P\cup Q\setminus(V(P)\cap V(Q))$. Note that $P_{0}^{\prime}=\emptyset$ if and
only if $P_{0}=\\{a_{1}\\}$, i.e., $L(P_{0})=0$. The same holds for
$Q_{0}^{\prime}$, $P_{\ell}^{\prime}$ and $Q_{\ell}^{\prime}$. On the other
hand if $i\notin\\{0,\ell\\}$, then $P_{i}^{\prime}=\emptyset$ if and only if
$L(P_{i})=1$ and the same holds for $Q_{i}^{\prime}$. Let
$X,Y\in\\{P_{i}^{\prime},Q_{i}^{\prime}:P_{i}^{\prime}\neq\emptyset,Q_{i}^{\prime}\neq\emptyset\\}.$
A direct connection between $X$ and $Y$ is a path $R$ from $X$ to $Y$
internally disjoint from $V(P)\cup V(Q)$. We write $X\sim Y$ or $X\sim_{R}Y$
(Note that this is not an equivalence relation). Clearly such a path $R$
cannot be part of a graph where $P$ and $Q$ are longest paths, if there is a
path in $P\cup Q\cup R$, which is longer than $P$ (or $Q$).
The case in which $\sigma=Id$, is the prototype of the large class of graphs
mentioned above. Our principal ideas and methods are already present in this
case, so we will give a detailed proof and describe an example with $\ell=7$.
###### Theorem 1.2.
Let $G$ be a simple graph, and let $P,Q$ be longest paths in $G$. Assume that
$a_{i}=b_{i}$ for all $i$ (which means that $\sigma=Id$) and that $V(P)\neq
V(Q)$. Then the complement of $V(P)\cap V(Q)$ cannot be connected.
###### Proof.
Assume by contradiction that it is connected. Since $V(P)\neq V(Q)$, there
exists $i_{0}$ such that $P_{i_{0}}^{\prime}\neq\emptyset$, and it follows
that $Q_{i_{0}}^{\prime}\neq\emptyset$, since $L(P_{i_{0}})=L(Q_{i_{0}})$.
Consequently there exist
$X_{1},\dots,X_{r}\in\\{P_{i}^{\prime},Q_{i}^{\prime}:P_{i}^{\prime}\neq\emptyset,Q_{i}^{\prime}\neq\emptyset\\},\quad\text{such
that $X_{i}\sim X_{i+1}$ and $X_{1}=P_{i_{0}}^{\prime}$ and
$X_{r}=Q_{i_{0}}^{\prime}$}.$
Let $\widehat{X}_{i}=\begin{cases}P_{j},&\mbox{if }X_{i}=Q_{j}\\\
Q_{j},&\mbox{if }X_{i}=P_{j}\end{cases}$. Set
$j_{0}=\min\\{j>1,\widehat{X}_{j}\in\\{X_{1},\dots,X_{j-1}\\}\\}.$
We can assume that $X_{j}\subset P$ for $j<j_{0}$. In fact, if
$X_{j}=Q_{i}^{\prime}$, we redefine $P$ as $P_{<i}+Q_{i}+P_{>i}$ and $Q$ as
$Q_{<i}+P_{i}+Q_{>i}$.
There exists $i_{1}$ such that $X_{j_{0}}=Q_{i_{1}}^{\prime}$ and so there is
a path $R$ from $P_{i_{1}}^{\prime}$ to $Q_{i_{1}}^{\prime}$ internally
disjoint from $Q$ and from $P_{i_{1}}$. The endpoints of $R$ split the
subpaths $Q_{i_{1}}$ and $P_{i_{1}}$ into two subpaths each, which we name
$P_{i_{1},1}$, $P_{i_{1},2}$,$Q_{i_{1},1}$,$Q_{i_{1},2}$. The two paths
$Q_{<i_{1}}+Q_{i_{1},1}+R+P_{i_{1},2}+Q_{>i_{1}}\quad\text{and}\quad
Q_{<i_{1}}+P_{i_{1},1}+R+Q_{i_{1},2}+Q_{>i_{1}}$
$P_{1}$$P_{i_{1}-1}$$P_{i_{1},2}$$P_{\ell-1}$$P_{\ell}$$Q_{1}$$Q_{i_{1}-1}$$Q_{i_{1},1}$$Q_{\ell-1}$$Q_{\ell}$$R$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$
$P_{1}$$P_{i_{1}-1}$$P_{i_{1},1}$$Q_{i_{1},2}$$P_{\ell-1}$$P_{\ell}$$Q_{1}$$Q_{i_{1}-1}$$Q_{\ell-1}$$Q_{\ell}$$R$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$
have lengths that sum $2L(Q)+2L(R)$, a contradiction that concludes the proof.
∎
###### Example 1.3.
We will illustrate the proof of the theorem in an example with $\ell=7$. The
green path is $P$, the red path is $Q$ and the blue paths are in
$G\setminus(P\cup Q)$, with endpoints in $P\cup Q\setminus(V(P)\cap V(Q))$.
$P_{0}$$Q_{0}$$P_{2}$$P_{3}$$P_{4}$$P_{5}$$P_{6}$$Q_{6}$$Q_{2}$$Q_{4}$
In this example $i_{0}=2$, since the additional blue paths connect $P_{2}$
with $Q_{2}$. Moreover
$X_{1}=P_{2}^{\prime},\quad X_{2}=P_{4}^{\prime},\quad
X_{3}=P_{3}^{\prime},\quad X_{4}=P_{5}^{\prime},\quad
X_{5}=Q_{6}^{\prime},\quad X_{6}=Q_{4}^{\prime},\quad X_{7}=Q_{2}^{\prime},$
and so
$\widehat{X}_{1}=Q_{2}^{\prime},\quad\widehat{X}_{2}=Q_{4}^{\prime},\quad\widehat{X}_{3}=Q_{3}^{\prime},\quad\widehat{X}_{4}=Q_{5}^{\prime},\quad\widehat{X}_{5}=P_{6}^{\prime},\quad\widehat{X}_{6}=P_{4}^{\prime},\quad\widehat{X}_{7}=P_{2}^{\prime}.$
We have $j_{0}=6$, since
$\widehat{X}_{2}\notin\\{X_{1}\\},\quad\widehat{X}_{3}\notin\\{X_{1},X_{2}\\},\quad\widehat{X}_{4}\notin\\{X_{1},X_{2},X_{3}\\},\quad\widehat{X}_{5}\notin\\{X_{1},X_{2},X_{3},X_{4}\\},$
but
$\widehat{X}_{6}=P_{4}^{\prime}=X_{2}\in\\{X_{1},X_{2},X_{3},X_{4},X_{5}\\}.$
Note that we also have
$\widehat{X}_{7}=P_{2}^{\prime}=X_{1}\in\\{X_{1},X_{2},X_{3},X_{4},X_{5},X_{6}\\}$.
Since $X_{5}=Q_{6}^{\prime}$, we redefine $P$ and $Q$, swapping $Q_{6}$ with
$P_{6}$.
$P_{0}$$Q_{0}$$P_{2}$$P_{3}$$P_{4}$$P_{5}$$Q_{6}$$P_{6}$$Q_{2}$$Q_{4}$
Then $i_{1}=4$, since $X_{j_{0}}=X_{6}=Q_{4}^{\prime}$, and we arrive at the
following situation, where $P_{i_{1}}^{\prime}=P_{4}^{\prime}$ is connected
with $Q_{i_{1}}^{\prime}=Q_{4}^{\prime}$ via a path $R$ that is internally
disjoint from $Q$ and from $P_{4}\cup Q_{4}$.
$P_{0}$$Q_{0}$$P_{2}$$P_{3}$$P_{4}$$P_{5}$$Q_{6}$$P_{6}$$Q_{2}$$Q_{4}$$R$
As in the proof of the theorem, now we can construct the two paths, whose
length sum $2L(Q)+2L(R)$.
We want to generalize the theorem in order to apply to other permutations, and
not only for $\sigma=Id$. Note that in the proof of the theorem we use the
following properties of the pair $\\{P_{i},Q_{i}\\}$:
* •
$L(P_{i})=L(Q_{i})$,
* •
They share both endpoints, when $1<i<\ell$, and one endpoint when
$i\in\\{0,\ell\\}$.
* •
Thus they can be interchanged, i.e., $P_{<i}+Q_{i}+P_{>i}$ and
$Q_{<i}+P_{i}+Q_{>i}$ are two longest paths having the same intersection and
the same union as $P$ and $Q$.
* •
An internally disjoint path $R$ between $P_{i}^{\prime}$ and $Q_{i}^{\prime}$
when $0<i<\ell$, enables to construct the two paths $P_{i,1}+R+Q_{i,2}$ and
$Q_{i,1}+R+P_{i,2}$ such that both paths connect the endpoints of $P_{i}$.
Moreover, the lengths of the new paths sum $2L(P_{i})+2R>L(Q_{i})+L(P_{i})$.
## 2 Bi-traceable graphs
We want to consider the graph which is the union of the two longest paths $P$
and $Q$. In this section we assume the same notations as in Notation 1.1.
###### Definition 2.1.
1. (1)
A bi-traceable graph (BT-graph) is a simple graph that has two longest paths
such that the union is the whole graph.
2. (2)
The representation of a bi-traceable graph associated to $P$ and $Q$ is a two
colored graph $BT(P,Q)$, obtained from the BT-graph $P\cup Q$ by the following
process. First we assign to all the edges of $P$ one color. Then the edges of
$Q$ are colored with the other color. If the paths share an edge, then this
edge is duplicated, and each copy is colored with one of the colors. The
vertices are not colored.
Bi-traceable graph$BT(P,Q)$ with $P$ in green and $Q$ in red
Note that the resulting graph is no longer simple, but $P$ and $Q$ are still
longest paths in it. We also have $BT(P,Q)=BT(Q,P)$.
3. (3)
The subpaths $P_{i}$, $Q_{i}$ are called the completed components of
$BT(P,Q)$, or simply the completed components, if $P$ and $Q$ are clear from
the context. Note that each of the extremal completed components $P_{0}$,
$Q_{0}$, $P_{\ell}$ and $Q_{\ell}$ may be only a point.
Note that some components $P_{i}^{\prime}$ or $Q_{i}^{\prime}$ might be empty.
###### Definition 2.2.
Two components $X,Y$ are said to be not directly connectable (NDC), if one of
them is empty, or if $X\sim_{R}Y$ implies that there exist two paths
$\widehat{P}$ and $\widehat{Q}$ in $BT(P,Q)\cup R$ such that
$L(\widehat{P})+L(\widehat{Q})=2L(R)+L(P)+L(Q)=2L(P)+2L(R).$
The following result is one of the main ingredients in the proof of the
Hippchen conjecture for $k=4$ in [G], and the idea was already present in
[H]*Lemma 2.2.3.
###### Lemma 2.3 ([G]*Lemma 4.1).
If two completed components of different colors are adjacent (which means that
they have a common vertex), then they are NDC.
###### Proof.
See the diagram, or see also [G]*Lemma 4.1.
$Q$$P$$R$$\dots$$\dots$$\dots$$\dots$$\widehat{Q}$$\dots$$\dots$$\dots$$\dots$$\widehat{P}$$\dots$$\dots$$\dots$$\dots$
∎
###### Lemma 2.4.
Two extremal components of different colors are NDC. The extremal components
are $P_{0}^{\prime},P_{\ell}^{\prime},Q_{0}^{\prime},Q_{\ell}^{\prime}$.
###### Proof.
Assume for example $P_{0}^{\prime}\sim_{R}Q_{\ell}^{\prime}$. Let
$P_{0,1}+P_{0,2}=P_{0}$ such that $P_{0,1}\cap P_{0,2}$ is one endpoint of $R$
and such that $P_{0,1}$ and $P$ share one endpoint. Similarly define
$Q_{\ell,1}$ and $Q_{\ell,2}$, with $Q_{\ell,2}$ having a common endpoint with
$Q$. Then
$\widehat{P}=Q_{\ell,2}+R+P_{0,2}+P_{1}+\dots+P_{\ell}\quad\text{and}\quad\widehat{Q}=P_{0,1}+R+Q_{\ell,1}+Q_{\ell-1}+\dots+Q_{1}+Q_{0}$
$P_{0,1}$$Q_{\ell,2}$$P$ in green, $Q$ in red, $R$ in blue$\widehat{P}$ in
blueThe path $\widehat{Q}$ in blue
are two paths whose lengths sum $2L(P)+2L(R)$, as desired. ∎
Let $G$ be a graph, $P$ and $Q$ be longest paths. Assume that $BT(P,Q)$ has at
least two non empty components of different colors. If all pairs of different
colors in $BT(P,Q)$ are NDC, then $V(P)\cap V(Q)$ is a separator of $G$.
Verifying this condition for all pairs of different colors in small graphs is
a manageable task, but in order to obtain results generalizing Theorem 1.2, we
need a notion of “connectable” with a broader scope. For this we formalize the
process of swapping colors of some completed components in the proof of
Theorem 1.2, and we analyze the block structure of $BT(P,Q)$.
###### Definition 2.5.
A swap unit in $BT(P,Q)$ is a set of completed components of two colors, such
that if we swap the colors in the given set of completed components, then we
obtain a new representation $BT(\widetilde{P},\widetilde{Q})$ of the original
BT-graph.
###### Definition 2.6.
An internal building block (IBB) of $BT(P,Q)$ is a 2-connected subgraph which
is the union of two subpaths $\widetilde{P}$ of $P$ and $\widetilde{Q}$ of
$Q$, such that $\widetilde{P}$ and $\widetilde{Q}$ have the same endpoints,
and don’t intersect other subpaths of $P$ and $Q$ other than in these
endpoints, which are called the endpoints of the IBB.
An elementary IBB is the union of two completed components that share both
endpoints.
For example, if the $P$ and $Q$ share one edge, then the duplicated edge is an
elementary IBB.
Elementary IBBIBB with embeddedelementary IBBIBB with 5 embedded IBB’s
Note that every IBB is a swap unit. In fact, write $P=P_{1}+P_{2}+P_{3}$ and
$Q=Q_{1}+Q_{2}+Q_{3}$, where $P_{2}$ and $Q_{2}$ are the subpaths spanning the
building block. Swapping the colors generates two longest paths
$\widetilde{P}=P_{1}+Q_{2}+P_{3}\quad\text{and}\quad\widetilde{Q}=Q_{1}+P_{2}+Q_{3},$
such that the union is still $BT(P,Q)$ and so we obtain a new representation
$BT(\widetilde{P},\widetilde{Q})$. Hence, if $B$ is an IBB, the subpath of $P$
and the subpath of $Q$ have the same length, which we call the length of the
block $L(B)$.
###### Remark 2.7.
An IBB with endpoints $a$ and $b$, can be defined independently of $BT(P,Q)$
as the representation of a bitraceable graph with fixed endpoints $a$ and $b$,
where such a graph is generated by two paths from $a$ to $b$ that have the
same length, and such that there is no longer path from $a$ to $b$.
If the intersection of two IBB’s is one common endpoint, then the
concatenation of the IBB’s is their union, and similarly we can concatenate
three or more IBB’s.
###### Definition 2.8.
An extremal building unit of $BT(P,Q)$ is a subgraph which is the union of two
extremal subpaths $\widetilde{P}$ of $P$ and $\widetilde{Q}$ of $Q$, such that
$\widetilde{P}$ and $\widetilde{Q}$ have exactly one endpoint in common, and
this common endpoint is neither an endpoint of $P$ nor of $Q$. We also require
that $\widetilde{P}$ and $\widetilde{Q}$ don’t intersect other subpaths of $P$
and $Q$.
An extremal building block (EBB) of $BT(P,Q)$, is a minimal extremal building
unit. This means that it is an extremal building unit, which is not the
concatenation of one or more IBB’s with another extremal building unit.
An elementary EBB is an EBB, which is the union of two extremal completed
components that share one endpoint.
A building block (BB) is an IBB, an EBB or all of $BT(P,Q)$, if $BT(P,Q)$ is
not the concatenation of IBB’s and/or EBB’s, and $P$ and $Q$ have no common
endpoints.
EBB with two embedded IBB’sElementary EBB$BT(P,Q)$ is a BB
Note that an IBB can be contained in another BB, and that BB’s are also swap
units. Note also that in the case $\sigma=Id$, for each $0<i<\ell$ the union
of $P_{i}$ and $Q_{i}$ is an IBB, and the union of $P_{0}$ and $Q_{0}$ and the
union of $P_{\ell}$ and $Q_{\ell}$ are the EBB’s.
###### Definition 2.9.
Given a building block, the exterior swap unit (ESU) associated with the
building block, is the union of all completed components in the building
block, that are not contained in any embedded IBBs.
Note that the ESU of an elementary IBB is the whole IBB and the same holds for
an elementary EBB.
The following representation of a BT-graph is the concatenation of three
IBB’s, two of them are elementary IBB’s and the middle one contains five
elementary IBBs. Hence $BT(P,Q)$ has seven ESU’s of two completed components
each and one ESU consisting of 8 completed components (it has ten edges).
BT- graphRepresentation7 ESU’s with 2 edgesESU with 10 edges
Note that swapping the color of the edges in a swap unit doesn’t change the
intersection vertices, since each intersection vertex has two incident edges
of each color. After the swapping it must still have two incident edges of
each color, since otherwise one of the paths would visit this vertex twice.
###### Lemma 2.10.
In a swap unit, the sum of the lengths of the completed components of one
color equals the sum of the lengths of the completed components of the other
color. Moreover, the number of components of each color coincide.
###### Proof.
The sum of the lengths coincide, since otherwise one of the new paths
resulting from the swap would be longer. The same holds for the number of
components, since swapping the color of the edges doesn’t change the
intersection vertices, and so the number of components of each longest path
remains constant. ∎
###### Remark 2.11.
Note that the union of the ESU’s in $BT(P,Q)$ is $BT(P,Q)$ and that no
completed component is in two ESU’s at the same time.
###### Definition 2.12.
Let $X$ and $Y$ be components of different colors in an ESU. The pair $X,Y$ is
non connectable (NC), if a path $R$ that connects $X$ with $Y$, satisfying
$R\cap V(P)\cap(Q)=\emptyset$ and such that $R$ is internally disjoint from
the given ESU and in each of the other ESU’s touches at most one color, allows
to construct two paths $\widehat{P}$ and $\widehat{Q}$ in $BT(P,Q)\cup R$ such
that
$L(\widehat{P})+L(\widehat{Q})=2L(R)+L(P)+L(Q)=2L(P)+2L(R).$
The ESU is called NC, if all the pairs of component of the ESU of different
colors are NC.
###### Proposition 2.13.
The components of an elementary IBB are NC.
###### Proof.
Let $R$ be a path that connects one component of one color in the ESU with one
component of another color in the ESU, such that $R$ is internally disjoint
from the given ESU and in each of the other ESU’s touches at most one color.
$P_{j}$$Q_{i}$$R$$P$ in green, $Q$ in red, $R$ in blue
Then $R$ is internally disjoint from $\widetilde{Q}$, which is the path
obtained from $Q$ by swapping the colors in the ESU’s where $R$ touches only
the color of $Q$.
$P_{j}$$Q_{i}$$R$$\widetilde{Q}$$R$ is internally disjoint from the new path
$\widetilde{Q}$ in red
Write $\widetilde{Q}$ as $\widetilde{Q}_{1}+Q_{i}+\widetilde{Q}_{2}$, where
$Q_{i}$ is the subpath of $Q$ in the given ESU, and assume that $P_{j}$ is the
subpath of $P$ in the given ESU. The endpoints of $R$ split the subpaths
$Q_{i}$ and $P_{j}$ into two subpaths each, which we name $P_{j,1}$,
$P_{j,2}$,$Q_{i,1}$,$Q_{i,2}$. The two paths
$\widehat{Q}=\widetilde{Q}_{1}+Q_{i,1}+R+P_{j,2}+\widetilde{Q}_{2}\quad\text{and}\quad\widehat{P}=\widetilde{Q}_{1}+P_{j,1}+R+Q_{i,2}+\widetilde{Q}_{2}$
$P_{j,2}$$Q_{i,1}$$\widehat{Q}=\widetilde{Q}_{1}+Q_{i,1}+R+P_{j,2}+\widetilde{Q}_{2}$
in red
$P_{j,1}$$Q_{i,2}$$\widehat{P}=\widetilde{Q}_{1}+P_{j,1}+R+Q_{i,2}+\widetilde{Q}_{2}$
in green
have lengths that sum $2L(\widetilde{Q})+2L(R)=2L(P)+2L(R)$, as desired. ∎
###### Proposition 2.14.
Let $G$ be a graph and $P$,$Q$ longest paths. If all the ESU’s in $BT(P,Q)$
are NC, then there can be no path in $G\setminus(V(P)\cap V(Q))$ from a
component of one color in one ESU to a component of the other color in the
same ESU.
###### Proof.
Assume by contradiction that such a path from $P_{p}^{\prime}$ to
$Q_{q}^{\prime}$ exists. Then there exist
$X_{1},\dots,X_{r}\in\\{P_{i}^{\prime},Q_{i}^{\prime}:P_{i}^{\prime}\neq\emptyset,Q_{i}^{\prime}\neq\emptyset\\}$,
such that $X_{i}\sim X_{i+1}$ and $X_{1}=P_{p}^{\prime}$ and
$X_{r}=Q_{q}^{\prime}$. Let $i_{0}<j_{0}$ be such that $X_{j_{0}}$ and
$X_{i_{0}}$ are in the same ESU and have different colors, and such that
$j_{0}-i_{0}=\min\\{j-i:\ i<j,X_{j}\text{ and }X_{i}\text{ are in the same ESU
and have different colors}\\}.$
Then the subpath $R^{\prime}$ of $R$ which connects successively
$X_{i_{0}},X_{i_{0}+1},\dots,X_{j_{0}}$, is internally disjoint from the ESU
of $X_{j_{0}}$ and $X_{i_{0}}$ and in each of the other ESU’s touches at most
one color. Since the ESU of $X_{j_{0}}$ and $X_{i_{0}}$ is NC, by Definition
2.12 there exists a path in $BT(P,Q)\cup R$ that is longer than $P$. This path
determines a path of the same length in $G$, a contradiction that concludes
the proof. ∎
###### Remark 2.15.
Proposition 2.14 characterizes the large class of graphs $BT(P,Q)$, in which
all the ESU’s are NC. When $\\#(V(P)\cap V(Q))$ is small, in most cases
$BT(P,Q)$ is in this class, in particular we will show that this class
includes all $BT(P,Q)$ such that $\\#(V(P)\cap V(Q))\leq 4$.
###### Proposition 2.16.
Let $G$ be a graph and $P$,$Q$ longest paths. If $V(P)\neq V(Q)$ and all the
ESU’s in $BT(P,Q)$ are NC, then $V(P)\cap V(Q)$ is a separator of $G$.
###### Proof.
Since $V(P)\neq V(Q)$, there exists $i_{0}$ such that
$P_{i_{0}}^{\prime}\neq\emptyset$. Every $P_{i}^{\prime}$ is part of an ESU in
$BT(P,Q)$, and by Lemma 2.10 there is a $Q_{j_{0}}^{\prime}\neq\emptyset$ in
that ESU. By Proposition 2.14 these components cannot be connected in
$G\setminus(V(P)\cap V(Q))$, so $V(P)\cap V(Q)$ is a separator of $G$, as
desired. ∎
###### Definition 2.17.
* a)
Let $B$ be an BB in $BT(P,Q)$ and let $X,Y$ be components of different colors
of the corresponding ESU. The pair $X,Y$ is called locally non connectable
(LNC), if for any $x\in X$ and $y\in Y$, there exist two pairs of disjoint
paths in $B$: one pair of disjoint paths, $X_{1}$ from $x$ to one endpoint of
$B$, and $Y_{1}$ from $y$ to another endpoint of $B$; and another pair of
disjoint paths, $X_{2}$ from $x$ to one endpoint of $B$, and $Y_{2}$ from $y$
to another endpoint of $B$, such that
$X_{1}\cup Y_{1}\cup X_{2}\cup Y_{2}=B,$
and such that the intersection of each of $X_{1}$, $X_{2}$, $Y_{1}$, $Y_{2}$
with an embedded ESU is either empty, or is equal to the intersection of the
embedded ESU with $P$ or with $Q$.
$X_{1}$ and $Y_{1}$ thickened$X_{1}$$x$$Y_{1}$$y$$a$$b$$a$$b$$X_{2}$ and
$Y_{2}$ in blue$Y_{2}$$x$$X_{2}$$y$
* b)
The ESU of a building block $B$ is called LNC, if all the pairs of component
of the ESU of different colors are LNC. The building block $B$ is called LNC,
if its ESU and the ESU’s of all embedded IBB’s are LNC.
###### Remark 2.18.
Note that if one of the paths of the pair $X_{1}$, $Y_{1}$ has an edge
contained in one of the embedded IBB’s, then it goes from one endpoint of the
IBB to the other, since by definition an IBB can touch the rest of $BT(P,Q)$
only at its endpoints. Consequently, the other path in the pair cannot touch
this embedded IBB. The same holds for the pair $X_{2},Y_{2}$. Since the union
is the whole block, each embedded IBB has to be travelled twice, and so in
each of the two pairs one of the paths has to go through the given embedded
IBB.
###### Proposition 2.19.
If a pair of completed components of different colors in an ESU is LNC, then
it is NC. Consequently, if an ESU is LNC, then it is NC.
###### Proof.
Take a pair of components of different colors $X=P_{i}^{\prime}$ and
$Y=Q_{j}^{\prime}$ of a given ESU. Assume that there exists a path $R$ that
connects $X$ and $Y$, such that $R$ is internally disjoint from the given ESU
and in each of the other ESU’s touches at most one color. Then $R$ is
internally disjoint from $\widetilde{Q}$, which is the path obtained from $Q$
by swapping the colors in the ESU’s where $R$ touches only the color of $Q$.
Let $x$, $y$ be the endpoints of $R$ in $P_{i}$ and $Q_{j}$ respectively, and
let $X_{1},X_{2},Y_{1},Y_{2}$ be as in Definition 2.17(a).
Let $\widetilde{X}_{1}$ be the path obtained from $X_{1}$ by swapping the
ESU’s of the embedded IBB’s, where $R$ touches $X_{1}$. This means that if $R$
touches $X_{1}$ in the ESU of an embedded block in a certain color, then,
since all the completed components of $X_{1}$ in this ESU have this one color,
we can replace these completed components of $X_{1}$ with the completed
components of the ESU of the other color, which are not touched by $R$, and we
have still a path. Similarly we define $\widetilde{X}_{2}$,
$\widetilde{Y}_{1}$ and $\widetilde{Y}_{2}$. By Remark 2.18, these are paths
that have the same endpoints and the same length as the original ones,
$\widetilde{X}_{1}$ and $\widetilde{Y}_{1}$ are disjoint, and
$\widetilde{X}_{2}$ and $\widetilde{Y}_{2}$ are disjoint. Hence
$\widehat{P}=\widetilde{X}_{1}+R+\widetilde{Y}_{1}\quad\text{and}\quad\widehat{Q}=\widetilde{Y}_{2}+R+\widetilde{X}_{2},$
are two paths such that
$L(\widehat{P})+L(\widehat{Q})=2L(B)+2L(R),$
and such that the set of the endpoints of the paths coincides with the set of
the endpoints of $B$. If the building block $B$ is all of $BT(P,Q)$, then this
finishes the proof, since then $L(P)=L(B)$. If $B$ is an EBB or an IBB, then
we extend $\widehat{P}$ and $\widehat{Q}$ using $\widetilde{Q}$. For this we
write $\widetilde{Q}=\widetilde{Q}_{1}+\widetilde{Q}_{2}+\widetilde{Q}_{3}$,
where $\widetilde{Q}_{2}$ is the intersection of $\widetilde{Q}$ with the
given block. Then we set
$\widehat{\widehat{P}\,}:=\widetilde{Q}_{1}+\widehat{P}+\widetilde{Q}_{3}\quad\text{and}\quad\widehat{\widehat{Q}\,}:=\widetilde{Q}_{1}+\widehat{Q}+\widetilde{Q}_{3},$
in order to obtain two paths $\widehat{\widehat{P}\,}$ and
$\widehat{\widehat{Q}\,}$ such that
$L(\widehat{\widehat{P}\,})+L(\widehat{\widehat{Q}\,})=2L(P)+2L(R),$
which concludes the proof. ∎
###### Theorem 2.20.
Let $G$ be a graph and let $P$, $Q$ be two longest paths. If $V(P)\neq V(Q)$
and all the ESU’s in $BT(P,Q)$ are LNC, then $V(P)\cap V(Q)$ is a separator of
$G$.
###### Proof.
By Propositions 2.16 and 2.19. ∎
Our next goal is to construct recursively new LNC building blocks out of some
given LNC building blocks. For this we first generalize Lemmas 2.3 and 2.4.
###### Proposition 2.21.
If two completed components $X,Y$ of different colors in an ESU are adjacent,
then they are LNC.
###### Proof.
Let one of the subpaths spanning the block be
$\widetilde{P}_{1}+X+\widetilde{P}_{2}$ and the other
$\widetilde{Q}_{1}+Y+\widetilde{Q}_{2}$, where one common vertex is
$u=X\cap\widetilde{P}_{2}=Y\cap\widetilde{Q}_{1}.$
Let $x\in X$ and $y\in Y$ be internal vertices, and write
$X=\widetilde{X}_{1}+\widetilde{X}_{2}$, with
$x=\widetilde{X}_{1}\cap\widetilde{X}_{2}$ and similarly
$Y=\widetilde{Y}_{1}+\widetilde{Y}_{2}$. Then
$X_{1}=\widetilde{P}_{1}+\widetilde{X}_{1},\quad
X_{2}=\widetilde{Q}_{1}+\widetilde{X}_{2},\quad
Y_{1}=\widetilde{P}_{2}+\widetilde{Y}_{1}\quad\text{and}\quad
Y_{2}=\widetilde{Q}_{2}+\widetilde{Y}_{2},$
$\widetilde{X}_{1}$$\widetilde{X}_{2}$$\widetilde{Y}_{2}$$\widetilde{Y}_{1}$$\widetilde{P}_{1}$$\widetilde{Q}_{2}$$\widetilde{P}_{2}$$\widetilde{Q}_{1}$$x$$y$$u$$X=\widetilde{X}_{1}+\widetilde{X}_{2}$
and $Y=\widetilde{Y}_{1}+\widetilde{Y}_{2}$ are
adjacent$X_{1}$$Y_{2}$$Y_{1}$$X_{2}$$x$$y$$X_{1}$ and $Y_{1}$ in black,
$X_{2}$ and $Y_{2}$ in blue
satisfy the conditions of Definition 2.17. Note that if the common vertex is
an end point of the building block, then $\widetilde{P}_{2}$ and
$\widetilde{Q}_{1}$ have length zero. ∎
###### Proposition 2.22.
If two extremal components of $P$ and $Q$ are disjoint and in the same BB,
then they are LNC.
###### Proof.
The proof is similar to the proof of Lemma 2.4. Let
$\widetilde{P}+\widetilde{X}_{1}+\widetilde{X}_{2}\quad\text{and}\quad\widetilde{Q}+\widetilde{Y}_{2}+\widetilde{Y}_{1}$
$\widetilde{X}_{1}$$\widetilde{X}_{2}$$\widetilde{Y}_{2}$$\widetilde{Y}_{1}$$\widetilde{P}$$\widetilde{Q}$
be two paths spanning the BB, such that
$X=\widetilde{X}_{1}+\widetilde{X}_{2}$ and
$Y=\widetilde{Y}_{1}+\widetilde{Y}_{2}$ are the disjoint extremal components.
Then
$X_{1}=\widetilde{P}+\widetilde{X}_{1},\quad X_{2}=\widetilde{X}_{2},\quad
Y_{1}=\widetilde{Y}_{1}\quad\text{and}\quad
Y_{2}=\widetilde{Q}+\widetilde{Y}_{2}$
satisfy the conditions of Definition 2.17. ∎
###### Definition 2.23.
A locally non connectable internal building unit (LNC IBU) is a LNC IBB, or
the concatenation of LNC IBB’s.
###### Proposition 2.24.
The following four constructions yield LNC blocks, starting from some given
LNC IBU’s.
1. (1)
Let $B^{\prime}$ be a LNC IBU with endpoints $d$ and $e$, as represented in
the figure. We can embed $B^{\prime}$ as in the figure and obtain a LNC IBB
$B$ with endpoints $a$ and $b$. The four new subpaths $W$ from $a$ to $e$, $X$
from $d$ to $b$, $Y$ from $a$ to $d$ and $Z$ from $e$ to $b$ with the given
colors are the ESU of the new building block. Moreover,
$L(W)+L(X)=L(Y)+L(Z).$
$e$$d$The LNC IBU $B^{\prime}$$a$$b$$e$$d$$W$$Y$$Z$$X$The new IBB $B$
2. (2)
Let $B^{\prime}$ be a LNC IBU with endpoints $d$ and $e$, as represented in
the figure. We can embed $B^{\prime}$ as in the figure and obtain a LNC EBB
$B$ with common endpoint $c$ and with two other endpoints $a$ and $b$. The
four new subpaths with the given colors are the ESU of the new building block.
$e$$d$The LNC IBU $B^{\prime}$$c$$b$$a$$e$$d$The new EBB $B$
3. (3)
Let $B^{\prime}$ be an a LNC IBU with endpoints $d$ and $e$, as represented in
the figure. We can embed $B^{\prime}$ as in the figure and obtain a LNC IBB
$B$ with endpoints $a$ and $b$. The six new subpaths with the given colors are
the ESU of the new building block.
$d$$e$The LNC IBU $B^{\prime}$ $a$$b$$d$$e$The new IBB $B$
4. (4)
Let $B^{\prime}$ be an a LNC IBU with endpoints $d$ and $e$, as represented in
the figure. We can embed $B^{\prime}$ as in the figure and obtain a LNC EBB
$B$ with common endpoint $c$ and with two other endpoints $a$ and $b$. The six
new subpaths with the given colors are the ESU of the new building block.
$d$$e$The LNC IBU $B^{\prime}$ $c$$a$$b$$d$$e$The new EBB $B$
###### Proof.
(1) By assumption all the ESU’s in $B^{\prime}$ are LNC. All the pairs of
different colors in the new ESU are adjacent, so Proposition 2.21 concludes
the proof in this case.
(2) By assumption, all the ESU’s in $B^{\prime}$ are LNC. On the other hand
all the pairs of different colors of the new components are either adjacent or
both extremal, so Propositions 2.21 and 2.22 conclude the proof in this case.
(3) By assumption, all the ESU’s in $B^{\prime}$ are LNC. On the other hand
all the pairs of the new components of different color except one, are
adjacent, so Propositions 2.21 proves that they are LNC. The remaining pair is
proven to be LNC by the paths $X_{1},X_{2},Y_{1},Y_{2}$ in the following
diagrams, that satisfy the conditions of Definition 2.17.
$a$$b$$d$$e$The paths $X_{1}$ and $Y_{1}$$X_{1}$$x$$Y_{1}$$y$$a$$b$$d$$e$The
paths $X_{2}$ and $Y_{2}$ $Y_{2}$$x$$X_{2}$$y$
(4) By assumption, the ESU’s in $B^{\prime}$ are LNC. On the other hand all
the pairs of the new components of different color except one, are adjacent or
both extremal, so Propositions 2.21 and 2.22 prove that they are LNC. The
remaining pair is proven to be LNC by the paths in the following diagrams.
$c$$b$$a$$d$$e$The paths $X_{1}$ and
$Y_{1}$$X_{1}$$x$$Y_{1}$$y$$c$$b$$a$$d$$e$The paths $X_{2}$ and $Y_{2}$
$Y_{2}$$x$$X_{2}$$y$
∎
## 3 Bi-traceable graphs with $\\#(V(P)\cap V(Q))\leq 4$
In this section we will prove in Theorem 3.4 that if $G$ is a simple graph,
$P,Q$ are longest paths and $\\#V(P)\cap V(Q)\leq 4$, then $BT(P,Q)$ is either
a concatenation of LNC blocks, or it is totally disconnected (TD), according
to the following definition.
###### Definition 3.1.
$BT(P,Q)$ is totally disconnected (TD), if all pairs of components are NDC.
###### Remark 3.2.
If $BT(P,Q)$ is TD and $V(P)\neq V(Q)$, then $V(P)\cap V(Q)$ is a separator.
Clearly, if $\ell=\\#(V(P)\cap V(Q))=1$, then $BT(P,Q)$ is TD. When $\ell=2$,
then the representation of the resulting BT-graph is a concatenation of LNC
blocks.
When $\ell=3$, then there are two different representation types for
$BT(P,Q)$. Either it is the concatenation of two elementary IBB’s and
eventually up to two elementary EBB’s, or it is the concatenation of an
elementary EBB and an EBB as in Proposition 2.24(2). For this we write
$(i,j,k)$ for the permutation $\sigma$ with $\sigma(1)=i$, $\sigma(2)=j$ and
$\sigma(3)=k$, and similarly for $\ell>3$. Consider the equivalence relation
generated by $\sigma\sim\sigma^{-1}$ and $\sigma\sim\sigma^{\bot}$, where
$\sigma^{\bot}(j)=\sigma(\ell-j)$.
We will draw generic representations for each equivalence class of
permutations for $\ell=3,4,5$. This means that in each completed component the
number of vertices is left open. Note that two equivalent permutations
$(i_{1},j_{1},k_{1})\sim(i_{2},j_{2},k_{2})$ have the same generic
representation.
For the identity permutation $(1,2,3)$ we have
$(1,2,3)\sim(3,2,1)\text{ and the (generic) representation of $BT(P,Q)$ is \ \
}{\vbox{\hbox{ \leavevmode\hbox to51.62pt{\vbox
to14.06pt{\pgfpicture\makeatletter\hbox{\hskip-8.3359pt\lower 10.04306pt\hbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{0,1,0}{}\pgfsys@moveto{34.14365pt}{17.07182pt}\pgfsys@curveto{42.67957pt}{23.90057pt}{42.67957pt}{23.90057pt}{51.21548pt}{17.07182pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{17.07182pt}{17.07182pt}\pgfsys@moveto{17.67183pt}{17.07182pt}\pgfsys@curveto{17.67183pt}{17.4032pt}{17.4032pt}{17.67183pt}{17.07182pt}{17.67183pt}\pgfsys@curveto{16.74045pt}{17.67183pt}{16.47182pt}{17.4032pt}{16.47182pt}{17.07182pt}\pgfsys@curveto{16.47182pt}{16.74045pt}{16.74045pt}{16.47182pt}{17.07182pt}{16.47182pt}\pgfsys@curveto{17.4032pt}{16.47182pt}{17.67183pt}{16.74045pt}{17.67183pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{17.07182pt}{17.07182pt}\pgfsys@moveto{34.14365pt}{17.07182pt}\pgfsys@moveto{34.74365pt}{17.07182pt}\pgfsys@curveto{34.74365pt}{17.4032pt}{34.47502pt}{17.67183pt}{34.14365pt}{17.67183pt}\pgfsys@curveto{33.81227pt}{17.67183pt}{33.54364pt}{17.4032pt}{33.54364pt}{17.07182pt}\pgfsys@curveto{33.54364pt}{16.74045pt}{33.81227pt}{16.47182pt}{34.14365pt}{16.47182pt}\pgfsys@curveto{34.47502pt}{16.47182pt}{34.74365pt}{16.74045pt}{34.74365pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{34.14365pt}{17.07182pt}\pgfsys@moveto{51.21548pt}{17.07182pt}\pgfsys@moveto{51.81549pt}{17.07182pt}\pgfsys@curveto{51.81549pt}{17.4032pt}{51.54686pt}{17.67183pt}{51.21548pt}{17.67183pt}\pgfsys@curveto{50.88411pt}{17.67183pt}{50.61548pt}{17.4032pt}{50.61548pt}{17.07182pt}\pgfsys@curveto{50.61548pt}{16.74045pt}{50.88411pt}{16.47182pt}{51.21548pt}{16.47182pt}\pgfsys@curveto{51.54686pt}{16.47182pt}{51.81549pt}{16.74045pt}{51.81549pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{51.21548pt}{17.07182pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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}\pgfsys<EMAIL_ADDRESS>
For the remaining permutations we have
$(1,3,2)\sim(2,1,3)\sim(2,3,1)\sim(3,1,2)\text{ and the representation of the
BT-graph is \ \ }{\vbox{\hbox{ \leavevmode\hbox to34.54pt{\vbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
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}\definecolor[named]{pgffillcolor}{rgb}{0,1,0}{}\pgfsys@moveto{25.60774pt}{22.19342pt}\pgfsys@curveto{28.16853pt}{22.19342pt}{28.16853pt}{22.19342pt}{34.14365pt}{17.07182pt}\pgfsys@stroke\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
### The case $\ell=4$
Note that if $\ell=4$, then $\sigma^{\bot}=(4,3,2,1)\circ\sigma$, where
$(4,3,2,1)$ corresponds to $(14)(23)$ in the standard form, so our equivalence
relation is the same as in [CCP]. We have seven classes, which coincide with
the cases in [CCP], and are listed in the following table.
Case | Permutations | Representation | Conn.
---|---|---|---
1. | $(1,2,3,4),(4,3,2,1)$ | | LNC
2. | $(1,2,4,3),(3,4,2,1),(4,3,1,2),(2,1,3,4)$ | | LNC
3. | $(1,3,2,4),(4,2,3,1)$ | | LNC
4. | $\begin{array}[]{c}(1,3,4,2),(1,4,2,3),(2,3,1,4),(2,4,3,1),\\\ (3,1,2,4),(3,2,4,1),(4,1,3,2),(4,2,1,3)\end{array}$ | | LNC
5. | $(1,4,3,2),(2,3,4,1),(4,1,2,3),(3,2,1,4)$ | | LNC
6. | $(2,1,4,3),(3,4,1,2)$ | | LNC
7. | $(2,4,1,3),(3,1,4,2)$ | | TD
###### Theorem 3.3.
If $G$ is a simple graph, $P,Q$ are longest paths and $\\#V(P)\cap V(Q)\leq
4$, then $BT(P,Q)$ is either a concatenation of LNC blocks, or it is TD.
###### Proof.
For $\ell=1$ and $\ell=2$, we know by the discussion after Definition 3.1 that
$BT(P,Q)$ is TD. For $\ell=3$ and for the first five subcases of the case
$\ell=4$, $BT(P,Q)$ is the concatenation of LNC blocks. In fact, the first
case for $\ell=3$ and the first case for $\ell=4$ is a concatenation of
elementary building blocks which are LNC (for example by Proposition 2.21).
The second case for $\ell=3$ and the cases 2. and 5. for $\ell=4$ have
additional blocks corresponding to Proposition 2.24(2). The cases 3. and 4.
for $\ell=4$ have additional blocks corresponding to items (1) and (4) of
Proposition 2.24. So it remains to prove that for $\ell=4$ in Case 6.
$BT(P,Q)$ is LNC and in Case 7. it is TD. In the sixth case there is one BB,
with two embedded elementary IBB’s and an ESU with six completed components.
By Proposition 2.21, the two elementary ESU’s are LNC.
ESU with six edges$x$$y$$X_{1}$$Y_{1}$$Y_{2}$$X_{2}$$X_{1}$ and $Y_{1}$ in
black, $X_{2}$ and $Y_{2}$ in blue
In the ESU with six completed components all the pairs of the components of
different color except one, are adjacent or both extremal, so by Propositions
2.21 and 2.22 they are LNC. The remaining pair is proven to be LNC by the
paths in the second diagram.
Finally we prove that $BT(P,Q)$ in Case 7 is TD. We claim that adjacent
components are NDC, and that any pair of extremal components is also NDC. In
fact, since the graph is symmetric, we can assume that these pairs are of
different colors and so Lemmas 2.3 and 2.4 prove the claim. Again by symmetry,
we are left with two cases:
1. a)
Either one component is extremal and the other is not adjacent and internal,
or
2. b)
both components are internal and not adjacent.
The following diagrams show that in both cases the pairs are NDC.
Case a), with $\widehat{P}$ in green, $\widehat{Q}$ in redCase b), with
$\widehat{P}$ in green, $\widehat{Q}$ in red
This concludes Case 7 and thus finishes the proof. ∎
###### Corollary 3.4.
If $G$ is a simple graph, $P,Q$ are longest paths, $V(P)\neq V(Q)$ and
$\\#V(P)\cap V(Q)\leq 4$, then $V(P)\cap V(Q)$ is a separator.
###### Proof.
By Theorem 3.3 and Proposition 2.19 in the first six cases all the ESU’s in
$BT(P,Q)$ are NC. Hence, Proposition 2.16 implies that $V(P)\cap V(Q)$ is a
separator, as desired. Case 7 follows from Remark 3.2. ∎
###### Corollary 3.5.
Assume that $P$ and $Q$ are two longest paths in a $5$-connected simple graph
$G$. Then $\\#(V(P)\cap V(Q))\geq 5$.
###### Proof.
We know that $\\#V(P)\geq 5$, so we can assume that $V(P)\neq V(Q)$. Since a
$5$-connected graph is also $4$-connected, by [G]*Theorem 4.2 we know that
$\\#(V(P)\cap V(Q))\geq 4$. Assume by contradiction that $\\#(V(P)\cap
V(Q))<5$, i.e., that $\\#(V(P)\cap V(Q))=4$. Since $G$ is $5$-connected, the
complement of $V(P)\cap V(Q)$ is connected, which contradicts the fact that by
Corollary 3.4 $V(P)\cap V(Q)$ is a separator. This contradiction concludes the
proof. ∎
###### Proposition 3.6.
Let $P,Q,R$ be three longest paths in a graph $G$. If all the ESU’s in
$BT(P,Q)$ are NC, or $BT(P,Q)$ is TD, then the intersection of the three
longest paths $V(P)\cap V(Q)\cap V(R)$ is not empty.
###### Proof.
On one hand, $R$ must touch in at least one ESU the components of different
colors, since otherwise we could swap the colors conveniently and obtain a
longest path $\widetilde{Q}$ such that $\widetilde{Q}\cap R=\emptyset$, which
is impossible. On the other hand, it is impossible that $R$ touches two
components of different colors in an ESU. In fact, if all ESU’s are NC, then
this follows from Proposition 2.14, and if $BT(P,Q)$ is TD, then $R$ can touch
only one component of $BT(P,Q)$. ∎
###### Corollary 3.7.
If the intersection of three longest paths $P,Q,R$ is empty, then
$\\#(V(P)\cap V(Q))\geq 5$.
###### Proof.
If $\\#(V(P)\cap V(Q))<5$, then by Theorem 3.3 we know that in $BT(P,Q)$ all
ESU’s are NC, or $BT(P,Q)$ is TD. Proposition 3.6 concludes the proof. ∎
###### Remark 3.8.
The following representation of a BT-graph with $\ell=V(P)\cap V(Q)=5$ shows
that the method for $\ell\leq 4$ cannot be carried over to this case, since
the highlighted ESU is not NC. In fact, the blue path $R$ connects two
components of different colors in the same highlighted ESU, but cannot be
completed to two paths whose lengths sum $2L(R)+2L(P)$, contradicting
Definition 2.12.
$R$
However, we will prove in Corollary 5.3, that this graph is not a
counterexample to the Hippchen conjecture. We will even prove that in this
graph $V(P)\cap V(Q)$ is a separator in Theorem 5.7.
## 4 Bi-traceable graphs with $\\#(V(P)\cap V(Q))=5$
In this section we will prove in Theorem 4.2 that if $G$ is a simple graph,
$P,Q$ are longest paths and $\\#V(P)\cap V(Q)=5$, then $BT(P,Q)$ is either a
concatenation of LNC building blocks, or it is TD, or it is one of the three
cases 6., 13. or 14. in the following table.
Case | Permutations | Representation | Conn.
---|---|---|---
1. | $(1,2,3,4,5),(5,4,3,2,1)$ | | LNC
2. | $(1,2,3,5,4),(4,5,3,2,1),(2,1,3,4,5),(5,4,3,1,2)$ | | LNC
3. | $(1,2,4,3,5),(5,3,4,2,1),(1,3,2,4,5),(5,4,2,3,1)$ | | LNC
4. | $\begin{array}[]{c}(1,2,4,5,3),(3,5,4,2,1),(1,2,5,3,4),(4,3,5,2,1),\\\ (3,1,2,4,5),(5,4,2,1,3),(2,3,1,4,5),(5,4,1,3,2)\end{array}$ | | LNC
5. | $(1,2,5,4,3),(3,4,5,2,1),(3,2,1,4,5),(5,4,1,2,3)$ | | LNC
6. | $(1,3,2,5,4),(4,5,2,3,1),(2,1,4,3,5),(5,3,4,1,2)$ | | –
7. | $(1,3,4,2,5),(5,2,4,3,1),(1,4,2,3,5),(5,3,2,4,1)$ | | LNC
8. | $\begin{array}[]{c}(1,3,4,5,2),(2,5,4,3,1),(4,1,2,3,5),(5,3,2,1,4),\\\ (1,5,2,3,4),(4,3,2,5,1),(2,3,4,1,5),(5,1,4,3,2)\end{array}$ | | LNC
9. | $\begin{array}[]{c}(1,3,5,2,4),(4,2,5,3,1),(2,4,1,3,5),(5,3,1,4,2),\\\ (1,4,2,5,3),(3,5,2,4,1),(3,1,4,2,5),(5,2,4,1,3)\end{array}$ | | LNC
10. | $\begin{array}[]{c}(1,3,5,4,2),(2,4,5,3,1),(4,2,1,3,5),(5,3,1,2,4),\\\ (1,5,2,4,3),(3,4,2,5,1),(3,2,4,1,5),(5,1,4,2,3)\end{array}$ | | LNC
11. | $(1,4,3,2,5),(5,2,3,4,1)$ | | LNC
12. | $\begin{array}[]{c}(1,4,3,5,2),(2,5,3,4,1),(4,1,3,2,5),(5,2,3,1,4),\\\ (1,5,3,2,4),(4,2,3,5,1),(2,4,3,1,5),(5,1,3,4,2)\end{array}$ | | LNC
13. | $(1,4,5,2,3),(3,2,5,4,1),(3,4,1,2,5),(5,2,1,4,3)$ | | –
14. | $\begin{array}[]{c}(1,4,5,3,2),(2,3,5,4,1),(4,3,1,2,5),(5,2,1,3,4),\\\ (1,5,4,2,3),(3,2,4,5,1),(3,4,2,1,5),(5,1,2,4,3)\end{array}$ | | –
15. | $(1,5,3,4,2),(2,4,3,5,1),(4,2,3,1,5),(5,1,3,2,4)$ | | LNC
Case | Permutations | Representation | Con.
---|---|---|---
16. | $(1,5,4,3,2),(2,3,4,5,1),(4,3,2,1,5),(5,1,2,3,4)$ | | LNC
17. | $(2,1,3,5,4),(4,5,3,1,2)$ | | LNC
18. | $\begin{array}[]{c}(2,1,4,5,3),(3,5,4,1,2),(3,1,2,5,4),(4,5,2,1,3),\\\ (2,1,5,3,4),(4,3,5,1,2),(2,3,1,5,4),(4,5,1,3,2)\end{array}$ | | LNC
19. | $(2,1,5,4,3),(3,4,5,1,2),(3,2,1,5,4),(4,5,1,2,3)$ | | LNC
20. | $\begin{array}[]{c}(2,3,5,1,4),(4,1,5,3,2),(2,5,1,3,4),(4,3,1,5,2),\\\ (4,1,2,5,3),(3,5,2,1,4),(3,1,4,5,2),(2,5,4,1,3)\end{array}$ | | LNC
21. | $(2,4,1,5,3),(3,5,1,4,2),(3,1,5,2,4),(4,2,5,1,3)$ | | TD
22. | $\begin{array}[]{c}(2,4,5,1,3),(3,1,5,4,2),(3,5,1,2,4),(4,2,1,5,3),\\\ (4,1,5,2,3),(3,2,5,1,4),(3,4,1,5,2),(2,5,1,4,3)\end{array}$ | | LNC
23. | $(2,5,3,1,4),(4,1,3,5,2)$ | | TD
Now we will verify that for all cases in the table $BT(P,Q)$ is made of LNC
blocks, when the last entry is LNC, and that it is TD, when the last entry is
TD. We know by Proposition 2.13 that elementary blocks are LNC, so Case 1. is
clear. Cases 2., 5., 15., 16. and 17. follow from Proposition 2.24(2) with
different embedded LNC IBU’s, Cases 3. and 11. from Proposition 2.24(1), Cases
4. and 8. from Proposition 2.24(4), and Case 7. from Proposition 2.24(3).
Case 9.: In this case we have one elementary EBB and one EBB with 10
components and no embedded IBB.
$1$$2$$3$$4$$5$$a$$e$$d$$b$$c$ESU with 10 componentsThe case
1b$x$$y$$X_{2}$$Y_{2}$$Y_{1}$$X_{1}$The case
1e$x$$y$$X_{1}$$X_{2}$$Y_{1}$$Y_{2}$
One verifies directly that in this ESU all the pairs of components of
different colors except seven, are adjacent or both extremal, so Propositions
2.21 and 2.22 prove that they are LNC. The two pairs $1b$ and $1e$ are proven
to be LNC by the paths in the second and third diagrams above, where $X_{1}$
and $Y_{1}$ are in black and $X_{2}$ and $Y_{2}$ are in blue. The remaining
pairs $2e,3c,4a,5a,5d$ are proven to be LNC by the paths in the following
diagrams, where $X_{1}$ and $Y_{1}$ are in black and $X_{2}$ and $Y_{2}$ are
in blue.
The case 2e$x$$y$$X_{1}$$Y_{1}$$X_{2}$$Y_{2}$The case
3c$x$$y$$X_{2}$$X_{1}$$Y_{2}$$Y_{1}$The case
4a$x$$y$$X_{2}$$X_{1}$$Y_{2}$$Y_{1}$
The case 5a$x$$y$$X_{1}$$Y_{1}$$X_{2}$$Y_{2}$The case
5d$x$$y$$X_{1}$$Y_{1}$$X_{2}$$Y_{2}$
###### Remark 4.1.
From now on, when we identify in a graph $X_{1},X_{2},Y_{1},Y_{2}$ and $x,y$
satisfying Definition 2.17, then we will draw the paths $X_{1}$ and $Y_{1}$ in
black, the paths $X_{2}$ and $Y_{2}$ in blue and $x$ and $y$ in red. We don’t
need to write the names in the graph and we won’t do it.
Case 10.: In this case we have one elementary EBB, and one bigger EBB, that
has one embedded IBB and that has an ESU with 8 components:
abcd$1$$2$$3$$4$ESU with 8 components
One verifies directly that in this ESU all the pairs of components of
different colors except four, are adjacent or both extremal, so Propositions
2.21 and 2.22 prove that they are LNC. The remaining pairs $1d,2a,3d,4c$ are
proven to be LNC by the paths in the following diagrams. The paths $X_{1}$ and
$Y_{1}$ are in black and the paths $X_{2}$ and $Y_{2}$ are in blue.
The pair $1d$The pair $2a$The pair $3d$The pair $4c$
Case 12.: In this case we have one elementary EBB, and one bigger EBB, that
has one embedded IBB and that has an ESU with 8 components:
abcd$1$$2$$3$$4$ESU with 8 components
One verifies directly that in this ESU all the pairs of components of
different colors except four, are adjacent or both extremal, so Propositions
2.21 and 2.22 prove that they are LNC. The remaining pairs $1b,2a,3d,4a$ are
proven to be LNC by the paths in the following diagrams. The paths $X_{1}$ and
$Y_{1}$ are in black and the paths $X_{2}$ and $Y_{2}$ are in blue.
The pair 1bThe pair 2aThe pair 3dThe pair 4a
Case 18.: In this case we have one BB with two embedded elementary IBB’s and
an ESU with 8 components:
abcd$1$$2$$3$$4$ESU with 8 components
One verifies directly that in this ESU all the pairs of components of
different colors except four, are adjacent or both extremal, so Propositions
2.21 and 2.22 prove that they are LNC. The remaining pairs $1c,2b,2d,3a$ are
proven to be LNC by the paths in the following diagrams. The paths $X_{1}$ and
$Y_{1}$ are in black and the paths $X_{2}$ and $Y_{2}$ are in blue.
The pair 1cThe pair 2bThe pair 2dThe pair 3a
Case 19.: In this case we have one BB with three embedded elementary IBB’s and
an ESU with 6 components:
ESU with 6 components$X_{1}$ and $Y_{1}$ in black, $X_{2}$ and $Y_{2}$ in blue
One verifies directly that in this ESU all the pairs of components of
different colors except one, are adjacent or both extremal, so Propositions
2.21 and 2.22 prove that they are LNC. The remaining pair is proven to be LNC
by the paths in the second diagram, where the paths $X_{1}$ and $Y_{1}$ are in
black and the paths $X_{2}$ and $Y_{2}$ are in blue.
Case 20.: In this case we have one BB with one embedded elementary IBB and an
ESU with 10 components:
abcde$1$$2$$3$$4$$5$ESU with 10 componentsThe case $1b$The case $2b$
One verifies directly that in this ESU all the pairs of components of
different colors except seven, are adjacent or both extremal, so Propositions
2.21 and 2.22 prove that they are LNC. The two pairs $1b$ and $2b$ are proven
to be LNC by the paths in the second and third diagrams above, where $X_{1}$
and $Y_{1}$ are in black and $X_{2}$ and $Y_{2}$ are in blue. The remaining
pairs $2e,3a,3c,4a,5d$ are proven to be LNC by the paths in the following
diagrams, where $X_{1}$ and $Y_{1}$ are in black and $X_{2}$ and $Y_{2}$ are
in blue.
The case $2e$The case $3a$The case $3c$
The case $4a$The case $5d$
Case 21. and Case 23. In both cases we have the same bi-traceable graph, which
is TD. In order to see this, we first use Lemmas 2.3 and 2.4 and the symmetry
of the graph, in order to verify that adjacent pairs and pairs of extremal
components are NDC. Using again the symmetry of the graph, in order to check
the remaining pairs, it suffices to prove that the five pairs of components
$ab$, $ac$, $ad$, $be$ and $ce$ are NDC, which follows from the paths in the
diagrams below, where $X_{1}$ and $Y_{1}$ are in black and $X_{2}$ and $Y_{2}$
are in blue. In fact, the paths $\widehat{P}$ and $\widehat{Q}$ of Definition
2.2 are given in each case by
$\widehat{P}=X_{1}+R+Y_{1}\quad\text{and}\quad\widehat{Q}=X_{2}+R+Y_{2}.$
abcdeBi-traceable graphThe case $ab$The case $ac$
The case $ad$The case $be$The case $ce$
Case 22.: In this case we have one BB with one embedded elementary IBB and an
ESU with 10 components:
$5$$b$$2$$d$$e$$1$$c$$4$$3$$a$ESU with 10 componentsThe case $1b$The case $2e$
One verifies directly that in this ESU all the pairs of components of
different colors except seven, are adjacent or both extremal, so Propositions
2.21 and 2.22 prove that they are LNC. The two pairs $1b$ and $2e$ are proven
to be LNC by the paths in the second and third diagrams above, where $X_{1}$
and $Y_{1}$ are in black and $X_{2}$ and $Y_{2}$ are in blue. The remaining
pairs $3c,4a,4c,5b,5d$ are proven to be LNC by the paths in the following
diagrams, where $X_{1}$ and $Y_{1}$ are in black and $X_{2}$ and $Y_{2}$ are
in blue.
The case $3c$The case $4a$The case $4c$
The case $5b$The case $5d$
This finishes all the cases in the table, and thus we have proven the
following theorem.
###### Theorem 4.2.
If $G$ is a simple graph, $P,Q$ are longest paths and $\\#V(P)\cap V(Q)=5$,
then $BT(P,Q)$ is either a concatenation of LNC blocks, or it is TD, or it is
one of the three cases 6., 13. or 14. in the table above.
## 5 In the three exceptional cases $V(P)\cap V(Q)$ is a separator
In this section we will prove that none of the three exceptional cases of the
previous section is a counterexample to the Hippchen conjecture, which proves
the Hippchen conjecture for $k=6$. Then we prove that in the three cases
$V(P)\cap V(Q)$ is a separator.
###### Lemma 5.1.
Assume $G$ is $\ell+1$-connected and let
$a_{1},b_{1},P_{0}^{\prime},Q_{0}^{\prime}$ be as in Notation 1.1, in
particular we assume that $\\#(V(P)\cap V(Q)))=\ell$. If $a_{1}=b_{1}$, then
$P_{0}^{\prime}\neq\emptyset$ and $Q_{0}^{\prime}\neq\emptyset$.
###### Proof.
Since $a_{1}=b_{1}$, we can interchange $P_{0}$ and $Q_{0}$, and
$P_{0}^{\prime}=\emptyset$ if and only if $Q_{0}^{\prime}=\emptyset$. Assume
by contradiction that $P_{0}^{\prime}=Q_{0}^{\prime}=\emptyset$, and so
$a_{1}=b_{1}$ is an endpoint of $P$ and of $Q$. Since $G$ is
$\ell+1$-connected, there is an edge connecting $a_{1}$ with a point
$t\not\in\\{a_{2},\dots,a_{\ell}\\}=(V(P)\cap V(Q))$, which we call $a_{1}t$.
If $t\notin V(P)$, then $L(P+a_{1}t)>L(P)$ which contradicts the fact that $P$
is a longest path; and if $t\notin V(Q)$, then $L(Q+a_{1}t)>L(Q)$, which
contradicts the fact that $Q$ is a longest path, concluding the proof. ∎
###### Proposition 5.2.
In the three cases 6. 13. and 14., we can assume $a_{1}=b_{1}$. In any of the
three cases, if $P_{0}^{\prime}\neq\emptyset$ and
$Q_{0}^{\prime}\neq\emptyset$, then $V(P)\cap V(Q)$ is a separator.
###### Proof.
Note that in the three exceptional cases can assume $a_{1}=b_{1}$, changing
the directions of $P$ and/or $Q$, if necessary.
Case 6.: The following diagrams show that $P_{0}^{\prime}$ can connect
directly only with $Q_{3}^{\prime}$ and $P_{3}^{\prime}$, and that
$Q_{3}^{\prime}$ cannot be connected directly with $P_{3}^{\prime}$.
$P_{0}^{\prime}$$Q_{3}^{\prime}$$P_{3}^{\prime}$$P_{0}^{\prime}$$Q_{3}^{\prime}$$P_{3}^{\prime}$$P_{0}^{\prime}$$Q_{3}^{\prime}$$P_{3}^{\prime}$
In fact, the blue and black path show that the connections shown in the
diagrams are impossible. By symmetry of the elementary ESU’s, $P_{0}^{\prime}$
cannot connect with each of the other components of the elementary ESU’s.
Moreover, the adjacent components cannot be connected with $P_{0}^{\prime}$ by
Lemma 2.3 and the extremal components cannot be connected by Lemma 2.4. So
$Q_{3}^{\prime}$ and $P_{3}^{\prime}$ are the only left. By symmetry
$Q_{0}^{\prime}$ can connect directly only with $Q_{3}^{\prime}$ and
$P_{3}^{\prime}$. But $P_{0}^{\prime}$ and $Q_{0}^{\prime}$ form an NC pair,
so they cannot connect to the same component
$X\in\\{P_{3}^{\prime},Q_{3}^{\prime}\\}$. On the other hand, $Q_{3}^{\prime}$
can connect only with $P_{0}^{\prime}$ and $Q_{0}^{\prime}$, since any other
pair that contains $Q_{3}^{\prime}$ but not $P_{3}^{\prime}$, is either
adjacent, or can be discarded by the paths in the following diagrams.
The symmetric argument shows that $P_{3}^{\prime}$ can connect only with
$P_{0}^{\prime}$ and $Q_{0}^{\prime}$. Hence $P_{0}^{\prime}$ and
$Q_{0}^{\prime}$ cannot be connected in $G\setminus(V(P)\cap V(Q))$, which
shows that $V(P)\cap V(Q)$ is a separator, as desired.
Case 13.: The following diagrams show that $P_{0}^{\prime}$ cannot be
connected to any other component.
$P_{0}^{\prime}$$P_{0}^{\prime}$The graph is symmetric
In fact, the first two diagrams show paths that prevent $P_{0}^{\prime}$ to be
connected to two components, and by symmetry also to the other component of
the elementary ESU. The third diagram shows that the graph is symmetric, and
so we can discard three more components. The remaining components are either
adjacent to $P_{0}^{\prime}$ or are extremal, so they cannot be connected to
$P_{0}^{\prime}$ by Lemmas 2.3 and 2.4. Hence there cannot be a path from
$P_{0}^{\prime}$ to $Q_{0}^{\prime}$ in $G\setminus(V(P)\cap V(Q))$, and so
$V(P)\cap V(Q)$ is a separator, as desired.
Case 14.: The following diagrams show that $P_{0}^{\prime}$ cannot be
connected to any other component.
$P_{0}^{\prime}$$P_{0}^{\prime}$$P_{0}^{\prime}$$P_{0}^{\prime}$
In fact, the diagrams show paths that prevent $P_{0}^{\prime}$ to be connected
to four components, and by symmetry also to the other component of the
elementary ESU’s. The remaining components are either adjacent to
$P_{0}^{\prime}$ or are extremal, so they cannot be connected to
$P_{0}^{\prime}$ by Lemmas 2.3 and 2.4. Hence there cannot be a path from
$P_{0}^{\prime}$ to $Q_{0}^{\prime}$ in $G\setminus(V(P)\cap V(Q))$, and so
$V(P)\cap V(Q)$ is a separator, as desired. This finishes the three cases and
concludes the proof. ∎
###### Corollary 5.3.
Assume that $P$ and $Q$ are two longest paths in a $6$-connected simple graph
$G$. Then $\\#(V(P)\cap V(Q))\geq 6$.
###### Proof.
We know that $\\#V(P)\geq 6$, so we can assume that $V(P)\neq V(Q)$. Since a
$6$-connected graph is also $5$-connected, by Corollary 3.5 we know that
$\\#(V(P)\cap V(Q))\geq 5$. Assume by contradiction that $\\#(V(P)\cap
V(Q))<6$, i.e., that $\\#(V(P)\cap V(Q))=5$. Since $G$ is $6$-connected, the
complement of $V(P)\cap V(Q)$ is connected, which contradicts the fact that by
Lemma 5.1 and Proposition 5.2 we know that $V(P)\cap V(Q)$ is a separator.
This contradiction concludes the proof. ∎
Now we prove that in the three exceptional cases $V(P)\cap V(Q)$ is always a
separator.
###### Lemma 5.4.
Let $P_{0}^{\prime}$, $Q_{0}^{\prime}$, $P_{\ell}^{\prime}$ and
$Q_{\ell}^{\prime}$ be as in Notation 1.1, and assume that one of
$P_{0}^{\prime}$, $Q_{0}^{\prime}$, $P_{\ell}^{\prime}$ or $Q_{\ell}^{\prime}$
is empty. Then the two adjacent partial paths of the other longest path have
length $1$.
###### Proof.
Clear, since otherwise one can extend $P$ (respectively $Q$) using the first
part of the adjacent partial path. ∎
We will show that in each of the three cases, if
$P_{0}^{\prime}=Q_{0}^{\prime}=\emptyset$, then $BT(P,Q)$ is weakly
disconnected (WD), according to the following definition.
###### Definition 5.5.
The graph $BT(P,Q)$ is called weakly disconnected (WD) if each pair $X,Y$ of
components of different colors is either NDC, or is weakly non directly
connectable (WNDC), which means that for every path $R$ that connects $X$ with
$Y$, such that $R$ is internally disjoint from $BT(P,Q)$, we can construct a
path $\widehat{P}$ and a cycle $C$ in $BT(P,Q)\cup R$ such that
$L(\widehat{P})+L(C)=2L(R)+L(P)+L(Q)=2L(P)+2L(R).$
Clearly in a WD graph with $V(P)\neq V(Q)$, the set $V(P)\cap V(Q)$ is a
separator, since the length of $\widehat{P}$ and the length of the opened
cycle $\widetilde{C}$ sum $L(\widehat{P})+L(C)-1=2L(P)+2L(R)-1>2L(P)$, which
shows that no pair of components of different colors can be connected.
###### Proposition 5.6.
Consider the three cases 6. 13. and 14. and assume $a_{1}=b_{1}$. If
$P_{0}^{\prime}=Q_{0}^{\prime}=\emptyset$, then $V(P)\cap V(Q)$ is a
separator.
###### Proof.
Case 6.: The paths in the following diagrams show that the graph is TD.
$P_{1}$$P_{5}^{\prime}$$Q_{1}$$P_{1}$$Q_{1}$$P_{5}^{\prime}$$P_{1}$$Q_{1}$$P_{1}$$Q_{1}$
In fact, the two first diagrams show that one cannot connect $P_{5}^{\prime}$
with any other component. Some components are either adjacent, or extremal,
$P_{1}^{\prime}$ and $Q_{1}^{\prime}$ are empty, and for the three remaining
components, the two diagrams show that they cannot be connected. Note that two
of them are components in an elementary ESU, so by symmetry it suffices to
show it for one of them. By symmetry the same holds for $Q_{5}^{\prime}$, and
the remaining pairs that are not adjacent are shown to be NDC in the last two
diagrams. Note that by symmetry it suffices to prove it for one horizontal
connection.
Case 13.: The paths and cycles in the following diagrams show that the graph
is WD.
$P_{5}^{\prime}$Cycle $C$ in black$P_{5}^{\prime}$Cycle $C$ in blackCycle $C$
in blackCycle $C$ in black
In fact, the two first diagrams show that one cannot connect $P_{5}^{\prime}$
with any other component. Some components are either adjacent, or extremal,
$P_{1}^{\prime}$ and $Q_{1}^{\prime}$ are empty, and for the three remaining
components, the two diagrams show that they cannot be connected. Note that two
of them are components in an elementary ESU, so by symmetry it suffices to
show it for one of them. By symmetry the same holds for $Q_{5}^{\prime}$, and
the remaining pairs that are not adjacent are shown to be WNDC in the last two
diagrams. Note that by symmetry it suffices to prove it for one pair of
components of different elementary ESU’s.
Case 14.: The paths and cycles in the following six diagrams show that the
graph is WD.
In fact, one uses the fact that $P_{1}^{\prime}$ and $Q_{1}^{\prime}$ are
empty, discard all adjacent pairs and pairs with two extremal components, and
the remaining pairs are connected in the six diagrams above, where by symmetry
we consider the connection with only one of the components of each elementary
ESU.
Thus in all three exceptional cases $V(P)\cap V(Q)$ is a separator, as
desired. ∎
###### Theorem 5.7.
If $G$ is a simple graph, $P,Q$ are longest paths, $V(P)\neq V(Q)$ and
$\\#V(P)\cap V(Q)\leq 5$, then $V(P)\cap V(Q)$ is a separator.
###### Proof.
If $\\#V(P)\cap V(Q)\leq 4$, then it is true by Corollary 3.4. If $\\#V(P)\cap
V(Q)=5$ and $BT(P,Q)$ is none of the exceptional cases, then by Theorem 4.2
the graph $BT(P,Q)$ is a concatenation of LNC BB, or it is TD. Hence all ESU’s
are NC, and so by Proposition 2.16 and Theorem 2.20 the set $V(P)\cap V(Q)$ is
a separator. Finally, if $BT(P,Q)$ is in one of the exceptional cases, then we
can assume $a_{1}=b_{1}$. If $P_{0}^{\prime}\neq\emptyset$ and
$Q_{0}^{\prime}\neq\emptyset$, then Proposition 5.2 yields the result.
Otherwise necessarily $P_{0}^{\prime}=Q_{0}^{\prime}=\emptyset$, and then
Proposition 5.6 concludes the proof. ∎
Note that Theorem 5.7 implies Corollary 5.3.
###### Corollary 5.8.
Assume that $P$ and $Q$ are two longest paths in a simple graph $G$. If
$V(Q)\neq V(P)$ and $n=|V(G)|\leq 7$ then $V(Q)\cap V(P)$ is a separator.
###### Proof.
Since $V(Q)\neq V(P)$ and $|V(G)|\leq 7$, it follows that $\\#(V(P)\cap
V(Q))\leq 5$, and the result follows from the previous theorem. ∎
## 6 Three longest paths in the exceptional cases
In this section we will show that in none of the three exceptional cases 6.
13. and 14., we can have three disjoint longest paths. In order to do this, we
will analyze the number of times and the sequential order in which a third
longest path $R$ intersects the components of $BT(P,Q)$. We will show that in
the three cases there is only one pair of components of different colors in
$BT(P,Q)$ connected directly by $R$, and in the next lemma we prove that this
is impossible.
###### Lemma 6.1.
Assume $P,Q,R$ are disjoint longest paths. Then it is impossible that there is
only one pair of components of different colors in $BT(P,Q)$ connected
directly by $R$.
###### Proof.
We will prove the following three statements.
1. (1)
Given two components of different colors in $BT(P,Q)$, there cannot be two
disjoint subpaths of $R$ joining directly one component with the other.
2. (2)
If there is only one pair of components of different colors that are connected
directly by $R$, then there is only one subpath of $R$ connecting directly $P$
and $Q$.
3. (3)
It is impossible that there is only one subpath of $R$ connecting directly $P$
and $Q$.
Clearly the lemma follows from statements (2) and (3).
(1) The corresponding result for cycles is well known (See for example
[Ch]*p.145, Claim 1). Assume by contradiction that $R_{1}$ has one endpoint
$x_{1}$ in $P$ and the other endpoint $y_{1}$ in $Q$, and $R_{2}$ has one
endpoint $x_{2}$ in $P$ and the other endpoint $y_{2}$ in $Q$. Then
$R_{1}+Q_{[y_{1},y_{2}]}+R_{2}$ is internally disjoint from $P$ and
$R_{1}+P_{[x_{1},x_{2}]}+R_{2}$ is internally disjoint from $Q$. Then
$\widehat{P}=P_{\leq x_{1}}+R_{1}+Q_{[y_{1},y_{2}]}+R_{2}+P_{\geq
x_{2}}\quad\text{and}\quad\widehat{Q}=Q_{\leq
y_{1}}+R_{1}+P_{[x_{1},x_{2}]}+R_{2}+Q_{\geq y_{2}}$
$x_{1}$$x_{2}$$y_{1}$$y_{2}$$P$ in green, $Q$ in red, $R_{1}$, $R_{2}$ in
blue$x_{1}$$x_{2}$$y_{1}$$y_{2}$$\widehat{P}$ in
blue$x_{1}$$x_{2}$$y_{1}$$y_{2}$$\widehat{Q}$ in blue
are two paths that have lengths that sum $L(P)+L(Q)+2L(R_{1})+2L(R_{2})$, a
contradiction that proves the first statement.
(2) By item (1) the only other possibility is that two consecutive subpaths of
$R$ go back and forth between the components of the only pair connected by
$R$. But then $R$ intersects one of the paths $P$ or $Q$ in only one point,
which is impossible, for example by Corollary 3.7 applied to $V(R)\cap V(P)$
or $V(R)\cap V(Q)$.
(3) Assume there is only one subpath $R_{2}$ of $R$ connecting directly $P$
and $Q$. Let the endpoint of that subpath of $R$ in $P$ be $a$ and the other
endpoint in $Q$ be $b$. Then $a$ and $b$ partition $R$ into three subpaths
$R_{1},R_{2},R_{3}$, the point $a$ partitions the path $P$ into $P_{1}$,
$P_{2}$ and $b$ partitions the path $Q$ into $Q_{1}$, $Q_{2}$. Interchanging
if necessary $P$ with $Q$, and $Q_{1}$ with $Q_{2}$, we can assume without
loss of generality that $L(R_{1})\geq L(R_{3})$ and that $L(Q_{1})\geq
L(Q_{2})$. But then the path $R_{1}+R_{2}+Q_{1}$ has length
$L(R_{1})+L(R_{2})+L(Q_{1})>L(R)/2+L(Q)/2=L(Q)=L(R),$
$R_{1}$$R_{2}$$R_{3}$$a$$b$$P_{1}$$P_{2}$$Q_{1}$$Q_{2}$$L(R_{1})\geq L(R_{3})$
and $L(Q_{1})\geq L(Q_{2})$
contradicting that $R$ is a longest path and thus proving statement (3). ∎
###### Proposition 6.2.
If $P,Q,R$ are disjoint longest paths, then $BT(P,Q)$ cannot have the
configuration of Case 6 in the table of section 4.
###### Proof.
We can assume $a_{1}=b_{1}$.
$P_{0}$$Q_{0}$$Q_{1}$$P_{1}$$P_{3}$$Q_{3}$$Q_{5}$$P_{5}$$Q_{4}$$P_{4}$$Q_{2}$$P_{2}$
From the proof of Proposition 5.2 we know that if $R$ touches
$P_{0}^{\prime}$, then it can touch only $P_{3}^{\prime}$ and
$Q_{3}^{\prime}$, and that $P_{3}^{\prime}$ cannot be connected directly with
$Q_{3}^{\prime}$, and that $Q_{3}^{\prime}$ and $P_{3}^{\prime}$ can be
connected only with $P_{0}^{\prime}$ or $Q_{0}^{\prime}$. Since
$P_{0}^{\prime}$ and $Q_{0}^{\prime}$ are NC, if $R$ touches one of
$P_{0}^{\prime}$, $Q_{0}^{\prime}$, $P_{3}^{\prime}$ or $Q_{3}^{\prime}$, then
the only possibilities for the set of components touched by $R$ are four sets
with two elements and two sets with three elements, which are connected in the
following way:
$P_{0}^{\prime}\sim P_{3}^{\prime},\quad Q_{0}^{\prime}\sim
Q_{3}^{\prime},\quad Q_{0}^{\prime}\sim P_{3}^{\prime},\quad
P_{0}^{\prime}\sim Q_{3}^{\prime},\quad P_{3}^{\prime}\sim Q_{0}^{\prime}\sim
Q_{3}^{\prime},\quad\text{or}\quad P_{3}^{\prime}\sim P_{0}^{\prime}\sim
Q_{3}^{\prime}.$
In all cases there is at most one pair of components of different colors
connected by $R$, which is impossible by Lemma 6.1. Hence $R$ cannot touch
$P_{0}^{\prime}$, $Q_{0}^{\prime}$, $P_{3}^{\prime}$ nor $Q_{3}^{\prime}$.
We are searching for pairs of components that can be connected directly by a
subpath of $R$. We have discarded all pairs that contain $P_{0}^{\prime}$,
$Q_{0}^{\prime}$, $P_{3}^{\prime}$ or $Q_{3}^{\prime}$, and we can also
discard all pairs that are adjacent and have different colors, and all pairs
of extremal components of different colors. Using the paths in the following 3
diagrams,
and using that by symmetry, if you cannot connect one component of an
elementary ESU, then you cannot connect the other; we can discard all pairs
except the following four.
$P_{1}^{\prime}\sim P_{5}^{\prime}$$P_{1}^{\prime}\sim
Q_{5}^{\prime}$$Q_{1}^{\prime}\sim Q_{5}^{\prime}$$Q_{1}^{\prime}\sim
P_{5}^{\prime}$
The blue and the black paths in the following diagram, whose lengths sum more
than $2L(P)$, show that $R$ cannot connect simultaneously $P_{1}^{\prime}$
with $Q_{5}^{\prime}$ and $Q_{1}^{\prime}$ with $P_{5}^{\prime}$.
We cannot have $P_{1}^{\prime}\sim Q_{5}^{\prime}$ and $Q_{1}^{\prime}\sim
P_{5}^{\prime}$
Thus there is only one pair of components of different colors in $BT(P,Q)$
connected by $R$, which contradicts Lemma 6.1 and concludes the proof. ∎
###### Proposition 6.3.
If $P,Q,R$ are disjoint longest paths, then $BT(P,Q)$ cannot have the
configuration of Case 13 in the table of section 4.
###### Proof.
We can assume $a_{1}=b_{1}$.
$P_{0}$$Q_{0}$$P_{1}$$Q_{1}$$Q_{3}$$Q_{2}$$Q_{5}$$P_{4}$$P_{5}$$Q_{4}$$P_{2}$$P_{3}$
As in the proof of Proposition 5.2 we know that $P_{0}^{\prime}$ cannot be
connected with any other component, and by symmetry the same holds for
$Q_{0}^{\prime}$. The paths in the following three diagrams show that the
components in the elementary ESU’s cannot be connected to any other component.
Note that we use the symmetry of the graph. We discard the adjacent pairs and
the pair with two extremal components, and are left with 9 possible pairs that
$R$ can connect directly. There are six pairs with the same color
$P_{1}^{\prime}\sim P_{3}^{\prime},\quad P_{1}^{\prime}\sim
P_{5}^{\prime},\quad P_{5}^{\prime}\sim P_{3}^{\prime},\quad
Q_{1}^{\prime}\sim Q_{3}^{\prime},\quad Q_{1}^{\prime}\sim
Q_{5}^{\prime},\quad Q_{5}^{\prime}\sim Q_{3}^{\prime},$
and three pairs with different colors
$P_{1}^{\prime}\sim Q_{5}^{\prime},\quad P_{3}^{\prime}\sim
Q_{3}^{\prime},\quad P_{5}^{\prime}\sim Q_{1}^{\prime}.$
The paths in the following diagrams show that $R$ can connect only one pair of
different colors.
$Q_{1}^{\prime}\sim P_{5}^{\prime}$ and $P_{3}^{\prime}\sim
Q_{3}^{\prime}$$P_{1}^{\prime}\sim Q_{5}^{\prime}$ and $P_{3}^{\prime}\sim
Q_{3}^{\prime}$$P_{1}^{\prime}\sim Q_{5}^{\prime}$ and $P_{5}^{\prime}\sim
Q_{1}^{\prime}$
Thus Lemma 6.1 concludes the proof. ∎
###### Proposition 6.4.
If $P,Q,R$ are disjoint longest paths, then $BT(P,Q)$ cannot have the
configuration of Case 14 in the table of section 4.
###### Proof.
We can assume $a_{1}=b_{1}$.
$Q_{0}$$P_{0}$$P_{1}$$P_{2}$$P_{3}$$P_{4}$$P_{5}$$Q_{1}$$Q_{2}$$Q_{3}$$Q_{4}$$Q_{5}$
As in the proof of Proposition 5.2 we know that $P_{0}^{\prime}$ cannot be
connected with any other component, and by symmetry the same holds for
$Q_{0}^{\prime}$. We discard also the pairs of adjacent components and if both
components are extremal. There are 14 possible connections left, (where we
count the connection to the two components of an elementary ESU only once),
and 8 of them are discarded by the paths in the following diagrams.
So we are left with 6 possible connections.
$P_{2}^{\prime}\sim P_{5}^{\prime}$$P_{3}^{\prime}\sim
Q_{5}^{\prime}$$Q_{3}^{\prime}\sim Q_{5}^{\prime}$
$Q_{1}^{\prime}\sim Q_{4}^{\prime}$$P_{1}^{\prime}\sim
P_{3}^{\prime}$$P_{1}^{\prime}\sim Q_{3}^{\prime}$
Since $P_{2}$ and $Q_{4}$ are the components of an elementary ESU, we also
have the connections $P_{2}^{\prime}\sim Q_{1}^{\prime}$ and
$Q_{4}^{\prime}\sim P_{5}$. So we have two possibilities: either $R$ touches
some of $P_{2}^{\prime},P_{5}^{\prime},Q_{1}^{\prime},Q_{4}^{\prime}$, or $R$
touches some of $P_{1}^{\prime},P_{3}^{\prime},Q_{3}^{\prime},Q_{5}^{\prime}$.
In the first case, since $P_{2}^{\prime}$ and $Q_{4}^{\prime}$ are NC, they
cannot be connected simultaneously by $R$ to the same component, thus we have
two possibilities
$Q_{1}^{\prime}\sim P_{2}^{\prime}\sim P_{5}^{\prime}\quad\text{or}\quad
Q_{1}^{\prime}\sim Q_{4}^{\prime}\sim P_{5}^{\prime}.$
But both cases are impossible by Lemma 6.1.
So $R$ must touch some of
$P_{1}^{\prime},P_{3}^{\prime},Q_{3}^{\prime},Q_{5}^{\prime}$. The paths in
the following diagram show that $R$ cannot connect simultaneously
$P_{1}^{\prime}\sim Q_{3}^{\prime}$ and $P_{3}^{\prime}\sim Q_{5}^{\prime}$.
$P_{1}^{\prime}\sim Q_{3}^{\prime}$ and $P_{3}^{\prime}\sim Q_{5}^{\prime}$
Since $P_{1}$ and $Q_{5}$ are adjacent and $P_{3}$ and $Q_{3}$ are also
adjacent, there is only one pair of components of different colors in
$BT(P,Q)$ connected by $R$, which contradicts Lemma 6.1 and concludes the
proof. ∎
###### Theorem 6.5.
If the intersection of three longest paths $P,Q,R$ is empty, then
$\\#(V(P)\cap V(Q))\geq 6$.
###### Proof.
Assume that the intersection of three longest paths $P,Q,R$ is empty, then
$\\#(V(P)\cap V(Q))\geq 5$ by Corollary 3.7. If $\\#(V(P)\cap V(Q))=5$, then
we know by Theorem 4.2 that in $BT(P,Q)$ all the ESU’s are NC or $BT(P,Q)$ is
TD, or it has the representation type of one of the cases 6., 13 or 14. If all
the ESU’s are NC or $BT(P,Q)$ is TD, then Proposition 3.6 yields a
contradiction. Else Propositions 6.2, 6.3 and 6.4 lead to contradictions and
thus finish the proof. ∎
###### Remark 6.6.
In each of the cases that couldn’t be discarded constructing longer paths in
the three exceptional cases, we proved that $R$ connects only one pair of
components of different colors of $BT(P,Q)$, which is impossible by Lemma 6.1.
It would be interesting to characterize the configurations in which this
strategy works. One could start trying to find all configurations (in which
not all ESU’s are NC), in which $R$ can connect only one pair of components of
different colors of $BT(P,Q)$.
## References
|
remarkRemark hypothesisHypothesis claimClaim Numerical procedure for hybrid
systemsR. Pytlak and D. Suski
# Numerical procedure for optimal control of hybrid systems with sliding
modes, Part II
RadosŁaw Pytlak Faculty of Mathematics and Information Science, Warsaw
University of Technology, 00-665 Warsaw, Poland<EMAIL_ADDRESS>Damian Suski Institute of Automatic Control and Robotics, Warsaw University of
Technology, 02-525 Warsaw, Poland<EMAIL_ADDRESS>
###### Abstract
This paper concerns the numerical procedure for solving hybrid optimal control
problems with sliding modes. A sliding mode is coped with differential-
algebraic equations (DAEs) and that guarantees accurate tracking of the
sliding motion surface. In the second part of the paper we demonstrate the
correspondence between the discrete adjoint equations and the discretized
version of the continuous adjoint equations in the case of system equations
described by DAEs. We show that the discrete adjoint state trajectories
converge to their continuous counterparts. Next, we describe the application
of the proposed procedure to three optimal control problems. The first problem
concerns optimal control of a simple mechanical system with dry friction. The
second problem is related to the planning of a haemodialysis process. The
third problem concerns the optimal steering of a racing car.
###### keywords:
optimal control problems, hybrid systems, sliding modes, implicit Runge–Kutta
method, adjoint equations
65L80, 49M37, 65K10
## 1 Introduction
The second part of the paper continues the description of the numerical
procedure for optimal control problems for hybrid system exhibiting sliding
motion behavior. The procedure is the implementation of the algorithm for
these problems introduced in [9]. In the first part of the paper we
concentrated on describing the efficient way of evaluating reduced gradients
of the optimization problem. We paid the attention to the case the hybrid
system does not enter the sliding mode and its dynamics is described by ODEs.
In the second part we supplement our considerations by analyzing the sliding
motion case.
Let us consider the hybrid system evolving according to the following scheme.
For $t\in\left[t_{0},t_{t}\right]$ the system evolves according to ODEs
(1) $x^{\prime}=f^{1}(x,u),$
at a switching time $t_{t}$ system state reaches the switching surface
(2) $g(x(t_{t}))=0$
and enters the sliding mode. For $t\in\left[t_{t},t_{f}\right]$ the state
trajectory is the solution of DAEs
(3) $\displaystyle x^{\prime}$ $\displaystyle=$ $\displaystyle
f_{F}(x,u)+g_{x}^{T}(x)z=f^{2}(x,z,u)$ (4) $\displaystyle 0$ $\displaystyle=$
$\displaystyle g(x)$
We now aim at finding the reduced gradient formula for the endpoint functional
([10])
(5) $\bar{F}_{0}(u)=\phi(x^{u}(t_{f}))$
with respect to controls. This formula can be found with the help of adjoint
equations. We now recall the appropriate formulation of adjoint equations in
the considered case. The consistent terminal values of the adjoint variables
$\lambda_{f}(t_{f})$ and $\lambda_{g}(t_{f})$ can be found by solving the
following set of equations with respect to the variables
$\lambda_{f}(t_{f}),\lambda_{g}(t_{f}),\nu_{1}$ ([9])
(6) $\displaystyle\phi_{x}^{T}(x(t_{f}))+\lambda_{f}(t_{f})$ $\displaystyle=$
$\displaystyle\nu_{1}g_{x}^{T}(x(t_{f}))$ (7) $\displaystyle 0$
$\displaystyle=$ $\displaystyle g_{x}(x(t_{f}))\lambda_{f}(t_{f})$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left(g_{x}(x(t_{f}))\right)^{\prime}\lambda_{f}(t_{f})-$
$\displaystyle
g_{x}(x(t_{f}))\left(f_{F}\right)_{x}^{T}(x(t_{f}),u(t_{f}))\lambda_{f}(t_{f})-$
$\displaystyle
g_{x}(x(t_{f}))\left(g_{x}^{T}(x(t_{f}))z(t_{f})\right)_{x}^{T}\lambda_{f}(t_{f})+$
$\displaystyle g_{x}(x(t_{f}))g_{x}^{T}(x(t_{f}))\lambda_{g}(t_{f})$
where $\nu_{1}$ is some real number.
Having $\lambda_{f}(t_{f})$ and $\lambda_{g}(t_{f})$ we then solve DAEs (the
meaning of $t_{t}^{-}$ and $t_{t}^{+}$ is explained in [10])
$\displaystyle(\lambda_{f}^{T})^{\prime}(t)$ $\displaystyle=$
$\displaystyle-\lambda_{f}^{T}(t)(f_{F})_{x}(x(t),u(t))$
$\displaystyle-\lambda_{f}^{T}(t)(g_{x}^{T}(x(t))z(t))_{x}+\lambda_{g}^{T}(t)g_{x}(x(t))$
(10) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\lambda_{f}^{T}(t)g_{x}^{T}(x(t)),\ t\in[t_{t}^{+},t_{f}].$
backwards in time. At the transition time $t_{t}$ the adjoint variable
$\lambda_{f}$ undergoes a jump. To calculate the value of
$\lambda_{f}(t_{t}^{-})$ the following system of equations have to be solved
for the variables $\lambda_{f}(t_{t}^{-}),\ \pi$
(11) $\displaystyle\lambda_{f}(t_{t}^{-})$ $\displaystyle=$
$\displaystyle\lambda_{f}(t_{t}^{+})-\pi g_{x}^{T}(x(t_{t}))$
$\displaystyle\lambda_{f}^{T}(t_{t}^{-})f^{1}(x(t_{t}^{-}),u(t_{t}^{-}))$
$\displaystyle=$
$\displaystyle\lambda_{f}^{T}(t_{t}^{+})f_{F}(x(t_{t}^{+}),u(t_{t}^{+}))+$
$\displaystyle\lambda_{f}^{T}(t_{t}^{+})g_{x}^{T}(x(t_{t}^{+}))z(t_{t}^{+})-\lambda_{g}^{T}(t_{t}^{+})g(x(t_{t}^{+})).$
Eventually we solve adjoint ODEs
(13)
$(\lambda_{f}^{T})^{\prime}(t)=-\lambda_{f}^{T}(t)(f^{1})_{x}(x(t),u(t)),\
t\in[t_{0},t_{t}^{-}).$
Now the directional derivative of the functional $\bar{F}_{0}(u)$ with respect
to the control variation $d$ can be calculated from
$\displaystyle\langle\nabla\bar{F}_{0}(u),d\rangle$ $\displaystyle=$
$\displaystyle-\int_{t_{0}}^{t_{t}^{-}}\lambda_{f}^{T}(t)(f^{1})_{u}(x(t),u(t))d(t)dt$
$\displaystyle-\int_{t_{t}^{+}}^{t_{f}}\lambda_{f}^{T}(t)(f_{F})_{u}(x(t),u(t))d(t)dt.$
In the next section (Section 2) we will analyze the discrete time versions of
equations (1)–(10) and the approximation to the second term in (1). The
present analysis complements the analysis given in the first part of the paper
[10] which concerned equations (13) and the first term in (1). In addition in
Section 3 we will examine how the discretization of adjoint equations
influences the jump conditions (1). Section 4 presents the efficiency of the
proposed numerical procedure in solving three optimal control problems.
## 2 Numerical calculation of reduced gradients
Let us consider the control system described by higher index differential-
algebraic equations, we focus on index 2 DAEs in Hessenberg form ([3])
(15) $\displaystyle x^{\prime}(t)$ $\displaystyle=$ $\displaystyle
f(x(t),z(t),u(t)),$ (16) $\displaystyle 0$ $\displaystyle=$ $\displaystyle
g(x(t)),$
where $x\in\mathbb{R}^{n_{x}}$ is a differential state and
$z\in\mathbb{R}^{n_{z}}$ is an algebraic state. The differentiation index of a
system (15)-(16) is 2 provided that the matrix $g_{x}(x)f_{z}(x,z,u)$ is
nonsingular ([3]). To integrate DAEs, the following Runge-Kutta scheme can be
used ([3])
(17) $\displaystyle x_{i}^{\prime}(k+1)$ $\displaystyle=$ $\displaystyle
f\left(x_{i}(k+1),\ z_{i}(k+1),u(k)\right),$ (18) $\displaystyle 0$
$\displaystyle=$ $\displaystyle g\left(x_{i}(k+1)\right),$ (19) $\displaystyle
x_{i}(k+1)$ $\displaystyle=$ $\displaystyle
x(k)+h(k)\sum_{j=1}^{s}a_{ij}x_{j}^{\prime}(k+1),$ (20) $\displaystyle
z_{i}(k+1)$ $\displaystyle=$ $\displaystyle
z(k)+h(k)\sum_{j=1}^{s}a_{ij}z_{j}^{\prime}(k+1)$
for $i=1,...,s$ and
(21) $\displaystyle x(k+1)$ $\displaystyle=$ $\displaystyle
x(k)+h(k)\sum_{i=1}^{s}b_{i}x_{i}^{\prime}(k+1),$ (22) $\displaystyle z(k+1)$
$\displaystyle=$ $\displaystyle
z(k)+h(k)\sum_{i=1}^{s}b_{i}z_{i}^{\prime}(k+1).$
It is possible to get rid of variables $x_{i}^{\prime}(k+1)$ and
$z_{i}^{\prime}(k+1)$ from (17)-(22), provided that the Runge-Kutta matrix
$A=(a_{ij})$ is invertible. In such case, from (20) we have
(23)
$z_{i}^{\prime}(k+1)=\frac{1}{h(k)}\sum_{j=1}^{s}a_{ij}^{-}(z_{j}(k+1)-z(k)),\
i=1,\dots,s,$
where $a_{ij}^{-}$ are the coefficients of the matrix $A^{-1}$. Now the system
(17)-(22) can be rewritten as
(24) $\displaystyle x_{i}(k+1)$ $\displaystyle=$ $\displaystyle
x(k)+h(k)\sum_{j=1}^{s}a_{ij}f\left(x_{j}(k+1),\ z_{j}(k+1),u(k)\right),$ (25)
$\displaystyle 0$ $\displaystyle=$ $\displaystyle g\left(x_{i}(k+1)\right),$
for $i=1,...,s$ and
(26) $\displaystyle x(k+1)$ $\displaystyle=$ $\displaystyle
x(k)+h(k)\sum_{i=1}^{s}b_{i}f\left(x_{i}(k+1),\ z_{i}(k+1),u(k)\right),$
$\displaystyle z(k+1)$ $\displaystyle=$ $\displaystyle
z(k)+\sum_{i=1}^{s}\sum_{j=1}^{s}b_{i}a_{ij}^{-}(z_{j}(k+1)-z(k)).$
$\displaystyle=$ $\displaystyle
z(k)+\sum_{i=1}^{s}b_{i}^{-}(z_{i}(k+1)-z(k)),$
where
(28) $b_{i}^{-}=\sum_{i=1}^{s}b_{j}a^{-}_{ji},\ i=1,\dots,s.$
At each step of the Runge–Kutta scheme the nonlinear system (24)-(25) is first
solved for variables $x_{i}(k+1),z_{i}(k+1)\ i=1,\dots,s$ and then (26)-(2)
are used to calculate $x(k+1)$ and $z(k+1)$. Let us notice that $z(k)$ is
required to calculate $z(k+1)$, but not for solving (24)-(25).
The adjoint equations for the control system (15)-(16) are ([9]):
(29) $\displaystyle\lambda_{f}^{\prime}(t)$ $\displaystyle=$ $\displaystyle-
f_{x}^{T}(x(t),z(t),u(t))\lambda_{f}(t)-g_{x}^{T}(x(t))\lambda_{g}(t),$ (30)
$\displaystyle 0$ $\displaystyle=$ $\displaystyle-
f_{z}^{T}(x(t),z(t),u(t))\lambda_{f}(t).$
For the adjoint DAEs (29)-(30) to be properly defined, the term
$f_{z}^{T}(x,z,u)\lambda_{f}$ should be differentiable. The algebraic state
$z(t)$ and the control $u(t)$ are not continuous, so to keep the
differentiability we assume that $f_{z}^{T}(x(t),z(t),u(t))$ does not depend
neither on $z(t)$ nor on $u(t)$. Let us notice that the adjoint equations
formulated for the sliding motion satisfy that condition. From now we will
consider the adjoint equations of the form
(31) $\displaystyle\lambda_{f}^{\prime}(t)$ $\displaystyle=$ $\displaystyle-
f_{x}^{T}(x(t),z(t),u(t))\lambda_{f}(t)-g_{x}^{T}(x(t))\lambda_{g}(t),$ (32)
$\displaystyle 0$ $\displaystyle=$ $\displaystyle-
f_{z}^{T}(x(t))\lambda_{f}(t).$
The adjoint equations are index 2 DAEs ([9]), so the Runge-Kutta scheme can be
used to numerically integrate them ([3]). The Runge-Kutta scheme for adjoint
equations (31)-(32) is
$\displaystyle\lambda_{fi}(k)=$
$\displaystyle\lambda_{f}(k+1)-\bar{h}(k+1)\sum_{j=1}^{\bar{s}}\bar{a}_{ij}\left[-f_{x}^{T}\left(x\left(\bar{t}_{j}(k)\right),z\left(\bar{t}_{j}(k)\right),\bar{u}(k+1)\right)\lambda_{fj}(k)\right.$
(33) $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\ \
\left.-g_{x}^{T}\left(x\left(\bar{t}_{j}(k)\right)\right)\lambda_{gj}(k)\right]=$
$\displaystyle\lambda_{f}(k+1)+\bar{h}(k+1)\sum_{j=1}^{\bar{s}}\bar{a}_{ij}\left[f_{x}^{T}\left(x\left(\bar{t}_{j}(k)\right),z\left(\bar{t}_{j}(k)\right),\bar{u}(k+1)\right)\lambda_{fj}(k)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\ \
\left.+g_{x}^{T}\left(x\left(\bar{t}_{j}(k)\right)\right)\lambda_{gj}(k)\right],$
(34) $\displaystyle
0=-f^{T}_{z}\left(x\left(\bar{t}_{i}(k)\right)\right)\lambda_{fi}(k)$
for $i=1,\dots,\bar{s}$ and
$\displaystyle\lambda_{f}(k)=$
$\displaystyle\lambda_{f}(k+1)-\bar{h}(k+1)\sum_{i=1}^{\bar{s}}\bar{b}_{i}\left[-f_{x}^{T}\left(x\left(\bar{t}_{i}(k)\right),z\left(\bar{t}_{i}(k)\right),\bar{u}(k+1)\right)\lambda_{fi}(k)\right.$
(35) $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\ \
\left.-g_{x}^{T}\left(x\left(\bar{t}_{i}(k)\right)\right)\lambda_{gi}(k)\right]=$
$\displaystyle\lambda_{f}(k+1)+\bar{h}(k+1)\sum_{i=1}^{\bar{s}}\bar{b}_{i}\left[f_{x}^{T}\left(x\left(\bar{t}_{i}(k)\right),z\left(\bar{t}_{i}(k)\right),\bar{u}(k+1)\right)\lambda_{fi}(k)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\ \
\left.+g_{x}^{T}\left(x\left(\bar{t}_{i}(k)\right)\right)\lambda_{gi}(k)\right],$
$\displaystyle\lambda_{g}(k)=$ (36)
$\displaystyle\lambda_{g}(k+1)+\sum_{i=1}^{\bar{s}}\sum_{j=1}^{\bar{s}}\bar{b}_{i}\bar{a}_{ij}^{-}(\lambda_{gj}(k)-\lambda_{g}(k+1))=$
$\displaystyle\lambda_{g}(k+1)+\sum_{i=1}^{\bar{s}}\bar{b}_{i}^{-}(\lambda_{gi}(k)-\lambda_{g}(k+1)),$
where for the sake of the shorter notation we denote
(37) $\bar{t}_{i}(k)=t(k+1)-\bar{c}_{i}\bar{h}(k+1),\ i=1,\dots,s.$
At each step of the Runge-Kutta scheme, the system of equations (33)-(34) is
first solved for variables $\lambda_{fi}(k),\lambda_{gi}(k),\
i=1,\dots,\bar{s}$ and then (35)-(36) are used to calculate $\lambda_{f}(k)$
and $\lambda_{g}(k)$.
In the case of the sliding motion the differential of the endpoint functional
$\phi(t_{f})$ is of the form
(38)
$d\phi(x(t_{f}))=\int_{t_{0}}^{t_{f}}d(t)^{T}f_{u}^{T}(x(t),u(t))\lambda_{f}(t)dt.$
Two important remarks need to be emphasized. First we assume that $f_{u}(x,u)$
does not depend on $z$. That is another limitation lied on the form of the
function $f(x,z,u)$, but it is satisfied by the sliding motion equations.
Second, the integral depends only on the adjoint variable $\lambda_{f}$, but
not on $\lambda_{g}$. Under that conditions the reduced gradients for DAEs are
calculated the same way as it was for ODEs (see [10])
(39)
$\frac{d\phi(x(t_{f}))}{du_{n}}\simeq\sum_{k=k_{n-1}}^{k_{n}-1}\tilde{h}(k)\sum_{i=1}^{\tilde{s}}\tilde{b}_{i}f_{u}^{T}\left(x\left(\tilde{t}(k)+\tilde{c}_{i}\tilde{h}(k)\right),u_{n}\right)\lambda_{f}\left(\tilde{t}(k)+\tilde{c}_{i}\tilde{h}(k)\right),$
where for the discrete steps $k=k_{n-1},\ldots,k_{n}-1$ the control is
$u_{n}$.
Similarly to the ODEs case presented in [10], we utilize discrete adjoint
equations and discrete reduced gradients to avoid the necessity of calculation
of states and adjoint variables at arbitrary time moments. The Runge-Kutta
scheme (24)-(26) can be rewritten to a vector form (discrete step argument
omitted)
(40)
$\left(\begin{array}[]{c}x_{1}-x-h\sum_{j=1}^{s}a_{1j}f\left(x_{j},z_{j},u\right)\\\
-g\left(x_{1}\right)\\\ \vdots\\\
x_{s}-x-h\sum_{j=1}^{s}a_{sj}f\left(x_{j},z_{j},u\right)\\\
-g\left(x_{s}\right)\\\
x^{+}-x-h\sum_{i=1}^{s}b_{i}f\left(x_{i},z_{i},u\right)\end{array}\right)=\left(\begin{array}[]{c}0\\\
0\\\ \vdots\\\ 0\\\ 0\\\ 0\end{array}\right).$
If we now define the augmented state vector $X(k)$ as
(41)
$X(k)=\left(x_{1}(k)^{T},z_{1}(k)^{T},\ldots,x_{s}(k)^{T},z_{s}(k)^{T},x(k)^{T}\right)^{T},$
then (40) can be presented in a form of the implicit discrete time state
equation [8] (see also [12])
(42) $F\left(X(k+1),X(k),u(k)\right)=0.$
The partial derivatives matrices of the discrete state equation are
(43)
$F_{X^{+}}(k)=\left(\begin{array}[]{ccccccc}I-ha_{11}f_{x1}&-ha_{11}f_{z1}&\ldots&-ha_{1s}f_{xs}&-ha_{1s}f_{zs}&0\\\
-g_{x1}&0&\ldots&0&0&0\\\ \vdots&\vdots&&\vdots&\vdots&\vdots\\\
-ha_{s1}f_{x1}&-ha_{s1}f_{z1}&\ldots&I-ha_{ss}f_{xs}&-ha_{ss}f_{zs}&0\\\
0&0&\ldots&-g_{xs}&0&0\\\
-hb_{1}f_{x1}&-hb_{1}f_{z1}&\ldots&-hb_{s}f_{xs}&-hb_{s}f_{zs}&I\end{array}\right)$
and
(44) $F_{X}(k)=\left(\begin{array}[]{ccccccc}0&0&\ldots&0&0&-I\\\
0&0&\ldots&0&0&0\\\ \vdots&\vdots&&\vdots&\vdots&\vdots\\\
0&0&\ldots&0&0&-I\\\ 0&0&\ldots&0&0&0\\\ 0&0&\ldots&0&0&-I\end{array}\right),$
where for the sake of a shorter notation we denote
(45) $f_{xi}=f_{x}(x_{i},z_{i},u),f_{zi}=f_{z}(x_{i}),g_{xi}=g_{x}(x_{i}).$
Let us denote
$\displaystyle\Lambda(k)$ $\displaystyle=$
$\displaystyle\left(l_{f1}(k)^{T},l_{g1}(k)^{T},\ldots,l_{fs}(k)^{T},l_{gs}(k)^{T},\lambda_{f}(k)^{T}\right)^{T},$
$\displaystyle\Lambda(k+1)$ $\displaystyle=$
$\displaystyle\left(l_{f1}^{+}(k)^{T},l_{g1}^{+}(k)^{T},\ldots,l_{fs}^{+}(k)^{T},l_{gs}^{+}(k)^{T},\lambda_{f}^{+}(k)^{T}\right)^{T},$
$\displaystyle R(k)$ $\displaystyle=$
$\displaystyle\left(r_{f1}(k)^{T},r_{g1}(k)^{T},\ldots,r_{fs}(k)^{T},r_{gs}(k)^{T},r_{f}(k)^{T}\right)^{T}.$
Now the discrete adjoint equations ([8])
(46) $\displaystyle F_{X^{+}}^{T}(k)R(k)$ $\displaystyle=$
$\displaystyle\Lambda(k+1),$ (47) $\displaystyle\Lambda(k)$ $\displaystyle=$
$\displaystyle-F_{X}^{T}(k)R(k),$
at a discrete time step $k$ take the form
(48)
$\left(\begin{array}[]{cccccc}I-ha_{11}f^{T}_{x1}&-g^{T}_{x1}&\ldots&-ha_{s1}f^{T}_{x1}&0&-hb_{1}f_{x1}^{T}\\\
-ha_{11}f^{T}_{z1}&0&\ldots&-ha_{s1}f^{T}_{z1}&0&-hb_{1}f_{z1}^{T}\\\
\vdots&\vdots&&\vdots&\vdots&\vdots\\\
-ha_{1s}f^{T}_{xs}&0&\ldots&I-ha_{ss}f^{T}_{xs}&-g^{T}_{xs}&-hb_{s}f_{xs}^{T}\\\
-ha_{1s}f^{T}_{zs}&0&\ldots&-ha_{ss}f^{T}_{zs}&0&-hb_{s}f_{zs}^{T}\\\
0&0&\ldots&0&0&I\\\ \end{array}\right)\left(\begin{array}[]{c}r_{f1}\\\
r_{g1}\\\ \vdots\\\ r_{fs}\\\ r_{gs}\\\
r_{f}\end{array}\right)=\left(\begin{array}[]{c}l^{+}_{f1}\\\ l^{+}_{g1}\\\
\vdots\\\ l^{+}_{fs}\\\ l^{+}_{gs}\\\ \lambda^{+}_{f}\end{array}\right)$
(49) $\left(\begin{array}[]{c}l_{f1}\\\ l_{g1}\\\ \vdots\\\ l_{fs}\\\
l_{gs}\\\
\lambda_{f}\end{array}\right)=-\left(\begin{array}[]{cccccc}0&0&\ldots&0&0&0\\\
0&0&\ldots&0&0&0\\\ \vdots&\vdots&&\vdots&\vdots&\vdots\\\ 0&0&\ldots&0&0&0\\\
0&0&\ldots&0&0&0\\\
-I&0&\ldots&-I&0&-I\end{array}\right)\left(\begin{array}[]{c}r_{f1}\\\
r_{g1}\\\ \vdots\\\ r_{fs}\\\ r_{gs}\\\ r_{f}\end{array}\right)$
If we rewrite (48) as a system of equations we obtain
(50) $\displaystyle r_{fi}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{s}ha_{ji}f_{xi}^{T}r_{fj}+g_{xi}^{T}r_{gi}+hb_{i}f_{xi}^{T}r_{f}+l^{+}_{fi}$
(51) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{s}ha_{ji}f_{zi}^{T}r_{fj}+hb_{i}f_{zi}^{T}r_{f}+l_{gi}^{+}$
for $i=1,\ldots,s$ and
(52) $r_{f}=\lambda_{f}^{+}$
Using (52), (50)-(51) can be written as
(53) $\displaystyle r_{fi}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{s}ha_{ji}f_{xi}^{T}r_{fj}+g_{xi}^{T}r_{gi}+hb_{i}f_{xi}^{T}\lambda_{f}^{+}+l^{+}_{fi}$
(54) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{s}ha_{ji}f_{zi}^{T}r_{fj}+hb_{i}f_{zi}^{T}\lambda_{f}^{+}+l_{gi}^{+}$
If we rewrite (49) as a system of equations we get
(55) $\displaystyle l_{fi}$ $\displaystyle=$ $\displaystyle 0$ (56)
$\displaystyle l_{gi}$ $\displaystyle=$ $\displaystyle 0$
for $i=1,\ldots,s$ and
(57) $\displaystyle\lambda_{f}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{s}r_{fi}+r_{f}.$
Using (52), (57) gives
(58) $\displaystyle\lambda_{f}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{s}r_{fi}+\lambda_{f}^{+}.$
From (55)-(56) we have that
$l_{f1}(k)=0,l_{g1}(k)=0,\ldots,l_{fs}(k)=0,l_{gs}(k)=0$ for steps
$k=0,...,K-1$. During the analysis we also assume that
$l_{f1}(K)=0,l_{g1}(K)=0,\ldots,l_{fs}(K)=0,l_{gs}(K)=0$. Under that
assumption (53)-(54) is equivalent to
(59) $\displaystyle r_{fi}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{s}ha_{ji}f_{xi}^{T}r_{fj}+g_{xi}^{T}r_{gi}+hb_{i}f_{xi}^{T}\lambda_{f}^{+}$
(60) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{s}ha_{ji}f_{zi}^{T}r_{fj}+hb_{i}f_{zi}^{T}\lambda_{f}^{+}$
Let us now introduce the auxiliary variables
(61) $\displaystyle\lambda_{fi}$ $\displaystyle=$
$\displaystyle\lambda_{f}^{+}+\sum_{j=1}^{s}\frac{a_{ji}}{b_{i}}r_{fj},$ (62)
$\displaystyle\lambda_{gi}$ $\displaystyle=$
$\displaystyle\frac{r_{gi}}{hb_{i}}$
for $i=1,\ldots,s$. (59)-(60) can then be transformed to
(63) $\displaystyle r_{fi}$ $\displaystyle=$ $\displaystyle
hb_{i}\left(f_{xi}^{T}\lambda_{fi}+g_{xi}^{T}\lambda_{gi}\right),$ (64)
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
hb_{i}f_{zi}^{T}\lambda_{fi}.$
If we put (63) into (61) we obtain
(65) $\displaystyle\lambda_{fi}$ $\displaystyle=$
$\displaystyle\lambda_{f}^{+}+h\sum_{j=1}^{s}\frac{a_{ji}b_{j}}{b_{i}}\left(f_{xj}^{T}\lambda_{fj}+g_{xj}^{T}\lambda_{gj}\right).$
Dividing (64) by $hb_{i}$ results in
(66) $\displaystyle 0$ $\displaystyle=$ $\displaystyle
f_{zi}^{T}\lambda_{fi}.$
If we put (63) into (58) we obtain
(67) $\displaystyle\lambda_{f}$ $\displaystyle=$
$\displaystyle\lambda_{f}^{+}+h\sum_{i=1}^{s}b_{i}\left(f_{xi}^{T}\lambda_{fi}+g_{xi}^{T}\lambda_{gi}\right).$
Let us rewrite equations (65)-(67) by introducing the discrete step argument
(68) $\displaystyle\lambda_{fi}(k)$ $\displaystyle=$
$\displaystyle\lambda_{f}(k+1)+h(k)\sum_{j=1}^{s}\frac{a_{ji}b_{j}}{b_{i}}\left[f_{x}^{T}\left(x_{j}\left(k+1\right),z_{j}\left(k+1\right),u(k)\right)\lambda_{fj}(k)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\ \
\left.+g_{x}^{T}\left(x_{i}\left(k+1\right)\right)\lambda_{gj}(k)\right],$
(69) $\displaystyle 0$ $\displaystyle=$ $\displaystyle
f^{T}_{z}\left(x_{i}\left(k+1\right)\right)\lambda_{fi}(k)$
for $i=1,\dots,s$ and
(70) $\displaystyle\lambda_{f}(k)$ $\displaystyle=$
$\displaystyle\lambda_{f}(k+1)+h(k)\sum_{i=1}^{s}b_{i}\left[f_{x}^{T}\left(x_{j}\left(k+1\right),z_{i}\left(k+1\right),u(k)\right)\lambda_{fi}(k)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\ \
\left.+g_{x}^{T}\left(x_{i}\left(k+1\right)\right)\lambda_{gi}(k)\right]$
and augment them by the equation for $\lambda_{g}(k)$
(71) $\displaystyle\lambda_{g}(k)$ $\displaystyle=$
$\displaystyle\lambda_{g}(k+1)+\sum_{i=1}^{s}b_{i}^{-}(\lambda_{gi}(k)-\lambda_{g}(k+1)).$
We now consider the Runge-Kutta scheme (33)-(36) under the assumption that the
discrete steps are the same as in the forward scheme and the Runge-Kutta
scheme coefficients satisfy
$\bar{a}_{ij}=\frac{a_{ji}b_{j}}{b_{i}},\bar{b}_{i}=b_{i},\bar{c}_{i}=1-c_{i}$
(72) $\displaystyle\lambda_{fi}(k)$ $\displaystyle=$
$\displaystyle\lambda_{f}(k+1)+h(k)\sum_{j=1}^{s}\frac{a_{ji}b_{j}}{b_{i}}\left[f_{x}^{T}\left(x\left(t_{j}(k)\right),z\left(t_{j}(k)\right),u(k)\right)\lambda_{fj}(k)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\ \
\left.+g_{x}^{T}\left(x\left(t_{j}(k)\right)\right)\lambda_{gj}(k)\right],$
(73) $\displaystyle 0$ $\displaystyle=$
$\displaystyle-f^{T}_{z}\left(x\left(t_{i}(k)\right)\right)\lambda_{fi}(k)$
for $i=1,\dots,s$ and
(74) $\displaystyle\lambda_{f}(k)$ $\displaystyle=$
$\displaystyle\lambda_{f}(k+1)+h(k)\sum_{i=1}^{s}b_{i}\left[f_{x}^{T}\left(x\left(t_{i}(k)\right),z\left(t_{i}(k)\right),u(k)\right)\lambda_{fi}(k)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\ \
\left.+g_{x}^{T}\left(x\left(t_{i}(k)\right)\right)\lambda_{gi}(k)\right],$
(75) $\displaystyle\lambda_{g}(k)$ $\displaystyle=$
$\displaystyle\lambda_{g}(k+1)+\sum_{i=1}^{s}b_{i}^{-}(\lambda_{gi}(k)-\lambda_{g}(k+1)),$
where
(76) $t_{i}(k)=t(k)+c_{i}h(k).$
Now the Runge-Kutta scheme (72)-(75) for adjoint equations is almost identical
to discrete adjoint equations (68)-(71). The only difference is that in
discrete equations $x_{i}(k+1)$ and $z_{i}(k+1)$ are used instead of
$x(t(k)+c_{i}h(k))$ and $z(t(k)+c_{i}h(k))$. The convergence of (72)-(75) is
guaranteed if only the Runge-Kutta scheme $\bar{a}_{ij},\ \bar{b}_{i},\
\bar{c}_{i},\ i,j=1,\dots,s$ satisfies appropriate conditions. We will also
justify that the usage of $x_{i}(k+1)$ and $z_{i}(k+1)$ instead of
$x(t(k)+c_{i}h(k))$ and $z(t(k)+c_{i}h(k))$ does not destroy the convergence
of (68)-(71) solutions to continuous adjoint trajectory.
The partial derivative $F_{u}(k)$ is
(77) $F_{u}(k)=\left(\begin{array}[]{c}-h\sum_{j=1}^{s}a_{1j}f_{uj}\\\ 0\\\
\vdots\\\ -h\sum_{j=1}^{s}a_{sj}f_{uj}\\\ 0\\\
-h\sum_{i=1}^{s}b_{i}f_{ui}\end{array}\right),$
where for the sake of the shorter notation we omitted the discrete step
argument and introduced
(78) $f_{ui}=f_{u}(x_{i},u).$
Let us now inspect $-F_{u}^{T}(k)R(k)$:
$\displaystyle-F_{u}^{T}(k)R(k)$ $\displaystyle=$
$\displaystyle-\left(\begin{array}[]{c}-h\sum_{j=1}^{s}a_{1j}f_{uj}\\\ 0\\\
\vdots\\\ -h\sum_{j=1}^{s}a_{sj}f_{uj}\\\ 0\\\
-h\sum_{i=1}^{s}b_{i}f_{ui}\end{array}\right)^{T}\left(\begin{array}[]{c}r_{f1}\\\
r_{g1}\\\ \vdots\\\ r_{fs}\\\ r_{gs}\\\ r_{f}\end{array}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{s}h\left(\sum_{j=1}^{s}a_{ij}f^{T}_{uj}\right)r_{fi}+h\sum_{i=1}^{s}b_{i}f^{T}_{ui}r_{f}.$
From (52) and (63) we get
(92) $\displaystyle-F_{u}^{T}(k)R(k)=$
$\displaystyle\sum_{i=1}^{s}h\left(\sum_{j=1}^{s}a_{ij}f^{T}_{uj}\right)hb_{i}\left(f_{xi}^{T}\lambda_{fi}+g_{xi}^{T}\lambda_{gi}\right)+h\sum_{i=1}^{s}b_{i}f^{T}_{ui}\lambda_{f}^{+}=$
$\displaystyle
h\sum_{j=1}^{s}f^{T}_{uj}\left(h\sum_{i=1}^{s}a_{ij}b_{i}\left(f_{xi}^{T}\lambda_{fi}+g_{xi}^{T}\lambda_{gi}\right)\right)+h\sum_{i=1}^{s}b_{i}f^{T}_{ui}\lambda_{f}^{+}=$
$\displaystyle
h\sum_{j=1}^{s}f^{T}_{uj}b_{j}\left(h\sum_{i=1}^{s}\frac{a_{ij}b_{i}}{b_{j}}\left(f_{xi}^{T}\lambda_{fi}+g_{xi}^{T}\lambda_{gi}\right)\right)+h\sum_{i=1}^{s}b_{i}f^{T}_{ui}\lambda_{f}^{+}.$
From (65) we get that
$\displaystyle-F_{u}^{T}(k)R(k)$ $\displaystyle=$ $\displaystyle
h\sum_{j=1}^{s}f^{T}_{uj}b_{j}\left(\lambda_{fj}-\lambda^{+}_{f}\right)+h\sum_{i=1}^{s}b_{i}f^{T}_{ui}\lambda_{f}^{+}$
$\displaystyle=$ $\displaystyle
h\sum_{i=1}^{s}f^{T}_{ui}b_{i}\left(\lambda_{fi}-\lambda^{+}_{f}\right)+h\sum_{i=1}^{s}b_{i}f^{T}_{ui}\lambda_{f}^{+}$
$\displaystyle=$ $\displaystyle h\sum_{i=1}^{s}b_{i}f^{T}_{ui}\lambda_{fi}.$
The discrete reduced gradient now takes the form
(94)
$\frac{\phi(X(K))}{du_{n}}=\sum_{k=k_{n-1}}^{k_{n}-1}h(k)\sum_{i=1}^{s}b_{i}f_{u}^{T}(x_{i}(k+1),u_{n})\lambda_{fi}(k).$
Under the assumption that the discrete steps are the same as in the forward
scheme and the quadrature coefficients satisfy
$\tilde{b}_{i}=b_{i},\tilde{c}_{i}=c_{i}$, the reduced gradients formula (39)
is
(95)
$\frac{d\phi(x(t_{f}))}{du_{n}}\simeq\sum_{k=k_{n-1}}^{k_{n}-1}h(k)\sum_{i=1}^{s}b_{i}f_{u}^{T}\left(x\left(t(k)+c_{i}h(k)\right),u_{n}\right)\lambda_{f}\left(t(k)+c_{i}h(k)\right).$
Now the quadrature scheme (95) for continuous reduced gradients is almost
identical to discrete reduced gradients (94). The only difference is that in
(94) $x_{i}(k+1)$ and $\lambda_{fi}(k)$ are used instead of
$x(t(k)+c_{i}h(k))$ and $\lambda_{f}(t(k)+c_{i}h(k))$. The convergence of (95)
is guaranteed if only the quadrature with coefficients $b_{i},\ c_{i},\
i=1,\dots,s$ satisfies appropriate conditions. We will also justify that the
usage of $x_{i}(k+1)$ and $\lambda_{fi}(k)$ instead of $x(t(k)+c_{i}h(k))$ and
$\lambda_{f}(t(k)+c_{i}h(k))$ does not destroy the convergence of discrete
reduced gradients (94) to continuous reduced gradients.
Now we want to justify the convergence of discrete adjoint trajectories and
discrete reduced gradients to their continuous counterparts. To obtain the
right error order estimated we always assume that the system functions
$f(x,z,u)$ and $g(x)$ and their partial derivatives are Lipschitz continuous
functions. To achieve that goal we use the theorems presented in [3]. That
theorems are formulated under the constant step size assumption,
(96) $h(k)=h,\ k=1,\ldots,K,$
and we also carry out our analysis under that assumption. The variable step
size case can be cumbersome to analyze, and in this section we will make a
short note on that problem.
We assume that the system equations are integrated using RADAU IIA scheme for
$s=3$. The coefficients that appear in discrete adjoint equations
$\bar{a}_{ij}=\frac{a_{ji}b_{j}}{b_{i}},\bar{b}_{i}=b_{i},\bar{c}_{i}=1-c_{i}$
define the RADAU IA scheme (see [10]). The global error of the differential
state $x(t)$ is ([3], Theorem 4.4 p. 36 and Theorem 5.9 p. 67)
(97) $x(k)-x(t(k))=O(h^{p}).$
and the global error of the algebraic state $z(t)$ is ([3], Theorem 4.6 p. 40)
(98) $z(k)-z(t(k))=O(h^{q}).$
where for the RADAU IIA scheme and $s=3$ we have [3]
(99) $p=5,\ q=3.$
We emphasize that in the DAEs case the invertibility of the matrix of
coefficients $A=(a_{ij})$ is essential for convergence of the scheme. Also the
value of so called radius of stability
(100) $R(\infty)=1-b^{T}A^{-1}\mathbf{1}.$
plays an important role. For RADAU IA and RADAU IIA schemes $A$ matrices are
invertible and $R(\infty)=0$, so the assumptions of the appropriate theorems
from [3] are satisfied.
To derive the subsequent results we consider the unperturbed Runge – Kutta
scheme
(101) $\displaystyle x_{i}^{n}(k+1)$ $\displaystyle=$ $\displaystyle
x^{n}(k)+h\sum_{j=1}^{s}a_{ij}f\left(x_{j}^{n}(k+1),\
z_{j}^{n}(k+1),u(k)\right),$ (102) $\displaystyle 0$ $\displaystyle=$
$\displaystyle g\left(x_{i}^{n}(k+1)\right),$
for $i=1,...,s$ and
(103) $\displaystyle x^{n}(k+1)$ $\displaystyle=$ $\displaystyle
x^{n}(k)+h\sum_{i=1}^{s}b_{i}f\left(x_{i}^{n}(k+1),\
z_{i}^{n}(k+1),u(k)\right)$ (104) $\displaystyle z^{n}(k+1)$ $\displaystyle=$
$\displaystyle z^{n}(k)+\sum_{i=1}^{s}b_{i}^{-}(z_{i}^{n}(k+1)-z^{n}(k)).$
and the perturbed Runge-Kutta scheme
(105) $\displaystyle x_{i}^{p}(k+1)$ $\displaystyle=$ $\displaystyle
x^{n}(k)+h\sum_{j=1}^{s}a_{ij}f\left(x_{j}^{p}(k+1),\
z_{j}^{p}(k+1),u(k)\right)+\delta_{i},$ (106) $\displaystyle 0$
$\displaystyle=$ $\displaystyle g\left(x_{i}^{p}(k+1)\right),$
for $i=1,...,s$ and
(107) $\displaystyle x^{p}(k+1)$ $\displaystyle=$ $\displaystyle
x^{n}(k)+h\sum_{i=1}^{s}b_{i}f\left(x_{i}^{p}(k+1),\
z_{i}^{p}(k+1),u(k)\right)+\delta_{s+1}.$ (108) $\displaystyle z^{p}(k+1)$
$\displaystyle=$ $\displaystyle
z^{n}(k)+\sum_{i=1}^{s}b_{i}^{-}(z_{i}^{p}(k+1)-z^{n}(k))+\delta_{s+2}.$
From ([3], Theorem 4.2 p. 33) we have
(109) $\displaystyle x_{i}^{p}(k+1)-x_{i}^{n}(k+1)$ $\displaystyle=$
$\displaystyle O(\delta)$ (110) $\displaystyle z_{i}^{p}(k+1)-z_{i}^{n}(k+1)$
$\displaystyle=$ $\displaystyle\frac{1}{h}O(\delta)$
where $\delta=max\\{\delta_{1},\dots,\delta_{s}\\}$. Contrary to the ODEs
case, using the variable step size for integration of DAEs may be cumbersome,
because of the factor $\frac{1}{h}$ that occurs in (110). The estimate (110)
indicates that using very small step sizes may lead to big errors. On the
other hand, the results for the constant step size can be extended to the
variable step size case if we assume that the following condition holds
(111) $h_{m}\geq h(k)\geq\mu h_{m},\ k=1,\ldots,K$
where $h_{m}$ is the maximum step size and $\mu\in(0,1)$ is a constant
independent from $h_{m}$.
The main step errors can be estimated as
(112) $\displaystyle x^{p}(k+1)-x^{n}(k+1)$ $\displaystyle=$ $\displaystyle
O(\delta)+O(\delta_{s+1})$ (113) $\displaystyle z^{p}(k+1)-z^{n}(k+1)$
$\displaystyle=$ $\displaystyle\frac{1}{h}O(\delta)+O(\delta_{s+2})$
Let us now derive the estimates of the global errors
$x_{i}(k+1)-x(t(k)+c_{i}h(k))$ and $z_{i}(k+1)-z(t(k)+c_{i}h(k))$. The nominal
Runge-Kutta scheme is formulated assuming that the exact state value is known
at $t(k)$
(114) $\displaystyle x_{i}^{n}(k+1)$ $\displaystyle=$ $\displaystyle
x(t(k))+h\sum_{j=1}^{s}a_{ij}f\left(x_{j}^{n}(k+1),\
z_{j}^{n}(k+1),u(k)\right),$ (115) $\displaystyle 0$ $\displaystyle=$
$\displaystyle g\left(x_{i}^{n}(k+1)\right),$
for $i=1,...,s$. The following local error estimates are valid for RADAU IIA
scheme ([3], Lemma 4.3 p. 34)
(116) $\displaystyle x^{n}_{i}(k+1)-x(t(k)+c_{i}h)$ $\displaystyle=$
$\displaystyle O(h^{q+1})$ (117) $\displaystyle z^{n}_{i}(k+1)-z(t(k)+c_{i}h)$
$\displaystyle=$ $\displaystyle O(h^{q})$
In this case the perturbed Runge-Kutta scheme is actually the regular Runge-
Kutta scheme, with the approximation of the state $x(k)$ at time $t(k)$ used
$\displaystyle x_{i}(k+1)=$ (118) $\displaystyle
x(k)+h\sum_{j=1}^{s}a_{ij}f\left(x_{j}(k+1),\ z_{j}(k+1),u(k)\right)=$
$\displaystyle x(t(k))+h\sum_{j=1}^{s}a_{ij}f\left(x_{j}(k+1),\
z_{j}(k+1),u(k)\right)+(x(k)-x(t(k)),$ (119) $\displaystyle
0=g\left(x_{i}(k+1)\right),$
for $i=1,...,s$, so the perturbations are
(120) $\delta_{i}=\delta=x(k)-x(t(k)),\ i=1,\dots,s.$
From (97) we have $\delta=x(k)-x(t(k))=O(h^{p})$. From (109) and (110) we
obtain
(121) $\displaystyle x_{i}(k+1)-x^{n}_{i}(k+1)$ $\displaystyle=$
$\displaystyle O(h^{p}).$ (122) $\displaystyle z_{i}(k+1)-z^{n}_{i}(k+1)$
$\displaystyle=$ $\displaystyle O(h^{p-1}).$
By combining (116)-(117) and (121)-(122) we obtain the required global error
estimates
(123) $\displaystyle x_{i}(k+1)-x(t(k)+c_{i}h)$ $\displaystyle=$
$\displaystyle O(h^{q+1})+O(h^{p})=O(h^{min\\{p,q+1\\}})$ $\displaystyle=$
$\displaystyle O(h^{q_{x}^{g}})$ (124) $\displaystyle
z_{i}(k+1)-z(t(k)+c_{i}h)$ $\displaystyle=$ $\displaystyle
O(h^{q})+O(h^{p-1})$ $\displaystyle=$ $\displaystyle O(h^{q_{z}^{g}})$
For the RADAU IIA scheme with $s=3$ we get
(125) $q_{x}^{g}=min\\{5,3+1\\}=4,\ q_{z}^{g}=min\\{5-1,3\\}=3.\\\ $
Let us now define the unperturbed Runge–Kutta scheme as the Runge-Kutta scheme
for continuous adjoint equations with the exact adjoint state known at time
$t(k+1)$
(126) $\displaystyle\lambda_{fi}^{n}(k)$ $\displaystyle=$
$\displaystyle\lambda_{f}(t(k+1))+h\sum_{j=1}^{s}\bar{a}_{ij}\left[f_{x}^{T}\left(x\left(t_{j}(k)\right),z\left(t_{j}(k)\right),u(k)\right)\lambda_{fj}^{n}(k)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\ \
\left.+g_{x}^{T}\left(x\left(t_{j}(k)\right)\right)\lambda_{gj}^{n}(k)\right],$
(127) $\displaystyle 0$ $\displaystyle=$
$\displaystyle-f^{T}_{z}\left(x\left(t_{i}(k)\right)\right)\lambda_{fi}^{n}(k)$
for $i=1,\dots,s$ and
(128) $\displaystyle\lambda_{f}^{n}(k)$ $\displaystyle=$
$\displaystyle\lambda_{f}(t(k+1))+h\sum_{i=1}^{s}\bar{b}_{i}\left[f_{x}^{T}\left(x\left(t_{i}(k)\right),z\left(t_{i}(k)\right),u(k)\right)\lambda_{fi}^{n}(k)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\ \
\left.+g_{x}^{T}\left(x\left(t_{i}(k)\right)\right)\lambda_{gi}^{n}(k)\right],$
(129) $\displaystyle\lambda_{g}^{n}(k)$ $\displaystyle=$
$\displaystyle\lambda_{g}(t(k+1))+\sum_{i=1}^{s}\bar{b}_{i}^{-}(\lambda_{gi}^{n}(k)-\lambda_{g}(t(k+1))),$
The coefficients $\bar{a}_{ij},\bar{b}_{i},\bar{c}_{i}$ constitutes RADAU IA
scheme ([10]), so the following local error estimates are valid ([3], Theorem
4.3 p. 43)
(130) $\displaystyle\lambda^{n}_{fi}(k)-\lambda_{f}(t(k)+c_{i}h)$
$\displaystyle=$ $\displaystyle O(h^{\bar{q}+1}),\ i=1,\dots,s\ ,$ (131)
$\displaystyle\lambda^{n}_{gi}(k)-\lambda_{g}(t(k)+c_{i}h)$ $\displaystyle=$
$\displaystyle O(h^{\bar{q}}),\ i=1,\dots,s\ ,$ (132)
$\displaystyle\lambda^{n}_{f}(k)-\lambda_{f}(t(k))$ $\displaystyle=$
$\displaystyle O(h^{\bar{q}+1}),$ (133)
$\displaystyle\lambda^{n}_{g}(k)-\lambda_{g}(t(k))$ $\displaystyle=$
$\displaystyle O(h^{\bar{q}}).$
where for the RADAU IA scheme with $s=3$ we have ([3])
(134) $\bar{p}=5,\ \bar{q}=2.$
The perturbed Runge–Kutta scheme is defined as the discrete adjoint equations
with the exact adjoint state known at time $t(k+1)$
(135) $\displaystyle\lambda_{fi}^{p}(k)$ $\displaystyle=$
$\displaystyle\lambda_{f}(t(k+1))+h\sum_{j=1}^{s}\bar{a}_{ij}\left[f_{x}^{T}\left(x_{j}(k+1),z_{j}(k+1),u(k)\right)\lambda_{fj}^{p}(k)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\ \
\left.+g_{x}^{T}\left(x_{j}(k+1)\right)\lambda_{gj}^{p}(k)\right],$ (136)
$\displaystyle 0$ $\displaystyle=$
$\displaystyle-f^{T}_{z}\left(x_{i}(k+1)\right)\lambda_{fi}^{p}(k)$
for $i=1,\dots,s$ and
(137) $\displaystyle\lambda_{f}^{p}(k)$ $\displaystyle=$
$\displaystyle\lambda_{f}(t(k+1))+h\sum_{i=1}^{s}\bar{b}_{i}\left[f_{x}^{T}\left(x_{i}(k+1),z_{i}(k+1),u(k)\right)\lambda_{fi}^{p}(k)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\ \
\left.+g_{x}^{T}\left(x_{i}(k+1)\right)\lambda_{gi}^{p}(k)\right],$ (138)
$\displaystyle\lambda_{g}^{p}(k)$ $\displaystyle=$
$\displaystyle\lambda_{g}(t(k+1))+\sum_{i=1}^{s}\bar{b}_{i}^{-}(\lambda_{gi}^{p}(k)-\lambda_{g}(t(k+1))),$
The perturbed system can be rewritten as
(139) $\displaystyle\lambda_{fi}^{p}(k)$ $\displaystyle=$
$\displaystyle\lambda_{f}(t(k+1))+h\sum_{j=1}^{s}\bar{a}_{ij}\left[f_{x}^{T}\left(x(t_{j}(k)),z(t_{j}(k)),u(k)\right)\lambda_{fj}^{p}(k)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\ \
\left.+g_{x}^{T}\left(x(t_{j}(k))\right)\lambda_{gj}^{p}(k)\right]+\delta_{i},$
(140) $\displaystyle 0$ $\displaystyle=$
$\displaystyle-f^{T}_{z}\left(x(t_{i}(k))\right)\lambda_{fi}^{p}(k)$
for $i=1,\dots,s$ and
(141) $\displaystyle\lambda_{f}^{p}(k)$ $\displaystyle=$
$\displaystyle\lambda_{f}(t(k+1))+h\sum_{i=1}^{s}\bar{b}_{i}\left[f_{x}^{T}\left(x(t_{i}(k)),z(t_{i}(k)),u(k)\right)\lambda_{fi}^{p}(k)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\ \
\left.+g_{x}^{T}\left(x(t_{i}(k))\right)\lambda_{gi}^{p}(k)\right]+\delta_{s+1},$
(142) $\displaystyle\lambda_{g}^{p}(k)$ $\displaystyle=$
$\displaystyle\lambda_{g}(t(k+1))+\sum_{i=1}^{s}\bar{b}_{i}^{-}(\lambda_{gi}^{p}(k)-\lambda_{g}(t(k+1))),$
where the perturbations result from differences
$x_{i}(k+1)-x\left(t(k)+c_{i}h(k)\right)=O(h^{q_{x}^{g}})=O(h^{4})$ and
$z_{i}(k+1)-z\left(t(k)+c_{i}h(k)\right)=O(h^{q_{z}^{g}})=O(h^{3})$. The
perturbations satisfy
(143) $\delta_{i}=O(h^{q_{z}^{g}}),\ i=1,\dots,s+1.$
From (109)-(113) we obtain the error estimates
(144) $\displaystyle\lambda^{p}_{fi}(k)-\lambda^{n}_{fi}(k)$ $\displaystyle=$
$\displaystyle O(h^{q_{z}^{g}}),$ (145)
$\displaystyle\lambda^{p}_{gi}(k)-\lambda^{n}_{gi}(k)$ $\displaystyle=$
$\displaystyle O(h^{q_{z}^{g}-1}),$ (146)
$\displaystyle\lambda^{p}_{f}(k)-\lambda^{n}_{f}(k)$ $\displaystyle=$
$\displaystyle O(h^{q_{z}^{g}}),$ (147)
$\displaystyle\lambda^{p}_{g}(k)-\lambda^{n}_{g}(k)$ $\displaystyle=$
$\displaystyle O(h^{q_{z}^{g}-1}).$
By combining (130)-(133) with (144)-(147) we obtain the following local error
estimates
(148) $\displaystyle\lambda^{p}_{fi}(k)-\lambda_{f}(t(k)+c_{i}h)$
$\displaystyle=$ $\displaystyle
O(h^{min\\{q_{z}^{g},\bar{q}+1\\}})=O(h^{\bar{q}_{f}^{l}}),\ i=1,\dots,s\ ,$
(149) $\displaystyle\lambda^{p}_{gi}(k)-\lambda_{g}(t(k)+c_{i}h)$
$\displaystyle=$ $\displaystyle
O(h^{min\\{q_{z}^{g}-1,\bar{q}\\}})=O(h^{\bar{q}_{g}^{l}}),\ i=1,\dots,s\ ,$
(150) $\displaystyle\lambda^{p}_{f}(k)-\lambda_{f}(t(k))$ $\displaystyle=$
$\displaystyle O(h^{min\\{q_{z}^{g},\bar{q}+1\\}})=O(h^{\bar{p}_{f}^{l}}),$
(151) $\displaystyle\lambda^{p}_{g}(k)-\lambda_{g}(t(k))$ $\displaystyle=$
$\displaystyle O(h^{min\\{q_{z}^{g}-1,\bar{q}\\}})=O(h^{\bar{p}_{g}^{l}}).$
For the RADAU IA scheme with $s=3$ we have
(152) $\displaystyle\bar{q}_{f}^{l}=\bar{p}_{f}^{l}$ $\displaystyle=$
$\displaystyle min\\{q_{z}^{g},\bar{q}+1\\}=min\\{3,2+1\\}=3$ (153)
$\displaystyle\bar{q}_{g}^{l}=\bar{p}_{g}^{l}$ $\displaystyle=$ $\displaystyle
min\\{q_{z}^{g}-1,\bar{q}\\}=min\\{3-1,2\\}=2$
The global error of $\lambda^{p}_{f}(k)$ can be obtained from ([3], Theorem
4.4 p. 36)
(154)
$\lambda_{f}(k)-\lambda_{f}(t(k))=O(h^{\bar{p}_{f}^{l}-1})=O(h^{\bar{p}_{f}^{g}}).$
For RADAU IIA scheme with $s=3$ we obtain
$\bar{p}_{f}^{g}=min\\{q_{z}^{g},\bar{q}+1\\}-1=min\\{3,2+1\\}-1=2$. Let us
now derive the global error estimate of
$\lambda_{fi}(k)-\lambda_{f}(t(k)+c_{i}h(k))$. Using the global error estimate
(154) and the local error estimate (144) we can repeat the reasoning for
discrete system equations and derive the global error estimate
(155)
$\lambda_{fi}(k)-\lambda_{f}(t(k)+c_{i}h(k))=O(h^{min\\{\bar{p}_{f}^{g},\bar{q}_{f}^{l}\\}})=O(h^{\bar{q}_{f}^{g}}).$
For RADAU IIA scheme with $s=3$ we obtain $\bar{q}_{f}^{g}=$
$min\\{\bar{p}_{f}^{g},\bar{q}_{f}^{l}\\}$ $=min\\{2,3\\}=2$.
Having the global error estimates (123) for $x_{i}(k+1)-x(t_{i}(k))$ and (155)
for $\lambda_{fi}(k)-\lambda_{f}(t_{i}(k))$, we can repeat the derivation of
the reduced gradients calculation order presented in the first part of the
paper (for ODEs case) to obtain
(156)
$\frac{d\phi(X(K))}{du_{n}}-\frac{d\phi(x(t_{f}))}{du_{n}}=O(h^{\tilde{p}_{d}})$
where $\tilde{p}_{d}=min\\{\tilde{p},q_{x}^{g},\bar{q}_{f}^{g}\\}$. For RADAU
IIA scheme we obtain
(157) $\tilde{p}_{d}=min\\{5,4,2\\}=2.$
This result confirms that the discrete reduced gradients converge to the
continuous reduced gradients with an order at least $\tilde{p}_{d}=2$ (for
RADAU IIA scheme with $s=3$). The discrete reduced gradients provide therefore
an efficient and reliable method for approximation of continuous reduced
gradients.
In Table 1 we summarize the proven minimal integration orders for system and
adjoint equations derived in this paper and in [10]. It follows that if a
state trajectory includes sections with sliding modes then the reduced
gradient can be approximated with the accuracy at least $O(h^{2})$, otherwise
the accuracy is at least $O(h^{3})$. This accuracy estimate takes into account
the fact that if a discrete state changes then we have to start the
integration of system and adjoint equations with the perturbed initial values
of system and adjoint variables respectively—the influence of that
perturbation on the accuracy of system and adjoint variables determination
follows from Theorem 4.3 in [4] (cf. (120)–(121) and (139) in [10]), or from
Theorem 4.2 in [3] (cf. (112)–(113) and (144)–(147)).
Table 1: Integration orders of convergence if RADAU IIA is applied to systems equations Equations | Orders $(x,z)$ | Orders $(\lambda_{f},\lambda_{g})$ | gradient orders
---|---|---|---
$x^{\prime}=f(x,u)$ | $p=5$ | $\bar{q}_{f}=4$ | $p_{d}=3$
$x^{\prime}=f(x,u)+g_{x}(x)z$ | | |
$0=g(x)$ | $p=5$, $q=3$ | $\bar{q}_{f}=2$, $\bar{q}_{g}=2$ | $p_{d}=2$
## 3 Calculating adjoint variables jumps
In this section we want to discuss the correspondence between jump conditions
for discrete and continuous adjoint equations at transition times. Let us
consider the hybrid system trajectory described by (1)-(4). As a result of the
numerical integration we obtain the discrete state equations (42).
In Section 1 we have stated that at a transition time $t_{t}$ adjoint
variables undergo jumps, the extent of the jump depends on the sequence of
discrete variables before and after the jump. The analysis which follows
concerns the case of the equations (1)-(4), in that case the extent of the
jump can be determined according to the equations (11)–(1)
These equations can be solved with respect to $\pi_{t}$ and
$\lambda_{f}(t_{t}^{-})$ giving:
$\displaystyle{\displaystyle\pi_{t}=-\frac{\lambda(t_{t}^{+})^{T}\left(f^{2}(x(t_{t}^{+}),z(t_{t}^{+}),u(t_{t}^{+}))-f^{1}(x(t_{t}^{-}),u(t_{t}^{-}))\right)}{g_{x}(x(t_{t}))f^{1}(x(t_{t}^{-}),u(t_{t}^{-}))}}$
(we have taken into account that $g(x(t_{t}^{+}))=0$ ).
Between transitions, a system of differential-algebraic equations is
integrated with the help of an appropriate numerical integration scheme. The
numerical integration scheme is represented by the equation
(158) $F(X(k+1),X(k),u(k),h(k))=0$
in which the discrete step $k$ corresponds to a time instant $t(k)$.
During the numerical integration of a hybrid trajectory, a possible violation
of invariant set conditions has to be monitored. This task is realized by
checking the sign changes of
$g(x(k))$
in subsequent steps, where $x(k)=x(t(k))$. When a sign change of $g(x(k))$
between discrete steps $k$ and $k+1$ is detected, the following problem is
solved
(159) ${\rm find}\ \ t_{t}\in[t(k),t(k+1)],\ {\rm s.t.}\ \ \hat{g}(t_{t})=0$
where $\hat{g}(\cdot)$ is a function, which approximates $g(\cdot))$ on a time
interval $[t(k),t(k+1)]$. When a transition time $t_{t}$ is found, the actual
iteration of numerical integration is repeated but with a step-size
$h(k)=t_{t}-t(k)$ instead of $h(k)=t(k+1)-t(k)$. The discrete step at which
the transition takes place we denote by $k_{t}$.
We assume that at discrete times $0,\ldots,k_{t}$ the system evolves according
to the equations:
(160) $F^{1}(X^{1}(k+1),X^{1}(k),u(k),h(k))=0,$
and at times $k_{t},\ldots,K-1$ by the equations
(161) $F^{2}(X^{2}(k+1),X^{2}(k),u(k),h(k))=0$
The optimal control problem with that system was investigated in [11].
Therein, the adjoint equations for the functional $\phi(x(t_{f}))$ were
established, herein parts of these equations are presented to expose jumps in
adjoint variables.
$\displaystyle{\rm for}\ k=N-1,\ldots,k_{t}+1$ (162a)
$\displaystyle\Lambda^{2}(k)$
$\displaystyle=-F^{2}_{X}(k_{t})^{T}\left[F^{2}_{X^{+}}(k_{t})\right]^{-T}\Lambda^{2}(k+1)$
(162b) $\displaystyle\Lambda^{2+}(k_{t})$
$\displaystyle=-F^{2}_{X}(k_{t})^{T}\left[F^{2}_{X^{+}}(k_{t})\right]^{-T}\Lambda^{2}(k_{t}+1)$
(162c)
$\displaystyle\tilde{\Lambda}^{1-}(k_{t})+\pi(k_{t})\left(g_{x}(k_{t})\right)^{T}$
$\displaystyle=\tilde{\Lambda}^{2+}(k_{t})$ (162d)
$\displaystyle\Lambda^{1-}(k_{t})^{T}\left[F^{1}_{X^{+}}(k_{t}-1)\right]^{-1}F^{1}_{h}(k_{t}-1)$
$\displaystyle=\Lambda^{2}(k_{t}+1)^{T}\left[F^{2}_{X^{+}}(k_{t})\right]^{-1}F^{2}_{h}(k_{t}),$
(162e) $\displaystyle\Lambda^{1}(k_{t}-1)$
$\displaystyle=-F^{1}_{X}(k_{t})^{T}\left[F^{1}_{X^{+}}(k_{t})\right]^{-T}\Lambda^{1-}(k_{t})$
$\displaystyle{\rm for}\ k=k_{t}-2,\ldots,1$ (162f)
$\displaystyle\Lambda^{1}(k)$
$\displaystyle=-F^{1}_{X}(k_{t})^{T}\left[F^{1}_{X^{+}}(k_{t})\right]^{-T}\Lambda^{1}(k+1).$
Here, $\tilde{\Lambda}^{1-}(k_{t})$ and $\tilde{\Lambda}^{2+}(k_{t})$ are
parts of vectors $\Lambda^{1}(k_{t})$ and $\Lambda^{2}(k_{t})$ respectively,
defined in such a way to be able to extract the essential part of Eq. (32d) in
[11]. Notice that
$\displaystyle\Lambda^{2}(k)$ $\displaystyle=$
$\displaystyle\left(\left(l_{f1}^{2}(k)\right)^{T},\left(l_{g1}^{2}(k)\right)^{T},\ldots,\left(l_{fs}^{2}(k)\right)^{T},\left(l_{gs}^{2}(k)\right)^{T},\left(\lambda_{f}^{2}(k)\right)^{T}\right)^{T},$
for $k=k_{t}+1,\ldots,K$, but
(163) $\displaystyle l_{fi}^{2}(k)$ $\displaystyle=$ $\displaystyle 0$ (164)
$\displaystyle l_{gi}^{2}(k)$ $\displaystyle=$ $\displaystyle 0$
for $i=1,\ldots,s$ and $k=k_{t}+1,\ldots,K$ (the justification of that is
given in Section 2), and similarly
$\displaystyle\Lambda^{1}(k)$ $\displaystyle=$
$\displaystyle\left(\left(l_{1}^{1}(k)\right)^{T},\ldots,\left(l_{s}^{1}(k)\right)^{T},\left(\lambda^{1}(k)\right)^{T}\right)^{T},$
for $k=0,\ldots,k_{t}-1$ and
(165) $\displaystyle l_{i}^{1}(k)$ $\displaystyle=$ $\displaystyle 0$
for $i=1,\ldots,s$ and steps $k=0,...,k_{t}$ (the justification for that is
given by Eq. (78) of Section 5 in [10]).
Therefore, we can take
$\displaystyle\tilde{\Lambda}^{1-}(k_{t})$ $\displaystyle=$
$\displaystyle\lambda^{1-}(k_{t})$ $\displaystyle\tilde{\Lambda}^{2+}(k_{t})$
$\displaystyle=$ $\displaystyle\lambda_{f}^{2+}(k_{t})$
and then (162c) becomes
(166)
$\displaystyle\lambda^{1-}(k_{t})+\pi(k_{t})\left(g_{x}(k_{t})\right)^{T}$
$\displaystyle=$ $\displaystyle\lambda_{f}^{2+}(k_{t})$
Our aim is to determine $\pi(k_{t})$ on the basis of equations (166) and
(162d). To this end we need analytical formula for $F^{1}_{X}(k_{t})$,
$F^{1}_{X^{+}}(k_{t})$, $F^{2}_{X}(k_{t})$, $F^{2}_{X^{+}}(k_{t})$,
$F^{1}_{h}(k_{t})$, $F^{2}_{h}(k_{t})$. The mappings $F^{1}$ and $F^{2}$ are
stated in Eq. (62) (in [10]) and (40) respectively. Furthermore, matrices
$F^{1}_{X+}$, $F^{1}_{X}$ are given by Eqns (70)–(71) in [10] and matrices
$F^{2}_{X+}$, $F^{2}_{X}$ by (43)–(44). It remains to provide formula for
$F^{1}_{h}(k_{t})$ and $F^{2}_{h}(k_{t})$. According to Eq. (62) (in [10]) and
(40) we have
(171) $\displaystyle F^{1}_{h}(k)=\frac{\partial
F^{1}(X^{1}(k+1),X^{1}(k),u(k),h(k))}{\partial h(k)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\sum_{j=1}^{s}a_{1j}f^{1}(x_{j},u)\\\
\vdots\\\ -\sum_{j=1}^{s}a_{sj}f^{1}(x_{j},u)\\\
-\sum_{i=1}^{s}b_{i}f^{1}(x_{i},u)\end{array}\right)$ (178) $\displaystyle
F^{2}_{h}(k)=\frac{\partial F^{2}(X^{2}(k+1),X^{2}(k),u(k),h(k))}{\partial
h(k)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\sum_{j=1}^{s}a_{1j}f^{2}(x_{j},z_{j},u)\\\
0\\\ \vdots\\\ -\sum_{j=1}^{s}a_{sj}f^{2}(x_{j},z_{j},u)\\\ 0\\\
-\sum_{i=1}^{s}b_{i}f^{2}(x_{i},z_{i},u)\end{array}\right).$
In order to derive analytical formula for $\pi(k_{t})$ we have to figure out
two vectors:
$w(k_{t})=\left[F^{2}_{X^{+}}(k_{t})\right]^{-1}F^{2}_{h}(k_{t})$,
$w(k_{t}-1)=\left[F^{1}_{X^{+}}(k_{t}-1)\right]^{-1}F^{1}_{h}(k_{t}-1)$ — both
these vectors are solutions to linear equations:
(191)
$\displaystyle{\displaystyle\left(\begin{array}[]{ccccccc}I-ha_{11}f_{x1}^{2}&-ha_{11}f_{y1}^{2}&\ldots&-ha_{1s}f_{xs}^{2}&-ha_{1s}f_{ys}^{2}&0\\\
-g_{x1}^{2}&0&\ldots&0&0&0\\\ \vdots&\vdots&&\vdots&\vdots&\vdots\\\
-ha_{s1}f_{x1}^{2}&-ha_{s1}f_{y1}^{2}&\ldots&I-ha_{ss}f_{xs}^{2}&-ha_{ss}f_{ys}^{2}&0\\\
0&0&\ldots&-g_{xs}^{2}&0&0\\\
-hb_{1}f_{x1}^{2}&-hb_{1}f_{y1}^{2}&\ldots&-hb_{s}f_{xs}^{2}&-hb_{s}f_{ys}^{2}&I\end{array}\right)\left(\begin{array}[]{c}w_{f}^{1}\\\
w_{g}^{1}\\\ \vdots\\\ w_{f}^{s}\\\ w_{g}^{s}\\\ w_{f}\end{array}\right)=}$
(198)
$\displaystyle{\displaystyle\left(\begin{array}[]{c}-\sum_{j=1}^{s}a_{1j}f^{2}(x_{j},z_{j},u)\\\
0\\\ \vdots\\\ -\sum_{j=1}^{s}a_{sj}f^{2}(x_{j},z_{j},u)\\\ 0\\\
-\sum_{i=1}^{s}b_{i}f^{2}(x_{i},z_{i},u)\end{array}\right),}$
$\left(\begin{array}[]{cccc}I-ha_{11}f_{x1}^{1}&\ldots&-ha_{1s}f_{xs}^{1}&0\\\
\vdots&&\vdots&\vdots\\\ -ha_{s1}f_{x1}^{1}&\ldots&I-ha_{ss}^{1}f_{xs}&0\\\
-hb_{1}f_{x1}^{1}&\ldots&-hb_{s}f_{xs}^{1}&I\end{array}\right)\left(\begin{array}[]{c}w^{1}\\\
\vdots\\\ w^{s}\\\
w\end{array}\right)=\left(\begin{array}[]{c}-\sum_{j=1}^{s}a_{1j}f^{1}(x_{j},u)\\\
\vdots\\\ -\sum_{j=1}^{s}a_{sj}f^{1}(x_{j},u)\\\
-\sum_{i=1}^{s}b_{i}f^{1}(x_{i},u)\end{array}\right).$
Eventually, to transform equation (162d) to its useful form we take into
account (163)–(164) and (165), then the equation (162d) becomes
$\displaystyle{\displaystyle\lambda^{1-}(k_{t})^{T}\left(h(k_{t}-1)\sum_{i=1}^{s}b_{i}f^{1}_{xi}(k_{t}-1)w^{i}(k_{t}-1)\right.}$
$\displaystyle{\displaystyle\left.-\sum_{i=1}^{s}b_{i}f^{1}(x_{i}(k_{t}),u(k_{t}-1))\right)}$
$\displaystyle{\displaystyle=\lambda_{f}^{2}(k_{t}+1)^{T}\left(h(k_{t})\sum_{i=1}^{s}b_{i}f^{2}_{xi}(k_{t})w^{i}_{f}(k_{t})+h(k_{t})\sum_{i=1}^{s}b_{i}f^{2}_{yi}(k_{t})w^{i}_{g}(k_{t})\right.}$
(199)
$\displaystyle{\displaystyle\left.-\sum_{i=1}^{s}b_{i}f^{2}(x_{i}(k_{t}+1),z_{i}(k_{t}+1),u(k_{t}))\right)}$
By plugging $\lambda^{1-}(t_{k})$ from equations (199) into equation (166) one
will get
$\displaystyle{\displaystyle\lambda_{f}^{2+}(k_{t})^{T}\left(h(k_{t}-1)\sum_{i=1}^{s}b_{i}f^{1}_{xi}(k_{t}-1)w^{i}(k_{t}-1)\right.}$
$\displaystyle{\displaystyle\left.-\sum_{i=1}^{s}b_{i}f^{1}(x_{i}(k_{t}),u(k_{t}-1))\right)}$
$\displaystyle{\displaystyle-\pi
g_{x}(k_{t})\left(h(k_{t}-1)\sum_{i=1}^{s}b_{i}f^{1}_{xi}(k_{t}-1)w^{i}(k_{t}-1)\right.}$
$\displaystyle{\displaystyle\left.-\sum_{i=1}^{s}b_{i}f^{1}(x_{i}(k_{t}),u(k_{t}-1))\right)}$
$\displaystyle{\displaystyle=\lambda^{2}_{f}(k_{t}+1)^{T}\left(h(k_{t})\sum_{i=1}^{s}b_{i}f^{2}_{xi}(k_{t})w^{i}_{f}(k_{t})+h(k_{t})\sum_{i=1}^{s}b_{i}f^{2}_{yi}(k_{t})w^{i}_{g}(k_{t})-\right.}$
(200)
$\displaystyle{\displaystyle\left.\sum_{i=1}^{s}b_{i}f^{2}(x_{i}(k_{t}+1),z_{i}(k_{t}),u(k_{t}))\right)}$
and eventually
$\displaystyle{\displaystyle\pi(k_{t})=\left[\lambda^{2}_{f}(k_{t}+1)^{T}\left(-h(k_{t})\sum_{i=1}^{s}b_{i}f^{2}_{xi}(k_{t})w^{i}_{f}(k_{t})-h(k_{t})\sum_{i=1}^{s}b_{i}f^{2}_{yi}(k_{t})w^{i}_{g}(k_{t})\right.\right.}$
$\displaystyle{\displaystyle\left.+\sum_{i=1}^{s}b_{i}f^{2}(x_{i}(k_{t}+1),z_{i}(k_{t}),u(k_{t}))\right)-\lambda_{f}^{2+}(k_{t})^{T}\times}$
$\displaystyle{\displaystyle\left.\left(-h(k_{t}-1)\sum_{i=1}^{s}b_{i}f^{1}_{xi}(k_{t}-1)w_{i}(k_{t}-1)+\sum_{i=1}^{s}b_{i}f^{1}(x_{i}(k_{t}),u(k_{t}-1))\right)\right]\bigg{/}}$
$\displaystyle{\displaystyle
g_{x}(k_{t})\left(h(k_{t}-1)\sum_{i=1}^{s}b_{i}f^{1}_{xi}(k_{t}-1)w_{i}(k_{t}-1)-\sum_{i=1}^{s}b_{i}f^{1}(x_{i}(k_{t}),u(k_{t}-1))\right).}$
(201)
We assume that all functions values and their partial derivatives: $f^{1}$,
$f^{2}$, $f^{1}_{xi}$, $f^{2}_{xi}$, $f^{2}_{yi}$ are uniformly bounded in a
neighborhood of the considered solution (cf. assumptions (H1) and (H2) in
[9]). Furthermore, in the neighborhood, matrices $F^{1}_{X+}$, $F^{2}_{X+}$
are invertible and their elements uniformly bounded. Therefore, there exists
$0<L<+\infty$ such that $\|z\|\leq L$ and $\|w\|\leq L$.
We consider RADAU IIA scheme which has the property $\sum_{i=1}^{s}b_{i}=1$.
That together with the boundedness of $z$ and $w$ show that
(202) $\displaystyle{\displaystyle\pi(t_{k})\rightarrow\pi_{t}}$
provided that $h(k)\rightarrow 0$.
However that property can be not sufficient to guarantee high accuracy of
solutions to a continuous optimal control problem with hybrid system when a
Runge–Kutta method with variable step sizes is applied. For that method we
prefer a scheme for adjoint variables jumps which is determined by the formula
$\pi(k_{t})$ with the property
(203) $\displaystyle{\displaystyle|\pi(k_{t})-\pi_{t}|\leq
O(h^{\bar{q}_{\pi}}),}$
where the order of convergence $\bar{q}$ can be estimated on the basis of the
integration orders for state and adjoint variables.
That can be achieved for many new formula for $\pi(k_{t})$ which does not
dispatch far away from the formula (201). For example, consider the formula
$\displaystyle{\displaystyle\hat{\pi}(k_{t})=\frac{\lambda_{f}^{2+}(k_{t})^{T}\left(f^{2}(x(k_{t}),z(k_{t}),u(k_{t}-1))-f^{1}(x(k_{t}),u(k_{t}-1))\right)}{g_{x}(k_{t})f^{1}(x(k_{t}),u(k_{t}-1))}.}$
(204)
Then, we will have
$\displaystyle{\displaystyle\hat{\pi}(t_{k})-\pi_{t}=}$
$\displaystyle{\displaystyle
g_{x}(x(t_{t}))f^{1}(x(t_{t}^{-}),u(t_{t}^{-}))\lambda_{f}^{2+}(k_{t})^{T}\left(f^{2}(x(k_{t}),z(k_{t}),u(k_{t}-1))-\right.}$
$\displaystyle{\displaystyle\left.f^{1}(x(k_{t}),u(k_{t}-1))\right)\bigg{/}g_{x}(k_{t})f^{1}(x(k_{t}),u(k_{t}-1)){g_{x}(x(t_{t}))f^{1}(x(t_{t}^{-}),u(t_{t}^{-}))}-}$
$\displaystyle{\displaystyle
g_{x}(k_{t})f^{1}(x(k_{t}),u(k_{t}-1))\lambda(t_{t}^{+})^{T}\left(f^{2}(x(t_{t}^{+}),z(t_{t}^{+}),u(t_{t}^{+}))-\right.}$
$\displaystyle{\displaystyle\left.f^{1}(x(t_{t}^{-}),u(t_{t}^{-}))\right)\bigg{/}g_{x}(k_{t})f^{1}(x(k_{t}),u(k_{t}-1))g_{x}(x(t_{t}))f^{1}(x(t_{t}^{-}),u(t_{t}^{-})).}$
Since we have to assume that in the neighborhood of $x(t_{t})$ the following
regularity assumption must hold (cf. the assumption (H3) in [9])
(205) $\displaystyle{\displaystyle\left|g_{x}(x)f^{1}(x,u(t_{t}))\right|\geq
M>0,}$
and the product of Lipschitz functions is a Lipschitz function on the bounded
domain, after some transformations we will arrive at the relation
$\displaystyle\left|\hat{\pi}(t_{k})-\pi_{t}\right|$ $\displaystyle\leq$
$\displaystyle
L\left(\|g_{x}(x(k_{t}))-g_{x}(x(t_{t}))\|+\|\lambda_{f}^{2+}(k_{t})-\lambda(t_{t}^{+})\|+\right.$
$\displaystyle\|f^{1}(x(k_{t}),u(k_{t}))-f^{1}(x(t_{t}),u(t_{t}))\|+$
$\displaystyle\left.\|f^{2}(x(k_{t}),z(k_{t}),u(k_{t}))-f^{2}(x(t_{t}),z(t_{t}),u(t_{t}))\|\right)$
Since
$\displaystyle\lambda_{f}^{2+}(k_{t})-\lambda(t_{t}^{+})$ $\displaystyle=$
$\displaystyle O(h^{\bar{q}_{f}})$ $\displaystyle x(k_{t})-x(t_{t})$
$\displaystyle=$ $\displaystyle O(h^{p})$ $\displaystyle z(k_{t})-z(t_{t})$
$\displaystyle=$ $\displaystyle O(h^{q})$
($\bar{q}_{f}$, $p$ and $q$ are the appropriate integration orders) and if we
assume that functions $g_{x}$, $f^{1}$, $f^{2}$ are Lipschitz continuous then
(203) holds with $\bar{q}_{\pi}=min\\{\bar{q}_{f},p,q\\}$. From (166) the
adjoint variable $\lambda^{1-}(k_{t})$ is calculated with the error
$O(h^{\bar{q}_{f}})$. $\lambda^{1-}(k_{t})$ states an initial condition for
the calculation of the adjoint variables in the non-sliding phase. The order
of the global error of the adjoint variables in the non-sliding phase is
therefore reduced relative to the estimates presented in Table 1. The order of
convergence for reduced gradients is therefore equal to $\tilde{p}_{d}=2$ for
both sliding and non sliding phase.
## 4 Numerical results
In this section we present results of solving three optimal control problems
with hybrid systems by using the methods discussed in this paper and in the
papers accompanying it ([9],[10]). The results have been obtained by our
preliminary software based on its two core subroutines: the first one aimed at
solving optimal control problems with piecewise constant approximations to
control functions ([7]); the second one for evaluating solutions to
differential–algebraic equations and their corresponding adjoint equations.
To solve the reported problems we applied the SQP code described in [7],
however instead of using range–space active set method for solving QP
subproblems the new version of the SQP code is based on the implementation of
the interior point method as described in [2]. The matrices $H_{k}$ used in
the direction finding subproblems ${\bf P_{c}(u)}$ ([10]) were evaluated
according to BFGS updates with the Powell’s modifications ([6])—see the
discussion in Section 4 of Chapter 5 of [7].
We have used RADAU5 subroutine, which is the Fortran implementation of the
RADAU IIA scheme. The subroutine does not have the facility for locating
switching points and then restarting the procedure with a new description of
differential–algebraic equations. In order to enhance RADAU5 procedure
applicability to DAEs with hybrid description we incorporated into it the
subroutine ROOTS (for finding a root of nonlinear algebraic equations with a
secant method) from the SUNDIALS package. Since the ROOTS subroutine was
implemented with multistep integration methods in mind (in which polynomial
approximations of state variables $(x,y)$ between mesh points are provided by
methods themselves) we had to work out an approach to the interpolation of
state variables between mesh points which would be suitable for Runge–Kutta
methods.
In our implementation switching points $t_{t}$ are evaluated with the help of
$x_{i}(k)$, $i=1,\ldots,s$ determined at the intermediate points $t(k)$,
$t(k)+c_{i}h(k)$, $i=1,\ldots,s$. Having $c_{i}$ and $x_{i}(k)$ we evaluate
derivatives at these points:
$\displaystyle{\displaystyle
x_{i}^{{}^{\prime}}(k)=f^{1}(x_{i}(k),u(k-1),t(k)+c_{i}h(k)),\ i=1,\ldots,s}$
and then construct the Hermite polynomials by taking into accounts points
$x_{i}(k)$ and their derivatives $x_{i}^{{}^{\prime}}(k)$, $i=1,\ldots,s$. In
our current implementation we use two vectors (and their corresponding
derivatives) to build the polynomial approximation to vector $x$ on the time
interval $[t(k),t(k)+h(k)]$ ([1]):
(206) $\displaystyle x(t)$ $\displaystyle=$
$\displaystyle(2\tau^{3}-3\tau^{2}+1)x(k)+h(k)(\tau^{3}-2\tau^{2}+\tau)x^{{}^{\prime}}(k)+$
$\displaystyle(-2\tau^{3}+3\tau^{2})x_{s}(k)+h(k)(\tau^{3}-\tau^{2})x_{s}^{{}^{\prime}}(k),$
That interpolating scheme guarantees the accuracy $O(h(k)^{4})$ ([1]). We
notice, that the order of the interpolating scheme is lower than the numerical
integration order $p=5$ and affects the integration order after the discrete
transition. Nevertheless, the estimated orders for reduced gradients
calculation presented in Table 1 remains valid if we assume integration order
$p=4$.
The use of interpolating polynomials in locating switching times can influence
also the accuracy with which these times are determined. Suppose that on an
open interval $(a,b)$, such that $t_{t}\in(a,b)$, the state trajectory $x(t)$
is perturbed by $\delta x(t)$. To that state perturbation corresponds the
perturbation of the switching time, $t_{t}+\delta t_{t}$. We will have
(207) $\displaystyle{\displaystyle g(x(t_{t}+\delta t_{t})+\delta
x(t_{t}+\delta t_{t}))=0.}$
Evaluating $g$ around $(t_{t},0)$ will result in
(208) $\displaystyle 0$ $\displaystyle=$ $\displaystyle g(x(t_{t}+\delta
t_{t})+\delta x(t_{t}+\delta t_{t}))=g_{x}(x(t_{t}))\delta x(t_{t})+$
$\displaystyle g_{x}(x(t_{t}))f^{1}(x(t_{t}),u(t_{t}))\delta t_{t}+o(\delta
t_{t},\delta x).$
Eventually, we have
(209) $\displaystyle{\displaystyle\delta
t_{t}=-\frac{1}{g_{x}(x(t_{t}))f^{1}(x(t_{t}),u(t_{t}))}g_{x}(x(t_{t}))\delta
x(t_{t})+o(\delta t_{t},\delta x).}$
where $\tau=(t-t(k))/h(k)$.
This together with (205) imply that to the perturbation of $x$ by $O(h^{p})$
corresponds the perturbation of the switching time $t_{t}$ of the same order
$O(h^{p})$. It is straightforward to show that this perturbation will have the
same effect on order of convergence of integrating procedures as the
perturbation of initial conditions of the same order.
The next issue concerning the implementation of adjoint equations evaluation
scheme is that related to the jump formula. To obtain the numerical results
reported in the paper we have applied the formula (201). Since it guarantees
convergence of the adjoint equations Runge–Kutta scheme to their continuous
counterpart under the assumption $h(k)\rightarrow 0$ we set absolute and
relative tolerances to $10^{-9}$ in RADAU5 procedure. Consequently our
tolerances for satisfying optimality conditions had to be essentially less
demanding. Our optimization procedure stopped iterating when the following set
of conditions was satisfied:
$\displaystyle\sigma_{c_{k}}^{H_{k}}(u_{k})$ $\displaystyle\geq$
$\displaystyle-\varepsilon,$ $\displaystyle\left|g^{1}_{i}(u_{k})\right|$
$\displaystyle\leq$ $\displaystyle\varepsilon,\ i\in E,$ $\displaystyle
g^{2}_{j}(u_{k})$ $\displaystyle\leq$ $\displaystyle\varepsilon,\ j\in I.$
We set $\varepsilon=10^{-6}$ in our calculations.
We show the results of application of our numerical procedures to three
optimal control problems with nonlinear differential equations. In the first
two examples functions $g(x)$ defining switching surface are linear, in the
third example it is nonlinear.
Example 1 The first example concerns the Coulomb–Stribeck friction model. A
mass $m$ is attached to inertial space with a spring $k$. The mass is riding
on a belt, which itself is moving with a constant velocity $v_{dr}$ (see
Figure 6.4a in [5]).The relative velocity of the mass with respect to the belt
is equal to $v_{rel}=v-v_{dr}$. Between the mass and the belt there is the dry
friction with a friction force $F_{T}$. In the slip phase it is the function
of $v_{rel}$ and is given by the relation
(210) $\displaystyle{\displaystyle
F_{T}=-\frac{\mu_{s}}{1+\delta|v_{rel}|}F_{N}{\rm sign}(v_{rel}).}$
Here, $F_{N}=mg$. Furthermore, in the stick phase the friction force is
limited by the relation $|F_{T}|\leq F_{s}=\mu_{s}mg$. The functions $f_{1}$,
$f_{2}$ and $g$ of the hybrid system are
(213) $\displaystyle f^{1}(x,u)=\left[\begin{array}[]{c}x_{2}\\\
-\frac{k}{m}x_{1}+\frac{1}{m}\frac{F_{s}}{1+\delta|x_{2}-v_{dr}|}+x_{3}\\\
u\end{array}\right],$ (216) $\displaystyle
f^{2}(x,u)=\left[\begin{array}[]{c}x_{2}\\\
-\frac{k}{m}x_{1}-\frac{1}{m}\frac{F_{s}}{1+\delta|x_{2}-v_{dr}|}+x_{3}\\\
u\end{array}\right]$
and
(217) $\displaystyle{\displaystyle g(x)=x_{2}-v_{dr}.}$
Here, $x_{1}$ corresponds to the mass position, $x_{2}$ to its velocity and
$x_{3}$ influences the mass movement through the control $u$.
The optimal control problem is as follows
(218) $\displaystyle\min_{u\in{\mathcal{U}}}x_{2}(t_{f})$ (219)
$\displaystyle{\rm s.\ t.}$ (220) $\displaystyle\ x(t_{0})=x_{0}$ (221)
$\displaystyle x^{\prime}$ $\displaystyle=$ $\displaystyle f^{1}(x,u),\
\text{if }g(x)<0$ (222) $\displaystyle x^{\prime}$ $\displaystyle=$
$\displaystyle f^{2}(x,u),\ \text{if }g(x)>0$ (223) $\displaystyle x^{\prime}$
$\displaystyle=$ $\displaystyle f_{F}(x,u)+g_{x}^{T}(x)z,\ \text{if sliding
mode occurs}$ $\displaystyle 0$ $\displaystyle=$ $\displaystyle g(x)$
and
(224) $\displaystyle{\displaystyle x_{1}(t_{f})-0.6=0,}$
with the additional constraint on the control signal value $u$
$\displaystyle{\displaystyle-2.5\leq u(t)\leq 2.5\ t\in[t_{0},t_{f}],}$
and the time interval endpoints $t_{0}=0$ $t_{f}=1$.
The optimal trajectories are shown in Fig. 1(a) and the optimal control is
shown in Fig. 1(b). We used $N=100$ in piecewise constant approximations of
control functions. One can observe that from the time $t\approx 0.8$ the
system is in the sliding mode. The program needed $4$ iterations to find the
solution.
(a) Optimal state trajectories.
(b) Optimal control.
Figure 1: Mass–spring example optimal solution
Example 2 The example concerns the application of our approach to the problem
of planning a haemodialysis process. The problem is fully described in [13].
In this paper we present some results obtained for a variant of the problem.
The system equations that determine the concentrations of urea and phosphorus
in intracellular fluid – $C_{IC}^{urea}$, $C_{IC}^{PO_{4}}$, urea and
phosphorus concentrations in extracellular fluid – $C_{EC}^{urea}$,
$C_{EC}^{PO_{4}}$ and ultrafiltration volume – $UFR$ are as follows
$\displaystyle\frac{dC_{EC}^{urea}}{dt}$ $\displaystyle=$
$\displaystyle\frac{K_{IE}^{urea}\cdot(C_{IC}^{urea}-C_{EC}^{urea})-C_{EC}^{urea}\cdot(K_{D}^{urea}+K_{r}^{urea}+K^{ufr})}{0.34\cdot
V(0)-UFR}$ (226) $\displaystyle\frac{dC_{IC}^{urea}}{dt}$ $\displaystyle=$
$\displaystyle\frac{K_{IE}^{urea}\cdot(C_{EC}^{urea}-C_{IC}^{urea})+G^{urea}}{0.66\cdot
V(0)}$ (227) $\displaystyle\frac{dC_{EC}^{PO_{4}}}{dt}$ $\displaystyle=$
$\displaystyle\frac{K_{IE}^{PO_{4}}\cdot(C_{IC}^{PO_{4}}-C_{EC}^{PO_{4}})-K_{D}^{PO_{4}}\cdot
C_{EC}^{PO_{4}}}{0.34\cdot V(0)-UFR}+K_{3}^{PO_{4}}+K_{4}^{PO_{4}}$ (228)
$\displaystyle\frac{dC_{IC}^{PO_{4}}}{dt}$ $\displaystyle=$
$\displaystyle\frac{K_{IE}^{PO_{4}}\cdot(C_{EC}^{PO_{4}}-C_{IC}^{PO_{4}})}{0.66\cdot
V(0)}$ (229) $\displaystyle\frac{dUFR}{dt}$ $\displaystyle=$ $\displaystyle
U_{ufr}$ (230) $\displaystyle K_{4}^{PO_{4}}$ $\displaystyle=$
$\displaystyle\alpha\cdot\max\left(C_{min}^{PO_{4}}-C_{IC}^{PO_{4}},0\right)$
(231) $\displaystyle K_{3}^{PO_{4}}$ $\displaystyle=$
$\displaystyle\beta\cdot\max\left(C_{max}^{PO_{4}}-C_{IC}^{PO_{4}},0\right)$
where
(232) $\displaystyle{\displaystyle
K_{D}^{urea}=\frac{e^{K_{0}A\left(Q_{D}-Q_{B}\right)/\left(Q_{B}Q_{D}\right)}-1}{e^{K_{0}A\left(Q_{D}-Q_{B}\right)/\left(Q_{B}Q_{D}\right)}-Q_{D}/Q_{B}}}.$
The model coefficients are explained in [13]—we used the model parameters as
stated for the first run of optimization. The algebraic equations are
responsible for hybrid behavior of the system equations. We have
$0<C_{min}^{PO_{4}}-C_{IC}^{PO_{4}}<C_{max}^{PO_{4}}<+\infty$ and two
switching functions:
(233) $\displaystyle g_{1}(C_{IC}^{PO_{4}})$ $\displaystyle=$ $\displaystyle
C_{IC}^{PO_{4}}-C_{min}^{PO_{4}}$ (234) $\displaystyle g_{1}(C_{IC}^{PO_{4}})$
$\displaystyle=$ $\displaystyle C_{IC}^{PO_{4}}-C_{max}^{PO_{4}}.$
Having combined kinetic models of urea and phosphorus we look for proper
concentrations of urea and phosphorus at the end of the haemodialysis process
by controlling the parameters $Q_{B}$, $Q_{D}$ and $U_{ufr}$ (see [13] for
details). In other words, by solving the optimal control problem we want to
choose a proper dialysis membrane in order to achieve final parameters of
haemodialysis.
The optimization problem is as follows:
(235) $\min_{Q_{B},Q_{D},U_{ufr}}C_{EC}^{urea}(t_{f})$
subject to the constraints (LABEL:dialysis1)–(231), the following constraints
at final time
(236) $\displaystyle C_{IC}^{urea}(t_{f})$ $\displaystyle\leq L^{urea}_{IC}$
(237) $\displaystyle L_{min}^{UFR}\leq$ $\displaystyle UFR(t_{f})$
$\displaystyle\leq L_{max}^{UFR},$
and the constraints on the control variables
(238) $\displaystyle Q_{B}^{min}\leq$ $\displaystyle Q_{B}(t)$
$\displaystyle\leq Q_{B}^{max}$ (239) $\displaystyle Q_{D}^{min}\leq$
$\displaystyle Q_{D}(t)$ $\displaystyle\leq Q_{D}^{max}$ (240) $\displaystyle
U_{ufr}^{min}\leq$ $\displaystyle U_{ufr}(t)$ $\displaystyle\leq
U_{ufr}^{max}.$
on $[t_{0},t_{f}]$. Parameters for the inequality constraints are the same as
in [13] with the exception that now we allow $U_{ufr}$ to vary by assuming:
$U_{ufr}^{min}=0$, $U_{ufr}^{max}=75$.
Some of the optimal trajectories are shown in Figs. 2–3. One can observe that
from the time $t\approx 125$ the system is in the sliding mode. Optimal
control variable $U_{ufr}$ is shown in Fig. 4, the other control variables
reached their allowed maximum values—we used $N=10$ in the piecewise constant
approximations of control variables. Optimization procedure needed $69$
iterations to find the solution with the specified accuracy.
Figure 2: State trajectories for the urea $C_{IC}^{urea}$ and $C_{EC}^{urea}$.
Figure 3: State trajectories for the phosphorus $C_{IC}^{PO_{4}}$ and
$C_{EC}^{PO_{4}}$ (rebound of $C_{EC}^{PO_{4}}$ can be seen in 125 minute of
haemodialysis). Figure 4: Control trajectory for the $U_{ufr}$.
Example 3
The example concerns the optimal control of a of a race car discussed in [14]
and depicted in Fig.5.
$n(\theta)$$t(\theta)$$v$$Fn(\theta)$$x_{2}$$x_{1}$$\theta$ Figure 5: Race car
model – mathematical description
We introduce the following notation
* •
$x=(x_{1},x_{2})^{T}\in\mathbb{R}^{2}$ is a position of a car in the plane
* •
$v=(v_{1},v_{2})^{T}\in\mathbb{R}^{2}$ is a velocity of a car
* •
$\theta\in\mathbb{R}$ is an angle of the car orientation
* •
$t(\theta)=(\cos\theta,\sin\theta)^{T}$ is an unit vector pointing in the
direction $\theta$
* •
$n(\theta)=(-\sin\theta,\cos\theta)^{T}$ is an unit vector normal to
$t(\theta)$
* •
$a(t)\in\mathbb{R}$ is an acceleration force
* •
$s(t)\in\mathbb{R}$ is a steering control
The equations of motion are [14]
(241) $\displaystyle x^{\prime}$ $\displaystyle=$ $\displaystyle v,$ (242)
$\displaystyle v^{\prime}$ $\displaystyle=$ $\displaystyle
a(t)t(\theta)+Fn(\theta),$ (243) $\displaystyle\theta^{\prime}$
$\displaystyle=$ $\displaystyle s(t)(t(\theta)^{T}v),$
where $Fn(\theta)$ is a friction force vector. $F$ is given by
(244) $F=-\mu N{\rm sign}(n(\theta)^{T}v),$
where $\mu$ and $N$ are a friction coefficient and a normal contact force
respectively. The switching surface is therefore defined as a solution set for
an equation
(245) $n(\theta)^{T}v=-v_{1}\sin\theta+v_{2}\cos\theta=0.$
During the non-sliding motion the amplitude of a friction force vector is
constant and equal to $\mu N$. The friction force is normal to $t(\theta)$ and
it acts such that it decreases the component of a velocity which is normal to
a direction $t(\theta)$. In other words the friction force always attempts to
reduce the angle between $t(\theta)$ and $v$ to zero. Physically, the non-
sliding motion corresponds to a drift of a car.
We face the the sliding motion when the following conditions are satisfied
(246) $\displaystyle n(\theta)^{T}v$ $\displaystyle=$ $\displaystyle 0,$ (247)
$\displaystyle-\mu N-s(t)(t(\theta)^{T}v)^{2}$ $\displaystyle<$ $\displaystyle
0,$ (248) $\displaystyle\mu N-s(t)(t(\theta)^{T}v)^{2}$ $\displaystyle>$
$\displaystyle 0.$
During the sliding motion the car is always oriented in the direction of a car
velocity ($t(\theta)$ and $v$ are collinear), there is no drift and the
friction force prevents the motion in a direction normal to $t(\theta)$.
We solved the following optimization problem:
(249) $\displaystyle{\displaystyle\min_{a,s}x_{1}(t_{f})}$
subject to the system equations (241)–(243), the endpoint constraints
(250) $\displaystyle x_{2}(t_{f})$ $\displaystyle=$ $\displaystyle 0$ (251)
$\displaystyle-v_{1}(t_{f})$ $\displaystyle\leq$ $\displaystyle 0$ (252)
$\displaystyle v_{2}(t_{f})$ $\displaystyle=$ $\displaystyle 0$ (253)
$\displaystyle\theta(t_{f})$ $\displaystyle=$ $\displaystyle 0,$
and the constraints on controls
(254) $\displaystyle-0.3\leq$ $\displaystyle a(t)$ $\displaystyle\leq 0.3$
(255) $\displaystyle-1.0\leq$ $\displaystyle s(t)$ $\displaystyle\leq 1.0$
on an interval $[t_{0},t_{f}]$ ($t_{f}=3.0$). We assumed $\mu N=0.5$ and
initial values: $x_{1}(0)=0$, $x_{2}(0)=1$, $v_{1}(0)=1$, $v_{2}(0)=0$.
Our program needed 52 iterations to find an approximation to an optimal
solution with the required accuracy, we assumed $N=10$ in the piecewise
constant approximations of control functions. The optimal controls, positions
and velocities are presented in Fig.6, 7(a) and 7(b) respectively.
Figure 6: Optimal controls for the racing car problem.
(a) Optimal trajectories of positions.
(b) Optimal trajectories of velocities.
Figure 7: Racing car – optimal solution
The optimal orientation trajectory as well as the value of $n^{T}(\theta)v$
along the trajectory are presented in Fig.8(a).
Near the end of the time interval the condition $n^{T}(\theta)v=0$ becomes
satisfied, so the system enters the sliding mode. The relevant fragment of
system trajectories is presented in Fig.8(b).
(a) $\theta$ and $n^{T}(\theta)v$ trajectories.
(b) $\theta$ and $n^{T}(\theta)v$ near the transition to the sliding mode.
Figure 8: Racing car – optimal solution
## 5 Conclusions
The paper presents the computational approach to hybrid optimal control
problems with sliding modes. It seems to be the first method for optimal
control problems with hybrid systems which can exhibit sliding modes. Our
computational method is based on a Runge–Kutta method for integrating system
and adjoint equations. The possible improvements of our code are related to
the approximating rules for the jumps of adjoint variables (such as (204)) and
polynomial approximations of state variables used in the procedure of the
switching times $t_{t}$ localization (cf. (206)). These issues are the subject
of our current research.
## References
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* [6] M. J. D. Powell, ”The convergence of variable metric methods for nonlinearly constrained optimization calculations”, Technical Memorandum No. 315, Argonne National Laboratory, Argonne, Illinois, 1977.
* [7] R. Pytlak, Numerical Methods for Optimal Control Problem With State Constraints, Lecture Notes in Mathematics, vol. 1707, Springer–Verlag, (1999), .
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|
# Boundedness and concentration of random singular integrals defined by
wavelet summability kernels
Hugo Aimar and Ivana Gómez
###### Abstract.
We use Cramér-Chernoff type estimates in order to study the Calderón-Zygmund
structure of the kernels
$\sum_{I\in\mathcal{D}}a_{I}(\omega)\psi_{I}(x)\psi_{I}(y)$ where $a_{I}$ are
subgaussian independent random variables and $\\{\psi_{I}:I\in\mathcal{D}\\}$
is a wavelet basis where $\mathcal{D}$ are the dyadic intervals in
$\mathbb{R}$. We consider both, the cases of standard smooth wavelets and the
case of the Haar wavelet.
###### Key words and phrases:
Singular integrals; Wavelets; Subgaussian random variable.
This work was supported by the MINCYT in Argentina: CONICET and Agencia I+D+i;
and UNL
## 1\. Introduction
Set $\mathcal{D}$ to denote the family of all dyadic intervals in
$\mathbb{R}$. Then $\mathcal{D}=\cup_{j\in\mathbb{Z}}\mathcal{D}^{j}$ with
$\mathcal{D}^{j}=\\{I^{j}_{k}=[k2^{-j},(k+1)2^{-j}):k\in\mathbb{Z}]\\}$ the
sequence of all dyadic intervals with length $2^{-j}$. We shall use the
notation $\psi_{I}(x)=\psi_{j,k}(x)=2^{j/2}\psi(2^{j}x-k)$ with
$I=I^{j}_{k}\in\mathcal{D}$, to denote the orthonormal wavelet basis of
$L^{2}(\mathbb{R})$ generated by the wavelet function $\psi$.
The basic kernels associated to the unconditional character of
$\\{\psi_{I}:I\in\mathcal{D}\\}$ as a basis for $L^{p}(\mathbb{R})$ with
$1<p<\infty$, are given by series of the form
$K(x,y)=\sum_{I\in\mathcal{D}}\omega_{I}\psi_{I}(x)\psi_{I}(y)$ (1.1)
with $\omega_{I}=\pm 1$ for each $I\in\mathcal{D}$.
Under some mild conditions on $\psi$ the kernels $K(x,y)$ become
Calderón–Zygmund type kernels and the induced operators are bounded in
$L^{p}(\mathbb{R})$ for $1<p<\infty$.
The standard use of the Calderón-Zygmund theory in the proof of the
unconditionality of wavelet bases in Lebesgue and Sobolev spaces (see
[Mey90]), is based on the estimates
$\sum_{I\in\mathcal{D}}\left|\psi_{I}(x)\right|\left|\psi_{I}(y)\right|\leq\frac{C}{\left|x-y\right|}$
and
$\sum_{I\in\mathcal{D}}\left|\frac{d\psi_{I}}{dx}(x)\right|\left|\psi_{I}(y)\right|\leq\frac{C}{\left|x-y\right|^{2}}.$
See Chapter 9 in [Dau92]. Nevertheless, when
$\left|\psi(x)\right|+\left|\psi^{\prime}(x)\right|\leq\frac{C}{(\left|x\right|+1)^{1+\varepsilon}}$,
these estimates only work for the original kernel
$K_{\sigma}(x,y)=\sum_{I\in\mathcal{D}}\sigma_{I}\psi_{I}(x)\psi_{I}(y)$, with
$\left|\sigma_{I}\right|=1$ or for bounded sequences $\sigma_{I}$. Actually,
in the application to the proof of unconditionally, $\sigma_{I}$ is the
Rademacher sequence of independent identically distributed random variables
which take only the values $+1$ and $-1$.
On the other hand, a simple classical case which is not covered by this
approach is the case of the Haar wavelet. The size estimate of the kernel
holds, nevertheless there is not enough regularity. As shown in [AG18], we
recover size and regularity estimates of Calderón-Zygmund type, after changing
the underlying metric. The right metric is the dyadic distance, instead of the
Euclidean one.
For a sequence of independent, unbounded random variables, $a_{I}(\omega)$,
$I\in\mathcal{D}$, defined on a probability space $(\Omega,\mathscr{P})$, the
kernel
$K(x,y;\omega)=\sum_{I\in\mathcal{D}}a_{I}(\omega)\psi_{I}(x)\psi_{I}(y),$
with $\psi$ a good wavelet as in Chapter 9 of [Dau92], is not even well
defined. We aim to use Cramér-Chernoff method in order to prove that the
kernels $K(x,y;\omega)$ are Calderón-Zygmund kernels valued on
$L^{2}(\Omega,\mathscr{P})$ when $a_{I}$ are independent subgaussian random
variables with variance factors bounded above.
The main results of this paper are the following. First we prove the almost
sure convergence of the series
$\sum_{I\in\mathcal{D}}a_{I}(\omega)\psi_{I}(x)\psi_{I}(y)$ for $x\neq y$,
$a_{I}$ independent and uniformly subgaussian. Second, we prove that for
$a_{I}$ independent and uniformly subgaussian and for
$\left|\psi(x)\right|+\left|\psi^{\prime}(x)\right|\leq\frac{C}{(1+\left|x\right|)^{1+\varepsilon}}$,
$\varepsilon>0$, the operator
$T:f\to\sum_{I\in\mathcal{D}}a_{I}(\omega)\left<f,\psi_{I}\right>\psi_{I}$ is
bounded from $L^{2}(\mathbb{R})$ to $L^{2}(L^{2}(\Omega,d\mathscr{P});dx)$.
Third, we show that under the same assumptions, $K(x,y;\cdot)$ is a Calderón-
Zygmund kernel valued in $L^{2}(\Omega,d\mathscr{P})$ and hence that $T$ is
bounded from $L^{p}(\mathbb{R})$ to $L^{p}(L^{2}(\Omega,d\mathscr{P});dx)$.
Then, we extend the above to the Haar case, with the size and smoothness
estimates for the kernel provided by the dyadic distance $\delta(x,y)$ in
$\mathbb{R}^{+}$ instead of $\left|x-y\right|$. As a byproduct we prove
concentration type inequalities for the random kernels about their mean value
kernels, and for the random operators about the operator induced by these mean
value kernel.
Section 2 is devoted to introduce the basic result regarding Cramér-Chernoff
method and subgaussian random variables. We also review in this section a
classical theorem due to Kolmogorov, the so called “Three Series Theorem” that
we shall use in the prove of the almost sure convergence of the series
defining the kernels.
In Section 3 we deal with the problem of convergence of the series for almost
every $\omega$ when the $a_{I}(\omega)$ are independent and uniformly
subgaussian.
In Section 4 we prove the $L^{2}$ boundedness of $T$ and the Calderón-Zygmund
estimates of the kernel with respect to the norm of
$L^{2}(\Omega,d\mathscr{P})$. Section 5 is devoted to the case of the Haar
system. Finally in Section 6 we consider the concentration inequalities.
## 2\. The Cramér-Chernoff bounding method and subgaussian random variables
For the sake of completeness, we shall briefly review in this section our main
tool. Namely the Cramér-Chernoff method ([Cra38], [Che52]). In doing so we
shall follow the lines of [BLM13]. The starting point is Markov’s inequality
for the distribution of a random variable with finite expected value.
Let $(\Omega,\mathscr{F},\mathscr{P})$ be a probability space and let $X$ be a
random variable with $\mathscr{E}\left|X\right|<\infty$. In other words $X\in
L^{1}(\Omega,d\mathscr{P})$. In the search of estimates for the tail
probabilities of $X$ about its mean $\mathscr{E}X$, we have to consider, for
$t>0$, the two probabilities
$\mathscr{P}\\{X-\mathscr{E}X\geq t\\}\textrm{ and
}\mathscr{P}\\{\mathscr{E}X-X\geq t\\}.$
Since for $\lambda>0$ fixed, the function of $t>0$ given by $e^{\lambda t}$ is
increasing, from Markov’s inequality we obtain
$\mathscr{P}\\{X-\mathscr{E}X\geq t\\}\leq e^{-\lambda
t}\mathscr{E}e^{\lambda(X-\mathscr{E}X)}$
and
$\mathscr{P}\\{\mathscr{E}X-X\geq t\\}\leq e^{-\lambda
t}\mathscr{E}e^{\lambda(\mathscr{E}X-X)}.$
The logarithmic moment-generating function
$\eta_{X-\mathscr{E}X}(\lambda)=\log\mathscr{E}e^{\lambda(X-\mathscr{E}X)}$
plays an important role in Cramér-Chernoff argument and provide an easy way to
generalize normality. Following [BLM13] we say that an integrable random
variable $X$ belongs to $\mathscr{G}(\nu)$ or that $X$ is subgaussian with
variance factor $\nu>0$ if the inequality
$\eta_{X-\mathscr{E}X}(\lambda)\leq\lambda^{2}\frac{\nu}{2}$ holds for every
$\lambda\in\mathbb{R}$.
###### Proposition 2.1.
If $X\in\mathscr{G}(\nu)$, then
$\mathscr{P}\\{X-\mathscr{E}X\geq t\\}\leq e^{-\frac{t^{2}}{2\nu}}$
and
$\mathscr{P}\\{\mathscr{E}X-X\geq t\\}\leq e^{-\frac{t^{2}}{2\nu}},$
for every $t>0$.
###### Proof.
Let us consider the first estimate. Since
$\mathscr{P}\\{X-\mathscr{E}X\geq t\\}\leq e^{-\lambda
t}\mathscr{E}e^{\lambda(X-\mathscr{E}X)}$
for every $\lambda\geq 0$, then
$\log\mathscr{P}\\{X-\mathscr{E}X\geq t\\}\leq-\lambda
t+\eta_{X-\mathscr{E}X)}(\lambda)\leq-\lambda t+\lambda^{2}\frac{\nu}{2},$
for every $\lambda\geq 0$. Hence
$\log\mathscr{P}\\{X-\mathscr{E}X\geq t\\}\leq\inf_{\lambda\geq
0}\left(\lambda^{2}\frac{\nu}{2}-\lambda t\right)=-\frac{t^{2}}{2\nu}$
and we are done. ∎
Notice that every normally distributed random variable is subgaussian. Observe
also that the Rademacher random variables are all subgaussian with $\nu=1$.
This fact follows from Hoeffding’s Lemma ([Hoe63]) that shows that every
bounded random variable is subgaussian. But, of course, not every subgaussian
random variable is bounded since normal random variables are subgaussian.
###### Proposition 2.2.
Assume that $X_{1},\ldots,X_{n}$ are independent random variables with
$\mathscr{E}\left|X_{j}\right|<\infty$ for every $j=1,\ldots,n$ and that
$X_{j}\in\mathscr{G}(\nu_{j})$. Then $S=\sum_{j=1}^{n}X_{j}$ belongs to
$\mathscr{G}\left(\sum_{j=1}^{n}\nu_{j}\right)$.
###### Proof.
Since $\mathscr{E}S=\sum_{j=1}^{n}\mathscr{E}X_{j}$, from independence we have
that
$\displaystyle\eta_{S-\mathscr{E}S}(\lambda)$
$\displaystyle=\log\mathscr{E}e^{\lambda(S-\mathscr{E}S)}$
$\displaystyle=\log\mathscr{E}e^{\lambda\sum_{j=1}^{n}(X_{j}-\mathscr{E}X_{j})}$
$\displaystyle=\log\mathscr{E}\prod_{j=1}^{n}e^{\lambda(X_{j}-\mathscr{E}X_{j})}$
$\displaystyle=\log\prod_{j=1}^{n}\mathscr{E}e^{\lambda(X_{j}-\mathscr{E}X_{j})}$
$\displaystyle=\sum_{j=1}^{n}\log\mathscr{E}e^{\lambda(X_{j}-\mathscr{E}X_{j})}$
$\displaystyle=\sum_{j=1}^{n}\eta_{X_{j}-\mathscr{E}X_{j}}(\lambda)$
$\displaystyle\leq\frac{\lambda^{2}}{2}\left(\sum_{j=1}^{n}\nu_{j}\right),$
as desired. ∎
The above result extends to series of independent random variables with
convergence in the $L^{2}(\Omega,d\mathscr{P})$ sense, provided that the
series $\sum_{j\geq 1}\mathscr{E}\left|X_{j}\right|$ and $\sum_{j\geq
1}\nu_{j}$ both converge.
###### Proposition 2.3.
Let $\\{X_{j}:j\geq 1\\}$ be a sequence of independent random variables with
$\sum_{j\geq 1}\mathscr{E}\left|X_{j}\right|<\infty$,
$X_{j}\in\mathscr{G}(\nu_{j})$, and $\sum_{j\geq 1}\nu_{j}=\nu<\infty$. Then,
the series $\sum_{j\geq 1}X_{j}$ converges in $L^{2}(\Omega,d\mathscr{P})$ to
a random variable $S$. Moreover,
$\left\|S-\mathscr{E}S\right\|^{2}_{L^{2}(\Omega,d\mathscr{P})}\leq 2\nu$.
###### Proof.
For $n\geq 1$, set $S_{n}=\sum_{j=1}^{n}X_{j}$. Then, with $n>m\geq 1$ we have
that
$\left\|S_{n}-S_{m}\right\|_{L^{2}(\Omega,d\mathscr{P})}=\left\|\sum_{j=m+1}^{n}X_{j}\right\|_{L^{2}(\Omega,d\mathscr{P})}\leq\left\|\sum_{j=m+1}^{n}(X_{j}-\mathscr{E}X_{j})\right\|_{L^{2}(\Omega,d\mathscr{P})}+\sum_{j=m+1}^{n}\mathscr{E}\left|X_{j}\right|.$
Since $\sum_{j=m+1}^{n}\mathscr{E}\left|X_{j}\right|<\infty$, the second term
above tends to zero for $m\to\infty$. For the first term we use Propositions
2.1 and 2.2
$\displaystyle\left\|\sum_{j=m+1}^{n}(X_{j}-\mathscr{E}X_{j})\right\|_{L^{2}(\Omega,d\mathscr{P})}^{2}$
$\displaystyle=\int_{\Omega}\left|\sum_{j=m+1}^{n}(X_{j}-\mathscr{E}X_{j})\right|^{2}d\mathscr{P}$
$\displaystyle=\int_{\Omega}\left(\int_{0}^{\left|\sum_{j=m+1}^{n}(X_{j}-\mathscr{E}X_{j})\right|^{2}}dt\right)d\mathscr{P}$
$\displaystyle=\int_{0}^{\infty}\mathscr{P}\left\\{\left|\sum_{j=m+1}^{n}(X_{j}-\mathscr{E}X_{j})\right|>\sqrt{t}\right\\}dt$
$\displaystyle\leq\int_{0}^{\infty}e^{-\tfrac{t}{2\sum_{j=m+1}^{n}\nu_{j}}}dt$
$\displaystyle=2\sum_{j=m+1}^{n}\nu_{j},$
which tends to zero for $m$ tending to infinity. Notice that
$\left\|S-\mathscr{E}S\right\|_{L^{2}(\Omega,d\mathscr{P})}=\lim_{n\to\infty}\left\|S_{n}-\mathscr{E}S_{n}\right\|_{L^{2}(\Omega,d\mathscr{P})}\leq\limsup_{n\to\infty}2\sum_{j=1}^{n}\nu_{j}=2\nu.$
∎
The above proposition extends to higher order moments of $S-\mathscr{E}S$.
This fact together with Theorem 2.1 on page 25 in [BLM13] will allow to show
that $S$ is also a subgaussian random variable. Theorem 2.1 in [BLM13] proves
that for a random variable $X$ with $\mathscr{E}X=0$, we have that
$X\in\mathscr{G}(4C)$ provided that $\mathscr{E}X^{2k}\leq k!C^{k}$ for every
$k=1,2,3,\ldots$
###### Proposition 2.4.
Let $\\{X_{j}:j\geq 1\\}$ be a sequence of independent random variables with
$\sum_{j\geq 1}\mathscr{E}\left|X_{j}\right|<\infty$,
$X_{j}\in\mathscr{G}(\nu_{j})$ and $\sum_{j\geq 1}\nu_{j}=\nu<\infty$. Then,
the series $\sum_{j\geq 1}X_{j}$ converges in $L^{2k}(\Omega,d\mathscr{P})$ to
random variable $S$ for every integer $k\geq 1$ and
$\left\|S-\mathscr{E}S\right\|^{2k}_{L^{2k}(\Omega,d\mathscr{P})}\leq
k!(2\nu)^{k}.$
###### Proof.
Set as before $S_{n}=\sum_{j=1}^{n}X_{j}$. Again, the Cauchy character of
$S_{n}$ in $L^{2k}(\Omega,d\mathscr{P})$ is determined by the behavior of the
tail norms
$\displaystyle\left\|\sum_{j=m+1}^{n}(X_{j}-\mathscr{E}X_{j})\right\|^{2k}_{L^{2k}(\Omega,d\mathscr{P})}$
$\displaystyle=\int_{\Omega}\left(\int_{0}^{\left|\sum_{j=m+1}^{n}(X_{j}-\mathscr{E}X_{j})\right|^{2k}}dt\right)d\mathscr{P}$
$\displaystyle\leq\int_{0}^{\infty}\mathscr{P}\left\\{\left|\sum_{j=m+1}^{n}(X_{j}-\mathscr{E}X_{j})\right|>t^{\tfrac{1}{2k}}\right\\}dt$
$\displaystyle\leq\int_{0}^{\infty}e^{-\frac{(t^{1/2k})^{2}}{2\sum_{m+1}^{n}\nu_{j}}}dt$
$\displaystyle=\Bigl{(}2\sum_{m+1}^{n}\nu_{j}\Bigr{)}^{k}\int_{0}^{\infty}e^{-s}s^{k-1}ds$
$\displaystyle=k\Gamma(k)\Bigl{(}2\sum_{m+1}^{n}\nu_{j}\Bigr{)}^{k}$
$\displaystyle=k!\Bigl{(}2\sum_{m+1}^{n}\nu_{j}\Bigr{)}^{k}.$
This estimate proves both, the convergence of the series in
$L^{2k}(\Omega,d\mathscr{P})$ and the inequality
$\left\|S-\mathscr{E}S\right\|^{2k}_{L^{2k}(\Omega,d\mathscr{P})}\leq
k!(2\nu)^{k}.$
∎
###### Proposition 2.5.
Let $\\{X_{j}:j\geq 1\\}$ be a sequence of independent random variables with
$\sum_{j\geq 1}\mathscr{E}\left|X_{j}\right|<\infty$,
$X_{j}\in\mathscr{G}(\nu_{j})$ and $\sum_{j\geq 1}\nu_{j}=\nu<\infty$. Then
$S=\sum_{j\geq 1}X_{j}$ converges in $L^{p}(\Omega,d\mathscr{P})$ for every
$p<\infty$ and $S\in\mathscr{G}(8\nu)$.
###### Proof.
Follows from Proposition 2.4 and Theorem 2.1 in [BLM13]. ∎
For completeness we finish this section with a well known result in
Probability Theory that shall be used in the further development of the main
results. From [Chu01] we take the following statement of Kolmogorov’s Three
Series Theorem regarding the a.e. convergence of series of independent random
variables.
###### Theorem (Kolmogorov’s Theorem ([Chu01], Chapter 5)).
If $(X_{n})$ is a sequence of independent random variables and $A$ is a
positive number, the almost everywhere convergence of the series
$\sum_{n}X_{n}$ is equivalent to the simultaneous convergence of the following
three numerical series
1. (i)
$\sum_{n}\mathscr{P}\\{X_{n}\neq Y_{n}\\}$;
2. (ii)
$\sum_{n}\mathscr{E}(Y_{n})$ and
3. (iii)
$\sum_{n}\sigma^{2}(Y_{n})$,
with
$Y_{n}=\begin{cases}X_{n}&if\left|X_{n}\right|\leq A\\\
0&if\left|X_{n}\right|>A.\end{cases}$
For the distribution, mean and variance of truncations we have the following
straightforward result.
###### Lemma 2.6.
Let $X:\Omega\to\mathbb{R}$ be a random variable with finite variance. Let $A$
be a given positive number and
$X_{A}(\omega)=\begin{cases}X(\omega)&if\left|X(\omega)\right|\leq A\\\
0&if\left|X(\omega)\right|>A.\end{cases}$
Then the distribution measure $\mu_{X_{A}}$ of $X_{A}$ is related to the
distribution measure $\mu_{X}$ of $X$ by
$\mu_{X_{A}}(B)=\mu_{X}((B\cap[-A,A])\setminus\\{0\\})+(\mu_{X}(\\{0\\})+\mu_{X}([-A,A]^{c}))\delta_{0}(B)$,
where $\delta_{0}$ is the Dirac delta at the origin, $[-A,A]$ is the closed
interval $-A\leq x\leq A$, $[-A,A]^{c}=\mathbb{R}\setminus[-A,A]$ and $B$ is a
one dimensional Borel set. Hence
$\mathscr{E}(X_{A})=\int_{[-A,A]}xd\mu_{X}(x),$
and
$Var(X_{A})=\int_{[-A,A]}x^{2}d\mu_{X}(x)-\left(\int_{[-A,A]}x\mu_{X}(x)\right)^{2}.$
## 3\. The almost everywhere convergence of the series $K(x,y;\omega)$
The main result of this section is the almost sure convergence of the series
$\sum_{I\in\mathcal{D}}a_{I}(\omega)\psi_{I}(x)\psi_{I}(y)=K(x,y;\omega)$
for $x\neq y$,
$\left|\psi(x)\right|\leq\frac{C}{(1+\left|x\right|)^{1+\varepsilon}}$ and
$\\{a_{I}:I\in\mathcal{D}\\}$ independent random variables in
$\mathscr{G}(\nu)$ for some $\nu>0$.
###### Theorem 3.1.
Let $\psi$ be such that there exist positive constants $C$ and $\varepsilon$
with $\left|\psi(x)\right|\leq C(1+\left|x\right|)^{-1-\varepsilon}$ for every
$x\in\mathbb{R}$. Assume that $\widetilde{a}=\\{a_{I}:I\in\mathcal{D}\\}$ is a
sequence of independent random variables on the probability space
$(\Omega,\mathscr{F},\mathscr{P})$ such that
$\\{a_{I}:I\in\mathcal{D}\\}\subset\mathscr{G}(\nu)$ for some positive $\nu$
and $\sum_{I}\mathscr{E}\left|a_{I}\right|<\infty$. Then for every $x\neq y$
the series
$K_{\widetilde{a}}(x,y;\omega)=\sum_{I\in\mathcal{D}}a_{I}(\omega)\psi_{I}(x)\psi_{I}(y)$
converges for almost every $\omega\in\Omega$.
###### Proof.
Fix $x\neq y$ both in $\mathbb{R}$. We shall use the Three Series Theorem of
Kolmogorov in order to prove the desired convergence. Notice first that since
$\sum_{I}\mathscr{E}\left|a_{I}\right|$ converges it is enough to prove the
convergence of the series
$\sum_{I}(a_{I}(\omega)-\mathscr{E}a_{I})\psi_{I}(x)\psi_{I}(y)$ for almost
every $\omega\in\Omega$. Set
$X_{I}(x,y)(\omega)=(a_{I}(\omega)-\mathscr{E}a_{I})\psi_{I}(x)\psi_{I}(y)$.
Take $A>0$ fixed. Define
$Y_{I}(x,y)(\omega)=\begin{cases}X_{I}(x,y)(\omega)&if\left|X_{I}(x,y)(\omega)\right|\leq
A\\\ 0&if\left|X_{I}(x,y)(\omega)\right|>A.\end{cases}$
Let us start by checking (i) in Kolmogorov’s Theorem. For fixed
$I\in\mathcal{D}$ we have
$\displaystyle\mathscr{P}\\{X_{I}(x,y)\neq Y_{I}(x,y)\\}$
$\displaystyle=\mathscr{P}\\{\left|X_{I}(x,y)\right|>A\\}$
$\displaystyle=\mathscr{P}\\{\left|a_{I}-\mathscr{E}a_{I}\right|\left|\psi_{I}(x)\right|\left|\psi_{I}(y)\right|>A\\}$
$\displaystyle=\begin{cases}0&if\left|\psi_{I}(x)\right|\left|\psi_{I}(y)\right|=0,\\\
\mathscr{P}\\{\left|a_{I}-\mathscr{E}a_{I}\right|>\frac{A}{\left|\psi_{I}(x)\right|\left|\psi_{I}(y)\right|}\\}&if\left|\psi_{I}(x)\right|\left|\psi_{I}(y)\right|\neq
0\end{cases}$ $\displaystyle\leq
e^{-\tfrac{1}{2\nu}\tfrac{A^{2}}{\left|\psi_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2}}},$
the last inequality follows from Proposition 2.1 since the random variables
$a_{I}$ are uniformly in $\mathscr{G}(\nu)$. Now, since the estimates in
[Dau92], we have that
$\sum_{I\in\mathcal{D}}\left|\psi_{I}(x)\right|\left|\psi_{I}(y)\right|$
converges for $x\neq y$, so does the series
$\sum_{I\in\mathcal{D}}\mathscr{P}\\{X_{I}(x,y)\neq
Y_{I}(x,y)\\}\leq\sum_{I\in\mathcal{D}}e^{-\tfrac{1}{2\nu}\tfrac{A^{2}}{\left|\psi_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2}}}.$
The series in (ii) of Kolmogorov’s Theorem converges since
$\displaystyle\sum_{I\in\mathcal{D}}\mathscr{E}\left|Y_{I}(x,y)\right|$
$\displaystyle\leq\sum_{I\in\mathcal{D}}\mathscr{E}\left|X_{I}(x,y)\right|$
$\displaystyle=\sum_{I\in\mathcal{D}}\mathscr{E}\left|a_{I}-\mathscr{E}a_{I}\right|\left|\psi_{I}(x)\right|\left|\psi_{I}(y)\right|$
$\displaystyle\leq
2\sum_{I\in\mathcal{D}}\mathscr{E}\left|a_{I}\right|\left|\psi_{I}(x)\right|\left|\psi_{I}(y)\right|$
$\displaystyle\leq M\frac{1}{\left|x-y\right|},$
for some $M>0$. Let us finally check the convergence of the third series of
Kolmogorov. In fact
$\displaystyle\sigma^{2}(Y_{I}(x,y))$
$\displaystyle=\mathscr{E}(Y_{I}(x,y))^{2}-(\mathscr{E}Y_{I}(x,y))^{2}$
$\displaystyle\leq\mathscr{E}(X_{I}(x,y))^{2}+(\mathscr{E}\left|X_{I}(x,y)\right|)^{2}$
$\displaystyle=\mathscr{E}((a_{I}-\mathscr{E}a_{I})\psi_{I}(x)\psi_{I}(y))^{2}+(\mathscr{E}\left|X_{I}(x,y)\right|)^{2}.$
From the estimate for (ii) we see that
$\sum_{I\in\mathcal{D}}(\mathscr{E}\left|X_{I}(x,y)\right|)^{2}$ is finite for
$x\neq y$. Let us use the fact that $a_{I}\in\mathscr{G}(\nu)$ to estimate the
first term above. Write
$\displaystyle\mathscr{E}((a_{I}-\mathscr{E}a_{I})\psi_{I}(x)\psi_{I}(y))^{2}$
$\displaystyle=\int_{\Omega}(a_{I}-\mathscr{E}a_{I})^{2}\psi_{I}^{2}(x)\psi_{I}^{2}(y)d\mathscr{P}(\omega)$
$\displaystyle=\psi_{I}^{2}(x)\psi_{I}^{2}(y)\int_{\Omega}\left(\int_{0}^{(a_{I}-\mathscr{E}a_{I})^{2}}dt\right)d\mathscr{P}$
$\displaystyle=\psi_{I}^{2}(x)\psi_{I}^{2}(y)\int_{0}^{\infty}\mathscr{P}\\{\left|a_{I}(\omega)-\mathscr{E}a_{I}\right|>\sqrt{t}\\}dt$
$\displaystyle\leq\psi_{I}^{2}(x)\psi_{I}^{2}(y)\int_{0}^{\infty}e^{-\tfrac{t}{2\nu}}dt$
$\displaystyle=2\nu\psi_{I}^{2}(x)\psi_{I}^{2}(y).$
So that
$\sum_{I\in\mathcal{D}}\mathscr{E}((a_{I}-\mathscr{E}a_{I})\psi_{I}(x)\psi_{I}(y))^{2}\leq
2\nu\sum_{I\in\mathcal{D}}\left|\psi_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2}\leq
2\nu\frac{c^{2}}{\left|x-y\right|^{2}},$
and we are done. ∎
Let us point out that the above result does not involve any assumption of
smoothness on the wavelet $\psi$. Hence the result holds also for the Haar
wavelet, since being
$h_{0}^{0}(x)=\mathcal{X}_{[0,1/2)}(x)-\mathcal{X}_{[1/2.1)}(x)$ compactly
supported, certainly satisfies the estimate $\left|h_{0}^{0}(x)\right|\leq
C(1+\left|x\right|)^{-1-\varepsilon}$.
## 4\. $K(x,y;\omega)$ as on $L^{2}(\Omega,d\mathscr{P})$ valued Calderón-
Zygmund kernel. The case of $\psi$ smooth
For the main result of this section the wavelet function $\psi$ is assumed to
satisfy the classical condition
$\left|\psi(x)\right|+\left|\psi^{\prime}(x)\right|\leq\frac{C}{(1+\left|x\right|)^{1+\varepsilon}}$
(4.1)
for every $x\in\mathbb{R}$ and some positive constants $C$ and $\varepsilon$.
We shall also assume that $\\{a_{I}(\omega):I\in\mathcal{D}\\}$ is a sequence
of random variables in $(\Omega,\mathscr{F},\mathscr{P})$ such that
* (4.2.$i$)
the $a_{I}$’s are independent random variables;
* (4.2.$e$)
$\sum_{I\in\mathcal{D}}\mathscr{E}\left|a_{I}\right|<\infty$;
* (4.2.$\sigma$)
$\\{a_{I}:I\in\mathcal{D}\\}\subset\mathscr{G}(\nu)$ for some $\nu>0$.
Let us start by the $L^{2}(\mathbb{R},dx)$ theory. Notice that in general the
operator
$T:f\longrightarrow\sum_{I\in\mathcal{D}}a_{I}(\omega)\left<f,\psi_{I}\right>\psi_{I}(x),$
for a given $\omega\in\Omega$, is not bounded on $L^{2}(\mathbb{R})$, since
$a_{I}(\omega)$ can be unbounded as a sequence on $\mathcal{D}$. Nevertheless
we have the following result.
###### Theorem 4.1.
Assume that the sequence $\\{a_{I}:I\in\mathcal{D}\\}$ satisfies $(4.2.i)$,
$(4.2.e)$ and $(4.2.\sigma)$. Assume also that
$\left|\psi(x)\right|\leq\tfrac{C}{(1+\left|x\right|)^{1+\varepsilon}}$ for
some $C>0$, some $\varepsilon>0$ and every $x\in\mathbb{R}$. Then $T$ is
bounded as an operator from $L^{2}(\mathbb{R},dx)$ to
$L^{2}(L^{2}(\Omega,d\mathscr{P});dx)$.
###### Proof.
Let us denote with
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{2}$
the norm in $L^{2}(L^{2}(\Omega,d\mathscr{P});dx)$ and
$\left\|\cdot\right\|_{2}$ the $L^{2}(dx)$ norm. Then
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|Tf\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{2}^{2}=\int_{\mathbb{R}}\left(\int_{\Omega}\left|\sum_{I\in\mathcal{D}}a_{I}(\omega)\left<f,\psi_{I}\right>\psi_{I}(x)\right|^{2}d\mathscr{P}(\omega)\right)dx.$
For fixed $x\in\mathbb{R}$ we can estimate
$\int_{\Omega}\left|\sum_{I\in\mathcal{D}}a_{I}(\omega)\left<f,\psi_{I}\right>\psi_{I}(x)\right|^{2}d\mathscr{P}(\omega)$
using the fact that the random variables $a_{I}$ are subgaussian. In the
perspective of Proposition 2.5 in Section 2 above, set
$X_{I}(\omega)=a_{I}(\omega)\left<f,\psi_{I}\right>\psi_{I}(x)$, for
$I\in\mathcal{D}$. Since $a_{I}\in\mathscr{G}(\nu)$, we have that
$\eta_{a_{I}-\mathscr{E}a_{I}}(\lambda)\leq\lambda^{2}\frac{\nu}{2}.$
Hence
$\displaystyle\eta_{X_{I}-\mathscr{E}X_{I}}(\lambda)$
$\displaystyle=\log\mathscr{E}e^{\lambda\left<f,\psi_{I}\right>\psi_{I}(x)(a_{I}-\mathscr{E}a_{I})}$
$\displaystyle=\eta_{a_{I}-\mathscr{E}a_{I}}(\lambda\left|\left<f,\psi_{I}\right>\right|\left|\psi_{I}(x)\right|)$
$\displaystyle\leq\frac{\nu}{2}\lambda^{2}\left|\left<f,\psi_{I}\right>\right|^{2}\left|\psi_{I}(x)\right|^{2}.$
So that $\\{X_{I}:I\in\mathcal{D}\\}$ is a sequence of independent random
variables with
$\sum_{I\in\mathcal{D}}\mathscr{E}\left|X_{I}\right|=\left|\left<f,\psi_{I}\right>\right|\left|\psi_{I}(x)\right|\sum_{I\in\mathcal{D}}\mathscr{E}\left|a_{I}\right|<\infty.$
Also, from the above estimate for $\eta_{X_{I}-\mathscr{E}X_{I}}(\lambda)$, we
see that
$X_{I}\in\mathscr{G}(\nu\left|\left<f,\psi_{I}\right>\right|^{2}\left|\psi_{I}(x)\right|^{2})$.
Since
$\int_{\mathbb{R}}\sum_{I\in\mathcal{D}}\nu\left|\left<f,\psi_{I}\right>\right|^{2}\left|\psi_{I}(x)\right|^{2}dx=\nu\sum_{I\in\mathcal{D}}\left|\left<f,\psi_{I}\right>\right|^{2}\left\|\psi_{I}\right\|^{2}=\nu\left\|f\right\|^{2}$,
we have, except for a null set in $\mathbb{R}$, that the series
$\sum_{I\in\mathcal{D}}\nu\left|\left<f,\psi_{I}\right>\right|^{2}\left|\psi_{I}(x)\right|^{2}=\nu\sum_{I\in\mathcal{D}}\left|\left<f,\psi_{I}\right>\right|^{2}\left|\psi_{I}(x)\right|^{2}$
converges. Then from Proposition 2.5 we get that
$\sum_{I\in\mathcal{D}}X_{I}(\omega)$ converges in
$L^{2}(\Omega,d\mathscr{P})$ to a sum $S$ that belongs to
$\mathscr{G}\left(8\nu\sum_{I\in\mathcal{D}}\left|\left<f,\psi_{I}\right>\right|^{2}\left|\psi_{I}(x)\right|^{2}\right)$.
Briefly,
$\eta_{\sum_{I\in\mathcal{D}}(a_{I}-\mathscr{E}a_{I})\left<f,\psi_{I}\right>\psi_{I}(x)}(\lambda)\leq
e^{-\tfrac{8\nu}{2}\left(\sum_{I\in\mathcal{D}}\left|\left<f,\psi_{I}\right>\right|^{2}\left|\psi_{I}(x)\right|^{2}\right)\lambda^{2}}.$
So that, from Proposition 2.1,
$\mathscr{P}\left\\{\left|\sum_{I\in\mathcal{D}}(a_{I}-\mathscr{E}a_{I})\left<f,\psi_{I}\right>\psi_{I}(x)\right|>t\right\\}\leq
2e^{-\tfrac{t^{2}}{4\nu\sum_{I\in\mathcal{D}}\left|\left<f,\psi_{I}\right>\right|^{2}\left|\psi_{I}(x)\right|^{2}}}.$
(4.2)
With this last estimate in mind we are in position to obtain an upper bound
for
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|Tf\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}^{2}_{2}$.
In fact, notice first that for $x$ fixed as above,
$\displaystyle\int_{\Omega}\left|\sum_{I\in\mathcal{D}}(a_{I}-\mathscr{E}a_{I})\left<f,\psi_{I}\right>\psi_{I}(x)\right|^{2}d\mathscr{P(\omega)}$
$\displaystyle=\int_{\Omega}\left(\int_{0}^{\left|\sum_{I\in\mathcal{D}}(a_{I}-\mathscr{E}a_{I})\left<f,\psi_{I}\right>\psi_{I}(x)\right|^{2}}dt\right)d\mathscr{P}(\omega)$
$\displaystyle=\int_{0}^{\infty}\left(\int_{\\{\omega\in\Omega:\left|\sum_{I\in\mathcal{D}}(a_{I}-\mathscr{E}a_{I})\left<f,\psi_{I}\right>\psi_{I}(x)\right|>\sqrt{t}\\}}d\mathscr{P}(\omega)\right)dt$
$\displaystyle\leq
2\int_{0}^{\infty}e^{-\tfrac{t}{4\nu\sum_{I\in\mathcal{D}}\left|\left<f,\psi_{I}\right>\right|^{2}\left|\psi_{I}(x)\right|^{2}}}dt$
$\displaystyle=8\nu\sum_{I\in\mathcal{D}}\left|\left<f,\psi_{I}\right>\right|^{2}\left|\psi_{I}(x)\right|^{2}.$
And
$\displaystyle{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|Tf-\Bigl{(}\sum_{I\in\mathcal{D}}\mathscr{E}a_{I}\Bigr{)}f\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}^{2}_{2}$
$\displaystyle=\int_{\mathbb{R}}\left(\int_{\Omega}\left|\sum_{I\in\mathcal{D}}(a_{I}(\omega)-\mathscr{E}a_{I})\left<f,\psi_{I}\right>\psi_{I}(x)\right|^{2}d\mathscr{P}(\omega)\right)dx$
$\displaystyle\leq
8\nu\sum_{I\in\mathcal{D}}\left|\left<f,\psi_{I}\right>\right|^{2}\int_{\mathbb{R}}\left|\psi_{I}(x)\right|^{2}dx$
$\displaystyle=8\nu\left\|f\right\|_{2}^{2}.$
Hence
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|Tf\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{2}\leq\left\|Tf-\Bigl{(}\sum_{I\in\mathcal{D}}\mathscr{E}a_{I}\Bigr{)}f\right\|_{2}+\Bigl{(}\sum_{I\in\mathcal{D}}\mathscr{E}a_{I}\Bigr{)}\left\|f\right\|_{2}\leq\Bigl{(}\sqrt{8\nu}+\sum_{I\in\mathcal{D}}\mathscr{E}\left|a_{I}\right|\Bigr{)}\left\|f\right\|_{2}.$
∎
The kernel of the operator $T$ is given by
$K(x,y;\omega)=\sum_{I\in\mathcal{D}}a_{I}(\omega)\psi_{I}(x)\psi_{I}(y)$. We
shall think $K(x,y;\cdot)$ as a kernel defined in $\mathbb{R}^{2}$ with values
in $L^{2}(\Omega,d\mathscr{P})$. The next result contain the basic estimates
showing that $K(x,y;\cdot)$ is an $L^{2}(\Omega,d\mathscr{P})$ valued
Calderón-Zygmund kernel.
###### Theorem 4.2.
Assume that the wavelet $\psi$ satisfies (4.1) and that the $a_{I}$’s satisfy
(4.2.i), (4.2.e) and (4.2.$\sigma$). Then there exists a constant $B$ such
that
1. (4.2.a)
$\left\|K(x,y,\cdot)\right\|_{L^{2}(\Omega,d\mathscr{P})}\leq\frac{B}{\left|x-y\right|}$,
$x,y\in\mathbb{R}$;
2. (4.2.b)
$\left\|\frac{\partial K}{\partial
x}(x,y,\cdot)\right\|_{L^{2}(\Omega,d\mathscr{P})}+\left\|\frac{\partial
K}{\partial
y}(x,y,\cdot)\right\|_{L^{2}(\Omega,d\mathscr{P})}\leq\frac{B}{\left|x-y\right|^{2}}$,
$x,y\in\mathbb{R}$.
###### Proof.
Since
$K(x,y;\omega)=\Sigma(x,y;\omega)+\sum_{I\in\mathcal{D}}\mathscr{E}a_{I}\psi_{I}(x)\psi_{I}(y)$
with
$\Sigma(x,y;\omega)=\sum_{I\in\mathcal{D}}(a_{I}(\omega)-\mathscr{E}a_{I})\psi_{I}(x)\psi_{I}(y)$,
and from the classical result in [Dau92] we have that
$\left|\sum_{I\in\mathcal{D}}\mathscr{E}a_{I}\psi_{I}(x)\psi_{I}(y)\right|$
and its partial derivatives satisfy the desired estimates, it is enough to
prove (4.2.a) and (4.2.b) with $\Sigma(x,y;\omega)$ instead of
$K(x,y;\omega)$. Let us start proving (4.2.a) for $\Sigma(x,y,\cdot)$. Let us
use again Proposition 2.5. Take now
$X_{I}(\omega)=(a_{I}(\omega)-\mathscr{E}a_{I})\psi_{I}(x)\psi_{I}(y)$ for
$x\neq y$ both fixed. Notice first that
$\displaystyle\sum_{I\in\mathcal{D}}\mathscr{E}\left|X_{I}\right|$
$\displaystyle=\sum_{I\in\mathcal{D}}\mathscr{E}(\left|a_{I}(\omega)-\mathscr{E}a_{I}\right|)\left|\psi_{I}(x)\right|\left|\psi_{I}(y)\right|$
$\displaystyle\leq
2\sum_{I\in\mathcal{D}}\mathscr{E}\left|a_{I}\right|\left|\psi_{I}(x)\right|\left|\psi_{I}(y)\right|$
$\displaystyle\leq
2\sup_{I\in\mathcal{D}}\mathscr{E}\left|a_{I}\right|\frac{c}{\left|x-y\right|}<\infty.$
Also
$\displaystyle\eta_{X_{I}}(\lambda)$
$\displaystyle=\eta_{(a_{I}(\omega)-\mathscr{E}a_{I})\psi_{I}(x)\psi_{I}(y)}(\lambda)$
$\displaystyle=\log\mathscr{E}e^{\lambda\psi_{I}(x)\psi_{I}(y)(a_{I}(\omega)-\mathscr{E}a_{I})}$
$\displaystyle=\eta_{a_{I}-\mathscr{E}a_{I}}(\lambda\left|\psi_{I}(x)\right|\left|\psi_{I}(y)\right|)$
$\displaystyle\leq\frac{\nu}{2}\lambda^{2}\left|\psi_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2}$
so that $\\{X_{I}:I\in\mathcal{D}\\}$ is a sequence of independent random
variables with $\sum_{I\in\mathcal{D}}\mathscr{E}\left|X_{I}\right|<\infty$
and
$X_{I}\in\mathscr{G}(\nu\left|\psi_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2})$.
On the other hand, the estimates in [Dau92] show that the series
$\sum_{I\in\mathcal{D}}\nu\left|\psi_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2}=\nu\sum_{I\in\mathcal{D}}\left|\psi_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2}$
converges. Then, from Proposition 2.5, we have that
$\Sigma(x,y;\omega)\in\mathscr{G}(8\nu\sum_{I\in\mathcal{D}}\left|\psi_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2})$.
Now, from Proposition 2.1, we get
$\mathscr{P}\\{\left|\Sigma(x,y;\omega)\right|>t\\}\leq
2e^{-\frac{t^{2}}{4\nu\sum_{I\in\mathcal{D}}\left|\psi_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2}}}.$
(4.3)
Hence
$\displaystyle\left\|\Sigma(x,y;\cdot)\right\|^{2}_{L^{2}(\Omega,d\mathscr{P})}$
$\displaystyle=\int_{\Omega}\left|\Sigma(x,y;\omega)\right|^{2}d\mathscr{P}$
$\displaystyle=\int_{\Omega}\left(\int_{0}^{\left|\Sigma(x,y;\omega)\right|^{2}}dt\right)d\mathscr{P}$
$\displaystyle=\int_{0}^{\infty}\mathscr{P}\\{\left|\Sigma(x,y;\omega)\right|^{2}>t\\}dt$
$\displaystyle\leq
2\int_{0}^{\infty}e^{-\frac{t}{4\nu\sum_{I\in\mathcal{D}}\left|\psi_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2}}}dt$
$\displaystyle=8\nu\sum_{I\in\mathcal{D}}\left|\psi_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2}$
$\displaystyle\leq\frac{8\nu C}{\left|x-y\right|^{2}},$
and (4.2.a) is proved for $\Sigma$.
Let us now prove (4.2.b). It suffices to show that
$\left\|\frac{\partial\Sigma}{\partial
x}(x,y,\cdot)\right\|_{L^{2}(\Omega,d\mathscr{P})}\leq\frac{B}{\left|x-y\right|^{2}}$.
With the arguments in the proof of Theorem 3.1 and the assumptions on $\psi$
and $\psi^{\prime}$, we have that the series
$\sum_{I\in\mathcal{D}}(a_{I}(\omega)-\mathscr{E}a_{I})\left|I\right|^{-2}\psi^{\prime}(2^{j(I)}x-k(I))\psi(2^{j(I)}y-k(I))$
converges for almost every $\omega\in\Omega$. Then
$\frac{\partial\Sigma}{\partial
x}(x,y;\omega)=\sum_{I\in\mathcal{D}}(a_{I}(\omega)-\mathscr{E}a_{I})\left|I\right|^{-1}\widetilde{\psi}_{I}(x)\psi_{I}(y),$
where $\widetilde{\psi}=\frac{d\psi}{dx}$. Since $\psi$ and $\widetilde{\psi}$
have the same size estimate, we can proceed as in the proof of (4.2.a). In
fact, we shall use again Proposition 2.5 with
$X_{I}(\omega)=(a_{I}(\omega)-\mathscr{E}a_{I})\left|I\right|^{-1}\widetilde{\psi}_{I}(x)\psi_{I}(y)$,
for $x\neq y$. Now
$\displaystyle\sum_{I\in\mathcal{D}}\mathscr{E}\left|X_{I}\right|$
$\displaystyle\leq
2\sum_{I\in\mathcal{D}}\mathscr{E}\left|a_{I}\right|\left|I\right|^{-1}\widetilde{\psi}_{I}(x)\psi_{I}(y)$
$\displaystyle\leq
2\sup_{I\in\mathcal{D}}\mathscr{E}\left|a_{I}\right|\left(\sum_{I\in\mathcal{D}}\left|I\right|^{-1}\widetilde{\psi}_{I}(x)\psi_{I}(y)\right)$
$\displaystyle\leq
2\sup_{I\in\mathcal{D}}\mathscr{E}\left|a_{I}\right|\frac{C}{\left|x-y\right|^{2}}.$
Also
$\displaystyle\eta_{X_{I}}(\lambda)$
$\displaystyle=\log\mathscr{E}e^{\lambda\left|I\right|^{-1}\widetilde{\psi}_{I}(x)\psi_{I}(y)(a_{I}-\mathscr{E}a_{I})}$
$\displaystyle=\eta_{a_{I}-\mathscr{E}a_{I}}(\lambda\left|I\right|^{-1}\left|\widetilde{\psi}_{I}(x)\right|\left|\psi_{I}(y)\right|)$
$\displaystyle\leq\frac{\nu}{2}\lambda^{2}\left|I\right|^{-2}\left|\widetilde{\psi}_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2}.$
Hence, from Proposition 2.5,
$\frac{\partial\Sigma}{\partial
x}(x,y;\omega)\in\mathscr{G}\left(8\nu\sum_{I\in\mathcal{D}}\left|I\right|^{-2}\left|\widetilde{\psi}_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2}\right).$
Then
$\displaystyle\left\|\frac{\partial\Sigma}{\partial
x}(x,y;\cdot)\right\|^{2}_{L^{2}(\Omega,d\mathscr{P})}$
$\displaystyle=\int_{\Omega}\left|\frac{\partial\Sigma}{\partial
x}(x,y;\omega)\right|^{2}d\mathscr{P}$ $\displaystyle\leq
2\int_{0}^{\infty}e^{-\frac{t}{4\nu\sum_{I\in\mathcal{D}}\left|I\right|^{-2}\left|\widetilde{\psi}_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2}}}dt$
$\displaystyle=8\nu\sum_{I\in\mathcal{D}}\left|I\right|^{-2}\left|\widetilde{\psi}_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2}$
$\displaystyle\leq\frac{8\nu C}{\left|x-y\right|^{4}},$
the last estimate follows again as in [Dau92]. ∎
Now the boundedness properties of $T$ follow from the general results on
vector valued singular integrals in [RRT86] or [GLY09].
###### Theorem 4.3.
Assume that the wavelet $\psi$ satisfies (4.1) and that the $a_{I}$’s satisfy
(4.2.i), (4.2.e) and (4.2.$\sigma$). Then for $1<p<\infty$,
$Tf=\sum_{I\in\mathcal{D}}a_{I}\left<f,\psi_{I}\right>\psi_{I}$ is bounded as
an operator from $L^{p}(\mathbb{R},dx)$ to
$L^{p}(L^{2}(\Omega,d\mathscr{P});dx)$. Moreover,
$\left|\\{x\in\mathbb{R}:\left\|Tf(x)\right\|_{L^{2}(\Omega,d\mathscr{P})}>t\\}\right|\leq\frac{C}{\lambda}\left\|f\right\|_{L^{1}(\mathbb{R},dx)}.$
## 5\. $K(x,y;\omega)$ as an $L^{2}(\Omega,d\mathscr{P})$ valued Calderón-
Zygmund kernel defined in the space of homogeneous type
$(\mathbb{R}^{+},\delta,\left|\cdot\right|)$. The case of the Haar wavelet
Let us observe first that since the function
$\psi(x)=\mathcal{X}_{[0,1/2)}(x)-\mathcal{X}_{[1/2,1)}(x)$ satisfies the
basic size estimate
$\left|\psi(x)\right|\leq\frac{C}{(1+\left|x\right|)^{1+\varepsilon}}$, all
the results in the previous section which do not involve smoothness holds for
the Haar wavelet. In this section we shall briefly sketch the results for the
Haar wavelet following the lines in [AG18] where a natural metric structure in
$\mathbb{R}^{+}$ allows to use the general theory of Calderón-Zygmund Singular
Integrals.
In particular Theorem 3.1 and Theorem 4.1 hold for the Haar function. The only
results that needs to be considered is an analogous of Theorem 4.2.
Set $\mathbb{R}^{+}$ to denote the set of nonnegative real numbers and
$\mathcal{D}^{+}$ the set of dyadic intervals in $\mathbb{R}^{+}$. The set
$\mathbb{R}^{+}$ with Lebesgue measure and the dyadic distance
$\delta(x,y)=\inf\\{\left|I\right|:x,y\in I,I\in\mathcal{D}^{+}\\}$ is a space
of homogeneous type. Actually $(\mathbb{R}^{+},\delta,\left|\cdot\right|)$ is
a $1$-Ahlfors regular or normal space. Moreover, the kernel $K(x,y,\cdot)$
valued in $L^{2}(\Omega,d\mathscr{P})$ is a Calderón-Zygmund kernel in this
space of homogeneous type.
###### Theorem 5.1.
Let $\psi$ be the Haar wavelet. Assume that the $a_{I}$’s satisfy (4.2.i),
(4.2.e) and (4.2.$\sigma$). Then there exists a constant $B$ such that
* (5.1.a)
$\left\|K(x,y,\cdot)\right\|_{L^{2}(\Omega,d\mathscr{P})}\leq\frac{B}{\delta(x,y)}$,
$x,y\in\mathbb{R}^{+}$;
* (5.1.b.i)
$\left\|K(x^{\prime},y;\cdot)-K(x,y;\cdot)\right\|_{L^{2}(\Omega,d\mathscr{P})}\leq
B\frac{\delta(x^{\prime},x)}{\delta(x,y)^{2}}$, when
$2\delta(x^{\prime},x)\leq\delta(x,y)$;
* (5.1.b.ii)
$\left\|K(x,y^{\prime};\cdot)-K(x,y;\cdot)\right\|_{L^{2}(\Omega,d\mathscr{P})}\leq
B\frac{\delta(y,y^{\prime})}{\delta(x,y)^{2}}$, when
$2\delta(y^{\prime},y)\leq\delta(x,y)$.
Let us point out here that once the above results is proved, the analogous of
Theorem 4.3 for the Haar system follow from the general setting of the
Calderón-Zygmund theory given in [GLY09].
###### Proof of Theorem 5.1.
Let us start with (5.1.a). Notice that (4.1.a) holds since only th size
condition on $\psi$ is used in its proof. Nevertheless, since
$\delta(x,y)\geq\left|x-y\right|$ but there metrics are not equivalent,
(5.1.a) is a better estimate for the size of $K$ which can not be directly
obtained from (4.2.a). For $x\neq y$ both in $\mathbb{R}^{+}$, with the
notation in the proof of Theorem 4.2, we have that inequality (4.3) holds
mutatis mutandis for $\psi$ the Haar wavelet. Then
$\left\|\Sigma(x,y;\cdot)\right\|^{2}_{L^{2}(\Omega,d\mathscr{P})}\leq
8\nu\sum_{I\in\mathcal{D}}\left|\psi_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2}.$
Let us estimate the series in the right hand side of the above inequality. Let
$I(x,y)$ be the smallest dyadic interval in $\mathcal{D}^{+}$ containing both,
$x$ and $y$. Set $I^{l}$ to denote the $l^{th}$ ancestor of $I(x,y)$.
Precisely $I^{0}=I(x,y)$, $I^{1}$ the only interval in $\mathcal{D}^{+}$
containing $I^{0}$ with $\left|I^{1}\right|=2\left|I^{0}\right|$. For $l=2$,
$I^{2}\subset I^{1}$, $I^{2}\in\mathcal{D}^{+}$ and
$\left|I^{2}\right|=2\left|I^{1}\right|$ and so on. Notice that for each
$I\subsetneq I^{0}$ we have that
$\left|\psi_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2}=0$ since $x$ or $y$
does not belong to $I$, being $I^{0}$ the smallest interval in
$\mathcal{D}^{+}$ containing $x$ and $y$. Hence
$\displaystyle\sum_{I\in\mathcal{D}}\left|\psi_{I}(x)\right|^{2}\left|\psi_{I}(y)\right|^{2}$
$\displaystyle=\sum_{l\geq
0}\left|\psi_{I^{l}}(x)\right|^{2}\left|\psi_{I^{l}}(y)\right|^{2}$
$\displaystyle=\sum_{l\geq 0}\left|I^{l}\right|^{-2}$
$\displaystyle=\sum_{l\geq 0}(2^{l}\left|I^{0}\right|)^{-2}$
$\displaystyle=\left|I^{0}\right|^{-2}\sum_{l\geq 0}4^{-l}$
$\displaystyle=\frac{4}{3}\frac{1}{\left|I(x,y)\right|^{2}}$
$\displaystyle=\frac{4}{3}\frac{1}{\delta(x,y)^{2}},$
and (5.1.a) is proved.
Let us prove (5.1.b.i). The second estimate can be handled in a similar way.
With the above notation, for fixed $\omega\in\Omega$ we have that
$\displaystyle K(x^{\prime},y;\omega)-K(x,y;\omega)$
$\displaystyle=\sum_{I\in\mathcal{D}}a_{I}(\omega)(\psi_{I}(x^{\prime})-\psi_{I}(x))\psi_{I}(y)$
$\displaystyle=\sum_{l\geq
0}a_{I^{l}}(\omega)(\psi_{I^{l}}(x^{\prime})-\psi_{I^{l}}(x))\psi_{I^{l}}(y).$
Now, since $\left|I^{0}\right|=\left|I(x,y)\right|=\delta(x,y)\geq
2\delta(x,x^{\prime})$, $x$ and $x^{\prime}$ must belong to the same half of
$I(x,y)$. And hence $x$ and $x^{\prime}$ belong to the same half of each
$I^{l}$. Then $\psi_{I^{l}}(x)=\psi_{I^{l}}(x^{\prime})$ and
$K(x^{\prime},y;\omega)=K(x,y;\omega)$. And we are done. ∎
## 6\. Concentration
In all the results of the previous sections we have been dealing with a
sequence of random variables $\\{a_{I}:I\in\mathcal{D}\\}$ satisfying (4.2.i),
(4.2.e) and (4.2.$\sigma$). In particular the kernels $K(x,y;\omega)$ and the
induced operators $T_{\omega}$, have mean values given by
$K(x,y)=\sum_{I\in\mathcal{D}}\mathscr{E}a_{I}\psi_{I}(x)\psi_{I}(y)$
and
$Tf(x)=\sum_{I\in\mathcal{D}}\mathscr{E}a_{I}\left<f,\psi_{I}\right>\psi_{I}(x).$
Since from (4.2.e) the sequence $\\{\mathscr{E}a_{I}:I\in\mathcal{D}\\}$ is
bounded, $K$ is a Calderón-Zygmund kernel and $T$ a Calderón-Zygmund operator
both scalar valued. Inequalities (4.3) and (4.2) give estimates for the
concentration of $K(x,y;\omega)$ about $K(x,y)$ and of $T_{\omega}$ about $T$.
In particular the subgaussian character of the distribution
$\mathscr{P}\\{\omega:\left|K(x,y;\omega)-K(x,y)\right|>t\\}$
reveals as a variance factor the reciprocal of the underlying metric in the
space.
Even when the main steps have already been proved in Theorems 4.1, 4.2 and
5.1, let us state these estimates.
###### Theorem 6.1.
1. (A)
Let $\\{a_{I}:I\in\mathcal{D}\\}$ be a sequence of random variables satisfying
(4.2.i), (4.2.e) and (4.2.$\sigma$). Let $\psi$ be a wavelet function
satisfying $\left|\psi(x)\right|\leq C(1+\left|x\right|)^{-1-\varepsilon}$.
Then, for every $t>0$,
$\mathscr{P}\\{\left|K(x,y;\omega)-K(x,y)\right|>t\\}\leq
2e^{-\tfrac{C^{2}}{4\nu}\left|x-y\right|^{2}t^{2}}.$
2. (B)
Let $\\{a_{I}:I\in\mathcal{D}^{+}\\}$ be a sequence of random variables
satisfying (4.2.i), (4.2.e) and (4.2.$\sigma$). Let
$\psi(x)=\mathcal{X}_{[0,1/2)}(x)-\mathcal{X}_{[1/2,1)}(x)$ be the Haar
wavelet. Then, for every $t>0$,
$\mathscr{P}\\{\left|K(x,y;\omega)-K(x,y)\right|>t\\}\leq
2e^{-\tfrac{C^{2}}{4\nu}\delta(x,y)^{2}t^{2}}.$
3. (C)
For $\\{a_{I}\\}$ as before and $\psi$ satisfying (4.1) or with $\psi$ the
Haar wavelet, we have
$\mathscr{P}\\{\left|T_{\omega}f(x)-Tf(x)\right|>t\\}\leq
2e^{-\tfrac{t^{2}}{4\nu\sum_{I\in\mathcal{D}}\left|\left<f,\psi_{I}\right>\right|^{2}\left|\psi_{I}(x)\right|^{2}}}.$
###### Proof.
(A) follows from (4.3) and the standard estimates in [Dau92]. (B) follows from
(4.3) and the estimate in the proof of Theorem 5.1. (C) follows from (4.1). ∎
Let us finally observe that in the case of Rademacher random variables the
above concentration estimates hold with $\nu=1$, $K(x,y)\equiv 0$ and $T\equiv
0$.
## References
* [AG18] Hugo Aimar and Ivana Gómez, _On the Calderón-Zygmund structure of Petermichl’s kernel_ , C. R. Math. Acad. Sci. Paris 356 (2018), no. 5, 509–516. MR 3790422
* [BLM13] Stéphane Boucheron, Gábor Lugosi, and Pascal Massart, _Concentration inequalities: A nonasymptotic theory of independence_ , Oxford Scholarship, 2013.
* [Che52] Herman Chernoff, _A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations_ , Ann. Math. Statistics 23 (1952), 493–507. MR 57518
* [Chu01] Kai Lai Chung, _A course in probability theory_ , third ed., Academic Press, Inc., San Diego, CA, 2001. MR 1796326
* [Cra38] Harald Cramér, _Sur un nouveau théorème-limite de la théorie des probabilités_ , Actualités scientifiques et industrielles 736 (1938), 2–23.
* [Dau92] Ingrid Daubechies, _Ten lectures on wavelets_ , CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107
* [GLY09] Loukas Grafakos, Liguang Liu, and Dachun Yang, _vector-valued singular integrals and maximal functions on spaces of homogeneous type_ , Mathematica Scandinavica 104 (2009), no. 2, 296–310.
* [Hoe63] Wassily Hoeffding, _Probability inequalities for sums of bounded random variables_ , J. Amer. Statist. Assoc. 58 (1963), 13–30. MR 144363
* [Mey90] Yves Meyer, _Ondelettes et opérateurs. I_ , Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990, Ondelettes. [Wavelets]. MR 1085487
* [RRT86] José L. Rubio de Francia, Francisco J. Ruiz, and José L Torrea, _Calderón-Zygmund theory for operator-valued kernels._ , Adv. Math. 62 (1986), 7–48.
Affiliations: Instituto de Matemática Aplicada del Litoral, UNL, CONICET.
Address: CCT CONICET Santa Fe, Predio “Alberto Cassano”, Colectora Ruta Nac.
168 km 0, Paraje El Pozo, S3007ABA Santa Fe, Argentina.
E-mail address<EMAIL_ADDRESS>`ivanagomez@santafe-
conicet.gov.ar`
|
# Classification of COVID-19 X-ray Images Using a Combination of Deep and
Handcrafted Features
###### Abstract
Coronavirus Disease 2019 (COVID-19) demonstrated the need for accurate and
fast diagnosis methods for emergent viral diseases. Soon after the emergence
of COVID-19, medical practitioners used X-ray and computed tomography (CT)
images of patients’ lungs to detect COVID-19. Machine learning methods are
capable of improving the identification accuracy of COVID-19 in X-ray and CT
images, delivering near real-time results, while alleviating the burden on
medical practitioners. In this work, we demonstrate the efficacy of a support
vector machine (SVM) classifier, trained with a combination of deep
convolutional and handcrafted features extracted from X-ray chest scans. We
use this combination of features to discriminate between healthy, common
pneumonia, and COVID-19 patients. The performance of the combined feature
approach is compared with a standard convolutional neural network (CNN) and
the SVM trained with handcrafted features. We find that combining the features
in our novel framework improves the performance of the classification task
compared to the independent application of convolutional and handcrafted
features. Specifically, we achieve an accuracy of 0.988 in the classification
task with our combined approach compared to 0.963 and 0.983 accuracy for the
handcrafted features with SVM and CNN respectively.
Index Terms— COVID-19, Deep learning, SVM, Feature extraction, Classification
## 1 Introduction
Coronavirus disease 2019 (COVID-19) is an infectious disease caused by Severe
Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2). Since the emergence in
Wuhan, China in December 2019, it has spread worldwide and has caused a severe
pandemic. The COVID-19 infection causes mild symptoms in the initial stage,
but may lead to severe acute symptoms like multi-organ failure and systemic
inflammatory response syndrome [1, 2]. As of December 2020, there have been
more than 1.8 million COVID-19 related deaths around the world and daily new
cases of the disease are still rising. Currently, reverse transcription
polymerase chain reaction test (RT-PCR) is the most accurate diagnostic test.
However, it requires specialized materials, equipment, personnel, and takes at
least 24 hours to obtain a result. It may also require a second RT-PCR or a
different test to confirm the diagnosis. Therefore, radiological imaging
techniques like X-ray and CT-scan can serve as a complement to improve
diagnosis accuracy [3].
In recent years, machine learning has been used extensively for automatic
disease diagnosis in the healthcare sector [4, 5]. Various standard supervised
learning algorithms such as logistic regression, random forests, and support
vector machines (SVM) have been applied in detecting COVID-19 in X-ray and CT
images of patients’ lungs [6, 7, 8, 9]. The convolutional neural network (CNN)
is a deep learning algorithm that can extract features from images through a
combination of convolutional, pooling, and fully connected layers. It has been
used extensively for image recognition, classification, and object detection.
Recent works [10, 11, 12, 13, 14] show that it can also provide accurate
results in detecting COVID-19 in images. These recent works present some
insightful thoughts and valuable opinions. However, the lack of publicly
available image databases and the limited amount of patient data are
inevitable challenges for training a CNN.
In this study, we propose a fusion model that classifies X-ray images from a
combination of handcrafted features and CNN deep features. The model is
trained and tested on a large dataset with 1,143 COVID-19 cases, 2,000 normal
cases and 2,000 other pneumonia cases collected from [15, 16]. The feature
fusion classifier has been shown as an effective way of boosting the
performance of CNN models in face recognition [17] and biomedical image
classifications [18, 19]. Handcrafted and deep features extract different
information from the same input image, so the fusion of these two systems has
the potential to outperform the standard approaches [20]. Our key interest is
whether a fusion model can also surpass the standard CNN and SVM for COVID-19
detection. The paper is organized as follows: The methodology and feature
extraction techniques are presented in Section 2, the comparative
classification performances are given in Section 3, and the final conclusions
are made in Section 4.
## 2 Methodology
The proposed COVID-19 classifier is trained and tested on a collective dataset
with 5,143 X-ray images categorized into three cases: COVID-19, Normal and
Pneumonia. All the images are resized to 224 $\times$ 224 pixels and the local
contrast is enhanced by an adaptive histogram equalization algorithm during
the preprocessing stage. Several preprocessed example images are show in
Figure 1. Both handcrafted features and VGG16/ResNet50 deep features are
extracted from the dataset, then combined and fed into an SVM classifier. The
entire process is shown in Figure 2.
(a) COVID-19
(b) Normal
(c) Pneumonia
Fig. 1: Sample images after preprocessing Fig. 2: Methodology
### 2.1 Handcrafted Features
Handcrafted features seek to characterize each image by computing properties
using the information directly present in each image. These handcrafted
features are computed for each image and used as input into the SVM. There are
308 features computed on each image by evaluating 14 different statistical
measures on the output of each image under different transformations. The
transformations can be categorized into six groups: Texture, Gray-Level Co-
Occurrence Matrix (GLCM), Gray Level Difference Method (GLDM), Fast Fourier
Transform, Wavelet transform, and Local Binary Pattern. The features are
computed by applying the following same 14 statistical measures on the outputs
from the aforementioned six transformations: area, mean, standard deviation,
skewness, kurtosis, energy, entropy, maximum, mean absolute deviation, median,
minimum, range, root mean square, and uniformity as used in a COVID-19 image
classifier that used handcrafted features only [21]. Of the aforementioned 14
measures, the following 10 are all calculated using the standard definitions:
Mean, standard deviation, maximum, minimum, median, range, root mean square,
skewness, mean absolute deviation, and kurtosis. Energy was calculated using
the following definition:
$Energy:=\sum_{i=1}^{length(p)}p_{i}^{2}$ (1)
where $p_{i}$ is the $i^{th}$ value from the output vector of a
transformation. Area here is defined as the sum of all of the components of
the output vector. Entropy is calculated by first taking the frequency of each
unique intensity via the numpy function unique() and then normalizing that
vector. From there the entropy is directly calculated by taking the
elementwise sum of that normalized vector times the base 2 log of itself.
Uniformity is also calculated from this normalized vector. For clarity, the
pseudo-code is reproduced below:
$p$ (vector of output from a transformation)
$value,counts=unique(p,returncounts=True)$
$counts=counts/(\sum_{i}^{length(p)}counts_{i})$
$entropy=-\sum_{i}^{length(p)}counts_{i}*log_{2}(counts_{i})$
$uniformity=\sum_{i}^{length(p)}counts_{i}^{2}$
* •
Texture The texture features are calculated by considering each input image as
a single row vector and then calculating each of the above metrics on the
vector. For example the texture feature corresponding to the mean is simply
the sum of all of the pixel values (integer from 0 to 255) divided by the
number of pixels in the image. This results in a total of 14 features
computed.
* •
GLCM The GLCM transform characterizes an image by creating a histogram of co-
occurring greyscale values at a given offset and direction over an image [22].
In this specific implementation of GLCM, features are determined by applying
the greycomatrix() function from the skimage library directly on each image
with an offset of 1 and in four different directions (0, $\pi$/4, $\pi$/2,
3$\pi$/4). This function returns a 4-D array corresponding to each direction.
Each dimension is evaluated on the 14 statistical measures as before,
resulting in a total of 56 features.
* •
GLDM GLDM is a method that characterizes an image by creating a distribution
of the absolute differences of pixel intensity to the pixel intensity of
surrounding pixels at a given distance and direction [23]. In this
implementation, GLDM is computed in four directions (0, $\pi$/2, $pi$,
3$\pi$/2) with a distance of 10 pixels. Each of the four directions gives an
output vector and the 14 statistical measures are computed on each output
resulting in 56 features
* •
FFT The FFT features are evaluated on each image by transforming the image via
a Fast Fourier Transform. Each image is input into the numpy fft.fft2()
function resulting in a vector of output values that are then put into the
numpy fft.fftshift() function. Next, the numpy floor() function is used to
convert to a vector of integers which is the final output that is used to
compute the 14 statistical measures which are the FFT features.
* •
Wavelet The wavelet features are computed by applying the pywt package’s
dwt2() function directly on each image [24]. The output of this function gives
4 different arrays and the 14 statistical measures are computed on each array
resulting in 56 features. The first array from the dwt2() output is then put
back into the dwt2() function as input, resulting in another 4 matrices.
Again, these 4 matrices are used to compute 14 statistical measures each for
another 56 features resulting in 112 features in total from the wavelet
transforms.
* •
LBP LBP works by looking at points surrounding each pixel within a given
distance and tests whether the points are greater than or less than the
central point resulting in a binary output [25]. In this implementation,
scikit-image’s local_binary_pattern function is used to compute the LBP
outputs with distances of 2,3,5, and 7. The resulting four LBPs are then used
to compute the 14 statistical measures resulting in 56 features.
### 2.2 Deep Features
Deep features are extracted from two CNN models, VGG16 [26] and ResNet50 [27].
More specifically, only the feature extraction layers of the model are
utilized which are positioned prior to dense layers meant for the
classification task. The model weights are pre-trained on the ImageNet dataset
[28] which contains millions of images belonging to 1,000 classes. An
important note is that no fine tuning is done to the models, meaning that the
model weights are fixed and no further training is done.
The VGG16 CNN architecture contains 16 layers with trainable weights (with 5
being dense layers that are not used for feature extraction) consisting of 5
blocks that include convolutional and pooling layers which can be seen in
Figure 3. The input of the model accepts RGB images of size 224 $\times$ 224
$\times$ 3 pixels. To maintain compatibility with the model, the grayscale the
X-ray images are converted to to have three color channels by simply
duplicating the pixel values and having each color channel be identical.
Additionally, each image is zero-centered with respect to the ImageNet dataset
without scaling. For each X-ray image the resulting feature output is of
dimension 7 $\times$ 7 $\times$ 512 with subsequent flattening producing a
vector containing 25,088 features.
Fig. 3: VGG16 feature extraction layers
As opposed to CNN architectures such as VGG, ResNets can have more layer depth
with increasing accuracy while at the same time having less overall
complexity. This is achieved by utilizing shortcut connections allowing
residual mapping that may skip one or more layers and performing identity
mapping which can alleviate the problem of vanishing gradients. A residual
block of this type is shown in Figure 4. The ResNet50 model contains 50 layers
with trainable weights (of which a single dense layer is not used for feature
extraction). As with VGG16 the input of the model accepts RGB images of size
224 $\times$ 224 $\times$ 3 pixels. Again, the grayscale X-ray images are
converted to duplicated three channel RGB before being zero-centered with
respect to the ImageNet dataset without scaling. Each X-ray image results in a
feature output of dimension 7 $\times$ 7 $\times$ 2048 and is flattened into a
vector of 100,352 features.
Fig. 4: Residual block
After features are extracted from the models, kernel principal component
analysis (PCA) is applied to reduce the dimensionality of the deep features.
The number of components after the transformation is selected to be 1,000 as
this number of features is near the order of magnitude as the number of
handcrafted features that are extracted.
### 2.3 Classifier
A linear SVM using one-vs-all approach is applied to classify the combined
features. Despite the fact that most deep learning models employ the softmax
activation function for classification task, it was shown that SVM works
better on several standard datasets like MNIST,CIFAR-10,and the ICML 2013
Representation Learning Workshop’s face expression recognition challenge [29].
## 3 Results and Discussions
To evaluate the performance of the method outlined above, it was important to
compare the performance of combined deep features and handcrafted features in
an SVM classifier with baseline individual CNNs in addition to solely using
the handcrafted features in an SVM.
Both the VGG and ResNet CNNs were evaluated again with the feature extraction
layers frozen with pre-trained ImageNet weights. Two layers were added to the
models, a 1,000 neuron dense layer with a rectified linear activation function
and a three neuron output layer with a softmax activation function. The
objective of the addition of the layers is to allow the classification of
three classes to be possible in addition to increasing the number of trainable
parameters as the feature extraction layers are frozen. Additionally, during
training both models use categorical cross-entropy while employing the Adam
optimizer [30] with a learning rate of 0.005.
A parametric study was performed on the handcrafted features to evaluate which
configuration created most accurate results as inputs into the SVM. Results in
Table 1 show that by itself, the Wavelet features resulted in the highest
classification accuracy followed by GLDM and GLCM. The lowest performing
feature group was the texture features with an accuracy of 0.762. It was found
that inputting all features (308) into the SVM resulted in the highest
accuracy and F-1 Score. A 95% confidence interval is given for all values in
Table 1.
For each classification model outlined above, the dataset of 5,143 was divided
into the same train and test subsets with an 80/20 split. This resulted in
4,114 training images and 1,029 test images. The results of each
classification model can be seen in Table 2. All the metrics listed in the
table are unweighted averages of the statistics of each class with a 95%
confidence interval. From these results its is clear that all models that
incorporate deep features clearly performed better than the SVM that only uses
handcrafted features. The two models utilizing both deep features and
handcrafted featured with an SVM classifier slightly outperform the
conventional VGG16 and ResNet50 CNNs. Additionally, the confusion matrices of
the combined deep features and handcrafted features SVM models are seen in
Figure 5. Both combined feature models achieve the same low false negative and
false positive rates of 0.41% and 0.13% respectively.
Table 1: Performance of X-ray image classification using SVM with handcrafted features only Handcrafted Features | Accuracy | F1-Score
---|---|---
Texture | 0.762 $\pm$ 0.026 | 0.771 $\pm$ 0.026
GLCM | 0.896 $\pm$ 0.019 | 0.880 $\pm$ 0.020
GLDM | 0.900 $\pm$ 0.018 | 0.894 $\pm$ 0.019
FFT | 0.818 $\pm$ 0.024 | 0.809 $\pm$ 0.024
Wavelet | 0.940 $\pm$ 0.015 | 0.934 $\pm$ 0.015
LBP | 0.874 $\pm$ 0.020 | 0.880 $\pm$ 0.020
All Features Combined | 0.963 $\pm$ 0.012 | 0.957 $\pm$ 0.012
Table 2: Performance of X-ray image classification models Classification Model | Accuracy | F1-Score
---|---|---
Handcrafted Features (SVM) | 0.963 $\pm$ 0.012 | 0.957 $\pm$ 0.012
VGG16 | 0.982 $\pm$ 0.008 | 0.983 $\pm$ 0.008
ResNet50 | 0.983 $\pm$ 0.008 | 0.984 $\pm$ 0.008
VGG16 DF + HF (SVM) | 0.988 $\pm$ 0.007 | 0.989 $\pm$ 0.006
ResNet50 DF + HF (SVM) | 0.987 $\pm$ 0.007 | 0.988 $\pm$ 0.007
(a) VGG16 DF + HF
(b) ResNet50 DF + HF
Fig. 5: Confusion matrices
## 4 Conclusion
This work demonstrated the use of a combined handcrafted and deep feature
approach for classifying COVID-19, pneumonia, and healthy patients in
radiological images. This new approach was compared to 7 handcrafted feature
classifiers and two CNN architectures. With respect to all performance
metrics, the combination of deep features and handcrafted features surpassed
that of handcrafted features or deep features alone. Notably, the proposed
architecture achieved an accuracy of 0.988 by combining VGG16 deep features
and handcrafted features. The next best accuracy of an approach without
combining deep and handcrafted features was 0.983 for ResNet50.
## References
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|
# Approximating the identity of convolution with random mean and random
variance
Hugo Aimar and Ivana Gómez
###### Abstract.
We provide sufficient conditions on the profile $\varphi$, on the sequence of
random variables $\varepsilon_{j}>0$ and on the sequence of random vectors
$y_{j}\in\mathbb{R}^{n}$ such that
$\mathscr{E}\left(\frac{1}{\varepsilon_{j}^{n}(\omega)}\int_{z\in\mathbb{R}^{n}}\varphi\left(\frac{\left|x-z-
y_{j}(\omega)\right|}{\varepsilon_{j}(\omega)}\right)f(z)dz\right)\underset{j\to\infty}{\longrightarrow}f(x)$
for almost every $x\in\mathbb{R}^{n}$, $f\in L^{p}(\mathbb{R}^{n})$, $1\leq
p\leq\infty$, where $\mathscr{E}$ denotes the expectation, $\varepsilon_{j}$
tends to $0\in\mathbb{R}$ in law and $y_{j}$ tends to
$\mathbf{0}\in\mathbb{R}^{n}$ in law.
###### Key words and phrases:
Approximate identity; convolution; differentiation theorem
###### 2010 Mathematics Subject Classification:
Primary 42B25, 44A35
## 1\. Introduction
Classical Harmonic Analysis is strongly related to the two basic geometric
operations of translation and dilation. A paradigmatic analytic problem
involving these basic operators is the pointwise approximation of the identity
of convolution. The basic question is the almost everywhere convergence of
$K_{\varepsilon}*f(x)=\int_{y\in\mathbb{R}^{n}}K_{\varepsilon}(x-y)f(y)dy$ to
$f(x)$ when $\varepsilon\to 0^{+}$, with
$K_{\varepsilon}(x)=\tfrac{1}{\varepsilon^{n}}K(\tfrac{x-y}{\varepsilon})$,
$\int_{\mathbb{R}^{n}}K(x)dx=1$ and $f$ belongs to $L^{p}(\mathbb{R}^{n})$,
$1\leq p\leq\infty$. The most general conditions on $K$ in order to solve the
almost everywhere convergence problem goes back to the 1976 results due to
Felipe Zó (see [4] or [2]). It is worthy noticing at this point that the main
result in [4] can also be obtained from the vector valued approach to the
Calderón-Zygmund theory of singular integrals. The classical results (see [3])
of decreasing kernels are covered by this general approach.
In order to introduce the problem that we consider in this note, let us recall
that the basic property of a nonnegative kernel
$K_{\varepsilon}(x,y)=\tfrac{1}{\varepsilon^{n}}K(\tfrac{x-y}{\varepsilon})$
in order to produce an approximation to the Dirac delta, is that
$\int_{x\in\mathbb{R}^{n}}K(x)dx=1$. In other words $K$ and hence each
$K_{\varepsilon}(\cdot,y)$ are probability densities in $\mathbb{R}^{n}$.
Assume that $K$ has finite second moments, i.e.
$\int_{x\in\mathbb{R}^{n}}\left|x\right|^{2}K(x)dx<\infty$. An important case
of the above is that of $K(x)=\varphi(\left|x\right|)$. The symmetry of the
kernel shows immediately that the expected value of the probability
distribution $K_{\varepsilon}(\cdot,y)$ is $y$. Moreover, its variance is
$\varepsilon^{2}$. Hence $\varepsilon$ is its standard deviation.
In this note we aim to consider the pointwise convergence of the mean when the
standard deviation $\varepsilon$ and the expected value $y$ of
$K_{\varepsilon}(\cdot-y)$ are random variables that converge “in law” to the
corresponding Dirac deltas at the origins of $\mathbb{R}$ and of
$\mathbb{R}^{n}$ respectively.
Let us precise the above. Let $K(x)=\varphi(\left|x\right|)$. Let
$(\Omega,\mathscr{F},\mathscr{P})$ be a probability space. Let
$\varepsilon_{j}:\Omega\to\mathbb{R}^{+}$ be a sequence of random variables
distributed according to the one dimensional probability measures $\nu_{j}$.
Let $y_{j}:\Omega\to\mathbb{R}^{n}$ be a sequence of random vectors
distributed according to the $n$ dimensional probability measures $\mu_{j}$.
Assume that $\nu_{j}\to\delta_{0}$ and $\mu_{j}\to\delta_{\mathbf{0}}$ for
$j\to\infty$ vaguely where $\delta_{0}$ and $\delta_{\mathbf{0}}$ denote the
Dirac deltas at $0\in\mathbb{R}$ and at $\mathbf{0}\in\mathbb{R}^{n}$
respectively. Set $\Pi_{j}$ to denote the joint distribution of
$\varepsilon_{j}$ and $y_{j}$. We consider sufficient conditions on $\varphi$,
$\nu_{j}$, $\mu_{j}$ and $\Pi_{j}$ in order to have
$\mathscr{E}\left(\frac{1}{\varepsilon_{j}^{n}(\omega)}\int_{z\in\mathbb{R}^{n}}\varphi\left(\frac{\left|x-z-
y_{j}(\omega)\right|}{\varepsilon_{j}(\omega)}\right)f(z)dz\right)\underset{j\to\infty}{\longrightarrow}f(x)$
for almost every $x\in\mathbb{R}^{n}$, $f\in L^{p}(\mathbb{R}^{n})$, $1\leq
p\leq\infty$. Here $\mathscr{E}(X)$ denotes the expected value of the random
variable $X$. We may write the above expression for the expectation using
translation and mollification operators $(\tau_{y}g)(x)=g(x-y)$ and
$M_{\varepsilon}g(x)=\tfrac{1}{\varepsilon^{n}}g(\tfrac{x}{\varepsilon})$, as
$\mathscr{E}(\tau_{y_{j}(\omega)}M_{\varepsilon_{j}(\omega)}K*f(x))$ with
$K(x)=\varphi(\left|x\right|)$. Which at least formally can be written as a
convolution operator in $\mathbb{R}^{n}$ with kernel
$\int_{\Omega}\tau_{y_{j}(\omega)}M_{\varepsilon_{j}(\omega)}K(x)d\mathscr{P}(\omega)$.
In general, this new kernel is not of mollification type and the general
results in [4] are useful to deal with them. In other words, the Calderón-
Zygmund decomposition methods apply under some conditions on $\varphi$ and on
the distributions of $\varepsilon_{j}$ and $y_{j}$.
The paper is organized as follows. Section 2 is devoted to the pointwise
convergence for continuous functions and to the pointwise convergence for
general functions under the assumption of the weak type (1,1) of the maximal
operators. In Section 3 we deal with the estimates and boundedness properties
of the maximal operator under different assumptions on $\varphi$, $\nu_{j}$,
$\mu_{j}$ and $\Pi_{j}$ the joint distribution of $\varepsilon_{j}(\omega)$
and $y_{j}(\omega)$.
## 2\. Pointwise convergence
Given a probability space $(\Omega,\mathscr{P})$, a sequence of random
variables $X_{j}$ defined in $\Omega$ is said to converge in distribution, in
law, or weakly to the random variable $X$ if the distribution
$\mu_{j}=\mathscr{P}\circ X^{-1}_{j}$ converge vaguely to
$\mu=\mathscr{P}\circ X^{-1}$, the distribution of $X$. The vague convergence
coincides with the convergence of $\int g(x)d\mu_{j}(x)$ to $\int g(x)d\mu(x)$
for every bounded and continuous function $g(x)$. For real valued random
variables, their distributions are probability measures on $\mathbb{R}$. In
this case the vague convergence is equivalent to the convergence of
$\mu_{j}((a,b])$ to $\mu((a,b])$ for $a$ and $b$ in a dense subset of
$\mathbb{R}$. (See [1]). In particular, a sequence $\varepsilon_{j}$ of random
variables tends to $0$ in law when the distribution $\nu_{j}$ of
$\varepsilon_{j}$ tends to $\delta_{0}$ vaguely. And the sequence of random
vectors $y_{j}$ tends to $\mathbf{0}\in\mathbb{R}^{n}$ in law when $\mu_{j}$,
the $n$ dimensional distribution sequence, tends to $\delta_{\mathbf{0}}$
vaguely.
Let us precise the above. For $j=1,2,\ldots$, let
$\varepsilon_{j}:\Omega\to\mathbb{R}^{+}=\\{\varepsilon>0\\}$ be a sequence of
positive random variables. Let
$\nu_{j}(B)=\mathscr{P}(\varepsilon_{j}^{-1}(B))$, $B$ a Borel set in
$\mathbb{R}$, be the distribution of $\varepsilon_{j}$. The convergence in law
of $\varepsilon_{j}$ to $0$ is equivalent to the weak or vague convergence of
$\nu_{j}$ to $\delta_{0}$, the Dirac unit mass at the origin $0\in\mathbb{R}$.
Since the distribution $\mu_{j}$ of $y_{j}$ is given by
$\mu_{j}(B)=\mathscr{P}(y_{j}^{-1}(B))$ for each Borel subset $B$ of
$\mathbb{R}^{n}$, the convergence in law of $y_{j}$ to
$\mathbf{0}\in\mathbb{R}^{n}$ is equivalent to the weak or vague convergence
of $\mu_{j}$ to $\delta_{\mathbf{0}}$, the Dirac unit mass at the origin
$\mathbf{0}\in\mathbb{R}^{n}$. The joint distribution of $\varepsilon_{j}$ and
$y_{j}$ is given by
$\Pi_{j}(E)=\mathscr{P}(\left\\{\omega\in\Omega:(\varepsilon_{j}(\omega),y_{j}(\omega))\in
E\right\\})$ for every Borel sets $E$ of $\mathbb{R}^{n+1}$.
The integrability condition on the kernel profile
$\varphi:\mathbb{R}^{+}\to\mathbb{R}^{+}\cup\left\\{0\right\\}$ is
$\omega_{n}\int_{0}^{\infty}\rho^{n-1}\varphi(\rho)d\rho=1$, where
$\omega_{n}$ is the surface measure of the unit sphere in $\mathbb{R}^{n}$.
Set $K(x)=\varphi(\left|x\right|)$. Since we are dealing with the
differentiation problem, the operators involved are positive, hence we can
consider nonnegative measurable functions $f:\mathbb{R}^{n}\to\mathbb{R}$ in
order to have well defined, for each $x\in\mathbb{R}^{n}$, the mean
$m_{j}(f)(x)=\mathscr{E}[(\tau_{y_{j}(\omega)}M_{\varepsilon_{j}(\omega)}K)*f(x)],$
which could be infinite. Notice that in particular, if $f$ is continuous and
bounded. Since for each $\omega\in\Omega$, $\varepsilon_{j}(\omega)>0$ we have
that
$m_{j}(f)(x)=\mathscr{E}\left[\int_{z\in\mathbb{R}^{n}}\varphi(\left|z\right|)f(x-y_{j}(\omega)-\varepsilon_{j}(\omega)z)dz\right]\leq\left\|f\right\|_{\infty},$
for every $x$.
###### Proposition 2.1.
Assume that $\Pi_{j}$, the joint distribution of $(\varepsilon_{j},y_{j})$,
converges vaguely to $\delta_{(0,\mathbf{0})}$, the Dirac unit mass at
$(0,\mathbf{0})\in\mathbb{R}^{n+1}$. Let $g$ be a continuous and bounded
function defined on $\mathbb{R}^{n}$. Then
$m_{j}g(x)\to g(x)$
for every $x\in\mathbb{R}^{n}$, when $j\to\infty$.
###### Proof.
Since $\Pi_{j}$ is the distribution of the random vector
$(\varepsilon_{j},y_{j})$ we have that
$\displaystyle\left|m_{j}g(x)-g(x)\right|$
$\displaystyle\leq\mathscr{E}\left[\int_{z\in\mathbb{R}^{n}}\varphi(\left|z\right|)\left|g(x-y_{j}(\omega)-\varepsilon_{j}(\omega)z)-g(x)\right|dz\right]$
$\displaystyle=\int_{(s,y)\in\mathbb{R}^{n+1}}\left(\int_{z\in\mathbb{R}^{n}}\varphi(\left|z\right|)\left|g(x-y-
sz)-g(x)\right|dz\right)d\Pi_{j}(s,y).$
Notice now that the function
$G(s,y)=\int_{z\in\mathbb{R}^{n}}\varphi(\left|z\right|)\left|g(x-y-
sz)-g(x)\right|dz$ is continuous, bounded and $G(0,\mathbf{0})=0$. Hence
$\left|m_{j}(g)(x)-g(x)\right|\to 0$ as $j\to\infty$ from the weak convergence
of $\Pi_{j}$ to $\delta_{(0,\mathbf{0})}$. ∎
The next result contains an elementary sufficient condition for the vague
convergence of $\Pi_{j}$ to $\delta_{(0,\mathbf{0})}$.
###### Lemma 2.2.
Let $(\Omega,\mathscr{F},\mathscr{P})$ be a probability space. Let
$\varepsilon_{j}(\omega)$ be a positive random variable distributed by
$\nu_{j}$ for each $j=1,2,3,\dots$ Assume that $\nu_{j}$ converges vaguely to
$\delta_{0}$, the Dirac delta at $0$. Let $y_{j}(\omega)$ be an
$\mathbb{R}^{n}$ valued random vector distributed by $\mu_{j}$. If there
exists a positive constant $C$ such that the inequality
$\left|y_{j}(\omega)\right|\leq C\varepsilon_{j}(\omega)$ almost surely for
every $j$. Then the distribution $\Pi_{j}$ of the
$\mathbb{R}^{+}\times\mathbb{R}^{n}$ valued random vector
$Z_{j}(\omega)=(\varepsilon_{j}(\omega),y_{j}(\omega))$ converges vaguely to
$\delta_{(0,\mathbf{0})}$, the Dirac delta in $\mathbb{R}^{n+1}$. Precisely,
for any continuous and bounded function $g$ in $\mathbb{R}^{n+1}$ we have that
$\int_{\mathbb{R}}\idotsint_{\mathbb{R}^{n}}g(s,y)d\Pi_{j}(s,y)\longrightarrow
g(0;\mathbf{0})$
when $j$ tends to infinity.
###### Proof.
Let $D$ be a dense subset of $\mathbb{R}$ such that
$\nu_{j}((a,b])\to\delta_{0}((a,b])$ for every choice of $a$ and $b$ in $D$
with $a<b$. Hence $\nu_{j}((a,b])\to 1$ if $a<0\leq b$ and $\nu_{j}((a,b])\to
0$ if $b<0$ or $a\geq 0$. Let $h_{1}<0<h_{2}$ be two real numbers in $D$. Then
$\displaystyle\int_{\mathbb{R}}\idotsint_{\mathbb{R}^{n}}\left|g(s,y)-g(0,\mathbf{0})\right|d\Pi_{j}(s,y)$
$\displaystyle\leq\int_{(h_{1},h_{2}]}\idotsint_{\overline{B}(\mathbf{0},Ch_{2})}\left|g(s,y)-g(0,\mathbf{0})\right|d\Pi_{j}(s,y)+2\left\|g\right\|_{\infty}\Pi_{j}[\mathbb{R}^{n+1}\setminus(h_{1},h_{2}]\times\overline{B}(\mathbf{0},Ch_{2})],$
where $\overline{B}(\mathbf{0},Ch_{2})]$ is the closure of teh ball
$B(\mathbf{0},Ch_{2})]$. For $\left|h_{1}\right|$ and $h_{2}$ small enough,
the first term on the right is as small as desired uniformly in $j$ because
$g$ is continuous. For the second term, since $\Pi_{j}$ is the joint
distribution of the random vector $(\varepsilon_{j},y_{j})$, we have
$\displaystyle\Pi_{j}[\mathbb{R}^{n+1}\setminus(h_{1},h_{2}]\times\overline{B}(\mathbf{0},Ch_{2})]$
$\displaystyle=\mathscr{P}(\\{(\varepsilon_{j},y_{j})\notin(h_{1},h_{2}]\times\overline{B}(\mathbf{0},Ch_{2})\\})$
$\displaystyle=\mathscr{P}(\\{\varepsilon_{j}\notin(h_{1},h_{2}]\textrm{ or
}y_{j}\notin\overline{B}(\mathbf{0},Ch_{2})\\})$
$\displaystyle\leq\mathscr{P}(\\{\varepsilon_{j}\notin(h_{1},h_{2}]\\}),$
since $\left|y_{j}\right|>Ch_{2}$ implies $\varepsilon_{j}>h_{2}$. In other
words,
$\Pi_{j}[\mathbb{R}^{n+1}\setminus(h_{1},h_{2}]\times\overline{B}(\mathbf{0},Ch_{2})]\leq
1-\nu_{j}((h_{1},h_{2}]))$
which tends to zero for $j\to+\infty$. ∎
In order to concentrate in Section 3 the more technical aspects regarding the
boundedness of the maximal operators, let us state and sketch the proof of the
convergence result assuming the weak type of the underlying maximal operator.
Let $\varphi:\mathbb{R}^{+}\to\mathbb{R}^{+}\cup\\{0\\}$ with
$\omega_{n}\int_{0}^{\infty}\rho^{n-1}\varphi(\rho)d\rho=1$. Let
$\varepsilon_{j}$ be a sequence of positive random variables and $y_{j}$ a
sequence of $n$ dimensional random vectors. For any measurable function $f$
the sublinear and positively homogeneous maximal operator
$\displaystyle\mathscr{M}f(x)$ $\displaystyle=\sup_{j\geq
1}\mathscr{E}[\tau_{y_{j}}M_{\varepsilon_{j}}K*\left|f\right|(x)]$
$\displaystyle=\sup_{j\geq
1}\mathscr{E}\left[\int_{z\in\mathbb{R}^{n}}\frac{1}{\varepsilon_{j}^{n}(\omega)}\varphi\left(\frac{\left|x-y_{j}(\omega)-z\right|}{\varepsilon_{j}(\omega)}\right)\left|f(z)\right|dz\right]$
is well defined. Of course with the above general conditions it could be
identically equal to $+\infty$.
As usual we say that $\mathscr{M}$ is of weak type (1,1) if there exists a
constant $A>0$ such that
$\left|\left\\{x\in\mathbb{R}^{n}:\mathscr{M}f(x)>\lambda\right\\}\right|\leq\frac{A}{\lambda}\left\|f\right\|_{L^{1}(\mathbb{R}^{n})}$
for every $\lambda>0$. Here the vertical bars denote the Lebesgue measure in
$\mathbb{R}^{n}$. For the sake of completeness we state and sketch the proof
of the convergence theorem under the assumption of weak type (1,1) of
$\mathscr{M}$.
###### Theorem 2.3.
Let $\varphi\geq 0$ with $\int_{\mathbb{R}^{n}}\varphi(\left|x\right|)dx=1$.
Assume that the joint distribution $\Pi_{j}$ of $\varepsilon_{j}$ and $y_{j}$,
$j\in\mathbb{Z}^{+}$, converges weakly to $\delta_{(0,\mathbf{0})}$ and that
$\mathscr{M}$ is of weak type (1,1). Then, for every $f\in
L^{1}(\mathbb{R}^{n})$ we have that
$m_{j}(f)(x)\underset{j\to\infty}{\longrightarrow}f(x)$
for almost every $x\in\mathbb{R}^{n}$.
###### Proof.
For $f\in L^{1}(\mathbb{R}^{n})$ and $g$ any continuous and compactly
supported function on $\mathbb{R}^{n}$, from Proposition 2.1 we have that the
exceptional set of convergence for $f$ is the same as the exceptional set of
convergence of $f-g$, hence
$\displaystyle\left|\left\\{\right.\right.x\in\mathbb{R}^{n}:$
$\displaystyle\,\mathscr{E}[(\varphi_{\varepsilon_{j}}*f)(x+y_{j})]\nrightarrow
f(x)\left.\right\\}\left.\right|$
$\displaystyle=\left|\left\\{x\in\mathbb{R}^{n}:\mathscr{E}[(\varphi_{\varepsilon_{j}}*(f-g))(x+y_{j})]\nrightarrow(f-g)(x)\right\\}\right|$
$\displaystyle=\left|\left\\{x\in\mathbb{R}^{n}:\limsup_{j}\left|\mathscr{E}[(\varphi_{\varepsilon_{j}}*(f-g))(x+y_{j})]-(f-g)(x)\right|>0\right\\}\right|$
$\displaystyle=\left|\bigcup_{k\geq
1}\left\\{x\in\mathbb{R}^{n}:\limsup_{j}\left|\mathscr{E}(\varphi_{\varepsilon_{j}}*(f-g))(x+y_{j})-(f-g)(x)\right|>\frac{1}{k}\right\\}\right|.$
On the other hand, for each $k=1,2,3,\ldots$ from the weak type of
$\mathscr{M}$ and Chebyshev inequality
$\displaystyle\left|\left\\{x\in\mathbb{R}^{n}:\limsup_{j}\left|\mathscr{E}(\varphi_{\varepsilon_{j}}*(f-g))(x+y_{j})-(f-g)(x)\right|>\frac{1}{k}\right\\}\right|$
$\displaystyle\leq\left|\left\\{x\in\mathbb{R}^{n}:\mathscr{M}(f-g)(x)>\frac{1}{2k}\right\\}\right|+\left|\left\\{x\in\mathbb{R}^{n}:\left|f(x)-g(x)\right|>\frac{1}{2k}\right\\}\right|$
$\displaystyle\leq 2k(A+1)\left\|f-g\right\|_{L^{1}(\mathbb{R}^{n})},$
since the continuous functions are dense in $L^{1}(\mathbb{R}^{n})$ we have
that each set
$\left\\{x\in\mathbb{R}^{n}:\limsup_{j}\left|\mathscr{E}[(\varphi_{\varepsilon_{j}}*(f-g))(x+y_{j})-(f-g)(x)]\right|>\frac{1}{k}\right\\}$
has Lebesgue measure equal to zero and the theorem is proved. ∎
## 3\. The maximal operator
In this section we deal with the estimates leading to the weak type (1,1)
inequality for $\mathscr{M}$. Since $\mathscr{M}$ is bounded in
$L^{\infty}(\mathbb{R}^{n})$ Marcinckiewicz interpolation will provide the
$L^{p}(\mathbb{R}^{n})$ boundedness of $\mathscr{M}$ also for $1<p\leq\infty$.
The next result contains some basic properties of the kernel of the operators
$m_{j}(f)$.
###### Proposition 3.1.
Let $\varphi:\mathbb{R}^{+}\cup\\{0\\}\to\mathbb{R}^{+}$ with
$\int_{0}^{\infty}\rho^{n-1}\varphi(\rho)d\rho<\infty$. Let $\Pi_{j}$ be the
distribution measure of $(\varepsilon_{j},y_{j})$. Then,
1. (3.1.a)
(random variance and random mean)
$m_{j}(f)(x)=\int_{z\in\mathbb{R}^{n}}K_{j}(x-z)f(z)dz$, with
$K_{j}(x)=\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\frac{1}{s^{n}}\varphi\left(\frac{\left|x-y\right|}{s}\right)d\Pi_{j}(s,y);$
2. (3.1.b)
(random variance only) if $y_{j}\equiv 0$ for every $j\in\mathbb{Z}^{+}$,
$K_{j}(x)=\psi_{j}(\left|x\right|)$ with
$\psi(t)=\int_{\mathbb{R}^{+}}\frac{1}{s^{n}}\psi(\frac{t}{s})d\nu_{j}(s)$;
3. (3.1.c)
(random mean only) if $\varepsilon_{j}\equiv 0$ for every
$j\in\mathbb{Z}^{+}$,
$m_{j}(f)(x)=\int_{y\in\mathbb{R}^{n}}f(x-y)d\mu_{j}(y)$;
4. (3.1.d)
(the case of self-similarity of $\Pi_{j}$) if $\Pi_{j}(E)=\Pi(jE)$ for every
$j\in\mathbb{Z}^{+}$ and every Borel set $E$ in $\mathbb{R}^{n+1}$,
$K_{j}(x)=j^{n}K(jx)$ with
$K(x)=\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\frac{1}{s^{n}}\varphi(\frac{\left|x-y\right|}{s})d\Pi(s,y)$.
###### Proof.
In order to prove (3.1.a) we only have to compute the expectation of the
random variable
$\frac{1}{\varepsilon_{j}^{n}(\omega)}\varphi\left(\frac{\left|x-y_{j}(\omega)\right|}{\varepsilon_{j}(\omega)}\right)$
in term of the distribution $\Pi_{j}$ of the $n+1$ dimensional random vector
$(\varepsilon_{j}(\omega),y_{j}(\omega))$. The formula in (3.1.b) follows from
(3.1.a) with $\Pi_{j}=\nu_{j}\times\delta_{\mathbf{0}}$. The formula in
(3.1.c) follows directly from the definition of $m_{j}$.
Let us prove (3.1.d). Notice first that the identity $\Pi_{j}(E)=\Pi(jE)$ can
be written as
$\iint_{\mathbb{R}^{n+1}}\mathcal{X}_{E}(s,y)d\Pi_{j}(s,y)=\iint_{\mathbb{R}^{n+1}}\mathcal{X}_{E}(\frac{s}{j},\frac{y}{j})d\Pi(s,y)$.
From this identity and standard arguments we have that
$\iint_{\mathbb{R}^{n+1}}g(s,y)d\Pi_{j}(s,y)=\iint_{\mathbb{R}^{n+1}}g(\frac{s}{j},\frac{y}{j})d\Pi(s,y)$,
for $g\geq 0$ measurable. Taking, for fixed $x\in\mathbb{R}^{n}$,
$g(s,y)=\frac{1}{s^{n}}\varphi(\frac{\left|x-y\right|}{s})$ we have
$\displaystyle K_{j}(x)$
$\displaystyle=\iint_{\mathbb{R}^{n+1}}\frac{1}{s^{n}}\varphi\left(\frac{\left|x-y\right|}{s}\right)d\Pi_{j}(s,y)$
$\displaystyle=\iint_{\mathbb{R}^{n+1}}\left(\frac{j}{s}\right)^{n}\varphi\left(\frac{\left|x-\frac{y}{j}\right|}{\frac{s}{j}}\right)d\Pi(s,y)$
$\displaystyle=j^{n}\iint_{\mathbb{R}^{n+1}}\frac{1}{s^{n}}\varphi\left(\frac{\left|jx-y\right|}{s}\right)d\Pi(s,y)$
$\displaystyle=j^{n}K(jx).$
∎
Since of the main results in [4] are the central tools of our analysis, we
shall briefly recall them here.
The operator $\sup_{\alpha\in\Lambda}\left|K_{\alpha}*f\right|$ is of weak
type (1,1) if the integrals
$\int_{\mathbb{R}^{n}}\left|K_{\alpha}(x)\right|dx$ are uniformly bounded for
$\alpha\in\Lambda$ and $\int_{\left|x\right|\geq
2\left|z\right|}\sup_{\alpha\in\Lambda}\left|K_{\alpha}(x-z)-K_{\alpha}(x)\right|dx$
are uniformly bounded in $z\in\mathbb{R}^{n}$. The classical conditions for
$K_{\lambda}(x)=\lambda^{n}K(\lambda x)$ with $K$ positive and integrable, is
given in terms of the size of the gradient of $K$; $\left|\nabla
K(x)\right|\leq\frac{C}{\left|x\right|^{n+1}}$. On the other hand, the result
contained in Theorem 4 in [4] provides an integrable function $f$ for each non
lacunary sequence $\lambda_{j}$ such that
$\limsup_{j\to\infty}K_{\lambda_{j}}(f)(x)=+\infty$ almost everywhere when $K$
is unbounded and increasing in the interval $[0,1]$. In particular,
singularity of the kernel outside the origin provide unbounded maximal
operators.
Let us start with the case of random variance (only) with probability measures
$\Pi_{j}=\nu_{j}\times\delta_{\mathbf{0}}$ with $\nu_{j}$ self-similar,
$j\in\mathbb{Z}^{+}$.
###### Theorem 3.2.
Let $\varphi:\mathbb{R}^{+}\to\mathbb{R}^{+}\cup\\{0\\}$ be a
$\mathscr{C}^{1}$ function with
$\int_{0}^{\infty}\rho^{n-1}\varphi(\rho)d\rho<\infty$ and
$\left|\varphi^{\prime}(\rho)\right|\leq\frac{B}{\rho^{n+1}}$ for some
constant $B$ and every $\rho$ positive. Let $\varepsilon_{j}$ be a sequence of
random variables distributed according $\nu_{j}$. Assume that
$\\{\nu_{j}:j\in\mathbb{Z}^{+}\\}$ are self-similar. Then the maximal operator
$\mathscr{M}f(x)=\sup_{j\geq 0}m_{j}(\left|f\right|)(x)$
is of weak type (1,1).
###### Proof.
The self-similarity of $\nu_{j}$ implies the self-similarity
$\Pi_{j}=\nu_{j}\times\delta_{\mathbf{0}}$. Hence from (3.1.d),
$K_{j}(x)=j^{n}K_{1}(jx)$. In order to apply the gradient criteria stated
above, we are led to prove the integrability of $K_{1}$,
$\displaystyle\int_{\mathbb{R}^{n}}\left|K_{1}(x)\right|dx$
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{+}}\frac{1}{s^{n}}\varphi\left(\frac{\left|x\right|}{s}\right)d\nu_{1}(s)dx$
$\displaystyle=\omega_{n}\int_{0}^{\infty}\rho^{n-1}\int_{\mathbb{R}^{+}}\frac{1}{s^{n}}\varphi\left(\frac{\rho}{s}\right)d\nu_{1}(s)d\rho$
$\displaystyle=\omega_{n}\int_{\mathbb{R}^{+}}\frac{1}{s^{n}}\left(\int_{0}^{\infty}\rho^{n-1}\varphi\left(\frac{\rho}{s}\right)d\rho\right)d\nu_{1}(s)$
$\displaystyle=\omega_{n}\int_{0}^{\infty}\rho^{n-1}\varphi(\rho)d\rho$
which is finite. Let us check that the gradient of $K_{1}$ has the desired
size estimate. Let $i=1,2,\ldots,n$, for $\left|x\right|\neq 0$ we can take
the $i$-th partial derivative of $K_{1}(x)$ inside the integral. Hence
$\frac{\partial K_{1}}{\partial
x_{i}}(x)=\int_{\mathbb{R}^{+}}\frac{1}{s^{n}}\varphi^{\prime}\left(\frac{\left|x\right|}{s}\right)\frac{1}{s}\frac{x_{i}}{\left|x\right|}d\nu_{1}(s).$
Then
$\left|\nabla K_{1}(x)\right|\leq
B\int_{\mathbb{R}^{+}}\frac{1}{s^{n}}\left(\frac{s}{\left|x\right|}\right)^{n+1}\frac{1}{s}d\nu(s)=\frac{B}{\left|x\right|^{n+1}}$
as desired. ∎
The next result shows that the hypothesis of self-similarity of $\nu_{j}$ in
Theorem 3.2 above can be avoided.
###### Theorem 3.3.
Let $\varphi$ be as in Theorem 3.2. Let $\varepsilon_{j}(\omega)$ be a
positive random variable distributed according to $\nu_{j}$ and $y_{j}\equiv
0$, $j\in\mathbb{Z}^{+}$. Then the maximal operator $\mathscr{M}$ is of weak
type (1,1).
###### Proof.
Let us show that
1. (a)
$\int_{\mathbb{R}^{n}}\left|K_{j}(x)\right|dx$ are uniformly bounded in $j$;
and
2. (b)
$\int_{\left|x\right|\geq 2\left|z\right|}\sup_{j\geq
1}\left|K_{j}(x-z)-K_{j}(x)\right|dx\leq A<\infty$ uniformly in
$z\in\mathbb{R}^{n}$.
Recall that
$K_{j}(x)=\int_{\mathbb{R}^{+}}\frac{1}{s^{n}}\varphi\left(\frac{\left|x\right|}{s}\right)d\nu_{j}(s)$.
Let us check (a),
$\displaystyle\int_{\mathbb{R}^{n}}\left|K_{j}(x)\right|dx$
$\displaystyle=\int_{\mathbb{R}^{+}}\frac{1}{s^{n}}\int_{\mathbb{R}^{n}}\varphi\left(\frac{\left|x\right|}{s}\right)dxd\nu_{j}(s)$
$\displaystyle=\int_{\mathbb{R}^{n}}\varphi(\left|x\right|)dx$
$\displaystyle=\omega_{n}\int_{0}^{\infty}\rho^{n-1}\varphi(\rho)d\rho.$
In order to prove (b), fix $z\in\mathbb{R}^{n}$ with $\left|z\right|>0$.
Notice that for $\left|x\right|\geq 2\left|z\right|$ we have that any $\xi$ in
the segment joining $x-z$ and $x$ has length larger than or equal to
$\tfrac{\left|x\right|}{2}$. In fact, set $\xi=\theta(x-z)+(1-\theta)x$, with
$0<\theta<1$, then
$\displaystyle\left|x\right|$ $\displaystyle=\left|\theta
x+(1-\theta)x\right|\leq\left|\theta(x-z)+(1-\theta)x\right|+\theta\left|z\right|=\left|\xi\right|+\theta\left|z\right|$
$\displaystyle\leq\left|\xi\right|+\left|z\right|\leq\left|\xi\right|+\frac{\left|x\right|}{2}.$
Hence for $\left|x\right|\geq 2\left|z\right|$ we have
$\displaystyle\left|K_{j}(x-z)-K_{j}(x)\right|$
$\displaystyle\leq\int_{\mathbb{R}^{+}}\frac{1}{s^{n}}\left|\varphi\left(\frac{\left|x-z\right|}{s}\right)-\varphi\left(\frac{\left|x\right|}{s}\right)\right|d\nu_{j}(s)$
$\displaystyle\leq\int_{\mathbb{R}^{+}}\frac{1}{s^{n}}\left|\varphi^{\prime}\left(\frac{\left|\xi(x,z)\right|}{s}\right)\right|\frac{\left|z\right|}{s}d\nu_{j}(s)$
$\displaystyle\leq
B\left|z\right|\int_{\mathbb{R}^{+}}\frac{1}{s^{n+1}}\left(\frac{s}{\left|\xi(x,z)\right|}\right)^{n+1}d\nu_{j}(s)$
$\displaystyle\leq 2^{n}B\frac{\left|z\right|}{\left|x\right|^{n+1}}.$
Since $\int_{\left|x\right|\geq
2\left|z\right|}\frac{\left|z\right|}{\left|x\right|^{n+1}}dx=\omega_{n}\left|z\right|\int_{2\left|z\right|}^{\infty}\frac{d\rho}{\rho^{2}}=\frac{\omega_{n}}{2}$
we have (b) and the proof of the theorem. ∎
Let us now consider some particular instances of non identically vanishing
means $y_{j}(\omega)$.
Let us start from the case (3.1.c) in Proposition 3.1. In this case the
operator $m_{j}(f)$ is given by
$m_{j}(f)(x)=\int_{\mathbb{R}^{n}}f(x-y)d\mu_{j}(y)$ where
$\mu_{j}=\mathscr{P}\circ y_{j}^{-1}$ is the $n$ dimensional distribution of
the random means $y_{j}$. Of course that there is a large class of cases of
sequences $\left\\{\mu_{j}:j\in\mathbb{Z}^{+}\right\\}$ that produce a well
behaved maximal operator. But in general the behavior of the maximal operator
in this case is far away from being good. At this point Theorem 4 in [4] helps
to build self-similar probability measures such that the differentiation
$m_{j}(f)(x)\to f(x)$ a.e., fails. For the sake of completeness let us
rephrase Theorem 4 in [4] in our current notation.
###### Theorem 3.4.
Let $n=1$. Assume that $\mu_{j}(B)=\mu_{1}(jB)$; $j\in\mathbb{Z}^{+}$ with
$d\mu_{1}(y)=\gamma(y)dy$ with $\gamma\in L^{1}(\mathbb{R})$, $\gamma$
unbounded and nondecreasing in $(0,1)$. Then, there exists an integrable
function $f$ in $\mathbb{R}$ such that
$\limsup_{j\to\infty}m_{j}(f)(x)=+\infty$
almost everywhere.
The above results show that in some sense the random character of the variance
is less restrictive that the random character of the mean for the
differentiation properties of
$\mathscr{E}[\tau_{y_{j}}M_{\varepsilon_{j}}K*f]$. Let us consider now some
cases where both $\\{\varepsilon_{j}\\}$ and $\\{y_{j}\\}$ are nontrivial. We
start by providing some sufficient conditions on the joint distribution of
$\varepsilon_{j}$ and $y_{j}$ for the weak type of $\mathscr{M}$ when the
profile $\varphi(t)=\mathcal{X}_{(0,1)}(t)$ is the standard for the Lebesgue
differentiation theorem through centered Euclidean balls.
###### Theorem 3.5.
Let $\varphi(\rho)=\mathcal{X}_{(0,1)}(\rho)$. Assume that
$\Pi_{j}(E)=\Pi_{1}(jE)$, $j\in\mathbb{Z}^{+}$ that
$\operatorname{supp}\Pi_{1}\subset[0,\alpha]\times\overline{B}(0,\beta)$ and
that $d\Pi_{1}(s,y)=\gamma(s,y)dsdy$ with $\gamma\in
L^{1}(\mathbb{R}^{+},L^{\infty}(\mathbb{R}^{n}))$. Then $\mathscr{M}(f)$ is
dominated by the standard Hardy-Littlewood maximal function $f^{*}$ defined on
the centered balls of $\mathbb{R}^{n}$.
###### Proof.
From (3.1.d) in Proposition 3.1 we have to estimate the kernel $K_{1}$. From
(3.1.a)
$\displaystyle K_{1}(x)$
$\displaystyle=\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\frac{1}{s^{n}}\varphi\left(\frac{\left|x-y\right|}{s}\right)d\Pi_{1}(s,y)$
$\displaystyle=\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\frac{1}{s^{n}}\mathcal{X}_{(0,1)}\left(\frac{\left|x-y\right|}{s}\right)d\Pi_{1}(s,y)$
$\displaystyle=\int_{\mathbb{R}^{+}}\frac{1}{s^{n}}\int_{B(x,s)}\gamma(s,y)dyds$
$\displaystyle=\int_{0}^{\alpha}\frac{1}{s^{n}}\int_{B(x,s)\cap
B(0,\beta)}\gamma(s,y)dyds.$
So that $K_{1}(x)\leq
V_{n}\left\|\gamma\right\|_{L^{1}(\mathbb{R}^{+},L^{\infty}(\mathbb{R}^{n}))}$
with $V_{n}$ the volumen of unit ball of $\mathbb{R}^{n}$. On the other hand,
for $\left|x\right|>\alpha+\beta$ we have that $B(x,s)\cap
B(0,\beta)=\emptyset$ and $K_{1}(x)=0$. Hence
$K_{1}(x)\leq\left\|\gamma\right\|_{L^{1}(\mathbb{R}^{+},L^{\infty}(\mathbb{R}^{n}))}\mathcal{X}_{(0,\alpha+\beta)}(\left|x\right|)$
and the result is proved. ∎
Again, the self-similarity of the measures $\Pi_{j}$ can be substituted by
more general conditions. Notice that under the hypotheses of Theorem 3.5 we
have that $d\Pi_{j}(s,y)=\gamma_{j}(s,y)dsdy$ with
$\gamma_{j}(s,y)=j^{n+1}\gamma(js,jy)$. So that the boundedness of the density
$\gamma$ implies that
$\left\|\gamma_{j}\right\|_{\infty}=j^{n+1}\left\|\gamma\right\|_{\infty}$ and
$\operatorname{supp}\Pi\subseteq[0,\alpha]\times\overline{B}(0,\beta)$ implies
$\operatorname{supp}\Pi_{j}\subseteq[0,\tfrac{\alpha}{j}]\times\overline{B}(0,\tfrac{\beta}{j})$.
###### Theorem 3.6.
Let $\varphi=\mathcal{X}_{(0,1)}$. Assume that $d\Pi_{j}=\gamma_{j}dsdy$,
$\operatorname{supp}\gamma_{j}\subseteq[0,r_{j}]\times B(0,r_{j})$ and
$\left\|\gamma_{j}\right\|_{\infty}\leq\frac{A}{r_{j}^{n+1}}$. Then
$\mathscr{M}(f)(x)$ is bounded by a constant time $f^{*}(x)$ for every
$x\in\mathbb{R}^{n}$.
###### Proof.
Since $K_{j}(x)=\int_{0}^{r_{j}}\frac{1}{s^{n}}\int_{B(x,s)\cap
B(0,r_{j})}\gamma_{j}(s,y)dyds$, we see that if $\left|x\right|\geq 2r_{j}$,
then $K_{j}(x)=0$. That is, $\operatorname{supp}K_{j}\subseteq B(0,2r_{j})$.
On the other hand,
$K_{j}(x)\leq\left\|\gamma_{j}\right\|_{\infty}\int_{0}^{r_{j}}\frac{1}{s^{n}}\left|B(x,s)\cap
B(0,r_{j})\right|ds\leq
V_{n}\frac{A}{r_{j}^{n+1}}r_{j}=\frac{V_{n}A}{r_{j}^{n}},$
for every $x\in\mathbb{R}^{n}$. Hence
$\sup_{j\geq 1}(K_{j}*f)(x)\leq V_{n}A\sup_{j\geq
1}\frac{1}{r_{j}^{n}}\int_{B(x,2r_{j})}\left|f(z)\right|dz,$
and $\mathscr{M}f(x)\leq 2^{n}AV_{n}^{2}f^{*}(x)$. ∎
Let us finally provide sufficient conditions on $\varepsilon_{j}$, $y_{j}$ in
order to have the weak type of $\mathscr{M}$ when $\varphi$ in not localized,
now with $y_{j}\neq 0$. In the proof of our result we shall use the following
corollary of Theorem 3.6.
###### Corollary 3.7.
Let $\varphi=\varphi_{\alpha,\beta}=\alpha\mathcal{X}_{(0,\beta)}$ with
$\alpha$ and $\beta$ positive. Assume that $\Pi_{j}$ satisfies the hypotheses
of Theorem 3.6. Then
$\mathscr{M}(f)(x)\leq\alpha\beta^{n}(\beta+1)^{n}A\,V_{n}^{2}f^{*}(x)$
for every $x\in\mathbb{R}^{n}$.
###### Proof.
Set $K_{j}^{\alpha,\beta}(x)$ to denote the kernel defined by
$\varphi_{\alpha,\beta}$ and $\Pi_{j}$. Then
$\displaystyle K_{j}^{\alpha,\beta}(x)$
$\displaystyle=\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\frac{1}{s^{n}}\alpha\mathcal{X}_{(0,\beta)}\left(\frac{\left|x-y\right|}{s}\right)d\Pi_{j}(s,y)$
$\displaystyle\leq\alpha\left\|\gamma_{j}\right\|_{\infty}\int_{0}^{r_{j}}\int_{B(x,s\beta)\cap
B(0,r_{j})}\frac{1}{s^{n}}dyds$
$\displaystyle\leq\alpha\left\|\gamma_{j}\right\|_{\infty}\int_{0}^{r_{j}}\frac{\left|B(x,s\beta)\right|}{s^{n}}ds$
$\displaystyle=\alpha\beta^{n}V_{n}\left\|\gamma_{j}\right\|_{\infty}r_{j}$
$\displaystyle\leq\alpha\beta^{n}V_{n}\frac{A}{r_{j}^{n}}.$
On the other hand, $K_{j}^{\alpha,\beta}(x)$ vanishes for
$\left|x\right|\geq(\beta+1)r_{j}$. In fact, for this values of $x$ we have
that $B(x,s\beta)\cap B(0,r_{j})=\emptyset$. Then
$K_{j}^{\alpha,\beta}(x)\leq\alpha\beta^{n}V_{n}\frac{A}{r_{j}^{n}}\mathcal{X}_{B(0,(\beta+1)r_{j})}(x)=\alpha\beta^{n}(\beta+1)^{n}V_{n}^{2}A\frac{1}{\left|B(0,(\beta+1)r_{j})\right|}\mathcal{X}_{B(0,(\beta+1)r_{j})}(x)$
and we are done. ∎
###### Theorem 3.8.
Let $\varphi:\mathbb{R}^{+}\to\mathbb{R}^{+}\cup\\{0\\}$ be bounded and non
increasing with
$\int_{\mathbb{R}^{n}}\left|x\right|^{n}\varphi(\left|x\right|)dx<\infty$. Let
$d\Pi_{j}=\gamma_{j}dsdy$ with
$\operatorname{supp}\gamma_{j}\subseteq[0,r_{j}]\times\overline{B}(0,r_{j})$
and $\left\|\gamma_{j}\right\|_{\infty}\leq\frac{A}{r_{j}^{n+1}}$. Then
$\mathscr{M}(f)(x)$ is dominated by the Hardy-Littlewood maximal function
$f^{*}(x)$.
###### Proof.
From the integrability and monotonicity properties of $\varphi$ we have
$\displaystyle\int_{\mathbb{R}^{n}}\left|x\right|^{n}\varphi(\left|x\right|)dx$
$\displaystyle\geq\omega_{n}\int_{1}^{\infty}\rho^{2n-1}\varphi(\rho)d\rho$
$\displaystyle=\omega_{n}\sum_{l\geq
0}\int_{2^{l}}^{2^{l+1}}\rho^{2n}\varphi(\rho)\frac{d\rho}{\rho}$
$\displaystyle\geq\omega_{n}\sum_{l\geq
0}\varphi(2^{l+1})2^{2ln}\int_{2^{l}}^{2^{l+1}}\frac{d\rho}{\rho}$
$\displaystyle=\omega_{n}\log 2\sum_{l\geq 0}\varphi(2^{l+1})2^{2ln}.$
Hence Now $(K_{j}*\left|f\right|)(x)=(K_{j}^{0}*\left|f\right|)(x)+\sum_{l\geq
1}(K_{j}^{l}*\left|f\right|)(x)$ with
$K_{j}^{0}(x)=\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\frac{1}{s^{n}}\varphi\left(\frac{\left|x-y\right|}{s}\right)\mathcal{X}_{(0,1)}\left(\frac{\left|x-y\right|}{s}\right)d\Pi_{j}(s,y)$
and
$K_{j}^{l}(x)=\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\frac{1}{s^{n}}\varphi\left(\frac{\left|x-y\right|}{s}\right)\mathcal{X}_{(2^{l-1},2^{l})}\left(\frac{\left|x-y\right|}{s}\right)d\Pi_{j}(s,y).$
Since $\varphi$ is bounded $K_{j}^{0}*f$ is uniformly dominated by $f^{*}$
because of Theorem 3.6. On the other hand, since for $l\geq 1$ we have
$K_{j}^{l}(x)\leq\varphi(2^{l-1})\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\frac{1}{s^{n}}\mathcal{X}_{(0,2^{l})}\left(\frac{\left|x-y\right|}{s}\right)d\Pi_{j}(s,y).$
Then, from Corollary 3.7, with $\alpha=\varphi(2^{l-1})$ and $\beta=2^{l}$,
$\sup_{j\geq
1}(K_{j}^{l}*\left|f\right|)(x)\leq\varphi(2^{l-1})2^{nl}(2^{l}+1)^{n}V_{n}^{2}Af^{*}(x)$
and we are done since
$\displaystyle\sum_{l\geq 1}\varphi(2^{l-1})2^{nl}(2^{l}+1)^{n}$
$\displaystyle\leq\varphi(1)2^{n}6^{n}+\sum_{l\geq 0}\varphi(2^{l+1})2^{2ln}$
$\displaystyle\leq\varphi(1)2^{n}6^{n}+\frac{1}{\omega_{n}\log
2}\int_{\mathbb{R}^{n}}\left|x\right|^{n}\varphi(\left|x\right|)dx$
$\displaystyle<\infty.$
∎
Notice that the conditions on $\varphi$ in Theorem 3.3 allow heavy tails for
$\varphi$. And also singularity of $\varphi$ at the origin. Instead in Theorem
3.8 the profile $\varphi$ is bounded and heavy tails like Poisson type kernels
are not covered.
## References
* [1] Kai Lai Chung, _A course in probability theory_ , third ed., Academic Press, Inc., San Diego, CA, 2001. MR 1796326
* [2] Miguel de Guzmán, _Real variable methods in Fourier analysis_ , North-Holland Mathematics Studies, vol. 46, North-Holland Publishing Co., Amsterdam-New York, 1981, Notas de Matemática [Mathematical Notes], 75. MR 596037
* [3] Elias M. Stein, _Singular integrals and differentiability properties of functions_ , Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
* [4] Felipe Zó, _A note on approximation of the identity_ , Studia Math. 55 (1976), no. 2, 111–122. MR 423013
## Acknowledgements
This work was supported by the Ministerio de Ciencia, Tecnología e Innovación-
MINCYT in Argentina: Consejo Nacional de Investigaciones Científicas y
Técnicas-CONICET and Agencia Nacional de Promoción Científica y Técnica-
ANPCyT, (Grant PICT 2015-3631).
Affiliations: Instituto de Matemática Aplicada del Litoral, UNL, CONICET.
Address: CCT CONICET Santa Fe, Predio “Alberto Cassano”, Colectora Ruta Nac.
168 km 0, Paraje El Pozo, S3007ABA Santa Fe, Argentina.
E-mail address<EMAIL_ADDRESS>`ivanagomez@santafe-
conicet.gov.ar`
|
††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Antivax movement and epidemic spreading in the era of social networks:
nonmonotonic effects, bistability and network segregation
Marcelo A. Pires1 Andre L. Oestereich2 Nuno Crokidakis3 Sílvio M. Duarte
Queirós1,4,5 1Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro/RJ,
Brazil
2Unidade Central De Educação Faem Faculdade, Itapiranga/SC, Brazil
3Instituto de Física, Universidade Federal Fluminense, Niterói/RJ, Brazil
4National Institute of Science and Technology for Complex Systems, Brazil
5i3N, Universidade de Aveiro, Campus de Santiago, 3810-193 Aveiro, Portugal
###### Abstract
In this work, we address a multicoupled dynamics on complex networks with
tunable structural segregation. Specifically, we work on a networked epidemic
spreading under a vaccination campaign with agents in favor and against the
vaccine. Our results show that such coupled dynamics exhibits a myriad of
phenomena such as nonequilibrium transitions accompanied by bistability.
Besides we observe the emergence of an intermediate optimal segregation level
where the community structure enhances negative opinions over vaccination but
counterintuitively hinders - rather than favoring - the global disease
spreading. Thus, our results hint vaccination campaigns should avoid policies
that end up segregating excessively anti-vaccine groups so that they
effectively work as echo chambers in which individuals look to confirmation
without jeopardising the safety of the whole population.
## I Introduction
In such a complex contemporary society where elements – people and events –
influence one another and feedback at different scales [1], the application of
tools set forth in Statistical Physics in order to cope with collective
phenomena has gained prominence in other areas such as Biology and Medicine,
Social Sciences, and Humanities, which have put quantitative tools in the
methodologies they apply [2, 3, 4, 5, 6, 7].
Accordingly, phenomena in which there is a change in the collective behavior
displayed by a social system have turn into an appropriate field for the
application of such techniques [8, 9, 10]; among the several different
instances we can find important contributions within the spreading of
epidemics as well as opinion dynamics (see eg [11] and Sec. II). In spite of
the fact that the two subject-matters are not related at first, the
dissemination of a causal relationship between neurological disorders and
vaccinations [12] has prompted an urban myth that ultimately has jeopardized
the elimination of the disease in countries with a very high Human Development
Index as the USA [13].
Next the manuscript is organized as follows: in Sec. II, we establish the
state-of-the-art of the problem; in Sec. III, we introduce our model for the
combined dynamics of opinion and contagion; in Sec. IV, we discuss the results
for the model; and in Sec. V, we present our final observations on the work
and future perspectives about it.
## II Literature review
Modular networks [14] are generated by an algorithm that leads to networks
with an architecture of communities. A given node in each community can be
connected to nodes of the same community (intracommunity links) and/or to
nodes of the other community (intercommunity links).
The impact of the network modularity in spreading processes has been
investigated in recent years. Since the results introduced in Ref. [14], a
series of works were published regarding the subject of optimal network
modularity; therein, the authors showed that modular structure may have
counterintuitive effects on information diffusion. Indeed, it was discussed
that the presence of strong communities in modular networks can facilitate
global diffusion by improving local intracommunity spreading.
Still in relation to modular networks, it was recently found that an optimal
community structure that maximizes spreading dynamics which can pave the way
to rich phase diagrams with exhibiting first-order phase transitions [15].
Within the same context, the authors in Ref. [16] discussed the impact of
social reinforcement in information diffusion. They also found optimal multi-
community network modularity for information diffusion, i.e., depending on the
range of the parameters the multi-community structure can facilitate
information diffusion instead of hindering it.
Regarding biological systems, it was recently found there is a nonlinear
relation between modularity and global efficiency in animal networks, with the
latter peaking at intermediate values of the former [17]. In addition, in
neural networks there exists an optimal modularity for memory performance,
where a balance between local cohesion and global connectivity is established,
allowing optimally modular networks to remember longer [18].
The authors in Ref. [19] studied the importance of close and ordinary social
contacts in promoting large-scale contagion and found an optimal fraction of
ordinary contacts for outbreaks at a global scale. With respect to
correlations in complex networks, it was found that constraining the mean
degree and the fraction of initially informed nodes, the optimal structure can
be assortative (modular), core-periphery or even disassortative [20]. Other
recent works leading with optimal modularity in networks can be found in [21,
22].
In a recent work [23], it was proposed a model of disease spreading in a
structural modular complex network and studied how the number of bridge nodes
$n$ that connect communities affects disease spreading. It was verified that
near the critical point as $n$ increases, the disease reaches most of the
communities, but each community has only a small fraction of recovered nodes.
Moreover, a combination of social networks with game theory was studied in
Refs. [24, 25].
Disease information can spark strong emotions like fear — or even panic — that
would affect behavior during an epidemic. The authors in [26] considered an
agent-based model that assumes that agents can obtain a complete picture of
the epidemic via information from local daily contacts or global news
coverage. Those results helped conclude that such model can be used to mimic
real-world epidemic situations and explain disease transmission, behavior
changes, and distribution of prevalence panic. Game theory was also considered
to reproduce the decision-making process of individuals during the evolution
of a disease. In [27] a spatial evolutionary game was coupled to a SIR model,
and the results showed that protective behaviors decrease the numbers of
infected individuals and delay the peak time of infection. The study also
concluded that increased numbers of risk-averse individuals and preemptive
actions can more effectively mitigate disease transmission; however, changes
in human behavior require a high social cost (such as avoidance of crowded
places leading to absences in schools, workplaces, or other public places).
A recent work considered a coupled behavior-change and infection in a
structured population characterized by homophily and outgroup aversion [28].
It was found that homophily can either increase or decrease the final size of
the epidemic depending on its relative strength in the two groups. In
addition, homophily and outgroup aversion can also produce a ‘second wave’ in
the first group that follows the peak of the epidemic in the second group.
Models of opinion dynamics were applied in the context of opinions about
vaccination (pro versus anti-vaccine) without coupling an epidemic process
[29]. Later, kinetic opinion dynamics were coupled to classical epidemic
models in order to study the feedback among risk perception, opinions about
vaccination, and the disease spreading. In [30] it was found that the
engagement of the pro-vaccine individuals can be crucial for stopping the
epidemic spreading. On the other hand, the work [31] found counterintuitive
outcomes like the fact that an increment in the initial fraction of the
population that is pro-vaccine can lead to smaller epidemic outbreaks in the
short term, but it also contributes to the survival of the chain of infections
in the long term.
Recently, the anti-vaccine sentiment was treated as a cultural pathogen. The
authors in [32] modeled it as a ’infection’ dynamics. The authors showed that
interventions to increase vaccination can potentially target any of three
types of transitions - decreasing sentiment transmission to undecided
individuals, increasing pro-vaccine decisions among undecided individuals, or
increasing sentiment switching among anti-vaccine individuals.
We previously cited anti-vaccine opinions, thus it is important to mention
some recent discussion about the global anti-vaccine movement. Since the
online discussions dominate the social interactions in our modern world, the
propagation of such anti-vaccine opinions is growing fast. A recent report
noted that 31 million people follow anti-vaccine groups on Facebook, with 17
million people subscribing to similar accounts on YouTube [33]. The authors in
[34] recently pointed that if the current trends continue, anti-vaccine views
will dominate online discussion in 10 years. The importance of anti-vaccine
movement is fundamental for the evolution of COVID-19 outbreak. Indeed, the
authors in [35] called attention to the fact that it is a key point to
qualitatively assess how the administration of a vaccine could affect the
COVID-19 outbreak, taking into account of the behavioral changes of
individuals in response to the information available on the status of the
disease in the community. According to a study published in August 2020,
nearly one in four adults would not get a vaccine for COVID-19 [36] and in
some countries, more than half of the population would not get it, including
Poland and France [37]. In September 2020, it was verified that only 42
percent of Americans said yes to receiving a future COVID-19 vaccine, across
all political sides. It means that even in a best-case scenario where a future
high performing vaccine is 95$\%$ effective in an individual, it would only
impact 42x95$\approx 40\%$ of the population, which is way below predicted
thresholds for herd immunity [38].
Figure 1: Examples of modular networks with $N=100$, $\langle k\rangle=10$ for
different values of $\mu$. The parameter $\mu$ is the community
interconnectivity: small values of $\mu$ means few intercommunities bridges
which implies strong community structure, ie strong modularity/segregation. In
these examples we can see the strengthening of the community structure for
lower values of $\mu$. Figure 2: Coupled vaccination and continuous opinion
dynamics. Figure 3: Steady-state for the spreading measure $I_{i}$ and
collective opinion $m_{i}$ for each community $i=\\{1,2\\}$. Symbols are the
steady-state outcome for each sample, i.e., each symbol is the result from
each Monte Carlo realization. Results for $\mu=0.1$. For this high level of
segregation, each community ends up preserving the sign of its initial
opinion. Besides, the chain of contagion starts in the community $2$ and
cannot become permanent in the community $1$. Figure 4: Steady-state for the
spreading measure $I_{i}$ and collective opinion $m_{i}$ for each community
$i=\\{1,2\\}$. Symbols are the steady-state outcome for each sample, ie, each
symbol is the result from each Monte Carlo realization. Results for $\mu=0.2$.
For this intermediate level of segregation, there is the possibility for a
switch of opinion in the community $2$ (seed). The epidemic spreading does not
survive at the global and local levels. Figure 5: Steady-state for the
spreading measure $I_{i}$ and collective opinion $m_{i}$ for each community
$i=\\{1,2\\}$. Symbols are the steady-state outcome for each sample, i.e.,
each symbol is the result from each Monte Carlo realization. Results for
$\mu=0.3$. For this low level of segregation, there is the possibility for a
switch of opinion in both communities. The epidemic dynamics can survive if
the contagion is not too aggressive (intermediate values of $\lambda$) .
Figure 6: Steady-state for the spreading measure $I_{i}$ and collective
opinion $m_{i}$ for each community $i=\\{1,2\\}$. Symbols are the steady-state
outcome for each sample, i.e., each symbol is the result from each Monte Carlo
realization. Results for $w=0.1$ and $\lambda=0.8$.
## III Model
### III.1 I: Opinion dynamics
Even though payoff-based models have been employed to address the problem of
vaccination dynamics (for instance see [24, 25, 39] and the references
therein), there is an alternative approach that is based on the coupling of
epidemic and psychosocial factors that have been provided a successful
modelling of phenomena related to vaccination dynamics [40, 41, 42, 28, 43,
32, 44]. In this work, we follow such second methodology. Specifically, based
on Refs. [31],[45] we consider an agent-based dynamics in which the opinion
about vaccination, $o_{i}\in[-1,1]$, of each agent, $i$, evolves with
$o_{i}(t+1)=o_{i}(t)+\epsilon o_{j}(t)+wI_{n(i)}(t),$ (1)
A negative (positive) value of $o_{i}$ represents an individual $i$ supporting
anti-vaccine (pro-vaccine) opinion. Equation (1) takes into account the
agent’s opinion, $o_{i}(t+1)$, depends on multiple factors: (i) his previous
opinion $o_{i}(t)$; (ii) the peer pressure exerted by a randomly selected
neighbor, $j$, modulated by a stochastic heterogeneity $\epsilon$, uniformly
distributed in the interval $[0,1]$; (iii) the proportion of infected
neighbors, $I_{n(i)}(t)$, modulated by a risk perception parameter, $w$.
Notice that, in Eq. (1), if the value of the opinion exceeds (falls below) the
value $1(-1)$, then it adopts the extreme value $1(-1)$ [45].
The opinion dynamics regarding the vaccination campaign is coupled with the
epidemic dynamics, due to the factor $I_{n(i)}(t)$ in Eq. (1).
### III.2 II: epidemics-vaccination dynamics
Based on [30, 31] (and references therein), we define the transitions among
the epidemic compartments as follows:
* •
$S\stackrel{{\scriptstyle g_{i}}}{{\rightarrow}}R$: a Susceptible agent $i$
becomes Vaccinated with probability $g_{i}$;
* •
$S\stackrel{{\scriptstyle(1-g_{i})\lambda}}{{\rightarrow}}I$: a Susceptible
agent $i$ becomes Infected with probability $(1-g_{i})\lambda$ if he is in
contact with an Infected agent;
* •
$I\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}S$: an Infected agent $i$
recovers with probability $\alpha$;
* •
$R\stackrel{{\scriptstyle\phi}}{{\rightarrow}}S$: a immune agent $i$ becomes
Susceptible again with the resusceptibility probability $\phi$. We assume that
Vaccinated and Recovered agents are in the same compartment[46, 47, 48, 49,
50].
The vaccination probability $g_{i}$ of an agent $i$ is proportional to his
opinion about vaccination $-1\leq o_{i}\leq 1$:
$\displaystyle g_{i}(t)=\frac{1+o_{i}(t)}{2}\in[0,1]$ (2)
Despite the differences, the modeling of the coupling between disease and
opinion evolution is still a open subject. In this work, we consider the two
dynamics have the same time scale. An overview of our model is shown in Fig.2.
An element in this problem which is still focus of debate concerns the
timescale of each dynamics, epidemic and opinion. On the one hand, it is often
assumed in the epidemiological literature [41, 43, 42, 51] that the two
timescales are equivalent. At first, this can be understood as a
simplification it captures the mass vaccination campaigns governments swiftly
implement in order to avoid disease outbreaks. On the other hand, it is
possible to assume different timescales of evolution of the diseases and
opinions about the disease [52, 53]. In this work we consider the first
approach of equality between the two timescales.
### III.3 Community structure
Based on Ref. [54] and related literature, we start by picking the first
$N_{1}=N/2$ of the $N$ nodes and attaching them to the community $1$, and
assigning the other $N_{2}=N-N_{1}$ nodes to community $2$. We then proceed by
randomly assigning $(1-\mu)M$ connections among pairs of nodes from the same
community and $\mu M$ connections are randomly distributed among pairs of
nodes that belong to distinct communities, where $M=N\,k/2$ and $k$ is the
network average degree [54].
The parameter $\mu$ regulates the community strength: large values of $\mu$
means more ties between the two communities consequently a weaker community
organization. Another way to control the network structure – especially in the
formation of the echo chambers – is by considering rewiring [55].
### III.4 Initial condition
We consider that community $1$ holds a positive stance on vaccination, whereas
the community $2$ holds a negative opinion about that. We also assume the
chain of infections starts in community $2$, because $o_{i}<0$ leads to a low
propensity for the agents to get vaccinated, which is naturally more relevant.
If the epidemic started in community $1$, pro-vaccine opinions, $o_{i}>0$,
would induce a higher probability for an agent to get vaccinated that
ultimately would end up disrupting the chain of contagions.
Let $U(a,b)$ be a single random value from a uniform distribution in the range
$[a,b]$.
At $t=0$, we set:
* •
For i in $0\ldots N/2-1$: (community 1: $o_{i}>0$; $0\%$ infected)
* –
$o_{i}\sim U(0,1)$
* –
status(i) = S
* •
For i in $N/2\ldots N-1$: (community 2: $o_{i}<0$; $1\%$ of infected)
* –
$o_{i}\sim U(-1,0)$
* –
status(i) = S with probability $0.99$
* –
status(i) = I with probability $0.01$
## IV Results and Discussion
Figure 7: Steady-state for the spreading measure $I_{i}$ for community 1,
above, and 2, bellow. The colors indicate the average of non zero steady-state
outcomes of all 200 samples. Results for $w=0.1$.
Figure 8: Steady-state for the spreading measure $I_{i}$ for community 1,
above, and 2, bellow. The colors indicate the average of non zero steady-state
outcomes of all 200 samples. Results for $w=0.2$.
In this section, we present our results come from Monte Carlo simulations of
networks with $N=10^{4}$ nodes and k=20. In all simulations, we set
$\alpha=0.1$ and $\phi=0.01$, without loss of generality. In Figs. 3-6, we
show the steady-state density of infected agents in the community $u$,
$I_{u}$. We also depict the behavior of the stationary opinion in the
community $u$, $m_{u}$. In turn, $I_{tot}$ and $m_{tot}$ refer to the global
proportion of infected individuals and global mean opinion, respectively.
The results in Fig. 3 show that in the community $2$ — the seed community —
there is a transition from the absorbing phase (extinction of the epidemic) to
the epidemic survival phase. In the community $1$, there is no survival of the
chain of infections in the long term. In this setting with $\mu=0.1$ – which
can be understood as yielding a weak modular structure because of the small
value of the parameter – the seed community remains with the negative opinion
about vaccination, which weakens the vaccination campaign and thus facilitates
the local permanence of the disease. Similarly, there is a persistence of the
initial opinion in the community $1$, which in this case is pro-vaccine and
therefore favors the vaccine uptake that makes the epidemic spreading
unsustainable. This means that a low number of intercommunity ties hinders the
change in the community stance over vaccination; that creates a strong
distinction in the epidemic spread between both communities with community $1$
being unfavorable to epidemic spreading since $m_{1}>0$, and community $2$
being favorable since $m_{2}<0$.
In Fig. 4, it is notable that an intermediate community strength leads to the
elimination of the epidemic transmission in both communities even when there
is a dominance of the negative opinion about vaccination in the community $2$.
The epidemic contagion spreading is halted in the community $2$, even though
the agents have a negative opinion about the vaccination, due to the
intermediate number of bridges, $\mu=0.2$, to the other community. These
bridges are just strong enough to drain the infected agents of the community
$2$, but not strong enough to change its average opinion.
In Fig. 5, with $\mu=0.3$ there is a high number of intercommunity links. This
additional connectivity between communities weakens the initial epidemic
spreading in the community $2$, but it is sufficient to introduce the
possibility of a wide opinion change in the community $1$. The opinion change
in the community $1$ facilitates the epidemic spreading in that community.
This effect is limited because we can see for high infection probabilities
$\lambda>0.8$ the epidemic spread vanishes. So, we have a counterintuitive
effect, because for higher transmissibility the epidemic spread vanishes. The
reason behind this is the risk perception, $wI$ in Eq. (1), which promotes
vaccination, so higher transmissibility leads to a bigger outbreak that in
turn results in better opinions about vaccination which ends up stopping the
epidemic outbreak.
The emergence of an intermediate range of $\mu$ that blocks the local and
global epidemic spreading is visible in Fig. 6. Regarding the opinion
dynamics, an initial increase in $\mu$ leads to a decrease in $m_{1}$ and an
increase in $m_{2}$, that is the collective opinions tend to be less extremist
for an initial rise in the amount of inter-communities routes. Then a further
increase in $\mu$ promotes a sudden rise in $m_{1}$ and $m_{2}$ which means a
speed up in the switch of opinions in the community $2$. A further rise in
$\mu$ leads to a bistable behavior in both communities.
This intermediary range of inter-community connectivity that promotes a
minimal epidemic spreading seems to also come from a perceived increment in
the probability of an infected individual having a vaccinated neighbor. The
increment of the bridges between communities the initially infected agents
have a bigger probability of having a neighbor that was vaccinated because
initially most of the infected people are in the community $2$ and most agents
with a positive opinion about vaccination are in the community $1$. This
effect does not persist for higher values of $\mu$ because then both
communities tend to adopt the same average opinion about vaccination and this
opinion can, in some cases, be negative. A negative global opinion about
vaccines does not guarantee that the epidemic spread will persist, as can be
seen in some cases for $\mu\approx 0.23$ where all samples had no infected
individual but some of them had negative opinions about the vaccination. This
can occur due to the fact that the number of infected agents can become zero
before the negative global consensus about vaccines is reached.
While in Figs. 3-4 there is a single stable steady-state (either extinction or
persistence), Fig. 5 displays bistable solutions depending on the randomness
’embedded’ in the dynamics. Moreover, the results in Fig. 3 suggest the
absorbing-active epidemic transition is continuous for strong communities
(such as $\mu=0.1$) whereas the results shown in Fig. 5 signalize this
extinction-persistence epidemic transition is discontinuous for weak
communities (such as $\mu=0.3$). Therefore, the structural factors present in
the modular networks can induce the emergence of bistability in the epidemic-
vaccination-opinion dynamics as well as a change in the nature of the
absorbing-active transitions.
An overall look into Figs. 3-6 reveals a sudden transition can emerge from
structural factors (increasing $\mu$) or epidemiological factors (increasing
$\lambda$). The transitions from the Disease-Free phase to the active phase
and vice-versa (epidemic resurgence) highlight the nonmonotonic behavior of
the full dynamics with the transmissibility $\lambda$.
Comparing with other works, we see that while in [14] there is an optimal
modularity for enhancing information spreading, here there is an optimal
modularity for hindering epidemic spreading.
In figs. 7 and 8 we can see a wide range of results for two different settings
of risk perception, i.e. $w=0.1$ and $w=0.2$. These results are similar but
they show how increasing the risk perception reduces the range of parameters
that present an endemic state. Other than that, we can also see that in
community 1 the endemic state is more prevalent for higher values of
modularity $\mu$, this is to be expected since initially only community 2 has
infected agents. In community 2 the increment in modularity initially reduces
the fraction of infected agents, but at a certain point when the endemic state
appears in community 1 it surges back in community 2. This further reinforces
that optimal modularity reduces the epidemic spreading.
## V Final Remarks
In previous work, namely Ref. [40], it was shown with a binary opinion
dynamics that the spread of opinions against vaccination is one of the
potential responsible for the large outbreaks of vaccine-preventable diseases
in many high-income countries. In this work, we have gone farther afield to
show the emergence of a networked SIRSV model that the spectrum of scenarios
arising from the competition of pro- vs anti-vaccine views during an epidemic
spreading is highly complex.
The several outcomes shown in Figs.3-8 point out that our model produces a
diverse phenomenology where the social and biological scenarios exhibit a
nonmonotonic dependence with spreading rate $\lambda$. From the perspective of
the dynamical systems, our results provide a new mechanism for bistability in
a biological-social setting. From a practical point of view, our work offers
new perspectives for the development of novel strategies for halting epidemic
spreading based on tuning the modularity to an optimal degree.
Some pro-vaccine strategies can have as side effect the segregation between
individuals with conflicting views about the vaccines and clustering of
similars. In [56] the authors found that in scenarios with effective vaccines,
the impact of clustering and correlation of belief systems become stronger.
Alternatively, the authors in Ref. [57] shown that segregation of anti-vaxxers
can potentially extend the duration of an epidemic spreading, whereas in Ref.
[58] it was found that an increase in the contact between vaccine refusers and
the rest of the society can lead to a scenario where vaccination alone may not
be able to prevent an outbreak. Here we show that too much or too low
segregation of anti-vaxxers favors the chain of contagion, but an intermediate
level of segregation disfavor the epidemic spreading. Therefore, our results
indicate that vaccination campaigns should avoid strategies that have as a
side effect too much informational segregation of anti-vaccine groups so that
reliable pro-vaxx information can reach those groups whilst enforcing a
minimum degree of physical distancing as it occurs in countries where
childhood vaccination is required at some degree, namely school entry [59].
Our work produces a thought-provoking analogy. In a small-world architecture,
there is an intermediate number of long-range bridges that lead the full
network to have unusual properties such as high clustering and low path
lengths. Here, a structure with an intermediate number of inter-community ties
leads the dynamics in the full network to produce an interesting outcome,
namely the suppression of the epidemics. Thus, it would be interesting to
consider further sophisticated network architectures, like multiplex networks.
Despite the rich phenomenology we observed in our model, some limitations can
be discussed which can be targeted in future work. The structure social
contacts’ structure of modular networks, presenting communities, is relevant
to study several dynamical processes [14]-[23]. However, it could be more
realistic to consider two distinct layers, one for the spreading of each
dynamics (epidemic and opinion ones), but with each dynamics influencing the
other. Such multiplex network structure can model better the coupled opinion-
epidemic dynamics. Other rules for the opinion dynamics, distinct of the
kinetic exchanges, could also be considered.
Besides, it will be worthwhile to consider the interplay between several
sources of heterogeneity in agent’s bias, namely plurality and polarization
[60].
## Acknowledgments
The authors acknowledge financial support from the Brazilian funding agencies
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq),
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and
Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro
(FAPERJ).
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|
# First principles feasibility assessment of a topological insulator at the
InAs/GaSb interface
Shuyang Yang Department of Materials Science and Engineering, Carnegie Mellon
University, Pittsburgh, PA 15213, USA Derek Dardzinski Department of
Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA
15213, USA Andrea Hwang Department of Materials Science and Engineering,
Carnegie Mellon University, Pittsburgh, PA 15213, USA Dmitry I. Pikulin
Microsoft Quantum, Redmond, WA 98052, USA Microsoft Quantum, Microsoft
Station Q, University of California, Santa Barbara, California 93106-6105, USA
Georg W. Winkler Microsoft Quantum, Microsoft Station Q, University of
California, Santa Barbara, California 93106-6105, USA Noa Marom
<EMAIL_ADDRESS>Department of Materials Science and Engineering,
Carnegie Mellon University, Pittsburgh, PA 15213, USA Department of
Chemistry, Carnegie Mellon University, Pittsburgh, PA 15213, USA Department
of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA
###### Abstract
First principles simulations are conducted to shed light on the question of
whether a two-dimensional topological insulator (2DTI) phase may be obtained
at the interface between InAs and GaSb. To this end, the InAs/GaSb interface
is compared and contrasted with the HgTe/CdTe interface. Density functional
theory (DFT) simulations of these interfaces are performed using a machine-
learned Hubbard U correction [npj Comput. Mater. 6, 180 (2020)]. For the
HgTe/CdTe interface our simulations show that band crossing is achieved and an
inverted gap is obtained at a critical thickness of 5.1 nm of HgTe, in
agreement with experiment and previous DFT calculations. In contrast, for
InAs/GaSb the gap narrows with increasing thickness of InAs; however the gap
does not close for interfaces with up to 50 layers (about 15 nm) of each
material. When an external electric field is applied across the InAs/GaSb
interface, the GaSb-derived valence band maximum is shifted up in energy with
respect to the InAs-derived conduction band minimum until eventually the bands
cross and an inverted gap opens. Our results show that it may be possible to
reach the topological regime at the InAs/GaSb interface under the right
conditions. However, it may be challenging to realize these conditions
experimentally, which explains the difficulty of experimentally demonstrating
an inverted gap in InAs/GaSb.
††preprint: APS/123-QED
## I INTRODUCTION
Two-dimensional topological insulators (2DTIs) have attracted increasing
attention in recent years owing to the emergence of helical edge states and
backscattering-free edge currents relevant for applications in spintronics and
quantum computing [1, 2, 3]. 2DTIs were first proposed based on a theoretical
model of graphene incorporating spin-orbit interactions [4]. However, the
required type of spin-orbit coupling in graphene is too weak to observe the
quantum spin Hall effect (QSHE) experimentally [5]. Later, a proposal for 2DTI
was made based on a HgTe/CdTe quantum well (QW) [6] and the signatures of the
QSHE were experimentally demonstrated [7]. When the thickness of the HgTe in
the QWs is varied, the band structure changes from a trivial insulator to a
2DTI with an inverted gap when a critical thickness is reached [6, 7, 8, 9].
2DTIs have been proposed in additional materials systems, some of which have
shown promising signs [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].
In the present work we focus on another QW structure, InAs/GaSb. It has been
proposed that a 2DTI may be realized in InAs/GaSb QWs because the band lineup
of coupled InAs/GaSb QWs could lead to the coexistence of electrons and holes
at the charge neutrality point [23, 24]. The topological insulator phase would
arise if the band ordering were inverted and coupling between electron and
hole states opened a hybridization gap which is necessarily topological due to
the orbital structure of the hybridized bands [25]. Such band ordering could
potentially be achieved by choosing appropriate QW thickness and by applying
an external electric field [5]. InAs/GaSb QWs are in the family of well-
studied III-V compounds and have thus attracted considerable experimental
interest [26, 27, 28, 29, 30, 31, 32, 33, 34]. The experiments have provided
some encouraging signs of edge conductance in the material. However, a phase
diagram showing a clear topological transition accompanied by edge state
formation has yet to be demonstrated. Here, we use first principles
simulations to investigate whether it would be possible to realize a 2DTI at
the InAs/GaSb interface and under what conditions.
The HgTe/CdTe and InAs/GaSb interfaces have been studied theoretically using a
variety of methods. This includes the k$\cdot$p method [35, 9, 36, 37, 38,
39], pseudopotential models [40, 41], and tight-binding [42, 43]. The drawback
of these semi-empirical methods is that the fitting to experimental data
largely determines the extent of their predictive capability. Atomistic ab
initio simulations may provide a more accurate representation of the
electronic properties and their dependence on the structure of the interface.
First principles studies based on density functional theory (DFT) have
investigated the influence of thickness on the edge states of HgTe/CdTe(100)
[44, 45]. Using different exchange-correlation functionals and different
thicknesses of CdTe, Ref. [44] predicted a critical thickness of 4.6 nm of
HgTe, whilst Ref. [45] predicted a critical thickness 6.5 nm of HgTe. Both
results are close to the experimental critical thickness of 6.3 nm [6].
For InAs and GaSb, local and semi-local exchange-correlation functionals
severely underestimate the band gaps to the point that they reduce to zero
[46], due to the self-interaction error (SIE). Some DFT studies of InAs/GaSb
have applied an empirical correction to the DFT band gaps [47, 48]. Others
have used hybrid functionals, which mitigate the effect of SIE by including a
fraction of exact exchange [49]. An alternative approach, which has been used
to obtain more accurate band gaps for InAs/GaSb is many-body perturbation
theory within the $GW$ approximation, where $G$ stands for the one-particle
Green’s function and $W$ stand for the screened Coulomb interaction [50].
Although hybrid DFT functionals and the $GW$ approximation produce
significantly improved band gaps, their high computational cost limits their
applicability to relatively small system sizes. Therefore, these methods have
been used only for periodic heterostructures of InAs/GaSb with very few layers
[49, 50]. DFT studies of large interface slab models with vacuum regions have
not been conducted. All previous ab initio studies of InAs/GaSb have not
reported the band structure and band alignment at the interface and have not
shown an inverted band gap. Furthermore, previous studies have not considered
the effect of applying an electric field, which plays an important role in
experiments, and therefore should be considered computationally.
Recently, we have introduced a new method of DFT with a machine-learned
Hubbard U correction, which can provide a solution for accurate and efficient
simulations of InAs and GaSb [51]. Within the Dudarev formulation of DFT+U
[52] the effective Hubbard U is defined as $U_{eff}=U-J$, where $U$ represents
the on-site Coulomb repulsion, and $J$ represents the exchange interaction.
For a given material, the $U_{eff}$ parameters of each element are machine-
learned using Bayesian optimization (BO). The BO algorithm finds the optimal
$U_{eff}$ values that maximize an objective function formulated to reproduce
as closely as possible the band gap and the qualitative features of the band
structure obtained with a hybrid functional. The DFT+U(BO) method allows for
negative $U_{eff}$ values. Negative $U_{eff}$ values are theoretically
permissible when the exchange term, $J$, is larger than the on-site Coulomb
repulsion, $U$ [53, 54, 55, 56, 57]. We have found that negative $U_{eff}$
values are necessary to produce band gaps for narrow-gap semiconductors, such
as InAs and GaSb. Because the reference hybrid functional calculation is
performed only once for the bulk material to determine the optimal $U_{eff}$
values, the computational cost of DFT+U(BO) calculations for interfaces is
comparable to semi-local DFT.
In this work, we use the DFT+U(BO) method to study the HgTe/CdTe and InAs/GaSb
interfaces. For the HgTe/CdTe interface, we obtain band crossing at a critical
thickness of 5.1 nm of HgTe, and subsequently an inverted gap is observed. Our
results are in agreement with experiment and previous DFT studies, thus
validating the DFT+U(BO) method. For the InAs/GaSb interface, we find that
increasing the thickness of InAs leads to gap narrowing. However, band
crossing is not obtained up to the largest number of layers calculated here.
When an external electric field is applied across the InAs/GaSb interface, the
GaSb-derived valence band maximum is shifted up in energy compared to the
InAs-derived conduction band minimum. Band crossing is achieved at a critical
field, followed by an inverted gap which widens and shifts higher above the
Fermi level as the field is increased. Our results indicate that it may be
possible to reach the topological regime in InAs/GaSb QWs. However, doing so
would require a combination of careful interface engineering, a considerable
electric field across the interface, and gating to tune the position of the
Fermi level. This explains the difficulty of experimentally demonstrating an
inverted gap in InAs/GaSb.
## II METHODS
### II.1 Computational details
DFT calculations were performed using the Vienna ab initio simulation package
(VASP) [58] with the projector augmented wave method (PAW) [59, 60]. The
generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE)
[61, 62] was used with a Hubbard U correction [52] determined by Bayesian
optimization [51], as detailed below. Spin-orbit coupling (SOC) [63] was
included throughout and the energy cutoff was set to 400 eV. For bulk band
structure calculations a 8$\times$8$\times$8 k-point grid was used to sample
the Brillouin zone. For interface calculations a 8$\times$8$\times$1 k-point
grid was used to sample the interface Brillouin zone and dipole corrections
[64] were included. Bulk unfolding [65] was applied to project interface band
structures onto the primitive cell, as described in the SI.
### II.2 Performance of PBE+U(BO)
Figure 1: Performance of different DFT functionals for CdTe and HgTe: PBE
band structures of (a) CdTe and (b) HgTe; HSE band structures of (c) CdTe and
(d) HgTe; PBE+U(BO) band structures of (e) CdTe and (f) HgTe; The
contributions of the Cd/Hg $s$, Cd/Hg $d$, and Te $p$ states are indicated by
the red, green, and yellow dots, respectively. $A$ is the point along
$X-\Gamma$ in the bulk’s Brillouin zone with the coordinates (0.1, 0.1, 0).
$A-\Gamma-A$ is mapped to $\overline{A}-\overline{\Gamma}-\overline{A}$ in the
(001) direction.
The PBE functional fails to provide an adequate description of the band
structures of the materials studied here. The cases of InAs and GaSb have been
discussed in detail in [51]. For CdTe, Fig. 1a shows that PBE severely
underestimates the bad gap compared to the experimental value of 1.60 eV [66].
This is because the Cd 4$d$ states, which contribute significantly to the top
of the valence band, are pushed up in energy due to the SIE [67]. For HgTe,
Fig. 1b shows that PBE produces an incorrect band shape and band ordering at
the $\Gamma$ point with the Hg $s$ orbitals and Te $p$ orbitals inverted
around 1 eV below the Fermi level [45]. These issues are rectified by the
Heyd-Scuseria-Ernzerhof (HSE) [68, 69] hybrid functional, as shown in Fig. 1c
for CdTe and Fig. 1d for HgTe. However, the computational cost of HSE is too
high for simulations of large interface models.
To achieve a balance between accuracy and efficiency, a Hubbard $U$ correction
was applied to the $p$ orbitals of In, As, Ga and Sb and the $d$ orbitals of
Hg and Cd within the Dudarev approach [52]. For each orbital, the optimal
value of Ueff was machine learned by Bayesian optimization [51]. The objective
function was formulated to reproduce as closely as possible the band structure
produced by HSE:
${f(\vec{U})}=-{\alpha_{1}}(\text{E}_{\text{g}}^{\text{HSE}}-\text{E}_{\text{g}}^{\text{PBE+U}})^{2}-{\alpha_{2}}(\Delta\text{Band})^{2}$
(1)
Here, $\vec{U}$ = [$U^{1}$, $U^{2}$,…,$U^{n}$] is the vector of $U_{eff}$
values applied to different atomic species and $U^{i}\in[-10,10]$ eV.
$\Delta\text{Band}$ is defined as the mean squared error of the PBE+U band
structure with respect to HSE:
${\Delta\text{Band}}=\sqrt{\frac{1}{N_{E}}\sum^{N_{k}}_{i=1}\sum^{N_{b}}_{j=1}(\epsilon_{HSE}^{j}[k_{i}]-\epsilon_{PBE+U}^{j}[k_{i}])^{2}}$
(2)
$N_{E}$ represents the total number of eigenvalues, $\epsilon$, included in
the comparison, $N_{k}$ is the number of $k$-points, and $N_{b}$ is the number
of bands selected for comparison. To avoid double counting the band gap
difference in the calculation of $\Delta\text{Band}$, the valence band maximum
(VBM) and conduction band minimum (CBM) are shifted to zero for both the PBE+U
and HSE band structures. Hence, $\Delta\text{Band}$ captures differences in
the qualitative features of the band structures produced by PBE+U vs. HSE,
independently of the difference in the band gap. The coefficients $\alpha_{1}$
and $\alpha_{2}$ may be used to assign different weights to the band gap vs.
the band structure. The default values are 0.25 and 0.75, respectively. For
CdTe, we set $\alpha_{1}=\alpha_{2}=0.5$ to assign a higher weight to the band
gap term. For HgTe, which is a metal, we set $\alpha_{1}=0$ and
$\alpha_{2}=1$.
For InAs and GaSb the optimal values of Ueff have been found to be:
U${}_{eff}^{In,p}$ = -0.5 eV, U${}_{eff}^{As,p}$ = -7.5 eV, U${}_{eff}^{Ga,p}$
= 0.8 eV, U${}_{eff}^{Sb,p}$ = -6.9 eV, as reported in [51]. With these
parameters, DFT+U(BO) yields a band gap of 0.31 eV for InAs, in good agreement
with the experimental value of 0.41 eV [70], and a band gap of 0.45 eV for
GaSb, which is somewhat underestimated compared to the experimental value of
0.81 eV [70]. For CdTe, BO produces an optimal value of U${}_{eff}^{Cd,d}$ =
8.3 eV, somewhat higher than the value of 7 eV used in [67]. This results in a
band gap of 0.87 eV, which is closer to experiment than previous ab initio
calculations [71, 72]. The qualitative features of the PBE+U(BO) band
structure are in agreement with HSE, as shown in Fig. 1e, however the gap and
the band width are still somewhat underestimated. For HgTe, BO produces a
value of U${}_{eff}^{Hg,d}$ = 8.4 eV, somewhat lower than the value of 9.4 eV
used in Ref. [45]. The band structure, shown in Fig. 1d, has the correct band
shape, comparable to the HSE band structure, and is in agreement with Ref.
[45]. To demonstrate the transferability of the U${}_{e}ff$ values obtained by
BO from bulk materials to interfaces, we compare the band structures produced
by PBE+(BO) and HSE for an InAs/GaSb interface with 5 layers of InAs and 5
layers of GaSb, constructed as detailed below. Fig 2 shows that overall good
agreement is obtained between PBE+U(BO) and HSE, however PBE+U(BO) somewhat
underestimates the band gap and the band width. We note that the Ueff values
obtained here are based on the implementation of the Dudarev formalism in
VASP. Different DFT+U implementations may yield different results [73, 74].
Figure 2: The band structure of an InAs/GaSb interface with 5 layers of InAs
and 5 layers of GaSb obtained with (a) HSE and (b) DFT+U(BO). Orange and green
dots indicate the contributions of InAs and GaSb, respectively.
### II.3 Interface model construction
For the HgTe/CdTe(100) interface, we constructed periodic heterostructures,
similar to Ref. [45]. However, we used a larger number of CdTe layers to
ensure convergence, as detailed below. The thickness of HgTe was varied to
study the evolution of the electronic structure. The experimental lattice
constants of 6.45 $\mathrm{\SIUnitSymbolAngstrom}$ for HgTe and 6.48
$\mathrm{\SIUnitSymbolAngstrom}$ for CdTe are closely matched [75]. We assumed
that an epitaxially matched HgTe film would grow on top of a CdTe substrate
with the experimental lattice constant of 6.48
$\mathrm{\SIUnitSymbolAngstrom}$.
For the InAs/GaSb interface, we constructed two types of interface slab
models: The InSb-type interface has In and Sb as the terminal atoms at the
surfaces and interface. The GaAs-type interface has Ga and As as the terminal
atoms. The experimental lattice constants of 6.058
$\mathrm{\SIUnitSymbolAngstrom}$ for InAs and 6.096
$\mathrm{\SIUnitSymbolAngstrom}$ for GaSb [70] are closely matched. We assumed
that an epitaxially matched InAs film would grow on top of GaSb with the
lattice constant of 6.096 $\mathrm{\SIUnitSymbolAngstrom}$, based on the
experiment in Ref. [34]. To study the effect of the InAs and GaSb thickness,
interface models were constructed with the number of layers of each material
varying from 10 to 50. The notation ”A/B” is used to describe an InAs/GaSb
interface with A layers of InAs and B layers of GaSb. A vacuum region of about
40 $\mathrm{\SIUnitSymbolAngstrom}$ was added to the interface model to
prevent spurious interactions between periodic images (for the purpose of band
unfolding the closest integer number of primitive cells to 40
$\mathrm{\SIUnitSymbolAngstrom}$ was used [65]). In order to terminate
dangling bonds, In and Ga atoms on the surface were passivated by pseudo
hydrogen atoms with 1.25 fractional electrons, whereas As and Sb atoms on the
surface were passivated by pseudo hydrogen atoms with 0.75 fractional
electrons. Structural relaxation was performed for the surface atoms and
passivating pseudo-hydrogen atoms until the change of the all forces was below
10-3 eV/$\mathrm{\SIUnitSymbolAngstrom}$.
The number of layers included in slab models needs to be converged to the bulk
limit to avoid quantum size effects. For semiconductors the band gap is
typically used as a the convergence criterion [76, 65]. Fig. 3 shows the band
gap as a function of the number of layers for InAs(100), GaSb(100), and
CdTe(100). We note that here ”layer” is defined as one atomic layer. In each
iteration, the number of layers was increased by 8 for InAs and GaSb and by 6
for CdTe. If the band gap difference between the current iteration and the
previous iteration was within $1\times 10^{-2}$ eV, the current number of
layers was regarded as converged. For InAs and GaSb surfaces, 50 layers are
required, whereas for CdTe 40 layers are required to converge the band gap.
The converged band gap values are close to the bulk values. The size of the
interface models used to simulate the effect of an electric field was limited
to 10 layers of InAs with 10 layers of GaSb due to convergence issues, as
detailed in the SI.
Figure 3: The band gap obtained with PBE+U(BO) as a function of the number of
layers for InAs(100), GaSb(100), and CdTe(100) surface slabs.
## III RESULTS AND DISCUSSION
### III.1 HgTe/CdTe
To validate the DFT+U(BO) method, we begin by applying it to the well-studied
HgTe/CdTe interface. Bulk-unfolded band structures of HgTe/CdTe
heterostructures with 40 layers of CdTe and a varying number of HgTe layers
are shown in Fig. 4. The red dots indicate the contributions from Hg $s$
orbitals and the blue dots indicate the contributions from Te $p$ orbitals.
The band gap value as a function of the number of HgTe layers is shown in Fig.
5. Negative values indicate an inverted band gap. A drastic change is observed
with the thickness of HgTe. When the number of layers is below 16, the
interface behaves as a trivial insulator, with the Hg $s$ orbitals forming the
bottom of the conduction band and the Te $p$ orbitals forming the top of the
valence band. When the number of HgTe layers reaches 16, a transition point
from a trivial insulator to a topological insulator occurs. At this transition
point, both the CBM and VBM show a hybridized $sp$ character. When the number
of HgTe layers exceeds 16, an inverted gap opens, leading to the occurrence of
a topologically nontrivial phase, in which the VBM is dominated by Hg $s$
states and the CBM is dominated by Te $p$ states. The critical thickness of 16
layers, corresponds to 5.1 nm in good agreement with the experimental result
of 6.3 nm (around 19 layers) [6]. Our result is comparable to previous DFT
calculations, which used different functionals and considered structures with
fewer layers of CdTe. Ref. [44] obtained a critical thickness of 4.6 nm of
HgTe on top of 4 layers of CdTe using the modified Becke-Johnson (MBJ)
functional.Ref. [45] obtained a critical thickness of 6.5 nm of HgTe on top of
10 layers of CdTe using GGA+U for HgTe and GGA for CdTe. Thus, the DFT+U(BO)
method successfully describes the electronic structure of the HgTe/CdTe
interface and captures the transition from trivial to topological behavior.
Figure 4: Band structures of a HgTe/CdTe interface with 40 layers of CdTe and
(a) 4 layers, (b) 16 layers, and (c) 20 layers of HgTe. The red dots indicate
the contributions of Hg $s$ states and the blue dots indicate the
contributions of the Te $p$ states. Figure 5: The band gap of a HgTe/CdTe
interface with 40 layers of CdTe as a function of the number of HgTe layers.
Negative values indicate an inverted band gap.
### III.2 InAs/GaSb
#### III.2.1 Effect of layer thickness
Figure 6: Band gap values as a function of number of layers for 50-layer
InAs/X-layer GaSb and X-layer InAs/50-layer GaSb of InSb-type and GaAs-type
interface.
To investigate the influence of the thickness of InAs and GaSb on the band
gap, we conducted two series of calculations for InSb-type and GaAs-type
interfaces. In one series, the thickness of InAs was fixed at 50 layers InAs
and the number of GaSb layers (X) was varied. In the other series, the
thickness of GaSb was fixed at 50 layers and the number of InAs layers (X) was
varied. The results are shown in Fig. 6. For the InAs(50)/GaSb(X) series, the
band gap of the InSb-type interface increases with increasing GaSb thickness,
whereas the band gap of the GaAs-type interface does not change significantly.
For the InAs(X)/GaSb(50) series, the band gap decreases with increasing InAs
thickness for both interface types, although the gap of the InSb-type
interface remains smaller than that of the GaAs-type interface throughout. The
trend of the gap decreasing with the increase in InAs thickness is in
agreement with experimental observations [77]. The thickest interface we were
able to calculate comprises 50 layers, which corresponds to about 15 nm of
each material. The bulk-unfolded band structure of a 50/50 InSb-type interface
is shown in the SI. Because this interface still has a gap of over 0.2 eV, and
the rate of the gap narrowing decreases with increasing InAs thickness, as
shown in Fig. 6, we estimate that it would either require a significantly
thicker film of InAs for the gap to completely close or the gap would approach
a finite asymptotic limit rather than close. In addition to increasing the QW
thickness, strain engineering, which is not taken into account here, may also
help modulate the gap. [78, 79, 80].
We note that an analysis based on the empirical 8-band Kane model found band
inversion and the quantum spin Hall phase for an InAs thickness above 9 nm at
fixed 10 nm GaSb thickness [25]. However, this analysis was based on empirical
parameters for the material and interface properties and did not take the
atomic details of the interface structure into account. For example, in Ref.
[25] the band alignment at the interface was chosen such that the GaSb valence
band is 150 meV higher than the InAs conduction band leading to a band
inversion even for relatively thin layers. In contrast, within our first
principles approach, we find that band inversion is not achieved up to an InAs
thickness of 15 nm for a range of GaSb thicknesses including 10 nm.
Furthermore, we found that the atomic details of the interface, like the type
of bonds formed at the interface (InSb or GaAs), are relevant, which was
neglected in the effective theory of Ref. [25]. Finally, it should be noted
that experiments seem to indicate that an electric field is required to
achieve an inverted regime in InAs/GaSb heterostructures [81].
#### III.2.2 Effect of electric field
Figure 7: Electronic structure of a 10/10 InSb-type interface with different
external electric fields. a-c) Bulk unfolded band structures with the
contributions of the interface layers of InAs and GaSb colored in orange and
green, respectively.
The band alignment at the interface of InAs/GaSb can be manipulated by
applying external gate voltages. Ref.[81] has presented strong experimental
evidence that the gap closes when the external gate voltages reach a critical
value. Therefore, we performed DFT simulations for interface slabs in presence
of electric field. In the VASP code, an external electric field is simulated
by adding an artificial dipole sheet in the vacuum region of the unit cell
[64]. Due to screening effects and the electric susceptibility inside the
materials, the effective electric field at the interface may be significantly
smaller than the input electric field [64, 82]. To estimate the effective
electric field, we calculated the gradient of the potential in the InAs and in
GaSb, based on the electrostatic potential averaged over xy plane. The
averaged gradient is taken as the effective electric field. The full account
of the effective field estimation is provided in the SI. The electric field is
applied perpendicular to the plane of the interface and points from the GaSb
side to the InAs side. We note that in VASP only an external electric field
can be set, whereas in experiments the position of the Fermi level can be
independently controlled by applying front-gate and back-gate voltages. Owing
to convergence issues in DFT calculations with external electric fields (see
SI), the largest interfaces we were able to calculate comprise 10 layers of
InAs and 10 layers of GaSb.
Fig. 7 shows the band structure of a 10/10 InSb-type interface. When no
electric field is applied, the interface is in the trivial insulator state.
The CBM is dominated by the interface InAs layer (orange), whereas the
interface GaSb layer (green) contributes predominantly to the VBM. As the
electric field increases, the bands contributed by the GaSb shift upwards with
respect to the bands contributed by the InAs and the gap narrows. When the
input electric field reaches 0.25 V/$\mathrm{\SIUnitSymbolAngstrom}$, which
corresponds to an effective field of 0.014 V/$\mathrm{\SIUnitSymbolAngstrom}$,
the GaSb VBM overlaps with the InAs CBM, the gap closes, and band crossing
occurs. Our results are qualitatively in agreement with previous studies [83,
25, 81], which indicated that the band gap in InAs/GaSb could be closed via an
external electric field. When the electric field is increased further, an
inverted gap opens. As the electric field is increased, the inverted gap
expands, but also shifts higher above the Fermi level. Fig 8 shows the
position of the inverted gap above the Fermi level at the $\Gamma$ point as a
function of the electric field. With an input electric field of 0.35
V/$\mathrm{\SIUnitSymbolAngstrom}$, which corresponds to an effective field of
0.018 V/$\mathrm{\SIUnitSymbolAngstrom}$, the gap at the $\Gamma$ point is 65
meV and the bottom of the inverted gap is found 66 meV above the Fermi level.
For the GaAs-type interface, shown in the SI, the band gap also decreases as
the electric field increases. However, because the GaAs-type interface has a
larger band gap and the effect of the electric field is weaker than for the
InSb-type interface, the gap does not close even for an input electric field
as high as 0.55 V/$\mathrm{\SIUnitSymbolAngstrom}$.
Fig 9 shows the change in the band gap, $\Delta$, as a function of the input
electric field, $E_{in}$, for 10/10 GaAs-type and InSb-type interfaces:
$\Delta=Gap(E_{in})-Gap(E_{in}=0)$ (3)
The blue and orange dashed lines indicate the band gaps of the 10/10 GaAs-type
and InSb-type interfaces, respectively. The gap closes when the dashed line is
crossed. To estimate the input electric field that would be required for the
gap to close for a 50/50 interface, we assume that the change in the gap would
behave similarly to a 10/10 interface. The green and red dashed lines indicate
the band gaps of the 50/50 GaAs-type and InSb-type interfaces, respectively.
Based on this, we estimate that an input electric field of 0.19
V/$\mathrm{\SIUnitSymbolAngstrom}$, which corresponds to an effective electric
field of 0.012 V/$\mathrm{\SIUnitSymbolAngstrom}$, would be needed to close
the gap for a 50/50 InSb-type interface, as indicated by the red solid line.
For the GaAs-type interface an input electric field of 0.55
V/$\mathrm{\SIUnitSymbolAngstrom}$, which corresponds to an effective electric
field of 0.017 V/$\mathrm{\SIUnitSymbolAngstrom}$, would be needed to close
the gap, as indicated by the green solid line. We highlight that the effective
electric field of 0.017 V/$\mathrm{\SIUnitSymbolAngstrom}$ corresponds to a
potential drop of $2.55$ V over the 15 nm thickness of the QW in this case,
which is likely to make the material conducting well before the topological
transition. Our results indicate that while it may be possible to tune the
InAs/GaSb interface into the topological regime, it would not be trivial.
Figure 8: The inverted band gap at $\Gamma$ and its position above the Fermi
level as a function of the input electric field, where the shift is defined as
the energy difference between the position of the bottom of the inverted band
gap and the Fermi level at the $\Gamma$ point. Figure 9: The band gap
reduction, $\Delta$, as a function of electric field for 10/10 GaAs-type and
InSb-type interfaces.
## IV Conclusion
In summary, we have studied the HgTe/CdTe and InAs/GaSb quantum wells using
DFT with a Hubbard U correction determined by Bayesian optimization. DFT+U(BO)
produces band structures of comparable accuracy to a hybrid functional at the
computational cost of a semi-local functional. This enables us to conduct
simulations of large interface models with hundreds of atoms.
For the HgTe/CdTe interface we find that an inverted gap opens at a critical
thickness of 5.1 nm of HgTe, in agreement with experimental observations and
previous theoretical studies. For InAs/GaSb QWs with 50 layers (about 15 nm)
of GaSb we find that the gap narrows with increasing thickness of InAs in
agreement with the previous theory estimations. However, the gap does not
completely close with up to 50 layers (about 15 nm) of InAs. Based on the rate
of gap narrowing, we estimate that it would either require a significantly
thicker InAs film to close the gap or the gap would decay to a finite
asymptotic limit.
Simulations with an external electric field applied perpendicular to the
interface, pointing from GaSb to InAs, have been conducted for models with 10
layers of each material. We find that with increasing field strength the GaSb
VBM shifts upwards relative to the InAs CBM, leading to narrowing of the gap
at the interface. For the InSb-type interface, band crossing is observed at a
critical field and subsequently an inverted gap opens. As the electric field
increases the gap increases but also shifts higher in energy above the Fermi
level. Because the 10/10 interface has a larger gap due to the quantum size
effect, we estimate the reduced critical field that would be required to
achieve band inversion and reach the topological regime for thicker QWs
comprising 50 layers of each material.
Our results explain the difficulty of experimentally reaching the topological
regime in InAs/GaSb QWs. In principle, under the right conditions, an inverted
gap could be produced in this system. However, achieving this requires a
delicate balance between several parameters. To tune the initial gap, the
structure of the QWs must be precisely controlled, including the layer
thickness, the bonding configuration at the interface, and possibly also the
lattice strain. Even if a smaller zero-field gap is obtained by interface
engineering, a considerable electric field may still be required to obtain
band crossing and drive the system into the topological regime. Finally,
gating or doping may be required to tune the Fermi level position inside the
inverted gap.
The HgTe/CdTe QW does not suffer from this difficulty because HgTe has an
inverted band structure intrinsically. Therefore, no electric field is
necessary to achieve band inversion at the HgTe/CdTe interface and it is
easier to reach the topological regime. Thus, our results make a case for
limited applicability of InAs/GaSb quantum wells for 2DTI production and
suggest that alternative, more promising materials should be sought.
###### Acknowledgements.
We would like to thank Sergey Frolov from the University of Pittsburgh, Chris
Palmstrøm from the University of California, Santa Barbara, Vlad Pribiag from
the University of Minnesota, and Michael Wimmer from TU Delft for helpful
discussions. Work at CMU was funded by the National Science Foundation (NSF)
through grant OISE-1743717. This research used resources of the National
Energy Research Scientific Computing Center (NERSC), a DOE Office of Science
User Facility supported by the Office of Science of the U.S. Department of
Energy under contract no. DE-AC02-05CH11231.
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# On Radó’s theorem for polyanalytic functions
Abtin Daghighi
###### Abstract.
We prove versions of Radó’s theorem for polyanalytic functions in one variable
and also on simply connected $\mathbb{C}$-convex domains in $\mathbb{C}^{n}$.
Let $\Omega\subset\mathbb{C}$ be a bounded, simply connected domain and let
$q\in\mathbb{Z}_{+}.$ Suppose at least one of the following conditions holds
true: (i) $g\in C^{q}(\Omega).$ (ii) $g\in C^{\kappa}(\Omega),$ for
$\kappa=\min\\{1,q-1\\},$ such that $g$ is $q$-analytic on $\Omega\setminus
g^{-1}(0)$ and such that $\operatorname{Re}g$ ($\operatorname{Im}g$
respectively) is a solutions to the $p^{\prime}$-Laplace equation
($p^{\prime\prime}$-Laplace equation respectively) on $\Omega\setminus
g^{-1}(0)$, for some $p^{\prime},p^{\prime\prime}>1$. Then $g$ agrees
(Lebesgue) a.e. with a function that is $q$-analytic on $\Omega.$
In the process we give a simple proof of the fact that if $f\in C^{q}(\Omega)$
is $q$-analytic on $\Omega\setminus f^{-1}(0)$ then $f$ is $q$-analytic on
$\Omega.$ The extensions of the results to several complex variables are
straightforward using known techniques.
###### Key words and phrases:
Radó’s theorem, Polyanalytic functions, zero sets, $\alpha$-analytic functions
###### 2000 Mathematics Subject Classification:
Primary 35G05, 35J62; 32A99, 32V25
*Corresponding author<EMAIL_ADDRESS>
## 1\. Introduction
Radó’s theorem states that a continuous function on an open subset of
$\mathbb{C}^{n}$ that is holomorphic off its zero set extends to a holomorphic
function on the given open set. For the one-dimensional result see Radó [7],
and for a generalization to several variables, see e.g. Cartan [4].
###### Definition 1.1.
Let $\Omega\subset\mathbb{C}$ be an open subset. A function $f$ on $\Omega$ is
called _polyharmonic of order $q$_ if $\Delta^{q}f=0$ on $\Omega$, where
$\Delta$ denotes the Laplace operator.
###### Definition 1.2.
Let $\Omega\subseteq\mathbb{R}^{n}$ be an open subset. For a fixed $p>1,$ the
$p$-Laplace operator of a real-valued function $u$ on $\Omega$ is defined as
(1) $\Delta_{p}:=\mbox{div}(\left\lvert\nabla u\right\rvert^{p-2}\nabla u)$
The operator can also be defined for $p=1$ (it is then the negative of the so-
called mean curvature operator) and $p=\infty$ but we shall not concern
ourselves with such cases.
###### Remark 1.3.
Note the subtle similarity between the notation for the $p$-Laplace operator
(2) $\Delta_{p}=\mbox{div}(\left\lvert\nabla u\right\rvert^{p-2}\nabla u)$
and that of the $p$:th power of the Laplace operator $\Delta^{p}$. We have
that $\Delta_{2}=\Delta.$ More generally, we have
(3) $\Delta_{p}u=\left\lvert\nabla u\right\rvert^{p-4}\left(\left\lvert\nabla
u\right\rvert^{2}\Delta
u+(p-2)\sum_{i,j=1}^{n}\partial_{x_{i}}u\cdot\partial_{x_{j}}u\cdot\partial_{x_{i}}\partial_{x_{j}}u\right)$
Note that $\Delta_{p}$ is quasilinear. At least they both share the property
of being elliptic operators. In the case of $\Delta^{p}$ this is a direct
consequence of the fact that $\Delta$ is a elliptic operator and therefore any
finite power is also, in particular the elliptic regularity theorem applies to
$\Delta^{p}$ and to $\Delta_{p}$, and implies that any real-valued
distribution solution $u$ to $\Delta^{p}u=0$ (or to $\Delta_{p}$) on a domain
$\Omega\subset\mathbb{R}^{n}$ is Lebesgue a.e. equal to a $C^{\infty}$-smooth
solution $\tilde{u}$ to $\Delta^{p}\tilde{u}=0$ (or to
$\Delta_{p}\tilde{u}=0$) on $\Omega.$
Kilpeläinen [5] proved the following.
###### Theorem 1.4.
If $\omega\subset\mathbb{R}^{2}$ is a domain and if $u\in C^{1}(\Omega)$
satisfies the $p$-Laplace equation
$\mbox{div}(\left\lvert\nabla\right\rvert^{p-2}\nabla u)=0$ on
$\Omega\setminus u^{-1}(0)$ then $u$ is a solution to the $p$-Laplacian on
$\Omega.$
We mention that, more recently, Tarkhanov & Ly [6] proved the following
related result in higher dimension.
###### Theorem 1.5.
Let $\Omega\subseteq\mathbb{R}^{n}$ be an open subset. If $u\in
C^{1,\frac{1}{p-1}}(\Omega)$ such that
$\mbox{div}(\left\lvert\nabla\right\rvert^{p-2}\nabla u)=0$ on
$\Omega\setminus u^{-1}(0)$ then this holds true on all of $\Omega.$
We shall use the result of Kilpeläinen [5] in order to prove a natural version
of Radó’s theorem for polyanalytic functions. Avanissian & Traoré [1], [2]
introduced the following definition of polyanalytic functions of order
$\alpha\in\mathbb{Z}_{+}^{n}$ in several variables.
###### Definition 1.6.
Let $\Omega\subset\mathbb{C}^{n}$ be a domain, let
$\alpha\in\mathbb{Z}_{+}^{n}$ and let $z=x+iy$ denote holomorphic coordinates
in $\mathbb{C}^{n}$. A function $f$ on $\Omega$ is called _polyanalytic of
order $\alpha$_ if in a neighborhood of every point of $\Omega$,
$\left(\frac{\partial}{\partial\bar{z}_{j}}\right)^{\alpha_{j}}\\!f(z)=0,1\leq
j\leq n$.
###### Definition 1.7.
Let $\Omega\subset\mathbb{C}^{n}$ be an open subset and let
$(z_{1},\ldots,z_{n})$ denote holomorphic coordinates for $\mathbb{C}^{n}.$ A
function $f$, on $\Omega,$ is said to be separately $C^{k}$-smooth with
respect to the $z_{j}$-variable, if for any fixed
$(c_{1},\ldots,c_{n-1})\in\mathbb{C}^{n-1},$ chosen such that the function
$z_{j}\mapsto f(c_{1},\ldots,c_{j-1},z_{j},c_{j},\ldots,c_{n-1}),$
is well-defined (i.e. such that
$(c_{1},\ldots,c_{j-1},z_{j},c_{j},\ldots,c_{n-1})$ belongs to the domain of
$f$) is $C^{k}$-smooth with respect to
$\operatorname{Re}z_{j},\operatorname{Im}z_{j}$. For
$\alpha\in\mathbb{Z}_{+}^{n}$ we say that $f$ is separately $\alpha$-smooth if
$f$ is separately $C^{\alpha_{j}}$-smooth with respect to $z_{j}$ for each
$1\leq j\leq n$.
We shall need the following result.
###### Theorem 1.8.
(See [2, Theorem 1.3, p. 264]) Let $\Omega\subset\mathbb{C}^{n}$ be a domain
and let $z=(z_{1},\ldots,z_{n}),$ denote holomorphic coordinates in
$\mathbb{C}^{n}$ with $\real z=:x,\operatorname{Im}z=y$. Let $f$ be a function
which, for each $j$, is polyanalytic of order $\alpha_{j}$ in the variable
$z_{j}=x_{j}+iy_{j}$ (in such case we shall simply say that $f$ is separately
polyanalytic of order $\alpha$). Then $f$ is jointly smooth with respect to
$(x,y)$ on $\Omega$ and furthermore is polyanalytic of order
$\alpha=(\alpha_{1},\ldots,\alpha_{n})$ in the sense of Definition 1.6.
## 2\. Statement and proof of the result
Let us make the following first observation.
###### Proposition 2.1.
Let $\Omega\subset\mathbb{C}$ be a simply connected domain, let
$q\in\mathbb{Z}_{+}$ and let $f\in C^{q}(\Omega)$ be a $q$-analytic function
on $\Omega\setminus f^{-1}(0).$ Then $f$ is $q$-analytic on $\Omega.$
###### Proof.
If $f\equiv 0$ then we are done, so assume $f\not\equiv 0.$ Since $f$ is
$C^{q}$-smooth the function $\partial_{\bar{z}}^{q}f$ is continuous. By
assumption $\partial_{\bar{z}}^{q}f=0$ on $\Omega\setminus f^{-1}(0).$ Set
$Z:=(f^{-1}(0))^{\circ}$ (∘ denoting the interior) and $X:=\\{f\neq 0\\}\cup
Z.$ Now $f|_{Z}$ clearly satisfies $\partial_{\bar{z}}^{q}f=0$. Let
$p\in\partial X.$ If $p$ is an isolated zero of $f$, then by continuity we
have $\partial_{\bar{z}}^{q}f(p)=0.$ Suppose $p$ is a non-isolated zero. We
have for each sufficiently large $j\in\mathbb{Z}_{+}$ that $\\{\left\lvert
z-p\right\rvert<1/j\\}\cap X\neq\emptyset$. This implies that there exists a
sequence $\\{z_{j}\\}_{j\in\mathbb{Z}_{+}}$ of points $z_{j}\in X$ such that
$z_{j}\to p$ as $j\to\infty.$ By continuity we have
(4)
$\partial_{\bar{z}}^{q}f(p)=\lim_{j\to\infty}\partial_{\bar{z}}^{q}f(z_{j})=0$
This completes the proof. ∎
###### Theorem 2.2.
Let $\Omega\subset\mathbb{C}$ be a bounded, simply connected domain, let
$q\in\mathbb{Z}_{+}$ and let $f$ be a function $q$-analytic on
$\Omega\setminus f^{-1}(0)$. Suppose at least one of the following conditions
holds true:
(i) $f\in C^{\kappa}(\Omega),$ for $\kappa=\min\\{1,q-1\\},$ and
$\operatorname{Re}f$ ($\operatorname{Im}f$ respectively) is a solutions to the
$p^{\prime}$-Laplace equation ($p^{\prime\prime}$-Laplace equation
respectively) on $\Omega\setminus f^{-1}(0)$, for some
$p^{\prime},p^{\prime\prime}>1$.
(ii) $f\in C^{q}(\Omega).$
Then $f$ agrees (Lebesgue) a.e. with a function that is $q$-analytic on
$\Omega.$
###### Proof.
The case (ii) follows from Proposition 2.1. So suppose (i) holds true. If
$q=1$ the theorem is well-known and due to Radó [7], so assume $q\geq 2.$ Let
$f=u+iv$ where $u=\operatorname{Re}f,$ $v=\operatorname{Im}f.$ Now
$f^{-1}(0)=u^{-1}(0)\cap v^{-1}(0)$, whence $u$ (and $v$ respectively) is a
solution to the $p^{\prime}$-Laplace equation ($p^{\prime\prime}$-Laplace
equation respectively) on $\Omega\setminus u^{-1}(0)$ ($\Omega\setminus
v^{-1}(0)$ respectively). If $f\in C^{\kappa}(\Omega)$ and $q\geq 2$ then $u$
and $v$ respectively are at least $C^{1}$-smooth thus satisfy the conditions
of Theorem 1.4. Hence it follows that $u$ ($v$ respectively) are solutions to
the $p^{\prime}$-Laplace equation ($p^{\prime\prime}$-Laplace equation
respectively) on all of $\Omega$. By Remark 1.3 (in particular Elliptic
regularity) it follows that $u$ and $v$ respectively agree (Lebesgue) a.e. on
$\Omega$ with $C^{\infty}$-smooth functions $\tilde{u}$ and $\tilde{v}$
respectively. This implies that the function $\tilde{f}:=\tilde{u}+i\tilde{v}$
is $C^{\infty}$-smooth on $\Omega$ and agrees (Lebesgue) a.e. on $\Omega$ with
$f.$ Suppose there exists a point $p_{0}\in\Omega$ such that
$\partial_{\bar{z}}^{q}\tilde{f}(p_{0})\neq 0.$ Set $Z:=(f^{-1}(0))^{\circ}$
and $X:=\\{f\neq 0\\}\cup Z.$ By continuity there exists an open neighborhood
$U_{p_{0}}$ of $p_{0}$ in $\Omega$ such that
$\partial_{\bar{z}}^{q}\tilde{f}\neq 0$ on the open subset $U_{p_{0}}\cap X.$
By the definition of $\tilde{f}$ there exists a set $E$ of zero measure such
that on $V_{p_{0}}:=(X\cap U_{p_{0}})\setminus E$ we have that
$\partial_{\bar{z}}^{q}f$ exists (since $X$ contains no point of
$f^{-1}(0)\setminus Z$) and satisfies
$0=\partial_{\bar{z}}^{q}f=\partial_{\bar{z}}^{q}\tilde{f}$ on $V_{p_{0}},$
which could only happen if $V_{p_{0}}$ is empty which is impossible since $E$
cannot possess interior points. We conclude that
$\partial_{\bar{z}}^{q}\tilde{f}=0$ on $\Omega.$ This completes the proof. ∎
###### Theorem 2.3 (Radó’s theorem for polyanalytic functions in several
complex variables).
Let $\Omega\subset\mathbb{C}^{n}$ be a bounded $\mathbb{C}$-convex domain. Let
$\alpha\in\mathbb{Z}_{+}^{n}$. Suppose $f$ is $\alpha$-analytic on
$\Omega\setminus f^{-1}(0)$ such that one of the following conditions hold
true:
(i) For each $j=1,\ldots,n$, the function $f$ is separately
$C^{\kappa_{j}}$-smooth with respect to $z_{j}$ (i.e. for each fixed value of
the remaining variables $z_{k},$ $k\neq j$, $f$ becomes a
$C^{\kappa_{j}}$-smooth function of $z_{j}$),
$\kappa_{j}=\min\\{1,\alpha_{j}-1\\}$ and $\operatorname{Re}f$
($\operatorname{Im}f$ respectively) are solutions to the $p^{\prime}$-Laplace
equation ($p^{\prime\prime}$-Laplace equation respectively) for some
$p^{\prime},p^{\prime\prime}>1.$
(ii) For each $j=1,\ldots,n$, the function $f$ is separately
$C^{\alpha_{j}}$-smooth with respect to $z_{j}$.
Then $f$ agrees (Lebesgue) a.e. with a function that is $\alpha$-analytic on
$\Omega$.
###### Proof.
Denote for a fixed $c\in\mathbb{C}^{n-1}$,
$\Omega_{c,k}:=\\{z\in\Omega:z_{j}=c_{j},j<k,z_{j}=c_{j-1},j>k\\}$. Since
$\Omega$ is $\mathbb{C}$-convex, $\Omega_{c,k}$ is simply connected. Consider
the function $f_{c}(z_{k}):=$
$f(c_{1},\ldots,c_{k-1},z_{k},c_{k},\ldots,c_{n-1})$. Clearly, $f_{c}$ is
$\alpha_{k}$-analytic on $\Omega_{c,k}\setminus f^{-1}(0)$ for any
$c\in\mathbb{C}^{n-1}.$ Since $f^{-1}_{c}(0)\subseteq f^{-1}(0)$, Theorem 2.2
applies to $f_{c}$ meaning that $f$ agrees a.e. with a function $\tilde{f}$
that is separately polyanalytic of order $\alpha_{j}$ in the variable
$z_{j},1\leq j\leq n$. By Theorem 1.8 the function $\tilde{f}$ must be
polyanalytic of order $\alpha$ on $\Omega$. This completes the proof. ∎
###### Corollary 2.4.
Let $\Omega\subset\mathbb{C}$ be a bounded $\mathbb{C}$-convex domain and let
$\alpha\in\mathbb{Z}_{+}^{n}$. Suppose $f$ is separately
$C^{\alpha_{j}}$-smooth with respect to $z_{j},$ $j=1,\ldots,n.$ If $f$ is
$\alpha$-analytic on $\Omega\setminus f^{-1}(0),$ then $f$ agrees (Lebesgue)
a.e. with a function that is $\alpha$-analytic on $\Omega$.
## References
* [1] V. Avanissian, A. Traoré, Sur les fonctions polyanalytiques de plusiers variables, C.R. Acad. Sci. Paris Sér. A-B 286 (1978), no.17, A743-A746
* [2] V. Avanissian, A. Traoré, Extension des théorèmes de Hartogs et de Lindelöf aux fonctions polyanalytiques de plusieurs variables, C.R. Acad. Sci. Paris Sér. A-B 291 (1980), no. 4, A263-A265
* [3] M.B. Balk, Polyanalytic functions and their generalizations, Encyclopaedia of Mathematical Sciences (Eds: A.A. Gonchar, V.P. Havin, N.K. Nikolski), Complex Analysis I, Vol.85, p.197-253, Springer, 1997
* [4] H. Cartan, Sur une extension d’un theorème de Radó, Math. Ann. 125 (1952), 49-50
* [5] T. Kilpeläinen, A Radó type theorem for p-harmonic functions in the plane, Electron. J. Diff. Equ. 9 (1994), 1-4
* [6] I. Ly, N. Tarkhanov, A Radó theorem for p-harmonic functions, Bol. Soc. Mat. Mex. 22 (2016), 461-472
* [7] T. Radó, Über eine nicht fortsetzbare Riemannsche Mannigfaltigkeit, Math. Z. 20 (1924), 1-6
|
# Convolutional neural network for self-mixing interferometric displacement
sensing
Stéphane Barland 1,* and François Gustave2 1Université Côte d’Azur - CNRS,
Institut de Physique de Nice, 1361 route des Lucioles, F-06560, Valbonne,
France
2DOTA, ONERA, Université Paris-Saclay, F-91123, Palaiseau, France
<EMAIL_ADDRESS>
††journal: oe
Self mixing interferometry is a well established interferometric measurement
technique. In spite of the robustness and simplicity of the concept,
interpreting the self-mixing signal is often complicated in practice, which is
detrimental to measurement availability. Here we discuss the use of a
convolutional neural network to reconstruct the displacement of a target from
the self mixing signal in a semiconductor laser. The network, once trained on
periodic displacement patterns, can reconstruct arbitrarily complex
displacement in different alignment conditions and setups. The approach
validated here is amenable to generalization to modulated schemes or even to
totally different self mixing sensing tasks.
## 1 Introduction
Optical interferometric measurements are routinely used in science and
engineering and many schemes can be used to adapt the approach to the specific
measurement to be performed. One particularly interesting and well established
method is the so-called self-mixing interferometry, which consists in
realizing interference between the beam reflected by a target and a reference
beam inside the laser resonator emitting the reference beam (see eg [1, 2, 3,
4, 5] for reviews). For its simplicity and versatility, many applications have
been envisioned and perhaps the most immediate is that of displacement
measurement. Two limit regimes are considered [6]: that of very small
displacement (much smaller than the laser wavelength) or the opposite case
where the displacement takes place over a very large number of wavelengths. In
the first case, information about the target displacement can be retrieved
from fitting the shape of the interferometric signal. In the latter case, most
of the information is obtained by counting the fringes that are observed as a
sawtooth signal whose symmetry depends on the direction of the motion. Despite
its apparent simplicity, this analysis is often complicated since the exact
shape of the interferometric signal depends on many factors including bias
current, target reflectivity, alignment conditions [7], and modal structure of
the laser which may even lead to a double-peak structure in each fringe [8].
Furthermore, on diffusive targets, speckle leads to an effective variation of
the feebdack parameters and therefore a change of the signal shape in the
course of the measurement. In practice, all these effects tremendously affect
the availability of self-mixing measurement setups. This has led to a number
of hardware and software proposals to either improve the signal quality or the
retrieval of the displacement from the interferometric signal [9, 10, 11, 12,
13, 14].
Computer neural networks are one of the many architectures which can be used
for machine learning tasks, whereby a computer is used to infer rules from a
set of data and results instead of providing results on the basis of an input
and a priori known rules. The training of a neural network leads to the
formulation of kind of statistical model [15], able to predict new results on
the basis of new data. These neural networks are already very widely used in
everyday life and they are proving increasingly useful many areas of research
and technology. In the specific context of interferometry, very few attempts
exist to date. They have been used to identify and count fringes in [16, 17,
18]. In [19] and [20] they have been used to pre-process self mixing traces
and in [21] they are used as a part of a self-mixing blood pressure
measurement scheme.
In the following, we discuss the use of a convolutional neural network for the
direct reconstruction of a displacement signal across many different alignment
conditions. We address a particularly delicate regime which is the one of "few
wavelengths" displacement. The neural network is first trained on a set of
periodic data and in different alignment conditions. Its performance in
reconstructing the displacement of a target from a self-mixing interferometric
signal is then validated on aperiodic times series whose continuous spectrum
spans more than three octaves and under different alignment conditions, not
used during training. We also analyze the robustness of the reconstruction to
the presence of very strong detection noise. Finally, we observe that the
neural network can, without any tuning, provide a sensible reconstruction of
the displacement of a target obtained on a different experimental setup based
on the same operating principle. We then briefly discuss some details on the
operation of such a network and some further possible uses of this approach in
the context of self mixing. Thus, a reasonably simple neural network such as
the one used here can become one of the tools which contribute to the
robustness and high availability of self-mixing interferometric setups.
## 2 Neural network design and training
### 2.1 Experimental setup
Figure 1: The self-mixing experimental setup consists of a laser diode (LD),
short focal length collimator (Col.), laser diode current driver (LD driver)
and voltage amplifier (RF amp).
The experimental arrangement, presented in Fig. 1, consists of a single
transverse mode laser emitting at $\lambda=1310~{}nm$ (ML725B8F) whose
threshold current is about $6.5~{}mA$ . In all the experiments reported here
the laser is driven at a constant current of 9 mA. The laser beam is focused
by a high numerical aperture (NA=0.7) lens on the central region of a basic
computer speaker located at about 20 cm from the laser. This speaker is put in
motion via an electrical signal produced by the sound card of a computer which
can easily produce many kinds of patterns at a sampling rate of 44.1 kHz. The
linearity of the speaker response has been assessed over a range of
frequencies from 5 Hz to 100 Hz and over the range of voltage provided by the
sound card. Over this range, the speaker responds with constant 0 phase. Thus,
in the range where the linearity of the displacement has been checked, one can
use the voltage at the speaker as an independent measurement of the position
of the target with respect to some unknown origin. From that point on, we will
therefore use this voltage as a proxy for the target position. The self mixing
signal is measured as a voltage at the laser electrodes, which is amplified by
an AC-coupled amplifier with $10^{4}$ amplification factor and several MHz
bandwidth. We deliberately did not optimize the self-mixing signal quality,
exactly because one of our aims is to check that neural networks can help in
making the measurement work even in sub optimal conditions.
### 2.2 Network setup
The first thing to be noted is that in the "few wavelengths" range of
displacement, the self-mixing signal contains no information about the
absolute position. Although one can be tempted to consider that counting
fringes (or some other equivalent technique) will lead to knowledge of the
exact position with respect to some unknown arbitrary origin, this approach is
bound to diffuse in the long term: If for some reason a fringe is missed, the
measurement system has no way to recover from this error because the physics
of the system does not include this information. Thus, (independently of how
accurate the fringe counting is unless it is strictly perfect), a position
measurement will unavoidably loose accuracy at a rate proportional to
$\sqrt{t}$ where $t$ is the measurement duration. Therefore, our aim here is
to provide a measurement of the displacement within a prescribed time
interval, a velocity.
Once this is established, the setting of the architecture of the neural
network is strongly influenced by the specific question one wants to address.
Here we assume that the self-mixing signal is acquired at a much larger
sampling rate than the Nyquist frequency of the displacement signal one wants
to measure. This is a very reasonable requirement in this context since most
approaches address the question by counting fringes. Here we assume that the
signal is sampled at least 256 times faster than the Nyquist frequency of the
displacement to be measured. Therefore, the reconstruction of the trajectory
consists in inferring from 256 self-mixing signal points one single
instantaneous velocity corresponding to the displacement of the target during
the 256 points acquisition. Then, in terms of machine learning the problem is
reduced to a "regression" problem, where some algorithm must provide a single
number on the basis of the available information (a piece of time trace of
length 256).
The setting therefore consists in analyzing a sequence where temporal ordering
matters and therefore a recurrent neural network can be envisioned as a
suitable architecture. However, these networks are notoriously difficult to
train and convolutional neural networks are known to be an easier to train and
valid alternative alternative. Therefore, we build a network based on a stack
of 1-dimensional convolutional layers with pooling layers between two
convolutional layers. At the end of the stack, two fully connected layers
convert the features identified by the convolutional layers into a single
number which is the inferred velocity of the target during the measurement
sequence. More details are given in appendix, table 1. This global
architecture was chosen from first principles of neural network design [15]
and the model details where then determined empirically. The network was
implemented with the Keras library, which offers an excellent tradeoff in
terms of complexity and versatility for our purpose [22].
### 2.3 Network training
Once the network architecture is chosen, the network must be trained with
known data. In practice, that means providing the network a large number of
pairs $[s(t_{0},...,t_{0}+256~{}dt),v]$ where $s(t)$ is a self-mixing signal
acquired during 256 sampling times $dt$ and $v$ is the average velocity of the
target during the duration of the interval $256~{}dt$. One must underline that
neural networks are known to be able to represent arbitrary functions provided
a sufficient number of layers and cells are present in the network [23].
Therefore, given enough computer time for training, a sufficiently large
network will be able to perfectly reproduce the training data it has been
shown. This means that a model trained this way achieves excellent accuracy.
However, one of the key issues with self-mixing implementations is that the
alignment conditions are sometimes different from one measurement to the next.
Equivalently, speckle generated by the reflection of the beam on a diffusive
target will lead to effective variations of the feedback strength parameter in
the course of the measurement. Therefore, the network trained here must be
able to adapt to these changes. This is known as the capacity of the network
to generalize the features which were learnt and identify them in unseen data.
To achieve this, we train the network on a deliberately limited set of data
and observe the reconstruction of the network on a very different data set,
both in terms of the dynamics of the target (different displacement patterns)
and in terms of the alignment of the beam on the target. The training data
consists exclusively of measurements of the interferometric signal in response
to periodic displacement of the target. We record self mixing signals in six
different alignment conditions, in three of them a double peak is visible in
each fringe. For each of these alignment conditions, we record the self mixing
signal for a set of 19 frequencies evenly spaced between 10 and 100Hz. For
each frequency we record 5 different amplitude signals. For each of these
settings we record sinusoidal and triangular waveforms. The sampling rate of
the oscilloscope is set to 250 kHz so that $dt=4\mu$s. In total the network is
trained on about $1.95\times 10^{3}$ segments of 256 time steps, each of them
of duration $256*4\mu$s $=1.024$ ms. The operation regime of self-mixing
sensing setups is often characterized in terms of the $C$ feeback parameter.
Here, the alignment configurations we use are such that the system operates in
the weak feedback regime $C<1$ (we do not observe multistability). However, we
also avoid the weakest feeback regime $C<<1$ in which the interferometric
signal is symmetric since it does not carry the relevant information. As is
common in deep learning network training, the data is further augmented by
adding noise to the training set. Here we add a delta-correlated gaussian
noise on top of the measured interferometric signal. It is trained by
minimizing the mean squared error between a guess it provides and the known
measured displacement.
## 3 Results
After training, one will assess the performance of the neural network (also
"the model") by comparing the displacement reconstructed from the
interferometric signal and the voltage at the speaker’s ends, used as a proxy
of position. First, we check the model’s prediction accuracy in known settings
(periodic signals and known alignment conditions) and then in unseen settings.
Figure 2: Reconstruction of periodic displacements similar to those used to
train the network. Top row: interferometric signal corresponding to three
different alignment configurations. Bottom row: measured displacement signal
(blue continuous line) and trajectory predicted by the network. The orange
crosses are predicted by the model and the dashed line is a simple cubic
interpolation. The shaded areas correspond to the interferometric signals
shown on the top row.
### 3.1 Periodic signals, known alignment conditions
We show on Fig. 2 how the network reconstructs examples of periodic traces
after training. This exact sequence has not been used during training but
these alignment conditions were used during training and sequences with
identical frequencies and amplitude were used during training.
On the top row we show three examples of interferometric signal which
correspond to three of the alignment conditions used during the training of
the network and three different displacement frequencies. On the bottom row,
we show the displacement per time unit of the target, as it can be measured
from the voltage at the edges of the speaker (blue continuous line).
Independently of that voltage measurement, we use the trained neural network
to infer the displacement from the self-mixing signal. This is the orange
dashed line, which is almost perfectly superimposed to the actual displacement
measured from the voltage at the speaker’s ends. This almost perfect
reconstruction is not very surprising since, even if the network had not seen
this exact piece of time trace during training, it has seen periodic signals
at these frequencies, these amplitudes and in these exact alignment
conditions. That is however a confirmation that the training of the network
has worked to an excellent accuracy and under different alignment conditions.
### 3.2 Aperiodic signals, unknown alignments
We check the capacity of the statistical model to adapt to unseen situations
by preparing a completely different displacement pattern. This pattern is
obtained by applying a fifth order butterworth band-pass filter between 5 and
100 Hz to a delta-correlated gaussian random noise. This pattern is sent to
the speaker in two different alignment conditions, none of them corresponding
the the situations used during training. In one of the two situations, the
interferometric signal shows a double-peak structure. The two interferometric
signals are then concatenated into a single time series and we use the model
to reconstruct the displacement of the target corresponding to this
concatenated time series. The results are shown on Fig. 3.
Figure 3: Reconstruction of unknown and complex displacements in alignment
situations which are not in the training set. Top: the blue line is the
displacement measured from the speaker voltage, the orange line is the model’s
prediction. Middle row: zoom around the central area of the top row. The
discontinuity close to 100 ms is where we numerically connected the two time
traces (see text). Bottom row: interferometric signal corresponding to the
middle row.
As can be immediately appreciated, the reconstruction is excellent, the
prediction matching almost perfectly the independently measured displacement.
Of course the discontinuity close to 100 ms, where the two measurements are
artificially concatenated, cannot be predicted by the network since it is
absent in the measured interferometric signal. We just choose to emphasize
this region as it shows that the prediction is essentially insensitive to the
alignment conditions, which change abruptly in the middle of the trace. As is
evident from the lower panel of Fig. 3, reconstructing a trajectory from this
interferometric signal would be very difficult due to the presence of noise
and very widely varying fringe shapes and repetition rates.
From the above, one concludes that the model is able to generalize from its
learning set to provide an accurate reconstruction of the displacement in
unseen alignment conditions and for very complex time series, much more
difficult to analyze than the simple periodic time traces used during the
training phase.
To better appreciate the accuracy of the inference, one can plot the predicted
displacement as a function of the actual displacement as shown on Fig. 4. A
perfect reconstruction would be the one shown by the orange line where the
prediction is exactly equal to the truth. We can quantify the reconstruction
quality by the Pearson’s correlation coefficient between the reconstruction
and the ground truth which is here 0.90 and the absolute standard error which
is here $0.30\lambda/ms$. Specifically, one can notice that the prediction is
less good for the largest absolute values of displacement. This can be related
to the statistical properties of the training set as shown on the right panel
of Fig. 4. Here one can appreciate that absolute values of displacement larger
than 2.5$\lambda/ms$ have been seen by the network during training only a few
hundreds of times, while smaller displacements are much more frequent in our
training set. Thus, the large displacements are very under-represented in the
training set. This results in a lower precision of the reconstruction for
larger displacements, which can also be appreciated on the top panel of Fig. 3
where the largest displacements are in general under estimated.
Figure 4: The accuracy of prediction on unknown samples (at a given sampling
rate) is conditioned by the training set. Left: displacement inferred by the
model as a function of the true displacement. The correlation is excellent but
the model slightly underestimates the larger displacements (about 2.5
$\lambda$/ms). Right: histograms of the distribution of displacements in the
training sets. The larger displacements (about 2.5 $\lambda$/ms) are very
under-represented in the training set.
### 3.3 Noise sensitivity
One of the difficulties in reconstructing the displacement from the self
mixing signal also comes from the fact that simple Fourier filtering is often
not very efficient at separating the detection noise from the interferometric
signal (although neural networks have been proposed to alleviate this issue
[20]). Here we check that the statistical model is very robust to the addition
of noise on top of the interferometric signal. To assess this robustness, we
use the model to reconstruct the displacement corresponding to the complex
interferometric signal described in 3.2 after adding to this signal a gaussian
white noise. As we show on Fig. 5, the model predictions are extremely robust.
Since the added noise is $\delta-$ correlated, its standard deviation
$\sigma_{n}$ is a measure of its power density. Here, we normalize the
interferometric signal itself in absence of added noise to its standard
deviation $\sigma_{s}$ so that $\sigma_{s}=1$. We then vary the standard
deviation of the added noise $\sigma_{n}$ between 0 and 3 times $\sigma_{s}$.
On Fig. 5a), we observe that both the root mean squared and the absolute error
remain very low up to $\sigma_{n}=1$ where it grows significantly. On Fig.
5b), we plot the correlation coefficient between the reconstructed
displacement and the independently measured displacement. As for the error,
the correlation coefficient indicates an excellent reconstruction of the
displacement up to approximately $\sigma_{n}=1$. We show the prediction and
the noisy interferometric signal for $\sigma_{n}=0.7$ on Fig. 5 c) and d)
respectively. As can be easily observed, the interferometric signal would be
rather difficult to process by the usual means and the stochastic model
provides a very useful reconstruction.
Figure 5: Robustness against detection noise. a) Mean absolute (orange
continuous) and RMS (blue dashed) error of the reconstruction as function of
added noise in units of the interferometric signal’s standard deviation
$\sigma_{n}$. b) Pearson’s correlation coefficient between the prediction and
the signal as function of added noise. c) Example of prediction (orange line)
and true displacement (blue line) for an added noise power density
$\sigma_{n}=0.7\times\sigma_{n}$ where $\sigma_{n}$ is the interferometric
signal’s standard deviation. d) Interferometric signal in the same situation,
zoom over the 94-106 ms region (the same signal as Fig. 3, with added noise).
### 3.4 Unknown experiment
The analysis above has shown that the statistical model is able to reconstruct
the displacement from the self-mixing signal in a broad range of unknown
conditions. However, all of the above was realized on a single experimental
setup. Contrary to a physical model, which is constructed to capture only the
universal features of an experiment, an empirically constructed statistical
model such as the neural network we use may capture also non-universal and
system-specific features. Thus, it is interesting to check what the model can
predict on the basis of a different experiment, based on the same principle.
To address this question, we prepare an "almost-twin" experiment, based on the
same self mixing interferometry principle shown in Fig. 1 but featuring a
different laser (HL6323MG, $\lambda=639~{}nm$, driven at $I=75~{}mA$ for a
threshold current $I_{th}=45~{}mA$), a different speaker (with a different
range of linear response), a different voltage amplifier for the acquisition
of the laser diode voltage etc. Although this experiment is in principle the
same, it differs in many of the details which should not be relevant to the
physics, yet carry a significant risk of distortion of the interferometric
signal as compared to the one used in training set.
To check the model’s ability to analyze this new self mixing experiment, we
prepare a new displacement time series consisting of the sum of four different
frequencies $S=\sum_{i=1}^{4}\sin{2\pi f_{i}t}$ with
$f_{i}=425,718,808,1076~{}Hz$. This time series is therefore in a very
different (much higher) frequency band with respect to the training
experiment. In order to provide the model with comparable input data, the
acquisition of the self-mixing signal is performed at a ten times faster rate
than in the training experiment (2.5 MHz). The displacement per time unit of
the speaker is, as in the previous experiment, measured as a voltage at the
edges of the speaker. The comparison between the displacement estimated from
the speaker’s voltage and the displacement reconstructed from the
interferometric signal is shown on Fig. 6.
Figure 6: Reconstruction of the displacement on an unknown experiment. Top:
comparison between the displacement predicted by the model (orange line) and
the actual displacement (blue line). There are no free parameters. Middle:
zoom on one specific region. Bottom: self mixing signal corresponding to
middle panel. The green circles show where jumps between bistable states
occur.
The agreement between the prediction and the measurement is strikingly good,
especially taking into account that no free parameter exist: The model trained
on experiment 1 can immediately be used to infer displacements in units of
$\lambda/dt$ in experiment 2.
It is important to underline once more the robustness of this process with
respect to specific experimental conditions. For instance, in this experiment,
the interferometric signal shows clear signs of bistability between external
cavity modes in forms of very fast jumps between states (green circles on
bottom panel of Fig. 6). These features are absent from the training set. Here
one sees that the model essentially filters them out automatically. Besides
this, it is also worth noting that, as compared to Fig. 3 for instance, the
displacements and time scales are very different. This shows that, provided an
adequate sampling rate is chosen, the model can work at much higher frequency
than the band it was trained in and for much larger displacements per time
unit. This feature is not unexpected since the network knows only about
displacement "per $256\times dt$", without reference to the exact value of
$dt$. Therefore, with adequate sampling, the measurement range can be
tremendously extended with respect to the training range. This feature is
extremely useful since it allows training in an easily accessible range
(displacement and frequency band) and prediction in very different range for
more demanding applications.
## 4 Discussion
The results above clearly show that a convolutional neural network is a useful
tool in the reconstruction of a target’s displacement, very robust to unknown
displacement signal shapes, alignment conditions, electronic noise and even
whole setups. We have also verified that a network trained in the 10-100 Hz
frequency band can also meaningfully reconstruct displacements including
frequencies of hundreds of Hz, provided the measurement sampling rate is
adapted. Thus, the neural network strength lies less in the absolute precision
that it allows than in its robustness against detailed experimental conditions
and versatility across the "sub-wavelength/analog" and "beyond
wavelength/digital" classification [6] for arbitrarily complex waveforms.
One natural question which arises when preparing a neural network is that of
model capacity [15]. A network which does not possess enough cells or layers
may be unable to take into account all the complexity of the task. On the
other hand, a network with a very large number of cells and layers will sooner
or later learn features of the experiment which should not be significant (for
instance, all the details about an amplifier used in the setup). This prevents
the network from generalizing, ie accurately predicting unknown data. This is
in principle dealt with during the training phase [15] but it is only when the
network processes fully new data that this issue can be totally ruled out.
Here this issue has been taken care of by predicting arbitrarily complex
trajectories and also by using two different setups. In fact, the imperfect
reconstruction in the case of the unknown experiment is most probably due to
the model learning some system-specific features of the training experiment.
This can be mitigated by a minor retraining of the final layer of the model on
the new experiment (a procedure known as "fine tuning" in the deep learning
context). We have noticed that a larger network featuring more than $10^{5}$
coefficients instead of the $5.7\times 10^{4}$ used here does not lead to
better training and may even lead to worse predictions in the unknown
experiment.
The performance of the network is of course strongly related to the training
data set which is used. Here we deliberately use only a very limited set of
displacements during training in order to very clearly show the generalization
phenomenon, the network being able to predict correctly displacement shapes it
has never seen before. For real use beyond the proof of concept presented
here, more refined training is possible: A training set featuring a more
uniform distribution of displacements will provide a more accurate
reconstruction of the larger displacements for a given sampling rate. As an
alternative, simply increasing the sampling rate at the prediction time may
also be a sufficient solution to adapt the time series to the operating range
of the model as we have shown in 3.4. Care must be taken when training the
network that the correct operating range of the neural network is set by a
displacement per time unit, which includes limitations in terms of
displacement frequency and amplitude. Translating it in terms of counting
fringes, that means that the network will saturate beyond a certain number of
fringes during the $256\times dt$ measurement window. In terms of feedback
range, here we have used only $C<1$ in training, avoiding too low values of
$C$ where the interferometric signal is symmetric. At the prediction phase,
the model is robust to $C$ slightly overcoming unity but when multistability
becomes strong the information of few wavelengths displacements is lost and
the model has no chance to recover it. Similarly, we have checked that when
$C$ is so low that the signal is symmetric, the model cannot predict
accurately. A full characterization of performance degradation and the use of
multichannel measurements to mitigate this issue is beyond the scope of this
work.
One particularly interesting avenue to circumvent the limitations of an
experimental training set is to train the network on numerically generated
data [24]. One drawback is that the network will not learn more than what is
in the physical model used for the simulations, which may be hard in complex
settings such as multimode lasers [25, 26]. However, numerics may provide a
way to obtain a training set for which controlled laboratory experiments would
be very hard to realize such as hard shocks or high frequency and high
amplitude displacements. Experimental data may then be used to refine the
training by using different sampling times as described in the previous
paragraph.
Once trained, a convolutional neural network can be used in real time since no
pre-processing of the data is required and prediction over thousands of
interferometric measurements is very fast: As an example, 3 seconds of signal
(749056 interferometric data points) are processed in 0.16 seconds on a
standard laptop. In addition, after the initial training, a neural network can
relatively easily be repurposed by retraining only its final layers even with
a very limited set of data. For instance, it would be particularly interesting
to assess the performance of the network trained here on self-mixing schemes
which include bias current modulation towards some other sensing task such as
refractive index measurements. Alternatively, the input layer of the network
can also be reworked at minor cost to take into account multichannel
measurements and most interesting would probably be to integrate this approach
into multimodality imaging systems [27].
## 5 Conclusion
To conclude, we have presented a detailed analysis of how a reasonably simple
convolutional neural network can be used to reconstruct the displacement of a
target on the basis of self mixing interferometry. We believe that this
approach can become one of the many tools which can be used to tailor or
enhance self-mixing coherent sensing setups. Far from being limited to
displacement measurement and single mode settings, we believe that computer
neural networks can become an extremely useful element of many sensing
apparatus, especially self-mixing setups. Finally, we stress that there are
very few hard rules about the design of neural networks and this design is in
itself often an area of research. The architecture we use here is essentially
a simple starting point and many refinements are possible. More specifically,
one of the most immediate extensions of the work exposed above is to train a
network which can provide an estimation of the accuracy of the reconstruction.
This can be achieved by adding a calibrated regression stage on top of the
convolutional base prepared here [28]. Other extensions may include more
complex network topologies, perhaps mixing convolutional and recurrent layers
or including skip connections.
## Appendix A Network details
The key elements of the network are shown on table 1. We refer the reader to
deep learning fundamentals for background information [15]. The total number
of parameters (57 153) is much smaller than the number of time series segments
in the data set even before augmentation. Networks of identical architecture
with more cells per layer did not lead to significant improvements. A dropout
layer is used here mostly as "safety net" since the data set is very large
anyway which makes overfitting improbable. The training of the network takes
about ten minutes on a simple GPU (GeForce GTX 1060) and about four times more
on CPU (Intel Xeon 3.8GHz).
Network structure: sequential
---
Layer type | Main
hyperparameters | Trainable
parameters | Output shape
1D convolutional | kernel size: 7
filters: 16 | 128 | (250, 16)
Max Pooling | pool size: 2 | 0 | (125, 16)
1D convolutional | kernel size:7
filters: 32 | 3616 | (119, 32)
Max Pooling | pool size: 2 | 0 | (59, 32)
1D convolutional | kernel size: 7
filters 64 | 14400 | (53, 64)
Max Pooling | pool size: 2 | 0 | (26, 64)
Dropout | 10% | 0 | (26, 64)
1D convolutional | kernel size: 7
filters: 64 | 28736 | (20, 64)
Max Pooling | pool size: 2 | 0 | (10, 64)
Fully connected | units: 16 | 10256 | 16
Fully connected | units: 1 | 17 | 1
Table 1: Main parameters of the network used in this work. The network is a
sequence of 1-dimensional convolutional and dropout layers followed by two
fully connected layers for the final regression. The total number of trainable
parameters is 57 153.
## Acknowledgments
The authors thank Dr. L. Columbo, Dr. M. Dabbicco and Dr. F. Pedaci for
helpful discussions.
## Disclosures
The authors declare no conflicts of interest.
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|
# Direct evidence of ferromagnetism in MnSb2Te4
Wenbo Ge Department of Physics and Astronomy, Rutgers University, Piscataway,
New Jersey 08854, USA Paul M. Sass Department of Physics and Astronomy,
Rutgers University, Piscataway, New Jersey 08854, USA Pacific Northwest
National Laboratory, Richland, Washington, 99352, USA Jiaqiang Yan Materials
Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831, USA Department of Materials Science and Engineering,
University of Tennessee, Knoxville, Tennessee 37996, USA Seng Huat Lee 2D
Crystal Consortium, Materials Research Institute, Pennsylvania State
University, University Park, PA 16802, USA Department of Physics,
Pennsylvania State University, University Park, PA 16802, USA Zhiqiang Mao
2D Crystal Consortium, Materials Research Institute, Pennsylvania State
University, University Park, PA 16802, USA Department of Physics,
Pennsylvania State University, University Park, PA 16802, USA Weida Wu
<EMAIL_ADDRESS>Department of Physics and Astronomy, Rutgers
University, Piscataway, New Jersey 08854, USA
###### Abstract
We report the magnetic imaging of ferromagnetic domains in the van der Waals
single crystal MnSb2Te4 from two different sources using cryogenic magnetic
force microscopy. The magnetic field dependence of the domains reveals very
weak pinning of domain walls in MnSb2Te4, resulting in a negligibly small
magnetic hysteresis loop. The temperature dependence of the domain contrast
reveals a mean field like behavior, in good agreement with that of bulk
magnetization measurements.
## I Introduction
The interplay between magnetism and topological electronic states has led to
many fascinating quantum phenomena, such as the quantum anomalous Hall (QAH)
effect and axion insulator state. The QAH state was realized for the first
time in the magnetically doped topological insulator (Bi, Sb)2Te3 [1]. The
introduction of magnetic dopants breaks local time reversal symmetry and opens
an exchange gap at the Dirac point of the topological surface band, leading to
a dissipationless chiral edge mode with no external magnetic field [2, 3, 4,
1, 5]. However, the magnetic inhomogeneity inherent to doping limits the full
quantization to ultra-low temperatures [6, 7, 8, 9]. The recently synthesized
MnBi2Te4-family compounds are van der Waals materials that host intrinsic
magnetism and topologically nontrivial band structure [10, 11, 12, 13, 14, 15,
16, 17]. The magnetism of this compound comes from the long-range order of Mn
layers instead of magnetic doping. As an intrinsic magnetic TI, MnBi2Te4
provides a clean platform to realize the QAH effect at elevated temperatures
and to explore other exotic topological phases. The structure of the MnBi2Te4
septuple-layer can be viewed as a Mn-Te bilayer being inserted into the middle
of a quintuple layer of Bi2Te3. Each Mn2+ ion carries 5 $\mu_{\textrm{B}}$
magnetic moment in the localized magnetism approximation. The intra-plane
exchange coupling is ferromagnetic while the inter-plane coupling is
antiferromagnetic, which results in an A-type antiferromagnetic order with
uniaxial anisotropy [11, 10]. First principle calculations predict that an odd
number of septuple-layers can realize the QAH effect and an even number can
realize an axion insulator state [18]. Both phenomena were recently reported
in odd and even layers of exfoliated flakes of MnBi2Te4 single crystals above
1 K [19, 20]. However, the topological nature of the electronic phases of even
and odd layers remains controversial [21].
MnSb2Te4 is isostructural to MnBi2Te4 [22]. Its ideal magnetic ground state is
also A-type antiferromagnetic [12]. However, a recent study revealed that the
magnetism in MnSb2Te4 can be tuned from antiferromagnetic to ferromagnetic by
changing the Mn/Sb site mixing [23]. This study pointed out a potential route
to tune the magnetism in MnBi2Te4 to realize a robust QAH effect at high
temperature. In ferromagnetic MnSb2Te4, the Mn atoms mixed into the Sb sites
order antiparallel to neighboring Mn layers and mediate a ferromagnetic inter-
septuple-layer coupling. Density functional theory (DFT) calculations show
that ferromagetic MnSb2Te4 may host Weyl points despite the fact that
synthesized ferromagnetic MnSb2Te4 requires significant site mixing leading to
a topologically trivial band structure [23]. Engineering the magnetism and
maintaining the band topology of MnSb2Te4 and MnBi2Te4 requires further study
of their electronic band structure and magnetic properties. Neutron
diffraction, transport and magnetization studies have been reported on
ferromagnetic MnSb2Te4, yet, the direct visualization of the ferromagnetic
state in MnSb2Te4 is still lacking [23, 24].
Here, we report cryogenic magnetic force microscopy (MFM) of ferromagnetic
MnSb2Te4 single crystal samples from two different sources. Micron size
ferromagnetic domains are visualized on both samples after zero-field cooling
(ZFC) below $T_{\textrm{C}}\approx 33$ K. We observed typical soft
ferromagnetic domain behavior with very weak domain wall pinning. The magnetic
field dependence of domain population shows little hysteresis, in good
agreement with that of bulk magnetization. The temperature dependence of
domain contrast follows mean field behavior, in good agreement with that of
the saturation magnetization.
Figure 1: (Color Online) Field dependent MFM images of two samples synthesized
by different groups. (a)-(i) Sequential MFM images of S1 (Penn State) measured
at 6 K in various magnetic fields. The field value is shown on the upper left
corner of each image. The color scale is 12 Hz. (j)-(r) Sequential MFM images
of S2 (ORNL) measured at 10 K in various magnetic fields. The field value is
shown on the upper left corner of each image. The color scale is 30 Hz.
Qualitatively, the same domain behavior was observed in samples from both
sources.
## II materials and methods
Ferromagnetic MnSb2Te4 single crystals from Penn State were grown using a
self-flux method [10, 25]. The mixture of manganese powder (99.95%), antimony
shot (99.9999%), and tellurium ingot (99.9999+%) with the molar ratio of
Mn:Sb:Te = 1:10:16 were loaded into alumina crucible and sealed in double
evacuated quartz tubes. The mixture was heated up to 900 ∘C for 12 h and then
slowly cooled down to 630 ∘C at a rate of 2 ∘C/h and dwelt at 630 ∘C for 24 h.
Finally, the extra flux was removed by centrifuging at 630 ∘C. The phase and
crystallinity of single crystals are checked by x-ray diffraction. Single
crystals from Oak Ridge National Laboratory (ORNL) were grown with similar
methods [10]. The ferromagnetic order with Curie temperature of 33 K was
confirmed through magnetization measurements using a commercial
superconducting quantum interference device (SQUID) magnetometer.
Here, we denote S1 for single crystals from Penn State, and S2 for single
crystals from ORNL. Samples were cleaved in air before MFM experiments. The
MFM experiments were carried out in a homemade cryogenic magnetic force
microscope using commercial piezoresistive cantilevers (spring constant
$\approx 3$ N/m, resonant frequency $\approx 42$ kHz) [26]. The homemade MFM
is interfaced with a Nanonis SPM controller and a commercial phase-lock loop
(SPECS). Out-of-plane magnetic field was applied via a superconducting magnet.
MFM tips were prepared by depositing a nominally 100 nm Co film onto the bare
tips using sputtering. The MFM signal, the change of cantilever resonant
frequency, is proportional to the out-of-plane stray field gradient.
Electrostatic interaction was minimized by balancing the tip-surface potential
difference. Dark (bright) regions in MFM images represent attractive
(repulsive) magnetization, where magnetization are parallel (antiparallel)
with the tip polarization.
## III results and discussion
We performed magnetic field dependent MFM measurements on S1 (6 K) and S2 (10
K), respectively. The MFM images are shown in Fig. 1. In Fig. 1(a) and Fig.
1(j), randomly shaped domains form after ZFC from 40 K. This is clearly
different from the prior MFM results of antiferromagnetic MnSb2Te4 where
domain walls were visualized [26]. The typical domain size characterized by
the width of the curvilinear stripes in the MFM images is approximately 1.7
$\mu$m in S1 and 2.7 $\mu$m in S2. After ZFC, we observed equal populations of
up and down domains. After increasing the out-of-plane positive applied
magnetic field, the domain walls propagate in such a way that the up domains
(dark region) expand until they fully occupy the scanning area at around 0.1 T
[Figs. 1(a)-(e) and (j)-(n)]. Consistently, S2 is magnetically saturated above
0.1 T [23]. Down domains nucleate as the external field is reduced from the
saturation field [Figs. 1(f) and (o)] and their area increases as the field
decreases [Figs. 1(f)-(i) and (o)-(r)]. The populations of up and down domains
become nearly the same again at zero applied field. These results show that
both MnSb2Te4 samples are soft ferromagnets with extremely weak domain wall
pinning. Moreover, the domain contrast is relatively stronger at high fields
possibly due to the enhancement of the MFM tip moment. There exists subtle
difference in the domain shape between S1 and S2. Domains in S1 are more
stripy than domains in S2, suggesting that the magnetism in S1 is more
2D-like. This subtle difference might be related to the different correlation
length along the $c$-axis. Further studies are needed to understand their
subtle difference.
Figure 2: (Color Online) (a)-(i) Sequential MFM images of the same area of
MnSb2Te4 (S2) single crystal in different magnetic fields at 5 K. The magnetic
field values are labeled in the upper left corner of each image. The color
scale is 20 Hz. (j) Binarized version of MFM image (i). (k) Normalized
magnetization estimated from three sets of binarized field dependent MFM
images. Magnetization from SQUID is plotted for comparison (red curve).
To demonstrate that the observed ferromagnetic domain behavior is
representative of the bulk property, we performed detailed field dependent MFM
measurements and domain population analysis on the S2 (ORNL) sample. Fig.
2(a)-(i) shows a negative field sweep of the S2 sample at the same location as
in Fig. 1. The sample was saturated with large negative magnetic field at 5 K.
The MFM tip moment was initialized in the down state. A line-shaped up domain
emerges at $-0.08$ T. As the applied magnetic field further decreases, the
domain starts to wiggle and branches out at the bending point in order to
reduce the magnetostatic energy cost [Figs. 2(b)-(e)]. The up domain grows as
the field decreases further and the area finally reaches an approximately 50%
domain population at zero-field [Figs. 2(f)-(i)].
Figure 3: (Color Online) (a)-(i) Sequential zero-field MFM images of S2 at
different temperatures upon warming after 0.05 T field cooling from 40 K. The
temperature values are labeled in the upper left corner of each image. The
scales are 20 Hz for (a)-(e) and 4 Hz for (g)-(i). (j) RMS value of MFM images
as a function of temperature (red connected dots) in comparison with $M(T)$
under 0.1 T (blue curve). The Curie temperature $T_{\textrm{C}}$ extracted
from molar susceptibility is 33.2 K as labeled in the figure.
Detailed examination of the domain processes reveals weak domain wall pinning
in S2. To quantify the domain population and estimate the normalized
magnetization, we binarize our MFM images using a threshold value
approximately at the middle of the domain wall and assigning 1 ($-1$) to up
(down) domains. Here, we assume that the magnetic moment is uniform everywhere
inside each domain. As an example, Fig. 2(j) is the binarized version of Fig.
2(i). The normalized magnetization $M/M_{\textrm{S}}$ of the sample can then
be estimated from the binarized MFM images from
$(N_{\uparrow}-N_{\downarrow})/(N_{\uparrow}+N_{\downarrow})$ by counting
$N_{\uparrow}$ and $N_{\downarrow}$. Here, $M_{\textrm{S}}$ is the saturation
magnetization and $N_{\uparrow}$ ($N_{\downarrow}$) represents the population
of up (down) domains. Using the method described above, we obtain the
normalized $M(H)$ data as plotted in Fig. 2(k). The external field was swept
from 0 T to $-0.1$ T so that the sample was saturated, and then reduced back
to 0 T. No visible hysteresis was found in the normalized magnetization
$M/M_{\textrm{S}}$ vs. $\mu_{0}H$ data, confirming the very weak domain wall
pinning in MnSb2Te4. Note that the domain patterns are different on the up-
sweep and down-sweep of magnetic field, indicating random nucleation of
reversed domains. Our MFM results indicate that the domain pattern is
hysteretic while the total magnetization is not. For comparison, the $M(H)$
curve measured by SQUID is plotted in Fig. 2(k). Clearly, the estimated
magnetization from local MFM imaging is in good agreement with the global
magnetometry result.
To further establish the correspondence between local (domain) and global
(bulk magnetization) properties, we performed MFM measurements on sample S2 at
different temperatures upon warming in zero-field after 5 mT field cooling
from 40 K (above $T_{\textrm{C}}$). Bubble-shaped domains were observed, as
shown in Fig. 3(a). The domain pattern remains the same below 25 K while the
domain contrast decreases monotonically as the temperature increases, as shown
in Figs. 3(a)-(e). As the temperature approaches $T_{\textrm{C}}$, the domain
contrast becomes so weak that the color scale of those images (32 K, 33 K and
34 K) was reduced to reveal the domain pattern. Interestingly, the domains
start to deform [Fig. 3(f)] and break up into branches as shown in Fig. 3(g),
indicating further reduction of anisotropy energy. At 33 K, just below
$T_{\textrm{C}}$, more bubble-shaped domains form on the sample surface, which
indicates a reduction of domain wall energy [27]. The domain contrast
completely vanishes at and above 34 K. Thus, the Curie temperature is between
33 K and 34 K, in excellent agreement with $T_{\textrm{C}}\approx 33.2$ K
estimated from the inflection point of $M(H)$ (blue curve) from SQUID
measurements shown in Figs. 3(j).
In the MFM images, the domain contrast is proportional to the stray field
gradient, which scales with the saturation magnetization at each temperature
[28]. Since our MFM images are dominated by the domain contrast, the averaged
domain contrast is proportional to the root-mean-square (rms) value of the MFM
signal in each image. Fig. 3(j) shows the temperature dependence of the rms
value of MFM images Figs. 3(a)-(i). This curve exhibits mean field behavior
which again confirms the ferromagnetic ordering of MnSb2Te4. The blue curve in
Fig. 3(j) shows $M(T)$ measured by SQUID with a 0.1 T out-of-plane magnetic
field. Thus, the temperature dependence of the domain contrast at zero-field
is in excellent agreement with that of the saturation magnetization,
supporting the existence of long-range ordering of the ferromagnetism in
MnSb2Te4.
## IV conclusion
The magnetic imaging of domains in MnSb2Te4 single crystals from two different
sources provides direct evidence of long-range ferromagnetic ordering in
MnSb2Te4. The extremely weak pinning of domain walls observed in the field
dependence explains why little hysteresis was observed in bulk magnetization
measurements. The agreement between domain population and the bulk
magnetization demonstrates that the MFM measurements capture the
representative domain behavior. Also, the temperature dependence of domain
contrast agrees with the that of the saturation magnetization. The direct
evidence of ferromagnetism in MnSb2Te4 will encourage further exploration of
potential ferromagnetic Weyl physics in the MnBi2Te4-family.
###### Acknowledgements.
The MFM studies at Rutgers are supported by the Office of Basic Energy
Sciences, Division of Materials Sciences and Engineering, U.S. Department of
Energy under Award No. DE-SC0018153. Work at ORNL was supported by the U.S.
Department of Energy, Office of Science, Basic Energy Sciences, Materials
Sciences and Engineering Division. Support for crystal growth and
characterization at Penn State was provided by the National Science Foundation
through the Penn State 2D Crystal Consortium-Materials Innovation Platform
(2DCC-MIP) under NSF Cooperative Agreement DMR-1539916.
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|
These authors contributed equally to this work Ph.D. Program in Chemistry, The
Graduate Center of the City University of New York, New York, NY These authors
contributed equally to this work Ph.D. Program in Biochemistry, The Graduate
Center of the City University of New York, New York, NY These authors
contributed equally to this work Ph.D. Program in Biochemistry, The Graduate
Center of the City University of New York, New York, NY Ph.D. Program in
Chemistry, The Graduate Center of the City University of New York, New York,
NY Ph.D. Program in Biochemistry, The Graduate Center of the City University
of New York, New York, NY
# Alchemical Transfer Approach to Absolute Binding Free Energy Estimation
Joe Z. Wu Department of Chemistry, Brooklyn College of the City University of
New York, New York, NY Solmaz Azimi Department of Chemistry, Brooklyn
College of the City University of New York, New York, NY Sheenam Khuttan
Department of Chemistry, Brooklyn College of the City University of New York,
New York, NY Nanjie Deng Department of Chemistry and Physical Sciences, Pace
University, New York, New York, 10038 Emilio Gallicchio
<EMAIL_ADDRESS>Department of Chemistry, Brooklyn College of the
City University of New York, New York, NY
###### Abstract
The Alchemical Transfer Method (ATM) for the calculation of standard binding
free energies of non-covalent molecular complexes is presented. The method is
based on a coordinate displacement perturbation of the ligand between the
receptor binding site and the explicit solvent bulk, and a thermodynamic cycle
connected by a symmetric intermediate in which the ligand interacts with the
receptor and solvent environments with equal strength. While the approach is
alchemical, the implementation of ATM is as straightforward as for physical
pathway methods of binding. The method is applicable in principle with any
force field, it does not require splitting the alchemical transformations into
electrostatic and non-electrostatic steps, and it does not require soft-core
pair potentials. We have implemented ATM as a freely available and open-source
plugin of the OpenMM molecular dynamics library. The method and its
implementation are validated on the SAMPL6 SAMPLing host-guest benchmark set.
The work paves the way to streamlined alchemical relative and absolute binding
free energy implementations on many molecular simulation packages and with
arbitrary energy functions including polarizable, quantum-mechanical, and
artificial neural network potentials.
## 1 Introduction
The binding free energy of a molecular complex is a rigorous thermodynamic
measure of the degree of affinity of two molecules for each other.
Measurements of binding free energies are useful in a wide range of chemical
and medicinal applications ranging from drug discovery to chemical detection
and toxicology. The ability to estimate binding free energies by computational
modeling adds an important dimension to this probe.1 Relative binding free
energy models are widely used, for example, in drug lead optimization.2
This work is concerned in particular with atomistic models of the absolute
binding free energy. These methods can be divided into two classes that are
based on the nature of the thermodynamic path used to connect the bound and
unbound states of the molecular complex.3 Physical pathway methods define a
spatial coordinate along which the reversible work for bringing the two
molecules together is calculated. Conversely, alchemical methods connect the
bound and unbound states by a series of unphysical intermediate states. The
Single-Decoupling Method (SDM),4, 5 for example, is based on progressively
turning on the effective interaction between the ligand and the receptor with
an implicit representation of the solvent. The related Double-Decoupling
Method (DDM),6 which is widely used to estimate the absolute binding free
energy with an explicit representation of the solvent, estimates the binding
free energy as the difference between the free energies of coupling the ligand
to a pure solvent system and to the solvated receptor.
Physical and alchemical binding free energy methods are characterized by
distinct challenges and limitations. While it is broadly employed, the
alchemical DDM approach is known to suffer from poor convergence and strong
bias, especially for large and charged ligands, and these consequences are
characterized by large and compensating decoupling free energies that result
in the amplification of statistical and systematic errors in the binding free
energy estimate.7 Another significant shortcoming of DDM is the large
perturbation of the environment of the ligand in going from the solvated
states to the vacuum intermediate state. Unless the ligand is conformationally
restrained, the transition from a solvated state to vacuum can trigger
substantial intramolecular conformational changes that relax slowly to the
bound or unbound configurations. Other technical limitations of DDM stem from
differences in the composition and size of the molecular systems used for the
two decoupling legs,8 and from the inconsistent treatment of long-range
interactions.9, 10 Even for relatively simple systems, these and other
alchemical transformations have to be conducted with care to avoid
singularities and slow convergence. It is recommended for example, to couple
electrostatic interactions separately from other interactions,11 and to employ
customized soft-core pair potentials12 to avoid end-point singularities.
Because they are often based on modifying the parameters and the form of the
energy function, software implementations of alchemical binding free energy
methods also tend to be significantly complex and require in-depth knowledge
of the data structures of the target molecular simulation package. For
example, core energy routines are usually customized to implement the specific
modified pair potentials that represent the interaction of the ligand with the
rest of the system. These modified pair potentials depend on the alchemical
progress parameter (generally denoted by $\lambda$) which, together with other
alchemical variables, become additional system parameters. Methods such as
Thermodynamic Integration (TI)13, 14 require additional routines to implement
the calculations of the gradients of the energy function with respect to the
parameters that are alchemically transformed. The implementation of alchemical
transformations involving many-body potential terms, where the $\lambda$
dependence affects more than individual pair interaction energies, is
particularly challenging. These include alchemical applications with
polarizable potentials,15 Ewald long-range electrostatic treatments,16
implicit solvent models,17 as well as some conventional intramolecular
potential terms.18
Physical pathway methods19, 20, 7 address some the limitations of DDM by
physically moving the ligand from the solvent bulk to the binding site.
Physical pathway binding free energy calculations are seen as preferable over
DDM for large and charged ligands because they are typically performed as one
continuous path in a single solvent box without transferring the ligand to a
different phase. In addition, software implementations of physical pathway
methods do not require as much customization of the underlying molecular
simulation package as alchemical methods do. The primary limitation of
physical pathway methods is the high computational cost due to the requirement
of equilibrating the complex at many intermediate receptor-ligand separations
that might not be of interest. Because they require a physical exit and entry
channel, physical pathway methods are also not generally applicable to
occluded binding sites.21
Building upon on our Single Decoupling Method (SDM) for absolute binding free
energy estimation22, 23 implemented in OpenMM,24 we have been investigating
ways to streamline alchemical calculations with explicit solvation. SDM, which
has been designed for binding free energy calculations with implicit
solvation, computes the alchemical perturbation energy by translating the
ligand from the solvent medium to the receptor binding site, rather than
attempting to selectively turn-on and turn-off individual ligand-receptor
interactions. The approach treats all interactions in one concerted step and
employs the standard molecular mechanics force field without soft-core pair
potentials. End-point singularities are addressed by a suitable non-linear
alchemical energy function.23 We have recently shown that the same approach is
applicable to the estimation of the concerted hydration free energies of drug-
sized solutes in water droplets with explicit solvation.25
In this work, we extend this concerted alchemical scheme to the calculations
of absolute binding free energies with explicit solvation. In the resulting
alchemical scheme, called the Alchemical Transfer Method (ATM) (Figure 1B),
the unbound and bound states of the molecular complex are related by a
translation vector that brings the ligand from the solvent bulk to the
receptor binding site in a single solvent box. We show that the proposed
method addresses some of the aforementioned challenges of binding free energy
calculations by exploiting the best characteristics of the alchemical and
physical methods. Like DDM, the method is based on alchemical transformations
that aim conformational sampling only in the solvent bulk and the receptor
binding site, and like tphysical methods, ATM is based on moving the ligand in
physical space, in a single simulation box, and without transferring the
ligand to vacuum. The ATM method, implemented as a freely available plugin of
the OpenMM molecular simulation package, did not require any modifications of
the OpenMM core energy routines. We validate the ATM approach on a rigorous
benchmarking dataset developed by Rizzi et al.26
## 2 Theory and Methods
### 2.1 Alchemical Transformations
Free energy changes are estimated using alchemical transformations based on a
$\lambda$-dependent potential energy functions $U_{\lambda}(x)$ that brings
the system from an initial state at $\lambda=0$, described by the potential
function $U_{0}(x)$, to a final state at $\lambda=1$, corresponding to the
potential function $U_{1}(x)$.
For each transformation, the alchemical potential energy is expressed as
$U_{\lambda}(x)=U_{0}(x)+W_{\lambda}(u)$ (1)
where $x$ represents the set of atomic coordinates of the system,
$u(x)=U_{1}(x)-U_{0}(x)$ (2)
is the perturbation energy, and $W_{\lambda}(u)$ is the generalized softplus
alchemical perturbation function
$W_{\lambda}(u)=\frac{\lambda_{2}-\lambda_{1}}{\alpha}\ln\left[1+e^{-\alpha(u_{\rm
sc}(u)-u_{0})}\right]+\lambda_{2}u_{\rm sc}(u)+w_{0}.$ (3)
The parameters $\lambda_{2}$, $\lambda_{1}$, $\alpha$, $u_{0}$, and $w_{0}$
are functions of $\lambda$ (see Computational Details), and
$u_{\rm sc}(u)=\begin{cases}u&u\leq u_{0}\\\ (u_{\rm max}-u_{0})f_{\rm
sc}\left[\frac{u-u_{0}}{u_{\rm max}-u_{0}}\right]+u_{0}&u>u_{0}\end{cases}$
(4)
with
$f_{\text{sc}}(y)=\frac{z(y)^{a}-1}{z(y)^{a}+1}\,,$ (5)
and
$z(y)=1+2y/a+2(y/a)^{2}$ (6)
is the soft-core perturbation energy. The soft-core function is monotonic map
that avoids singularities near the initial state of the alchemical
transformation at $\lambda=0$ without affecting the distribution of
perturbation energies at the final state at $\lambda=1$. As expressed in Eq.
(4), the soft-core perturbation energy is designed to cap the perturbation
energy $u(x)$ to a maximum positive value $u_{\rm max}$ and to be equal to the
perturbation energy when this is below a cutoff value $u_{0}$. The $u_{0}$
cutoff is selected to be sufficiently large so that $u_{\rm sc}(u)=u$ for all
observed samples collected at the end state. The specific values of $u_{0}$,
$u_{\rm max}$ and of the scaling parameter $a$ used in this work are listed in
Computational Details.
In order to reproduce the desired end points, it is necessary that the
alchemical perturbation function is defined such that $W_{0}(u)=0$ and
$W_{1}(u)=u$ at $\lambda=0$ and $\lambda=1$, respectively. This requirement is
satisfied by the linear function, $W_{\lambda}(u)=\lambda u_{\rm sc}(u)$,
which is special case of the softplus function in Eq. (3) for which
$\lambda_{1}=\lambda_{2}=\lambda$. The linear function is the standard choice
for the alchemical perturbation function. As it can be verified from Eq. (3),
in general the end-point requirement is satisfied whenever
$\lambda_{1}=\lambda_{2}=0$ at $\lambda=0$ and $\lambda_{1}=\lambda_{2}=1$ at
$\lambda=1$.
### 2.2 The Alchemical Transfer Method for Binding Free Energy Estimation
Consider the non-covalent binding process between a receptor R and a ligand L.
The standard free energy of binding, $\Delta G^{\circ}_{b}$, is defined as the
difference in free energy between the bound complex and the unbound components
at the standard concentration of $C^{\circ}=1$ M,
$\Delta G^{\circ}_{b}=\Delta G^{\circ}_{\rm site}+\Delta G^{\ast}_{b}.$ (7)
where $\Delta G^{\ast}_{b}$ is the excess component, defined as the reversible
work for transferring the ligand to the binding site region of the receptor of
volume $V_{\rm site}$ from a region of the same volume in the solvent bulk
(Figure 1) plus a concentration-dependent term
$\Delta G^{\circ}_{\rm site}=-k_{B}T\ln C^{\circ}V_{\rm site}$ (8)
that corresponds to the free energy of transfer of a ligand molecule from an
ideal solution at concentration $C^{\circ}$ to a region of volume $V_{\rm
site}$ in the solvent.3 In the remainder, we will focus on the calculation of
the excess free energy component $\Delta G^{\ast}_{b}$ by alchemical molecular
simulations.
(A) (B)
Figure 1: Schematic illustrations of (A) the double decoupling (DDM) and (B)
the Alchemical Transfer (ATM) methods for the calculation of the binding free
energy between a molecular receptor (orange doughnut) and a ligand (black
circle). The dashed circle within the receptor represents the binding site
region. The blue boxes represent the solvent. The unbound and bound end states
for the two methods are considered thermodynamically equivalent. In both
methods, the end states are transformed to a common intermediate state and the
excess binding free energy is the difference of the free energy changes of the
two legs, $\Delta G^{\ast}_{b}=\Delta G_{2}-\Delta G_{1}$. Double decoupling
defines an intermediate state in which the ligand is in vacuum (white). The
alchemical transformations in the Alchemical Transfer Method are instead
performed in the same solvent box and the ligand does not leave the solvated
system.
The Double Decoupling Method (DDM)6 has emerged as one of the gold standards
for the alchemical calculation of absolute binding free energies in condensed
phases. DDM is based on the thermodynamic cycle illustrated in Fig. 1A,
whereby the bound and unbound states of the molecular complex are
thermodynamically linked by an intermediate state in which the ligand is
placed in vacuum. The excess binding free energy is expressed as the
difference of the free energies of alchemically decoupling the ligand from the
unbound (leg 1) and bound (leg 2) states into the intermediate vacuum state,
as
$\Delta G^{\ast}_{b}=\Delta G_{2}-\Delta G_{1}\,.$ (9)
The Alchemical Transfer Method (ATM) proposed here avoids the vacuum
intermediate and requires only one molecular system. As illustrated in Fig.
1B, ATM is based on an alchemical intermediate in which the ligand interacts
simultaneously with the solvent bulk and the receptor. We assume, without loss
of generality, that there is a suitable coordinate frame attached to the
receptor and that the binding site region (represented by the dashed circle in
Fig. 1) is fixed relative to this coordinate frame. Under these assumptions,
every point in the binding site region maps to a unique point into a region of
the same shape in the solvent bulk by means of a constant translation vector
$h$, also at rest relative to the receptor coordinate frame (Figure 2). The
bound state of the system is defined as any configuration in which the ligand
is in the binding site region. Conversely, the unbound state of the system is
defined as any configuration of the system in which the ligand is placed into
the bulk solvent region (Fig. 1B).
Under these assumptions, any configuration of the bound state maps to an
unique configuration of the unbound state by a rigid translation of the ligand
atoms by the vector $h$ (Figure 2). The reverse is also true. Any
configuration of the unbound system maps uniquely to a configuration of the
bound state by translation of the vector $-h$. Hence, the translation vector
$h$ can be used as a perturbation parameter to connect, in a statistical
thermodynamic sense, the bound and unbound states.
Figure 2: Illustration of the unbound state of the complex between the CB8
host (center) and the G3 guest (lower left). The small red spheres represent
the oxygen atoms of the water molecules. The displacement vector that
translates the ligand from the binding site to the bulk solvent position is
indicated.
For example, consider leg 1 of the ATM cycle in Fig. 1B. Denoting the
potential energy function of the system as $U(x)$, with $x=(x_{S},x_{L})$
being the coordinates of the bound system, where $x_{L}$ are the coordinates
of the ligand in the receptor binding site and $x_{S}$ are the coordinates of
the receptor and the solvent (the surroundings), the potential energy
functions $U_{0}$ and $U_{1}$ of the initial and final states of the leg (the
bound and unbound states in this case) are respectively
$U_{0}(x_{S},x_{L})=U(x_{S},x_{L})$ (10)
and
$U_{1}(x_{S},x_{L})=U(x_{S},x_{L}+h)\,.$ (11)
To connect the bound and unbound states, we consider the hybrid alchemical
potential of Eq. (1) with the perturbation energy
$u_{1}(x_{S},x_{L})=U(x_{S},x_{L}+h)-U(x_{S},x_{L}),$ (12)
which is defined as the change in the potential energy of the system for
rigidly translating the ligand atoms from the binding site region to the bulk
solvent region while all other degrees of freedom of the system remain
unchanged.
With these definitions, the alchemical hybrid potential used for the first leg
of the ATM cycle is:
${\rm Leg\ 1:}\quad
U_{\lambda}(x_{S},x_{L})=U(x_{S},x_{L})+W_{\lambda}(u_{1})\,,\quad
0<=\lambda<=1/2$ (13)
where the alchemical perturbation function $W_{\lambda}(u)$ is defined in Eq.
(3) and the perturbation energy $u_{1}$ is defined in Eq. (12). As indicated
in Eq. (13), the alchemical pathway for the first leg is terminated at
$\lambda=1/2$, where the ligand interacts with half strength with both the
receptor environment and the solvent bulk. This ensures that severe steric
clashes are not likely to occur at $\lambda=1/2$. The $u_{0}$ parameter of the
soft-core perturbation potential is set to a large enough value so that the
perturbation potential $u$ does not exceed $u_{0}$ at $\lambda=1/2$. Under
these conditions, the original and the soft-core perturbation potentials are
the same and it follows from Eq. (13) that the alchemical potential energy at
$\lambda=1/2$ is
$U_{1/2}(x_{S},x_{L})=\frac{1}{2}U(x_{S},x_{L})+\frac{1}{2}U(x_{S},x_{L}+h).$
(14)
Eq. (14) defines the potential energy function of the alchemical intermediate
of the ATM cycle in Fig. 1B. Accordingly, in the alchemical intermediate
ensemble the ligand interacts symmetrically with the receptor and bulk solvent
environments. Conversely, in this state the receptor atoms and the solvent
molecules interact with the ligand at half strength. The alchemical
calculation that corresponds to the alchemical potential Eq. (13) yields the
free energy change $\Delta G_{1}$ in going from the bound state to the ATM
alchemical intermediate.
To formulate the second leg of the ATM cycle connecting the unbound state to
the intermediate state (Figure 1), the role of the end states is reversed
relative to the first leg. As before, $x_{L}$ describes the coordinates of the
ligand in the receptor binding site and $x_{L}+h$ describes the coordinates of
the ligand in the solvent bulk and the initial and final states are now
defined by the potential energy functions of, respectively, the unbound and
bound states:
$U_{0}(x_{S},x_{L})=U(x_{S},x_{L}+h)$ (15)
and
$U_{1}(x_{S},x_{L})=U(x_{S},x_{L})\,.$ (16)
The alchemical potential energy function for the second leg of the ATM cycle
is
${\rm Leg\ 2:}\quad
U_{\lambda}(x_{S},x_{L})=U(x_{S},x_{L}+h)+W_{\lambda}(u_{2})\,,\quad
0<=\lambda<=1/2$ (17)
with the perturbation energy
$u_{2}(x_{S},x_{L})=U(x_{S},x_{L})-U(x_{S},x_{L}+h)$ (18)
that corresponds to the potential energy change of transferring the ligand
from the bulk to the receptor binding site. Note that under the same
assumptions that led to Eq. (14), Eq. (17) reaches $\lambda=1/2$ at the same
alchemical intermediate as Eq. (13). Thus, Eq. (9) holds for the ATM
thermodynamic cycle in Fig. 1B.
We end the presentation of the Alchemical Transfer Method by discussing the
requirement of splitting the alchemical path into two legs. Although the
symmetric hybrid potential Eq. (14) can formally cover the direct path from
the unbound to the bound states by extending the $\lambda$ range from $0$ to
$1$, in practice it suffers severe end-point singularities at both of the end
states. The potential energy, $U(x_{S},x_{L}+h)$, when the ligand is placed in
the solvent bulk region, is ill-defined near $\lambda=0$ due to the atomic
clashes that occur when ligand-solvent interactions are turned off.
Conversely, near $\lambda=1$, the potential energy $U(x_{S},x_{L})$, when the
ligand is placed in the receptor site, is ill-defined due to clashes between
ligand atoms and receptor atoms. The soft-core protocol we employ is based on
an asymmetric definition of the perturbation potential [see Eqs. (12) and
(18)] that can address the singularity at one end-point or the other but not
both simultaneously with only one continuous alchemical perturbation potential
energy function.
### 2.3 Software Implementation
The method is implemented as an integrator plugin
(github.com/rajatkrpal/openmm_sdm_plugin)23 of the OpenMM library.24 The
integrator is based on the Langevin thermostat and high-level routines that
displace the ligand, issue calls to energy and forces calculation routines,
and compute the alchemical potential energy [Eq. (3)] and its gradients by
combining the returned system energies and forces. For example for leg 1, at
each MD-step the plugin first computes and saves in temporary buffers the
potential energy and the forces when the ligand is in the binding site. Then
it displaces the ligand in the bulk by the displacement vector $h$ and
recalculates the energy and forces (except for the binding site restraining
potential, see below). The perturbation energy $u_{1}$ [Eq. (12)] and its
gradients are obtained by taking the corresponding differences before and
after the displacement. The gradients of the alchemical perturbation energy
(13) are derived from those of $u_{1}$ and of the undisplaced potential
$U(x_{S},x_{L})$ by application of the chain rule. The resulting forces are
then used to propagate atomic coordinates by one MD step. The same process is
used for leg 2, except that the ligand is initially placed in the solvent bulk
and it is translated into the binding site by reversing the displacement. In
each case the binding restraint potential (see below) is applied when the
ligand is in the binding site. No modifications of the core OpenMM energy
routines are applied.
### 2.4 Benchmark Systems
The benchmark systems here were drawn from the host-guest systems presented in
the SAMPL6 SAMPLing challenge26. The octa-acid (OA) and cucurbit[8]uril (CB8)
hosts are well-studied supramolecular systems that have been featured in
previous host-guest binding SAMPL challenges and the three guests resemble
conventional druglike small molecules and fragments. In total, the benchmark
systems presented here include 5-hexenoic acid (OA-G3) and 4-methylpentanoic
acid (OA-G6) for the OA host, and quinine (CB3-G3) for the CB3 host. Despite
the name, the guest G3 in CB8-G3 is distinct from the G3 guest in OA-G3. The
parametrized systems, including their solvent descriptions, partial charges,
and initial geometries, are provided at the github SAMPL6 site:
github.com/samplchallenges/SAMPL6/tree/master/host_guest/SAMPLing.
Host-guest systems are practical alternatives to protein-ligand systems
because of their minimal atom count and the improbability of undergoing major
conformational reorganization. These systems thus enable the investigation of
novel simulation techniques, such as longer timescales and appropriate
sampling of multiple equivalent binding modes. Each host-guest complex has
five conformations that differ in the position within the binding site, as
well as in torsion angles. The procedure of obtaining five replicate free
energy calculations allows for assessing the statistical uncertainty and
reproducibility of the methodologies. The benchmark systems employed here
create an accessible platform to improve both the predictive accuracy and
computational efficiency of free energy calculations.26
### 2.5 Computational Details
The ATM calculations employed the host-guest molecular complexes parametrized
using the GAFF1.8/AM1-BCC force-field and the TIP3P water model as provided by
Rizzi et al.26 from
(https://github.com/samplchallenges/SAMPL6/tree/master/host_guest/SAMPLing).
The Cartesian components of the displacement vector $h$ were set to
approximately half the dimensions of the simulation box in order to ensure
that the ligand is placed in the corner of the solvent box (Figure 2). This
position is the farthest from the host, which the solvent box is centered
around, and its periodic images. The complexes were energy minimized and
thermalized at 300 K. Then, using the ATM alchemical potential energy function
for leg 1 [Eq. (13)] and starting at the bound state at $\lambda=0$, the
systems were annealed to the symmetric intermediate $\lambda=1/2$ for $250$
ps. The purpose of this step is to obtain a suitable initial configuration
without severe unfavorable repulsion interactions at either end of the
alchemical paths in order to start the molecular dynamics replica exchange
alchemical calculations for each leg (see below). To limit the fluctuations of
the position of the bulk solvent region, which would impact convergence, the
position and orientation of the receptor was loosely restrained with a flat-
bottom harmonic potential of force constant 25.0 kcal/(mol Å2) and a tolerance
of 1.5 Å on all of the heavy atoms of the lower portion of the receptor (the
first 40 atoms of the host as listed in the provided files).
Polar hydrogen atoms with zero Lennard-Jones parameters were modified to
$\sigma_{\rm LJ}=0.1$ Å and $\epsilon_{\rm LJ}=10^{-4}$ kcal/mol to avoid
large attractive interactions between opposite charges at small distances in
nearly uncoupled states. The change in potential energy of the system in the
unbound, bound, and symmetric intermediate states due to this modification of
the Lennard-Jones parameters is below single floating point precision. Single
Decoupling alchemical calculations were prepared using the SDM workflow
(github.com/egallicc/openmm_sdm_workflow.git). MD calculations employed the
OpenMM24 MD engine and the SDM integrator plugins
(github.com/rajatkrpal/openmm_sdm_plugin.git) using the OpenCL platform. The
ASyncRE software,27 customized for OpenMM and SDM
(github.com/egallicc/async_re-openmm.git), was used for the Hamiltonian
Replica Exchange in $\lambda$ space for each ATM leg.
The linear alchemical perturbation potential, $W_{\lambda}(u)=\lambda u_{\rm
sc}(u)$, corresponding to Eq. (3) with $\lambda_{1}=\lambda_{2}=\lambda$, was
used for the octa-acid (OA) systems with $11$ $\lambda$-states uniformly
distributed between $\lambda=0$ and $1/2$ for each of the two ATM legs. The
ATM calculations for the more challenging CB8-G3 complex employed the softplus
perturbation potential Eq. (3) with $24$ $\lambda$-states and the parameters
listed in Tables 1 and 2 for each leg. The parameters of the softplus
perturbation potential were optimized using the scheme described previously23,
25 which involved running trial calculations with the linear potential,
fitting the analytical model of binding28 to each transformation, and
adjusting the parameters of the softplus potential to the resulting
$\lambda$-function23 to obtain a smooth alchemical transition.
The soft-core perturbation energy Eq. (4) was used for all calculations with
$u_{\rm max}=300$ kcal/mol, $u_{0}=100$ kcal/mol. The ligand was sequestered
within the binding site by means of a flat-bottom harmonic potential between
the centers of mass of the host and the ligand with a force constant of $25$
kcal/mol Å2 applied for separation greater than $4.5$ Å. Perturbation energy
samples and trajectory frames were saved every 5 ps. Hamiltonian replica
exchanges in $\lambda$-space were performed every 5 ps. The Langevin
thermostat with a time constant of 2 ps was used to maintain the temperature
at 300 K. Each replica was simulated for a minimum of 10 ns. Binding free
energies and the corresponding uncertainties were computed from the
perturbation energy samples using UWHAM29, discarding the first 5 ns of
trajectory, followed by the addition of the concentration-dependent term
$\Delta G^{\circ}_{\rm site}=-k_{B}T\ln C^{\circ}V_{\rm site}=0.87$ kcal/mol
that corresponds to $300$ K temperature and the volume $V_{\rm site}$ of a
sphere of radius $4.5$ Å. The replica exchange simulations were run on the
XSEDE Comet GPU HPC cluster at the San Diego Supercomputing Center each using
four GPUs per node.
Table 1: Alchemical schedule of the softplus perturbation function for leg 1
for the CB8/G3 complex.
$\lambda$ | $\lambda_{1}$ | $\lambda_{2}$ | $\alpha$a | $u_{0}$b | $w_{0}$b
---|---|---|---|---|---
0.000 | 0.000 | 0.000 | 0.250 | 230.0 | 0.000
0.022 | 0.000 | 0.050 | 0.250 | 230.0 | 0.000
0.043 | 0.000 | 0.100 | 0.250 | 220.0 | 0.000
0.065 | 0.000 | 0.150 | 0.250 | 215.0 | 0.000
0.087 | 0.000 | 0.200 | 0.250 | 210.0 | 0.000
0.109 | 0.000 | 0.250 | 0.250 | 205.0 | 0.000
0.130 | 0.000 | 0.300 | 0.250 | 200.0 | 0.000
0.152 | 0.000 | 0.350 | 0.250 | 195.0 | 0.000
0.174 | 0.000 | 0.400 | 0.250 | 188.0 | 0.000
0.196 | 0.000 | 0.400 | 0.250 | 178.0 | 0.000
0.217 | 0.000 | 0.400 | 0.200 | 170.0 | 0.000
0.239 | 0.000 | 0.400 | 0.150 | 160.0 | 0.000
0.261 | 0.000 | 0.400 | 0.150 | 152.0 | 0.000
0.283 | 0.000 | 0.425 | 0.140 | 145.0 | 0.000
0.304 | 0.000 | 0.450 | 0.130 | 140.0 | 0.000
0.326 | 0.000 | 0.475 | 0.140 | 135.0 | 0.000
0.348 | 0.000 | 0.500 | 0.150 | 128.0 | 0.000
0.369 | 0.000 | 0.500 | 0.200 | 120.0 | 0.000
0.391 | 0.000 | 0.500 | 0.200 | 112.0 | 0.000
0.413 | 0.100 | 0.500 | 0.200 | 110.0 | 0.000
0.435 | 0.200 | 0.500 | 0.150 | 110.0 | 0.000
0.457 | 0.300 | 0.500 | 0.150 | 110.0 | 0.000
0.478 | 0.400 | 0.500 | 0.100 | 110.0 | 0.000
0.500 | 0.500 | 0.500 | 0.100 | 110.0 | 0.000
aIn (kcal/mol)-1. bIn kcal/mol.
Table 2: Alchemical schedule of the softplus perturbation function for leg 2
for the CB8/G3 complex.
$\lambda$ | $\lambda_{1}$ | $\lambda_{2}$ | $\alpha$a | $u_{0}$b | $w_{0}$b
---|---|---|---|---|---
0.000 | 0.000 | 0.000 | 0.250 | 175 | 0.000
0.022 | 0.000 | 0.084 | 0.250 | 172.5 | 0.000
0.043 | 0.000 | 0.167 | 0.250 | 170 | 0.000
0.065 | 0.000 | 0.233 | 0.225 | 165 | 0.000
0.087 | 0.000 | 0.333 | 0.200 | 160 | 0.000
0.109 | 0.000 | 0.417 | 0.175 | 155 | 0.000
0.130 | 0.000 | 0.500 | 0.150 | 150 | 0.000
0.152 | 0.000 | 0.550 | 0.125 | 145 | 0.000
0.174 | 0.000 | 0.600 | 0.100 | 140 | 0.000
0.196 | 0.000 | 0.650 | 0.095 | 135 | 0.000
0.217 | 0.000 | 0.700 | 0.090 | 130 | 0.000
0.239 | 0.000 | 0.700 | 0.085 | 125 | 0.000
0.261 | 0.000 | 0.700 | 0.080 | 120 | 0.000
0.283 | 0.000 | 0.700 | 0.075 | 112 | 0.000
0.304 | 0.050 | 0.700 | 0.070 | 110 | 0.000
0.326 | 0.100 | 0.700 | 0.0675 | 102.5 | 0.000
0.348 | 0.150 | 0.700 | 0.065 | 105 | 0.000
0.369 | 0.200 | 0.700 | 0.060 | 100 | 0.000
0.391 | 0.250 | 0.650 | 0.055 | 100 | 0.000
0.413 | 0.300 | 0.625 | 0.050 | 100 | 0.000
0.435 | 0.350 | 0.600 | 0.045 | 100 | 0.000
0.457 | 0.400 | 0.575 | 0.040 | 100 | 0.000
0.478 | 0.450 | 0.550 | 0.035 | 100 | 0.000
0.500 | 0.500 | 0.500 | 0.030 | 100 | 0.000
aIn (kcal/mol)-1. bIn kcal/mol.
## 3 Results
The standard binding free energy estimates, $\Delta G_{b}^{\circ}$, for the
host-guest systems obtained using the ATM method are listed in Table 3
compared to the corresponding experimental measurements, $\Delta
G^{\circ}_{b}$(exp), and the reference computational estimates obtained
through the attach-pull-release (APR) methodology.26 Table 3 also lists the
number of energy and forces evaluations per replicate (a proxy for the
computational cost) for the reference APR calculations, $n_{\rm eval}$(ref),
and for the ATM calculations reported here ($n_{\rm eval}$). The APR method
was selected as representative for this comparison because APR, similar to ATM
and unlike DDM, displaces the ligand in a position in the solvent bulk at a
finite distance from the host.
The ATM binding free energy estimates reported in Table 3 are obtained as the
average of five replicates started from different initial conformations as
reported in Tables 4, 5, and 6. These tables also report the calculated free
energies, $\Delta G_{1}$ and $\Delta G_{2}$, of the two alchemical legs for
each replicate. The binding free energy of each replicate is the difference
between those of the two legs (the excess component) plus the standard state
term $\Delta G^{\circ}_{\rm site}$, which in this case measures out to be
approximately $0.87$ (see Computational Details). The statistical
uncertainties of the averages of each term reported in Tables 4, 5, and 6 are
expressed as 95% confidence interval of the mean based on the t-test
distribution with four degrees of freedom.26 The statistical fluctuations of
the binding free energies among the five conformations for each system were
consistently smaller than the those of each of the legs, suggesting some level
of systematic error cancellation.
The ATM results obtained for the OA-G3 and OA-G6 complexes are in good
agreement (within $0.5$ kcal/mol) with the reference values and well within
the range of estimates obtained with other methods.26 The ATM binding free
energy estimate for the more challenging CB-G3 complex deviates more
substantial from the APR reference ($2$ kcal/mol less favorable) and from
those of the other methods applied to this benchmark system.26 The origin of
this discrepancy is not obvious. However all ATM estimates appear to generally
underestimate binding affinities relative to the other methods bringing them,
perhaps coincidentally, closer to the experimental measurements. The
deviations between the experimental standard binding free energies and the ATM
estimates are $0.71$ kcal/mol, $1.35$ kcal/mol, and $2.08$ kcal/mol for,
respectively, the OA-G3, OA-G6, and CB8-G3 complexes, compared to $1.12$
kcal/mol, $1.83$ kcal/mol, and $4.05$ kcal/mol with APR.
The range of the spread between ATM replicates obtained here for the octaacid
systems is generally larger than with APR and other methods26 albeit at a
generally higher computational cost. For CB8-G3, ATM yields a spread similar
to APR and the other methods with significantly less computational cost ($480$
vs. $2,135$ million energy evaluations as compared to APR, Table 3).
In overall, ATM yields standard binding free energy estimates within the
general range displayed by the established methods tested on the SAMPL6
SAMPLing benchmark set at a similar computational expense.26 These initial
results confirm the validity of the ATM approach.
Table 3: Standard binding free energy estimates and corresponding
computational effort for the three host-guest complexes with the Alchemical
Transfer Method compared to experimental and reference computed values.
Complex | $\Delta G^{\circ}_{b}$(exp)a,b | $\Delta G^{\circ}_{b}$(ref)a,c,d | $n_{\rm eval}$(ref)c,e | $\Delta G^{\circ}_{b}$a,e | $n_{\rm eval}$f
---|---|---|---|---|---
OA-G3 | $-5.18\pm 0.02$ | $-6.3\pm 0.1$ | $458\times 10^{6}$ | $-5.89\pm 0.33$ | $220\times 10^{6}$
OA-G6 | $-4.97\pm 0.02$ | $-6.8\pm 0.1$ | $305\times 10^{6}$ | $-6.32\pm 0.21$ | $220\times 10^{6}$
CB8-G3 | $-6.45\pm 0.06$ | $-10.5\pm 0.6$ | $2135\times 10^{6}$ | $-8.53\pm 0.64$ | $480\times 10^{6}$
aIn kcal/mol. bFrom references 30 and 31. cFrom reference 26. dAPR method.
eThis work, from Tables 4, 5, and 6. fThis work.
Table 4: Free energy estimates for the two legs of the Alchemical Transfer
Method and corresponding standard binding free energy estimates for the OA-G3
complex starting with each of the the five initial SAMPL6 SAMPLing
conformations.
Conformation | $\Delta G_{1}$a | $\Delta G_{2}$a | $\Delta G^{\circ}_{\rm site}$a | $\Delta G^{\circ}_{b}$a
---|---|---|---|---
OA-G3-0 | $57.00$ | $50.47$ | $0.87$ | $-5.66$
OA-G3-1 | $57.79$ | $50.97$ | $0.87$ | $-5.95$
OA-G3-2 | $57.74$ | $50.79$ | $0.87$ | $-6.08$
OA-G3-3 | $57.65$ | $51.21$ | $0.87$ | $-5.57$
OA-G3-4 | $57.25$ | $50.19$ | $0.87$ | $-6.18$
Averageb | $57.49\pm 0.42$ | $50.73\pm 0.50$ | $0.87$ | $-5.89\pm 0.33$
aIn kcal/mol. bWith t-test 95% confidence intervals with 4 degrees of freedom
based on the standard deviation of the mean.
Table 5: Free energy estimates for the two legs of the Alchemical Transfer
Method and corresponding standard binding free energy estimates for the OA-G6
complex starting with each of the the five initial SAMPL6 SAMPLing
conformations.
Conformation | $\Delta G_{1}$a | $\Delta G_{2}$a | $\Delta G^{\circ}_{\rm site}$a | $\Delta G^{\circ}_{b}$a
---|---|---|---|---
OA-G6-0 | $57.80$ | $50.64$ | $0.87$ | $-6.29$
OA-G6-1 | $58.19$ | $50.78$ | $0.87$ | $-6.54$
OA-G6-2 | $58.27$ | $50.97$ | $0.87$ | $-6.43$
OA-G6-3 | $58.74$ | $51.70$ | $0.87$ | $-6.17$
OA-G6-4 | $58.27$ | $51.25$ | $0.87$ | $-6.14$
Averageb | $58.25\pm 0.41$ | $51.07\pm 0.52$ | $0.87$ | $-6.32\pm 0.21$
aIn kcal/mol. bWith t-test 95% confidence intervals with 4 degrees of freedom
based on the standard deviation of the mean.
Table 6: Free energy estimates for the two legs of the Alchemical Transfer
Method and corresponding standard binding free energy estimates for the CB8-G3
complex starting with each of the the five initial SAMPL6 SAMPLing
conformations.
Conformation | $\Delta G_{1}$a | $\Delta G_{2}$a | $\Delta G^{\circ}_{\rm site}$a | $\Delta G^{\circ}_{b}$a
---|---|---|---|---
CB8-G3-0 | $70.50$ | $61.06$ | $0.87$ | $-8.57$
CB8-G3-1 | $70.65$ | $60.81$ | $0.87$ | $-8.97$
CB8-G3-2 | $71.27$ | $61.40$ | $0.87$ | $-9.00$
CB8-G3-3 | $71.74$ | $62.96$ | $0.87$ | $-7.91$
CB8-G3-4 | $69.09$ | $60.02$ | $0.87$ | $-8.20$
Average | $70.65\pm 1.24$ | $61.25\pm 1.34$ | $0.87$ | $-8.53\pm 0.64$
aIn kcal/mol. bWith t-test 95% confidence intervals with 4 degrees of freedom
based on the standard deviation of the mean.
## 4 Discussion
The Alchemical Transfer Method (ATM) presented here implements a perturbation
potential based on rigidly displacing the coordinates of the ligand atoms from
a region in the solvent bulk to the receptor binding site or viceversa. Like
in smart-darting Monte Carlo32, this is accomplished using a displacement
vector that can be thought as connecting a unique pair of points of two
conformational macrostates. At each MD time-step the ligand disappears from
one place and appears in another in a way that is physically not achievable or
”alchemical”. The change in potential energy of the system due to the ligand’s
displacement is the perturbation energy of the $\lambda$-dependent alchemical
potential energy function that is used for conformational sampling and free
energy estimation. While not presented here, the method is applicable to the
calculation of the relative free energy of binding between two ligands13 by
swapping their positions in the bulk and in the receptor site. This work is
ongoing and will be reported in a forthcoming publication.
Similarly to physical pathway methods,19, 33, 21 ATM is relatively easy to
implement in molecular simulation packages because it does not require
modifications of system parameters nor customized single- and dual-topologies
setups that characterize conventional alchemical binding free energy methods.
The method is illustrated here using a plugin implementation on top of the
core OpenMM library.24 The method is agnostic of the underlying energy
function. It has been validated here with explicit solvation and Particle Mesh
Ewald (PME) long range electrostatics. It is conceivably applicable without
approximations to any kind of many-body potential function, including
polarizable,15, 34 quantum-mechanical,35, 36, 37 and artificial neural
network38, 39 potentials.
Unlike alchemical approaches such as double-decoupling,6 which requires two
systems, and dual-system single box alchemical methods,40, 41, 42 which
require dual topologies, ATM works with a single standard model of the
receptor-ligand complex solvated in a solvent box as in conventional molecular
dynamics applications. In addition, ATM does not require soft-core pair
potentials nor the splitting of the alchemical transformations into separate
electrostatic and steric/dispersion steps.23, 25
Similar to single-box alchemical approaches40, 41, 42 ATM avoids alchemical
transformations that place the ligand in vacuum,6, 43 which are particularly
problematic for large and charged ligands.21 The desolvation step of double-
decoupling, for example, removes all of the ligand-water interactions, even
though those of the solvent-exposed region of the ligand are likely to form
again in the subsequent coupling step with the receptor. With ATM, instead,
existing hydration interactions in the bulk are more likely to be replaced by
similar interactions with the ligand displaced into the binding site.
Moreover, unless the ligand is properly restrained, the vacuum intermediate is
likely to introduce hard to converge free energy terms related to the
reorganization of the ligand conformational ensemble from vacuum to the
solvated environment.
ATM has some drawbacks, some of which are technical in nature and are likely
to be addressed in the future. Because it calculates the system energy and
forces twice for each MD time-step,4 once with the ligand in the bulk and
again with the ligand in the receptor pocket, the method is a factor of two
slower per step than standard molecular dynamics. The two energy evaluations
are however independent and can be conceivably run in parallel on two attached
computational devices for added performance. Currently the method also
requires the recalculation of the non-bonded neighbor list after each ligand
displacement resulting in an additional 10 to 15% slow-down per step with
OpenMM for these systems. As we observed here for the CB8-G3 system, the
binding of bulky ligands requires optimized softplus alchemical perturbation
functions trained on trial calculations with the linear alchemical
potential.28, 23, 25 In future work, we plan to implement an adaptive
algorithm to refine the parameters of the alchemical potential function on the
fly.
Here we have validated ATM on the rigorous SAMPL6 SAMPLing dataset.26 The
dataset includes well-studied systems prepared with a single set of force
field parameters and in different initial conformations to probe both
systematic and statistical errors. The binding free energies of the systems
have been computed and validated with a diverse collection of approaches,
including alchemical and physical pathway methods.26 The ATM results for the
octacid systems obtained here are well within the range of estimates reported
in reference 26, thereby adding confidence that the ATM approach is sound and
that it has been implemented correctly. We observed in particular good
agreement with the Attach Pull Release (APR) method,44 a physical pathway
approach in which the guest is progressively displaced into the solvent bulk
to a comparable distance from the host as in this work. ATM yields a
statistically significant less favorable binding free energy estimate than the
other methods for the more challenging CB8-G3 system. The source of the
deviation is unclear, however, ATM appears to generally yield binding free
energies of smaller magnitude and closer to the experimental measurements than
the other methods. Taking into account the relative computational expense, the
statistical uncertainties obtained here indicate that ATM estimates have a
comparable level of reproducibility and computational efficiency as the
methods tested in reference 26.
## 5 Conclusions
We have presented the Alchemical Transfer Method (ATM) for the calculation of
standard binding free energies of non-covalent molecular complexes. The method
is based on a coordinate displacement perturbation of the ligand between the
receptor binding site and the bulk solvent and a thermodynamic cycle connected
by a symmetric intermediate in which the ligand interacts with the receptor
and solvent environments equally. While the approach is alchemical, ATM’s
implementation is as straightforward as physical pathway methods of binding.
ATM does not require splitting the alchemical transformations into
electrostatic and non-electrostatic steps and it does not employ soft-core
pair potentials. We have implemented ATM as a freely available and open-source
plugin of the OpenMM molecular dynamics library. The method and its
implementation have been validated on the SAMPL6 SAMPLing host-guest benchmark
set.
## 6 Acknowledgments
We acknowledge support from the National Science Foundation (NSF CAREER
1750511 to E.G.). Molecular simulations were conducted on the Comet GPU
supercomputer cluster at the San Diego Supercomputing Center supported by NSF
XSEDE award TG-MCB150001.
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|
# Evidence that Ultra-High-Energy Gamma Rays are a Universal Feature Near
Powerful Pulsars
A. Albert Physics Division, Los Alamos National Laboratory, Los Alamos, NM,
USA R. Alfaro Instituto de Física, Universidad Nacional Autónoma de México,
Ciudad de Mexico, Mexico C. Alvarez Universidad Autónoma de Chiapas, Tuxtla
Gutiérrez, Chiapas, México J.D. Álvarez Universidad Michoacana de San
Nicolás de Hidalgo, Morelia, Mexico J.R. Angeles Camacho Instituto de
Física, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico
J.C. Arteaga-Velázquez Universidad Michoacana de San Nicolás de Hidalgo,
Morelia, Mexico K.P. Arunbabu Instituto de Geofísica, Universidad Nacional
Autónoma de México, Ciudad de Mexico, Mexico D. Avila Rojas Instituto de
Física, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico
H.A. Ayala Solares Department of Physics, Pennsylvania State University,
University Park, PA, USA V. Baghmanyan Institute of Nuclear Physics Polish
Academy of Sciences, PL-31342 IFJ-PAN, Krakow, Poland E. Belmont-Moreno
Instituto de Física, Universidad Nacional Autónoma de México, Ciudad de
Mexico, Mexico S.Y. BenZvi Department of Physics & Astronomy, University of
Rochester, Rochester, NY , USA C. Brisbois Department of Physics, University
of Maryland, College Park, MD, USA K.S. Caballero-Mora Universidad Autónoma
de Chiapas, Tuxtla Gutiérrez, Chiapas, México T. Capistrán Instituto de
Astronomía, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico
A. Carramiñana Instituto Nacional de Astrofísica, Óptica y Electrónica,
Puebla, Mexico S. Casanova Institute of Nuclear Physics Polish Academy of
Sciences, PL-31342 IFJ-PAN, Krakow, Poland U. Cotti Universidad Michoacana
de San Nicolás de Hidalgo, Morelia, Mexico J. Cotzomi Facultad de Ciencias
Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico
S. Coutiño de León Instituto Nacional de Astrofísica, Óptica y Electrónica,
Puebla, Mexico E. De la Fuente Departamento de Física, Centro Universitario
de Ciencias Exactase Ingenierias, Universidad de Guadalajara, Guadalajara,
Mexico C. de León Universidad Michoacana de San Nicolás de Hidalgo, Morelia,
Mexico R. Diaz Hernandez Instituto Nacional de Astrofísica, Óptica y
Electrónica, Puebla, Mexico B.L. Dingus Physics Division, Los Alamos
National Laboratory, Los Alamos, NM, USA M.A. DuVernois Department of
Physics, University of Wisconsin-Madison, Madison, WI, USA M. Durocher
Physics Division, Los Alamos National Laboratory, Los Alamos, NM, USA J.C.
Díaz-Vélez Departamento de Física, Centro Universitario de Ciencias Exactase
Ingenierias, Universidad de Guadalajara, Guadalajara, Mexico R.W. Ellsworth
Department of Physics, University of Maryland, College Park, MD, USA K. Engel
Department of Physics, University of Maryland, College Park, MD, USA C.
Espinoza Instituto de Física, Universidad Nacional Autónoma de México, Ciudad
de Mexico, Mexico K.L. Fan Department of Physics, University of Maryland,
College Park, MD, USA M. Fernández Alonso Department of Physics,
Pennsylvania State University, University Park, PA, USA N. Fraija Instituto
de Astronomía, Universidad Nacional Autónoma de México, Ciudad de Mexico,
Mexico A. Galván-Gámez Instituto de Astronomía, Universidad Nacional
Autónoma de México, Ciudad de Mexico, Mexico J.A. García-González
Tecnologico de Monterrey, Escuela de Ingeniería y Ciencias, Ave. Eugenio Garza
Sada 2501, Monterrey, N.L., Mexico, 64849 F. Garfias Instituto de
Astronomía, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico
G. Giacinti Max-Planck Institute for Nuclear Physics, 69117 Heidelberg,
Germany M.M. González Instituto de Astronomía, Universidad Nacional Autónoma
de México, Ciudad de Mexico, Mexico J.A. Goodman Department of Physics,
University of Maryland, College Park, MD, USA J.P. Harding Physics Division,
Los Alamos National Laboratory, Los Alamos, NM, USA S. Hernandez Instituto
de Física, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico
B. Hona Department of Physics and Astronomy, University of Utah, Salt Lake
City, UT, USA D. Huang Department of Physics, Michigan Technological
University, Houghton, MI, USA F. Hueyotl-Zahuantitla Universidad Autónoma de
Chiapas, Tuxtla Gutiérrez, Chiapas, México P. Hüntemeyer Department of
Physics, Michigan Technological University, Houghton, MI, USA A. Iriarte
Instituto de Astronomía, Universidad Nacional Autónoma de México, Ciudad de
Mexico, Mexico A. Jardin-Blicq Max-Planck Institute for Nuclear Physics,
69117 Heidelberg, Germany Department of Physics, Faculty of Science,
Chulalongkorn University, 254 Phayathai Road, Pathumwan, Bangkok 10330,
Thailand National Astronomical Research Institute of Thailand (Public
Organization), Don Kaeo, MaeRim, Chiang Mai 50180, Thailand V. Joshi
Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-Universität
Erlangen-Nürnberg, Erlangen, Germany D. Kieda Department of Physics and
Astronomy, University of Utah, Salt Lake City, UT, USA A. Lara Instituto de
Geofísica, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico
W.H. Lee Instituto de Astronomía, Universidad Nacional Autónoma de México,
Ciudad de Mexico, Mexico J. Lee University of Seoul, Seoul, Rep. of Korea
H. León Vargas Instituto de Física, Universidad Nacional Autónoma de México,
Ciudad de Mexico, Mexico J.T. Linnemann Department of Physics and Astronomy,
Michigan State University, East Lansing, MI, USA A.L. Longinotti Instituto
Nacional de Astrofísica, Óptica y Electrónica, Puebla, Mexico Instituto de
Astronomía, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico
G. Luis-Raya Universidad Politecnica de Pachuca, Pachuca, Hgo, Mexico J.
Lundeen Department of Physics and Astronomy, Michigan State University, East
Lansing, MI, USA K. Malone Physics Division, Los Alamos National Laboratory,
Los Alamos, NM, USA V. Marandon Max-Planck Institute for Nuclear Physics,
69117 Heidelberg, Germany O. Martinez Facultad de Ciencias Físico
Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico J.
Martínez-Castro Centro de Investigación en Computación, Instituto Politécnico
Nacional, México City, México. J.A. Matthews Dept of Physics and Astronomy,
University of New Mexico, Albuquerque, NM, USA P. Miranda-Romagnoli
Universidad Autónoma del Estado de Hidalgo, Pachuca, Mexico J.A. Morales-Soto
Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico E. Moreno
Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de
Puebla, Puebla, Mexico M. Mostafá Department of Physics, Pennsylvania State
University, University Park, PA, USA A. Nayerhoda Institute of Nuclear
Physics Polish Academy of Sciences, PL-31342 IFJ-PAN, Krakow, Poland L.
Nellen Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de
Mexico, Ciudad de Mexico, Mexico M. Newbold Department of Physics and
Astronomy, University of Utah, Salt Lake City, UT, USA M.U. Nisa Department
of Physics and Astronomy, Michigan State University, East Lansing, MI, USA R.
Noriega-Papaqui Universidad Autónoma del Estado de Hidalgo, Pachuca, Mexico
L. Olivera-Nieto Max-Planck Institute for Nuclear Physics, 69117 Heidelberg,
Germany N. Omodei Department of Physics, Stanford University: Stanford, CA
94305–4060, USA A. Peisker Department of Physics and Astronomy, Michigan
State University, East Lansing, MI, USA Y. Pérez Araujo Instituto de
Astronomía, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico
E.G. Pérez-Pérez Universidad Politecnica de Pachuca, Pachuca, Hgo, Mexico
C.D. Rho University of Seoul, Seoul, Rep. of Korea Y.J. Roh University of
Seoul, Seoul, Rep. of Korea D. Rosa-González Instituto Nacional de
Astrofísica, Óptica y Electrónica, Puebla, Mexico E. Ruiz-Velasco Max-Planck
Institute for Nuclear Physics, 69117 Heidelberg, Germany H. Salazar Facultad
de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla,
Puebla, Mexico F. Salesa Greus Institute of Nuclear Physics Polish Academy
of Sciences, PL-31342 IFJ-PAN, Krakow, Poland Instituto de Física
Corpuscular, CSIC, Universitat de València, E-46980, Paterna, Valencia, Spain
A. Sandoval Instituto de Física, Universidad Nacional Autónoma de México,
Ciudad de Mexico, Mexico M. Schneider Department of Physics, University of
Maryland, College Park, MD, USA H. Schoorlemmer Max-Planck Institute for
Nuclear Physics, 69117 Heidelberg, Germany J. Serna-Franco Instituto de
Física, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico
A.J. Smith Department of Physics, University of Maryland, College Park, MD,
USA R.W. Springer Department of Physics and Astronomy, University of Utah,
Salt Lake City, UT, USA P. Surajbali Max-Planck Institute for Nuclear
Physics, 69117 Heidelberg, Germany M. Tanner Department of Physics,
Pennsylvania State University, University Park, PA, USA K. Tollefson
Department of Physics and Astronomy, Michigan State University, East Lansing,
MI, USA I. Torres Instituto Nacional de Astrofísica, Óptica y Electrónica,
Puebla, Mexico R. Torres-Escobedo Departamento de Física, Centro
Universitario de Ciencias Exactase Ingenierias, Universidad de Guadalajara,
Guadalajara, Mexico Tsung-Dao Lee Institute $\&$ School of Physics and
Astronomy, Shanghai Jiao Tong University, Shanghai, China R. Turner
Department of Physics, Michigan Technological University, Houghton, MI, USA
F. Ureña-Mena Instituto Nacional de Astrofísica, Óptica y Electrónica,
Puebla, Mexico L. Villaseñor Facultad de Ciencias Físico Matemáticas,
Benemérita Universidad Autónoma de Puebla, Puebla, Mexico T. Weisgarber
Department of Physics, University of Wisconsin-Madison, Madison, WI, USA E.
Willox Department of Physics, University of Maryland, College Park, MD, USA
H. Zhou Tsung-Dao Lee Institute $\&$ School of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai, China missing<EMAIL_ADDRESS>
###### Abstract
The highest-energy known gamma-ray sources are all located within 0.5 degrees
of extremely powerful pulsars. This raises the question of whether ultra-high-
energy (UHE; $>$ 56 TeV) gamma-ray emission is a universal feature expected
near pulsars with a high spin-down power. Using four years of data from the
High Altitude Water Cherenkov (HAWC) Gamma-Ray Observatory, we present a
joint-likelihood analysis of ten extremely powerful pulsars to search for
subthreshold UHE gamma-ray emission correlated with these locations. We report
a significant detection ($>$ 3$\sigma$), indicating that UHE gamma-ray
emission is a generic feature of powerful pulsars. We discuss the emission
mechanisms of the gamma rays and the implications of this result. The
individual environment, such as the magnetic field and particle density in the
surrounding area, appears to play a role in the amount of emission.
††journal: The Astrophysical Journal Letters
=500
Kelly Malone
## 1 Introduction
Ultra-high-energy (UHE; $>$ 56 TeV) gamma-ray emission can be created via
hadronic or leptonic processes. In the hadronic mechanism, a neutral pion
decays into two gamma rays. In the leptonic mechanism, a lower-energy photon
scatters off a relativistic electron via inverse Compton scattering. The
electron transfers part of its energy to the gamma ray, resulting in a higher-
energy photon.
Traditionally, it was thought that UHE gamma-ray sources would be hadronic in
nature, as leptonic emission is suppressed in this energy regime due to the
Klein-Nishina (KN) effect. However, present-day gamma-ray observatories have
the sensitivity required to detect leptonic sources above 56 TeV. See, for
example, the detections of the Crab Nebula as well as several known pulsar
wind nebulae (PWN) (Abeysekara et al., 2019; Amenomori et al., 2019;
Abeysekara et al., 2020; Abdalla et al., 2019). The astrophysical spectrum of
a leptonic source typically exhibits significant curvature due to the KN
effect (Moderski et al., 2005). Conversely, hadronic sources follow the
spectrum of their parent cosmic-ray population, which may or may not include a
cutoff or curvature.
The High Altitude Water Cherenkov (HAWC) Observatory is a gamma-ray
observatory with an wide instantaneous field-of-view ($\sim$2 steradians) and
sensitivity to energies between a few hundred GeV and a few hundred TeV. It is
sensitive to sources with declinations between -26 and +64 degrees (Smith,
2015; Abeysekara et al., 2017, 2019).
The first HAWC catalog of UHE sources (Abeysekara et al. (2020); hereafter
referred to as the “eHWC” catalog) contains nine sources emitting above 56
TeV, three of which continue above 100 TeV. The highest-energy sources all
exhibit curvature in the spectrum. Additionally, all nine sources are located
within 0.5 degrees of pulsars. For eight of the nine sources, the pulsar has
an extremely high spin-down power ($\dot{E}>10^{36}$ erg/s). This is much
higher than the number of high-$\dot{E}$ pulsars that would be expected to be
found near UHE gamma-ray sources (Abeysekara et al., 2020). Emission near
pulsars is expected to be powered by a PWN or TeV halo and is therefore
dominantly leptonic, even at the high energies studied here (Breuhaus et al.,
2020; Sudoh et al., 2021). While young pulsars are expected to have an
associated supernova remnant (SNR) with accompanying hadronic emission, there
has only been one detection of gamma-ray emission from an SNR to UHE, and
leptonic emission from this source has not been conclusively ruled out (Albert
et al., 2020a). SNR theory starts to run into technical problems accelerating
particles to these energies (Gabici, 2017). SNR acceleration to UHE energies
may not be possible (Zeng et al., 2021).
The proximity of these gamma-ray sources to the most powerful pulsars, along
with the curvature in their spectra, invites the question of whether UHE
gamma-ray emission is a generic feature expected near these sources. This is
investigated in this paper through a joint-likelihood analysis of pulsars with
$\dot{E}>10^{36}$ erg/s to search for subthreshold sources in the HAWC data.
While each source is too weak to be individually detected in HAWC’s standard
catalog search, analyzing the regions jointly may lead to a general detection
for this source class.
The paper is organized as follows: Section 2 describes the details of the
joint-likelihood method. Section 3 contains the results of the analysis.
Section 4 discusses implications of the results.
## 2 Analysis method
### 2.1 Joint-Likelihood Method
In this paper, we perform a joint analysis of several high-$\dot{E}$ pulsars.
The analysis uses a binned maximum-likelihood method that searches for a
gamma-ray excess above the background, using a simple model that describes the
UHE emission from pulsars based on various observables such as the distance
and pulsar age. This is performed using the HAWC Accelerated Likelihood
(HAL)111https://github.com/threeML/hawc_hal plugin to the Multi-Mission
Maximum Likelihood Framework (3ML) (Vianello et al., 2015). We describe the
background rejection and likelihood method in Abeysekara et al. (2019). HAWC’s
background comes from two main sources: the Galactic diffuse background
stemming from unresolved gamma-ray sources, and cosmic rays that are detected
by HAWC and survive gamma/hadron separation cuts. Note that neither background
is very large above 56 TeV. The Galactic diffuse background is largely due to
Inverse Compton scattering, which is not very prominent at these energies. The
fraction of cosmic rays surviving gamma/hadron separation cuts is also very
low: $\sim$0.001 (Abeysekara et al., 2017).
Data were collected over 1343 days between June 2015 and June 2019. The data
are binned in quarter-decade bins of reconstructed gamma-ray energy using the
“ground parameter” energy estimator, one of two energy estimators currently
used by HAWC (Abeysekara et al., 2019).
We use the last three quarter-decade energy bins from Abeysekara et al.
(2019), restricting the analysis to reconstructed energies between 56 and 316
TeV. These are the highest energies probed by HAWC. At these energies, there
is very little diffuse emission and less source confusion than at lower
energies.
The spectral model is a power law:
$\frac{dN}{dE}=A_{i}K\left(\frac{E}{E_{0}}\right)^{-\alpha},$ (1)
where $A_{i}$ accounts for the model-dependent relative flux of each source
(see Section 2.3), $K$ is the normalization, and $E_{0}$ is the pivot energy,
which is fixed at the center of each energy bin. The three values used for
$E_{0}$ in this analysis are 74.99 TeV, 133.35 TeV, and 237.14 TeV. Setting
the pivot energy at the center of each bin makes the analysis relatively
insensitive to the choice of $\alpha$, which is fixed at 2.5.
We fit each source (see Table 1) individually in each energy bin, placing a
step function at the boundaries of the bin to ensure that events that are mis-
reconstructed in energy are excluded. We assume the sources are spatially
extended with a Gaussian morphology. The width is fixed at $\sigma$=0.34∘. The
values of $\alpha$ and $\sigma$ are approximately the average values for the
highest-energy gamma-ray sources from the eHWC catalog (Abeysekara et al.,
2020).
To obtain a joint-likelihood result, the log-likelihood profiles for each
individual source, in each energy bin, are added and the value of $K$
(hereafter called $\hat{K}$) that optimizes this log-likelihood profile is
found. The total flux normalization, $\kappa$, for the ten sources combined is
then just:
$\kappa=\sum_{i=1}^{n_{sources}}\hat{K}A_{i}.$ (2)
Using this flux normalization, the total differential flux from all ten
sources is:
$\frac{dN}{dE}=\kappa\left(\frac{E}{E_{0}}\right)^{-\alpha}.$ (3)
We calculate a test statistic (TS) to show how significant the value of
$\kappa$ is:
$\mathrm{TS}=2\mathrm{ln}\frac{L_{S+B}(\kappa)}{L_{B}},$ (4)
where $L_{S+B}$ is the maximum likelihood for the signal-plus-background
hypothesis and $L_{B}$ is the likelihood for the background-only hypothesis.
After an overall best-fit value of $\kappa$ is determined, it is used as an
input to a Markov chain Monte Carlo and a distribution of the value is
determined. If the TS in a given energy bin is significant (TS $>4$), a
Bayesian credible interval (68$\%$ containment) is shown, obtained from the
estimated distribution of the parameter $\kappa$. The model only has one
degree of freedom, so using Wilks Theorem (Wilks, 1938), a TS value of 4
corresponds to 2$\sigma$. Otherwise, 95$\%$ credible interval quasi-
differential upper limit is plotted. A uniform prior is used in the Bayesian
analysis.
In the bins where an upper limit is determined, the range of expected upper
limits are obtained from simulations of Poisson-fluctuated background.
### 2.2 Source Selection and Dataset
Figure 1: $\sqrt{TS}$ for $\hat{E}>$ 56 TeV emission from HAWC, for the field-
of-view used in this analysis. We assume a point source morphology. The
locations of the ten sub-threshold pulsars used in this analysis are labeled.
For this analysis, we define the source list by selecting all pulsars from the
ATNF pulsar database, version
1.62222https://www.atnf.csiro.au/research/pulsar/psrcat/ (Manchester et al.,
2005) in the inner Galactic plane that are within HAWC’s field-of-view ($|b|<$
1∘, 5∘ $<l<$ 90∘) and have $\dot{E}>$ 1036 erg/s. There are 24 pulsars that
meet this criteria (see Table 1). Pulsars with a high $\dot{E}$ are centered
around the Galactic plane, so the choice to concentrate on $|b|<1^{\circ}$
only removes 3 additional pulsars from the analysis. We use this list of 24
pulsars to determine which models of emission are reasonable (see Section
2.3).
After making this determination, this list is then downselected to search for
sub-threshold gamma-ray sources. First, we remove all pulsars that are located
within a degree of sources from the eHWC catalog (Abeysekara et al., 2020).
This removes pulsars that already have significant UHE emission detected in
their vicinity. Since the gamma-ray emission is modeled as extended in nature,
this also removes pulsars whose associated emission may overlap with those
known sources, which would require more detailed modeling.
We remove three additional pulsars from the source selection. PSR J1813-1749
is removed because HAWC has added more data since the publication of
Abeysekara et al. (2020) and has detected a new UHE source (eHWC-J1813-176)
$\sim$0.2∘ away from the pulsar. PSR J1913+1011 and PSR J1930+1852 have been
removed because the known TeV emission in those regions is likely from a SNR,
not a PWN (Abdalla et al., 2018a, b). Since the majority of the emission at
these energies is expected to be from a PWN or TeV halo, this is done to
prevent the introduction of a different, predominantly hadronic source class
into the analysis.
The final list is composed of ten pulsars that are candidates for sub-
threshold gamma-ray emission. Figure 1 shows HAWC’s $>$ 56 TeV map with these
sources labeled.
PSR name | RA (∘) | Dec (∘) | Age ($\frac{P}{2\dot{P}}$) (kyr) | $\dot{E}$ ($\times$1036 erg/s) | Distance (kpc) | P (s) | $\dot{P}$ (ss-1) | Subthreshold?
---|---|---|---|---|---|---|---|---
B1800-21 | 270.96 | -21.62 | 15.8 | 2.2 | 4.40 | 0.134 | 1.35 $\times 10^{-13}$ | $\checkmark$
J1809-1917 | 272.43 | -19.29 | 51.3 | 1.8 | 3.27 | 0.083 | 2.55 $\times 10^{-14}$ |
J1811-1925 | 272.87 | -19.42 | 23.3 | 6.4 | 5.00 | 0.065 | 4.4 $\times 10^{-14}$ |
J1813-1749 | 273.40 | -17.83 | 5.6 | 56 | 4.70 | 0.045 | 1.27 $\times 10^{-13}$ |
J1826-1256 | 276.54 | -12.94 | 14.4 | 3.6 | 1.55 | 0.110 | 1.21 $\times 10^{-13}$ |
B1823-13 | 276.55 | -13.58 | 21.4 | 2.8 | 3.61 | 0.101 | 7.53 $\times 10^{-14}$ |
J1828-1101 | 277.08 | -11.03 | 77.1 | 1.6 | 4.77 | 0.072 | 1.48 $\times 10^{-14}$ | $\checkmark$
J1831-0952 | 277.89 | -9.87 | 128 | 1.1 | 3.68 | 0.067 | 8.32 $\times 10^{-15}$ | $\checkmark$
J1833-1034 | 278.39 | -10.57 | 4.85 | 34 | 4.10 | 0.062 | 2.02 $\times 10^{-13}$ | $\checkmark$
J1837-0604 | 279.43 | -6.08 | 33.8 | 2.0 | 4.77 | 0.096 | 45.1 $\times 10^{-15}$ |
J1838-0537 | 279.73 | -5.62 | 4.89 | 6.0 | 2.0333Pseudo-distance from Pletsch et al. (2012) | 0.146 | 4.72 $\times 10^{-13}$ |
J1838-0655 | 279.51 | -6.93 | 22.7 | 5.5 | 6.60 | 0.070 | 4.93 $\times 10^{-14}$ | $\checkmark$
J1844-0346 | 281.14 | -3.78 | 11.6 | 4.2 | 2.40444Pseudo-distance derived from Eq. 3 of Wu et al. (2018) | 0.113 | 1.55 $\times 10^{-13}$ |
J1846-0258 | 281.60 | -2.98 | 0.728 | 8.1 | 5.8 | 0.327 | 7.11 $\times 10^{-12}$ |
J1849-0001 | 282.23 | -0.02 | 42.9 | 9.8 | 7.0555Distance estimate from Gotthelf et al. (2011) | 0.039 | 1.42 $\times 10^{-14}$ |
J1856+0245 | 284.21 | 2.76 | 20.6 | 4.6 | 6.32 | 0.081 | 6.21 $\times 10^{-14}$ | $\checkmark$
J1907+0602 | 286.98 | 6.04 | 19.5 | 2.8 | 2.37 | 0.107 | 8.68 $\times 10^{-14}$ |
J1913+1011 | 288.33 | 10.19 | 169 | 2.9 | 4.61 | 0.036 | 3.37 $\times 10^{-15}$ |
J1928+1746 | 292.18 | 17.77 | 82.6 | 1.6 | 4.34 | 0.069 | 1.32 $\times 10^{-14}$ | $\checkmark$
J1930+1852 | 292.63 | 18.87 | 2.89 | 12 | 7.00 | 0.137 | 7.51 $\times 10^{-13}$ |
J1935+2025 | 293.92 | 20.43 | 20.9 | 4.7 | 4.60 | 0.080 | 6.08 $\times 10^{-14}$ | $\checkmark$
B1937+21 | 294.91 | 21.58 | 2.35e5 | 1.1 | 3.50 | 0.002 | 1.05$\times 10^{-19}$ | $\checkmark$
J2021+3651 | 305.27 | 36.85 | 17.2 | 3.4 | 1.8 | 0.104 | 9.57 $\times 10^{-14}$ |
J2022+3842 | 305.59 | 38.70 | 8.94 | 30 | 10.00 | 0.049 | 8.61 $\times 10^{-14}$ | $\checkmark$
Table 1: Information on the pulsars. All information comes from the ATNF
database, version 1.62 (Manchester et al., 2005) except for some distance
estimates which are not included in the pulsar database (see footnotes). Here,
$\dot{E}$ is the spin-down energy loss rate, $P$ is the barycentric period,
and $\dot{P}$ is the time derivative of the period. The checkmark in the last
column denotes the ten pulsars included in the sub-threshold analysis.
### 2.3 Models
The parameter $A_{i}$ in Equation 1 describes the relative contribution each
pulsar receives in the analysis. This parameter can be used to test different
models of gamma-ray emission near pulsars. For the models that rely on pulsar
parameters, the relevant quantities are taken from the ATNF pulsar catalog,
version 1.62 (Manchester et al., 2005). We consider a variety of different
models. In all descriptions, “emission” refers solely to gamma-ray emission
above 56 TeV.
* •
No model: Here, $A_{i}$ for all sources is set equal to 1. All sources are
treated equally and the emission is expected to be uncorrelated with pulsar
parameters such as distance.
* •
$\boldsymbol{1/d^{2}}$: In this model, $A_{i}$ is set to 1/$d^{2}$, where $d$
is the distance to the pulsar. This model assumes that closer sources will
produce observable emission.
* •
$\boldsymbol{\dot{E}/d^{2}}$: Here, the 1/$d^{2}$ model discussed above is
multiplied by the spin-down power of the pulsar. Therefore, closer, more-
energetic pulsars have more gamma-ray emission.
* •
Inverse age: In this model, $A_{i}$ is the inverse of the spin-down age. This
is defined as $P/(2\dot{P})$, where $P$ and $\dot{P}$ are the period and the
time derivative of the period, respectively. In this model, younger sources
are more likely to have detectable emission.
* •
Flux at 7 TeV: Here, $A_{i}$ is the HAWC flux at 7 TeV. This model assumes
that sources that are bright at multi-TeV energies should also give off
detectable emission above 56 TeV. The 0.5∘ extended source map from the third
catalog of HAWC sources (3HWC) (Albert et al., 2020b) is used to extract these
values. $A_{i}$ is computed by averaging the flux from all pixels within a 0.5
degree radius of the pulsar.
## 3 Results
We first run the joint-likelihood analysis (as described in Section 2) with
the full list of 24 pulsars to determine which models of emission are
reasonable. Table 2 gives the total TS for each scenario. For each model, we
can also calculate the expected gamma-ray flux above 56 TeV from an arbitrary
pulsar using Equation 1. The values of $K$, derived from Equation 2 are also
given in Table 2. In all cases, the TS is much higher than 5$\sigma$. Two
scenarios, the flux at 7 TeV and 1/$d^{2}$, perform better than the “no model
case”. The model based on the inverse age of the pulsar performs significantly
worse than the others, but the significance, $\sqrt{TS}$ is still well above
the 5$\sigma$ level.
Model | $K$ (56 $<$ E $<$ 100 TeV) | $K$ (100 $<$ E $<$ 177 TeV) | $K$ (177 $<$ E $<$ 316 TeV) | Units for $A_{i}$ | Total TS
---|---|---|---|---|---
No model | 2.55 $\times$ 10-16 | 4.65 $\times$ 10-17 | 9.46 $\times$ 10-18 | dimensionless | 633
1/$d^{2}$ | 2.44 $\times$ 10-16 | 4.50 $\times$ 10-16 | 8.12 $\times$ 10-17 | kpc-2 | 734
$\dot{E}/d^{2}$ | 4.06 $\times$ 10-13 | 6.88 $\times$ 10-14 | 1.04 $\times$ 10-14 | erg m-2 s-1 | 568
Inverse age | 6.36 $\times$ 10-13 | 1.10 $\times$ 10-13 | 1.65 $\times$ 10-14 | yr-1 | 152
Flux at 7 TeV | 5.42 $\times$ 10-3 | 9.11 $\times$ 10-4 | 1.90 $\times$ 10-4 | TeV-1 cm-2 s-1 | 886
Table 2: The proportionality constants needed to calculate the expected gamma-
ray flux near a given pulsar. The dimensions for $A_{i}K$ are energy-1
distance-1 time-1. With the value of $K$ known and $A_{i}$ in the units given
by the last column, the expected gamma-ray flux can easily be calculated using
Equation 1. $K$ is reported at the pivot energy in each bin. Proportionality
constants are desrived from the full sample of 24 pulsars.
Satisfied that the chosen models are adequate, we then run the joint-
likelihood analysis using the downselected list of ten pulsars (as described
in Section 2.2) to search for sub-threshold leptonic emission from pulsars.
Results are given in Table 3. Each of the models give a total TS value above
9, corresponding to a detection of more than 3$\sigma$. For some models, the
TS is much higher. The 1/$d^{2}$ model gives the highest TS: 41.3
(6.4$\sigma$). The same two models as before, 1/$d^{2}$ and the gamma-ray flux
at 7 TeV, perform better than the “no model” case, but in this case 1/$d^{2}$
performs the best, with a TS slightly higher than the “gamma-ray flux at 7
TeV” scenario.
Model | TS (56 $<$ E $<$ 100 TeV) | TS (100 $<$ E $<$ 177 TeV) | TS (177 $<$ E $<$ 316 TeV) | Total TS
---|---|---|---|---
No model | 27.9 | 8.33 | 1.59 | 37.9
1/$d^{2}$ | 31.9 | 9.08 | 1.29 | 41.3
$\dot{E}$/$d^{2}$ | 9.58 | 5.24 | 0.00 | 14.8
Inverse age | 9.19 | 3.79 | 0.03 | 13.0
Flux at 7 TeV | 26.3 | 9.61 | 3.62 | 39.6
Table 3: The test statistic for the joint-likelihood analysis for each model,
using the ten sub-threshold sources. Note that $\kappa$ is fit individually in
each energy bin so the last column is not a joint TS. Figure 2: The total
combined flux from the ten sub-threshold candidates for the scenario where the
pulsars have a relative contribution defined by 1/$d^{2}$, where $d$ is the
distance between the pulsar and the Earth. For the first two energy bins, the
TS $>$ 4 so Bayesian credible intervals (68$\%$ containment) are plotted. In
the last energy bin, there is no significant detection so a 95$\%$ upper limit
is plotted. The dark and light grey bands are 68$\%$ and 90$\%$ containment
for expected upper limits from Poisson fluctuated background. The dotted lines
are systematic uncertainties on the central value (i.e., the solid red line
may be as low or as high as the dotted line once systematic uncertainties are
included, see section 3.1)
Figure 2 shows the total flux from all ten sub-threshold candidates for this
best-case scenario. The figures for the other models can be seen in the
Appendix. Table 4, also located in the Appendix, gives the total flux
normalizations for each model.
The Appendix also includes a discussion of how often TS this high would be
expected from stacking random points on the sky.
### 3.1 Systematic Uncertainties
Systematic uncertainties are broken down into two categories: detector
systematics and modeling systematics. Detector systematics, described in
Abeysekara et al. (2019), stem from mis-modeling of detector quantities such
as the photomultiplier tube threshold and charge in simulated Monte Carlo
events. Each systematic is treated independently; the results are added in
quadrature to get a total uncertainty. This process is repeated in each energy
bin. Depending on the energy bin and model assumed, the detector systematic
ranges from 10$\%$ to 25$\%$.
Modeling systematic uncertainties investigate how the analysis would change if
the emission near the sub-threshold pulsars is different from what is assumed
in the main analysis. Several modeling systematics are considered:
* •
Spectral indices in the power law ($\alpha$ in Equation 1) of 2.0 and 3.0
* •
Replacing the power-law spectral model (Equation 1) with a power-law with an
exponential cutoff:
$\frac{dN}{dE}=A_{i}K\left(\frac{E}{E_{0}}\right)^{-\alpha}e^{-E/E_{cut}}.$
(5)
Here, $\alpha$ is fixed at 2.5 and $E_{cut}$ is fixed at 60 TeV, which are
average values from Abeysekara et al. (2020)
* •
Changing the Gaussian width to 0.23∘ and 0.45∘. These are $\pm$ 1 standard
deviation from the average extension of the sources from HAWC’s eHWC catalog
(Abeysekara et al., 2020).
The modeling systematics are larger than the detector systematics, driven
predominantly by the assumed source size. In the last energy bin, assuming a
power law with an exponential cutoff instead of a power law is also a dominant
effect.
Depending on energy and model, the modeling systematic ranges from 13$\%$ to
34$\%$. The sum of the detector and modeling systematics, added in quadrature,
are denoted with a dotted black line in Figure 2 and in Figures 3 through 5 in
the Appendix.
As described in Albert et al. (2020b), the absolute pointing uncertainty of
HAWC is declination-dependent and may be larger than previously thought;
perhaps as large as 0.3∘ at the edges of the field-of-view. This means that
the TS values presented in this paper may be underestimated; the size of this
effect is $\sim$10$\%$. The flux may also be underestimated. The flux
underestimate is a function of energy and ranges from 9$\%$ in the 56-100 TeV
bin, decreasing to a negligible amount (0.6$\%$) in the 177-316 TeV bin.
## 4 Discussion
Regardless of which model is used, it is clear that the areas around
high-$\dot{E}$ pulsars show hints of emitting at ultra-high energies ($>$ 56
TeV). Interestingly, some models based on the pulsar parameters, such as
$\dot{E}$ or inverse age, have much lower TS values than the “no model” case,
implying that they do not describe the emission well. Two models that perform
better than the “no model” case are the gamma-ray flux at 7 TeV and 1/$d^{2}$.
It is unclear at this time why some pulsar parameters do not seem to be good
predictors of emission. Some of these parameters have fairly large
uncertainties, which could be a contributing factor. For example, the
characteristic age of the pulsar can differ from the true age.
However, the uncertainties on the pulsar distances are relatively small. For
the ATNF pulsar database, 95$\%$ of pulsars will have a relative error of less
than a factor of 0.9 in their distance estimate (10$\%$ uncertainty) (Yao et
al., 2017).
Alternatively, the individual environment that each pulsar is in could play a
large factor in the amount of UHE emission. While the pulsar itself is the
particle accelerator, diffusion of electrons and positrons away from the
pulsar is strongly dependent on quantities such as the density and magnetic
field of the environment. Note that this may be only true for the
high-$\dot{E}$ pulsars studied here, which are relatively young; this means
the magnetic fields are likely to be affected by the pulsar age and SNR
interactions. For weaker, older pulsars, the magnetic field density and
environment are instead likely dominated by the ISM.
Also, the emission mechanisms are still uncertain. While it is commonly
assumed that the bulk of emission from the vicinity of a pulsar is from a PWN
and therefore predominantly leptonic, a hadronic contribution cannot be a
priori discarded. While emission from associated SNRs are unlikely, some have
raised the possibility of more exotic hadronic emission mechanisms in or near
PWN (Amato et al., 2003; Di Palma et al., 2017). Several of the HAWC sources
known to emit above 56 TeV, most notably eHWC J1908+063 and eHWC J1825-134,
have molecular clouds nearby (Duvidovich et al., 2020; Voisin et al., 2016).
These molecular clouds may be serving as a target for a portion of the gamma-
ray production.
Multi-wavelength and multi-messenger campaigns can help disentangle emission
mechanisms. A neutrino detection coincident with one of these pulsars would be
a smoking gun for hadronic emission mechanisms in or near PWN. However, a
recent stacked analysis looking for neutrino emission from PWN by IceCube did
not yield a detection (Aartsen et al., 2020).
Electromagnetic multi-wavelength studies could also be helpful. A leptonic
source emitting above 56 TeV will have a different signature at lower energies
than a hadronic one. For example, 100 TeV gamma rays imply a synchrotron peak
in the keV regime, assuming a 3 $\mu$G field (Hinton & Hofmann, 2009), with
the emission extending up to the MeV energy range. If the emission is instead
predominantly hadronic, there will be no such peak at these energies. Proposed
experiments such as AMEGO (McEnery et al., 2019) will be important in
distinguishing emission mechanisms.
## 5 Conclusions
In this study, we have searched for UHE gamma-ray emission in the vicinity of
pulsars with an $\dot{E}>10^{36}$ erg/s. We find, with high significance ($>$
3$\sigma$) , that UHE gamma-ray emission is a generic feature in the vicinity
of this class of pulsars. 1/$d^{2}$ is the model that gives the highest TS; a
source that is closer to us is more likely to have observed UHE emission.
Other pulsar parameters do not seem to be good predictors of emission. This
implies that the environments the pulsars are located in may play a role in
the amount of emission.
The TS values obtained are higher than would be expected from combining random
points in the Galactic plane. There is the possibility that all known gamma-
ray sources emit above this energy threshold, albeit at an extremely low flux.
Multimessenger and multiwavelength studies are needed to disentagle the origin
of the UHE emission from these pulsars.
We acknowledge the support from: the US National Science Foundation (NSF); the
US Department of Energy Office of High-Energy Physics; the Laboratory Directed
Research and Development (LDRD) program of Los Alamos National Laboratory;
Consejo Nacional de Ciencia y Tecnología (CONACyT), México, grants 271051,
232656, 260378, 179588, 254964, 258865, 243290, 132197, A1-S-46288,
A1-S-22784, cátedras 873, 1563, 341, 323, Red HAWC, México; DGAPA-UNAM grants
IG101320, IN111315, IN111716-3, IN111419, IA102019, IN110621; VIEP-BUAP; PIFI
2012, 2013, PROFOCIE 2014, 2015; the University of Wisconsin Alumni Research
Foundation; the Institute of Geophysics, Planetary Physics, and Signatures at
Los Alamos National Laboratory; Polish Science Centre grant,
DEC-2017/27/B/ST9/02272; Coordinación de la Investigación Científica de la
Universidad Michoacana; Royal Society - Newton Advanced Fellowship 180385;
Generalitat Valenciana, grant CIDEGENT/2018/034; Chulalongkorn University’s
CUniverse (CUAASC) grant. Thanks to Scott Delay, Luciano Díaz and Eduardo
Murrieta for technical support.
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### .1 Additional joint-likelihood results
Figures 3 through 5 are analogous to Figure 2 in the main text, but for the
other four models that have been investigated (described in Section 2.3). All
figures contain the combined flux for the joint-likelihood analysis of the ten
sub-threshold pulsars. Table 4 contains the total combined flux normalizations
for each of the models that have been considered.
Figure 3: Identical to Figure 2 from the main text, but for the “no model” case. Figure 4: Identical to Figure 2 from the main text, but for the $\dot{E}/d^{2}$ model. Figure 5: Identical to Figure 2 from the main text, but for the inverse age model. Figure 6: Identical to Figure 2 from the main text, but for the model defined by the gamma-ray flux at 7 TeV. Model | $\kappa$ (56 $<$ E $<$ 100 TeV) | $\kappa$ (100 $<$ E $<$ 177 TeV) | $\kappa$ (177 $<$ E $<$ 316 TeV)
---|---|---|---
No model | 8.47${{}_{-1.78}^{+1.88}}$ $\times$ 10-16 | 1.57${}_{-0.56}^{+0.63}$ $\times$ 10-16 | 7.56 $\times$ 10-17
1/$d^{2}$ | 8.38${}_{-1.72}^{+1.76}$ $\times$ 10-16 | 1.54${}_{-0.55}^{+0.60}$ $\times$ 10-16 | 7.07 $\times$ 10-17
$\dot{E}$/$d^{2}$ | 3.45${}_{-1.18}^{+1.27}$ $\times$ 10-16 | 7.41${}_{-3.24}^{+4.10}$ $\times$ 10-17 | 3.39 $\times$ 10-17
Inverse age | 3.99${}_{-1.39}^{+1.48}$ $\times$ 10-16 | 1.59 $\times$ 10-16 | 4.48 $\times$ 10-17
Flux at 7 TeV | 7.80${}_{-1.69}^{+1.78}$ $\times$ 10-16 | 1.56${}_{-0.55}^{+0.57}$ $\times$ 10-16 | 7.98 $\times$ 10-17
Table 4: The total combined flux normalization for the sub-threshold joint-
likelihood analysis for each of the models ($\kappa$ from Equation 3). The
units are TeV-1 cm-2 s-1. The flux normalization is reported at the pivot
energy, which is the center of each bin. Values without uncertainties are
upper limits, otherwise the values correspond to the 68$\%$ containment for
the Bayesian credible interval. Uncertainties are statistical only.
### .2 Testing of Random Backgrounds
We investigate how often randomly chosen non-source locations give TS values
as high as in the study presented in the main text. Sets of ten randomly
chosen points from the analysis region ($4^{\circ}$ $<l<$ 90∘;
$|b|<1^{\circ}$) are run through the joint-likelihood analysis. Points that
are within a degree of either a known high-energy source or any of the
selected ATNF pulsars are excluded. Because the gamma-ray emission is assumed
to be spatially extended, this is necessary to avoid contributions from the
known sources and the pulsars of interest.
This analysis is performed for 2,877 sets of ten random source locations. Only
five of these randomly chosen sets have a TS greater than 37.9. This is the TS
from the sub-threshold joint-likelihood “no model” case and is used as the
comparison because the other models require known pulsar information. This
means that the results that we have obtained are significant at the 3$\sigma$
level.
We also investigate sets of ten randomly chosen sources from HAWC’s third
catalog (3HWC) (Albert et al., 2020b) to see if any of HAWC’s previously
detected TeV sources emit at UHE. Once again, sources that are known to emit
above 56 TeV and sources within a degree of the ATNF pulsars used in the
nominal analysis are removed. Note that the majority of HAWC sources are
leptonic in origin (Linden et al., 2017).
Due to the relatively small number of remaining 3HWC sources after this
downselection (48 of the original 65 3HWC sources), it is not possible to run
thousands of sets of ten sources without repeating a large subset of the
sources. We instead run 100 sets of ten randomly chosen 3HWC sources.
None of these 100 trials have a TS above the value from the sub-threshold
joint-likelihood unmodeled case. The highest TS from this study is 26.9 and
the mean is 12.2 (standard deviation 5.5). This implies that known gamma-ray
sources that are not located near high-$\dot{E}$ pulsars are unlikely to emit
at high energies, or if they do, their fluxes are very low and combining ten
sources is not enough for a detection.
We also explore combining 20 3HWC sources at a time, instead of the ten that
were combined in the preceding paragraph. The entire TS distribution is
shifted to higher values. The highest TS is 45.9 (higher than in the “no
model” case); the mean is 24.6 and the standard deviation is 7.2. The higher
TS values when more 3HWC sources are combined implies that UHE emission may be
a generic feature of known gamma-ray sources, but with a very low flux. This
should be investigated, but may be hard to do with current-generation
experiments. Proposed experiments such as SWGO (Huentemeyer et al., 2019),
will be important here.
|
# Cross-domain Few-shot Learning with Unlabelled Data
Fupin Yao
iFlyTek-Surrey Joint Research Centre on Artificial Intelligence, UK
<EMAIL_ADDRESS>
###### Abstract
Few shot learning aims to solve the data scarcity problem. If there is a
domain shift between the test set and the training set, their performance will
decrease a lot. This setting is called Cross-domain few-shot learning.
However, this is very challenging because the target domain is unseen during
training. Thus we propose a new setting some unlabelled data from the target
domain is provided, which can bridge the gap between the source domain and the
target domain. A benchmark for this setting is constructed using DomainNet
[18]. We come up with a self-supervised learning method to fully utilize the
knowledge in the labeled training set and the unlabelled set. Extensive
experiments show that our methods outperforms several baseline methods by a
large margin. We also carefully design an episodic training pipeline which
yields a significant performance boost.
## 1 Introduction
Suppose you are given a project: given one labeled X-ray image for each kind
of several diseases, you are asked to diagnose diseases using X-ray images.
How will you solve this problem? Of course, you boss won’t be pleased if you
just say this is impossible. One solution is to utilize other labeled datasets
available, such as ImageNet or other medical images. Then you may try domain
adaptation or transfer learning methods. But wait, they will not work because
you can only touch very few images in the testing set. Domain adaptation and
transfer learning require a large amount of data otherwise the models will
overfit the very small dataset soon. Cross-domain few-shot learning aims to
solve this problem. However since the target domain is unseen during training,
you are given access to some unlabelled X-ray images of diseases which are
different from the diseases in the test set. This problem is called cross-
domain few-shot learning with unlabelled data. we will discuss it in detail in
this paper below.
Figure 1: We have 3 datasets in this problem: a labelled training set from the
source domain, an unlabelled training set from the target domain and a testing
set from the target domain [10] [1]
Deep learning become extremely successfully in the last decade. The image
recognition performance on ImageNet [5] has surpassed the performance of
humans []. Google and Apple built remarkable intelligent speech assistant
using speech recognition and speech synthesis technology. Self-driving cars
has been invested a lot and given high expectations. We can see them in the
near future if we are lucky. However, most successful deep learning systems
are supervised learning systems which means that they demand a large a mount
of labelled data. The consequence is that it will become extremely difficult
if you want to train your model on mobile devices and you can’t expect your
model to work well if labelling is too expensive.
Few-shot learning is proposed to alleviate the ”big data demand” problem in
deep learning and has achieved a lot during the last several years. In the
standard few-shot learning benchmark, there are only 1 or 5 images in each
class, and they are called one-shot learning and 5-shot learning respectively.
Most researchers address this few-shot problem using meta-learning. Meta-
learning, also known as learning to learn, aims to learn a model which learn
new skills or adapt to new environments rapidly [25]. The training set of
meta-learning includes lots of episodes each of which is a few shot learning
task. Lots of meta-learning algorithms [19] [6] [14] [4] has been proposed and
get better and better performance on several benchmarks. However, when the
test set comes from a different distribution, the performance of most meta-
learning algorithms will degrade a lot, which is first noticed by [4]. In
addition, the case where the test set comes from another distribution happens
a lot in the real world. When the model is deployed, we will always see the
testing samples vary greatly from the training distribution.
Cross-domain few-shot learning is dedicated to solve the above distribution
shift problem. There are several papers in this topic, and [21] [10] are
representatives of them. However, they assume there are several source
domains, which is a cumbersome requirement. And, with only 3-5 domains, there
is little hope that the model can generalize to another unseen domain. Most
importantly, it assumes you can’t touch the target domain which is not a valid
assumption and makes the problem extremely difficult to solve.
Cross-domain few-shot learning with unlabelled data is a new setting which is
proposed in this paper. A unlabelled dataset from the target domain is
provided to bridge the gap between the source domain and the target domain. In
most of the cases, unlabelled data is easy to collect since no labels are
required. In addition, the access to the target domain is not a problem in
most of real life problems. After some unlabelled data is added, we have a
better chance to solve the few shot learning problem in the target domain. So
this is a more realistic and feasible setting. We will show our arguments by
extensive experiments in later sections.
In summary, the contributions of this paper are as follows:
1\. We propose a more realistic and feasible setting.
2\. We construct a dataset for this setting so that algorithms under this
topic can be fairly compared.
3\. A novel self-supervised learning based algorithm is proposed for this
problem and it’s proven to be effective by extensive experiments.
## 2 Related work
### 2.1 Cross-domain few-shot learning
Cross-domain few-shot learning aims to deal with the domain shift problem in
few shot learning when the testing dataset is from another domain which is
different from the meta-training dataset. [21] first proposed a feature-wise
transformation architecture to solve this problem. A few feature-wise
transformation layers are inserted into the feature extractor to simulate
feature distributions extracted from the tasks in various domains. The whole
pipeline is a learning to learn approach and they hope to learn these hyper-
parameters of these feature-wise transformation layers. In the meta-testing
phase, they hope their models can generalize to unseen domains. [4] proposes a
more realistic setting where the domain shift between the training domains and
the testing domains are much larger. For example, the training images can be
natural images while the testing images can be medical images or satellite
images. They also proposed several baseline methods in the paper. Although
this is a more realistic setting, we claim that it’s too challenging since we
can’t tough the testing domains and those testing domain has a vast domain
shift from those training domains. The accuracies on medical image datasets
are very low.
### 2.2 Heterogeneous domain adaptation
Our problem setting is related to heterogeneous domain adaptation. Traditional
domain adaptation targets the cases where the testing data distribution is
different from the training domain. In heterogeneous domain adaptation
(classification), the label spaces of training set and the testing set are
also different. From example, in your testing datasets, you have some classes
which are unseen in the training dataset, which is very common in practice as
you can’t expect your classifier to work in the exact same environment when
you train it.
This is a less studied research topic and there are very few papers. [2]
proposes a method using a shared space between the source and target domain.
It trains the model with an unsupervised factorisation loss and a graph-based
loss. [7] adopts a deep clustering method for this problem.
### 2.3 Unsupervised domain adaptation for Person/object ReID
We also find our problem is related to unsupervised domain adaptation for
Person/object ReID. Researchers study the domain adaptation in person/object
ReID because it is observed that domain shift is also very common in
person/object re-identification where the testing images have very different
characteristics from the training images. For example, there maybe more
occlusions and worse illuminations and extreme weathers.
There are two types of methods in this field, domain translation based
methods,such as using [28] to translate images from the target domain to the
source domain and then train a model with translated images. Another type of
methods is pseudo-label based method in which clustering techniques are used
to extract pseudo-labels for the unlabeled target data and then unlabeled data
with pseudo-labels is used to train a model.
The person IDs in the target and source domain are disjoint, which is the same
as our problem. But unsupervised domain adaptation for person/object ReID is
targeted on person/object ReID only while are are studying a much broader
problem. In addition, they don’t have class labels but have person ids which
can be viewed as fine-grained class labels.
### 2.4 Self-supervised learning
Self-supervised learning becomes really hot these years which tries use the
supervision signals from the data itself instead of from labels. This reduce
the cost of labelling.
Contrastive learning is now widely used in self-supervised learning. In
contrastive learning we try to find positive and negative samples and force
the model to obtain a small distance between a reference sample and its
positive samples and big distance between the reference and negative sample.
For example, in SimCLR ([3]), augmented images of the same image are treated
as positive samples while other images are treated as negative samples. Using
this distance supervision signal we can learn meaningful representation even
though we may not have labels. Recent self-supervised methods are going to
surpass the performance of supervised methods. Our baseline method is based on
this type of method.
Context based self-supervised learning This type of self-supervised learning
utilize the information from the context of the current data. For example,
[15] predicts missing words according to neighboring words. [17] predicts the
relative positions of image blocks. [9] predicts the angles after images are
rotated. Our method is based this rotation based supervised learning method.
## 3 Cross-Domain Few-Shot Learning with unlabeled data
### 3.1 Motivation
Without touching the target domain, getting a good performance on the unseen
testing domain becomes extremely challenging. To bridge the gap between the
training and testing domain, we provide some unlabeled data from the target
domain during the training phase to make the problem less challenging. We have
this modification to the cross-domain few-shot learning for the following
reasons:
1\. Unlabeled data are easy to collect as you don’t need experts to label
them, Even for medical X-ray images, machine produce billions of images every
year and you don’t need doctors to label them one by one.
2\. Access to the target testing domain is easy to obtain in most cases. You
can easily collect data from the same domain where the target domain is from.
The only exception is the case where the data generation is very expensive.
For example, if you want to collect car accident images for auto-driving cars.
But these cases are very rare because the car accident rate is very low. We
don’t study this type of problem in this paper.
Therefore, our setting is more realistic as we have large domain shifts and
also more feasible since we have access to the target testing domain while
very less limitations in terms of applications.
### 3.2 Problem formulation
In our proposed setting, cross-domain few-shot learning with unlabeled data,
we have 3 datasets: $D_{train},D_{unlabelled},D_{D_{test}}$. $D_{train}$ is
the meta-training dataset from the training domain $A$, $D_{unlabelled}$ is
the unlabelled dataset from the testing domain $B$ and $D_{test}$ is the
testing dataset from the testing domain $B$. There three dataset also have
different label spaces: $L_{train},L_{unlabelled},L_{test}$.
Learning objective Our learning objective is get good performance on few-shot
tasks sampled from $D{test}$ after training models on $D_{train}$ and
$D_{unlabelled}$. Each few-shot task is 5-way 5-shot or 5-way 1-shot task same
as the standard few-shot learning. Here the number for shot means the number
of images in each class and the number of ways means the number of classes in
the task.
## 4 Proposed benchmark
### 4.1 Benchmark requirements
As we propose a new setting, we a new benchmark to compare the performance of
different algorithms. We have several requirements to satisfy for our setting:
1\. The domain shift between the testing dataset and the training dataset
should be large enough. This is to make our setting more realistic as you will
most always seen vast domain shift when your models are deployed.
2\. We should have multiple different domains to choose. The effectiveness of
a algorithm should not depend on the choice of domain and it should be
consistent across different domains. So we should have multiple different
domains.
3\. The number of images and the number of classes should not be too small or
large. If they are too small, the setting becomes less realistic. If they are
too large, most researchers will waste lots of computing resources which is
not worthwhile.
We have several choices at hand to choose. [10] proposed a benchmark where
there is only one source domain and four other testing domains which has
bigger and bigger domain shifts. The training dataset is ImageNet [5] and the
testing sets are crop disease dataset [16], Euro satellite dataset [11],
medical dataset with colors [20] and medical X-ray dataset [24]. We don’t
choose this because there is only one source dataset and some of those dataset
are too big.
[21] uses CUB [23], Cars [13], Places [27] and Plantae [22]. However, several
of these datasets are natural images and the domain shifts are not big enough
to be more realistic. And also, several of these datasets are too big.
In the end, we choose DomainNet [18]. This is one of the largest domain
adapation dataset. It has 6 domains and 345 classes for each domain. We
discard ’Infograph’ as it is too noisy and we can’t learn any effective
signals from it. We also discard ’quickdraw’ because images in it contain too
less information and it’s too challenging. Discarding these two domain is a
common practice in domain adaptation research and domain generalization [12].
In the end, we have 4 domain: real, painting, sketch and clipart. ’real’
contains natural images, ’painting’ are painting images, ’sketch’ includes
sketch images and ’clipart’ is a collection of clip art images. A quick
overview of DomainNet is in figure 2.
To keep the number of classes in each split as consistent as the standard few-
shot learning, we split the 345 classes as 64 classes for training, 261
classes for unlabelled dataset and 20 for the testing dataset.
Figure 2: DomainNet overview
## 5 Self-supervised learning for cross-domain few-shot learning
### 5.1 A baseline
Before we introduce our own method, we first introduce one of the baseline we
have experimented. We will use this as a baseline. The method is inspired by
([8]) The original method is designed for object re-identification and we re-
purpose it for our setting. First we use the clustering algorithm DBSCAN
(k-means can also be applied) to cluster the unlabeled target domain data.
Then we have 3 sets of samples: labeled source dataset, clustered unlabeled
target dataset and unclustered samples in the unlabeled target dataset. Many
methods in UDA Re-ID discard those unclustered data but we think it’s a waste
of data so we propose a unified framework to fully utilize all the data we
have:
$L_{f}=-log\frac{exp(<f,z^{+}>/\tau)}{\sum_{k=1}^{n^{s}}exp(<f,w_{k}>/\tau)+\sum_{k=1}^{n_{c}^{t}}exp(<f,c_{k}>/\tau)+\sum_{k=1}^{n_{o}^{t}}exp(<f,v_{k}>/\tau)}$
(1)
This is our loss function in which f is a feature vector of an image either
from the source dataset, clustered unlabeled target dataset or uncluster
outliers. $z^{+}$ indicates a positive prototype corresponding to $f$. $w_{k}$
is the centroid of all features in the class $k$, $c_{k}$ is the centroid of
all features in the cluster $k$ and $v_{k}$ is the $k^{th}$ outlier feature.
If f is from the source dataset, then $z^{+}=w_{k}$ is the centroid feature
vector of class k that f belongs to; if f is from the clustered dataset, then
$z^{+}=c_{k}$ is the centroid feature vector of the cluster $k$ that f belongs
to and if $f$ is from the unclusted outliers then $z^{+}=v_{k}$ is the outlier
feature that f belongs to.
Our algorithm alternates between updating the model using the above
comparative learning loss and producing pseudo-labels using clustering
methods. Detailed description is show in algorithm 1.
Result: Classification accuracy
Let N be the number of iterations for updating the model after each
clustering;
while _Traing_ do
Obtain pseudo-labels for clustered samples and unclustered samples using
DBSCAN algorithm with the unlabeled dataset;
$iter$ = 0;
while _$iter <N$_ do
$iter=iter+1$ ;
Updating the model using the loss function in formula 1.1 with all labeled
samples in the source dataset, clustered samples with their pseudo-labels and
unclustered samples;
Algorithm 1 Baseline
### 5.2 Self-supervised learning method
Inspired by recent progress in self-supervised learning and semi-supervised
learning [26], we proposed a new self-supervised learning method for our new
setting.
Rotation is a widely used technique in self-supervised learning. They first
rotation these images and then predict degrees of rotation. The intuition
behind this is that we should fully utilize the relative rotations among
images and use them as supervision signals which will greatly help us to learn
general image representation. These feature representations can be used for
any downstream tasks since they are not trained on any specific tasks during
self-supervised learning. All you need to do is fine-tuning these
representations.
Figure 3: Method overview: our method is based on rotation-based self-
supervised learning. We predict degrees for rotated images in both the labeled
training set and the unlabelled dataset and we also have another supervised
loss using class labels.
Our method adopts the rotation idea and works as following: we have two
training phases. The first one is feature extractor training and the second
one is episodic training. Two-phase training is a standard practice in few-
shot learning and it performs much better compared to one-phase training.
Phase 1: feature extractor training We rotate each image in the labeled
training set and the unlabelled dataset for 0, 90, 180 and 270 degrees and
then we use these degrees as labels to form the self-supervised loss. In
addition, since our labeled training set have labels, we have another
supervised loss. We train our model with these two loss functions together. In
this way, our model fully utilize the knowledge in both the labeled training
set and unlabelled set. Since the target domain has been seen during training,
we will seen a great performance boost during testing compare to cross-domain
few-shot learning without unlabelled data. The overview of feature extractor
training is in figure 3.
The self-supervised loss and supervised loss use the same feature extractor.
There is no need to have feature extractor with different parameters. Using
the feature extractor can reduce the number of parameters and make the feature
representation more generalized.
Phase 2: episodic training We follow the standard few-shot learning practice
to do the episodic training. We sample 1000 tasks in each epoch and train the
model for few hundreds of epochs. We use the mean centroid classifier, the
same as Prototypical neural networks [19]. In this way, our model learns to
deal with few shot tasks after only seeing a few training examples.
The whole pipeline is in algorithm 2.
Phase 1: feature extractor training
Let $N_{1}$ be the number of epochs;
$iter$ = 0;
while _$iter <N_{1}$_ do
Randomly sample images from the labeled set or the unlabelled dataset, rotate
them and get the self-supervised loss using rotation degrees as labels;
Use the above images from the labelled training set and class labels to form
the supervised loss;
Update the model using the two above losses;
$iter=iter+1$;
Phase 2: episodic traing Let $N_{2}$ be the number of epochs;
$iter$ = 0;
while _$iter <N_{2}$_ do
Randomly sample 5-way 1-shot and 5-way 5-shot tasks from the labeled training
set;
Use the tasks and the standard mean centroid classifier to train the model;
$iter=iter+1$;
Algorithm 2 Self-supervised learning for cross-domain few-shot learning with
unlabelled data
## 6 Experiments
### 6.1 Datasets
As dicussed in the section ’Proposed benchmark’, we use ’real’ as the source
domain where the training set belongs to and others as the target domains
which the unlabelled dataset and the testing dataset are sampled from. In each
domain, we have about 40000 images.
### 6.2 Model architecture and training details
There are four commonly used neural network architectures in standard few shot
learning: Conv4 (4 convolutional layers), ResNet 12, ResNet 18 and WideResNet.
The size of neural networks should not be too small as [4] points out that too
small neural networks can’t fully extract useful feature representations from
images and the differences among different architectures are much larger than
the differences among different few shot learning methods with the same
architecture. The size can’t also be too large. Otherwise they consume too
much resource to train models. So Resnet 18 is a good choice and we use it for
all of our experiments.
As mentioned earlier, we first train a feature extractor and then do the
episodic training. During the episodic training, we sample 100 hundreds
episodes for each epoch and train our model with 200 epochs. During testing,
to get accuracies with lower variance, we test our models on 10000 episodes.
### 6.3 Result
We compare our method with feature extractor trained on the ’real’ domain
(‘Backbone‘ in the table) and the constrative learning baseline (’Baseline’
and ’Baseline1’ in the table).
We can see from the table our method surpasses all the baselines on most of
target datasets and the margin is also big enough. This shows that our method
is very effective as we expected. Our result is shown in table 1.
| clipart | painting | sketch | average
---|---|---|---|---
Backbone | 65.17 | 59.13 | 53.62 | 59.31
Baseline | 66.18 | 58.60 | 56.38 | 60.39
Baseline1 | 69.78 | 60.94 | 58.76 | 63.16
Our method | 75.37 | 66.37 | 71.51 | 71.08
Table 1: Results of contrastive learning, ’Backbone’ means feature extractors
trained on the real domain without unlabeled data; ’Baseline’ is the
contrastive learning based approach we introduced earlier; ’Baseline1 is the
result after we increased the size of the unlabelled dataset. Our method uses
the increased dataset.’
### 6.4 Ablation study
The size of the unlabeled dataset We tried to investigate the effects of the
size of the unlabelled dataset as we have observed that the baseline method
performs better when the unlabelled dataset is bigger. The original unlabeled
target dataset has only about 2-3k images, much smaller than the source
dataset 20-40 k images.
We increase the size of unlabeled dataset for two reasons: First, in most
cases, unlabeled dataset are easy to build because no labels are needed and we
don’t need experts to label them. Second, most unsupervised learning methods
consume more samples compared to supervised learning methods. By increasing
the size of the unlabeled dataset, we have more room for these algorithms.
Our result is shown in table 1 after the size of unlabeled dataset is
increased. From the table we can see that the baseline shows a big improvement
on accuracies. However, our own method still outperforms this result by a
large margin.
## 7 Discussion and Future Work
In this paper, we have proposed a new new setting, which is cross-domain few-
shot learning with unlabelled data. We proved it’s significance and
importance. We hope more and more researchers to put more efforts on this
important research topic.
In addition, we established a new benchmark for this new research problem. Our
new benchmark is carefully designed and ideal for this problem. We hope it
will be widely adopted by the research community.
We also proposed an self-supervised learning base approach to fully utilize
the knowledge in the labelled training set and unlabelled dataset and bridge
the gap between the source domain and the target domain. Our experiments show
that our method is super effective and beat the baseline by a large margin.
While our method shows promising results,there is still large room to further
investigate the problem. For example, how much can the best self-supervised
learning algorithms help this problem? Does self-supervised learning really
reduce the domain gap or just learns more useful and generalized knowledge?
Nevertheless, we hope that this work inspires other researchers in the field
to continue exploring this problem.
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|
# Classifying Nearly Complete Intersection Ideals Generated in Degree Two
Charlie Miller<EMAIL_ADDRESS>and Branden Stone
<EMAIL_ADDRESS>
###### Abstract.
Nearly complete intersection ideals were introduced in [BS18] and defines a
special class of monomial ideals in a polynomial ring. These ideals were used
to give a lower bound of the total sum of betti numbers that appear a minimal
free resolution of a monomial ideal. In this note we give a graph theoretic
classification of nearly complete intersection ideals generated in degree two.
In doing so, we define a novel graph operation (the inversion) that is
motivated by the definition of this new class of ideals.
The first author was supported by McGowan Family Fund, a summer research award
for students at Hamilton College, Clinton, NY
## 1\. Introduction
Let $I$ be a homogeneous ideal in a polynomial ring $R$ over a field $\Bbbk$.
We denote the rank of the $i$-th free module in a minimal free resolution of
$R/I$ as $\beta_{i}(R/I)$. The long-standing conjecture of Buchsbaum-Eisenbud
[BE77] and Horrocks [Har79] states that if $I$ has height $c$, then
$\beta_{i}(R/I)\geqslant\binom{c}{i}.$
While the case when $c\geqslant 5$ is still open, the weaker statement
$\sum\beta_{i}(R/I)\geqslant 2^{c},$
known as the Total Rank Conjecture, has been completely solved for arbitrary
ideals (with char $k\not=2$) by M. Walker [Wal17]. At the same time, a special
case of this conjecture was independently shown for monomial ideals by A.
Boocher and J. Seiner [BS18]. In particular, they show that if $I$ is not a
complete intersection, then
$\sum\beta_{i}(R/I)\geqslant 2^{c}+2^{c-1}.$
In order to achieve this lower bound, the authors reduce to a special class of
ideals they define as nearly complete intersections (NCI) (see Definition
2.2). Our main theorem, Theorem 3.7, gives a complete characterization of NCI
ideals generated in degree 2 by examining the associated graph $G$. For
example, a squarefree monomial ideal $I$ generated in degree 2 is _not_ a
nearly complete intersection if $P_{5}$ is an induced subgraph of $G$. Section
2 gives the necessary background information. The main classification theorem
is proved in Section 3 as well as a new graph operation, the inversion of a
vertex (Definition 3.1), motivated by the definition of this new class of
ideals.
### Acknowledgments
This project is the result of undergraduate summer research supported by
Hamilton College. We are thankful to the faculty in the Math department for
their encouragement throughout the process. In particular, we are grateful to
Courtney Gibbons for her continual guidance and feedback. The first author
would also like to thank the organizers and speakers at the Thematic Program
in Commutative Algebra and its Interaction with Algebraic Geometry held at
Notre Dame in the summer of 2019. This workshop introduced the foundational
material relevant to this work.
## 2\. Preliminaries
Unless otherwise noted, we let $R=\Bbbk[x_{1},x_{2},\dots,x_{n}]$ be a
standard graded polynomial ring over a field $\Bbbk$ in $n$ variables. Given a
monomial ideal $I\subseteq R$, the support of $I$ (denoted $\text{Supp}(I)$)
will refer to the set of variables appearing in at least one minimal monomial
generator. The following fact about the support is helpful throughout the note
and we state it without proof.
###### Lemma 2.1.
Let $R=\Bbbk[x_{1},x_{2},\dots,x_{n}]$ and
$I=(m_{1},m_{2},\dots,m_{t})\subseteq R$ be a monomial ideal. If $m\in R$ is a
monomial such that
$\text{Supp}(m)\cap\left[\bigcup^{t}_{i=1}\text{Supp}(m_{i})\right]=\varnothing,$
then $\overline{m}\in R/I$ is a non-zero divisor.
Using notation defined in [BS18], $I(x=1)$ is the ideal defined by setting
$x=1$ for some variable $x$ in the support of $I$. As such, $I\subseteq
I(x=1)$. E.g., if $I=(ab,bc,ac)\subseteq\Bbbk[a,b,c]$, then $I(a=1)=(b,c)$ and
$I\subseteq I(a=1)$. The following was defined in [BS18] and is the main
object of study in this note.
###### Definition 2.2 ([BS18]).
A squarefree monomial ideal $I\subseteq R$ is a nearly complete intersection
if
1. (1)
it is generated in degree at least two,
2. (2)
is not a complete intersection, and
3. (3)
for each variable $x$ in the support of $I$, $I(x=1)$ is a complete
intersection.
For example, let $I=(ab,ac,bc)\subseteq\Bbbk[a,b,c]$. We see that $I$ is
generated in degree $2$ and is not a complete intersection. Further, for each
element of $\text{Supp}(I)$, $I(a=1)=(b,c)$, $I(b=1)=(a,c)$, and
$I(c=1)=(a,b)$ are complete intersections. Thus $I$ is a nearly complete
intersection.
The main result, Theorem 3.7, completely classifies the NCIs generated in
degree two via their associated graphs. Throughout, a finite graph $G$ is a
pair $G=(V(G),E(G))$ where $V(G)=\\{x_{1},x_{2},\dots,x_{n}\\}$ is the set of
vertices of $G$, and $E(G)$ is a collection edges of $G$ consisting of two
element subsets of $V(G)$. We will further assume all graphs are simple, i.e.
not allowing loops and multiple edges between vertices.
There exists a one-to-one correspondence between finite simple graphs and
monomial ideals generated degree two. In particular, given a graph $G$, the
edge ideal $I(G)$ is typically defined by
$I(G)=\left(x_{i}x_{j}\;|\;\\{x_{i},x_{j}\\}\in
E(G)\right)\subseteq\Bbbk[x_{1},x_{2},\dots,x_{n}],$
where $V(G)=\\{x_{1},x_{2},\dots,x_{n}\\}$. For bookkeeping reasons, we
slightly modify the standard definition of edge ideal to allow for singletons,
while at the same time preserving the one-to-one correspondence. In
particular, in this note the edge ideal of a graph $G$ is defined as
$I(G)=\left(x_{i}x_{j},x_{k}\;|\;\\{x_{i},x_{j}\\}\in E(G),x_{k}\in V(G)\text{
is a singleton}\right)\subseteq\Bbbk[x_{1},x_{2},\dots,x_{n}].$
For example, the graph $G$ below corresponds to the edge ideal
$I(G)=(ab,ac,bc,d)\subseteq\Bbbk[a,b,c,d]$.
$a$$b$$c$$d$$G$:
Abusing notation we will often refer to an element $u_{k}\in I(G)$ both as
$u_{k}=x_{i_{k}}x_{j_{k}}\in I(G)$ and $u_{k}=\\{x_{i_{k}},x_{j_{k}}\\}\in
E(G)$. Using this correspondence, we say a graph $G$ is a nearly complete
intersection (NCI) if the edge ideal $I(G)$ is a nearly complete intersection.
As such classifying the NCI graphs will in turn classify the NCI ideals
generated in degree two.
We end this section with a standard fact about graphs associated to complete
intersections.
###### Lemma 2.3.
Let $G$ be a simple graph and $I(G)\subseteq R$ be its edge ideal. Then $R/I$
is a complete intersection if and only if $G$ is a disjoint union of edges and
singletons.
###### Proof.
Assume that $E(G)=\\{u_{1},u_{2},\dots,u_{n}\\}$ and that $R/I(G)$ is a
complete intersection. Suppose to the contrary that there exists a vertex $v$
of degree 2 in $V(G)$. This implies that there exist edges $u_{i},u_{j}\in
E(G)$ such that $u_{i}\cap u_{j}=\\{v\\}$. Assuming $i=1$ and $j=2$, we have
that $\overline{u_{1}}\in R/(u_{2},\ldots,u_{n})$ is a zero-divisor. As such
there does not exist a vertex of degree two and $G$ must be a disjoint union
of edges and singletons.
Assume $G$ is a disjoint union of edges as well as singletons. With out loss
of generality, we can reduce to the case that $G$ does not contain any
singletons. Thus, the edge ideal of $G$ is
$I(G)=(x_{i}y_{i}\;|\;i=1,\ldots,n)$. Notice that
$\text{Supp}(x_{i}y_{i})\cap\left[\bigcup^{i-1}_{j=1}\text{Supp}(x_{j}y_{j})\right]=\varnothing$
for all $i=1,\ldots,n$. Therefore, $R/I(G)$ is a complete intersection by
Lemma 2.1. ∎
## 3\. Classifying NCIs
As mentioned in the previous section, a graph $G$ is an NCI if the edge ideal
$I(G)$ is a nearly complete intersection as defined in Definition 2.2. The
main result, Theorem 3.7, gives a complete classification of NCI ideals
generated in degree at most 2 using the above graph correspondence. Before we
can prove the result, we define a new graph operation necessary for the proof.
We denote the neighbors of a vertex $v$ in $V(G)$, $\operatorname{N}(v)$, and
the induced subgraph on a subset $V^{\prime}\subseteq V(G)$ as
$G[V^{\prime}]$.
###### Definition 3.1.
The inversion of a vertex $v$ in a graph $G$ is the graph defined by
$\mathscr{I}(v,G)=(V^{\prime},E^{\prime}),$
where $V^{\prime}=V\setminus\\{v\\}$ and
$E^{\prime}=E\left(G\left[V^{\prime}\setminus\mathrm{N}(v)\right]\right)$.
This operation is a direct translation of the operation $I(x=1)$ used in
Definition 2.2 (3) and is the main tool used in the classification of NCI
graphs. With it, we can further formalize NCI graphs with the following lemma
whose proof is a direct translation of definitions.
###### Lemma 3.2.
A graph $G$ is an NCI if and only if
1. (1)
$G$ is not a complete intersection, and
2. (2)
for each vertex $v\in V(G)$, $\mathscr{I}(v,G)$ is a complete intersection.
From this we have an immediate corollary.
###### Corollary 3.3.
NCI graphs are connected.
This corollary highlights the observations in Section 4 of [BS18]. In the next
example we can use Lemma 3.2 to determine if graphs are NCI or not.
###### Example 3.4.
Here we have a graph $G$ and two inversions at the vertices $c$ and $f$.
Notice that after the inversions we do not have a complete intersection (Lemma
2.3), hence $G$ is not an NCI since every inversion must be a complete
intersection (Lemma 3.2).
$G$$a$$b$$c$$d$$e$$f$$g$$\mathscr{I}(c,G)$$a$$b$$d$$e$$f$$g$$\mathscr{I}(f,G)$$a$$b$$c$$d$$e$$g$
Above shows that not all graphs are NCI. In fact the NCI property seems to be
quite rare. Below are examples of graphs that are NCI. Notice that any
inversion of a vertex will create a disjoint union of edges and singletons,
i.e. a complete intersection. Applying Lemma 3.2 shows they are NCI.
$C_{4}$$K_{5}$$C_{4}$$P_{3}$$S_{7}$$S_{7}$$S_{7}$$S_{7}$$S_{7}$$S_{7}$$S_{7}$
It’s natural to look at the families these graphs belong to. For example, the
family of paths are not all NCI. Indeed, if $n>4$, then the path $P_{n}$ is
not an NCI. To see this one only needs to invert a leaf of the graph and
notice the resulting graph is not a complete intersection, but another path
connecting at least three vertices. A similar result/argument holds for
cycles, i.e. if $n>5$, then a cycle $C_{n}$ is not an NCI. However, this is
not the case for complete graphs.
###### Proposition 3.5.
Any complete graph with more than 2 vertices is an NCI.
###### Proof.
Let $G=K_{n}$ be a complete graph on $n\geq 3$ vertices. If $v\in V=V(G)$, the
inversion of $v$ is given by
$\mathscr{I}(v,G)=\left(V^{\prime},E\left(G\left[V^{\prime}\setminus\operatorname{N}(v)\right]\right)\right),$
where $V^{\prime}=V\setminus\\{v\\}$. As $G$ is complete, we have that
$N(G)=V\setminus\\{v\\}=V^{\prime}$, and hence
$E\left(G\left[V^{\prime}\setminus\operatorname{N}(v)\right]\right)=\emptyset.$
Thus $\mathscr{I}(v,G)$ is a collection of singletons and hence a complete
intersection by Lemma 2.3. ∎
In the above path and cycle examples, we saw that the threshold for a graph to
be NCI was having $|V(G)|\leqslant 4$ and 5 respectively. It turns out that we
can explicitly state the NCI property for connected graphs with at most 4
vertices.
###### Proposition 3.6.
Let $G$ be a connected graph.
1. (1)
If $|V(G)|\leqslant 2$, then $G$ is not an NCI.
2. (2)
If $|V(G)|=3\text{ or }4$, then $G$ is an NCI.
###### Proof.
When $|V(G)|\leqslant 2$ the graph is a complete intersection by Lemma 2.3 and
hence cannot be an NCI by Lemma 3.2. When $|V(G)|=3\text{ or }4$, $G$ cannot
be a complete intersection due to the connected assumption, i.e. any vertex
$v\in V(G)$ must be connected to at least one other vertex. Thus
$\mathscr{I}(v,G)$ has at most one edge and is a complete intersection. This
forces $G$ to be an NCI. ∎
We are now ready to prove the main classification theorem. In the theorem, we
define the graph $T$ as the following.
$T$$v_{1}$
This graph, along with $P_{5}$, become the major obstructions to the NCI
property.
###### Theorem 3.7.
Let $G$ be a connected graph with $|V(G)|\geqslant 5$. The graph $G$ is not an
NCI if and only if there exist vertices $v_{1},v_{2},v_{3},v_{4},v_{5}\in
V(G)$ such that the following conditions hold:
1. (1)
the vertex $v_{1}$ is a leaf in $G[v_{1},v_{2},v_{3},v_{4},v_{5}]$;
2. (2)
the path $P_{5}$ or $T$ is a spanning tree of
$G[v_{1},v_{2},v_{3},v_{4},v_{5}]$ where the neighbors of $v_{1}$ all have
degree 2 in the spanning tree.
###### Proof.
Assume $G$ is not an NCI. As such, there exists $v\in V(G)$ such that
$\mathscr{I}(v,G)$ is not a complete intersection. In particular,
$\mathscr{I}(v,G)$ has a vertex $w\in V^{\prime}=V\setminus\\{v\\}$ of degree
two. As $G$ is connected, there must exist a path from $v$ to $w$ that passes
through the neighbors of $v$ in $G$. So there exists
$v_{2}\in\operatorname{N}_{G}(v)$ such that the path
(1) $v\longrightarrow v_{2}\longrightarrow
v_{3}\longrightarrow\cdots\longrightarrow w$
exists in $G$. Without losing generality, we can assume the vertices in the
path from $v_{3}$ to $w$ (inclusive) avoid $\operatorname{N}_{G}(v)$. Indeed
if there was a vertex $u\in\operatorname{N}_{G}(v)$ between $v_{3}$ and $w$,
we could replace $v_{2}$ with $u$, shortening the path. As such, we may assume
the path from $v_{3}$ to $w$ is completely contained in the subgraph
$G[V^{\prime}\setminus\operatorname{N}_{G}(v)]\subset G$. We now consider two
cases, $v_{3}=w$ and $v_{3}\not=w$, which can be visualized in the following
abstract representation of $G$.
$v$$v_{2}$$v_{3}$$w^{\prime}$$w$$w^{\prime\prime}$$\operatorname{N}_{G}(v)$III$G[V^{\prime}-\operatorname{N}_{G}(v)]$
### Case I
Assume $v_{3}=w$. Since $w$ is a degree two vertex in $\mathscr{I}(v,G)$,
there exist $w^{\prime},w^{\prime\prime}\in V^{\prime}$ such that
$ww^{\prime},ww^{\prime\prime}\in
E^{\prime}=E\left(G\left[V^{\prime}\setminus\mathrm{\operatorname{N}}_{G}(v)\right]\right)$.
In particular $w^{\prime},w^{\prime\prime}\notin\operatorname{N}_{G}(v)$. As
such, $v$ is a leaf in the induced subgraph
$H=G[v,v_{2},w,w^{\prime},w^{\prime\prime}]$, and by construction, $T$ is a
spanning tree of $H$ where $v_{2}$ is the only neighbor of $v$. Further, the
degree of $v_{2}$ is two in the spanning tree $T$, thus both of the desired
conditions are satisfied.
### Case II
Assume $v_{3}\not=w$. As $v_{3}$ and $w$ are distinct, we can reduce to the
case where there is a single vertex between them on the path (1), say
$w^{\prime}$. As $v_{3},w^{\prime},w\notin\operatorname{N}_{G}(v)$, we have
that $v$ is a leaf in the induced subgraph $H=G[v,v_{2},v_{3},w^{\prime},w]$.
In this case, by construction, $P_{5}$ is a spanning tree of $H$ where $v_{2}$
is the only neighbor of $v$. As the degree of $v_{2}$ is two in the spanning
tree $P_{5}$, we have our desired result.
Conversely, assume the conditions hold for a graph $G$ that is NCI. In this
scenario, there exists vertices $v_{1},v_{2},v_{3},v_{4},v_{5}\in V(G)$ such
that $v_{1}$ is a leaf in the induced subgraph
$G[v_{1},v_{2},v_{3},v_{4},v_{5}]$. In the situation where $P_{5}$ is a
spanning tree of $G[v_{1},v_{2},v_{3},v_{4},v_{5}]$, we can assume the vertex
labels of the path are as follows.
$P_{5}$:$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$
Since $v_{1}$ is a leaf in the induced subgraph, we know that
$v_{3},v_{4},v_{5}\notin\operatorname{N}_{G}(v)$. Hence the degree of $v_{4}$
is at least two in $\mathscr{I}(v_{1},G)$. This shows that
$\mathscr{I}(v_{1},G)$ is not a complete intersection, a contradiction of
Lemma 3.2. A similar argument holds for when $T$ is a spanning tree of
$G[v_{1},v_{2},v_{3},v_{4},v_{5}]$. ∎
Theorem 3.7, together with Proposition 3.6 give a complete classification of
NCI graphs. As a result, we have a graph theoretic classification of NCI
ideals generated in degree two. A natural desire is to extend this result to
NCIs with generators in higher degrees. One direction to consider is
classifying these ideals with hypergraphs. A Hypergraph is a pair $G=(V,E)$
where $V$ is the set of vertices of $G$ and the set of edges $E$ is a set of
nonempty subsets of $V$. In this scenario, more than two vertices can be
incident to a single edge. As with graphs, a similar correspondence exits
between hypergraphs and ideals and can be seen in the following example.
###### Example 3.8.
The left image below is an example of an NCI hypergraph $G$ on a vertex set
$V(G)=\\{a,b,c,d,e,f,g\\}$. Notice this hypergraph has three edges,
$\\{a,b,c\\}$, $\\{g\\}$, and $\\{d,e,f\\}$. Further, there is a natural
correspondence between these hypergraphs and monomial ideals in
$\Bbbk[a,b,c,d,e,f,g]$. In particular $I(G)$ is listed below $G$.
$a$$b$$c$$g$$d$$e$$f$$I(G)=(abc,def,ag,bg,cg,dg,eg,fg)$$b$$c$$g$$d$$e$$f$$\mathscr{I}(a,G)$
is CI
Lemma 3.2 can also be extended to this scenario as well as the definition of
inversion. Notice that inverting $a$ (or any vertex) will produce the complete
intersection on the right. It is worth noting that all the examples of NCI
hypergraphs we were able to construct were related to the above example. This
hints at the possibility that all higher degree NCI ideals are related to the
above hypergraph.
We end this section with an observation relating to the original result of
[BS18]. Let $I$ be a height $c$ monomial ideal in a polynomial ring $S$ that
is not a complete intersection. A. Boocher and J. Seiner show that
$\sum\beta_{i}(S/I)\geqslant 2^{c}+2^{c-1}$. In particular, equality holds if
and only if the generating function for $\beta_{i}(S/I)$ is either
$(1+3t+2t^{2})(1+t)^{c-2}\text{ or }(1+5t+5t^{2}+t^{3})(1+t)^{c-3}.$
When $c=2$ or $3$, respectively, the generating functions are defined by
ideals with the betti sequence $\\{1,3,2\\}$ and $\\{1,5,5,1\\}$,
respectively. We are able to retrieve these sequences from the obstructions
noted in Theorem 3.7. Indeed, the edge ideal $I(P_{5})$ and $I(T)$ both have
the betti sequence $\\{1,4,4,1\\}$. However, if we connect the end points of
the path $P_{5}$ to create a 5-cycle, the betti sequence becomes
$\\{1,5,5,1\\}$. Similarly, removing a leaf of either $P_{5}$ or $T$ can
create the path $P_{4}$, obtaining the betti sequence $\\{1,3,2\\}$.
## References
* [BE77] David A Buchsbaum and David Eisenbud. Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3. American Journal of Mathematics, 99(3):447–485, 1977.
* [BS18] Adam Boocher and James Seiner. Lower bounds for Betti numbers of monomial ideals. J. Algebra, 508:445–460, 2018.
* [Har79] Robin Hartshorne. Algebraic vector bundles on projective spaces: a problem list. Topology, 18(2):117–128, 1979.
* [Wal17] Mark E. Walker. Total Betti numbers of modules of finite projective dimension. Ann. of Math. (2), 186(2):641–646, 2017.
|
# Soliton Solutions to the Curve Shortening Flow on the 2-dimensional
hyperbolic plane
Fábio Nunes da Silva111Universidade de Brasília, Department of Mathematics,
70910-900, Brasília-DF, Brazil<EMAIL_ADDRESS>Keti Tenenblat222
Universidade de Brasília, Department of Mathematics, 70910-900, Brasília-DF,
Brazil<EMAIL_ADDRESS>Partially supported by CNPq Proc.
312462/2014-0, Ministry of Science and Technology, Brazil and FAPDF/Brazil
grant 0193.001346/2016.
###### Abstract
We show that a curve is a soliton solution to the curve shortening flow if and
only if its geodesic curvature can be written as the inner product between its
tangent vector field and a fixed vector of the 3-dimensional Minkowski space.
We use this characterization to provide a qualitative study of the solitons.
We show that for each fixed vector there is a 2-parameter family of soliton
solution to the curve shortening flow on the 2-dimensional hyperbolic space.
Moreover, we prove that each soliton is defined on the entire real line, it is
embedded and its geodesic curvature converges to a constant at each end.
_Keywords_ : Curve shortening flow; solitons solutions.
## 1 Introduction
A family of curves $\hat{X}^{t}:I\longrightarrow M$, $t\in[0,T)$, on a
2-dimensional Riemannian manifold $M^{2}$ is said to be a solution to the
Curve Shortening Flow (CSF) with initial condition
$\hat{X}^{0}(\cdot)=X(\cdot)$, if it satisfies the following equation
$\left\\{\begin{array}[]{ll}\displaystyle{\frac{\partial}{\partial
t}\hat{X}^{t}(\cdot)}=\hat{k}^{t}(\cdot)\hat{N}^{t}(\cdot)\\\
\hat{X}^{0}(\cdot)=X(\cdot),\end{array}\right.$ (1.1)
where $\hat{k}^{t}(\cdot)$ is the geodesic curvature and $\hat{N}^{t}(\cdot)$
is the unit vector field normal to $\hat{X}^{t}(\cdot)$ for each $t\in[0,T)$.
Epstein and Gage [6] showed that when $M^{2}=\mathbb{R}^{2}$, the CSF is
geometrically the same if tangential components are added to the right hand
side of the differential equation (1.1). Therefore, one can define that a
1-parameter family of curves $\hat{X}^{t}:I\rightarrow\mathbb{R}^{2}$,
$t\in[0,T)$ is a solution to the CSF in $\mathbb{R}^{2}$ with initial
condition $\hat{X}^{0}(\cdot)=X(\cdot)$, if it satisfies
$\left\langle\displaystyle\frac{\partial}{\partial
t}\hat{X}^{t}(\cdot),\hat{N}^{t}(\cdot)\right\rangle=\hat{k}^{t}(\cdot),$
(1.2)
where $\langle\cdot,\cdot\rangle$ is the cannonical inner product on
$\mathbb{R}^{2}$, $\hat{k}^{t}(\cdot)$ is the curvature and
$\hat{N}^{t}(\cdot)$ is the unit vector field normal to $\hat{X}^{t}(\cdot)$
for each $t\in[0,T)$. The name curve shortening flow is justified by the fact
that when the curves of the family $\hat{X}^{t}$ are closed, then the length
of the curves decreases along the flow, i.e., it is a gradient type of flow
for the arc length functional. Grayson [12] observed that the CSF is also
known as the curvature flow or the heat flow for isometric immersions.
According to Epstein and Gage [6], the original motivation for studying
equation (1.1) was to find a new and maybe more natural proof of the existence
of closed geodesics on Riemannian manifolds. However, the first results in
this direction were obtained by Grayson [13] em 1989. But equation (1.1) for
the Euclidean plane was investigated earlier by several authors [2], [7], [8],
[9], [10] e [12].
An important class of solutions to the CSF are those that evolve by isometries
or homotheties. Such solutions are called self-similar solutions and solitons
if they evolve just by isometries. On the Euclidean plane, the straight lines
are not affected by the flow and they are considered to be trivial solutions.
Circles evolve homothetically to a point in finite time. The Grim Reaper curve
given by the graph of the function $f(s)=ln(cos(s))$ evolves by a flow of
translations. Giga [11] proved that this is the unique curve on the plane that
evolves by translations. An example of a plane curve that evolves by
isometries of the plane is the yin-yang spiral. Abresch-Langer [2] and
Epstein-Weinstein [7] investigated the closed curves, not necessarily simple,
that evolve by homotheties. Halldorsson [14] concluded the description of all
self-similar solutions on the plane.
Some authors studied classes of solutions to the CSF on the plane and proved
that after a certain time the flow evolves into a self-similar one. First,
Gage [8] [9] showed that closed convex curves on $\mathbb{R}^{2}$ evolve into
circular curves after a certain time. Then Gage and Hamilton [10] showed that
convex closed curves collapse into a point. Grayson [12] proved that closed
embedded curves evolve to circular curves and then they collapse into a point
at a finite time. Moreover, Angenent [3], under more general conditions,
proved that the CSF evolves in a sense into a self-similar flow, showing the
importance of self-similar solutions.
One should point out that, when the ambient space is not the Euclidean plane,
there are very few results on self-similar solutions to the CSF. In 2015,
Halldorsson [15] classified all the self-similar solutions on the Minkowski
plane. Dos Reis and Tenenblat [5] caracterized and described all the soliton
solutions on the sphere $\mathbb{S}^{2}$. Some results on the CSF for
Riemannian manifolds different from the plane can be found in [10], [13], [16]
and [20], among others. Moreover, Angenent, [4] studied the topology of the
closed geodesics on compact surfaces by using the CSF.
In this paper, we will study the soliton solutions of curve shortening flow on
the 2-dimensional hyperbolic space $\mathbb{H}^{2}\subset\mathbb{R}_{1}^{3}$,
where $\mathbb{R}_{1}^{3}$ is the 3-dimensional Minkowski space.
## 2 Main Results
We consider the 3-dimensional Minkowki space as
$\mathbb{R}_{1}^{3}=(\mathbb{R}^{3},\langle,\rangle)$, where $\mathbb{R}^{3}$
is the 3-dimensional vector space and $\langle,\rangle$ is the Minkowski
metric defined by
$\langle u,v\rangle=-u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}.$
Let $\displaystyle
X:I\subset\mathbb{R}\rightarrow\mathbb{H}^{2}\subset\mathbb{R}_{1}^{3}$ be a
regular curve parametrized by arc length $s$. We denote by
$T(s)=X^{\prime}(s)$ the tangent vector field, $N(s)=X(s)\times T(s)$ the unit
normal vector field and $k(s)=\langle T^{\prime}(s),N(s)\rangle$ the geodesic
curvature of $X$. A one parameter family of curves
$\displaystyle\hat{X}:I\times J\rightarrow\mathbb{H}^{2}$ is called a curve
shortening flow (CSF) with initial condition $X$, if
$\left\\{\begin{array}[]{ll}\displaystyle\left\langle\frac{\partial}{\partial
t}\hat{X}(s,t),\hat{N}(s,t)\right\rangle=\hat{k}(s,t),\\\
\hat{X}(s,0)=X(s),\end{array}\right.$ (2.1)
where $\hat{k}^{t}(\cdot)=\hat{k}(\cdot,t)$ is the geodesic curvature and
$\hat{N}^{t}(\cdot)=\hat{N}(\cdot,t)$ is the unit normal vector field of
$\hat{X}^{t}(\cdot)=\hat{X}(\cdot,t)$. When $X$ is a geodesic i.e. $k=0$, then
the family $\hat{X}^{t}(s)=X(s)$ gives a trivial solution to the CSF. Our goal
is to study the case when $\hat{X}^{t}(s)$ evolves by a 1-parameter family of
isometries of $\mathbb{H}^{2}$.
###### Definition 2.1.
Let $\displaystyle\hat{X}:I\times
J\rightarrow\mathbb{H}^{2}\subset\mathbb{R}_{1}^{3}$ be a solution to the
curve shortening flow (2.1) on $\mathbb{H}^{2}$, with initial condition
$\displaystyle X:I\rightarrow\mathbb{H}^{2}$. We say that $X$ is a soliton
solution to the curve shortening flow if there is a 1-parameter family of
isometries $M(t):\mathbb{H}^{2}\rightarrow\mathbb{H}^{2}$ such that $M(0)=Id$
and
$\hat{X}^{t}(s)=M(t)X(s)$
for all $t\in J$, where $Id$ is the identity map.
We remark that an isometry of $\mathbb{H}^{2}$ is an element of the Lie group
$O_{1}(3)=\\{M\in GL(3,\mathbb{R}):M^{T}\epsilon M=\epsilon\\}$ that preserves
$\mathbb{H}^{2}$, where $M^{T}$ is the transpose of $M$ and
$\epsilon=\left(\begin{array}[]{lll}-1&0&0\\\ 0&1&0\\\
0&0&1\end{array}\right).$
###### Theorem 2.2.
Let $\displaystyle X:I\rightarrow\mathbb{H}^{2}$ be a regular curve
parametrized by arc length. Then $X(s)$, $s\in I$, is a soliton solution to
the curve shortening flow if, and only if, there is a vector
$v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$ such that
$\langle T(s),v\rangle=k(s),$ (2.2)
where $T(s)$ is the unit tangent vector field and $k(s)$ is the geodesic
curvature of $X$.
We observe that when $X$ is a geodesic of
$\mathbb{H}^{2}\subset\mathbb{R}^{3}_{1}$, then it is a planar curve and hence
there exists a vector $v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$ such that
$\displaystyle\langle T,v\rangle=0.$
The following theorem describes the qualitative behaviour of the soliton
solutions to the CSF in $\mathbb{H}^{2}.$
###### Theorem 2.3.
For any $v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$, there is a 2-parameter
family of non-trivial soliton solutions to the curve shortening flow on the
2-dimensional hyperbolic space. Each soliton solution is an embedded curve
$X(s)$ on $\mathbb{H}^{2}$, defined for all $s\in\mathbb{R}$. Moreover, at
each end, the curvature function $k(s)$ tends to one of the following
constants $\\{-1,0,1\\}$.
## 3 Proofs of the main results
In this section we prove our main results.
Proof of Theorem 2.2. Suppose that $X(s)$ is parametrized by arc length $s$.
If $X$ is a soliton solution to the CSF, then
$\displaystyle\hat{X}^{t}(s)=M(t)X(s)$ is solution to (2.1), where $M(t)$ is a
family of isometries of $\mathbb{H}^{2}$.Taking the derivative of
$\displaystyle\hat{X}(s,t)$ at $t$, we have
$\frac{\partial}{\partial t}\hat{X}(s,t)=M^{\prime}(t)X(s).$
It follows from definition of the CSF that
$\hat{k}(s,t)=\left\langle\frac{\partial}{\partial
t}\hat{X}(s,t),\hat{N}(s,t)\right\rangle=\left\langle
M^{\prime}(t)X(s),M(t)N(s)\right\rangle.$
In particular, for $t=0$, we have
$k(s)=\langle M^{\prime}(0)X(s),N(s)\rangle.$
$M^{\prime}(0)$ is an element of the Lie algebra $\mathfrak{o}_{1}(3)$ of the
Lie group $O_{1}(3)$. Let ${A_{1},A_{2},A_{3}}$ be a basis of
$\mathfrak{o}_{1}(3)$, where
$\displaystyle A_{1}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\
0&-1&0\end{array}\right),\,\,\,A_{2}=\left(\begin{array}[]{ccc}0&0&1\\\
0&0&1\\\
1&-1&0\end{array}\right)\,\,\,\,\text{e}\,\,\,\,\,A_{3}=\left(\begin{array}[]{ccc}0&1&0\\\
1&0&0\\\ 0&0&0\end{array}\right).$
Then $M^{\prime}(0)=c_{1}A_{1}+c_{2}A_{2}+c_{3}A_{3}$, for real numbers
$c_{i}$.
By simple computations, we can prove that $\displaystyle\langle
M^{\prime}(0)X(s),N(s)\rangle=\langle T(s),v\rangle$, where $X(s)\times
N(s)=-T(s)$ and $v=(c_{1}+c_{2},c_{2},-c_{3})$. Therefore,
$k(s)=\langle T(s),v\rangle.$
Conversely, let $X(s)$ be a curve in $\mathbb{H}^{2}\subset\mathbb{R}^{3}_{1}$
parametrized by arc length $s$, such that $\langle T(s),v\rangle=k(s)$ for a
vector $v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$. Without loss of generality,
up to isometries of $\mathbb{H}^{2}$, we can consider $v$ to be a multiple of
$w_{1}=(1,0,0)$ if $v$ is a timelike vector, a multiple of $w_{2}=(1,1,0)$ if
$v$ is a lightlike vector and a multiple of $w_{3}=(0,0,-1)$ if $v$ is a
spacelike vector. Thus, depending on the type of the vector $v$, we can write
the curvature as $k_{i}(s)=\langle T(s),v_{i}\rangle$ where $v_{i}=aw_{i}$,
$a>0$ and $i=1,2,3$.
Now, we define the evolution of $X$ in $\mathbb{H}^{2}$ to be
$\hat{X}_{i}(s,t)=M_{i}(t)X(s)$, where
$\displaystyle M_{1}(t)$ $\displaystyle:=$
$\displaystyle\left(\begin{array}[]{lll}1&0&0\\\
0&cos(\varphi_{1}(t))&sen(\varphi_{1}(t))\\\
0&-sen(\varphi_{1}(t))&cos(\varphi_{1}(t))\end{array}\right),\,\,\,M_{2}(t):=\left(\begin{array}[]{ccc}1+\frac{(\varphi_{2}(t))^{2}}{2}&-\frac{(\varphi_{2}(t))^{2}}{2}&\varphi_{2}(t)\\\
\frac{(\varphi_{2}(t))^{2}}{2}&1-\frac{(\varphi_{2}(t))^{2}}{2}&\varphi_{2}(t)\\\
\varphi_{2}(t)&-\varphi_{2}(t)&1\end{array}\right),$ (3.8) $\displaystyle
M_{3}(t)$ $\displaystyle:=$
$\displaystyle\left(\begin{array}[]{lll}cosh(\varphi_{3}(t))&senh(\varphi_{3}(t))&0\\\
senh(\varphi_{3}(t))&cosh(\varphi_{3}(t))&0\\\ 0&0&1\end{array}\right),$
(3.12)
and $\varphi_{i}(t)=at$ for each $i=1,2,3.$
A straightforward computation shows that
$\displaystyle\left\langle
M^{\prime}_{i}(t)X(s),M_{i}(t)N(s)\right\rangle=-\varphi^{\prime}_{i}(t)\langle
X(s)\times
N(s),w_{i}\rangle=-\varphi^{\prime}_{i}(t)\langle-T(s),w_{i}\rangle.$
Thus,
$\displaystyle\left\langle\frac{\partial}{\partial
t}\hat{X}_{i}(s,t),\hat{N}_{i}(s,t)\right\rangle$ $\displaystyle=$
$\displaystyle\left\langle M^{\prime}_{i}(t)X(s),M_{i}(t)N(s)\right\rangle$
$\displaystyle=$
$\displaystyle-\varphi^{\prime}_{i}(t)\langle-T(s),w_{i}\rangle\;=\;\langle
T(s),v_{i}\rangle$ $\displaystyle=$ $\displaystyle
k_{i}(s)\;=\;\hat{k}_{i}(s,t),$
where the last equality follows from the fact that isometries preserve
geodesic curvature. Therefore, $X$ is a soliton solution to the CSF.
$\square$
It follows from Theorem 2.2 that the study of the solitons solutions to the
CSF on the 2-dimensional hyperbolic space is reduced to describing the curves
that satisfy Equation (2.2) for some vector
$v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$. Up to isometries of $\mathbb{H}^{2}$
we consider $v$ as being $v_{i}=ae_{i}$, where $a\in\mathbb{R}^{+}$,
$e_{1}=(-1,0,0)$ if $v$ is a timelike vector, $e_{1}=(-1,1,0)$ if $v$ is a
lightlike vector and $e_{1}=(0,0,1)$ if $v$ is a spacelike vector. Our next
result characterizes (2.2) in terms of a system of differential equations.
###### Proposition 3.1.
Let $\displaystyle X:I\rightarrow\mathbb{H}^{2}$ be a regular curve
parametrized by arc length $s$. Consider the vectors
$e_{1}=(-1,0,0),\qquad e_{2}=(-1,1,0)\qquad e_{3}=(0,0,1).$ (3.13)
For each $i\in\\{1,2,3\\}$, define the functions
$\alpha_{i}(s)=\langle X(s),e_{i}\rangle,\qquad\tau_{i}(s)=\langle
T(s),e_{i}\rangle\qquad\eta_{i}(s)=\langle N(s),e_{i}\rangle,$
where $T$ and $N$ are the unit vector fields tangent and normal to $X$,
respectively. For a fixed $a>0$,
$k_{i}(s)=a\tau_{i}(s)$
is satisfied, for all $s\in I$ if, and only if, the functions $\alpha_{i}(s)$,
$\tau_{i}(s)$ and $\eta_{i}(s)$ satisfy the system
$\left\\{\begin{array}[]{lll}\alpha^{\prime}_{i}(s)=\tau_{i}(s),\\\
\tau^{\prime}_{i}(s)=a\tau_{i}(s)\eta_{i}(s)+\alpha_{i}(s),\\\
\eta^{\prime}_{i}(s)=-a\tau^{2}_{i}(s),\end{array}\right.$ (3.14)
with initial condition $(\alpha_{i}(0),\tau_{i}(0),\eta_{i}(0))$ satisfying
$-\alpha^{2}_{i}(0)+\tau^{2}_{i}(0)+\eta^{2}_{i}(0)=\left\\{\begin{array}[]{cll}-1,&\text{if}&i=1,\\\
0,&\text{if}&i=2,\\\ 1,&\text{if}&i=3.\end{array}\right.$ (3.15)
For such functions, the expression
$-\alpha^{2}_{i}(s)+\tau^{2}_{i}(s)+\eta^{2}_{i}(s)$ is equal to the right
hand side of (3.15), for all $s\in I$. Moreover, $\eta_{i}(s)$ is a decreasing
function.
###### Proof.
The vector fields $X$, $T$ and $N$ satisfy the following system of equations
$\left\\{\begin{array}[]{lll}X^{\prime}(s)=T(s),\\\
T^{\prime}(s)=k(s)N(s)+X(s),\\\ N^{\prime}(s)=-k(s)T(s).\end{array}\right.$
(3.16)
Taking the inner product with $e_{i}$, we get that $\alpha_{i}(s)$,
$\tau_{i}(s)$ and $\eta_{i}(s)$ satisfy the system of equations
$\left\\{\begin{array}[]{lll}\alpha^{\prime}_{i}(s)=\tau_{i}(s),\\\
\tau^{\prime}_{i}(s)=k_{i}(s)\eta_{i}(s)+\alpha_{i}(s),\\\
\eta^{\prime}_{i}(s)=-k_{i}(s)\tau_{i}(s),\end{array}\right.$ (3.17)
Suppose that $k_{i}(s)=a\tau_{i}(s)$ for all $s\in I$. Then substituting into
(3.17), we obtain (3.14). Note that,
$e_{i}=\alpha_{i}(s)X(s)+\tau_{i}(s)T(s)+\eta_{i}N(s).$
Therefore, $\langle
e_{i},e_{i}\rangle=-\alpha^{2}_{i}(s)+\tau^{2}_{i}(s)+\eta^{2}_{i}(s)$ is
constant for all $s\in I$. In particular, for $s=0$, we obtain (3.17).
Moreover, it follows from the third equation of the system (3.14) that the
function $\eta_{i}(s)$ is decreasing.
Conversely, suppose that the functions $\alpha_{i}(s)$, $\tau_{i}(s)$ and
$\eta_{i}(s)$ satisfy (3.14) and (3.15) for each $i\in\\{1,2,3\\}$. Since
(3.17) holds, we have
$\displaystyle\left\\{\begin{array}[]{l}a\tau_{i}(s)\eta_{i}(s)+\alpha_{i}(s)=k_{i}(s)\eta_{i}(s)+\alpha_{i}(s),\\\
-(a\tau_{i}(s))\tau_{i}(s)=-k_{i}(s)\tau_{i}(s),\end{array}\right.$
i.e.,
$\displaystyle\left\\{\begin{array}[]{l}\left[a\tau_{i}(s)-k_{i}(s)\right]\eta_{i}(s)=0,\\\
\left[a\tau_{i}(s)-k_{i}(s)\right]\tau_{i}(s)=0,\end{array}\right.$
for all $s\in I$. For eaxch $i$, in order to conclude that
$k_{i}(s)=a\tau_{i}(s)$, for all $s$, we will assume that $k_{i}(s)\neq
a\tau_{i}(s)$ at some point $s_{0}$. Then this will occur on some interval
$J\subset I$ around $s_{0}$. Hence $\eta_{i}(s)=\tau_{i}(s)=0$ for $s\in J$.
Therefore, $e_{i}$ will be orthogonal to $T(s)$ and $N(s)$ for all $s\in J$.
Thus, $e_{i}$ will be parallel to $X(s)$ for all $s\in J$. But $e_{i}$ is a
constant vector for each $i$, so this can only happen at some isolated points
of a curve $X$ in $\mathbb{H}^{2}$, which is a contradiction. Therefore,
$k_{i}(s)=a\tau_{i}(s)$ for all $s\in I$ and for each $i\in\\{1,2,3\\}$. ∎
Our next proposition shows how a solution of the system (3.14), with initial
conditions satisfying (3.15), is related to a soliton solution to the CSF.
###### Proposition 3.2.
Given a solution $(\alpha(s),\tau(s),\eta(s))$ to the system (3.14) on some
interval $J$ with fixed $a>0$ and initial conditions
$(\alpha(0),\tau(0),\eta(0))$ satisfying
$-\alpha^{2}(0)+\tau^{2}(0)+\eta^{2}(0)=-1$ (resp. $0$ and $1$), there exists
a smooth curve $\displaystyle X:I\rightarrow\mathbb{H}^{2}$ parametrized by
arc length $s$, such that its tangent and normal unit vector fields $T$ and
$N$ satisfy
$\alpha(s)=\langle X(s),e\rangle,\hskip 14.22636pt\tau(s)=\langle
T(s),e\rangle\hskip 14.22636pt\text{and}\hskip 14.22636pt\eta(s)=\langle
N(s),e\rangle,$ (3.20)
where $e=(-1,0,0)$ (resp. $e=(-1,1,0)$ and $e=(0,0,1)$).
###### Proof.
Define $k(s)=a\tau(s)$. Thus, up to isometries of $\mathbb{H}^{2}$, there
exists an unique curve $\displaystyle X:I\rightarrow\mathbb{H}^{2}$, whose
curvature is $k(s)$ i.e. $X(s)$ and its tangent and normal unit vector fields
$T(s)$ and $N(s)$ satisfy the system (3.16). The curve $X(s)$ is uniquely
determined by the initial conditions $X(0)$, $T(0)$ and $N(0)$, that can be
chosen such that
$-\alpha(0)X(0)+\tau(0)T(0)+\eta(0)N(0)=e,$
where $e=(-1,0,0)$ (resp. $e=(-1,1,0)$ and $e=(0,0,1)$). A straightforward
computations shows that (3.15) and (3.16) imply
$\displaystyle\frac{d}{ds}\left[-\alpha(s)X(s)+\tau(s)T(s)+\eta(s)N(s)\right]=0.$
Therefore, (3.20) is satisfied. ∎
###### Remark 3.3.
Let $\displaystyle X:I\rightarrow\mathbb{H}^{2}$ be a regular curve
parametrized by arc length $s$ given by $X(s)=(x_{1}(s),x_{2}(s),x_{3}(s))$.
The function $\alpha(s)$ defined by (3.20) has the following geometric
interpretation.
* •
If $e=(-1,0,0)$ (timelike vector), then $\alpha(s)=x_{1}(s)>0$ for all $s\in
I$. Moreover, $\alpha(s)$ is the height function with respect to the vector
$(1,0,0).$
* •
If $e=(-1,1,0)$ (lightlike vector), then $\alpha(s)=x_{1}(s)+x_{2}(s)>0$ for
all $s\in I$. Moreover, $\alpha(s)$ is the height function with respect to the
vector $(1,1,0).$
* •
If $e=(0,0,1)$ (spacelike vector), then $\alpha(s)=x_{3}(s)$ for all $s\in I$.
Moreover, $\alpha(s)$ is the height function (with sign) with respect to the
vector $(0,0,1).$
Figure 1a (resp. 1b and 1c) provides a geometric illustration of the function
$\alpha(s)$ as a height function with respect to vector $(-1,0,0)$ (resp.
$e=(-1,1,0)$ and $e=(0,0,1)$.)
(a)
(b)
(c)
Figure 1: Geometric interpretation of the functions $\alpha(s)$.
As we have seen in Propositions 3.1, 3.2 and Remark 3.3, the investigation of
the soliton solutions to the CSF on the 2-dimensional hyperbolic space is
equivalent to studying the solutions $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ of
the system
$\left\\{\begin{array}[]{lll}\alpha^{\prime}(s)=\tau(s)\\\
\tau^{\prime}(s)=a\tau(s)\eta(s)+\alpha(s)\\\
\eta^{\prime}(s)=-a\tau^{2}(s),\end{array}\right.$ (3.21)
for each constant $a>0$ and initial condition $\displaystyle\psi(0)\in H\cup
C\cup S\subset\mathbb{R}^{3},$ where
$\begin{array}[]{l}H:=\\{(\alpha,\tau,\eta)\in\mathbb{R}^{3}:-\alpha^{2}+\tau^{2}+\eta^{2}=-1,\alpha>0\\},\\\
C:=\\{(\alpha,\tau,\eta)\in\mathbb{R}^{3}\setminus\\{0\\}:-\alpha^{2}+\tau^{2}+\eta^{2}=0,\alpha>0\\},\\\
S:=\\{(\alpha,\tau,\eta)\in\mathbb{R}^{3}:-\alpha^{2}+\tau^{2}+\eta^{2}=1\\}.\end{array}$
(3.22)
These are disjoint sets and if the initial condition $\psi(0)\in H$ (resp. $C$
or $S$) then the solution $\psi(s)$ defined on the maximal interval $I$ will
be contained in $H$ (resp. $C$ or $S$) for all $s\in I$.
From now on, using (3.21), we will prove a series of lemmas that will provide
the proof of the main result (Theorem 2.3). Namely, we will prove that for any
initial condition, the solutions $\psi(s)$ of (3.21) and hence the associated
soliton solutions to the hyperbolic space are defined on the whole
$\mathbb{R}$. Moreover, we will analize the behaviour of the curvature
function of the solitons at each end.
In the first lemma we will study the solution of (3.21) such that the function
$\tau(s)$ is constant. As we will see such solutions (that will be called
trivial) only exit on $S$.
###### Lemma 3.4.
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non null solution of (3.21)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial
condition $\psi(0)\in H\cup C\cup S$, where $H$, $C$ and $S$ are given by
(3.22). Then the function $\tau(s)=b$, $s\in I$, where $b$ is a real constant
if, and only if, $b\in\\{-1,0,1\\}$, $I=\mathbb{R}$ and $\psi(s)\in S$ for all
$s\in\mathbb{R}.$ Moreover,
* i)
If $b=0$, then $\psi(s)=(0,0,\pm 1)$ are singular solutions of (3.21) in $S$.
* ii)
If $b^{2}=1$, then $a=1$ and $\psi(s)=(\pm s+\alpha(0),\,\pm
1,\,-s\pm\alpha(0)).$
###### Proof.
If $b=0$, it follows from (3.21) that $\alpha(s)=0$ for all $s\in I$. Using
the equation $-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=\gamma$, where
$\gamma\in\\{-1,0,1\\}$, we obtain $\eta^{2}(s)=1$ for all $s\in I$. Hence,
$\psi(s)=(0,0,\pm 1)$, $\forall\;s\in\mathbb{R}$ are singular solutions of
(3.21) in $S$.
If $\psi(s)$ is not a singular solution, then $b\neq 0$ and it follows from
(3.21) that
$a\,b\,\eta(s)=-\alpha(s)\hskip 28.45274pt\text{and}\hskip
28.45274pt\eta(s)=-ab^{2}s+\eta(0),$
for all $s\in\mathbb{R}$. Using the relation
$-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=\gamma$, where
$\gamma\in\\{-1,0,1\\}$, we conclude that
$[-a^{2}\,b^{2}+1]\eta^{2}(s)=\gamma-b^{2},$
and hence the function $[-a^{2}b^{2}+1]\alpha^{2}(s)$ is also constant. Since
$\psi(s)$ is not a singular solution, it follows that $\displaystyle
a^{2}b^{2}=1$ and $b^{2}=\gamma$. Therefore, $\gamma=1$, $b=\pm 1$, $a=1$ and
$\alpha(s)=\mp\eta(s)$, for all $s\in\mathbb{R}$. This concludes the proof. ∎
It follows from Lemma 3.4 that when $\tau(s)$ is a constant function then the
functions $\alpha(s)$ and $\eta(s)$ are linear in $s$ and its corresponding
soliton solutions to the CSF are curves of constant curvature i.e. geodesics
when $k(s)=\tau(s)=0$ or planar curves with curvature $k(s)=\tau(s)=\pm 1$.
In this context, we define a trivial solution
$\psi(s)=(\alpha(s),\tau(s),\eta(s))$ of (3.21), when $\tau(s)$ is a constant
function. From now on, we will study only non trivial solutions of (3.21). It
follows from Lemma 3.4 that there are no trivial solutions of (3.21) in $H\cup
C$.
On our next lemmas we will study the solutions $\psi(s)$ of (3.21) contained
in $H\cup C$ and those contained in $S$ separately.
###### Lemma 3.5.
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21) defined on
the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial condition
$\psi(0)\in H\cup C$, where $H$ and $C$ are given by (3.22).
* i)
If $\alpha(s)$ has a critical point then it is a global minimum point of
$\alpha$. Moreover, there exists always $\overline{s}\in I$ such that
$\alpha(s)$ is strictly monotone on the intervals $(\omega_{-},\overline{s}]$
and $[\overline{s},\omega_{+})$.
* ii)
If $s_{0}$ is a critical point of $\tau(s)$, then $a^{2}\tau^{2}(s_{0})>1$ and
$s_{0}$ is a local minimum (resp. maximum) point of $\tau(s)$ if, and only if,
$\tau(s_{0})<0$ (resp. $\tau(s_{0})>0$).
###### Proof.
i) Let $s_{0}$ be a critical point of $\alpha(s)$. Note that $\alpha(s)>0$ for
all $s\in I$ whenever $\psi(0)\in H\cup C$. Taking the second derivative of
$\alpha(s)$ and using (3.21) at $s=s_{0}$, we have
$\alpha^{\prime\prime}(s_{0})=\tau^{\prime}(s_{0})=a\tau(s_{0})\eta(s_{0})+\alpha(s_{0})=\alpha(s_{0})>0.$
Hence, $s_{0}$ is a global minimum point of $\alpha(s)$. Therefore,
$\alpha(s)$ has at most one critical point. If there are no critical points
then $\alpha(s)$ is trictly monotone on $I$.
ii) Let $s_{0}$ be a critical point of $\tau(s)$. Then
$\tau^{\prime}(s_{0})=a\eta(s_{0})\tau(s_{0})+\alpha(s_{0})=0$ and
$\eta(s_{0})\tau(s_{0})\neq 0$ because $\alpha(s)>0$ for all $s\in I$. Since
$\psi(0)\in H\cup C$, it follows that
$-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=\delta\leq 0$ for all $s\in I$, where
$\delta\in\\{-1,0\\}$. Thus,
$\delta=-a^{2}\tau^{2}(s_{0})\eta^{2}(s_{0})+\tau^{2}(s_{0})+\eta^{2}(s_{0})=\eta^{2}(s_{0})[-a^{2}\tau^{2}(s_{0})+1]+\tau^{2}(s_{0})\leq
0.$
Hence, $-a^{2}\tau^{2}(s_{0})+1<0$. Taking the second derivative of $\tau(s)$
and using (3.21) at $s=s_{0}$, we have
$\tau^{\prime\prime}(s_{0})=a\tau^{\prime}(s_{0})\eta(s_{0})+a\tau(s_{0})\eta^{\prime}(s_{0})+\alpha^{\prime}(s_{0})=\tau(s_{0})\left[-a^{2}\tau^{2}(s_{0})+1\right].$
(3.23)
This concludes the proof of ii). ∎
###### Lemma 3.6.
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21) defined on
the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial condition
$\psi(0)\in C$, where $C$ is given by (3.22).
* i)
If $\tau(s)>0$ in $I$, then $\alpha(s)$ is strictly increasing in $I$,
$\tau(s)$ is bounded and it has at most one critical point in $I$. Moreover,
$\omega_{-}=-\infty$, $\displaystyle\lim_{s\to-\infty}\psi(s)=(0,0,0)$ and
$\displaystyle\lim_{s\to\omega_{+}}-\eta(s)=\lim_{s\to\omega_{+}}\alpha(s)=+\infty.$
* ii)
If $\tau(s)<0$ in $I$, then $\alpha(s)$ is strictly decreasing in $I$,
$\tau(s)$ is bounded and it has at most one critical point in $I$. Moreover,
$\omega_{+}=+\infty$, $\displaystyle\lim_{s\to+\infty}\psi(s)=(0,0,0)$ and
$\displaystyle\lim_{s\to\omega_{-}}\eta(s)=\lim_{s\to\omega_{-}}\alpha(s)=+\infty.$
###### Proof.
i) If $\tau(s)>0$, then it follows from Lemma 3.5 that $\tau(s)$ has only
local maximum points i.e. $\tau(s)$ has at most one critical point. The
positive function $\alpha(s)$ is bounded and there exists $\overline{s}\in I$
such that $\alpha$ is strictly increasing on $(\omega_{-},\overline{s})$.
Thus, it follows from equation $\alpha^{2}(s)=\tau^{2}(s)+\eta^{2}(s)$ that we
can take $\overline{s}$ such that $\tau(s)$ and $\eta(s)$ are bounded and
monotone on $(\omega_{-},\overline{s})$. The interval $I$ is maximal, hence
$\omega_{-}=-\infty.$ Since the limits
$\displaystyle\lim_{s\to-\infty}\alpha(s)$,
$\displaystyle\lim_{s\to-\infty}\alpha^{\prime}(s)=\lim_{s\to-\infty}\tau(s)$
and $\displaystyle\lim_{s\to-\infty}\tau^{\prime}(s)$ exist, we obtain that
$\displaystyle\lim_{s\to-\infty}\tau(s)=0$ and
$\displaystyle\lim_{s\to-\infty}\alpha(s)=\lim_{s\to-\infty}\eta(s)=0$. Using
the fact that the function $\eta(s)$ is decreasing, we get that $\eta(s)<0$
for all $s\in I.$
We claim that $\alpha(s)$ is ununbounded on $(\overline{s},\omega_{+})$. In
fact, assume by contradiction that the strictly increasing function
$\alpha(s)$ is bounded on $(\overline{s},\omega_{+})$. Thus, it follows from
equation $\alpha^{2}(s)=\tau^{2}(s)+\eta^{2}(s)$ that we can take
$\overline{s}$ such that $\tau(s)$ and $\eta(s)$ are bounded and monotone on
$(\overline{s},\omega_{+})$. Hence, there exists $p\in C$ such that
$\displaystyle\lim_{s\to\omega_{+}}(\alpha(s),\tau(s),\alpha(s))=p$ and $p$ is
a singular (trivial) solution in $C$, which contradicts Lemma 3.4. Therefore,
$\displaystyle\lim_{s\to\omega_{+}}\alpha(s)=+\infty.$
Now, assume by contradiction that the strictly decreasing and negative
function $\eta(s)$ is bounded on $(\overline{s},\omega_{+})$. Since
$\tau^{2}(s)+\eta^{2}(s)=\alpha^{2}(s)$, it follows that the function
$\tau(s)$ is unbounded and positive on $(\overline{s},\omega_{+})$, because we
showed that $\alpha(s)$ is unbounded on $(\overline{s},\omega_{+})$. Thus, we
can choose $\overline{s}$ such that $2\tau(s)<a\tau^{2}(s)$ for all
$s>\overline{s}$. Using the equations of (3.21), we obtain
$2\alpha(s)-2\alpha(\overline{s})=2\int_{\overline{s}}^{s}\tau(s)ds<\int_{\overline{s}}^{s}a\tau^{2}(s)ds=-\eta(s)+\eta(\overline{s}).$
(3.24)
Hence
$2\alpha(s)<-\eta(s)+\eta(\overline{s})+2\alpha(\overline{s}),$
for each $s\in(\overline{s},\omega_{+})$. But this contradicts the fact that
$\alpha(s)$ is unbounded. Therefore, $\eta(s)$ is unbounded and
$\displaystyle\lim_{s\to\omega_{+}}\eta(s)=-\infty$.
Finally, if $\tau(s)$ is unbounded on $(\overline{s},\omega_{+})$,
$\overline{s}\in I$, then we can choose again $\overline{s}$ such that
$2\tau(s)<a\tau^{2}(s)$ for all $s>\overline{s}$. Using (3.24), we obtain
$\alpha(s)+\eta(s)<2\alpha(\overline{s})-\eta(\overline{s})-\alpha(s),$
this is a contradiction, because $\alpha^{2}(s)=\tau^{2}(s)+\eta^{2}(s)$ i.e.
$\alpha(s)>-\eta(s)$ and
$\displaystyle\lim_{s\to\omega_{+}}\alpha(s)=+\infty$.
ii) This proof is analogous to the proof of item i). ∎
###### Lemma 3.7.
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21) defined on
the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial condition
$\psi(0)\in H$, where $H$ is given by (3.22). Then there exists a unique
$s_{0}$ such that $\alpha^{\prime}(s_{0})=\tau(s_{0})=0$.
###### Proof.
Assume by contradiction that such an $s_{0}$ does not exist. Then, either
$\tau(s)<0$ or $\tau(s)>0$ for all $s\in I$. If $\tau(s)<0$, then $\alpha(s)$
is strictly decreasing. Taking $\overline{s}\in I$ we have
$1\leq\alpha(s)\leq\alpha(\overline{s})$ for all $s>\overline{s}$. Since
$-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=-1$, then
$\tau^{2}(s)+\eta^{2}(s)<\alpha^{2}(s)<\alpha^{2}(\overline{s})$ for all
$s>\overline{s}$. Hence, the functions $\alpha(s),\eta(s)$ are bounded and
monotone in $[\overline{s},\omega_{+})$, and
$\displaystyle\lim_{s\to\omega_{+}}\tau(s)$ exists, because
$\tau^{2}(s)=-1+\alpha^{2}(s)-\eta^{2}(s)$. Thus, there exists a point $p\in
H$ such that
$\displaystyle\lim_{s\to\omega_{+}}(\alpha(s),\tau(s),\alpha(s))=p$.
Therefore, $\displaystyle\omega_{+}=+\infty$ and $p$ is a singular (trivial)
solution of (3.21), which contradicts Lemma 3.4. In a similar way, we can
prove that $\tau(s)<0$ for all $s\in I$ cannot occur.
Therefore, there is an $s_{0}\in I$ such that
$\alpha^{\prime}(s_{0})=\tau(s_{0})=0$. It follows from Lemma 3.5, that
$s_{0}$ is a global minimum of the function $\alpha(s)$. Hence, $s_{0}$ is
unique. ∎
In the next three lemmas, we will suppose that $\psi(0)\in H\cup C$ and that
$\alpha(s)$ has only one critical point. Note that, this hypothesis only
excludes the case presented in Lemma 3.6 because it follows from Lemma 3.4
that $\alpha(s)$ has at most one critical point when $\psi(0)\in H\cup C$ and
Lemma 3.7 shows that $\alpha(s)$ has a unique critical point, when $\psi(0)\in
H$.
###### Lemma 3.8.
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21) defined on
the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial condition
$\psi(0)\in H\cup C$, where $H$ and $C$ are given by (3.22). If $\alpha(s)$
has one critical point, then
$\displaystyle\lim_{s\to\omega_{-}}\alpha(s)=\lim_{s\to\omega_{+}}\alpha(s)=\infty$.
###### Proof.
Let $s_{0}$ be the global minimum point of $\alpha(s)$. Thus, $\alpha(s)$ is
monotone on the intervals $(\omega_{-},s_{0}]$ and $[s_{0},\omega_{+})$.
Assume by contradiction that $\alpha(s)$ is bounded on the intervals
$(\omega_{-},s_{0}]$ and $[s_{0},\omega_{+})$. Since the functions
$\alpha(s)$, $\tau(s)$ and $\eta(s)$ satisfy
$\tau^{2}(s)+\eta^{2}(s)=\delta+\alpha^{2}(s)\leq\alpha^{2}(s)$, where
$\delta\in\\{-1,0\\}$, then the functions $\alpha(s)$ and $\eta(s)$ are
bounded and monotone on $(\omega_{-},s_{0}]$ and $[s_{0},\omega_{+})$. The
limits $\displaystyle\lim_{s\to\omega_{-}}\tau(s)$ and
$\displaystyle\lim_{s\to\omega_{+}}\tau(s)$ exist, because
$\tau^{2}(s)=\delta+\alpha^{2}(s)-\eta^{2}(s)$. Thus, there are points $p_{1}$
and $p_{2}\in H\cup C$ such that
$\displaystyle\lim_{s\to\omega_{-}}(\alpha(s),\tau(s),\alpha(s))=p_{1}$ and
$\displaystyle\lim_{s\to\omega_{+}}(\alpha(s),\tau(s),\alpha(s))=p_{2}$.
Hence, $\omega_{-}=-\infty$, $\omega_{+}=+\infty$ and $\\{p_{1},p_{2}\\}$ is a
set of singular (trivial) solutions of (3.21), which contradicts Lemma 3.4.
Therefore, we conclude that the function $\alpha(s)$ is unbounded on the
intervals $(\omega_{-},s_{0}]$ and $[s_{0},\omega_{+})$, and moreover
$\displaystyle\lim_{s\to\omega_{-}}\alpha(s)=\lim_{s\to\omega_{+}}\alpha(s)=\infty$.
∎
###### Lemma 3.9.
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21) defined on
the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial condition
$\psi(0)\in H\cup C$, where $H$ and $C$ are given by (3.22). If $\alpha(s)$
has one critical point, then
$\displaystyle\lim_{s\to\omega_{-}}\eta(s)=\infty$ and
$\displaystyle\lim_{s\to\omega_{+}}\eta(s)=-\infty$.
###### Proof.
Let $s_{0}$ be the global minimum point of $\alpha(s)$. Then $\tau(s)<0$ for
all $s<s_{0}$ and $\tau(s)>0$ for all $s>s_{0}$. Moreover, $\alpha(s)$ is
unbounded and monotone on the intervals $(\omega_{-},s_{0}]$ and
$[s_{0},\omega_{+})$.
Assume by contradiction that the function $\eta(s)$ is bounded on
$(\omega_{-},s_{0}]$. Since $\tau^{2}(s)+\eta^{2}(s)=\delta+\alpha^{2}(s)$,
where $\delta\in\\{-1,0\\}$, then it follows from Lemma 3.7 that the function
$\tau(s)$ is unbounded and negative on $(\omega_{-},s_{0})$ i.e. there exists
$s_{1}\in(\omega_{-},s_{0}]$ such that $a\tau(s)<-1$ and
$-a\tau^{2}(s)<\tau(s)$ for all $s\in(\omega_{-},s_{1}]$. Thus, using (3.21)
for each $s\in(\omega_{-},s_{1}]$, we obtain
$\alpha(s)-\alpha(s_{1})=-\int_{s}^{s_{1}}\tau(s)ds<\int_{s}^{s_{1}}a\tau^{2}(s)ds=\eta(s)-\eta(s_{1}),$
that is,
$\alpha(s)<\eta(s)-\eta(s_{1})+\alpha(s_{1}),$
for each $s\in(\omega_{-},s_{1}]$ which contradicts Lemma 3.8. Therefore,
$\eta(s)$ is unbounded on $(\omega_{-},s_{1}]$.
In a similar way, we can prove that the function $\eta(s)$ is unbounded on
$[s_{0},\omega_{+})$.
Since $\eta(s)$ is decreasing on $(\omega_{-},\omega_{+})$, it follows that
$\displaystyle\lim_{s\to\omega_{-}}\eta(s)=\infty$ and
$\displaystyle\lim_{s\to\omega_{+}}\eta(s)=-\infty$. ∎
###### Lemma 3.10.
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21), with $a>0$,
defined on the maximal interval $I=(\omega_{-},\omega_{+})$ and initial
condition $\psi(0)\in H\cup C$, where $H$ and $C$ are given by (3.22). If
$\alpha(s)$ has one critical point, then the function $\displaystyle\tau(s)$
is unbounded on $I$ and it has only two critical points.
###### Proof.
Let $s_{0}\in I$ be the global minimum point of $\alpha(s)$. The arguments
consist in studying the existence and the properties of the critical points of
$\tau(s)$.
Claim. If $\tau(s)$ does not have any critical point on $I$, then
$-1<a\tau(s)<0$ on $(\omega_{-},s_{0})$ and $0<a\tau(s)<1$ on
$(s_{0},\omega_{+})$. In fact, suppose that $\tau^{\prime}(s)\neq 0$ for all
$s\in I$. At $s_{0}$, $\tau(s_{0})=0$ and
$\tau^{\prime}(s_{0})=\alpha(s_{0})>0$. Moreover,
$\tau^{\prime}(s)=a\tau(s)\eta(s)+\alpha(s)>0$ i.e. $\tau(s)$ is an increasing
function on $I$, $\displaystyle\lim_{s\to\omega_{-}}\tau(s)\neq 0$ and
$\displaystyle\lim_{s\to\omega_{+}}\tau(s)\neq 0$. It follows from Lemma 3.9
that $\displaystyle\lim_{s\to\omega_{-}}\eta(s)=+\infty$ and
$\displaystyle\lim_{s\to\omega_{+}}\eta(s)=-\infty$. We also know that
$\tau(s)$ is negative on $(\omega_{-},s_{0})$ and positive on
$(s_{0},\omega_{+})$. Thus, there are $s_{1}\in(\omega_{-},s_{0})$ and
$s_{2}\in(s_{0},\omega_{+})$ such that $-\alpha(s)<a\tau(s)\eta(s)<0$ for all
$s\in I\setminus[s_{1},s_{2}]$. Hence,
$-\alpha^{2}(s)<-a^{2}\tau^{2}(s)\eta^{2}(s)$ for all $s\in
I\setminus[s_{1},s_{2}]$ and
$\delta=-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)<-a^{2}\tau^{2}(s)\eta^{2}(s)+\tau^{2}(s)+\eta^{2}(s),$
i.e.,
$a^{2}\tau^{2}(s)\eta^{2}(s)<-\delta+\tau^{2}(s)+\eta^{2}(s).$
Therefore,
$1<\frac{-\delta}{a^{2}\eta^{2}(s)\tau^{2}(s)}+\frac{1}{a^{2}\eta^{2}(s)}+\frac{1}{a^{2}\tau^{2}(s)}$
for all $s\in I\setminus[s_{1},s_{2}]$.
Taking the limit when $s\to\omega_{+}$ and $s\to\omega_{-}$, using the fact
that $\tau(s)$ is increasing and Lemma 3.9, we obtain that
$\lim_{s\to\omega_{+}}\frac{1}{a^{2}\tau^{2}(s)}>1\hskip
28.45274pt\text{and}\hskip
28.45274pt\lim_{s\to\omega_{-}}\frac{1}{a^{2}\tau^{2}(s)}>1,$
i.e.,
$\lim_{s\to\omega_{+}}a^{2}\tau^{2}(s)<1\hskip 28.45274pt\text{and}\hskip
28.45274pt\lim_{s\to\omega_{-}}a^{2}\tau^{2}(s)<1.$
Thus, using that $\tau(s)$ is increasing, we conclude that $-1<a\tau(s)<0$ on
$(\omega_{-},s_{0})$ and $0<a\tau(s)<1$ on $(s_{0},\omega_{+})$. This proves
our Claim.
Still assuming that $\tau(s)$ does not have any critical point, we define the
positive functions $f(s)=\alpha(s)+\eta(s)$ and $g(s)=\alpha(s)-\eta(s)$
(observe that $-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=\delta\leq 0$). Taking
the derivatives of $f$ and $g$ and using (3.21), we obtain
$f^{\prime}(s)=\tau(s)[1-a\tau(s)]$ and $g^{\prime}(s)=\tau(s)[1+a\tau(s)]$.
It follows from our Claim that the functions $f$ and $g$ are decreasing when
$\tau(s)<0$ and they are increasing when $\tau(s)>0$ and
$\displaystyle 0<f(s)\cdot g(s)=-\delta+\tau^{2}(s)<\frac{-\delta
a^{2}+1}{a^{2}}$
for all $s\in I$, where $\delta\in\\{-1,0\\}$. Hence, the functions $f$ and
$g$ are positive, monotone and bounded on the interval $(\omega_{-},s_{0})$.
Similarly one shows that the functions $f$ and $g$ are positive, monotone and
bounded on $(s_{0},\omega_{+})$. Thus, we conclude that there exist
$M_{1},M_{2}\in\mathbb{R}$ such that
$\left\\{\begin{array}[]{ll}\alpha(s)+\eta(s)\leq M_{1},\\\
\alpha(s)-\eta(s)\leq
M_{2},\end{array}\right.\qquad\forall\;s\in(\omega_{-},s_{0})\cup(s_{0},\omega_{+}).$
Hence,
$2\alpha(s)\leq
M_{1}+M_{2}\,\,\forall\,\,s\,\,\in\,\,(\omega_{-},s_{0})\,\,\text{and}\,\,2\alpha(s)\leq
M_{1}+M_{2}\,\,\forall\,\,s\,\,\in\,\,(s_{0},\omega_{+}).$
These inequalities contradict Lemma 3.8. Hence, the function $\tau(s)$ has at
least one critical point on each interval $(\omega_{-},s_{0})$ and
$(s_{0},\omega_{+})$. From item ii) of Lemma 3.5 we have that $\tau(s)$ has
only one local minimum $s_{1}\in(\omega_{-},s_{0})$ and it has only one local
maximum $s_{2}\in(s_{0},\omega_{+})$ i.e.
$\tau(s_{1})\leq\tau(s)\leq\tau(s_{2})$ for all $s\in I$ and $\tau(s)$ is
bounded on the interval $I$. This concludes the proof. ∎
###### Lemma 3.11.
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21) defined on
the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial condition
$\psi(0)\in H\cup C$, where $H$ and $C$ are given by (3.22). Then
$I=\mathbb{R}$.
###### Proof.
From Lemmas 3.6 and 3.10 we have that $\tau(s)$ is bounded. Let $M>0$ be such
that $|\tau(s)|\leq M$, for all $s\in I$. Using (3.21), we obtain,
$|\alpha(s)-\alpha(s_{0})|=\left|\int_{s_{0}}^{s}\tau(s)ds\right|\leq
M|s-s_{0}|$ (3.25)
for each $s\in I$. If $\alpha(s)$ has a global minimum point, then it follows
from Lemma 3.8 that
$\displaystyle\lim_{s\to\omega_{-}}\alpha(s)=\lim_{s\to\omega_{+}}\alpha(s)=\infty$.
Hence, $I=\mathbb{R}$.
If $\alpha(s)$ does not have any critical point and $\tau(s)<0$ (resp.
$\tau(s)>0$) for all $s\in I$, then it follows from Lemma 3.6 that
$\omega_{+}=+\infty$ and $\displaystyle\lim_{s\to\omega_{-}}\alpha(s)=+\infty$
(resp. $\omega_{-}=-\infty$ and
$\displaystyle\lim_{s\to\omega_{+}}\alpha(s)=+\infty$). Using (3.25), we
conclude that $\omega_{-}=-\infty$ (resp. $\omega_{+}=+\infty$). Therefore,
$I=\mathbb{R}$. ∎
We will now study the solutions of the system (3.21), with initial conditions
on $S$. We will first classify the singular points of the system that are in
the set $S$.
###### Lemma 3.12.
Let $\Phi:S\rightarrow TS\subset\mathbb{R}^{3}$ be the differential vector
field given by
$\Phi(\alpha,\tau,\eta)=\left(\tau,a\tau\eta+\alpha,-a\tau^{2}\right),$
where $a>0$. Then $p=(0,0,1)$ and $-p=(0,0,-1)$ are the singular points of
$\Phi$ and the eigenvalues of $d\Phi_{p}$ and $d\Phi_{-p}$ are given
respectively by
$\lambda_{p}=\frac{a\pm\sqrt{a^{2}+4}}{2},\hskip
14.22636pt\lambda_{-p}=\frac{-a\pm\sqrt{a^{2}+4}}{2}.$ (3.26)
###### Proof.
Note that, if $\Phi(\alpha,\tau,\eta)=0$, then $\alpha=\tau=0$ and $\eta=\pm
1$. Hence, $p=(0,0,1)$ and $-p=(0,0,-1)$ are the singular points of $\Phi$.
The tangent plane in each singular point is defined by $T_{\pm
p}S\approx\\{(\alpha,\tau,\eta)\in\mathbb{R}^{3}:\eta=0\\}$. Thus,
$d\Phi=\left(\begin{array}[]{ccc}0&1&0\\\ 1&a\eta&a\tau\\\
0&-2a\tau&0\end{array}\right).$
Hence $\lambda$ is an eigenvalue of $d\Phi_{\pm p}$ if there is a non null
vector $w=(w_{1},w_{2},0)\in T_{\pm p}S$ such that $d\Phi_{\pm p}(w)=\lambda
w$ i.e.
$\left\\{\begin{array}[]{cc}w_{2}=\lambda w_{1},\\\ w_{1}\pm aw_{2}=\lambda
w_{2}.\end{array}\right.$
Hence, $\lambda$ satisfies $\lambda^{2}\mp a\lambda-1=0$, which gives (3.26).
∎
In Lema 3.12, we saw that $(0,0,\pm 1)$ are saddle points for the vector field
$\Phi$ on $S$, i.e., $\psi(s)=(0,0,\pm 1)$, $s\in I$ are singular solutions of
(3.21). If the functions $\alpha$ and $\tau$ are identically zero then the
corresponding curve $X(s)$ is the intersection of the upper half hyperboloid
with the plane going through the origin, orthogonal to $(0,0,1)$. Hence both
singular solutions of the system correspond to the same curve.
In order to study the non trivial solutions of the system (3.21), we consider
the singular point $p=(0,0,1)$ and $\psi(s,q)$ a solution of the system with
initial condition $q\in S$. Since the eigenvalues of the linearized system at
the singular point are not zero, it follows that the local behavior of the
system (3.21) is equivalent to the linearized one. Hence, there exist initial
conditions $q,\overline{q}\in S\setminus\\{p\\}$ such that
$\displaystyle\lim_{s\to-\infty}\psi(s,q)=p$ and
$\displaystyle\lim_{s\to+\infty}\psi(s,\overline{q})=p$. We define the
unstable and stable sets as
$\displaystyle W^{u}(p)=\\{q\in
S:\lim_{s\to-\infty}\psi(s,q)=p\\}\,\,\;\;\text{and }\,\,\;\;\displaystyle
W^{s}(p)=\\{q\in S:\lim_{s\to+\infty}\psi(s,q)=p\\}.$ (3.27)
From Lemma 3.4 we know that, if the function $\tau$ is a non zero
constant,i.e., $\tau(s)=b$, $b\in\mathbb{R}\setminus\\{0\\}$ for all $s\in I$,
then $b^{2}=a=1$. Our next result provides two non trivial solutions of the
system (3.21), $a=b^{2}=1$, defined on $\mathbb{R}$, with initial conditions
on the set $S$. They are particular cases of the solutions obtained in Lemma
3.4 with the constant of integration being zero. Moreover, we also obtain the
soliton solutions corresponding to these soltions.
###### Proposition 3.13.
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21) defined on
the maximal interval $I$, $a=1$ and $\psi(0)\in S$, where $S$ is given in
(3.22). Then, $\psi(s)=(-s,-1,-s)$ (resp. $\overline{\psi}(s)=(s,1,-s)$),
$s\in I=\mathbb{R}$ satisfy (3.21) with initial condition $(0,-1,0)$ (resp.
$(0,1,0)$). Moreover,
* i)
The curve
$X(s)=\left(\frac{1+l^{2}+s^{2}}{2l},\frac{l^{2}-1-s^{2}}{2l},-s\right),$
(3.28)
where $l>0$ is the soliton solution to the CSF in $\mathbb{H}^{2}$ which
corresponds to the solution $\psi(s)=(-s,-1,-s)$ of (3.21);
* ii)
The curve
$\overline{X}(s)=\left(\frac{1+l^{2}+s^{2}}{2l},\frac{l^{2}-1-s^{2}}{2l},s\right),$
where $l>0$ is the soliton solution to the CSF in $\mathbb{H}^{2}$ which
corresponds to the solution $\psi(s)=(s,1,-s)$ of (3.21)
###### Proof.
Straightforward computations show that $\psi(s)$ and $\overline{\psi}(s)$
satisfy (3.21) with initial conditions $(0,-1,0)$ and $(0,1,0)$ respectively.
i) Note that, if a curve $X(s)$ is given by (3.28), then
$\displaystyle T(s)$ $\displaystyle=$
$\displaystyle\left(\frac{s}{l},-\frac{s}{l},1\right),$ $\displaystyle N(s)$
$\displaystyle=$ $\displaystyle X(s)\times
T(s)=\left(\frac{-1+l^{2}+s^{2}}{2l},\frac{l^{2}+1-s^{2}}{2l},-s\right),$
$\alpha(s)=\langle X(s),(0,0,1)\rangle=s$, $\tau(s)=\langle
T(s),(0,0,1)\rangle=1$ and $\eta(s)=\langle N(s),(0,0,1)\rangle=-s$.
ii) The proof is similar to the case i). ∎
In Figure 2, we illustrate the curve $X(s)$ in $\mathbb{H}^{2}$ given by
(3.28) with $l=1$.
Figure 2: Soliton solution to the CSF with $a=1$, fixed vector $v=(0,0,1)$ and
constant curvature ($k(s)=1$).
In the following lemmas, we study the behavior of the functions $\alpha(s)$,
$\tau(s)$ and $\eta(s)$ when $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ is a non
trivial solution of the system (3.21) and initial condition $\psi(0)\in S$.
###### Lemma 3.14.
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (3.21)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial
condition $\psi(0)\in S$, where $S$ is given in (3.22). If $s_{0}\in I$ is a
critical point of $\alpha(s)$, then $s_{0}$ is the global minimum (resp.
maximum) of $\alpha(s)$ if, and only if, $\alpha(s_{0})>0$ (resp.
$\alpha(s_{0})<0$). Moreover, there exists always $\overline{s}$ such that the
function $\displaystyle\alpha(s)$ is monotone on the intervals
$(\omega_{-},\overline{s})$ and $(\overline{s},\omega_{+}).$
###### Proof.
Let $s_{0}$ be a critical point of $\alpha(s)$, then $\tau(s_{0})=0$. It
follows from (3.21) that
$\alpha^{\prime\prime}(s_{0})=\alpha(s_{0}).$ (3.29)
Note that, if $\alpha(s_{0})=\tau(s_{0})=0$ and $\eta(s_{0})=1$, then
$\alpha(s)$ is constant, because $(0,0,1)$ is a singular (trivial) solution of
(3.21).
Therefore, it follows from (3.29) that $s_{0}$ is a local minimum point of
$\alpha(s)$ if $\alpha(s_{0})>0$. If there is another critical point $s_{1}$
of $\alpha(s)$ such that $s_{0}$ and $s_{1}$ are consecutive, then
$\alpha(s_{1})>\alpha(s_{0})>0$, because $s_{0}$ is a local minimum point.
Thus, $\alpha^{\prime\prime}(s_{1})=\alpha(s_{1})>0$ and $s_{1}$ is a local
minimum point $\alpha(s)$, this is a contradiction. Therefore, if
$\alpha(s_{0})>0$, then $s_{0}$ is a global minimum of $\alpha(s)$.
If $\alpha(s_{0})<0$, it follows from (3.29) that $s_{0}$ a local maximum
point of $\alpha(s)$. The proof that $s_{0}$ is a global maximum point of
$\alpha(s)$ is analogue to the previous case.
Since the function $\alpha(s)$ has at most one critical point, there exists
always $\overline{s}$ such that the function $\displaystyle\alpha(s)$ is
monotone on the intervals $(\omega_{-},\overline{s})$ and
$(\overline{s},\omega_{+}).$ ∎
We observe that if for a solution $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ of
(3.21) with $\psi(0)\in S$, the function $\alpha(s)$ does not have a crtitical
point, then $\alpha(s)$ is monotone on $I$. Moreover, it follows from Lemma
3.14, that for any solution of (3.21) with $\psi(0)\in W^{u}\cup
W^{s}\setminus\\{(0,0,1)\\}$, the functions $\alpha(s)$ and $\eta(s)$ do not
have critical points.
###### Lemma 3.15.
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (3.21)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial
condition $\psi(0)\in S$, where $S$ is given by (3.22). Consider $W^{u}(p)$
and $W^{s}(p)$ given by (3.27). If $\psi(0)\in S\setminus W^{u}(p)$ (resp.
$S\setminus W^{s}(p)$), then
$\displaystyle\lim_{s\to\omega_{-}}|\alpha(s)|=+\infty$ (resp.
$\displaystyle\lim_{s\to\omega_{+}}|\alpha(s)|=+\infty$).
###### Proof.
It follows from Lemma 3.14 that there exits $\overline{s}\in I$ such that
$\alpha(s)$ is monotone on the intervals $(\omega_{-},\overline{s})$ and
$(\overline{s},\omega_{+})$. If $\psi(0)\in S\setminus W^{u}(p)$ assume by
contradiction that $\alpha(s)$ is bounded on $(\omega_{-},\overline{s})$.
Since $\tau^{2}(s)+\eta^{2}(s)=\alpha^{2}(s)+1$, it follows that the functions
$\alpha(s)$ and $\eta(s)$ are bounded and monotone on
$(\omega_{-},\overline{s})$ and the limit
$\displaystyle\lim_{s\to\omega_{-}}\tau(s)$ exists. Hence, there exists
$q\in\mathbb{R}_{1}^{3}$ such that
$\displaystyle\lim_{s\to\omega_{-}}(\alpha(s),\tau(s),\alpha(s))=q$,
$\omega_{-}=-\infty$ and $q$ is a singular solution of (3.21). But the system
(3.21) does not have any singular solution on the set $S\setminus W^{u}(p)$.
Therefore, $\displaystyle\lim_{s\to\omega_{-}}|\alpha(s)|=+\infty$. Similarly,
when $\psi(0)\in S\setminus W^{s}(p)$ one proves that
$\displaystyle\lim_{s\to\omega_{+}}|\alpha(s)|=+\infty$. ∎
###### Lemma 3.16.
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (3.21)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial
condition $\psi(0)\in S$, where $S$ is given in (3.22).
* i)
If $s_{0}$ is a critical point of $\tau(s)$. Then $\tau^{2}(s_{0})\neq 1$ and
$a^{2}\tau^{2}(s_{0})\neq 1$. If $\tau(s_{0})>0$, then $s_{0}$ is a local
minimum (resp. maximum) point of $\tau(s)$ if, and only if, $\tau(s_{0})>1$
(resp. $0<\tau(s_{0})<1$). If $\tau(s_{0})<0$, then $s_{0}$ is a local minimum
(resp. maximum) point of $\tau(s)$ if, and only if, $\tau(s_{0})<-1$ (resp.
$-1<\tau(s_{0})<0$).
* ii)
The function $\tau(s)$ has at the most a finite number of critical points.
###### Proof.
i) Let $s_{0}$ be a critical point of $\tau(s)$. If $\tau^{2}(s_{0})=1$, it
follows from $-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=1$ that
$\alpha^{2}(s_{0})=\eta^{2}(s_{0})$. Thus, $0=\tau^{\prime}(s_{0})=\pm
a\eta(s_{0})+\alpha(s_{0})$ i.e. $a=1$. Hence, it follows from Lemma 3.4 that
the solution $\psi(s)$ of (3.21) with initial condition $\psi(s_{0})$ is a
trivial solution, which contradicts the hypothesis. Therefore,
$\tau(s_{0})\neq 1$. If $a^{2}\tau^{2}(s_{0})=1$, then
$0=\tau^{\prime}(s_{0})=\pm\eta(s_{0})+\alpha(s_{0})$ and from
$-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=1$ we have that $\tau(s_{0})=1$, which
also contradicts the hypothesis. Hence, $\tau^{2}(s_{0})\neq 1$ and
$a^{2}\tau^{2}(s_{0})\neq 1$. Moreover, taking the second derivative of
$\tau(s)$ at $s=s_{0}$, we obtain (3.23). This concludes the proof of the item
i).
ii) Note that, it follows from Lemma 3.14 that there exists always
$\overline{s}$ such that $\tau(s)$ does not change sign on each interval
$(\omega_{-},\overline{s})$ and $(\overline{s},\omega_{+})$. We will prove for
the interval $(\overline{s},\omega_{+})$, since similar arguments can be used
for the interval $(\omega_{-},\overline{s})$. If $\psi(0)\in W^{s}(p)$ i.e.
$\displaystyle\lim_{s\to+\infty}\tau(s)=0$, then it follows from item i) that
$\tau(s)$ has at most a finite number of critical points on
$(\overline{s},\omega_{+})$. If $\psi(0)\in S\setminus W^{s}(p)$, then it
follows from Lemma 3.15 that
$\displaystyle\lim_{s\to\omega_{+}}|\alpha(s)|=+\infty$.
Assume that $\tau(s)>0$ for all $s\in(\overline{s},\omega_{+})$, then there
exists $s_{1}$ such that $\alpha(s)>0$ for all $s>s_{1}$. If the function
$\eta(s)$, which is monotone, is always positive, then $\tau^{\prime}(s)>0$
for all $s>s_{1}$, i.e. $\tau$ has no critical on $(s_{1},\omega_{+})$. Now,
consider $s_{1}$ such that $\eta(s)<0$ for all $s>s_{1}$. Assume by
contradiction that there are $s_{2},s_{3}\in(s_{1},\omega_{+})$, two local
maximum points of $\tau(s)$. From item i), we obtain that there are
$b,d\in(s_{2},s_{3})$ such that $b<d$, $\tau(b)=\tau(d)=1$,
$\tau^{\prime}(b)<0$, $\tau^{\prime}(d)>0$. It follows from
$-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=1$ that $\eta^{2}(b)=\alpha^{2}(b)$
and $\eta^{2}(d)=\alpha^{2}(d)$. Thus, from $\tau^{\prime}(b)<0$ we obtain
$a\eta(b)<-\alpha(b)<0$ i.e. $a>1$ and from $\tau^{\prime}(d)>0$ we conclude
$\alpha(d)>-a\eta(d)>0$ i.e. $a<1$, this is a contradiction. Therefore, $\tau$
has at most one local maximum point on the interval $(s_{1},\omega_{+})$.
Analogously, assume that $\tau(s)<0$ for all $s\in(\overline{s},\omega_{+})$,
then there exists $s_{1}$ such that $\alpha(s)<0$ for all $s>s_{1}$. If the
monotone function $\eta(s)$ is always positive, then $\tau^{\prime}(s)<0$ for
all $s>s_{1}$, and hence $\tau$ has no critical point on $(s_{1},\omega_{+})$.
Now, we consider $s_{1}$ such that $\eta(s)<0$ for all $s>s_{1}$. Assume by
contradiction that there are $s_{2},s_{3}\in(s_{1},\omega_{+})$, two local
maximum points of $\tau(s)$. From item i) we obtain that there are
$b,d\in(s_{2},s_{3})$ such that $b<d$, $\tau(b)=\tau(d)=-1$,
$\tau^{\prime}(b)<0$, $\tau^{\prime}(d)>0$. It follows from
$-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=1$ that $\eta^{2}(b)=\alpha^{2}(b)$
and $\eta^{2}(d)=\alpha^{2}(d)$. Thus, from $\tau^{\prime}(b)<0$ we obtain
that $\alpha(b)<a\eta(b)<0$ i.e $a<1$ and from $\tau^{\prime}(d)>0$ we
conclude that $0>\alpha(d)>a\eta(d)$ i.e. $a>1$, this is a contradiction.
Therefore, $\tau$ has at most one local maximum point on the interval
$(s_{1},\omega_{+})$.
Similar arguments for the interval $(\omega_{-},\overline{s})$ imply that
$\tau(s)$ has at most a finite number of critical points on $I$. ∎
The following lemma shows that the function $\tau(s)$ is bounded and hence the
curvature of the soliton $X(s)$ on $\mathbb{H}^{2}$ is bounded.
###### Lemma 3.17.
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (3.21)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ with
initial condition $\psi(0)\in S$, where $S$ is given by (3.22). Then the
function $\tau(s)$ is bounded on $I$.
###### Proof.
It follows from Lemma 3.14 that there exists $\overline{s}\in I$ such that
$\alpha(s)$ is monotone on the intervals $(\omega_{-},\overline{s})$ and
$(\overline{s},\omega_{+})$. Moreover, from Proposition 3.2 we have that
$\eta(s)$ is monotone.´
If $\psi(0)\in W^{s}(p)$, where $p$ is a singular point, then
$\displaystyle\lim_{s\to+\infty}\tau(s)=0$ and $\tau(s)$ is bounded on
$(\overline{s},+\infty)$ for any $\overline{s}\in I$ fixed. Similarly, if
$\psi(0)\in W^{u}(p)$, then $\displaystyle\lim_{s\to-\infty}\tau(s)=0$ and
$\tau(s)$ is bounded on $(-\infty,\overline{s})$, $\overline{s}\in I$ fixed.
We will now consider the cases when the initial condition $\psi(0)$ belongs to
$S\setminus W^{u}(p)$ or $S\setminus W^{s}(p)$. If $\psi(0)\in S\setminus
W^{u}(p)$ assume by contradiction that $\tau(s)$ is unbounded on
$(\omega_{-},\overline{s})$, then it follows from Lemma 3.16 that there exists
$s_{1}\in(\omega_{-},\overline{s})$ such that $|\tau(s)|>1$ and $a|\tau(s)|>2$
for all $s\in(\omega_{-},s_{1})$. Thus, $|\alpha(s)|>|\eta(s)|$, because
$\alpha^{2}(s)-\eta^{2}(s)=\tau^{2}(s)-1>0$ and $a\tau^{2}(s)>2|\tau(s)|$ for
all $s\in(\omega_{-},s_{1})$. From Lemma 3.14 we have that $\tau(s)$ does not
sign on $(\omega_{-},s_{1})$.
If $\alpha^{\prime}(s)=\tau(s)<0$ on $(\omega_{-},s_{1})$, then $\alpha(s)$ is
strictly decreasing on this interval. Thus, it follows from Lemma 3.15, that
$\displaystyle\lim_{s\to\omega_{-}}\alpha(s)=+\infty$. Hence, $s_{1}$ can be
chosen so that $\alpha(s)$ is decreasing and positive for all $s<s_{1}$.
Therefore, using (3.21) and the fact that $a\tau^{2}(s)>2|\tau(s)|$, we obtain
$2\alpha(s)-2\alpha(s_{1})=-2\int^{s_{1}}_{s}\tau(s)ds<\int^{s_{1}}_{s}a\tau^{2}(s)ds=\eta(s)-\eta(s_{1}),$
i.e.,
$0<\alpha(s)-\eta(s)<2\alpha(s_{1})-\eta(s_{1})-\alpha(s),$
which contradicts Lemma 3.15, because
$\displaystyle\lim_{s\to\omega_{-}}\alpha(s)=+\infty$. Hence, $\tau(s)$ is
bounded on $(\omega_{-},s_{1})$.
If $\alpha^{\prime}(s)=\tau(s)>0$ on $(\omega_{-},s_{1})$, then $\alpha(s)$ is
strictly increasing on this interval. Thus, it follows from Lemma 3.15 that
$\displaystyle\lim_{s\to\omega_{-}}\alpha(s)=-\infty$. Hence, $s_{1}$ can be
chosen so that $\alpha(s)$ is increasing and negative for all $s<s_{1}$.
Therefore, using (3.21) and the fact that $a\tau^{2}(s)>2|\tau(s)|$, we obtain
$2\alpha(s_{1})-2\alpha(s)=2\int^{s_{1}}_{s}\tau(s)ds<\int^{s_{1}}_{s}a\tau^{2}(s)ds=-\eta(s_{1})+\eta(s),$
i.e. ,
$0<-\alpha(s)-\eta(s)<-2\alpha(s_{1})-\eta(s_{1})+\alpha(s),$
which contradicts Lemma 3.15, because
$\displaystyle\lim_{s\to\omega_{-}}\alpha(s)=-\infty$. Hence, $\tau(s)$ is
bounded on $(\omega_{-},\overline{s})$.
When $\psi(0)\in S\setminus W^{s}(p)$, the similar arguments show that
$\tau(s)$ is bounded on $(\overline{s},\omega_{+})$.
Therefore, the function $\tau(s)$ is bounded on $I$. ∎
Our next lemma provides the behavior of the function $\eta(s)$.
###### Lemma 3.18.
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (3.21)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial
condition $\psi(0)\in S$, where $S$ is given by (3.22). Consider $W^{u}(p)$
and $W^{s}(p)$ given by (3.27). If $\psi(0)\in S\setminus W^{u}(p)$ (resp.
$S\setminus W^{s}(p)$), then
$\displaystyle\lim_{s\to\omega_{-}}\eta(s)=+\infty$ (resp.
$\displaystyle\lim_{s\to\omega_{+}}\eta(s)=-\infty$).
###### Proof.
The proof follows from Lemmas 3.15 and 3.17 and the fact that
$\tau^{2}(s)+\eta^{2}(s)=\alpha^{2}(s)+1$. ∎
###### Lemma 3.19.
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (3.21)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial
condition $\psi(0)\in S$, where $S$ is given by (3.22). Then $I=\mathbb{R}$.
###### Proof.
It follows from Lemma 3.17 that $\tau(s)$ is bounded on the interval $I$. Let
$M>0$ be such that $|\tau(s)|\leq M$ for all $s\in I$. Using (3.21), we obtain
that
$|\alpha(s)-\alpha(s_{0})|=\left|\int_{s_{0}}^{s}\tau(s)ds\right|\leq
M|s-s_{0}|$ (3.30)
for all $s\in I$. We will now show that $\omega_{-}=-\infty$ and
$\omega_{+}=+\infty$.
If $\psi(0)\in S\setminus W^{u}(p)$, then from Lemma 3.15 we have that
$\alpha(s)$ is unbounded on $(\omega_{-},\overline{s})$ for any
$\overline{s}\in I$ fixed. Hence, it follows from (3.30) that
$\omega_{-}=-\infty$. It follows from the definition of $W^{u}(p)$ that
$\omega_{-}=-\infty$ when $\psi(0)\in W^{u}(p)$. Since $\displaystyle
S=W^{u}(p)\cup\left[S\setminus W^{u}(p)\right]$, we conclude that
$\omega_{-}=-\infty$.
If $\psi(0)\in S\setminus W^{s}(p)$, then from Lemma 3.15 we have that
$\alpha(s)$ is unbounded on $(\overline{s},\omega_{+})$ for any
$\overline{s}\in I$ fixed. Hence, it follows from (3.30) that
$\omega_{+}=+\infty$. It follows from the definition of $W^{s}(p)$ that
$\omega_{+}=+\infty$ when $\psi(0)\in W^{s}(p)$. Since $\displaystyle
S=W^{s}(p)\cup\left[S\setminus W^{s}(p)\right]$, then $\omega_{+}=+\infty$.
Therefore, $I=\mathbb{R}$. ∎
###### Lemma 3.20.
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (3.21),
with $a>0$ and initial condition $\psi(0)\in H\cup C\cup S$, where $H$, $C$
and $S$ are given by (3.22). Then $\psi(s)$ and the corresponding soliton
solution $X(s)$ to the CSF on $\mathbb{H}^{2}$ are defined for all
$s\in\mathbb{R}$. Moreover, at each end the curvature $k(s)$ of $X$ converges
to one of the following constants $\\{-1,0,1\\}$.
###### Proof.
Since $X(s)$ is a soliton solution to the CSF corresponding to $\psi(s)$, then
$k(s)=a\tau(s)$. Thus, Lemmas 3.6, 3.10, 3.14 and 3.17 imply that $k(s)$ is
bounded on $\mathbb{R}$ and it has at most a finite number of critical points.
Thus, the limits
$\displaystyle\lim_{s\to\pm\infty}k(s)=\displaystyle\lim_{s\to\pm\infty}a\tau(s)$
exist. In particular, when $\psi(0)\in W^{u}(p)$ then
$\displaystyle\lim_{s\to-\infty}\tau(s)=0$. Similarly, when $\psi(0)\in
W^{s}(p)$ then $\displaystyle\lim_{s\to+\infty}\tau(s)=0$. In these cases, the
curvature function converges to zero at $-\infty$ and $+\infty$, respectively.
If $\displaystyle\lim_{s\to\pm\infty}\tau(s)\neq 0$, then
$\displaystyle\lim_{s\to\pm\infty}|\alpha(s)|=+\infty$ and it follows from
$-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=\delta$, where $\delta\in\\{-1,0,1\\}$
that
$\displaystyle\lim_{s\to\pm\infty}\frac{\eta^{2}(s)}{\alpha^{2}(s)}=\displaystyle\lim_{s\to\pm\infty}\left(\frac{-\tau^{2}(s)+\delta}{\alpha^{2}(s)}+1\right)=1.$
Using (3.21), Lemmas 3.8, 3.9, 3.15, 3.18 and L’Hospital rule, we obtain
$\displaystyle\lim_{s\to\pm\infty}-\frac{\eta(s)}{\alpha(s)}=\lim_{s\to\pm\infty}a\tau(s)=\lim_{s\to\pm\infty}k(s).$
Therefore, $\displaystyle\lim_{s\to-\infty}k(s)=\pm 1$ and
$\displaystyle\lim_{s\to+\infty}k(s)=\pm 1$. ∎
Finally, we will prove our main theorem.
Proof of Theorem 2.3. For any vector $v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$,
without loss generality we can consider $v=ae$, where $a>0$ and
$\displaystyle e=\left\\{\begin{array}[]{clll}(-1,0,0)&\text{if}&v&\text{is a
timelike vector},\\\ (-1,1,0)&\text{if}&v&\text{is a lightlike vector },\\\
(0,0,1)&\text{if}&v&\text{is a spacelike vector.}\end{array}\right.$
Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21) defined on
the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial condition
$\psi(0)\in\mathbb{R}^{3}$ satisfying
$\displaystyle-\alpha^{2}(0)+\tau^{2}(0)+\eta^{2}(0)=\left\\{\begin{array}[]{clll}-1&\text{if}&v&\text{is
a timelike vector},\\\ 0&\text{if}&v&\text{is a lightlike vector},\\\
1&\text{if}&v&\text{is a spacelike vector,}\end{array}\right.$
i.e., $\psi(0)\in H\cup C\cup S$, where $H$, $C$ and $S$ are the disjoint sets
given by (3.22). Moreover, it follows from Proposition 3.2 that there is a
soliton solution $X(s)$ to the CSF, with curvature $k(s)=a\tau(s)$, such that
the relations
$\alpha(s)=\langle X(s),e\rangle,\hskip 8.5359pt\tau(s)=\langle
T(s),e\rangle\hskip 8.5359pt\text{and}\hskip 8.5359pt\eta(s)=\langle
N(s),e\rangle,$
are satisfied, where $T$ and $N$ are the unit vector fields tangent and normal
to $X$.
Thus, the initial conditions of (3.21), which are given by two constants,
determine the soliton solution in each case. Therefore, for each fixed vector
$v\in\mathbb{R}_{1}^{3}\setminus\\{0\\}$ there is a 2-parameter family of non
trivial soliton solutions to the CSF in $\mathbb{H}^{2}$. Moreover, it follows
from Lemmas 3.4, 3.11 and 3.19 that each soliton solution is defined for all
$s\in\mathbb{R}$, i.e. $I=\mathbb{R}$ and Lemma 3.20 shows that the curvature
at each end converges to one of the following constants $\\{-1,0,1\\}$.
Note that, from Lemmas 3.4, 3.5 and 3.14 we know that there exists
$\overline{s}\in\mathbb{R}$ such that $\alpha(s)$ is strictly monotone on the
intervals $(-\infty,\overline{s})$ and $(\overline{s},+\infty)$. Since
$\alpha(s)$ describes the Euclidean height of $X(s)$ with respect to a fixed
plane, then $X(s)$ does not have self-intersections in each one of the
intervals$(-\infty,\overline{s})$ and $(\overline{s},+\infty)$. Therefore,
$X(s)$ is embedded if $\alpha(s)$ is monotone in $\mathbb{R}$.
If $\alpha(s)$ is not monotone in $\mathbb{R}$ then $\alpha(s)$ has only one
critical point. Suppose that $X(s)$ has some self-intersection and consider
$\Sigma$ the simple region bounded by $X([s_{1},s_{2}])$ with
$X(s_{1})=X(s_{2})$, $s_{1}<\overline{s}<s_{2}$ and $\theta$ the external
angle between the tangent vectors $T(s_{1})$ and $T(s_{2})$, which is at the
most $\pi$. By Gauss-Bonnet’s theorem, we obtain
$\displaystyle 0<2\pi\chi(\Sigma)-\theta$ $\displaystyle=$
$\displaystyle\int_{\Sigma}\kappa d\sigma+\int_{X([s_{1},s_{2}])}k(s)ds$
$\displaystyle=$
$\displaystyle-\int_{\Sigma}d\sigma+\int_{X([s_{1},s_{2}])}a\tau(s)ds$
$\displaystyle=$
$\displaystyle-\int_{\Sigma}d\sigma+a[\alpha(s_{2})-\alpha(s_{1})]$
$\displaystyle=$ $\displaystyle-\int_{\Sigma}d\sigma<0.$
This is a contradiction. Hence, the soliton solution $X(s)$ to the CSF in
$\mathbb{H}^{2}$ does not admit self-intersections. Note that, $X(s)$ is
already embedded on the intervals $(-\infty,\overline{s})$ and
$(\overline{s},+\infty)$ and from Lemmas 3.8 and 3.15 we have that the two
ends of the curve are unbounded. Therefore, $X(s)$ is an embedded curve.
$\square$
## 4 Visualizing some Soliton Solutions to the CSF on $\mathbb{H}^{2}$
In this setion, we visualize some examples of soliton solutions to the CSF on
the hyperbolic space. In order to do so, we use the following parametrization.
for $\mathbb{H}^{2}$
$\chi(u,w)=(\sqrt{1+u^{2}+w^{2}},u,w)$
If a curve of $\mathbb{H}^{2}$ $X(s)=\chi(u(s),w(s))$ is parametrized by arc
lentgh, then
$T(s)=\left(\frac{u(s)u^{\prime}(s)+w(s)w^{\prime}(s)}{\sqrt{1+u^{2}(s)+w^{2}(s)}},u^{\prime}(s),w^{\prime}(s)\right)$
the functions $u(s)$ and $w(s)$ satisfy the following system of ODEs
$\left\\{\begin{array}[]{ll}(u^{\prime})^{2}+(w^{\prime})^{2}+\left(u^{\prime}w-uw^{\prime}\right)^{2}=1+u^{2}+w^{2},\\\
w^{\prime\prime}u^{\prime}-u^{\prime\prime}w^{\prime}+uw^{\prime}-u^{\prime}w=k(s)\sqrt{1+u^{2}+w^{2}},\end{array}\right.$
(4.1)
where $k(s)$ is the curvature of $X(s)$. The first equation follows from the
fact that the curve is parametrized by arc lentgh and the second one from the
expression of the curvature of $X$.
In Theorem 2.2, we saw that the curvature of a soliton solution to the CSF on
$\mathbb{H}^{2}$ is determined by its tangent vector field and a non zero
fixed vector $v$. We use (4.1) and the software Maple to plot examples of such
solitons. In each example, we visualize the curve on the three models of the
2-dimensional hyperbolic space, namely the hyperboloid, the Poincaré disk and
the upper half space.
In Figure 3 a), the blue curve on the hyperboloid provides the visualization
of a soliton solution $X(s)$ to the CSF on $\mathbb{H}^{2}$ whose curvature is
given by $\displaystyle k(s)=\langle T(s),(-1,0,0)\rangle$ and $a=1$. The red
curve is the Euclidean orthogonal projection of $X(s)$ on the plane that
contains the origin and it is orthogonal to the vector $(-1,0,0)$. In Figures
3 b) and c) we visualize the same soliton on the Poincaré disk and on the half
space model respectively.
(a)
(b)
(c)
Figure 3: Soliton solution to the CSF on $\mathbb{H}^{2}$ with fixed vector
$v=(-1,0,0)$ and $a=1$.
In Figure 4 a), the blue curve provides the visualization of a soliton
solution $X(s)$ to the CSF on $\mathbb{H}^{2}$ whose curvature is given by
$\displaystyle k(s)=\langle T(s),(-1,1,0)\rangle$ and $a=1$. In Figures 4 b)
and c) we visualize the same soliton on the Poincaré disk and on the half
space model respectively.
(a)
(b)
(c)
Figure 4: Sóliton do fluxo FRC em $\mathbb{H}^{2}$ com vetor fixado $(-1,1,0)$
e $a=1$.
Finally, in Figure 5 a), the blue curve provides the visualization of a
soliton solution $X(s)$ to the CSF on $\mathbb{H}^{2}$ whose curvature is
given by $\displaystyle k(s)=\langle T(s),(0,0,1)\rangle$ and $a=1$. In
Figures 4 b) and c) we visualize the same soliton on the Poincaré disk and on
the half space model respectively. We point out that this soliton has non
constant curvature and hence it is different from the one given in Proposition
3.13. In fact, in order to obtain Figure 5, we used initial condition
$u(0)=w(0)=0$ and $\displaystyle
u^{\prime}(0)=-w^{\prime}(0)=-\,\frac{1}{\sqrt{2}}$ for the system (4.1),
i.e., $\displaystyle
T(0,0)=\left(0,-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$ and
$\displaystyle\tau(0)=\frac{1}{\sqrt{2}}\neq\pm 1$. Hence the curvature is not
constant. In Figures 5 b) and c) we visualize the soliton given in Figure 5 a)
on the Poincaré disk and on the half space model respectively.
(a)
(b)
(c)
Figure 5: Sóliton do fluxo FRC em $\mathbb{H}^{2}$ com vetor fixado $(0,0,1)$
e $a=1$.
Aknowlegment: The first author aknowledges the support given by the
Universidade Federal do Oeste da Bahia during his graduate program at the
Universidade de Brasília, when this research was undertaken.
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|
# Possible periodic activity in the short bursts of SGR 1806-20: connection
to fast radio bursts
G. Q. Zhang School of Astronomy and Space Science, Nanjing University, Nanjing
210093, China Zuo-Lin Tu School of Astronomy and Space Science, Nanjing
University, Nanjing 210093, China F. Y. Wang School of Astronomy and Space
Science, Nanjing University, Nanjing 210093, China Key Laboratory of Modern
Astronomy and Astrophysics (Nanjing University), Ministry of Education,
Nanjing 210093, China F. Y. Wang<EMAIL_ADDRESS>
###### Abstract
Magnetars are highly magnetized neutron stars that are characterized by
recurrent emission of short-duration bursts in soft gamma-rays/hard X-rays.
Recently, FRB 200428 were found to be associated with an X-ray burst from a
Galactic magnetar. Two fast radio bursts (FRBs) show mysterious periodic
activity. However, whether magnetar X-ray bursts are periodic phenomena is
unclear. In this paper, we investigate the period of SGR 1806-20 activity.
More than 3000 short bursts observed by different telescopes are collected,
including the observation of RXTE, HETE-2, ICE and Konus. We consider the
observation windows and divide the data into two sub-samples to alleviate the
effect of unevenly sample. The epoch folding and Lomb-Scargle methods are used
to derive the period of short bursts. We find a possible period about
$398.20\pm 25.45$ days. While other peaks exist in the periodograms. If the
period is real, the connection between short bursts of magnetars and FRBs
should be extensively investigated.
Magnetar; Soft gamma repeaters; fast radio bursts
## 1 Introduction
Soft gamma repeaters (SGRs) are associated with extremely magnetized neutron
stars (named magnetars, Kouveliotou et al. (1998); Kaspi & Beloborodov
(2017)). Magnetars undergo occasional random outbursts during which time their
persistent emission increases significantly while simultaneously emitting
bursts (or intermediate flares), in the hard X-ray or soft gamma-ray energy
regime. More than 20 SGRs have been discovered and multiple bursts have been
detected from each source. Recently, SGR 1935+2154 reached its active phase
and produced a burst forest. Among these bursts, there is a very special burst
that is associated with FRB 200428 (The CHIME/FRB Collaboration et al., 2020;
Bochenek et al., 2020; Lin et al., 2020; Li et al., 2020; Mereghetti et al.,
2020).
Fast radio bursts (FRBs) are millisecond duration radio bursts with high
dispersion measures and brightness temperatures (Lorimer et al., 2007; Petroff
et al., 2019; Cordes & Chatterjee, 2019; Zhang, 2020a; Xiao et al., 2021).
Among the observed FRBs, repeating FRBs are more interesting. They show
multiple bursts, which indicates a non-catastrophe central engine, such as the
flaring activity of magnetars (Lyubarsky, 2014; Kulkarni et al., 2014; Katz,
2016; Murase et al., 2016; Metzger et al., 2017; Beloborodov, 2017; Wang et
al., 2020), the cosmic combing (Zhang, 2017, 2018), the collision between
neutron stars and asteroids (Geng & Huang, 2015; Dai et al., 2016), etc
(Platts et al., 2019). Moreover, the statistical properties of the repeating
bursts are consistent with those of Galactic magnetar bursts (Wang & Yu, 2017;
Wadiasingh & Timokhin, 2019; Cheng et al., 2020). Recently, FRB 200428 has
been detected to be originated from the Galactic SGR 1935+2154 (The CHIME/FRB
Collaboration et al., 2020). This observation supports the model that FRBs
origin from magnetars. The burst time of FRB 200428 is consistent with that of
an X-ray burst (Li et al., 2020; Mereghetti et al., 2020).
An intriguing property of repeating FRBs is the mysterious periodic activity.
FRB 180916, the second localized repeating FRB, has been found with a period
of $16.35\pm 0.15$ days (CHIME/FRB Collaboration et al., 2020). Later, Rajwade
et al. (2020) found a possible period 156 days for FRB 121102, which was
confirmed by Cruces et al. (2020). So far, there is no similar period behavior
that have been found in other repeating FRBs. This may be caused by the small
number of observed bursts. Whether all repeating FRBs are periodic is still
unknown.
Due to the connection between radio bursts of FRBs and X-ray bursts of
magnetars, it is natural to consider whether X-ray bursts of magnetars have
similar periodic behavior. Although only one radio burst has been observed
from SGR 1935+2154, many short X-ray bursts have been detected from this
source. A possible periodic behavior has been found in SGR 1935+2154 (Grossan,
2020). The reported period is about 232 days, which is similar to that of FRB
121102. The spin period of SGR 1935+2154 is about 3.2 s (Israel et al., 2016),
which is much shorter than 232 days. Thus, this active cycle must be caused by
other processes.
There are some models to explain the periodic behavior of FRBs. The first one
is a binary system containing a magnetar (Ioka & Zhang, 2020; Lyutikov et al.,
2020; Gu et al., 2020). The periods of FRBs origin from the orbital periods of
the binaries. The second model is the free precession of magnetars (Levin et
al., 2020; Zanazzi & Lai, 2020). In this scenario, the strong magnetic field
deforms the magnetar, which induces the free precession with a period from
weeks to months. Similar to the free precession, there are also some works to
investigate the force precession. Sob’yanin (2020) suggested that the forced
precession is natural and can be used to explain the period of FRBs. The
fallback disk and the orbit motion may also induce the force precession (Tong
et al., 2020; Yang & Zou, 2020). Besides, the magnetar-asteroid impact model
also is proposed to explain the observed periodicity (Dai & Zhong, 2020).
Although a possible periodic behavior has been found in SGR 1935+2154, it is
still unclear that whether the periodic behavior is common in magnetars. If
the active cycle is unique for SGR 1935+2154, the origin of this behavior may
be associated with the birth of FRBs, as suggested by some binary models (Ioka
& Zhang, 2020). While if the period behavior is common for magnetars, the
mechanism of this period maybe also valid for FRBs. SGR 1806-20 is a typical
magnetar, which was discovered in 1979. Until to now, thousands of bursts have
been detected from this source (Ulmer et al., 1993; Aptekar et al., 2001;
Nakagawa et al., 2007; Prieskorn & Kaaret, 2012). We investigate the periodic
behavior of this active source.
This letter is organized as follows. In Section 2, we compile the observations
of SGR 1806-20 which are used to derive the period. In Section 3, two methods
are used to derive the period. We discuss the possible relationship between
FRB period and SGR period in Section 4. Finally, conclusions are given in
Section 5.
## 2 The data sample
After the first detection in 1979, SGR 1806-20 has been observed by many
telescopes. Thousands of bursts have been reported (Ulmer et al., 1993;
Aptekar et al., 2001; Nakagawa et al., 2007; Prieskorn & Kaaret, 2012). The
spin period of this source is 7.55 s and the spin-down rate is $4.95\times
10^{-10}$ s/s (Woods et al., 2007). Among the observed magnetars, the surface
magnetic field of SGR 1806-20 is the strongest, which is about $2\times
10^{15}$ G 111http://www.physics.mcgill.ca/~pulsar/magnetar/main.html(Olausen
& Kaspi, 2014). The strong dipole field is capable to drive strong bursts. As
an example, a giant flare has been detected from this source (Palmer et al.,
2005). No FRBs-like event was detected to associated with this giant flare
(Tendulkar et al., 2016).
The observations of SGR 1806-20 from different telescopes are collected. This
source is close to the ecliptic, so it is difficult to observe in December and
January. The unevenly sample may induce the false periodic signal. To
alleviate this effect, we also collect the observation windows of different
telescopes. The collected data includes the following four sub-samples.
* •
The observation of Rossi X-ray Timing Explorer (RXTE). We use the catalog
reported by Prieskorn & Kaaret (2012), which contains over 3040 bursts from
SGR 1806-20. This catalog collects the bursts observed from Nov. 1996 to Sep.
2009. The timeline of RXTE is recorded in XTEMASTER
222https://heasarc.gsfc.nasa.gov/W3Browse/all/xtemaster.html and
XTESLEW333https://heasarc.gsfc.nasa.gov/W3Browse/xte/xteslew.html catalog. We
collect the observation windows from these two catalogs.
* •
The observation of The High Energy Transient Explorer (HETE-2). 50 bursts are
recorded in this sub-sample (Nakagawa et al., 2007). The observation last for
five years, from 2001 to 2005. Among these bursts, 41 bursts are detected in
2004 and 2005. HETE-2 is always point in the anti-solar direction. Therefore,
the bursts observed by HETE-2 are concentrated on the summer season. The
timelines of HETE-2 are listed in
HETE2TL444https://heasarc.gsfc.nasa.gov/W3Browse/all/hete2tl.html catalog.
However, this catalog only lists the observation time, not the duration of
observation. In our calculation, we only consider the most important
observation windows. We assume that the observation during summer is
continuous and use the timeline recorded in HETE2TL to set the start time and
end time of each year.
* •
The observation of Konus-Wind. This sub-sample includes 25 bursts (Aptekar et
al., 2001). In this sample, only one burst occurred in 1979, the other bursts
occurred in 1996-1999. We delete the 1979 burst from this sub-sample because
it was observed by another telescope. The entire celestial is monitored by
Konus-Wind with a duty cycle of 95%. In our calculation, we assume that the
observation of Konus-Wind is continuous555V. D. Pal’shin private
communication. The earliest burst and the latest burst are taken as the start
time and end time of this observation window.
* •
The observation of Internal Cometary Explorer (ICE). It contains 134 bursts
from 1979 to 1984 (Ulmer et al., 1993). Most of the observed bursts occurred
in 1983. This telescope is designed to continuously observe the Sun. It can
continuously observe any source closed in the ecliptic plane. Laros et al.
(1987) estimated the effective coverage from 1978 Aug. to 1983 Dec. to be 75%
$\pm$ 25% and after 1984 Jan. to less than 20%. This telescope has expired
many years. We are unable to obtain the detailed observation windows. For
simplification, we assume the observation is continuous until Jun. 1984. This
hypothesis covers the duration with high effective coverage and contains all
the bursts.
We divide these sub-samples into two classes: sample A, the observation of
RXTE and sample B, the other three observations. The sample A contains more
bursts and has a clear observation window, which is used to derive the period
of SGR 1806-20. The sample B is used to examine the reliability of the period
derived from the sample A.
## 3 Methods
Two methods are used to search the period of SGR 1806-20, including the epoch
folding and the Lomb-Scargle periodogram. In our calculation, MJD 43840 is
taken as phase 0. This choice is a little arbitrary, but it does not
significantly affect our results if the period is real.
### 3.1 Epoch Folding
The epoch folding method has been used to derive the period of FRB 180916
(CHIME/FRB Collaboration et al., 2020). We try to find the active period of
SGR 1806-20 using this method. The burst time of SGR 1806-20 can be folded
into different phases through
$\psi=\frac{T-T_{0}}{P}-floor(\frac{T-T_{0}}{P}),$ (1)
where $\psi$ is the folded phase, $T$ is the burst time, $T_{0}$ is the start
point (MJD 43840), $P$ is a given period, and $floor$ is a function which
returns the floor of the input number. The folded phases $\psi$ are grouped
into different phase bins. For sample A, we use 20 phase bins in our
calculation. The number of bursts in sample B is less, so we only use 10 phase
bins. The classical Pearson $\chi^{2}$ is used to examine the derivation from
uniformity. It can be calculated as
$\chi^{2}=\sum^{20}_{i}\frac{(N_{i}-pT_{i})^{2}}{pT_{i}},$ (2)
where $N_{i}$ is the observed number of bursts in the $i$th bin, $p$ is the
average burst rate, and $T_{i}$ is the observation time of the $i$th phase
bin. The peak of $\chi^{2}$ indicates the period of the source.
We search the period from 50 days to 500 days with the step $\Delta
f=0.1/T_{span}$ in frequency, where $T_{span}$ is the longest time between the
first observed burst and the last one. The reduced $\chi^{2}$ of these two
samples are shown in the top panels of Figure 1 and Figure 2, respectively.
The vertical green dashed line indicates the peak of the reduce $\chi^{2}$.
Using sample A, we derive the period is about 398.20 days. However, there is a
peak around 430 days with similar significance. We use the vertical red dashed
line to indicate this peak. This period is caused by the observation window,
we will prove it using Lomb-Scargle periodogram in the following section. The
peak of the reduced $\chi^{2}$ of sample B is 395.86 days, which is consistent
with that of sample A. We use the vertical green dashed line to indicate this
peak and use the vertical red dashed line to denote a similar high peak, which
is caused by the observation window. In these two samples, the high reduced
$\chi^{2}$ of the peak suggest that this peak is almost impossible to be
caused by chance. But it can not rule out the case that these peaks are caused
by the observation windows.
### 3.2 Lomb-Scargle periodogram
The Lomb-Scargle periodogram can be used to deal with unevenly sampled
observations (Lomb, 1976; Scargle, 1982; VanderPlas, 2018). Cruces et al.
(2020) used this method to verify the period of FRB 121102. We use the
LombScargle function provided by astropy666https://www.astropy.org/ to
calculate the periodogram of these two samples. The period is searched from 50
days to 500 days. The periodograms are shown as blue solid line in the middle
panels of Figures 1 and 2, respectively. The vertical green dashed lines
indicate the periods derived from the epoch folding method. In each figure,
the green dashed line is coincided with one peak of the Lomb-Scargle
periodogram. However, there are other peaks in each periodogram. The
periodogram of sample A has a maximum peak about 76.96 days, which is caused
by the observation window. The peak of sample B is 395.86 days, but there are
some peaks similar significance.
We also check the false alarm probabilities of the peaks, which are
$2.11\times 10^{-84}$ for the peak 398.20 days in sample A and $1.45\times
10^{-40}$ for the peak 395.86 day in sample B. This low false alarm
probability suggests that these peaks are unlikely to have occurred by chance.
However, it may be caused by the observation window, not the internal period
of SGR 1806-20.
### 3.3 Simulation
The observation windows have a strong impact on the period search. To
understand its effect, we simulate a series of points that are uniformly
distributed in the observation windows. The Lomb-Scargle method is used to
deal with these simulated points. The interval of simulated points is 0.1
days, which is much shorter than the possible period of SGR 1806-20.
Therefore, the internal period of simulated points would not affect the
results. The peaks in the simulated periodogram are caused by the unevenly
sample. We show the periodogram of simulated points in the middle panels of
Figure 1 and 2 with blue dot-dashed lines. In Figure 1, many peaks of the
observed data coincide with the peaks of the simulated points. In this figure,
the vertical red dashed line is the second peak derived from epoch-folding.
This line is coincided with a peak of the simulated periodogram, which
supports that this peak is caused by the observation window. While the green
line agrees with a bottom of the simulated periodogram. Therefore, this peak
is unlikely to be caused by the observation window. We normalize the observed
periodogram and the simulated periodogram with maximum values and subtract the
simulated periodogram from the observed periodogram. The result is shown in
the bottom panel of Figure 1 as blue solid line. In this figure, the vertical
green dashed line is the period derived from epoch folding and vertical red
dashed line is the second peak of epoch folding results. The peak of this
periodogram is 398.20 days, which is consistent with the period derived from
epoch-folding. We also check the Lomb-Scargle periodogram caused by the
observation window of sample B. The simulated periodogram is shown in the
middle panel of Figure 2 with blue dot-dashed line. This periodogram has
several peaks near 360 days. Like sample A, we subtract the normalized
simulated periodogram from the normalized observed periodogram and show the
result in the bottom panel of Figure 2. This periodogram also has a peak about
395 day, but the maximum peak is about 278 days. This may be caused by two
reasons. The first one is inappropriate subtraction. We normalize these two
periodograms with maximum values and perform the subtraction. It can tell us
which peak is caused by the observation window, but can’t give the
significance of this peak. The second reason is the incomplete observation
window. The detailed observation windows of HETE-2 and ICE are unclear. We
assume the observation is continuous in a specific window. This assumption
imports some uncertainties.
According to these two methods, the period is about $398.20\pm 25.45$ days for
sample A and $395.86\pm 3.92$ days for sample B. The error is derived using
the method in CHIME/FRB Collaboration et al. (2020). It can be derived as
$\sigma_{P}=PW_{active}/T_{span}$, where $P$ is the period and $W_{active}$ is
the number of active days. The periods of these two samples are consistent
with each other, but a slight different. This may be caused by the variation
of period, which will be discussed in the next section. Taking MJD 43840 as
the phase 0, we show the folded phase histogram in Figure 3. The period is
chosen as $395.86$ day, because the burst time of sample B is closer to phase
0. In this figure, the blue histogram is the distribution of sample A and the
red line is the kernel density estimation of sample B. The distribution of
these two samples both show a peak around phase 0.58. But in the case of
sample A, the distribution has other peaks, which are about 0.12, 0.77, and
0.87. While in sample B, the bursts are concentrated on phase 0.5-0.6. The
number of burst located in other phases is very small. The peak of phase
distribution of these two different samples is similar, which enhances the
reliable of the derived period. Besides, the phase distribution of SGR 1806-20
is different from those of FRB 180916 and FRB 121102. The phase distributions
of these two FRBs are concentrated on a small region, while SGR 1806-20 has
multiple peaks and spreads on the whole phase. We also show the burst time and
phase in Figure 4. The different colored points denote the bursts observed by
different telescopes. The gray regions represent the main peak and the latter
two peaks of the phases. Most of the points are located in the gray region.
Due to the existence of the first peak, there are some points outside the gray
region.
Although this period exists in these two samples, the significance of this
peak is not strong enough. There are multiple peaks around 50-150 days in the
Lomb-Scargle periodogram of sample A. Even considering the effects of the
observation window, there are still several peaks that cannot be explained,
such as the peaks about 76 days and 131 days. The reduce $\chi^{2}$ of epoch-
folding also show several peaks about 149 days and 199 days. These peaks are
difficult to understand. The results of sample B are much worse. The reduce
$\chi^{2}$ and the Lomb-Scargle periodogram both have other significant peaks,
and these peaks cannot be explained by the observation window. Sample B only
contains 208 bursts, which is much smaller than the bursts in sample A. The
observation windows of sample B are not determined very well. Besides, the
burst time of sample B spans a large range, from 1979 to 2005. The period has
undergone evolution during this long epoch. All of these factors can have
impacts on the results. In our results, 398 day is the most possible period of
SGR 1806-20, but it is not significant enough.
## 4 Discussions
The association between FRB 200428 and SGR 1935+2154 supports the conjecture
that FRBs origin from magnetars and FRBs are accompanied with X-ray bursts.
Therefore, the periods of FRB and SGR may be correlated.
Some theoretical models have been proposed to explain the periodic activity of
FRBs. For example, the binary model has been proposed to explain the periods
of FRB 180916 and FRB 121102 (Ioka & Zhang, 2020; Lyutikov et al., 2020).
Although this model can give a reasonable explanation of the period of FRBs,
it is difficult to apply this model to SGRs. There is no evidence to support
the existence of a companion for SGR 1806-20. It is difficult to observe it
due to the large distance. More importantly, unlike the radio emission, the
X-ray bursts would not be absorbed by stellar winds. Thus, the periodic
activity of SGR 1806-20 can not be caused by the orbital motion.
Another promising model of FRBs period is the free precession of magnetars
(Levin et al., 2020; Zanazzi & Lai, 2020). The free precession originates from
the non-sphericity of magnetars, which may be caused by the strong internal
magnetic field or the misaligned between the principal axis of the elastic
crust and the angular velocity (Zanazzi & Lai, 2020). Although the superfluid
vortices insides the magnetar can suppress the free precession (Shaham, 1977),
Levin et al. (2020) proposed that hyperactive magnetars are likely hot enough
to quell superfluid vortices. Besides, the force precession model also is
discussed by some works (Sob’yanin, 2020; Tong et al., 2020; Yang & Zou,
2020). The torque can come from electromagnetic field of magnetars (Sob’yanin,
2020), the companion (Yang & Zou, 2020), or the fallback disk (Tong et al.,
2020). This torque can enhance the precession and lead to a large period.
In order to explain the periodic activity of FRBs, the free precession model
requires that the radio bursts tend to occur in a specific location. It is
believed that X-ray bursts of SGRs are generated by starquakes of magnetars
(Thompson & Duncan, 1995). But the trigger mechanism of these bursts is a
mystery. Whether burst emissions locate in a small region or a large area is
unclear at present. Some models suggest that the bursts tend to occur in a
small region (Gourgouliatos et al., 2015; Lander et al., 2015). In this case,
the period of SGR 1806-20 can be explained by the free precession. The
precession period of magnetar is given by (Levin et al., 2020)
$P_{\mathrm{pr}}\simeq
396\left(\frac{k}{0.01}\right)^{-1}\left(\frac{B_{\text{int}}}{7\times
10^{15}\mathrm{G}}\right)^{-2}\frac{B_{\text{dip}}}{2\times
10^{15}\mathrm{G}}\left(\frac{t}{240~{}\mathrm{yr}}\right)^{1/2}\text{days},$
(3)
where $k$ is numerical coefficient, $B_{\rm int}$ is the internal magnetic
field of magnetar, $B_{\rm dip}$ is the surface dipole magnetic field, and $t$
is the age of magnetar. If the magnetic field is fully coherent and purely
toroidal, $k$ would approach 1. The value of $k$ will be reduced if the field
is tangled. The surface dipole magnetic field of SGR 1806-20 is $2\times
10^{15}$ G and the age is about 240 yr (Olausen & Kaspi, 2014). If the
internal magnetic field is about $7\times 10^{15}$ G and $k=0.01$, the
precession period is about 396 days, which is very close to the burst period
of SGR 1806-20.
The precession model can also explain the wide span of bursts in the phase
space. First, these bursts are tend to occur in a special region, but also can
occur in other positions. This will affect the period determination.
Therefore, the bursts can span a wide range in the phase space. Second, from
equation (3), we can see that the precession period depends on the strength of
magnetic field and age of magnetars. Therefore, the period can evolve. Because
of the long observational time of SGR 1806-20, from 1979 to 2011, and the
young age of SGR 1806-20, the period could evolve significantly, which causes
some bursts outside the grey region in Figure 4. The multiple peaks in Figure
1 and Figure 2 may also be caused by the evolution of period. Considering the
association between FRBs and X-ray bursts, if the precession explanation is
correct, we predict that the periods of FRBs evolve with time. Some works
suggested the ages of central magnetars of FRB 121102 and FRB 180916 are young
(Metzger et al., 2017; Cao et al., 2017; Marcote et al., 2020; Wu et al.,
2020; Zhao et al., 2020). Future long-term monitoring is required to test this
prediction.
## 5 Conclusion
The period behavior of FRBs is still a mystery. Given the connection between
FRBs and X-ray bursts of SGRs, we investigate the period behavior of SGR
1806-20. Three methods are used to derive the period, including the epoch
folding method, the Lomb-Scargle periodogram and the QMIEU periodogram. To
alleviate the effect of unevenly sample, we divide the observation into two
samples. The sample A contains the observation of RXTE and the sample B
includes the observation of ICE, HETE-2 and Konus. We find the period
$398.20\pm 25.26$ days for all the cases. The phase distribution is shown in
Figure 3. The blue histogram is the distribution of sample A and the red line
is the kernel density estimate of sample B. The phase distribution is
consistent with each other at the main peak $\psi\simeq 0.58$. There are other
peaks in sample A, but these peaks are invisible in sample B. Although the
peak about 398 days is visible in both sample A and sample B, the existence of
other peaks suggests that this period is not significant enough. This period
may be caused by observational window, not the internal period of SGR 1806-20.
We discuss possible physical mechanisms for the periodic behavior. If the
triggers of bursts are tended to be localized in a small region, the
precession model can explain the periodic behavior. Considering the
association between FRBs and bursts of SGRs, the physical mechanism of
periodic behavior may be same. The free precession model also predicts that
the period evolves with time, which can be tested with long-term monitoring.
The unstable period can also explain the multiple peaks in the periodogram and
the bursts outside the main phase peak.
The association between SGR and FRB periods may be complex. From observations,
29 bursts of SGR 1935+2154 were not associated with FRBs (Lin et al., 2020).
One most possible reason is that the FRB emission is much more beamed than SGR
burst (Zhang, 2020b). Radio bursts from SGR J1935+2154 discovered by FAST
(Zhang et al., 2020) and the BSA LPI radio telescope (Alexander & Fedorova,
2020) may be due to beaming. If this situation is common in SGRs, the periods
of FRB activity and SGR activity are not the same. If the duty cycle of SGR
bursts is large, there may be no relevant FRB period. On the other hand,
several radio bursts were observed overlapping with X-ray monitoring, without
an associated X-ray burst detection (Zhang et al., 2020; Kirsten et al.,
2020). Given the energy ratio between FRB 200428 and the XRB from SGR
1935+2154, the flux of X-ray burst is too low for current X-ray telescopes.
## acknowledgements
We thank the anonymous referee for constructive comments. We thank V. D.
Pal’shin, for helpful discussion on the observation window of Konus-Wind. We
thank Z. Prieskorn and P. Kaaret for their kindness to share the burst catalog
of SGR 1806-20. This work is supported by the National Natural Science
Foundation of China (grant U1831207).
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Figure 1: The period of SGR 1806-20 derived from sample A. This sample
contains the observation of RXTE, which has over 3040 bursts. The top panel is
the result of epoch-folding method. The Lomb-Scargle periodogram of observed
data and simulated data are shown in the middle panel with blue solid line and
blue dot-dashed line, respectively. The simulated data are uniformly
distributed in the observation windows, so these peaks are caused by the
observation windows. We normalized these two periodgrams with the maximum
value and subtract the simulated periodogram from the observed one. The result
is shown in the bottel panel with blue solid line. The green dashed line
indicates the period derived from epoch-folding, which is 398.20 day. This
period is consistent with a peak of Lomb-Scargle method. Considering the peak
caused by the observation window, the maximum peak is also consistent with
this period. The red dashed line is another peak in the reduce $\chi^{2}$,
which is consistent with a peak of simulated points. Therefore, this peak is
caused by the observation window. Figure 2: The period of SGR 1806-20 derived
from sample B. This sample contains the observations of HETE-2, Konus and ICE.
It contains 208 bursts. We show the results derived from epoch-folding and
Lomb-Scargle periodogram. We show the reduce $\chi^{2}$ in the top panel. The
vertical green line is the derived period 395.86 days. In the middle panel,
the blue solid line is the observed periodogram and the blue dot-dashed line
is the simulated periodogram. We normalized these two periodgrams with the
maximum value and subtract the simulated periodogram from the observed one.
The result is shown in the bottel panel. The green line indicates the best
result of epoch-folding. It is consistent well with the period given by the
Lomb-Scargle method. The vertical red dashed line is another significant peak
of reduce $\chi^{2}$. This line is consistent with a peak of simulated
periodogram, so it is caused by the observation window. Figure 3: The folded
phase of SGR 1806-20. The blue histogram is the distribution of sample A and
the red line is the kernel density estimate of sample B. These two
distributions are consistent with each other at the main peak $\psi\simeq
0.58$. In sample A, there are three minor peaks, which are invisible in sample
B. Figure 4: The MJD and phase of SGR 1806-20. The gray regions denote the
main peak and the later two minor peaks in Figure 3. Due to the existence of
other minor peaks, there are some points outside the gray regions.
|
# ALMA Observations of Massive Clouds in the Central Molecular Zone:
Ubiquitous Protostellar Outflows
Xing Lu (吕行) National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka,
Tokyo 181-8588, Japan Shanghuo Li Korea Astronomy and Space Science
Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Republic of Korea
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan
Road, Shanghai 200030, P. R. China University of Chinese Academy of Sciences,
19A Yuquanlu, Beijing 100049, P. R. China Adam Ginsburg Department of
Astronomy, University of Florida, P.O. Box 112055, Gainesville, FL 32611, USA
Steven N. Longmore Astrophysics Research Institute, Liverpool John Moores
University, IC2, 146 Brownlow Hill, Liverpool, L3 5RF, United Kingdom J. M.
Diederik Kruijssen Astronomisches Rechen-Institut, Zentrum für Astronomie der
Universität Heidelberg, Mönchhofstraße 12-14, D-69120 Heidelberg, Germany
Daniel L. Walker Department of Physics, University of Connecticut, 196A
Auditorium Road, Storrs, CT 06269, USA Siyi Feng National Astronomical
Observatories, Chinese Academy of Science, Beijing 100101, P. R. China
Academia Sinica Institute of Astronomy and Astrophysics, No. 1, Section 4,
Roosevelt Road, Taipei 10617, Taiwan National Astronomical Observatory of
Japan, 2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan Qizhou Zhang Center for
Astrophysics — Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138,
USA Cara Battersby Department of Physics, University of Connecticut, 196A
Auditorium Road, Storrs, CT 06269, USA Thushara Pillai Institute for
Astrophysical Research, 725 Commonwealth Ave, Boston University Boston, MA
02215, USA Elisabeth A. C. Mills Department of Physics and Astronomy,
University of Kansas, 1251 Wescoe Hall Dr., Lawrence, KS 66045, USA Jens
Kauffmann Massachusetts Institute of Technology, 99 Millstone Road, Haystack
Observatory, Westford, MA 01886, USA Yu Cheng Department of Astronomy,
University of Virginia, Charlottesville, VA 22904, USA Shu-ichiro Inutsuka
Department of Physics, Graduate School of Science, Nagoya University, Nagoya
464-8602 , Japan
(Received - -, –; Revised - -, –; Accepted - -, –)
###### Abstract
We observe 1.3 mm spectral lines at 2000 AU resolution toward four massive
molecular clouds in the Central Molecular Zone of the Galaxy to investigate
their star formation activities. We focus on several potential shock tracers
that are usually abundant in protostellar outflows, including SiO, SO, CH3OH,
H2CO, HC3N, and HNCO. We identify 43 protostellar outflows, including 37
highly likely ones and 6 candidates. The outflows are found toward both known
high-mass star forming cores and less massive, seemingly quiescent cores,
while 791 out of the 834 cores identified based on the continuum do not have
detected outflows. The outflow masses range from less than 1 $M_{\odot}$ to a
few tens of $M_{\odot}$, with typical uncertainties of a factor of 70. We do
not find evidence of disagreement between relative molecular abundances in
these outflows and in nearby analogs such as the well-studied L1157 and
NGC7538S outflows. The results suggest that i) protostellar accretion disks
driving outflows ubiquitously exist in the CMZ environment, ii) the large
fraction of candidate starless cores is expected if these clouds are at very
early evolutionary phases, with a caveat on the potential incompleteness of
the outflows, iii) high-mass and low-mass star formation is ongoing
simultaneously in these clouds, and iv) current data do not show evidence of
difference between the shock chemistry in the outflows that determines the
molecular abundances in the CMZ environment and in nearby clouds.
Galatic: center — stars: formation — ISM: clouds
††journal: ApJ††software: CASA (McMullin et al., 2007), APLpy (Robitaille &
Bressert, 2012), Astropy (Astropy Collaboration et al., 2013)
## 1 INTRODUCTION
Star formation in the Central Molecular Zone (CMZ; the inner 500 pc of our
Galaxy) has been a controversial topic. With more than $10^{7}$ $M_{\odot}$
gas of mean density at $10^{4}$ cm-3 (Morris & Serabyn, 1996; Ferrière et al.,
2007; Longmore et al., 2013a), the CMZ exhibits about 10 times less efficient
star formation than expected by dense gas-star formation relations that have
been tested toward nearby molecular clouds and external galaxies (Longmore et
al., 2013a; Kruijssen et al., 2014; Barnes et al., 2017). Massive clouds in
the CMZ have been suggested to be progenitors of young massive star clusters
(Longmore et al., 2013b; Rathborne et al., 2015; Walker et al., 2016), but
observations reveal inefficient star formation in these clouds (Kauffmann et
al., 2017; Walker et al., 2018; Lu et al., 2019a, b), with an overall dearth
of compact dense cores across much of the CMZ (Battersby et al., 2020;
Hatchfield et al., 2020).
In Lu et al. (2020, hereafter Paper I), we reported ALMA Band 6 continuum
observations toward four clouds, including the 20 km s-1 cloud, the 50 km s-1
cloud, Sgr B1-off, and Sgr C, which are some of the most massive clouds in the
CMZ and show signs of embedded star formation (Kauffmann et al., 2017). We
identified hundreds of 2000 AU-scale cores in three of the clouds (the
exception being the 50 km s-1 cloud), and suggested that the three clouds will
likely form OB associations that contain less than 20 high-mass stars and have
a spatial extent of $\sim$5–10 pc. In the 50 km s-1 cloud, no cores above the
5$\sigma$ level and larger than the synthesized beam were found, likely
because this cloud has evolved to a much later phase when cold cores vanish
and H ii regions dominate (Mills et al., 2011). At sub-0.1 pc scales, we found
evidence of thermal Jeans fragmentation and a similar core mass function as in
Galactic disk clouds, which may hint at similar star formation processes at
small spatial scales taking place in the CMZ and elsewhere in the Galaxy.
Figure 1: Typical 1.3 mm spectra captured by ALMA, toward a star forming hot
core and a chemically quiescent region spatially offset from star forming
regions in the 20 km s-1 cloud, respectively. The corresponding positions
where the spectra are extracted are denoted by blue arrows and labeled in
Figure 2. The lines plotted in Figure 2 are highlighted by vertical dashed
lines. Most of the other lines detected toward the hot core are from
rotational transitions of complex organic molecules. The inset shows the 13CO
and C18O spectra toward the hot core along the $V_{\text{lsr}}$ axis.
Absorption at $-$55, $-$30, and $-$5 km s-1 owing to foreground gas is seen.
However, it is unclear whether these cores are prestellar or protostellar
(i.e., whether there are already embedded protostars). In Lu et al. (2019a,
b), we used H2O masers, class II CH3OH masers, and ultra-compact H ii regions
to trace high-mass star formation and identified a few high-mass star forming
regions in these clouds. Yet, the relatively poor resolution of those
observations, $\sim$4″ ($\sim$33,000 AU), prevents us from associating a
particular high-mass star formation tracer with any of the 2000 AU-scale cores
in Paper I. Further, the ultra-compact H ii regions trace a later evolutionary
phase of high-mass star formation, such that we may miss low to intermediate-
mass star formation or early evolutionary phases of high-mass star formation.
The masers are able to reveal low to intermediate-mass star formation, but
suffer from potentially low detection rates and contamination from masing
sources other than star forming regions.
To this end, molecular outflows associated with the 2000 AU-scale cores are a
promising tracer of star formation. Outflows are ubiquitously found in star
forming regions, and are detected around both low-mass and high-mass
protostars across a wide range of evolutionary phases as long as gas accretion
is underway (e.g., Zhang et al., 2005; Arce et al., 2010; Li et al., 2019,
2020). Several molecular lines that are potential outflow tracers were
observed along with the continuum data in Paper I. Therefore, in this paper we
use the lines to search for direct evidence of star formation in the form of
protostellar outflows.
In addition, the shock chemistry in protostellar outflows in the CMZ is poorly
constrained. For one thing, only a handful of protostellar outflows have been
detected in the CMZ, which are mostly in the most actively star forming
region, Sgr B2 (Qin et al., 2008; Higuchi et al., 2015). More recently, with
the advent of high resolution ALMA observations, more outflows are being
detected outside of Sgr B2 (e.g., D. Walker et al. submitted, 2020). On the
other hand, the chemistry of the molecular gas at pc scales in the CMZ seems
to be distinct from that in the solar neighborhood, with noticeable
enhancement of complex organic molecules and shock tracers throughout the CMZ
(Martín-Pintado et al., 1997; Requena-Torres et al., 2006, 2008; Menten et
al., 2009), likely caused by the extreme conditions such as widespread shocks,
high gas temperatures, high cosmic ray fluxes, and high X-ray fluxes (Mills &
Morris, 2013; Ginsburg et al., 2016; Henshaw et al., 2016; Padovani et al.,
2020; Bykov et al., 2020). Once we obtain a large sample of outflows, we will
be able to systematically compare the relative abundances of the molecules in
these outflows to those in nearby clouds, to investigate whether the shock
chemistry differs between the CMZ and elsewhere in the Galaxy at sub-0.1 pc
scales.
In the following, we first introduce our ALMA observation and data reduction
strategies, as well as an assessment of the missing flux issue in Section 2.
Then in Section 3 we summarize our observational results, including an
overview of the line emission and a visual identification of outflows. In
Section 4, we estimate column densities, molecular abundances, and masses of
the outflows, and discuss the implications to chemistry and star formation. We
conclude our paper in Section 5. In Appendix A, we introduce the procedures to
estimate column densities using the molecular line data. In Appendix B, we
list properties of the identified outflows in a detailed table. Throughout the
paper we adopt a distance of 8.178 kpc to the CMZ (Gravity Collaboration et
al., 2018).
## 2 OBSERVATIONS AND DATA REDUCTION
### 2.1 ALMA Observations
The ALMA observations were carried out in the C40-3 and C40-5 configurations
in April and July of 2017 (project code: 2016.1.00243.S). Details of the
sample selection, observation setup, and data calibration can be found in
Paper I. The four clouds in the sample are listed in Table 1. The covered
fields are chosen based on Submillimeter Array (SMA) and VLA observations that
revealed potential sites of high-mass star formation including H2O masers and
massive dense cores of 0.2 pc scale (Lu et al., 2019a). We imaged the Band 6
(1.3 mm) spectral lines using CASA 5.4.0. The covered frequencies range from
217–221 GHz to 231–235 GHz with a uniform channel width of 0.977 MHz (1.3 km
s-1). The effective spectral resolution is 1.129 MHz (1.5 km s-1) after a
Hanning smoothing done by the observatory.
We first manually identified line-free channels in the visibility data, and
fed them to the uvcontsub task to subtract the continuum baseline. Then we
used the tclean task to image the spectral lines, with Briggs weighting and a
robust parameter of 0.5, and multi-scale algorithm with scales of [0, 5, 15,
50, 150] pixels and a pixel size of 0$\farcs$04\. The image reconstruction was
carried out in a two-step manner: first, the auto-masking algorithm with the
recommended
parameters111https://casaguides.nrao.edu/index.php/Automasking_Guide was
employed in tclean to automatically identify and clean signals; then the
tclean task was restarted using clean models and residuals from the previous
step as input, and all pixels within 20% primary beam response included in the
clean mask, to a threshold of $\sim$5$\sigma$ (8 mJy beam-1 per channel), in
order to clean any residual significant signals. In a few cases where strong
spatially diffuse emission is detected (e.g., H2CO in the 20 km s-1 cloud), a
threshold of 8 mJy beam-1 may be too low and causes the clean algorithm to
diverge. We elevated the threshold to 10 mJy beam-1 for these lines. We have
compared images produced by our automatic approach with those produced by a
fine-tuned manual tclean of several lines, and found that the images are
almost identical.
The resulting synthesized beam is on average 0$\farcs$28$\times$0$\farcs$19
(equivalent to 2200 AU$\times$1500 AU) but slightly varies between lines. The
largest recoverable angular scale is 10″ ($\sim$0.4 pc) with a shortest
baseline length of 17 m. The achieved spectral line rms is between 1.6–2.0 mJy
beam-1 (0.8–1.0 K in brightness temperatures) per 1.3 km s-1 channel depending
on frequencies and regions.
Table 1: Properties of the clouds.
Cloud | $V_{\text{lsr}}$ | No. of cores | No. of outflows | Fraction with outflows | $\bar{X}$(SiO) | $\bar{X}$(SO) | $\bar{X}$(CH3OH) | $\bar{X}$(H2CO) | $\bar{X}$(HC3N) | $\bar{X}$(HNCO)
---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
The 20 km s-1 cloud | 12.5 | 471 | 20 | 0.042 | 1.69$\times$10-9 | 7.10$\times$10-9 | 0.99$\times$10-7 | 0.80$\times$10-8 | 0.65$\times$10-9 | 4.40$\times$10-9
Sgr B1-off | 31.1 | 89 | 5 | 0.056 | 1.39$\times$10-9 | 6.19$\times$10-9 | 0.86$\times$10-7 | 1.09$\times$10-8 | 1.24$\times$10-9 | 4.32$\times$10-9
Sgr C | $-$52.6 | 274 | 18 | 0.066 | 2.39$\times$10-9 | 9.87$\times$10-9 | 1.33$\times$10-7 | 1.84$\times$10-8 | 1.48$\times$10-9 | 5.44$\times$10-9
All three clouds | $\cdots$ | 834 | 43 | 0.052 | 2.05$\times$10-9 | 8.49$\times$10-9 | 1.15$\times$10-7 | 1.40$\times$10-8 | 1.16$\times$10-9 | 4.95$\times$10-9
The 50 km s-1 cloud | 48.6 | 0 | 0 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
Note. — Column (1): cloud name. Column (2): $V_{\text{lsr}}$ of the cloud
(Kauffmann et al., 2017). Column (3): number of the identified cores (Paper
I). Column (4): number of the identified outflows in this work. Column (5):
fraction of the cores that have identified outflows. Columns (6–11): mean
molecular abundances of all the outflows in the cloud. Only the abundances
with independent measurements are considered (entries without notes in the
last column in Table 3; see Section 4.1.2).
### 2.2 Assessing the Missing Flux
By their nature, interferometric observations do not recover structures on
size scales larger than their largest recoverable angular scale (10″ or 0.4 pc
for these observations). If structures larger than this exist in the field,
the flux captured by interferometers will be smaller than the true flux, which
is referred to as the missing flux problem. Spatially extended ($\gtrsim$0.4
pc) structures including outflows and filaments are seen in the spectral line
images. In principle, we can combine our data with shorter baseline as well as
single-dish data to recover any spatially extended emission. Such data are
available from the SMA and the Caltech Submillimeter Observatory (CSO)
observations published in Lu et al. (2017, 2019a) and Battersby et al. (2020).
However, we note several issues that prevent us from efficiently combining the
data: i) The sensitivity of the SMA data is not optimal for the combination.
For several regions, e.g., Sgr C, the SMA observation recovers an even smaller
flux than the ALMA data, suggesting that some weaker emission is missed by the
SMA data due to its lower sensitivity. ii) Several regions of particular
interest (e.g., the western part of Sgr C) are not well sampled by existing
SMA observations.
We attempted to combine the ALMA and SMA data, by concatenating the visibility
data and imaging them together. The resulting image is not improved compared
to the ALMA image, e.g., the rms becomes higher, and the integrated fluxes of
several spatially extended structures do not increase significantly (in the
extreme case of Sgr C, the fluxes even decrease).
Therefore, we conclude that the imaging of diffuse structures does not benefit
from the addition of the SMA data. This does not rule out the possibility that
better shorter baseline data may help. A longer SMA observation, or an ACA
observation that provides better sensitivity than the SMA, would be necessary
to be combined with the ALMA data to recover spatially extended emission.
Meanwhile, we compared the integrated fluxes recovered by the CSO and ALMA,
focusing on the SiO emission in Sgr C which is the most spatially extended in
our data and thus the most affected by the missing flux. In a circle of 50″
diameter centered in between the two clusters in Sgr C, the ALMA/CSO SiO
integrated fluxes are measured to be 120/200 Jy km s-1. The ALMA data recover
60% of the flux observed by the CSO. We thus estimate an upper limit of 40%
for the missed flux in our ALMA data. In Section 4.1.4, we will see that this
does not affect our estimate of outflow properties, as the dominant
uncertainty is the molecular abundance that is unconstrained by several orders
of magnitude.
Figure 2: Molecular line emission in the 20 km s-1 cloud. The inner and outer
dashed loops in all panels demonstrate the 50% and 30% primary beam responses
of the ALMA mosaics. The first panel shows a three-color image made from
Spitzer IRAC 3.6, 4.5, and 5.8 µm bands, with yellow contours overlaid
illustrating the ALMA 1.3 mm continuum emission at levels of
[5,25,45]$\times$40 $\mu$Jy beam-1. Positions of H2O masers are marked by
crosses, among which those with AGB star counterparts (Lu et al., 2019a) are
colored in magenta. The other panels show integrated intensities of molecular
lines in a logarithmic scale, which are integrated in a velocity range of
[$-20$,$40$] km s-1, except for CH3OH and CH3CN where this range is adjusted
to avoid confusion with adjacent lines. The colorbars are in unit of Jy beam-1
km s-1. In selected panels, black boxes show regions where outflows are
identified, with zoomed-in views presented in Figures 6–22. Figure 2:
Continued. Figure 3: Molecular line emission in the 50 km s-1 cloud. The
layout of panels and symbols is the same as in Figure 2. Figure 4: Molecular
line emission in Sgr B1-off. The layout of panels and symbols is the same as
in Figure 2. Figure 4: Continued. Figure 5: Molecular line emission in Sgr C.
The layout of panels and symbols is the same as in Figure 2. Figure 5:
Continued.
## 3 RESULTS
### 3.1 Overview of the Line Emission
Typical spectra toward a chemically active core and a relatively quiescent
region spatially offset from any cores but with line emission are shown in
Figure 1. The former shows characteristic hot molecular core chemistry, while
the latter represents regions that are likely influenced by pc scale shocks
prevailing in the CMZ. Note that there exist even more chemically active cores
(e.g., the two UC H ii regions in Sgr C), with many more spectral lines
detected, mostly from complex organic molecules. Here, we focus on the
spatially extended spectral line emission detected outside of the hot cores or
UC H ii regions, and leave the discussion of the line-rich chemically active
cores to a future paper.
We identified spatially extended line species, and plotted their integrated
intensities in Figures 2–5. The line species include the CO isotopologue C18O,
a group of potential shock tracers (SiO, SO, HNCO, CH3OH), and several dense
gas tracers that are sometimes found in outflows (H2CO, H213CO, CH3CN, c-C3H2,
HC3N). 13CO emission is more spatially extended than C18O and is not plotted.
Three features are clearly seen: i) Linear structures spatially associated
with dust emission are prominent, which may be outflow lobes (black boxes in
the figures). ii) Multiple lines are tracing similar filamentary structures
that are spatially offset from any dust emission, including some that are more
typically confined to hot cores in environments outside of the CMZ (e.g.,
CH3CN), whose nature is unclear. An example is marked by the blue arrow in
Figure 2. iii) Point-like SiO emission with large linewidths ($>$20 km s-1) is
found toward two H2O masers that have known AGB star counterparts (magenta
crosses in Figures 2 & 4; see Lu et al. 2019a), with no associated dust
emission within a radius of 0.1 pc, which is probably originated from the
atmosphere of AGB stars (González Delgado et al., 2003). Here we focus on the
first feature, potential outflows, while leaving the discussion of the other
features to future papers.
The two CO isotopologue lines, 13CO and C18O, present strong absorption at
velocities of $-$55, $-$30, and $-$5 km s-1 against strong continuum emission
(see the inset in Figure 1). These are consistent with the absorption features
seen in other line observations toward the Galactic Center (e.g., Jones et
al., 2012), and are attributed to foreground gas in spiral arms along the line
of sight. In particular, the absorption at $-$55 km s-1 is close to the cloud
velocity of Sgr C, which complicates the interpretation of the two lines in
Sgr C. Many of the lines, including H2CO, CH3OH, SiO, and the two CO
isotopologues themselves, also present absorption features at a few km s-1
blue-shifted with respect to the cloud velocity toward continuum emission
peaks (e.g., the absorption at $\sim$12 km s-1 in the inset of Figure 1).
These features are likely owing to a combination of missing flux as a result
of interferometer observations and self absorption when the lines become
optically thick.
### 3.2 Identification of Outflows
Signatures of protostellar outflows have been detected toward Sgr B2(M) and
(N), the two high-mass protoclusters in the Sgr B2 complex (Qin et al., 2008;
Higuchi et al., 2015; Bonfand et al., 2017). Outside of Sgr B2, detecting
protostellar outflows has been challenging in the CMZ, because of the lack of
resolution to spatially resolve outflows and the prevalence of broad linewidth
gas produced by phenomena other than outflows (Henshaw et al., 2019; Sormani
et al., 2019). Previous Submillimeter Array (SMA) observations have detected
widespread emission of potential shock tracers (e.g., SiO, SO, CH3OH) at 0.2
pc scales in CMZ clouds (Kauffmann et al., 2013; Lu et al., 2017; Battersby et
al., 2020). However, limited by the angular resolution and the imaging
sensitivity, it was unclear whether the emission seen by the SMA is owing to
protostellar outflows or pc-scale shocks prevailing in the CMZ. Only recently,
ALMA observations with high resolution and high sensitivity start to detect
outflows in the CMZ, e.g., in the G0.253$+$0.025 cloud (D. Walker et al.
submitted, 2020).
Our high angular resolution ALMA observations resolved substructures of 2000
AU scale in the molecular line emission, enabling us to search for
protostellar outflows. The integrated intensity maps in Figures 2–5 already
reveal collimated emission spatially associated with cores, indicating the
existence of outflows.
We then examined potential shock tracers detected by ALMA, including SiO, SO,
H2CO, HNCO, HC3N, and CH3OH. All these tracers have been previously found to
be enhanced by at least one order of magnitude in shocked regions in
protostellar outflows (e.g., Bachiller & Pérez Gutiérrez, 1997). We applied
the following criteria to identify outflows:
i) We used the H2O masers from Lu et al. (2019a) as a guidance to search for
associated shock tracer emission. First, we made integrated intensity maps of
SiO across the full velocity range, and searched for linear structures
spatially associated with the masers. If linear emission is found, then, we
made integrated intensity maps of blue and red shifted components based on the
$V_{\text{lsr}}$ of the cloud (see Table 1), and checked whether the linear
structures show symmetric blue and red shifted emission with respective to the
masers. Finally, we determined the systematic velocity of each individual
outflow driving source, by using dense gas tracers in the ALMA data toward the
maser position (CH3CN, HC3N, CH3OH, or C18O, in a decreasing order of
preference; C18O was used only once, towards the 20 km s-1 cloud-F #1), and
remade integrated intensity maps of blue and red shifted components based on
the individual $V_{\text{lsr}}$. If the shock tracer emission exhibits blue
and red shifted components with respect to the systematic velocity at opposite
positions to the maser position, it is considered as an outflow.
ii) In cases where H2O masers are not present, we checked the emission of the
six tracers around the 2000-AU scale cores from Paper I following the same
procedures, to search for blue and/or red shifted line emission spatially
offset from the cores. We require the blue and/or red shifted features to be
seen in at least two shock tracers, including the canonical shock tracer SiO,
plus any of the five supplemental tracers.
We identified 43 outflows, and marked regions where they are detected with
boxes in Figures 2–5. The zoomed-in views are in Figures 6–22, in which we
plot the red/blue shifted shock tracer emission with respect to the systematic
velocity and highlight individual outflows with arrows. The position-velocity
diagrams of the 43 outflows made from the SiO line are displayed in Figure 23.
The numbers of outflows identified in the individual clouds are listed in
Table 1.
In several cases, lobes from different outflows spatially overlap with each
other (e.g., outflows #5–#9 in region C of the 20 km s-1 cloud; see Figure 8),
but we were able to separate them apart unambiguously based on velocities (see
Figure 23). Most of the other outflows are easily distinguished spatially from
nearby outflows and the diffuse emission. All these outflows are considered as
‘highly likely’. However, there exist cases where the outflows cannot be
robustly separated from other outflows or the diffuse emission, either
spatially or kinematically. One example is the two blue-shifted lobes in
region B of the 20 km s-1 cloud, where the lobes overlap in both projected
locations and velocities (see Figures 7 & 23). Following the definitions in Li
et al. (2020), we classified these ambiguous identifications as ‘candidates’.
The classifications are noted with asterisks in Table 3, Figures 6–22, and
Figure 23. Among the 43 outflows, 37 are highly likely, and 6 are candidates.
We stress that this visual identification is likely to be incomplete.
Potential outflows could have been missed if they cannot be distinguished from
the background emission or other outflows, or if their emission is too weak.
The actual (in)completeness, however, is difficult to quantify, as the
identification is based on visual inspection and is subjective in nature.
Recent ALMA surveys toward infrared dark clouds in the Galactic disk using CO
lines as the primary outflow tracer yield detection rates of 14%–22% (e.g., 62
out of 280 cores in Kong et al. 2019 and 41 out of 301 cores in Li et al. 2020
are identified with outflows). The detection rate using SiO as the primary
tracer in this paper is 4–7% (Table 1), which is much lower. It is infeasible
to directly compare, e.g., the outflow mass sensitivities of previous surveys
and ours, considering that the observations use different lines as outflow
tracers and assume different abundances. Assuming that the observations are
sufficiently sensitive to detect all exiting outflows, the lower detection
rate in our sample may suggest that we have missed a substantial number of
outflows that are not traced by SiO, or may reflect the variation of outflow
occurrence rates along the evolutionary stages.
Meanwhile, we also note that due to the complicated environment in the CMZ
(e.g., the wide-spread shock tracer emission; Martín-Pintado et al., 1997) and
possible contamination from the foreground, it is likely that false positive
identifications exist in our sample if such large scale shock tracer emission
accidentally lies upon cores. But such accidental spatial coincidence should
be rare, as the velocities of the identified outflows and cores have a
continuous overlap (Figure 23). Note that the cores associated with the
outflows are all likely in the CMZ, given that the velocities of their line
emission (e.g., C18O; see Figure 1 inset) are consistent with the overall
velocity field in the CMZ (e.g., Henshaw et al., 2016), while spiral arm
clouds along the line of sight are mostly seen as absorption in 13CO and C18O
(Figure 1 inset), indicating that the overlapping spiral arm clouds consist of
low-density gas and hence are unlikely to have dense cores.
SiO and SO seem to be the best outflow tracers among the six molecules. Their
emission is usually well separated from the background, and is usually
collimated as expected for outflows. CH3OH, H2CO, and HNCO often suffer from
contamination from the background or foreground emission, and thus are not
tracing the outflows as well as SiO and SO. HC3N traces both cores and
outflows. Its emission is weaker than the other molecules, therefore is not an
optimized outflow tracer either. As pointed out by several previous studies,
CH3OH, H2CO, HC3N, and HNCO may be released into the gas phase by slow
($\lesssim$20 km s-1) shocks that evaporate ice mantles of dust or be produced
by gas phase reactions in post-shock regions, therefore probe the widespread
low-velocity shocks in the CMZ (Lu et al., 2017; Tanaka et al., 2018;
Taniguchi et al., 2018). SiO and SO, on the other hand, may be released from
the dust by sputtering of the grain core, thus better probe fast ($\gtrsim$20
km s-1) shocks induced by the outflows.
Figure 6: Outflows in region A in the 20 km s-1 cloud. The gray scale image in
all panels shows the ALMA 1.3 mm continuum emission, with the scale bar at the
top in unit of mJy beam-1 in a logarithmic scale. The green crosses mark
positions of the H2O masers in Lu et al. (2019a). In each panel, the blue and
red contours illustrate the blue and red shifted line emission integrated
within the specified velocity ranges, at levels of [3,6,12]$\sigma$, where
$\sigma=\sqrt{N}\sigma_{c}v_{c}$, $N$ being the number of channel,
$\sigma_{c}$ the rms of individual channels, and $v_{c}$ the channel width in
km s-1. When there are multiple outflows in the panel, the velocity ranges are
chosen to highlight the bipolar morphologies of all the outflows, but for
individual outflows, the blue and red shifted lobes may extend beyond these
velocity ranges or be contaminated by diffuse gas, which can be better
visualized in Figure 23. Identified outflows are highlighted by blue and red
arrows, but note that some outflows are not seen in all the lines (in the case
shown here, outflow #1 is not detected in HNCO). The thick black crosses mark
the reference positions we choose to derive column densities of the molecules
based on which we estimate molecular abundances. Figure 7: Outflows in region
B in the 20 km s-1 cloud. Figure 8: Outflows in region C in the 20 km s-1
cloud. Figure 9: Outflows in region D in the 20 km s-1 cloud. Figure 10:
Outflows in region E in the 20 km s-1 cloud. Figure 11: Outflows in region F
in the 20 km s-1 cloud. Figure 12: Outflows in region G in the 20 km s-1
cloud. Figure 13: Outflows in region A in Sgr B1-off. Figure 14: Outflows in
region B in Sgr B1-off. Figure 15: Outflows in region C in Sgr B1-off. Figure
16: Outflows in region A in Sgr C. Figure 17: Outflows in region B in Sgr C.
Figure 18: Outflows in region C in Sgr C. Figure 19: Outflows in region D in
Sgr C. Figure 20: Outflows in region E in Sgr C. Figure 21: Outflows in region
F in Sgr C. Figure 22: Outflows in region G in Sgr C. Figure 23: Position-
velocity diagrams of the 43 outflows using the SiO line. The slices are taken
along the arrows shown in Figures 6–22, from the blue-shifted to red-shifted
side, averaged within a width of 5000 AU. The background image and the cyan
contours show the SiO emission, with contour levels starting at 2$\sigma$ in
step of 2$\sigma$ where $\sigma$$\approx$1 mJy beam-1. The x axis is the
spatial offset along the slices, and the y axis is $V_{\text{lsr}}$. The
vertical and horizontal dashed lines mark the position and $V_{\text{lsr}}$ of
the cores, respectively.
## 4 DISCUSSION
### 4.1 Estimate of Physical Properties of the Outflows
In this section, we calculate the column densities of the six molecules
detected toward the identified outflows (Section 4.1.1), introduce the method
to estimate the molecular abundances in the outflows (Section 4.1.2), and
estimate the outflow masses, and where possible, the outflow energetics
(Section 4.1.3). The uncertainties involved in the estimate of column
densities, abundances, and masses are discussed in Section 4.1.4. We need an
order-of-magnitude estimate on these parameters to discuss implications for
astrochemistry and star formation in following sections, so the unavoidable
significant uncertainties in these results (one to two orders of magnitude, as
detailed in Section 4.1.4) are acceptable.
#### 4.1.1 Column Densities of the Shock Tracers
We measure the molecular line integrated fluxes of the outflows in the primary
beam corrected maps within a contour level of 3$\sigma$ for each line, and
list the results in Table 3. The velocity range of the integration is chosen
to start from one channel ($\sim$1.3 km s-1) away from $V_{\text{lsr}}$ of the
core to avoid diffuse emission around the system velocity, and end at the
channel where the emission drops below 2$\sigma$. The separation of 1.3 km s-1
should be able to role out most of the diffuse component as the FWHM linewidth
at 0.1 pc scale in these clouds drops to this value and the linewidth at even
smaller scales should be narrower (Kauffmann et al., 2017). The integrated
fluxes should be lower limits given limited sensitivities and potential missed
flux by the interferometer.
We then derive column densities with the measured fluxes, using the calcu
toolkit222https://github.com/ShanghuoLi/calcu (Li et al., 2020). Local
thermodynamic equilibrium (LTE) conditions and optically thin line emission
has been assumed. A constant excitation temperature of 70 K, which is the
characteristic gas kinetic temperature in the CMZ (Ao et al., 2013; Ginsburg
et al., 2016; Krieger et al., 2017), is assumed for all the lines. Note that
the adopted temperature is different from that assumed for the dust in the
cores in Paper I, 20 K. Observations show that the gas temperature at
$\sim$0.1 pc scale in CMZ clouds is 70 K or even higher (Mills & Morris, 2013;
Lu et al., 2017), while the dust temperature at this scale is largely
unconstrained (see discussion in Paper I, ). Nevertheless, in Section 4.1.4 we
find that the derived column densities are not sensitive to the choice of
temperatures. Details of the column density calculation can be found in
Appendix A.
We also select reference positions in the outflow lobes, and calculate the
column densities. The reference positions must have detectable HC3N emission,
which we choose as the anchor molecule in the next section. The positions are
adjusted to include emission of as many shock tracers as possible. A circle of
0$\farcs$3 across, comparable to the synthesized beam size, is used to define
the area. The reference positions are marked by black crosses in Figures 6–22,
and the derived column densities are listed in Table 3. The results are used
to infer molecular abundances in the next section, and are further discussed
in Section 4.2 in terms of astrochemical implications.
#### 4.1.2 Molecular Abundances
In order to obtain the masses of the outflows, we must adopt an abundance for
each shock tracer. However, the abundances of these molecules are known to be
highly variable, especially toward the environment of our outflow sample with
both strong shocks and complicated factors in the CMZ. One example is SiO,
whose abundance has been found to vary by a factor of $>$$10^{5}$ across
different regions (e.g., Martín-Pintado et al., 1997; Sanhueza et al., 2012;
Csengeri et al., 2016; Li et al., 2019).
The dust emission is not always detected toward the outflows, so anchor
molecules with relatively well-constrained abundances are often used to
determine abundances of other molecules. One commonly used anchor molecule is
CO and its isotopologues (e.g., Feng et al., 2016). By assuming a canonical
12CO abundance of 10-4 with respect to H2 (Blake et al., 1987), one might in
principle use CO emission to derive the H2 column density at a reference
position in the outflow, and then determine the abundances of other molecules
by dividing their column densities by the H2 column density. However, in our
data, the CO lines suffer from strong absorption and missing flux, and more
importantly, morphologically they are not tracing the outflows seen in other
molecules (Figures 2–5), probably because they are optically thick thus
tracing the surface of the clouds instead of the outflows in the interior.
Therefore, we choose to anchor our estimate of molecular abundances on HC3N,
which is detected toward both cores and outflows and has shown a relatively
stable abundance between the core/outflow environments in other observations
(e.g., a factor of $\sim$30 enhancement from cores to outflow lobes around
low-mass and high-mass protostars; Bachiller & Pérez Gutiérrez, 1997; Feng et
al., 2016; Mendoza et al., 2018). The other molecules either show up only
toward the outflows (e.g., SiO, SO) or suffer from contamination by pc-scale
diffuse emission or strong absorption toward the cores (e.g., CH3OH, H2CO, and
HNCO), and thus are not appropriate as the anchor tracer.
First, we compare the column densities of HC3N and of H2 toward the cores, and
estimate abundances of HC3N in the cores. The column densities of HC3N is
derived following the procedures in Section 4.1.1. The H2 column densities are
derived using the dust continuum from Paper I. Then we assume an enhancement
of factor 30 (Bachiller & Pérez Gutiérrez, 1997; Feng et al., 2016), and
obtain the abundances of HC3N in the outflows. Finally we derive the H2 column
density in the outflows and use it to calibrate the abundances of the other
shock tracers. The adopted molecular abundances with respect to molecular
hydrogen are listed in Table 3, and the mean values of individual clouds are
given in Table 1.
The estimated HC3N abundances in the outflows are consistent in terms of the
order of magnitude with previous result toward the 20 km s-1 cloud using
multiple HC3N transitions ($10^{-9}$–$10^{-8}$ depending on the assumptions;
Walmsley et al., 1986).
We note that the estimated abundances of the molecules span a large range. For
example, the SiO abundance with respect to H2 in our outflow sample ranges
from 10-10 to 10-8, with a mean value of 2.05$\times$10-9. This justifies our
choice of estimating molecular abundances case by case, rather than assuming a
constant abundance, for the latter case will bias the mass estimates
significantly.
There are several cases where we cannot directly determine the abundance of a
molecule in an outflow, and then we have to circumvent them by making
reasonable assumptions: i) One lobe of an outflow has a well-defined reference
position and the abundance can be determined from a scaling of the HC3N
emission, while the other lobe does not. In this case, we assume that the
abundances of the blue/red-shifted lobes of the same outflow are identical and
adopt the abundance of the other lobe. ii) An outflow has well-defined
reference positions in multiple molecular line emission including HC3N, but
the core associated with it does not have detectable HC3N emission, and
therefore the abundance of HC3N in the outflow cannot be determined. In this
case, we adopt a mean HC3N abundance of all the other outflows in the region,
and then use it to determine abundances of other molecules in this outflow.
iii) An outflow has well-defined reference positions in multiple molecular
line emission, but not in HC3N. We have to adopt mean molecular abundances of
all the other outflows in the region for each of the detected molecules in the
outflow. All these cases are explicitly noted in Table 3.
Figure 24: Outflow masses derived from different molecules, plotted against
masses of the cores where the outflows originate. The outflow masses are the
sum of those of the blue and red lobes, color-coded by molecular tracers based
on which the masses are estimated. The systematic uncertainty of factor 70 in
the outflow masses is not plotted here. The black crosses denote the mean
outflow masses of the different molecules.
#### 4.1.3 Masses and Energetics of the Outflows
After the molecular column densities and abundances are derived, the outflow
masses are estimated by assuming uniform abundances within each outflow, using
the calcu toolkit. The results are listed in Table 3, and plotted in Figure
24. For the same outflow lobe, multiple shock tracers could be detected, in
which case we derive more than one outflow masses. All of these masses are
deemed to be worth reporting, as the different molecules may trace different
components of the same outflow with different chemical environments or
excitation conditions. Nevertheless, the masses of the same outflow are
consistent within an order of magnitude as demonstrated in Figure 24.
In a few cases where the outflow emission can be unambiguously separated from
contaminations and the lobes are well collimated, we are able to measure the
projected length and estimate a dynamical time-scale. Then the energetics of
the outflows, e.g., the outflow rate and the outflow mechanical force, can be
estimated following procedures in Li et al. (2020). One example of such well-
defined outflows is the blue-shifted lobe of the Sgr C region F #2 outflow.
With a projected angular scale of 20″ ($\sim$0.8 pc), and an outflow terminal
velocity (the maximum velocity difference between the outflow and the core, as
noted by the velocity range listed in Table 3) of 33.7 km s-1, the dynamical
time-scale is $3.2$$\times$$10^{4}$/$\cos$($\theta$) yr where $\theta$ is the
inclination angle of the outflow lobe with respect to the plane of the sky.
The outflow mass rate is then $\sim$$6$$\times$$10^{-4}$$\cos$($\theta$)
$M_{\odot}$ yr-1. If the inclination angle is not too large ($\theta$$<$80°),
the outflow mass rate is $\gtrsim$$10^{-4}$ $M_{\odot}$ yr-1, which is usually
found toward outflows around high-mass young stellar objects (Maud et al.,
2015). It is also possible to estimate the accretion rate, with the same
assumptions in Li et al. (2020), e.g., a wind speed of 500 km s-1 from the
disk and a ratio between the accretion rate and the mass ejection rate of 3,
which leads to a value of $2.5$$\times$$10^{-5}$$\cos$($\theta$) $M_{\odot}$
yr-1. However, there are significant uncertainties in the outflow masses (see
next section) and the true physical scale of the outflow lobes (contamination,
potentially missed weak emission, inclination angle), on top of the
unconstrained assumptions made in the calculation of the accretion rate (Li et
al., 2020). We expect the uncertainty in the estimated outflow mass rate and
accretion rate to be potentially three orders of magnitude or even greater,
which is comparable to the dynamical range of our data (Figure 24). Therefore,
we will not discuss these results further.
#### 4.1.4 Uncertainties in the Estimated Parameters
There are several significant uncertainties in the estimate of column
densities and outflow masses. First, we have assumed LTE conditions and a
constant excitation temperature of 70 K when calculating the column densities.
If we consider subthermal excitation, then the excitation temperature would be
substantially lower than the kinetic temperature. If the temperature is varied
between 20 and 100 K, a range that has been observed toward the CMZ (Ao et
al., 2013; Ginsburg et al., 2016; Lu et al., 2017) and toward outflows in
nearby clouds (Lefloch et al., 2012), then the resulting mass will vary by
60%, or 0.2 dex. Considering that the temperature could be even higher in
post-shock regions ($\sim$103 K; Tafalla et al., 2013), this uncertainty will
only be larger. Second, we have assumed optically thin emission for all the
molecular lines, while this may be invalid especially for CH3OH and H2CO,
whose optical depths are higher than the other molecular lines as evidenced by
the strong self absorption toward the cores. As shown in Figure 24, the masses
of the same outflow estimated from different molecules are usually consistent
within an order of magnitude. Therefore, the uncertainty in the outflow masses
stemming from different optical depths of the molecular lines is estimated to
be half of this range at most, or 0.5 dex. Third, as discussed in Section 2.2,
the missing flux issue of the ALMA data may lead to an underestimate of 40%
for the measured flux, or 0.15 dex. The three uncertainties above go into the
calculation of column densities.
For outflow masses, the molecular abundances must be taken into account
additionally. The abundances of these shock tracers hinge on that of HC3N,
which is again based on: i) the total molecular column densities in the cores
that are derived from the dust continuum, with dependences on the assumed dust
temperature, the dust opacity, and the gas-to-dust mass ratio (for detailed
discussion, see Paper I, ), ii) the LTE condition assumed for the calculation
of HC3N column densities in the cores, which may not hold given the non-LTE
excitation found in several CMZ clouds (Mills et al., 2018), and iii) the
assumed enhancement of factor 30 of HC3N from the core to the outflows. The
estimated molecular abundances in the outflows usually span two orders of
magnitude (Table 3). Therefore, we assign an uncertainty of one order of
magnitude to the abundances, and note that the uncertainty is likely even
greater.
Taken together, we estimate an uncertainty of 0.85 dex (a factor of 7) in the
column densities, and an uncertainty of 1.85 dex (a factor of 70) in the
outflow masses. These estimates are typical uncertainties for the individual
outflows, though for particular outflows the uncertainties may be smaller or
larger (e.g., for outflows where molecular abundances cannot be determined so
mean abundances of the region are adopted, the uncertainties in the abundance
and therefore the masses would be larger; for spatially compact outflows, the
underestimate of the fluxes because of the missing flux issue would be less
significant; for outflows with temperatures higher than 100 K, the
uncertainties in the column densities and masses would be larger).
Figure 25: Absolute abundances of molecules with respect to H2 in the outflows
are plotted in (a), and those normalized with respect to the abundance of HC3N
are plotted in (b). Here we only consider the abundances with independent
measurements but exclude those guessed from other outflows (i.e., entries with
notes in the last column in Table 3). The boxes denote the first to third
quartiles while the caps mark the full range of abundances in our outflow
sample. The median of abundances of each molecule is marked by a horizontal
orange line. The abundances of the low-mass outflow in L1157 and the high-mass
outflow in NGC7538S are also plotted. The systematic uncertainties in the
abundances are not plotted here.
### 4.2 Shock Chemistry in the Outflows
We compare the relative abundances of the six shock tracers and investigate
the shock chemistry in the outflows. This is the first spatially resolved
astrochemical study toward outflows in the CMZ, and one of the very few such
studies even including works toward Galactic disk targets.
The abundances of the six shock tracers are presented in Figure 25. For each
molecule, we plot the median of its abundances in all the outflows as a
horizontal orange line.
To understand the relative abundances of the six molecules, we compare our
result with similar studies toward nearby star forming clouds. However, we
note that spatially resolved observations toward outflows that include all the
six molecules, even for popular targets such as the Orion molecular cloud, are
rare. For example, we are not able to find a paper that reports SiO, H2CO, or
HC3N column densities or abundances in the explosive outflow around Orion KL,
even though results of SO, CH3OH, and HNCO are available (Feng et al., 2015).
In the end, we are able to find only two representative outflows in nearby
clouds: L1157, a well-studied, prototypical low-mass outflow around a low-mass
protostar (e.g., Bachiller & Pérez Gutiérrez, 1997; Rodríguez-Fernández et
al., 2010; Podio et al., 2017; Holdship et al., 2019), and NGC7538S, a
prototypical massive outflow around a high-mass protostar (e.g., Naranjo-
Romero et al., 2012; Feng et al., 2016).
We take the abundances of the six molecules toward L1157-B1 and B2 (two
reference positions in the blue-shifted lobe) from Bachiller & Pérez Gutiérrez
(1997) and Rodríguez-Fernández et al. (2010). For NGC7538S, we take the
results from JetS, a reference position on the red-shifted side of the
protostar, from Feng et al. (2016), except for SiO that is not observed by the
authors. We instead obtain the SiO abundance at the reference position using
data from Naranjo-Romero et al. (2012) (L. Zapata, private communications).
The JetN position in Feng et al. (2016) does not show HC3N emission, and
therefore only an upper limit of its abundance can be detected, which prevents
an appropriate comparison to our results as we use HC3N to infer the
abundances of the other molecules. All the results from the above publications
have assumed LTE conditions and optically thin line emission. The abundances
are plotted in Figure 25(a).
In addition, we note that the abundances toward L1157-B1/B2 are estimated
based on CO lines, which may become optically thick and therefore result in
overestimated abundances for other molecules (Bachiller & Pérez Gutiérrez,
1997). This may explain the systematically higher abundances for all the six
molecules in L1157-B1/B2 than the other targets in Figure 25(a). To eliminate
such biases, we normalize the abundances with respect to that of HC3N in the
outflows, and plot the relative abundances of the six molecules in Figure
25(b).
By comparing the two samples in Figure 25(b), the CMZ clouds vs. the two
nearby clouds, we do not find clear evidence of difference between the
relative abundances of the six shock tracers in the outflows. The relative
abundances of the two nearby clouds usually fall within an order of magnitude
apart from the medians of our CMZ outflow sample. However, given the
significant uncertainty of the abundances and a limited sample from nearby
clouds, it is premature to conclude any consistency between the two samples.
### 4.3 Implications for Star Formation and Chemistry
Protostellar outflows are ubiquitously detected in star forming regions,
suggesting active gas accretion around protostars (e.g., Shang et al., 2007;
Bally, 2016). Here we investigate star formation and chemistry in the four
massive clouds in the CMZ based on our observations of the outflows.
The first implication, obviously, is that protostellar accretion disks
ubiquituously exist in these clouds, as protostellar outflows are supposedly
driven by disks (Shang et al., 2007). Direct observational evidence of
protostellar accretion disks in the CMZ has been limited to the hot cores in
the Sgr B2 cloud (e.g., Hollis et al., 2003; Higuchi et al., 2015), and even
for these cases the evidence is ambiguous given the complicated kinematic
environments in Sgr B2. More recently, D. Walker et al. (submitted, 2020)
reported detection of protostellar outflows in G0.253$+$0.025 in the CMZ based
on ALMA observations. Our finding of a large population of outflows (except in
the 50 km s-1 cloud, which is likely in a more evolved phase of star
formation; Mills et al., 2011; Lu et al., 2019a), suggests that active
accretion is ongoing around protostars in these CMZ clouds.
The second implication concerns the evolutionary phases of star formation in
these clouds based on the statistics of the cores with or without star
formation signatures. In Paper I, we identify 834 cores at 2000 AU scales in
the three CMZ clouds. Among them, only 43 are found to be associated with
outflows. The remaining 791 cores are not associated with other signatures of
star formation (masers, H ii regions) either, and therefore are candidates of
starless cores (gravitationally bound and prestellar, or simply unbound).
However, as mentioned in Section 3.2, the outflow sample is very likely to be
incomplete because of the subjectivity of the identification, thus potential
outflows, even those with sufficiently strong emission, may have been missed.
In addition, deeper observations may reveal more signatures of star formation
such as weaker outflows or new masers. Therefore, the fraction of protostellar
and starless cores is highly uncertain. For individual clouds, the fractions
of cores associated with outflows range from 0.04 to 0.07 (Table 1), although
this is unlikely to suggest any evolutionary trend among the three clouds
given the small numbers of the outflow detections and the potential
incompleteness of the outflow sample.
If we base our analysis on the current observations, i.e., 4–7% of the cores
identified in Paper I are protostellar (Table 1), then we may put a constraint
on the evolutionary phase of the clouds. The time scale needed to enter the
protostellar phase is of the order 1–2 Myr for both low-mass and high-mass
star forming cores (Enoch et al., 2008; Könyves et al., 2015; Battersby et
al., 2017). This time scale is similar to the proposed lifetime of molecular
clouds in the CMZ (Jeffreson et al., 2018; Barnes et al., 2020). Assuming that
all the cores we detected will eventually evolve into the protostellar phase
in a time scale of 1–2 Myr, the small fraction of the currently identified
protostellar cores may suggest an age of star formation in these clouds as
short as $\sim$0.05–0.1 Myr. In such case, star formation may have started
only recently, if the clouds just condensed out of a more diffuse state,
possibly driven by tidal compression during their arrival in the CMZ or on
their orbit around the Galactic Center (Longmore et al., 2013b; Kruijssen et
al., 2015, 2019), or by the impact of adjacent expanding H ii regions (Kendrew
et al., 2013; Barnes et al., 2020). Again, we stress that this estimate
depends on the (in)completeness of the outflow sample, and the age of star
formation in these clouds is likely longer as the fraction of protostellar
cores is potentially higher.
The third implication is related to the result presented in Figure 24, where
we find outflows from both high-mass ($>$100 $M_{\odot}$) and low-mass ($<$5
$M_{\odot}$) cores. Here the core masses have a strong dependence on the
unconstrained dust temperature and may be overestimated by a factor of 3, as
demonstrated in Paper I. Several of the high-mass cores are known to be
forming high-mass protostars, with UC H ii regions and class ii CH3OH masers
(Lu et al., 2019a, b). The low-mass cores, on the other hand, are only capable
of forming low-mass stars with the current mass even assuming a high star
formation efficiency of 50%. Meanwhile, the majority of the outflow masses lie
in the range of 1–10 $M_{\odot}$, albeit with a large uncertainty of factor
70. This mass range is characteristically found around high-mass protostars
(Zhang et al., 2005; Lu et al., 2018). Some of the outflows have lower masses
of $<$1 $M_{\odot}$, which are typical for low-mass star forming regions (Arce
et al., 2010). Therefore, considering the mass ranges of the cores and the
outflows, the detected outflows likely trace a mix of high-mass and low-mass
star formation. Simultaneous low-mass and high-mass star formation has been
observed ubiquitously in massive clouds in the Galactic disk (e.g., Cyganowski
et al., 2017; Pillai et al., 2019; Sanhueza et al., 2019), which we now
confirm to take place in the CMZ as well.
The last implication, as discussed in Section 4.2, is about the shock
chemistry in the outflows. Given the large uncertainties involved in the
abundances (at least one order of magnitude), we are not able to conclude
consistency between the shock chemistry in the CMZ clouds and in nearby
analogs, but we do not find evidence of difference either. This is in contrast
to the situation on the cloud scale of a few pc, where the chemistry in the
CMZ is distinctly different from that in nearby clouds, e.g., an anomalous
enhancement of complex organic molecules and shock tracers as compared to
those in nearby clouds, suggesting the presence of wide-spread low-velocity
shocks (Martín-Pintado et al., 1997; Requena-Torres et al., 2006; Menten et
al., 2009). Several previous studies have pointed out that at the sub-0.1 pc
scale, physical processes such as gas fragmentation and turbulent linewidths
in the CMZ and in nearby regions may start to converge (e.g., Kauffmann et al.
2017; Lu et al. 2019a; Paper I; D. Walker et al. submitted, 2020) despite
distinct properties on larger scales. Our results set the first step toward a
similar comparison of the shock chemistry in protostellar outflows in the CMZ
and in nearby clouds. Multi-transition spectral line observations toward the
CMZ outflows that enable a more robust estimate of column densities and
abundances, and a larger sample of resolved astrochemical studies toward
outflows in nearby clouds, will help clarify whether the shock chemistry in
the two environments are consistent or not.
## 5 CONCLUSIONS
As a follow-up of our Paper I, in which we used ALMA 1.3 mm continuum emission
to study cores of 2000 AU scale in four massive clouds in the CMZ, we further
use 1.3 mm molecular lines to identify protostellar outflows and investigate
star formation activities associated with the cores. We choose six commonly
used shock tracer molecules, including SiO, SO, CH3OH, H2CO, HC3N, and HNCO.
In three clouds (the 20 km s-1 cloud, Sgr B1-off, and Sgr C), we identify 43
outflows traced by the six molecules, including 37 highly likely ones and 6
less likely ones that are considered as candidates. This is by far the largest
sample of protostellar outflows identified in the CMZ. Then we estimate
molecular abundances and masses of the outflows. Based on these findings and
our previous studies (Lu et al. 2019a; Paper I), we conclude that:
* •
We find no evidence of differences between the physics (existence of accretion
disks, Jeans fragmentation) and shock chemistry (relative abundances of the
six shock tracer molecules in the outflows) in the sub-0.1 pc scale in the CMZ
and in nearby clouds. Although on the cloud scale of a few pc, gas in the CMZ
exhibits extraordinary physical and chemical properties as compared to gas in
the Galactic disk or in nearby clouds, such as large turbulence linewidth,
strong magnetic fields, and enhancements of particular molecules, in the
smaller scale of $<$0.1 pc where gas starts to be self-gravitating, observed
gas properties, and therefore physics and chemistry of the interstellar
medium, may start to converge.
* •
Based on the identified star formation signatures associated with the cores,
the fraction of protostellar cores in these clouds may be as low as $\sim$5%,
which would indicate a short age of star formation of $\ll$1 Myr and a very
early evolutionary phase for the three clouds, but this time scale is likely
underestimated as our outflow sample is likely incomplete.
* •
Some of the identified outflows have small masses of $\lesssim$1 $M_{\odot}$
and are associated with low-mass cores of $\lesssim$5 $M_{\odot}$, and
therefore likely trace low-mass star formation. Several high-mass outflows are
associated with high-mass cores with known evidence of high-mass star
formation. Therefore, low-mass and high-mass star formation are ongoing
simultaneously in these clouds.
We thank the anonymous referee for helpful comments. X.L. thanks Yuxin Lin,
Luis Zapata, Luca Matrà, and Hauyu Baobab Liu for helpful discussions. X.L.
thanks his family, Qinyu E and Xiaoe Lyu, for their support during the
COVID-19 outbreak during which this manuscript was prepared. X.L. was
supported by JSPS KAKENHI grants No. 18K13589 & 20K14528. J.M.D.K. gratefully
acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) through an Emmy Noether Research Group (grant number
KR4801/1-1), the DFG Sachbeihilfe (grant number KR4801/2-1), and the SFB 881
“The Milky Way System” (subproject B2), as well as from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme via the ERC Starting Grant MUSTANG (grant agreement number 714907).
C.B. and D.W. gratefully acknowledge support from the National Science
Foundation under Award No. 1816715. This paper makes use of the following ALMA
data: ADS/JAO.ALMA#2016.1.00243.S. ALMA is a partnership of ESO (representing
its member states), NSF (USA) and NINS (Japan), together with NRC (Canada),
MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ. Data analysis was in part carried out on the open-use data analysis
computer system at the Astronomy Data Center (ADC) of NAOJ. This research has
made use of NASA’s Astrophysics Data System. , pvextractor
(http://pvextractor.readthedocs.io), calcu (Li et al., 2020)
## Appendix A Calculation of Molecular Column Densities
Assuming optically thin emission, negligible background, Rayleigh-Jeans
approximation, and local thermodynamic equilibrium (LTE) conditions, the beam-
averaged column density of a molecule can be derived following Mangum &
Shirley (2015):
$N_{\text{tot}}=\frac{8\pi
k_{\text{B}}\nu^{2}}{hc^{3}A_{ul}}\frac{Q_{\text{rot}}}{g_{J}g_{K}g_{I}}\exp\left(\frac{E_{u}}{k_{\text{B}}T_{\text{ex}}}\right)\int
T_{B}\text{d}v,$ (A1)
where $k_{\text{B}}$ is the Boltzmann constant, $\nu$ is the rest frequency of
the transition,$h$ is the Planck constant, $c$ is the speed of light, $A_{ul}$
is the spontaneous emission coefficient from the upper state $u$ to the lower
state $l$, $Q_{\text{rot}}$ is the partition function of the molecule, $g_{i}$
($i=J$, $K$, or $I$) are the degeneracies, $E_{u}$ is the energy of the upper
state above the ground level, $T_{\text{ex}}$ is the excitation temperature,
and $\int T_{B}\text{d}v$ is the integrated brightness temperature of the
transition along the velocity axis. The spontaneous emission coefficient of
the transition is
$A_{ul}=\frac{64\pi^{4}\nu^{3}}{3hc^{3}}S\mu^{2},$ (A2)
where $S$ is the line strength and $\mu$ is the permanent dipole moment of the
molecule. Here we directly take the values of $A_{ul}$ from the LAMDA database
(Schöier et al., 2005).
The partition functions $Q_{\text{rot}}$ are approximated using the equations
in Mangum & Shirley (2015), as listed in Table 2. The rotation constants that
are used in the calculation of $Q_{\text{rot}}$ are also listed in Table 2.
Then the column densities of the molecules are derived using Equation A1. The
calculations are implemented in the calcu toolkit (Li et al., 2020).
Table 2: Transition spectral parameters of the outflow tracers. Molecule | Transition | Frequency (GHz) | $E_{u}/k_{\text{B}}$ (K) | $A_{ul}$ (s-1) | $g_{J}$/$g_{K}$/$g_{I}$ | $Q_{\text{rot}}$ | Rotation Constants (MHz)
---|---|---|---|---|---|---|---
SiO | 5–4 | 217.104919 | 31.3 | 5.197$\times$$10^{-4}$ | 11/1/1 | $\frac{k_{\text{B}}T_{\text{ex}}}{hB_{0}}+\frac{1}{3}$ | $B_{0}=21711.96$
HC3N | 24–23 | 218.324723 | 131.0 | 0.826$\times$$10^{-3}$ | 49/1/1 | " | $B_{0}=4549.059$
SO | 6(5)–5(4) | 219.949442 | 35.0 | 1.335$\times$$10^{-4}$ | 13/1/0.5 | " | $B_{0}=21523.556$
H2CO | 3(0,3)–2(0,2) | 218.222192 | 21.0 | 2.818$\times$$10^{-4}$ | 7/1/0.25 | $\frac{1}{2}\left(\frac{\pi k_{\text{B}}^{3}T_{\text{ex}}^{3}}{h^{3}A_{0}B_{0}C_{0}}\right)^{0.5}$ | $A_{0}=281970.56$
| | | | | | | $B_{0}=38833.987$
| | | | | | | $C_{0}=34004.244$
CH3OH | 4(2) –3(1)E1 | 218.440063 | 45.5 | 4.686$\times$$10^{-5}$ | 9/1/0.25 | $\frac{1}{3}\left(\frac{\pi k_{\text{B}}^{3}T_{\text{ex}}^{3}}{h^{3}A_{0}B_{0}C_{0}}\right)^{0.5}$ | $A_{0}=127523.4$
| | | | | | | $B_{0}=24690.2$
| | | | | | | $C_{0}=23759.7$
HNCO | 10(0,10)–9(0,9) | 219.798274 | 58.0 | 1.510$\times$$10^{-4}$ | 21/1/1 | $\left(\frac{\pi k_{\text{B}}^{3}T_{\text{ex}}^{3}}{h^{3}A_{0}B_{0}C_{0}}\right)^{0.5}$ | $A_{0}=912711.4$
| | | | | | | $B_{0}=11071.00$
| | | | | | | $C_{0}=10910.57$
## Appendix B Observational and Physical Properties of the Outflows
The observational and physical properties of the identified outflows are
listed in Table 3.
Table 3: Properties of the outflows.
ID | $V_{\text{lsr}}$ | $M_{\text{core}}$ | Lobes | $\Delta v$ | $F_{\text{int}}$ | $N_{\text{ref}}$ | $X$ | $M_{\text{out}}$ | Notes
---|---|---|---|---|---|---|---|---|---
| (km s-1) | ($M_{\odot}$) | | (km s-1) | (Jy km s-1) | (cm-2) | | ($M_{\odot}$) |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
20 km s-1-A #1 | 24.7 | 13.1 | SiO-blue | [1.5, 23.4] | 2.83 | 1.12$\times$1014 | 1.29$\times$10-9 | 3.36 |
| [CH3CN] | | SO-blue | [3.4, 23.5] | 1.31 | 3.54$\times$1014 | 4.08$\times$10-9 | 3.46 |
| | | CH3OH-blue | [1.7, 23.4] | 2.30 | 7.68$\times$1015 | 8.84$\times$10-8 | 4.73 |
| | | H2CO-blue | [2.8, 23.4] | 1.89 | 6.32$\times$1014 | 7.27$\times$10-9 | 4.79 |
| | | HC3N-blue | [5.7, 23.4] | 0.97 | 9.19$\times$1013 | 1.06$\times$10-9 | 3.96 | $X$(HC3N) of #2
| | | HNCO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | SiO-red | [25.8, 42.1] | 0.79 | 5.14$\times$1013 | 1.17$\times$10-9 | 1.03 |
| | | SO-red | [26.0, 31.5] | 0.38 | 2.00$\times$1014 | 4.54$\times$10-9 | 0.90 |
| | | CH3OH-red | [26.0, 35.6] | 0.73 | 2.92$\times$1015 | 6.64$\times$10-8 | 2.00 |
| | | H2CO-red | [25.9, 36.8] | 0.61 | 2.76$\times$1014 | 6.28$\times$10-9 | 1.78 |
| | | HC3N-red | [26.0, 36.9] | 0.22 | 4.65$\times$1013 | 1.06$\times$10-9 | 0.94 | $X$(HC3N) of #2
| | | HNCO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
20 km s-1-A #2 | 27.1 | 197.6 | SiO-blue | [$-$5.2, 25.9] | 2.45 | 1.87$\times$1014 | 1.01$\times$10-9 | 3.72 |
| [CH3CN] | | SO-blue | [2.0, 25.8] | 2.03 | 1.20$\times$1015 | 6.54$\times$10-9 | 3.34 |
| | | CH3OH-blue | [$-$5.0, 25.8] | 4.06 | 1.56$\times$1016 | 8.46$\times$10-8 | 8.71 |
| | | H2CO-blue | [4.2, 26.0] | 0.36 | 7.23$\times$1014 | 3.92$\times$10-9 | 1.68 |
| | | HC3N-blue | [5.7, 26.1] | 1.69 | 1.95$\times$1014 | 1.06$\times$10-9 | 6.79 |
| | | HNCO-blue | [20.7, 25.8] | 0.28 | 2.04$\times$1014 | 1.11$\times$10-9 | 4.50 |
| | | SiO-red | [28.4, 74.5] | 2.75 | 1.22$\times$1014 | 1.40$\times$10-9 | 3.01 |
| | | SO-red | [28.4, 51.5] | 1.91 | 5.92$\times$1014 | 6.80$\times$10-9 | 3.01 |
| | | CH3OH-red | [28.4, 38.3] | 1.19 | 5.30$\times$1015 | 6.09$\times$10-8 | 3.56 |
| | | H2CO-red | [28.4, 39.5] | 1.13 | 7.49$\times$1014 | 8.60$\times$10-9 | 2.42 |
| | | HC3N-red | [28.4, 43.6] | 1.21 | 9.21$\times$1013 | 1.06$\times$10-9 | 4.91 |
| | | HNCO-red | [28.4, 36.8] | 0.33 | 1.46$\times$1014 | 1.68$\times$10-9 | 3.57 |
20 km s-1-B #1* | 16.6 | 13.8 | SiO-blue | [$-$36.3, 15.3] | 4.61 | 3.11$\times$1014 | 3.08$\times$10-10 | 22.92 |
| [CH3OH] | | SO-blue | [$-$11.3, 15.5] | 1.34 | 3.58$\times$1014 | 3.54$\times$10-10 | 40.51 |
| | | CH3OH-blue | [$-$19.9, 15.3] | 3.48 | 2.59$\times$1016 | 2.56$\times$10-8 | 24.69 |
| | | H2CO-blue | [$-$18.8, 15.3] | 2.50 | 1.49$\times$1015 | 1.47$\times$10-9 | 31.29 |
| | | HC3N-blue | [$-$3.8, 15.3] | 0.60 | 1.32$\times$1014 | 1.31$\times$10-10 | 19.85 |
| | | HNCO-blue | [2.0, 15.5] | 0.55 | 5.30$\times$1014 | 5.24$\times$10-10 | 37.40 |
| | | SiO-red | [17.7, 39.4] | 4.46 | 1.04$\times$1014 | 1.89$\times$10-10 | 36.19 |
| | | SO-red | [17.9, 28.8] | 1.42 | 3.84$\times$1014 | 6.96$\times$10-10 | 21.84 |
| | | CH3OH-red | [17.9, 24.8] | 0.68 | 2.55$\times$1015 | 4.62$\times$10-9 | 26.84 |
| | | H2CO-red | [17.7, 26.0] | 1.58 | 3.18$\times$1014 | 5.77$\times$10-10 | 50.26 |
| | | HC3N-red | [17.8, 26.1] | 0.33 | 7.20$\times$1013 | 1.31$\times$10-10 | 10.69 |
| | | HNCO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
20 km s-1-B #2* | 15.1 | 27.9 | SiO-blue | [$-$7.9, 13.8] | 0.76 | $\cdots$ | 5.81$\times$10-10 | 2.01 | $X$(SiO) of the red lobe
| [CH3CN] | | SO-blue | [8.7, 13.8] | 0.04 | $\cdots$ | 2.28$\times$10-9 | 0.18 | $X$(SO) of the red lobe
| | | CH3OH-blue | [$-$7.7, 14.0] | 0.93 | $\cdots$ | 2.76$\times$10-8 | 6.09 | $X$(CH3OH) of the red lobe
| | | H2CO-blue | [$-$10.7, 13.8] | 0.89 | $\cdots$ | 4.18$\times$10-9 | 3.92 | $X$(H2CO) of the red lobe
| | | HC3N-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HNCO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | SiO-red | [16.4, 35.3] | 1.73 | 1.39$\times$1014 | 5.81$\times$10-10 | 4.58 |
| | | SO-red | [16.4, 32.8] | 0.96 | 5.46$\times$1014 | 2.28$\times$10-9 | 4.53 |
| | | CH3OH-red | [16.4, 27.5] | 0.65 | 6.60$\times$1015 | 2.76$\times$10-8 | 4.24 |
| | | H2CO-red | [16.4, 31.4] | 1.06 | 9.99$\times$1014 | 4.18$\times$10-9 | 4.64 |
| | | HC3N-red | [16.4, 26.1] | 0.18 | 9.11$\times$1013 | 3.81$\times$10-10 | 2.10 |
| | | HNCO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
20 km s-1-C #1 | 13.9 | 35.8 | SiO-blue | [$-$18.7, 12.6] | 0.59 | $\cdots$ | 1.44$\times$10-9 | 0.62 | Mean $X$(SiO) of region C
| [CH3OH] | | SO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | CH3OH-blue | [7.1, 12.6] | 0.26 | $\cdots$ | 8.41$\times$10-8 | 0.56 | Mean $X$(CH3OH) of region C
| | | H2CO-blue | [1.5, 12.6] | 0.26 | $\cdots$ | 6.67$\times$10-9 | 0.71 | Mean $X$(H2CO) of region C
| | | HC3N-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HNCO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | SiO-red | [15.0, 31.3] | 0.67 | $\cdots$ | 1.44$\times$10-9 | 0.72 | Mean $X$(SiO) of region C
| | | SO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | CH3OH-red | [15.2, 20.7] | 0.67 | $\cdots$ | 8.41$\times$10-8 | 1.44 | Mean $X$(CH3OH) of region C
| | | H2CO-red | [15.0, 23.2] | 0.35 | $\cdots$ | 6.67$\times$10-9 | 0.97 | Mean $X$(H2CO) of region C
| | | HC3N-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HNCO-red | [15.2, 18.2] | 0.03 | $\cdots$ | 2.24$\times$10-9 | 0.46 | Mean $X$(HNCO) of region C
20 km s-1-C #2* | 16.6 | 17.1 | SiO-blue | [$-$14.7, 15.3] | 2.18 | 7.03$\times$1013 | 8.44$\times$10-10 | 3.96 |
| [CH3OH] | | SO-blue | [6.0, 15.5] | 0.29 | 2.00$\times$1014 | 2.40$\times$10-9 | 1.31 |
| | | CH3OH-blue | [$-$0.9, 15.3] | 2.94 | 9.65$\times$1015 | 1.16$\times$10-7 | 4.59 |
| | | H2CO-blue | [0.1, 15.3] | 1.03 | 4.02$\times$1014 | 4.83$\times$10-9 | 3.93 |
| | | HC3N-blue | [4.3, 15.3] | 0.22 | 4.19$\times$1013 | 5.03$\times$10-10 | 1.99 |
| | | HNCO-blue | [$-$0.6, 15.5] | 0.75 | 2.92$\times$1014 | 3.51$\times$10-9 | 3.79 |
| | | SiO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | SO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | CH3OH-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | H2CO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HC3N-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HNCO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
20 km s-1-C #3 | 13.9 | 5.9 | SiO-blue | [$-$13.3, 12.6] | 1.09 | $\cdots$ | 1.42$\times$10-9 | 1.18 | $X$(SiO) of the red lobe
| [CH3OH] | | SO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | CH3OH-blue | [8.5, 12.6] | 0.02 | $\cdots$ | 4.10$\times$10-8 | 0.10 | $X$(CH3OH) of the red lobe
| | | H2CO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HC3N-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HNCO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | SiO-red | [15.0, 74.5] | 2.14 | 1.17$\times$1014 | 1.42$\times$10-9 | 2.31 |
| | | SO-red | [15.2, 30.2] | 0.46 | 1.25$\times$1014 | 1.52$\times$10-9 | 3.26 |
| | | CH3OH-red | [15.2, 24.8] | 0.25 | 3.38$\times$1015 | 4.10$\times$10-8 | 1.12 |
| | | H2CO-red | [15.0, 25.9] | 0.30 | 4.20$\times$1014 | 5.10$\times$10-9 | 1.07 |
| | | HC3N-red | [15.1, 34.2] | 0.18 | 4.15$\times$1013 | 5.04$\times$10-10 | 1.59 | Mean $X$(HC3N) of region C
| | | HNCO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
20 km s-1-C #4* | 11.2 | 4.8 | SiO-blue | [$-$43.0, 9.9] | 4.70 | 3.21$\times$1014 | 1.29$\times$10-9 | 5.58 |
| [CH3OH] | | SO-blue | [$-$42.0, 10.1] | 4.38 | 1.17$\times$1015 | 4.74$\times$10-9 | 9.92 |
| | | CH3OH-blue | [$-$9.1, 9.9] | 2.10 | 7.08$\times$1015 | 2.85$\times$10-8 | 13.37 |
| | | H2CO-blue | [$-$9.3, 9.9] | 1.75 | 1.07$\times$1015 | 4.31$\times$10-9 | 7.47 |
| | | HC3N-blue | [$-$6.5, 9.9] | 0.75 | 1.25$\times$1014 | 5.04$\times$10-10 | 6.35 | Mean $X$(HC3N) of region C
| | | HNCO-blue | [$-$6.0, 10.1] | 0.76 | 3.92$\times$1014 | 1.58$\times$10-9 | 8.59 |
| | | SiO-red | [12.3, 61.0] | 1.67 | 7.80$\times$1013 | 1.16$\times$10-9 | 2.20 |
| | | SO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | CH3OH-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | H2CO-red | [12.3, 25.9] | 0.36 | 2.46$\times$1014 | 3.66$\times$10-9 | 1.81 |
| | | HC3N-red | [12.4, 15.2] | 0.02 | 3.39$\times$1013 | 5.04$\times$10-10 | 0.40 | Mean $X$(HC3N) of region C
| | | HNCO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
20 km s-1-C #5 | 12.5 | 18.7 | SiO-blue | [$-$39.0, 11.2] | 1.03 | 9.12$\times$1013 | 4.37$\times$10-10 | 3.62 |
| [CH3CN] | | SO-blue | [3.4, 11.5] | 0.33 | 2.16$\times$1014 | 1.03$\times$10-9 | 3.42 |
| | | CH3OH-blue | [3.1, 11.3] | 1.01 | 7.29$\times$1015 | 3.49$\times$10-8 | 5.24 |
| | | H2CO-blue | [1.5, 11.2] | 0.13 | 2.91$\times$1014 | 1.39$\times$10-9 | 1.71 |
| | | HC3N-blue | [1.6, 11.2] | 0.20 | 8.65$\times$1013 | 4.14$\times$10-10 | 1.93 |
| | | HNCO-blue | [6.0, 11.5] | 0.43 | 3.40$\times$1014 | 1.63$\times$10-9 | 4.74 |
| | | SiO-red | [13.7, 42.1] | 2.12 | 1.43$\times$1014 | 9.30$\times$10-10 | 3.49 |
| | | SO-red | [13.8, 24.8] | 1.25 | 2.36$\times$1014 | 1.54$\times$10-9 | 8.75 |
| | | CH3OH-red | [13.8, 22.1] | 1.07 | 8.52$\times$1015 | 5.54$\times$10-8 | 3.52 |
| | | H2CO-red | [13.7, 25.9] | 1.55 | 7.95$\times$1014 | 5.17$\times$10-9 | 5.50 |
| | | HC3N-red | [13.8, 32.8] | 0.73 | 6.37$\times$1013 | 4.14$\times$10-10 | 7.73 |
| | | HNCO-red | [13.8, 19.5] | 0.68 | 3.37$\times$1014 | 2.19$\times$10-9 | 5.54 |
20 km s-1-C #6 | 12.5 | 89.0 | SiO-blue | [$-$44.4, 11.2] | 2.08 | 3.01$\times$1014 | 3.94$\times$10-10 | 8.12 |
| [CH3CN] | | SO-blue | [$-$32.7, 11.5] | 1.15 | 9.86$\times$1014 | 1.29$\times$10-9 | 9.54 |
| | | CH3OH-blue | [$-$13.2, 11.3] | 1.26 | 1.49$\times$1016 | 1.95$\times$10-8 | 11.74 |
| | | H2CO-blue | [$-$26.9, 11.2] | 0.86 | 1.49$\times$1015 | 1.95$\times$10-9 | 8.06 |
| | | HC3N-blue | [$-$13.3, 11.2] | 0.62 | 2.07$\times$1014 | 2.71$\times$10-10 | 9.82 |
| | | HNCO-blue | [$-$2.0, 11.5] | 0.25 | 5.11$\times$1014 | 6.69$\times$10-10 | 6.67 |
| | | SiO-red | [13.7, 34.0] | 0.96 | 1.01$\times$1014 | 2.66$\times$10-10 | 5.53 |
| | | SO-red | [13.8, 24.8] | 0.31 | 5.56$\times$1013 | 1.46$\times$10-10 | 22.54 |
| | | CH3OH-red | [13.8, 35.6] | 0.71 | 9.70$\times$1015 | 2.55$\times$10-8 | 5.06 |
| | | H2CO-red | [13.7, 34.1] | 1.09 | 1.10$\times$1015 | 2.89$\times$10-9 | 6.91 |
| | | HC3N-red | [13.8, 30.1] | 0.24 | 1.03$\times$1014 | 2.71$\times$10-10 | 3.69 |
| | | HNCO-red | [13.8, 16.8] | 0.06 | 9.40$\times$1013 | 2.47$\times$10-10 | 4.13 |
20 km s-1-C #7 | 15.1 | 29.2 | SiO-blue | [$-$1.2, 13.8] | 1.25 | 7.24$\times$1013 | 9.64$\times$10-10 | 1.98 |
| [CH3CN] | | SO-blue | [0.7, 13.8] | 0.94 | 4.48$\times$1014 | 5.96$\times$10-9 | 1.69 |
| | | CH3OH-blue | [4.4, 14.0] | 2.05 | 1.23$\times$1016 | 1.64$\times$10-7 | 2.27 |
| | | H2CO-blue | [2.8, 13.8] | 0.24 | 3.21$\times$1014 | 4.27$\times$10-9 | 1.01 |
| | | HC3N-blue | [5.7, 13.9] | 0.32 | 8.40$\times$1013 | 1.12$\times$10-9 | 1.21 |
| | | HNCO-blue | [10.1, 13.8] | 0.22 | 2.28$\times$1014 | 3.04$\times$10-9 | 1.31 |
| | | SiO-red | [16.4, 46.1] | 3.52 | 9.16$\times$1013 | 1.45$\times$10-9 | 3.72 |
| | | SO-red | [16.4, 30.2] | 2.21 | 6.94$\times$1014 | 1.10$\times$10-8 | 2.16 |
| | | CH3OH-red | [16.4, 27.5] | 1.68 | 3.64$\times$1015 | 5.77$\times$10-8 | 5.29 |
| | | H2CO-red | [16.4, 30.0] | 3.65 | 8.18$\times$1014 | 1.30$\times$10-8 | 5.17 |
| | | HC3N-red | [16.4, 22.0] | 0.13 | 7.06$\times$1013 | 1.12$\times$10-9 | 0.50 |
| | | HNCO-red | [16.4, 23.5] | 0.66 | 1.36$\times$1014 | 2.15$\times$10-9 | 5.48 |
20 km s-1-C #8 | 15.1 | 10.4 | SiO-blue | [$-$10.6, 13.8] | 1.36 | 8.97$\times$1013 | 4.12$\times$10-9 | 0.51 |
| [CH3CN] | | SO-blue | [$-$10.0, 13.8] | 1.31 | 4.74$\times$1014 | 2.18$\times$10-8 | 0.65 |
| | | CH3OH-blue | [$-$7.7, 14.0] | 0.64 | 5.34$\times$1015 | 2.45$\times$10-7 | 0.48 |
| | | H2CO-blue | [$-$9.3, 13.8] | 0.20 | 2.69$\times$1014 | 1.23$\times$10-8 | 0.30 |
| | | HC3N-blue | [9.7, 13.9] | 0.07 | 7.45$\times$1012 | 3.42$\times$10-10 | 0.88 |
| | | HNCO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | SiO-red | [16.4, 47.5] | 3.54 | 1.33$\times$1014 | 2.46$\times$10-9 | 2.20 |
| | | SO-red | [16.4, 38.2] | 1.82 | 5.36$\times$1014 | 9.90$\times$10-9 | 1.97 |
| | | CH3OH-red | [16.4, 27.5] | 0.50 | 3.22$\times$1015 | 5.95$\times$10-8 | 1.55 |
| | | H2CO-red | [16.4, 38.1] | 1.41 | 7.75$\times$1014 | 1.43$\times$10-8 | 1.81 |
| | | HC3N-red | [16.4, 22.0] | 0.12 | 1.85$\times$1013 | 3.42$\times$10-10 | 1.46 |
| | | HNCO-red | [16.4, 20.8] | 0.18 | 1.61$\times$1014 | 2.98$\times$10-9 | 1.10 |
20 km s-1-C #9 | 11.1 | 10.4 | SiO-blue | [$-$9.3, 9.8] | 0.60 | 1.26$\times$1014 | 1.05$\times$10-9 | 0.89 |
| [HC3N] | | SO-blue | [2.0, 10.1] | 0.08 | 2.18$\times$1014 | 1.82$\times$10-9 | 0.48 |
| | | CH3OH-blue | [3.1, 9.9] | 0.23 | 5.77$\times$1015 | 4.81$\times$10-8 | 0.87 |
| | | H2CO-blue | [1.5, 9.8] | 0.30 | 5.64$\times$1014 | 4.70$\times$10-9 | 1.17 |
| | | HC3N-blue | [7.0, 9.8] | 0.03 | 4.51$\times$1013 | 3.76$\times$10-10 | 0.32 |
| | | HNCO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | SiO-red | [12.3, 44.8] | 0.75 | 1.45$\times$1014 | 2.92$\times$10-9 | 0.39 |
| | | SO-red | [12.4, 44.8] | 0.99 | 8.00$\times$1014 | 1.61$\times$10-8 | 0.66 |
| | | CH3OH-red | [12.4, 20.7] | 0.21 | 4.93$\times$1015 | 9.91$\times$10-8 | 0.38 |
| | | H2CO-red | [12.3, 27.3] | 0.16 | 4.28$\times$1014 | 8.60$\times$10-9 | 0.35 |
| | | HC3N-red | [12.4, 16.6] | 0.03 | 1.87$\times$1013 | 3.76$\times$10-10 | 0.37 |
| | | HNCO-red | [12.4, 16.8] | 0.16 | 1.84$\times$1014 | 3.70$\times$10-9 | 0.76 |
20 km s-1-D #1 | 11.1 | 56.2 | SiO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| [CH3CN] | | SO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | CH3OH-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | H2CO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HC3N-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HNCO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | SiO-red | [12.3, 35.3] | 1.35 | $\cdots$ | 3.83$\times$10-9 | 0.54 | Mean $X$(SiO) of region D
| | | SO-red | [12.4, 16.8] | 0.25 | $\cdots$ | 1.70$\times$10-8 | 0.16 | Mean $X$(SO) of region D
| | | CH3OH-red | [12.4, 18.0] | 0.40 | $\cdots$ | 1.74$\times$10-7 | 0.42 | Mean $X$(CH3OH) of region D
| | | H2CO-red | [12.3, 19.2] | 0.63 | $\cdots$ | 1.68$\times$10-8 | 0.69 | Mean $X$(H2CO) of region D
| | | HC3N-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HNCO-red | [12.4, 15.5] | 0.17 | $\cdots$ | 1.18$\times$10-8 | 0.53 | Mean $X$(HNCO) of region D
20 km s-1-D #2 | 11.1 | 56.2 | SiO-blue | [$-$21.4, 9.8] | 8.95 | 2.03$\times$1014 | 1.01$\times$10-9 | 13.58 |
| [CH3CN] | | SO-blue | [$-$20.7, 9.8] | 8.28 | 6.64$\times$1014 | 3.30$\times$10-9 | 26.94 |
| | | CH3OH-blue | [$-$10.5, 9.9] | 7.00 | 3.83$\times$1015 | 1.91$\times$10-8 | 66.50 |
| | | H2CO-blue | [$-$14.7, 9.8] | 7.39 | 8.66$\times$1014 | 4.31$\times$10-9 | 31.52 |
| | | HC3N-blue | [$-$6.5, 9.8] | 1.92 | 1.24$\times$1014 | 6.18$\times$10-10 | 13.33 |
| | | HNCO-blue | [$-$6.0, 9.8] | 1.42 | 3.78$\times$1014 | 1.88$\times$10-9 | 26.99 |
| | | SiO-red | [12.3, 17.8] | 0.47 | 5.56$\times$1013 | 3.97$\times$10-10 | 1.81 |
| | | SO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | CH3OH-red | [12.4, 18.0] | 1.28 | 9.72$\times$1015 | 6.93$\times$10-8 | 3.34 |
| | | H2CO-red | [12.3, 16.5] | 0.99 | 4.10$\times$1014 | 2.92$\times$10-9 | 6.23 |
| | | HC3N-red | [12.4, 17.9] | 0.16 | 8.66$\times$1013 | 6.18$\times$10-10 | 1.13 |
| | | HNCO-red | [12.4, 16.8] | 0.39 | 2.43$\times$1014 | 1.73$\times$10-9 | 8.07 |
20 km s-1-D #3 | 9.9 | 35.1 | SiO-blue | [$-$26.8, 8.6] | 2.74 | 2.14$\times$1014 | 2.31$\times$10-9 | 1.82 |
| [CH3OH] | | SO-blue | [$-$16.6, 8.8] | 1.36 | 6.98$\times$1014 | 7.52$\times$10-9 | 1.95 |
| | | CH3OH-blue | [$-$1.0, 8.6] | 1.10 | 4.58$\times$1015 | 4.94$\times$10-8 | 4.03 |
| | | H2CO-blue | [$-$2.6, 8.6] | 0.59 | 3.39$\times$1014 | 3.65$\times$10-9 | 2.97 |
| | | HC3N-blue | [$-$1.1, 8.5] | 0.46 | 8.20$\times$1013 | 8.84$\times$10-10 | 2.24 | Mean $X$(HC3N) of region D
| | | HNCO-blue | [$-$0.6, 8.8] | 0.62 | 2.77$\times$1014 | 2.99$\times$10-9 | 7.44 |
| | | SiO-red | [11.2, 38.0] | 2.09 | 1.13$\times$1014 | 3.96$\times$10-9 | 0.81 |
| | | SO-red | [11.2, 23.5] | 0.63 | 2.56$\times$1014 | 8.98$\times$10-9 | 0.75 |
| | | CH3OH-red | [11.2, 21.2] | 0.14 | 1.24$\times$1015 | 4.35$\times$10-8 | 0.59 |
| | | H2CO-red | [11.0, 19.2] | 0.11 | 2.10$\times$1014 | 7.37$\times$10-9 | 0.27 |
| | | HC3N-red | [11.1, 21.1] | 0.05 | 2.52$\times$1013 | 8.84$\times$10-10 | 0.23 | Mean $X$(HC3N) of region D
| | | HNCO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
20 km s-1-D #4 | 11.1 | 38.7 | SiO-blue | [$-$7.9, 9.8] | 2.03 | 9.71$\times$1013 | 3.53$\times$10-9 | 0.88 |
| [CH3CN] | | SO-blue | [$-$3.3, 9.8] | 0.68 | 3.84$\times$1014 | 1.40$\times$10-8 | 0.53 |
| | | CH3OH-blue | [$-$1.0, 9.9] | 1.26 | 9.10$\times$1015 | 3.31$\times$10-7 | 0.69 |
| | | H2CO-blue | [$-$9.3, 9.8] | 0.82 | 4.88$\times$1014 | 1.78$\times$10-8 | 0.84 |
| | | HC3N-blue | [3.0, 9.8] | 0.08 | 3.17$\times$1013 | 1.15$\times$10-9 | 0.31 |
| | | HNCO-blue | [$-$0.6, 9.8] | 0.25 | 3.90$\times$1014 | 1.42$\times$10-8 | 0.62 |
| | | SiO-red | [12.3, 38.0] | 1.62 | 1.07$\times$1014 | 1.04$\times$10-8 | 0.24 |
| | | SO-red | [12.4, 22.2] | 0.71 | 3.46$\times$1014 | 3.36$\times$10-8 | 0.23 |
| | | CH3OH-red | [12.4, 18.0] | 0.52 | 2.86$\times$1015 | 2.77$\times$10-7 | 0.34 |
| | | H2CO-red | [12.3, 21.9] | 0.77 | 4.33$\times$1014 | 4.20$\times$10-8 | 0.34 |
| | | HC3N-red | [12.4, 13.9] | 0.05 | 1.19$\times$1013 | 1.15$\times$10-9 | 0.21 |
| | | HNCO-red | [12.4, 19.5] | 0.72 | 3.01$\times$1014 | 2.92$\times$10-8 | 0.88 |
20 km s-1-E #1 | 11.2 | 5.8 | SiO-blue | [$-$34.9, 9.9] | 7.00 | $\cdots$ | 1.69$\times$10-9 | 6.35 | Mean $X$(SiO) of the cloud
| [CH3OH] | | SO-blue | [$-$10.0, 10.1] | 2.78 | $\cdots$ | 7.10$\times$10-9 | 4.20 | Mean $X$(SO) of the cloud
| | | CH3OH-blue | [5.8, 9.9] | 0.24 | $\cdots$ | 9.86$\times$10-8 | 0.45 | Mean $X$(CH3OH) of the cloud
| | | H2CO-blue | [$-$0.1, 9.9] | 1.49 | $\cdots$ | 7.99$\times$10-9 | 3.42 | Mean $X$(H2CO) of the cloud
| | | HC3N-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HNCO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | SiO-red | [12.3, 24.5] | 0.88 | $\cdots$ | 1.69$\times$10-9 | 0.79 | Mean $X$(SiO) of the cloud
| | | SO-red | [12.5, 20.8] | 1.27 | $\cdots$ | 7.10$\times$10-9 | 1.92 | Mean $X$(SO) of the cloud
| | | CH3OH-red | [12.5, 18.0] | 0.55 | $\cdots$ | 9.86$\times$10-8 | 1.02 | Mean $X$(CH3OH) of the cloud
| | | H2CO-red | [12.3, 20.5] | 1.11 | $\cdots$ | 7.99$\times$10-9 | 2.57 | Mean $X$(H2CO) of the cloud
| | | HC3N-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HNCO-red | [12.5, 22.5] | 0.56 | $\cdots$ | 4.40$\times$10-9 | 4.54 | Mean $X$(HNCO) of the cloud
20 km s-1-F #1 | 6.1 | 6.4 | SiO-blue | [$-$21.4, 4.8] | 3.39 | $\cdots$ | 1.69$\times$10-9 | 3.08 | Mean $X$(SiO) of the cloud
| [C18O] | | SO-blue | [$-$2.0, 4.8] | 1.54 | $\cdots$ | 7.10$\times$10-9 | 2.34 | Mean $X$(SO) of the cloud
| | | CH3OH-blue | [$-$1.0, 4.8] | 1.26 | $\cdots$ | 9.86$\times$10-8 | 2.32 | Mean $X$(CH3OH) of the cloud
| | | H2CO-blue | [$-$3.9, 4.8] | 1.51 | $\cdots$ | 7.99$\times$10-9 | 3.48 | Mean $X$(H2CO) of the cloud
| | | HC3N-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HNCO-blue | [$-$2.0, 4.8] | 1.25 | $\cdots$ | 4.40$\times$10-9 | 10.12 | Mean $X$(HNCO) of the cloud
| | | SiO-red | [7.4, 29.9] | 3.01 | $\cdots$ | 1.69$\times$10-9 | 2.73 | Mean $X$(SiO) of the cloud
| | | SO-red | [7.4, 17.4] | 0.82 | $\cdots$ | 7.10$\times$10-9 | 1.24 | Mean $X$(SO) of the cloud
| | | CH3OH-red | [7.4, 17.4] | 0.22 | $\cdots$ | 9.86$\times$10-8 | 0.40 | Mean $X$(CH3OH) of the cloud
| | | H2CO-red | [7.4, 16.5] | 0.76 | $\cdots$ | 7.99$\times$10-9 | 1.74 | Mean $X$(H2CO) of the cloud
| | | HC3N-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HNCO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
20 km s-1-G #1* | 8.5 | 17.8 | SiO-blue | [$-$12.0, 7.2] | 1.28 | 6.93$\times$1013 | 1.28$\times$10-9 | 1.54 |
| [CH3CN] | | SO-blue | [$-$0.6, 7.5] | 1.03 | 3.40$\times$1014 | 6.28$\times$10-9 | 1.76 |
| | | CH3OH-blue | [0.4, 7.2] | 1.12 | 1.00$\times$1016 | 1.85$\times$10-7 | 1.09 |
| | | H2CO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HC3N-blue | [$-$4.3, 7.2] | 0.29 | 7.72$\times$1013 | 1.43$\times$10-9 | 0.87 |
| | | HNCO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | SiO-red | [9.6, 29.9] | 2.14 | 5.64$\times$1013 | 1.34$\times$10-9 | 2.44 |
| | | SO-red | [9.8, 11.5] | 0.15 | 9.36$\times$1013 | 2.22$\times$10-9 | 0.75 |
| | | CH3OH-red | [9.8, 16.7] | 0.60 | 6.68$\times$1015 | 1.59$\times$10-7 | 0.69 |
| | | H2CO-red | [9.6, 17.8] | 0.74 | 3.58$\times$1014 | 8.52$\times$10-9 | 1.60 |
| | | HC3N-red | [9.7, 16.6] | 0.46 | 6.00$\times$1013 | 1.43$\times$10-9 | 1.37 |
| | | HNCO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
Sgr B1-off-A #1 | 28.4 | 55.2 | SiO-blue | [3.2, 27.2] | 2.43 | 1.81$\times$1014 | 1.82$\times$10-9 | 2.06 |
| [CH3CN] | | SO-blue | [4.7, 27.1] | 1.24 | 7.34$\times$1014 | 7.40$\times$10-9 | 1.81 |
| | | CH3OH-blue | [$-$2.3, 27.1] | 1.72 | 1.02$\times$1016 | 1.03$\times$10-7 | 3.05 |
| | | H2CO-blue | [4.2, 27.3] | 1.56 | 1.11$\times$1015 | 1.12$\times$10-8 | 2.58 |
| | | HC3N-blue | [9.7, 27.4] | 1.38 | 1.97$\times$1014 | 1.99$\times$10-9 | 2.98 |
| | | HNCO-blue | [22.1, 27.1] | 0.18 | 1.64$\times$1014 | 1.65$\times$10-9 | 1.93 |
| | | SiO-red | [29.7, 51.5] | 1.75 | 6.73$\times$1013 | 1.50$\times$10-9 | 1.79 |
| | | SO-red | [29.7, 50.2] | 0.90 | 4.78$\times$1014 | 1.06$\times$10-8 | 0.92 |
| | | CH3OH-red | [29.7, 49.1] | 1.31 | 6.57$\times$1015 | 1.46$\times$10-7 | 1.64 |
| | | H2CO-red | [29.7, 51.6] | 1.02 | 6.77$\times$1014 | 1.51$\times$10-8 | 1.26 |
| | | HC3N-red | [29.7, 39.6] | 0.72 | 8.93$\times$1013 | 1.99$\times$10-9 | 1.55 |
| | | HNCO-red | [29.7, 39.5] | 0.64 | 2.91$\times$1014 | 6.47$\times$10-9 | 1.79 |
Sgr B1-off-B #1 | 28.8 | 14.5 | SiO-blue | [17.7, 27.5] | 0.15 | $\cdots$ | 1.39$\times$10-9 | 0.16 | Mean $X$(SiO) of the cloud
| [CH3OH] | | SO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | CH3OH-blue | [22.0, 27.5] | 0.09 | $\cdots$ | 8.63$\times$10-8 | 0.20 | Mean $X$(CH3OH) of the cloud
| | | H2CO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HC3N-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HNCO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | SiO-red | [29.9, 39.4] | 0.12 | $\cdots$ | 1.39$\times$10-9 | 0.13 | Mean $X$(SiO) of the cloud
| | | SO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | CH3OH-red | [30.1, 35.6] | 0.12 | $\cdots$ | 8.63$\times$10-8 | 0.26 | Mean $X$(CH3OH) of the cloud
| | | H2CO-red | [29.9, 40.8] | 0.11 | $\cdots$ | 1.09$\times$10-8 | 0.19 | Mean $X$(H2CO) of the cloud
| | | HC3N-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HNCO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
Sgr B1-off-C #1 | 30.0 | 30.6 | SiO-blue | [8.3, 28.7] | 0.46 | $\cdots$ | 1.66$\times$10-9 | 0.42 | $X$(SiO) of the red lobe
| [HC3N] | | SO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | CH3OH-blue | [22.0, 28.9] | 0.15 | $\cdots$ | 5.39$\times$10-8 | 0.50 | $X$(CH3OH) of the red lobe
| | | H2CO-blue | [11.0, 28.7] | 0.52 | $\cdots$ | 1.11$\times$10-8 | 0.87 | $X$(H2CO) of the red lobe
| | | HC3N-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HNCO-blue | [24.7, 28.8] | 0.09 | $\cdots$ | 5.41$\times$10-9 | 0.29 | $X$(HNCO) of the red lobe
| | | SiO-red | [31.2, 51.5] | 0.55 | 8.64$\times$1013 | 1.66$\times$10-9 | 0.51 |
| | | SO-red | [31.3, 47.5] | 0.19 | 2.84$\times$1014 | 5.44$\times$10-9 | 0.38 |
| | | CH3OH-red | [31.3, 39.7] | 0.24 | 2.81$\times$1015 | 5.39$\times$10-8 | 0.81 |
| | | H2CO-red | [31.3, 47.5] | 0.63 | 5.78$\times$1014 | 1.11$\times$10-8 | 1.05 |
| | | HC3N-red | [31.3, 38.2] | 0.17 | 3.40$\times$1013 | 6.52$\times$10-10 | 1.10 |
| | | HNCO-red | [31.3, 39.5] | 0.16 | 2.82$\times$1014 | 5.41$\times$10-9 | 0.53 |
Sgr B1-off-C #2 | 29.7 | 230.4 | SiO-blue | [20.5, 28.6] | 0.51 | 5.13$\times$1013 | 1.01$\times$10-9 | 0.78 |
| [CH3CN] | | SO-blue | [20.7, 28.4] | 0.77 | 4.52$\times$1014 | 8.86$\times$10-9 | 0.94 |
| | | CH3OH-blue | [19.3, 28.4] | 0.27 | 4.90$\times$1015 | 9.61$\times$10-8 | 0.51 |
| | | H2CO-blue | [11.0, 28.7] | 0.44 | 4.61$\times$1014 | 9.04$\times$10-9 | 0.90 |
| | | HC3N-blue | [24.6, 28.4] | 0.20 | 5.34$\times$1013 | 1.05$\times$10-9 | 0.80 |
| | | HNCO-blue | [24.7, 28.4] | 0.17 | 1.52$\times$1014 | 2.98$\times$10-9 | 1.02 |
| | | SiO-red | [31.0, 35.3] | 0.27 | $\cdots$ | 1.01$\times$10-9 | 0.42 | $X$(SiO) of the blue lobe
| | | SO-red | [31.0, 36.8] | 0.95 | 2.20$\times$1014 | 2.72$\times$10-9 | 3.78 |
| | | CH3OH-red | [31.0, 35.6] | 0.98 | 3.80$\times$1015 | 4.70$\times$10-8 | 3.79 |
| | | H2CO-red | [31.0, 40.8] | 0.64 | 4.69$\times$1014 | 5.80$\times$10-9 | 2.04 |
| | | HC3N-red | [31.0, 38.2] | 0.27 | 8.47$\times$1013 | 1.05$\times$10-9 | 1.09 |
| | | HNCO-red | [31.0, 36.8] | 0.66 | 3.29$\times$1014 | 4.07$\times$10-9 | 2.91 |
Sgr B1-off-C #3 | 30.0 | 32.6 | SiO-blue | [17.7, 28.7] | 0.28 | 5.22$\times$1013 | 9.50$\times$10-10 | 0.46 |
| [HC3N] | | SO-blue | [19.4, 28.8] | 0.16 | 2.32$\times$1014 | 4.22$\times$10-9 | 0.42 |
| | | CH3OH-blue | [17.9, 28.9] | 0.32 | 4.16$\times$1015 | 7.57$\times$10-8 | 0.78 |
| | | H2CO-blue | [17.7, 28.7] | 0.36 | 6.18$\times$1014 | 1.12$\times$10-8 | 0.60 |
| | | HC3N-blue | [21.9, 28.8] | 0.12 | 7.08$\times$1013 | 1.29$\times$10-9 | 0.38 |
| | | HNCO-blue | [20.7, 28.8] | 0.27 | 3.08$\times$1014 | 5.61$\times$10-9 | 0.86 |
| | | SiO-red | [31.2, 46.0] | 1.38 | 9.48$\times$1013 | 1.42$\times$10-9 | 1.50 |
| | | SO-red | [31.3, 44.9] | 1.01 | 2.72$\times$1014 | 4.06$\times$10-9 | 2.69 |
| | | CH3OH-red | [31.3, 42.4] | 0.92 | 5.52$\times$1015 | 8.25$\times$10-8 | 2.04 |
| | | H2CO-red | [31.3, 42.2] | 1.81 | 8.52$\times$1014 | 1.27$\times$10-8 | 2.64 |
| | | HC3N-red | [31.3, 39.6] | 0.49 | 8.62$\times$1013 | 1.29$\times$10-9 | 1.62 |
| | | HNCO-red | [31.3, 38.2] | 0.73 | 2.72$\times$1014 | 4.07$\times$10-9 | 3.20 |
Sgr C-A #1 | $-$51.6 | 10.9 | SiO-blue | [$-$74.4, $-$52.8] | 4.94 | 5.40$\times$1013 | 1.60$\times$10-9 | 4.71 |
| [CH3CN] | | SO-blue | [$-$64.6, $-$52.9] | 1.48 | 2.12$\times$1014 | 6.28$\times$10-9 | 2.53 |
| | | CH3OH-blue | [$-$58.9, $-$52.9] | 0.57 | 2.04$\times$1015 | 6.05$\times$10-8 | 1.69 |
| | | H2CO-blue | [$-$64.6, $-$52.9] | 0.62 | 6.26$\times$1014 | 1.86$\times$10-8 | 0.61 |
| | | HC3N-blue | [$-$57.7, $-$52.9] | 0.59 | 3.73$\times$1013 | 1.11$\times$10-9 | 2.27 |
| | | HNCO-blue | [$-$63.3, $-$52.9] | 0.33 | 4.42$\times$1014 | 1.31$\times$10-8 | 0.45 |
| | | SiO-red | [$-$50.3, $-$17.7] | 4.86 | 1.92$\times$1014 | 1.55$\times$10-9 | 4.79 |
| | | SO-red | [$-$50.3, $-$28.5] | 2.25 | 5.36$\times$1014 | 4.32$\times$10-9 | 5.58 |
| | | CH3OH-red | [$-$50.3, $-$41.2] | 0.84 | 5.04$\times$1015 | 4.07$\times$10-8 | 3.75 |
| | | H2CO-red | [$-$50.3, $-$38.8] | 2.96 | 5.95$\times$1014 | 4.81$\times$10-9 | 11.26 |
| | | HC3N-red | [$-$50.3, $-$40.0] | 1.04 | 1.37$\times$1014 | 1.11$\times$10-9 | 3.99 |
| | | HNCO-red | [$-$50.3, $-$44.5] | 0.28 | 4.93$\times$1013 | 3.98$\times$10-10 | 12.26 |
Sgr C-A #2 | $-$51.6 | 5.2 | SiO-blue | [$-$81.2, $-$52.8] | 2.06 | 1.28$\times$1014 | 6.65$\times$10-9 | 0.47 |
| [CH3CN] | | SO-blue | [$-$72.6, $-$52.9] | 1.02 | 3.92$\times$1014 | 2.04$\times$10-8 | 0.53 |
| | | CH3OH-blue | [$-$61.6, $-$52.9] | 1.46 | 8.41$\times$1015 | 4.37$\times$10-7 | 0.60 |
| | | H2CO-blue | [$-$68.6, $-$52.9] | 1.93 | 1.09$\times$1015 | 5.67$\times$10-8 | 0.62 |
| | | HC3N-blue | [$-$60.4, $-$52.9] | 0.34 | 4.91$\times$1013 | 2.55$\times$10-9 | 0.57 |
| | | HNCO-blue | [$-$63.3, $-$52.9] | 0.25 | 3.37$\times$1014 | 1.75$\times$10-8 | 0.25 |
| | | SiO-red | [$-$50.3, 9.3] | 4.40 | 2.48$\times$1014 | 7.59$\times$10-9 | 0.89 |
| | | SO-red | [$-$50.3, $-$37.8] | 0.49 | 1.81$\times$1014 | 5.54$\times$10-9 | 0.95 |
| | | CH3OH-red | [$-$50.3, $-$45.3] | 0.28 | 2.59$\times$1015 | 7.93$\times$10-8 | 0.65 |
| | | H2CO-red | [$-$50.3, $-$33.3] | 1.64 | 2.15$\times$1014 | 6.58$\times$10-9 | 4.58 |
| | | HC3N-red | [$-$50.3, $-$42.7] | 0.33 | 8.34$\times$1013 | 2.55$\times$10-9 | 0.56 |
| | | HNCO-red | [$-$50.3, $-$45.8] | 0.34 | 1.39$\times$1014 | 4.25$\times$10-9 | 1.43 |
Sgr C-B #1* | $-$53.4 | 11.0 | SiO-blue | [$-$59.6, $-$54.7] | 0.12 | $\cdots$ | 1.68$\times$10-9 | 0.11 | $X$(SiO) of the red lobe
| [CH3OH] | | SO-blue | [$-$64.6, $-$54.7] | 0.14 | $\cdots$ | 5.44$\times$10-9 | 0.28 | $X$(SO) of the red lobe
| | | CH3OH-blue | [$-$60.2, $-$54.7] | 0.31 | $\cdots$ | 8.17$\times$10-8 | 0.69 | $X$(CH3OH) of the red lobe
| | | H2CO-blue | [$-$68.6, $-$54.7] | 0.31 | $\cdots$ | 8.78$\times$10-9 | 0.64 | $X$(H2CO) of the red lobe
| | | HC3N-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HNCO-blue | [$-$56.6, $-$54.7] | 0.02 | $\cdots$ | 2.15$\times$10-9 | 0.21 | $X$(HNCO) of the red lobe
| | | SiO-red | [$-$52.1, $-$39.3] | 1.70 | 7.96$\times$1013 | 1.68$\times$10-9 | 1.55 |
| | | SO-red | [$-$52.1, $-$45.8] | 0.50 | 2.58$\times$1014 | 5.44$\times$10-9 | 0.98 |
| | | CH3OH-red | [$-$52.1, $-$46.6] | 0.80 | 3.87$\times$1015 | 8.17$\times$10-8 | 1.77 |
| | | H2CO-red | [$-$52.4, $-$40.1] | 1.13 | 4.16$\times$1014 | 8.78$\times$10-9 | 2.36 |
| | | HC3N-red | [$-$52.3, $-$45.4] | 0.18 | 4.07$\times$1013 | 8.59$\times$10-10 | 0.88 |
| | | HNCO-red | [$-$52.1, $-$48.5] | 0.09 | 1.02$\times$1014 | 2.15$\times$10-9 | 0.75 |
Sgr C-B #2 | $-$54.9 | 2.0 | SiO-blue | [$-$66.3, $-$56.2] | 1.17 | 4.64$\times$1013 | 3.37$\times$10-9 | 0.53 |
| [HC3N] | | SO-blue | [$-$63.3, $-$56.2] | 0.68 | 2.40$\times$1014 | 1.74$\times$10-8 | 0.42 |
| | | CH3OH-blue | [$-$64.3, $-$56.1] | 1.18 | 6.38$\times$1015 | 4.63$\times$10-7 | 0.46 |
| | | H2CO-blue | [$-$75.4, $-$56.2] | 2.05 | 7.78$\times$1014 | 5.65$\times$10-8 | 0.66 |
| | | HC3N-blue | [$-$63.1, $-$56.2] | 0.44 | 3.41$\times$1013 | 2.47$\times$10-9 | 0.77 |
| | | HNCO-blue | [$-$60.6, $-$56.2] | 0.24 | 8.27$\times$1013 | 6.00$\times$10-9 | 0.71 |
| | | SiO-red | [$-$53.6, $-$31.2] | 0.72 | 8.91$\times$1013 | 3.84$\times$10-9 | 0.29 |
| | | SO-red | [$-$53.9, $-$37.8] | 0.40 | 4.44$\times$1014 | 1.91$\times$10-8 | 0.22 |
| | | CH3OH-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | H2CO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | HC3N-red | [$-$53.6, $-$44.0] | 0.15 | 5.74$\times$1013 | 2.47$\times$10-9 | 0.26 |
| | | HNCO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
Sgr C-B #3 | $-$50.2 | 27.9 | SiO-blue | [$-$140.6, $-$51.4] | 4.16 | 4.17$\times$1014 | 1.62$\times$10-9 | 3.92 |
| [CH3CN] | | SO-blue | [$-$126.0, $-$51.2] | 2.88 | 2.34$\times$1015 | 9.12$\times$10-9 | 3.38 |
| | | CH3OH-blue | [$-$64.3, $-$51.5] | 1.24 | 8.23$\times$1015 | 3.21$\times$10-8 | 7.00 |
| | | H2CO-blue | [$-$114.6, $-$51.5] | 2.88 | 3.66$\times$1015 | 1.43$\times$10-8 | 3.69 |
| | | HC3N-blue | [$-$75.3, $-$51.5] | 0.56 | 2.22$\times$1014 | 8.65$\times$10-10 | 2.78 |
| | | HNCO-blue | [$-$68.6, $-$51.2] | 0.93 | 3.99$\times$1014 | 1.55$\times$10-9 | 10.63 |
| | | SiO-red | [$-$48.9, 45.8] | 3.53 | 3.27$\times$1014 | 1.12$\times$10-9 | 4.82 |
| | | SO-red | [$-$48.9, 15.6] | 1.88 | 1.12$\times$1015 | 3.86$\times$10-9 | 5.22 |
| | | CH3OH-red | [$-$48.9, $-$23.6] | 0.84 | 1.51$\times$1016 | 5.18$\times$10-8 | 2.93 |
| | | H2CO-red | [$-$48.9, 4.5] | 3.02 | 3.51$\times$1015 | 1.20$\times$10-8 | 4.61 |
| | | HC3N-red | [$-$48.9, $-$14.3] | 0.72 | 2.52$\times$1014 | 8.65$\times$10-10 | 3.57 |
| | | HNCO-red | [$-$48.9, $-$45.8] | 0.10 | 1.32$\times$1014 | 4.53$\times$10-10 | 3.86 |
Sgr C-B #4 | $-$50.8 | 5.8 | SiO-blue | [$-$75.8, $-$52.1] | 0.58 | 1.47$\times$1014 | 2.30$\times$10-9 | 0.39 |
| [HC3N] | | SO-blue | [$-$67.3, $-$52.1] | 0.18 | 2.90$\times$1014 | 4.54$\times$10-9 | 0.43 |
| | | CH3OH-blue | [$-$60.2, $-$52.0] | 0.41 | 3.82$\times$1015 | 5.98$\times$10-8 | 1.23 |
| | | H2CO-blue | [$-$68.6, $-$52.1] | 0.31 | 7.24$\times$1014 | 1.13$\times$10-8 | 0.50 |
| | | HC3N-blue | [$-$59.0, $-$52.1] | 0.04 | 4.17$\times$1013 | 6.52$\times$10-10 | 0.27 |
| | | HNCO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | SiO-red | [$-$49.5, $-$32.5] | 0.64 | 7.55$\times$1013 | 7.27$\times$10-10 | 1.35 |
| | | SO-red | [$-$49.5, $-$37.8] | 0.20 | 2.50$\times$1014 | 2.40$\times$10-9 | 0.91 |
| | | CH3OH-red | [$-$49.5, $-$41.2] | 0.52 | 3.85$\times$1015 | 3.71$\times$10-8 | 2.53 |
| | | H2CO-red | [$-$49.7, $-$38.8] | 0.80 | 6.85$\times$1014 | 6.60$\times$10-9 | 2.22 |
| | | HC3N-red | [$-$49.5, $-$41.3] | 0.22 | 6.77$\times$1013 | 6.52$\times$10-10 | 1.46 |
| | | HNCO-red | [$-$49.5, $-$47.2] | 0.06 | 1.15$\times$1014 | 1.11$\times$10-9 | 0.93 |
Sgr C-B #5 | $-$50.7 | 1.7 | SiO-blue | [$-$97.4, $-$52.0] | 0.98 | 1.72$\times$1014 | 7.01$\times$10-9 | 0.21 |
| [CH3OH] | | SO-blue | [$-$71.3, $-$52.0] | 0.16 | 2.56$\times$1014 | 1.04$\times$10-8 | 0.16 |
| | | CH3OH-blue | [$-$58.9, $-$52.0] | 0.40 | 4.40$\times$1015 | 1.79$\times$10-7 | 0.40 |
| | | H2CO-blue | [$-$68.6, $-$52.0] | 0.50 | 7.14$\times$1014 | 2.91$\times$10-8 | 0.32 |
| | | HC3N-blue | [$-$61.7, $-$52.0] | 0.14 | 6.31$\times$1013 | 2.57$\times$10-9 | 0.23 |
| | | HNCO-blue | [$-$56.6, $-$52.0] | 0.17 | 1.12$\times$1014 | 4.56$\times$10-9 | 0.68 |
| | | SiO-red | [$-$49.4, $-$35.2] | 0.20 | 5.53$\times$1013 | 4.05$\times$10-9 | 0.08 |
| | | SO-red | [$-$49.4, $-$41.8] | 0.05 | 1.24$\times$1014 | 9.12$\times$10-9 | 0.06 |
| | | CH3OH-red | [$-$49.4, $-$42.5] | 0.20 | 3.62$\times$1015 | 2.65$\times$10-7 | 0.14 |
| | | H2CO-red | [$-$49.7, $-$41.5] | 0.30 | 5.62$\times$1014 | 4.12$\times$10-8 | 0.13 |
| | | HC3N-red | [$-$49.5, $-$44.0] | 0.06 | 3.51$\times$1013 | 2.57$\times$10-9 | 0.09 |
| | | HNCO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
Sgr C-C #1 | $-$52.9 | 31.4 | SiO-blue | [$-$67.7, $-$54.1] | 6.92 | 1.35$\times$1014 | 5.26$\times$10-9 | 2.01 |
| [CH3CN] | | SO-blue | [$-$65.9, $-$54.2] | 5.44 | 7.42$\times$1014 | 2.88$\times$10-8 | 2.02 |
| | | CH3OH-blue | [$-$65.6, $-$54.2] | 8.34 | 1.05$\times$1016 | 4.09$\times$10-7 | 3.69 |
| | | H2CO-blue | [$-$67.3, $-$54.2] | 10.36 | 1.47$\times$1015 | 5.72$\times$10-8 | 3.32 |
| | | HC3N-blue | [$-$65.8, $-$54.2] | 1.75 | 9.24$\times$1013 | 3.60$\times$10-9 | 2.07 |
| | | HNCO-blue | [$-$65.9, $-$54.2] | 1.92 | 5.81$\times$1014 | 2.26$\times$10-8 | 1.51 |
| | | SiO-red | [$-$51.6, $-$6.9] | 8.44 | 1.19$\times$1014 | 5.73$\times$10-9 | 2.25 |
| | | SO-red | [$-$51.6, $-$13.8] | 7.00 | 9.62$\times$1014 | 4.64$\times$10-8 | 1.62 |
| | | CH3OH-red | [$-$51.6, $-$35.8] | 4.37 | 7.39$\times$1015 | 3.56$\times$10-7 | 2.22 |
| | | H2CO-red | [$-$51.6, $-$21.2] | 10.13 | 1.56$\times$1015 | 7.51$\times$10-8 | 2.47 |
| | | HC3N-red | [$-$51.6, $-$42.7] | 0.58 | 7.47$\times$1013 | 3.60$\times$10-9 | 0.69 |
| | | HNCO-red | [$-$51.6, $-$40.5] | 1.28 | 3.42$\times$1014 | 1.65$\times$10-8 | 1.38 |
Sgr C-C #2 | $-$48.1 | 9.9 | SiO-blue | [$-$86.6, $-$49.4] | 5.04 | 1.69$\times$1014 | 3.43$\times$10-9 | 2.24 |
| [HC3N] | | SO-blue | [$-$69.9, $-$49.4] | 2.79 | 5.16$\times$1014 | 1.05$\times$10-8 | 2.85 |
| | | CH3OH-blue | [$-$65.6, $-$49.3] | 3.02 | 1.14$\times$1016 | 2.31$\times$10-7 | 2.36 |
| | | H2CO-blue | [$-$67.3, $-$49.4] | 3.91 | 1.19$\times$1015 | 2.42$\times$10-8 | 2.96 |
| | | HC3N-blue | [$-$60.4, $-$49.4] | 1.13 | 1.19$\times$1014 | 2.42$\times$10-9 | 2.00 |
| | | HNCO-blue | [$-$59.3, $-$49.4] | 1.05 | 2.83$\times$1014 | 5.74$\times$10-9 | 3.25 |
| | | SiO-red | [$-$46.8, $-$42.0] | 0.10 | 2.77$\times$1013 | 4.46$\times$10-9 | 0.04 |
| | | SO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | CH3OH-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | H2CO-red | [$-$47.0, $-$45.5] | 0.06 | 8.27$\times$1013 | 1.33$\times$10-8 | 0.09 |
| | | HC3N-red | [$-$46.8, $-$42.7] | 0.07 | 1.50$\times$1013 | 2.42$\times$10-9 | 0.13 |
| | | HNCO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
Sgr C-C #3 | $-$50.7 | 58.7 | SiO-blue | [$-$143.3, $-$52.0] | 2.05 | 3.33$\times$1014 | 5.53$\times$10-10 | 5.67 |
| [CH3OH] | | SO-blue | [$-$65.9, $-$52.0] | 0.61 | 3.32$\times$1014 | 5.52$\times$10-10 | 11.74 |
| | | CH3OH-blue | [$-$61.6, $-$52.0] | 1.03 | 8.08$\times$1015 | 1.34$\times$10-8 | 13.94 |
| | | H2CO-blue | [$-$82.2, $-$52.0] | 1.38 | 1.21$\times$1015 | 2.01$\times$10-9 | 12.55 |
| | | HC3N-blue | [$-$73.9, $-$52.0] | 0.49 | 1.55$\times$1014 | 2.57$\times$10-10 | 8.12 |
| | | HNCO-blue | [$-$59.3, $-$52.0] | 0.30 | 3.65$\times$1014 | 6.06$\times$10-10 | 8.64 |
| | | SiO-red | [$-$49.4, $-$46.0] | 0.20 | 1.81$\times$1013 | 2.49$\times$10-10 | 1.19 |
| | | SO-red | [$-$49.4, $-$47.2] | 0.12 | 1.13$\times$1014 | 1.56$\times$10-9 | 0.82 |
| | | CH3OH-red | [$-$49.4, $-$43.9] | 0.79 | 4.93$\times$1015 | 6.78$\times$10-8 | 2.12 |
| | | H2CO-red | [$-$49.7, $-$45.5] | 1.85 | 7.84$\times$1014 | 1.08$\times$10-8 | 3.14 |
| | | HC3N-red | [$-$49.5, $-$46.7] | 0.09 | 1.87$\times$1013 | 2.57$\times$10-10 | 1.42 |
| | | HNCO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
Sgr C-D #1 | $-$49.5 | 6.3 | SiO-blue | [$-$87.9, $-$50.8] | 0.76 | 9.09$\times$1013 | 9.70$\times$10-10 | 1.19 |
| [HC3N] | | SO-blue | [$-$55.3, $-$50.8] | 0.05 | 2.26$\times$1013 | 2.42$\times$10-10 | 2.34 |
| | | CH3OH-blue | [$-$56.2, $-$50.7] | 0.44 | 3.76$\times$1015 | 4.01$\times$10-8 | 1.98 |
| | | H2CO-blue | [$-$64.6, $-$50.8] | 0.29 | 4.04$\times$1014 | 4.31$\times$10-9 | 1.24 |
| | | HC3N-blue | [$-$59.0, $-$50.8] | 0.16 | 5.62$\times$1013 | 6.00$\times$10-10 | 1.11 |
| | | HNCO-blue | [$-$55.3, $-$50.8] | 0.14 | 1.40$\times$1014 | 1.49$\times$10-9 | 1.64 |
| | | SiO-red | [$-$48.2, $-$27.1] | 0.46 | 8.08$\times$1013 | 2.34$\times$10-9 | 0.30 |
| | | SO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | CH3OH-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | H2CO-red | [$-$48.3, $-$30.6] | 0.26 | 6.76$\times$1014 | 1.96$\times$10-8 | 0.24 |
| | | HC3N-red | [$-$48.2, $-$46.7] | 0.02 | 2.07$\times$1013 | 6.00$\times$10-10 | 0.15 |
| | | HNCO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
Sgr C-D #2 | $-$53.4 | 9.1 | SiO-blue | [$-$73.1, $-$54.7] | 0.76 | 7.57$\times$1013 | 1.54$\times$10-9 | 0.76 |
| [CH3OH] | | SO-blue | [$-$60.6, $-$54.7] | 0.17 | 1.38$\times$1014 | 2.80$\times$10-9 | 0.64 |
| | | CH3OH-blue | [$-$64.3, $-$54.7] | 0.39 | 4.86$\times$1015 | 9.87$\times$10-8 | 0.72 |
| | | H2CO-blue | [$-$63.2, $-$54.7] | 0.45 | 3.33$\times$1014 | 6.76$\times$10-9 | 1.22 |
| | | HC3N-blue | [$-$61.7, $-$54.7] | 0.20 | 5.59$\times$1013 | 1.13$\times$10-9 | 0.75 |
| | | HNCO-blue | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
| | | SiO-red | [$-$52.1, $-$5.5] | 1.80 | 2.19$\times$1014 | 1.88$\times$10-9 | 1.46 |
| | | SO-red | [$-$52.1, $-$27.1] | 0.68 | 5.52$\times$1014 | 4.74$\times$10-9 | 1.52 |
| | | CH3OH-red | [$-$52.1, $-$45.3] | 1.05 | 4.92$\times$1015 | 4.23$\times$10-8 | 4.48 |
| | | H2CO-red | [$-$52.4, $-$40.1] | 1.25 | 4.64$\times$1014 | 3.99$\times$10-9 | 5.72 |
| | | HC3N-red | [$-$52.3, $-$38.6] | 0.44 | 1.32$\times$1014 | 1.13$\times$10-9 | 1.65 |
| | | HNCO-red | [$-$52.1, $-$47.2] | 0.07 | 1.44$\times$1014 | 1.24$\times$10-9 | 0.97 |
Sgr C-D #3 | $-$51.6 | 57.7 | SiO-blue | [$-$93.3, $-$52.8] | 3.28 | 1.54$\times$1014 | 1.32$\times$10-9 | 3.80 |
| [CH3CN] | | SO-blue | [$-$79.3, $-$52.9] | 1.56 | 4.08$\times$1014 | 3.50$\times$10-9 | 4.77 |
| | | CH3OH-blue | [$-$64.3, $-$52.9] | 1.48 | 1.83$\times$1015 | 1.57$\times$10-8 | 16.99 |
| | | H2CO-blue | [$-$72.7, $-$52.9] | 1.08 | 5.49$\times$1014 | 4.70$\times$10-9 | 4.22 |
| | | HC3N-blue | [$-$72.5, $-$52.9] | 0.99 | 1.17$\times$1014 | 1.00$\times$10-9 | 4.22 |
| | | HNCO-blue | [$-$56.6, $-$52.9] | 0.26 | 6.39$\times$1013 | 5.47$\times$10-10 | 8.49 |
| | | SiO-red | [$-$50.3, $-$19.0] | 1.88 | 1.01$\times$1014 | 1.00$\times$10-9 | 2.87 |
| | | SO-red | [$-$50.3, $-$35.1] | 0.53 | 2.64$\times$1014 | 2.62$\times$10-9 | 2.17 |
| | | CH3OH-red | [$-$50.3, $-$43.9] | 0.20 | 4.31$\times$1015 | 4.27$\times$10-8 | 0.83 |
| | | H2CO-red | [$-$50.3, $-$32.0] | 1.75 | 8.47$\times$1014 | 8.40$\times$10-9 | 3.82 |
| | | HC3N-red | [$-$50.3, $-$40.0] | 0.60 | 1.01$\times$1014 | 1.00$\times$10-9 | 2.54 |
| | | HNCO-red | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |
Sgr C-D #4 | $-$51.6 | 33.2 | SiO-blue | [$-$59.6, $-$52.8] | 1.56 | 2.15$\times$1013 | 4.51$\times$10-10 | 5.28 |
| [CH3CN] | | SO-blue | [$-$82.0, $-$52.9] | 1.77 | 4.96$\times$1014 | 1.04$\times$10-8 | 1.82 |
| | | CH3OH-blue | [$-$68.3, $-$52.9] | 2.14 | 6.42$\times$1015 | 1.35$\times$10-7 | 2.87 |
| | | H2CO-blue | [$-$72.7, $-$52.9] | 2.60 | 9.16$\times$1014 | 1.92$\times$10-8 | 2.48 |
| | | HC3N-blue | [$-$60.4, $-$52.9] | 0.34 | 5.13$\times$1013 | 1.08$\times$10-9 | 1.34 |
| | | HNCO-blue | [$-$56.6, $-$52.9] | 0.06 | 5.02$\times$1013 | 1.05$\times$10-9 | 0.96 |
| | | SiO-red | [$-$50.3, $-$36.6] | 2.32 | 4.18$\times$1013 | 5.60$\times$10-10 | 6.31 |
| | | SO-red | [$-$50.3, $-$32.5] | 1.51 | 3.42$\times$1014 | 4.58$\times$10-9 | 3.54 |
| | | CH3OH-red | [$-$50.3, $-$42.5] | 1.30 | 6.18$\times$1015 | 8.27$\times$10-8 | 2.84 |
| | | H2CO-red | [$-$50.3, $-$34.7] | 4.61 | 6.46$\times$1014 | 8.65$\times$10-9 | 9.77 |
| | | HC3N-red | [$-$50.3, $-$40.0] | 0.78 | 8.04$\times$1013 | 1.08$\times$10-9 | 3.07 |
| | | HNCO-red | [$-$50.3, $-$44.5] | 0.31 | 1.31$\times$1014 | 1.75$\times$10-9 | 3.18 |
Sgr C-E #1 | $-$63.6 | 8.5 | SiO-blue | [$-$79.8, $-$64.9] | 0.29 | 6.93$\times$1013 | 2.37$\times$10-9 | 0.19 |
| [CH3CN] | | SO-blue | [$-$67.3, $-$64.9] | 0.06 | 1.08$\times$1014 | 3.70$\times$10-9 | 0.18 |
| | | CH3OH-blue | [$-$67.0, $-$64.9] | 0.06 | 3.20$\times$1015 | 1.10$\times$10-7 | 0.10 |
| | | H2CO-blue | [$-$70.0, $-$64.9] | 0.15 | 3.99$\times$1014 | 1.37$\times$10-8 | 0.20 |
| | | HC3N-blue | [$-$74.9, $-$64.9] | 0.07 | 4.36$\times$1013 | 1.49$\times$10-9 | 0.21 |
| | | HNCO-blue | [$-$67.3, $-$64.9] | 0.03 | 1.65$\times$1014 | 5.65$\times$10-9 | 0.08 |
| | | SiO-red | [$-$62.3, $-$42.0] | 0.68 | 5.88$\times$1013 | 1.82$\times$10-9 | 0.57 |
| | | SO-red | [$-$62.3, $-$45.8] | 0.30 | 3.84$\times$1014 | 1.19$\times$10-8 | 0.27 |
| | | CH3OH-red | [$-$62.3, $-$42.5] | 0.22 | 8.97$\times$1015 | 2.77$\times$10-7 | 0.15 |
| | | H2CO-red | [$-$62.3, $-$38.8] | 0.18 | 4.84$\times$1014 | 1.50$\times$10-8 | 0.22 |
| | | HC3N-red | [$-$62.3, $-$58.9] | 0.12 | 4.83$\times$1013 | 1.49$\times$10-9 | 0.35 |
| | | HNCO-red | [$-$62.3, $-$43.2] | 0.13 | 2.06$\times$1014 | 6.37$\times$10-9 | 0.35 |
Sgr C-F #1 | $-$48.9 | 154.8 | SiO-blue | [$-$81.2, $-$50.1] | 6.88 | 6.55$\times$1013 | 8.17$\times$10-10 | 12.87 |
| [CH3CN] | | SO-blue | [$-$84.6, $-$50.2] | 10.91 | 7.88$\times$1014 | 9.82$\times$10-9 | 11.89 |
| | | CH3OH-blue | [$-$61.6, $-$50.2] | 13.13 | 6.84$\times$1015 | 8.53$\times$10-8 | 27.85 |
| | | H2CO-blue | [$-$72.7, $-$50.2] | 3.72 | 1.07$\times$1015 | 1.33$\times$10-8 | 5.12 |
| | | HC3N-blue | [$-$63.1, $-$50.2] | 6.19 | 1.14$\times$1014 | 1.42$\times$10-9 | 18.58 |
| | | HNCO-blue | [$-$69.9, $-$50.2] | 5.29 | 5.65$\times$1014 | 7.05$\times$10-9 | 13.33 |
| | | SiO-red | [$-$47.6, $-$28.5] | 3.85 | 6.59$\times$1013 | 4.73$\times$10-10 | 12.43 |
| | | SO-red | [$-$47.6, $-$32.5] | 6.55 | 5.04$\times$1014 | 3.62$\times$10-9 | 19.36 |
| | | CH3OH-red | [$-$47.6, $-$39.8] | 2.33 | 1.47$\times$1015 | 1.06$\times$10-8 | 39.76 |
| | | H2CO-red | [$-$47.6, $-$33.3] | 7.57 | 9.81$\times$1014 | 7.04$\times$10-9 | 19.71 |
| | | HC3N-red | [$-$47.6, $-$31.9] | 2.00 | 1.98$\times$1014 | 1.42$\times$10-9 | 6.01 |
| | | HNCO-red | [$-$47.6, $-$33.8] | 3.22 | 2.35$\times$1014 | 1.69$\times$10-9 | 33.89 |
Sgr C-F #2 | $-$50.2 | 303.7 | SiO-blue | [$-$83.9, $-$51.4] | 23.10 | 8.27$\times$1013 | 1.74$\times$10-9 | 20.27 |
| [CH3CN] | | SO-blue | [$-$78.0, $-$51.2] | 35.94 | 1.16$\times$1015 | 2.44$\times$10-8 | 15.76 |
| | | CH3OH-blue | [$-$58.9, $-$51.5] | 39.10 | 1.36$\times$1016 | 2.86$\times$10-7 | 24.73 |
| | | H2CO-blue | [$-$70.0, $-$51.5] | 28.04 | 8.35$\times$1014 | 1.76$\times$10-8 | 29.18 |
| | | HC3N-blue | [$-$63.1, $-$51.5] | 13.80 | 9.71$\times$1013 | 2.04$\times$10-9 | 28.83 |
| | | HNCO-blue | [$-$69.9, $-$51.2] | 18.06 | 6.99$\times$1014 | 1.47$\times$10-8 | 21.83 |
| | | SiO-red | [$-$48.9, $-$20.4] | 8.12 | 2.36$\times$1014 | 2.16$\times$10-9 | 5.74 |
| | | SO-red | [$-$48.9, $-$28.5] | 10.41 | 2.26$\times$1015 | 2.06$\times$10-8 | 5.41 |
| | | CH3OH-red | [$-$48.9, $-$39.8] | 2.04 | 8.28$\times$1015 | 7.58$\times$10-8 | 4.87 |
| | | H2CO-red | [$-$48.9, $-$30.6] | 6.23 | 2.10$\times$1015 | 1.92$\times$10-8 | 5.94 |
| | | HC3N-red | [$-$48.9, $-$31.9] | 4.22 | 2.23$\times$1014 | 2.04$\times$10-9 | 8.82 |
| | | HNCO-red | [$-$48.9, $-$33.8] | 2.60 | 6.43$\times$1014 | 5.88$\times$10-9 | 7.85 |
Sgr C-G #1 | $-$59.6 | 37.2 | SiO-blue | [$-$73.1, $-$60.9] | 0.48 | 7.04$\times$1013 | 5.26$\times$10-10 | 1.38 |
| [CH3CN] | | SO-blue | [$-$74.0, $-$60.9] | 0.35 | 3.72$\times$1014 | 2.78$\times$10-9 | 1.36 |
| | | CH3OH-blue | [$-$67.0, $-$60.9] | 0.63 | 6.00$\times$1015 | 4.48$\times$10-8 | 2.56 |
| | | H2CO-blue | [$-$70.0, $-$60.9] | 1.18 | 1.32$\times$1015 | 9.86$\times$10-9 | 2.19 |
| | | HC3N-blue | [$-$68.5, $-$60.9] | 0.17 | 7.11$\times$1013 | 5.31$\times$10-10 | 1.33 |
| | | HNCO-blue | [$-$64.6, $-$60.9] | 0.22 | 1.69$\times$1014 | 1.26$\times$10-9 | 3.08 |
| | | SiO-red | [$-$58.3, $-$40.6] | 0.46 | 1.08$\times$1014 | 7.67$\times$10-10 | 0.91 |
| | | SO-red | [$-$58.3, $-$37.8] | 0.48 | 6.64$\times$1014 | 4.72$\times$10-9 | 1.10 |
| | | CH3OH-red | [$-$58.3, $-$45.3] | 0.14 | 4.88$\times$1015 | 3.47$\times$10-8 | 0.76 |
| | | H2CO-red | [$-$58.3, $-$42.8] | 0.35 | 1.00$\times$1015 | 7.10$\times$10-9 | 0.90 |
| | | HC3N-red | [$-$58.3, $-$50.8] | 0.13 | 7.48$\times$1013 | 5.31$\times$10-10 | 1.01 |
| | | HNCO-red | [$-$58.3, $-$51.2] | 0.12 | 2.52$\times$1014 | 1.79$\times$10-9 | 1.19 |
Note. — Column (1): outflow ID. Entries marked with asterisks are candidates,
and those without asterisks are highly likely outflows. Column (2):
$V_{\text{lsr}}$ of the core, and the line used to determine the
$V_{\text{lsr}}$. Column (3): mass of the core (Paper I). Column (4): outflow
lobe identifier. Column (5): velocity range of the outflow lobe. Column (6):
integrated intensity of molecular emission. Column (7): column density at the
reference positions, which are marked by black crosses in Figures 6–22. Column
(8): adopted molecular abundance with respect to H${}_{\text{2}}$. Column (9):
outflow mass. Column (10): notes on the selection of molecular abundances.
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|
# Modeling the interplay between seasonal flu outcomes and individual
vaccination decisions
Irena Papst Center for Applied Mathematics, Cornell University, Ithaca, NY
14853 Kevin P. O’Keeffe Senseable City Laboratory, Massachusetts Institute
of Technology, Cambridge, MA 02139 Steven H. Strogatz Department of
Mathematics, Cornell University, Ithaca, NY 14853
###### Abstract
Seasonal influenza presents an ongoing challenge to public health. The rapid
evolution of the flu virus necessitates annual vaccination campaigns, but the
decision to get vaccinated or not in a given year is largely voluntary, at
least in the United States, and many people decide against it. In early
attempts to model these yearly flu vaccine decisions, it was often assumed
that individuals behave rationally, and do so with perfect
information—assumptions that allowed the techniques of classical economics and
game theory to be applied. However, the usual assumptions are contradicted by
the emerging empirical evidence about human decision-making behavior in this
context. We develop a simple model of coupled disease spread and vaccination
dynamics that instead incorporates experimental observations from social
psychology to model annual vaccine decision-making more realistically. We
investigate population-level effects of these new decision-making assumptions,
with the goal of understanding whether the population can self-organize into a
state of herd immunity, and if so, under what conditions. Our model agrees
with established results while also revealing more subtle population-level
behavior, including biennial oscillations about the herd immunity threshold.
*Corresponding author<EMAIL_ADDRESS>
Keywords: seasonal influenza, vaccination, decision-making, social
psychology, SIR model
## 1 Introduction
Annual influenza epidemics are a significant public health challenge, with up
to 650,000 individuals dying from respiratory diseases associated with the flu
each year (World Health Organization 2017). In the United States alone, the
total economic burden of seasonal influenza, including direct medical costs
and lost earnings due to illness or death, has been estimated as $26.8 billion
annually (Molinari et al. 2007).
One of the main challenges of controlling seasonal influenza spread is that
the viruses evolve quickly (on the same time scale as the annual epidemics),
with multiple strains circulating concurrently. A key adaptation mechanism,
antigenic drift, gives rise to new influenza strains by randomly changing
segments of viral surface proteins. Given that a host’s immune system uses
these surface proteins to identify the virus so that it may be neutralized
(Taubenberger and Kash 2010), antigenic drift thus acts as an evolutionary
countermeasure. It helps the flu evade the immune system and thereby promotes
its spread through the host population.
The rapid evolution of the flu results in the constant threat of a pandemic,
and it also makes it challenging to develop effective, long-lasting vaccines.
The seasonal flu vaccine is updated every year to protect against the strains
that seem to pose the largest upcoming threat. Seasonal influenza vaccination
is largely voluntary in the United States, so individuals must decide whether
or not to vaccinate each year.
For many people, this decision is not easy. It involves many quantities that
are effectively impossible for an individual to estimate accurately, such as
their likelihood of being vaccinated successfully, the probability of an
adverse reaction to the vaccine, as well as their increased risk of catching
the flu by foregoing the vaccine. Yet although the decision may be difficult
at an individual level, at a societal level the benefits of vaccinating can be
immense; if a critical mass of individuals choose to immunize themselves,
“herd immunity” can be achieved. In this desirable state, the density of
susceptible individuals is so low that an infection chain cannot be sustained
and so an epidemic cannot occur (Fine 1993).
Somewhat paradoxically, the possibility of achieving herd immunity makes an
individual’s decision to vaccinate or not even more complex, at least when
viewed through the lens of classical game theory (wherein agents are assumed
to be purely self-interested, and to behave rationally with perfect
information about the situation at hand). The issue is that as vaccination
coverage increases, individuals are increasingly incentivized _not_ to
vaccinate. Each person would do better relying on others to bear any burden
associated with vaccination, while everyone reaps the benefits of widespread
immunity. Such “free-riding” logic makes it impossible to ever actually
achieve herd immunity with rational hosts.
The free-riding problem is common in models where agents rationally weigh a
delayed collective group benefit against immediate individual costs.
Individuals are assumed to make decisions by selecting the strategy that
maximizes an objective, individual payoff function, as prescribed by classical
economics (Hardin 2013). Some early models of voluntary vaccination decisions
involve such assumptions and inevitably yield agents that utilize free-riding
logic, which precludes herd immunity (Geoffard and Philipson 1997; Bauch and
Earn 2004).
However, a more recent empirical study suggests that free-riding logic is
uncommon when individuals specifically consider whether or not to get the
seasonal influenza vaccine (Parker et al. 2013). In fact, this study finds
that the majority of individuals surveyed do not account for the vaccination
decisions of others when making their own decision. Nevertheless, despite
increasing evidence that assumptions from classical economics may not
appropriately capture human behavior in the context of infectious disease
spread, some behavior-disease models continue to be built on such foundations
(Verelst et al. 2016).
Other studies have challenged these assumptions by replacing them with those
from behavioral economics, which leverages social psychology in its models of
human decision-making. Voinson et al. (2015) develop a behavior-disease model
that incorporates cognitive biases and differing vaccine opinions among
individuals to study vaccination coverage over time. Oraby and Bauch (2015)
study pediatric acceptance of vaccines by incorporating prospect theory into
their disease model.
In this paper, we consider a simplified model for the interplay between annual
vaccination decisions and seasonal influenza spread, in which individual
voluntary vaccination decisions are informed by observed social psychology in
this context. Unlike previous work, we model _repeated_ vaccination decisions
to reflect the annual vaccination decision necessitated by the rapid evolution
of influenza viruses. We investigate population-level effects of these new
decision-making assumptions, with the goal of understanding whether the
population can self-organize into a state of herd immunity, and if so, under
what conditions. Despite our model’s idealized nature, we find that its
results align with those utilizing assumptions based on classical economics,
although our model also predicts more nuanced population-level behaviors, such
as oscillations in and out of herd immunity on a biennial basis.
## 2 Conceptual Model
Decision theory and social psychology suggest that, in general, individuals
tend to use heuristics, or rules of thumb, rather than a “rational” cost-
benefit analysis in complex decision-making (Tversky and Kahneman 1974).
Moreover, decision-making tends to obey the law of inertia: choices generally
remain unchanged but are sensitive to both small nudges and unfavorable
resulting outcomes (Thaler and Sunstein 2009). Our model is based on both of
these ideas.
For simplicity, assume that each year, an individual chooses whether or not to
receive the seasonal influenza vaccine based solely on evaluating their most
recent outcome with both the vaccine and the disease. The vaccine carries a
cost, or risk, of adverse reaction, which can be interpreted as a cost to an
individual’s health due to vaccine side effects (morbidity), a direct economic
cost from paying for the vaccine, and/or an indirect economic cost such as
taking unpaid leave from work to get vaccinated. In what follows, we will
interpret vaccine cost as morbidity, but the modeling framework is flexible
enough to accommodate other interpretations. There is also some probability
that vaccination successfully confers immunity upon the recipient.
Before we present the model in detail in Section 3, let us first describe it
intuitively. In any given year, individuals make the decision of whether or
not to vaccinate, then follow through with their choice, and then the flu
season occurs. The epidemic resolves itself, each individual assesses their
personal outcome from the past year, and then decides whether or not to get
vaccinated prior to next year’s flu season. The decision rule is a simple
heuristic: if a person “won” last year (did not get sick and did not have an
adverse reaction to the vaccine), they stick to their vaccination choice and
make the same decision the following year. If they “lost” (got sick or had a
bad reaction to the vaccine), they are nudged to change their behavior (switch
from vaccinating to not, or vice versa) in hopes of eliciting a better during
the next flu season.
[capbesideposition=right,center,capbesidewidth=0.45]figure[]
Figure 1: All possible decisions and outcomes for an individual in year
$\bm{n}$, leading to different decisions for year $\bm{n}+\bm{1}$. Boxes with
solid borders denote decisions and possible intermediate repercussions in year
$n$. Boxes with dashed and dotted borders denote final repercussions that
determine the decision in year $n+1$: vaccinate (box with dashed border) or do
not vaccinate (box with dotted border). In the model, $p_{n}$ is the
proportion of vaccinators in year $n$, $r$ is the probability that the vaccine
induces a cost, $s$ is the probability that the vaccine succeeds, $\phi$ is
the final size of the epidemic (normalized as a fraction of the whole
population), and $f(\phi)=\phi(sp_{n})/(1-sp_{n})$ is the fraction of the
susceptible population that was infected in the epidemic occurring in year
$n$. A “bad reaction” results when one incurs a cost from the vaccine (_e.g.,_
vaccine side effects on health, or economic burden). See Section 3 for model
details.
Figure 1 shows a schematic of the model. As an example of how to flow through
the chart, let us first consider an individual who has decided to vaccinate in
a given year (a decision made by a proportion $p_{n}$ of the population in
year $n$). This decision corresponds to the top left fork of the tree in
Figure 1. Next, suppose the vaccine succeeds in conferring immunity (which
occurs with probability $s$), and the vaccine does not elicit an adverse
reaction (which occurs with probability $1-r$). This favorable outcome does
not push the individual away from their default (winning) strategy of
vaccinating, so they decide to vaccinate again the following year; the vaccine
seems to have succeeded in protecting them from the flu. (We are assuming here
that only the failure of the vaccine can be observed by the individual, and
only if they happen to get ill that year. Otherwise, the success of the
vaccine is presumed, since there is no evidence to the contrary.)
On the other hand, an individual may choose _not_ to vaccinate in a given year
(a decision made by a proportion $1-p_{n}$ of the population, shown by
following the top right fork in Figure 1). If such an individual then happens
to contract the flu (which occurs with a probability that we will calculate
below), then since this non-vaccinating individual’s choice was a “losing”
strategy, they decide to vaccinate the following year. The various other paths
through the tree can be understood similarly.
In the next section, we write down the governing equations for our model. We
begin by recalling some standard results for a simple epidemiological model
and then couple that model to a social psychological model for individual
vaccination decisions.
## 3 Mathematical Model
### 3.1 Epidemic model
We choose the susceptible-infected-removed (SIR) model of infectious disease
dynamics (Kermack and McKendrick 1927), both because it is a well-established
epidemiological model and because there exists an analytical expression for
the final size of epidemics predicted by the model. The SIR model is adequate
for modeling each annual influenza outbreak individually, though we note that
a more realistic flu model could be substituted into our vaccination coverage
model, provided that the final size could at least be calculated numerically.
Suppose $S(t)$ and $I(t)$ represent proportions of a population that are
susceptible and infected, respectively, at a time $t$. Infected individuals
are assumed to be immediately infectious; there is no latency period in this
model. There may also be individuals that are removed from the infection
process as they have already recovered from the illness (denoted by proportion
$R(t)$), but since we assume this disease propagates in a closed population,
we have $R(t)=1-S(t)-I(t)$, which means we do not need to track the removed
individuals explicitly.
The SIR model is defined by a set of two coupled, nonlinear, ordinary
differential equations:
$\displaystyle\frac{dS}{dt}$ $\displaystyle=-\beta SI,$ (3.1a)
$\displaystyle\frac{dI}{dt}$ $\displaystyle=\beta SI-\gamma I,$ (3.1b)
where $\beta$ is the disease transmission rate and $\gamma$ is the disease
recovery rate. The derived quantity $\mathcal{R}_{0}=\beta/\gamma$ is the
basic reproduction number of the disease; it gives the average number of
secondary cases generated by an infectious individual in a fully susceptible
population over the course of their illness.
In our model, we assume the initial conditions
$S(0)=1-sp,\quad 0<I(0)<<1,\quad R(0)=sp$ (3.2)
to incorporate a vaccine uptake level at proportion $p$ with success
probability $s$.
### 3.2 Final size of the epidemic
For this model, one can derive an implicit equation for the final size of the
epidemic $\phi(x)$, where $x$ is the proportion initially immune to the
disease (Ma and Earn 2006):
$\phi(x)=(1-x)(1-e^{-\mathcal{R}_{0}\phi(x)}).$ (3.3)
The solution to Equation 3.3 can be written in terms of the principal branch
of the product log function (_i.e.,_ the Lambert W-function), denoted by
$W[\cdot]$:
$\phi(x)=1-x+\frac{1}{\mathcal{R}_{0}}W\left[-\mathcal{R}_{0}(1-x)e^{-\mathcal{R}_{0}(1-x)}\right]$
(3.4)
This expression for the final size of the yearly influenza epidemic is used
later in Equation 3.7 to complete the model.
### 3.3 Critical vaccination threshold for herd immunity
An epidemic cannot be sustained if the average number of secondary cases
provoked by an infected individual in the population is below one (since this
infected individual cannot even replace themselves in the infection chain, let
alone generate further infections). In other words, for the population to
achieve herd immunity, the _effective_ reproduction number (the basic
reproduction number times the proportion currently susceptible),
$\mathcal{R}_{\rm eff}=\mathcal{R}_{0}(1-sp)$, must be driven below 1. Thus
the critical vaccination threshold, $p_{\rm crit}$, satisfies the equation
$\mathcal{R}_{0}(1-sp_{\rm crit})=1$, and so
$p_{\rm crit}=\frac{1}{s}\left(1-\frac{1}{\mathcal{R}_{0}}\right).$ (3.5)
### 3.4 Estimating $\mathcal{R}_{0}$ for seasonal influenza
Estimates of $\mathcal{R}_{0}$ for vary depending on year, location, and
influenza subtype since the basic reproduction number depends not only on the
immunological properties of the virus, but also on the social behavior of the
host population. A systematic review by Biggerstaff et al. (2014) catalogues
many estimates of both the basic and effective reproduction numbers of
pandemic, zoonotic, and seasonal influenza.
The most relevant estimates for our study are those for the 1976-1981 outbreak
of H1N1/H3N2/B in the USA. Two studies were performed to estimate the basic
reproduction number in this outbreak, and they both use serologically
confirmed infections for their data, which make these estimates particularly
reliable111It is difficult to distinguish seasonal influenza from other upper
respiratory tract infections by symptoms alone, so studies based purely on
reported symptoms may not yield a good estimate for the $\mathcal{R}_{0}$ of
seasonal influenza. Instead, studies based on serologically confirmed
infections are more reliable.. One study found $\mathcal{R}_{0}=1.70$
(Ferguson et al. 2006), while another found $\mathcal{R}_{0}=1.16$ (Britton
and Becker 2000) for this outbreak. We average these two values and take
$\mathcal{R}_{0}=1.4$ as a reasonable estimate of the basic reproduction
number for seasonal influenza in a modern US population.
### 3.5 Vaccination decision model
Let $p_{n}$ be the proportion of the population that vaccinates in year $n$.
Our goal in this section is to derive a discrete map for the vaccine coverage
$p_{n+1}$ in year $n+1$.
In the following argument, we assume the vaccine is fully quantified by its
cost and its success rate. For the purposes of this discussion, we will think
of the cost in terms of vaccine morbidity (side effects to immunization),
though the cost could be interpreted as an economic one (for instance, if
individuals have to pay for the vaccine or take unpaid time off of work to
obtain it). We denote the cost, or probability of vaccine morbidity, by $0\leq
r\leq 1$, and the probability of vaccine success by $0\leq s\leq 1$.
To ease the notation in the derivation below, it proves useful to introduce a
function $f(\phi)$ to denote the proportion of all _susceptible_ individuals
who get sick during an epidemic of size $\phi$. To calculate $f$ in terms of
$\phi$, note that the fraction of the total population that is susceptible in
year $n$ is $1-sp_{n}$. Of these individuals, a fraction $f\cdot(1-sp_{n})$
will get infected, by definition of $f$. But since this fraction also equals
the number of infected individuals divided by the total population, it simply
equals $\phi$, the fractional size of the epidemic, as given by Equation 3.4.
Therefore, $\phi=f\cdot(1-sp_{n})$, from which we conclude that
$f(\phi(sp_{n}))=\frac{\phi(sp_{n})}{1-sp_{n}}.$ (3.6)
In other words, the proportion of susceptible individuals who end up infected
is simply the final size of the epidemic renormalized to the susceptible
population.
With these preliminaries out of the way, we can deduce the vaccine coverage
rate $p_{n+1}$ in year $n+1$ by considering all possible outcomes for an
individual based on their choice of whether or not to vaccinate in year $n$,
and by counting the proportion of the population flowing down each of the
branches in Figure 1 into vaccinating in year $n+1$. We assume that every
individual is susceptible to that year’s flu strain at the start of each flu
season, so everyone must make the choice of whether or not to vaccinate each
year.
First consider the group of non-vaccinating individuals, which make up a
proportion $1-p_{n}$ of the population. These individuals will only vaccinate
in year $n+1$ if they get sick in year $n$, an event that occurs to a fraction
$f=f(\phi(sp_{n}))$ of them. Thus, the equation for $p_{n+1}$ will include a
term $f\cdot(1-p_{n})$, which accounts for those that did not vaccinate and
got sick.
For vaccinating individuals, either the vaccine succeeds, with probability
$s$, or it does not, with probability $1-s$. If the vaccine succeeds, there is
still an independent chance that the individual will have side effects that
discourage them from vaccinating the following year, which occurs at the
vaccine morbidity rate, $r$. However, those for whom the vaccine successfully
conferred immunity and provoked no side effects will once again vaccinate the
following year since they have no reason to change strategy, which adds the
term $(1-r)\cdot s\cdot p_{n}$ to the equation for $p_{n+1}$.
If the vaccine fails for an individual (with probability $1-s$), but did not
cause any discouraging side effects (with probability $1-r$), the only reason
they would continue to vaccinate would be if they thought the vaccine
succeeded; that is, they happened not to get sick, even though they were not
successfully immunized. A proportion $1-f$ of susceptible individuals avoid
infection, so the final term of the equation for $p_{n+1}$ is
$(1-f)\cdot(1-r)\cdot(1-s)\cdot p_{n}$.
Putting all of these contributions together, we find that the discrete map for
$p_{n+1}$, the proportion of the population vaccinating in year $n+1$, is
given by
$p_{n+1}=\frac{\phi(sp_{n})}{1-sp_{n}}(1-p_{n})+(1-r)sp_{n}+\left[1-\frac{\phi(sp_{n})}{1-sp_{n}}\right](1-r)(1-s)p_{n},$
(3.7)
where the function $\phi$ is given by Equation 3.4. This map is biologically
sensible; if $0\leq p_{n}\leq 1$, one can check that $0\leq p_{n+1}\leq 1$.
Hence, as long as the initial condition is sensible ($0\leq p_{0}\leq 1$), all
subsequent iterations remain in $[0,1]$.
## 4 Results
### 4.1 Model predictions
The predictions of the model depend on the relative magnitudes of its
parameters: the vaccine parameters (morbidity or cost, $r$, and success, $s$),
and the disease parameter, $\mathcal{R}_{0}$. The basic reproduction number
$\mathcal{R}_{0}$ gives a sense of the “infectiousness” of the disease; in our
analysis, we estimate the basic reproduction number of seasonal influenza in a
modern US population to be $\mathcal{R}_{0}=1.4$ (see Section 3 for details),
indicating that a person infected with seasonal influenza will infect on
average 1.4 other people in a fully susceptible population. For the vaccine
parameters, we note that seasonal flu vaccines have very low morbidity
(Centers for Disease Control 2016), and their success varies from low to
moderate (Osterholm et al. 2012; World Health Organization 2015; Centers for
Disease Control 2017).
The best case would be for the population to self-organize into herd immunity,
by driving the proportion of the population vaccinated above the critical
threshold, $p_{\rm crit}$. When vaccine coverage meets or exceeds this
threshold, no epidemic occurs. One might expect the model to self-organize
into herd immunity if the population can collectively make use of the memory
of the previous flu seasons in a lasting way.
To our disappointment, we find that if there is any cost to the vaccine
($r>0$), our model cannot self-organize into lasting herd immunity. There are
two main regions of parameter space in this case (Figure 2): a large region
where the system eventually converges to vaccine levels below the herd
immunity threshold (region I), and a smaller region where the system
oscillates in and out of the herd immunity region on a yearly basis (region
II).
Figure 2: Long-term model behavior for $\bm{\mathcal{R}_{0}=1.4}$ and
$\bm{p_{0}=0}$, as a function of vaccine morbidity ($\bm{0<r\leq 1}$) and
vaccine success ($\bm{0\leq s\leq 1}$). Behavior in these regions of parameter
space was deduced by iterating the vaccine coverage map (Equation 3.7)
numerically until it converged to a fixed point. The majority of parameter
space is dominated by convergence to vaccine levels below the herd immunity
threshold, which results in no lasting herd immunity (region I: the system
converges to a period 1 fixed point, $p^{*}$, that satisfies $p^{*}<p_{\rm
crit}$). For higher vaccine success, there is a possibility of achieving herd
immunity every other year, provided that vaccine morbidity $r$ is sufficiently
large (region II: the system converges to a period-2 fixed point,
$(p_{1}^{*},p_{2}^{*})$). In this regime, the system oscillates between sub-
optimal vaccine coverage ($p_{1}^{*}<p_{\rm crit}$) and herd immunity with
overvaccination ($p_{2}^{*}>p_{\rm crit}$): see Figure 3.
Note that where game-theoretic models always predict free-riding that make
herd immunity impossible to achieve in the long term, our model predicts that
it is possible for the population to achieve herd immunity every other year,
even if there is a cost and moderate failure rate to the vaccine. This
oscillatory behavior (Figure 3) is the result of the system converging to a
state where it alternates between the population bearing a significant disease
burden (a large epidemic in the previous year encouraging vaccination in the
following year) and the population bearing a significant vaccine cost (which
incentivizes non-vaccination _en masse_ in the following year). When a vaccine
has a moderate-to-high success rate, and a sufficiently high cost, the system
is constantly balancing an illness-vaccine cost tradeoff.
[capbesideposition=left,center,capbesidewidth=0.25]figure[]
Figure 3: Vaccine coverage level over time in the regime where herd immunity
eventually occurs every other year ($\bm{\mathcal{R}_{0}=1.4,r=0.55,s=0.9}$).
The system converges to a state where the vaccine coverage level oscillates
asymmetrically about the critical vaccination threshold, $p=p_{\rm crit}$,
denoted by the dashed line.
Even in the case where there is no _lasting_ herd immunity, the system may
nevertheless spend a significant length of time in the herd immunity interval
before eventually dropping out (Figure 4). This effect is especially
pronounced when vaccine morbidity is low, and even when the vaccine is only
moderately successful, both of which are properties of the real seasonal
influenza vaccines. The transient herd immunity period increases as vaccine
success increases and/or vaccine cost decreases.
[capbesideposition=right,center,capbesidewidth=0.25]figure[]
Figure 4: Number of years spent in the herd immunity interval ($\bm{p>p_{\rm
crit}}$) during the transient period in the regime of no lasting herd immunity
($\bm{\mathcal{R}_{0}=1.4}$, $\bm{p_{0}=0}$). As the vaccine improves in
quality (either vaccine morbidity decreases, or vaccine success increases),
the time period spent in the herd immunity interval lengthens.
When there is no cost to the vaccine ($r=0$; Figure 5), the system can self-
organize into herd immunity in three ways (regions whose label contains
“lasting herd immunity”), in addition to yielding no lasting herd immunity as
before (region I). The system may start in the herd immunity region and
therefore stay in it indefinitely (region II), since there is no vaccine cost
to drive the coverage level down. Alternatively, the system may converge to
lasting herd immunity which is either inefficient as it involves
overvaccination (region III), or it may converge to optimal, lasting herd
immunity precisely at the herd immunity threshold (region IV).
Figure 5: Long-term model behavior with no vaccine cost ($\bm{r=0}$) for
$\bm{\mathcal{R}_{0}=1.4}$. Behavior in these regions of parameter space can
be deduced by iterating the vaccine coverage map (Equation 3.7) numerically
until it converges to a fixed point, but the regions correspond to the
analytical criteria detailed in the Appendix. There is still a region with no
lasting herd immunity (region I: the system converges to a fixed point,
$p^{*}$, that satisfies $p^{*}<p_{\rm crit}$); in this regime, the system
never achieves herd immunity ($p_{n}<p_{\rm crit}$ for all $n\geq 0$).
However, the system exhibits self-organized herd immunity when vaccine success
is sufficiently high, through a variety of mechanisms. The system may start
(and therefore stay) in the herd immunity interval (region II: $p_{0}\geq
p_{\rm crit}$), it may converge to “inefficient” lasting herd immunity (region
III: sustained overvaccination), or it may converge to “optimal” lasting herd
immunity (region IV: vaccination approaching the herd immunity threshold
$p_{\rm crit}$).
The mechanisms that drive the population to either inefficient or optimal
self-organized herd immunity are markedly different (Figure 6). In the case of
sustained overvaccination, the population starts at a relatively low level of
vaccination initially (lighter curve). A substantial epidemic occurs in the
first year, encouraging a large proportion of individuals to vaccinate in the
following year: too many, in fact. The population springs itself into the herd
immunity interval after that first year, and since there is no opposing force
pushing vaccination coverage down, overvaccination continues indefinitely. In
the case of optimal vaccination, the population may also start at a relatively
low level of initial vaccination, but the first epidemic sustained is not as
devastating as in the previous case (darker curve). A moderate proportion of
the population is affected by the disease and switches to vaccinating in the
following year. The epidemic sustained in this next year is not as large as
the one before it (thanks to the increase in vaccination), and encourages
another (smaller) group of individuals to switch to vaccinating next year.
This process gradually guides the population to the herd immunity threshold,
eventually achieving the optimal level of vaccination.
[capbesideposition=right,center,capbesidewidth=0.25]figure[]
Figure 6: Vaccine coverage level over time with no vaccine cost ($\bm{r=0}$)
in the regime of self-organized herd immunity
$\bm{(\mathcal{R}_{0}=1.4,s=0.6})$. If the initial population level is too low
(lighter curve), the population springs into the interior of the herd immunity
interval, $[p_{\rm crit},1]$, which results in sustained overvaccination. If
the population initially vaccinates at a more moderate level (darker curve),
vaccine coverage converges to the optimal herd immunity threshold, $p_{\rm
crit}$, in an asymptotic way.
## 5 Discussion
While the simple model studied here cannot self-organize into sustained herd
immunity when there is any cost to the vaccine, it may still achieve herd
immunity every other year. When the vaccine success rate is sufficiently large
for a given cost, there is an ongoing battle between the disease and the
vaccine. If the population undervaccinates in one year, it undergoes an
epidemic which drives the system to overvaccinate in the following year. A
non-trivial proportion of the population then bears some cost associated with
the vaccine, which discourages those individuals from getting vaccinated the
following year, driving the system back down to an undervaccinated state, and
the cycle repeats. Provided the amplitude of this (asymmetric) oscillation
about the herd immunity threshold is sufficiently small, this regime
effectively achieves herd immunity as any epidemic that occurs is relatively
small. While the goal of disease eradication has not strictly been achieved,
the resulting epidemics are so small in the model that a bit of stochasticity
may be enough to push the circulating flu strain into extinction (in a closed
population).
Although this promising biannual behavior is possible, the region of no
lasting herd immunity dominates vaccine parameter space, particularly for
vaccine morbidity and success levels that are realistic for the seasonal
influenza vaccine (cost near zero, success rate around 50%) (Figure 2). The
system may not drive itself to herd immunity asymptotically for these types of
vaccines, but a significant length of time is spent in the herd immunity
interval during the transient period (Figure 4). This effect opens the door
for other public health interventions (_e.g.,_ vaccination- and disease-
awareness campaigns) which have not been included in the model but may help
push the population into lasting herd immunity. Increases to the length of
time spent in the herd immunity interval can be achieved by improving the
vaccine, by increasing vaccine success, and/or by decreasing vaccine
morbidity.
In the case where there is no cost to the vaccine, the system can achieve
self-organized herd immunity that is either inefficient (due to
overvaccination) or optimal (at the critical vaccination threshold). If the
population initially vaccinates at a very low level, it undergoes a large
epidemic, and the following year, the population overreacts, propelling itself
into the herd immunity interval much like a diver on a springboard.
Overvaccination continues since there is no cost to the vaccine, and thus no
force pushing population vaccine coverage down. On the other hand, moderate
initial vaccination leads the population to converge to the optimal
vaccination level at the herd immunity threshold. In this case, the population
gradually learns from year to year through successively smaller epidemics.
Each such epidemic recruits smaller and smaller proportions of the population
to vaccinate until herd immunity is achieved. Notably, this result occurs even
with a moderately effective vaccine, like that of real seasonal influenza.
While it may not be realistic to assume that a vaccine can be considered
costless to an individual, this extreme case illustrates that if a vaccine can
be perceived as costless, the population can self-organize into sustained herd
immunity. Such a result is still possible even if the vaccine is only
moderately successful and even if the population does not immediately take to
getting vaccinated.
## 6 Conclusion
We have presented an intentionally simple model for seasonal influenza
vaccination that challenges the usual assumption that individuals make use of
perfect, global information completely rationally when making annual flu
vaccination decisions. The usual assumption wrongly predicts widespread use of
free-rider logic, which is not typically observed in this context. We make use
of established results in social psychology to inform our model, which gives
rise to both interesting and interpretable dynamics.
In the case where there is some cost to the vaccine, our model still predicts
regimes where vaccination coverage is below the herd immunity threshold in the
long term, which agrees with previous models. However, our model also predicts
new regions where herd immunity is achieved every other year: a result of the
population oscillating between vaccine-based and disease-based morbidity. When
we further assume that the vaccine has no cost, it is still possible for the
model to predict no lasting herd immunity. We also observe convergence to
enduring herd immunity, either at the optimal level, where the population
vaccinates exactly enough to reach this protected state, or inefficiently,
where the population overvaccinates.
Our disease-behavior model is deliberately simple as a first step, to focus on
the effect of incorporating a more realistic decision model on top of a well-
established model for disease spread. Future work should focus on making this
model more realistic and validating it with appropriate data.
Currently, both the decision-making and disease processes are deterministic; a
stochastic version of this model in either respect would be closer to reality.
Since agents only rely on the current state of the system to inform their next
decision, our model could easily be cast in a Markov chain framework. Modeling
the disease spread on a socio-spatial network would also provide greater
realism, by mimicking the way hosts interact and thus spread infectious
diseases like the flu (Chao et al. 2010).
Seasonal influenza is an immensely complex phenomenon, and we have not
accounted for issues such as that of multiple concurrently circulating strains
(Prosper et al. 2011), cross-reactivity of vaccines between strains (Iorio et
al. 2012; Moa et al. 2016), or waning vaccine immunity over the course of the
flu season (Rambhia and Rambhia 2018). Such additions to the model would also
serve to make it more realistic.
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## Appendix: Analytical criteria for long-term behavior in model with no cost
($r=0$)
Figure 5 can be produced by directly iterating the map to numerical
convergence, but it can be produced equivalently using analytical criteria
based on the existence and location of fixed points.
When $r=0$, the model map (Equation 3.7) reduces to
$p_{n+1}=\frac{\phi(sp_{n})}{1-sp_{n}}(1-p_{n})+sp_{n}+\left[1-\frac{\phi(sp_{n})}{1-sp_{n}}\right](1-s)p_{n}.$
(A1)
Fixed points, $p$, of this map must satisfy
$p=\frac{\phi(sp)}{1-sp}(1-p)+sp+\left[1-\frac{\phi(sp)}{1-sp}\right](1-s)p,$
(A2)
which simplifies to
$\frac{\phi(sp)}{1-sp}(p(2-s)-1)=0.$ (A3)
Equation A3 is satisfied when either (i) $\phi(sp)=0$ or (ii) $p(2-s)-1=0$.
Case (i) is satisfied by any $p\in[p_{\rm crit},1]$ since $\phi(sp)=0$ if and
only if $p\geq p_{\rm crit}$. Provided $p_{\rm crit}<1$, the ”herd immunity”
interval $[p_{\rm crit},1]$ is an invariant set of neutrally stable fixed
points that exists in the map’s domain of $[0,1]$. In other words, if
$p_{n}\in[p_{\rm crit},1]$ for any $n$, the trajectory is then trapped in the
herd immunity interval for all remaining time (precisely at the value
$p_{n}$).
Region II in Figure 5 is given by all $(p_{0},s)$ that satisfy $p_{0}\geq
p_{\rm crit}=\frac{1}{s}\left(1-\frac{1}{\mathcal{R}_{0}}\right)$. In other
words, the population starts at herd immunity and remains at herd immunity
indefinitely. Region III is given by all $(p_{0},s)$ such that $p_{0}<p_{\rm
crit}$ but $p_{1}\geq p_{\rm crit}$. This region represents populations whose
first epidemic was so large that it propels the population into herd immunity
immediately after the first year.
Case (ii) is satisfied by $p^{*}=1/(2-s)$. This fixed point is disjoint from
the herd immunity interval when
$\displaystyle p^{*}$ $\displaystyle<p_{\rm crit},$ (A4)
$\displaystyle\frac{1}{2-s}$
$\displaystyle<\frac{1}{s}\left(1-\frac{1}{\mathcal{R}_{0}}\right),$ (A5)
$\displaystyle s$ $\displaystyle<1-\frac{1}{2\mathcal{R}_{0}-1}.$ (A6)
Let us define $s_{\rm crit}=1-1/(2\mathcal{R}_{0}-1)$; when $s\geq s_{\rm
crit}$, the fixed point $p^{*}=1/(2-s)$ disappears into the herd immunity
interval.
Numerical simulations suggest that $p^{*}$ is stable when it exists disjoint
from the herd immunity interval (_i.e.,_ when $s<s_{\rm crit}$), with the
basin of attraction being $[0,p_{\rm crit})$; such trajectories will converge
to $p^{*}$ as $n\to\infty$. Region I is given by all $(p_{0},s)$ that satisfy
$p_{0}<p_{\rm crit}$ and $s<s_{\rm crit}$. In other words, vaccine coverage
starts below the herd immunity threshold and never surpasses it. Instead,
vaccine coverage converges to $p^{*}=1/(2-s)<p_{\rm crit}$.
Lastly, region IV is given by all $(p_{0},s)$ with both $p_{0}<p_{\rm crit}$
and $p_{1}<p_{\rm crit}$, but also $s\geq s_{\rm crit}$. In this case, the
fixed point $p=1/(2-s)$ does not exist distinct from the herd immunity
interval, but also the first epidemic is not strong enough propel vaccine
coverage over the herd immunity threshold ($p_{1}<p_{\rm crit}$). Under these
conditions, numerical simulations suggest that $\\{p_{0},p_{1},...\\}$ is a
monotonically increasing sequence where each $p_{n}<p_{\rm crit}$, but
$\lim_{n\to\infty}p_{n}=p_{\rm crit}$.
|
# Impacts of Earthquakes on Electrical Grid Resilience
Adam Mate 1* — and Travis Hagan 2 and
Eduardo Cotilla-Sanchez 2* — and Ted K. A. Brekken 2* — and
Annette Von Jouanne 3 1 The author is with the Advanced Network Science
Initiative at Los Alamos National Laboratory (LANL ANSI), Los Alamos, NM 87544
USA<EMAIL_ADDRESS>The authors are with the School of Electrical
Engineering and Computer Science, Oregon State University, Corvallis, OR 97331
USA. Email:{hagantr, ecs<EMAIL_ADDRESS>The author is with the
Energy Systems Department of Electrical and Computer Engineering, Baylor
University, Waco, TX 76706 USA<EMAIL_ADDRESS>versions of one or more of the figures in this paper are available online at
https://ieeexplore.ieee.org.
###### Abstract
One of the most complex and devastating disaster scenarios that the U.S.
Pacific Northwest region and the state of Oregon faces is a large magnitude
Cascadia Subduction Zone earthquake event. The region’s electrical grid lacks
in resilience against the destruction of a megathrust earthquake, a powerful
tsunami, hundreds of aftershocks and increased volcanic activity, all of which
are highly probable components of this hazard.
This research seeks to catalyze further understanding and improvement of
resilience. By systematizing power system related experiences of historical
earthquakes, and collecting practical and innovative ideas from other regions
on how to enhance network design, construction, and operation, important steps
are being taken toward a more resilient, earthquake-resistant grid. This paper
presents relevant findings in an effort to be an overview and a useful
guideline for those who are also working towards greater electrical grid
resilience.
###### Index Terms:
Disaster Preparedness, Earthquakes, Resilience, Power System Planning, Network
Operation.
## I Introduction
Since the 1980s, when researchers recognized the Cascadia Subduction Zone
(abbr. CSZ) as an active fault, the scientific community has been aware of the
major geological hazard that the state of Oregon and the U.S. Pacific
Northwest region (abbr. PNW) faces: the possibility of a tremendous earthquake
and tsunami caused by this fault, which can strike at any moment and can cause
enormous destruction thorough the region [1].
The Oregon Talent Council (abbr. OTC), in their Oregon Talent Plan [2],
identified a critical need for more and better prepared electrical power
systems engineers to serve Oregon’s energy technologies and utility industry.
There is an urgent need for agile engineers with expertise combining technical
education, applied skills, and work experience focused in disaster
preparedness and system resilience. Consequently, OTC decided to fund the
_Pacific Northwest Electrical System Resiliency and Disaster Preparedness
Training Project_ to address the existing talent gap; to conduct novel
research on electrical system resilience, to increase resilience awareness,
and to train Oregon electrical engineers for disaster preparedness with
particular attention to a CSZ event.
This paper presents some of the key findings of the OTC funded research. It
aims to systematize power system related experiences of historical
catastrophic earthquakes in order to gain knowledge that can help the PNW to
best prepare for a future CSZ event. The key contribution is a comprehensive
collection of proven practices and innovative ideas from other regions and
nations of the world, on how to improve power system design, construction and
operation, which can help to transform Oregon’s and the PNW’s electrical grid
into a more resilient, earthquake and tsunami resistant power system.
The remainder of this paper is organized as follows. Section II. gives a
comprehensive review on the topic’s background: information about the CSZ and
its future, current resilience in Oregon and efforts to increase it, and a
brief introduction of historical earthquakes that can serve as exemplar cases
to provide valuable input. Sections III., IV., and V. present findings related
to the generation, transmission and distribution levels of the electrical
grid, respectively. Section VI summarizes the research and concludes with
future steps and foreseeable difficulties.
## II Background
### II-A Next Cascadia Earthquake
The CSZ is an approx. 600 miles long offshore fault, which lies in the coastal
region of the PNW, stretching from Cape Mendocino in northern California –
through the states of Oregon and Washington – to the Brooks Peninsula in
southern British Columbia [1, 3, 4, 5]. The fault is part of a great arc of
subduction zones that surrounds the Pacific Ocean, creating a formation called
the “Ring of Fire,” and is a geologic mirror image of the subduction zone
lying east of Japan [1].
In the CSZ, three denser oceanic tectonic plates (namely the Explorer, Juan de
Fuca, and Gorda plates) are sliding from west to east and subducting beneath
the less dense continental plate (North American plate) that moves in a
general southwest direction, overriding the oceanic plates. Fig. 1 illustrates
the location of the subduction zone and these plates.
The movement of these plates is neither constant nor smooth: the plates stick,
building stress until the fault suddenly breaks and releases the accumulated
energy in the form of an earthquake(s) [3]. Thereafter, the plates start
moving again, and continue to move, until getting stuck again.
There is no doubt that another subduction earthquake will strike the PNW in
the future. In the last decade, research has confirmed that the CSZ has a long
history of great earthquakes. The most recent happened on January 26, 1700,
creating a magnitude 9.0 earthquake followed within minutes by a large
tsunami. Energy for the next earthquake is currently building up along the
fault, and has been since the last earthquake [1, 3, 5]. The time interval
between previous CSZ events varied from a few decades to many centuries, but
most intervals were shorter than the time elapsed since the last event in 1700
[1]. The calculated odds that the next earthquake will occur in the next 50
years range from 7-15% for a “great” earthquake affecting the entire PNW to
about 37% for a “very large” earthquake affecting southern Oregon and northern
California [1, 3, 4, 5].
Geologists assembled a ten-thousand-year record of past events, by studying
sediments in coastal marshes and on the ocean floor. This shows that half of
the past earthquakes have been “very large” (estimated magnitude of 8.3 to
8.6) and centered on the southern Oregon coast, while the other half have been
“great” (estimated magnitude 8.7 to 9.3) and extended along the full length of
the fault [1]. Although it is possible that the next CSZ earthquake(s) will be
a partial rupture of the fault, section by section, in a series of large
events over a period of years, it is strongly anticipated by many scientists
that it will be similar to the last event in 1700, and will be the result of
the entire fault rupturing, causing one great earthquake measuring magnitude
9.0, with ground shaking lasting 4-6 minutes [3, 5].
Figure 1: Schematic view of the tectonic plates in the CSZ area [5, 6].
### II-B Earthquake resilience in Oregon
After the discovery of the major threat, the rate of change to increase
Oregon’s resilience has been slow. The first-time that explicit seismic
provisions were adopted in Oregon’s building codes was in 1993 [1, 7]. Only
following the 2011 Tohoku, Japan earthquake and tsunami disaster, was special
attention given to the CSZ threat at the state governmental level. The Oregon
Seismic Safety Policy Advisory Commission, with the help of more than 150
volunteer professionals, prepared the Oregon Resilience Plan in 2013 [1]. This
plan was the first comprehensive study and had the following goals: assess the
state of resilience in Oregon, plan for the impacts of a CSZ event, and map a
path of policy and investment priorities for the next fifty years. Up until
today, two CSZ earthquake scenario documents have been created, [3] and [5]
(with multiple versions), with the goal to provide information for the public
on the hazards the PNW faces.
These three studies [1, 3, 5] identified the following key findings on the
current state of resilience of Oregon’s electrical grid:
* •
Electrical facilities and network components – including power plants,
substations, transmission lines – are seismically vulnerable to damage and
have significant risk due to ground shaking and ground failure, especially
landslides, soil liquefaction, lateral spreading and coastal subsidence
hazards.
* •
Most of Oregon’s critical and non-critical energy infrastructure has been
constructed with seismic design deficiencies, and not initially built to with-
stand earthquakes. Substantial improvements, investments, and a uniform set of
design and construction codes are needed to minimize extensive direct
earthquake damage, indirect losses, and possible ripple effects.
* •
A CSZ event will cause the failure of numerous power system components, and
over half of the region’s electrical grid may suffer medium to high damage on
all grid-levels. Outages and blackouts may occur not only within 100 miles of
the coastline, but even in areas that were not directly affected by the
earthquake. The destruction on the coastal areas may be severe enough as to
render the equipment and structures irreparable. In the I-5 corridor,
considerable damage on generation and distribution levels may result in the
loss of over half of the system’s capacity.
* •
Oregon’s liquid fuel supply is extremely vulnerable. Besides the high
dependency on Washington State – more than 90% of Oregon’s refined petroleum
products come from the Puget Sound area, which is also vulnerable to a CSZ
earthquake – the storage facilities along Oregon’s Willamette River lie on
liquefiable riverside soils. This significantly affects most sectors of the
economy critical to emergency response and economic recovery.
* •
Estimated restoration time of electrical grid after a CSZ event ranges from
1-3 months (in the Willamette Valley) to 3-6 months (Coastal region),
depending on the degree of destruction, available utility personnel,
contractors and road conditions.
Even though small steps – conducted studies, funded research, legislative
changes –- have been taken in the recent years to start and accelerate the
needed change in this issue, Oregon’s electrical grid today is still far from
resilient to the impacts of a CSZ event.
### II-C Notable Historical Earthquakes
TABLE I: Comparison of Historical Catastrophic Earthquakes
Other regions and nations of the world can provide plenty of valuable
firsthand experiences, useful practices and ideas for the PNW to utilize. Even
though environmental and geological conditions and occurring natural disasters
will be unique to those locations, there are methods, thoughts, tools and
developments that can be applied to make the PNW’s electric grid more
resilient.
Table I. presents historical catastrophic earthquakes (in chronological order)
that were investigated in this research. Besides comparing the main attributes
of these events to each other, the last line also compares them to the
upcoming CSZ earthquake. Thus, the magnitude of this future event becomes more
easily conceivable. The data in the table is compiled from sources listed in
the References section and publicly available information.
The PNW, and specifically the state of Oregon, is prone to earthquakes, just
like all other regions inside the Pacific Ring of Fire. In this research
special attention was paid to Japan, Chile and New Zealand. Earthquakes of all
sizes are much more common in these areas than anywhere else in the world, and
as such, these countries have become better prepared and possess years of
expertise in electrical grid resilience related questions. The Nisqually
earthquake is one of the few earthquakes in the PNW that was near a major
metropolitan area and a detailed analysis of the power grid had been
completed. The Nepal earthquake was included because it shows the effect of
earthquakes on mountainous regions – similar to some areas of the PNW – and
the additional considerations that must be made when dealing with this type of
geography.
The following three sections present findings that are divided into the
categories of generation, transmission, and distribution levels of the
electrical grid.
## III Generation Level
Generation plants across the PNW face a varying degree of threat from
landslides, soil liquefaction, ground settling, lateral spreading and ground
shaking. Each threat has the potential to take generation plants off-line for
months.
Landslides are more unique to the geography of the PNW. The relatively wet
winters combined with the number of hills and mountains creates an environment
where events of this nature are expected. The 7.8 magnitude earthquake that
struck Nepal in 2015, caused tens of thousands of landslides, and illustrates
the link between earthquakes and landslides [8]. Saturated soils can have
about half the strength than comparable dry soils [8]. This is concerning for
the Willamette Valley in Oregon, Puget Sound in Washington, coastal regions
and the Columbia River Gorge [8], which is where the majority of the
generation facilities lie.
The PNW has several hydroelectric dams spanning the U.S. states of Oregon,
Washington, Idaho, and in the Canadian Province of British Columbia. These
dams supply about 75% of the PNW’s power demands, and some of the dams lie in
areas prone to landslides. The Columbia River Gorge also contains most of the
PNW’s wind generation. The wind turbines, perched on hills above the river,
are also at risk from landslides from a large magnitude earthquake. A small
slide near the Columbia River has the potential to vastly affect and take down
generation capabilities.
The worst-case scenario is a slide of the same magnitude as the “Bonneville
slide”. This slide, which is believed to be the result of an earthquake
between 1550 and 1750, dammed the Columbia River near present day Bonneville
Dam, and created a lake spanning 100 miles upstream. Sometime later, the dam
breached and created the Cascades Rapids. This feature is now hidden by
Bonneville Reservoir [9]. If the same event occurred today, Bonneville Dam
would be completely buried, and the Dalles Dam and John Day Dam would have no
outflow. This would lead to a loss of nearly five gigawatts of nameplate
capacity directly, and several more gigawatts indirectly.
Soil liquefaction, settling, and lateral spreading, and the phenomena that
they cause, have great potential for damage in the PNW. They are best studied
and understood through analysis of earthquakes in other countries. In the
Nepal and Tohoku, Japan earthquakes, most of the settling –- induced by soil
liquefaction –- occurred near the coastline and in reclaimed soil [10]. As
such, generation sites that lie near water sources will experience a greater
amount of settling, leading to more damage at these facilities. This poses a
problem for generation along the Columbia River especially [8].
Ground shaking is the underlying cause behind landslides, soil liquefaction,
settling, and lateral spreading, but it also poses a direct risk to generation
sites. In Japan, many generation sources went offline due to the shifting of
vibration isolators [10]. Similar issues were encountered in the New Zealand
earthquakes where mercury safety switches disconnected transformers due to
shaking [11]. Ground shaking is particularly an issue in concrete structures,
which can crack under the repeated stress [11].
Another important consideration is the survivability of dams. Many of the dams
in the northwest were built prior to understanding the CSZ, and seismic
effects were not considered. Despite this, dams have been shown to hold up
well in earthquakes [12]. According to a PUD (Public Utility District) in
Washington State that oversees several of the dams, the dams are robust and
will survive an earthquake, but will be damaged [12]. Surprisingly, a study
published by FERC indicated that the most damaging force to these dams are the
local seismic activities and not a CSZ event [13]. Although a dam structure
may survive, significant damage inside can affect the generators, which would
require a lengthy repair process.
## IV Transmission Level
Transmission lines, substations, and substation equipment make-up a large
fraction of electrical infrastructure. These critical pieces of equipment
cross the PNW, covering great distances; further increasing the risk of damage
from an earthquake.
Landslides pose the greatest risk to transmission lines and towers that cross
mountain ranges: they are very likely to occur during an earthquake [8], and
given the length of some of the transmission lines the likelihood of lines
being down is a significant factor. This is especially true for the
connections crossing the Coastal Mountain range and Cascade Mountain range. An
examination of the Oregon coast shows that very little generation lies on the
coastal side of the mountains. This requires the power to be transferred
across at least one mountain range, and increases the likelihood of critical
line failures.
Another consideration of the transmission lines crossing mountain ranges, is
that a large earthquake would render them nearly impossible to access by
ground transportation. The Oregon Resilience Plan highlighted this as a
concern and the impacts were demonstrated in Nepal [1, 14]. Minimal damage
levels would damage bridges and shift or settle roads, making most routes
impassible. In extreme cases, landslides may completely block routes. Nepal
had several remote sites that were inaccessible following the earthquake,
which made repairing some of its generation facilities difficult [14].
Following a CSZ event, restoration efforts will be slowed due to blocked
routes.
Soil liquefaction, settling, and lateral spreading also threaten transmission
lines, towers, substations and substation equipment. In the Christchurch, New
Zealand earthquake, several sites showed signs of soil liquefaction and
settling [11]. Some of these sites were abandoned in favor of rebuilding in
new areas less susceptible to liquefaction. At one site, liquefaction tipped
transformers at a noticeable angle, leading to a complete loss of the
substation. Many of New Zealand’s substations were originally unreinforced
masonry, but a significant effort started years prior, reinforced the
substations. This is considered a significant factor in almost all the
substations surviving the earthquake [11].
Although many models are available for predicting soil liquefaction,
determining specific movements requires an in-depth site analysis. In the PNW,
a site analysis was performed at a substation near the coast. Two bore holes
were drilled to determine the depth of the bedrock. The bedrock was at
significantly different depths at opposite ends of the same substation. The
analysis concluded that soil liquefaction is a primary concern at this site,
and it would be a significant process to mitigate the risk. This is likely the
situation at most sites west of the Cascade Mountains, where ground shaking is
likely to be high from a CSZ earthquake.
## V Distribution Level
Threats to the distribution system are the most well documented, and often the
most heavily damaged in terms of downed pieces of equipment. Few significant
earthquakes have occurred in the PNW making a direct study difficult. The
Nisqually and New Zealand earthquakes, while lesser in magnitude, provide a
detailed look at the threat of a smaller earthquake.
Soil liquefaction, settling and lateral spreading proved to be the most
damaging to the distribution level equipment. In many areas, buried cables are
used instead of overhead power lines. Early reports in New Zealand stated that
4% of the 11kV buried cables failed, but this number was later increased to
15% because of continued failures after being reenergized [11]. The study of
the New Zealand earthquake showed that buried cables were particularly
susceptible to damage from liquefaction. In many of the instances in this
earthquake, concrete failed putting additional strain on the buried cables.
The strain caused high curvature bends and stretching. Later, when re-
energized, the weakened cables failed anywhere from seconds to months
following the earthquake [11]. The buried cables also made identifying and
repairing failures more difficult and time consuming, requiring a minimum of
12 hours to locate and repair each fault [11].
In the Nisqually earthquake, it was observed that most of the outages occurred
near the Bayfront, around Harbor Island and south to the Seattle International
Airport [15]. This area also experienced the greatest amount of ground
shaking. The report compared the recorded shaking to the city projections, and
showed issues created by artificial land and in saturated soils. Harbor Island
is a man-made island built from the hills that used to lie near the Bayfront.
In Japan, reclaimed and coastal areas experienced the greatest amount of
settling and soil liquefaction as well [10]. However, in areas of Japan where
liquefaction prevention measures were in place, liquefaction was far less of
an issue [10]. In Seattle, half of the feeders in the severely shaken regions
experienced failures. It was indicated that the failures were likely the
result of base movement in poles near the artificially created areas of Harbor
Island and the Duwamish River [15]. 37% of this damage fit the previously
mapped liquefaction zones for Seattle.
In larger earthquakes, like Chile, distribution level equipment had issues
beyond soil and ground shaking [16]. The main difficulties in restoring
distribution level power was that several buildings fell on lines and the
tsunami washed away lines and poles. Thus, power restoration in heavily
damaged areas took weeks as new poles and lines were put in place [16]. In
Chile and New Zealand, restoration efforts were helped significantly by
temporary generators [11, 16]. New Zealand used the diesel generators to power
critical loads, such as water facilities [11]. Chile, used them in more
isolated areas [16].
In both buried and above ground distribution networks, soil liquefaction
proved to be an issue. Buried lines in liquefiable soils experienced concrete
fracturing and ground displacement causing lines to fail. Above ground lines
suffered from having poles tip in poor soils or by being destroyed by falling
structures. The shaking itself did little harm to distribution level
equipment.
## VI Conclusion
In this paper, the current state of resilience in the state of Oregon was
presented, along with systematized power system related experiences of
historical earthquakes, proven practices and innovative ideas from affected
regions, in order to contribute to the preparation for a future CSZ event in
the PNW.
This paper is intended to encourage and catalyze the creation of a more
resilient U.S. PNW and Oregon electrical grid. Better resilience can be
achieved, but the threat first must to be recognized at all levels and
appropriate attention must be paid to the issue. Although the research is
still in early phases, the move toward addressing this critical issue has
fortunately already begun.
Acknowledgements This research was supported by the Oregon Talent Council
under the _“Pacific Northwest Electrical System Resiliency and Disaster
Preparedness Training Project”_. The authors would like to thank OTC for their
financial support, and would also like to thank project and industry partners:
Portland State University, Central Lincoln PUD, Portland General Electric, and
Pacific Power.
## References
* [1] “The Oregon Resilience Plan: Reducing Risk and Improving Recovery for the Next Cascadia Earthquake and Tsunami,” Oregon Seismic Safety Policy Advisory Commission, Report, Feb. 2013. [Online]. Available: https://www.oregon.gov/. [Accessed: Dec. 03, 2019].
* [2] P. Scruggs et al., “The Oregon Talent Plan: A Needs Assessment for Professional and Technical Occupations,” Nov. 2015. [Online]. Available: https://www.oregon.gov/. [Accessed: Dec. 03, 2019].
* [3] “Cascadia Subduction Zone Catastrophic Earthquake and Tsunami,” Washington and Oregon Whole Community Exercise Design Committee, Exercise Scenario, Jan. 2015. [Online]. Available: https://www.oregon.gov/. [Accessed: Dec. 03, 2019].
* [4] C. Goldfinger et al., “Turbidite Event History – Methods and Implications for Holocene Paleoseismicity of the Cascadia Subduction Zone,” _U.S. Geological Survey Professional Paper_ , 1661-F, 170 p.
* [5] “Cascadia Subduction Zone Earthquakes: A Magnitude 9.0 Earthquake Scenario,” Cascadia Region Earthquake Workgroup, Sep. 2013. [Online]. Available: https://www.dnr.wa.gov/. [Accessed: Dec. 03, 2019].
* [6] B.F. Atwater et al., “The Orphan Tsunami of 1700 – Japanese Clues to a Parent Earthquake in North America,” _U.S. Geological Survey_ , Professional Paper 1707, Sep. 2015.
* [7] “Earthquake Design History – A Summary of Requirements in the State of Oregon,” State of Oregon Building Codes Division, Feb. 2012. [Online]. Available: https://www.oregon.gov/. [Accessed: Dec. 03, 2019].
* [8] R.E.S. Moss et al., “Geotechnical Effects of the 2015 Magnitude 7.8 Gorkha, Nepal, Earthquake and Aftershocks,” _Seismological Research Letters_ , vol.: 86, no.: 6, pp.: 1514–1523, Nov. 2015.
* [9] R. Hill, “Geotechnical Effects of the 2015 Magnitude 7.8 Gorkha, Nepal, Earthquake and Aftershocks,” _Washington Geology_ , vol.: 29, no.: 12, pp.: 35–38, 2001.
* [10] M. Kazamaa et al., “Damage Statistics – Summary of the 2011 off the Pacific Coast of Tohoku Earthquake Damage,” _The Japanese Geotechnical Society - Soils and Foundations_ , Dec. 2012.
* [11] A. Kwasinski et al., “Performance of Electric Power Systems in the 2010–2011 Christchurch, New Zealand, Earthquake Sequence,” _Earthquake Spectra_ , vol.: 30, no.: 1, pp.: 205–230, Feb. 2014.
* [12] “Earthquake Study Raises Risk Potential Around Central Wash. Dams,” Northwest News Network, Aug. 2012. [Online]. Available: https://www.opb.org/news/article/n3-earthquake-study-raises-risk-potential-around-central-wash-dams/. [Accessed: Dec. 03, 2019].
* [13] “Probabilistic Seismic Hazard Analyses Project for the Mid-Columbia Dams,” Federal Energy Regulatory Commission, 2012.
* [14] K. Schneider, “Nepal Earthquake Damages At Least 14 Hydropower Dams,” May 2015. [Online]. Available: https://www.circleofblue.org/2015/world/nepal-earthquake-damages-at-least-14-hydropower-dams/. [Accessed: Dec. 03, 2019].
* [15] J. Park et al., “Nisqually Earthquake Electric Utility Analysis,” _Earthquake Spectra_ , vol.: 22, no.: 2, pp.: 491–509, May 2006.
* [16] J.C. Araneda et al., “Lessons from the 2010 Chilean Earthquake and its Impact on Electricity Supply,” IEEE International Conference on Power System Technology 2010, Oct. 2010.
|
# Air-Ground Collaborative Mobile Edge Computing: Architecture, Challenges,
and Opportunities
Zhen Qin, Hai Wang, Yuben Qu, Haipeng Dai, and Zhenhua Wei Z. Qin and H. Wang
are with the College of Communications Engineering, the Army Engineering
University of PLA, China.Y. Qu (corresponding author) is with the Department
of Computer Science and Engineering, Shanghai Jiao Tong University, China.H.
Dai is with the Department of Computer Science and Technology, Nanjing
University, China.Z. Wei is with the Xi’an Research Institute of High
Technology, China.
###### Abstract
By pushing computation, cache, and network control to the edge, mobile edge
computing (MEC) is expected to play a leading role in fifth generation (5G)
and future sixth generation (6G). Nevertheless, facing ubiquitous fast-growing
computational demands, it is impossible for a single MEC paradigm to
effectively support high-quality intelligent services at end user equipments
(UEs). To address this issue, we propose an air-ground collaborative MEC (AGC-
MEC) architecture in this article. The proposed AGC-MEC integrates all
potentially available MEC servers within air and ground in the envisioned 6G,
by a variety of collaborative ways to provide computation services at their
best for UEs. Firstly, we introduce the AGC-MEC architecture and elaborate
three typical use cases. Then, we discuss four main challenges in the AGC-MEC
as well as their potential solutions. Next, we conduct a case study of
collaborative service placement for AGC-MEC to validate the effectiveness of
the proposed collaborative service placement strategy. Finally, we highlight
several potential research directions of the AGC-MEC.
## I Introduction
Fifth-generation (5G) network has been deployed worldwide and commercially
available in 2020, which offers many more functions than previous generations
[1]. However, with the advancement of smart devices and Internet of Things
(IoT) technology, as well as diversified applications (_e.g.,_ smart city,
mobile augmented reality, face recognition, and autonomous driving), 5G
networks cannot completely meet future rapidly growing traffic demands.
Accordingly, sixth-generation (6G) have attracted increasing attention from
both industry and academia, which will be transformative and revolutionize the
wireless evolution form. 6G network is expected to effectively support high-
quality services and unlimited connectivity for a large number of intelligent
devices [2]. Meanwhile, it brings great challenges to the computing power of
centralized data center and intelligent terminals. The traditional cloud
computing cannot meet the requirements of massive data processing, and
computing power will be transferred from the network core to the network edge.
Mobile edge computing (MEC) [3] is an emerging computing paradigm that can
push mobile computing, cache, and network control to the edge in the close
proximity of mobile user equipments (UEs). MEC is envisioned to play a leading
role in 6G by operating as an intermediate layer that provides fast and
localized data processing for many critical and resource-constrained
applications [4].
With the help of MEC, computation-intensive and latency-sensitive tasks can be
offloaded for remote execution, which can enhance the computing ability, and
reduce energy consumption and latency. MEC servers are usually deployed in a
fixed fashion at the ground base stations (BSs), wireless access points (APs),
and roadside units (RSU). Nevertheless, such formed traditional terrestrial
infrastructure-based MEC system has its limitations, which may not work in
many critical applications, such as military, emergency relief, and disaster
response. In addition, since terrestrial MEC servers lacks mobility, it cannot
meet the computation and connectivity demands with the spatio-temporal
dynamics. In contrast to the terrestrial MEC network, due to the Line-of-Sight
(LoS) links, flexible deployment, and maneuverability, the aerial MEC network
consisting of unmanned aerial vehicles (UAVs), airships, and balloons equipped
with MEC servers might compensate those weaknesses. The aerial MEC network is
also faced with many challenges such as the limited battery life. As a result,
considering intelligent endogenous and ubiquitous computational power
requirements of 6G networks, it is impossible for a single MEC paradigm to
accomplish such a difficult task.
TABLE I: A comparison of different MEC paradigms in AGC-MEC. MEC Paradigms | UAV | Airship/Balloon | BS/AP/RSU | Vehicle | Powerful Mobile User
---|---|---|---|---|---
Location | Air | Air | Ground | Ground | Ground
Cost | Small | High | High | Medium | Small
Availability | Medium | High | High | Medium | Low
Reliability | Low | High | High | Medium | Low
Mobility | | Mobile
---
(3D)
| Quasi-Stationary
---
(3D)
| Static
---
(2D)
| Mobile
---
(2D and Restricted)
| Mobile
---
(2D and Very Restricted)
Energy Supply | Poor | Medium | Abundant | Medium | Poor
Coverage | Medium | Large | Large | Medium | Small
Computation Power | Weak$\sim$Medium | Medium$\sim$Strong | Strong | Medium | Very Weak
Communication Ability | Weak$\sim$Medium | Medium$\sim$Strong | Strong | Medium | Very Weak
Storage Capacity | Weak$\sim$Medium | Medium$\sim$Strong | Strong | Medium | Very Weak
(a) Collaborations between air and ground
(b) Typical use cases
Figure 1: AGC-MEC architecture.
To promote the development of 6G networks, we propose an air-ground
collaborative MEC (AGC-MEC) architecture, which actively explores the
complementary integration of computational powers from both air and ground
segments to provide intelligent deployment and management of different MEC
paradigms. The novelty of the proposed AGC-MEC lies in that it involves all
potentially available MEC servers in the air-ground integrated networks.
Specifically, the AGC-MEC architecture comprises a two-layer networking
architecture: aerial MEC and terrestrial MEC. In the aerial MEC, airships and
balloons are deployed as (near) static aerial MEC servers, while UAVs as
mobile aerial MEC servers. In the terrestrial MEC, BSs/APs/RSUs are deployed
as static terrestrial MEC servers, while vehicles as mobile terrestrial MEC
servers. Meanwhile, powerful mobile users can be used as opportunistic mobile
ground MEC servers. Through different cooperations between the air and ground,
the AGC-MEC architecture is more flexible than the single terrestrial MEC
network and more powerful than the single aerial MEC network.
There exist several studies that investigate air-ground integrated MEC
network. They can be roughly divided into two categories: one considers aerial
MEC servers only and the other considers both aerial and terrestrial MEC
servers. For the former one, Zhou _et al._ introduced three UAV-enabled MEC
architectures, which improve computation performance and reduce execution
latency by integrating UAV into MEC networks [5]. Cheng _et al._ proposed a
novel air-ground integrated mobile edge network (AGMEN), where UAVs are
flexibly deployed and scheduled, and assist the computing, caching, and
communication of the edge network [6]. For the latter one, Zhou _et al._
proposed an air-ground integrated MEC framework, where ground vehicles and
UAVs are envisaged as supplementary MEC servers for efficient service
provisioning [7]. Jiang _et al._ proposed a heterogeneous MEC (H-MEC)
architecture, which aims to address the key challenges of the H-MEC
architecture in dynamic environments using AI-based solutions [8]. However,
all the aforementioned works only consider partial MEC paradigms in the air-
ground integrated network. For example, [5, 6] consider the aerial MEC solely,
while [7] considers mobile ground vehicles and UAVs, and [8] additionally
considers the fixed ground BS based on [7]. In contrast, our proposed AGC-MEC
involves all potentially available MEC servers (both static and mobile) within
air and ground in the context of future 6G networks. Furthermore, AGC-MEC
considers how to enable different MEC servers collaborate effectively, rather
than merely integrate them.
In the rest of this article, we first present the proposed AGC-MEC
architecture and elaborate three typical use cases to illustrate its potential
values in Section II. Then, we analyze several main technical challenges of
the AGC-MEC in Section III. Next, we discuss three potential research
directions in Section V, while we conduct a case study to evaluate the
performance of the AGC-MEC in Section IV. Finally, we draw conclusions in
Section VI.
## II Air-Ground Collaborative Mobile Edge Computing (AGC-MEC)
In this section, we propose the AGC-MEC architecture, which integrates all
potentially available MEC servers within air and ground by a variety of
collaborative ways to deliver computation services at their best for user
equipments (UEs). Firstly, we introduce the AGC-MEC architecture, which
consists of aerial and terrestrial MEC networks. Secondly, in order to show
the potential of the AGC-MEC, we elaborate three typical use cases.
### II-A AGC-MEC Overview
Fig. 1 (a) illustrates the conceptual architecture of the proposed AGC-MEC,
which comprises a two-layer networking architecture following the general air-
ground integrated networks: aerial MEC and terrestrial MEC. Specifically, the
aerial MEC network is mainly composed of UAVs, airships, and balloons, while
the terrestrial MEC network is generally made up of BSs, APs, RSUs, vehicles,
and mobile users, all of which are equipped with edge computing servers. In
the aerial MEC, airships and balloons are deployed as (near) static aerial MEC
servers and UAVs as mobile aerial MEC servers. In the terrestrial MEC, there
are three types of MEC servers. Specifically, BSs/APs/RSUs and vehicles are
employed as static and mobile terrestrial MEC servers, while some powerful
mobile users can be used as opportunistic mobile terrestrial MEC servers. We
envision that all the aforementioned MEC servers can collaborate with each
other in the proposed AGC-MEC, which mainly focuses on the collaborations
between air and ground. As shown in Fig. 1 (a), the collaborations can be
divided into four categories as follows: A) static-air and static-ground
collaboration; B) static-air and mobile-ground collaboration; C) mobile-air
and static-ground collaboration; D) mobile-air and mobile-ground
collaboration. In practical, several collaborations could work together.
Different MEC paradigms have their particular features, which are partially
overlapping but also complementary. We present a comprehensive comparison of
different MEC paradigms in the AGC-MEC in Tab. 1. MEC servers are usually
deployed in a fixed fashion at the BSs, APs, RSUs, airships, and balloons.
Despite their high availability and reliability, they also bring higher costs
than UAVs, vehicles and powerful mobile user. Due to mobility and flexibility,
UAVs, vehicles can be deployed quickly on demand. UAVs move much faster than
vehicles, but with less computation, communication and storage resources. On
the contrary, vehicles move slower but hold more resources. In fact, BSs, APs,
RSUs, airships and balloons have the most available resources. Furthermore,
the coverage of fixed MEC server is large but remains unchanged. They cannot
exploit its mobility to move closer to UEs with computation-intensive tasks.
In contrast, UAVs, vehicles and powerful mobile users have limited coverage
and energy, but can move close to UEs to provide low-latency services and
communication. By collaborating different MEC paradigms of air and ground,
AGC-MEC aims to manage and control heterogeneous network resources smartly to
meet computing demands.
### II-B Typical Use Cases
As illustrated in Fig. 1 (b), the proposed AGC-MEC architecture is envisioned
to be useful particularly in several applications as follows.
#### II-B1 Ubiquitous Latency-Sensitive Applications
With the development of IoTs and various mobile applications, more and more
data is generated at the edge of the network. It is a general trend to process
and analyze the data in real time widely at the network edge, which needs
strong computing power support. By effectively collaborating different MEC
paradigms, the AGC-MEC architecture can provide efficient and flexible
computing services at the edge to meet the demands of ubiquitous latency-
sensitive applications. For example, intelligent transportation system (ITS)
requires low-latency communication and high computation capabilities. Static
ground RSUs equipped with MEC servers can process the local data, which not
only reduces the burden of network transmission, but also speeds up the data
processing speed. Nevertheless, there are situations where additional air and
ground MEC servers are required to handle temporary high traffic loads during
extreme traffic congestion or unexpected weather conditions. According to
congestion conditions and traffic events, UAVs can be dynamically deployed as
mobile aerial MEC servers. In addition, MEC can play a significant role in
connected healthcare systems by offering better insight of heterogeneous
healthcare content to support affordable and quality patient care [9].
Ubiquitous collaborative MEC servers can help patients choose better advice
from the right guardians in real time when some emergencies occur.
#### II-B2 Spatio-Temporal Dynamic Applications
In the daily life, mobile UEs have obvious group effect and usually form
different dense crowds with bursty requests over time, which results in a
spatio-temporal dynamic computing demand. The AGC-MEC can deploy various MEC
servers on demand to meet such a dynamic demand of UEs. One typical example
scenario is the stadium in a major sport or concert event. There are a huge
number of people gathered, who execute computation-intensive applications in
their mobile phones such as Virtual Reality (VR) and online gaming. In this
case, BSs may be overloaded and cannot support massive UEs. Fortunately, UAVs,
airships, balloons and vehicles can be temporary deployed to collaborate with
ground BSs to offload computation tasks and improve the user quality of
experience (QoE). Other typical example scenario is tourist attractions and
important transportation hubs. For example, on October 3, 2019, Tiananmen
received 2.68 million tourists throughout the day; the passenger flow reached
the peak of that day at the time of flag lowering, which was 180 thousand. To
meet the dynamic computing demands of tourists, UAVs can be deployed flexibly
to collaborate with ground MEC systems during holidays.
#### II-B3 Emergency and Military Applications
In some extreme cases such as emergency and military applications, the ground
infrastructure-based MEC system may be destroyed partially or completely and
thus cannot work properly. The AGC-MEC can assemble mobile MEC servers
including UAVs, airships, balloons, and vehicles to assist the existing
communication infrastructure, if any, more critically, in providing computing
services. For instance, in the event of natural disasters and earthquakes,
ground infrastructure may be damaged. The rescue crews may need mobile
augmented reality to search the area, which needs a significant amount of
computing resources. The AGC-MEC can provide the required computation
resources to increase the search scope and speed up the rescue. Moreover, in
military applications, there are massive computation-intensive reconnaissance
tasks (_e.g.,_ estimating the locations and the dynamics of the hostile
forces). For a large-scale reconnaissance area without infrastructure, UAVs
can collaborate with ground vehicles to expand the reconnaissance scope,
reduce cloud computing latency, and enhance the computing ability.
## III Challenges in AGC-MEC
Due to the features of high mobility, heterogeneity, frequent inevitable air-
ground interactions and time-varying channel conditions, the AGC-MEC
architecture is difficult in platform integration, network deployment,
resource management, and intelligence realization. In this section, we discuss
four challenges and their potential solutions.
### III-A Generic Computing Platform Integration
Generic computing platform integration is critical to ensure synergy between
different MEC paradigms. The AGC-MEC involves different MEC paradigms, which
exist in different segments and have distinct characteristics. MEC paradigms
are connected to each other through a communication protocol and communicate
with users and devices. However, different MEC paradigms have various
communication protocols, communication links, and interfaces, which
significantly limits interoperability [6]. Therefore, how to integrate the
computing resources of various MEC paradigms and build a generic computing
platform is our primary consideration.
Network Function Virtualization (NFV) is an emerging network technology, which
moves the network function from the original special equipment to the general
equipment. Specifically, it decouples network functions from specialized
hardware, and can be leveraged to flexibly implement network functions as
software instances in the network slices. Despite its great potential
benefits, NFV is also faced with some problems to be solved. For example,
under the virtualized network environment, which interfaces can be used
privately, and which interfaces need to be standardized, these issues remain
to be clarified.
### III-B On-Demand 3D Network Deployment
In the AGC-MEC architecture, a fundamental and critical issue is how to deploy
air/ground MEC servers, which includes two aspects. For one thing, based on
the demand, we need to determine when, where, what MEC paradigms and how many
MEC servers to deploy. For another thing, we need to determine how to
collaborate optimize the trajectories of mobile air/ground MEC servers.
There are many challenges for the on-demand deployment of edge servers. First,
due to the high mobility of users, heterogeneity of QoS requirements, and
stochastic characteristics of wireless network, the deployment of MEC servers
is exposed to extreme difficulties. Second, since AGC-MEC architecture
involves the cooperation between air and ground, the MEC servers need to be
deployed in 3D space. The critical design challenge is to adaptively adjust
the trajectories to meet the dynamic computing task requirements. The existing
work considers to use Deep Reinforcement Learning (DRL) algorithm, which can
learn optimal placement policies and plan trajectories of mobile MEC servers
intelligently [8]. However, DRL algorithm brings more computational cost than
traditional methods.
### III-C Computing-Oriented Resource Allocation
Resource allocation is an important guarantee for effective collaboration of
AGC-MEC architecture, which directly impacts network performance. In order to
effectively complete computing tasks, the AGC-MEC architecture should carry
out efficient, adaptive, and intelligent resource allocation. Air-ground
collaborative resource allocation involves various resources, which can be
mainly divided into the following two types: the communication resource
including bandwidth and channels, and the computation resource including
computing power and storage capacity. In particular, to provide personal
services for different users, MEC servers need to store corresponding data
including object databases, libraries and trained machine learning models,
associated with services. Service provisioning is an essential and critical
issue, _i.e.,_ how to determine where to store/place which MEC service to meet
various computing service demands. Furthermore, jointly optimizing resource
allocation, trajectory, and placement of MEC servers can obtain globally
optimal performance that would be more useful in practice.
There are two main challenges to deal with the above problem. First, the
problem includes many decision variables to be jointly optimized, _e.g.,_ the
continuous computation resource allocation and UAV trajectory variables, and
integral task offloading and service placement variables. Therefore, the
optimization problem is a non-convex mixed integer nonlinear programming
(MINLP) problem and is hard to solve in general. Secondly, computing demands
vary in time and space. In order to match resource provision with computing
tasks, resource allocation needs to be adjusted dynamically in a real-time
manner. In order to better adapt to the dynamic environment, the complicated
network behaviors can be analyzed in real time, and the online algorithm can
be used to flexibly adjust resource allocation.
### III-D Ubiquitous Edge Intelligence Realization
Edge intelligence (EI) is emerging as a promising key enabler for MEC to
fulfill the vision of ubiquitous intelligence, which pushes intelligence to
the network edge by running artificial intelligence (AI) algorithms on edge
devices. EI can be divided into two main types of technology: AI for edge and
AI on edge [10]. The former focuses on utilizing AI algorithms to provide
effective solutions for key problems in edge computing, and the latter focuses
on realizing AI model training and inference on the edge. However, it is
difficult to realize ubiquitous EI in the following aspects.
AI for AGC-MEC: It is devoted to provide better solutions to constrained
optimization problems, _e.g.,_ trajectory optimization, resource allocation,
network deployment, and real-time decision making. Despite its great potential
benefits, utilizing AI algorithms is also faced with some problems to be
solved. Firstly, since some MEC servers is resource-constrained, how to
balance optimality and efficiency of the AGC-MEC architecture is a great
challenge. Secondly, if we want to utilize AI algorithms to obtain solutions,
the formulated optimization problem and mathematical model need to be
restricted [10]. Therefore, the model establishment is a huge challenge.
AI on AGC-MEC: The computing and storage capacity of edge servers is far less
than that of cloud servers, which cannot meet the needs of a large number of
computing and storage resources for AI training. Fortunately, federated
learning (FL) can make multiple resource-constrained end devices to
collaboratively train effective learning models, which is an emerging
distributed learning architecture. However, it is challenging in learning-
oriented training configuration, and energy efficient training strategies.
## IV Case Study: Collaborative Service Placement for AGC-MEC
In this section, we conduct a case study of collaborative service placement
for AGC-MEC to validate the effectiveness of the proposed collaborative
service placement strategy. Firstly, we introduce the system model and problem
formulation. Secondly, we describe the simulation settings and comparison
algorithms. Finally, we show the simulation results.
### IV-A System Model and Problem Formulation
#### IV-A1 Network Model
We consider an air-ground collaborative MEC network, which is composed of one
UAV, one BS, and multiple UEs. The UAV and BS can collaborative with each
other to provide various types of computing services for UEs. The computation-
intensive tasks of UEs can be operated locally or be offloaded to the UAV or
BS.
#### IV-A2 Latency Model
The latency is composed of two parts: computation time taken to execute tasks,
communication time taken to offload tasks. The time for transmitting
computation results is usually ignored. The computation time is calculated by
the required number of CPU frequency cycles and allocated computing resources.
The communication time is calculated by the input data size and transmission
rate.
#### IV-A3 Energy Model
The energy consumption of the UE includes computation and communication energy
consumption. The computation energy is related to the effective switched
capacitance, local computing power and required number of CPU frequency
cycles. The communication energy is related to the communication time and
transmission power of the UE.
(a) Convergence result
(b) UAV trajectory
Figure 2: Convergence and optimized UAV trajectory.
#### IV-A4 Problem Formulation
We aim to minimize UEs’ overall energy consumption by jointly optimizing
service placement, task offloading, UAV trajectory and computation resource
allocation. The optimization problem can be formulated as:
* •
Optimization objective: the minimization of UEs’ overall energy consumption.
* •
Optimization variables: service placement, task offloading, UAV trajectory and
computation resource allocation.
* •
Constraint 1: storage capacities of the BS and UAV, which store corresponding
data including object databases, trained machine learning models, and
libraries, associated with services.
* •
Constraint 2: the restriction of maximum number of UEs associated with the BS
and UAV.
* •
Constraint 3: the coverage area of the UAV.
* •
Constraint 4: the limitation of flight distance.
* •
Constraint 5: offloading condition, _i.e.,_ the required service need to be
placed in the BS or UAV.
(a) Overall energy consumption _v.s._ UAV’s storage capacity
(b) Overall energy consumption _v.s._ UEs’ workload
Figure 3: Effects of UAV’s storage capacity and UEs’ workload.
The optimization problem is a MINLP problem, which includes continuous
computation resource allocation and UAV trajectory variables and integral task
offloading and service placement variables. To deal with this problem, we
exploit alternating optimization techniques to propose a suboptimal solution
with convergence guarantee. To be specific, we obtain the closed form of the
optimal computation resource allocation and iteratively solve the task
offloading and service placement subproblem by Branch and Bound (BnB), and UAV
trajectory subproblem by successive convex approximation (SCA).
### IV-B Simulation Settings and Comparison Algorithms
This case considers an air-ground collaborative MEC network with a mobile UAV,
a static BS and ten UEs in a ${\rm{200m}}\times{\rm{200m}}$ squared area. The
UAV and BS are equipped with MEC servers and UEs are randomly distributed in
the squared zone. We divide the whole mission period into 100 time slots, and
the length of time slot is $1s$. Each UE generates a computation task
constantly in each time slot. The number of required CPU cycles is in the
range of ${\rm{[1}}{{\rm{0}}^{\rm{8}}},{10^{9}}]$ cycles and the size of input
data is in the range of ${\rm{[100}},1000]$KB. We assume that the service
population of UEs’ requests are produced according to the popular Zipf
distribution with a skewness parameter value of $0.5$ form a service set with
size $30$ [11]. The storage size required by each services is uniformly chosen
from $[0.5,1]$. We assume that the storage capacity of the BS is twice that of
the UAV. The maximum transmission power of UEs equals to $0.1w$ and the
bandwidth is $1MHz$. For computation energy consumption model, we assume the
effective switched capacitance is ${10^{27}}$. UAV related parameters are set
as follows: the maximum ground coverage radius is $100m$, maximum flying
distance in a time slot is $30m$, maximum computation capacity is
${\rm{[5}},10]GHz$, and the maximum number of UEs associated with UAV is
${\rm{[3}},5]$.
To verify the performance of the proposed algorithm, we compare it with the
following three algorithms:
* •
Random: according to storage capacities of the BS or UAV, this algorithm
places services randomly;
* •
Greedy: this algorithm uses a greedy strategy to select the service with the
minimum required storage, which results in maximizing the number of services
placed at the BS or UAV;
* •
Local: this algorithm executes tasks locally, which obtains an upper bound of
UEs’ overall energy consumption.
### IV-C Simulation Results
Fig. 2 (a) shows the convergence performance of the proposed algorithm. It can
be seen that after three iterations, UEs’ overall energy consumption decreases
from 150.5956 J to 31.8719 J, which means the proposed algorithm converges
quickly. Fig. 2 (b) demonstrates the trajectory of the UAV optimized by the
proposed algorithm, where the initial and final horizontal positions of the
UAV are (0,0) and (200,200), respectively. As shown in Fig. 2 (b), the
proposed algorithm enables the UAV to fly close to UEs. This is because that
the UAV can provide better computing service in the close proximity of UEs.
Fig. 3 (a) shows the trend of UEs’ overall energy consumption under different
UAV’s storage capacity. Meanwhile, the storage capacity of the BS is also
changing, which is twice that of the UAV. With the increase of storage
capacity, UEs’ overall energy consumption by all algorithms except Local has
been decreased. This is because, the UAV and BS can store more MEC services
required by UEs with larger storage capacity. Compared with Random, Greedy and
Local, UEs’ overall energy consumption of the proposed algorithm can be
reduced by 65.96%, 49.35%, 87.89%, respectively. Fig. 3 (b) demonstrates the
trend of UEs’ overall energy consumption under different UEs’ workload
(_i.e.,_ the number of required CPU cycles). In this simulation, we use the
workload increasing coefficient to represent the change of workload. As shown
in Fig. 3 (b), compared with Random, Greedy and Local, UEs’ overall energy
consumption of the proposed algorithm can be reduced by 46.62%, 48.42%,
75.53%, respectively.
## V Potential Research Directions of AGC-MEC
In despite of its great potential, the study on the AGC-MEC architecture is
still in its infancy, where many critical problems should be solved. In this
section, we introduce three potential research directions that can help
translate the visions of the AGC-MEC into reality.
### V-A Network Control
As previously mentioned, the AGC-MEC network involves different MEC paradigms
belonging to different segments distributed in a wide area and thus needs to
be well managed. As is known to all, software defined networking (SDN)
separating the data plane and control plane introduces a unified control plane
interface and global view of the whole network. SDN is able to provide
centralized network control and flexible resource management of the
collaboration among different MEC paradigms. Still, the SDN-based AGC-MEC is
faced with many critical issues. For example, the placement of SDN controllers
is a key to the SDN-based AGC-MEC, which should take many factors into
consideration such as energy efficiency, load balancing, latency, scalability,
_etc_ [12].
### V-B Security and Privacy
Since MEC servers in the AGC-MEC are located at the network edge and have
frequent interactions, they lack effective backup and recovery measures of
their data, which is prone to be attacked or misused by malicious users.
Furthermore, compared with the cloud computing data center in the core
network, MEC can collect more high-value sensitive information of UEs,
including location information, lifestyle, social relations, even health
status, _etc._ Therefore, the security and privacy protection is critical to
the AGC-MEC architecture. Fortunately, Blockchain is a burgeoning distributed
ledger technology, which has the potential to ensure the data and resource
exchange among untrusted MEC nodes being safe in the AGC-MEC.
### V-C Intelligent Collaboration
For a long-lasting sophisticated mission, it may require multiple MEC servers
to provide services together. To be specific, some MEC servers may act as
controllers to determine offloading decisions. According to the decisions,
some MEC servers may be used as relays to offload task to servers with strong
computing power and rich resources. And then these MEC servers may cooperate
to provide computing services. Therefore, it is valuable to study how many MEC
servers are needed to complete task and how to intelligently collaborate.
## VI Conclusion
In this article, we have proposed the AGC-MEC architecture to explore the
complementary integration of all potentially available MEC servers within air
and ground by various collaborative ways to provide high-quality intelligent
services for future 6G network. We have described the AGC-MEC architecture and
three typical use cases. The challenging issues and their potential solutions
have also been discussed. Furthermore, we have conducted a case study of
collaborative service placement for AGC-MEC, and presented several potential
research directions for future study.
## References
* [1] M. Z. Chowdhury, M. Shahjalal, S. Ahmed, and Y. M. Jang, “6g wireless communication systems: Applications, requirements, technologies, challenges, and research directions,” _IEEE Open Journal of the Communications Society_ , vol. 1, pp. 957–975, 2020.
* [2] H. Yang, A. Alphones, Z. Xiong, D. Niyato, J. Zhao, and K. Wu, “Artificial-intelligence-enabled intelligent 6g networks,” _IEEE Network_ , vol. 34, no. 6, pp. 272–280, 2020.
* [3] Y. Mao, C. You, J. Zhang, K. Huang, and K. B. Letaief, “A survey on mobile edge computing: The communication perspective,” _IEEE Communications Surveys Tutorials_ , vol. 19, no. 4, pp. 2322–2358, 2017.
* [4] H. N. Mahmood, H. Alves, L. A. O. Loacute;pez, M. Shehab, P. M. D. Osorio, and M. Latva-aho, “Six key enablers for machine type communication in 6g,” _arXiv: Networking and Internet Architecture_ , 2019.
* [5] F. Zhou, R. Q. Hu, Z. Li, and Y. Wang, “Mobile edge computing in unmanned aerial vehicle networks,” _IEEE Wireless Communications_ , vol. 27, no. 1, pp. 140–146, 2020.
* [6] N. Cheng, W. Xu, W. Shi, Y. Zhou, N. Lu, H. Zhou, and X. Shen, “Air-ground integrated mobile edge networks: Architecture, challenges, and opportunities,” _IEEE Communications Magazine_ , vol. 56, no. 8, pp. 26–32, 2018.
* [7] Z. Zhou, J. Feng, L. Tan, Y. He, and J. Gong, “An air-ground integration approach for mobile edge computing in iot,” _IEEE Communications Magazine_ , vol. 56, no. 8, pp. 40–47, 2018.
* [8] F. Jiang, K. Wang, L. Dong, C. Pan, W. Xu, and K. Yang, “Ai driven heterogeneous mec system with uav assistance for dynamic environment: Challenges and solutions,” _IEEE Network_ , pp. 1–9, 2020.
* [9] G. Li, G. Xu, A. K. Sangaiah, J. Wu, and J. Li, “Edgelaas: Edge learning as a service for knowledge-centric connected healthcare,” _IEEE Network_ , vol. 33, no. 6, pp. 37–43, 2019.
* [10] S. Deng, H. Zhao, W. Fang, J. Yin, S. Dustdar, and A. Y. Zomaya, “Edge intelligence: The confluence of edge computing and artificial intelligence,” _IEEE Internet of Things Journal_ , vol. 7, no. 8, pp. 7457–7469, 2020.
* [11] V. Farhadi, F. Mehmeti, T. He, T. L. Porta, H. Khamfroush, S. Wang, and K. S. Chan, “Service placement and request scheduling for data-intensive applications in edge clouds,” in _IEEE INFOCOM 2019 - IEEE Conference on Computer Communications_ , 2019, pp. 1279–1287.
* [12] T. Das, V. Sridharan, and M. Gurusamy, “A survey on controller placement in sdn,” _IEEE Communications Surveys Tutorials_ , vol. 22, no. 1, pp. 472–503, 2020.
|
# MIT SafePaths Card (MiSaCa): Augmenting Paper Based Vaccination Cards with
Printed Codes
Joseph Bae1, 3, Rohan Sukumaran1, Sheshank Shankar1, Saurish Srivastava1,
Rohan Iyer1,
Aryan Mahindra1, Qamil Mirza1, Maurizio Arseni1, Anshuman Sharma1, Saras
Agrawal1,
Orna Mukhopadhyay1, Colin Kang1, Priyanshi Katiyar1, Apurv Shekhar1, Sifat
Hasan1,
Krishnendu Dasgupta1, Darshan Gandhi1, Sethuraman TV1, Parth Patwa1, Ishaan
Singh1,
Abhishek Singh2, Ramesh Raskar1,2
1PathCheck Foundation, 02139 Cambridge, USA.
2MIT Media Lab, 02139 Cambridge, USA.
3Renaissance School of Medicine, Stony Brook University, 11794 Stony Brook,
USA.
<EMAIL_ADDRESS>
###### Abstract
In this early draft, we describe a user-centric, card-based system for vaccine
distribution. Our system makes use of digitally signed QR codes and their use
for phased vaccine distribution, vaccine administration/record-keeping,
immunization verification, and follow-up symptom reporting. Furthermore, we
propose and describe a complementary scanner app system to be used by
vaccination clinics, public health officials, and immunization verification
parties to effectively utilize card-based framework. We believe that the
proposed system provides a privacy-preserving and efficient framework for
vaccine distribution in both developed and developing regions.
## 1 Introduction
Without an effective curative or preventative measure, the unprecedented
coronavirus disease 2019 (COVID-19) pandemic has led to a significant amount
of human deaths (1,900,000 at the time of publication (for Disease Control &
Prevention (2020c))). However, now with the advent of vaccines, we face the
challenges of strategic, equitable and privacy preserved ways for last-mile
vaccine distribution (Bae et al. (2020); MPH (2020)).
First, the vaccine recipients must be dynamically prioritized to ensure an
equitable reach, especially as multiple vaccines with different protocols are
approved in various areas. In addition, once a citizen’s first dose is
administered, they must follow through with their second dose as well. Also, a
communication plan must also be put in place to combat inevitable rumours,
misinformation, and conspiracy theories aiming to disrupt citizen engagement
in the vaccination process ( Morales et al. (2021); Omer (2021)). It must also
address the mistrust of vaccines in society (Palamenghi et al. (2020)),
especially within previously marginalized minority populations (Toner et al.
(2020)). This is why we must take a user-centric approach that preserves trust
— vaccines are meaningless if citizens aren’t willing to take them (M et al.
(2020)). Lastly, the health outcomes (effectiveness, safety, long-term
effects, etc) of the vaccines must be effectively monitored in a privacy-
preserving way (Borenstein & Weintraub (2020)).
In today’s society, multiple technological systems are being utilized by the
Center for Disease Control (CDC) to combat these challenges ( for Disease
Control & Prevention (2020a); Smith & Mennis (2020); for Disease Control &
Prevention (2020b)). For example, the Vaccine Administration Management System
(VAMS) streamlines the vaccine distribution process for jurisdictions,
employers, and healthcare providers. In addition, it’s an effective user-
centric system as it allows for vaccine recipients to schedule appointments,
receive records of their visit, and receive reminders for a second dose (for
Disease Control (CDC)). The Immunization Information Systems (IIS) are a group
of privacy preserving database systems that track all vaccinations within
various areas (for Disease Control (CDC)). Lastly, the Vaccine Adverse Event
Reporting System (VAERS) is the prominent system for the monitoring of health
outcomes ( for Disease Control (CDC); ADMINISTRATION (2019)).
In our previous work, we detail the MIT SafePaths app-based protocol for
vaccine distribution. In this paper, we introduce a separate user-centric card
protocol that uses printed codes as a supplement to traditional paper based
vaccination cards.
## 2 Card Flow
### 2.1 Overview
Here we present a vaccine distribution system utilizing physical SafePaths
cards and four digitally signed QR code stickers (henceforth termed Coupon,
Badge, Passkey, and Status). The digital signing of a QR code is simply a
secure process of verifying the authenticity of the information contained in
the QR code (Singh et al. (2020)). These QR code stickers are simply QR codes
printed onto adhesive stickers that can then be attached to a user’s physical
card.
Figure 1: The 4 digitally signed QR code stickers (Coupon, Badge, Status, and
Passkey) present on the SafePaths cards. Figure 2: MIT SafePaths Card Mockup
The digital signature of the QR codes take place as below Certificate =
(message, signature(messages))
For each sticker below the message is as follows -
* •
Coupon = (number, total, city, phase, (age, job, comorbidities/sick))
* •
Badge = (coupon, dose_info, Hash(passkey())
* •
Status = ((vaccinated = 0,1,2), Hash(passkey())
* •
Passkey = (name, DOB, salt)
= hash:sj2d8k8hy7j
Name | Equation | Description | Example
---|---|---|---
Coupon | | {m, sign(m)}
---
where m = (i, zip code,
job type)
| Coupon code is signed
---
by CDC and indicates the
zip code and job type of the
receiver
| {37, 5000, Springfield, 1B,
---
Teacher}
Badge | | {m, sign(m)}
---
where m = (dose_info,
coupon, hash(passkey))
| Badge is available after 2
---
doses and it gives the
information pertinent to
the vaccine shot.
| {[Pfizer, “1st Dose”,
---
1/1/2021], fe4c2,
3be33c20cc4c85a0c32f7bf5b4}
Status | | {m, sign(m)}
---
where m = (status,
hash(passkey))
| Contains bare minimum
---
information to prove the
user is vaccinated
| {vaccinated,
---
3be33c20cc4c85a0c32f7bf5b4}
Passkey | | ID = User_PII
---
Key = salt
| Key is the random salt
---
used for increasing the
entropy of hashed data
| {John Doe,
---
6363fe744f74ee8f280958}
Table 1: This outlines the four QR codes and what information is digitally
signed in it.
Our solution is intended to decouple the health information and personally
identifiable information (PII) in this process. Thereby, we are essentially
proposing to separate the eligibility of the vaccination from the distribution
of it. This way we can have the health information centralised, whilst the PII
information decentralised.
### 2.2 Vaccine eligibility confirmation
To accommodate the several-stage vaccination policies that countries have
begun to employ, SafePaths cards will be distributed containing one digitally-
signed Coupon QR code. This would be provided by a central government agency
such as the CDC and made available to users either by an employer or local
government location. A pseudo random identifier generated for this Coupon
serves as the identifying information for the user throughout the remaining
workflow. This Coupon would initially come with SafePaths cards while the
remaining three adhesive stickers must be obtained and placed onto the card
following vaccination events.
### 2.3 Vaccine administration
Check-in at a vaccination clinic would require the verification of a user’s
Coupon.
Upon vaccination, the vaccination clinic would create a digitally-signed
record of immunization and print it as a QR code on an adhesive sticker. This
adhesive sticker (henceforth referred to as the Badge) would contain
information regarding vaccine lot, manufacturer, and first/second dose
information. The Badge would also contain information regarding the time,
date, and location of vaccination.
The vaccination clinic would also create a unique encryption key to encrypt
the Badge. This key, as well as encrypted PII such as name, age, sex, etc.
would be stored on a Passkey QR code, printed onto a Passkey QR sticker. This
Passkey is required for decryption of PII and in-depth vaccination information
(time, date, location of vaccination).
At this stage, a vaccine recipient would then have Coupon, Badge, and Passkey
QR stickers.
### 2.4 Second Dose
When a user attempts to receive a second dose of a vaccine, the vaccination
clinic would utilize a user’s Badge to determine the appropriate vaccine type
and dose and the Passkey to confirm a user’s identity. Again, the user Passkey
contains information that solely exists on the physical card carried by a
user. Use of this sticker is required to decrypt in depth vaccination
information for a patient contained in the Badge (location of vaccination,
date, etc.). Once final vaccination has been performed, the vaccine clinic
would create a fourth and final Status QR code sticker for a recipient’s
SafePaths card, which would simply indicate whether or not a user has been
vaccinated. Status would not contain any further information and therefore
would be unencrypted.
### 2.5 Record-keeping
User vaccination records could be linked by anonymized upload to a centralized
system using a user’s pseudorandom identifier. The user’s Passkey, containing
their encryption key that decrypts their PII, would not be uploaded to the CDC
without consent. Alternatively, we propose an anonymous record keeping
function in our Scanner App section.
### 2.6 Vaccination verification
Verification of immunization status might be required in various scenarios
such as airline travel, return to school/work, etc. Vaccine verification at
these venues would follow the receipt of a second COVID-19 dose.
Information regarding an individual’s vaccination status would be digitally
signed by the vaccine clinic onto the Status sticker. When scanned, this
sticker would provide the verifier with information regarding whether or not
an individual has been vaccinated. If further verification of identity is
required, the verifier could make use of a consenting individual’s Passkey
sticker to decrypt the holder’s name. With this method, a user would have
multiple levels of information they can share, beginning with vaccination
status in the unencrypted Status sticker, basic personal information (i.e.
name) that must be decrypted using the Passkey sticker, and finally full
personal vaccination information encrypted in the Badge.
### 2.7 Safety and efficacy monitoring
Short and long-term monitoring of health outcomes would rely on self-
reporting. These cards could still facilitate the anonymous information upload
by interacting with existing centralized systems such as VAERS or V-Safe while
bypassing PII input. All health and symptom information could instead be tied
to a user’s pseudorandom ID. We also propose a scanner app solution in the
Scanner Flow section that could aggregate symptom reporting and vaccine record
data anonymously.
Figure 3: Card protocol workflow diagram.
## 3 Scanner Flow
### 3.1 Overview
Here we discuss the systems that must be built for vaccine clinics and
distributors in order to enable the use of the SafePaths card framework
presented above. We present several relevant protocols as well as the
functionality of a proposed vaccine distributor/verifier scanner app. This
scanner app would be necessary to function with the encrypted QR codes
described above.
### 3.2 Vaccine eligibility confirmation
Phased vaccination using the SafePaths card system requires the distribution
of SafePaths cards containing digitally signed Coupons to appropriate subsets
of the population during each stage of vaccination. There are several ways
that this might be achieved. We propose potential solutions below, though we
recognize that these strategies must be determined by individual jurisdictions
to meet the circumstances in different locations.
1. 1.
Disseminate to businesses to provide to employees (eg: hospitals, restaurants,
etc. as appropriate)
2. 2.
Make available at local government building (similar to DMV process of
obtaining a driver’s license)
3. 3.
Mail out to individuals based on employment/other factors (via background
check systems, centralized databases such as IRS)
Figure 4: Scanner app protocol workflow diagram.
### 3.3 Vaccine administration
To confirm an individual for vaccination scheduling/check-in, a clinic must
verify the authenticity of a vaccine recipient’s QR Coupons. The first
function of our proposed scanner app would be to scan a vaccine recipient’s
Coupon to determine authenticity and prevent the use of a single Coupon by
multiple individuals. This would be achieved by scanning the digital signature
present on a SafePaths Coupon and verifying its digital signature.
The second function of our proposed scanner app would be to create digitally
signed Badge and Passkey stickers for post vaccination. This would make use of
our previously described algorithm (Singh et al. (2020)) for secure recording
of vaccine information into a Badge sticker, encrypted using the encryption
key present in the Passkey. After creating these stickers, the proposed
scanner app would not store any information regarding a recipient’s encryption
key; that information would only exist within the Passkey sticker.
### 3.4 Second dose
Second dose administration functionality would be implemented into the scanner
app in the same manner as described in the previous section for ‘Vaccine
Administration’. A Status sticker would be created by the scanner app in a
similar manner to the Badge sticker, also drawing on the methods described in
our cryptographic protocol (Singh et al. (2020)).
### 3.5 Record-keeping
Another critical function of our scanner app would be the ability to integrate
with existing systems, such as VAMS in the United States. Ideally, our app
would be able to automatically provide vaccination record information to VAMS
while replacing PII with pseudo identifiers.
Alternatively, our scanner system would also have the capability to directly
aggregate vaccination record data in an anonymized fashion, retaining
population-level statistics such as vaccination prevalence in a given
jurisdiction that might be important for public health policy development.
Details concerning clinic location, vaccine dose, and vaccine manufacturer
could be stored by the scanner app and aggregated for public health official
viewing.
### 3.6 Vaccination verification
Our proposed scanner app would enable vaccination verification simply by
reading immunization status contained in a user’s Status sticker. For further
identity verification, a form of ID (such as driver’s license) can be compared
with the decrypted PII from the scanner app using an individual’s Passkey
sticker. The scanner app would not store this information following completion
of the immunization confirmation.
## 4 Conclusion
In this early draft, we present a complete protocol for a physical card-based
system for phased vaccine distribution, individual vaccination, second-dose
adherence, and symptom follow-up. Due to their physical nature and simplicity,
digitally-signed QR codes may be a convenient and non-intrusive modality for
some users seeking vaccination. Digitally-signed QR stickers enable
verification of authentically created immunization records, and the encryption
schema presented using a unique passkey sticker ensures that user PII can only
be decrypted with the user’s consent. This information is stored physically on
the user’s SafePaths card in a decentralized manner wherein a user must
provide their physical passkey sticker for decryption of PII. These cards also
extend privacy-focused protocols to low-resource areas and populations,
equalizing disparities in access to individual-centric solutions and
frameworks for COVID-19 vaccination. The centralised health data collected
(which is rid of all PIIs) can be used by the concerned authorities to have
population aggregated view of the vaccine adherence in a region. Furthermore,
such privacy preserving dashboards which show aggregated data can help the
authorities take informed decisions.
#### Acknowledgments
We are grateful to Riyanka Roy Choudhury, CodeX Fellow, Stanford University,
Adam Berrey, CEO of PathCheck Foundation, Dr. Brooke Struck, Research Director
at The Decision Lab, Canada, Vinay Gidwaney, Entrepreneur and Advisor,
PathCheck Foundation, and Paola Heudebert, co-founder of Blockchain for Human
Rights, Alison Tinker, Saswati Soumya, Sunny Manduva, Bhavya Pandey, and
Aarathi Prasad for their assistance in discussions, support and guidance in
writing of this paper.
## References
* ADMINISTRATION (2019) U.S. FOOD & DRUG ADMINISTRATION. Vaers overview, 2019. URL https://www.fda.gov/vaccines-blood-biologics/vaccine-adverse-events/vaers-overview.
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|
# A restless supermassive black hole in the galaxy J0437+2456
Dominic W. Pesce Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden
Street, Cambridge, MA 02138, USA Black Hole Initiative at Harvard University,
20 Garden Street, Cambridge, MA 02138, USA Anil C. Seth Department of Physics
and Astronomy, University of Utah, 115 South 1400 East, Salt Lake City, UT
84112, USA Jenny E. Greene Department of Astrophysics, Princeton University,
Princeton, NJ, USA James A. Braatz National Radio Astronomy Observatory, 520
Edgemont Road, Charlottesville, VA 22903, USA James J. Condon National Radio
Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA
Brian R. Kent National Radio Astronomy Observatory, 520 Edgemont Road,
Charlottesville, VA 22903, USA Davor Krajnović Leibniz-Institut für
Astrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany
###### Abstract
We present the results from an observing campaign to confirm the peculiar
motion of the supermassive black hole (SMBH) in J0437+2456 first reported in
Pesce et al. (2018). Deep observations with the Arecibo Observatory have
yielded a detection of neutral hydrogen (Hi) emission, from which we measure a
recession velocity of 4910 km s-1 for the galaxy as a whole. We have also
obtained near-infrared integral field spectroscopic observations of the
galactic nucleus with the Gemini North telescope, yielding spatially resolved
stellar and gas kinematics with a central velocity at the innermost radii
($0.1^{\prime\prime}\approx 34$ pc) of 4860 km s-1. Both measurements differ
significantly from the $\sim$4810 km s-1 H2O megamaser velocity of the SMBH,
supporting the prior indications of a velocity offset between the SMBH and its
host galaxy. However, the two measurements also differ significantly from one
another, and the galaxy as a whole exhibits a complex velocity structure that
implies the system has recently been dynamically disturbed. These results make
it clear that the SMBH is not at rest with respect to the systemic velocity of
the galaxy, though the specific nature of the mobile SMBH – i.e., whether it
traces an ongoing galaxy merger, a binary black hole system, or a
gravitational wave recoil event – remains unclear.
††facilities: Arecibo Observatory, Gemini North††software: AOIDL, dynesty
(Speagle, 2020), Montage555http://montage.ipac.caltech.edu, Gemini IRAF,
Kinemetry (Krajnović et al., 2006), pPXF (Cappellari & Emsellem, 2004)
## 1 Introduction
Given that nearly all galaxies are thought to harbor central supermassive
black holes (SMBHs; Magorrian et al., 1998), interactions between SMBHs have
long been recognized as a natural and perhaps inevitable byproduct of galaxy
mergers. The two primary dynamical states that result from such interactions
are SMBH binaries (Begelman et al., 1980; Roos, 1981) and gravitational recoil
events (Fitchett, 1983; Redmount & Rees, 1989), both of which predict
substantial nonequilibrium (“peculiar”) motion of the SMBH with respect to its
surrounding environment. Yet despite much theoretical attention and
observational effort, direct dynamical evidence for SMBH peculiar motion has
remained elusive (see, e.g., Eracleous et al., 2012; Popović, 2012; Komossa &
Zensus, 2016; Barack et al., 2019). In the absence of recent interactions with
comparable-mass objects, an SMBH is expected to be in kinetic equilibrium with
its surrounding environment (Merritt et al., 2007); for most SMBHs, the
equilibrium velocity is $\ll 1$ km s-1 with respect to the system barycenter.
Pesce et al. (2018, hereafter P18) presented a technique for using H2O
megamasers to measure SMBH peculiar motions. The key idea is that masers
residing in the accretion disks around SMBHs (on scales of $\sim$0.1 pc) act
as test particles whose dynamics can be used to probe the gravitational
potential around the black hole, and very long baseline interferometric (VLBI)
maps of the maser distribution enable precise (uncertainty ${\lesssim}10$ km
s-1) measurements of the SMBH’s line-of-sight velocity (e.g., Miyoshi et al.,
1995; Kuo et al., 2011; Gao et al., 2017). P18 compared the maser-derived SMBH
velocity measurements for 10 systems with independent estimates of their host
galaxy velocities to constrain relative motions. One galaxy from the P18
sample – SDSS J043703.67+245606.8, hereafter J0437+2456 – showed a
statistically significant (${>}5{\sigma}$) difference between the SMBH and
host galaxy line-of-sight velocities; P18 thus identified J0437+2456 as a
promising candidate for hosting either a recoiling or binary SMBH.
J0437+2456 is an approximately Sb-type spiral galaxy located at a distance of
${\sim}70$ Mpc (Greene et al., 2016; Pjanka et al., 2017). As measured by the
Sloan Digital Sky Survey (SDSS)111Here we quote the quantities compiled in the
NASA-Sloan Atlas, http://nsatlas.org/., J0437+2456 has an $r$-band absolute AB
magnitude of $M_{r}=-21.37$ and an estimated stellar mass of $7.2\times
10^{10}$ M⊙. The megamaser system in J0437+2456 was mapped by Gao et al.
(2017, hereafter G17), who also modeled the maser rotation curve and
determined an SMBH velocity of $4818\pm 10.5$ km s-1. P18 used a SDSS spectrum
to measure the recession velocity of J0437+2456 to be $4887.6\pm 7.1$ km s-1.
The apparent $69.6\pm 12.7$ km s-1 blueshift of the SMBH with respect to its
host galaxy constitutes the putative peculiar motion. However, given the
strong prior expectation for zero peculiar motion and the possibility that
systematic effects such as SDSS fiber misalignment could plausibly account for
a large fraction of the observed velocity difference, P18 cautioned that the
peculiar motion measurement should be regarded as tentative pending
corroborating observations.
In this paper we present the results from a followup observing campaign to
confirm the peculiar motion of the SMBH in J0437+2456. This paper is organized
as follows. In Section 2 we describe our observations and subsequent data
reduction procedures, and in Section 3 we detail the velocity measurements
made using these data. We discuss the results in Section 4, and we summarize
and conclude in Section 5. Unless noted otherwise, all velocities quoted in
this paper use the optical convention in the barycentric reference frame, and
we assume a distance to J0437+2456 of 70 Mpc.
## 2 Observations and data reduction
In quiescent systems, Hi provides an appealing recession velocity tracer
because it follows the global dynamics of the galaxy well outside of the SMBH
sphere of influence and does not suffer from reddening or extinction. P18
targeted J0437+2456 with the Very Large Array (VLA) to observe neutral
hydrogen (Hi), but no emission was detected within the six-hour integration
time. We have obtained followup Hi observations of J0437+2456 using the
Arecibo Observatory, which is much more sensitive than the VLA to low surface
brightness emission but which lacks the ability to spatially resolve the gas
distribution (see Figure 1). Our Arecibo observations are presented in Section
2.1.
Lacking Hi data, P18 measured the recession velocity for J0437+2456 using an
SDSS spectrum. Like the Arecibo spectrum, the kinematics contributing to the
SDSS spectrum are spatially unresolved within the 3-arcsecond aperture of the
optical fiber used to transport light from the focal plane to the spectrograph
(Gunn et al., 2006). However, because the aperture is smaller than the region
containing the emitting material, the SDSS measurement is subject to an
unknown amount of systematic uncertainty associated with the relative
placement of the fiber center and the galactic nucleus. We have thus obtained
followup high-resolution integral field spectra taken using the Gemini North
telescope, which are able to spatially resolve the nuclear kinematics.
Additionally, dust absorption should be weaker in the NIFS near-infrared
waveband than at the SDSS optical wavelengths, so any velocity errors caused
by patchy dust absorption will be smaller. Our Gemini observations are
presented in Section 2.2.
Figure 1: False-color image of J0437+2456 made by combining the $i$-, $r$-,
and $g$-band observations from the SDSS Legacy Survey (York et al., 2000),
with the 2.4-arcminute Arecibo beam and the $3^{\prime\prime}\times
3^{\prime\prime}$ NIFS field of view overplotted in red and blue,
respectively.
### 2.1 Arecibo data
We performed Hi spectral-line observations of J0437+2456 over 6 nights using
the Arecibo Observatory L-wide receiver. The observations were position-
switched, with 5 minutes on and 5 minutes off source at matched elevation. We
used the Wideband Arecibo Pulsar Processor (WAPP) spectrometer backend in
single-polarization, 9-level autocorrelation mode with 4096 channels spanning
the bandwidth 1384.5–-1409.5 MHz (i.e., $\pm$2500 km s-1 centered on the Hi
line). We used two such boards, one per polarization. Calibration diodes were
observed at the end of every scan to determine the flux density scale.
Table 1 lists the on-source integration times for each of the 6 nights. With a
declination of $+25$ degrees, J0437+2456 passes through the Arecibo observing
window for ${\sim}2$ hours at a time. The position-switched observations thus
yielded roughly an hour of on-source time per night.
We reduced the Arecibo data using AO
IDL222http://outreach.naic.edu/ao/scientist-user-portal/astronomy/IDL-
Routines/Download-AO-IDL. We first converted the flux scale from K to Jy using
the regular gain curve monitoring
scans333http://www.naic.edu/p̃hil/sysperf/sysperfbymon.html performed by the
observatory (see Table 1). Each spectral scan was Hanning smoothed to mitigate
ringing, and a fourth-order polynomial fit to the emission-free regions of the
spectrum was subtracted off to remove low-frequency baseline ripples. We then
combined each scan and both polarizations using an RMS-weighted average.
Figure 2 shows the Arecibo spectrum, in which we strongly detect Hi emission
around the expected recession velocity range. The spectrum peaks at
${\sim}0.7$ mJy and has an integrated flux of ${\sim}$0.12 Jy km s-1,
consistent with the non-detection reported in P18. Assuming the Hi is
optically thin, the total Hi mass is given by (Haynes et al., 2011; Condon &
Ransom, 2016)
$M_{\text{H{i}}}=\left(2.356\times 10^{5}\text{
M}_{\odot}\right)\left(\frac{D}{\text{Mpc}}\right)^{2}\left(\frac{\int
S_{\nu}(v)dv}{\text{Jy\,km\,s${}^{-1}$}}\right),$ (1)
where $D$ is the distance to the galaxy and $S_{\nu}(v)$ is the flux density
as a function of velocity $v$. For a distance of 70 Mpc to J0437+2456, we
estimate $M_{\text{H{i}}}\approx 1.4\times 10^{8}$ M⊙ (see also Section 3).
Table 1: Arecibo observation details
| Integration | $\boldsymbol{T}_{\textbf{sys}}$ | Gain
---|---|---|---
Date | (min.) | (K) | (K Jy-1)
2019 Jan 22 | 60 | 26.6 | 6.9
2019 Jan 23 | 65 | 26.8 | 6.9
2019 Jan 24 | 65 | 26.9 | 6.8
2019 Feb 11 | 60 | 27.1 | 7.0
2019 Feb 14 | 50 | 27.4 | 6.9
2019 Feb 17 | 50 | 27.0 | 6.9
Note. — Observing dates, on-source integration times, system temperatures, and
gains for the Arecibo observations.
Figure 2: 1.4 GHz Arecibo spectrum towards J0437+2456. The spectrum is plotted
with its native spectral resolution in gray, and the spectrum after smoothing
by a 4-channel boxcar is shown in black. 1000 random posterior samples from a
3-component Gaussian model fit are overplotted in red, with the individual
components for the best-fit model shown in blue. The measured $V_{20}$
velocity (see Section 3.1) is marked by a vertical green line, and the peak-
to-peak velocity range is marked by a horizontal green line.
### 2.2 Gemini data
We obtained integral field spectra of a $3^{\prime\prime}\times
3^{\prime\prime}$ region centered on the nucleus of J0437+2456 using the
Gemini North Near-Infrared Integral Field Spectrometer (NIFS) on 2018 November
21 in natural seeing mode. The spectrometer grating was set for K-band, with a
central wavelength of 2.18 $\mu$m and spanning the range 1.99–2.41 $\mu$m.
Nine 500-second exposures were taken, with five dithered exposures on-source
and four offset to a blank sky location for subtraction. The observations were
performed at airmasses of 1.2–1.6 and seeing conditions corresponding to a
zenith-corrected point spread function of $\sim$0.3 arcseconds FWHM in
$K$-band.
The NIFS data were reduced using the Gemini version 1.13 IRAF packages, with
slight modifications to enable error array propagation and cube combination as
described in Ahn et al. (2018). The resulting final data cube has a central
signal-to-noise of $\sim$28, dropping to $\sim$10 at 0$\farcs$5 radius. Both
strong stellar aborption lines and excited H2 emission lines are seen, and
their velocities are described in more detail in Section 3.2.
Figure 3: Velocity maps derived from the Gemini NIFS data within the central
$2^{\prime\prime}\times 2^{\prime\prime}$ region. Left: Velocity map for the
stellar component, using Voronoi binning such that each bin has a signal-to-
noise ratio of at least 25. Right: H2 velocity map. In both panels, mean
continuum contours are overplotted at 5%, 10%, 20%, 40%, and 80% of the peak
value.
## 3 Analysis
In this section we describe the analysis procedures used to measure velocities
from the Arecibo spectrum (Section 3.1) and the Gemini spectra (Section 3.2).
### 3.1 Neutral hydrogen spectral decomposition and velocity measurements
Instead of the classic symmetric “double-horn” Hi profile (Roberts, 1978),
J0437+2456 shows a more unusual triple-peaked and asymmetric spectral
structure. Because our observations do not spatially resolve the Hi
kinematics, the association of individual spectral properties with distinct
dynamical components is ambiguous. While it is clear that a single double-horn
component cannot describe the observed spectral profile, there are a variety
of more complicated models that could potentially do so adequately. In
Appendix A we explore three plausible model extensions, from which we conclude
that the observed spectral structure is most conservatively and satisfactorily
modeled using a sum of Gaussian components. Using the dynesty nested sampling
routine (Speagle, 2020) to explore the posterior distribution, we find that
$N=3$ Gaussian components are sufficient to capture the spectral structure and
achieve a reduced-$\chi^{2}$ of $\sim$1 (see Figure 2); the velocities for
these components are reported in Table 2 along with their statistical
uncertainties. Our modeling procedure is described in more detail in Appendix
A.
We use the modeled Hi spectrum to make a measurement of $V_{20}$, defined to
be to the midpoint between the two points on the profile that rise to 20% of
the peak amplitude (see, e.g., Fouque et al., 1990). $V_{20}$ provides an
estimate of the galaxy recession velocity, and we find $V_{20}=4909.9\pm 1.9$
km s-1. For the associated width of the profile, $W_{20}$, we find
$W_{20}=326.0\pm 4.2$ km s-1, and for the total mass of Hi we find
$M_{\text{H{i}}}=(1.35\pm 0.18)\times 10^{8}$ M⊙.
Table 2: Velocity measurements for different components in J0437+2456
Source of velocity | Velocity (km s-1) | Uncertainty (km s-1) | Spatial scale (pc) | Reference
---|---|---|---|---
Maser rotation curve | 4818.0 | 10.5 | 0.2 | G17
Maser rotation curve re-analysis | 4809.3 | 10.0 | 0.2 | this work
NIFS stellar | 4857–4844 | $\sim$2$\sim$ | 34–340 | this work
NIFS H2 | 4858–4875 | $\sim$2$\sim$ | 34–170 | this work
NIFS integrated light | 4865–4853 | $\sim$2–4$\sim$ | 45–470 | this work
SDSS spectrum, emission lines | 4882.2 | 7.7 | $1.0\times 10^{3}$ | Pesce et al. (2018)
SDSS spectrum, stellar | 4921.4 | 19.1 | Pesce et al. (2018)
SDSS spectrum, average | 4887.6 | 7.1 | Pesce et al. (2018)
Hi spectrum, first component | 4774.6 | 3.2 | $4.9\times 10^{4}$ | this work
Hi spectrum, second component | 4870.2 | 0.8 | this work
Hi spectrum, third component | 4989.4 | 1.9 | this work
Hi spectrum, $V_{20}$ | 4909.9 | 1.9 | this work
Note. — Velocity measurements considered in this paper and the spatial scales
on which they are measured, assuming a distance of 70 Mpc to J0437+2456. For
the NIFS velocities, we quote the range of values corresponding to the
systemic velocities at the innermost and outermost annuli in which
measurements were made; note that for the NIFS stellar measurements, the
systemic velocity measured from the outer annulus is smaller than that
measured from the inner annulus. For the NIFS stellar and H2 velocity
measurements, the uncertainties are dominated by an overall calibration
systematic of $\sim$2 km s-1.
### 3.2 Systemic velocities of the stellar and H2 components
The NIFS $K$-band spectra show both the strong stellar absorption lines of CO
at $\sim$2.3 $\mu$m and molecular hydrogen emission lines, including the
strong H2 1-0 S(1) line at rest wavelength 2.12 $\mu$m.
Stellar kinematics were derived by first Voronoi binning the data cube to
$S/N\geq 25$ (Cappellari & Copin, 2003), and then fitting the data with pPXF
(Cappellari & Emsellem, 2004) using high resolution stellar templates from
Wallace & Hinkle (1996). The resulting radial velocity map can be seen in
Figure 3. Errors on individual bins are determined through Monte Carlo
simulations and range from 5–10 km s-1. The velocity map was then analyzed
using the Kinemetry code (Krajnović et al., 2006) to determine the barycentric
systemic velocity as a function of radius from the observed photocenter; we
note that the appearance of the galaxy in the NIFS data cubes is very
symmetric. At the smallest radius (0$\farcs$05), the systemic velocity is
4856.8$\pm$1.6 km s-1. Kinemetry reveals that the velocity steadily declines
with radius – at 0$\farcs$5 it is 4844.5$\pm$0.6 km s-1. These measurements
are shown as orange dots in Figure 5. We note that the quoted errors are the
formal errors produced by the Kinemetry code and are smaller than the
systematic errors in our velocities discussed below.
To check the veracity of the systemic velocity shift with radius, we also
binned the spectra in circular annular bins, and we ran pPXF on the resulting
spectra. The mean velocity of the innermost spectrum ($<$0$\farcs$1) is
4865.4$\pm$2.9 km s-1, while the annulus between 0$\farcs$4 and 0$\farcs$6 has
a velocity of 4848.8$\pm$3.1 km s-1. Thus it seems quite clear that there is
indeed a blueshift in the systemic velocity of $\sim$15 km s-1 between the
center of the galaxy and the galaxy at radii of a few hundred parsecs. These
“integrated light” measurements are shown as red points in Figure 5.
We also determine the kinematics of H2 1-0 S(1) emission line. Because the
emission line strength does not follow the stellar emission distribution, for
this measurement we do not bin the data, and instead we measure the velocities
of the emission lines in each pixel with a Gaussian fit. We fit only lines
where the total flux in our fitting region is $>$10 times the surrounding
noise level. The result is shown in the right panel of Figure 3. Clear
rotation with the same position angle as the stellar kinematics is visible.
However, Kinemetry reveals that while the systemic velocities of the stellar
and H2 components are similar at the innermost radii, at larger radii the H2
systemic velocity is actually redshifted (not blueshifted like the stellar
kinematics), as shown by the blue points in Figure 5.
We note that the wavelength solution of the NIFS data was verified through
fitting of sky lines in the spectra; a standard deviation of 1.1 km/s from the
mean velocity was found from pixel to pixel, and an overall offset of -0.8
km/s was found. Thus the systematic errors on our velocity measurements are
$<$2 km/s, much smaller than the velocity gradient observed in the stellar and
gas kinematics.
### 3.3 The velocity of the SMBH
The velocity of the SMBH in J0437+2456 has previously been measured to be
$4818\pm 10.5$ km s-1 by G17, who used VLBI measurements of H2O megamasers in
the SMBH accretion disk to map out its rotation curve well within the
gravitational sphere of influence. By fitting this rotation curve with a thin-
disk Keplerian model, G17 were able to measure both the mass and velocity of
the central SMBH. In this section, we re-analyze the same VLBI dataset using
an updated maser disk model, which relaxes several of the assumptions made by
G17 and thus permits an improved assessment of the associated velocity
uncertainty.
The VLBI observations carried out by G17 resulted in position and velocity
measurements for each of the detected maser features, or “spots.” G17 fit
their rotation curve using a two-step procedure, in which the entire VLBI map
is first rotated and shifted such that the blueshifted and redshifted masers
lie on the horizontal axis, and then the maser spot velocities are fit as a
function of their measured one-dimensional positions along this horizontal
axis. The G17 rotation curve model contains three free parameters: the SMBH
mass, the one-dimensional SMBH position along the horizontal axis, and the
SMBH’s line-of-sight velocity.
For the present analysis we employ a modified version of the maser disk model
described in Pesce et al. (2020) to fit the J0437+2456 VLBI data. The primary
modification is the removal of acceleration measurements from the model
likelihood, as the available VLBI dataset does not contain any such
acceleration measurements. Our fitting approach differs from that of G17 in
several respects:
1. 1.
We take the maser velocities, rather than their positions, to be the
“independent” quantities; i.e., the model is essentially $r(v)$ rather than
$v(r)$. This strategy leverages the fact that the individual velocity
measurements are uncertain at a level comparable to a spectral channel width
(${\sim}$1–2 km s-1) and therefore much smaller than the orbital velocities of
several hundred km s-1, while the position uncertainties are comparatively
large ($\sim$0.01–0.1 mas) relative to the orbital radii of several tenths of
a milliarcsecond.
2. 2.
We do not perform any pre-rotation of the VLBI map, and instead we fit for the
two-dimensional location of the SMBH on the sky along with the position angle
of the disk.
3. 3.
We permit the disk inclination angle to be a free parameter in the fit.
4. 4.
We permit a warp in the position angle of the disk with radius.
5. 5.
We fit for systematic “error floor” parameters in the $x$ and $y$ maser
position measurements alongside the disk model parameters. These error floor
parameters describe the additional uncertainty that it would be necessary to
add into the measurements to ensure that the data are consistent with the
model; i.e., these parameters enforce a final reduced-$\chi^{2}$ value that is
consistent with unity.
Detailed descriptions of the model parameters, likelihood, and fitting
procedure are provided in Pesce et al. (2020). Following G17, we fit only to
the redshifted and blueshifted maser features because there are no available
acceleration measurements to constrain the systemic maser feature orbital
radii. The final model contains 9 parameters, which are listed in Table 3
along with their priors and best-fit values.
Table 3: Results from re-analysis of maser VLBI data
Parameter | Units | Prior | Best-fit value
---|---|---|---
v_0 | km s-1 | U(4500,5500) | 4809.3 ±10.0
M | $10^{6}$ $\text{M}_{\odot}$ | U(0,30) | 2.86 ±0.2
x_0 | mas | U(-0.5,0.5) | 0.096 ±0.005
y_0 | mas | U(-0.5,0.5) | 0.109 ±0.014
i_0 | deg. | U(70,110) | unconstrained
Ω_0 | deg. | U(0,180) | 16.9 ±2.5
Ω_1 | deg. mas-1 | U(-100,100) | 14 ±8
σ_x | $\mu$as | U(0,1000) | 6 ±1.5
σ_y | $\mu$as | U(0,1000) | ¡5
Note. — Results from fitting a thin Keplerian disk model to the J0437+2456
VLBI maser dataset from G17, as described in Section 3.3 and shown in Figure
4. The fitted model parameters are the SMBH velocity $v_{0}$, the SMBH mass
$M$, the SMBH position $(x_{0},y_{0})$, the disk inclination $i_{0}$, the disk
position angle $\Omega_{0}$, a first-order warp in the disk position angle
with radius $\Omega_{1}$, and two error floor parameters $\sigma_{x}$ and
$\sigma_{y}$ for the maser $x$\- and $y$-position measurements, respectively.
The notation $\mathcal{U}(a,b)$ denotes a uniform distribution on the range
$(a,b)$. For most parameters we report the posterior mean and standard
deviation, though we note that the $\sigma_{y}$ parameter has a best-fit value
that is consistent with zero and so we report the 95% upper limit instead. For
the $i_{0}$ parameter, the posterior distribution matches the prior
distribution, and so we do not report constraints on this parameter. A more
detailed description of the various model parameters can be found in Pesce et
al. (2020).
Our best-fit rotation curve and disk model are shown in Figure 4, from which
we determine the SMBH velocity to be $4809.3\pm 10.0$ km s-1. We find that the
uncertainty in the derived velocity matches well with the result from G17,
though our best-fit velocity itself is approximately 9 km s-1 smaller. Because
our disk model relies on fewer assumptions than that employed in G17, we
hereafter adopt $4809.3\pm 10.0$ km s-1 as the velocity measurement of the
SMBH in J0437+2456.
Figure 4: Results from fitting a thin Keplerian disk model to the J0437+2456
maser measurements from G17; the best-fit model parameters are listed in Table
3. The top left panel shows the on-sky projected radial separation from the
SMBH versus orbital velocity for each of the maser spots, with the best-fit
rotation curve plotted in black and 200 draws from the posterior distribution
plotted in gray. The points corresponding to individual maser features have
been colored by velocity group, with blue points denoting blueshifted maser
features and red points denoting redshifted maser features. The bottom left
panel shows the posterior distribution for the SMBH velocity that we obtain
from our fitting procedure (blue histogram) along with a Gaussian distribution
with the mean and standard deviation reported in G17 (black line). The right
panel shows the VLBI map of the maser system, with the best-fit warped disk
midplane plotted as a dashed line and 200 draws from the posterior
distribution plotted as solid gray lines; the 1$\sigma$ and 2$\sigma$ contours
for the SMBH location are shown as thick and thin black ellipses,
respectively.
## 4 Discussion
The recession velocity measurements considered in this paper are listed in
Table 2, and they are plotted against spatial scale in Figure 5. We find that
all velocity measurements fall within a $\sim$100 km s-1 range spanning
${\sim}$4820–4920 km s-1, and there is a general trend for measurements made
at larger spatial scales to recover larger recession velocities. In this
section we discuss the various measurements and consider some possible
interpretations.
Figure 5: The spatial scales on which the various velocity measurements
considered in this paper are made. For the Hi spike, we take the spatial scale
to be $\geq$600 pc as implied by the brightness temperature limit given in
Equation 2. The full peak-to-peak velocity range spanned by the Hi emission
(corresponding to the horizontal green line at the bottom of Figure 2) is
shown as a vertical line that is horizontally offset from the $V_{20}$
velocity for visual clarity. For the NIFS measurements, we plot the systemic
velocities as a function of annulus diameter and we include an overall 2 km
s-1 calibration systematic uncertainty on the error bars. All other velocities
are plotted with statistical error bars.
### 4.1 Velocity measurements in J0437+2456
The largest spatial scales are probed by the Hi emission, which traces gas
throughout the galaxy and out to the edge of the Arecibo beam (roughly
$\sim$50 kpc across). J0437+2456’s Hi profile is atypical in that it shows
three prominent spectral peaks rather than the usual two that are expected for
a simply-rotating system. Similar profiles have been classified as “anomalous”
by previous authors (e.g., UGC 2889 in Courtois et al., 2009), and they are
often attributed to spatial blending of galaxy pairs in single-dish spectra,
such as in the case of NGC 876 and NGC 877 (Bottinelli et al., 1982; Lee-
Waddell et al., 2014). However, for J0437+2456 we see neither evidence for a
companion galaxy within the Arecibo beam (see Figure 1) nor obvious signs of
morphological disturbance in Hubble Space Telescope (HST) images (Pjanka et
al., 2017). Nevertheless, the measured Hi central velocity of $V_{20}=4910$ km
s-1 is in agreement with the SDSS stellar velocity measured by P18, supporting
the notion that both measurements trace the recession velocity of J0437+2456.
Furthermore, the Hi central velocity is in ${\sim}10{\sigma}$ disagreement
with the SMBH velocity as measured from the maser rotation curve (Section
3.3), indicating that the black hole is blueshifted by roughly 100 km s-1 with
respect to the galaxy’s recession velocity.
The NIFS measurements probe spatial scales of $\sim$30–300 pc. The sense of
rotation for both the stellar and H2 components agrees with that of the maser
disk (G17), though the maser disk has a position angle of ${\sim}20^{\circ}$
while the outermost stellar and H2 components have position angles of
${\sim}40^{\circ}$.444We note that this ${\sim}20^{\circ}$ difference in
position angle is consistent with the offsets between maser disks and
circumnuclear structures seen in other galaxies (Greene et al., 2013) and
comparable to the ${\sim}16^{\circ}$ position angle difference between the
J0437+2456 maser disk and nuclear structure reported in Pjanka et al. (2017).
We find that the systemic velocities of the stellar and H2 rotation curves
agree with one another on the smallest scales ($\sim$30 pc), though they both
show a $\sim$4.7$\sigma$ redshift with the respect to the maser velocity. At
larger radii the stellar and H2 systemic velocity measurements diverge, with
the stellar systemic velocity showing a $\sim$30 km s-1 blueshift with respect
to the H2 on scales of $\sim$200 pc. Such large variations in the measured
stellar systemic velocity as a function of radius are rare; the typical
dispersion of ATLAS${}^{\text{3D}}$ galaxies between the central and $r=500$
pc velocities is only $\sim$3–4 km s-1 (Appendix B; Krajnović et al. 2011),
and most of the galaxies with substantially larger systemic velocity gradients
show evidence of interaction. We note, however, that such a relative velocity
offset could also be plausibly explained by a combination of geometric and
obscuration effects (e.g., if the stellar and $H_{2}$ emission arose from two
separate misaligned and mutually obscuring disks of material) while leaving
the system dynamically relaxed, and that the structure maps produced by Pjanka
et al. (2017) do show evidence of dust on $\sim$0.3′′ and larger scales.
The outermost H2 emission ($\sim$200 pc) has a systemic velocity of 4875 km
s-1 that matches well with the emission line velocity measured by P18 from the
SDSS spectrum, indicating that these two measurements may be tracing similar
material. These measurements are both also in agreement with the velocity of
the “anomalous” central Hi spike, which has a velocity of $\sim$4870 km s-1
and an amplitude of $S_{\nu}\approx 0.7$ mJy (see Table 4). If this Hi spike
represents a distinct dynamical subsystem (rather than, e.g., one “horn” of a
double-horn profile), then we can set a lower limit on the area of the
emission region by requiring that the Hi brightness temperature not exceed its
spin temperature of $T_{s}\approx 150$ K (Condon & Ransom, 2016),
$\Omega\geq\frac{S_{\nu}c^{2}}{2k\nu^{2}T_{s}}.$ (2)
Here, $k$ is the Boltzmann constant, $\nu=1.4$ GHz is the emitting frequency,
and $\Omega$ is the solid angle subtended by the emitting region. Equation 2
implies that the angular size of the region contributing the Hi spike is
${\sim}1.8^{\prime\prime}\approx 600$ pc. This spatial scale is similar to
that probed by the NIFS observations, and together with the coincident
velocities suggests that all three sources of emission – i.e., the outermost
H2, the SDSS emission lines, and the Hi spike – may be originating from
material with shared dynamics. The velocity of this material is significantly
different from that of both the SDSS stellar and the central Hi velocity
(i.e., $V_{20}$), perhaps indicating that there is a kinematically distinct
subsystem located in the centermost few hundred parsecs of J0437+2456.
However, we note that the observed FWHM of the Hi spike of only $\sim$55 km
s-1 (see Table 4) is in tension with this interpretation, because at several-
hundred parsec radii the material in this galaxy should display a FWHM of
$\sim$200 km s-1 (Figure 3; see also Noordermeer et al. 2007). It thus may not
be viable to interpret this Hi spike as a distinct kinematic component.
### 4.2 Uncertainty in the black hole velocity measurement
Our measurement of the SMBH velocity (see Section 3.3) relies on accurate VLBI
position measurements for each of the maser features, and if there are
unaccounted-for systematic uncertainties in these position measurements then
we would expect the velocity measurement to be correspondingly impacted. G17
considered the impact of phase referencing uncertainties on the J0437+2456
maser position measurements. The absolute sky location of the J0437+2456 peak
maser emission (i.e., the emission at a velocity of 4505.8 km s-1 used as a
reference feature) is known from phase-referenced VLBI measurements to a
precision of better than 2 mas. G17 estimate that the expected additional
positional uncertainties associated with this imperfectly-known reference
position, when propagated to the rest of the maser features, should be
${\lesssim}$5 $\mu$as. This expectation is consistent with the magnitudes of
the error floor parameters that we recover from our model fitting (see Table
3). Additionally, we note that there are no obvious systematic trends in the
residual dispersion about the best fit such as would be expected if poor phase
calibration were present at this level.
### 4.3 An offset black hole
In our own Galactic Center, we have high-precision evidence that the SMBH is
coincident with the dynamical center of the Galaxy (Reid & Brunthaler, 2020).
While we believe that a similar situation should generally hold for other
galaxies as well, a number of effects can at least temporarily knock the SMBH
out of this equilibrium position. At very low galaxy mass, it is possible that
SMBHs never settle at their galaxy center, given the very shallow galactic
potential (e.g., Bellovary et al., 2019; Reines et al., 2020). However, at
higher galaxy masses, it is most likely that mergers are responsible for SMBH
motions.
Relative motions and spatial offsets between SMBHs and their host galaxies
occur throughout the merger process. As galaxies merge, the SMBHs from each
galaxy will be offset both spatially and in velocity from the center of the
merger. This stage may be observable as velocity offset active galaxies (e.g.,
Comerford et al., 2009; Comerford & Greene, 2014) or as spatially resolved
pairs of active galactic nuclei (AGN; e.g., Komossa et al. 2003; Gerke et al.
2007). Further along in the merger process, when the two SMBHs become
gravitationally bound, one may hope to observe the signatures of orbital
motion for the bound pair (e.g., Eracleous et al., 2012; Shen et al., 2013; Ju
et al., 2013, see also Appendix C). Finally, if an SMBH merger occurs, then
any anisotropy in the radiated linear momentum will lead to a gravitational
wave recoil (Fitchett, 1983). These have been many observational recoil
candidates proposed, but all have their complications (see reviews in Komossa
2012 and Blecha et al. 2016).
The SMBH in the galaxy J0437+2456 is, to our knowledge, the most concrete case
of an SMBH in motion with respect to its galaxy. Because our initial search
focused on megamaser disk galaxies (P18), the sources were all within 200 Mpc
where detailed followup observations are possible; luminous AGN that have been
identified as recoil or binary SMBH candidates in the past are often much more
distant. Even in the case of J0437+2456, ambiguity remains about whether we
are seeing an SMBH making its way to the galaxy center for the first time,
SMBH binary orbital motion, or a recoil product. However, the fact that the
galaxy on large scales is apparently out of equilibrium provides indirect
evidence that we are observing the aftermath of a merger.
## 5 Summary and conclusion
Following the identification in P18 of the galaxy J0437+2456 as a candidate
for hosting a binary or recoiling SMBH, we have obtained Arecibo and Gemini
NIFS observations of the galaxy. Our new observations support the claim of a
velocity offset between the SMBH and its host galaxy. Furthermore, the
systemic velocity in J0437+2456 exhibits an apparent spatial scale dependence;
the overall picture looks something like the following:
1. 1.
On the smallest spatial scales ($<$1 pc), where the motion of gas is dominated
by the gravitational potential of the SMBH, H2O masers orbit with a central
velocity of $\sim$4810 km s-1. We associate this velocity with the SMBH
itself.
2. 2.
At the photocenter of the galaxy, within the central $\sim$30 pc and
coincident with the location of the SMBH, both the stars and H2 gas emission
lines have a systemic velocity of $\sim$4860 km s-1. However, on somewhat
larger scales ($\sim$30–200 pc), both gas and stars exhibit unusual systemic
velocity gradients of $\sim$15 km s-1 in opposite directions. In all cases,
these velocities are significantly offset from the SMBH velocity as traced by
the masers.
3. 3.
On the largest spatial scales ($\sim$1–10 kpc), the velocity of the Hi
emission is in agreement with the SDSS stellar velocity from P18. We find a
central Hi velocity of $V_{20}\approx 4910$ km s-1 that we associate with the
recession velocity of the galaxy as a whole, though we note that the
“anomalous” structure of the Hi spectral profile complicates this
interpretation.
Multiple lines of evidence – including the different inferred systemic
velocities on different spatial scales, the “anomalous” Hi spectral structure,
and the gradient in stellar systemic velocity with radius – point to the
conclusion that the galaxy J0437+2456 has been dynamically perturbed sometime
in the recent past, likely through an interaction with another galaxy. Of
particular interest is the apparent difference between the systemic velocity
of the SMBH and that of any other dynamical tracer, indicating that the SMBH
in this galaxy is in motion with respect to the surrounding material.
P18 explored plausible causes of such relative motion, ultimately settling on
three possibilities: (1) the SMBH originates from an external galaxy that is
in the process of merging with J0437+2456; (2) the SMBH is part of a binary
system, and the velocity offset we observe is the result of its orbital
motion; or (3) the SMBH is recoiling from a recent merger event. Though any of
these possibilities would be exciting, with the current data we are
unfortunately unable to distinguish between them. Additional observations are
required to ascertain the nature of the peculiar SMBH in J0437+2456.
We are grateful to Robert Minchin, Joan Schmelz, and Arun Venkataraman at
Arecibo Observatory for their help with data acquisition and reduction. This
paper makes use of observations taken using the Gemini Observatory under
program GN-2018B-FT-110 and the Arecibo Observatory under programs A3241 and
A3300. The Arecibo Observatory is operated by SRI International under a
cooperative agreement with the National Science Foundation (AST-1100968), and
in alliance with Ana G. Méndez-Universidad Metropolitana, and the Universities
Space Research Association. Support for this work was provided by the NSF
through grants AST-1952099, AST-1935980, AST-1828513, and AST-1440254, and by
the Gordon and Betty Moore Foundation through grant GBMF-5278. This work was
supported in part by the Black Hole Initiative at Harvard University, which is
funded by grants from the John Templeton Foundation and the Gordon and Betty
Moore Foundation to Harvard University. ACS acknowledges support from NSF
AST-1350389. This research made use of Montage, which is funded by the
National Science Foundation under Grant Number ACI-1440620, and was previously
funded by the National Aeronautics and Space Administration’s Earth Science
Technology Office, Computation Technologies Project, under Cooperative
Agreement Number NCC5-626 between NASA and the California Institute of
Technology. Funding for the SDSS and SDSS-II has been provided by the Alfred
P. Sloan Foundation, the Participating Institutions, the National Science
Foundation, the U.S. Department of Energy, the National Aeronautics and Space
Administration, the Japanese Monbukagakusho, the Max Planck Society, and the
Higher Education Funding Council for England. The SDSS Web Site is
http://www.sdss.org/. The SDSS is managed by the Astrophysical Research
Consortium for the Participating Institutions. The Participating Institutions
are the American Museum of Natural History, Astrophysical Institute Potsdam,
University of Basel, University of Cambridge, Case Western Reserve University,
University of Chicago, Drexel University, Fermilab, the Institute for Advanced
Study, the Japan Participation Group, Johns Hopkins University, the Joint
Institute for Nuclear Astrophysics, the Kavli Institute for Particle
Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of
Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute
für Astronomy (MPIA), the Max-Planck-Institute für Astrophysics (MPA), New
Mexico State University, Ohio State University, University of Pittsburgh,
University of Portsmouth, Princeton University, the United States Naval
Observatory, and the University of Washington.
## Appendix A Hi spectral modeling
Here we describe three different models we use to fit the Hi spectrum from
Section 3.1. For each model, we use a Gaussian likelihood given by
$\ln\left(\mathcal{L}\right)=-\frac{1}{2}\sum_{j}\left[\left(\frac{S_{\nu}(v_{j})-\hat{S}_{\nu}(v_{j})}{\sigma}\right)^{2}+\ln\left(2\pi\sigma^{2}\right)\right],$
(A1)
where $S_{\nu}(v_{j})$ is the model flux density for a spectral channel with
velocity $v_{j}$, $\hat{S}_{\nu}(v_{j})$ is the observed flux density in that
channel, $\sigma$ is the flux density uncertainty in a single channel, and the
sum is taken over all channels. This likelihood assumes that every spectral
channel contains independent Gaussian-distributed noise with a standard
deviation $\sigma$ that we treat as a model parameter in each of our fits. We
use the dynesty nested sampling code (Speagle, 2020) for posterior
exploration. The best-fit values and uncertainties for all model parameters
are listed in Table 4.
### A.1 Modeling the profile using a sum of Gaussian components
The model we use in our primary analysis (Section 3.1) describes the Hi
spectral structure using a sum of Gaussian components,
$S_{\nu}(v)=\sum_{i=1}^{N}A_{i}\exp\left[-\frac{1}{2}\left(\frac{v-v_{i}}{\sigma_{i}}\right)^{2}\right],$
(A2)
where the model parameters are the amplitude $A_{i}$, central velocity
$v_{i}$, and width $\sigma_{i}$ for each component. The total number of model
parameters is $3N+1$, where $N$ is the number of Gaussian components; in this
paper, we use $N=3$. We impose uniform priors on all model parameters, in the
range $[0,1]$ mJy for Gaussian component amplitudes, $[0,500]$ km s-1 for all
Gaussian component standard deviations, $[4500,5300]$ km s-1 for all Gaussian
component central velocities, and $[0,1]$ mJy for $\sigma$. The posterior
distribution is trivially multimodal upon pairwise swaps of Gaussian
components, but the modes are widely separated in parameter space and so we
isolate a single mode when reporting parameter statistics. A plot of the
resulting fit to the spectrum is shown in the left panel of Figure 6.
Table 4: Results from Hi spectral modeling
Model description | Parameter description | Units | Best-fit value
---|---|---|---
three Gaussian components | central velocity of first component | km s-1 | 4774.6 ±3.2
central velocity of second component | km s-1 | 4870.2 ±0.8
central velocity of third component | km s-1 | 4989.4 ±1.9
FWHM of first component | km s-1 | 76.4 ±6.4
FWHM of second component | km s-1 | 54.6 ±2.3
FWHM of third component | km s-1 | 129.0 ±4.1
amplitude of first component | mJy | 0.20 ±0.01
amplitude of second component | mJy | 0.66 ±0.02
amplitude of third component | mJy | 0.45 ±0.01
thermal noise level (RMS) | mJy | 0.068 ±0.002
two double-horn components | central velocity of first component | km s-1 | 4810.7 ±2.0
central velocity of second component | km s-1 | 4955.9 ±2.8
velocity width of first component | km s-1 | 139.6 ±4.2
velocity width of second component | km s-1 | 234.0 ±9.4
flux of first component | mJy km s-1 | 51.7 ±3.5
flux of second component | mJy km s-1 | 97.6 ±2.6
asymmetry of first component | unitless | 0.46 ±0.09
asymmetry of second component | unitless | 0.38 ±0.07
solid-body fraction of first component | unitless | 0.03 ±0.03
solid-body fraction of second component | unitless | 0.89 ±0.07
velocity dispersion of first component | km s-1 | 12.0 ±1.2
velocity dispersion of second component | km s-1 | 11.0 ±3.1
thermal noise level (RMS) | mJy | 0.064 ±0.002
one double-horn component and one Gaussian component | central velocity of double-horn component | km s-1 | 4904.1 ±3.7
velocity width of double-horn component | km s-1 | 312.5 ±10.2
flux of double-horn component | mJy km s-1 | 124.3 ±3.2
asymmetry of double-horn component | unitless | 0.61 ±0.05
solid-body fraction of double-horn component | unitless | 0.63 ±0.14
velocity dispersion of double-horn component | km s-1 | 18.5 ±5.9
central velocity of Gaussian component | km s-1 | 4870.9 ±0.7
FWHM of Gaussian component | km s-1 | 38.1 ±2.1
amplitude of Gaussian component | mJy | 0.47 ±0.02
thermal noise level (RMS) | mJy | 0.066 ±0.002
Note. — Results from fitting the three different models described in Appendix
A to the Arecibo Hi spectrum; these fits are shown in Figure 6. For each
parameter, we quote the posterior mean and standard deviation. The thermal
noise has been determined per 61 kHz ($\approx$1.33 km s-1) spectral channel.
### A.2 Modeling the profile using a sum of double-horn components
Given that the galaxy J0437+2456 shows signs of dynamical disturbance
(potentially indicating a recent merger) and that it exhibits an “anomalous”
Hi profile (Figure 2), it is natural to ask whether a combination of double-
horn profiles could give rise to the observed spectral structure. We have thus
performed an alternative analysis using a sum of two double-horn components,
each described using the parameterization developed by Stewart et al. (2014)
for each component.
Figure 6: Similar to Figure 2, but showing the results of fits using the three
different classes of model described in Appendix A; the best-fit parameter
values for each model are listed in Table 4. In the left panel we show a more
detailed breakdown of the fit from Figure 2 using three Gaussian components,
with the individual best-fit Gaussian model components plotted in blue, green,
and violet. Their corresponding best-fit velocities are marked by the vertical
lines underneath each component. In the center panel we show a similar
breakdown for the fit using two double-horn components, and in the right panel
the fit using one double-horn component and one Gaussian component. In all
panels, the spectrum is plotted at its native spectral resolution in gray, the
spectrum after smoothing by a 4-channel boxcar is shown in black, and 1000
random posterior samples are overplotted in red. The bottom row of plots shows
the residuals (i.e., the difference between the data and best-fitting model)
for each fit.
The Stewart et al. (2014) model describes a double-horn profile using six
parameters: the total flux, the central velocity, the velocity width, an
asymmetry parameter, a parameter describing what fraction of the emission
comes from solid-body rotation, and a velocity dispersion. Because we model
the spectrum as a sum of $N$ such double-horn components, and because we
additionally model the channel uncertainty $\sigma$, the total number of model
parameters is $6N+1$; in this paper, we use $N=2$. We impose uniform priors on
all model parameters, in the range $[0,1]$ Jy km s-1 for the total flux,
$[4500,5300]$ km s-1 for the central velocity, $[0,600]$ km s-1 for the
velocity width, $[-1,1]$ for the asymmetry parameter, $[0,1]$ for the solid-
body fraction, $[0,100]$ km s-1 for the velocity dispersion, and $[0,1]$ mJy
for $\sigma$.
The results of fitting this alternative model to the Hi data are shown in the
central panel of Figure 6. We find that the best-fit model prefers only one of
the two components to exhibit a standard double-horn profile, while the other
component is dominated by the solid-body contribution and so has only a
single, wide spectral peak. This model struggles to fit the central Hi spike,
as evidenced by the large residual flux excess near $\sim$4850 km s-1, so we
disfavor it compared to the model composed of three Gaussian components.
### A.3 Modeling the profile using a sum of double-horn and Gaussian
components
Motivated by the appearance of the Hi spectrum, we also attempt to model it
using a sum of one double-horn component (parameterized as in Stewart et al.
2014 and Section A.2) and one Gaussian component. The resulting parameter
values are listed in Table 4 and the best-fit spectrum is plotted in Figure 6.
We again find that even the best-fit model struggles to fit the observed
spectral profile, with a substantial flux excess seen in the residuals around
$\sim$4750 km s-1. We thus disfavor this model compared to the model composed
of three Gaussian components.
## Appendix B ATLAS${}^{\text{3D}}$ systemic velocity curves
The ATLAS${}^{\text{3D}}$ project has collected integral field spectroscopic
measurements for a sample of 260 early-type galaxies in the local Universe
(Cappellari et al., 2011). This sample provides a reference against which we
can gauge the behavior of the NIFS stellar systemic velocity measurements for
J0437+2456, which show a systematic trend with radius (see Section 3.2).
Figure 7 shows the radial profile of the J0437+2456 stellar systemic velocity
measurements plotted alongside the same quantity measured for the “fast
rotator” galaxies from ATLAS${}^{\text{3D}}$. The ATLAS${}^{\text{3D}}$ sample
is made up of early-type galaxies, while J0437+2456 is a spiral, so for
comparison we select only fast rotators from the ATLAS${}^{\text{3D}}$ sample
because they are galaxies with high angular momentum (Emsellem et al., 2011),
stellar disks, and ordered (i.e., disk-like) stellar kinematics (Krajnović et
al., 2011, 2013). We note that unlike the $\sim$0.3-arcsecond seeing of our
NIFS observations (see Section 2.2), many of the ATLAS${}^{\text{3D}}$
observations were carried out under $\sim$1–2-arcsecond seeing conditions
(Emsellem et al., 2004) and so the innermost radial points of each profile in
Figure 7 may suffer accordingly. Nevertheless, we see that the steep rise of
the systemic velocity with radius, as well as the large difference in systemic
velocities as measured at small and large radii, are both considerably more
extreme in J0437+2456 than in the majority of ATLAS${}^{\text{3D}}$ galaxies.
At about 200 pc from the center the systemic velocity of a typical fast
rotator deviates by only $\sim$2-3 km s-1 from the systemic velocity measured
near the center. This trend does not change substantially with increasing
radius.
There are a few galaxies in the ATLAS${}^{\text{3D}}$ sample that have
systemic velocity deviations similar to or even larger than those seen in
J0437+2456, albeit at larger radii. The galaxies with the top four largest
deviations are labeled in Figure 7: NGC 4753, UGC 09519, NGC 4342 and NGC
3665. NGC 4753, which has the largest difference in the systemic velocity,
also shows clear morphological evidence of a recent merger and contains
complex dust filaments (Krajnović et al., 2011; Bílek et al., 2020),
indicating that it is likely not in equilibrium. UGC 09519 might be dusty in
the center, and it also has an unusual large-scale stellar disk characterised
by blue colours and low surface brightness (Duc et al., 2015). NGC 3665 has a
well defined nuclear dust and gas disk (Onishi et al., 2017), as well as
asymmetric outer isophotes (Bílek et al., 2020). NGC 4342 shows no evidence
for disturbances in morphology or kinematics, except harbouring a central
nuclear stellar disk (Scorza & van den Bosch, 1998), though Cretton & van den
Bosch (1999) note that this galaxy has a remarkably large central velocity
dispersion for its size and luminosity. The ATLAS${}^{\text{3D}}$ sample is
perhaps not an ideal comparison sample, as it is made of early-type galaxies
and the observations do not probe the same spatial scales as the NIFS data of
J0437+2456. Nevertheless, it is clear that the majority of
ATLAS${}^{\text{3D}}$ galaxies do not show strong variations in systemic
velocity with radius, and there is some evidence that those with strong
variations tend to exhibit other indications of kinematic disturbance.
Figure 7: The radial profile of the J0437+2456 systemic velocity as measured
from the NIFS stellar emission (plotted in black; see Section 3.2) compared
against similar profiles for “fast rotator” galaxies from
ATLAS${}^{\text{3D}}$ (plotted in blue). For each galaxy, we have subtracted
off the systemic velocity measured at the smallest radii ($V_{\text{sys},0}$)
and then taken an absolute value of the difference to aid comparison.
## Appendix C Observational constraints on the properties of a hypothetical
binary SMBH system in J0437+2456
Figure 8: Observational constraints on the space of secondary SMBH mass and
binary separation for J0437+2456. The blue shaded region is excluded by the
requirement that the observed maser disk be tidally undisrupted, the red
shaded region is excluded by the requirement that the observed SMBH exhibit
the measured velocity offset, and the gray shaded region is excluded by the
lack of an astrometric offset seen between the SMBH and the galactic center.
The remaining unshaded region indicates the permitted range of secondary SMBH
mass and binary separation in the presence of these constraints. For the red
and gray shaded regions, the solid and dashed lines represent 50% and 90%
probability bounds, respectively, determined as described in Appendix C.
We observe the SMBH in J0437+2456 to have a velocity offset with respect to
its host galaxy, as determined using various different systemic velocity
tracers (see Section 4.1). One possible explanation for this velocity offset
is that the observed SMBH is part of a binary black hole system with a second,
unseen SMBH. In this case, we have several observational constraints on the
properties that such a binary system must have; these constraints are
illustrated in Figure 8.
Our first constraint comes from the fact that the observed SMBH in J0437+2456
is surrounded by an accretion disk, which is traced by H2O maser emission to
extend out to radii of $\sim$0.3 pc ( G17, see also the right panel of Figure
4). If a second SMBH is present outside of this accretion disk666A second SMBH
located within the innermost observed edge of the accretion disk would likely
go undetected by the maser measurements (such a tight binary system would
appear to the maser system as a single SMBH with a mass equal to the combined
masses of both SMBHs), but it would not by itself lead to an observed velocity
offset between the maser measurements and the systemic velocity of the host
galaxy., then its mass and separation from the observed SMBH must be such that
it avoids tidally disrupting the accretion disk. This condition is roughly
equivalent to requiring that the outer edge of the accretion disk lie within
the Hill sphere of the observed SMBH. If we denote the mass of the observed
SMBH as $m_{1}$, the mass of the second SMBH as $m_{2}$, their separation as
$r$, and the Hill sphere radius as $r_{H}$, then we can cast this condition as
an upper bound on $m_{2}$ of
$m_{2}\leq\frac{m_{1}}{r_{H}^{2}}\left[\frac{1}{\left(r-r_{H}\right)^{2}}-\frac{1}{r^{2}}\right]^{-1}.$
(C1)
The blue shaded region in Figure 8 shows the combinations of $m_{2}$ and $r$
that are excluded by this criterion. We use the measured value of
$m_{1}=2.9\times 10^{6}$ M⊙ from G17 for the mass of the observed SMBH and the
aforementioned value of $r_{H}=0.3$ pc from the VLBI map.
Our second constraint comes from the observed velocity offset of the SMBH with
respect to the host galaxy. If this SMBH is participating in a binary system,
then its line-of-sight velocity $v$ is related to the parameters of the binary
orbit via (see, e.g., Murray & Dermott, 1999)
$v=m_{2}\sin(i)\big{[}\cos(\omega+f)+e\cos(\omega)\big{]}\sqrt{\frac{2G}{r\left(1-e^{2}\right)\left(m_{1}+m_{2}\right)}}.$
(C2)
Here, $i$ is the inclination of the orbital plane, $\omega$ is its argument of
pericenter, $f$ is the true anomaly of the observed SMBH, and $e$ is the
orbital eccentricity; $m_{1}$, $m_{2}$, and $r$ are the same as in Equation
C1. We do not currently have any ability to constrain the geometric parameters
of the orbit, so we instead treat them probabilistically; we assume that the
orbital plane is oriented randomly on the sphere (i.e., $\omega$ is
distributed uniformly on $[0,2\pi]$ and $\cos(i)$ is distributed uniformly on
$[-1,1]$), that $f$ is oriented randomly on the circle, and that $e$ is
distributed uniformly in the range $[0,1]$. The solid and dashed red lines in
Figure 8 show the 50% and 90% probability contours, respectively, for the
combined constraints on $m_{2}$ and $r$ given these assumptions about the
distribution of possible orbit geometries. For the purposes of this
constraint, we estimate the orbital velocity of the SMBH in J0437+2456 to be
$48\text{\,km\,s${}^{-1}$}\leq v\leq 101$ km s-1 based on the measurements
presented in this paper (see Table 2).
Our third and final constraint comes from the apparent lack of an astrometric
offset between the SMBH in J0437+2456 and the center-of-light, determined in
P18 to be $\lesssim$0.05 arcseconds. If we take this value to be an upper
limit on the SMBH binary on-sky separation, then we can convert it into a
constraint on the SMBH binary absolute separation. We are again faced with the
fact that we do not have any handle on the orientation of the binary orbit, so
we assume that the orbit is randomly distributed on the sphere and plot 50%
and 90% probability regions in Figure 8 (shown as light and dark gray shaded
regions, respectively).
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|
# Noise Learning Based Denoising Autoencoder
Woong-Hee Lee, Mustafa Ozger, Ursula Challita, and Ki Won Sung, This work was
supported by a Korea University Grant. This research was supported by the BK21
FOUR(Fostering Outstanding Universities for Research) funded by the Ministry
of Education(MOE, Korea) and National Research Foundation of Korea(NRF). This
work was partly funded by the European Union Horizon 2020 Research and
Innovation Programme under the EU/KR PriMO-5G project with grant agreement No
815191. (Corresponding author: Ki Won Sung.)W.-H. Lee is with the Department
of Control and Instrumentation Engineering, Korea University, Republic of
Korea (e-mail: woongheelee@korea.ac.kr).U. Challita is with Ericsson Research,
Stockholm, Sweden (e-mail: ursula.challita@ericsson.com).M. Ozger and K. W.
Sung are with the School of Electrical Engineering and Computer Science, KTH
Royal Institute of Technology, Stockholm, Sweden (e-mail:{ozger and
sungkw}@kth.se).
###### Abstract
This letter introduces a new denoiser that modifies the structure of denoising
autoencoder (DAE), namely noise learning based DAE (nlDAE). The proposed nlDAE
learns the noise of the input data. Then, the denoising is performed by
subtracting the regenerated noise from the noisy input. Hence, nlDAE is more
effective than DAE when the noise is simpler to regenerate than the original
data. To validate the performance of nlDAE, we provide three case studies:
signal restoration, symbol demodulation, and precise localization. Numerical
results suggest that nlDAE requires smaller latent space dimension and smaller
training dataset compared to DAE.
###### Index Terms:
machine learning, noise learning based denoising autoencoder, signal
restoration, symbol demodulation, precise localization.
## I Introduction
Machine learning (ML) has recently received much attention as a key enabler
for future wireless communications [1, 2, 3]. While the major research effort
has been put to deep neural networks, there are enormous number of Internet of
Things (IoT) devices that are severely constrained on the computational power
and memory size. Therefore, the implementation of efficient ML algorithms is
an important challenge for IoT devices, as they are energy and memory limited.
Denoising autoencoder (DAE) is a promising technique to improve the
performance of IoT applications by denoising the observed data that consists
of the original data and the noise [4]. DAE is a neural network model for the
construction of the learned representations robust to an addition of noise to
the input samples [vincent2008extracting, 5]. The representative feature of
DAE is that the dimension of the latent space is smaller than the size of the
input vector. It means that the neural network model is capable of encoding
and decoding through a smaller dimension where the data can be represented.
The main contribution of this letter is to improve the efficiency and
performance of DAE with a modification of its structure. Consider a noisy
observation $Y$ which consists of the original data $X$ and the noise $N$,
i.e., $Y=X+N$. From the information theoretical perspective, DAE attempts to
minimize the expected reconstruction error by maximizing a lower bound on
mutual information $I(X;Y)$. In other words, $Y$ should capture the
information of $X$ as much as possible although $Y$ is a function of the noisy
input. Additionally, from the manifold learning perspective, DAE can be seen
as a way to find a manifold where $Y$ represents the data into a low
dimensional latent space corresponding to $X$. However, we often face the
problem that the stochastic feature of $X$ to be restored is too complex to
regenerate or represent. This is called the curse of dimensionality, i.e., the
dimension of latent space for $X$ is still too high in many cases.
What can we do if $N$ is simpler to regenerate than $X$? It will be more
effective to learn $N$ and subtract it from $Y$ instead of learning $X$
directly. In this light, we propose a new denoising framework, named as noise
learning based DAE (nlDAE). The main advantage of nlDAE is that it can
maximize the efficiency of the ML approach (e.g., the required dimension of
the latent space or size of training dataset) for capability-constrained
devices, e.g., IoT, where $N$ is typically easier to regenerate than $X$ owing
to their stochastic characteristics. To verify the advantage of nlDAE over the
conventional DAE, we provide three practical applications as case studies:
signal restoration, symbol demodulation, and precise localization.
The following notations will be used throughout this letter.
* •
$\text{Ber},\text{Exp},\mathcal{U},\mathcal{N},\mathcal{CN}$: the Bernoulli,
exponential, uniform, normal, and complex normal distributions, respectively.
* •
$\mathbf{x},\mathbf{n},\mathbf{y}\in\mathbb{R}^{P}$: the realization vectors
of random variables $X,N,Y$, respectively, whose dimensions are $P$.
* •
$P^{\prime}(<P)$: the dimension of the latent space.
* •
$\mathbf{W}\in\mathbb{R}^{P^{\prime}\times
P},\mathbf{W}^{\prime}\in\mathbb{R}^{P\times P^{\prime}}$: the weight matrices
for encoding and decoding, respectively.
* •
$\mathbf{b}\in\mathbb{R}^{P^{\prime}},\mathbf{b}^{\prime}\in\mathbb{R}^{P}$:
the bias vectors for encoding and decoding, respectively.
* •
$\mathcal{S}$: the sigmoid function, acting as an activation function for
neural networks, i.e., $\mathcal{S}(a)=\frac{1}{1+e^{-a}}$, and
$\mathcal{S}(\mathbf{a})=(\mathcal{S}(\mathbf{a}[1]),\cdots,\mathcal{S}(\mathbf{a}[P]))^{T}$
where $\mathbf{a}\in\mathbb{R}^{P}$ is an arbitrary input vector.
* •
$f_{\theta}$: the encoding function where the parameter $\theta$ is
$\\{\mathbf{W},\mathbf{b}\\}$, i.e.,
$f_{\theta}(\mathbf{y})=\mathcal{S}(\mathbf{W}\mathbf{y}+\mathbf{b})$.
* •
$g_{\theta^{{}^{\prime}}}$: the decoding function where the parameter
$\theta^{{}^{\prime}}$ is $\\{\mathbf{W}^{\prime},\mathbf{b}^{\prime}\\}$,
i.e.,
$g_{\theta^{{}^{\prime}}}(f_{\theta}(\mathbf{y}))=\mathcal{S}(\mathbf{W}^{\prime}f_{\theta}(\mathbf{y})+\mathbf{b}^{\prime})$.
* •
$M$: the size of training dataset.
* •
$L$: the size of test dataset.
## II Method of nlDAE
(a) Training phase
(b) Test phase
Figure 1: An illustration of the concept of nlDAE. Figure 2: A simple example
of comparison between DAE and nlDAE: reconstruction error according to
$\sigma_{N}$.
In the traditional estimation problem of signal processing, $N$ is treated as
an obstacle to the reconstruction of $X$. Therefore, most of the studies have
focused on restoring $X$ as much as possible, which can be expressed as a
function of $X$ and $N$. Along with this philosophy, ML-based denoising
techniques, e.g., DAE, have also been developed in various signal processing
fields with the aim of maximizing the ability to restore $X$ from $Y$. Unlike
the conventional approaches, we hypothesize that, if $N$ has a simpler
statistical characteristic than $X$, it will be better to subtract from $Y$
after restoring $N$.
We first look into the mechanism of DAE to build neural networks. Recall that
DAE attempts to regenerate the original data $\mathbf{x}$ from the noisy
observation $\mathbf{y}$ via training the neural network. Thus, the parameters
of a DAE model can be optimized by minimizing the average reconstruction error
in the training phase as follows:
$\theta^{*},\theta^{{}^{\prime}*}=\operatorname*{arg\,min}_{\theta,\theta^{{}^{\prime}}}\frac{1}{M}\sum_{i=1}^{M}\mathcal{L}\big{(}\mathbf{x}^{(i)},g_{\theta^{{}^{\prime}}}(f_{\theta}(\mathbf{y}^{(i)}))\big{)},$
(1)
where $\mathcal{L}$ is a loss function such as squared error between two
inputs. Then, the $j$-th regenerated data $\tilde{\mathbf{x}}^{(j)}$ from
${\mathbf{y}}^{(j)}$ in the test phase can be obtained as follows for all
$j\in\\{1,\cdots,L\\}$:
$\tilde{\mathbf{x}}^{(j)}=g_{\theta^{{}^{\prime}*}}(f_{\theta^{*}}(\mathbf{y}^{(j)})).$
(2)
It is noteworthy that, if there are two different neural networks which
attempt to regenerate the original data and the noise from the noisy input,
the linear summation of these two regenerated data would be different from the
input. This means that either $\mathbf{x}$ or $\mathbf{n}$ is more effectively
regenerated from $\mathbf{y}$. Therefore, we can hypothesize that learning
$N$, instead of $X$, from $Y$ can be beneficial in some cases even if the
objective is still to reconstruct $X$. This constitutes the fundamental idea
of nlDAE.
The training and test phases of nlDAE are depicted in Fig. 1. The parameters
of nlDAE model can be optimized as follows for all $i\in\\{1,\cdots,M\\}$:
$\theta_{nl}^{*},{\theta}_{nl}^{{}^{\prime}*}=\operatorname*{arg\,min}_{\theta,\theta^{{}^{\prime}}}\frac{1}{M}\sum_{i=1}^{M}\mathcal{L}\big{(}\mathbf{n}^{(i)},g_{\theta^{\prime}}(f_{\theta}(\mathbf{y}^{(i)}))\big{)}.$
(3)
Notice that the only difference from (1) is that $\mathbf{x}^{(i)}$ is
replaced by $\mathbf{n}^{(i)}$. Let $\tilde{\mathbf{x}}_{nl}^{(j)}$ denote the
$j$-th regenerated data based on nlDAE, which can be represented as follows
for all $j\in\\{1,\cdots,L\\}$:
$\tilde{\mathbf{x}}_{nl}^{(j)}=\mathbf{y}^{(j)}-g_{\theta_{nl}^{{}^{\prime}*}}(f_{\theta_{nl}^{*}}(\mathbf{y}^{(j)})).$
(4)
To provide the readers with insights into nlDAE, we examine two simple
examples where the standard deviation of $X$ is fixed as 1, i.e.,
$\sigma_{X}=1$, and that of $N$ varies. $Y=X+N$ is comprised as follows:
* •
Example 1: $X\sim\mathcal{U}(0,2\sqrt{3})$ and
$N\sim\mathcal{N}(0,\sigma_{N})$.
* •
Example 2: $X\sim\text{Exp}(1)$ and $N\sim\mathcal{N}(0,\sigma_{N})$.
Fig. 2 describes the performance comparison between DAE and nlDAE in terms of
mean squared error (MSE) for the two examples111Throughout this letter, the
squared error and the scaled conjugate gradient are applied as the loss
function and the optimization method, respectively.. Here, we set $P=12$,
$P^{\prime}=9$, $M=10000$, and $L=5000$. It is observed that nlDAE is superior
to DAE when $\sigma_{N}$ is smaller than $\sigma_{X}$ in Fig. 2. The gap
between nlDAE and DAE widens with lower $\sigma_{X}$. This implies that the
standard deviation is an important factor when we select the denoiser between
DAE and nlDAE.
These examples show the consideration of whether $X$ or $N$ is easier to be
regenerated, which is highly related to differential entropy of each random
variable, $H(X)$ and $H(N)$ [marsh2013introduction]. The differential entropy
is normally an increasing function over the standard deviation of the
corresponding random variable, e.g., $H(N)=\log(\sigma_{N}\sqrt{2\pi e})$.
Naturally, it is efficient to reconstruct a random variable with a small
amount of information, and the standard deviation can be a good indicator.
## III Case Studies
To validate the advantage of nlDAE over the conventional DAE in practical
problems, we provide three applications for IoT devices in the following
subsections. We assume that the noise follows Bernoulli and normal
distributions, respectively, in the first two cases, which are the most common
noise modeling. The third case deals with noise that follows a distribution
expressed as a mixture of various random variables. For all the studied use
cases, we select the DAE as the conventional denoiser as a baseline for
performance comparison. We present the case studies in the first three
subsections. Then, we discuss the experimental results in Sec. III-D.
### III-A Case Study I: Signal Restoration
In this use case, the objective is to recover the original signal from the
noisy signal which is modeled by the corruptions over samples.
#### III-A1 Model
The sampled signal of randomly superposed sinusoids, e.g., the recorded
acoustic wave, is the summation of samples of $k$ damped sinusoidal waves
which can be represented as follows:
$\mathbf{x}=\Big{\\{}\sum_{l=1}^{k}V_{l}e^{-\gamma_{l}n\Delta t}\cos(2\pi
f_{l}n\Delta t)\Big{\\}}_{n=0}^{P-1},$ (5)
where $V_{l}$, $\gamma_{l}$, and $f_{l}$ are the peak amplitude, the damping
factor, and the frequency of the $l$-th signal, respectively. Here, the time
interval for sampling, $\Delta t$, is set to satisfy the Nyquist theorem,
i.e., $\frac{1}{2\Delta t}>\max\\{f_{1},\cdots,f_{k}\\}$. To consider the
corruption of $\mathbf{x}$, let us assume that the probability of corruption
for each sample follows the Bernoulli distribution $\text{Ber}(p_{cor})$,
which indicates the corruption with the probability $p_{cor}$. In addition,
let $\mathbf{b}\in\\{0,1\\}^{P}$ denote the realization of
$\text{Ber}(p_{cor})$ over $P$ samples. Naturally, the corrupted signal,
$\mathbf{y}\in\mathbb{R}^{P}$, can be represented as follows:
$\mathbf{y}=\mathbf{x}+C\mathbf{b},$ (6)
where $C$ is a constant representing the sample corruption.
#### III-A2 Application of nlDAE
Based on (6), the denoised signal $\tilde{\mathbf{x}}_{nl}^{(j)}$ can be
represented by
$\tilde{\mathbf{x}}_{nl}^{(j)}=\mathbf{x}^{(j)}+C\mathbf{b}^{(j)}-g_{\theta_{nl}^{{}^{\prime}*}}(f_{\theta_{nl}^{*}}(\mathbf{x}^{(j)}+C\mathbf{b}^{(j)}),$
(7)
where
$\theta_{nl}^{*},{\theta}_{nl}^{{}^{\prime}*}=\operatorname*{arg\,min}_{\theta,\theta^{{}^{\prime}}}\frac{1}{M}\sum_{i=1}^{M}\mathcal{L}\big{(}C\mathbf{b}^{(i)},g_{\theta^{\prime}}(f_{\theta}(\mathbf{x}^{(i)}+C\mathbf{b}^{(i)}))\big{)}.$
#### III-A3 Experimental Parameters
We evaluate the performance of the proposed nlDAE in terms of the MSE of
restoration. For the experiment, the magnitude of noise $C$ is set to 1 for
simplicity. In addition, $V_{l}$, $\gamma_{l}$, and $f_{l}$ follow
$\mathcal{N}(0,1)$, $\mathcal{U}(0,10^{3})$, and $\mathcal{U}(0,10\text{
kHz})$, respectively, for all $l$. The sampling time interval $\Delta t$ is
set to $0.5\times 10^{-4}$ second, and the number of samples $P$ is $12$. We
set $P^{\prime}=9$, $p_{cor}=0.9$, and $M=10000$ unless otherwise specified.
### III-B Case Study II: Symbol Demodulation
Here, the objective is to improve the symbol demodulation quality through
denoising the received signal that consists of channel, symbols, and additive
noise.
#### III-B1 Model
Consider an orthogonal frequency-division multiplexing (OFDM) system with $P$
subcarriers where the subcarrier spacing is expressed by $\Delta f$. Let
$\mathbf{d}\in\mathbb{C}^{P}$ be a sequence in frequency domain.
$\mathbf{d}[n]$ is the $n$-th element of $\mathbf{d}$ and denotes the symbol
transmitted over the $n$-th subcarrier. In addition, let $K$ denote the pilot
spacing for channel estimation. Furthermore, the channel impulse response
(CIR) can be modeled by the sum of Dirac-delta functions as follows:
$h(t,\tau)=\sum_{l=0}^{L_{p}-1}\alpha_{l}\delta(t-\tau_{l}),$ (8)
where $\alpha_{l}$, $\tau_{l}$, and $L_{p}$ are the complex channel gain, the
excess delay of $l$-th path, and the number of multipaths, respectively. Let
$\mathbf{x}\in\mathbb{C}^{P}$ denote the discrete signal obtained by $P$-point
fast Fourier transform (FFT) after the sampling of the signal experiencing the
channel at the receiver, which can be represented as follows:
${}\mathbf{x}=\mathbf{d}\odot\mathbf{h}=\\{\mathbf{d}[n]\sum_{l=0}^{L_{p}-1}\alpha_{l}e^{-j2\pi
n\Delta f\tau_{l}}\\}_{n=0}^{P-1},$ (9)
where $\odot$ denotes the operator of the Hadamard product. Here,
$\mathbf{h}\in\mathbb{C}^{P}$ is the channel frequency response (CFR), which
is the $P$-point FFT of $h(t,\tau)$. In addition, let
$\mathbf{n}\in\mathbb{C}^{P}$ denote the realization of the random variable
$N\sim\mathcal{CN}(0,\sigma_{N})$. Finally,
$\mathbf{y}(=\mathbf{d}\odot\mathbf{h}+\mathbf{n})$ is the noisy observed
signal.
Our goal is to minimize the symbol error rate (SER) over $\mathbf{d}$ by
maximizing the quality of denoising $\mathbf{y}$. We assume the method of
channel estimation is fixed as the cubic interpolation [6] to focus on the
performance of denoising the received signal.
(a)
(b)
(c)
(d)
Figure 3: Case study I (signal restoration): MSE according to (a) the
dimension of latent space; (b) the size of training dataset; (c) $p_{cor}$;
and (d) the depth of neural networks.
(a)
(b)
(c)
(d)
Figure 4: Case study II (symbol demodulation): SER according to (a) the
dimension of latent space; (b) the size of training dataset; (c) SNR; and (d)
the depth of neural networks.
(a)
(b)
(c)
(d)
Figure 5: Case study III (precise localization): Localization error according
to (a) the dimension of latent space; (b) the size of training dataset; (c)
$p_{NLoS}$; and (d) the depth of neural networks.
#### III-B2 Application of nlDAE
To consider the complex-valued data, we separate it into real and imaginary
parts. $\Re$ and $\Im$ denote the operators capturing real and imaginary parts
of an input, respectively. Thus, $\tilde{\mathbf{x}}_{nl}^{(j)}$ is the
regenerated $\mathbf{d}^{(j)}\odot\mathbf{h}^{(j)}$ by denoising
$\mathbf{y}^{(j)}$, which can be represented by
$\begin{split}\tilde{\mathbf{x}}_{nl}^{(j)}&=\Re(\mathbf{y}^{(j)})-g_{\theta_{nl,R}^{{}^{\prime}*}}(f_{\theta_{nl,R}^{*}}(\Re(\mathbf{y}^{(j)})))\\\
&+i(\Im(\mathbf{y}^{(j)})-g_{\theta_{nl,I}^{{}^{\prime}*}}(f_{\theta_{nl,I}^{*}}(\Im(\mathbf{y}^{(j)})))),\end{split}$
(10)
where
$\begin{split}\theta_{nl,{R}}^{*},{\theta}_{nl,{R}}^{{}^{\prime}*}=\operatorname*{arg\,min}_{\theta,\theta^{{}^{\prime}}}\frac{1}{M}\sum_{i=1}^{M}\mathcal{L}\big{(}\Re(\mathbf{n}^{(i)}),g_{\theta^{\prime}}(f_{\theta}(\Re(\mathbf{y}^{(i)})))\big{)},\\\
\theta_{nl,{I}}^{*},{\theta}_{nl,{I}}^{{}^{\prime}*}=\operatorname*{arg\,min}_{\theta,\theta^{{}^{\prime}}}\frac{1}{M}\sum_{i=1}^{M}\mathcal{L}\big{(}\Im(\mathbf{n}^{(i)}),g_{\theta^{\prime}}(f_{\theta}(\Im(\mathbf{y}^{(i)})))\big{)}.\end{split}$
Finally, the receiver estimates $\mathbf{h}$ with the predetermined pilot
symbols, i.e., $\mathbf{d}[nK+1]$ where $n=0,1,\cdots$, and demodulates
$\mathbf{d}$ based on the estimate of $\mathbf{h}$ and the regenerated
$\tilde{\mathbf{x}}_{nl}$.
#### III-B3 Experimental Parameters
The performance of the proposed nlDAE is evaluated when $L=5000$. For the
simulation parameters, we set $4$ QAM, $P=12$, $\Delta f=15$ kHz, $L_{p}=4$,
and $K=3$. We further assume that $\alpha\sim\mathcal{CN}(0,1)$ and
$\tau\sim\mathcal{U}(0,10^{-6})$. Furthermore, $P^{\prime}=9$, SNR$=5$ dB, and
$M=10000$ unless otherwise specified. We also provide the result of non-ML
(i.e., only cubic interpolation).
### III-C Case Study III: Precise Localization
The objective of this case study is to improve the localization quality
through denoising the measured distance which is represented by the quantized
value of the mixture of the true distance and error factors.
#### III-C1 Model
Consider a 2-D localization where $P$ reference nodes and a single target node
are randomly distributed. We estimate the position of the target node with the
knowledge of the locations of $P$ reference nodes. Let
$\mathbf{x}\in\mathbb{R}^{P}$ denote the vector of true distances from $P$
reference nodes to the target node when $X$ denotes the distance between two
random points in a 2-D space. We consider three types of random variables for
the noise added to the true distance as follows:
* •
$N_{N}$: ranging error dependent on signal quality.
* •
$N_{U}$: ranging error due to clock asynchronization.
* •
$N_{B}$: non line-of-sight (NLoS) event.
We assume that $N_{N}$, $N_{U}$, $N_{B}$ follow the normal, uniform, and
Bernoulli distributions, respectively. Hence, we can define the random
variable for the noise $N$ as follows:
$N=N_{N}+N_{U}+R_{NLoS}N_{B},$ (11)
where $R_{NLoS}$ is the distance bias in the event of NLoS. Note that $N$ does
not follow any known probability distribution because it is a convolution of
three different distributions. Besides, we assume that the distance is
measured by time of arrival (ToA). Thus, we define the quantization function
$\mathcal{Q}_{B}$ to represent the measured distance with the resolution of
$B$, e.g., $\mathcal{Q}_{10}(23)=20$. In addition, the localization method
based on multi-dimensional scaling (MDS) is utilized to estimate the position
of the target node [7].
#### III-C2 Application of nlDAE
In this case study, we consider the discrete values quantized by the function
$\mathcal{Q}_{B}$. Here, $\tilde{\mathbf{x}}_{nl}^{(j)}$ can be represented as
follows:
$\tilde{\mathbf{x}}_{nl}^{(j)}={\mathcal{Q}_{B}(\mathbf{y}}^{(j)})-g_{\theta_{nl,R}^{{}^{\prime}*}}(f_{\theta_{nl,R}^{*}}({\mathcal{Q}_{B}(\mathbf{y}}^{(j)}))),$
(12)
where
$\theta_{nl,{R}}^{*},{\theta}_{nl,{R}}^{{}^{\prime}*}=\operatorname*{arg\,min}_{\theta,\theta^{{}^{\prime}}}\frac{1}{M}\sum_{i=1}^{M}\mathcal{L}\big{(}\mathcal{Q}_{B}(\mathbf{n}^{(i)}),g_{\theta^{\prime}}(f_{\theta}(\mathcal{Q}_{B}(\mathbf{y}^{(i)})))\big{)}.$
Thus, $\tilde{\mathbf{x}}_{nl}$ is utilized for the estimation of the target
node position in nlDAE-assisted MDS-based localization.
#### III-C3 Experimental Parameters
The performance of the proposed nlDAE is evaluated via $L=5000$. In this
simulation, $12$ reference nodes and one target node are uniformly distributed
in a $100\times 100$ square. We assume that
$N_{N}\sim\mathcal{N}(0,10),N_{U}\sim\mathcal{U}(0,20),N_{B}\sim\text{Ber}(0.2)$,
and $R_{NLoS}=50$. The distance resolution $B$ is set to $10$ for the
quantization function $\mathcal{Q}_{B}$. Note that $P^{\prime}=9$,
$p_{NLoS}=0.2$, and $M=10000$ unless otherwise specified. We also provide the
result of non-ML (i.e., only MDS based localization).
### III-D Analysis of Experimental Results
Fig. 3(a), Fig. 4(a), and Fig. 5(a) show the performance of the three case
studies with respect to $P^{\prime}$, respectively. nlDAE outperforms non-ML
and DAE for all ranges of $P^{\prime}$. Particularly with small values of
$P^{\prime}$, nlDAE continues to perform well, whereas DAE loses its merit.
This means that nlDAE provides a good denoising performance even with an
extremely small dimension of latent space if the training dataset is
sufficient.
The impact of the size of training dataset is depicted in Fig. 3(b), Fig.
4(b), and Fig. 5(b). nlDAE starts to outperform non-ML with $M$ less than 100.
Conversely, DAE requires about an order higher $M$ to perform better than non-
ML. Furthermore, nlDAE converges faster than DAE, thus requiring less training
data than DAE.
In Fig. 3(c), Fig. 4(c), and Fig. 5(c), the impact of a noise-related
parameter for each case study is illustrated. When the noise occurs according
to a Bernoulli distribution in Fig. 3(c), the performance of ML algorithms
(both nlDAE and DAE) exhibits a concave behavior. This is because the variance
of $\text{Ber}(p)$ is given by $p(1-p)$. Similar phenomenon is observed in
Fig. 5(c) because the Bernoulli event of NLoS constitutes a part of
localization noise. As for non-ML, the performance worsens as the probability
of noise occurrence increases in both cases. Fig. 4(c) shows that the SER
performance of nlDAE improves rapidly as the SNR increases. In all
experiments, nlDAE gives superior performance than other schemes.
Thus far, the experiments have been conducted with a single hidden layer. Fig.
3(d), Fig. 4(d), and Fig. 5(d) show the effect of the depth of the neural
network. The performance of nlDAE is almost invariant, which suggests that
nlDAE is not sensitive to the number of hidden layers. On the other hand, the
performance of DAE worsens quickly as the depth increases owing to overfitting
in two cases.
In summary, nlDAE outperforms DAE over the whole experiments. nlDAE is
observed to be more efficient for the underlying use cases than DAE because it
requires smaller latent space and less training data. Furthermore, nlDAE is
more robust to the change of the parameters related to the design of the
neural network, e.g., the network depth.
## IV Conclusion and Future Work
We introduced a new denoiser framework based on the neural network, namely
nlDAE. This is a modification of DAE in that it learns the noise instead of
the original data. The fundamental idea of nlDAE is that learning noise can
provide a better performance depending on the stochastic characteristics
(e.g., standard deviation) of the original data and noise. We applied the
proposed mechanism to the practical problems for IoT devices such as signal
restoration, symbol demodulation, and precise localization. The numerical
results support that nlDAE is more efficient than DAE in terms of the required
dimension of the latent space and the size of training dataset, thus rendering
it more suitable for capability-constrained conditions. Applicability of nlDAE
to other domains, e.g., image inpainting, remains as a future work.
Furthermore, information theoretical criteria of decision making for the
selection between or a combination of DAE and nlDAE is an interesting further
research.
## References
* [1] U. Challita, H. Ryden, and H. Tullberg, “When machine learning meets wireless cellular networks: Deployment, challenges, and applications,” _IEEE Communications Magazine_ , vol. 58, no. 6, pp. 12–18, 2020.
* [2] M. Chen, U. Challita, W. Saad, C. Yin, and M. Debbah, “Artificial neural networks-based machine learning for wireless networks: A tutorial,” _IEEE Communications Surveys & Tutorials_, vol. 21, no. 4, pp. 3039–3071, 2019.
* [3] A. Azari, M. Ozger, and C. Cavdar, “Risk-aware resource allocation for URLLC: Challenges and strategies with machine learning,” _IEEE Communications Magazine_ , vol. 57, no. 3, pp. 42–48, 2019.
* [4] Y. Sun, M. Peng, Y. Zhou, Y. Huang, and S. Mao, “Application of machine learning in wireless networks: Key techniques and open issues,” _IEEE Communications Surveys & Tutorials_, vol. 21, no. 4, pp. 3072–3108, 2019.
* [5] Y. Bengio, L. Yao, G. Alain, and P. Vincent, “Generalized denoising auto-encoders as generative models,” _Advances in neural information processing systems_ , pp. 899–907, 2013.
* [6] S. Coleri, M. Ergen, A. Puri, and A. Bahai, “Channel estimation techniques based on pilot arrangement in OFDM systems,” _IEEE Transactions on broadcasting_ , vol. 48, no. 3, pp. 223–229, 2002.
* [7] I. Dokmanic, R. Parhizkar, J. Ranieri, and M. Vetterli, “Euclidean distance matrices: Essential theory, algorithms, and applications,” _IEEE Signal Processing Magazine_ , vol. 32, no. 6, pp. 12–30, 2015.
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# Identifying First-order Lowpass Graph Signals using Perron Frobenius Theorem
###### Abstract
This paper is concerned with the blind identification of graph filters from
graph signals. Our aim is to determine if the graph filter generating the
graph signals is _first-order lowpass_ without knowing the graph topology.
Notice that lowpass graph filter is a common pre-requisite for applying graph
signal processing tools for sampling, denoising, and graph learning. Our
method is inspired by the Perron Frobenius theorem, which observes that for
first-order lowpass graph filter, the top eigenvector of output covariance
would be the _only eigenvector_ with elements of the same sign. Utilizing this
observation, we develop a simple detector that answers if a given data set is
produced by a first-order lowpass graph filter. We analyze the effects of
finite-sample, graph size, observation noise, strength of lowpass filter, on
the detector’s performance. Numerical experiments on synthetic and real data
support our findings.
Index Terms— lowpass graph signals, graph learning, Perron Frobenius theorem
## 1 Introduction
A recent trend in data science is to develop tools for analyzing and inference
from signals defined on graph, a.k.a. graph signals. Examples are financial
and social networks data where graph structures can be leveraged to improve
inference [1, 2]. These premises motivated researches on graph signal
processing (GSP) [3, 4] which extends signal processing models to graph
signals.
To study graph signals, an important concept of GSP is to model them as
outputs from exciting a graph filter. The graph filter captures the
social/physical process that generates the observations we collect. Examples
are heat diffusion [5], dynamics of functional brain activities [6],
equilibrium seeking in network games [7], etc.. Similar to its linear time
invariant counterpart, the graph filters can be classified as _lowpass_ ,
_bandpass_ , _highpass_ according to their frequency responses computed
through the Graph Fourier transform [8]. Among others, lowpass graph filters
and signals are important to many GSP tools, e.g., sampling, denoising, graph
topology learning [9]. Importantly, a number of practical social/physical
models can naturally lead to lowpass graph signals as studied in [9].
Although lowpass graph filters/signals are common, most prior works took the
lowpass property as a default assumption without further verification. This
paper takes a different approach, where we address the validity of the lowpass
assumption for a given set of graph signals. Our approach is _data-driven_ ,
and we aim to provide certificates on whether certain GSP tools are applicable
to a dataset whose social/physical models are unknown, as well as determining
the type of social/physical process which supports the observed data.
Our work is related to the recent developments in _graph inference or
learning_ which aim at learning the graph topology blindly [10, 11, 12, 13, 7,
14, 15]. For instance, [10, 11, 12, 13] infer topology from smooth graph
signals which can be given by lowpass graph filtering, [7, 14, 15] consider
blind inference of communities and centrality from lowpass graph signals; see
[16, 17]. However, these works rely on the lowpass property as _an assumption_
on the graph signals, while the aim of this paper is to verify the latter
property. Recent works also considered the joint inference of graph topology
and network dynamics [18, 19], inferring the type of physical process [20], or
the identification of graph filters in [21, 22]. These works either require
the graph topology is known a priori, or the network dynamics is parameterized
with a specific model template.
Our contributions are three-fold. First, in Section 2, we demonstrate that for
first-order lowpass graph signals, the Perron Frobenius theorem shows that the
top eigenvector of the covariance matrix must have elements of the same sign.
Furthermore, this is the only such eigenvector. Second, in Section 3, we
design a simple detector for lowpass graph signals and analyze the effects of
finite-sample on its performance. Importantly, we show that for strong lowpass
graph filters, the detection performance is robust to the graph size and/or
number of samples. Third, in Section 4, we apply our detector on three real
dataset (S&P500 stock, US Senate’s voting record, number of new COVID-19
cases) and confirm if the underlying graph filter is lowpass. Our work
provides the first step towards blind identification of network dynamics
without knowing graph topology.
## 2 Graph Signals and Graph Filters
This section describes the graph signal model and introduces necessary
notations. We consider an undirected graph denoted by $G=(V,E)$, with $n$
nodes described in $V=\\{1,...,n\\}$ and $E\subseteq V\times V$ is the edge
set. We assume $(i,i)\notin E$ such that $G$ has no self-loop. Define the
weighted adjacency matrix ${\bm{A}}\in\Re^{n\times n}$ as a non-negative,
symmetric matrix with $A_{ij}>0$ if and only if $(i,j)\in E$. The Laplacian
matrix is defined as ${\bm{L}}={\rm Diag}({\bm{A}}{\bf 1})-{\bm{A}}$.
A graph signal [3] on $G$ is a scalar function defined on $V$, i.e.,
$x:V\rightarrow\mathbb{R}$, and it can be represented as an $n$-dimensional
vector ${\bm{x}}\in\mathbb{R}^{n}$. We use graph filters to describe an
unknown process on $G$, capturing phenomena such as information exchange,
diffusion, etc.. A linear graph filter can be expressed as a $T$th order
matrix polynomial:
$\textstyle{\cal
H}({\bm{S}})=\sum_{t=0}^{T-1}h_{t}{\bm{S}}^{t},\vspace{-.0cm}$ (1)
where $\\{h_{t}\\}_{t=0}^{T-1}$ is the set of filter weights and ${\bm{S}}$ is
the graph shift operator (GSO) which is a symmetric matrix that respects the
graph topology. In this paper, the GSO ${\bm{S}}$ can be either adjacency
matrix ${\bm{A}}$, or Laplacian matrix ${\bm{L}}$. In both cases, we consider
its eigendecomposition as ${\bm{S}}={\bm{V}}\bm{\Lambda}{\bm{V}}^{\top}$,
where ${\bm{V}}$ is orthogonal and $\bm{\Lambda}={\rm
Diag}(\lambda_{1},...,\lambda_{n})$ is a diagonal matrix of the eigenvalues of
${\bm{S}}$. The $i$th column of ${\bm{V}}$ satisfies
${\bm{S}}{\bm{v}}_{i}=\lambda_{i}{\bm{v}}_{i}$. Define the frequency response
function:
$\textstyle h(\lambda):=\sum_{t=0}^{T-1}h_{t}\lambda^{t}.\vspace{-.0cm}$ (2)
Alternatively, the graph filter can be written in terms of its frequency
response as ${\cal H}({\bm{S}})={\bm{V}}h(\bm{\Lambda}){\bm{V}}^{\top}$, where
$h(\bm{\Lambda})$ is a diagonal matrix with
$[h(\bm{\Lambda})]_{ii}=h(\lambda_{i})$.
In GSP, a _low frequency_ graph signal is one that varies little along the
edges on $G$, i.e., with a small graph $\ell_{2}$ total variation (graph TV)
${\bm{x}}^{\top}{\bm{L}}{\bm{x}}$ [4]. For Laplacian matrix ${\bm{L}}$, it is
known that for small eigenvalue $\lambda_{i}$, the corresponding eigenvector
${\bm{v}}_{i}$ is a low frequency graph signal, e.g.,
${\bm{L}}(1/\sqrt{n}{\bm{1}})=0\cdot(1/\sqrt{n}{\bm{1}})$. This observation
will be reversed for adjacency matrix ${\bm{A}}$, where the eigenvector with a
large $\lambda_{i}$ corresponds to a low frequency graph signal.
With a slight abuse of notations, we adopt the convention that when
${\bm{S}}={\bm{L}}$, the eigenvalues (a.k.a. graph frequencies) are ordered as
$0=\lambda_{1}\leq\ldots\leq\lambda_{n}$; while when ${\bm{S}}={\bm{A}}$, the
eigenvalues are ordered as $\lambda_{1}\geq\ldots\geq\lambda_{n}$. We now
define the lowpass graph filter for a general GSO ${\bm{S}}$ [9] as follows:
###### Definition 1
Let $K\in\\{1,...,n\\}$. A graph filter ${\cal H}({\bm{S}})$ is lowpass with
cutoff frequency at $\lambda_{K}$ and the lowpass ratio $\eta_{K}<1$ if
$\max\\{|h(\lambda_{K+1})|,...,|h(\lambda_{n})|\\}=\eta_{K}\min\\{|h(\lambda_{1})|,...,|h(\lambda_{K})|\\},\vspace{-.1cm}$
where the frequency response function $h(\cdot)$ was defined in (2).
The lowpass ratio $\eta_{K}$ determines the strength of the lowpass filter as
it quantifies the degree of attenuation beyond cutoff frequency. We say ${\cal
H}({\bm{S}})$ is strong (resp. weak) lowpass if $\eta_{K}\ll 1$ (resp.
$\eta_{K}\approx 1$).
Our task is to identify whether a set of graph signals was generated from a
lowpass graph filter, i.e., if they are _lowpass graph signals_. The observed
graph signals are modeled as the noisy filter outputs from ${\cal
H}({\bm{S}})$ subjected to excitations $\\{{\bm{x}}_{\ell}\\}_{\ell=1}^{m}$.
We have:
${\bm{y}}_{\ell}=\overline{\bm{y}}_{\ell}+{\bm{w}}_{\ell},~{}\text{where}~{}~{}\overline{\bm{y}}_{\ell}={\cal
H}({\bm{S}}){\bm{x}}_{\ell},~{}\ell=1,...,m,\vspace{-.1cm}$ (3)
such that ${\bm{w}}_{\ell}$ is the observation noise, and
${\bm{x}}_{\ell},{\bm{w}}_{\ell}$ are zero mean random vectors with the
covariances $\mathbb{E}[{\bm{x}}_{\ell}{\bm{x}}_{\ell}^{\top}]={\bm{I}}$,
$\mathbb{E}[{\bm{w}}_{\ell}{\bm{w}}_{\ell}^{\top}]=\sigma^{2}{\bm{I}}$. In the
above, the signal term
$\overline{\bm{y}}_{\ell}=\sum_{t=0}^{T-1}h_{t}{\bm{S}}^{t}{\bm{x}}_{\ell}$ is
a weighted sum of the shifted versions of ${\bm{x}}_{\ell}$, i.e., it is the
output of the graph filter under the excitation ${\bm{x}}_{\ell}$. Note that
we consider a _completely blind identification_ problem where it is not known
a priori if the GSO is a Laplacian matrix or an adjacency matrix.
### 2.1 Perron Frobenius Theorem and 1st-Order Lowpass Filter
While the general problem is to identify lowpass graph filters of _any order_
, this paper focuses on the _first-order lowpass graph filters_ whose cutoff
frequency is $\lambda_{1}$ with the lowpass ratio of $\eta_{1}<1$ [cf.
Definition 1]. Notice that this is a sufficient condition to enable graph
inference such as blind centrality estimation [14, 15], as well as ensuring
that the graph signal is smooth [11, 10]. Formally, our task is to distinguish
between the following two hypothesis:
$\begin{split}{\cal T}_{0}&:\text{the graph filter ${\cal H}({\bm{S}})$ is
first-order lowpass}\\\ {\cal T}_{1}&:\text{the graph filter ${\cal
H}({\bm{S}})$ is \emph{not} first-order lowpass}\\\\[-2.84544pt] \end{split}$
(4)
Note that ${\cal T}_{1}$ include highpass filters where Definition 1 is not
satisfied for any $K\in\\{1,...,n\\}$, as well as lowpass filters with a
higher cutoff frequency at $\lambda_{K}$, $K\geq 2$.
Our next endeavor is to study the spectral property of the covariance of
$\\{{\bm{y}}_{\ell}\\}_{\ell=1}^{m}$. For simplicity, let us focus on the
signal term $\overline{\bm{y}}_{\ell}$ with the following population
covariance matrix:
${\rm
Cov}(\overline{\bm{y}}_{\ell})=\mathbb{E}[\overline{\bm{y}}_{\ell}(\overline{\bm{y}}_{\ell})^{\top}]={\cal
H}({\bm{S}})^{2}={\bm{V}}h(\bm{\Lambda})^{2}{\bm{V}}^{\top}.\vspace{-.1cm}$
(5)
Let $\overline{\bm{v}}_{i}$ be the top $i$th eigenvector of ${\rm
Cov}(\overline{\bm{y}}_{\ell})$ with $\|\overline{\bm{v}}_{i}\|=1$. Under the
assumption that ${\cal H}({\bm{S}})$ is first-order lowpass [cf. Definition
1], it is obvious that we have $\overline{\bm{v}}_{1}={\bm{v}}_{1}$, i.e., the
lowest frequency graph signal that varies little along the edges on $G$. The
above suggests that one could detect if ${\cal H}({\bm{S}})$ is a first-order
lowpass graph filter by evaluating the graph TV of $\overline{\bm{v}}_{1}$ (or
its approximation computed from $\\{{\bm{y}}_{\ell}\\}_{\ell=1}^{m}$) and
applying a threshold detector. However, it is not possible to evaluate the
graph TV $\overline{\bm{v}}_{1}^{\top}{\bm{L}}\overline{\bm{v}}_{1}$ since
${\bm{L}}$ is unknown.
Instead of computing the graph TV of $\overline{\bm{v}}_{1}$, we inspect the
eigenvector ${\bm{v}}_{1}$ of the lowest frequency of the GSO. Assume that:
###### H 1
The graph $G$ has one and only one connected component.
1 is common and it allows us to characterize the lowest graph frequency
eigenvector of ${\bm{L}}$, ${\bm{A}}$. Since ${\bm{A}}$ is non-negative, and
${\bm{L}}$ has non-positive off-diagonal elements, applying the Perron-
Frobenius theorem shows that the lowest graph frequency eigenvectors must be
positive [23]. In particular, we observe:
###### Lemma 1
Under 1, it holds that:
1. 1.
For Laplacian matrix ${\bm{L}}$, the (smallest) eigenvalue $\lambda_{1}=0$ has
multiplicity one with the eigenvector ${\bm{v}}_{1}={\bf 1}/\sqrt{n}$. For
adjacency matrix ${\bm{A}}$, the (largest) eigenvalue $\lambda_{1}$ has
multiplicity one with the eigenvector ${\bm{v}}_{1}$, which is a positive
vector.
2. 2.
In addition, ${\bm{v}}_{1}$ is the only positive eigenvector111Note that both
${\bm{v}}_{1},-{\bm{v}}_{1}$ are eigenvectors with the eigenvalue
$\lambda_{1}$. We assume ${\bm{v}}_{1}>{\bm{0}}$ to avoid such ambiguity. of
${\bm{L}}$ or ${\bm{A}}$. For $i\neq 1$, the eigenvector
${\bm{v}}_{i}\neq{\bm{0}}$ must have at least one positive and one negative
element.
_Proof_ : The first statement is a consequences of 1 and the spectral graph
theory for ${\bm{L}}$ [24, Theorem 7.1.2], or the Perron-Frobenius theorem for
${\bm{A}}$ [23, Theorem 8.4.4]. Notice that the positivity of ${\bm{v}}_{1}$
for ${\bm{L}}$ can alternatively be shown by the Perron-Frobenius theorem
through studying $\upsilon{\bm{I}}-{\bm{L}}$ with sufficiently large
$\upsilon>0$.
To show the second statement, we use the orthogonality of ${\bm{V}}$ which
implies ${\bm{v}}_{i}^{\top}{\bm{v}}_{1}=0$ for any $i\neq 1$. For the sake of
contradiction, assume ${\bm{v}}_{i}\neq{\bm{0}}$, ${\bm{v}}_{i}\geq{\bm{0}}$
(resp. ${\bm{v}}_{i}\leq{\bm{0}}$). Since ${\bm{v}}_{1}>{\bm{0}}$, we must
have ${\bm{v}}_{i}^{\top}{\bm{v}}_{1}>0$ (resp.
${\bm{v}}_{i}^{\top}{\bm{v}}_{1}<0$), leading to a contradiction. $\square$
Consider the _null hypothesis_ ${\cal T}_{0}$, using 1 and (5), we observe the
top eigenvector $\overline{\bm{v}}_{1}$ of ${\rm Cov}({\bm{y}}_{\ell})$ must
be a _positive vector_. For the _alternative hypothesis_ ${\cal T}_{1}$, we
observe from (5) that $\overline{\bm{v}}_{1}$ will be taken as one of the
other eigenvectors, ${\bm{v}}_{i}$, of ${\bm{S}}$ with $i\neq 1$. By 1,
$\overline{\bm{v}}_{1}$ must have at least one negative and positive element.
## 3 Identifying Low-pass Graph Signals
In this section we propose heuristics to detect first-order lowpass graph
signals and provide insights into its performance.
The discussions from the previous section suggest that one could distinguish
between ${\cal T}_{0},{\cal T}_{1}$ by inspecting whether all elements of the
top eigenvector of ${\rm Cov}({\bm{y}}_{\ell})$ have the same sign. We define:
$\textstyle\widehat{\bm{C}}_{y}^{m}\mathrel{\mathop{:}}=(1/m)\sum_{\ell=1}^{m}{\bm{y}}_{\ell}({\bm{y}}_{\ell})^{\top},~{}~{}\widehat{\bm{v}}_{i}=\textsf{$i$th-
EV}(\widehat{\bm{C}}_{y}^{m}),\vspace{-.1cm}$ (6)
such that $\widehat{\bm{C}}_{y}^{m}$ is the sampled covariance and
$\widehat{\bm{v}}_{i}$ is the latter’s eigenvector with the $i$th largest
eigenvalue. We further define the scoring function:
$\Gamma({\bm{v}})\mathrel{\mathop{:}}=\min\big{\\{}\|{\bm{v}}-({\bm{v}})_{+}\|_{2},\|{\bm{v}}+(-{\bm{v}})_{+}\\}\|_{2}\big{\\}}\vspace{-.1cm}$
(7)
where $({\bm{v}})_{+}=\max\\{{\bm{v}},{\bm{0}}\\}$ is the elementwise maximum
of ${\bm{v}}$ and ${\bm{0}}$. In both cases, we observe that
${\bm{v}}=({\bm{v}})_{+}$ if ${\bm{v}}$ is positive, and
${\bm{v}}=-(-{\bm{v}})_{+}$ if ${\bm{v}}$ is negative. As such,
$\Gamma({\bm{v}})=0$ if and only if ${\bm{v}}$ is a positive or negative
vector; otherwise, $\Gamma({\bm{v}})>0$.
Based on the scoring function (7), we propose the following heuristic to
detect if a set of graph signals are first-order lowpass filtered. Let
$\widehat{\cal T}$ be the detector output, we have:
$\widehat{\cal T}=\begin{cases}{\cal
T}_{0}&,~{}\text{if}~{}\Gamma(\widehat{\bm{v}}_{1})\leq\min_{i=2,...,n}\Gamma(\widehat{\bm{v}}_{i}),\\\
{\cal T}_{1}&,~{}\text{otherwise}.\end{cases}\vspace{-.1cm}$ (8)
Note that the detector is inspired by the observation in 1 that under ${\cal
T}_{0}$, the top eigenvector of the signal term’s population covariance must
be positive and it is the only such eigenvector.
### 3.1 Insights from Performance Analysis
Analyzing the performance of the detector (8) is challenging as it involves
the order statistics of $\\{\Gamma(\widehat{\bm{v}}_{i})\\}_{i=1}^{n}$. We
instead provide insights on the performance of (8) through analyzing the
effects of finite-sample, graph size and noise variance.
We assume $\widehat{\bm{v}}_{j}^{\top}\overline{\bm{v}}_{j}\geq 0$ without
loss of generality. We adopt the Davis-Kahan theorem from [25, Corollary 3] as
follows:
###### Lemma 2
Let $j\in\\{1,...,n\\}$, consider the eigenvectors $\overline{\bm{v}}_{j}$,
$\widehat{\bm{v}}_{j}$ from the population, sampled covariance. Let
$\widehat{\bm{v}}_{j}^{\top}\overline{\bm{v}}_{j}\geq 0$, it holds
$\|\widehat{\bm{v}}_{j}-\overline{\bm{v}}_{j}\|_{2}\leq\frac{2^{3/2}\|\widehat{\bm{C}}_{y}^{m}-{\rm
Cov}(\overline{\bm{y}}_{\ell})\|_{2}}{\min\\{\beta_{j-1}-\beta_{j},\beta_{j}-\beta_{j+1}))}\vspace{-.1cm}$
(9)
where $\beta_{j}$ denotes the $j$th largest eigenvalue of the population
covariance ${\rm Cov}(\overline{\bm{y}}_{\ell})$ with the convention
$\beta_{0}=\infty,\beta_{n+1}=-\infty$.
Let $r={\rm Tr}({\rm Cov}(\overline{\bm{y}}_{\ell}))/\|{\rm
Cov}(\overline{\bm{y}}_{\ell})\|_{2}$, the denominator in the r.h.s. of (9)
can be bounded with high probability [26, Remark 5.6.3]:
$\begin{split}\|\widehat{\bm{C}}_{y}^{m}-{\rm
Cov}(\overline{\bm{y}}_{\ell})\|_{2}&\textstyle\leq\sigma^{2}+\|\widehat{\bm{C}}_{y}^{m}-(1/m)\sum_{\ell=1}^{m}{\bm{y}}_{\ell}{\bm{y}}_{\ell}^{\top}\|_{2}\\\
&=\sigma^{2}+{\cal O}(\|{\rm
Cov}(\overline{\bm{y}}_{\ell})\|_{2}\sqrt{r/m}).\\\\[-2.84544pt] \end{split}$
(10)
Note that $r$ is the effective rank of ${\rm Cov}(\overline{\bm{y}}_{\ell})$
which is close to 1 for strong lowpass filters with $\eta_{1}\ll 1$, yet for
weak lowpass filter with $\eta_{1}\approx 1$, one has $r\approx n$. In both
situations, this shows $\widehat{\bm{v}}_{j}\approx\overline{\bm{v}}_{j}$ with
small noise and large number of samples.
For $j\neq 1$, it is known that $\|{\bm{v}}_{j}-({\bm{v}}_{j})_{+}\|_{2}>0$
where ${\bm{v}}_{j}$ is the $j$th eigenvector of ${\bm{S}}$. However, it is
not clear how large should this value be. To this end, we state the following
conjecture:
###### Conjecture 1
For $j\neq 1$, we have $\|{\bm{v}}_{j}-({\bm{v}}_{j})_{+}\|_{2}=\Theta(1)$.
The conjecture states that the magnitude of
$\|{\bm{v}}_{j}-({\bm{v}}_{j})_{+}\|_{2}$ is independent of the graph size
$n$. Our rationale is that the vector ${\bm{v}}_{j}$ is ‘non-localized’ whose
energy is evenly spread and there are ${\cal O}(n)$ negative elements, e.g.,
see [27] for insights behind the conjecture.
Case ${\cal T}_{0}$. We consider the null hypothesis when ${\cal H}({\bm{S}})$
is a first-order lowpass filer. Observe $\overline{\bm{v}}_{1}>{\bm{0}}$ and
we have
$\begin{split}\Gamma(\widehat{\bm{v}}_{1})&=\|\widehat{\bm{v}}_{1}-(\widehat{\bm{v}}_{1})_{+}\|_{2}\leq
2\|\widehat{\bm{v}}_{1}-\overline{\bm{v}}_{1}\|_{2}+\|\overline{\bm{v}}_{1}-(\overline{\bm{v}}_{1})_{+}\|_{2}\\\\[-2.84544pt]
\end{split}$
where we have applied
$\|({\bm{v}})_{+}-({\bm{v}}^{\prime})_{+}\|_{2}\leq\|{\bm{v}}-{\bm{v}}^{\prime}\|_{2}$
in the first inequality. Furthermore, we have
$\begin{split}\beta_{1}-\beta_{2}&\textstyle=|h(\lambda_{1})|^{2}-\max_{i=2,...,n}|h(\lambda_{i})|^{2}=|h(\lambda_{1})|^{2}(1-\eta_{1}).\end{split}$
Combining 2 and (10) yields the upper bound:
$\Gamma(\widehat{\bm{v}}_{1})\leq
2^{5/2}(1-\eta_{1})^{-1}\big{\\{}\sigma^{2}+{\cal O}(\sqrt{r/m})\big{\\}}.$
(11)
where we used $\|{\rm
Cov}(\overline{\bm{y}}_{\ell})\|_{2}=|h(\lambda_{1})|^{2}$ to simplify the
expression.
On the other hand, for any $j\neq 1$, we have the following lower bound to the
scoring function $\Gamma(\widehat{\bm{v}}_{j})$:
$\begin{split}\Gamma(\widehat{\bm{v}}_{j})&=\|\widehat{\bm{v}}_{j}-(\widehat{\bm{v}}_{j})_{+}\|_{2}\geq\|\overline{\bm{v}}_{j}-(\overline{\bm{v}}_{j})_{+}\|_{2}-2\|\widehat{\bm{v}}_{j}-\overline{\bm{v}}_{j}\|_{2}\\\
&\geq\Theta(1)-\frac{2^{5/2}(\sigma^{2}+{\cal
O}(|h(\lambda_{1})|^{2}\sqrt{r/m}))}{\min\\{\beta_{j-1}-\beta_{j},\beta_{j}-\beta_{j+1}\\}}.\end{split}$
(12)
where the last inequality is due to 1 and 2.
Case ${\cal T}_{1}$. For the alternative hypothesis where ${\cal H}({\bm{S}})$
is _not a first-order lowpass filter_. We observe that $\overline{\bm{v}}_{1}$
is no longer equal to the lowest frequency eigenvector of ${\bm{S}}$, i.e.,
${\bm{v}}_{1}$. Similar to (12), this yields the following lower bound of
$\Gamma(\widehat{\bm{v}}_{1})$:
$\Gamma(\widehat{\bm{v}}_{1})\geq\Theta(1)-2^{5/2}(\beta_{1}-\beta_{2})^{-1}(\sigma^{2}+{\cal
O}(|h(\lambda_{1})|^{2}\sqrt{r/m}))$ (13)
On the other hand, there exists $j\neq 1$,
$\overline{\bm{v}}_{j}={\bm{v}}_{1}$. Similar to (11), we observe
$\Gamma(\widehat{\bm{v}}_{j})\leq\frac{2^{5/2}(\sigma^{2}+{\cal
O}(|h(\lambda_{1})|^{2}\sqrt{r/m}))}{\min\\{\beta_{j-1}-\beta_{j},\beta_{j}-\beta_{j+1}\\}}.$
(14)
Comparing the observations in (11)–(14) shows that the detector (8) has a low
error rate when (i) the observation noise $\sigma$ is small, (ii) the number
of samples $m$ is large, which are the expected behaviors. On the other hand,
the effects of graph size $n$ is not immediately clear. We observe from (11)
that the detection performance is insensitive to $n,m,\sigma^{2}$ with a
strong lowpass filter that has $\eta_{1}\ll 1$ (and thus $r\approx 1$). On the
other hand, the detection performance may degrade with a weak lowpass filter
since $\eta_{1}\approx 1$ (and thus $r\approx n$).
$10^{1}$$10^{2}$$10^{3}$$0$$0.2$$0.4$$0.6$$0.8$Sample size $m$Scoring fct.
$\Gamma(\widehat{\bm{v}}_{1})$Weak L. Lap.Weak L. Adj.Strong L. Lap.Strong L.
Adj.
$10^{1}$$10^{2}$$10^{3}$$0$$0.2$$0.4$$0.6$$0.8$Sample size $m$Weak H. Lap.Weak
H. Adj.Strong H. Lap.Strong H. Adj.
$10^{1}$$10^{2}$$0$$0.2$$0.4$$0.6$Graph size $n$Scoring fct.
$\Gamma(\widehat{\bm{v}}_{1})$Weak L. Lap.Weak L. Adj.Strong L. Lap.Strong L.
Adj.
$10^{1}$$10^{2}$$0$$0.2$$0.4$$0.6$$0.8$Graph size $n$Weak H. Lap.Weak H.
Adj.Strong H. Lap.Strong H. Adj.
Fig. 1: Effects of $m,n$ on the scoring function
$\Gamma(\widehat{\bm{v}}_{1})$: (Left) lowpass filter (${\cal T}_{0}$) and
(Right) high pass filter (${\cal T}_{1}$). L./H. means low/highpass and
Lap./Adj. means Laplacian /Adjacency.
$10^{1}$$10^{2}$$0$$0.2$$0.4$Graph size $n$Error rateWeak Lap. Weak Adj.
Strong Lap. Strong Adj.
$10^{1}$$10^{2}$$10^{3}$$0$$0.2$$0.4$Sample size $m$
$10^{-1}$$10^{0}$$0$$0.2$$0.4$Noise variance $\sigma^{2}$
12102099$0$$0.2$$0.4$$0.6$$0.8$Graph frequency $i$Scoring
fct.$\Gamma(\widehat{\bm{v}}_{i})$$\Gamma(\widehat{\bm{v}}_{i})_{\infty}$
Fig. 2: Error rate against (Left) the number of samples $m$, (Middle-left) the
graph size $n$, and (Middle-right) the noise variance $\sigma^{2}$. Lap./Adj.
means Laplacian/Adjacency. Dashed lines are the detector with scoring function
$\Gamma({\bm{v}})_{\infty}\mathrel{\mathop{:}}=\|{\bm{v}}-({\bm{v}})_{+}\|_{\infty}\wedge\|{\bm{v}}+(-{\bm{v}})_{+}\\}\|_{\infty}$
with the same respective color/marker. (Right) Scoring function
$\Gamma(\widehat{\bm{v}}_{i})$ and $\Gamma(\widehat{\bm{v}}_{i})_{\infty}$
against the graph frequency of the Stock data’s sample covariance.
## 4 Numerical Experiments
Synthetic Data. For the experiments below, the $n$-node graph $G$ is generated
as an undirected Erdos-Renyi (ER) graph with connection probability of
$p=2\log(n)/n$. Each observed graph signal ${\bm{y}}_{\ell}\in\Re^{n}$ is
independently generated as ${\bm{y}}_{\ell}={\cal
H}({\bm{S}}){\bm{x}}_{\ell}+{\bm{w}}_{\ell}$ with excitation
${\bm{x}}_{\ell}\sim N({\bm{0}},{\bm{I}})$ and noise ${\bm{w}}_{\ell}\sim
N({\bm{0}},\sigma^{2}{\bm{I}})$. ${\cal H}({\bm{S}})$ is a graph filter with
two possible settings (a) ${\bm{S}}={\bm{L}}$ is Laplacian matrix or (b)
${\bm{S}}={\bm{A}}$ is binary adjacency matrix.
We verify the analysis in Section 3 by experimenting with four types of graph
filters ${\cal H}({\bm{S}})$ and we take an example of highpass filter to
represent the alternative hypothesis ${\cal T}_{1}$. The first set of
experiments considers pairs of _weak_ lowpass/highpass filters as ${\cal
H}({\bm{S}})$:
$\begin{split}&\text{Setting (a)}\ \ \ ({\cal T}_{0}):\
({\bm{I}}+\alpha{\bm{L}})^{-1},\ ({\cal T}_{1}):\ {\bm{I}}+\alpha{\bm{L}},\\\
&\text{Setting (b)}\ \ \ ({\cal T}_{0}):\ ({\bm{I}}-\alpha{\bm{A}})^{-1},\
({\cal T}_{1}):\ {\bm{I}}-\alpha{\bm{A}},\vspace{-0.1cm}\end{split}$ (15)
where $\alpha=0.5/d_{\max}$ with $d_{\max}$ being the highest degree of graph
$G$. The second set of experiments considers pairs of _strong_
lowpass/highpass filter as ${\cal H}({\bm{S}})$: let $\tau=10/d_{max}$,
$\begin{split}&\text{Setting (a)}\ \ \ ({\cal T}_{0}):\ e^{-\tau{\bm{L}}},\
({\cal T}_{1}):\ e^{\tau{\bm{L}}},\\\ &\text{Setting (b)}\ \ \ ({\cal
T}_{0}):\ e^{\tau{\bm{A}}},\ ({\cal T}_{1}):\
e^{-\tau{\bm{A}}}.\\\\[-2.84544pt] \end{split}$ (16)
It can be shown that the above filters under ${\cal T}_{0}$ are first-order
lowpass with $\eta_{1}\approx 1$ in (15), and $\eta_{1}\ll 1$ in (16).
In Fig. 1, we illustrate the effects of number of samples $m$ and graph size
$n$ on $\Gamma(\widehat{\bm{v}}_{1})$ [cf. (7)]. Fixing the noise variance at
$\sigma^{2}=0.01$, the averaged values of $\Gamma(\widehat{\bm{v}}_{1})$ over
1000 trials is plotted. In Fig. 1 (Top), we fix the graph size at $n=100$ and
observe that under ${\cal T}_{0}$, the scoring function
$\Gamma(\widehat{\bm{v}}_{1})$ decays as the number of samples $m$ grows. On
the other hand, in Fig. 1 (Bottom), we fix the sample size at $m=1000$ and
observe that under ${\cal T}_{0}$, $\Gamma(\widehat{\bm{v}}_{1})$ may increase
as the graph size $n$ grows for weak lowpass filters. In both comparisons,
under ${\cal T}_{1}$, the scoring function floats around a constant value
$\geq 0.4$ irrespective of $m,n$. The above findings are consistent with (11),
(13), which predicts that under ${\cal T}_{0}$, the scoring function may
increase as the graph size $n$ grows, thereby leading to the degraded
detection performance with (8) as we shall illustrate next.
We next examine the detection performance measured in terms of the error rate
with equal number of data samples coming from pairs of weak and strong
low/highpass filter in (15), (16). We define:
$\text{\sf Error rate}\mathrel{\mathop{:}}=0.5\cdot{\cal P}(\widehat{\cal
T}={\cal T}_{1}|{\cal T}_{0})+0.5\cdot{\cal P}(\widehat{\cal T}={\cal
T}_{0}|{\cal T}_{1}),\vspace{-.1cm}$ (17)
and evaluate the averaged error rate from 1000 trials. For benchmarking
purpose, we also simulate a similar detector as (8) but replace the scoring
function in (7) with one that is computed by the $\ell_{\infty}$ norm, e.g.,
$\Gamma({\bm{v}})_{\infty}\mathrel{\mathop{:}}=\|{\bm{v}}-({\bm{v}})_{+}\|_{\infty}\wedge\|{\bm{v}}+(-{\bm{v}})_{+}\\}\|_{\infty}$.
The results are presented in Fig. 2. We compare the averaged error rate
against graph size $n$, sample size $m$, noise variance $\sigma^{2}$, while
fixing $(m,\sigma^{2})=(1000,0.01)$, $(n,\sigma^{2})=(100,0.01)$,
$(n,m)=(100,1000)$, respectively. For each parameter setting in the
experiments, we generate $m$ samples for each hypothesis, i.e., with
lowpass/highpass filter, in order to evaluate (17). To distinguish between
weak lowpass/highpass filter [cf. (15)], a larger $m$ reduces the error rate
while a larger $n$ raises the error rate. Moreover, the error rate reduces
with a small noise variance. For the experiments with pairs of strong
lowpass/highpass filter [cf. (16)], the detection performance is almost
invariant with $n,m,\sigma^{2}$, i.e., the error rate is close to 0 for all
cases. Lastly, we observe that the $\ell_{2}$ norm scoring function has
consistently outperformed its $\ell_{\infty}$ norm counterpart. The above
results are consistent with the prediction in Section 3.
Real Data. We consider identifying first-order lowpass signals from 3 real
datasets. The first dataset (Stock) is the daily return from S&P100 stocks in
May 2018 to Aug 2019 with $n=99$ stocks, $m=300$ samples, collected from
https://www.alphavantage.co/. The second dataset (Senate) contains $m=696$
votes grouped by $n=50$ states at the US Senate in 2007 to 2009, collected
from https://voteview.com. The third dataset (COVID-19) is the daily increment
of COVID-19 confirmed cases in the US from May 5th 2020 to Oct 15th 2020 with
$n=44$ states, $m=164$ samples, collected from https://covidtracking.com/.
12 102050$0$$0.2$$0.4$$0.6$$0.8$Graph frequency $i$Scoring
fct.$\Gamma(\widehat{\bm{v}}_{i})$$\Gamma(\widehat{\bm{v}}_{i})_{\infty}$
12102041$0$$0.2$$0.4$$0.6$$0.8$Graph frequency
$i$$\Gamma(\widehat{\bm{v}}_{i})$$\Gamma(\widehat{\bm{v}}_{i})_{\infty}$
Fig. 3: Scoring function $\Gamma(\widehat{\bm{v}}_{i})$,
$\Gamma(\widehat{\bm{v}}_{i})_{\infty}$ against the graph frequency order.
(Left) Senate data (Right) COVID-19 data. Fig. 4: Estimated centrality of 44
states in COVID-19 data. Deeper color indicates the state has a higher
centrality.
Fig. 2 (Right), Fig. 3 plots the scoring functions
$\Gamma(\widehat{\bm{v}}_{i})$ against the eigenvalue order $i$ for the 3
dataset considered. We first observe that the Stock data would satisfy ${\cal
T}_{0}$ since $\Gamma(\widehat{\bm{v}}_{1})<\Gamma(\widehat{\bm{v}}_{j})$ for
all $j\neq 1$, indicating that it is likely to be a set of first-order lowpass
graph signals. This certifies that lowpass GSP tools can be applied to the
dataset. For example, applying the method from [14, 15] ranks the centrality
of stocks in the decreasing order as: NVDA, NFLX, AMZN, ADBE, PYPL, CAT, MA,
GOOG, GOOGL, BA; see [14] for details. For the Senate data, we have
$\Gamma(\widehat{\bm{v}}_{2})<\Gamma(\widehat{\bm{v}}_{j})$ for all $j\neq 2$
which suggests that the data may not be first-order lowpass. However, since
the minimum occur at $\Gamma(\widehat{\bm{v}}_{2})$, it is plausible that the
data is generated from a lowpass graph filter with cutoff frequency at
$\lambda_{2}$. For the COVID-19 data, we see that
$\Gamma(\widehat{\bm{v}}_{1})<\Gamma(\widehat{\bm{v}}_{j})$ for $2\leq j\leq
40$, yet $\Gamma(\widehat{\bm{v}}_{j})$ are small again for $j=41,...,44$.
This suggests the sampled covariance has more than one (close-to) positive
eigenvector. We suspect that this abnormally is due to outliers such as the
beginning wave of a COVID-19 infection event at a state.
Inspired by the above, we model the COVID-19 data as a set of first-order
lowpass graph signals (with outliers). Again, we can apply the blind
centrality estimation method from [14, 15] to rank the centrality of states.
The results are illustrated in Fig. 4. The top states ranked in decreasing
centrality are FL, TX, CA, GA, LA, TN, SC, AL, NC, NY. Some of these states
are the transportation hubs.
Conclusions. This paper utilizes the Perron-Frobenius theorem to design a
simple, data-driven detector for identifying first-order lowpass graph
signals. The detector can be used to provide certificates for applying lowpass
GSP tools, and to make inference about the type of network dynamics. Future
works include verifying 1, designing detectors for _higher-order_ lowpass
graph signals.
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|
Serena Dipierro & Enrico ValdinociUniversity of Western Australia,
35 Stirling Highway,
Crawley WA 6009, Australia<EMAIL_ADDRESS>
Elliptic partial differential equations
from an elementary viewpoint
CHAPTER: PREFACE
more to life than mathematics”,
Joan said. “But not much more”.
Greg Egan, Glory.
These notes are the outcome of some courses taught to undergraduate and graduate students
from the University of Western Australia,
the Pontifícia Universidade Católica do Rio de Janeiro and
the Indian Institute of Technology Gandhinagar
in 2021.
Far from aiming at being all-encompassing, the following pages wish to
shed some light on a number of selected topics in the theory
of elliptic partial differential equations with a style that should be
accessible to third-year undergraduate students, possibly under an
inspired mentorship, but might
also provide some interest to more advanced students
and possibly professional researchers. While all the
topics presented are of classical flavor,
the exposition and the way the material are organized is
perhaps rather original, with the intention of addressing several quite
difficult points with a style that is as self-contained as possible,
rigorous as well as intuitive,
and approachable without major prerequisites (indeed,
we only assume prior knowledge of the “basic” analysis, giving also
reference to useful results and theorems whenever the less advanced readers
may need to consolidate their backgrounds).
We stress again that the list of topics covered here is far from being exhaustive,
since we mainly dealt with classical problems related to the Laplace operator
and their natural counterpart for equations in nondivergence form
(the divergence structure case is not really covered here,
though the Divergence Theorem is obviously utilized ubiquitously, the variational structure of the Laplace-Beltrami
operator is discussed quite in detail
and several pages are devoted to the theory of capacity which possesses a variational essence).
Also, we do not address here nonlinear, singular/degenerate elliptic operators, fully nonlinear equations
and fractional/nonlocal elliptic equations.
We do not even linger too much on explicit solutions
and cheap tricks to find them (e.g., we do not repeat over and over very specific methods such as the separation of variables),
since, after all, it is very unlikely that one can solve explicitly[Moreover, explicit solutions
may provide a handy resource to develop an initial theory, but they usually do not exhaust the complexity
of the problem. For example, according to <cit.>,
it is quite common to run into “one of the serious problems with such exact solutions [...]: namely, they often do not determine all possible solutions
and indeed, may not even give the most relevant one”.]
a partial differential equation,
instead it is often more useful to understand the qualitative and quantitative properties
of the solution without solving explicitly the corresponding equation.
The many parts of the theory of elliptic partial differential equations which are missing from these pages
are by no means less important than the topics covered: the issue is that we had to make
a rather harsh selection of topics just to be able to collect all the material into finitely many pages
(arguably, Jorge Luis Borges would remark that a comprehensive treatment of elliptic equations
can be found in the Library of Babel <cit.>).
Some additional fun for the interested reader could come from
the treatment of a few natural problems with geometric flavor, such as
the Soap Bubble Theorem and Serrin's overdetermined problem. To treat them,
but this is also a general approach used everywhere in the book,
we tried to provide different perspectives and different
points of view, also providing physical motivations
to trigger creativity and imagination, and favoring a dynamic interplay between geometric and analytic arguments.
These notes also collect a number of classical, albeit not always well known,
topics, which we believe can serve as an excellent training camp to develop
some familiarity with elliptic equations.
Here are
a couple of features of these notes that we hope can be appealing for many readers. First, all is built in an artisanal way, perhaps not by following the most elegant, concise or general approach, but aiming at a possibly slow, but conceptually clear, sequence of strategic steps. Second, this set of notes is planned in a way that one can read starting almost any place and freely jump from one topic to another just by following a personal stream of thoughts.
Though a list of classical and modern references is given at the end,
this is not a book focused on the history
of mathematics, hence we will not address topics such as priority
of discoveries or progress of subject over the course of time.
As usual, these notes may contain errors or inconsistencies: if you find
any, please let us know. In general, we will be happy to receive
comments and criticisms to possibly improve this work.
After all, scientific knowledge is based on a dynamic flow
of information and we
will certainly cherish and treasure readers' feedback and advice.
All right, enough chitchat, please fasten your PFD,
it's time to start our journey together.
Serena and Enrico
Courtesy of Burbuqe Shaqiri.
CHAPTER: MOTIVATIONS
§ THE LAPLACIAN COMING ON THE SCENE
Without aiming at being exhaustive, we present here some classical circumstances in which partial differential
equations, and especially those relying on the Laplace operator, naturally arise to model interesting natural phenomena.
The objective is not much to use mathematics to find the Answer to the Ultimate Question of Life, the Universe, and Everything
(which is well known to be $42$ anyway) but rather to develop some familiarity with some of the basic features
of partial differential equations, to see them in action and to build an intuition and an instinctive feeling
of the problems,
which will turn out handy in our voyager through the mighty jungle of technical and seemingly abstract mathematics.
§.§ The heat equation
Among the many occurrences in which elliptic equations naturally arise in nature, one of the most widely popularized comes from the stationary state of the heat equation.
In its most basic formulation, the setting of this equation is due to
Jean-Baptiste Joseph Fourier who[Interestingly, while changing the course of mathematics and physics, Fourier was acting as a full time politician, since in 1801 Napoléon Bonaparte decided to appoint him as Prefect of the Department of Isère, in the Alps. And perhaps it was not a completely trivial task to upset the First Consul (Emperor of the French from 1804) and dedicate oneself to mathematical investigations; in any case, playing around
with the heat equation at that point was not really part of Fourier's workload model.
See Figure <ref>
for a watercolor caricature of Fourier (by Julien-Léopold Boilly).] presented for the first time a partial differential equation to describe conductive diffusion of heat (and, to study this equation, he also introduced a marvelous instrument that was going to change forever the History of the Universe, namely, the Fourier Series).
Caricature of Joseph Fourier (Public Domain image from
The ansatz[“Ansatz” (plural: “Ansätze”) is a masculine German name.
As quite customary in German, this name is obtained by a preposition (“an”,
meaning “at”, or “on”, or “to”, or “against”, or pretty much whatever one likes) and another name
(“Satz”, plural: “Sätze”, meaning “sentence”, or “statement”). And as quite customary in German,
a precise translation goes well beyond the expertise of modest mathematicians of our caliber.
The “ansatz” could be literally a “starting point”. In jargon, it is frequently used to denote
an “educated guess” about a problem,
typically an additional simplifying assumption set at the beginning of an argument,
possibly to be verified later on (if it produces any result worthy of further consideration).
Maybe mathematics is not quite a deductive discipline, after all. Perhaps it is mostly the art of making good guesses.
Of course, anybody can make guesses. But for a good guess one needs to become acquainted with the problem
and to have developed that sort of familiarity that permits to separate the essential aspects from
the minor details (and, in trying to do this, sometimes
a little bit of luck doesn't hurt).] made by Fourier is that, at some[The space variable $x$ here is supposed to be in $\R^n$. Though the models presented refer to physical spaces, in which one can focus on the case $n=3$ (or even $n=2$, when dealing with plates, or $n=1$, when dealing with ropes of negligible thickness), whenever possible we prefer to work in (Euclidean) spaces of arbitrary dimension. When one does that, the notation becomes usually more effective and the essential treats of the main ideas are very often more transparent. Also, when it will be needed to focus our attention on the special case of particular dimensions, one will be able to understand why and to what extent these dimensions are different from the general case. In dealing with the arbitrary dimension case, we can't help quoting the first “general working principle” in the Preface of <cit.>: “PDE theory is (mostly) not restricted to two independent variables. Many texts describe PDE as if functions of the two variables $(x, y)$ or $(x, t)$ were all that matter. This emphasis seems to me misleading, as modern discoveries concerning many types of equations, both linear and nonlinear, have allowed for the rigorous treatment of these in any number of dimensions. I also find it unsatisfactory to classify partial differential equations: this is possible in two variables, but creates the false impression that there is some kind of general and useful classification scheme available in general”.]
point $x\in\R^n$, at an instant of time $t\in\R$, the variation of temperature $u(x,t)$ of some body is determined by the heat flow around the point under consideration, plus possibly additional heat[For the sake of precision, we should probably say here “heating or cooling sources”,
e.g., fireplaces, bonfires, stoves, radiators, gas heaters, infrared heaters, but also refrigerators, air conditioners, swamp boxes, chilled beams, or whatever heating or cooling device Fourier might have possessed at that time.
With this understanding, positive values of $f$ would correspond to heating systems and negative values to cooling systems.]
sources. Let us denote by $B(x,t)$ the heat
flux vector and by $f(x,t,u(x,t))$ the scalar intensity of the external[To make things simpler, one can suppose that $f$ depends only on $x$, as a heat source which is permanently switched on or off, or on $x$ and $t$, in case the source is turned on and off or its intensity is modified over time. The additional dependence on $u$ itself models the interesting case of a thermostat controlling a heating or cooling system.] heat sources.
A precise measure of the temperature precisely at a point $x$ is certainly rather difficult from a practical point of view, so it is useful sometimes to consider instead an average[As usual, we use
here (and repeatedly over this set of notes) the integral notation for averagesaverage, that is
$$ \fint_\Omega u(x,t)\,dx:=\frac1{|\Omega|}\,\int_\Omega u(x,t)\,dx,$$
being $|\Omega|$ the Lebesgue measure of $\Omega$.]
temperature, measured in some region of the space $\Omega$,
$$ U(t;\Omega):=\fint_\Omega u(x,t)\,dx.$$
In this setting, up to physical constants that we omit, we can equate the variation of $U$
with the heat flux through $\partial\Omega$, possibly adding to it the effect of the heat sources (if any) in $\Omega$,
namely we write that
\begin{equation}\label{DAGB-ADkrVoiweLL4re2346ytmngrrUj}
\partial_t U(t;\Omega)=-\frac1{|\Omega|}\int_{\partial\Omega} B(x,t)\cdot\nu(x)\,d{\mathcal{H}}^{n-1}_x+\fint_\Omega f(x,t,u(x,t))\,dx.\end{equation}
In terms of notation, we are denoting here (and we will mostly keep this notation throughout the all set of notes)
by $\nu(x)$ the unit normal vector at $x\in\partial\Omega$ pointing outwards from $\Omega$ and
by $d{\mathcal{H}}^{n-1}_x$ the surface element[In many textbooks, the surface
elementsurface element is denoted by $ds$, or $dS$, or $d\Sigma$. The
notation $d{\mathcal{H}}^{n-1}_x$ comes from the fact that we are denoting by ${\mathcal{H}}^k$ the $k$-dimensional
Hausdorff measure, and, for smooth $(n-1)$-dimensional surfaces, the surface measure itself is precisely
equal to ${\mathcal{H}}^{n-1}$. See e.g. <cit.> for a thorough presentation of the Hausdorff measure.
The notation $d{\mathcal{H}}^{n-1}_x$ used here is perhaps a bit heavier than $ds$, or $dS$, or $d\Sigma$, or similar ones, but it has the benefit of being clearer and more explicit. Also, it allows us to consider surface integrals along less regular objects without having to introduce a new notation on a case-by-case basis.] on $\partial\Omega$. The minus sign appearing on the right hand side of (<ref>) is due to the fact that the normal $\nu$ points towards the exterior, while we are computing there the flux coming into the region $\Omega$.
By differentiating under the integral sign[In this chapter about motivations, the arguments are developed at a formal level, we feel free to exchange derivatives and integrals, we do not keep track of lower order terms in the expansions, we do not discuss convergence issues and existence of limits, etc. This is a rather customary approach when one deals with providing convincing, but not necessarily circumstantial, motivations for a problem with the objective of developing some intuition about it. We promise to try to be more rigorous from the next chapter on.] in (<ref>), we find that
\begin{equation*}\fint_\Omega\partial_t u(x,t)\,dx=-\frac1{|\Omega|}\int_{\partial\Omega} B(x,t)\cdot\nu(x)\,d{\mathcal{H}}^{n-1}_x+\fint_\Omega f(x,t,u(x,t))\,dx.\end{equation*}
Hence, by the Divergence[Typically reserve the notation “$\nabla$” for the vectors of the derivatives with respect to the space variables and “$\div$” for the corresponding divergence. No confusion should arise with respect to derivatives with respect to the time variable, which is usually denoted here by “$\partial_t$”.]
\begin{equation}\label{DAGB-ADkrVoiweLL4re2346ytmngrrUj2}\fint_\Omega\partial_t u(x,t)\,dx=-\fint_{\Omega} \div B(x,t)\,dx+\fint_\Omega f(x,t,u(x,t))\,dx.\end{equation}
From this, Fourier took a further step by realizing that to make this identity manageable one needs to
take a constitutive law about the flux vector $B$, relating it to the temperature. Fourier's new ansatz was to suppose that the
flux of heat between two adjacent (infinitesimal) regions is proportional to the (infinitesimal) difference of their temperatures, namely
\begin{equation}\label{DAGB-ADkrVoiweLL4re2346ytmngrrUj3}
B(x,t)=-\kappa(x,t)\,\nabla u(x,t).
\end{equation}
The proportional coefficient $\kappa$ is sometimes called “heat conduction coefficient”heat conduction coefficient. We take $\kappa$ to be positive:
with respect to this, we stress that the minus sign in (<ref>) is motivated by the fact that the heat flows from hotter regions to colder ones (hence in the opposite direction of the growth of the temperature function $u$).
The relation in (<ref>) is also sometimes[Actually, the same presentation
here could have been done to describe the transport of mass through diffusive means.
In this setting, one argue that the flux goes from regions of high concentration to regions of low concentration.
The ansatz corresponding to (<ref>) would be that the
magnitude of the flux is proportional to the concentration gradient.
In the context of mass diffusion, $\kappa$ is sometimes called “diffusion coefficient”
and (<ref>) is sometimes refereed to with the name of
“Fick's Law”Fick's Law after the German physician and physiologist
Adolf Eugen Fick.
Alternative choices for the constitutive relation are possible. For instance, one can replace (<ref>) with a more general equation in which the flux depends possibly in a nonlinear way
from the gradient of the temperature (in the case of Fourier's model, or of the transported mass in the case of Fick's model),
that is one could suppose that $B=-\Phi(\nabla u)$, for some function $\Phi:\R^n\to\R^n$. A typical example is the case in which $\Phi(\nabla u)=|\nabla u|^{p-2}\nabla u$, that is $B=-\kappa|\nabla u|^{p-2}\nabla u$,
meaning that the flux is proportional to a power of the gradient of $u$ (more precisely, the vector $B$ has the same direction of the gradient of $u$ and its magnitude is proportional to a power of the magnitude of the gradient of $u$):
this setting would lead to the so-called “$p$-Laplace equation”.
Another setting of interest is the one in which the flux is related to a pressure drop
(this framework is usually related to the so-called Darcy's LawDarcy's Law), say $B=-\kappa\nabla P$, where $P$ has some physical meaning of pressure. In this sense, the case $P=u$ reduces to (<ref>),
but other situations are of interest. For instance, one could assume that $P$ and $u$ are related by
a state equation of the form $P(x)=p(u(x))$. A natural possibility is to take the function $p$ to be a
power of $u$, e.g. $p=u^m$. This choice would lead to the so-called porous medium equation.
These notes are of elementary nature, hence we will not address the cases of the $p$-Laplace equation,
or of the porous medium equation, or other types of “anomalous diffusions”, such as the type of diffusion that takes into
account mass transfer from remote regions due to long-range interactions.
The reader interested in these more advanced topics may look e.g. to <cit.> and the references therein.
For the readers interested in using Darcy's Law to have a proper cup of coffee, see e.g. <cit.>
and the references therein.] called “Fourier's Law”.
Portrait of Pierre-Simon Laplace by Johann Ernst Heinsius (image from
licensed under the Creative Commons Attribution-Share Alike 4.0 International license).
By substituting (<ref>) into (<ref>) we find that
\begin{equation}\label{DAGB-ADkrVoiweLL4re2346ytmngrrUj4}
\fint_\Omega\partial_t u(x,t)\,dx=\fint_{\Omega} \div \big(
\kappa(x,t)\,\nabla u(x,t)
\big)\,d{\mathcal{H}}^{n-1}_x+\fint_\Omega f(x,t,u(x,t))\,dx.\end{equation}
Since this identity holds true for all regions $\Omega$, we find a pointwise counterpart of (<ref>) by writing
\begin{equation}\label{DAGB-ADkrVoiweLL4re2346ytmngrrUj6}
{\partial_t u(x,t)}= \div \big(
\kappa(x,t)\,\nabla u(x,t)
\big)+f(x,t,u(x,t)).\end{equation}
The case of homogeneous media in which $\kappa$ is constant (say, equal to $1$ up to a renormalization of units of measure)
is of particular interest: in this case (<ref>) boils down[We exploit the
standard notation for the LaplacianLaplacian given by
$$ \Delta u:=\div(\nabla u)=\sum_{j=1}^n \frac{\partial^2 u}{\partial x_j} .$$
In some textbook, the use of “$\Delta$” is replaced by “$\nabla^2$”
or by “$|\nabla|^2$”. Of course all notations are good. Personally, we have a preference for
the $\Delta$ notation since we find it simpler to read. Also, in a sense, it highlights the fact that the Laplace operator
has a “dignity” which is “independent from the one of the gradient”.
This “philosophical” point will be perhaps clarified by the geometric interpretation of the Laplacian that we will discuss
in the forthcoming Theorem <ref>.
The name of the Laplace operator comes from Pierre-Simon, marquis de Laplace, who introduced it
while studying celestial mechanics, in connection with the gravitational potential (this approach will
be exploited here when dealing with the fundamental solution in Section <ref>.
See Figure <ref>
for a portrait of Laplace (by Johann Ernst Heinsius) when he was 35 years old, with an easel that quite resembles
a blackboard, working instrument and sign of distinction of every passionate mathematician.
Let us remark that a slightly different approach to the derivation of the heat equation is possible,
by considering “heat” instead of “temperature” as the main building block of the equation.
The disadvantage of having heat, rather than temperature, in a pivotal role is that it is perhaps a more vague and less intuitive concept
than temperature (just because temperature seems a notion we are so familiar with, e.g. in view of weather forecasts).
However, the notion of heat relates more directly to thermal energy and constitutes an extensive (i.e., additive)
property. In this perspective,
the heat equation reflects an energy budget
in which the heat change in time is equal to heat produced (source) plus heat entering through the boundary (flux).
Namely, if one is willing to take heat, instead of temperature, as the building block for the heat equation,
the balance equation in (<ref>) reads at the level of
variation of heat, instead of averaged temperature
(and note that the constitutive relation in the Fourier's Law (<ref>)
is already a relation between the notions of heat and temperature, since it assumes the
heat flux to be proportional to the gradient of temperature).
The final heat equation in (<ref>) would remain essentially
unchanged since the changes of heat content $Q$ directly relates to the changes in temperature
via the relation $\partial_t Q= c \rho \partial_tu$, where $\rho$ is mass density and $c$ is the specific heat capacity (a constant that depends on the internal properties of the material, which can be tabulated through experiments).]
\begin{equation}\label{DAGB-ADkrVoiweLL4re2346ytmngrrUj7}
\partial_t u(x,t) = \Delta u(x,t)+f(x,t,u(x,t)).\end{equation}
Equation (<ref>) (or variations of it) is typically[Often,
equations as in (<ref>) are referred to as “parabolic”.
The name comes from the following classification method.
A general linear differential operator of second order in the variables $(X_1,\dots,X_N)$ has the form
$$ \sum _{i,j=1}^{N} a_{i,j}{\frac {\partial ^{2}}{\partial X_{i}\partial X_{j}}}.$$
Then one classifies the operator, and the corresponding partial differential equation,
depending on the sign of the eigenvalues of the coefficient matrix $a_{i,j}$.
More specifically, when all the eigenvalues of $a_{i,j}$ have the same sign (either are all strictly
positive or all strictly negative), the equation is named “elliptic”elliptic.
When one eigenvalue is zero and all the other eigenvalues have the same sign
(either all strictly
positive or all strictly negative), the equation is referred to as “parabolic”parabolic.
When all the eigenvalues have the same sign except one which has the opposite sign
(i.e. if either there is only one negative eigenvalue and all the rest are positive,
or there is only one positive eigenvalue and all the rest are negative),
the equation is named “hyperbolic”hyperbolic.
This classification is not exhaustive: the remaining cases not classified here are typically quite
hard to study, require specific techniques and no general theory is available for those.
In the case of the heat equation in (<ref>), one has $N=n+1$, $X=(x,t)$ and
$$ a_{ij}=\begin{dcases}1 & {\mbox{ if }}i=j\in\{1,\dots,n\},\\
0&{\mbox{ otherwise }}.\end{dcases}$$
The corresponding eigenvalues are therefore $0$, with multiplicity $1$, and $1$, with multiplicity $n$,
hence the equation in (<ref>) is parabolic.]
called “the heat equation”heat equation.
Suppose now that $f$ in (<ref>) is independent of time, i.e. $f=
f(x,u(x,t))$. In this framework,
of particular interest are certainly stationary solutions of (<ref>),
i.e. solutions $u=u(x)$ which are also independent of time: these solutions are actually equilibria of (<ref>), since they provide solutions of (<ref>)
which remain the same at every instant of time. Of course, when $f$ and $u$ are independent of time,
equation (<ref>) boils down to[With respect to the classification
presented in footnote <ref>, equation (<ref>),
(<ref>) and (<ref>) are of elliptic type
(actually, they provide us with the “paradigmatic” form of elliptic equations).
To check their ellipticity, one exploits the setting of footnote <ref>
with $N=n$, $X=x$ and
1 & {\mbox{ if }}i=j,\\
0 &{\mbox{ if }}i\ne j.
\end{dcases}$$
In this scenario, all the eigenvalues of the coefficient matrix
are equal to $1$, hence strictly positive.
Further insight on the notion of ellipticity will be given on pages CLASSIFICATIONFOOTN2
and CLASSIFICATIONFOOTN3.
The above definition of $\delta_{ij}$ will be also used throughout all these notes (sometimes
this definition of $\delta_{ij}$ is referred to with the name of Kronecker notationKronecker notation).]
\begin{equation}\label{DAGB-ADkrVoiweLL4re2346ytmngrrUj8}
\Delta u(x)+f(x,u(x))=0.\end{equation}
In particular, when $f$ is independent of $u$, equation (<ref>) reduces to
\begin{equation}\label{DAGB-ADkrVoiweLL4re2346ytmngrrUj9}
\Delta u(x)+f(x)=0,\end{equation}
which is sometimes called Poisson's equation.
The case in which $f$ in (<ref>) vanishes identically produces
\begin{equation}\label{DAGB-ADkrVoiweLL4re2346ytmngrrUj10}
\Delta u(x)=0,\end{equation}
which is called Laplace's equationLaplace's equation.
Solutions of (<ref>) are named “harmonic functions”harmonic function.
The study of harmonic functions is essential, one way or another, in virtually any aspect of mathematics
(including mathematical analysis, mathematical physics, complex analysis, geometry, probability, finance, statistics, you name it). Also, the theory of harmonic functions is a clear example of adamantine beauty and elegance
in which human creativity has reached one of its highest peaks ever.
In the forthcoming pages we will do our best to present some bits of this theory, but we definitely invite
the reader to look to as many other sources as possible to collect the broadest possible amount of information and
develop their own point of view on this topic of paramount importance.
§.§ Population dynamics, chemotaxis and random walks
Another fascinating situation in which partial differential equations surface very often
occurs in the description of biological populations. This is a very captivating field of study,
which is truly crossdisciplinary in spirit and collects fundamental questions from (at least) mathematics, physics,
biology, ethology and social sciences.
For the purpose of these pages, we limit ourselves to a simple description of a
biological species exhibiting at a point $x\in\R^n$ and at a time $t\in\R$ a population density of the form $u(x,t)$.
We assume that a fraction $\rho\in[0,1]$ of
the population has the tendency of moving randomly, while the rest of the population
(corresponding to a fraction $\mu:=1-\rho$ of the totality of individuals) is driven
towards regions of higher concentration of a given attractant (e.g. a chemical signal)
distributed according to a function $w(x,t)$ (the
movement of an organism in response to a chemical stimulus and specifically
in dependence of the increasing or decreasing concentration of a particular substance
is called in jargon “chemotaxis”chemotaxis).
Additionally, we suppose that the population is subject to a driftdrift (e.g. due to the wind, or the stream, or the tide)
in a given direction $b(x,t)$.
The simpler cases in which there is no chemotactic effect (corresponding to $\mu:=0$),
or no random motion (corresponding to $\rho:=0$), or no drift (corresponding to $b:=0$)
are interesting special situations of the complex phenomena for which we now present a mathematical model
(the random component of this discussion will also be further generalized on page 0uojf29249-45kpkfdSmd11493839429efv
to more elaborated environments).
In outline, we consider a small spacial scale $h$ and a small time scale $\tau$.
We will choose in what follows a suitable relation between space and time scales to provide
coherent asymptotics in the formal limit.
In this setting,
at every unit of time the variation of the density of
the population is influenced by the previous factors in a rather explicit way. For the moment, let us focus on the simple case in which there is no chemotaxis and no drift. In this situation, $\rho:=1$, $\mu:=0$, $b$ vanishes identically
and the only drive for the population comes from a random walk.
To get to the bottom of this case,
we consider, at a given point $x$, at time $t+\tau$, the new random population $\rho u(x,t+\tau)$
(which actually coincides with the whole population $u(x,t+\tau)$, but let us keep the fraction $\rho$
for future use, even if for the moment $\rho=1$). Then, we have that $\rho u(x,t+\tau)$
is produced by the random population at the previous time $t$ which is located at some point $x+h\omega$, for a given direction $\omega\in\partial B_1$, times the probability
that these individuals move from $x+he$ to $x$ in the unit of time. If all directions are equally probable, this says that the new random population $\rho u(x,t+\tau)$ is produced by a term of the form $\rho\fint_{\partial B_1} u(x+h\omega,t)\,d{\mathcal{H}}^{n-1}_\omega$, that is
\begin{equation}\label{JOSN-Oohpiklfoigheigohbo-gherpoguev}
\rho u(x,t+\tau)
=\rho\fint_{\partial B_1} u(x+h\omega,t)\,d{\mathcal{H}}^{n-1}_\omega.
\end{equation}
Subtracting $\rho u(x,t)$ from both sides, dividing by $\tau$ and choosing $h:=\sqrt\tau$,
\begin{equation}\label{7tnfesewuxttau-x21s46}\begin{split}
\rho \frac{u(x,t+\tau)-u(x,t)}\tau\,&
=\,\frac1{\tau}\left[\rho\fint_{\partial B_1} u(x+h\omega,t)\,d{\mathcal{H}}^{n-1}_\omega-\rho u(x,t)\right]\\&
=\,\frac{\rho}{h^2}\fint_{\partial B_1} \Big(u(x+h\omega,t)-u(x,t)\Big)\,d{\mathcal{H}}^{n-1}_\omega.\end{split}
\end{equation}
Now we aim at showing that, for small $h$,
\begin{equation}\label{KMS:RUIDKVODIG}
\fint_{\partial B_1} \Big(u(x+h\omega,t)-u(x,t)\Big)\,d{\mathcal{H}}^{n-1}_\omega=
\frac{c h^2}2 \Delta u(x,t) +o(h^2),
\end{equation}
for some $c>0$ depending on the dimension $n$ (see formula (<ref>) below).
For this, we
consider the formal Taylor expansion
$$ u(x+h\omega,t)=u(x,t)+h\nabla u(x,t)\cdot \omega+\frac{h^2}2\,D^2u(x,t) \omega\cdot \omega+o(h^2).$$
We observe, by symmetry, that
$$ \int_{\partial B_1} \nabla u(x,t)\cdot \omega\,d{\mathcal{H}}^{n-1}_\omega=0,$$
because[Spotting these types of cancellations will be often an essential ingredient of our arguments.
After all, one way to obtain significant information about a complex problem is to understand
which quantities do not play much of a role since they “average out”.
At this stage, we discuss these simplifications at an intuitive level. Later on, see e.g. (<ref>),
we will give more rigorous arguments to justify them, but for the moment the main goal is to try to visualize the
possibility of these cancellations and, especially, to appreciate their importance.]
any (positive or negative) contribution to such surface integral coming from a point $\omega\in\partial B_1$
is canceled precisely by the (negative or positive) contribution coming from $-\omega$.
From these observations we arrive at
\begin{eqnarray*} &&\fint_{\partial B_1} \Big(u(x+h\omega,t)-u(x,t)\Big)\,d{\mathcal{H}}^{n-1}_\omega
=\fint_{\partial B_1} \left(h\nabla u(x,t)\cdot \omega+\frac{h^2}2\,D^2u(x,t) \omega\cdot \omega\right)\,d{\mathcal{H}}^{n-1}_\omega+o(h^2)\\&&\qquad=
\frac{h^2}2\,\fint_{\partial B_1}D^2u(x,t) \omega\cdot \omega\,d{\mathcal{H}}^{n-1}_\omega+o(h^2)=\frac{h^2}2
\sum_{i,j=1}^n\fint_{\partial B_1}\partial_{ij} u(x,t) \omega_i\omega_j\,d{\mathcal{H}}^{n-1}_\omega+o(h^2)
We now identify an additional simplification by observing that if $i\ne j$ then
$$ \fint_{\partial B_1}\partial_{ij} u(x,t) \omega_i\omega_j\,d{\mathcal{H}}^{n-1}_\omega=0,$$
because whatever contribution comes from $\omega_i$ is canceled precisely by that coming from $-\omega_i$
(notice how helpful was for all these cancellations that we are integrating over such a symmetric domain as $\partial B_1$,
which remains invariant under all these reflections).
$$ \fint_{\partial B_1} \Big(u(x+h\omega,t)-u(x,t)\Big)\,d{\mathcal{H}}^{n-1}_\omega=\frac{h^2}2
\sum_{i=1}^n\fint_{\partial B_1}\partial_{ii} u(x,t) \omega_i^2\,d{\mathcal{H}}^{n-1}_\omega+o(h^2).
Now, once again, we understand the symmetries of the problem under consideration. For this, we notice that,
for every $i$, $j\in\{1,\dots,n\}$
$$ \fint_{\partial B_1} \omega_i^2\,d{\mathcal{H}}^{n-1}_\omega
=\fint_{\partial B_1} \omega_j^2\,d{\mathcal{H}}^{n-1}_\omega,$$
because[Once again, we aim here at developing arguments in the most intuitive way possible.
This idea with be more rigorously retaken later on, see (<ref>).]
the role played by the $i$th coordinate in the sphere is precisely the same as the one played by the $j$th coordinate.
Therefore, we can define
\begin{equation}\label{CDEGLDDEFC} c:=\fint_{\partial B_1} \omega_i^2\,d{\mathcal{H}}^{n-1}_\omega\end{equation}
and we stress that this quantity does not depend on $i$.
$$ \fint_{\partial B_1} \Big(u(x+h\omega,t)-u(x,t)\Big)\,d{\mathcal{H}}^{n-1}_\omega=\frac{c h^2}2
\sum_{i=1}^n \partial_{ii} u(x,t) +o(h^2),
from which we obtain (<ref>), as desired.
Hence we insert (<ref>) into (<ref>) to find that
\begin{equation}\label{y8oqiwhfesjd8923wfyug3iuweLuhonwfdpinrgNqikedOKS}\begin{split}&
\rho \frac{u(x,t+\tau)-u(x,t)}\tau
=\frac{c\rho}{2}\Delta u(x,t) +o(1).
\end{split}
\end{equation}
By sending $\tau\searrow0$ (and thus $h\searrow0$), we thereby find that
\begin{equation}\label{HJA:A89MAOL8789892}
\rho \partial_t u(x,t)
= \frac{c\rho}{2}\Delta u(x,t) ,
\end{equation}
which coincides, up to constant, with the heat equation[See
e.g. <cit.>
for more information about the strong connections between
the random walk and the heat equation.] presented in (<ref>)
(here, with no source term).
Hard-working scientists (Public Domain image from
This computation was not only instructive from the technical point of view, but it also unveiled one of the mainsprings
of the mathematical theory of diffusion, revealing the strong conceptual connection
of a substance (or a population) moving randomly according to a Brownian motionBrownian motion and the way in which
heat dissipates: after all, the spreading of temperature all over the region subject to diffusion is, in some sense, nothing else[The link between the random motion of molecules and the
macroscopic phenomenon of diffusion is indeed a deep feature. While the Brownian motion was discovered
in 1827 by botanist Robert Brown by looking at dust grains floating in water, Albert Einstein
in his “annus mirabilis” 1905
gave a quantitative model for the motion of floating particles as being moved by individual water molecules <cit.>. Not only this article
founded the statistical physics analysis of Brownian motion, but also it
provides a way to determine the mass and the dimensions of the atoms involved in the
process, thus changing atomic theory from a controversial set of conjectures
into an established fact of science. We will come back to the theory proposed by Einstein in Section <ref>.
See Figure <ref>
depicting Albert Einstein together with Niels Bohr.
By the way, Niels Bohr's brother was Harald Bohr, who was a mathematician, pioneering almost periodic functions, and a soccer player. The Denmark national soccer team in which he was playing took part in the 1908 Summer Olympics, where football was an official event for the first time. Harald Bohr scored two goals in the first match and played in the semifinal (Denmark 17 - France 1, which remains the Olympic record of the most goals scored by one team). After that, the Denmark team lost the final and won the silver medal. In all the subsequent editions of the Olympic Games, Denmark soccer team won two more silver medals (in 1912 and 1960) and a bronze (in 1948); hence Harald Bohr's team also holds the record (ex aequo) of best placement for a Denmark soccer team in the Olympics.
See Figure <ref>
for a photo of the Danish soccer team in the 1908 Olympic Games: Harald Bohr is the second player from the left in the top row.
Niels Bohr was also a passionate soccer player. He played goalkeeper in the Copenhagen-based team Akademisk Boldklub (at the time, one of the best clubs in Denmark; actually, the two brothers played several matches together in this team). One can wonder however why Niels never made it to the national team. Well, according to https://www.theguardian.com/football/2005/jul/27/theknowledge.panathinaikos
in a match against a German team, one of the midfielders of the opposite team launched a very long shot and Niels, leaning against the post, did not react at all, missing an easy save and letting the German team score. After the game, Niels admitted that on that occasion he had been distracted by a mathematical problem he was thinking about. Strangely enough, Niels did not play for Akademisk Boldklub after that season.]
but the tendency of random movements to distribute mass in average from regions of high density
towards regions of small density. In both cases, the process
has the strong tendency to “balance out” differences:
as time goes, hot spots lose their temperature in favor of cold spots which are heated up by
the temperature of their neighbors, as well as under a random walk
the regions with low density get occupied by the population
coming from the highly populated regions.
This “democratic” tendency of averaging out differences is typical of the Laplace operator and it
will be the underlying feature of all the “regularity theories” for elliptic equations that we will present in the forthcoming
Danish soccer team at the 1908 Olympic Games (Public Domain image from
Having well understood the case in which the biological species is subject solely on a random motion,
we now retake the interesting case in which a chemotactic agent
also comes into play, for a fraction $\mu$ of the population (for the moment, we still suppose that there is no additional drift, hence $b$ vanishes identically). In this situation,
the fraction $\rho$ of the population performing the random walk would be still described
by (<ref>) (that was the reason to include $\rho$ in the previous
computation, even if before $\rho$ was just equal to $1$) and we focus now on
the fraction $\mu$ of the population following the attractant $w$.
To appreciate the effect of the chemotactic factor, we suppose that such a fraction $\mu$ of the population with density $u$ does not move completely randomly, but, at each time step, picks a direction of motion with a probability $\lambda$ that is proportional to values of the attractant having density $w$.
More explicitly, given $\omega\in\partial B_1$,
we suppose for the chemotactic population that a jump from the point $x$ to the point $x+h\omega$
occurs with probability
\begin{equation}\label{RISUGIBNSUPJDMD} \lambda(x,\omega,t):=\frac1{{\mathcal{H}}^{n-1}(\partial B_1)}+w(x+h\omega,t)-w(x-h\omega,t).\end{equation}
In this setting, we are assuming that the oscillations of $w$ are sufficiently small[More precisely,
one could rewrite (<ref>) as
\[ \lambda(x,\omega,t):=\frac1{{\mathcal{H}}^{n-1}(\partial B_1)}+\e_0\Big(w(x+h\omega,t)-w(x-h\omega,t)\Big),\]
where $\e_0>0$ is a parameter that takes into account the “sensibility” of the chemotactic population
to the chemical attractant. Just to be consistent with the probability scenario, it is convenient
to think that $\lambda\ge0$ in view of the smallness of $\e_0$ with respect to the oscillations
of the attractor's density $w$.]
to make the above quantity positive.
In this sense, the probability $\lambda$ differs from that of the classical random walk (corresponding to $w$
being constant) since it increases in favor of the directions $\omega$ in which the concentration of the chemical
attractant is higher. We stress that the above setting provides a normalized probability,
since the total probability of
jumping to the sphere of radius $h$ around the given point $x$ is $1$, because
\begin{equation*}
\int_{\partial B_1}\lambda(x,\omega,t)\,d{\mathcal{H}}_\omega^{n-1}
=1+\int_{\partial B_1}\Big(w(x+h\omega,t)-w(x-h\omega,t)\Big)\,d{\mathcal{H}}_\omega^{n-1}=1,
\end{equation*}
due to odd symmetry cancellations.
We also point out that, in the above setting, the probability of a jump from a point $a$ to
a point $b$ with $|a-b|=h$ is given by
\begin{eqnarray*} \lambda\left(a,\frac{b-a}{h},t\right)&=&\frac1{{\mathcal{H}}^{n-1}(\partial B_1)}+w\left(a+h\frac{b-a}{h},t\right)-w\left(a-h\frac{b-a}{h},t\right)\\&=&
\frac1{{\mathcal{H}}^{n-1}(\partial B_1)}+w(b,t)-w(2a-b,t).\end{eqnarray*}
In consequence, given $\omega\in\partial B_1$,
taking $a:=x+h\omega$ and $b:=x$, we find that
the jump from $x+h\omega$ to $x$ occurs with probability
$$ \lambda (x+h\omega,-\omega,t)=
\frac1{{\mathcal{H}}^{n-1}(\partial B_1)}+w(x,t)-w(x+2h\omega,t).$$
This being so, we can detect the chemotactic counterpart of (<ref>).
Indeed, since the chemotactic population $\mu u(x,t+\tau)$ is produced by a term of the form $\mu\int_{\partial B_1} \lambda (x+h\omega,-\omega,t) u(x+h\omega,t)\,d{\mathcal{H}}^{n-1}_\omega$, we have that
\begin{equation*} \begin{split}
\mu u(x,t+\tau)&=\mu\int_{\partial B_1} \lambda (x+h\omega,-\omega,t) u(x+h\omega,t)\,d{\mathcal{H}}^{n-1}_\omega\\
&=\mu\int_{\partial B_1} \left(\frac1{{\mathcal{H}}^{n-1}(\partial B_1)}+w(x,t)-w(x+2h\omega,t)\right)u(x+h\omega,t)\,d{\mathcal{H}}^{n-1}_\omega\\&=\mu\fint_{\partial B_1} u(x+h\omega,t)\,d{\mathcal{H}}^{n-1}_\omega
+\mu\int_{\partial B_1} \Big(w(x,t)-w(x+2h\omega,t)\Big)u(x+h\omega,t)\,d{\mathcal{H}}^{n-1}_\omega
Thus, we can subtract to both sides $\mu u(x,t)$, divide by $\tau=h^2$ and recall (<ref>)
and (<ref>) to find that
\begin{eqnarray*}
&&\mu \frac{u(x,t+\tau)-u(x,t)}\tau\\ &=&
\frac1{h^2}\left[\mu\fint_{\partial B_1}\Big( u(x+h\omega,t)-u(x,t)\Big)\,d{\mathcal{H}}^{n-1}_\omega+
\mu\int_{\partial B_1} \Big(w(x,t)-w(x+2h\omega,t)\Big)u(x+h\omega,t)\,d{\mathcal{H}}^{n-1}_\omega
\right]\\&=& \frac{c\mu}{2}\Delta u(x,t) +
\frac{\mu}{h^2}\int_{\partial B_1} \Big(w(x,t)-w(x+2h\omega,t)\Big)u(x+h\omega,t)\,d{\mathcal{H}}^{n-1}_\omega
+o(1)\\&=& \frac{c\mu}{2}\Delta u(x,t)-
\frac{\mu}{h^2}\int_{\partial B_1} \Big(
2h\nabla w(x,t)\cdot\omega
\Big)\Big(u(x,t)+h\nabla u(x,t)\cdot\omega\Big)\,d{\mathcal{H}}^{n-1}_\omega\\&&\qquad\qquad\qquad
+o(1)\\&=& \frac{c\mu}{2}\Delta u(x,t) -
\frac{\mu}{h^2}\int_{\partial B_1} \Big(
2h^2\nabla w(x,t)\cdot\omega\,\nabla u(x,t)\cdot\omega
+2h^2D^2w(x,t)\omega\cdot\omega\, u(x,t)\Big)\,d{\mathcal{H}}^{n-1}_\omega
+o(1)\\&=& \frac{c\mu}{2}\Delta u(x,t) -
2\mu \int_{\partial B_1} \Big(
\nabla w(x,t)\cdot\omega\,\nabla u(x,t)\cdot\omega
+D^2w(x,t)\omega\cdot\omega\, u(x,t)\Big)\,d{\mathcal{H}}^{n-1}_\omega\\&&\qquad\qquad\qquad
+o(1)\\&=& \frac{c\mu}{2}\Delta u(x,t) -
2\mu \sum_{i=1}^n\int_{\partial B_1} \Big(\partial_i w(x,t)\partial_i u(x,t)\omega_i^2
+\partial_{ii}w(x,t) u(x,t)\omega_i^2\Big)\,d{\mathcal{H}}^{n-1}_\omega
\\&=& \frac{c\mu}{2}\Delta u(x,t) -
2\mu \sum_{i=1}^n\int_{\partial B_1} \partial_i\big(\partial_i w(x,t) u(x,t)\big)\omega_i^2
\,d{\mathcal{H}}^{n-1}_\omega
+o(1)\\&=& \frac{c\mu}{2}\Delta u(x,t) -
\widetilde{c}\mu \sum_{i=1}^n \partial_i\big(\partial_i w(x,t) u(x,t)\big)
+o(1)\\&=& \frac{c\mu}{2}\Delta u(x,t) -
\widetilde{c}\mu \div\big(u(x,t) \nabla w(x,t)\big)
\end{eqnarray*}
for a suitable $
\widetilde{c}>0$.
Notice that here above we have repeatedly taken advantage of odd symmetry cancellations.
Gathering this and (<ref>) we thereby find that
\begin{equation}\label{DRIFPAG2}\begin{split}&
\frac{u(x,t+\tau)-u(x,t)}\tau=
\rho \frac{u(x,t+\tau)-u(x,t)}\tau +\mu \frac{u(x,t+\tau)-u(x,t)}\tau\\&\qquad
=\frac{c\rho}{2}\Delta u(x,t) +\frac{c\mu}{2}\Delta u(x,t) -
\widetilde{c}\mu \div\big(u(x,t) \nabla w(x,t)\big)+o(1)\\&\qquad
=\frac{c}{2}\Delta u(x,t) -
\widetilde{c}\mu \div\big(u(x,t) \nabla w(x,t)\big)+o(1).\end{split}\end{equation}
Sending $\tau\searrow0$ (and thus $h\searrow0$) we obtain
\begin{equation}\label{CHEMLKSDEQ}
\partial_t u(x,t)=\frac{c}{2}\Delta u(x,t) -
\widetilde{c}\mu \div\big(u(x,t) \nabla w(x,t)\big).\end{equation}
For constant distributions $w$ of the chemical attractant, equation (<ref>)
can be seen as a special case (up to normalizing constants) of the heat equation in (<ref>).
Stationary solutions of (<ref>) solve
\begin{equation}\label{OJS-PJDN-0IHGDOIUGDBV02ujrf}
\frac{c}{2}\Delta u(x,t) -
\widetilde{c}\mu \div\big(u(x,t) \nabla w(x,t)\big)=0.\end{equation}
This is the equation solved by the equilibria corresponding to a biological population in the presence of
a chemotactic factor.
Let us now introduce one last complication in the model by accounting for a possible drift
(as mentioned on page DRIFPAG). In this situation, we may suppose that, in the time unit,
the population is also moved by the velocity vector $b$ by a space displacement $b\tau$.
Regarding this, the population density $u(x+2\tau,t)$
has in fact moved to the location $x+b\tau$: correspondingly, to take into account this drift,
one can observe that $ u(x,t+2\tau)=u(x-b(x,t)\tau,t+\tau)$, and accordingly, recalling (<ref>)
and using that
$$u\big(x-b(x,t)\tau,t+\tau\big)=u(x,t )
-\tau b(x,t)\cdot\nabla u (x,t )+\tau\partial_t u(x,t)
we have that
\begin{eqnarray*}
&& \frac{u(x,t+2\tau)-u(x,t)}\tau= \frac{u(x,t+2\tau)-u(x,t+\tau)}\tau+ \frac{u(x,t+\tau)-u(x,t)}\tau
\\&&\qquad
\frac{c}{2}\Delta u(x,t) -
\widetilde{c}\mu \div\big(u(x,t) \nabla w(x,t)\big)+o(1)\\&&\qquad=
\frac{u(x,t )
-\tau b(x,t)\cdot\nabla u (x,t )+\tau\partial_t u(x,t)-
u(x,t)-\tau\partial_t u(x,t) +o(\tau)
\\&&\qquad\qquad+
\frac{c}{2}\Delta u(x,t) -
\widetilde{c}\mu \div\big(u(x,t) \nabla w(x,t)\big)+o(1)\\&&\qquad=
-b(x,t)\cdot\nabla u (x,t )+
\frac{c}{2}\Delta u(x,t) -
\widetilde{c}\mu \div\big(u(x,t) \nabla w(x,t)\big)+o(1)
\end{eqnarray*}
Thus, we can pass to the limit and obtain
\begin{equation}\label{OJS-PJDN-0IHGDOIUGDBV02ujrf220}
\partial_t u(x,t)=\frac{c}{2}\Delta u(x,t) -
\widetilde{c}\mu \div\big(u(x,t) \nabla w(x,t)\big)-b(x,t)\cdot\nabla u(x,t).\end{equation}
Stationary solutions of (<ref>) give rise to the equation
\begin{equation}\label{OJS-PJDN-0IHGDOIUGDBV02ujrf22}
\Delta u(x,t) -
\div\big(u(x,t) \nabla w(x,t)\big)-b(x,t)\cdot\nabla u(x,t)=0,\end{equation}
up to normalizing constants that we have omitted for the sake of simplicity.
In this spirit, (<ref>) models[The equation
in (<ref>), together with the diffusion of the attractant,
reveals the complexity
of the model under consideration which, depending on different parameter thresholds, exhibit a variety of
different stable geometrical patterns, colony formation, blowup mechanisms, flocculation and aggregation phenomena.
All these traits play a fundamental role for life as we know it, since these configurations can arise
from equations such as the ones presented here and possess quite a precise counterpart in nature, when
the corresponding behaviors occur, for example, in response to external stress,
or to lack of resources (sometimes the chemical attractant is produced by the biological species itself,
e.g. by amoebae in case of food scarceness) or in response to predation
(autoaggregation provides protection and defense against predators). These aggregation processes often produce slimy yet durable coatings, called biofilms. Even blowup phenomena
can be exploited by bacteria to enhance their possibility of surviving in hostile environments
(e.g., the individual on the top of the tower formed by the blowup have more chances to be picked up, say, by the wind
or by an external factor, and be deposited possibly in a less hostile environment), so all in all the variety
of patterns exhibited by the solutions of a relatively simple differential equation is in some sense the mathematical counterpart
of the variety of ways in which “life finds a way” (as Ian Malcolm utters in “Jurassic Park”).] the equilibria reached by a biological species possibly subject
to chemotaxis and to an external drift.
Notice that (<ref>) reduces to (<ref>)
in the absence of drift.
We have written (<ref>) in a form which emphasizes its “divergence structure”:
if instead one wants to highlight the presence of a second order
differential operator, it suffices to notice that $\div(u \nabla w)=u\Delta w+\nabla w\cdot\nabla u$ to recast (<ref>) into
\begin{equation}\label{OJS-PJDN-0IHGDOIUGDBV02ujrf22NDF}
\Delta u(x,t) -\big( b(x,t)+\nabla w(x,t)\big)\cdot\nabla u(x,t)- u(x,t)\Delta w(x,t)=0,\end{equation}
which is also a telling expression since it is revealing that the effect of the chemotaxis
is to alter the diffusion with an additional drift term $\nabla w$
which is favoring the movements towards regions with higher concentrations
of the the attractant.
We refer to <cit.> and the references therein for further information about chemotaxis and its biological implications.
§.§ Pattern formation, or how the leopard gets its spots
We know well that the natural world offers a variety of amazing
regular patterns, such as symmetries, tessellations, spirals, spots, etc.,
see e.g. Figure <ref>.
These patterns visible in nature
are determinant factors in the processes of natural and sexual selection,
providing organisms with structures recognizable by their conspecifics (thus favoring social
interactions and reproduction) and often realizing optimal configurations for mobility, hunt or camouflage (see e.g. <cit.> and the references therein).
Leopard mating dance (image from
Wikipedia, licensed under the Creative Commons Attribution 2.0 Generic license;
see Steve Jurvetson's website https://www.flickr.com/photos/jurvetson/5913330010/
for the complete series of shots).
The biological process leading to the development of the specific shape
of an organism is dubbed “morphogenesis” morphogenesis
and a scientific investigation of the formation of patterns in nature
is relatively recent.
One of the pioneer scientists interested in the mathematical analysis of the growth, form and evolution of plants and animals
was Sir D'Arcy Wentworth Thompson: his 1116 pages book <cit.>
(793 pages in the first edition of 1917) combined classical natural philosophy,
biology and mathematics to give insight on a number of biological shapes
and analyzed the differences in the forms in nature in the light of mathematical transformations.
A modern mathematical treatment of morphogenesis was initiated
by[Not only the father of mathematical morphogenesis,
computer science and artificial intelligence, Alan Turing was also an exceptional long-distance runner,
capable of world-class marathon standards. See https://kottke.org/18/04/alan-turing-was-an-excellent-runner
for a nice photo of Turing during one of his running performances.
During World War II, Turing worked as a code breaker for the British Intelligence,
devising an electromechanical machine, named “Bombe”,
to decipher the encrypted messages created by the German cipher device “Enigma”.
The headquarter of the British code breakers was located in a
mansion named Bletchley Park which hosts today a statue of Turing
made with slates (by Stephen Kettle). In this statue, Turing is depicted seated and looking at a German Enigma machine,
see Figure <ref>.
Turing was 39 years old when he was charged with “gross indecency” on the basis of a homosexual relationship
under Section 11 of the Criminal Law Amendment Act 1885.
The case, Regina v Turing and Murray, was brought to trial on 31 March 1952
and Turing pleaded guilty, insisting that he saw nothing wrong with his actions.
He was convicted and given a choice between imprisonment and chemical castration via synthetic estrogens.
As a result of choosing the second option, Turing suffered of severe mutations of his body,
was barred from his work for Government Communication Headquarters and
was denied entry into the United States.
On June 8 1954, Turing was found dead of cyanide poisoning with a half-eaten apple
next to his bed lay. The inquest into his death recorded a verdict of suicide, but
the apple was not tested for cyanide.
Turing's favorite fairy tale was Snow White and the Seven Dwarfs.]
Alan Mathison Turing in 1952, see <cit.>.
Turing's brilliant idea is that simple physical laws were sufficient to justify
the shaping of complex patterns, such as
animal markings and arrangement of leaves and florets in plants.
In a nutshell, Turing proposed that the process of morphogenesis
is regulated by the interaction of chemical substances, called morphogens, which
diffuse through a tissue: these substances could be
hormones, or genes, or any other essence which can act and react in presence of another one.
In practice, see e.g. <cit.>, one of the morphogens may act as an
“activator”, which is self-sustaining and introduces positive feedback, while the other
may play the role of an “inhibitor” which tends to suppress the self-amplification of the activator.
In this interplay, the pattern may be created by the different speeds of diffusion
of the two substances:
namely, the faster diffusion of the inhibitor
can catch up with the activator's self-replication (that is, roughly speaking, on the one side the activator's capacity of self-replicating could be strong enough to produce
local patches, but the predominant speed of the inhibitor could avoid that these patches grow
incessantly). All in all, the whole process can be thereby considered as a “diffusion driven instability”. See <cit.> for a very clear explanation of the roles of activators and inhibitors, via an example with sweating grasshoppers.
Statue of Alan Turing (photo by Jon Callas; image from
Wikipedia, licensed under the Creative Commons Attribution 2.0 Generic license.).
The mathematical formulation of Turing's idea combines the notion of diffusion,
as modeled for instance by the heat equation (<ref>),
and that of reaction, taking into account that each morphogen can chemically react with the others
and the effect of this interaction can depend on the concentration of the diffusing substances in the tissue. The combination of reaction and diffusion in this type
of systems of partial differential equations justified the name
of “reaction-diffusion
equations”. reaction-diffusion equation
For instance, one can consider the case of two morphogens with density $u$ and $v$ respectively
and a system of reaction-diffusion equations of the form
\begin{equation}\label{SYS:READI}
\begin{dcases}
&\partial_t u=\mu\Delta u+f(u,v),\\
&\partial_t v=\nu\Delta v+g(u,v),
\end{dcases}
\end{equation}
where $\mu$ and $\nu$ are positive coefficients which model the speed of diffusion
of the chemical substances with density $u$ and $v$ respectively.
Now, the formation of patterns as an outcome of (<ref>)
is, in a sense, not completely intuitive: on the contrary, given the
“democratic” tendency of the Laplace operator (as discussed on page DEEFFNVST)
one may imagine that the solution $(u(t),v(t))$, as $t\to+\infty$, will evolve spontaneously towards
some constant values $(u_0,v_0)$, which are just the common zeros of $f$ and $g$
(that is, such that $f(u_0,v_0)=g(u_0,v_0)=0$). And this is indeed one of the possible
destination for the solutions of (<ref>). However, there is also a more intriguing possibility:
the constant state $(u_0,v_0)$ may well exist (and it may also be “stable” for small
perturbations when $\mu:=0$ and $\nu:=0$), but it may be triggered off by random disturbances.
In this situation, when $\mu>0$ and $\nu>0$, the solutions may end up drifting away
from the constant $(u_0,v_0)$. Thus (unless for some reasons the solution diverges)
several interesting patterns may arise, such as oscillations between equilibria,
stationary waves, moving fronts, etc.
Though a full understanding of Turing's theory of morphogenesis goes well beyond the scopes of this
set of notes, following <cit.> one can at least grasp some of the ideas involved. For instance,
one can consider a simple subcase of (<ref>) in which $f$ and $g$ are linear functions
(this simplification can also be inspiring to treat the general case, since, in the vicinity of
the constant equilibrium $(u_0,v_0)$,
one can try to “linearize the equation” to obtain information on its dynamics).
That is, let us consider the system of equations
\begin{equation}\label{52568-345256890hro3}
\begin{dcases}
&\partial_t u=\mu\Delta u+a(u-u_0)+b(v-v_0),\\
&\partial_t v=\nu\Delta v+c(u-u_0)+d(v-v_0),
\end{dcases}
\end{equation}
for some $a$, $b$, $c$, $d\in\R$. For simplicity, let us also suppose that the problem is set on a circle,
namely $x\in\R$ and $u$ and $v$ are periodic functions of period $2\pi$.
One can thus look for solutions of (<ref>) in Fourier Series of the form
\begin{equation}\label{52568-345256890hro4} u(x,t)=u_0+\sum_{j\in\Z} U_j(t)\,e^{ijx}\qquad{\mbox{and}}\qquad
v(x,t)=v_0+\sum_{j\in\Z} V_j(t)\,e^{ijx},\end{equation}
with $U_j$ and $V_j$ to be determined.
Substituting (<ref>) into (<ref>) we have that
\begin{eqnarray*}&&
\sum_{j\in\Z} \dot U_j(t)\,e^{ijx}=
\partial_t u=\mu\Delta u+a(u-u_0)+b(v-v_0)\\&&\qquad
= -\mu\sum_{j\in\Z} j^2U_j(t)\,e^{ijx}
+a\sum_{j\in\Z} U_j(t)\,e^{ijx}+b\sum_{j\in\Z} V_j(t)\,e^{ijx}
\end{eqnarray*}
and similarly
\[ \sum_{j\in\Z} \dot V_j(t)\,e^{ijx}=
-\nu\sum_{j\in\Z} j^2V_j(t)\,e^{ijx}
+c\sum_{j\in\Z} U_j(t)\,e^{ijx}+d\sum_{j\in\Z} V_j(t)\,e^{ijx}.\]
From these equations we arrive at
\begin{equation}\label{52568-345256890hro4-1} \left(\begin{matrix} \dot{U}_j(t)\\ \dot{V}_j(t)
\end{matrix}\right)=
\left( \begin{matrix}a-j^2\mu & b\\
c& d-j^2\nu\end{matrix}\right)
\left(\begin{matrix} {U}_j(t)\\ {V}_j(t)
\end{matrix}\right).
\end{equation}
This is a first order ordinary differential equation with constant coefficients.
Hence, we suppose for simplicity that the (possibly complex) eigenvalues of the matrix $\left( \begin{matrix}a-j^2\mu & b\\
c& d-j^2\nu\end{matrix}\right)$, which we denote by $\lambda_j$ and $\Lambda_j$, are distinct.
We also denote by $w_j$ and $W_j\in \C^2$ the corresponding eigenvectors.
With this notation,
we find (see e.g. <cit.>) that the solutions of (<ref>)
are of the form
\begin{equation}\label{52568-345256890hro4-2} \left(\begin{matrix} {U}_j(t)\\ {V}_j(t)\end{matrix}\right)
=\xi_j \,e^{\lambda_j t} w_j+\Xi_j\, e^{\Lambda_j t} W_j
\end{equation}
for some $\xi_j$, $\Xi_j\in\C$.
Strem plot of the system of ordinary differential equations in (<ref>).
It is convenient to use the vector notation
$$ \xi_j=\left(\begin{matrix} \xi_{j1}\\ \xi_{j2}\end{matrix}\right)\qquad{\mbox{and}}\qquad
\Xi_j=\left(\begin{matrix} \Xi_{j1}\\ \Xi_{j2}\end{matrix}\right)$$
and let
$$ A_j:=\xi_j w_{j1},\qquad B_j:=\Xi_j W_{j1},\qquad
C_j:=\xi_j w_{j2}\qquad{\mbox{and}}\qquad D_j:=\Xi_j W_{j2}.$$
In this way, (<ref>) yields that
\begin{equation}\label{UJ:lao-myND9ona9nyuton-0201}
U_j(t)=A_j e^{\lambda_j t}+B_j e^{\Lambda_j t}\qquad{\mbox{and}\qquad}
V_j(t)=C_j e^{\lambda_j t}+D_j e^{\Lambda_j t}.\end{equation}
\begin{equation*}\begin{split}&\left( \begin{matrix}a-j^2\mu & b\\c& d-j^2\nu\end{matrix}\right)
\left( \begin{matrix}A_j\\ C_j\end{matrix}\right)=
\left( \begin{matrix}a-j^2\mu & b\\c& d-j^2\nu\end{matrix}\right)
\left( \begin{matrix}\xi_jw_{j1} \\ \xi_jw_{j2}\end{matrix}\right)
\\&\qquad=\xi_j\left( \begin{matrix}a-j^2\mu & b\\c& d-j^2\nu\end{matrix}\right)w_j=
\xi_j\lambda_jw_j=\lambda_j\left( \begin{matrix}A_j\\C_j\end{matrix}\right),
\end{split}\end{equation*}
leading to
\begin{equation}\label{UJ:lao-myND9ona9nyuton-0202}
(a-j^2\mu-\lambda_j)A_j+b C_j=0.
\end{equation}
\begin{equation}\label{UJ:lao-myND9ona9nyuton-0203}
(a-j^2\nu-\Lambda_j)B_j+b D_j=0.\end{equation}
In jargon, (<ref>) and (<ref>)
are sometimes dubbed “dispersion relations”dispersion relation:
their interest lies in the fact that they relate the speed of oscillation in the time variable
(quantified in (<ref>)
by the eigenvalues $\lambda_j$ and $\Lambda_j$) with the spatial periodicity of the medium
(characterized by the eigenvalues $-j^2$ of the one-dimensional Laplacian and modulated
by the speeds of diffusion $\mu$ and $\nu$).
To recap briefly, from the system of reaction-diffusion equations in (<ref>)
one arrives at the solutions introduced in (<ref>),
with $U_j$ and $V_j$ as in (<ref>),
\begin{equation}\label{o2re2s3p4on21di21ng214ly}
{\mbox{$\lambda_j$ and~$\Lambda_j$ are (distinct)
complex eigenvalues of the matrix }}\left( \begin{matrix}a-j^2\mu & b\\
c& d-j^2\nu\end{matrix}\right),\end{equation}
and with the parameters satisfying (<ref>)
and (<ref>).
As detailed in <cit.>
and <cit.>, this explicit mathematical construction
has a number of important biological consequence and presents a sufficiently rich structure
to account for many patterns visible in nature. To see these features, one may focus
on the case in which one of the eigenvalues has the largest real part (roughly speaking,
one expects that the other modes are dominated by this one). Also,
it is convenient to distinguish between the case in which the dominant eigenvalue is real
from the one in which it is complex and with nonzero imaginary part: indeed, real eigenvalues will be related to stationary states
and complex eigenvalues to oscillatory cases.
More specifically, suppose that
\begin{equation}\label{CMOPSDEArajeke24w}
{\mbox{$\Lambda_{j_0}$ is the eigenvalue with largest real part.}}\end{equation}
We notice that also $\Lambda_{-j_0}=\Lambda_{j_0}$, since
\begin{equation}\label{INVAMAJJ}
{\mbox{the matrix }}\left( \begin{matrix}a-j^2\mu & b\\ c& d-j^2\nu\end{matrix}\right) {\mbox{ remains the same if we exchange~$j$ with~$-j$}}.\end{equation}
Hence, dropping the higher order terms, we can assume that the dynamics of the solutions in (<ref>) is governed by the following long-time asymptotics:
\begin{equation}\label{INVAMAJJ2}
\begin{split}&
u(x,t)\simeq u_0+ U_{j_0}(t)\,e^{i{j_0}x}+U_{-j_0}(t)\,e^{-i{j_0}x}
\simeq u_0+ e^{\Lambda_{j_0} t}\Big( B_{j_0} e^{i{j_0}x}+B_{-j_0} e^{-i{j_0}x}\Big)\\
v(x,t)\simeq v_0+ e^{\Lambda_{j_0} t}\Big( D_{j_0} e^{i{j_0}x}+D_{-j_0} e^{-i{j_0}x}\Big).\end{split}
\end{equation}
The invariance in (<ref>) also suggests that if $A_j$, $B_j$, $C_j$ and $D_j$ are solutions of (<ref>) and (<ref>), then so are $A_{-j}$, $B_{-j}$, $C_{-j}$ and $D_{-j}$: for this reason, we can suppose that $B_{-j_0}=B_{j_0}$
and $D_{-j_0}=D_{j_0}$
in (<ref>), obtaining that
\begin{equation}\label{INVAMAJJ3}
\begin{split}&
u(x,t)\simeq u_0+ B_{j_0} e^{\Lambda_{j_0} t}\Big( e^{i{j_0}x}+e^{-i{j_0}x}\Big)=
u_0+ 2B_{j_0} e^{\Lambda_{j_0} t}\cos( j_0x)\\
v(x,t)\simeq v_0+2D_{j_0} e^{\Lambda_{j_0} t}\cos( j_0x).\end{split}
\end{equation}
Without aiming at exhausting all the possible patterns included in (<ref>),
let us now show a concrete case of interest.
For instance, let $\vartheta\in\N$,
\begin{equation}\label{G1A2MbdRbnU2U8n4ngVA9}\gamma:=
\frac{33\,\vartheta^2}{13\sqrt{3}-9}
\end{equation}
and also
\begin{equation}\label{MSUOJS5UIHJ6OR6YUFH8YTFUYHGJFDTYG}
u_0:=0,\qquad v_0:=0, \qquad a:=\gamma, \qquad b:=-2\gamma,\qquad c:=2\gamma\qquad {\mbox{and}}\qquad d:=-2\gamma. \end{equation}
In this case,
the system in (<ref>) describes an activator with density $u$
which is an autocatalytic activator: that is, such a substance stimulates the production
of itself (since $a>0$) and also activates the production of the substance with density $v$ (since also $c>0$). Also, the substance with density $v$ corresponds to a self-degrading inhibitor:
indeed, higher concentrations of this reactant are noxious for itself (since $d<0$)
and for the activator with density $u$ (since $b<0$).
Interestingly, the origin, which corresponds to the equilibrium $(u_0,v_0)$, is a stable sink for the system of ordinary differential equations corresponding to (<ref>) when $\mu:=0$ and $\nu:=0$. See Figure <ref> for a sketch of the trajectories
\begin{equation}\label{STREAMPLO}
\begin{dcases}
&\partial_t u= \gamma u -2\gamma v,\\
&\partial_t v=2\gamma u-2\gamma v.
\end{dcases}
\end{equation}
Quite remarkably, as discovered by Turing, the stability of (<ref>) can be destroyed by random fluctuations arising from the diffusivity of the chemical reactant. To appreciate this, given $u_0$, $v_0$, $a$, $b$, $c$ and $d$ as in (<ref>),
we take $\mu:=1$ and $\nu:=12$ in (<ref>). Note that this corresponds to a situation in which the diffusion of the inhibitor is faster than the one of the activator.
This scenario gives that
$$ \left( \begin{matrix}a-j^2\mu & b\\ c& d-j^2\nu\end{matrix}\right)=
\left( \begin{matrix}\gamma-j^2 & -2\gamma\\ 2\gamma& -2\gamma- 12j^2\end{matrix}\right),$$
which possesses eigenvalues of the form
\begin{equation}\label{CMOPSDEArajeke24w2}-\frac12\left(\gamma+13 j^2 \pm \sqrt{(11j^2+7\gamma)(11j^2-\gamma)}\right).\end{equation}
By (<ref>), we aim at detecting the greatest possible real part in (<ref>). To this end, note that when $(11j^2+7\gamma)(11j^2-\gamma)\le0$
then the real part in (<ref>) is equal to $-\frac12(\gamma+13 j^2)<0$.
Instead, if $(11j^2+7\gamma)(11j^2-\gamma)>0$ the largest possible real part in (<ref>) is equal to
\begin{equation}\label{LESOCHEDIBMECOHowrj-0}\begin{split}&
\sup_{j\in\Z}\frac12\left(\sqrt{(11j^2+7\gamma)(11j^2-\gamma)}
-\gamma-13 j^2\right)\\=\,&\frac\gamma2\sup_{j\in\Z}\left(
\sqrt{\left(\frac{11j^2}\gamma+7\right)\left(\frac{11j^2}\gamma-1\right)
}-1-\frac{13 j^2}\gamma\right)\\=\,&
\frac\gamma2\sup_{j\in\Z}\Phi\left(\frac{j^2}\gamma\right),\end{split}\end{equation}
$$ \Phi(\tau):=\sqrt{(11\tau+7)(11\tau-1)}-1-13 \tau.$$
Using elementary calculus, one checks that
$$ \max_{\tau\ge0}\Phi(\tau)= \frac4{11} \left(7 -4\sqrt{3}\right)
thanks to (<ref>).
This observation and (<ref>) give that the eigenvalues with largest possible real part in (<ref>) correspond to the choice $j:=\vartheta$ and are of the form
$$ \frac{2\gamma}{11} \left(7 -4\sqrt{3}\right)
=\frac{66\left(7 -4\sqrt{3}\right)\,\vartheta^2}{11\left(13\sqrt{3}-9\right)}
Hence, up to constants, the corresponding setting in (<ref>) takes the form
\begin{equation}\label{LESOCHEDIBMECOHowrj}
\begin{split}&
u(x,t)\simeq \exp\left(\frac{66\left(7 -4\sqrt{3}\right)\,\vartheta^2}{11\left(13\sqrt{3}-9\right)}\,t\right)\cos(\vartheta x)\\
v(x,t)\simeq \exp\left(\frac{66\left(7 -4\sqrt{3}\right)\,\vartheta^2}{11\left(13\sqrt{3}-9\right)}\,t\right)\cos (\vartheta x) .\end{split}
\end{equation}
See Figure <ref>
for an example with $\vartheta:=2$.
Plot of the function $(x,t)\mapsto\exp\left(\frac{264\left(7 -4\sqrt{3}\right)}{11\left(13\sqrt{3}-9\right)}\,t\right)\cos (2 x)$.
Of course, the asymptotics in (<ref>) are divergent as $t\to+\infty$,
which would correspond to the chemical substances to reach infinite density, which is certainly unfeasible in practice,
hence the meaning of (<ref>) has to be understood only in the vicinity of the equilibrium $(u_0,v_0)$,
which was set to be the origin for simplicity. Indeed, in practice the linear system in (<ref>)
must be considered as an efficient linearization only in the vicinity of the equilibrium $(u_0,v_0)$,
while the general situation is more accurately described by a nonlinear system as in (<ref>).
For practical purposes, the nonlinear sources $f$ and $g$ would force a bound on the densities $u$ and $g$
and possibly favor the convergence of the solution for large times to steady solutions $u=u(x)$ and $v=v(x)$ of
\begin{equation}\label{SYS:READI:STEAD}
\begin{dcases}
&\mu\Delta u(x)+f(u(x),v(x))=0,\\
&\nu\Delta v(x)+g(u(x),v(x))=0.
\end{dcases}
\end{equation}
That is, for small times, the linear mechanism identified in (<ref>) is helpful to detect how diffusion can drive
instability and place the kinetics of the system out of the “trivial” state $(u_0,v_0)$; then, at a longer time scale,
the nonlinear structure of (<ref>) becomes instrumental to confine the solution and lead it towards spatially
inhomogeneous patterns, as described by the steady solutions of (<ref>)
(as another option, the evolution of the equation may be stopped after a certain time in case
the release of the chemical substances stops; this could be the case in which
the pattern is formed at an embryonic stage for the animal due to chemical substances that are released only
during specific periods of the early stage development of an organism).
We also remark that, for the sake of simplicity, here we confined ourselves to the case in which the spatial domain
is a circle (i.e., the real line with periodic assumptions in $x$): in general, if one considers more complicated domains
(say, closer to biological situations of specific interest) then it is convenient to replace $e^{ijx}$ in (<ref>)
with the eigenfunctions
of the Laplacian in the domain of interest (with the corresponding boundary conditions).
In like manner, the dispersion relations in (<ref>) and (<ref>)
must take into account the corresponding eigenvalues in the place of $-j^2$.
These eigenvalues replace $-j^2$ in the matrix in (<ref>) too, and the diffusion eigenvalues $\lambda_j$ and $\Lambda_j$ must be modified accordingly.
The diffusion eigenvalues corresponding to diffusive instability still correspond to the ones with positive real part,
and one can focus for concreteness on the diffusive eigenvalue with largest real part in (<ref>).
The structural difference in the general case is however that the excited modes, that is the diffusive
eigenvalues with positive real part, depend on the domain. Since their region of positive and negative values
create a visible pattern in the dynamics of the solution (compare with the positive and negative values of the solution
depicted in Figure <ref>), it is conceivable that these regions have a connection with the visible
patterns in nature. For instance, two-dimensional
rectangular domains present eigenfunctions of the type $\cos\left(\frac{2\pi j_1 x_1}{\ell_1}\right)\cos\left(\frac{2\pi j_2 x_2}{\ell_1}\right)$, where $\ell_1$ and $\ell_2$ account for the lengths of the side of the rectangle
and $j_1$, $j_2\in\N$. In this scenario, excited states corresponding to $j_1=0$ give rise to horizontal stripes,
the ones corresponding to $j_2=0$ to vertical stripes, and the ones with $j_1\ne0$ and $j_2\ne0$ to maculate patterns, see
e.g. Figure <ref>.
$\quad$ $\quad$
Level sets of $(x_1,x_2)\mapsto2\cos x_2$, $(x_1,x_2)\mapsto2\cos x_1$ and $(x_1,x_2)\mapsto2\cos x_1\cos x_2$.
With respect to this, notice also that different excite modes correspond to different size of the pattern (such as width of the
or possible elongation of the spots). Of course, the analysis of simple rectangular regions is insufficient to capture the whole complexity of animal patterns: yet, it is suggestive to “approximate” an animal's coat with rectangular regions and
observe how for instance the orientation of stripes “locally” follow the proportion of the approximating rectangles,
see e.g. Figure <ref>
which clearly shows a change between vertical and horizontal stripes patterns in zebras
at the junctions between the body and the legs and between the body and the tail.
Mutually grooming zebras (photo by Duvignau Alain from
Wikipedia, available
under the Creative Commons CC0 1.0 Universal Public Domain Dedication).
Animal patterns may also bifurcate from stripes to spots in different regions of the organism, possibly in response of a variation
of the “local geometry”, compare e.g. with the tail markings of the cheetah in which the typical spots of the animal
become stripes at the tip of the tail, see Figure <ref>.
Caress of the Sphinx, by Fernand Khnopff (Public Domain image from
After <cit.>,
Turing's ideas about morphogenesis have become a cornerstone
in mathematical biology and led to a number of fantastic accomplishments, see e.g.
the mathematical reconstruction of very sophisticated animal patterns
presented in <cit.>.
Also, the name of Turing patternTuring pattern
is nowadays commonly used to denote biological structures
and specifically those arising by the imbalances between diffusion rates of different chemical agents which make a stable system sensitive to perturbations.
See <cit.>, <cit.>, <cit.>
and the references therein for a thorough analysis of diffusion driven pattern formation,
with several remarkable examples inspired by concrete natural phenomena.
See also <cit.> and the reference therein for more information about
reaction-diffusion equations.
§.§ Space invaders
A cane toad (bufo marinus) and the spread of cane toads in Australia from 1940 to 1980 in five-year intervals
(photo by Bill Waller and animated map by Froggydarb; images from
Wikipedia, licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license).
A topical argument in mathematical biology consists in the study of invasive species and in their territorial colonization ability,
see e.g. Figure <ref>.
To introduce ourselves to this subject, it is first opportune to understand the so-called logistic equation
logistic equation. This model was introduced by Pierre François Verhulst,
see Figure <ref> for his portrait,
and describes the evolution in time of a biological population.
The number of individuals $N(t)$ is supposed to grow at an intrinsic growth rate, parameterized by a given $\rho>0$,
which somewhat accounts for the ideal birth rate of the population ($N$ individuals would give rise to $\rho N$ newborns in the unit of time). Additionally, the population undergoes an intraspecific competition which causes the death of some individuals
due to possible overcrowding (assuming, for instance
that the environmental resources only allow a maximum number of individuals $K$). The combination of these effects
lead to the logistic ordinary differential equation
\begin{equation}\label{HJA:A89MAOL8789892BIOEQLOG}
\frac{dN}{dt}(t)=\rho N(t)\left(1-\frac{N(t)}K\right).
\end{equation}
This equation considers all the population located basically at the same place, but, for many practical purposes,
it is also convenient to describe biological individuals in a spatial environment, in which case the function $N$
depends on the time variable $t$ and also on the space variable $x$. If the population performs some kind of random walk,
as discussed in Section <ref> (see in particular equation (<ref>)), combining (<ref>) with the random diffusive tendency of the population we arrive at the equation
\begin{equation}\label{HJA:A89MAOL8789892BIOEQLOG-2}
\partial_t N(x,t)=c\Delta N(x,t)+\rho N(x,t)\left(1-\frac{N(x,t)}K\right),
\end{equation}
for some diffusion coefficient $c>0$.
We can actually simplify (<ref>) via a simple rescaling: namely, defining
\begin{equation}\label{HJA:A89MAOL8789892BIOEQLOG-4}
equation (<ref>) reduces to
\begin{equation}\label{HJA:A89MAOL8789892BIOEQLOG-3}
\partial_t u=\Delta u+u(1-u),
\end{equation}
which is dubbed[Equation (<ref>)
is named after Sir Ronald Aylmer Fisher <cit.>,
Andrey Nikolaevich Kolmogorov, Ivan Georgievich Petrovsky and Nikolai Semenovich Piskunov <cit.>.
Though biological invasions and
the spread of a genetic trait were
the prime motivations for the study of (<ref>),
nowadays this equation is also intensively studied for its applications
in combustion and flame propagation: that's the unifying power of
the good mathematics!
Besides the introduction of equation (<ref>) and some questionable
studies in eugenics, Fisher is considered one of the founders of population genetics and of modern statistical science.
His name is also linked to the so-called
Fisherian runaway sexual selection mechanism, aiming at an explanation of
some exaggerated, costly and apparently maladaptive male ornamentation in nature (seemingly in conflict with natural selection)
as dictated by persistent female choice (possibly triggered initially by the
ornament signaling greater potential fitness, hence likelihood of leaving more descendants):
the classical example of Fisherian runaway is the elaborate peacock plumage, see Figure <ref>.
A classical story about Fisher's contribution towards the standardization of statistical experiments is the “lady tasting tea” <cit.>. In a nutshell, once Fisher offered a cup of tea to phycologist Blanche Muriel Bristol.
She politely declined it, saying that she preferred the flavor when the milk was poured into the cup before (not after) the tea. So we have an intriguing dilemma: can the order of pouring milk affect the flavor of the tea? and, specifically, can Bristol notice the difference?
To test this, Fisher provided Bristol with eight randomly ordered cups of tea: four of which were prepared by first pouring the tea and then adding milk, and four by first pouring the milk and then adding the tea. Bristol had to identify the different cups.
Fisher had to interpret the data to distinguish the case in which Bristol could actually tell the difference in the pouring order
and the one in which she could identify correctly some cups, but just by chance.
For this,
with a bit of combinatorics, Fisher computed that there are ${{8}\choose{4}}=\frac{8!}{4! (8-4)!}=70$ possible combinations of cups.
Since Bristol was aware that there were four cups of each type, her answer would have included four of each.
There is of course only one possibility of making precisely the right identification: hence achieving a complete identification
just by chance occurs with probability $\frac{1}{70}=0.01428571428...$;
there are instead 16 possibilities of making exactly one error (indeed, one error comes from a single swap of two cups of different type, say the $i$th cup of the first type with the $j$th cup of the second type, with $i$, $j\in\{1,2,3,4\}$).
Hence, there are $1+16=17$ possibilities of making at most one mistake
and therefore, if we adopt a conventional probability criterion of success to be below the $5\%$ threshold, the ability of Bristol to properly categorize the cups of tea would have been confirmed if, and only if, she was able to correctly identify all of them, without making any mistake.
This story is relevant since it became an example of a randomized experiment for checking a “null hypothesis” in statistics and it contributed to a scientific establishment of randomization analysis of experimental data.
For the record, Muriel Bristol allegedly succeeded in classifying all eight cups correctly.
See <cit.> for further reading on the lady tasting tea experiment.
See also Figure <ref>
for a picture of Kolmogorov and footnote <ref>
on page KOLMONOTE for more information about the prominent figure of Kolmogorov.]
in jargon the Fisher-Kolmogorov-Petrovsky-Piskunov equation
(or Fisher-KPP equation for short). Fisher-Kolmogorov-Petrovsky-Piskunov equation
Interestingly, this equation is a reaction-diffusion equation reaction-diffusion equation
in the setting introduced on page REDI6NFHARLROA7KOLM789GIJ7solFUMHDNOJHNFOJED231.
Pierre Verhulst (Public Domain image from
The question of determining the invasivity of biological species is very related to (<ref>).
In this setting, in light of (<ref>), the level $u=0$ corresponds to the absence of the population
and the level $u=1$ to the maximal density permitted by environmental resources.
Remarkably, for any given $v\in[2,+\infty)$
\begin{equation}\label{BEVEPAVKCiuneA3OMAkac}
\begin{split}&
{\mbox{equation~\eqref{HJA:A89MAOL8789892BIOEQLOG-3} admits a solution~$u$ describing the full environmental invasion}}\\ &
{\mbox{of the biological species with velocity~$v$.}}\end{split}\end{equation}
More precisely, given any $v\in[2,+\infty)$ and any $\omega\in \partial B_1$, there exists a smooth function $U:\R\to(0,1)$, with
\begin{equation}\label{BEVEPAVKCiuneA3OMAkac22} \lim_{\tau\to+\infty}U(\tau)=0\qquad{\mbox{and}}\qquad\lim_{\tau\to-\infty}U(\tau)=1,\end{equation}
and a solution $u$ of (<ref>) of the form
\begin{equation}\label{SOLIWA} u(x,t):=U(\omega\cdot x-vt).\end{equation}
The solutions[The concrete interest of the very special solutions in (<ref>) is given by the fact that
under suitable initial conditions, “reasonable” solutions of (<ref>) evolve into a solitary wave,
see <cit.>. Roughly speaking, wells in the initial conditions
are quickly reabsorbed by the evolving dynamics, since regions with $u$ close to $1$ tend to grow
and eat up adjacent regions with $u$ close to $0$, producing, for large times, a monotonic traveling wave with constant speed.
This phenomenon makes solitary waves an essential building block for understanding the dynamics of the
Fisher-Kolmogorov-Petrovsky-Piskunov equation.] of this form are often called “solitary waves”. solitary wave
See Figure <ref> for a representation of a solitary wave.
A blue peacock presenting its feathers (image by from
Wikipedia, available under the Creative Commons Attribution-Share Alike 3.0 Unported).
To establish (<ref>),
we consider[As remarked in <cit.>, when $v:=\frac{5}{\sqrt {6}}=2.04124145232...$,
one can find explicit solutions of (<ref>) in the form
$$ U(\tau)=\frac1{\displaystyle\left(1+C\exp\left(\frac\tau{\sqrt{6}}\right)\right)^{2}},$$
for all $C>0$.
Figure <ref>
was obtained with this function with $C:=1$ and the frames of the second image there are obtained
by taking times $t\in\{0,2,4,6,8\}$.
See also <cit.> for more information about this explicit solution.
We are not aware of other solutions of (<ref>) which
can be written explicitly in a nice and closed form.]
the ordinary differential equation
\begin{equation}\label{BEVEPAVKCiuneA3OMAkac2}
\frac{d^2U}{d\tau^2}(\tau)+v\frac{dU}{d\tau}(\tau)+U(\tau)\big(1-U(\tau)\big)=0.\end{equation}
By setting $V(\tau):=\frac{dU}{d\tau}(\tau)$, we obtain the system of equations
\begin{equation}\label{BEVEPAVKCiuneA3OMAkac3b}
\left(\begin{matrix}
\displaystyle\frac{dU}{d\tau} \\
\\
\displaystyle\frac{dV}{d\tau}
\end{matrix}\right)=\left(\begin{matrix}
V \\
\\
\end{matrix}\right).\end{equation}
The equilibria of (<ref>) are $E_0:=(0,0)$ and $E_1:=(1,0)$.
The linearized dynamics in the vicinity of $E_1$ is described by the matrix $M_1:=\left(
\begin{matrix}
0 & 1\\
1 & -v
\end{matrix}\right)$, which possesses eigenvalues $\frac12\left(-v \pm \sqrt{v^2 + 4}\right)$
with eigenvectors $\left(\frac12 \left(v\pm\sqrt{v^2+4}\right), 1\right)$.
In particular, $E_1$ is a hyperbolic saddle for all values of $v\in\R$. So we consider the unstable manifold
(see e.g. <cit.>) of $E_1$ in the direction $-\left(\frac12 \left(v+\sqrt{v^2+4}\right), 1\right)$.
This provides us with an orbit $(U(\tau),V(\tau))$ such that
\begin{equation}\label{JMNSRAPPORA1}
\lim_{\tau\to-\infty}(U(\tau),V(\tau))=E_1\end{equation}
\begin{equation}\label{JMNSRAPPORA}
\lim_{\tau\to-\infty}
\left(\frac{dU}{d\tau} (\tau),\frac{dV}{d\tau}(\tau)\right)= -
\left(\frac12 \left(v+\sqrt{v^2+4}\right), 1\right),\end{equation}
see Figure <ref>.
Given $v\in[2,+\infty)$, we now consider the planar region
$$ {\mathcal{R}}:=\left\{{\mbox{$(U,V)\in\R^2$ \;s.t.\; $U<1$, \;$V\in\left(-\displaystyle\frac1v,0\right)$\;
and \; $\displaystyle\frac{v+\sqrt{v^2-4}}{2}\,U+V>0$}}\right\}.$$
A traveling wave of a species invading the spacial domain from left to right.
Notice that, in light of (<ref>) and (<ref>),
there exists $\tau_\star\in\R$ such that $(U(\tau),V(\tau))\in{\mathcal{R}}$ for all $\tau\in(-\infty,\tau_\star)$.
We claim that
\begin{equation}\label{CONGFIuynU6mnemt}
{\mbox{$(U(\tau),V(\tau))\in{\mathcal{R}}$\; for all \;$\tau\in(-\infty,+\infty)$.}}\end{equation}
Indeed, suppose not. Then, there exists $\tau_0\ge \tau_\star$ such that $(U(\tau),V(\tau))\in{\mathcal{R}}$ for all $\tau\in(-\infty,\tau_0)$
but either $V(\tau_0)=-1/v$, or $V(\tau_0)=0$
or $U(\tau_0)=1$, or $\frac{v+\sqrt{v^2-4}}{2}\,U(\tau_0)+V(\tau_0)=0$. We exclude all these cases to prove (<ref>).
First, we show that $V(\tau_0)=-1/v$ cannot hold. This is because otherwise $
\frac{dV}{d\tau}(\tau_0)\le0$, but we know from (<ref>)
$$ \frac{dV}{d\tau}(\tau_0)=-vV(\tau_0)-U(\tau_0)\big(1-U(\tau_0)\big)\ge-vV(\tau_0)-\frac14=1-\frac14>0.$$
Similarly, $V(\tau_0)=0$ cannot happen, otherwise
$$ 0\le\frac{dV}{d\tau}(\tau_0)=-vV(\tau_0)-U(\tau_0)\big(1-U(\tau_0)\big)\le-vV(\tau_0)=0.$$
This entails that either $U(\tau_0)=0$ or $U(\tau_0)=1$, whence either $(U(\tau_0),V(\tau_0))=E_0$
or $(U(\tau_0),V(\tau_0))=E_1$, which is impossible, being both $E_0$ and $E_1$ equilibria for the system.
Also $U(\tau_0)=1$ cannot hold true, otherwise, by (<ref>),
$$ 0\le\frac{dU}{d\tau}(\tau_0)=V(\tau_0)\le0,$$
that gives that $(U(\tau_0),V(\tau_0))=E_1$, which is impossible.
Finally, the case $\frac{v+\sqrt{v^2-4}}{2}\,U(\tau_0)+V(\tau_0)=0$ cannot occur as well, otherwise we argue as follows.
$$ F(\tau):=\frac{v+\sqrt{v^2-4}}{2}\,U(\tau)+V(\tau).$$
We note that $F(\tau)>0$ for $\tau\in(-\infty,\tau_0)$ and $F(\tau_0)=0$. As a result, from (<ref>) we arrive at
\begin{eqnarray*} 0&\ge&\frac{dF}{d\tau}(\tau_0)\\&=&
\frac{v+\sqrt{v^2-4}}{2}\,\frac{dU}{d\tau}(\tau_0)+\frac{dV}{d\tau}(\tau_0)\\&=&
\frac{v+\sqrt{v^2-4}}{2}\,V(\tau_0)-vV(\tau_0)-U(\tau_0)+U^2(\tau_0)\\&=&
\frac{\sqrt{v^2-4}-v}{2}\,V(\tau_0)-U(\tau_0)+U^2(\tau_0)\\&=&-
\frac{\big(\sqrt{v^2-4}-v\big)\big(v+\sqrt{v^2-4}\big)}{4}\,U(\tau_0)-U(\tau_0)+U^2(\tau_0)\\&=&-
\frac{(v^2-4)-v^2}{4}\,U(\tau_0)-U(\tau_0)+U^2(\tau_0)\\&=&
\end{eqnarray*}
This gives that $U(\tau_0)=0$, whence $V(\tau_0)=0$, but this contradicts the fact that $E_0$ is an equilibrium for the system.
The proof of (<ref>) is thereby complete.
As a byproduct of (<ref>), we also have that, for all $t\in\R$,
\begin{equation}\label{CONGFIuynU6mnemtLI9}
\end{equation}
Now we claim that
\begin{equation}\label{CONGFIuynU6mnemtLI}
\lim_{\tau\to+\infty}(U(\tau),V(\tau))=E_0.
\end{equation}
Indeed, by (<ref>) we know that $V(\tau)<0$ for all $\tau\in\R$, therefore by (<ref>), $U(\tau)$ is decreasing.
Since also by (<ref>) we have that $U(\tau)\in(0,1)$ for all $\tau\in\R$, we infer that
there exists $U_\infty\in[0,1)$ such that $U(\tau)\to U_\infty$ as $\tau\to+\infty$.
Now, let $\delta>0$, to be taken as small as we wish here below. Using the “dots” to denote derivatives, recalling (<ref>) and (<ref>) we observe that
\begin{equation}\label{KAQUINLtyY93l2}\begin{split}
\le \frac1\delta\left|\int_\tau^{\tau+\delta}\big(V(\tau)-V(\theta)\big)\,d\theta\right|+
\frac1\delta\left|\int_\tau^{\tau+\delta} V(\theta)\,d\theta\right|\\&=
\frac1\delta\left|\int_\tau^{\tau+\delta}\left(\int_\theta^\tau\dot{V}(\zeta)\,d\zeta\right)\,d\theta\right|+
\frac1\delta\left|\int_\tau^{\tau+\delta} \dot{U}(\theta)\,d\theta\right|\\&=
\frac1\delta\left|\int_\tau^{\tau+\delta}\left(\int^\theta_\tau vV(\zeta)+U(\zeta)\big(1-U(\zeta)\big)\,d\zeta\right)\,d\theta\right|+
\frac1\delta\left|U(\tau+\delta)-U(\tau)\right|\\&\le
\frac2\delta\int_\tau^{\tau+\delta} \left(\int^\theta_\tau \,d\zeta\right)\,d\theta+
\frac1\delta\left|U(\tau+\delta)-U(\tau)\right|\\&=
\delta+
\frac1\delta\left|U(\tau+\delta)-U(\tau)\right|.\end{split}
\end{equation}
From this we arrive at
$$ \limsup_{\tau+\infty}|V(\tau)|\le\limsup_{\tau+\infty}
\delta+
\frac1\delta\left|U(\tau+\delta)-U(\tau)\right|=\delta+
\frac1\delta\left|U_\infty-U_\infty\right|=\delta$$
and accordingly, by choosing $\delta$ arbitrarily small,
$$ \lim_{\tau+\infty} V(\tau)=0.$$
As a result, we have that $(U(\tau),V(\tau))\to(U_\infty,0)\in[0,1)\times\{0\}$.
Since the only equilibrium of (<ref>) on $[0,1)\times\{0\}$
is $E_0$, the proof of (<ref>) is complete.
Phase portrait and heteroclinc connection for the dynamical system in (<ref>) with $v:=5/2$.
Hence, in view of (<ref>), (<ref>), (<ref>) and (<ref>),
the trajectory $U(\tau)$ provides a solution of (<ref>) satisfying the asymptotics in (<ref>). Then, by (<ref>),
\begin{eqnarray*}&&
-\partial_t u(x,t)+\Delta u(x,t)+u(x,t)\big(1-u(x,t)\big)\\&&\qquad=
v \frac{dU}{d\tau}(\omega\cdot x-vt)
+\frac{d^2U}{d\tau^2}U(\omega\cdot x-vt)
+U(\omega\cdot x-vt)\big(1-U(\omega\cdot x-vt)\big)=0.
\end{eqnarray*}
This shows that $u$ is the desired solitary wave solution of the Fisher-Kolmogorov-Petrovsky-Piskunov equation in (<ref>).
For completeness, we show also that biologically interesting solitary waves of the Fisher-Kolmogorov-Petrovsky-Piskunov equation
do not exist when $v\in[0,2)$, since
\begin{equation}\label{1343g43678EPAVKCiuneA3OMAkI536}
\begin{split}&
{\mbox{when~$v\in[0,2)$ any solution~$U:\R\to(-\infty,1]$ of~\eqref{BEVEPAVKCiuneA3OMAkac22} and~\eqref{BEVEPAVKCiuneA3OMAkac2}}}\\&{\mbox{necessarily attains also negative values.}}\end{split}\end{equation}
We recall indeed that biologically interesting values of the solutions are the ones modeling a population density,
hence they are confined in the interval $[0,1]$, due to (<ref>).
To check the claim in (<ref>),
suppose by contradiction that a solution $U:\R\to(0,1]$ of (<ref>) and (<ref>)
exists for some $v\in[0,2)$ and let $V:=\frac{dU}{d\tau}$ to find the system of equations in (<ref>).
We claim that, for all $\tau\ge0$,
\begin{equation}\label{KAQUINLtyY93l}
|V(\tau)|< |V(0)|+\frac1v.
\end{equation}
Indeed, suppose not. Then, there exists $\underline\tau>0$ such that $|V(\tau)|<|V(0)|+\frac1v$
for all $\tau\in[0,\underline\tau)$ and $|V(\underline\tau)|=|V(0)|+\frac1v$. Two cases can occur,
either $V(\underline\tau)\ge0$ or $V(\underline\tau)<0$. In the first case, we have that
$$ 0\le\frac{dV}{d\tau}(\underline\tau)=-vV(\underline\tau)-U(\underline\tau)\big(1-U(\underline\tau)\big)
\le-1,$$
which is a contradiction.
If instead $V(\underline\tau)<0$, then
$$ 0\ge\frac{dV}{d\tau}(\underline\tau)=-vV(\underline\tau)-U(\underline\tau)\big(1-U(\underline\tau)\big)=
which is a contradiction too, hence (<ref>) is proved.
Using this and computing as in (<ref>), for all $\tau\ge0$ and $\delta>0$ we arrive at
\begin{eqnarray*}
\frac1\delta\left|\int_\tau^{\tau+\delta}\left(\int^\theta_\tau vV(\zeta)+U(\zeta)\big(1-U(\zeta)\big)\,d\zeta\right)\,d\theta\right|+
\frac1\delta\left|U(\tau+\delta)-U(\tau)\right|\\&\le&
C\delta +
\frac1\delta\left|U(\tau+\delta)-U(\tau)\right|,
\end{eqnarray*}
with $C:=v\left(|V(0)|+\frac1v\right)+1$.
|V(\tau)|\le C\delta +
\frac1\delta\left|0-0\right|=C\delta$$
and accordingly, taking $\delta$ as small as we wish, we find that $V(\tau)\to0$ as $\tau\to+\infty$.
In this way, we have established that $(U(\tau),V(\tau))\to(0,0)$ as $\tau\to+\infty$.
Hence, we pick $\eta>0$, to be chosen conveniently small in what follows and we find $\tau_\eta\in\R$ such that $|(U(\tau),V(\tau))|\le\eta$ for all $\tau\ge\tau_\eta$. Up to a time shift, i.e. up to replacing $(U(\tau),V(\tau))$ with $(U(\tau-\tau_\eta),V(\tau-\tau_\eta))$, we can assume from now on that $\tau_\eta=0$ and write that
\begin{equation}\label{L-X2343545y678we34eS}
\big|(U(\tau),V(\tau))\big|\le\eta\qquad{\mbox{for all }}\,\tau\ge0.
\end{equation}
Now we perform an argument related to the nonlinear stability of equilibria in planar dynamical systems.
Namely, we take $\varrho(\tau)>0$ and $\varphi(\tau)\in\R$ such that
$$ U(\tau)=\varrho(\tau)\,\cos(\varphi(\tau))\qquad {\mbox{and}}\qquad
Actually, since $U(\tau)>0$, we have that
\begin{equation}\label{922i3k2e944racrhfo}
{\mbox{we can pick~$\varphi(\tau)$ in the interval }}\,\left(-\frac\pi2,\frac\pi2\right).\end{equation}
Thus, denoting by the dot the derivative with respect to $\tau$,
$$ \dot U=\dot\varrho\cos\varphi- \varrho\dot\varphi\sin\varphi
\qquad {\mbox{and}}\qquad
\dot V=\dot\varrho\sin\varphi+\varrho\dot\varphi\cos\varphi.$$
From this we arrive at
\begin{eqnarray*}
\dot U\sin\varphi-\dot V\cos\varphi&=&
\big(\dot\varrho\cos\varphi- \varrho\dot\varphi\sin\varphi\big)\sin\varphi-
\big(\dot\varrho\sin\varphi+\varrho\dot\varphi\cos\varphi\big)\cos\varphi
\\&=&
\dot\varrho\sin\varphi\cos\varphi- \varrho\dot\varphi\sin^2\varphi-
\dot\varrho\sin\varphi\cos\varphi-\varrho\dot\varphi\cos^2\varphi
\\&=&-\varrho\dot\varphi.
\end{eqnarray*}
Since, by (<ref>),
\begin{eqnarray*}
\dot U\sin\varphi-\dot V\cos\varphi&=&
\varrho\cos^2\varphi-\varrho^2\cos^3\varphi\\&=&\varrho+\frac{v\varrho}2\sin(2\varphi)
\end{eqnarray*}
we conclude that
\[ -\varrho\dot\varphi=\varrho+\frac{v\varrho}2\sin(2\varphi)
and accordingly
\[ -\dot\varphi=1+\frac{v}2\sin(2\varphi)
In particular, using (<ref>) and the assumption that here $v\in[0,2)$,
\[ -\dot\varphi\geq 1-\frac{v}2-\varrho\ge 1-\frac{v}2-\eta
\ge\frac{1-\displaystyle\frac{v}2}{2}>0,\]
as long as $\eta$ is small enough.
This gives that there exists $\tau_\star>0$ such that $\varphi(\tau_\star)<-\frac{\pi}{2}$,
in contradiction with (<ref>).
This completes[Here is an alternative argument
to prove (<ref>), relying on the additional simplifying assumptions that
$$ \lim_{\tau\to+\infty}\dot{U}(\tau)=0$$
and the following limit exists
$$ \ell:=\lim_{\tau\to+\infty} \frac{\ddot{U}(\tau)}{\dot{U}(\tau)}.$$
From this, if $U(\tau)\in(0,1)$ for all $\tau\in\R$ and $U(\tau)\to0$ as $\tau\to+\infty$, we can apply L'Hôpital's Rule and deduce that
$$ \lim_{\tau\to+\infty} \frac{\dot{U}(\tau)}{U(\tau)}=
\lim_{\tau\to+\infty} \frac{\ddot{U}(\tau)}{\dot{U}(\tau)}=\ell.$$
As a result, dividing (<ref>) by $U(\tau)$ and taking the limit we find that
\begin{eqnarray*}
&=&\lim_{\tau\to+\infty}\frac{\ddot{U}(\tau)}{\dot U(\tau)}\;
\frac{\dot{U}(\tau)}{U(\tau)}+v\frac{\dot{U}(\tau)}{U(\tau)}+1\\
&=&\left( \ell+\frac{v}2\right)^2+1-\frac{v^2}4
\\&\ge&1-\frac{v^2}4,
\end{eqnarray*}
which gives that $v^2\ge4$, hence (in the convention that $v\ge0$) necessarily $v\ge2$ to allow invading fronts.
These computations are also interesting since for more general equations such as $\ddot{U}+v\dot{U}+f(U)=0$ with $f$ smooth, $f(r)>0$ for all $r\in(0,1)$, $f(0)=f(1)=0$ and $f'(1)<0$, a calculation as above leads to
\begin{eqnarray*}&& 0=\ell^2+v\ell+\lim_{\tau\to+\infty}\frac{f(U(\tau))}{U(\tau)}=
\ell^2+v\ell+f'(0)=\left( \ell+\frac{v}2\right)^2+f'(0)-\frac{v^2}4\ge f'(0)-\frac{v^2}4,
\end{eqnarray*}
leading to the necessary condition for invasion $v\ge 2\sqrt{f'(0)}$.
This is quite interesting, since it also points out that the speed of the invasive front is dictated by its low density fringes not by the high density regions (namely, the velocity of invasion is completely determined by the forcing term $f$
and specifically by the values of $f$ in the vicinity of $0$, not in the vicinity of $1$).
A function $f$ with the above structure is often called a bistable nonlinearity bistable nonlinearity.
For further details on bistable nonlinearities and plane wave solutions, see <cit.>.]
the proof of (<ref>).
For additional information on the Fisher-Kolmogorov-Petrovsky-Piskunov, see <cit.>,
<cit.>, <cit.>
and the references therein.
§.§ Equations from hydrodynamics
A classical arena for partial differential equations is provided by fluid mechanics and hydrodynamics.
Let us present some of the situations that naturally occur in this framework.
Let us consider a fluid with density $\rho(x,t)$, vectorial velocity $v(x,t)$ and pressure $p(x,t)$.
The position of a parcel of fluid at time $t$ is given by $x(t)$ and therefore such a parcel
travels according to the velocity prescription
\begin{equation}\label{0uojfSANMFOSJFEL}
\dot x(t)=v(x(t),t).
\end{equation}
The title page to Euler's original article “Principes généraux du mouvement des fluides”,
published in Mémoires de l'Académie des Sciences de Berlin in 1757 (Public Domain source from
Internet Archive).
Also, the total mass of a fluid element occupying a region $\Omega$ of the space at a given time $t_0$
is given by the quantity $\int_{\Omega} \rho(x,t_0)\,dx$. This amount of fluid will move around in a small time $\tau$
possibly occupying a different region, that we name $\Omega_\tau$ which collects all the evolution trajectories $x(t_0+\tau)$
of the fluid parcels such that $x(t_0)\in\Omega$. Since we are assuming that matter is neither created nor disappear in this process
we deduce a conservation law given by the equation
$$ \int_{\Omega} \rho(x,t_0)\,dx=\int_{\Omega_\tau} \rho(x,t_0+\tau)\,dx$$
or equivalently
\begin{equation}\label{0ojCOMAS} \frac{d}{d\tau}\int_{\Omega_\tau} \rho(x,t_0+\tau)\,dx=0.\end{equation}
Notice that we can change variable in the previous integral simply by flowing back the fluid.
Specifically, given a point $q\in\R^n$ we denote by $\Phi^t(q)$ the evolution according to the fluid vector field $v$
with initial position at time $t_0$ equal to $q$, or more formally we define $\Phi^t(q)$ as the solution of the following
Cauchy problem for ordinary differential equations:
$$ \begin{dcases}
\frac{d}{dt} \Phi^t(q)=v\big(\Phi^t(q),t),\\
\Phi^{t_0}(q)=q.
\end{dcases}$$
This is nothing else as a rewriting of (<ref>), but we are keeping track here explicitly of the initial position.
With this notation, we have that $\Omega_\tau=\{\Phi^\tau(y)$, $y\in\Omega\}$.
As a result, one can perform the change of variable
$$x=\Phi^\tau(y)=y+\tau v(y,t_0)+o(\tau),$$
which produces
$$ |\det \partial_y x|=1+\tau \div v(y,t_0)+o(\tau)$$
and therefore
\begin{eqnarray*}&&
\int_{\Omega_\tau} \rho(x,t_0+\tau)\,dx\\&=&
\int_{\Omega} \rho(y+\tau v(y,t_0),t_0+\tau)\,\big(1+\tau \div v(y,t_0)\big)\,dy+o(\tau)
\\&=&\int_{\Omega}
\Big(\rho(y,t_0)+\tau\nabla\rho(y,t_0)\cdot v(y,t_0)
+\tau\partial_t \rho(y,t_0)\Big)
\,\big(1+\tau \div v(y,t_0)\big)\,dy+o(\tau)\\&=&\int_{\Omega}
\Big(\rho(y,t_0)+\tau\nabla\rho(y,t_0)\cdot v(y,t_0)
+\tau\partial_t \rho(y,t_0)+
\tau\rho(y,t_0)\div v(y,t_0)\Big)\,dy+o(\tau)\\&=&\int_{\Omega}
\Big(\rho(y,t_0)
+\tau\partial_t \rho(y,t_0)+
\tau\div\big(\rho(y,t_0)\, v(y,t_0)
\big)\Big)\,dy+o(\tau).
\end{eqnarray*}
This and (<ref>) formally lead to
\begin{equation}\label{D23q4wy5OPACLPASIJMDNIAKS279uyt3M}
\begin{split}0=\left.
\frac{d}{d\tau}\int_{\Omega_\tau} \rho(x,t_0+\tau)\,dx\right|_{\tau=0}=
\int_{\Omega}
\Big(\partial_t \rho(y,t_0)+\div\big(\rho(y,t_0)\, v(y,t_0)\big)\Big)\,dy.\end{split}\end{equation}
Since this holds true for all domains $\Omega$ and all times $t_0$, we have obtained the equation
\begin{equation}\label{OJS-PJDN-0IHGDOIUGDBV02ujrfMTE}
\partial_t \rho+\div(\rho v)=0.
\end{equation}
This is called[It is interesting to observe that in the framework of equation (<ref>)
one can reinterpret the divergence term in (<ref>)
as a transport term with velocity $\nabla w$ (that is, the effect of chemotaxis
is to produce a transport of the biological population with a velocity proportional to the gradient of the chemical attractant).
Moreover, in chemistry one often models the spreading of some substrate with concentration $\rho$
in a fluid with velocity $v$: in this context equation (<ref>)
is often called “convection equation”convection equation.]
in jargon “mass transport equation”mass transport equation
(or simply “transport equation”) transport equation
or “continuity equation”continuity equation.
Of course, transport of mass, corresponding to the conservation of matter, is not the only ingredient to accurately
describe the dynamics of a fluid, hence one has to complement (<ref>) with additional information. In particular, one can take into account Newton's Law regarding momentum conservation.
In this setting, the change of fluid momentum must correspond to the forces acting on the fluid
(for simplicity, we assume here that the fluid is only subject to gravity and to its own pressure force
with no other external forces to be exerted).
Notice that the momentum corresponding to the mass of the fluid in a region of space $\Omega$
is given by the quantity $\int_\Omega \rho(x,t) v(x,t)\,dx$.
The corresponding gravity force is provided by $-ge_n\int_\Omega \rho(x,t)\,dx$, being $g$ the gravity acceleration constant
(and $e_n=(0,\dots,0,1)$, supposing the gravity acting downwards in the vertical direction).
As for the pressure force, it arises from $p$ in the normal direction
along the surface $\partial\Omega$, with a conventional
minus sign (so that a resistance to velocity increasing occurs when moving towards regions with high pressures, while
low pressure regions produce a sucking effect). These considerations translate
Newton's Law about momentum conservation into the formula
\begin{equation} \label{90uyoihfs83wEUSlNOE}
\frac{\partial}{\partial t}\int_\Omega \rho(x,t) v(x,t)\,dx=
-ge_n\int_\Omega \rho(x,t)\,dx-\int_{\partial\Omega} p(x,t) \nu(x)\,d{\mathcal{H}}^{n-1}_x,\end{equation}
being, as usual, $\nu(x)$ the unit external normal at $x\in\partial\Omega$.
Furthermore, in light of (<ref>)
(applied here to $\rho v_j$ instead of $\rho$) we have that, for all $j\in\{1,\dots,n\}$,
\begin{eqnarray*}
&&\frac{\partial}{\partial t}\int_\Omega \rho(x,t) v_j(x,t)\,dx\\
&=& \int_{\Omega}
\Big(\partial_t (\rho v_j)(x,t)+\div\big((\rho v_j)(x,t)\, v(x,t)\big)\Big)\,dx\\&=&
\int_{\Omega}
\Big(\rho (x,t)\partial_t v_j(x,t)+\rho(x,t) v(x,t)\cdot\nabla v_j(x,t)\Big)\,dx,
\end{eqnarray*}
where (<ref>) has been exploited in the latter step.
Thus, using the notation $v\cdot\nabla$ to denote the operator $\displaystyle\sum_{i=1} v_i\partial_i$,
possibly applied to a vector valued function componentwise, we can write that
\begin{eqnarray*}
&&\frac{\partial}{\partial t}\int_\Omega \rho(x,t) v(x,t)\,dx=
\int_{\Omega}
\Big(\rho (x,t)\partial_t v(x,t)+\rho(x,t) \big(v(x,t)\cdot\nabla\big) v(x,t)\Big)\,dx.
\end{eqnarray*}
From this and (<ref>)
we obtain that
\begin{equation} \label{90uyoihfs83wEUSlNOE-VXBNC}\begin{split}&
\int_{\Omega}
\Big(\rho (x,t)\partial_t v(x,t)+\rho(x,t) \big(v(x,t)\cdot\nabla\big) v(x,t)\Big)\,dx\\&\qquad\qquad=
-ge_n\int_\Omega \rho(x,t)\,dx-\int_{\partial\Omega} p(x,t) \nu(x)\,d{\mathcal{H}}^{n-1}_x.\end{split}\end{equation}
Also, from the Divergence Theorem, for all $j\in\{1,\dots,n\}$,
$$ \int_{ \Omega} \partial_j p(x,t)\,dx=
\int_{ \Omega} \div\big( p(x,t) e_j\big)\,dx
=\int_{\partial\Omega} p(x,t) e_j\cdot\nu(x)\,d{\mathcal{H}}^{n-1}_x$$
and in consequence
\[ \int_{ \Omega} \nabla p(x,t)\,dx=\int_{\partial\Omega} p(x,t) \nu(x)\,d{\mathcal{H}}^{n-1}_x.\]
From these considerations and (<ref>)
we arrive at
\Big(\rho (x,t)\partial_t v(x,t)+\rho(x,t) \big(v(x,t)\cdot\nabla\big) v(x,t)\Big)\,dx=
-ge_n\int_\Omega \rho(x,t)\,dx-\int_{ \Omega} \nabla p(x,t)\,dx$$
and accordingly
\begin{equation}\label{EUMSD-OS-D}
\rho \partial_t v+\rho (v\cdot\nabla) v=
-ge_n \rho-\nabla p.
\end{equation}
The system containing the mass conservation equation in (<ref>)
and the momentum balance equation in (<ref>) (plus possibly a constitutive relation
linking pressure and density) constitutes what in jargon[The fountainhead of these equations
came indeed from the work of Leonhard Euler,
see Figure <ref>.
See also Figure <ref>
for Euler's iconic portrait (by Jakob Emanuel Handmann).
Many of us wonder why Euler is depicted in a pijama with a towel on his head.
Arguably, it was not a pijama but a fashionable banyan, not a towel but a silk cloth
and possibly at that time it was believed that loose and informal
dresses (the banyan, no wig)
contribute to exercise of the faculties of the mind (and indeed we also find it much more comfortable
to do mathematics wearing very informal clothes).
Note that Euler is portrayed as facing left,
possibly to mitigate the impact on the spectator of a right eyelid ptosis and a divergent strabismus.
At age 31, Euler became almost blind in his right eye <cit.>
(and this might be a possible explanation for which Euler was nicknamed “the cyclops”
by Frederick II, King of Prussia).
This partial loss of vision did not discourage Euler from producing the finest possible
mathematics (actually, he allegedly stated “Now I will have fewer distractions”).
At age 59, a surgical restoration for a cataract in his left eye rendered Euler
almost totally blind. Yet, Euler's passion and talent for mathematics remained undefeated
and he kept producing a vast number of impressive works which changed the course of mathematics
till he died at age 76. In his eulogy, Marquis de Condorcet wrote
“il cessa de calculer et de vivre” (French for “he ceased to calculate and to live”).]
are called “Euler fluid flow equations”Euler equation.
Portrait of Leonhard Euler (Public Domain image from
In a sense, these equations are obtained for a rather ideal situation and several modifications
of the previous setting can be performed to account for more complex models. Among these
modifications, one of the most popular one consists in accounting for some friction between the fluid molecules
which creates a “viscosity”viscosity resisting to the change of fluid velocity. More specifically,
let us quantify[This idea of confronting the pointwise value of a function
with the average nearby will be extensively used in Chapter <ref> and it will be
one of the leitmotifs of the study of harmonic functions.]
the difference of velocity between a fluid parcel at $x$ and that of the parcels nearby
by the difference of $v(x,t)$ and its average in a small ball, say of radius $h$ centered at $x$, namely
$$ {\mathcal{D}}(x,t):= v(x,t)-\fint_{B_h(x)}v(y,t)\,dy.$$
By exploiting (<ref>) and polar coordinates, we find that
\begin{equation*}
\begin{split}
{\mathcal{D}}(x,t)&= \fint_{B_h(x)}\big( v(x,t)-v(y,t)\big)\,dy\\
&=-\frac{1}{|B_h|} \int_0^h \left( \int_{\partial B_1}
r^{n-1} \big(v(x+r\omega,t)-v(x,t)\big)\,d{\mathcal{H}}^{n-1}_\omega\right)\,dr\\
&=-\frac{|B_1|}{|B_h|} \int_0^h
r^{n-1} \left(
\frac{c h^2}2 \Delta v(x,t) +o(h^2)
\right)\,dr\\
&=-\mu h^2 \Delta v(x,t) +o(h^2),
\end{split}\end{equation*}
for some $\mu>0$.
Then, we suppose that the velocity of the fluid is reduced when ${\mathcal{D}}$ is positive (since the parcel at $x$
is faster than the ones in its vicinity, and therefore it is dragged back by them)
and is enhanced when ${\mathcal{D}}$ is negative (since the parcel at $x$
is slower than the ones in its vicinity, and therefore it is pulled forward by them).
For simplicity, we can therefore assume that such an additional acceleration term is proportional to ${\mathcal{D}}$
(say, to make it a finite quantity in the limit, proportional to ${\mathcal{D}}/h^2$). By taking $h\searrow0$,
we thus obtain an additional acceleration (or deceleration) given by a term of the form $\mu \Delta v$.
By incorporating this correction into (<ref>) we thus find the equation
\begin{equation}\label{EUMSD-OS-DNS}
\rho \partial_t v+\rho (v\cdot\nabla) v=\mu\Delta v
-ge_n \rho-\nabla p,
\end{equation}
which is called in jargon[Equation (<ref>) (possibly complemented with
other structural equations of the fluid)
come from Claude Louis Marie Henri Navier and Sir George Gabriel Stokes, 1st Baronet.
Our experience with the Laplacian as a “democratic” operator (recall the discussion on page DEEFFNVST)
may suggest that the Laplacian in (<ref>) is providing a smoothing effect.
This may be the case, but our present understanding of the solutions of (<ref>)
is unsatisfactory and highly incomplete. In particular, it is not yet known if smooth (say,
globally defined and meeting some natural conditions)
solutions always exist. If the reader finds an answer to this question,
they do not only earn immortal fame among the circle of PDEs enthusiasts, but also obtain a
substantial amount of money, since the problem is listed in the Millennium Prize Problems
and a $10^6$ US $ award is offered to the first person providing a solution
(the case of incompressible fluids in $\R^3$ is hard enough). See e.g. <cit.> for further details about the most difficult way to become rich.]
the Navier-Stokes equationNavier-Stokes equation.
We stress that (<ref>) is a vectorial equation (or, equivalently, a system of equations in each component of $v$). There are also interesting situations in which (<ref>) reduces to a scalar equation,
such as the one in which the fluid is constrained within an infinite pipe with a given direction. For instance,
to make things as simple as possible,
let us suppose that the fluid is incompressibleincompressibility: hence, we have that the density of the fluid remains constant in the flow
(namely, a change in the density over time would imply that the fluid had either compressed or expanded). That is, for incompressible fluids the quantity $\rho(x(t),t)$ must be constant in time, leading to
\begin{equation*} 0=\frac{d}{dt}\rho(x(t),t)=\nabla\rho(x(t),t)\cdot \dot x(t)+\partial_t\rho(x(t),t)=
\nabla\rho(x(t),t)\cdot v(x(t),t)+\partial_t\rho(x(t),t).\end{equation*}
Combining this with (<ref>),
\begin{equation*} \rho\div v=\div(\rho v)-\nabla\rho\cdot v=\div(\rho v)+\partial_t\rho(x(t),t)=0,\end{equation*}
\begin{equation}\label{INCOMPRE}
\div v=0,\end{equation}
that is the velocity is a divergence-free vector field.
A horizontal pipe in direction $e_1$ with cross section $\Omega$.
For this reason, recalling (<ref>), stationary solutions of incompressible viscous fluids
are solutions of
\begin{equation}\label{EUMSD-OS-DNSSER}
\begin{dcases}&
\mu\Delta v-\rho (v\cdot\nabla)v-ge_n \rho-\nabla p=0,\\&
\div v=0.
\end{dcases}
\end{equation}
Suppose now that the above fluid moves in parallel streamlines
through a horizontal straight pipe of a given cross sectional form.
Up to rotations which leave invariant the vertical directions, we can assume that the pipe has the form $\R\times\Omega$
for some $\Omega\subset\R^{n-1}$ and $v=(u,0,\dots,0)$ for some scalar function $u$ (that is, the pipe directing the fluid is oriented towards the $e_1$-direction, see Figure <ref>). In this situation,
we have that $\Delta v=\Delta u \,e_1$,
$(v\cdot\nabla)v=u\partial_1u\,e_1$ and $\div v=\partial_1u$.
Consistently with these observations,
(<ref>) reduces to
\begin{equation}\label{EUMSD-OS-DNSSE234R}
\begin{dcases}&
\mu\Delta u\,e_1-\rho u\partial_1u\,e_1-ge_n \rho-\nabla p=0,\\&
\partial_1u=0.
\end{dcases}
\end{equation}
Taking the components $\{2,\dots,n-1\}$ of the first equation in (<ref>) we see that $\partial_i p=0$
for each $i\in\{2,\dots,n-1\}$, giving that the fluid pressure is constant in these directions.
Taking the last component instead, we find $\partial_n p=-g\rho$.
The most interesting information however comes from the first component of the first equation in (<ref>) since
this provides an elliptic equation for $u:\Omega\to\R$. Indeed, focusing on this aspect, we infer from (<ref>) that $u$ is a solution of
\begin{equation*}
\begin{dcases}&
\mu\Delta u-\rho u\partial_1u-\partial_1 p=0,\\&
\partial_1u=0.
\end{dcases}
\end{equation*}
Hence, plugging the second equation into the first one,
\begin{equation}\label{EUMSD-OS-32456i7DNSSE234R}
\mu\Delta u=\partial_1 p \qquad{\mbox{ in }}\,\Omega.
\end{equation}
A special case of interest, which constitutes one of the main motivations for the theory that will be presented
in Sections <ref> and <ref>, is when:
\begin{equation}\label{NK090SOJDKHJNZ}\begin{split}
&{\mbox{the pressure rate is constant,}}\\
&{\mbox{the velocity on the boundary of the pipe is constant}}\\
&{\mbox{and the tangential stress on the pipe is constant.}}
\end{split}\end{equation}
The problem discussed in detail in Sections <ref> and <ref>, and dating back
to James Serrin <cit.>, is precisely
to identify the possible shape of the pipe that allows such special configurations (that is, to determine all the cross sections $\Omega$ that give rise to solutions of (<ref>)
which are compatible with the prescriptions in (<ref>)):
this is certainly a problem of great practical importance, since ideally one would like to project pipes which
do not produce excessive tangential stress, to reduce the wearing of the material and the consequential formation of defects, holes
and licking (for this, maintaining a constant stress along the pipe could be a nice way to balance
the wearing somewhat uniformly along the whole surface of the pipe).
To efficiently address a question of this type, we first need to translate the physical
requests described in (<ref>)
into a mathematical framework. In order to achieve this,
we point out that the prescription that the pressure rate is constant simply says that $\partial_1 p=c_1$, for some $c_1\in\R$.
Also, the assumption that the velocity on the boundary of the pipe is constant means that $u=c_2$ along $\partial\Omega$,
for some $c_2\in\R$. As a matter of fact, since equation (<ref>) remains invariant
if we replace $u$ with $u-c_2$, we can simply assume[Equivalently, one can argue that the invariance of the form of the description of physical problems
among mutually translating reference frames allows us to choose an inertial frame of reference
moving at constant speed $c_2$. In this frame the previous prescription reduces to that
the velocity on the boundary of the pipe is equal to zero.]
that $c_2=0$, hence $u=0$ along $\partial\Omega$.
As for the stress on the pipe, we can suppose that this is due to the fact that fluid parcels near the pipe may move faster or slower
that the ones directly adhering to the pipe (that is, in the normalization above, the fluid parcel in the vicinity of the pipe
could have strictly positive or strictly negative velocity). We can additionally suppose that the wearing of the pipe is directly
influenced by this change of velocity (say, some particles of the pipe get removed by the fluid parcels moving either forwards or backwards). In this spirit, one can model the stress along the pipe as proportional to the normal derivative of the fluid velocity.
With this, the prescription of having a constant stress along the pipe is translated into $\partial_\nu u=c_3$ along $\partial\Omega$,
for some $c_3\in\R$.
In consideration of this, and normalizing constants for simplicity,
we write (<ref>) and (<ref>) as
\begin{equation}\label{EUMSD-OS-32456i7DNSSE234R-BNM}
\begin{dcases}
\Delta u=1&{\mbox{ in }}\Omega,\\
u=0&{\mbox{ on }}\partial\Omega,\\
\partial_\nu u=c&{\mbox{ on }}\partial\Omega.
\end{dcases}\end{equation}
This equation will be indeed[We observe that (<ref>)
does possess a solution when $\Omega$ is a ball. Specifically, given $x_0\in\R^n$ and $r>0$, the function $u(x):=\frac{|x-x_0|^2-r^2}{2n}$ is a solution of (<ref>)
in the ball $B_r(x_0)$
(and note that this solution corresponds to our physical intuition of the problem,
since it corresponds to a fluid reaching its maximal speed at the center of a circular pipe, with the velocity of the fluid being minimal at the boundary of the pipe, due to the viscous effects).
We will discuss in Sections <ref> and <ref>
whether this is the only type of possible solution
or whether there are domains different from the ball which allow different kinds of solutions.]
the starting point of the topics presented in
Sections <ref> and <ref>. A different model leading to the same mathematical
problem will be also presented in Section <ref>.
We end this section by recalling a simple, but helpful, use of the “principle of inertia” in the fluid dynamics setting.
Namely, a case of special interest is provided by a rigid body (say, with a given shape described by a set $\Omega$) moving in a fluid with constant speed. In this situation, the equations presented in this section hold outside the moving domain described, for instance, by $\Omega(t):=\{x\in\R^3$ s.t. $x+v_0 t\in\Omega\}$, for a given constant vector $v_0\in\R^3$.
Since fluid dynamics is already very difficult in a given domain, it is usually more agreeable to change the inertial frame to describe the motion by following the moving body (and we expect by Newton's First Law that this new choice of coordinates does not alter the physical description of the phenomena).
More explicitly, it comes in handy to define
\begin{equation}\label{ISUJONPRENsyhfngTNAbaujnfRGbhndfIKAR}
\widetilde v(x,t):=v(x-v_0 t,t)+v_0,\qquad
\widetilde p(x,t):=p(x-v_0 t,t)\qquad{\mbox{and}}\qquad\widetilde \rho(x,t):=\rho(x-v_0 t,t)\end{equation}
and observe that
equations (<ref>), (<ref>), (<ref>)
and (<ref>) are all preserved by this transformation, since $\div\widetilde v=\div v$,
\begin{eqnarray*}
\partial_t \widetilde\rho+\div(\widetilde\rho \widetilde v)&=&
\partial_t \rho-v_0\cdot\nabla\rho+\div\big(\rho( v+v_0)\big)\\&=&\partial_t \rho+\div(\rho v)
\end{eqnarray*}
\begin{eqnarray*}&&
\widetilde\rho \partial_t \widetilde v+\widetilde\rho (\widetilde v\cdot\nabla) \widetilde v-\mu\Delta \widetilde v+ge_n \widetilde\rho+\nabla \widetilde p\\&=&
\rho \big(\partial_t v-v_0\cdot\nabla v\big)
+\rho \big((v+v_0)\cdot\nabla) v-\mu\Delta v+ge_n \rho+\nabla p\\&=&
\rho \partial_t v
+\rho v\cdot\nabla v-\mu\Delta v+ge_n \rho+\nabla p.
\end{eqnarray*}
Also, the new domain of reference is now simply the complement of $\Omega$.
Notice that the transformation in (<ref>)
is simply replacing the situation of a rigid body in constant speed motion in a fluid at rest at infinity with the one of a still body in a fluid with constant speed at infinity. This is precisely the idea leading to the construction of wind tunnels: to replicate the aerodynamic interactions between a moving object and the surrounding air, it is common to construct large tubes with air blowing through them against a static model of the object.
In the forthcoming Sections <ref> and <ref>, where we will describe rigid bodies moving at a constant speed, we will tacitly assume to have performed the transformation in (<ref>) to reduce ourselves to the case of still objects and domains that do not vary with time.
§.§ Irrotational fluids
Among all possible fluids, a class deserving some special attention is provided by those which present “no vortex”.
In light of Stokes' Theorem, this notion is made precise by the mathematical concept of curl, that is
we say that a fluid is irrotational if the curl of its velocity field vanishes.
A natural question in this setting is whether “vortexes can be produced out of nothing”:
for instance if the initial conditions of a fluid present no vortexes, is it possible that vortexes arise at a later stage?
To increase our familiarity with multivariate calculus and partial differential equations,
we show that this is impossible, at least for inviscid, incompressible and barotropic flows
(in this setting, a flow is said to be barotropic if its density is a function of the pressure only, say $\rho=g(p)$
for some positive function $g$). More precisely, we consider a barotropic solution in the absence of gravity of
the momentum balance equation in (<ref>)
satisfying the incompressibility condition in (<ref>), that is
\begin{equation}\label{CURLDESE1}
\begin{dcases}
\rho \partial_t v+\rho (v\cdot\nabla) v=-\nabla p,\\
\div v=0,\\
\rho=g(p)
\end{dcases}\end{equation}
and we show that
\begin{equation}\label{CURLDESE}
{\mbox{if~$\curl v=0$ at time~$t=0$, then~$\curl v=0$ at every time~$t$.}}\end{equation}
To achieve this goal, we first recall the vector calculus identity, valid for all smooth vector fields $V=(V_1,V_2,V_3):\R^3\to\R^3$,
\begin{equation}\label{VCISTPOK}
(V\cdot\nabla )V=\frac{1}{2}\nabla |V|^2-
V\times (\curl V),\end{equation}
having denoted by $\times$ the vector product operation.
To prove (<ref>), up to exchanging the order of the coordinates, we can concentrate our attention on the first
coordinate: hence we aim at proving that
\begin{equation}\label{VCISTPOK-7}
\sum_{j=1}^3 V_j\partial_j V_1=\frac{1}{2}\partial_1 |V|^2-
\big(V\times (\curl V)\big)_1.\end{equation}
To check this, we calculate:
\begin{eqnarray*}&&
\big(V\times (\curl V)\big)_1-\frac{1}{2}\partial_1 |V|^2\\&=&V_2(\curl V)_3-V_3(\curl V)_2-
V\cdot\partial_1 V\\
&=& V_2\big( \partial_1 V_2-\partial_2 V_1\big) - V_3 \big( \partial_3 V_1-\partial_1 V_3\big)-V_1\partial_1 V_1
-V_2\partial_1 V_2-V_3\partial_1 V_3\\
&=& - V_2\partial_2 V_1 - V_3 \partial_3 V_1-V_1\partial_1 V_1,
\end{eqnarray*}
which proves (<ref>) and thus (<ref>).
We also recall another vector calculus identity, valid for all smooth vector fields $V=(V_1,V_2,V_3)$, $W=(W_1,W_2,W_3):\R^3\to\R^3$:
\begin{equation}\label{nhfhgtfredwM3S54E4y5C236terdhrCHE}
\curl(V\times W)
= \div W\;V- \div V \;W+(W\cdot\nabla)\, V- (V\cdot\nabla)\,W.\end{equation}
To prove this, we can write $V=V_1e_1+V_2e_2+V_3 e_3$ and notice that, since (<ref>) is linear in $V$, it suffices to prove it when $V$
reduces to each of the components $V_j e_j$.
Thus, up to reordering the coordinates, we can focus on the case in which $V$ is actually $V_1 e_1$.
Hence we calculate
\begin{eqnarray*}
&&\curl((V_1e_1)\times W)+ \div( V_1e_1) \;W-(W\cdot\nabla)\, V_1e_1+( (V_1e_1)\cdot\nabla)\,W
\\&=&
\curl(0,-V_1W_3,V_1W_2)
+ \partial_1 V_1\,W-\sum_{j=1}^3W_j\partial_j V_1e_1+V_1\partial_1 W\\
&=&\big( \partial_2(V_1W_2)+\partial_3(V_1W_3),\,-\partial_1(V_1W_2),\,-\partial_1(V_1W_3)\big)
+ \partial_1 V_1\,W-\sum_{j=1}^3W_j\partial_j V_1e_1+V_1\partial_1 W
\\&=&
\big(
\partial_2V_1\,W_2+V_1\partial_2 W_2+\partial_3 V_1\,W_3+V_1\partial_3 W_3+\partial_1 V_1\,W_1
0,\,0\big)-\sum_{j=1}^3W_j\partial_j V_1e_1\\&=&
\big(V_1\partial_2 W_2+V_1\partial_3 W_3
\\&=& \div W\;V_1e_1
\end{eqnarray*}
and this completes the proof of (<ref>).
The geometric argument used to prove (<ref>).
It is also useful to recall the vector calculus identity
\begin{equation}\label{YY08ojew9oik903hyto3298ythgfjb8ihyri-1urX} \div(\curl V)=0.\end{equation}
For this, we take any ball $B\subset\R^3$ and we consider the spherical surface $S:=\partial B$.
We split $S$ into two hemispheres $S^+$ and $S^-$
with a common equator $\eta$, which is the intersection between the boundaries of the surfaces $S^+$ and $S^-$.
Looking at $\eta$ as a curve, the natural direction of travel along $\eta$ when considered as the boundary of $S^+$
is opposite to the one obtained when considering it as the boundary of $S^-$, see Figure <ref>
(here, we are endowing $S^+$ and $S^-$ with the external unit vector field of $\partial B$).
As a result, the circulation of a vector field $V$ along $\eta$ considered as the boundary of $S^+$ (that we denote by ${\mathcal{C}}^+(V)$)
is opposite to the circulation of $V$ along $\eta$ considered as the boundary of $S^-$
(that we denote by ${\mathcal{C}}^-(V)$), that is
$$ {\mathcal{C}}^+(V)=-{\mathcal{C}}^-(V).$$
In addition, by Stokes' Theorem, we know that ${\mathcal{C}}^\pm(V)$ agrees with the flux
of $\curl V$ through the surface $S^+$.
Using these bits of information and the Divergence Theorem, we conclude that
\begin{eqnarray*}
\int_{S^+}\curl V\cdot\nu+\int_{S^-}\curl V\cdot\nu=\int_{\partial B}\curl V\cdot\nu=
\int_{B}\div(\curl V).
\end{eqnarray*}
From the arbitrariness of the ball $B$, we obtain (<ref>), as desired.
We also point out that, for any smooth function $\psi$,
\begin{equation}\label{08ojew9oik903hyto3298ythgfjb8ihyri-1urX} \curl\nabla\psi=0.\end{equation}
This can be checked by employing Stokes' Theorem. Indeed, if $\Sigma$ is any smooth oriented element of surface in $\R^3$
with boundary $\partial \Sigma $, the flux of $\curl\nabla\psi$ through the surface $\Sigma$
is equal to the circulation of $\nabla\psi$ over the loop $\partial\Sigma$. The latter object, if we describe $\partial\Sigma$
by a closed curve $\gamma:[0,L]\to\R^3$ with arc length parameterization, can be written as
$$ \int_0^L \nabla\psi(\gamma(\tau))\cdot\dot\gamma(\tau)\,d\tau=
\int_0^L \frac{d}{d\tau}\psi(\gamma(\tau))\,d\tau=\psi(\gamma(L))-\psi(\gamma(0))=0.$$
Since this is valid for an arbitrary surface $\Sigma$, the proof of (<ref>) is complete.
Now, to prove (<ref>), we let $\omega:=\curl v$, we use the momentum balance equation
and the barotropicity in (<ref>)
in combination with (<ref>) and we see that
\begin{equation}\label{dui39urteifjewgtlie4y9548796575903549873204327957608497685490549030594}
\begin{split}
0\,&=\curl\left( \partial_t v+ (v\cdot\nabla) v+\frac{\nabla p}{\rho}\right)\\&=\curl\left( \partial_t v+ \frac{1}{2}\nabla |v|^2-
v\times (\curl v)+\frac{\nabla p}{g(p)}\right)\\
&=\partial_t \omega+\curl\left( \frac{1}{2}\nabla |v|^2-
v\times \omega+\frac{\nabla p}{g(p)}\right).
\end{split}\end{equation}
Now we observe that
\begin{eqnarray*} &&
\curl\left(\frac{\nabla p}{g(p)}\right)= \left(\partial_2 \left(\frac{\partial_3 p}{g(p)}\right)-
\partial_3 \left(\frac{\partial_2 p}{g(p)}\right),\partial_3 \left(\frac{\partial_1 p}{g(p)}\right)-
\partial_1 \left(\frac{\partial_3 p}{g(p)}\right),
\partial_1 \left(\frac{\partial_2 p}{g(p)}\right)-
\partial_2 \left(\frac{\partial_1 p}{g(p)}\right)
\right)\\&&\qquad=\left( \frac{\partial_{23} p}{g(p)}-\frac{\partial_3p\,g'(p)\,\partial_2p}{g^2(p)}
-\frac{\partial_{23} p}{g(p)} + \frac{\partial_2 p\,g'(p)\,\partial_3p}{g^2(p)},\right.\\ &&\qquad\qquad\qquad\left.
\frac{\partial_{13} p}{g(p)}-\frac{\partial_1 p\,g'(p)\,\partial_3p}{g^2(p)}
-\frac{\partial_{13} p}{g(p)}+\frac{\partial_3 p\,g'(p)\,\partial_1p}{g^2(p)},\right.\\ &&\qquad\qquad\qquad\left.
\frac{\partial_{12} p}{g(p)}-\frac{\partial_2 p\,g'(p)\,\partial_1p}{g^2(p)}
-\frac{\partial_{12} p}{g(p)}+\frac{\partial_1 p\,g'(p)\,\partial_2p}{g^2(p)}
\right)\\&&\qquad=(0,0,0).
\end{eqnarray*}
Hence, from this, (<ref>) and (<ref>),
\begin{equation*}
\begin{split}
0=\partial_t \omega-\curl\left( v\times \omega\right).
\end{split}\end{equation*}
This and (<ref>) lead to
\begin{equation*}
\partial_t \omega=
\curl(v\times \omega)
= \div \omega\;v- \div v \; \omega+( \omega\cdot\nabla)\, v- (v\cdot\nabla)\, \omega.\end{equation*}
Thus, combining the incompressibility condition in (<ref>)
and (<ref>),
\begin{equation*}
\partial_t \omega=( \omega\cdot\nabla)\, v- (v\cdot\nabla)\, \omega\end{equation*}
and therefore, following the fluid parcel $x(t)$,
\begin{equation*}\begin{split}&
\frac{d}{dt} \omega(x(t),t)=
\partial_t \omega(x(t),t)+(\dot x(t)\cdot\nabla)\omega(x(t),t)\\&\qquad
=\partial_t \omega(x(t),t)+\big(v( x(t),t)\cdot\nabla\big)\omega(x(t),t)
=\big( \omega(x(t),t)\cdot\nabla\big)\, v(x(t),t).
\end{split}\end{equation*}
As a result, if $Z(t)=(Z_1(t),Z_2(t),Z_3(t)):=\omega(x(t),t)$, we have that $Z$ is a solution of the Cauchy problem
for ordinary differential equations
$$ \begin{dcases}\displaystyle
\dot Z(t)=\sum_{k=1}^3 Z_k(t)\, \partial_k v(x(t),t),\\
\end{dcases}$$
By the uniqueness result for solutions of ordinary differential equations we thereby infer that $Z(t)=0$ for all $t\in\R$
and this completes[As a technical detail, we point out that we are freely assuming that if, for a given function $f$,
we know that $f(x(t),t)=0$ for all times $t$, then we can infer that $f(x,t)=0$
for all spatial positions $x$ and times $t$. This is because we are implicitly assuming that the flow
of the fluid exists for all times: hence, given any position $x$ we can “flow $x$ backwards for a time $t$”,
that is consider the fluid parcel evolution starting at some point $x_0$ at time $0$ arriving at $x$ at time $t$
(this gives us the possibility of choosing $x(t)=x$).]
the proof of (<ref>).
§.§ Propagation of sound waves
Now we retake the set of Euler fluid flow equations for the velocity $v$ of an inviscid fluid,
namely we look at the mass conservation equation in (<ref>), at
the momentum balance equation in (<ref>) and at a constitutive relation
linking the pressure $p$ and the density $\rho$, namely (neglecting the gravity effects)
\begin{equation}\label{EUMSD-OS-D-TUT}
\begin{dcases}&
\partial_t \rho+\div(\rho v)=0,\\&
\rho \partial_t v+\rho (v\cdot\nabla) v=-\nabla p,\\&
\end{dcases}
\end{equation}
for some function $f$. In practice, it is useful to take $f$ to be strictly increasing (the higher the density of the fluid, the higher the pressure produced; this would also give that the fluid is barotropic in the setting of Section <ref>).
We can think of (<ref>) as a very simple model for a gas
and we aim here at understanding the propagation of sound waves in such a medium.
For this objective, we first point out that a solution of (<ref>)
is provided by $(v,p,\rho)=(0,p_0,\rho_0)$, for every $\rho_0\in(0,+\infty)$ and $p_0:=f(\rho_0)$.
This configuration corresponds to the physical situation of a gas at rest, with constant density and pressure.
Suppose now that we perturb this configuration by creating a small variation of the gas pressure,
for instance by singing, or by playing[To play guitar, see the forthcoming Section <ref>.]
a guitar. The idea is thus to look for (at least approximate) solutions of the form
$$ v(x,t)=\e v_1(x,t),\qquad p(x,t)=p_0+\e p_1(x,t)\qquad{\mbox{and}}\qquad \rho(x,t)=\rho_0+\e\rho_1(x,t).$$
In this setting, $\e$ is a small parameter and our goal is to determine the functions $(v_1,p_1,\rho_1)$ in order to satisfy the set of equations in (<ref>), formally up to negligible errors in $\e$. To this end, we observe that
$$ p_0+\e p_1=p=f(\rho)=f(\rho_0+\e \rho_1)=f(\rho_0)+\e f'(\rho_0) \rho_1+o(\e)=p_0+\e f'(\rho_0) \rho_1+o(\e),$$
leading to the choice
\begin{equation*}
p_1=f'(\rho_0) \rho_1,
\end{equation*}
up to higher orders in $\e$ that we here sloppily disregard.
A plane wave.
\begin{equation*}
0=\partial_t \rho+\div(\rho v)=\e\partial_t \rho_1+\e\rho_0\div v_1+o(\e)
\end{equation*}
\begin{equation*}
0= \rho \partial_t v+\rho (v\cdot\nabla) v+\nabla p=\e\rho_0 \partial_t v_1+\e\nabla p_1+o(\e)=
\e\rho_0 \partial_t v_1+\e f'(\rho_0)\nabla \rho_1+o(\e).
\end{equation*}
These observations give
\begin{equation*}
\begin{dcases}
&\partial_t \rho_1+ \rho_0\div v_1=0,\\
&\rho_0 \partial_t v_1+f'(\rho_0)\nabla \rho_1=0.
\end{dcases}\end{equation*}
As a consequence,
\begin{equation*}
\partial_{tt} \rho_1=-\partial_t\big(\rho_0\div v_1\big)
=-\div \big(\rho_0\partial_t v_1\big)=
\div\big(
f'(\rho_0)\nabla \rho_1
\big)
=f'(\rho_0)\Delta \rho_1
That is, the perturbed density $\rho_1$
satisfies the equation
\begin{equation}\label{WAYEBJJD121t416JH098327uyrhdhc832nbcM}
\partial_{tt}u=c^2 \Delta u
\end{equation}
with $c:=\sqrt{f'(\rho_0)}>0$.
Equation (<ref>) is called[In terms of the classification
mentioned in footnote <ref>
on page CLASSIFICATIONFOOTN, equation (<ref>) is hyperbolic.
Indeed, we can take here $N=n+1$, $X=(x,t)$ and
$$ a_{ij}=\begin{dcases}
c^2 & {\mbox{ if }}i=j\in\{1,\dots,n\},\\
-1 & {\mbox{ if }}i=j=n+1,\\
0 & {\mbox{ otherwise,}}
\end{dcases}$$
thus producing $n$ strictly positive, and one strictly negative, eigenvalues.]
in jargon the “wave equation”wave equation.
To have a feeling of the “propagation of waves” encoded in equation (<ref>),
we can consider a smooth function $u_\star:\R\to\R$ and a direction $\omega\in\partial B_1$,
and thus define $u_\omega(x,t):=u_\star(\omega\cdot x-ct)$. Notice that $u_\omega$ is indeed a solution of (<ref>),
physically corresponding to a traveling plane waveplane wave.
Indeed, its evolution in time corresponds to a translation of $u_\star$ along the direction $\omega$
with speed $c$. Also, for $\kappa\in\R$, the parallel hyperplanes $\{\omega\cdot x-ct=\kappa\}$
(again, traveling with constant speed $c$ in direction $\omega$)
correspond to the level sets $\{ u_\omega=\kappa_\star\}$, where $\kappa_\star:=u_\star(\kappa)$
(the level sets of a wave are often called “wavefronts”
and they have special importance since they correspond to the surfaces on which,
at a given moment of time, all particles of the medium undergo the same motion).
See Figure <ref> for a series of frames of a traveling plane wave
(the picture is obtained by choosing $u_\star(r):=e^{-r^2}$, $\omega:=\left(\frac{\sqrt2}{2},\frac{\sqrt2}{2}\right)$, $c=1$
and instants of time $t\in\{0,1,2,3\}$).
See also Figure <ref>
for the pictures of the corresponding level sets of the wave
(i.e., of the corresponding wavefronts).
Wavefronts of the plane wave in Figure <ref>.
§.§ Hydrodynamics doesn't always work right
Now that we built some confidence in the equations of hydrodynamics,
it is time to question and challenge our own knowledge, to appreciate its power as well as its limitation.
After all, it's not so important to know, but rather to know of not knowing.
In particular, the great Euler (“Master of us all” according to Laplace) allowing,
we will recall here a rather striking, and perhaps surprising, flaw of the hydrodynamic equations presented in Section <ref> in the description of aerodynamic forces.
We will focus here on the notion of “drag”drag, which is the
force acting in the opposite direction
to an object moving in a fluid (say, the air resistance, for example).
This is not just a mathematical curiosity: for instance, according to Wikipedia
“induced drag tends to be the most important component for airplanes during take-off or landing flight”
and, more importantly,
“in the physics of sports, the drag force is necessary to explain the motion of balls, javelins, arrows and frisbees and the performance of runners and swimmers”.
Portrait of Jean Le Rond d'Alembert (Public Domain image from
The issue about the drag that we discuss here was discovered by Jean le Rond d'Alembert in 1752.
Often, this remarkable discovery is called with the name of
“d'Alembert's paradox”d'Alembert's paradox;
see Figure <ref>
for a portrait of d'Alembert's intense glance (by Maurice Quentin de La Tour). In a nutshell, we will prove that
\begin{equation}\label{PARADDA}
\begin{split}
&{\mbox{if a body is moving with constant velocity}}\\&{\mbox{in an incompressible, inviscid, irrotational and steady flow,}}\\
&{\mbox{the corresponding drag force is zero.}}
\end{split}
\end{equation}
Of course, we may wonder how such a statement can be coherent with our own
everyday experience, in which the effect of air resistance is usually apparent and quite decisive
(at least when we catch a flight or we kick a ball). Though the full understanding
of hydrodynamics is likely way above the present possibilities of science,
one common explanation of the discrepancy between the claim in (<ref>) and real life situations
relies on remarking that the fluid's description in (<ref>) is too idealistic.
In particular, the occurrence of the paradox is usually attributed
to the neglected effects of viscosity[The importance of viscosity actually reflects the importance of the Laplace operator in the model: compare equations (<ref>) and (<ref>).
Though we do not explicitly use this here, for completeness we point out that
an incompressible and irrotational fluid is automatically inviscid
(hence, the lack of viscosity is also a byproduct of two assumptions, namely the incompressibility and
the lack of vortexes, which, together, entail very restrictive byproducts): indeed, using the vector calculus
$$ \nabla (\div V)-\curl(\curl V)=\Delta V,$$
(see e.g. (<ref>) for a proof) we have that for incompressible and irrotational fluid it holds that
$$ \Delta v=\nabla (\div v)-\curl(\curl v)=0-0=0,$$
hence the Navier-Stokes equation in (<ref>)
boils down to the Euler equation in (<ref>).]
(roughly speaking, one of the effects of viscosity <cit.> is to produce
thin boundary layersboundary layer near the surface of the moving object, which may entail
friction, flow separation and a low-pressure wake, leading to pressure drag).
However, the official resolution of the question is possibly debatable in its full generality:
for instance, in relation to the paradoxical features of (<ref>),
Garrett Birkhoff <cit.> states[Garrett Birkhoff's statements
were in turn criticized by James J. Stoker <cit.>
(who was concerned about the possibility that readers were
“very likely to get wrong ideas about some of the important
and useful achievements in hydrodynamics”
being misled by the “negative aspects of the theory”). Some of the original statements in <cit.>
were revised in the second edition <cit.>.
All in all, much more knowledge has to be built before we reach a complete understanding of
fluid mechanics and hydrodynamics.]
that: “I think that to attribute them all to the neglect of viscosity is an unwarranted oversimplification. The root lies deeper, in lack of precisely that deductive rigor whose importance is so commonly minimized by physicists and engineers”.
Now we work out the mathematical details needed to prove the claim in (<ref>).
To achieve this goal, we recall
the Euler equation in (<ref>)
and we complement it with the mass transport equation in (<ref>),
the incompressibility condition in (<ref>)
and an irrotational condition. We also assume that the flow is “steady”:
this word usually refers to a situation in
which the fluid properties at a point in the system do not change over time
(they only depend on the point); concretely, to avoid ambiguities, we will suppose here that the density $\rho$
is only a function of $x$ (no dependence on the time $t$).
In this setting, the velocity $v$
of an incompressible fluid, subject to neither viscosity nor gravity effects, with steady density $\rho$ and pressure $p$
and presenting no vortexes can be described by the set of equations (valid outside the body $\Omega$)
\begin{equation}\label{VCISTPOK-2}
\begin{dcases}
\rho \partial_t v+\rho (v\cdot\nabla) v=-\nabla p,\\
\div(\rho v)=0,\\
\div v=0,\\
\curl v=0.
\end{dcases}
\end{equation}
Now, combining (<ref>) with the irrotationality condition in (<ref>) we deduce that
\begin{equation}\label{VCISTPOK-3}
(v\cdot\nabla )v=\frac{1}{2}\nabla |v|^2.\end{equation}
We also consider the differential form $\omega:=v_1\,dx_1+v_2\,dx_2+v_3\,dx_3$ and we observe that
\begin{eqnarray*} d\omega&=&-\partial_2 v_1\,dx_1\wedge dx_2-
\partial_3 v_1\,dx_1\wedge dx_3+\partial_1 v_2\,dx_1\wedge dx_2\\&&\qquad
-\partial_3 v_2\,dx_2\wedge dx_3+\partial_1v_3\,dx_1\wedge dx_3
+\partial_2v_3\,dx_2\wedge dx_3\\&=&(\curl v)_3\,dx_1\wedge dx_2-(\curl v)_2\,dx_1\wedge dx_3+
(\curl v)_1\,dx_1\wedge dx_2\\&=&0,
\end{eqnarray*}
thanks again to the irrotationality condition in (<ref>).
Accordingly, the differential form $\omega$ is exact, and therefore closed, thanks to the
Poincaré Lemma (see e.g. <cit.>). This gives that there exists a velocity potential function $\varphi$
satisfying $\omega=d\varphi$, that is
\begin{equation}\label{jiaknfujsuE0ujkjHAHLoefvujs98ou38jhPD98jhdjhfX8}
v=\nabla \varphi. \end{equation}
We stress that $\varphi$ is harmonic outside $\Omega$, since $0=\div v=\div\nabla\varphi=\Delta\varphi$.
Moreover, the first equation in (<ref>),
and (<ref>) entail that
\begin{equation}\label{kSUDSpREdfskljhREHGkozWI23f3aziw}\begin{split}&
0= \partial_t v+(v\cdot\nabla) v+\frac{\nabla p}\rho=
\partial_t v+\frac{1}{2}\nabla |v|^2+\frac{\nabla p}\rho\\&\qquad\quad=
\partial_t \nabla \varphi+\frac{1}{2}\nabla |\nabla\varphi|^2+\frac{\nabla p}\rho=
\nabla\left( \partial_t \varphi+\frac{1}{2}|\nabla\varphi|^2\right)+\frac{\nabla p}{\rho}.\end{split}
\end{equation}
Now we recall the assumption in (<ref>) that the object is moving with constant speed, say $v_0\in\R^3\setminus\{0\}$.
That is, if we denote by $x(t)$ the position at a given time $t$ of a parcel of the fluid, we have that $\dot x(t)=v_0$
and accordingly $v(x(t),t)=v_0$ for every time $t$.
From this observation we infer that
$$ 0=\frac{d}{dt}v(x(t),t)=\partial_t v(x(t),t)+(v_0\cdot\nabla) v(x(t),t).$$
and (<ref>) (recall also
footnote <ref> on page LAFOPRI)
give that
$$0= \partial_t \nabla\varphi+(v_0\cdot\nabla) (\nabla\varphi)=
\nabla\Big(\partial_t \varphi+(v_0\cdot\nabla)\varphi\Big)
Consequently, the function $\partial_t \varphi+(v_0\cdot\nabla)\varphi$ is constant in space
and therefore there exists a function of time, say $R=R(t)$, such that
\begin{equation*} \partial_t \varphi+(v_0\cdot\nabla)\varphi=R.\end{equation*}
From this and (<ref>) we arrive at
\begin{equation}\label{kSUDSpREdfskljhREHGkozWI23f3aziw2}\frac{\nabla p}{\rho}=
-\nabla\left( R-(v_0\cdot\nabla)\varphi+\frac{1}{2}|\nabla\varphi|^2\right)=
\nabla\left( (v_0\cdot\nabla)\varphi-\frac{1}{2}|\nabla\varphi|^2\right).
\end{equation}
Furthermore, using the conservation of mass and the incompressibility condition in (<ref>),
\begin{equation}\label{kSUDSpREdfskljhREHGkozWI23f3aziw7}
0=\div(\rho v)=\nabla\rho\cdot v+ \rho\div v=
\nabla\rho\cdot v=\nabla\rho\cdot v_0.
\end{equation}
With this, we can now compute the drag that the air exerts on the moving body.
For this, we recall that the force $F$ of the air is produced by the pressure, acting normally on the surface
of the body (see (<ref>)). Hence, calling the moving object $\Omega$ and $\nu$ its outer normal, we have that
\begin{equation}\label{FEUYT90hfbvjjxJ} F=-\int_{\partial\Omega} p\nu.\end{equation}
The minus sign here above is just recalling that positive pressures are assumed to go in the opposite direction
with respect to the exterior normal.
Since the drag $D$ is the force acting in the direction of motion of the body, using the Divergence Theorem we thus conclude that
\begin{equation}\label{EXDIVTHMQU} D=F\cdot\frac{v_0}{|v_0|}
=-\int_{\partial\Omega} p\frac{v_0\cdot\nu}{|v_0|}=\int_{\R^3\setminus\Omega}
\div\left(p\frac{v_0}{|v_0|}\right)=\int_{\R^3\setminus\Omega} \frac{\nabla p\cdot v_0}{|v_0|}
Notice that to apply the Divergence Theorem here to the infinite region $\R^3\setminus\Omega$
(that corresponds to the region occupied by the fluid) we are implicitly supposing that the pressure decays sufficiently fast at infinity
(that is, the pressure disturbance is essentially localized in the vicinity of the moving object, which is a reasonable[Once again, in this chapter, if an argument sounds convincing we buy it! We will be much more skeptical about heuristic reasonings from next chapter on. For instance, when we need to integrate on exterior domains in Section <ref>, the reader will appreciate how careful we will be in checking explicitly the appropriate decay properties of the functions involved (which is perhaps a delicate and annoying, albeit absolutely necessary, detail to be taken care of to avoid nonsensical computations).
In any case, let us mention that the possibility of utilizing the Divergence Theorem in an exterior domain
in (<ref>) is quite a delicate matter (see e.g. the thoughtful argument in <cit.>)
and it is a byproduct of the decay at infinity of the fluid velocity and pressure which
is modeled on that of harmonic functions.
Arguably, in the setting of these notes, alternative arguments to the ones in <cit.>
could be provided by using
the Green's Representation Formula in (<ref>)
(say, in the domain $B_R\setminus\overline{\Omega}$,
sending $R\to+\infty$): for this one would need to assume that the velocity potential
has a limit at infinity (thus, from Cauchy's Estimates
in Theorem <ref>, one can also bound the derivative of the potential along $\partial B_R$).
In any case, the decay estimates of the velocity fields rely
on the fact that the fluid cannot penetrate inside the object (as formalized in (<ref>)),
since this information allows one to get rid of one order of magnitude in the corresponding estimates. This is an interesting
physical feature since it reveals the “global” character of fluid dynamics, in which some features
in the proximity of the moving body have a significant influence on the velocity field at infinity, and vice versa.
In general, the decay analysis for solutions of fluid mechanics equations
is a delicate matter, see e.g. <cit.> and the references therein.]
physical assumption, though quantifying it precisely and checking it rigorously would require some technical skills).
Hence, recalling (<ref>),
\begin{equation*}
\begin{split} D\,&= \int_{\R^3\setminus\Omega} \frac{\rho v_0}{|v_0|}\cdot\nabla\left( (v_0\cdot\nabla)\varphi-\frac{1}{2}|\nabla\varphi|^2\right)\\
&=\int_{\R^3\setminus\Omega} \left\{\div\left[ \left( (v_0\cdot\nabla)\varphi-\frac{1}{2}|\nabla\varphi|^2\right)\frac{\rho v_0}{|v_0|}\right]-
\frac{\nabla\rho \cdot v_0}{|v_0|}\left( (v_0\cdot\nabla)\varphi-\frac{1}{2}|\nabla\varphi|^2\right)\right\}.
\end{split}\end{equation*}
Combining this and (<ref>),
\begin{equation}\label{inHTKMSue21567536uhtrf}
D=\int_{\R^3\setminus\Omega}\div\left[ \left( (v_0\cdot\nabla)\varphi-\frac{1}{2}|\nabla\varphi|^2\right)\frac{\rho v_0}{|v_0|}\right].
\end{equation}
We now stress that since the air cannot penetrate inside the object, necessarily
\begin{equation}\label{PSLmoLMSxHNS-AR}
{\mbox{the normal component
of the fluid velocity vanish along~$\partial\Omega$,}}\end{equation}
namely $v_0\cdot\nu=0$
on $\partial\Omega$.
As a result, using (<ref>) and the Divergence Theorem again,
\begin{eqnarray*}&&D=
\int_{\R^3\setminus\Omega} \div\left[ \left( (v_0\cdot\nabla)\varphi-\frac{1}{2}|\nabla\varphi|^2\right)\frac{\rho v_0}{|v_0|}\right]
=-\int_{\partial\Omega} \left( (v_0\cdot\nabla)\varphi-\frac{1}{2}|\nabla\varphi|^2\right)\frac{\rho v_0}{|v_0|}\cdot\nu=0.
\end{eqnarray*}
This completes the proof of d'Alembert's paradox in (<ref>).
See e.g. <cit.> and the references therein for additional information
on d'Alembert's paradox.
Portrait of Nikolai Yegorovich Joukowski,
Museum of Moscow Aviation Institute (image by Just from
Wikipedia, licensed under the Creative Commons Attribution-Share Alike 4.0 International license).
§.§ Lift of an airfoil
To safely recover from the shock received in Section <ref>
by d'Alembert's paradox, we give now a positive result
in terms of the aerodynamic lift of an airfoilairfoil. This result was established independently
by Martin Kutta and Nikolai Yegorovich Joukowski in the early twentieth century and, in its simplest formulation,
can be stated[Though the theory that
we present in these pages is way too crude to
comprise aerodynamics in its full complexity, the importance of a statement like (<ref>) in the theory of
flight appears to be paramount, as confirmed by the solid recognition bestowed to the scientists involved. For instance,
the Russian Air Force Academy and one of the airports in Moscow
were named after Joukowski. See also Figure <ref>.
In terms of real world applications, a common belief is indeed that the setting in (<ref>)
is too idealized to detect the intricate patterns produced by real fluids in the vicinity of a traveling object: however,
it is also believed that the circulation detected in (<ref>) does reflect significant physical
information since in many concrete situations
the flow around a thin airfoil is composed of a narrow viscous region near the body
(a sort of “boundary layer”) outside which the idealized
and inviscid description of the flow in (<ref>) turns out to be sufficiently realistic.
The gist is then to apply the setting in (<ref>) not quite to the traveling body but to the aggregate of the body and its own boundary layer:
in particular, the loop to compute the circulation in (<ref>) must be chosen outside this boundary layer.
With this, the boundary layer gives any traveling object an “effective shape” that may be different from its physical shape by accounting for the region in which the velocity changes from zero at the surface to the stream value away from the surface: the effectiveness of (<ref>) is thus to incorporate, as much as possible, the effects of turbulence and skin friction into the effective shape described by this boundary layer, remaining with a more mathematically treatable description away from it.
In this sense, the success of
the Kutta-Joukowski theory Kutta-Joukowski theory
consists in providing a simple, but not trivial, approach to aerodynamics which incorporates some aspects of viscous effects, while neglecting others.]
as follows:
\begin{equation}\label{JTTA}
\begin{split}&
{\mbox{an airfoil in relative motion with constant velocity~$-v_0$}}\\&
{\mbox{to an ambient inviscid homogeneous, irrotational fluid}}\\&{\mbox{has a lift force (that is the component of the force perpendicular to~$v_0$)}}\\&{\mbox{of magnitude }}
\rho \,|v_0| \Gamma, \\&{\mbox{where~$\Gamma$ is a circulation of~$v_0$ along the cross section of the airfoil.}}\end{split}\end{equation}
It is interesting to observe that (<ref>) explains for instance the generation of a lift on a wing as a result
of the contribution of the circulation $\Gamma$ of the velocity field around the wing.
Remarkably, the fact that (<ref>) takes into account the component of the force perpendicular to $v_0$
clearly states the importance of such a result for the theory of flight (that is, this lift force is what
allows, in principle
at least, an airplane to take off, provided they manage to create
a sufficiently large circulation $\Gamma$).
To model an airfoil, that is a “long wing” (actually, infinitely long for simplicity),
we consider a (nice, contractible) planar domain $\Omega\subset\R^2$
and the wing given by $\Omega_\star:=\Omega\times\R$, see Figure <ref>.
We reconsider the Euler equation in (<ref>) (in the absence of gravity) and the incompressibility condition in (<ref>) outside $\Omega_\star$
assuming that, by symmetry, the fluid parameters $v$ and $p$
are actually independent of $x_3$ and also on time (leading to a completely
steady solution of the problem).
In this way, we may consider
the following set of equations for $v=v(x,y)$ and $p=p(x,y)$, with $X=(x,y)\in\R^2\setminus\Omega$,
for a given constant $\rho\in(0,+\infty)$
(the constancy of the density being the mathematical translation
that the fluid is homogeneous):
\begin{equation}\label{DFRE}
\begin{dcases}&\rho (v\cdot\nabla) v=-\nabla p,\\
&\div v=0,\\&\curl v=0.\end{dcases}\end{equation}
Two-dimensional model of an airfoil, as inspired by a sailplane.
Notice that in this setting (thinking the vector field $(v_1(x,y),v_2(x,y),0)$ to be three-dimensional
but with a trivial last entry to compute the curl), we have that
\begin{equation}\label{CUR25t34yuD2}
0=\curl v=\big(\partial_xv_2-\partial_y v_1\big) e_3.
\end{equation}
Moreover, since, according to the principle of inertia
discussed on page PRINERY,
we are taking an inertial frame in which the body appears to be still,
\begin{equation}\label{ALLIVO0}
\lim_{|(x,y)|\to+\infty}v(x,y)=v_0.\end{equation}
Also, the Euler equation in (<ref>) and the assumption that the density is constant give that
\begin{eqnarray*}&& v\cdot\nabla\left(\frac{|v|^2}2+\frac{p}\rho\right)=
v\cdot\left(\sum_{k=1}^2 v_k\nabla v_k+\frac{\nabla p}\rho\right)=
v\cdot\left(\sum_{k=1}^2 v_k\nabla v_k-(v\cdot\nabla) v\right)
\\&&\qquad
=\sum_{j,k=1}^2 v_j v_k\partial_j v_k-
v\cdot\left(\sum_{j=1}^2 v_j\partial_j v\right)
=\sum_{j,k=1}^2 v_j v_k\partial_j v_k-
\sum_{j,k=1}^2 v_k v_j\partial_j v_k=0.
\end{eqnarray*}
As a consequence, if $\beta(X):= \frac{|v(X)|^2}2+\frac{p(X)}\rho$,
$$ v\cdot \nabla\beta=0,$$
\begin{equation*}\frac{d}{dt}\beta(X(t))=\nabla\beta(X(t))\cdot \dot X(t)=
\nabla\beta(X(t))\cdot v( X(t))=0.
\end{equation*}
As a result, $\beta(X(t))=\beta(X(0))$ for every time $t$, that is
\begin{equation}\label{BERNEQl}
\frac{|v(X(t))|^2}{2}+\frac{p(X(t))}{\rho}=
\frac{|v(X(0))|^2}{2}+\frac{p(X(0))}{\rho}.
\end{equation}
Equations such as (<ref>) are often referred to with the name of Bernoulli's Principle.
A topological argument used to take care of the fact that $\R^2\setminus\Omega$
has a hole.
Now we rely on the two-dimensional structure of the problem combined with the divergence free condition
in (<ref>) to see that the differential form $\zeta:= v_2\,dx-v_1\,dy$ is closed,
$$ d\zeta=-(\partial_y v_2+\partial_x v_1)\,dx\wedge dy=-\div v\,dx\wedge dy=0.$$
Now, the exterior of $\Omega$ is not simply connected in $\R^2$, however we can consider two
regions ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ which are simply connected and their union is the exterior of $\Omega$, see Figure <ref>.
In this setting, we can apply the Poincaré Lemma (see e.g. <cit.>)
to each of the regions ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$, finding two potentials $\psi_j:{\mathcal{R}}_j\to\R$ such that $d\psi_j=\zeta$ in ${\mathcal{R}}_j$, with $j\in\{1,2\}$.
In particular,
\begin{equation}\label{56STRE78} \partial_x\psi_j=v_2\qquad{\mbox{and}}\qquad\partial_y\psi_j=-v_1.\end{equation}
We observe that ${\mathcal{R}}_1\cap{\mathcal{R}}_2$ consists of two connected components which reach infinity,
say ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$. We observe that
\[ \nabla(\psi_1-\psi_2)=
\nabla\psi_1-\nabla\psi_2=(v_2,-v_1)-(v_2,-v_1)=0,\]
in ${\mathcal{S}}_1$, hence there exists $c_1\in\R$ such that $\psi_1-\psi_2=c_1$ in ${\mathcal{S}}_1$.
Similarly, there exists $c_2\in\R$ such that $\psi_1-\psi_2=c_2$ in ${\mathcal{S}}_2$.
Up to incorporating the constant into $\psi_2$ (which leaves (<ref>) invariant), we can suppose that $c_2=0$.
We claim that
\begin{equation}\label{Cik231ma3223y0-2tkhojnhoiuwhg8732gtpiuywfng2uahHBSne}
\end{equation}
For this, let $A\in(\partial\Omega)\cap\overline{{\mathcal{S}}_1}$ and $B\in(\partial\Omega)\cap\overline{{\mathcal{S}}_2}$
and let $\vartheta:[0,\ell]\to(\partial\Omega)\cap\overline{{\mathcal{R}}_1}$ be a curve, parameterized by its arc length, lying in $(\partial\Omega)\cap\overline{{\mathcal{R}}_1}$ and
joining $A$ to $B$, say with $\vartheta(0)=A$ and $\vartheta(\ell)=B$.
It follows from the impenetrability condition in (<ref>)
that the velocity field is tangential to $\partial\Omega$ and consequently
\begin{equation}\label{DEFSTTHiow9805e04938} v(\vartheta(\tau))
=\alpha(\tau)\,\dot\vartheta(\tau)\qquad{\mbox{ for all }}\tau\in[0,\ell],\end{equation}
for some scalar function $\alpha(\tau)$.
Thus, since $\vartheta$ lies in $\overline{{\mathcal{R}}_1}$, we can utilize (<ref>) with $j:=1$
and write that
$$ \Big( -\partial_y\psi_1\big(\vartheta(\tau)\big),\,\partial_x\psi_1\big(\vartheta(\tau)\big)\Big)=
\alpha(\tau)\,\dot\vartheta(\tau)$$
and therefore
$$ \nabla\psi_1\big(\vartheta(\tau)\big)=
\alpha(\tau)\,\big(\dot\vartheta_2(\tau),\,-\dot\vartheta_1(\tau)\big).$$
From this, we obtain that
$$ \frac{d}{d\tau}\psi_1\big(\vartheta(\tau)\big)=
\nabla\psi_1\big(\vartheta(\tau)\big)\cdot\dot\vartheta(\tau)
\alpha(\tau)\,\big(\dot\vartheta_2(\tau),\,-\dot\vartheta_1(\tau)\big)\cdot
\big(\dot\vartheta_1(\tau),\,\dot\vartheta_2(\tau)\big)=0
and consequently
$$ \psi_1(B)=\psi_1\big(\vartheta(\ell)\big)=\psi_1\big(\vartheta(0)\big)=\psi_1(A).$$
Similarly, by considering a curve lying in $(\partial\Omega)\cap\overline{{\mathcal{R}}_2}$, one obtains that
$$ \psi_2(B)=\psi_2(A).$$
As a result,
\begin{eqnarray*}
\end{eqnarray*}
and the proof of (<ref>) is complete.
Hence, we can now define, in the exterior of $\Omega$,
$$ \psi:=
\begin{dcases}
\psi_1 & {\mbox{ in }}{\mathcal{R}}_1,\\
\psi_2 & {\mbox{ in }}{\mathcal{R}}_2\\
\end{dcases}$$
and we stress that this is a fair definition, since $\psi_1=\psi_2$ in ${\mathcal{R}}_1\cap{\mathcal{R}}_2$,
owing to (<ref>).
Furthermore, by (<ref>), in the exterior of $\Omega$,
\begin{equation}\label{9uojhexterimega}
\partial_x\psi =v_2\qquad{\mbox{and}}\qquad\partial_y\psi =-v_1.\end{equation}
Note in particular that $|\nabla\psi|=v$. Moreover, $\psi$ is harmonic, thanks to the irrotationality
situation pointed out in (<ref>).
The function $\psi$ is often referred to with the name of
“stream function”.
This name comes from a physical intuition. Indeed, the streamlines, i.e. the trajectories of fluid particles, are described by the level sets of the stream function, because
\begin{equation}\label{STERTEAMSDN} \frac{d}{dt}\psi(X(t))=\nabla\psi(X(t))\cdot \dot X(t)=\nabla\psi(X(t))\cdot v(X(t))=0,\end{equation}
thanks to (<ref>).
Let us now reassess Bernoulli's Principle in (<ref>).
In our setting we need a refinement of (<ref>) along the boundary of our moving object, namely we claim that, for every $x\in\partial\Omega$,
\begin{equation}\label{BERNEQ}
\end{equation}
for some constant $p_0\in\R$.
To prove this, we take the simplifying assumption that the velocity field possesses at most finitely many zeros along $\partial\Omega$. We denote by $Z_1,\dots,Z_N$ these zeros (if any).
We pick $X_0\in(\partial\Omega)\setminus\{Z_1,\dots,Z_N\}$. Up to a rigid motion, we assume that $X_0=0$
and that the tangent vector of $\partial\Omega$ at $0$ is horizontal.
Say, we describe $\Omega$ near $0$ as the subgraph of a smooth function $f:\R\to\R$ with $f(0)=0$ and $f'(0)=0$.
We can also replace $\psi$ by $\psi-\psi(0)$ and thus assume additionally that $\psi(0)=0$.
In this setting, from the impenetrability condition in (<ref>), we have that the fluid vector field at the origin
is horizontal and thus, by (<ref>), we have that $\nabla\psi(0)$ is vertical and different from zero.
Let us suppose that
\begin{equation}\label{deltastreamline}
\partial_y\psi(0)>0,\end{equation}
the case $\partial_y\psi(0)<0$ being similar, just considering level sets
of $\psi$ corresponding to negative, instead of positive, values.
In this framework, near $0$, the level sets of $\psi$ can be written as graph of smooth functions: more precisely,
near the origin, for small $\delta>0$, the level set $\{\psi=\delta\}$ can be identified with the graph $\{y=f_\delta(x)\}$
for a suitable $f_\delta$. It follows from (<ref>) that $f_{\delta'}\ge f_\delta$ if $\delta'\ge\delta$.
Also, since, by (<ref>), $\{\psi=\delta\}$ describes the streamlines of the fluid,
we have that $\{\psi=\delta\}$ is contained in the complement of $\Omega$ for all $\delta>0$ and consequently,
near the origin, $\partial\Omega$ lies below the graph of $f_\delta$ for all $\delta>0$.
Thus, given the monotonicity of $f_\delta$, near the origin we can define
$$ f_0(x):=\lim_{\delta\searrow0}f_\delta(x)$$
and we have that $f\le f_0$. Moreover, since
$$ \psi(x,f_0(x))=\lim_{\delta\searrow0}\psi(x,f_\delta(x))=\lim_{\delta\searrow0}\delta=0=\psi(x,f(x)),$$
we deduce from (<ref>) that $f_0=f$, see Figure <ref>.
Foliation of streamlines in the vicinity of an airfoil.
This shows that the streamline emanating from the origin (that is, from every point of $\partial\Omega$ with nonzero velocity field)
remains on $\partial\Omega$.
Thus, if the velocity field does not vanish on $\partial\Omega$, then $\partial\Omega$ consists of a streamline
and then (<ref>) follows from (<ref>).
Partitioning $\partial\Omega$ as in (<ref>)
(not necessarily a realistic picture).
If instead
the velocity field vanishes at $Z_1,\dots,Z_N\in\partial\Omega$, we can decompose $\partial\Omega$
into open arcs $\Lambda_1,\dots,\Lambda_N$ such that
\begin{equation}\label{PTOP12DAILATENSIWASTbni9buj9unooREDItangeFIlAHOLEWFINASDJGIOSE2}\partial\Omega=\Lambda_1\cup\dots\cup\Lambda_N\cup\{Z_1,\dots,Z_N\},\end{equation}
in such a way that the velocity field does not vanish on each $\Lambda_j$ and $\Lambda_j$
is a streamline connecting $Z_j$ to $Z_{j+1}$ in infinite time (with $Z_{N+1}:=Z_1$), see Figure <ref>.
In this setting, we can consider a trajectory $X(t)$ starting at a given point of $\Lambda_j$ and deduce from (<ref>) that,
for every $x\in\Lambda_j$,
$$ |v(x)|^2+\frac{2p(x)}{\rho}=p_j,$$
for some $p_j\in\R$, and
$$ p_j=\lim_{t\to\pm\infty}|v(X(t))|^2+\frac{2p(X(t))}{\rho}$$
which leads to
$$ p_j=|v(Z_j)|^2+\frac{2p(Z_j)}{\rho}=|v(Z_{j+1})|^2+\frac{2p(Z_{j+1})}{\rho}.$$
This actually says that $p_1=\dots=p_N$ and thus (<ref>) plainly follows, as desired.
Now we describe $\partial\Omega$ as a smooth and closed curve, traveled counterclockwise and parameterized by its arc length, say $\gamma:[0,L]\to\C$, for some $L>0$ (in particular, notice that $\gamma(0)=\gamma(L)$).
We identify points of $\R^2$ with complex numbers.
In this way, given $\tau\in[0,L]$, a unit tangent vector to $\partial\Omega$ at $\gamma(\tau)$
is given by $\dot\gamma(\tau)$ and the unit normal vector $\nu(\gamma(\tau))$ pointing outwards is given by a clockwise rotation
of $\dot\gamma(\tau)$ by an angle $\frac\pi2$, corresponding to the complex multiplication by $e^{-i\pi/2}=-i$
(that is, the outward unit normal can be written as $-i\dot\gamma(\tau)$).
For this reason, in light of (<ref>), we have that
the force acting on the airfoil (after identifying $\R^2$ with $\C$) takes the form
\begin{equation*} F=-\int_{\partial\Omega} p\nu
=-\int_0^L p(\gamma(\tau))\,\nu(\gamma(\tau))\,d\tau=i\int_0^L p(\gamma(\tau))\,\dot\gamma(\tau)\,d\tau.
\end{equation*}
From this and (<ref>) we arrive at
\begin{equation}\label{BERNEQ2} \frac{2F}\rho
=i\int_0^L \frac{2p(\gamma(\tau))}\rho\,\dot\gamma(\tau)\,d\tau
\left(p_0-|\nabla\psi(\gamma(\tau))|^2\right)
\,\dot\gamma(\tau)\,d\tau.
\end{equation}
$$ \int_0^L \dot\gamma(\tau)\,d\tau=\gamma(L)-\gamma(0)=0,$$
we deduce from (<ref>) that
\begin{equation}\label{BERNEQ3} \frac{2F}\rho
=-i\int_0^L |\nabla\psi(\gamma(\tau))|^2\,\dot\gamma(\tau)\,d\tau.
\end{equation}
We also recall that, in light of the impenetrability condition in (<ref>)
combined with (<ref>),
we have that $\nabla\psi$ is orthogonal to
the unit tangent vector $\dot\gamma$.
Thus, we write that
\begin{equation}\label{BERNEQ4} \nabla\psi(\gamma(\tau))=\pm i
Also, in the complex variable setting, that is using the notation $z=x+iy$ and identifying $x+iy$ with $(x,y)$, we set
\begin{equation}\label{ENGBSC2O232M4P3AND5D} w(z)=w(x+iy):=\partial_x\psi(x,y)-i\partial_y\psi(x,y),\end{equation}
and it follows that $w$ is the complex conjugated of $\nabla\psi$ and $|w|=|\nabla\psi|$.
Hence, taking the complex conjugation in (<ref>),
\begin{equation*} w(\gamma(\tau))=
\pm \overline{i|w(\gamma(\tau))|\,\dot\gamma(\tau)}=
\mp i|w(\gamma(\tau))|\,\overline{\dot\gamma(\tau)}.\end{equation*}
Taking the square of this identity
\begin{equation*} w^2(\gamma(\tau))=-|w(\gamma(\tau))|^2\,(\overline{\dot\gamma(\tau)})^2
\end{equation*}
and as a result
\begin{equation*} \begin{split}&w^2(\gamma(\tau)) \dot\gamma(\tau)=-|w(\gamma(\tau))|^2\,(\overline{\dot\gamma(\tau)})^2{\dot\gamma(\tau)}
\end{equation*}
We thus combine this information with the complex conjugation of (<ref>) and we conclude that
\begin{equation}\label{wdotgammatau}
\begin{split}&\overline{\frac{2i F}\rho}
=\overline{\int_0^L |\nabla\psi(\gamma(\tau))|^2\,\dot\gamma(\tau)\,d\tau}=
\int_0^L |\nabla\psi(\gamma(\tau))|^2\,\overline{\dot\gamma(\tau)}\,d\tau\\&\quad\qquad\qquad\qquad=-\int_0^L w^2(\gamma(\tau)) \dot\gamma(\tau)\,d\tau=-
\oint_\gamma w^2(z)\,dz.\end{split}
\end{equation}
Also, using the harmonicity of $\psi$, we observe that
$$ \partial_x(\Re w)=\partial_{xx}\psi=-\partial_{yy}\psi=\partial_y(\Im w)$$
and moreover
$$ \partial_y(\Re w)=\partial_{xy}\psi(x,y)=-\partial_x(\Im w).$$
These observations give that $w$ satisfy the Cauchy-Riemann equations
and therefore $w$ is holomorphic outside $\Omega$.
We can therefore take $R>0$ large enough such that $\Omega\Subset B_R$
and employ Laurent's Theorem (see e.g. <cit.>, applied here with $R_1:=R$, $R_2:=+\infty$
and $z_0:=0$): in this way we
obtain that, for all $z$ in the exterior of $B_R$,
the following series representation holds
true for suitable $a_k$, $b_k\in \C$:
$$ w(z)=w_0(z)+w_\star(z)\qquad{\mbox{with}}\qquad
w_0(z):=\sum_{k=0}^{+\infty} a_k z^k\qquad{\mbox{and}}\qquad
w_\star(z):=\sum_{k=1}^{+\infty}\frac{b_k}{ z^k},$$
where the power series representing $w_0$ converges in $\C$ and the one representing $w_\star$
converges in the exterior of $B_R$.
In particular, $w_0$ is holomorphic in the whole of $\C$
and the convergence of the power series defining $w_\star$ entails (see e.g. <cit.>) that
$$ \limsup_{k\to+\infty} |b_k|^{1/k}\le{2R}.$$
As a result, we can take $k_0\in\N\cap[2,+\infty)$ sufficiently large such that $|b_k|\le(3R)^k$ for every $k> k_0$
and therefore
\begin{eqnarray*}&& \limsup_{|z|\to+\infty}|w_\star(z)|\le
\limsup_{|z|\to+\infty}\sum_{k=1}^{k_0}\frac{|b_k|}{| z|^k}+\limsup_{|z|\to+\infty}\sum_{k=k_0+1}^{+\infty}\frac{|b_k|}{| z|^k}
\le0+\limsup_{|z|\to+\infty}\sum_{k=k_0+1}^{+\infty}\left(\frac{3R}{| z|}\right)^k\\&&\qquad\qquad\qquad\qquad
\left(\frac{3R}{| z|}\right)^{k_0+1}\frac{1}{1-\displaystyle\frac{3R}{| z|}}=0.
\end{eqnarray*}
For this reason, recalling (<ref>),
$$ |v_0|=\limsup_{|(x,y)|\to+\infty}|\nabla\psi (x,y)|=\limsup_{|z|\to+\infty}|w(z)|=
\limsup_{|z|\to+\infty}|w_0(z)+w_\star(z)|=\limsup_{|z|\to+\infty}|w_0(z)|.$$
As a consequence, $w_0$ is bounded in the whole of $\C$ and therefore, by Liouville's Theorem
(see e.g. <cit.>) we have that $w_0$ is constant.
That is, by (<ref>), $w_0(z)=i \overline{v_0}$ for all $z\in\C$, whence, for all $z$ in the exterior of $B_R$,
\begin{equation}\label{EXTE6B-pjrmfR} w(z)=i \overline{v_0}+w_\star(z)=
i \overline{v_0}+\sum_{k=1}^{+\infty}\frac{b_k}{ z^k}.\end{equation}
\begin{eqnarray*}&&\limsup_{M\to+\infty}\left|\oint_{\partial B_M}\sum_{k=k_0+1}^{+\infty}\frac{b_k}{ z^k}\,dz\right|\le
\limsup_{M\to+\infty}\sum_{k=k_0+1}^{+\infty}\frac{2\pi|b_k|}{M^{k-2}}\\&&\quad
\le\limsup_{M\to+\infty}\sum_{k=k_0+1}^{+\infty}\frac{2\pi M^2(3R)^k}{M^{k}}
2\pi M^2\left(\frac{3R}{M}\right)^{k_0+1}\frac{1}{1-\displaystyle\frac{3R}{M}}=0.\end{eqnarray*}
Thus, by (<ref>) and Cauchy's Theorem (see e.g. <cit.>),
\begin{equation}\label{SunmgeDediamJofeMSy0r987wqugf7wegbDEDS}\begin{split}&
\oint_\gamma w(z)\,dz=
\lim_{M\to+\infty}\oint_{\partial B_M} w(z)\,dz\\&\quad=
\lim_{M\to+\infty}\oint_{\partial B_M} \left(i \overline{v_0}+\sum_{k=1}^{+\infty}\frac{b_k}{ z^k}\right)\,dz=
\lim_{M\to+\infty}\oint_{\partial B_M} \left(i \overline{v_0}+\sum_{k=1}^{k_0}\frac{b_k}{ z^k}\right)\,dz.\end{split}
\end{equation}
We stress that the function in the latter integrand is holomorphic in $B_M$ and we can thereby employ
Cauchy's Residue Theorem (see e.g. <cit.>), thus deducing from (<ref>) that
\begin{equation}\label{SunmgeDediamJofeMSy0r987wqugf7wegbDEDS-2}
\oint_\gamma w(z)\,dz=
\lim_{M\to+\infty}\oint_{\partial B_M} w(z)\,dz=2\pi i b_1.
\end{equation}
In addition,
\begin{equation*}
\begin{split}&
\oint_\gamma w(z)\,dz=\int_0^L
\Big(\partial_x\psi(\gamma(\tau))-i\partial_y\psi(\gamma(\tau))\Big)
\Big( \dot\gamma_1(\tau)+i\dot\gamma_2(\tau)\Big)\,d\tau\\&\quad=
\int_0^L
\Big(\partial_x\psi(\gamma(\tau)) \dot\gamma_1(\tau) +\partial_y\psi(\gamma(\tau))\dot\gamma_2(\tau)-i\partial_y\psi(\gamma(\tau))\dot\gamma_1(\tau)
\Big)\,d\tau\\&\quad=
\int_0^L
\left( \frac{d}{d\tau}\psi(\gamma(\tau)) -i\partial_y\psi(\gamma(\tau))\dot\gamma_1(\tau)
\right)\,d\tau\\&\quad=\psi(\gamma(L))-\psi(\gamma(0))-i
\int_0^L
\left( \partial_y\psi(\gamma(\tau))\dot\gamma_1(\tau)-\partial_x\psi(\gamma(\tau))\dot\gamma_2(\tau)
\right)\,d\tau\\&\quad=
\int_0^L
\left( \partial_y\psi(\gamma(\tau))\dot\gamma_1(\tau)-\partial_x\psi(\gamma(\tau))\dot\gamma_2(\tau)
\right)\,d\tau.
\end{split}\end{equation*}
Therefore, recalling the definition of $\Gamma$ in (<ref>) and the relations in (<ref>),
\begin{equation*}
\begin{split}&
\oint_\gamma w(z)\,dz=i
\int_0^L
\left( v_1(\gamma(\tau))\dot\gamma_1(\tau)+v_2(\gamma(\tau))\dot\gamma_2(\tau)
\right)\,d\tau=i\Gamma.
\end{split}\end{equation*}
As a result of this and (<ref>) we deduce that
$$ b_1=\frac{\Gamma}{2\pi }.$$
This and (<ref>) lead to
$$ w^2(z)=\left(
i \overline{v_0}+\frac{\Gamma}{2\pi z}+\sum_{k=2}^{+\infty}\frac{b_k}{ z^k}
\right)^2=-\overline{v_0}^2+\frac{i \overline{v_0} \Gamma}{\pi z}+\sum_{k=2}^{+\infty}\frac{c_k}{ z^k},$$
for suitable $c_k\in\C$.
This, Cauchy's Residue Theorem (see e.g. <cit.>) and (<ref>) give that
\begin{equation*}
\overline{\frac{2i F}\rho}
=2 \overline{v_0} \Gamma
\end{equation*}
and therefore
\begin{equation*}
{\frac{2i F}\rho}
=\overline{2\overline{v_0} \Gamma}=2v_0 \Gamma.
\end{equation*}
$$ F=-i\rho v_0 \Gamma$$
and this establishes the claim in (<ref>).
The level sets of the potential function in (<ref>).
A natural question now is however if it is possible to produce a nonzero circulation in (<ref>).
This is in general quite an intriguing problem: here we just provide a simple and explicit[Though
we do not really exploit this fact here, the example is inspired by a rotating cylinder producing
a Magnus effect, that is what soccer players use to make the ball curve during flight. See https://www.youtube.com/watch?v=XdL7EDKr_rk for a famous application of
the Magnus effect in soccer.] example.
For this, we let
\begin{equation}\label{HARGIRA4AXELHARLROA7789GIJ7soloDItangeFI2A}\Omega:=B_1\end{equation}
\begin{equation}\label{HARGIRA4AXELHARLROA7789GIJ7soloDItangeFI2} \psi(x,y):= y\left(1-\frac1{x^2+y^2}\right)+\ln(x^2+y^2).\end{equation}
See Figure <ref> for the level sets of $\psi$.
These level sets will correspond to streamlines, since we will consider the vector field $v=(v_1,v_2)$ with
\begin{equation}\label{HARGIRA4AXELHARLROAGIJ7soloDItangeFI2}
\begin{split}
\frac{2 x^2}{(x^2 + y^2)^2 }-\frac{1 + 2 y}{x^2 + y^2}-1
\\ {\mbox{and}}\qquad&
v_2(x,y):=\partial_x\psi(x,y)=\frac{2 x}{x^2 + y^2} +\frac{2 x y}{(x^2 + y^2)^2}.\end{split}
\end{equation}
See Figure <ref> for a sketch
of the vector field $v$.
The vector field in (<ref>).
The complex
analysis enthusiasts will also enjoy the fact that the setting in (<ref>)
is also available by posing
\begin{equation}\label{COMPLEXPOTE}w:=\frac2z + i \left(\frac1{z^2} - 1\right).\end{equation}
Notice that $w$ is holomorphic away of the origin. And the knowledge of this complex
map is pretty much all that is needed to produce the fluid velocity field in (<ref>).
The action of the above map $w$ is sketched in Figure <ref>.
We observe that $v$ is tangential to $\partial\Omega$ since
$$ v(\cos\theta,\sin\theta)\cdot(\cos\theta,\sin\theta)=0$$
and this is consistent with the impenetrability condition in (<ref>).
Additionally, we have that $\Delta\psi=0$ outside $\Omega$, which leads that $\div=0$
outside $\Omega$ as well.
Furthermore, recalling (<ref>),
\begin{equation}\label{8ihTGBAlFer3ONmdfZ345tyIjdmvREBSLv02}
\curl v=\big(\partial_xv_2-\partial_y v_1\big) e_3=\big(\partial_x(\partial_x\psi)+\partial_y \partial_y\psi\big) e_3=0
\end{equation}
\begin{eqnarray*}
\lim_{|(x,y)|\to+\infty} v(x,y)=
\lim_{|(x,y)|\to+\infty}\left(
\frac{2 x^2}{(x^2 + y^2)^2 }-\frac{1 + 2 y}{x^2 + y^2}-1,
\;\frac{2 x}{x^2 + y^2} +\frac{2 x y}{(x^2 + y^2)^2}\right)
The complex map $w$ in footnote <ref>.
Thus, to check that this example is indeed a solution of our fluid dynamics problem, it remains
to show that the first equation in (<ref>) holds true.
For this, we choose $\rho:=1$ and, inspired by the Bernoulli's Principle
in (<ref>),
$$ p(x,y):=\frac{|v_0|^2-|v(x,y)|^2}{2}=\frac{1-|v(x,y)|^2}{2}$$
and we combine the vectorial identity in (<ref>) with (<ref>)
to calculate that
\begin{eqnarray*}&&\rho (v\cdot\nabla) v+\nabla p=
(v\cdot\nabla) v-\nabla\frac{|v|^2}{2}=-
v\times (\curl v)
\end{eqnarray*}
This shows that the vector field in (<ref>)
is consistent with the fluid dynamics setting in (<ref>). To show the interest of (<ref>)
it thus remains to check that this vector field provides a nontrivial circulation $\Gamma$: to this end, we compute that
\begin{eqnarray*}
\Gamma&=&\int_0^{2\pi} v(\cos\theta,\sin\theta)\cdot(-\sin\theta,\cos\theta)\,d\theta\\
&=&2\int_0^{2\pi} (\cos^2\theta-\sin\theta-1,\;\cos\theta +\sin\theta\cos\theta)\cdot(-\sin\theta,\cos\theta)\,d\theta\\&=&2\int_0^{2\pi} (1+\sin\theta)\,d\theta\\&=&4\pi
\\&\ne&0,
\end{eqnarray*}
as desired.
But hold on a sec, the skeptical reader (who is always very welcome) will complain we've been cheating on them:
we have promised this section was devoted to airfoils, but then we tried to sell in (<ref>) the disk as an example of airfoils.
Come on, airfoils are objects as the ones in Figure <ref>, nothing in Figures <ref>, <ref> or <ref>
looks like an airfoil, just because the disk is not an airfoil at all!
Joukowski airfoil
(image by Krishnavedala from
Wikipedia, licensed under the Creative Commons Attribution-Share Alike 4.0 International license).
Well, in fact, to a certain degree, it is: this is the beauty of mathematics, and of complex analysis in particular.
Indeed, one of the brilliant ideas of the pioneers of aerodynamics is that the potential in (<ref>)
describes all the fluid dynamics outside the disk of the example that we have discussed explicitly in detail,
therefore other examples can be constructed via conformal transformations.
In particular, using dilations, translations and maps of the form $z+\frac1z$, one can transform a disk
into an airfoil shaped as in Figure <ref>.
The set of conformal transformations linking circles and airfoils can be visualized for instance via the Wolfram Demonstrations Projects
or via the GeoGebra application
or likely via a number of IT resources.
Interestingly, on
one can also plot the resulting flow around the airfoil: the result that we have obtained
using this application is reported in Figure <ref>,
which nicely shows how a disk is mapped into an airfoil.
From a disk to an airfoil, and the corresponding fluid flow (images produced
by the online application in
For further readings about the Kutta-Joukowski theory and the analysis of airfoils see e.g. <cit.>
and the references therein.
See also <cit.> for an extensive treatment of fluid dynamics from a complex analysis perspective.
We cannot avoid mentioning that one of the most spectacular applications of fluid dynamics is probably showcased
by the motion[The origin of the name boomerang is a bit uncertain: some references link it to an extinct Aboriginal language of New South Wales, others to the language of the Turuwal people (a sub-group of the Darug) of the Georges River (Tucoerah River).
Besides the well-known traditional employment by some Aboriginal Australian peoples for hunting, see Figure <ref>, it seems that boomerangs have been used also in ancient Europe, Egypt, and North America (ancient Egyptian boomerangs have been tested and seemed to work well as returning boomerangs and a boomerang discovered in the Carpathian Mountains in Poland dated back to about $30\times10^3$ years ago.
Interestingly, a boomerang
was used to set the Guinness World Record
for the longest throw of any object by a human: namely, in 2005, at Murarrie Recreation Ground, in Queensland,
David Schummy
performed a throw of 427.2 metres, see https://www.youtube.com/watch?v=ly3nCEbcQig
Currently, long distance boomerangs are mostly
shaped as a question mark and often have
a beveled edge, to facilitate the pitch
and lower the drift, since the boomerang in this case
is usually thrown almost horizontally (indeed, in the above mentioned
record, the objective
was not to make the boomerang come back to the throw location;
actually the boomerang ended up on a tree).] of a boomerang.
Otto Jungarryi Sims with a boomerang sitting inside a cave in the Northern Territory of Australia
(photo by Ed Gold; image from Wikipedia for free use under ticket #2020101210010454).
Roughly speaking, each wing of a boomerang is shaped as an airfoil section,
allowing the airflow over the wings to create a significant lift.
If the boomerang is thrown nearly upright, the rotating blades generate more lift at the top than the bottom,
because at the top the speed of the rotation adds up to the forward speed, while at the bottom
the speed of the rotation subtracts from the forward speed, see Figure <ref>
in which the forward speed is represented by the yellow arrow.
This additional lift from the top produces a torque (represented by the blue arrow in Figure <ref>)
whose effect is to make the rotation plane turn around: we point out that this torque is typically not sufficient
to tilt the boomerang around its axis of travel, given its high
angular momentum (the spinning of the boomerang being represented by the red arrow in Figure <ref>):
hence, the stability of the rotating plane ensured by the gyroscopic precession combines with the aerodynamic torque
and leads to the curved trajectory sketched by the green arrow in Figure <ref>.
Why do boomerangs return?
§.§ Surfing the waves
A topical problem in the dynamics of fluids consists in the description of waves in shallow water,
since this is a typical case arising in the proximity of land. Though a large number of different models are available for this goal,
in this pages we recall the classical equation
Students picnic on the Union Canal in 1922
(copyright The University of Edinburgh, available for public use; image from
licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license).
\begin{equation}\label{KDV-or}
a\partial _{t}\phi +b\partial _{x}^{3}\phi +c\,\phi \,\partial _{x}\phi =0
\end{equation}
for $a$, $b$, $c\in\R\setminus\{0\}$, with $\phi=\phi(x,t)$, $x\in\R$ and $t\in(0,+\infty)$.
Equation (<ref>) is called[Equation (<ref>)
is named after Diederik Johannes Korteweg and Gustav de Vries <cit.>, though it was
introduced by Joseph Valentin Boussinesq <cit.>.
De Vries completed his PhD under Korteweg's supervision and then worked all his life
as a high school teacher in Haarlem in the Netherlands.
The strong interest in the formation and propagation of waves in canals was possibly the outcome of a direct observation by
John Scott Russell <cit.> which took place in the Union Canal
(a canal in Scotland, running from Falkirk to Edinburgh).
Russell's own words have been repeated in virtually all papers and books which even remotely discuss water wave problems and we have no intention of breaking this consolidated tradition, hence here is his report on the astounding experience of meeting a traveling wave for the first time: “I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped – not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. [...] Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation”.
See Figure <ref> for a historical picture
of the Union Canal.] the
Korteweg-de Vries equation (or KdV equation for short). Korteweg-de Vries equation
The physical content of equation (<ref>) is that $\phi$ models the shape of a wave in a canal.
The position on the canal is given by $x\in\R$ and $t$ stands for the time variable.
The amplitude of the wave and the depth of the channel are supposed to be small
(as will be discussed below in the approximations leading to the derivation of the equation),
hence the validity of (<ref>) has to be limited to small amplitude waves in shallow waters.
Though equation (<ref>) should be considered as a very simplified model, not able to capture the complexity
of oscillatory phenomena and waves in the real world, it is interesting to notice that the Korteweg-de Vries equation
can describe some interesting features such as traveling waves
and provide interesting information on the shape of the waves, as we now discuss.
The traveling wave solution for the Korteweg-de Vries equation equation, as found in
in (<ref>).
First of all (recalling the presentation on page 01ojehniTNasIplamnewa) one can seek for
traveling wave solutions of (<ref>) in the form
\begin{equation}\label{BSNEGIAaon}
\phi(x,t)=\phi_0(x-vt)\end{equation}
for some velocity $v$.
In this setting, equation (<ref>) reduces to
\begin{equation}\label{KAlyhchtvdhr4htFAn}
0=-av\phi_0' +b\phi'''_0 +c\,\phi_0 \,\phi_0' =-av\phi_0' +b\phi'''_0 +\frac{c}2\,(\phi_0^2)'=\left(-av\phi_0 +b\phi''_0 +\frac{c}2\,\phi_0^2\right)'.
\end{equation}
We integrate this equation by assuming that at infinity $\phi_0$
and its derivatives converge to zero, finding that
\begin{equation*}0=
-av\phi_0 +b\phi''_0 +\frac{c}2\,\phi_0^2.
\end{equation*}
As a result,
\begin{equation*}
0=\left( -av\phi_0 +b\phi''_0 +\frac{c}2\,\phi_0^2\right)\phi_0'=
\left(-\frac{av}2\phi_0^2 +\frac{b}2 (\phi'_0)^2 +\frac{c}{6}\,\phi_0^3 \right)'.
\end{equation*}
Integrating this as above, we deduce that
\begin{equation*} 0
=-\frac{av}2\phi_0^2 +\frac{b}2 (\phi'_0)^2+\frac{c}{6}\,\phi_0^3.\end{equation*}
This ordinary differential equation
can be solved by separation of variables (assuming $\phi_0\ge0$, $c>0$ and $abv>0$) leading,
up to a translation, to
\begin{equation}\label{DIKDVSOLIFI-EQ} \phi_0(r)=
\frac{6 a v}{c \left( \cosh\left(r \sqrt{\frac{a v}b} \right)+ 1\right)}.\end{equation}
See Figure <ref> for a sketch[See also http://lie.math.brocku.ca/$\sim$sanco/solitons/kdv_solitons.php
for several animations of (possibly interacting) traveling waves
of the Korteweg-de Vries equation.] of this traveling wave (recall (<ref>))
when $a:=1$, $b:=1$, $c:=1$ and $v:=1$.
It is also interesting to observe that the highest crest of the wave in (<ref>) is attained for $r:=0$
and it is equal to
\begin{equation}\label{d3ae9ikjm3f82KSMpf334}
\frac{3 a v}{c}.\end{equation}
This suggests that the fastest waves (i.e., waves with larger velocity $v$) correspond to the
highest ones.
Wave trains crossing in front of Île de Ré, in the Atlantic Ocean
(photo by Michel Griffon; image from
licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license).
Another nice feature captured by the Korteweg-de Vries equation in (<ref>)
is a sufficiently accurate description of the shape of a set of waves: it is indeed a common experience that
“real” waves exhibit sharper crests and flatter troughs than those of sine and cosine functions,
see Figures <ref> and <ref>.
To understand the shape of these waves, it is convenient to define, given $\varphi\in\R$ and $m\in[0,1)$,
\begin{equation}\label{NokmsLPSo8s0} U(\varphi,m):=\int _{0}^{\varphi }{\frac{d\theta }{\sqrt {1-m\sin ^{2}\theta }}}.\end{equation}
We observe that $\frac{\partial U}{\partial \varphi}>0$, hence, given $m\in(0,1)$, the function $\varphi\mapsto U(\varphi,m)$ is invertible
and, with a slight abuse of notation, we call $\varphi(U,m)$ the inverse function.
In this setting, one defines the elliptic cosine (in Latin, cosinus amplitudinis) by elliptic cosine
\begin{equation}\label{NokmsLPSo8s0BIS} \operatorname{cn} (U,m):=\cos (\varphi(U,m)).\end{equation}
Now, to find periodic solutions of the Korteweg-de Vries equation in (<ref>)
and to compare their shapes with Figure <ref> we go back to (<ref>)
and we integrate it, now not assuming that $\phi_0$ decays at infinity: in this way, we have that
$$ -av\phi_0 +b\phi''_0 +\frac{c}2\,\phi_0^2=\kappa_1,$$
for some constant $\kappa_1\in\R$.
As a consequence,
$$0=\Big(-av\phi_0 +b\phi''_0 +\frac{c}2\,\phi_0^2-\kappa_1\Big)\phi_0'=
\left(-\frac{av}2\phi_0^2 +\frac{b}2 (\phi'_0)^2 +\frac{c}6\,\phi_0^3-\kappa_1\phi_0\right)'
from which a further integration produces
\begin{equation}\label{MS-pqewkrfciaub} -\frac{av}2\phi_0^2 +\frac{b}2 (\phi'_0)^2 +\frac{c}6\,\phi_0^3-\kappa_1\phi_0=\kappa_2,\end{equation}
for some constant $\kappa_2\in\R$.
We observe that we can write (<ref>) as
\begin{equation}\label{MS-pqewkrfciaub2} (\phi'_0)^2={\mathcal{P}}(\phi_0),\end{equation}
for a suitable polynomial ${\mathcal{P}}$ of degree $3$. The setting that we consider now
is that in which ${\mathcal{P}}$ possesses three distinct real roots $\beta_1>\beta_2>\beta_3$.
With this, we rewrite (<ref>) as
\begin{equation}\label{MS-pqewkrfciaub-x}
where $\kappa:=c/(3b)$, that we assume to be positive.
Now it turns out to be useful to seek solutions in the form
\begin{equation}\label{MS-pqewkrfciaub-x98ytr} \phi_0(r)=\beta_1\cos^2 (\Phi(r))+\beta_2\sin^2 (\Phi(r)), \end{equation}
where the function $\Phi$ is supposed to be increasing and has to be determined.
We observe that
$$ (\phi_0')^2=\Big( -2\beta_1\cos\Phi\,\sin\Phi\, \Phi'+2\beta_2\sin\Phi \,\cos\Phi\,\Phi'\Big)^2=
4(\beta_1-\beta_2)^2 \sin^2\Phi\,\cos^2\Phi\,(\Phi')^2.$$
\begin{eqnarray*}
\phi_0-\beta_1=\beta_1\big( \cos^2 \Phi -1\big)+\beta_2\sin^2 \Phi=-(\beta_1-\beta_2)\sin^2 \Phi
\end{eqnarray*}
\begin{eqnarray*}
\phi_0-\beta_2=\beta_1 \cos^2 \Phi +\beta_2\big(\sin^2 \Phi-1)=(\beta_1-\beta_2)\cos^2 \Phi.
\end{eqnarray*}
These observations and (<ref>) yield that
\begin{equation*}
\end{equation*}
Thus, since
$$ \phi_0-\beta_3=\beta_1\Big(1-\sin^2 \Phi\Big)+\beta_2\sin^2 \Phi-\beta_3=(\beta_1-\beta_3)-
we conclude that
\begin{equation}\label{MS-pqewkrfciaub-x89} \Phi'=\frac{\sqrt\kappa}2\,\sqrt{ (\beta_1-\beta_3)-(\beta_1-\beta_2)\sin^2\Phi}=
\kappa_0\,\sqrt{ 1-m\sin^2\Phi}
$$ \kappa_0:=\frac{\sqrt{\kappa\,(\beta_1-\beta_3)}}2\qquad{\mbox{and}}\qquad
We can thus utilize the separation of variable method in (<ref>)
and, in view of (<ref>), obtain that
$$ \kappa_0 r=\int_{0}^{\Phi(r)}\frac{d\vartheta}{\sqrt{ 1-m\sin^2\vartheta}}=U\big(\Phi(r),m\big),
where we normalized the picture so that $\Phi(0)=0$.
From this and (<ref>) it follows that
$$ \operatorname{cn}(\kappa_0 r)=\operatorname{cn}\big( U\big(\Phi(r),m\big)\big)=\cos(\Phi(r)).
Going back to (<ref>) we thereby conclude that
\begin{equation}\label{EQcnoidal-waves} \begin{split}&\phi_0(r)=\beta_1\cos^2 (\Phi(r))+\beta_2\big(1-\cos^2 (\Phi(r))\big)
=(\beta_1-\beta_2)\cos^2 (\Phi(r))+\beta_2\\&\qquad\quad\qquad=(\beta_1-\beta_2)\operatorname{cn}^2(\kappa_0 r )+\beta_2=
A\operatorname{cn}^2(\kappa_0 r)+B,\end{split}
\end{equation}
where $A:=\beta_1-\beta_2$ and $B:=\beta_2$.
The profile of these functions, for suitable choices of parameters, resembles the shape of waves
in Figures <ref> and <ref> (which are often
called “cnoidal waves” cnoidal wave due to the presence of the elliptic cosine $\operatorname{cn}$
in their expression). To have a feeling on how the different parameters
change the shape of a cnoidal waves see Figure <ref>.
Cnoidal wave solution to the Korteweg-de Vries equation (image by Kraaiennest from
licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license).
It is also interesting to take note of the fact that, in spite of its (relative) simplicity, the Korteweg-de Vries equation
in (<ref>) is capable of capturing complex phenomena such as the formation of oscillatory waves, with crests
traveling at different speeds according to their heights: these complicated situations may even arise from very simple
initial data, see Figure <ref>.
Furthermore, we point out that the Korteweg-de Vries equation often turns in handy
in the description of natural phenomena that seem to be well beyond its range of applicability.
For example (see e.g. <cit.> and the references therein), the Korteweg-de Vries equation
has been exploited in the description[Although it is debatable whether the artists
intended their works to be interpreted in that way, it is customary to
refer to the artworks in Figure <ref>
in connection with tsunamis.]
of tsunamis tsunami
even though the extreme depth of oceans may appear to be incompatible with an equation which was originally
designed for shallow waters. The reason for the success of the Korteweg-de Vries equation even in this setting
is, at least, twofold. First of all, though oceans are very deep, tsunami waves can reach a spacial
extent of even larger size: therefore, when the depth of the ocean is sufficiently small with respect to the breadth of the wave
the shallow water theory may still be applied, since what counts is the relative (and not absolute) sizes of the length parameters.
Secondly, while tsunamis in the deep ocean consists typically of very long waves with quite small amplitudes, as they approach shallow waters the nonlinear effects become predominant and they change significantly
the shape and velocity of the leading waves, for which the Korteweg-de Vries equation can also provide an approximate, but often
effective, model.
Now we give a brief motivation for the Korteweg-de Vries equation
in (<ref>). While its expression may appear rather mysterious
at first sight, especially in view of the third derivative appearing as a leading term, we will
see now that equation (<ref>) arises naturally from the equations of fluid dynamics
presented in Section <ref>.
More specifically, we consider the Euler fluid flow equation in (<ref>).
Here we take coordinates $(x,z)\in\R^2$ where $x\in\R$ represents the direction of a canal
and $z\in\R$ is the vertical displacement. The fluid velocity $v$ will be written in components as $v=(\overline u,\overline w)$,
with $\overline u$ and $\overline w$ scalar functions. Hence, being $\rho$ the density of the fluid (assumed to be constant)
and $p$ the pressure, equation (<ref>) reads
\begin{equation}\label{Cm-PAPMam}
\rho \partial_t (\overline u,\overline w)+\rho ((\overline u,\overline w)\cdot(\partial_x,\partial_z)) (\overline u,\overline w)=
-\rho g(0,1) -(\partial_x,\partial_z) p.
\end{equation}
Hence, setting
\begin{equation}\label{Cm-PAPMam20} \overline P:=\frac{p}\rho+gz,
\end{equation}
we rewrite (<ref>) as the following system of two equations:
\begin{equation}\label{Cm-PAPMam2}
\begin{dcases}
\partial_t \overline u+
\overline u\partial_x\overline u+\overline w \partial_z \overline u=- \partial_x\overline P,
\\
\partial_t \overline w+
\overline u \partial_x\overline w
+\overline w \partial_z \overline w=
-\partial_z\overline P.
\end{dcases}
\end{equation}
We suppose that the bottom of the canal is $z=0$ and the fluid at rest has height $h_0\in(0,+\infty)$.
The wave is thus described as a perturbation of the rest fluid surface $z=h_0$
described by the graph of a function $\overline\eta=\overline\eta(x,t)$, modulated by a small amplitude $\alpha\in(0,+\infty)$:
namely, at every instant of time $t$ the fluid in the channel corresponds to the region of the points $(x,z)\in\R^2$
with $0<z<h_0+\alpha\overline\eta(x,t)$. That is, the equations in (<ref>) are set in the domain
\begin{equation}\label{Cm-PAPMam30} \Omega^{(t)}:=\Big\{ (x,z)\in\R^2{\mbox{ s.t. }}x\in\R{\mbox{ and }}z\in\big(0,h_0+\alpha\overline\eta(x,t)\big)\Big\}.\end{equation}
We also assume that the fluid is incompressible, hence, by (<ref>),
\begin{equation}\label{Cm-PAPMam3}
0=\div (\overline u,\overline w)=\partial_x\overline u+\partial_z\overline w.\end{equation}
The equations in (<ref>) and (<ref>) are complemented by boundary conditions
corresponding to the bottom of the canal and the free surface of the fluid.
Indeed, at the bottom $z=0$ we suppose that the floor of the canal is impenetrable, hence the vertical component
of the fluid velocity vanishes, which reads
\begin{equation}\label{Cm-PAPMam4}
\overline w(x,0,t)=0.\end{equation}
As for the free surface, we assume that fluid parcels staying at the top remain at the top:
hence if $(x(t),z(t))$ represents the trajectory of a fluid particle on the upper surface of the fluid at time $t$ we have that
$$ z(t)=h_0+\alpha\overline\eta(x(t),t).$$
Thus, taking the time derivative of this expression and recalling that the fluid parcel velocity $(\dot x(t),\dot z(t))$ agrees with $(\overline u,\overline w)$,
\begin{equation}\label{Cm-PAPMam44}
\overline w\big(x(t),h_0+\alpha\overline\eta(x(t),t),t\big)=\dot z(t)=\alpha\partial_x\overline\eta(x(t),t)\dot x(t)+\alpha\partial_t\overline\eta(x(t),t).\end{equation}
The cnoidal waves in (<ref>)
with $A:=1$, $B:=3$ and $(\kappa_0,m)\in\big\{ (1/7,0.9),\,(1/6,0.99),\,(1/5,0.999),\,(1/3,0.999999)
\big\}$.
The free surface $z=h_0+\alpha\overline\eta$ also presents an additional boundary condition related to pressure.
Indeed, the pressure on the top surface of the fluid is balanced by the atmospheric pressure, that we denote by $p_0$
and assume to be constant. Hence, by (<ref>),
\begin{equation}\label{Cm-PAPMam45} \overline P\big(x,h_0+\alpha\overline\eta(x,t),t\big)=\frac{p_0}\rho+g\,\big(h_0+\alpha\overline\eta(x,t)\big).\end{equation}
We now consider a regime of shallow waters and small amplitudes.
That is,
we consider the quantities
\begin{equation}\label{EPSIDELKDV}
\e:=\frac{\alpha}{h_0}\qquad{\mbox{and}}\qquad\delta:=\frac{h_0}{g}.\end{equation}
We suppose that $\e$ and $\delta$ are small and also, up to normalizing constants, that
\begin{equation}\label{sawqewet54b5494pppppplllllkkkk}
\delta=\e.\end{equation}
We define
\begin{equation}\label{RTArasmsImPNSd-2}\begin{split}& u^\star(x,z,t):=\frac{\overline u\big(\sqrt{gh_0} x,h_0z,t\big)}{\sqrt{gh_0}},\qquad
w^\star(x,z,t):=\frac{\overline w\big(\sqrt{gh_0} x,h_0z,t\big)}{h_0},\\&\qquad
\eta^\star(x,t):= {\overline\eta(\sqrt{gh_0} x,t)}
\qquad{\mbox{and}}\qquad
P^\star(x,z,t):=\frac{\overline P\big(\sqrt{gh_0} x,h_0z,t\big)}{{gh_0}}.
\end{split}\end{equation}
In view of (<ref>), the above functions are defined in the region
\begin{equation}\label{NEWPD-do1}
\begin{split}
\Omega_{ t }^\star&:=
\Big\{ (x,z)\in\R^2{\mbox{ s.t. }}(\sqrt{gh_0} x,h_0z)\in\Omega^{(t)}\Big\}\\&
=\Big\{ (x,z)\in\R^2{\mbox{ s.t. }}x\in\R{\mbox{ and }} z\in\big(0,1+\e\eta^\star( x,t)\big)\Big\}.\end{split}
\end{equation}
We thus collects the equations in (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), exploiting also (<ref>) and (<ref>), by writing
\begin{equation}\label{w78uhdbc-384irton2rve7g}
\begin{dcases}
\partial_t u^\star+
u^\star\partial_xu^\star+w^\star \partial_z u^\star=- \partial_x P^\star
& {\mbox{ in }}\Omega_{ t }^\star,
\\
\e\big( \partial_t w^\star+
u^\star \partial_x w^\star
+ w^\star \partial_z w^\star\big)=
-\partial_z P^\star & {\mbox{ in }}\Omega_{ t }^\star,\\ \partial_xu^\star+\partial_zw^\star=0& {\mbox{ in }}\Omega_{ t }^\star,\\
P^\star=\displaystyle\frac{p_0}{\rho gh_0}+{1+\e\eta^\star }
& {\mbox{ on }} z=1+\e\eta^\star ,\\
& {\mbox{ on }} z=1+\e\eta^\star ,\\
w^\star=0& {\mbox{ on }} z=0.
\end{dcases}
\end{equation}
We let $c_0:=\frac{p_0}{\rho gh_0}+1$.
It is now convenient to “surf the wave” and look at the new variable $\xi:=x-t$
(roughly speaking, using this variable one is moving with a wave that travels at unit[Interestingly,
moving at unit speed for $\eta^\star$ corresponds to moving at speed $\sqrt{gh_0}$
for the original profile $\overline\eta$, thanks to the change of spatial variables in (<ref>).
The quantity $\sqrt{gh_0}$ happens indeed to be one of the characteristic velocities of small waves
in shallow waters (relating the velocity to $\sqrt{gh_0}$
also tells us that the speed of these waves decreases when the height of the canal
is smaller and, conversely, their speed increases when moving from very shallow to slightly deeper water).] speed).
It is also appropriate to look at large times by setting $\tau:=\e t$. Thus, we rephrase our physical quantities with
respect to these new variables by defining
\begin{equation}\label{RTArasmsImPNSd-1}
\begin{split}&
u(\xi,z,\tau):=u^\star\left(\xi+\frac\tau\e,z,\frac\tau\e\right),\qquad w(\xi,z,\tau)=
\eta^\star\left(\xi+\frac\tau\e,\frac\tau\e\right)
\qquad{\mbox{and}}\qquad P(\xi,z,\tau):=
\end{split}\end{equation}
By (<ref>), these functions are defined in the region
\begin{eqnarray*} \Omega_\tau&:=&\Big\{ (\xi,z)\in\R^2{\mbox{ s.t. $\xi=x-t$, $\tau=\e t$ and~$(x,z)\in\Omega_{ t }^\star$}}\Big\}\\&
=&\Big\{ (\xi,z)\in\R^2{\mbox{ s.t. }}\xi\in\R{\mbox{ and }} z\in\big(0,1+\e\eta( \xi,\tau)\big)\Big\}
In this setting, one can rewrite (<ref>) in the form
\begin{equation}\label{KDV-0m-001}
\begin{dcases}
-\partial_\xi u+\e\partial_\tau u+u\partial_\xi u+w\partial_z u=-\partial_\xi P&{\mbox{ in }}\Omega_\tau,\\
\e\big( -\partial_\xi w+\e\partial_\tau w+u\partial_\xi w+w\partial_z w\big)=-\partial_z P&{\mbox{ in }}\Omega_\tau,\\
\partial_\xi u+\partial_z w=0&{\mbox{ in }}\Omega_\tau,\\
P=c_0+\e\eta & {\mbox{ on }}z=1+\e\eta,\\
w=\e(\e\partial_\tau\eta+(u-1)\partial_\xi\eta)& {\mbox{ on }}z=1+\e\eta,\\
w=0 & {\mbox{ on }}z=0.
\end{dcases}
\end{equation}
Numerical solution of the KdV equation $\partial_t u + u\partial_x u + (0.022)^2\partial_x^3u = 0$
with initial condition $ u(x, 0) = \cos(\pi x)$.
The initial cosine wave evolves into a train of solitary-type waves <cit.>
(by Ta2o; image from
licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license).
It is now convenient to consider formal Taylor
expansions of the physical quantities involved in powers of $\e$ by writing
\begin{eqnarray*}&& u(\xi,z,\tau)=\sum_{k=0}^{+\infty}\e^k u_k(\xi,z,\tau),\qquad
w(\xi,z,\tau)=\sum_{k=0}^{+\infty}\e^k w_k(\xi,z,\tau)\\&&
P(\xi,z,\tau)=\sum_{k=0}^{+\infty}\e^k P_k(\xi,z,\tau)
\qquad{\mbox{and}}\qquad
\eta(\xi,\tau)=\sum_{k=0}^{+\infty}\e^k \eta_k(\xi,\tau).\end{eqnarray*}
Substituting these expressions into (<ref>) we obtain that
\begin{equation}\label{KDV-0m-002}
\begin{dcases}
-\displaystyle\sum_{k=0}^{+\infty}\e^k\partial_\xi u_k+\displaystyle\sum_{k=0}^{+\infty}\e^{k+1}\partial_\tau u_k+
\displaystyle\sum_{k,m=0}^{+\infty}\e^{k+m}
u_k\partial_\xi u_m+\displaystyle\sum_{k,m=0}^{+\infty}\e^{k+m}
w_k\partial_z u_m=-\displaystyle\sum_{k=0}^{+\infty}\e^k\partial_\xi P_k&{\mbox{ in }}\Omega_\tau,\\
-\displaystyle\sum_{k=0}^{+\infty}\e^{k+1}\partial_\xi w_k+
\displaystyle\sum_{k=0}^{+\infty}\e^{k+2}\partial_\tau w_k+
\displaystyle\sum_{k,m=0}^{+\infty}\e^{k+m+1}u_k\partial_\xi w_m+
\displaystyle\sum_{k,m=0}^{+\infty}\e^{k+m+1}w_k\partial_z w_m=-\displaystyle\sum_{k=0}^{+\infty}\e^k\partial_z P_k&{\mbox{ in }}\Omega_\tau,\\
\displaystyle\sum_{k=0}^{+\infty}\e^k(\partial_\xi u_k+\partial_z w_k)=0&{\mbox{ in }}\Omega_\tau,\\
P\left( \xi, 1+\displaystyle\sum_{k=0}^{+\infty}\e^{k+1}\eta_k,\tau\right)=c_0+\displaystyle\sum_{k=0}^{+\infty}\e^{k+1}\eta_k, & \\
w\left( \xi, 1+\displaystyle\sum_{k=0}^{+\infty}\e^{k+1}\eta_k,\tau\right)=
\displaystyle\sum_{k=0}^{+\infty}\e^{k+2}\partial_\tau\eta_k+
\displaystyle\sum_{k,m=0}^{+\infty}\e^{k+m+1}u_k\partial_\xi\eta_m-
\displaystyle\sum_{k=0}^{+\infty}\e^{k+1}\partial_\xi\eta_k,& \\
w(\xi,0,\tau)=0. &
\end{dcases}
\end{equation}
The strategy is now to approximate (<ref>) by neglecting the terms of size $\e^k$ with $k\ge3$.
For this, we consider the orders in $\e^k$ in (<ref>) with $k\in\{0,1,2\}$ and we formally equate the corresponding
coefficients. Namely, by formally considering the coefficients related to $\e^0$ in (<ref>) we see that
\begin{equation*}
\begin{dcases}
-\partial_\xi u_0+u_0\partial_\xi u_0+w_0\partial_z u_0=-\partial_\xi P_0&{\mbox{ in }}\Omega_\tau,\\
0=\partial_z P_0&{\mbox{ in }}\Omega_\tau,\\
\partial_\xi u_0+\partial_z w_0=0&{\mbox{ in }}\Omega_\tau,\\
P_0( \xi, 1,\tau)=c_0, & \\
w_0( \xi, 1,\tau)=0,& \\
w_0(\xi,0,\tau)=0. &
\end{dcases}
\end{equation*}
Since we are interested in small perturbations of steady states, we also take $u_0:=0$ and $w_0:=0$, therefore
\begin{equation}\label{KDV-0m-003}
\begin{dcases}
0=-\partial_\xi P_0&{\mbox{ in }}\Omega_\tau,\\
0=\partial_z P_0&{\mbox{ in }}\Omega_\tau,\\
P_0( \xi, 1,\tau)=c_0. &
\end{dcases}
\end{equation}
From the second and third lines in (<ref>) it follows that
\begin{equation}\label{KDV-0m-005}P_0(\xi,z,\tau)=c_0.\end{equation}
The Great Wave off Kanagawa, famous woodblock print by the ukiyo-e artist Katsushika Hokusai
(H. O. Havemeyer Collection, Bequest of Mrs. H. O. Havemeyer, 1929, Metropolitan Museum of Art; Public Domain
image from Wikipedia). Right: Die Woge, sculpture by Tobias Stengel in Dresden, Germany (photo by Christoph Münch,
Press Section of the city of Dresden's official homepage; image from
licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license).
Now we consider the coefficients related to $\e^1$ in (<ref>) and we obtain that
\begin{equation}\label{KDV-0m-004}
\begin{dcases}
-\partial_\xi u_1=-\partial_\xi P_1&{\mbox{ in }}\Omega_\tau,\\
0=-\partial_z P_1&{\mbox{ in }}\Omega_\tau,\\
\partial_\xi u_1+\partial_z w_1=0&{\mbox{ in }}\Omega_\tau,\\
P_1(\xi,1,\tau)+\partial_z P_0(\xi,1,\tau)\eta_0=\eta_0, & \\
w_1(\xi,1,\tau)=-\partial_\xi\eta_0,& \\
w_1(\xi,0,\tau)=0. &
\end{dcases}
\end{equation}
Owing to (<ref>),
the fourth line in (<ref>) can be written as $P_1(\xi,1,\tau)=\eta_0(\xi,\tau)$, which
combined with the second line gives that
\begin{equation}\label{KDV-0m-009}
Hence, from the first line,
\begin{equation}\label{TH:EVE}
\partial_\xi u_1=\partial_\xi\eta_0.\end{equation}
This and the third line give that
$$ \partial_z (w_1+z\partial_\xi\eta_0)=\partial_z w_1+\partial_\xi\eta_0=-\partial_\xi u_1+\partial_\xi\eta_0=0
and therefore, by the last line in (<ref>), we infer that
\begin{equation}\label{KDV-0m-006}w_1=-z\partial_\xi\eta_0,\end{equation}
which is also in agreement with the second last line in (<ref>).
Additionally, from (<ref>), we have that $u_1(\xi,\tau,z)=\eta_0(\xi,\tau)+\upsilon(\tau,z)$,
for an auxiliary function $\upsilon$.
As a matter of fact, if we have a solution of (<ref>)
and we add to $u_1$ a function of $(\tau,z)$ we obtain another solution. Therefore,
we choose the “simplest possible” solution by picking $\upsilon$ identically equal to zero, thus obtaining
\begin{equation}\label{TH:EVE:2}
\end{equation}
Now we consider the order related to $\e^2$ in the expansion in (<ref>)
(the higher orders being formally neglected). In this way, we find that
\begin{equation}\label{KDV-0m-008}
\begin{dcases}
-\partial_\xi u_2+\partial_\tau u_1+
u_1\partial_\xi u_1+w_1\partial_z u_1=-\partial_\xi P_2&{\mbox{ in }}\Omega_\tau,\\
-\partial_\xi w_1=-\partial_z P_2&{\mbox{ in }}\Omega_\tau,\\
\partial_\xi u_2+\partial_z w_2=0&{\mbox{ in }}\Omega_\tau,\\
\partial_zP_0(\xi,1,\tau)\eta_1+
\displaystyle\frac12\partial_{zz}P_0(\xi,1,\tau)\eta_0^2+\partial_zP_1(\xi,1,\tau)\eta_0+P_2(\xi,1,\tau)=
\eta_1, & \\
\partial_zw_1(\xi,1,\tau)\eta_0+w_2(\xi,1,\tau)=
\partial_\tau\eta_0+u_1\partial_\xi\eta_0-\partial_\xi\eta_1,& \\
w_2(\xi,0,\tau)=0. &
\end{dcases}
\end{equation}
Recalling (<ref>) and (<ref>), the fourth line in (<ref>) becomes
$$ P_2(\xi,1,\tau)=
\eta_1(\xi,\tau).
Also, from the second line in (<ref>) and (<ref>),
$$ \partial_zP_2=-\partial_\xi(z\partial_\xi\eta_0)=-z\partial_{\xi \xi}\eta_0.
These observations entail that, for all $z\in(0,1)$,
\begin{eqnarray*}&&\eta_1(\xi,\tau)-P_2(\xi,z,\tau)=
\int_z^1\partial_zP_2(\xi,z',\tau)\,dz'\\&&\qquad=-\int_z^1 z'\partial_{\xi \xi}\eta_0(\xi,\tau)\,dz'=-\frac{1-z^2}2\partial_{\xi \xi}\eta_0(\xi,\tau)
\end{eqnarray*}
and therefore
$$ P_2(\xi,z,\tau)=\eta_1(\xi,\tau)+\frac{1-z^2}2\partial_{\xi \xi}\eta_0(\xi,\tau).$$
From this and the first equation in (<ref>) we infer that
\begin{eqnarray*}
\partial_\xi u_2&=&
\partial_\tau u_1+u_1\partial_\xi u_1+w_1\partial_z u_1+\partial_\xi P_2\\
\partial_\tau u_1+u_1\partial_\xi u_1+w_1\partial_z u_1+\partial_\xi \eta_1+\frac{1-z^2}2\partial_{\xi\xi \xi}\eta_0.
\end{eqnarray*}
Notice the appearance of a third derivative for $\eta_0$. Thus, recalling
the third equation in (<ref>),
\begin{eqnarray*}
-\partial_z w_2=
\partial_\xi u_2=
\partial_\tau u_1+u_1\partial_\xi u_1+w_1\partial_z u_1+\partial_\xi \eta_1+\frac{1-z^2}2\partial_{\xi\xi \xi}\eta_0.
\end{eqnarray*}
Combining this with the last two equations in (<ref>) we conclude that
\begin{eqnarray*}&&
\partial_\tau\eta_0(\xi,\tau)+u_1(\xi,1,\tau)\partial_\xi\eta_0(\xi,\tau)-\partial_\xi\eta_1(\xi,\tau)-\partial_zw_1(\xi,1,\tau)\eta_0(\xi,\tau)\\&=&w_2(\xi,1,\tau)\\
&=& w_2(\xi,1,\tau)-w_2(\xi,0,\tau)
\\&=&\int_0^1 \partial_zw_2(\xi,z,\tau)\,dz\\&=&
-\int_0^1 \Bigg(
\partial_\tau u_1(\xi,z,\tau)+u_1(\xi,z,\tau)\partial_\xi u_1(\xi,z,\tau)+w_1(\xi,z,\tau)\partial_z u_1(\xi,z,\tau)\\&&
\qquad\qquad+\partial_\xi \eta_1(\xi,\tau)+\frac{1-z^2}2\partial_{\xi\xi \xi}\eta_0(\xi,\tau)
\Bigg)\,dz
\end{eqnarray*}
This and (<ref>) yield that
\begin{eqnarray*}&&
\partial_\tau\eta_0(\xi,\tau)+u_1(\xi,1,\tau)\partial_\xi\eta_0(\xi,\tau)-\partial_\xi\eta_1(\xi,\tau)+\partial_\xi\eta_0(\xi,\tau)\eta_0(\xi,\tau)\\&=&
-\int_0^1 \left(
\partial_\tau u_1(\xi,z,\tau)+u_1(\xi,z,\tau)\partial_\xi u_1(\xi,z,\tau)-z\partial_\xi\eta_0(\xi,\tau)\partial_z u_1(\xi,z,\tau)
\right)\,dz\\&&\qquad\qquad-\partial_\xi \eta_1(\xi,\tau)-\frac13\partial_{\xi\xi \xi}\eta_0(\xi,\tau).
\end{eqnarray*}
Then, we can simplify the term $-\partial_\xi\eta_1(\xi,\tau)$ and exploit (<ref>), finding that
\begin{eqnarray*}&&
\partial_\tau\eta_0(\xi,\tau)+\eta_0(\xi,\tau)\partial_\xi\eta_0(\xi,\tau)+\partial_\xi\eta_0(\xi,\tau)\eta_0(\xi,\tau)\\&=&
-\int_0^1 \left(
\partial_\tau \eta_0(\xi,\tau)+\eta_0(\xi,\tau)\partial_\xi \eta_0(\xi,\tau)
\right)\,dz-\frac13\partial_{\xi\xi \xi}\eta_0(\xi,\tau)\\&=&
-\partial_\tau \eta_0(\xi,\tau)-\eta_0(\xi,\tau)\partial_\xi \eta_0(\xi,\tau)
-\frac13\partial_{\xi\xi \xi}\eta_0(\xi,\tau),
\end{eqnarray*}
which is the Korteweg-de Vries equation
\begin{equation}\label{VJlaVDa8jr3} \partial_\tau\eta_0+ \frac32\partial_\xi\eta_0\eta_0+\frac16\partial_{\xi\xi \xi}\eta_0=0,\end{equation}
to be compared with (<ref>), as desired.
It is now instructive to recover equation (<ref>) in terms of the original variables. For this, owing to (<ref>) and (<ref>),
\begin{equation}\label{VJlaVDa8jr3dra6mo9}\eta(\xi,\tau)=
\eta^\star\left(\xi+\frac\tau\e,\frac\tau\e\right)=\overline\eta\left(\sqrt{gh_0} \left(\xi+\frac\tau\e\right),\frac\tau\e\right).\end{equation}
Moreover, since $\overline\eta(x,t)$ was the profile of the free surface of the fluid
in the canal and $\eta_0(\xi,\tau)$ the leading order of this profile in the new variables $(\xi,\tau)$
in a shallow water and low amplitude approximation, in light of (<ref>) we can consider the leading order of
the original profile $\overline\eta_0$ as defined implicitly by the relation
$$ \eta_0(\xi,\tau)=\overline\eta_0\left(\sqrt{gh_0} \left(\xi+\frac\tau\e\right),\frac\tau\e\right).$$
Therefore (<ref>) becomes
\begin{eqnarray*}\frac{
\sqrt{gh_0}}\e\partial_x\overline\eta_0+\frac1\e\partial_t\overline\eta_0+ \frac{3\sqrt{gh_0}}2\partial_x\overline\eta_0\overline\eta_0+\frac{ (gh_0)^{3/2}}6\partial_{x}^3\overline\eta_0=0
\end{eqnarray*}
and thus, recalling (<ref>),
\begin{eqnarray*}
\partial_t\overline\eta_0+\frac{ \sqrt{g} h_0^{5/2}}{6}\partial_{x}^3\overline\eta_0
+ \frac{3h_0^{3/2}}{2\sqrt{g}}\partial_x\overline\eta_0\overline\eta_0+\sqrt{gh_0} \partial_x\overline\eta_0=0.
\end{eqnarray*}
In this expression, one can get rid of the term $\partial_x\overline\eta_0$ by a vertical translation of $\overline\eta_0$,
that is, if
\begin{equation}\label{ZETATRAVEKDG32}\zeta_0(x,t):=\alpha\left(\overline\eta_0(x,t)+\frac{2g}{3h_0}\right)
=\frac{h_0^2}g \left(\overline\eta_0(x,t)+\frac{2g}{3h_0}\right),\end{equation}
leading to
\begin{eqnarray*}
\partial_t\zeta_0+\frac{ \sqrt{g} h_0^{5/2}}{6}\partial_{x}^3\zeta_0
+ \frac{3\sqrt{g}}{2\sqrt{h_0}}\partial_x\zeta_0\zeta_0=0,
\end{eqnarray*}
which corresponds[Retaking (<ref>), we know that
the velocity of the traveling wave $\phi_0$ is proportional to $\frac{c}a$ times the amplitude of the wave.
It is suggestive to compare this information with the numerology of the coefficients
and the relation with the physical shape of the wave. Namely, in this case we have that $\frac{c}a=c=\frac{3\sqrt{g}}{2\sqrt{h_0}}$
and, by (<ref>), the amplitude of the traveling wave would formally correspond to $
\frac{h_0^2}g \left(\max\overline\eta_0+\frac{2g}{3h_0}\right)=\frac{h_0^2}g \max\overline\eta_0+\frac{2h_0}{3}$.
The velocity of the traveling wave would thus be proportional to $\frac{3\sqrt{g}}{2\sqrt{h_0}}\left(
\frac{h_0^2}g \max\overline\eta_0+\frac{2h_0}{3}\right)=
\frac{3h_0^{3/2}}{2\sqrt{g}} \max\overline\eta_0+{\sqrt{gh_0}}
$. Once again, this suggests that the speed of small amplitude waves in a shallow canal is dictated by the quantity ${\sqrt{gh_0}}$,
plus an increasing term in the amplitude of the wave.
Of course, we do not aim here at exhausting the study of the
complicated topic of water waves' speed. See e.g. <cit.>
and the references therein for further details on this subject.]
to (<ref>) with $a:=1$, $b:=\frac{ \sqrt{g} h_0^{5/2}}{6}$ and $c:=\frac{3\sqrt{g}}{2\sqrt{h_0}}$.
For further information[For simplicity, we did not discuss the role of the surface tension
in the derivation of the Korteweg-de Vries equation, but we mention that this additional
term can be also easily included in the coefficient $b$ in (<ref>).] about the Korteweg-de Vries equation
and related topics see e.g. <cit.>
and the references therein.
§.§ Plasma physics
A state of matter is one of the distinct forms in which matter can exist in nature.
Classically, nature was classified
into three states of matter: solid, liquid, and gas.
Nowadays it is common[Also, a large number of
intermediate states are nowadays known to exist, such as
liquid crystals,
Bose-Einstein condensates,
Fermionic condensates,
neutron-degenerate matter, quark–gluon plasma, superfluids, superconductors, etc., but
we do not investigate these states here.
From the historical point of view,
plasma was first identified in laboratory by Sir William Crookes in 1879
(though more systematic studies of plasma only began in the twentieth century).
Besides discovering plasma, Crookes is also credited with discovering
thallium, inventing a radiometer and, quite conveniently, some $100\%$
ultraviolet blocking sunglass lens.
He was also interested in spiritualism and became president of the Society for Psychical Research,
whose purpose is to understand psychic and paranormal events.
See Figure <ref>
for a caricature of Sir William Crookes (by caricaturist Sir Leslie Ward).]
to consider also a fourth state
of matter, namely plasmaplasma.
Though our everyday experience with plasma appears to be rather limited
(we are experienced to plasma mostly only by
neon tubes, plasma lamps and lightnings),
plasma is allegedly the most abundant form of ordinary matter in the universe
(excluding dark matter and dark energy) and likely amounts to about $
99\%$ of the total mass of the visible universe.
It consists of a gas of ions and free electrons, usually produced by
very high temperatures ($10^4$ degrees or more)
which make electrons leave
their orbit around the corresponding nuclei. In this way,
if the initial status of the gas is overall in an electrical neutral state,
the plasma consists in a mixture of charged particles, namely the free electrons
and the corresponding ions.
Caricature of Sir William Crookes (Public Domain image from
The mathematical description of a plasma is a highly advanced subject but,
in a nutshell, there is a hierarchy of models accounting for the
the evolution of a plasma.
The most intuitive one would be to describe the plasma
by detecting positions and velocities of all its particles (ions and electrons) at a given time.
This approach is very precise but often impractical, given the high number of particles involved,
and therefore intermediate
models (often called “kinetic models”) have been introduced
to describe the plasma “in average” through the statistical analysis of the particle
Moreover, at a large scale, efficient models for a plasma leverage the knowledge of
fluid dynamics equations (somewhat close in the spirit to the ones that we
presented in Sections <ref> and <ref>):
namely, at a macroscopic level, close to thermodynamic equilibrium,
one can identify
each species of particles of a plasma (ions and electrons) with a fluid
and describe the corresponding density,
velocity and energy via a set of[To confirm the importance of partial differential
equations in our understanding of plasma, let us mention for instance that the
laboratory specialized in plasma and located in Toulouse, France,
is named LAboratoire PLAsma et Conversion d'Energie (LAPLACE).] partial differential equations.
To make this model concrete, we denote by an index $j\in\{1,2\}$
the species of particles of a given plasma (e.g., $j=1$ corresponding to ions and $j=2$ corresponding to
electrons). We suppose that the plasma, being neutral in average
but composed at a small scale by charged particles, is subject to its own magnetic and electric fields (denoted by $B$
and $E$, respectively).
We recall that charged particles are influenced by both the electric and the magnetic fields: more specifically,
the so-called
Lorentz force acting on a single charged particle (say, with charge $q_j$ and velocity $v_j$)
is of the form $q_j(E+v_j\times B)$. Therefore the total contribution of the Lorentz force acting on a portion of plasma with
particle density $\mu_j$ takes the form
\begin{equation}\label{TOTAFO-1}
\mu_jq_j(E+v_j\times B).\end{equation}
Additionally, the possible variations of density exert a force on the plasma similar to a pressure. Thus, we denote by
\begin{equation}\label{TOTAFO-2}
-\nabla p_j\end{equation}
this type of pressure (the minus sign
manifesting the fact that higher densities oppose the motion).
Differently from the case of classical fluids, in the description of a plasma we have also to account for
the effect of possible particle collisions. Since we are looking here at a macroscopic description
of the plasma, we suppose that the total momentum of the particles of species $j$
comes from the average over all particles of that species and therefore
like particle collisions do not change the total momentum (say, if two electrons collide,
one may slow down after collision and the other can be accelerated, but the total momentum
is preserved, thanks to Newton's Law; same if two ions collide).
We have therefore to focus on the collisions between unlike particles,
which allow momentum to be exchanged between the species:
in this situation, when an electron and a ion collide, the momentum loss of one particle entails
a corresponding momentum gain of the other particle:
that is if the first particle experiences a variation of momentum given by $P_1$,
the second particle experiences a variation of momentum given by $P_2=-P_1$.
Computing the momentum exchange of colliding particles can be in general a rather complicated task.
A simplifying assumption in this framework is to assume that the particles are reduced to a point.
An additional simplification arises if we are willing to suppose that the collisions are perfectly elastic (no dissipation
of energy due to particle collisions).
Furthermore, we can also reduce the problem to a simpler one if we are willing to use the fact that the mass
of the electrons is typically much smaller than the mass of the ions and take therefore
the mass of the ions as the leading term of a more complex approximation.
To make these ideas work, at least in this simplified setting, we recall that the elastic collision
of two point masses, say with masses $m_1$ and $m_2$,
initial (before collision) velocity $v_{1i}$ and $v_{2i}$
and final (after collision) velocity $v_{1f}$ and $v_{2f}$
is fully described by the equations
\begin{dcases}v_{1f}=\displaystyle{\frac {(m_{1}-m_{2})v_{1i}+2m_{2}v_{2i}}{m_{1}+m_{2}}},\\v_{2f}=\displaystyle{\frac {(m_{2}-m_{1})v_{2i}+2m_{1}v_{1i}}{m_{1}+m_{2}}},\end{dcases}
see e.g. <cit.>.
Hence, since we consider the index $j=1$ as corresponding to the electrons and we assume that $m_1\ll m_2$,
we have that the momentum exchange for the colliding electron is
\begin{eqnarray*}
m_1\left(\frac {(m_{1}-m_{2})v_{1i}+2m_{2}v_{2i}}{m_{1}+m_{2}}-v_{1i}\right)\\&=&
\frac {2m_1m_2(v_{2i}-v_{1i})}{m_{1}+m_{2}}\\
\end{eqnarray*}
In the framework of plasma collisions, this would give that $P_1=2\mu_1 m_1(v_2-v_1)$
and accordingly $P_2=2\mu_1 m_1(v_1-v_2)$.
Thus, if we denote by $\phi$ the collision frequency between the two species
(for simplicity, we are disregarding here the collisions between ions, or electrons,
with the remaining neutral gas),
the species corresponding to $j=1$ changes its momentum by a quantity $\phi\mu_1 m_1(v_2-v_1)$
and the species corresponding to $j=2$ changes its momentum by a quantity $\phi\mu_2 m_2(v_1-v_2)$
(where constants are omitted for the sake of simplicity).
The first durable colour photographic image (Public Domain image from Wikipedia).
From this, (<ref>) and (<ref>), by Newton's Law of momentum balance have that
\begin{eqnarray*} \mu_j m_j\partial_t v_j +\mu_j m_j(v_j\cdot\nabla)v_j&=& \mu_jm_j\frac{d}{dt} (v_j)\\&=&
\mu_jq_j(E+v_j\times B)-\nabla p_j+\phi\mu_j m_j(v_{k_j}-v_j),
\end{eqnarray*}
$$ k_j:=\begin{dcases}
1 & {\mbox{ if }}j=2,\\
2 & {\mbox{ if }}j=1.
\end{dcases}$$
This equation is usually complemented by a continuity equation (corresponding to the conservation
of the total number of particles, which can be seen as the counterpart here of the mass transport equation
in (<ref>)),
an equation of state (that is a constitutive law relating pressure and particle density,
which can be seen as the counterpart of the barotropic flow description given on page BAROFL,
usually taken of the form $p_j =\kappa \mu_j^\gamma$, for suitable positive constants $\kappa$ and $\gamma$)
and the classical Maxwell's equations[James Clerk Maxwell
founded the theory of electromagnetism, proving that electric and magnetic fields travel through space as waves moving at the speed of light and argued that light itself is an undulation causing electric and magnetic phenomena. This great conceptual unification of light and electrical phenomena made Maxwell one of the nineteenth century scientists having the greatest impact on the subsequent developments of relativity and quantum field theory.
Maxwell also worked on the kinetic theory of gases and pioneered the early days of control theory. Also, he established that the rings of Saturn were made of numerous small particles and he is credited for presenting the first durable colour photograph. For this, he had the idea of superimposing on a screen shots taken with red, green and blue filters: his result, depicting a tartan ribbon, is shown in Figure <ref>.
The photographic plates available at the time were almost insensitive to red and barely sensitive to green, hence the results in themselves were perhaps far from perfect, but Maxwell clearly knew this was after all just a rather minor detail in the Big Picture of Science
and he wrote
“by finding photographic materials more sensitive to the less refrangible rays, the representation of the colours of objects might be greatly improved”.] for the electric and magnetic fields:
omitting structural constants, this set of equations thus takes the form[Sometimes the setting in (<ref>)
is called in jargon “Euler-Maxwell equations”Euler-Maxwell equations, since it mixes the classical Euler's formulation
of fluid dynamics presented in Section <ref> (after suitable corrections
to deal with particle collision) with the Maxwell equationsMaxwell equations for electromagnetism.]
\begin{equation}\label{MAXWEUL}
\begin{dcases}
\mu_j m_j\partial_t v_j +\mu_j m_j(v_j\cdot\nabla)v_j=\mu_jq_j(E+v_j\times B)-\nabla p_j+\phi\mu_j m_j(v_k-v_j),\\
\partial_t\mu_j+\div(\mu_jv_j),\\
p_j =\kappa \mu_j^\gamma,\\
\div B=0,\\
\div E=\mu_1q_1+\mu_2q_2,\\
\curl E=-\partial_t E,\\
\curl B=\mu_1q_1v_1+\mu_2q_2v_2+\partial_t E,
\end{dcases}
\end{equation}
with $j\in\{1,2\}$.
For additional information about plasma physics, see e.g. <cit.> and the references therein.
§.§ Galaxy dynamics
Interestingly, a suitable modification of the model used to describe the motion of fluids
presented in Section <ref>
can represent the dynamics of galaxies and globular clusters
(see e.g. Figure <ref> for
a fascinating picture).
The ansatz of the model that we describe here is that the stars forming the galaxies interact only
by the gravitational field that they create collectively
and they can be described by parcels of a self-gravitating gas.
Neglecting collisional and relativistic effects, and disregarding the physical and chemical reactions
that continuously modify the internal structure of the stars, one
can provide a suitable set of equations describing the large scale galaxy motion.
These are named Jeans equationsJeans equations after Sir James Hopwood Jeans
(though were probably first derived by James Clerk Maxwell
and can be seen as a collisionless version of the Boltzmann equations
that describe the statistical behavior of a thermodynamic system not in equilibrium)
and are as follows:
\begin{equation}\label{STARS}
\begin{dcases}
\partial_t f+v\cdot\nabla_x f-\nabla u\cdot\nabla_v f=0,\\
\Delta u=\rho
\end{dcases}
\end{equation}
where $f=f(x,t,v)$ denotes the phase-space[By
phase-space here we mean the collection of position and velocity variables.]
density of stars, that is the density of stars corresponding to time $t$, position $x\in\R^3$ and velocity $v\in\R^3$,
$u=u(x,t)$ is the gravitational potential induced collectively by the stars,
$$ \rho=\rho(x,t):=\int_{\R^3}f(x,t,v)\,dv$$
is the star density corresponding to time $t$ and position $x$.
Hubble Space Telescope picture of the galaxies NGC 2207 (left) and IC 2163 (right) (Public Domain image from Wikipedia).
To understand the rationale of equation (<ref>) we argue as follows.
In analogy with the dynamics of fluid parcels described in (<ref>),
we assume that star dust trajectories follow the law $\dot x(t)=v(x(t),t)$
and that their acceleration follows the gravitational law $\ddot x(t)=-\nabla u(x(t),t)$.
One can therefore obtain an analogous of the continuity equation in (<ref>)
by assuming that, in absence of encounters, collisions
and collapses, the amount of stars is preserved by the flow.
Namely, the total quantity of stars occupying a region $Z$ of the phase-space $\R^3\times\R^3$ at a given time $t_0$
is given by the quantity $\int_{Z} f(x,t_0,v)\,dx\,dv$
and, in a small time $\tau$,
possibly occupying a different region, that we name $Z_\tau$ which collects all the evolution trajectories
in phase space given by
\begin{eqnarray*}
&& \Big(x(t_0+\tau),v\big(x(t_0+\tau),t_0+\tau\big)\Big)=
\big(x(t_0+\tau),\dot x(t_0+\tau)\big)=
(x(t_0),\dot x(t_0))+\tau (\dot x(t_0),\ddot x(t_0))+o(\tau)\\&&\qquad=
\Big(x(t_0),v\big(x(t_0),t_0\big)\Big)+\tau \Big(v\big( x(t_0),t_0\big),-\nabla u(x(t_0),t_0)\Big)+o(\tau).\end{eqnarray*}
Hence, the assumption that stars are conserved thus translates into
\begin{eqnarray*}
0&=&\left.\frac{d}{d\tau}\int_{Z_\tau} f(x,t_0+\tau,v)\,dx\,dv\right|_{\tau=0}\\
&=&\left.\frac{d}{d\tau}\left(\int_{Z} f\Big(x+\tau v(x,t_0), t_0+\tau,v-\nabla u(x,t_0)\Big)\,dx\,dv+o(\tau)\right)\right|_{\tau=0}\\
&=& \int_{Z}
\Big[ \nabla_x f(x,t_0,v)\cdot v(x,t_0)+\partial_t f(x,t_0,v)
-\nabla_v f(x,t,v)\cdot\nabla u(x,t_0)
\Big]\,dx\,dv.
\end{eqnarray*}
The arbitrariness of the phase-space domain $Z$ thus leads to
$$ \nabla_x f(x,t_0,v)\cdot v(x,t_0)+\partial_t f(x,t_0,v)
-\nabla_v f(x,t,v)\cdot\nabla u(x,t_0)=0,$$
which is the first equation in (<ref>).
Additionally, by Gauß's Flux Law for gravity,
we know that the flux of the gravitational field through a surfaces
balances the total mass enclosed within the surface: namely,
neglecting dimensional constants,
for every domain $\Omega\subset\R^3$,
$$ \int_{\partial\Omega} \nabla u(x)\cdot\nu(x)\,d{\mathcal{H}}^{n-1}_x=\int_\Omega \rho(x,t)\,dx.$$
From this and the Divergence Theorem we arrive at
$$ \int_{\Omega} \Delta u(x)\,dx=\int_\Omega \rho(x,t)\,dx,$$
and consequently, since $\Omega$ is arbitrary,
we find that $\Delta u=\rho$, which
is the second equation in (<ref>).
For additional information about galaxy dynamics see e.g. <cit.> and the references therein.
§.§ How to count what we cannot see
In 1812, Lorenzo Romano Amedeo Carlo Avogadro, Count of Quaregna and Cerreto,
hypothesized that the volume of a gas at a given pressure and temperature is proportional to the number of atoms or molecules.
That is, equal volumes of gases at the same temperature and pressure have the same number of molecules,
regardless of the nature of the gas.
In particular,
by Avogadro's Law, the number of molecules or atoms in a given volume of ideal gas is independent of their size.
While this prescription only holds true for ideal gases,
and real gases show instead small deviations from this ideal configuration,
Avogadro's Law is often a very useful approximation and it always provides a great conceptual tool,
since it detects a universal quantity that only depends on reasonable macroscopic parameters, such as
temperature and pressure, and is independent of the specific situation under consideration, namely the type of gas
that one is taking into account.
See Figure <ref> for a portrait of Avogadro (by C. Sentier).
Portrait of Amedeo Avogadro (Public Domain image from
It is perhaps worth stressing that Avogadro's Law is not so intuitive.
For instance, if one has three units of volumes of hydrogen
and one unit of volume of nitrogen, after they combine together and produce
ammonia, how many units of ammonia do we expect (assuming that
pressure and temperature are maintained constant)?
One unit? Three units? Four units?
The correct answer according to Avogadro's Law is two units of volume.
This is because three volumes of hydrogen contain $3k$ molecules of hydrogen $H_2$ (for some $k\in\N$)
and one volume of nitrogen contains $1k$ molecules of nitrogen $N_2$
(and the proportionality factor $k$ is the same for both hydrogen
and nitrogen, by Avogadro's Law).
Thus, from the reaction
\begin{equation}\label{REACCHI} 3H_2+1N_2=2NH_3,\end{equation}
the combination of $3k$ molecules of hydrogen
and $1k$ molecules of nitrogen produces $2k$ molecules
of ammonia $NH_3$. And again by Avogadro's Law
this corresponds to two volumes of ammonia. See Figure <ref>
for a sketch of this phenomenon.
Also, in chemistry, an obvious practical issue is given by the fact that molecules are small
while in the lab one has to deal with macroscopic quantities.
Thus, while it is desirable from the theoretical point of view to found the concept of
“amount of a given substance” on the number of “elementary entities”, i.e., atoms or molecules, present in the substance
(because these elementary entities are the ones which will play a role in the chemical reactions),
for practical purposes it is often convenient to relate the notion of “amount of a given substance”
to the ratio of measured macroscopic quantities (because measuring a huge number of microscopic entities
is typically unfeasible).
We remark that the equivalence between these two approaches is a notable consequence of
Avogadro's Law. Indeed, if, at a given pressure and temperature, we take, say, one gram of
atomic hydrogen $H$ (or, more simply, two grams of the common molecule of hydrogen $H_2$)
and we measure its volume, and then we consider the same volume of another gas, such as the carbon dioxide $CO_2$,
we know from Avogadro's Law that the latter contains the same number of molecules of $CO_2$
as the number of atoms of hydrogen $H$ (or the number of molecules of $H_2$) in the previous container.
If we measure the weight of the carbon dioxide $CO_2$ in the second container,
we find that it amounts to $44$ grams: we can accordingly conclude that
the mass of a molecule of carbon dioxide $CO_2$
is $44$ times bigger than the mass of an atom of hydrogen $H$
(or $22$ times bigger than the mass of a molecule of hydrogen $H_2$).
That is, we have measured, with a minimal effort, the microscopical weight of any molecule of any gases
in terms of a given “unit of measure”, such as the atomic weight of $H$ (or the molecular weight of $H_2$).
Hence, it has become a common practice to adopt
the notion of “mole” to denote the mass of a given substance
which contains the same number of molecules (or atoms in case of
pure atomic elements) as one gram of atomic hydrogen $H$.
In the previous example, since the mass of a molecule of carbon dioxide $CO_2$
happens to be $44$ times bigger than the mass of an atom of hydrogen $H$,
we infer that
\begin{equation}\label{LAMOLE}
{\mbox{a mole of carbon dioxide weights~$44$ grams.}}\end{equation}
The algebra in (<ref>)
with the corresponding gas volumes according to Avogadro's Law.
Moreover, by Avogadro's Law,
a mole of any gas at a given temperature and pressure occupies the same volume, making it a quite convenient
way to measure the (relative) amount of substance just by comparing volumes of gases.
The natural question is thus, given a certain chemical compound, how many molecules
of it correspond to a mole? How does this number depend on the specific chemical compound?
The answer is easy: by construction this number is the same for all substances and equals
the number of hydrogen atoms $H$ necessary to form one gram of hydrogen, and this
is called[We stress that
Avogadro's Number is not a “pure number” but a physical constant of dimension
“one over moles”. See also <cit.> for additional information on Avogadro's Number.]
Avogadro's Number ${\mathcal{N}}_{\!\mathcal{A}}$. Avogadro's Number
Notice that the knowledge of ${\mathcal{N}}_{\!\mathcal{A}}$ also permits to “pass from the ratios
to the real quantities”. For instance from (<ref>) we obtain that each molecule of
carbon dioxide weights $\frac{44}{ {\mathcal{N}}_{\!\mathcal{A}} }$ grams:
in general, the knowledge of Avogadro's Number entails the knowledge of atomic and molecular masses
as a byproduct of macroscopic measurements.
Therefore, it is highly desirable to have a precise[It turns
out (possibly with some mild approximation)
that ${\mathcal{N}}_{\!\mathcal{A}}=6.02214076\times 10^{23}$. Several people noticed
a similarity between this number and $2^{79}$ (which is half of the so-called yobibyte $2^{80}$):
indeed, it is a rather surprising coincidence that
$$ \frac{{\mathcal{N}}_{\!\mathcal{A}}-2^{79}}{{\mathcal{N}}_{\!\mathcal{A}}+2^{79}}$$
is almost zero.
There is nothing special in this, coincidences happen.
For instance, it is a mere, and quite remarkable, coincidence that the most important physical constant, namely the speed
of light, is almost equal to $300.000.000$ m/s.
It is also a coincidence that
the angular diameter of the Sun seen from Earth is quite close to that of the Moon
making it possible to see a full solar eclipse.
It is also a coincidence that
$$ \left(\frac{\pi^e-e^\pi}{\pi^e+e^\pi}\right)^{\pi+e}$$
turns out to be almost equal to zero.]
quantification of ${\mathcal{N}}_{\!\mathcal{A}}$. This is not a mathematical problem per se, since
to quantify ${\mathcal{N}}_{\!\mathcal{A}}$ one can only measure it through experiments.
However, one needs a brilliant piece of mathematics to contrive
a practical procedure to experimentally calculate ${\mathcal{N}}_{\!\mathcal{A}}$
and we recall here the method devised by Albert Einstein in <cit.>
(at that time, he was working as a patent clerk in Bern, Switzerland,
and the very notion of atoms and molecules was still a subject of controversy and intense scientific debate, since individual
atoms and molecules were, for the instruments available in 1905, simply
“too small” and “too fast”).
To measure Avogadro's Number, Einstein utilized a molecular theory of heat
more or less in the lines of the Brownian motion described here
in Section <ref> and leading to the heat equation in (<ref>).
For this, Einstein thought of some ideal spherical particles of radius $a$
subject to a Brownian motion in an ideal (for simplicity, one-dimensional and infinitely long) pipe
containing some fluid with viscosity coefficient equal to $\mu$.
In this setting, according to Einstein's calculation,
\begin{equation}\label{EINAVOGEAFG}
{\mathcal{N}}_{\!\mathcal{A}}=\frac{RTt}{3\pi \mu a\,X^2(t)},
\end{equation}
where $T$ denotes the temperature (or, more precisely, the thermodynamic temperature, that is the temperature measured in Kelvin), $t$ is any given time and $X^2(t)$ is the averaged square distance traveled by a Brownian particle in an interval
of time equal to $t$.
Equation (<ref>) is quite remarkable since it equates
Avogadro's Number with quantities which can be measured experimentally, thus leading
to a precise determination of ${\mathcal{N}}_{\!\mathcal{A}}$
(that's the power of mathematics!).
Jean Perrin (Public Domain image from
The experiment suggested by Einstein was indeed carried through by Jean Baptiste Perrin (see Figure <ref>)
and his team of research students
in 1909. The set up of the test consisted in a camera lucida with a microscope to
observe and record the motion of suspended gamboge[Gamboge
is a yellowish pigment.
The word gamboge comes from gambogium, the Latin word for the pigment, which in turns derives from Gambogia, the Latin word for Cambodia.
Besides its strong laxative properties, which did not play a role in Perrin's experiment,
gamboge also possessed the convenient feature of producing, with appropriate alcoholic additives and after a selective centrifuge,
almost perfect spherules with equal radius, thus providing the ideal suspension particles for the experiment
proposed by Einstein (this was not a cheap preparation,
about one kilo of gamboge and several months of work produced few decigrams of useful spherules).]
particles in
a liquid of a given viscosity and constant temperature.
Marking the particle's position on a piece of
graph paper at timed intervals, and repeating the test under different conditions
(such as different viscosity, temperature, radius of the gamboge grains, etc.), Perrin
obtained consistent results about the right hand side of (<ref>),
thus producing a rather accurate measurement of Avogadro's Number ${\mathcal{N}}_{\!\mathcal{A}}$
through a wealth of measurements that could not be contested.
This experiment also provided experimental confirmation of Einstein's equation (<ref>) and
raised atoms and molecules from the status of hypothetical objects to concrete and observable entities,
thus offering the ultimate confirmation to the atomic nature of matter
and concluding the struggle regarding the physical reality of molecules.
For these achievements, Perrin was awarded with the Nobel Prize for Physics in 1926.
It is now time to go back to Einstein's equation (<ref>)
to understand its roots in the theory of partial differential equations.
To this end, we recall that the first ingredient towards (<ref>) is Stokes' LawStokes' Law
about the viscosity force acting on a small sphere moving through a viscous fluid.
Namely, studying the incompressible steady flow of the Navier-Stokes equation (see Section <ref>), George Gabriel Stokes in 1851 had proposed the formula
\begin{equation}\label{SYO-EDCSJDKA235athcma}
F=6\pi \mu a v\end{equation}
to describe the frictional force $F$
acting on a spherical particle of radius $a$ with relative velocity $v$ in a fluid with viscous coefficient equal to $\mu$.
The second ingredient towards (<ref>) is
the van 't Hoff equationvan 't Hoff equation for[Equation (<ref>)
is named after Jacobus Henricus van 't Hoff Jr.,
first winner of the Nobel Prize in Chemistry, see Figure <ref>. Allegedly, van 't Hoff
chose to study chemistry
against the wishes of his father. Also, it seems he came up with equation (<ref>)
after a chance encounter and a quick chat with a botanist friend during a walk in a park in Amsterdam,
thinking about an analogy to the law for ideal gases.] the osmotic pressure $p$
of a solution containing $ n$ moles of solute particles in a solution of volume $ V $,
given by
\begin{equation}\label{BVNBSstVASNnsrNA}
in which $R$ is the universal gas constant.
Henry van 't Hoff (Public Domain image from
The third ingredient towards (<ref>) is the assumption that
the Brownian motion of the suspension particles can be effectively described by the heat equation,
as discussed in Section <ref>. Hence, we suppose that the density $u$ of the particles fulfills the equation
\begin{equation}\label{EINHEAT}
\partial_t u= c\partial_{xx} u,
\end{equation}
for some constant diffusion coefficient $c>0$ (here, we are assuming that the motion of the particles is confined
to a linear pipe, hence the variable $x$ is one-dimensional).
We can now retake van 't Hoff equation (<ref>)
by considering that the number of floating particles $N$ over the volume $V$
equals the density $u$ and, by definition, a mole contains ${\mathcal{N}}_{\!\mathcal{A}}$ particles.
These observations lead to
\begin{equation}\label{EDCSJDKA235S6IMfujNMimfr0}
p=\frac{NRT}{{\mathcal{N}}_{\!\mathcal{A}}\,V}=\frac{RT u}{{\mathcal{N}}_{\!\mathcal{A}}}.
\end{equation}
Also, the force $F$ acting on each floating particle arises from a pressure gradient, namely the gradient pressure force with respect to the unit of mass is $\frac{\partial_x p}u$.
This remark and (<ref>) give that
\begin{equation*}
F u=\partial_xp= \frac{RT\,\partial_x u }{{\mathcal{N}}_{\!\mathcal{A}}}.
\end{equation*}
At equilibrium, the force must correspond to the frictional force in (<ref>), therefore we obtain that
\begin{equation}\label{EDCSJDKA235S6IMfujNMimfr02}
6\pi \mu a v u=\frac{RT\,\partial_x u}{{\mathcal{N}}_{\!\mathcal{A}}}.\end{equation}
Now we compute the flux of floating particles through an ideal cross section of the pipe
by using two possible strategies. On the one hand, we can consider the model of particles traveling with velocity $v$,
hence we can quantify the flux by the product between $v$ and the particle density $u$.
On the other hand, the flux at some cross section $x$ can also be obtained by the variation in time
of the mass preceding $x$,
that is $\partial_t \int_{-\infty}^x u(\xi,t)\,d\xi$. These observations and (<ref>) give that
$$ v(x,t) u(x,t)=\partial_t \int_{-\infty}^x u(\xi,t)\,d\xi=
\int_{-\infty}^x \partial_t u(\xi,t)\,d\xi=c\int_{-\infty}^x \partial_{xx} u(\xi,t)\,d\xi=
c\partial_x u(x,t),$$
where, as usual, we have performed formal calculations and assumed some convenient decay of the solution at $-\infty$.
From this and (<ref>) we arrive at
\begin{equation}\label{EDCSJDKA235S6IMfujNMimfr03}
{\mathcal{N}}_{\!\mathcal{A}}=\frac{RT\,\partial_x u}{6\pi \mu a v u}=\frac{RT}{6\pi \mu a c}
Now we observe that
$$g (x,t):=\frac {1}{\sqrt {4\pi ct}}\exp \left(-{\frac {x^{2}}{4ct}}\right)$$
is a solution of (<ref>) concentrating at the origin when $t=0$, hence
we can suppose that the general solution $u$ of (<ref>) occurs as a superposition[The formalization
of this superposition method is the core of the notion of fundamental solution and, in the elliptic setting, it will
play a decisive role in Section <ref> and in particular in the forthcoming
Proposition <ref>.]
of traslations of $g$ of the form $g(x-x_0,t)$ (notice that each of these translations
would correspond to a particle located at $x_0$ when $t=0$).
The average squared displacement at time $t$ of the particle starting at the origin is therefore provided by the quantity
$$ \int_\R x^2 g(x,t)\,dx=\int_\R\frac {x^2}{\sqrt {4\pi ct}}\exp \left(-{\frac {x^{2}}{4ct}}\right)\,dx=2ct.
Since this displacement should not depend on the original position, we infer that it is the same for all
possibilities of initial location of the particles, namely
\begin{equation}\label{EINAVOGEAFG2} X^2(t)=2ct.\end{equation}
This is conceptually an important result since it suggests that velocities (namely ratios of $X(t)$ over $t$)
of Brownian motions behave sort of $\frac{\sqrt{2ct}}{t}=\frac{\sqrt{2c}}{\sqrt{t}}$,
hence quite irregularly[For a formalization of this heuristic and imprecise discussion see e.g. <cit.>.] for short times. That is,
measuring the velocity of the floating particles in an extremely short interval of time
just produces a result that approaches infinity (in a sense,
attempted experiments in this direction would simply end up measuring the wrong quantity).
One of the merit of Einstein's approach is thus to have bypassed this hindrance by detecting
the relevant physical quantities which were both consistent from the mathematical point of view and experimentally
measurable in laboratories. Notice indeed that by plugging (<ref>)
into (<ref>) we obtain (<ref>), as desired.
§.§ Vibrating strings
Another classical occurrence for partial differential equations arises in the description
of vibrating stringsvibrating string (e.g., the string of a guitar, see
Figure <ref>). Different models
are possible, taking into account different physical assumptions (see e.g. <cit.>),
but we focus here on the simplest possible (indeed linear) equation.
In our description, the string at rest is modeled by the segment $[0,L]\times\{0\}$
for some $L>0$ which represents the length of the string in the absence of further external forces.
One can assume that the string is constrained at the extrema
and can be deformed into the graph of a function $u$: namely, the position
of the vibrating string at each moment of time $t$ is described by the graph $(x,u(x,t))\in[0,L]\times\R$
and the constraints for the string correspond to the boundary prescription $u(0)=u(L)=0$.
Jimi Hendrix (photo by Steve Banks, image from
licensed under the Creative Commons Attribution-Share Alike 4.0 International license).
We suppose that the string is subject to gravity (acting downwards in the vertical direction)
and to its own “tension”tension (that is the internal force of the string acting between its elements
providing an elastic reaction to the external forces). The magnitude of this tension
at each point of the string $(x,u(x,t))$ will be denoted by $T(x,t)$.
We also consider the unit tangent vector to the string, as given by
$$ \tau(x,t):=\frac{\big(1,\partial_x u(x,t)\big)}{\sqrt{1+\big(\partial_x u(x,t)\big)^2}}.$$
The vibrating string model then consists in the assumption that, given a small $\delta>0$,
the tension forces acting at the string point $(x,u(x,t))$ are the byproduct of
a force $F_-$ acting on $(x-\delta,u(x-\delta,t))$ and a force $F_+$ acting on $(x+\delta,u(x+\delta,t))$
whose magnitude is proportional to the tension $T(x,t)$ and opposite tangential directions
(namely, the direction of $F_-$ is $-\tau(x-\delta,t)$ and
the direction of $F_+$ is $\tau(x+\delta,t)$), see Figure <ref>.
Therefore, the total tension force acting on the string at the point $(x,u(x,t))$ takes the form
\begin{equation}\label{TOTFOSTR}\begin{split}&F_+-F_-=\kappa T(x)\big(\tau(x+\delta,t)-\tau(x-\delta,t)\big)=
2\kappa \delta T(x)\partial_x\tau(x,t)+o(\delta)\\&\qquad\qquad\qquad=
2\kappa \delta T(x)\partial_x\left(
\frac{\big(1,\partial_x u(x,t)\big)}{\sqrt{1+\big(\partial_x u(x,t)\big)^2}}\right)
The constant $\kappa>0$ depends on the material of which the string is made
and accounts for the elastic properties of the string.
The total force acting on the string, taking into account gravity, is therefore
\begin{equation*} F=(F_1,F_2):=F_+-F_- -(0,m_\delta g)=
2\kappa\delta T(x)\partial_x\left(
\frac{\big(1,\partial_x u(x,t)\big)}{\sqrt{1+\big(\partial_x u(x,t)\big)^2}}\right)-(0,m_\delta g)
where $m_\delta$ is the mass of the string located between $x-\delta$ and $x+\delta$ and $g$ denotes
the gravity acceleration.
If we suppose that the density of the string is constant, say equal to some $\rho>0$, the mass $m_\delta$
is the product of $\rho$ and the length of the string located between $x-\delta$ and $x+\delta$, namely
\begin{equation}\label{MEPSI} m_\delta= \rho \int_{x-\delta}^{x+\delta} \sqrt{1+\big(\partial_x u(\zeta,t)\big)^2}\,d\zeta
=2\rho\delta \sqrt{1+\big(\partial_x u(x,t)\big)^2}+o(\delta).\end{equation}
Furthermore, by Newton's Second Law, the vertical component of the force is equal to the mass times
the vertical acceleration of the string, that is
$$ F_2=m_\delta \partial_{tt} u(x,t).$$
In light of these considerations, we find that[From the geometric point
of view, it is interesting to observe that the term $\partial_x\left(
\frac{ \partial_x u}{\sqrt{1+\big(\partial_x u\big)^2}}\right)$ in (<ref>)
detects the curvature of the string
(remarks of this type will be formalized in a general setting
in Theorem <ref>).]
for small $\delta$
\begin{equation}\label{CUSTRJMSONGJMSD}
\begin{split}
\partial_{tt} u\,&= \frac{F_2}{m_\delta}
\\&=\frac1{m_\delta}\left[
2\kappa\delta T\partial_x\left(
\frac{ \partial_x u}{\sqrt{1+\big(\partial_x u\big)^2}}\right)- m_\delta g
\right]\\&=
\frac{2\kappa\delta T}{m_\delta}
\partial_x\left(
\frac{ \partial_x u}{\sqrt{1+\big(\partial_x u\big)^2}}\right)- g\\&=
\frac{\kappa T}{
\rho \sqrt{1+\big(\partial_x u(x,t)\big)^2}+o(1)}
\partial_x\left(
\frac{ \partial_x u}{\sqrt{1+\big(\partial_x u\big)^2}}\right)- g\\
\frac{\kappa T}{
\rho \sqrt{1+\big(\partial_x u(x,t)\big)^2}}
\partial_x\left(
\frac{ \partial_x u}{\sqrt{1+\big(\partial_x u\big)^2}}\right)- g+o(1).
\end{split}\end{equation}
By formally sending $\delta\searrow0$ we thus obtain the partial differential equation
\begin{equation}\label{CUSTRJMSONGJMSD-2}
\partial_{tt} u=
\frac{\kappa T}{
\rho \sqrt{1+\big(\partial_x u(x,t)\big)^2}}
\partial_x\left(
\frac{ \partial_x u}{\sqrt{1+\big(\partial_x u\big)^2}}\right)- g.
\end{equation}
Several different models can be consider to describe explicitly the tension $T$
and thus find an expression for (<ref>)
solely depending on the shape of the string, on its density and elastic constant
and on the gravity acceleration.
One commonly accepted model is to consider the curve as “inextensible”
and $T$ as a constant (the tension being distributed uniformly along the whole string).
Another possibility is to consider the string as a superposition of infinitesimal elastic springs and thus take $T$ as proportional to the string's elongation.
Both these models however share the common treat that for small elongations $T$
is constant. Therefore, in the small elongation approximation, we take $T:=1$
and we disregard quadratic terms in $u$ in (<ref>)
(the ansatz being that for small $u$ the linear terms
will prevail against the quadratic ones). With this simplification in mind,
one can reduce (<ref>) to
\begin{equation}\label{CUSTRJMSONGJMSD-3}
\partial_{tt} u=
\frac{\kappa }{
\rho }
\partial_{xx}u- g.
\end{equation}
In particular, in the absence of gravity, the linear approximation of the vibrating string can be seen
as a particular case of the wave equation in (<ref>).
Interestingly, the speed of propagation $c$ in (<ref>)
corresponds here to $\sqrt{\frac{\kappa }{
\rho }}$. Namely the higher the elastic parameter of the string, the higher the
speed of propagation; the higher the density (equivalently, the heavier the string),
the slower the speed of propagation.
A string subject to its own tension.
The dependence of the solution of (<ref>) upon the structural parameters
of the string is not a mere mathematical curiosity: on the contrary it has a deep impact on music,
since these parameters precisely determine the pitch of the sound that the string generates.
Let us see, for instance, how understanding partial differential equations may turn out
to be useful when tuning a guitar. For accomplishing this goal, we neglect the gravity effect and summarize (<ref>) and the boundary conditions of the string
into the following mathematical setting:
\begin{equation}\label{CwUSTRJMSONGJMSD-4}
\begin{dcases}
\partial_{tt} u(x,t)=\displaystyle{\frac{\kappa }{
\rho }}\partial_{xx}u(x,t)\qquad{\mbox{for all }}(x,t)\in(0,L)\times(0,+\infty),\\
u(x,0)=\eta\displaystyle\sin\frac{\pi x}{L},\\
\partial_t u(x,0)=0,\\
u(0,t)=u(L,t)=0\qquad{\mbox{for all }}t\in(0,+\infty).
\end{dcases}
\end{equation}
The prescriptions in (<ref>) model a guitar string
with rest length $L$ and null initial velocity which is initially displaced (say by
an expert fingerpicking) as a sinusoidal graph (for the sake of simplicity, though more complicated
initial situations could be taken into account). The small parameter $\eta>0$
has been introduced here above just to be consistent with the small elongation approximation.
It can be readily checked that the function
\begin{equation}\label{CwUSTRJMSONGJMSD-5} u(x,t)=\eta\cos\frac{\sqrt\kappa\,\pi t}{\sqrt\rho\,L}
\sin\frac{\pi x}{L}\end{equation}
solves (<ref>).
This solution is quite telling regarding the pitch played by the string.
Indeed, in light of (<ref>),
the string vibrates with a frequency
\begin{equation}\label{CwUSTRJMSONGJMSD-6}
\omega:=
\frac{\sqrt\kappa\,\pi}{\sqrt\rho\,L}.\end{equation}
In particular:
* Strings with a higher elastic coefficient produce a higher pitch (and we already know this by experience, because tightening the tuning pegs of a guitar enhances the string's tension and correspondingly raises the pitch),
* Longer strings produce lower pitches and shorter strings produce higher pitches (and we already know this by experience too, since, in playing, one changes the length of the vibrating string by holding it firmly against the fingerboard with a finger and shortening the string, that is stopping it on a higher fret, gives higher pitch),
* Strings with higher density produce lower pitches, that is more massive strings vibrate more slowly (and this is also in agreement with our experience since, for instance, on classical guitars, the low density nylon strings are used for the high pitches and the higher density wire-wound strings are employed for low pitches).
To appreciate even more the impact of mathematics on guitar playing,
we recall that an octave[The name “octave” comes from the Latin adjective “octava”, meaning “eighth”. This etymology however is possibly a bit confusing, since it only represents the interval between one musical pitch and another with double its frequency: hence, an octave is more related to the number $2$ than to the number $8$.
Also, in the common western musical composition, the octave is composed by $7$ notes, or $12$ semitones, hence the octave seems to relate more to the numbers $7$ and $12$ rather than $8$. The fact is that when talking about intervals of notes the tradition is to count from the first note to the final note included (which makes $8$ notes, somewhat justifying the name of octave).]octave
is the distance between one pitch and another with half or double its frequency:
thus, from (<ref>),
holding the string against the fingerboard halving its length has the effect of doubling the frequency of the root note (consistently with the fact that each fret on the guitar's fretboard corresponds[The choice of
dividing the octave in twelve semitonessemitone is also mathematically well grounded, since it corresponds
to frequency ratios of the form $2^{\frac{j}{12}}$ for $j\in\{0,\dots,12\}$.
On the one hand, these are irrational numbers (except for $j=0$ and $j=12$).
On the other, notes sound harmonious to our ear if the frequency of the notes is close to a simple interval:
for instance a frequency ratios such as $\frac32$ (the “perfect fifth”
in musical jargon), $\frac43$ (the “fourth”) and $\frac54$ (the “major third”).
These two observations seem contradictory, till we realize
how “close to rationals” the irrational numbers of the form $2^{\frac{j}{12}}$ are
(for $j\in\{1,\dots,11\}$ these numbers are “almost” $\frac{16}{15}$,
$\frac98$, $\frac65$, $\frac54$, $\frac43$, $\frac75$, $\frac32$, $\frac85$, $\frac53$, $\frac{16}9$
and $\frac{15}8$).
For example,
$$ \frac32-2^{\frac{7}{12}}=1.5-1.49830707688...=0.00169292312...$$
showing how close the seventh semitone is to the perfect fifth.
While other tuning scales are possible and are indeed employed in several contexts,
the equal temperament one (i.e., the one splitting the scale into equal intervals)
presents several advantages, such as the possibility of transposing a tune into other keys
thus allowing instruments to play in all keys
(or singers to sing a song in a key which is more congenial to their voice)
with minimal flaws in intonation.
The twelve-tone equal temperament based on
the division of the octave into twelve equally spaced parts
on a logarithmic scale appears the most widespread system in music today.
Likely, the common adoption of this specific temperament
also influenced the composition of music in order to accommodate the system and minimize dissonance
coming from irrational approximation of rational ratios.]
to one semitone,
the distance of twelve semitones is the octave, and the position of the twelfth fret
corresponds to a string half as long as the original one).
Gustav Kirchhoff
(Public Domain image from
Wikipedia, source Smithsonian Libraries).
For completeness, let us also mention a slightly different model for vibrating strings proposed[Besides the nonlocal equations for vibrating strings related to (<ref>),
Gustav Kirchhoff provided fundamental contributions
to the theory of electrical circuits, spectroscopy, and black-body radiation.
Which creates quite a confusion, because the names of
Kirchhoff's Law and Kirchhoff equation end up being used in all these topics, with different meanings.
By the way, Kirchhoff's home town was Königsberg, a
historic Prussian city that is now Kaliningrad, in Russia's Kaliningrad Oblast
(a small exclave of the vast Russian state
providing an interesting counterexample to the conjecture that political states are connected
regions, up to additional islands). The reason for which we mention Königsberg is because
one classical and very famous mathematical problem that was settled by Euler is related to the bridges of
Königsberg (see e.g. <cit.> to know more about the Königsberg bridges).] by Gustav Robert Kirchhoff in <cit.>, see Figure <ref>.
The gist of this model is that the elastic coefficient $\kappa$ in (<ref>)
is assumed to be a positive constant but, in reality, the elastic properties of the string may
depend on its elongation (according to the material, the string could either become “stiff”
or “slack off” for large
elongations). To account for such a possibility, Kirchhoff proposed that $\kappa$
instead depends on the length of the deformed string, namely on
\begin{equation}\label{leng2tht346y356ef7o8om-ely}
{\mbox{\footnotesize{\calligra{L}}}}\;\,(t):=\int_0^L \sqrt{1+\big(\partial_x u(x,t)\big)^2}\,dt.\end{equation}
In this model, $\kappa$ in (<ref>) must be interpreted as $\kappa\left(
{\mbox{\footnotesize{\calligra{L}}}}\;\,(t)\right)$, that is a function of the length of the string
(the corresponding equation is thus called Kirchhoff equation). Kirchhoff equation
Notice that not only in this situation $\kappa$ varies with time, but also
it depends on a “nonlocal” quantity. Indeed, the length of the deformed string in (<ref>) is a “global” object: to calculate it, it is not sufficient to
know the shape of the string at a given point, instead full information about its large scale geometry
is required. And, of course, mathematical problems requiring the knowledge of nonlocal
quantities become structurally harder (but often quite interesting, since they aim at
capturing the Big Picture).
§.§ Elastic membranes
The description of elastic membraneselastic membrane can be seen as a multi-dimensional version
of the one of vibrating strings presented in Section <ref>.
A simple model could be that of considering the membrane as constituted by
a web of infinitesimal strings in the coordinate directions.
In this way, one may think that the vertical force exerted
at a point $(x,u(x))\in\R^n\times\R$ of the membrane
is the sum of the forces produced by the tension of the infinitesimal strings
in each direction, plus gravity.
Recalling (<ref>), we may think that, for each $i\in\{1,\dots,n\}$,
the vertical force exerted by the tension of the infinitesimal string
located in direction $e_i$ (say, located from $x-\delta e_i$ and $x+\delta e_i$) is equal to
$$ 2\kappa \delta T(x)\partial_i\left(
\frac{\partial_i u(x,t)}{\sqrt{1+\big(\partial_i u(x,t)\big)^2}}\right),$$
up to higher order in $\delta$.
Hence the total vertical force acting on the membrane at $(x,u(x))\in\R^n\times\R$
is given by
$$ 2\kappa \delta T(x)\sum_{i=1}^n\partial_i\left(
\frac{\partial_i u(x,t)}{\sqrt{1+\big(\partial_i u(x,t)\big)^2}}\right)-m_\delta g,$$
where, in view of (<ref>),
=2\rho\delta \sum_{i=1}^n\sqrt{1+\big(\partial_i u(x,t)\big)^2},$$
up to higher orders in $\delta$, being $\rho$ the linear densities of the infinitesimal strings in each direction
(that we suppose to be constant and independent of the direction).
In this setting, Newton's Second Law gives that
\begin{equation}\label{CUSTRJMSONGJMSD-287729ou35t4jrg}
\begin{split}
\partial_{tt}u&=\frac{2\kappa \delta T}{m_\delta}\sum_{i=1}^n\partial_i\left(
\frac{\partial_i u}{\sqrt{1+(\partial_i u)^2}}\right)-g\\
&=\frac{\kappa T}{\displaystyle
\rho\sum_{i=1}^n\sqrt{1+(\partial_i u)^2}}
\sum_{i=1}^n\partial_i\left(
\frac{\partial_i u}{\sqrt{1+(\partial_i u)^2}}\right)-g,
\end{split}\end{equation}
which can be considered the higher dimensional[From the geometric point
of view, it is interesting to observe that the term $\displaystyle
\sum_{i=1}^n\partial_i\left(
\frac{\partial_i u}{\sqrt{1+(\partial_i u)^2}}\right)$ in (<ref>)
detects the mean curvature of the membrane
(remarks of this type will be formalized in a general setting
in Theorem <ref>).]
version of (<ref>).
In the small elongation approximation we thus consider $T:=1$ and disregard the terms that
are quadratic in the displacement $u$, thus reducing (<ref>)
\begin{equation}\label{CUSTRJMSONGJMSD-287729ou35t4jrg24qwerty} \partial_{tt}u=\frac{ \kappa }\rho\Delta u-g,\end{equation}
which is the higher dimensional version of (<ref>).
In particular, an elastic membrane in equilibrium is a stationary solution of (<ref>) and thus satisfies
\begin{equation}\label{CUSTRJMSONGJMSD-287729ou35t4jrg24qwerty44} \frac{ \kappa }\rho\Delta u=g,\end{equation}
which is a form of the so-called Poisson's equation (compare with (<ref>)).
In the absence of gravity, (<ref>) reduces to
the so-called Laplace's equation (compare with (<ref>))
\begin{equation*}
\Delta u(x)=0.\end{equation*}
This gives that (at least in the linear approximation) elastic membranes constrained at their boundaries
are described by the graph of harmonic functions.
Once again, this fact brings us a useful hint, suggesting a smoothing effect
for harmonic functions, somewhat inherited from the fact that elastic membranes
do not develop spikes (the elasticity would indeed reduce the peak
to find a balance): this intuition will be mathematically formalized, developed
and consolidated in Chapter <ref>.
§.§ Elasticity theory and torsion of bars
A topical argument in material sciences deals with the determination of the shape of objects
subject to external forces and to the understanding of the relations between the forces
applied and the resulting deformations. Providing a comprehensive account of this elasticity theory goes well beyond
the scopes of these notes (see e.g. <cit.> and the references therein for further readings):
here we just introduce, in a rather simplified form, some basic concepts which will lead us to the study
of an interesting[Actually, the final equation that we will obtain in (<ref>)
is precisely the same as the one arising from fluid dynamics in (<ref>). This coincidence shows
once again the extraordinary unifying power of mathematics.]
elliptic equation (which actually will be studied in detail in Sections <ref> and <ref>).
For concreteness, we focus on the three-dimensional case and we consider an infinitesimal portion of the material under consideration,
modeled as a very tiny cubes with faces oriented along the coordinate axes, see Figure <ref>.
If the cube has the form $(0,\delta)^3$, for some small $\delta>0$, we denote by $S_1$, $S_2$ and $S_3$
three facets of the cubes in direction $e_1$, $e_2$ and $e_3$ respectively, i.e.
$$ S_1:=\{\delta\}\times(0,\delta)\times(0,\delta),\qquad
For each $i$, $j\in\{1,2,3\}$ we denote by $\sigma_{ij}$ the force per unit of area applied in direction $e_i$ to the surface $S_j$.
In jargon, the matrix $\{\sigma_{ij}\}_{i,j\in\{1,2,3\}}$ is sometimes called “stress tensor”stress tensor.
The stress tensor.
By Newton's Third Law, at equilibrium the forces acting on the above ideal infinitesimal cube
must balance, therefore the forces acting on the facets of the cubes opposite to $S_1$ (or $S_2$, or $S_3$) are the ones discussed above, but with a sign changed.
In particular, given $i\in\{1,2,3\}$, at equilibrium the total forces per unit of area applied in direction $e_i$ must balance out
and therefore (noticing that if $x\in S_j$ then $x-\delta e_j$ belongs to the facet opposite to $S_j$)
$$ 0=\sum_{j=1}^3\Big(\sigma_{ij}(x)-\sigma_{ij}(x-\delta e_j)\Big)=
\delta \sum_{j=1}^3\partial_j\sigma_{ij}(x)+o(\delta),$$
leading to the following balance prescription for the stress tensor:
\begin{equation}\label{356um1ijkialivj}
\sum_{j=1}^3\partial_j\sigma_{ij}=0\qquad{\mbox{for all }}\;i\in\{1,2,3\}.
\end{equation}
Additionally, the angular momentum contribution $\vartheta_1$ coming from $S_1$ is proportional to the vector product
between the vectorial force per unit of area on $S_1$, corresponding to $\sigma_{11}e_1+\sigma_{21}e_2+\sigma_{31}e_3$,
and the vector joining normally the center of the cube to $S_1$, namely $e_1$. Hence, up to normalizing constants,
$$ \vartheta_1=\det\left(\begin{matrix}
e_1 & e_2 & e_3\\
\sigma_{11} & \sigma_{21} & \sigma_{31}\\
1 & 0 &0
\end{matrix}\right)=\sigma_{31}e_2-\sigma_{21}e_3.
the angular momentum contribution $\vartheta_2$ coming from $S_2$ is proportional to the vector product
between the vectorial force per unit of area on $S_2$, corresponding to $\sigma_{12}e_1+\sigma_{22}e_2+\sigma_{32}e_3$,
and the vector joining normally the center of the cube to $S_2$, namely $e_2$, whence
$$ \vartheta_2=\det\left(\begin{matrix}
e_1 & e_2 & e_3\\
\sigma_{12} & \sigma_{22} & \sigma_{32}\\
0 & 1 &0
\end{matrix}\right)=-\sigma_{32}e_1+\sigma_{12}e_3.
And also the angular momentum contribution $\vartheta_3$ coming from $S_3$ is proportional to the vector product
between the vectorial force per unit of area on $S_3$, corresponding to $\sigma_{13}e_1+\sigma_{23}e_2+\sigma_{33}e_3$,
and the vector joining normally the center of the cube to $S_2$, namely $e_3$, whence
$$ \vartheta_3=\det\left(\begin{matrix}
e_1 & e_2 & e_3\\
\sigma_{13} & \sigma_{23} & \sigma_{33}\\
0 & 0 &1
\end{matrix}\right)=\sigma_{23}e_1-\sigma_{13}e_2.
On this account, the
conservation of the total angular momentum entails that, at equilibrium,
$$ 0=\vartheta_1+\vartheta_2+\vartheta_3=(\sigma_{23}-\sigma_{32})e_1
This establishes that the stress tensor is symmetric, namely
\begin{equation}\label{SYMME}
\sigma_{ij}=\sigma_{ji}\qquad{\mbox{for all }}\,i,j\in\{1,2,3\}.
\end{equation}
Now, so as to model the distortion of the material caused by these forces, we consider a displacement vector $v$
describing at any point the change in the configuration of the given body.
The displacement vector itself
may not be the most interesting object to take into account in an elasticity theory, since
translations and rotations do produce significant displacements without affecting the shape of the body under consideration.
It is therefore common to try to detect the deformation effects that displacements produce
by accounting for relative displacements. Namely, if a point $x$ is moved to $S(x):=x+v(x)$, given $\omega\in\partial B_1$,
the point $y:=x+\delta\omega$ is moved to ${\mathcal{P}}(y)=y+v(y)=x+\delta\omega+v(x+\delta\omega)$. To compute the corresponding
relative displacement, one checks how $y-x$ is affected by this transformation, namely we point out that
\begin{eqnarray*}
{\mathcal{P}}(y)-{\mathcal{P}}(x)={\mathcal{P}}(x+\delta \omega)-{\mathcal{P}}(x)=\delta\,D{\mathcal{P}}(x)\omega+O(\delta^2)=D{\mathcal{P}}(x)(y-x)+O(\delta^2).\end{eqnarray*}
Namely, the “deformation gradient” $D{\mathcal{P}}$ is a linear approximation measuring relative displacements.
If we want to measure the change of square length, we have that
\begin{equation}\label{UJSvherEDIMAPOMKE}\begin{split}& |{\mathcal{P}}(y)-{\mathcal{P}}(x)|^2-|y-x|^2=|D{\mathcal{P}}(x)(y-x)+O(\delta^2)|^2-|y-x|^2\\&\qquad=
\big(D{\mathcal{P}}(x)(y-x)+O(\delta^2)\big)\cdot\big(D{\mathcal{P}}(x)(y-x)+O(\delta^2)\big)-|y-x|^2\\&\qquad=
\big(D{\mathcal{P}}(x)(y-x)\big)\cdot\big(D{\mathcal{P}}(x)(y-x)\big)-|y-x|^2+O(\delta^3)\\&\qquad=
\big(D{\mathcal{P}}(x)\big)^\top D{\mathcal{P}}(x)(y-x)\cdot(y-x)-|y-x|^2+O(\delta^3)\\&\qquad=\Big(
\big(D{\mathcal{P}}(x)\big)^\top D{\mathcal{P}}(x)-{\rm Id}\Big)(y-x)\cdot(y-x)+O(\delta^3),
\end{split}\end{equation}
where ${\rm Id}$ is the identity matrix and the superscript “$\top$” denotes matrix transposition.
It is also useful to observe that
\begin{eqnarray*}&&
D{\mathcal{P}}^\top D{\mathcal{P}}-{\rm Id}=\big({\rm Id}+Dv\big)^\top\big({\rm Id}+Dv)-{\rm Id}=
Dv+Dv^\top+Dv^\top Dv.
\end{eqnarray*}
In particular, if one assumes that the displacements are small, the quadratic term $Dv^\top Dv$
is negligible with respect to the “symmetric gradient” $Dv+Dv^\top$ which is of linear type.
Therefore, it is customary to consider the approximation $
D{\mathcal{P}}^\top D{\mathcal{P}}-{\rm Id}\simeq Dv+Dv^\top$ and rewrite (<ref>) as
\begin{equation*}
|{\mathcal{P}}(y)-{\mathcal{P}}(x)|^2-|y-x|^2\simeq \Big(Dv(x)+\big(Dv(x)\big)^\top\Big)
\end{equation*}
or equivalently
\begin{equation}\label{STRTENDE}
|{\mathcal{P}}(y)-{\mathcal{P}}(x)|^2-|y-x|^2\simeq \sum_{i,j=1}^3 \e_{ij}(x)
\end{equation}
\begin{equation}\label{STRTENDE-2}
\e_{ij}(x):=\Big(Dv(x)+\big(Dv(x)\big)^\top\Big)_{ij}
=\partial_i v_j(x)+\partial_j v_i(x).
\end{equation}
In jargon, $\e_{ij}$ is called the “strain tensor”strain tensor (and notice the symmetry feature that $\e_{ij}=\e_{ji}$)
and (<ref>) expresses in a quantitative way (though after some simplifying approximation)
the relative change in the position of points under the deformation.
We now relate strain and stress.
To this end, once again,
the simplest possible ansatz, leading to a linear theory of elasticity, is now that
deformations (or, more precisely, strains, to avoid translations and rotations) are directly proportional
to the forces excerpt (this is indeed a possible rephrasing of Hooke's Law for linear elastic materials).
That is, neglecting proportionality constants, we will suppose that
\begin{equation}\label{STRESTAP}
\e_{ij}=\sigma_{ij}.\end{equation}
It is interesting to remark that this identity is also algebraically compatible with the fact that
both the tensors above are symmetric.
We now apply the general framework of the linear theory of elasticity showcased so far
to the special situations of straight bars subject to torsion. This will be indeed one of the founding motivations
in Sections <ref> and <ref>. To model a straight bar, we consider
a body of the form
\begin{equation}\label{HANXTHECRPR4RO4O}
\Omega\times[0,+\infty),\end{equation}
with $\Omega\subset\R^{2}$
smooth and contractible, see Figure <ref>.
A bar subject to torsion.
We assume that such a bar is constrained at the level $\{x_3=0\}$ but it gets “twisted” from the top.
The effect of the twist would be to make the cross section rotate by some angle, depending on the level that we take into account.
Assuming again a linear behavior, we suppose that the rotation performed at level $\{x_3=\zeta\}$
is proportional to $\zeta$ (i.e., the angle of rotation increases linearly with the height,
which is not so unreasonable, at least for small heights, since the bar is constrained at ground zero).
Accordingly, we model the twist at level $\{x_3=\zeta\}$ by a rotation of angle $\Theta_\zeta= c_0\zeta$, for some $c_0>0$.
The corresponding displacement sends therefore the point $x=(x_1,x_2,\zeta)$
to a point $z:=x+v(x)$, with the horizontal component of $v$ being prescribed by the above rotation by the angle $\Theta_\zeta$,
namely, for small heights $\zeta$,
\begin{eqnarray*} \left( \begin{matrix} z_1\\z_2\end{matrix}\right)&=&
\left( \begin{matrix} \cos\Theta_\zeta&\sin\Theta_\zeta\\-\sin\Theta_\zeta&\cos\Theta_\zeta\end{matrix}\right)
\left( \begin{matrix} x_1\\x_2\end{matrix}\right)\\&=&
\left( \begin{matrix} \cos(c_0\zeta)&\sin(c_0\zeta)\\-\sin(c_0\zeta)&\cos(c_0\zeta)\end{matrix}\right)
\left( \begin{matrix} x_1\\x_2\end{matrix}\right)\\&=&
\left( \begin{matrix} 1+O(\zeta^2)& c_0\zeta+O(\zeta^2)\\-c_0\zeta+O(\zeta^2)&1+O(\zeta^2)\end{matrix}\right)
\left( \begin{matrix} x_1\\x_2\end{matrix}\right).
\end{eqnarray*}
This gives that
\begin{equation*}
\left( \begin{matrix} v_1(x)\\v_2(x)\end{matrix}\right)=
\left( \begin{matrix} z_1\\z_2\end{matrix}\right)-
\left( \begin{matrix} x_1\\x_2\end{matrix}\right)
\left( \begin{matrix} O(\zeta^2)& c_0\zeta+O(\zeta^2)\\-c_0\zeta+O(\zeta^2)&O(\zeta^2)\end{matrix}\right)
\left( \begin{matrix} x_1\\x_2\end{matrix}\right)=
\left( \begin{matrix} c_0\zeta x_2\\-c_0\zeta x_1\end{matrix}\right)+O(\zeta^2).
\end{equation*}
Hence, by sloppily[This just because here we are playing around with motivations.
From next chapter on, this kind of sloppiness will no longer be tolerated!] neglecting the higher order terms, and using the fact that $x=(x_1,x_2,x_3)\in\{x_3=\zeta\}$,
we will simply write that
\begin{equation}\label{STRESTAP-3}
v_1(x)= c_0 x_2 x_3\qquad{\mbox{and}}\qquad v_2(x)=-c_0 x_1 x_3.
\end{equation}
As for the vertical component of the displacement, a common ansatz is to consider it independent of the height:
roughly speaking, the torsion of the bar may exhibit a distortion of the cross section out of its own plane
(known in jargon as “warping deformation”warping deformation), but usually this effect is less visible than other types of deformations
and we are assuming here that each layer of the bar (i.e., each horizontal cross section) “equally pushes
the next one”, regardless of its height (once again, this is not a completely unrealistic assumption
especially if we are restricting our attention to the layers close to ground zero). To model that this warping effect
does not depend on the height, we assume that the vertical component of the displacement vector is independent of $x_3$, that is
\begin{equation}\label{STRESTAP-4} v_3(x)=w(x_1,x_2),\end{equation}
for some function $w$.
Now, from (<ref>), (<ref>), (<ref>) and (<ref>) we arrive at
$$ \sigma_{13}=\e_{13}=\partial_1 v_3+\partial_3 v_1 =\partial_1 w+c_0x_2.$$
$$ \sigma_{23}=\e_{23}=\partial_2 v_3+\partial_3 v_2 =\partial_2 w-c_0x_1$$
$$ \sigma_{33}=\e_{33}=2\partial_3 v_3=0.$$
Interestingly, the above expressions for $\sigma_{i3}$ with $i\in\{1,2,3\}$ are all independent of the height $x_3$
(which was possibly not obvious from the beginning). Furthermore, recalling the
balance prescription in (<ref>) and the symmetry property in (<ref>),
\begin{eqnarray*}&& 0=\sum_{j=1}^3\partial_j\sigma_{3j}=\sum_{j=1}^3\partial_j\sigma_{j3}=\partial_1\sigma_{13}+\partial_2\sigma_{23}.
\end{eqnarray*}
That is, the differential form $\sigma_{23}\,dx_1-\sigma_{13}\,dx_2$ is exact, and therefore closed, thanks to the
Poincaré Lemma (see e.g. <cit.>). This yields that there exists a function $\phi=\phi(x_1,x_2)$,
sometimes refereed to with the name of “warping potential”warping potential, such that $d\phi=\sigma_{23}\,dx_1-\sigma_{13}\,dx_2$.
This gives that $\partial_1\phi=\sigma_{23}$ and $\partial_2\phi=-\sigma_{13}$. Consequently,
\begin{equation}\begin{split}\label{TRUNPDOmrfgEPPJO}&
\Delta\phi=\partial_1\sigma_{23}-\partial_2\sigma_{13}=
\partial_1(\partial_2 w-c_0x_1)-\partial_2(\partial_1 w+c_0x_2)\\&\qquad\qquad\qquad=
\partial_{12} w-c_0-\partial_{12} w-c_0=-2c_0.\end{split}
\end{equation}
Another notion of interest for engineering purposes is the “traction”traction ${\mathcal{T}}$ along the boundary of the bar,
which is defined as the normal component of the vertical stress. More explicitly, if we define $\sigma_3:=(\sigma_{31},\sigma_{32},\sigma_{33})$ and we consider the outer unit normal $\nu^\star$ of the bar, we set
\begin{equation}\label{CA2LT} {\mathcal{T}}:=\sigma_3\cdot\nu^\star.\end{equation}
As a matter of fact, since the straight bar has the form given in (<ref>),
we can identify $\nu^\star\in\R^3$ with the vector $(\nu,0)$, being $\nu\in\R^2$ the
outer unit normal at $\partial\Omega$ and consequently
$$ {\mathcal{T}}=\sigma_{31} \nu_1+\sigma_{32}\nu_2=\sigma_{13} \nu_1+\sigma_{23}\nu_2=
where $\tau:=(-\nu_1,\nu_2)$ can be taken as the unit tangent vector along $\partial\Omega$ (say, clockwise oriented).
In this spirit, we find that the traction is the tangential derivative of the warping potential.
In the special situation in which the bar is subject to no traction, it follows that $\nabla\phi\cdot\tau=0$ on $\partial\Omega$. This gives that $\phi$ is constant along $\partial\Omega$, because if $\partial\Omega$ is locally parameterized by a curve $\gamma:(-1,1)\to\R^2$ with $\dot\gamma$ proportional to $\tau$, we have that
$$ \frac{d}{dt}\phi(\gamma(t))=\nabla\phi(\gamma(t))\cdot\dot\gamma(t)=0.$$
Thus, if the traction of the bar vanishes, we can write that
\begin{equation}\label{TRUNPDOmrfgEPPJO2}
{\mbox{$\phi=c_2$ on~$\partial\Omega$,}}\end{equation}
for some $c_2\in\R$.
Recalling (<ref>), we call the magnitude of the traction on the surface of the bar $\partial\Omega$ the scalar $|\sigma_3|$.
We remark that
$$ |\nabla\phi|=\sqrt{ (\partial_1\phi)^2+(\partial_2\phi)^2}=\sqrt{\sigma_{23} ^2+\sigma_{13}^2}=
\sqrt{\sigma_{23} ^2+\sigma_{13}^2+\sigma_{33}^2}=|\sigma_3|$$
therefore the magnitude of the traction coincides also with the norm of the gradient of the warping potential.
Thus, a consequence of (<ref>) is that $|\partial_\nu\phi|=|\nabla\phi|$,
hence in this situation the magnitude of the traction coincides also (possibly up to a sign) with the normal
derivative of the warping potential along the surface of the bar.
As a consequence, in the special situation in which the bar is subject to no traction and
the magnitude of the traction is constant, we have that
\begin{equation}\label{TRUNPDOmrfgEPPJO2b}
{\mbox{$\partial_\nu\phi=c_3$ on~$\partial\Omega$,}}\end{equation}
for some $c_3\in\R$.
Thus, if we define
$$ u:=\frac{c_2-\phi}{2c_0},$$
we deduce from (<ref>),
(<ref>) and (<ref>)
that for a straight bar
subject to torsion and no traction, if the magnitude of the traction along the surface of the bar is constant then there exists a solution of
\begin{equation}\label{EUMSD-OS-32456i7DNSSE234R-BNMBAR}
\begin{dcases}
\Delta u=1&{\mbox{ in }}\Omega,\\
u=0&{\mbox{ on }}\partial\Omega,\\
\partial_\nu u=c&{\mbox{ on }}\partial\Omega.
\end{dcases}\end{equation}
Notice the perfect coincidence[As noticed in footnote <ref>
on page OJSLLAFOr5O3OSMEr3r,
equation (<ref>) possesses a solution when $\Omega$
is a disk, that is when the bar has a circular cross section.
We will discuss in Sections <ref> and <ref>
whether bars of other shapes maintain the same property of presenting no traction and constant
traction magnitude along their surface.]
of this set of prescriptions with that in (<ref>),
which also reveals a possibly unexpected connection between the dynamics of viscous fluids
and the elastic reactions of bars subject to torsion.
Bending steel beams with bare hands by the power of $\e$.
§.§ Bending beams and plates
Now we deal with deflection of beams, which is a classical topic in all superhero comics, see Figure <ref>.
Besides, the question has obvious applications in engineering, see e.g. Figure <ref>.
The beam equation beam equation
\begin{equation}\label{BARREbi96}
\end{equation}
describes[The linear model of elasticity accounting for the load-carrying and deflection of beams
is often called “Euler-Bernoulli beam theory”.
No confusion should arise between (<ref>) and the model presented
in Section <ref> where instead the torsion theory
of bars, and not the deformation under a load, was taken into account.]
the small deflection $u$ of a homogeneous beam at equilibrium in terms of the applied load $q$.
In (<ref>), $u$ and $q$ are functions of one variable $x\in\R$ (the beam is assumed to be infinitely long).
In a nutshell, the derivation of (<ref>) from prime principles relies on balance of forces and moments.
Roughly speaking, the bending of the beam produces a compression force on one side
of the beam and a tensile stress on the other side: these deformation
stresses will be modeled via an elastic Hooke's Law and produce
turning forces which, at equilibrium, together with the corresponding moments,
need to be balanced with the external load on the beam.
Narrows Bridge, Perth (image by Speddie23 from
licensed under the Creative Commons Attribution-Share Alike 4.0 International license).
The details go as follows. We assume that the beam is displayed along the horizontal axis and slightly deformed
into a graph of the form $z=u(x)$ (the beam may well be three-dimensional, in which case we assume
that its
transversal sections possess some given shape that remains essentially invariant under small bending, and the three-dimensional
coordinates are denoted, as usual, by $(x,y,z)$).
More precisely, the graph $z=u(x)$ describes the “neutral fiber” of the beam,
that is a fiber which maintains infinitesimally the original length that it possessed at rest, see Figure <ref>.
The transversal sections of the beam that were perpendicular to the neutral fiber before the beam deforms are assumed to remain perpendicular to the neutral fiber after the bending.
We stress that this neutral fiber may not be located at the mid-height of the beam. In any case, we assume that on one side
of this neutral fiber compression takes place and on the other side tension occurs
(specifically, in Figure <ref>, the upper fibers are in compression and the lower fibers are under tension).
To efficiently describe the elastic forces produced by this interplay of compression and tension, it is profitable to consider
the osculating circle at a given point of the neutral fiber, see Figure <ref>.
Considering an infinitesimal quantity $\e>0$ and an infinitesimal element of the beam between $x$ and $x+\e$,
we thereby replace the graph of $u$ with a small portion of the osculating circle of the graph of $u$ tangent to $(x,u(x))$.
We denote by $\varrho(x)$ the radius of this osculating circle and by $\vartheta$ the corresponding polar angle infinitesimally
joining $(x,u(x))$ to $(x+\e,u(x+\e))$.
A deflected beam in the $(x,z)$-plane with an osculating circle.
We consider an upper fiber located at distance $z$ above the neutral fiber and we
try to quantify the force produced by compression.
To this end, we consider the ratio $\upsilon(z)$
between the variation of infinitesimal
length of the fiber and its original length before bending
and we suppose that the corresponding compression force is linear with respect to $\upsilon(z)$
(linearity with respect to length variations is indeed the main ingredient of
elasticity according to Hooke's Law, and note that the bigger the distance $z$ from the neutral fiber the bigger
the compression). We also observe that the infinitesimal length of this fiber after bending (in the osculating circle approximation)
is given by $(\varrho(x)-z)\vartheta$ while its original length before bending was equal to that of the neutral fiber,
which is $\varrho(x)\vartheta$.
and accordingly the compression force related to an upper fiber at distance $z$
from the neutral fiber is proportional to $\frac{z}{\varrho(x)}$.
Similarly, the lower fiber located at distance $z$ below the neutral
one would produce by tension a force which is
proportional to $\frac{z}{\varrho(x)}$
(notice that the forces produced by tension are opposed to the ones produced by compression).
Thus, if we suppose that the elastic modulus of the material is the same for compression and tension
(which is another classical ansatz in the elasticity theory according to Hooke's Law) and we use the convention that a positive $z$
corresponds to the upper fibers and a negative $z$
corresponds to the lower fibers, the resulting elastic force $F$ is approximately oriented along the tangential axis and equals to $\frac{Ez}{\varrho(x)}$, where $E>0$ denotes the elastic modulus of the material (that we suppose to be constant).
We stress that the fact that the force $F$ changes sign with $z$ (accordingly, that compression and tension produce forces
in opposite directions above and below the neutral fiber) produces a total torque $M(x)$.
Indeed, to compute this torque at the point $(x,u(x))$, for each $z$ we take into consideration
the vector product between the position vector and the corresponding force, namely
if $(y,z)$ belongs to the cross section $A$ of the beam corresponding to $(x,u(x))$,
the magnitude of the torque at $(y,z)$ is approximately given by $zF=\frac{Ez^2}{\varrho(x)}$.
Loads and moments of a beam.
The magnitude of the total torque $M(x)$ is thereby the corresponding integral for $(y,z)\in A$, that is
\begin{equation}\label{uJNSEMM0olSdfg2} M(x)=\iint_A \frac{Ez^2}{\varrho(x)}\,dy\,dz.\end{equation}
It is also customary[Of course, the axial second moment of area depends on the shape of the cross section of the beam.
For instance, for a circular cross section we have that $A=\{(y,z)\in\R^2$ s.t. $y^2+z^2<r^2\}$, for some $r>0$, and therefore
in this case
$$I=\iint_{\{ y^2+z^2<r^2\}} z^2\,dy\,dz=2\int_{-r}^{r}z^2\sqrt{r^2-z^2} \,dz=\frac{\pi r^4}{4}.
Instead, if the cross section is a rectangle $(-a,a)\times(-b,b)$ for some $a$, $b>0$,
$$I=\iint_{(-a,a)\times(-b,b) } z^2\,dy\,dz=\frac{4ab^3}{3}.
to define the axial second moment of area axial second moment of area as
$$ I:=\iint_A z^2\,dy\,dz.$$
With this notation, we can write (<ref>) as
\begin{equation}\label{uJNSEMM0olSdfg223} M(x)=\frac{EI}{\varrho(x)}.\end{equation}
We also recall that the osculating radius $\varrho(x)$ can be written as $\frac{(1+(u'(x))^2)^{3/2}}{u''(x)}$,
see e.g. <cit.>.
Hence, for small deformations, the osculating radius $\varrho(x)$ can be approximated by $\frac{1}{u''(x)}$
and then, with this approximation, equation (<ref>) reduces to
\begin{equation}\label{uJNSEMM0olSdfg224} M(x)= EI\,u''(x).\end{equation}
To carry on with the derivation of (<ref>),
we now take into consideration the balance of moments, see Figure <ref>.
For this, we assume that the beam is subject to a vertical distributed load of magnitude $q=q(x)$
and we compute its torque at the reference point corresponding to $x+\e$. For this,
given $\ell\in(0,\e)$, we calculate the magnitude of the vector product between the distributed load at the point $x+\ell$,
which is $(0,-q(x+\ell))$ and the vector joining the reference point to the point corresponding to $x+\ell$, which is $\big(x+\ell,u(x+\ell)\big)-\big(x+\e,u(x+\e)\big)=\big( \ell-\e, u(x+\ell)-u(x+\e)\big)$, thus finding the quantity $ q(x+\ell)(\ell-\e)$.
Correspondingly, the magnitude of the total torque produced by the external load on the beam between $x$ and $x+\e$ corresponds to
\begin{equation}\label{BAMO-MAHNdfo-21} \int_0^\e q(x+\ell)(\ell-\e)\,d\ell=
\int_0^\e \big(q(x)+O(\e)\big)(\ell-\e)\,d\ell=
\begin{equation}\label{BAMO-MAHNdfo-22} M(x+\e)-M(x)=\e \partial_xM (x)+\frac{\e^2}2 \partial_x^2 M(x)+O(\e^3).\end{equation}
The balance of moments at the second order in $\e$ between (<ref>) and (<ref>)
leads[The first order in $\e$ in (<ref>) reveals the existence of a shear force
acting on the faces of the beam: namely, the first order in $\e$ moments balance would produce that this shear force equates
the term $\partial_xM$ (possibly up to a sign convention).]
$$ \partial_x^2 M(x)= q(x).$$
By combining this with (<ref>) we obtain
$$ q(x)=\partial_x^2( EI\,u''(x))=EI\,u''''(x),$$
which corresponds to the beam equation in (<ref>) (up to normalizing constants).
See e.g. <cit.> for further details on the beam equation and on related topics.
Augustus Edward Hough Love (Public Domain image from
It is interesting to generalize the model discussed so far to comprise the case of[The mathematical model commonly used to describe stresses and deformations in thin plates
is sometimes called “Kirchhoff-Love theory”,
after Gustav Robert Kirchhoff and Augustus Edward Hough Love.
We have already met Kirchhoff on page SIMKIRAoGREENFIDItangeFI.
Besides his work in elasticity theory, Love dedicated himself to the study of
the spin-orbit locking, that is the phenomenon occurring when an astronomical body
always has the same face toward the object it is orbiting
(for example, up to some minor variability,
the same side of the Moon always faces the Earth).
See Figure <ref> for a wood engraved print representing Love taken from the British Newspaper The Graphic.]
bending plates. For instance, one could model a thin plate as a graph $x_{n+1}=u(x_1,\dots,x_{n})$, with $x=(x_1,\dots,x_n)\in\R^n$
(of course, bearing in mind that concretely $n=2$ and the corresponding plate is two-dimensional).
In this setting, the beam equation in (<ref>) leads to the plate equation plate equation
\begin{equation}\label{BARREbi96-PLA}
\Delta^2 u=q,
\end{equation}
where $\Delta^2$ is the Laplace operator applied twice.
To deduce the plate equation in (<ref>) from the beam equation in (<ref>) one can argue as follows.
Let us suppose that the vertical bending force of a horizontal plate is described, for small deformations, by a linear differential operator ${\mathcal{L}}$ of the form
\begin{equation}\label{BARREbi96-PLA-DAD2} {\mathcal{L}}:=\sum_{{\alpha\in\N^n}\atop{|\alpha|\le m}}c_\alpha\partial^\alpha,\end{equation}
for some $m\in\N$ and $c_\alpha\in\R$. Under this assumption, the force balance
would lead to the equation ${\mathcal{L}}u=q$, being $q$ the external load.
Then, to check (<ref>), one needs to show that
\begin{equation}\label{BARREbi96-PLA-DAD}
{\mathcal{L}}=\Delta^2 .
\end{equation}
For this, we assume that, consistently with the model of beam deformation in (<ref>),
the operator ${\mathcal{L}}$ acts as a fourth derivative on any one-dimensional function.
Namely, let us suppose that if $u_\omega(x):=u_0(\omega\cdot x)$ for some $u_0:\R\to\R$ and $\omega\in\partial B_1$
\begin{equation}\label{BARREbi96-PLA-DAD3} {\mathcal{L}}u_\omega(x)=u_0''''(\omega\cdot x).\end{equation}
We observe that, for all $j\in\{1,\dots,n\}$,
$$ \partial_j u_\omega(x)=\omega_j \,u_0'(\omega\cdot x)$$
and therefore, if $\alpha=(\alpha_1,\dots,\alpha_n)\in\N^n$,
$$ \partial^\alpha u_\omega(x)=\partial_1^{\alpha_1}\dots\partial_n^{\alpha_n} u_\omega(x)
=\omega_1^{\alpha_1}\dots\omega_n^{\alpha_n} \,u_0^{(|\alpha|)}(\omega\cdot x)=\omega^\alpha\,u_0^{(|\alpha|)}(\omega\cdot x).
Hence, by (<ref>) and (<ref>),
$$ u_0''''(\omega\cdot x)={\mathcal{L}}u_\omega(x)=\sum_{{\alpha\in\N^n}\atop{|\alpha|\le m}}c_\alpha\, \omega^{\alpha} \,u_0^{(|\alpha|)}(\omega\cdot x)
Since $u_0$ is arbitrary, we infer that $c_\alpha=0$ unless $|\alpha|=4$ and
$$ 1=\sum_{{\alpha\in\N^n}\atop{|\alpha|=4}}c_\alpha\, \omega^{\alpha}
Now, given $X\in\R^n\setminus\{0\}$ we let $\rho:=|X|$ and $\omega:=\frac{X}{|X|}$ and we deduce that
\begin{eqnarray*}&& \sum_{i,j=1}^n X_i^2 X_j^2=\left(\sum_{i=1}^n X_i^2\right)^2
=\big(|X|^2\big)^2=|X|^4=|X|^4\sum_{{\alpha\in\N^n}\atop{|\alpha|=4}}c_\alpha\, \omega^{\alpha}\\&&\qquad=
\sum_{{\alpha\in\N^n}\atop{|\alpha|=4}}c_\alpha\, \rho^{|\alpha|}\omega^{\alpha}=\sum_{{\alpha\in\N^n}\atop{|\alpha|=4}}c_\alpha\,\big( \rho \omega\big)^{\alpha}=\sum_{{\alpha\in\N^n}\atop{|\alpha|=4}}c_\alpha\,X^{\alpha}
Consequently, by the Identity Principle for polynomials,
$$ c_\alpha=\begin{dcases}
1 & {\mbox{ if $\alpha=2e_i+2e_j$ for some }}i,j\in\{1,\dots,n\},\\
0&{\mbox{ otherwise.}}
\end{dcases}$$
From this and (<ref>) we arrive at
$$ {\mathcal{L}}=\sum_{i,j=1}^n \partial^2_i\partial_j^2=\sum_{i=1}^n \partial^2_i\Delta=\Delta^2$$
and this establishes (<ref>), as desired.
§.§ Gravitation and electrostatics
One of the reasons for which the Laplace operator became very popular, especially in the eighteenth andgravitation
nineteenth centuries, is its pivotal role in the description of the gravity and electromagneticelectrostatics potentials.
The physical intuition of these phenomena was actually very helpful for the beautiful minds of those ages
in understanding a number of very deep concepts which paved the way to the modern theory of
partial differential equations and which are now widely used in many branches of mathematics, physics
and engineering.
We give here a couple of motivations[Also, a different but related
perspective on electrostatics and magnetism will be given in the forthcoming
Section <ref>.] relating the gravity field and harmonic functions
(similar arguments would link the electrostatic field and harmonic functions as well,
just changing the notion of mass with that of electric charge, and possibly allowing
a sign change, given the fact that unlike charges attract each other, but
like charges repel).
One observation is that the gravity force (in $\R^3$) is inversely proportional to the square of the distance,
namely a point mass located at $x$ is attracted by a Permanent Center of Gravity located at the origin
via a force $f(x)$ equal (up to dimensional constant) to $-\frac{x}{|x|^3}$, with the minus
sign to stress that this force is attractive, hence tends to reduce the distance of
the mass located in $x$ to the origin.
It is readily seen that $f$ is originated by a gravity potential $u$: more precisely, if we set $u(x):=\frac{1}{|x|}$,
we have that $\nabla u(x)=-\frac{x}{|x|^3}=f(x)$. Remarkably, the gravity potential is harmonic away[We will learn in equation (<ref>)
a more handy way to perform the calculation in (<ref>).]
the origin (here it is important to work in dimension $3$) since, if $x=(x_1,x_2,x_3)\in\R^3\setminus\{0\}$,
\begin{equation}\label{WRBSY0perIKSMaismoBBeJPREB}
\Delta u(x)=-\sum_{i=1}^3 \partial_i \frac{x_i}{|x|^3}
=-\sum_{i=1}^3 \left(\frac{1}{|x|^3}
-\frac{3 x_i^2}{|x|^5}\right)=-\frac{3}{|x|^3}
+\frac{3 |x|^2}{|x|^5}=0.
\end{equation}
These considerations can be recast in a possibly more general, and more “geometric”,
framework, also allowing a higher dimensional presentation.
For this, we first employ the fact that the
work done by the gravitational force depends only on initial and final positions, and not on the path between them: this gives that the gravitational vector field $f$ is conservative and thus can be written as the gradient of some function $u$, which is usually called the gravitational potential.
Furthermore, Gauß's Flux Law states that the flux of the gravity
field through an arbitrary closed surface enclosing no masses (or no electric) charges is necessarily zero (roughly speaking,
the field lines going into the region enclosed by the surface balance exactly the ones coming out).
Thus, from the Divergence Theorem, for every bounded region $\Omega\subset\R^n$ (say, with smooth boundary)
$$ 0=\int_{\partial\Omega} f(x)\cdot\nu(x)\,d{\mathcal{H}}_x^{n-1}=
\int_{\Omega} \div f(x)\,dx=\int_{\Omega} \div (\nabla u(x))\,dx=\int_{\Omega} \Delta u(x)\,dx.$$
Since $\Omega$ is arbitrary we thus conclude that the gravity potential (as well as the electrostatic potential) is harmonic away from the masses that are present in the environment.
The study of functions that are harmonic away from a point singularity will be
the main topic of the forthcoming Section <ref>. There we will also appreciate
how different dimensions affect the precise expression of the above potentials.
§.§ Classical electromagnetism
The theory of classical electromagnetismelectromagnetism has its roots in the so-called Maxwell's equationsMaxwell equations
that describe the behavior and mutual interaction of electric and magnetic fields.
These equations are somewhat a unified version of several specific information arising from
Gauß's Flux Law, Faraday's Law and
Ampère's Law.
In the vacuum (hence, in a region of $\R^3$ with no electric charges and no electric currents)
Maxwell's equations take the form
\begin{equation}\label{KMSJUNDNSMDEJDN} \div E =0,\qquad \curl E=-{\frac {\partial B }{\partial t}},\qquad
\div B=0
\qquad{\mbox{and}}\qquad\curl B=\frac1{c^2}{\frac {\partial E }{\partial t}}.\end{equation}
Here above, $E$ is the electric field,
$B$ is the magnetic field and $c$ is a physical constant (corresponding to the speed of light in vacuum).
We also observe that, for all smooth vector fields $v:\R^3\to\R^3$,
\begin{equation}\label{KMSJUNDNSMDEJDN2}
\nabla (\div v)-\curl(\curl v)=\Delta v.
\end{equation}
To check this classical identity in an elementary way, given $v=(v_1,v_2,v_3)$,
we write $v=v^{(1)}+v^{(2)}+v^{(3)}$,
where $v^{(1)}:=(v_1,0,0)$, $v^{(2)}:=(0,v_2,0)$ and $v^{(3)}:=(0,0,v_3)$,
and we remark that, thanks to the linear structure of (<ref>),
it suffices to check it for the vector fields $v^{(1)}$, $v^{(2)}$ and $v^{(3)}$.
We can focus our attention on $v^{(1)}$, up to reordering coordinates.
In this situation, since, for every smooth vector field $V:\R^3\to\R^3$,
$$ \curl V=\left(
{\frac {\partial V_3}{\partial x_2}}-{\frac {\partial V_2}{\partial x_3}} ,\;
{\frac {\partial V_1}{\partial x_3}}-{\frac {\partial V_3}{\partial x_1}},\;
{\frac {\partial V_2}{\partial x_1}}-{\frac {\partial V_1}{\partial x_2}}\right),$$
we have that
$$ W:=\curl v^{(1)}=\big(0 ,\partial_3 v_1, -\partial_2 v_1\big)$$
and subsequently
\begin{eqnarray*}&&
\nabla (\div v^{(1)})-\curl(\curl v^{(1)})\\&=&
\nabla(\partial_1 v_1)
-\curl W\\
&=&\left(\partial_{11} v_1,\,
\partial_{12} v_1,\,
\partial_{13} v_1
\right)
\left(
{\frac {\partial W_3}{\partial x_2}}-{\frac {\partial W_2}{\partial x_3}} ,\;-
{\frac {\partial W_3}{\partial x_1}},\;
{\frac {\partial W_2}{\partial x_1}}\right)\\
&=&\left(\partial_{11} v_1,\,
\partial_{12} v_1,\,
\partial_{13} v_1
\right)-
\left(
-\partial_{22} v_1-\partial_{33} v_1 ,\,
\partial_{12} v_1,\,\partial_{13} v_1\right)\\&=&
\left(\partial_{11} v_1+
\partial_{22} v_1+\partial_{33} v_1 ,\,0,\,0\right)\\&=&\Delta v^{(1)},
\end{eqnarray*}
thus establishing (<ref>).
Now, in the light of (<ref>) and (<ref>),
\begin{equation}\label{EFEDLFIemfdL}
\partial_{tt}E=c^2\partial_t(\curl B)=c^2\curl(\partial_t B)=-c^2\curl(\curl E)
=c^2\big(\Delta E-\nabla (\div E)\big)=c^2\Delta E
\end{equation}
and similarly
\begin{equation*}
\partial_{tt}B=-\partial_t(\curl E)=-\curl(\partial_t E)=-c^2\curl(\curl B)=
c^2\big(\Delta B-\nabla (\div B)\big)=c^2\Delta B.
\end{equation*}
These observations show that
both the electric and the magnetic fields propagate as waves do, since they satisfy
the wave equation in (<ref>).
§.§ Quantum-mechanical systems
A popular equation governing the wave function of a quantum-mechanical system
was proposed by Erwin Schrödinger
(this equation formed the basis for the work that resulted in
Schrödinger's 1933 Nobel Prize).
In a nutshell, the Schrödinger equation reads
\begin{equation}\label{SCHREQOR}
-i\hbar\partial_t\psi=\frac{\hbar^2 \Delta \psi}{2m}-V\psi,
\end{equation}
where $i=\sqrt{-1}$, $\psi=\psi(x,t)$ is the wavefunction[Roughly speaking, a wavefunction is
a function that assigns a complex number to each point $x\in\R^3$ at each time $t\in\R$.
The magnitude squared of this function
represents the probability density of measuring the particle as being at the point $x$ at a given time $t$.]
of a particle of mass $m$ subject to a potential $V$
and $\hbar$ is a suitable positive physical constant (called the “reduced Planck constant”reduced Planck constant).
Particularly interesting solutions of (<ref>)
are the so-called standing waves, namely solutions which oscillate in time but whose amplitude profile does not change in space.
From the mathematical point of view, these solutions are of the form
$$ \psi(x,t)=u(x)\,e^{i\phi t},$$
where $u$ is a real valued function and $\phi>0$ is the time frequency of oscillation.
Since for standing waves we have that
$$ \partial_t\psi(x,t)=i\phi\,u(x)\,e^{i\phi t}
\qquad{\mbox{and}}\qquad
\Delta \psi(x,t)=\Delta u(x)\,e^{i\phi t},$$
these special solutions of (<ref>) are actually solutions[According to the classification
presented in footnote <ref>
on page CLASSIFICATIONFOOTN, equation (<ref>)
is of elliptic type. When $V=V(x)$ equation (<ref>)
is linear in $u$. More complex situations arise when $V$ also depends on the magnitude of the wavefunction,
say $V=V(x,u(x))$ or even $V=V(u(x))$, since in this situation equation (<ref>)
is not anymore linear in $u$ (though it is linear with respect to the Hessian of $u$). A study of these “semilinear” equations semilinear equation
will be started on page Pohozaev Identity.
Further comments on semilinear equations will be given in footnote <ref>
on page TRIS Pohozaev Identity.]
\begin{equation}\label{CLASSIFICATIONFOOTNSV}
\hbar \phi u
=\frac{\hbar^2 \Delta u }{2m}-V u.
\end{equation}
Going back to the initial discussion of this section,
one may wonder why and how Schrödinger introduced the equation in (<ref>) above.
This would be a rather long and complicated story and a short answer
was given in <cit.>:
“Where did we get that from? Nowhere. It’s not possible to derive it from anything you know. It came out of the mind of Schrödinger, invented in his struggle to find an understanding of the experimental observations of the real world”.
However, several quite convincing motivations of the Schrödinger equationSchrödinger equation are possible (see e.g. <cit.> and the references therein)
and we present one of them here below, based on the notion of canonical quantizationcanonical quantization.
To this end, we recall that the energy ${\mathcal{E}}$ of a photon is proportional to its temporal frequency $\phi$, according to the so-called Planck-Einstein energy-frequency relation
$$ {\mathcal{E}}=\hbar\phi.$$
Also, Einstein's relativistic energy formula reads
\begin{equation}\label{ENEJHND8iujMNSID-77} {\mathcal{E}}=\sqrt{ (|p|\, c)^2 + (mc^2)^2},\end{equation}
being $p$ the relativistic momentum, $m$ the mass at rest and $c$ the speed of light. For massless particles, such as photons, this reduces to
\begin{equation}\label{ENEJHND8iujMNSID} {\mathcal{E}}= |p|\,c,\end{equation}
from which we obtain
\begin{equation}\label{PPho} |p|=\frac{\mathcal{E}}c=\frac{\hbar\phi}{c}.\end{equation}
Now, we relate a photon to its electric field $E$ traveling through space and manifesting itself as a solution of the wave equation (recall (<ref>)). That is, we consider a direction of propagation $\varpi\in\partial B_1$, a spatial frequency $\kappa$ and a temporal frequency $\phi$ and we suppose that,
for large distances from the photon's source, the field is modeled by a simple plane wave, say
$$ E=E_0 e^{i(\kappa\varpi\cdot x-\phi t)},$$
where $E_0\in(0,+\infty)$.
More precisely, by (<ref>),
$$ -\phi^2 E_0 e^{i(\kappa\varpi\cdot x-\phi t)}=
\partial_{tt} E=c^2\Delta E=-c^2 \kappa^2 E_0 e^{i(\kappa\varpi\cdot x-\phi t)}$$
\begin{equation}\label{PPho22536-0i2rkjfmMS}
\phi=c \kappa,\end{equation}
thus relating temporal and spatial frequencies of the photon.
We thus obtain from (<ref>)
\begin{equation}\label{PPho22536}
and therefore
\begin{equation}\label{PPho2}|p|E=\hbar\kappa E=-i\hbar\nabla E\cdot\varpi.\end{equation}
Taking the direction of the momentum to coincide with the spatial direction $\varpi$ of the wave, i.e. taking $p=|p|\varpi$,
we rewrite (<ref>) in the form
\begin{equation}\label{PPho3}pE=-i\hbar\nabla E.\end{equation}
We can also reconsider (<ref>) in view of (<ref>) and (<ref>) and write that
$${\mathcal{E}}= |p|\,c=\hbar c\kappa=\hbar\phi$$
and consequently
\begin{equation}\label{ENEJHND8iujMNSID-2krwjeg}
{\mathcal{E}}E=\hbar \phi E=i\hbar\partial_t E.
\end{equation}
With this, without the aim of being exhaustive, we can recall some of the main ideas leading to canonical quantization.
This procedure, originally introduced by Paul Dirac in his 1926 PhD thesis, wishes to recast a classical theory into a quantum
one by preserving (as much as possible) its symmetries
and formal structures. Since the Hamiltonian formalism is one of the key tools to understand symmetries of classical mechanical systems, a natural idea in this framework is to try to (at least partially) preserve the Hamiltonian structure in the quantum description of nature.
To this end, a simple observation is that the Hamiltonian formalism relies on conjugated variables $q$ and $p$, with the physical meaning of position and momentum. The first goal of the canonical quantization is therefore to rephrase position and momentum in a way that is compatible with the quantum description of the observables.
In particular, in quantum mechanics,
all significant features
of a particle are contained in a certain state $\psi$.
The observables are represented by operators acting on states. For instance, the eigenvalues of an operator represent the values
of the measurements of the corresponding fundamental states of a particle (namely, the eigenfunctions, or eigenstates);
since any state is represented as a linear combination of eigenstates, the application of an operator to the state $\psi$
corresponds to the determination of a measurable parameter and the physical act of measuring corresponds to make the
values of the state “collapse” from a superposition of eigenstates to a single eigenstate due to the
interaction with the external world.
With these basic principia in mind, we therefore aim at replacing the classical position and momentum variables $q$ and $p$ with two operators, say $Q$ and $P$, by preserving the original structure as much as possible.
The most natural setting is therefore to consider the position operator $Q$ as the operator that corresponds to the position evaluation of a state (simply, applying $Q$ to a state $\psi$ being the evaluation of the function $\psi$ at a given point in space).
As for the corresponding momentum operator, in light of (<ref>), a natural choice is to take $P$ as the differential operator $-i \hbar \nabla$, thus obtaining a quantum analogue of (<ref>) via the relation
\begin{equation}\label{ENEJHND8iujMNSID-2krwjeg-21-ktg22} P\psi=-i\hbar\nabla\psi.\end{equation}
Pushing this analogy a bit further, we can also obtain a quantum counterpart ${\mathcal{H}}$ of
the total energy ${\mathcal{E}}$ (say, the Hamiltonian) in (<ref>).
In this setting, the operator analogue of (<ref>) would then be
\begin{equation}\label{ENEJHND8iujMNSID-2krwjeg-21-ktg}
{\mathcal{H}}\psi=-i\hbar\partial_t \psi.
\end{equation}
While this canonical formalism for quantum mechanics was obtained by analyzing the special case of the wave produced by a photon,
we can believe that the same protocol governs the evolution of the wave function of a particle (not necessarily a photon)
subject to a given potential $V$. For instance, for a particle
of mass $m>0$ one can push the previous construction by considering the quantum analogue of the mechanical energy
(perhaps neglecting for the moment some relativistic effects,
which we will briefly discuss in the forthcoming footnote <ref>).
Indeed, in classical mechanics the total energy would be given by the sum of the kinetic energy
of a particle (equal to $\frac{m|v|^2}{2}$, being $v$ its velocity) and the potential energy $V$. That is,
recalling the classical momentum definition $p=mv$,
$$ {\mathcal{E}}=\frac{m|v|^2}{2}+V=\frac{|p|^2}{2m}+V.$$
By formally applying the quantization in (<ref>) and (<ref>), we thus obtain its quantum counterpart as
$$ i\hbar\partial_t={\mathcal{H}}=\frac{|P|^2}{2m}+V=
\frac{P\cdot P}{2m}+V=-
\frac{\hbar^2 \nabla\cdot\nabla}{2m}+V=-\frac{\hbar^2 \Delta}{2m}+V,
that is the Schrödinger equation[It is interesting to point out that variations
in the previous arguments lead to other equations of interest: for instance,
thinking over the full relativistic energy formula in (<ref>) and formally inserting the quantum mechanical operator in (<ref>), the quantum analogue of the kinetic energy becomes
$$ \sqrt{ -\hbar^2\, c^2\Delta + (mc^2)^2},$$
that is the “square root of the Laplacian”square root of the Laplacian (see e.g. <cit.>
for a basic introduction to this very interesting object). Thus,
in the presence of an external potential $V$, the conservation of the full energy and the quantization
in (<ref>) lead to the balance
$$-i\hbar\partial_t =\sqrt{ -\hbar^2\, c^2\Delta + (mc^2)^2}-V,$$
which produces the equation
$$ -i\hbar\partial_t \psi=\sqrt{ -\hbar^2\, c^2\Delta \psi+ (mc^2)^2}-V\psi.$$
This equation and variants of it are usually
called “relativistic Schrödinger equations”relativistic Schrödinger equation, see e.g. equation (1.4) in <cit.>.
Another approach for taking into account relativistic effects consists in taking the square of the
relativistic energy formula in (<ref>), thus writing that
$$ {\mathcal{E}}^2= (|p|\, c)^2 + (mc^2)^2.$$
In the absence of external potentials (i.e., if this represents the square of the total energy of the system), one can think about exploiting the quantization procedure in (<ref>)
and (<ref>), formally finding that
$$ -\hbar^2\partial_{tt}= -\hbar^2 c^2\Delta+ (mc^2)^2.$$
This leads to the equation
$$ \partial_{tt}\psi= c^2\Delta\psi-\frac{ m^2c^4\psi}{\hbar^2},$$
which is often referred to with the name of “Klein-Gordon equation”Klein-Gordon equation,
see e.g. equation (34) in <cit.>.] in (<ref>).
§.§ Who wants to be a millionaire?
One way to become a millionaire is to solve a Millennium Prize Problem (see pages 106106
and 106106-bisPE), but this is probably by far the most difficult way to get rich.
Also, one should take into consideration the possibility that mathematics may not make us rich,
but it makes us happy, which is inestimable.
James Harris Simons in 2007
(image from
licensed under the
Creative Commons Attribution-Share Alike 2.0 Germany license).
There are however truly outstanding mathematicians, such as Jim Simons (see Figure <ref>)
who, after having revolutionized the theory of minimal surfaces and the topological quantum field theory,
and also after having broken codes for the NSA
during the Cold War,
decided to improve his portfolio:
allegedly, Simons' net worth is estimated to be $25.2\times 10^9$ US $,
making him the 66th-richest person in the world, and likely the richest[Simons has not forgotten
his love for mathematics. Not only because he owns a motor yacht named Archimedes,
but also because he
co-founded the Simons Foundation,
supporting projects related to scientific research, education and health, and
he sponsored the Simons Institute for the Theory of Computing
and the Mathematical Sciences Research Institute at Berkeley.
Here is an interesting story by the way. While Simons was working for the Institute for Defense Analyses by cracking Russian codes, his boss General Maxwell Taylor wrote an article in the New York Times Magazine strongly supporting the Vietnam War. Then, Simons published a counter-editorial in the Times, claiming that not everyone who worked for Taylor subscribed to his views. This resulted in an interview of Simons from Newsweek. When Simons told his superiors about this interview, they fired him right away.
Allegedly, Simons said “Getting fired once can be a good experience. You just don't want to make a habit of it”.]
among the mathematicians.
His financial success is related to the foundation of
a quantitative hedge fund (see below for the notion of hedging)
which trades using quantitative mathematical models, so mathematics can be sometimes useful after all.
One of the chief uses of partial differential equations
in mathematical finance is related to the Black-Scholes equationBlack-Scholes equation
\begin{equation}\label{B:LA:CK}
\frac{\partial V}{\partial t}+{\frac {1}{2}}\sigma ^{2}S^{2}
\frac{\partial ^{2}V}{\partial S^2}=rV-rS\frac {\partial V}{\partial S},\end{equation}
named after Fischer Black and Myron Scholes.
The meaning of this equation is the following.
Suppose that we have a stockstock (that is, a security that represents the ownership of a fraction of a corporation)
whose price is denoted by $S$.
The price of a stock fluctuates based on supply and demand
(if more people want to buy than sell it, the price will rise) hence we take $S=S(t)$ to be a function of time.
To model the fluctuations of $S$ in the simplest possible way, we can make two assumptions.
On the one hand, we can assume that the stock refers to some company
with revenues, earnings, dividends, etc., and we can estimate the growth rate
of the company and of the corresponding stocks: for this, we assume that the variation of $S$
in the unit of time is proportional, by some factor $\mu$, to the value of $S$
(the more stocks one possesses, and the higher the coefficient $\mu$, the more one receives from the growth of the company's revenues; actually positive values of $\mu$ correspond to a growth
and negative ones to a degrowth).
On the other hand, the market presents a number of uncertainties which are almost
impossible to fully take into account, such as economic crises, political developments,
real estate bubbles, technological innovations, pandemics, landing of aliens from outer space, etc.
If we have no specific information on the matter, we can just assume that this volatility
is modeled by a Brownian motion (pretty much as the random walk presented on page RANDOW).
This is accounted for by a process $W(t)$ that “randomly wiggles up and down” and
that quantifies the source of uncertainty in the price. This term is modulated by a volatility coefficient $\sigma\in[0,+\infty)$ (the higher this coefficient, the bigger the impact of the uncertainties in the price oscillation).
Assuming that these two effects contribute somewhat consistently to the variation in time of the stock,
we therefore write that[Equation (<ref>)
is sometimes called the “geometric Brownian motion”geometric Brownian motion
equation. It is a very popular tool in quantitative finance since it relies
on the simple, but often reasonable, assumption that the relative variation $\frac{dS}S$
of a value follows a diffusive process.]
\begin{equation}\label{3456yuioiuhgfdr5678iokmnbvfdr5678ijhgfde45678ijhgfdety3hfe2}
dS=\mu S\,dt+\sigma S\, dW.\end{equation}
Strictly speaking, this is not really a differential equation,
since we cannot “divide both terms by $dt$”, since Brownian motions
are “not differentiable”: see e.g. <cit.>
for a precise statement and for an elegant introduction to stochastic calculus.
Indeed, (<ref>)
is a stochastic differential equationstochastic differential equation,
which is a type of equations that we do not treat here,
dealing with it just at an intuitive level. Basically, the only bit of information that
we need from stochastic calculus is the so called[Kiyosi Itô
was the pioneer of the stochastic calculus that brings his name.
See Figure <ref>
for a picture of Kiyosi Itô
at age 22 (Kiyosi is the second from the left; the first from the left is his brother
Seizô, who also became a mathematician).]
Itô's Chain RuleItô's Chain Rule (see e.g. Sections 1.3, 4.3, 4.4 and in particular the theorem on page 80 of <cit.>) according to which if $dX=F\,dt+G\,dW$,
then the random variable $H(X(t),t)$ satisfies
\begin{equation} \label{ITOPIOTNNRRRESIDEKKA}
dH(X,t)=\left( \partial_t H +F \partial_X H+\frac{G^2}{2} \partial_X^2 H\right)\,dt+G\partial_X H\,dW.\end{equation}
Let us now come back to our financial setting.
Suppose now that we perceive the buying of a given stock as too risky
and we prefer instead to buy only “the possibility of buying the stock
in the future”, if this turns out to be convenient.
This type of contract is called “option”option
and conveys its owner
the right, but not the obligation, to buy the stock at a specified strike price either
prior to or on a specified date.
For simplicity, let us consider the case in which the option allows, but not obliges, the holder
to buy the stock for a specified price $K$ (called “strike price”strike price
in jargon)
at a specific time $T$, that is on the date of option maturity:
these options are called in jargon “European options”European option;
the options which instead
can be exercised any time up to and including the date of expiration
are called “American options”American option
(perhaps the name originated
by the alleged fact that one type of option was commonly
traded in London and the other in New York).
The Itô family (Public Domain image from
We denote by $V$ the price of this option
and we suppose that $V$ depends on time and on the value $S$ of the underlying stock, that is $V=V(t,S)$, for $t$ less than the maturity time $T$.
For instance, at the maturity date the value of this option is either $S(T)-K$,
namely the difference between the actual value of the stock $S$ at time $T$
and the strike price $K$ that was agreed at the beginning, if $S(T)-K$ is positive
(in which case the option turned out to be a convenient investment),
or null, if $S(T)-K$ is negative (in this case the value of the stock is lower than the strike price,
so no need to use the option to pay more, if one is interested in the stock can just forget
about the option and buy the stock at the market prince). More specifically, these observations yield that[For American options, one also has that
$$ V(t, S(t))\ge\max\{S(t)-K,0\} $$
for all $ t\in[0,T]$,
the inequality coming from the fact that one can, but is not obliged to, utilize the option at time $t$
and this provides an “obstacle problem”obstacle problem
for equation (<ref>), since it forces solutions of (<ref>)
to stay above a constraint
(or to drop the equation when they meet the constraint;
this situation is significantly more complicated than that of a single equation, and this is the reason for which
we limited ourselves here to the case of European options).
With respect to (<ref>), we may consider it as a “terminal condition”terminal condition
for equation (<ref>).
Interestingly, with respect to the classification discussed in footnote <ref> on page CLASSIFICATIONFOOTN,
we have that equation (<ref>) is of parabolic type, but the sign in front of the time derivative
is opposite to the case of the heat equation (compare with (<ref>);
a closer relation
between the Black-Scholes equation and the heat equation will be discussed on page BSDAGB-ADkrVoiweLL4re2346ytmngrrUj72).
In this sense, equation (<ref>) shares some similarity with heat type equations,
but going backwards in time. This makes sense, since while the heat equation
smooths out differences as time flows, the value of an option becomes quite rigid close to its maturity
\begin{equation}\label{TERM01-2pwf} V(T,S)=\max\{S-K,0\}.\end{equation}
Our objective is thus to understand the time evolution of the value $V$ of the option:
this is important, since if one possesses an option, may decide to sell it at some time $t$,
knowing just the strike price $K$ agreed at the beginning and the present value of the stock $S(t)$,
hence it is relevant to establish a fair price $V(t,S(t))$ for the option
under this information (since one would like to answer this question for all possible values
of the stocks, this reduces to understand $V(t,S)$ just knowing the strike price $K$,
with this one would substitute $S=S(t)$ to obtain the fair price of the option).
To model the time evolution of $V$, and thus provide a convincing derivation of
the Black-Scholes equation in (<ref>), we
use a method called in finance “delta hedging”delta hedging,
namely we adopt a trading strategy that reduces, or “hedges”, the risk related to the volatility
of the process.
For this, we suppose to buy a quantity $\alpha=\alpha(t,S(t))$ of options and some quantity $\beta=\beta(t,S(t))$ of the stocks.
The goal is to choose $\alpha$ and $\beta $ to fully cover the risk
(or, at least, to cover the risk to the best of our possibilities). More precisely, we consider a portfolio $P:=\alpha V+\beta S$
and we compare this investment with another one that carries zero risk,
that is some operation which ensure a certain (or, more realistically, almost certain)
future return and (virtually) no possibility of loss: for instance a treasury bill, or treasury bond,
namely a bond in which the face value is repaid at the time of maturity with some interest.
We denote by $r\ge0$ the risk-free interest rate (this will be precisely the parameter $r$
appearing in (<ref>)). The value of this zero risk portfolio $P_0$ is thus
described by ${dP_0}=r P_0\,dt$.
Since, via a suitable choice of $\alpha$ and $\beta$, the portfolio $P$ is set up to have zero risk, it must be equivalent
to the portfolio $P_0$, otherwise there would be a better choice to gain money at zero risk! This says that
also $ dP=rP\,dt$ and therefore
\begin{equation}\label{93456yuioiuhgfdr5678iokmnbvfdr5678ijhgfde45678ijhgfdety3hfe3CP8}
\big(r\alpha V+r\beta S\big)\,dt=
rP\,dt =d\Big(\alpha V+\beta S \Big).\end{equation}
We stress that the portfolio $P$ is assumed to be self-financingself-financing portfolio,
namely there are no inflows or outflows of money. More specifically, at every instant of time,
the purchase of a new option or stock must be financed by the sale of an old one:
hence, since there is no external infusion or withdrawal of money,
the variation of the value of the portfolio corresponds precisely to the variation of the values
of the options and stocks possessed, that is
\begin{equation}\label{93456yuioiuhgfdr5678iokmnbvfdr5678ijhgfde45678ijhgfdety3hfe3CP89} dP=\alpha\, dV+\beta \,dS.\end{equation}
We stress that this does not mean necessarily that $\alpha$ and $\beta$ are constants,
just that the market value of the portfolio $P$ at a given time
equals the purchase value of the new portfolio.
Namely, if in the infinitesimal interval of time $dt$ the value of the options has increased respectively by $dV$
and the value of the stocks by $dS$, then the gain obtained would correspond to the number of options
possessed times the options' increment value (that is $\alpha$ times $dV$) plus the number of stocks
possessed times the stocks' increment value (that is $\beta$ times $dS$): this gives
that the total gain in the infinitesimal interval of time $dt$ is $\alpha\, dV+\beta \,dS$, which is precisely
the right hand side in (<ref>).
Accordingly, the balance prescribed by (<ref>)
states that this gain is used precisely to reinvest to enlarge the portfolio itself
(this, in the optimistic scenario that there is a gain; if there is a loss then the right hand side of (<ref>) is negative and
the portfolio needs to be correspondingly shrunk).
Now we let $\varpi$ be the ratio[In most of the literature,
this ratio is denoted by either $\Delta$ or $\delta$, whence the name
of delta hedging. Here, for typographical convenience, and to avoid confusion with
the Laplace operator and with increments and derivatives, we preferred to call it $\varpi$.] between $\beta$ and $\alpha$, namely we take $\varpi=\varpi(t,S(t))$ such that $\beta=\alpha\varpi$.
Then (<ref>)
can be stated in the form
\begin{equation*}
\big(r\alpha V+r\alpha\varpi S\big)\,dt=
\big(r\alpha V+r\beta S\big)\,dt=dP
\end{equation*}
and therefore (assuming for simplicity $\alpha\ne0$)
\begin{equation}\label{93456yuioiuhgfdr5678iokmnbvfdr5678ijhgfde45678ijhgfdety3hfe38}
\big(r V+r\varpi S\big)\,dt
\end{equation}
We also point out that
\begin{equation*}
dV=\left( \partial_t V +\mu S \partial_S V+\frac{\sigma^2 S^2}{2} \partial_S^2 V\right)\,dt+\sigma S \partial_S V\,dW
thanks to (<ref>).
From this, (<ref>) and (<ref>) we infer that
\begin{equation}\label{0olLur2345609iuygfadTbSan2sDMrote}
\begin{split}
\big(rV+r\varpi S\big)\,dt\\
\left( \partial_t V +\mu S\partial_S V+\frac{\sigma^2 S^2}{2} \partial_S^2 V\right)\,dt+\sigma S\partial_S V\,dW
\big(rV+r\varpi S\big)\,dt\\&=
\left( \partial_t V +\mu S\partial_S V+\frac{\sigma^2 S^2}{2} \partial_S^2 V
-rV-r\varpi S
\right)\,dt+\sigma S\partial_S V\,dW
+\varpi\,\big(\mu S\,dt+\sigma S\, dW\big)
\\&=
\left( \partial_t V +\mu S\big(\partial_S V+\varpi\big)+\frac{\sigma^2 S^2}{2} \partial_S^2 V
-rV-r\varpi S
\right)\,dt+ \sigma S\big(\partial_S V+\varpi \big)\,dW.\end{split}
\end{equation}
Consequently, since $\varpi$ is chosen to make the portfolio free from the risk caused by the stochastic term
this gives that one must choose $\varpi$ such that $\partial_SV+\varpi=0$
(note that this cancels in (<ref>)
the term involving $W$ and the linear term in $\sigma$). That is, we take
\begin{equation*}
\varpi=-{\partial_SV}
\end{equation*}
and we thus plug this information into (<ref>),
concluding that
\begin{equation*}
0=\left( \partial_t V +\frac{\sigma^2 S^2}{2} \partial_S^2 V
which leads to the the Black-Scholes equation in (<ref>), as desired.
It is also interesting to revisit the Black-Scholes equation in (<ref>)
in view of the financial model that it describes. We note in particular that the
right hand side of (<ref>) is described only in terms
of the risk-free interest rate $r$: this is therefore the risk-free part of the investment
and it consists of the superposition of a long term strategy dictated by the option $V$ (which will become effective only at its maturity time) and a short term one embodied by the stock $S$ (which fluctuates
in a market subject to randomness).
On the left hand side of (<ref>) instead we see two terms. The first is the time derivative
of the option. The second term on the left hand side of (<ref>)
is more “geometric” since it reflects the convexity properties of the dependence of the option
on the underlying stock.
It is also useful to relate the Black-Scholes equation in (<ref>) to
the classical heat equation. As already observed on page BSDAGB-ADkrVoiweLL4re2346ytmngrrUj7,
to make this connection one needs to revert the arrow of time, hence it is convenient to look at a new time $\tau:=T-t$ which transforms the terminal condition (<ref>) into an initial condition.
Furthermore, it comes in handy to introduce the nonlinear transformation $x=\ln \left({\frac {S}{K}}\right)+\left(r-{\frac {\sigma^{2}}{2}}\right)\tau$ in order to simplify the right hand side of (<ref>). More specifically, one sets
$$ u(x,\tau):=e^{r\tau} \,V\left(T-\tau,Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau}\right)$$
and observes that if $V$ solves the Black-Scholes equation in (<ref>) then,
using the notation $S:=Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau}$ and $t:=T-\tau$,
\begin{eqnarray*}&&
\frac{\sigma^2}2 \partial_{xx} u(x,\tau)-\partial_\tau u(x,\tau)\right]\\&=&
\frac{\sigma^2}2 \partial_{x}
\left[ Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau}\partial_S
-r V\left(T-\tau,Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau}\right)\\&&\qquad
+Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau} \left(r-\frac{\sigma^2}2\right)\partial_S
+\partial_t V\left(T-\tau,Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau}\right)
\\&=&
\frac{\sigma^2}2
\left[
\left(Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau}\right)^2\partial_{SS}
\right]\\&&\qquad
-r V\left(T-\tau,Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau}\right)
+Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau} \left(r-\frac{\sigma^2}2\right)\partial_S
+\partial_t V\left(T-\tau,Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau}\right)\\
\left[ S\partial_S
\right]-r V(t,S)+S \left(r-\frac{\sigma^2}2\right)\partial_S
V(t,S)+\partial_t V(t,S)\\
&=&\frac{\sigma^2 S^2}2\partial_{SS}V(t,S)-r V(t,S)+rS\partial_S
V(t,S)+\partial_t V(t,S)\\&=&0.
\end{eqnarray*}
$$ u(x,0)=V\left(T,Ke^{x}\right).$$
Consequently, the Black-Scholes equation in (<ref>) and the terminal condition (<ref>)
produce the following heat diffusion problem for time $\tau>0$ with initial datum:
\begin{equation}\label{CALOCA}
\begin{dcases}
&\partial_\tau u(x,\tau)=\displaystyle\frac{\sigma^2}2 \partial_{xx} u(x,\tau),\\
\end{dcases}
\end{equation}
where, as usual, we used the notation for the characteristic function of a set, namely
$$ \chi_A(x):=\begin{dcases}1 & {\mbox{ if }}x\in A,\\
0 &{\mbox{ otherwise.}}\end{dcases}$$
Plot of a solution of (<ref>)
(with $K:=1$ and $\sigma:=\sqrt2$).
The plot of a solution of (<ref>) is sketched in Figure <ref>.
The advantage of this formulation is that one can focus on the solution of the classical heat equation in (<ref>)
and deduce useful information on the solution of the Black-Scholes equation in (<ref>)
by transforming back to the original variables.
See e.g. <cit.> for additional readings on the Black-Scholes equation[Of course,
as every equation, the Black-Scholes equation
must be taken with a pinch of salt.
For instance, according to Ian Stewart's article appeared in the British newspaper
The Observer
the Black-Scholes equation was “one ingredient in a rich stew of financial irresponsibility, political ineptitude,
perverse incentives and lax regulation” that ultimately led to
the 2007 financial crisis.
“The equation itself wasn't the real problem. It was useful, it was precise, and its limitations were clearly stated [...]. The trouble was its potential for abuse”. In particular, we stress that the equation relies on the knowledge of the market volatility $\sigma$.
“This is a measure of how erratically its market value changes. The equation assumes that the asset's volatility remains the same for the lifetime of the option, which needs not be correct. Volatility can be estimated by statistical analysis of price movements but it can't be measured in a precise, foolproof way, and estimates may not match reality [...].
The equation also assumes that there are no transaction costs, no limits on short-selling and that money can always be lent and borrowed at a known, fixed, risk-free interest rate. Again, reality is often very different”.] and related models.
§.§ Diffusion of transition probabilities
In a stochastic model, one is often interested in the so-called transition probability,transition probability
namely the likelihood of transitioning from one state to another.
In particular, Markov processesMarkov process are stochastic models “without memory”, i.e.
in which predictions about future outcomes can be done based solely on the knowledge of its present state.
In this setting, if $P(x_0,t_0| x,t)$ denotes the probability of going from the point $x_0$ at time $t_0$
to the point $x$ at time $ t$ (say, with $t>t_0$),
we have that such a probability can be expressed as the superposition of all the
probabilities of going first to a point $y$ at some time $s\in(t_0,t)$
and from there reaching $x$ at time $t$: more explicitly, given $s\in (t_0,t)$, it holds that
\begin{equation}\label{CHAKOL}
P(x_0,t_0|x,t)=\int_{\R^n} P(x_0,t_0|y,s)\,
\end{equation}
This relation is often called
Chapman-Kolmogorov equationChapman-Kolmogorov equation,
after[Chapman was a mathematician and geophysicist. At age 16 he entered the University of Manchester with a scholarship (he was the last student selected) and graduated with an engineering degree, but his passion for mathematics drove him to study for one further year to take a mathematics degree.
Besides his eminent contributions to stochastic processes, Chapman contributed to the theory of geomagnetism, investigating the beautiful phenomenon of aurorae (see Figure <ref>) in relation to the Earth's magnetic field and the solar wind,
contributing to the understanding of the photochemical mechanisms that produce the ozone layer,
and predicting the existence of the magnetosphere (confirmed experimentally 30 years later).
Kolmogorov's scientific contributions are paramount
and comprise basically all fields of mathematics, including probability, topology, logic, mathematical physics, harmonic analysis, mathematical biology and numerical analysis. His mathematical talent appeared quite early in his life: at the age of five, he wrote his first mathematical paper, published in the journal of his school. The content of the paper was the formula
$$ \sum_{k=0}^{N-1} (2k+1)=N^2,$$
which is easy peasy for all of us, but not quite a piece of cake for a five-year-old boy (by the way, the little kid also became editor of the mathematical section of the school journal).
His former student and prominent mathematician Vladimir Igorevich Arnold used to place Kolmogorov in his top 5 mathematicians list (with Poincaré, Gauß, Euler and Newton, see Figure <ref>
for a picture of Kolmogorov preparing his talk during a conference in Tallin in 1973).
A mark on Kolmogorov's reputation is however produced by his involvement in the so-called "Luzin Affair" during the Great Purge in 1936
(see also footnote <ref> on page PURFO: on this occasion, Nikolai Nikolaevich Luzin, prominent mathematical analyst and point-set topologist and Kolmogorov's former doctoral advisor (and author of the classical Luzin's Theorem, see e.g. <cit.>) became a target of Stalin's regime. Luzin was accused of plagiarism, nepotism, of being an enemy of the Soviet people and a servant to “fascistoid science” (whatever it means), as confirmed by the fact that Luzin published some of his results in foreign journals.
In view of this allegation, Luzin lost his academic position (not too bad after all,
his advisor Dimitri Fyodorovich Egorov, author of Egorov's Theorem, see <cit.>, was arrested during a previous purge and died after a hunger strike initiated in prison).
Regrettably, Kolmogorov took part in the hearing at the Commission of the Academy of Sciences of the USSR, where the allegations against Luzin were formalized. The question of whether Kolmogorov was coerced by the police into testifying against Luzin, possibly
using as a threat an alledged long-lasting homosexual relationship,
remains a topic of speculation, see e.g. <cit.>. In any case, a sad story of envy, violence, indifference, discrimination and brutality.
Which is good to know, since
those who fail to learn from the mistakes done in the past
are doomed to repeat them.]
Sydney Chapman and Andrey Nikolaevich Kolmogorov.
Collection of pictures of aurorae from around the world (photos by
Mila Zinkova, Samuel Blanc, Joshua Strang, Varjisakka and Jerry Magnum Porsbjer;
images from
licensed under the Creative Commons Attribution-Share Alike 1.0 Generic license).
It is clearly desirable, given an initial point $x_0$ at an initial time $t_0$,
to know more about the likelihood of being somewhere else at some future time, that is to have more information on
the function $p(x,t):=P(x_0,t_0|x,t)$.
Getting ready for a big presentation (photo by Terrence L. Fine; image from
licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license).
This is obtained
through a partial differential equation of the type[Equation (<ref>)
is usually called Fokker-Planck equationFokker-Planck equation.
The name comes after Adriaan Daniël Fokker and Max Karl Ernst Ludwig Planck.
Fokker was a physicist and a musician, inventor of
a 31-tone equal-tempered organ
(we recall that
the 12-tone equal temperament
is the most common musical system today, recall the musical digression on page JS-PGAGPA).
A picture of Fokker playing his organ is available at https://upload.wikimedia.org/wikipedia/en/a/ae/AdriaanFokker.jpg
Planck is the famous originator of quantum theory: fortunately, he did not follow the advice of
his professor Philipp von Jolly at the
Ludwig-Maximilians-Universität München, according to whom it was not worth to go into physics
since “in this field, almost everything is already discovered, and all that remains is to fill a few holes”.
The Planck family lived for many years in a villa in Berlin-Grunewald, located in Wangenheimstraße 21,
who became a gathering place for other professors, such as Albert Einstein and theologian Adolf von Harnack
(brother of the mathematician portrayed on Figure <ref>
on page HARAXELHARLROAGIJ7soloDItangeFIPAGI).
Planck family suffered hard times due to the deplorable political situation around.
At the onset of World War I, Planck was one of the signatories of the so-called
Manifesto of the Ninety-Three
stating an unequivocal support of German military actions; among the other ninety-two prominent signatories,
Fritz Haber, Adolf von Harnack, Philipp Lenard (Nobel Prize for cathode rays research,
future supporter of the Nazi ideology and future crusader against Einstein's relativity theory)
and Felix Klein (the outstanding mathematician who devise the “the Klein bottle”). During World War I,
Planck's oldest son Karl was killed in action at the Battle of Verdun
and his second son Erwin was taken prisoner by the French.
When the Nazis came to power, Planck was 74 and, when asked
to gather distinguished professors to issue a public proclamation against the expulsion of Jewish academics,
Planck replied, “If you are able to gather today 30 such gentlemen, then tomorrow 150 others will come and speak against it, because they are eager to take over the positions of the others”.
However, Planck tried to avoid the expulsion of Fritz Haber
(who had pioneered chemical warfare
by introducing the use of poisonous gases during World War I and
personally overseeing the first use of chlorine gas during the Second Battle of Ypres,
and who had received the Nobel Prize in Chemistry in 1918 for the invention of a process to synthesize ammonia from nitrogen and hydrogen gases, with broad application in the synthesis of fertilizers and explosives).
Planck's support in this circumstance was unsuccessful
and Haber died in exile the following year.
Planck's son Erwin was later arrested by the Gestapo following the attempted assassination of Hitler on 20 July 1944,
sentenced to death and executed by hanging.
On a positive note, see Figure <ref>
for a social dinner
among Nobel Laureates:
from left to right: Walther Nernst (1920 Nobel Prize for Chemistry), Albert Einstein
(1921 Nobel Prize for Physics), Max Planck (1918 Nobel Prize for Physics),
Robert A. Millikan (1923 Nobel Prize for Physics)
and Max von Laue (1914 Nobel Prize for Physics).]
\begin{equation}\label{FOKPLA}
\partial_t p(x,t)\,dx=-\div\big(\alpha(x,t)\,p(x,t)\big)+
\sum_{i,j=1}^n \partial_{ij}\big(\beta_{ij}(x,t)\,p(x,t)\big)
where $\alpha$ and $\beta_{ij}$ are called drift vector and diffusion tensor, respectively.
Nobel Laureates only
(Public Domain image from
We now give a motivation for equation (<ref>).
Let $\varphi$ be a smooth and compactly supported function in $\R^n$.
Using the Chapman-Kolmogorov equation in (<ref>), we know that, given $s\in (t_0,t)$,
\begin{eqnarray*}&&
\int_{\R^n}\varphi(x)\,p(x,t)\,dx=
\int_{\R^n}\varphi(x)\,P(x_0,t_0|x,t)\,dx\\&&\qquad=\iint_{\R^n\times\R^n} P(x_0,t_0|y,s)\,
\end{eqnarray*}
and, as a result, differentiating in $t$,
\begin{equation}\label{frasuetartialarphi}
\int_{\R^n}\varphi(x)\,\partial_t p(x,t)\,dx=\iint_{\R^n\times\R^n } P(x_0,t_0|y,s)\,
\partial_t P(y,s|x,t)\,\varphi(x)\,dx\,dy
It is also useful to recall that the total probability is normalized to $1$, namely
\begin{equation*} \int_{\R^n} P(y,s|x,t)\,dx=1,\end{equation*}
whence, taking a derivative in $t$,
\begin{equation}\label{BASDBAefTTrotlk4453} \int_{\R^n} \partial_t P(y,s|x,t)\,dx=0.\end{equation}
Given $y\in\R^n$, we also use a Taylor expansion to write that, when $|x-y|$ is small,
$$ \varphi(x)=\varphi(y)+\nabla\varphi(y)\cdot(x-y)+\frac12 D^2\varphi(y)(x-y)\cdot(x-y)+O(|x-y|^3).$$
This and (<ref>) give that
\begin{equation}\label{poe0586uyrhqwertyuiolkjhgfdszxcvbnm0987654}
\begin{split}
&\int_{\R^n} \partial_t P(y,s|x,t)\,\varphi(x)\,dx\\
=&\int_{\R^n} \partial_t P(y,s|x,t)\,\left(\varphi(y)+\nabla\varphi(y)\cdot(x-y)+\frac12 D^2\varphi(y)(x-y)\cdot(x-y)+O(|x-y|^3)
\right)\,dx\\
=&\int_{\R^n} \partial_t P(y,s|x,t)\,\left(\nabla\varphi(y)\cdot(x-y)+\frac12 D^2\varphi(y)(x-y)\cdot(x-y)+O(|x-y|^3)
\right)\,dx\\
=&\,\alpha(y,s,t)\cdot\nabla\varphi(y)+\frac12 \sum_{i,j=1}^n\beta_{ij}(y,t)\partial_{ij}\varphi(y)+
\int_{\R^n} \partial_t P(y,s|x,t)\,O(|x-y|^3)\,dx,
\end{split}\end{equation}
where[To be precise, we note that
the latter integral in (<ref>)
occurs only in the support of $\varphi$, which is a bounded set.]
\begin{eqnarray*}
&& \alpha(y,s,t):=
\int_{\R^n} \partial_t P(y,s|x,t)\,(x-y)\,dx\\
{\mbox{and }}&&
\beta_{ij}(y,s,t):=\frac12\int_{\R^n} \partial_t P(y,s|x,t)\,(x_i-y_i)(x_j-y_j)\,dx.
\end{eqnarray*}
Combining this and (<ref>), we find that
\begin{equation}\label{NEHDF-HSJDJI2}
\begin{split}&
\int_{\R^n}\varphi(x)\,\partial_t p(x,t)\,dx\\
\Bigg[
\alpha(y,s,t)\cdot\nabla\varphi(y)+ \sum_{i,j=1}^n\beta_{ij}(y,s,t)\partial_{ij}\varphi(y)\\&\qquad\qquad+
\int_{\R^n} \partial_t P(y,s|x,t)\,O(|x-y|^3)\,dx\Bigg]\,p(y,s)\,dy \\
\int_{\R^n} \alpha(y,s,t)\cdot\nabla\varphi(y)\,p(y,s)\,dy\\&\qquad\qquad+\int_{\R^n}
\sum_{i,j=1}^n\beta_{ij}(y,s,t)\partial_{ij}\varphi(y)\,p(y,s)\,dy +{\mathcal{R}}(s,t),\end{split}\end{equation}
\begin{eqnarray*}{\mathcal{R}}(s,t):=
\iint_{\R^n\times\R^n} \partial_t P(y,s|x,t)\,O(|x-y|^3)\,dx\,dy.
\end{eqnarray*}
We suppose additionally that the stochastic process is homogeneous in time, namely
the transition probability depends only on the points $x_0$ and $x$ and on the elapsed time $t-t_0$, that is
$$ P(x_0,t_0+\tau|x,t+\tau)=P(x_0,t_0|x,t)$$
for all $\tau\in\R$.
Differentiating in $\tau$, this gives that
$$ \partial_{t_0} P(x_0,t_0|x,t)+\partial_{t} P(x_0,t_0|x,t)=0.$$
As a consequence, observing that $
when $x\ne y$ (unless one is provided with the “gift of ubiquity”),
\begin{eqnarray*}
\alpha(y,t)&:=&\lim_{\e\searrow0}
\frac1\e\int_{t-\e}^t\alpha(y,s,t)\,ds\\&=&\lim_{\e\searrow0}
\frac1\e
\iint_{\R^n\times(t-\e,t)} \partial_t P(y,s|x,t)\,(x-y)\,dx\,ds\\&
\frac1\e\iint_{\R^n\times(t-\e,t)} \partial_s P(y,s|x,t)\,(x-y)\,dx\,ds\\&
\frac1\e\int_{\R^n}\Big( P(y,t-\e|x,t)-P(y,t|x,t)\Big)
\,(x-y)\,dx\\&
\frac1\e\int_{\R^n}P(y,t-\e|x,t)\,(x-y)\,dx
\\&=&\lim_{\e\searrow0}
\int_{\R^n}\frac{P(y,0|x,\e)\,(x-y)}\e\,dx.
\end{eqnarray*}
\[ \beta_{ij}(y,t):=\lim_{\e\searrow0}
\frac1\e\int_{t-\e}^t\beta_{ij}(y,s,t)\,ds=\lim_{\e\searrow0}
\frac12\int_{\R^n} \frac{ P(y,0|x,\e)}{\e}\,(x_i-y_i)(x_j-y_j)\,dx\]
\begin{equation}\label{NEHDF-HSJDJI}
\frac1\e\int_{t-\e}^t{\mathcal{R}}(s,t)\,ds=\lim_{\e\searrow0}
\iint_{\R^n\times\R^n} \frac{ P(y,0|x,\e)}{\e}\,O(|x-y|^3)\,dx\,dy.
\end{equation}
Formally, we also have that
\begin{eqnarray*}&&
\lim_{\e\searrow0}\frac1\e
\iint_{\R^n\times(t-\e,t)} \alpha(y,s,t)\cdot\nabla\varphi(y)\,p(y,s)\,dy\,ds\\&=&
\lim_{\e\searrow0}\frac1\e\left(
\iint_{\R^n\times(t-\e,t)} \alpha(y,s,t)\cdot\nabla\varphi(y)\,p(y,t)\,dy\,ds\right.\\&&\qquad\quad\left.+
\iint_{\R^n\times(t-\e,t)} \alpha(y,s,t)\cdot\nabla\varphi(y)\,\big(p(y,s)-p(y,t)\big)\,dy\,ds\right)\\&=&
\int_{\R^n} \alpha(y,t)\cdot\nabla\varphi(y)\,p(y,t)\,dy+
\lim_{\e\searrow0}\frac1\e
\iint_{\R^n\times(t-\e,t)} \alpha(y,s,t)\cdot\nabla\varphi(y)\,\big(p(y,s)-p(y,t)\big)\,dy\,ds
\\&=&
\int_{\R^n} \alpha(y,t)\cdot\nabla\varphi(y)\,p(y,t)\,dy+
\lim_{\e\searrow0}\frac1\e
\iint_{\R^n\times(t-\e,t)} |\nabla\varphi(y)|\,O(\e)\,dy\,ds\\&=&
\int_{\R^n} \alpha(y,t)\cdot\nabla\varphi(y)\,p(y,t)\,dy
\end{eqnarray*}
and, in a similar way,
\begin{eqnarray*}
\lim_{\e\searrow0}\frac1\e
\iint_{\R^n\times(t-\e,t)} \beta_{ij}(y,s,t)\partial_{ij}\varphi(y)\,p(y,s)\,dy
=\int_{\R^n} \beta_{ij}(y,t)\partial_{ij}\varphi(y)\,p(y,t)\,dy.\end{eqnarray*}
Therefore, integrating (<ref>) in $s\in(t-\e,\e)$, dividing by $\e$ and sending $\e\searrow0$,
we see that
\begin{equation}\label{NEHDF-HSJDJI2COSetyt6}
\begin{split}&
\int_{\R^n}\varphi(x)\,\partial_t p(x,t)\,dx
\int_{\R^n} \alpha(y,t)\cdot\nabla\varphi(y)\,p(y,t)\,dy\\&\qquad\qquad\qquad+\int_{\R^n}
\sum_{i,j=1}^n\beta_{ij}(y,t)\partial_{ij}\varphi(y)\,p(y,t)\,dy +{\mathcal{R}}(t).\end{split}\end{equation}
Leonard Ornstein and George Uhlenbeck (Public Domain images from
Now, for a sufficiently small time $\e$, if the stochastic process is continuous
we expect the probability of going from $y$ to $x$ in the elapsed time $\e$
to be “rather small” except when $|x-y|$ is small. It is therefore customary to consider the quantity ${\mathcal{R}}(t)$ in (<ref>) as a “negligible term”
(one may also want to recall the observation in footnote <ref>
to get rid of the contribution in ${\mathcal{R}}(t)$ coming from infinity).
Hence, taking the liberty of formally dropping it from the equation, we reduce (<ref>) to
\begin{equation*}\begin{split}&
\int_{\R^n}\varphi(x)\,\partial_t p(x,t)\,dx=\int_{\R^n} \alpha(y,t)\cdot\nabla\varphi(y)\,p(y,t)\,dy\\&\qquad\qquad+ \int_{\R^n}
\sum_{i,j=1}^n\beta_{ij}(y,t)\partial_{ij}\varphi(y)\,p(y,t)\,dy.\end{split}\end{equation*}
Integrating by parts (and renaming $y$ into $x$ in the variable of integration) we obtain
\begin{equation*}\begin{split}&
\int_{\R^n}\varphi(x)\,\partial_t p(x,t)\,dx\\&\qquad=-\int_{\R^n} \varphi(x)\,\div\big(\alpha(x,t)\,p(x,t)\big)\,\,dx+\int_{\R^n}
\sum_{i,j=1}^n \varphi(x)\,\partial_{ij}\big(\beta_{ij}(x,t)\,p(x,t)\big)\,dx,\end{split}\end{equation*}
which, by the arbitrariness of $\varphi$, produces the
Fokker-Planck equation in (<ref>), as desired.
Leonard Ornstein mural (image by Hansmuller from
licensed under the Creative Commons Attribution-Share Alike 4.0 International license).
Another possible derivation of the Fokker-Planck equation in (<ref>) comes from
stochastic differential equations (we present this argument in dimension $1$ for the sake of simplicity, but the same
ideas would carry over in higher dimensions too). Namely, if $dX=F\,dt+G\,dW$
we can use Itô's Chain Rule in (<ref>)
to every random variable $H(X(t),t)$, integrate and find that
\begin{equation*}
\int_0^T \,dH=
\int_0^T \left[
\left( \partial_t H +F \partial_X H+\frac{G^2}{2} \partial_X^2 H\right)\,dt+G\partial_X H \,dW\right].\end{equation*}
In particular, if $H=H(X(t))$,
\begin{equation}\label{WIENE-01}
\int_0^T \left[
\left(F \partial_X H+\frac{G^2}{2} \partial_X^2 H\right)\,dt+G\partial_X H \,dW\right].\end{equation}
Now we take the expected value ${\mathbb{E}}$ of this identity (say, at a given time $T$). For this, it is useful to recall that,
for a “reasonable” $\phi$, we have that
\begin{equation}\label{WIENE-02} {\mathbb{E}}\left(\int_0^T \phi\,dW\right)=0,\end{equation}
see e.g. <cit.>
(the bottom line of this formula being that $W$ presents independent increments hence the expectation of a function $\phi$
that is adapted to the stochastic process will vanish).
Thus, it follows from (<ref>) and (<ref>) that
\begin{equation}\label{WIENE-05} {\mathbb{E}}\Big(
\int_0^T
\left( F \partial_X H+\frac{G^2}{2} \partial_X^2 H\right)\,dt\right).\end{equation}
Now we recall the definition of expected value of a random variable $Y$ (see e.g. <cit.>), according to which
$$ {\mathbb{E}}(Y)=\int_{\R^n} Y(x) \,p(x,T)\,dx.$$
In this way, equation (<ref>) becomes
\begin{equation*}\begin{split}& \int_{\R^n} \Big(
\iint_{\R^n\times(0,T)}
\left( F(x,t) \partial_x H(x)+\frac{G^2(x,t)}{2} \partial_x^2 H(x)\right)\,p(x,T)\,dx\,dt.\end{split}\end{equation*}
We now write this identity for $T+\e$, subtract the one for $T$, divide by $\e$
and formally send $\e\searrow0$:
namely, recalling (<ref>),
\begin{eqnarray*}&&
\int_{\R^n} H(x)\,\partial_t p(x,T)\,dx\\&=&
\int_{\R^n} \Big(
H(x)-H(X(0))\Big)\,\partial_t p(x,T)\,dx\\
\int_{\R^n} \Big(
\lim_{\e\searrow0}\frac1\e
\left[\int_{\R^n} \Big(
\int_{\R^n} \Big(
\Bigg[\iint_{\R^n\times(0,T+\e)}
\left( F(x,t) \partial_x H(x)+\frac{G^2(x,t)}{2} \partial_x^2 H(x)\right)\,p(x,T+\e)\,dx\,dt\\&&\qquad
\left( F(x,t) \partial_x H(x)+\frac{G^2(x,t)}{2} \partial_x^2 H(x)\right)\,p(x,T)\,dx\,dt\Bigg]\\&=&\lim_{\e\searrow0}
\Bigg[\iint_{\R^n\times(0,T+\e)}
\left( F(x,t) \partial_x H(x)+\frac{G^2(x,t)}{2} \partial_x^2 H(x)\right)\,\frac{p(x,T+\e)-p(x,T)}\e\,dx\,dt\\&&\qquad
\left( F(x,t) \partial_x H(x)+\frac{G^2(x,t)}{2} \partial_x^2 H(x)\right)\,p(x,T)\,dx\,dt\Bigg]\\&=&
\left( F(x,T) \partial_x H(x)+\frac{G^2(x,T)}{2} \partial_x^2 H(x)\right)\,p(x,T)\,dx,
\end{eqnarray*}
$$ {\mathcal{S}}(T):=\iint_{\R^n\times(0,T)}
\left( F(x,t) \partial_x H(x)+\frac{G^2(x,t)}{2} \partial_x^2 H(x)\right)\,\partial_t p(x,T)\,dx\,dt.$$
The Ornstein-Uhlenbeck process, detail from the Leonard Ornstein mural (image by Hansmuller from
licensed under the Creative Commons Attribution-Share Alike 4.0 International license).
Since the term ${\mathcal{S}}(T)$ is small when $T$ is small, if we neglect it from the previous
computation we reduce the problem to
\[ \int_{\R^n} H(x)\,\partial_t p(x,T)\,dx=
\int_{\R^n}
\left( F(x,T) \partial_x H(x)+\frac{G^2(x,T)}{2} \partial_x^2 H(x)\right)\,p(x,T)\,dx.\]
Integrating by parts in $x$, we thus conclude that
\[ \int_{\R^n} H(x)\,\partial_t p(x,T)\,dx=
\int_{\R^n}
\left( -\partial_x\Big(F(x,T) \,p(x,T)\Big)+\frac12\partial_x^2\Big(
{G^2(x,T)}\, p(x,T)\Big)\right)\,H(x)\,dx.\]
From the arbitrariness of $H$
we thus obtain
\begin{equation}\label{KMS76ygiTSg12rZasdG} \partial_t p=-\partial_x \big(F p\big)+\frac12\partial_x^2\big({G^2} p\big),\end{equation}
which is the Fokker-Planck equation (<ref>) in this setting:
in the expression (<ref>) one can directly related the drift coefficient of the Fokker-Planck equation
with the “deterministic” term of the stochastic equation and the diffusion coefficient of the Fokker-Planck equation
with the “random” term of the stochastic equation.
A classical application for the Fokker-Planck equation in (<ref>) is given by the
so-called[The process in (<ref>) is named after
Leonard Salomon Ornstein and George Eugene Uhlenbeck, see Figure <ref>.
Compare also the photo of Ornstein in his lab with
Figure <ref>, showing a very elegant
mural in Utrecht, The Netherlands, depicting Ornstein,
the random walk (embodied by a drunkard, check the bottle of
booze in his left hand) and the Ornstein–Uhlenbeck
process (note the formula next to Ornstein's head,
as enlarged in Figure <ref>).
The mural has been painted by the Dutch painting collective
De Strakke Hand, see Figure <ref>.
See also http://www.destrakkehand.nl/ for other beautiful pieces of art in the urban landscape.
Sometimes, models similar to (<ref>) are also referred to with the name
of Langevin equation, Langevin equation
after Paul Langevin.] Ornstein-UhlenbeckOrnstein-Uhlenbeck process stochastic differential equation.
In this case, one considers in the background a stochastic differential equation of the form
\begin{equation}\label{ORST-ULHL-S-04} dX=-\vartheta \,X\,dt+\sigma \,dW,
\end{equation}
for suitable positive constant parameters $\vartheta$ and $\sigma$.
De Strakke Hand Team (image from De Strakke Hand website
To develop an intuition of the process in (<ref>), we can imagine an
elastic spring in the presence of a large dumping and thermal fluctuations. In a nutshell, Hooke's Law would prescribe
an equation of motion of the type $\ddot X=-\kappa X$, with $\kappa>0$ being the elastic constant
of the spring. In presence of friction, the previous equation becomes $\ddot X=-\kappa X-\gamma \dot X$,
being $\gamma$ the friction coefficient. Rewriting this equation in the form $\frac{\ddot X}{\gamma}=-\vartheta X-\dot X$,
with $\vartheta:=\kappa/\gamma$, for large values of elastic and dumping coefficients we can reduce ourselves
to $0=-\vartheta X-\dot X$, that we can formally write as $dX=-\vartheta X\,dt$. In this sense,
the Ornstein-Uhlenbeck equation in (<ref>) consists simply in adding to this model a random fluctuation
(e.g. due to thermal deviations)
and therefore it can be considered as describing a noisy relaxation process of an oscillator.
Notice that the setting in (<ref>) corresponds to that in (<ref>)
with the choices $F:=-\vartheta \,X$ and $G:=\sigma$, whence the corresponding
Fokker-Planck equation for the Ornstein-Uhlenbeck process takes the form
\begin{equation}\label{ORST-ULHL-S-05} \partial_t p=\vartheta\partial_x \big(x p\big)+\frac{\sigma^2}2\partial_x^2 p.\end{equation}
It is interesting to observe that the drift term modulated by $\vartheta$ in (<ref>)
corresponds to an attraction towards the center $x=0$ for the transition probability $p$.
To convince ourselves of this fact one can just compare (<ref>)
with (<ref>), which was describing the evolution of a biological population following a chemical attractant:
notice that $w$ in (<ref>) would correspond, up to constants, to $-\vartheta x^2$ in (<ref>),
hence $p$ is “attracted” towards the higher values of $-\vartheta x^2$ (namely, $x=0$) in the same way
as a biological population is attracted towards the higher density regions of the chemotactic factor.
Plotting the evolution of the transition probability for the Ornstein-Uhlenbeck process
according to the Fokker-Planck equation in (<ref>).
A plot of the solution $p$ of (<ref>), for a suitable choice of the parameters $\vartheta$ and $\sigma$, is sketched in Figure <ref>:
the initial datum at $t=0$ corresponds to the probability of finding the random particle located at $x=1$
(and for this reason the picture exhibits a blowup at $(x,t)=(1,0)$) and we can appreciate that, as time flows,
the solution has the tendency to move its mass towards $x=0$.
Evolution of the transition probability for the Ornstein-Uhlenbeck process
at subsequent times.
See also Figure <ref> for
frames of this solution at times $t=0.1$, $1$, $2$, $10$.
The situation in which the general tendency of reaching an objective (such as the origin
in Figures <ref>
and <ref>)
is weighted against the possibility of diffusing around and missing it
is actually quite common in our everyday experience, hence
the plots in Figures <ref>
and <ref>
should be somewhat close to our intuition. One can think for instance
to the case of rogaining, in which, in principle compass bearings point you straight to
the next control, but in reality one has to account for some fluctuation around the goal
(due to some miscalculations of angles and distances for which the teammate has to be blamed, see e.g. Figure <ref>,
to the roughness of the terrain, to possible modifications of the landscape, to possible inaccuracies of the map, etc.): accordingly, the evolution of an initial position is realistically not given
by a single point (that is, a Dirac Delta Function located at a moving point)
but rather by a probability distribution describing how likely one is exactly at that point.
In this scenario, the origin in Figures <ref>
and <ref> represents the ideal destination corresponding to the maximal value of the probability distribution, but the tails of this distribution represent the possibility of having missed the control during navigation
due to whatever fluctuation (at that point, the rogainer will have to seek the control patiently
and accurately; knowing the probability density would be perhaps a good indication on how far one should search).
For additional information about Brownian motion,
random disturbances and the Fokker-Planck equation,
see <cit.> and the references therein.
The difference between planning routes on a map and getting there.
§.§ Phase coexistence models
A topical problem in material sciences is the understanding of the separation
patterns between different phases of a given substance.
The two phases could be related to molecule orientation, magnetization,
multi-component alloy systems, state of the matter (including superconductive
or superfluid states). In this context, the description of a physical system often relies on the understanding of suitable “order parameters”, that are suitable quantities capable of distinguishing different phases of a given system.
For instance, the magnetization vector $M$ expresses the density of magnetic dipole moment in a magnetic material. Roughly speaking, at each point $x$ of a given magnetic material, the direction of $M(x)$ reproduces the direction of the magnetic field near $x$ and the magnitude $|M(x)|$ represents the amount of magnetization.
Another classical example of order parameter arises in nematic crystals. nematic crystalThese materials consist of long and thin molecules that prefer to align with one another. In this situation, the two ends of the elongated molecules are essentially indistinguishable, therefore an efficient order parameter for these crystals is given by a vector $N(x)$ modulo a sign (that is, an element of the projective plane, or equivalently a vector on a hemisphere with opposing points along the equator identified). That is, while magnetic materials tend to displace the magnetic vectors in a parallel fashion, the nematic vectors can be equally displayed either in a parallel or antiparallel form (the technological advantage is thus that these crystals can be aligned by an external magnetic or electric field, as it happens for instance in the liquid crystal displays for video games).
While the above mentioned $M$ and $N$ are vectorial order parameters,
there are also ideal cases in which a scalar order parameter is sufficiently efficient to
describe some features of a system. For example, nematic crystals
may exhibit at some sufficiently large scale a “preferred direction” along which most
of its elements are aligned, which is called in jargon “local nematic director”. local nematic director In this context, scalar state parameters which are often adopted for practical purposes
are the functions of the angle $\vartheta$ between the liquid-crystal molecular axis and the local nematic director.
The simplest of this function may be constructed as follows. We take, up to a rotation, the
local nematic director to be the first vector $e_1$ of the Euclidean basis
and we consider a state parameter $S(\vartheta)$ such that:
(i). $S$ is $2\pi$-periodic (consistently with the fact that $\vartheta$ is an angle),
(ii). $S(\vartheta)=S(\vartheta+\pi)$ (consistently with the fact that
the two endings of the nematic crystals are indistinguishable),
(iii). $S(\vartheta)=S(-\vartheta)$ (consistently with the fact that this
order parameter should account for the “angular distance” of a molecule to the local nematic director),
(iv). $S(\vartheta)\le 1=S(0)$ (normalizing the maximal of the state parameter with $1$,
corresponding to the full alignment case),
(v). $\displaystyle\fint_{\partial B_1} S\big(\arccos(\omega\cdot e_1)\big)\,d{\mathcal{H}}^{n-1}_\omega=0$
(normalizing at $0$ the case of random directions, using the notation $\omega\cdot e_1=\cos\vartheta$),
see Figure <ref>.
A scalar order parameter using the local nematic director.
By (i), one can seek $S$ in the form of a Fourier Series, say
$$ S(\vartheta)=\frac{a_0}2+\sum_{j=1}^{+\infty} \left(a_j\cos(j\vartheta)+b_j\sin(j\vartheta)\right).$$
By (iii), we have that $b_j=0$ for all $j\in\{1,2,\dots\}$. Hence, using (ii),
$$ 0=S(\vartheta+\pi)-S(\vartheta)
=\sum_{j=1}^{+\infty} a_j\left( \cos(j(\vartheta+\pi))-\cos(j\vartheta)\right)
=-2\sum_{{j\ge1}\atop{j{\tiny\mbox{ odd}}}} a_j \cos(j\vartheta)
and therefore $a_j=0$ for all $j$ odd.
Thus, neglecting higher orders, one can take
$$ S(\vartheta)=\frac{a_0}2+a_2\cos(2\vartheta).$$
By (iv), we have that $\frac{a_0}2=1-a_2$, whence, in view of (v),
\begin{eqnarray*}&&
0=\fint_{\partial B_1} S(\arccos(\omega\cdot e_1))\,d{\mathcal{H}}^{n-1}_\omega
=1-a_2+a_2\fint_{\partial B_1} \cos(2\arccos\omega_1)\,d{\mathcal{H}}^{n-1}_\omega\\&&\qquad\qquad=
1-a_2+a_2\fint_{\partial B_1} \big(2\cos^2(\arccos\omega_1)-1\big)\,d{\mathcal{H}}^{n-1}_\omega\\&&\qquad\qquad
=1-2a_2+2a_2\fint_{\partial B_1} \omega_1^2\,d{\mathcal{H}}^{n-1}_\omega\\&&\qquad\qquad
\end{eqnarray*}
which, in the concrete case $n=3$, yields that $a_2=3/4$.
As a result,
$$ S(\vartheta)=\frac14+\frac34\cos(2\vartheta)=\frac12\big(3\cos^2\vartheta-1\big).$$
Interestingly, we will discover that this coincides with the
second Legendre polynomialLegendre polynomial
(see footnote <ref> on page LISTACON).
Coming back to the study of phase transitions,
Landau Landau theory
theory[This theory is named after Lev Davidovich Landau,
1962 Nobel Prize in Physics for his mathematical theory of superfluidity.
In 1938, Landau wrote a leaflet, joint with Mosey Korets, condemning
Stalin and the People's Commissariat for Internal Affairs (the forerunner of
MVD and KGB) on the occasion of the Great Purge in the Soviet Union
(see also footnote <ref> on page PURFO2).
According to the leaflet,
Stalin and his acolytes
“can only beat defenseless prisoners, catch unsuspecting innocent people, plunder national wealth and invent ridiculous trials against nonexistent conspiracies”.
The authors of the leaflet were sure that
“Proletariat [...] will throw off the fascist dictator and his clique”.
Things didn't work out as expected and Landau was arrested and held in the Lubyanka prison for more
than one year, see Figure <ref>.
There were however other features which possibly compensated Landau's dissident leaflet
in the eyes of Stalin. Landau led a team of mathematicians supporting Soviet atomic and hydrogen bomb development, thus receiving for his work the Stalin Prize in 1949 and 1953
and the title Hero of Socialist Labor in 1954. We should be always suspicious about ourselves
when we receive too many prizes and awards, because,
as all Spider-Man fans know, “with great power comes great responsibility”.
By the way, according to Wikipedia,
“Landau believed in free love rather than monogamy and encouraged his wife
and his students to practise free love. However, his wife was not enthusiastic”.]
attempts to understand the phase transitions related to continuous order parameters. Its main ansatz is that the equilibria of the system should come from a “free energy” produced by the order parameter under consideration. This energy may also be sensitive to the temperature of the system. For instance, one can consider an order parameter $\eta$ for a given system at temperature $T$ and describe the energy $E$ of the system as a function of $\eta$ and $T$ (for the sake of simplicity,
we take here $\eta$ to be a scalar order parameter, but the case
of vector valued order parameters can be treated in a similar manner).
Up to a normalization, one can suppose that the energy corresponding to $\eta=0$ is zero. Moreover, in many concrete cases, the energy of the physical system is invariant if we exchange $\eta$ with $-\eta$ (this is the case, for instance, of magnetic materials, since exchanging the North with the South pole should not alter the energy state of the system, and it is also the case of nematic crystals, since the ending of their molecules is symmetric, recall e.g. point (iii) here above).
These considerations lead to an energy $E(\eta,T)$ which is even in $\eta$ and such that $E(0,T)=0$.
The corresponding Taylor expansion
must therefore contain only even terms and takes the form
$$ E(\eta,T)=a(T)\eta^2+b(T)\eta^4+\dots$$
and we will indeed neglect higher order terms, thus reducing simply to the case
\begin{equation}\label{ENEFRE} E(\eta,T)=a(T)\eta^2+b(T)\eta^4.\end{equation}
The energy coefficients $a(T)$ and $b(T)$ depend on the temperature $T$
and their signs play a decisive role in the formation and separation of phases. More specifically,
one usually considers stable solutions in correspondence to minimal energy levels:
therefore, to avoid minimizers corresponding to an energy equal to $-\infty$, a natural structural assumption
is to suppose that $b(T)>0$ for every temperature $T$. Instead, to allow for
possible phase changes, one can assume that $a(T)$ changes sign above and below some critical temperature $T_c$,
for instance taking $a(T)>0$ when $T<T_c$ and $a(T)<0$ when $T>T_c$.
Plot of the function $\eta\mapsto\frac{(\eta^2-1)^2-1}4$.
In this setting, below the critical temperature the free energy exhibits the null value of the order parameter $\eta$
as its only minimizer, but above the critical temperature the stable
phase corresponds to $\pm\eta_0$ where $\eta_0$ is the minimizer for $E$ given by
$$ \eta_0:=\sqrt{-\frac{a}{2b}},$$
see Figure <ref>.
In particular, if given $T>T_c$, one wishes to normalize the stable phases at the levels $\pm1$
(as done for instance in point (iv) here above), up to normalizing factors one can choose $a(T)=-\frac12$ and $b(T)=\frac14$,
so that (<ref>) reduces to
\begin{equation}\label{DWHADEKLTSTMONAKN2}E(\eta)=-\frac12\eta^2+\frac14\eta^4=\frac{(\eta^2-1)^2-1}4\end{equation}
and the stable phases are the zeros of
\begin{equation}\label{DWHADEKLTSTMONAKN}
It is interesting to remark that this analysis of the free energy is useful to detect
the stable phases, namely $\pm1$ in (<ref>),
but it does not give any information on the separation between the two possible phases:
indeed, all configurations only attaining the stable phases $\pm1$ would indeed be
zeros of (<ref>), as well as minimizers of (<ref>).
However, in many concrete situations, one expects the separation between phases to
be somewhat minimal as well: one often experiences situations in which the two phases
occupy two separate “bulks” of the material which are separated by an interface, thus showing
a distinctive coarsening and phase separation.
This phenomenon is possibly the outcome of a “ferromagnetic”
effect, which tends to align the direction of the molecules of the material,
hence avoiding oscillations of the order parameter.
To include this phenomenon into the phase separation model, one
can modify the energy in (<ref>) by adding
a small penalization term which charges the formation of interfaces.
The simplest version of this procedure is typically to add to (<ref>)
a small “gradient term” (a gradient indeed detects the “local oscillation” of phases), i.e.
replace (<ref>) by
\begin{equation}\label{JohnWCahnandSam AllenE} {\mathcal{G}}(\eta)=\frac{\e}2\int_\Omega|\nabla\eta(x)|^2\,dx+\int_\Omega
\frac{(\eta^2(x)-1)^2-1}4\,dx,\end{equation}
for a small parameter $\e>0$ (where $\Omega$ is the region occupied by the material).
Looking for the corresponding minimal points of the full energy ${\mathcal{G}}$ leads to the equation
\begin{equation}\label{JohnWCahnandSam Allen}\e
\Delta\eta=\eta^3-\eta,
\end{equation}
which is often referred[Equation (<ref>)
is named after John Cahn and Sam Allen, who presented
a theoretical treatment of metal alloys <cit.>.
With respect to the classification
presented in footnote <ref> on page CLASSIFICATIONFOOTN,
we see that equation (<ref>) is of elliptic type.
We think that it is interesting to relate the “democratic” features of the Laplace operator,
according to the discussions on pages DEEFFNVST
and DEEFFNVSTBIS, to the ferromagnetic tendency of some materials which avoids
oscillations in spins and molecule alignments.
Furthermore, equation (<ref>) provides one of the chief examples
of “semilinear equations”, within a structure that we will investigate further semilinear equation
from page Pohozaev Identity on. See also footnote <ref>
on page BIS Pohozaev Identity.
One can also appreciate the link between minimizers of the energy functional in (<ref>)
and interfaces of minimal area, by using
the Cauchy-Schwarz Inequality and the Coarea Formula (see e.g. <cit.>), namely setting $W(\eta):=
\frac{(\eta^2-1)^2}4$ and
noticing that
\begin{eqnarray*}&&{\mathcal{G}}(\eta)+\frac{|\Omega|}4=
\frac{\e}2\int_\Omega|\nabla\eta(x)|^2\,dx+\int_\Omega W(\eta(x))\,dx
\ge2\int_\Omega \sqrt{ \frac{\e}2|\nabla\eta(x)|^2W(\eta(x))}\,dx\\&&\qquad=
\sqrt{ 2{\e}}\int_\Omega |\nabla\eta(x)|\sqrt{W(\eta(x))}\,dx=\sqrt{2{\e}}
\int_\R \left[\int_{\{\eta=\tau\}} \sqrt{W(\eta(x))}\,d{\mathcal{H}}^{n-1}_x
\right]\,d\tau.
\end{eqnarray*}
Hence, heuristically, it should be convenient for a minimizer $\eta_\e$ to sit “as much as possible”
into the minima of $W$ (which correspond to the stable phases $\pm1$): if we assume that the level sets of $\eta_\e$
are surfaces “more or less parallel” to an interface ${\mathcal{S}}$
and $\eta_\e$ approaches “quite fast” the stable phases $\pm1$, the previous
computation suggests that the minimization of ${\mathcal{G}}$ reduces, as a first approximation,
to the minimization of the area of the interface ${\mathcal{S}}$.
Of course, it is not easy to transform the above heuristic argument into a rigorous proof, since the setting in (<ref>) is that of a singular perturbation singular perturbation,
namely the small parameter $\e$ affects precisely the most significant term in equation (<ref>).
As a matter of fact, the coherent development of arguments of this sort required the introduction
of a novel notion of asymptotics called $\Gamma$-convergence, $\Gamma$-convergence
see <cit.>.]
to with the name of Allen-Cahn equation. Allen-Cahn equation
See e.g. <cit.> and the references therein for additional details
on phase separation models.
Lev Landau in prison, photo by the People's Commissariat for Internal Affairs (Public Domain image from
§.§ Growth of interfaces
A number of scientific problems of interest are associated with the growth of the profile of suitable surfaces which describe, for instance, tumors, flame fronts, clusters, etc.
Opening ceremony at the Italian Parliament (October 10, 2021;
attribution: Presidenza della Repubblica https://www.quirinale.it/elementi/60152#&gid=1&pid=10).
A typical model describing this phenomenon is given by the so-called[Equation (<ref>) is named after Mehran Kardar, Giorgio Parisi and Yi-Cheng Zhang, who introduced this model in <cit.>.
Parisi is also known for his contributions in
quantum fluctuations, spin glasses and whirling flocks of birds. See Figure <ref>
in which the Italian President Sergio Mattarella (on the right) congratulates Parisi (on the left)
for having received the 2021 Nobel Prize in Physics (they are wearing face masks due to
COVID regulations).]
Kardar-Parisi-Zhang equationKardar-Parisi-Zhang equation
\begin{equation}\label{KPZEQUA}
\partial_t h=\mu \Delta h+\lambda |\nabla h|^2+f,
\end{equation}
where $\mu>0$ is a diffusion parameter, $\lambda$ is a parameter related to the
surface growth and $f$ is a forcing term (which could be either a deterministic function
or a stochastic term, such as a white noise). Of course, when $\lambda=0$ the setting in (<ref>)
essentially boils down to that of the heat equation in (<ref>), but when $\lambda\ne0$
the equation presents a nonlinear term of geometric importance.
Indeed, the motivation underpinning (<ref>) goes as follows. Suppose
that we have some aggregate with an active zone of growth on its surface
which is described, at a given time $t$, by the graph of a certain function $y=g(x,t)$.
We consider an “infinitesimal” time step $\tau$ in which the surface growth takes
place and suppose that such a growth is regulated by three ingredients:
first, in time $\left(t,t+\frac\tau3\right)$, some particle is added to the surface,
then, in time $\left(t+\frac\tau3,t+\frac{2\tau}3\right)$ a forcing term kicks in,
finally, in time $\left(t+\frac{2\tau}3,t+\tau\right)$ some diffusion takes place (the order
in which these phenomena occurs is not really important, but we fix this order just to keep a concrete case in mind).
To describe the first step, we assume that spherical particles of diameter $\tau$
are added uniformly along the surface of the cluster, as described in Figure <ref>.
In this setting, if $\vartheta$ denotes the angle between the normal
of the spherical particle and the vertical direction, we have that the infinitesimal vertical surface growth
corresponds to $\frac\tau{\cos\vartheta}$. Hence we write that
\begin{equation}\label{KPZ-Eq-1} g\left(x,t+\frac\tau3\right)=g(x,t)+\frac\tau{\cos\vartheta}.\end{equation}
Actually, the normal of the spherical particle agrees with the normal of the original surface $\frac{\left(-\nabla g(x,t),1\right)}{\sqrt{1+|\nabla g(x,t)|^2}}$ and accordingly
$$\cos\vartheta=\frac{\left(-\nabla g(x,t),1\right)}{\sqrt{1+|\nabla g(x,t)|^2}}\cdot e_n=
\frac{1}{\sqrt{1+|\nabla g(x,t)|^2}}.$$
We substitute this information into (<ref>) and we find that
\begin{equation}\label{KPZ-Eq-2} g\left(x,t+\frac\tau3\right)=g(x,t)+ \tau \sqrt{1+|\nabla g(x,t)|^2}.\end{equation}
Now we describe the second step, namely the action of a forcing term: this effect is encoded in the equation
\begin{equation}\label{KPZ-Eq-3} g\left(x,t+\frac{2\tau}3\right)= g\left(x,t+\frac\tau3\right)+ \tau f\left(x,t+\frac\tau3\right).\end{equation}
As for the third step, we assume that diffusion takes place in the form
\begin{equation}\label{KPZ-Eq-4} g(x,t+\tau)= g\left(x,t+\frac{2\tau}3\right)+ \tau \Delta g\left(x,t+\frac{2\tau}3\right)
Thus, by collecting the observations in (<ref>), (<ref>) and (<ref>),
\begin{eqnarray*}
&&\frac{g(x,t+\tau)-g(x,t)}\tau\\&=&\frac{\displaystyle g(x,t+\tau)-g\left(x,t+\frac{2\tau}3\right)}\tau
+\frac{ \displaystyle g\left(x,t+\frac{2\tau}3\right)-g\left(x,t+\frac{\tau}3\right)}\tau+
\frac{\displaystyle g\left(x,t+\frac{\tau}3\right)-g(x,t)}\tau\\&=& \Delta g\left(x,t+\frac{2\tau}3\right)
+ f\left(x,t+\frac\tau3\right)+\sqrt{1+|\nabla g(x,t)|^2},
\end{eqnarray*}
whence, formally taking the limit as $\tau\searrow0$,
\begin{equation}\label{KPZ-Eq-5} \partial_t g(x,t)= \Delta g(x,t)
+ f(x,t)+\sqrt{1+|\nabla g(x,t)|^2}
When the slope of the growing interface is small, one can consider $|\nabla g|$ as a small perturbation
and employ the approximation $\sqrt{1+|\nabla g|^2}\simeq 1+\frac{|\nabla g|^2}2$.
In this framework, one reduces (<ref>) to
\begin{equation*} \partial_t g= \Delta g
+ f+1+\frac{1}{2}|\nabla g|^2.\end{equation*}
Considering the velocity shift $h(x,t):=g(x,t)-t$, we thus conclude that
\begin{equation*} \partial_t h=
\partial_t g-1=
\Delta g
+ f+\frac{1}{2}|\nabla g|^2
=\Delta h
+ f+\frac{1}{2}|\nabla h|^2,
\end{equation*}
providing a motivation for the
Kardar-Parisi-Zhang equation in (<ref>).
Time evolution of a growing interface.
We mention that, in spite of its apparent simplicity, equations such as the one in (<ref>)
do capture complex phenomena and lead to the study of important
universality classes sharing the same characteristic exponents and scale invariant limits,
see e.g. <cit.>.
Sketch of Harry Bateman (Public Domain image from
Furthermore, equation (<ref>) is also directly related to other classical equations.
In particular, defining $v:=\nabla h$, we have that
\begin{eqnarray*}&&
\partial_t v+v\cdot\nabla v-\mu\Delta v-\nabla f=\nabla\Big(
\partial_t h-\mu\Delta h-f\Big)+\sum_{j=1}^n \partial_jh\, \partial_j \nabla h\\&&\qquad=\nabla\Big(
\partial_t h-\mu\Delta h-f\Big)+\frac12\nabla \left(\sum_{j=1}^n |\partial_jh |^2\right)\\&&\qquad=\nabla\left(
\partial_t h-\mu\Delta h-f+\frac12|\nabla h|^2\right),
\end{eqnarray*}
from which it follows that the
Kardar-Parisi-Zhang equation in (<ref>) with $\lambda:=\frac12$
is equivalent to the vectorial equation
\begin{equation}\label{VERBURG}
\partial_t v+v\cdot\nabla v=\mu\Delta v+\nabla f,
\end{equation}
which is a version of the Navier-Stokes equation in (<ref>) in the absence of gravity
(in this setting, $v$ corresponds to the speed of a given fluid, $\mu$ is its viscosity coefficient
and $f$ is minus the pressure).
For constant pressure, in dimension $n=1$, equation (<ref>) reduces to the scalar equation
\begin{equation}\label{VERBURG2}
\partial_t v+v\partial_x v=\mu\partial_{xx} v,
\end{equation}
which is known in jargon[Equation (<ref>)
is named after Harry Bateman and Johannes Martinus Burgers.
See Figure <ref> for a nice drawing of Harry Bateman
from the 1931 yearbook of the California Institute of Technology.] as the Bateman-Burgers equation Bateman-Burgers equation,
which models the speed of a fluid in a thin ideal pipe.
In this context, it is interesting to point out that in the absence of viscosity equation (<ref>) boils down to
\begin{equation}\label{VERBURG3}
\partial_t v+v\partial_x v=0,
\end{equation}
known as the inviscid Bateman-Burgers equation. A feature of the solutions of (<ref>)
is that they may develop “shock waves”, shock wave
i.e. singularities at a breaking time. For instance,
it is readily seen that the relation $v(x,t)=\arctan\big( tv(x,t)-x\big)$
provides, via the Implicit Function Theorem, a solution of (<ref>) for small times, with $v(x,0)=-\arctan x$.
This solution exhibits a shock wave since otherwise, for $t>1$,
$$ |\partial_x v(0,t)|=\left|\frac{t\partial_x v(0,t)}{\cos^2\big(tv(0,t)\big)}\right|\ge | t\partial_x v(0,t)|>|\partial_x v(0,t)|,
which is a contradiction. See Figure <ref>
for a sketch of the formation of such a shock wave.
Implicit plot of $v=\arctan( tv-x)$ for $t\in\{0.5,\,0.75,\,0.99,\,2\}$.
See e.g. <cit.> for more information on shock waves
in the context of the Bateman-Burgers equation.
See also Figure <ref> for a rather dramatic view of
expanding spherical atmospheric shock waves from a gun firing on the surface of the water.
The battleship USS Iowa of the United States Navy
during a training exercise in Puerto Rico (Public Domain image from
§.§ Definitions come later on
Given $a$, $b$, $c\ge0$, the equation
\begin{equation}\label{HEATELEEQ0}
\frac{\partial ^{2} u}{\partial x^{2}}-a\frac{\partial ^{2} u}{\ \partial t^{2}}=
b \frac {\partial u }{\partial t} +c u
\end{equation}
is called[The inventor of equation (<ref>)
was Oliver Heaviside, see Figure <ref>.
Heaviside's uncle was Sir Charles Wheatstone, co-inventor of the first commercially successful telegraph. Following his uncle's advice, Heaviside became a telegraph operator and an electrician, continuing to study and do science while working.
His contributions to mathematics, physics and engineering were deep and remained as classical contributions: they include the use of complex numbers to solve circuit analysis differential equations, the calculus of the deformations of an electromagnetic field surrounding a moving charge and the derivation of the magnetic force on a moving charged particle (the first contribution towards the understanding of the Lorentz Force) and the prediction of the existence of a reflective layer of the ionosphere (which allowed radio waves radiated into the sky to return to Earth beyond the horizon).
Several works by Heaviside
were of great practical use, including the possible exploitation of loading coils in telephone and telegraph lines to increase their self-induction and correct the distortion which they suffered.
Several years later, some American telecommunications companies hired their own scientists to extend Heaviside's work and adapt the use of coils previously introduced by Heaviside. Some corporations later offered Heaviside money in exchange for his rights, but he declined to accept any money unless the company were to give him full recognition for his discoveries and inventions (it turns out that Heaviside remained for all his life chronically poor).
Heaviside was perhaps a bit eccentric too. He was a firm opponent of Einstein's theory of relativity, possibly beyond reasonable scientific arguments, and at the end of his life he developed a very strong aversion to meeting people and became a recluse. He died at age 74 after falling from a ladder.
Heaviside said: “Mathematics is an experimental science, and definitions do not come first, but later on. They make themselves, when the nature of the subject has developed itself”.]
the telegrapher's equation. telegrapher's equation
In equation (<ref>), we have that $u=u(x,t)$ with $x\in\R$, $t\in[0,+\infty)$.
When $a:=0$, $c:=0$ and $b>0$, equation (<ref>) boils down to
the heat equation in (<ref>).
When $b:=0$, $c:=0$ and $a>0$, it reduces[With respect to the terminology
in footnote <ref> on page CLASSIFICATIONFOOTN,
we note that equation (<ref>) is hyperbolic when $a>0$, parabolic when $a:=0<b$
and elliptic when $a:=0$ and $b:=0$.]
to the wave equation
in (<ref>).
When $a$, $b$, $c>0$ however the behavior of the solutions of (<ref>) are interestingly different from
those of the wave equation,
since solutions of the telegrapher's equation present damping and dispersion effects
that are not present in the wave equation.
See e.g. Figure <ref>
in which the evolution of the solution of the telegrapher's equation
exhibits a visible dispersion, in which the velocity of travel depends on the frequency thus
enlarging the support of the solution, and losses of intensity, causing the peaks of the traveling front to reduce over time.
The solution $u$ in equation (<ref>) describes the electric current intensity (or, in a similar manner,
the voltage) along a transmission line.
This transmission line can be a telegraph wire, a overhead electrical conductor, a telephone line, etc.,
see Figure <ref>. See also Figure <ref>
for a peculiar use of transmission lines.
To get to the bottom of the telegrapher's equation (<ref>), we consider
an electric line made of two parallel electrical wires.
There is a voltage difference between the wires (e.g., produced by an electric generator
“at the end of the wires”: since the wires in our idealized model are straight lines, the generator is essentially
“at infinity”).
Oliver Heaviside (Public Domain image from
Thinking at the two wires just as parallel straight lines would be however reductive,
* each straight line is actually made by a conductor which presents some
distributed electric resistance (say, modeled by a series resistor) and also some
distributed inductance (e.g., due to the magnetic field around the wires and to self-inductance, modeled by a series inductor),
* furthermore the dielectric material separating the two conductors is also not completely neutral from
the electric point of view, since it can carry
capacitance and conductance (modeled by
a shunt capacitor and resistor located between each “infinitesimal” portion $dx$ of the conductors).
To describe these phenomena,
as customary in the physical description of electric phenomena, we
reserve the name $I$ for the current, $V$ for the voltage, $R$ for the
resistance, $L$ for the inductance and $C$ for the capacity. We also use the letter $G$ to
denote the conductance, which is the reciprocal of a resistance.
The parameters $R$, $L$, $C$ and $G$ will be treated as structural constants
and we will be interested in the description of the functions $I=I(x,t)$
and $V=V(x,t)$ with respect to the position $x\in\R$ on the transmission line
and the time $t\ge0$.
More specifically, for concreteness we focus on the current in the upper conductor, see Figure <ref>: in this setting,
the distributed resistance in the infinitesimal elemental length $dx$ of the conductor
is denoted by $R\,dx$
and the distributed inductance is denoted by $L\,dx$.
The conductance of the dielectric material separating the two conductors
(accounting for bulk conductivity of the dielectric and dielectric loss) is denoted by $G\,dx$
and the capacity by $C\,dx$ (notice that, in “real life” there is no wire connecting the top and the bottom
cable, but Figure <ref> translates the behavior of the dielectric
between the two conductors into the language of electric circuits).
Now, to describe the current flowing through the upper wire, we denote by $I(x,t)$ the current intensity on the left end
of the elementary upper conductor in Figure <ref>
and by $I(x+dx,t)$ the one on the right end (say, with the convention that
the current is traveling from left to right).
Similarly, we denote by $V(x,t)$ the voltage on the left end
of the elementary upper conductor and by $V(x+dx,t)$ the one on the right end
(actually, $V$ would stand for the difference of voltage from the upper and the lower conductors;
for simplicity[Actually, telegraphs originally used two wires, utilizing a forward and a return paths
in a closed circuit to move energy along the transmission line.
But it was then noticed that Earth itself can be used as the return path. In this setting, the bottom wire is just the ground, which is normalized to be approximatively at constant zero volt (from practical purposes however, Earth is an adequate, but certainly not optimal, return path). Thus, in the situation in which the transmission line is modeled by one forward cable using Earth as a return path, the poles are also considered as dielectric and provide some capacitance and conductance, see Figure <ref>. In this case,
the vertical elements related to $G\,dx$ and $C\,dx$ in Figure <ref>
can be thought as “concrete objects”, such as the telegraph poles, located at an “infinitesimal” distance $dx$
at a large scale (in which the transmitter and the receiver are located “at infinity”).
See e.g. https://youtu.be/ySuUZEjARPY for a very thorough video about electric transmission.] one can think that the conductor at the bottom is at zero voltage).
Let also $\iota_1$ be the current between the nodes of the upper conductor (oriented left to right),
and $\iota_2$ and $\iota_3$ be the current through the
shunt capacitor $C\,dx$ and resistor $G\,dx$ respectively (oriented downwards).
With this notation, by Kirchhoff's Junction Rule (or simply by conservation of charge), we have that
$$ I(x,t)=\iota_1+\iota_3\qquad{\mbox{and}}\qquad
\iota_1=I(x+dx,t)+\iota_2.$$
Also, by the Laws of Ohm and Faraday,
$$ V(x,t)=\frac{\iota_3}{G\,dx}\qquad{\mbox{and}}\qquad
V(x+dx,t)-V(x,t)=-RI(x,t)\,dx-L\partial_t I(x,t)\,dx.$$
Additionally, by the definition of capacity,
Comparison between the evolution in time of the solution of the wave equation and of the
telegrapher's equation (author
Jacopo Bertolotti, https://twitter.com/j_bertolotti/status/1172517281374572551,
images from
Wikipedia, licensed under the Creative Commons CC0 1.0 Universal Public Domain Dedication).
These considerations lead to
\begin{eqnarray*}
\partial_x V(x,t)\,dx\simeq V(x+dx,t)-V(x,t)=-RI(x,t)\,dx-L\partial_t I(x,t)\,dx
\end{eqnarray*}
\begin{eqnarray*}&&
\partial_x I(x,t)\,dx\simeq
=-\iota_2-\iota_3=-C\partial_t V(x,t)\,dx-GV(x,t)\,dx.
\end{eqnarray*}
\begin{equation}\label{ihkbdDAvbsdewI089JHGSFDCokwfIUyb-X1}
\begin{dcases}
\partial_x V=-L \partial_t I-R I,\\
\partial_x I=-C\partial_t V-GV.
\end{dcases}
\end{equation}
We can take the derivative with respect to $x$ of the first equation
and compare with the derivative with respect to $t$ of the second equation
in (<ref>), finding that
\begin{equation}\label{ihkbdDAvbsdewI089JHGSFDCokwfIUyb-X2}
\partial_x^2 V-LC \partial_t^{2} V=(RC+GL) \partial_t V+GRV.
\end{equation}
Left: utility pole for a telephone line; center: electrical wires;
right: overhead lines in Queensland
(images from
Wikipedia, Public Domain for the first, photo by Novoklimov, licensed under the Creative Commons Attribution-Share Alike 4.0 International, licensed under the Unported license
for the second, photo by Pytomelon87 licensed under the Creative Commons Attribution-Share Alike 4.0 International license for the third).
Similarly, we can take the derivative with respect to $t$ of the first equation
and compare with the derivative with respect to $x$ of the second equation
in (<ref>), which yields that
\begin{equation}\label{ihkbdDAvbsdewI089JHGSFDCokwfIUyb-X3}
\partial_x^{2} I-LC\partial_t^{2}I=(RC+GL) \partial_t I+GRI.
\end{equation}
We stress that, up to replacing voltage and current, the structure of (<ref>)
and (<ref>) is the same.
Also, (<ref>) and (<ref>)
give that both voltage and current are solutions of the telegrapher's equation in (<ref>).
White storks nesting on an utility pole
(photo by Myrabella; image from
Wikipedia, licensed under the
Creative Commons Attribution-Share Alike 3.0 Unported license).
It is interesting to note that, in this situation, the structural parameters
in the telegrapher's equation in (<ref>)
correspond to $a:=LC$, $b:=RC+GL$ and $c:=GR$.
Thus, a lossless transmission in which the roles of resistances can be neglected corresponds to $R:=0$ and $G:=0$
(this is a perfect conductor offering no resistance and a perfect electrical insulator as a dielectric allowing for no
conductance), which in turn produce $b:=0$ and $c:=0$ in (<ref>), thus
reducing to the wave equation.
With this observation in mind, one can also have a second look at Figure <ref> to appreciate how the damping and dispersion effects
are the outcome of the imperfect conductors and dielectric in terms of resistance and electrical insulation.
Schematic for an elemental length $dx$ of transmission line (image by Omegatron from
Wikipedia, licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license).
Transmission line for a telegraph, using Earth as return path.
§.§ Nerd sniping
Nerd sniping is a new sport invented in https://xkcd.com/356/
in which the value of mathematicians is frightfully appreciated,
see Figure <ref>. Here, we discuss the solution[The infinite resistor problem
happened to be studied quite in detail, see in particular <cit.>,
$\phantom{123456789}$ on-this-infinite-grid-of-resistors-whats-the-equivalent-resistance
and the references therein.
Strictly speaking, without additional information, the problem is possibly not uniquely posed, but in these pages we confine ourselves
to the case in which the solution is assumed to decay “fast enough” at infinity to validate the formal computations
showcased here, though we gloss over any technical aspect related to uniqueness, decay, convergence
and infinite cancellations (but let us mention, for instance, that if $\Gamma(k)$ is a solution of (<ref>), then
so is $\Gamma(k)+k_1$ and the equivalent resistance $R$ in (<ref>) would thus
be affected by an additional term $-4$, showing a uniqueness issue unless we add some decay assumption on $\Gamma$).
Also, the webcomic xkcd was created by Randall Patrick Munroe, see Figure <ref>.]
of the infinite resistor problem presented in the fourth cartoon of Figure <ref>
and its strong relation with elliptic partial differential equations (we mean, not the relation between nerds and
elliptic partial differential equations, but the one between the infinite resistor problem
and a discrete variant of elliptic partial differential equations). Actually, the answer to the problem is
\begin{equation}\label{NERD:ANS}
\frac4\pi-\frac12,\end{equation}
but you'll see, it's about the journey, not the destination.
We follow here an approach based on the fundamental solution
of a suitable operator reminiscent of the Laplacian. The methods related to fundamental solutions will be refined in
Section <ref>. For simplicity, we work here in dimension $2$, but many of the arguments that we present are applicable in higher dimensions and further generality, see <cit.> and the references therein. Given $\e>0$ and a function $u:\Z^2\to\R$, for all $x\in\Z^2$ we define
$$ {\mathcal{L}} u(x):=\sum_{j=1}^2\Big( u(x+e_j)+u(x-e_j)-2u(x)\Big).$$
Notice that the operator ${\mathcal{L}}$ can be seen as a discrete version[For more information
on the discretization of
the Laplace operator (and for the opportunities and dangers entailed by discretization), see e.g. <cit.>.]
of the Laplacian. Our main goal will be to determine the fundamental solution of this operator, namely a function $\Gamma:\Z^2\to\R$ such that
\begin{equation}\label{FU:NE:023} {\mathcal{L}}\,\Gamma =\delta_0,\end{equation}
where $\delta_0$ is the Dirac Delta Function at the origin in the “discrete sense”, meaning that
$$ \sum_{k\in\Z^2} \varphi(k)\,\delta_0(k)=\varphi(0)$$
for all $\varphi:\Z^2\to\R$.
The convenience of using this fundamental solution is due to the following observation. We put coordinates in $\R^2$ such that the nodes of the infinite resistor problem presented in the fourth cartoon of Figure <ref> correspond to the lattice $\Z^2$.
We can also suppose that the two red points in the lattice
of Figure <ref> correspond to $(0,0)$ and $(2,1)$.
Then, to test the resistors, we aim at constructing a distribution of voltage $V:\Z^2\to\R$ which produces a current with unit intensity from $(0,0)$ to $(2,1)$ and no further net current in the circuit.
Hence, we apply Ohm's Law at each node $k$ of the lattice, with respect to its first neighbors $k\pm e_j$, for $j\in\{1,2\}$.
Namely, by Ohm's Law, since all resistors of Figure <ref> are unitary, the current flowing into the node $k$ from its neighbors $k\pm e_j$ is equal to
$$ \sum_{j=1}^2 (V({k+ e_j})-V(k))+\sum_{j=1}^2 (V({k- e_j})-V(k)),$$
which in turn equals to ${\mathcal{L}}V(k)$.
Hence, the voltage distribution flowing one unit of current from $(0,0)$ to $(2,1)$ is such that
\begin{equation}\label{12wrt13kdelta0}
\delta_{(2,1)}(k)-\delta_{(0,0)}(k)
\end{equation}
If we found such a voltage distribution $V$, we can apply Ohm's Law again
and determine the requested equivalent resistance $R$ between
the two red nodes in Figure <ref> via the relation
\begin{equation}\label{12wrt13kdelta1} R=V(0,0)-V(2,1).\end{equation}
But to find such a $V$, it comes in handy to use the fundamental solution $\Gamma$:
indeed, if we find $\Gamma$ as in (<ref>) it suffices to set
\begin{equation}\label{12wrt13kdelta2} V(k):=\Gamma(k-(2,1))-\Gamma(k)\end{equation}
and observe that this is a solution of (<ref>).
Summarizing, in view of (<ref>) and (<ref>), once we determine the fundamental solution $\Gamma$ we can also find the desired equivalent resistance $R$ by the formal relation
\begin{equation} \label{12wrt13kdelta7} R=V(0,0)-V(2,1)=
\Big( \Gamma(-2,-1)-\Gamma(0,0)\Big)
- \Big( \Gamma(0,0)-\Gamma(2,1)\Big)
Nerd sniping https://xkcd.com/356/,
licensed under the Creative Commons Attribution-NonCommercial 2.5 License.
Hence, we focus now on determining the fundamental solution $\Gamma$. To this end, we deduce from (<ref>) that, for every $\xi\in\R^2$,
\begin{equation}\label{PMSTABSNdavsRAFSDs}\begin{split}&
1=\sum_{k\in\Z^2} \delta_0(k)\,e^{2\pi ik\cdot\xi} =\sum_{k\in\Z^2}{\mathcal{L}}\,\Gamma (k)\,e^{2\pi ik\cdot\xi}=
\sum_{{j\in\{1,2\}}\atop{k\in\Z^2}}\Big(\Gamma(k+e_j)+\Gamma(k-e_j)-2\Gamma(k)\Big)\,e^{2\pi ik\cdot\xi}\\&\qquad\qquad=
\sum_{{j\in\{1,2\}}\atop{m\in\Z^2}} \Gamma(m)\,e^{2\pi i(m-e_j)\cdot \xi}+
\sum_{{j\in\{1,2\}}\atop{m\in\Z^2}}\Gamma(m)\,e^{2\pi i(m+e_j)\cdot \xi}-2\sum_{{j\in\{1,2\}}\atop{m\in\Z^2}}\Gamma(m)\,e^{2\pi im\cdot \xi}\\&\qquad\qquad=
\sum_{{j\in\{1,2\}}\atop{m\in\Z^2}} \Big(e^{-2\pi i\xi_j}+e^{2\pi i\xi_j}-2\Big)\,\Gamma(m)\,e^{2\pi im\cdot \xi} =-\mu(\xi)
\sum_{{m\in\Z^2}} \Gamma(m)\,e^{2\pi im\cdot \xi}
$$ \mu(\xi):=2\sum_{j=1}^2\big( 1-\cos(2\pi \xi_j)\big).$$
We remark that $\mu$ is a periodic function, namely $\mu(\xi+\ell)=\mu(\xi)$ for every $\ell\in\Z^2$ and $\xi\in\R^2$.
Thus, also $\frac1\mu$ is periodic and then we can write this function as a Fourier Series.
For this, to be precise, since $\mu$ is nonnegative but may vanish,
it is convenient to pick $\delta>0$ and define $\mu_\delta(\xi):=\max\{\delta,\mu(\xi)\}$,
expand the function $\frac1{\mu_\delta}$ in Fourier Series and then pass $\delta\searrow0$ one way or another.
Namely, we have that
$$ \frac1{\mu_\delta(\xi)}=\sum_{m\in\Z^2} c_{m,\delta} \,e^{2\pi i m\cdot\xi},$$
$$ c_{m,\delta}:=\int_{Q} \frac{e^{-2\pi i m\cdot\eta}}{\mu_\delta(\eta)}\,d\eta\;\qquad{\mbox{and}}\qquad \; Q:=\left(-\frac12,\frac12\right)^2.$$
Thus, if $\phi:\R^2\to\R$ is a smooth and periodic function
$$ \widehat\phi_m:=\int_{Q} \phi(\xi)\,e^{-2\pi im\cdot \xi}\,d\xi$$
is the Fourier coefficient of $\phi$, we formally deduce from (<ref>) that
\begin{equation}\label{N2A3S3DwedabN023VhwVaq}\begin{split}&
\lim_{\delta\searrow0}
\sum_{m\in\Z^2} c_{m,\delta} \,\widehat\phi_{-m}=
\lim_{\delta\searrow0}\int_{Q}
\sum_{m\in\Z^2} c_{m,\delta} \,\phi(\xi)\,e^{2\pi i m\cdot\xi}\,d\xi=
\lim_{\delta\searrow0}\int_{Q} \frac{\phi(\xi)}{\mu_\delta(\xi)}\,d\xi=
\int_{Q} \frac{\phi(\xi)}{\mu(\xi)}\,d\xi\\&\qquad\qquad=
-\int_{Q}\sum_{{m\in\Z^2}} \Gamma(m)\,e^{2\pi im\cdot \xi}\,\phi(\xi)\,d\xi=-\sum_{{m\in\Z^2}} \Gamma(m)\,
\widehat\phi_{-m}.
\end{split}\end{equation}
We can thereby apply (<ref>)
to the function $\psi(\xi):=\phi(\xi)-\phi(0)$ and find that
\begin{equation}\label{76t54wedMSKAMS0987654DIJMD2r3teg}
\lim_{\delta\searrow0}
\sum_{m\in\Z^2}\int_{Q} \widehat\psi_{-m} \frac{e^{-2\pi i m\cdot\eta}\,d\eta}{\mu_\delta(\eta)}=
\lim_{\delta\searrow0}
\sum_{m\in\Z^2} c_{m,\delta} \,\widehat\psi_{-m}=
-\sum_{{m\in\Z^2}} \Gamma(m)\,
\widehat\psi_{-m}.
\end{equation}
Also, since
$$ \sum_{m\in\Z^2} \widehat\psi_{-m}=\sum_{m\in\Z^2} \widehat\psi_m=
\sum_{m\in\Z^2} \widehat\psi_m\,e^{2\pi im\cdot 0}=\psi(0)=0,$$
we have that
\begin{eqnarray*}
\int_{Q}\sum_{m\in\Z^2} \widehat\psi_{-m}\,\frac{d\eta}{\mu_\delta(\eta)}=0.
\end{eqnarray*}
This and (<ref>) lead to
\begin{equation}\label{76t54wedMSKAMS0987654DIJMD2r3teg3}
\lim_{\delta\searrow0}
\sum_{m\in\Z^2}\int_{Q} \widehat\psi_{-m} \frac{(e^{-2\pi i m\cdot\eta}-1)\,d\eta}{\mu_\delta(\eta)}=
-\sum_{{m\in\Z^2}} \Gamma(m)\,
\widehat\psi_{-m}.
\end{equation}
We also point out that $\mu_\delta$ is an even function and therefore
$$ \int_{Q} \frac{\sin(2\pi m\cdot\eta)\,d\eta}{\mu_\delta(\eta)}=0.$$
From this and (<ref>) it follows that
\begin{equation}\label{76t54wedMSKAMS0987654DIJMD2r3teg2}
\lim_{\delta\searrow0}
\sum_{m\in\Z^2}\int_{Q} \widehat\psi_{-m} \frac{\big(\cos(2\pi m\cdot\eta)-1\big)\,d\eta}{\mu_\delta(\eta)}=
-\sum_{{m\in\Z^2}} \Gamma(m)\,
\widehat\psi_{-m}.
\end{equation}
Randall Munroe and his signature (images from
Wikipedia, the first by re:publica/Jan Zappner,
licensed under the Creative Commons Attribution 2.0 Generic license, the second in the Public Domain).
The advantage of (<ref>) with respect to (<ref>) is that the singularity at the denominator
in the first integral is compensated by the vanishing of the corresponding numerator, thus allowing us
to pass to the limit inside the integral and write that
\begin{equation}\label{76t54wedMSKAMS0987654DIJMD2r3teg5}
\sum_{m\in\Z^2}\int_{Q} \widehat\psi_{-m} \frac{\big(\cos(2\pi m\cdot\eta)-1\big)\,d\eta}{\mu(\eta)}=
-\sum_{{m\in\Z^2}} \Gamma(m)\,
\widehat\psi_{-m}.
\end{equation}
We also observe that
\widehat\phi_0-\sum_{m\in\Z^2}\widehat\phi_m e^{2\pi i m\cdot0}=\widehat\phi_0-\sum_{m\in\Z^2}\widehat\phi_{-m}
and $
\widehat\psi_m=\widehat\phi_m$ for all $m\in\Z^2\setminus\{0\}$. Plugging this information into (<ref>) we find that
\begin{equation*}
\sum_{m\in\Z^2}\int_{Q} \widehat\phi_{-m} \frac{\big(\cos(2\pi m\cdot\eta)-1\big)\,d\eta}{\mu(\eta)}=
\sum_{{m\in\Z^2}} \big(\Gamma(0)-\Gamma(m)\big)\,
\widehat\phi_{-m}.
\end{equation*}
The arbitrariness of $\phi$ thus gives that
\begin{equation*}
\begin{split}
\Gamma(m)-\Gamma(0)&=-\int_{Q} \frac{\big(\cos(2\pi m\cdot\eta)-1\big)\,d\eta}{\mu(\eta)}\\&=\frac12
\iint_{(-1/2,1/2)\times(-1/2,1/2)} \frac{ 1-\cos(2\pi m_1\eta_1+2\pi m_2\eta_2) }{2-\cos(2\pi \eta_1)-\cos(2\pi \eta_2)}\,d\eta_1\,d\eta_2.
\end{split}\end{equation*}
We insert this into the equivalent resistance relation (<ref>) and we arrive at
\begin{equation} \label{76t54wedMSKAMS0987654DIJMD2r3teg5b90} R=
\iint_{(-1/2,1/2)\times(-1/2,1/2)} \frac{ 1-\cos(4\pi \eta_1+2\pi \eta_2) }{2-\cos(2\pi \eta_1)-\cos(2\pi \eta_2)}\,d\eta_1\,d\eta_2
The good news is that this is an integral involving simple trigonometric functions, so we should feel confident to solve it explicitly thanks to our consolidated calculus skills. For instance, we can proceed as follows. We note that
\begin{equation} \label{76t54wedMSKAMS0987654DIJMD2r3teg5b}\begin{split}\cos(4\pi \eta_1+2\pi \eta_2)\,&=
\cos(4\pi \eta_1)\cos(2\pi \eta_2)
-\sin(4\pi \eta_1)\sin(2\pi \eta_2)\\&=
\Big( 1-2\sin^2(2\pi \eta_1)\Big)\cos(2\pi \eta_2)
-2\sin(2\pi \eta_1)\cos(2\pi \eta_1)\sin(2\pi \eta_2).
\end{split}\end{equation}
Hence, if $\tau_1:=\tan(\pi \eta_1)$ and $\tau_2:=\tan(\pi \eta_2)$, using in (<ref>) the “tangent half-angle formulas”
$$ \sin(2\pi \eta_j)=\frac{2\tau_j}{1+\tau_j^2}\qquad{\mbox{and}}\qquad
\cos(2\pi \eta_j)=\frac{1-\tau_j^2}{1+\tau_j^2},$$
we find that
\begin{equation}\label{76t54wedMSKAMS0987654DIJMD2r3teg5c}1-\cos(4\pi \eta_1+2\pi \eta_2)=\frac{2 (2 \tau_1 + \tau_2 - \tau_1^2 \tau_2)^2}{(1 + \tau_1^2)^2 \,(1 + \tau_2^2)}.
\end{equation}
\begin{equation}\label{76t54wedMSKAMS0987654DIJMD2r3teg5d}
2-\cos(2\pi \eta_1)-\cos(2\pi \eta_2)=
\frac{2 (\tau_1^2 + \tau_2^2 + 2 \tau_1^2 \tau_2^2)}{(1+\tau_1^2) (1+\tau_2^2)}.
\end{equation}
Thus, since
$$ d\eta_j=\frac{d\tau_j}{\pi (1+\tau_j^2)},$$
we infer from (<ref>) and (<ref>) that
\begin{eqnarray*}
\frac{ 1-\cos(4\pi \eta_1+2\pi \eta_2) }{2-\cos(2\pi \eta_1)-\cos(2\pi \eta_2)}\,d\eta_1\,d\eta_2=
\frac{ (\tau_1^2 \tau_2 - 2 \tau_1 - \tau_2)^2}{\pi^2 (1+\tau_1^2 )^2 \,(1+\tau_2^2 ) \,(2 \tau_1^2 \tau_2^2 + \tau_1^2 + \tau_2^2)}\,d\tau_1\,d\tau_2
\end{eqnarray*}
This and (<ref>) lead to
\begin{equation*} \begin{split}
\iint_{\R\times\R}
\frac{(\tau_1^2 \tau_2 - 2 \tau_1 - \tau_2)^2}{\pi^2
(1+\tau_1^2 )^2 \,(1+\tau_2^2 ) \,(2 \tau_1^2 \tau_2^2 + \tau_1^2 + \tau_2^2)}\,d\tau_1\,d\tau_2\\&=\iint_{\R\times\R}
\frac{(x y^2 -x - 2 y)^2}{\pi^2 (1+x^2 ) \,(1+y^2 )^2 \,(2 x^2 y^2 + x^2 + y^2)}\,dx\,dy
where we simplified notation using the variables $(x,y)$ in place of $(\tau_2,\tau_1)$.
Since the latter integral remains invariant under the map $(x,y)\mapsto(-x,-y)$ we can reduce the previous equation to
\begin{equation}\label{76t54wedMSKAMS0987654DIJMD2r3teg5dwqr326tygh}
\frac{2 (x y^2 -x - 2 y)^2}{\pi^2
(1+x^2 ) \,(1+y^2 )^2 \,(2 x^2 y^2 + x^2 + y^2)}\,dx\,dy
Now one can check that a primitive of the function $\frac{(1+y^2)(x y^2 -x - 2 y)^2}{(1+x^2 ) (2 x^2 y^2 + x^2 + y^2)} $ in the variable $x$ is
\begin{eqnarray*} \Phi(x,y)&:=&2 y (y^2 - 1) \ln\frac{1+x^2}{2 x^2 y^2 + x^2 + y^2} + (y^4 - 6 y^2 + 1) \arctan x \\&&\qquad+ \frac{y ( 10 y^2-y^4 + 3) \arctan\frac{x \sqrt{2 y^2 + 1}}{y}}{\sqrt{2 y^2 + 1}}. \end{eqnarray*}
Hence, for every $y\in(0,+\infty)$,
\begin{eqnarray*} \Phi(\pm\infty,y)&=&2 y (y^2 - 1) \ln\frac{1}{2 y^2 + 1} \pm\frac{\pi}{2}\left((y^4 - 6 y^2 + 1) +\frac{y ( 10 y^2-y^4 + 3) }{\sqrt{2 y^2 + 1}}\right). \end{eqnarray*}
As a result, for all $y\in(0,+\infty)$,
\begin{eqnarray*}
\int_\R \frac{(1+y^2)(x y^2 -x - 2 y)^2}{(1+x^2 ) (2 x^2 y^2 + x^2 + y^2)} \,dx&=&
\Phi(+\infty,y)-\Phi(-\infty,y)\\&=&
\pi (y^4 - 6 y^2 + 1) +\frac{\pi y ( 10 y^2-y^4 + 3) }{\sqrt{2 y^2 + 1}}\end{eqnarray*}
and consequently
\begin{eqnarray*}
\int_\R \frac{2 (x y^2 -x - 2 y)^2}{\pi^2
(1+x^2 ) \,(1+y^2 )^2 \,(2 x^2 y^2 + x^2 + y^2)}\,dx &=&
\frac{2(y^4 - 6 y^2 + 1)}{\pi (1+y^2)^3} +
\frac{2 y ( 10 y^2-y^4 + 3)}{\pi (1+y^2)^3\sqrt{2 y^2 + 1}}.
\end{eqnarray*}
Combining this with (<ref>) we find that
\begin{eqnarray*}
\frac{2(y^4 - 6 y^2 + 1)}{\pi (1+y^2)^3} \,dy+\int_{0}^{+\infty}
\frac{2 y ( 10 y^2-y^4 + 3)}{\pi (1+y^2)^3\sqrt{2 y^2 + 1}}\,dy
\\&=& \left. \frac{2(y - y^3)}{\pi (1 + y^2)^2}\right|_{y=0}^{y=+\infty}
-\left. \frac2{\pi}\left(\frac{2 \sqrt{2 y^2 + 1}}{(1+y^2)^2} +\arctan\sqrt{2 y^2 + 1}\right)\right|_{y=0}^{y=+\infty}\\
-\arctan 1\right)\\&=&\frac{\pi}4-\frac12,
\end{eqnarray*}
in accordance with (<ref>), as desired.
§.§ Fighting a pandemic using differential equations
A topical subject nowadays consists in the spread of a virus among a given population, see Figure <ref>.
Mathematics has established a solid reputation in trying to understand the fundamental properties of
epidemic disease diffusion and has often helped taking farsighted decisions in the critical moments
of global pandemics.
In this set of notes, we certainly do not aim at presenting all possible mathematical models
that can be effectively used to describe infectious diseases, nor at giving a fully “realistic” description of
an epidemic in the real world. Just, we briefly recall a very classical approach which, in spite of its simplifications
and limitations, can already highlight the great potential of mathematics in dealing with epidemiology
and can obviously open the possibility of presenting more sophisticated and bespoke models to address more
realistically concrete cases of epidemics.
The setting that we recall here
builds on the research (among the others) of Sir Ronald Ross, Hilda Phoebe Hudson, William Ogilvy Kermack and Anderson Gray McKendrick and is called the SIR model “SIR model”, not because of Ross' formal honorific address, but
because it divides the whole population into three compartments:
* Susceptible: individuals who might become infected if exposed to the infectious agent (e.g., a virus),
* Infected (and Infectious): individuals who are currently infected and can transmit the
infection to susceptible individuals (e.g., by contact),
* Recovered (or, better to say, Removed): individuals who, after being
infected, become immune to the infection
(in this category, it is also common to place the people who died for the infection,
since, like the ones that recovered, they cannot contribute to the spread of the disease).
This is an example of compartmental model, compartmental model
since it splits a given population into “compartments” (three compartments, in this case,
that are labeled with the letters $S$, for susceptible, $I$, for infected, and $R$, for recovered),
and several types of related models are indeed broadly utilized in epidemiology.
For simplicity, we can consider a large population and denote by $S$, $I$ and $R$ the proportion
of susceptible, infected and recovered individuals, respectively: in this way, $S$, $I$, $R\in[0,1]$
(the value $1$ corresponding to $100\%$ of the population,
the value $\frac12$ to $50\%$, and so on). Also, possibly adopting the above mentioned convention of counting casualties
in $R$, we can suppose that the total population remains constant, hence
\begin{equation} \label{SIR:NCO}S+I+R=1.\end{equation}
The SIR model then specifies the evolution $S(t)$, $I(t)$ and $R(t)$
of the different individuals over time according to
transition rates between the different compartments.
First of all, in a unit of time, one assumes that some susceptible individual may become infected.
The ansatz in this situation is that the number of new infected people in the unit of time is proportional
to the number of susceptible individuals $S$ (the higher this number, the easier is that someone catches the disease)
and to the number of infected $I$ (the more the infected people, the easier for the epidemic to spread).
Accordingly, if we denote by $\alpha\ge0$ this proportionality coefficient,
in a unit of time, we have that $\alpha SI$ individuals transit from the compartment
of susceptible individuals to that of infected (the parameter $\alpha$ can be seen as
a “transmission rate”).
Natasha McClinton, a surgical nurse, prepares a patient for a procedure in a COVID-19 intensive care unit
(Public Domain image from
Concurrently, some infected people can recover. This number is taken to be proportional
to the number of infected:
e.g., if the medicine is effective in a given proportion of treatments,
the higher the number of people receiving the medical treatment, the higher the number of recovered patients
(also in the pessimistic scenario of counting casualties in this compartment,
the higher the number of infected individuals, the higher the number of possible deaths).
Accordingly, if we denote by $\beta\in[0,1]$ the proportionality coefficient involved in
this transition,
in a unit of time, we have that $\beta I$ individuals move from the compartment
of infected to that of recovered (the parameter $\beta$ thus plays the role of a “recovery rate”).
The transition between compartments described in this way is summarized in Figure <ref>.
It is certainly convenient to translate Figure <ref> into a mathematical formulation:
this is done by writing the system of ordinary differential equations corresponding to this compartmental transit,
\begin{equation}\label{SIR-MOD}
\begin{dcases}
\dot{S}=-\alpha SI,\\
\dot{I}=\alpha SI-\beta I,\\
\dot{R}=\beta I,
\end{dcases}
\end{equation}
where the dot represents the derivative with respect to time,
$S=S(t)$, $I=I(t)$ and $R=R(t)$. The system in (<ref>)
is usually taken as the mathematical description of the SIR model.
It is interesting to observe that, by (<ref>),
$$ \frac{d}{dt}(S+I+R)=
\dot{S}+\dot{I}+\dot{R}=-\alpha SI+(\alpha SI-\beta I)+\beta I=0,$$
consistently with (<ref>).
Similarly, one can also focus only on the first two equations in (<ref>)
to determine the time evolution of the number of susceptible and infected individuals
and then obtain as a byproduct the number of recovered individuals by using (<ref>).
In this way, one can reduce (<ref>) to the system of two ordinary differential equations
\begin{equation}\label{SIRRIDO}
\begin{dcases}
\dot{S}=-\alpha SI,\\
\dot{I}=\alpha SI-\beta I.
\end{dcases}
\end{equation}
In spite of its exceptional conceptual simplicity (after all, we are just saying that
susceptible individuals may become infected and
infected individuals may recover, or possibly die), the SIR model already showcases some
very interesting information about the spread of a disease.
First of all, the first equation in (<ref>) already suggests that
the infection occurs by the contact between infectious and susceptible
people (since the quantity $SI$ is a good model for “random encounters” between $S$ and $I$).
As a consequence, the parameter $\alpha$ in (<ref>) can be considered as
a “contact rate”.
For this reason, measures of
social distancing and lockdowns aim at reducing contacts between people, hence at reducing the value of $\alpha$.
States in a SIR epidemic model and transition between compartments.
The parameter $\beta$ in the second equation in (<ref>) instead reduces the number of infected
people, correspondingly [Beware that since casualties are also counted among recoveries,
a tragic way to increase the number of recoveries consists in killing infectious people.
This is a classical topic for science fiction horror films, such as
“The Crazies” by George A. Romero.]
raising the number of recovers.
To increase the parameter $ \beta$, one effective way is clearly to have better medicines,
since more effective treatments facilitate healing and accelerate recovery rates.
Another useful measure relies on the early detection of infected people and their
isolation in hospitals or quarantine areas: in this way, infected people are de facto
removed from the dynamics of (<ref>)
and, from the mathematical point of view, transit to the $R$ compartment (even if they are not yet recovered
from the medical perspective).
See Figure <ref> to appreciate how decreasing the transmission rate $\alpha$
and increasing the recovery rate $\beta$
can help slowing an epidemic's spread.
In the description of epidemic diseases, it is also interesting to consider the basic reproduction number
(often called “R naught” in jargon), e.g. defined by R naught
\begin{equation}\label{miMNDfgkdaPKSMo0liRnou53} {\mathcal{R}}_o:=\frac\alpha\beta .\end{equation}
Its importance lies in the fact that, by (<ref>),
\begin{equation}\label{miMNDfgkdaPKSMo0li}
\dot{I}=\beta I\left(\frac\alpha\beta S-1\right)=\beta I\left({\mathcal{R}}_o\, S-1\right)
\end{equation}
and thus, since $S\in[0,1]$,
$$ \dot{I}\le\beta I\left({\mathcal{R}}_o-1\right).$$
Hence, when
\begin{equation}\label{2rterOPT14534yhSHE}
we find that
$$ \frac{d}{dt} \ln I(t)=\frac{\dot{I}(t)}{I(t)}\le \beta \left({\mathcal{R}}_o-1\right)=-\beta \left(1-{\mathcal{R}}_o\right),$$
leading to
$$ I(t)\le I(0)\, e^{-\beta \left(1-{\mathcal{R}}_o\right)t} ,$$
which means that the number of infected people decreases exponentially fast and the epidemic is
likely to be overcome sufficiently quickly (conversely, when ${\mathcal{R}}_o>1$
one can expect that the number of infected people will grow exponentially, with obvious tragic consequences).
Flattening the curve of infection: the parameters $(\alpha,\beta)$
are chosen here in $\{(0.7,0.05),\;(0.5,0.1),\;(0.45,0.15)\}$..
The computation in (<ref>) also reveals an interesting
information about the possibility of embanking the spread of the epidemic by reducing the
number of susceptible individuals: indeed, when
\begin{equation}\label{HERDI}{\mathcal{R}}_o\, S<1\end{equation}
a similar computation
as above would lead to an exponential decreasing of infected individuals.
This is related to the notion of “herd immunity” herd immunity,
namely in obtaining a community protection due to a critical proportion of the population having become immune to the disease:
notice indeed that condition (<ref>) prescribes that, to reach this situation,
at least a proportion ${\mathcal{H}}_o:=1-\frac{1}{{\mathcal{R}}_o}$ must have become immune.
If we wish to play with some numbers, an initial estimate of
the World Health Organization on ${\mathcal{R}}_o$ for COVID-19
took into account a possibility of $ {\mathcal{R}}_o\simeq 2.4$,
thus largely violating the optimistic threshold in (<ref>).
Also, with this value of ${\mathcal{R}}_o$, one would obtain a herd immunity threshold ${\mathcal{H}}_o\simeq
1-\frac1{2.4}=0.58\overline{3}$, giving that about $60\%$ of the population
must be immune to protect the community from an exponential spread of epidemic
(but not from the disease[We stress that reaching the herd immunity
does not stop the epidemic overnight.
It only makes the curve of newly infected
people bend down and approach to zero exponentially fast. But, according to
the SIR model, other individuals will get infected even after having reached
the status of herd immunity and the curve representing the total number
of infected people will keep increasing towards a horizontal asymptote.]
in itself).
Reaching such a threshold of immunization without a vaccination campaign is likely extremely dangerous
(since it entails that a vast majority of the population must catch the disease). Since, realistically, vaccines are also not $ 100\%$
effective in avoiding the dissemination of the virus, if one aims at reaching a herd immunity via a vaccination campaign,
the efficacy of the vaccine is an important parameter to take into account.
For instance, if $80\%$ of the population is vaccinated with a vaccine which is effective in $70\%$
of the cases in avoiding new infections, we have that $56\%$ of the population
is immune to the disease (and this number is still below the $60\%$ threshold
discussed above for herd immunity).
Of course, we are not proposing here concrete measures to deal with COVID-19,
and the numbers plugged in for the naive computations above must not be taken seriously,
simply we wanted to stress how important is to deal with epidemics using a scientific approach
and how useful mathematics can be for such a life-or-death situation.
Another striking application of the SIR model consists in the
prediction of the epidemic peak, namely of the maximal number $I_{\max}$ of people
that get simultaneously infected during the epidemic.
Indeed, one can deduce from (<ref>) that
\begin{equation}\label{IMAXESYT} I_{\max} \simeq\frac{{\mathcal{R}}_o- 1-\log {\mathcal{R}}_o}{{\mathcal{R}}_o}.\end{equation}
Notice that as ${\mathcal{R}}_o\to+\infty$ we have that $ I_{\max}\to1$, which corresponds to the total population
ending up being infected at the same time.
We also remark that estimating $I_{\max}$ is of great strategic importance,
since to accommodate patients, one always wants to have a number of available beds in the hospitals
commensurate to the number of people in need of hospitalization.
To prove (<ref>), one can pick a time $t_\star$ at which we expect the number of infected
individuals to be maximal, i.e. such that $I_{\max}=I(t_\star)$. Then, by (<ref>),
$$ 0=\frac{dI}{dt}(t_\star)=\alpha \,S(t_\star)\,I(t_\star)-\beta\, I(t_\star),$$
from which we infer that
$$ S(t_\star)=\frac\beta\alpha=\frac1{{\mathcal{R}}_o}.$$
Notice that this is not quite what (<ref>) is looking for, since $ S(t_\star)$ represents only
the number of susceptible individuals at the time maximizing the number of infected people.
Nevertheless, the structure of (<ref>) comes in handy now and provides that
\begin{eqnarray*} \frac{d(I+S)}{dt}&=& (\alpha SI-\beta I)-\alpha SI\\
&=&-\beta I\\
&=&-\frac{\alpha\beta SI}{\alpha S}\\
&=&\frac{\beta}{\alpha S}\frac{dS}{dt}\\
&=&\frac{1}{{\mathcal{R}}_o}\frac{d}{dt}(\log S).
\end{eqnarray*}
$$ I(t)+S(t)=I(0)+S(0)+\frac{\log(S(t))-\log(S(0))}{{\mathcal{R}}_o}$$
and, as a result,
\begin{eqnarray*}
&=&I(0)+S(0)+\frac{-\log {\mathcal{R}}_o-\log(S(0))}{{\mathcal{R}}_o}-\frac1{{\mathcal{R}}_o}\\&=&
\frac{1+\log {\mathcal{R}}_o+\log(S(0))}{{\mathcal{R}}_o}.\end{eqnarray*}
If we assume that
at the beginning of the epidemic there is nobody that is recovering, namely[No confusion should
arise between the initial number of recovered $R(0)$ and the structural parameter ${\mathcal{R}}_o$
in (<ref>).] that $R(0)=0$, we
deduce from (<ref>) that $I(0)+S(0)=1$,
and thus
$$ I_{\max}=1-
\frac{1+\log {\mathcal{R}}_o+\log(S(0))}{{\mathcal{R}}_o},$$
that completes the proof of (<ref>).
We also mention that explicit solutions of the SIR model are available by using
an implicit “time reparameterization”. Specifically, one considers
the integral of $I$ as a “new time” $\tau$ by setting
\begin{equation}\label{EXPLISIR2} \tau(t):=\int_0^t I(s)\,ds.\end{equation}
Note that $\frac{\tau}{t}$ represents the average number of
infected people at time $t$. In this new variable, the solution of (<ref>) becomes explicit
and takes the form
\begin{equation}\label{EXPLISIR}\begin{dcases}
\tilde S(\tau)=S(0)\,e^{-\alpha \tau},\\
\tilde I(\tau)=I(0)+S(0)(1-e^{-\alpha \tau})-\beta \tau,\\
\tilde R(\tau)=R(0)+\beta \tau.
\end{dcases}
\end{equation}
Of course, one of the limitations of this explicit formulation is that the new solutions $\tilde S$,
$\tilde I$ and $\tilde R$ are functions of an implicitly defined time $\tau$ through the relations $\tilde S(\tau(t))=S(t)$,
$\tilde I(\tau(t))=I(t)$ and $\tilde R(\tau(t))=R(t)$, whence the explicit representation in (<ref>)
is not always particularly pleasant for concrete calculations (see however the forthcoming equation (<ref>)
for a reformulation of the time parameter problem).
To check the validity of (<ref>) one can proceed by implicit differentiation,
observing that from (<ref>) it follows that
\begin{equation*}\begin{dcases}\displaystyle
\frac{d\tilde R}{d\tau}=\beta,\\ \displaystyle
\frac{d\tilde S}{d\tau}=-\alpha \tilde{S},
\end{dcases}
\end{equation*}
which are explicitly solvable leading to $\tilde S(\tau)$ and $\tilde R(\tau)$ in (<ref>).
Then, using
$$\frac{d\tilde I}{d\tau}=\alpha \tilde S-\beta ,$$
and exploiting the previous explicit solution for $\tilde S(\tau)$ one
obtains the expression of $\tilde I(\tau)$ in (<ref>).
Having completed the proof of (<ref>),
we also observe that, by (<ref>),
$$ \frac{d\tau}{dt}=\tilde I(\tau),$$
from which one obtains an explicit integral formula relating the old and the new times of the type
\begin{equation} \label{34EXPLISIR}
t=\int_0^\tau\frac{ds}{I(0)+S(0)(1-e^{-\alpha s})-\beta s}.\end{equation}
Once again, this time reparameterization is explicit, but the integral cannot be usually
written in terms of standard elementary mathematical functions.
We stress that the classical SIR model in (<ref>) does not take into account the possible
mobility of individuals (roughly speaking, the whole population in (<ref>)
is concentrated at the same place). There are of course a number of models available in the literature
which account for possible spatial displacements of the population: for instance, if one models
the space configuration by a variable $x$ and assumes that the population moves according to a random walk
as in Section <ref>, one may think about the possibility of replacing the ordinary differential
equations in (<ref>) with a system of partial differential equations of the type
\begin{equation}\label{9-42coMS02urjf}
\begin{dcases}
\partial_t{S}=\mu\Delta S-\alpha SI,\\
\partial_t{I}=\nu\Delta I+\alpha SI-\beta I.
\end{dcases}
\end{equation}
In this case, $S=S(x,t)$, $I=I(x,t)$ and the diffusion coefficients $\mu$, $\nu\in[0,+\infty)$
may also be different to account for the possibility of a different speed between susceptible and infected people
(e.g., in a situation in which the diffusivity of infected individuals is limited by the illness itself,
or by the responsibility of self-isolating individuals with symptoms).
Restrictions on travel are of course a consequential measure to avoid or limit the spread of an epidemic
due to the diffusive nature of the system in (<ref>).
See for instance <cit.> for additional information on the SIR models
and for many related models utilized in mathematical epidemiology.
Now, for completeness, also in view of the discussion presented in
Section <ref>,
we recall here a variation of the SIR model based on the evolution of
the probability density $p(x,y,t)$ corresponding to having, at time $t$,
a proportion $x\in[0,1]$ of susceptible individuals and $y\in[0,1]$ of infected ones,
as proposed in <cit.>.
The core of this model is to obtain a partial differential equation of the form
\begin{equation}\label{MR2782833:EQUA}
\partial_t p+\div(p F)=0,\end{equation}
where the above divergence is taken with respect to the variables $(x,y)$
and the vector field $F$ corresponds to the right hand side of
the SIR model in (<ref>), that is
\begin{equation}\label{MR2782833:EQUA2}
F(x,y):=\left( -\alpha xy,\; \alpha xy-\beta y\right).
\end{equation}
We remark that (<ref>) has the form of a
transport equation transport equation as in (<ref>).
To deduce (<ref>) from the principles of the SIR model one can argue as follows.
Given $n$, $m\in\N$ with $n+m\le N$, we consider the probability $P(n,m,t)$ of having $n$ susceptible and $m$ infected at time $t$, being $N$ the total number of individuals
(notice that, in view of the constancy of the population in (<ref>), the number of recovered is thus necessarily $N-n-m$). We argue that, given an “infinitesimal” time increment $\tau$, the probability $P(n,m,t+\tau)$ of having $n$ susceptible and $m$ infected at time $t+\tau$ is built by the superposition of three occurrences, namely:
* either, during the elapsed time $\tau$, an infected individual may have recovered: this corresponds to the probability $P(n,m+1,t)$ of having one more infected at time $t$, times the probability of recovering, which, inspired by Figure <ref>, we take to be equal to $\frac{\beta(m+1)}{N}$,
* or, during the elapsed time $\tau$, a susceptible individual may have got infected: this corresponds to the probability $P(n+1,m-1,t)$ of having one more susceptible and one less infected at time $t$, times the probability[In all this discussion, we are
implicitly assuming that $\frac{\beta(m+1)}{N}$, $\frac{\alpha (n+1)(m-1)}{N^2}\in[0,1]$,
consistently with the standard notion of probability. This hypothesis is justified, for instance,
if the number of infected $m$ is sufficiently small with respect to the total population $N$.]
of getting infected, which, inspired by Figure <ref>, we take to be equal to $\frac{\alpha (n+1)(m-1)}{N^2}$,
* or, during the elapsed time $\tau$, no susceptible individual got infected
and no infected got recovered: this corresponds to the probability $P(n,m,t)$ of having the same number of susceptible and infected individuals at time $t$, times the probability of having no infections or recovery, which is equal to the remaining probability $1-
\frac{\beta m}{N}-\frac{\alpha nm}{N^2}$.
Due to these considerations, we write that
\begin{equation}\label{SIRMOPDEMD}\begin{split}&P(n,m,t+\tau)=\frac{\beta(m+1)}{N}P(n,m+1,t)+\frac{\alpha (n+1)(m-1)}{N^2}P(n+1,m-1,t)\\&\qquad\qquad\qquad\qquad\qquad+
\left(1-\frac{\beta m}N-\frac{\alpha nm}{N^2}\right)P(n,m,t).\end{split}\end{equation}
Now we try to achieve a continuum model by taking the limit as $N\to+\infty$, corresponding to a large population. For this, for all $x$, $y\in[0,1]$
we define
In this way, we deduce from (<ref>) that
\begin{eqnarray*}&&\partial_t p(x,y,t)\\&\simeq&\frac{p(x,y,t+\tau)-p(x,y,t)}{\tau}\\&=&\frac{N}{\tau}\Big[ P(Nx,Ny,t+\tau)-P(Nx,Ny,t)\Big]\\&=&\frac{N}{\tau}\Bigg[ \frac{\beta(Ny+1)}{N}P(Nx,Ny+1,t)+
\frac{\alpha (Nx+1)(Ny-1)}{N^2}P(Nx+1,Ny-1,t)\\&&\qquad\qquad+
\left(1- \beta y-\alpha xy\right)P(Nx,Ny,t)-P(Nx,Ny,t)\Bigg]\\
\beta\left(y+\frac1{N}\right)p\left(x,y+\frac1N,t\right)+
\alpha \left(x+\frac1N\right)\left(y-\frac1N\right)p\left(x+\frac1N,y-\frac1N,t\right)\\&&\qquad\qquad-
\left( \beta y+ \alpha xy\right)p(x,y,t)
\Bigg]\\&=&
\frac{1}{\tau}\Bigg[B\left(x,y+\frac1N,t\right)+A\left(x+\frac1N,y-\frac1N,t\right)-B(x,y,t)-A(x,y,t)
\Bigg]\\&\simeq&
\frac{1}{\tau}\Bigg[\frac1N\partial_y B(x,y,t)+
\frac1N\left(\partial_x A(x,y,t)-\partial_yA(x,y,t)\right) \Bigg]
$$ A(x,y,t):=\alpha xy\, p(x,y,t)
\qquad{\mbox{and}}\qquad
B(x,y,t):=\beta y \,p(x,y,t).$$
Hence, by choosing a scaling between the time step and the total population such that $\tau N\simeq1$,
we reduce the infectious disease spreading problem to
$$ \partial_t p=\partial_x A+\partial_y (B-A)=
\partial_x (\alpha xy p)+\partial_y\big((\beta y-\alpha xy)p\big),
which, due to the choice of the vector field in (<ref>), corresponds to the desired
transport equation in (<ref>).
For further motivations about elliptic partial differential equations, see e.g. page 0uojf29249-45kpkfdSmd11493839429efv
here, Chapter 2 in <cit.>,
Chapter 4 in <cit.>,
Chapter 1 in <cit.>,
pages 160–168 in <cit.>,
the introduction of <cit.>,
Chapter 1 in <cit.>,
Section 1.4 in <cit.>, Chapter 1 in <cit.>,
Chapters 4 and 5 in <cit.>,
Section 1.2 in <cit.>,
the appendix of <cit.>
and the references therein.
§ WHAT IS THE LAPLACIAN, AFTER ALL?
We now dive into some mathematical setting
which turns out to be handy when dealing with partial differential equations.
Given an open set $\Omega\subseteq\R^n$,
a function $u\in C^2(\Omega)$ and a point $x\in\Omega$, we consider the “Laplace operator” $\Delta$
(which we have already informally met and used, from (<ref>) on)
defined by
\begin{equation}\label{LA:DIFF}
\Delta u(x):=\sum_{j=1}^n \frac{\partial^2u}{\partial x_j^2}(x).\end{equation}
We point out that the Laplacian is invariant under translation,
\begin{equation}\label{TRAL} \Delta \big(u(x+y)\big)=\Delta u(x+y)\qquad{\mbox{for every~$x\in\Omega$ and~$y\in\R^n$ such that~$x+y\in\Omega$,}}\end{equation}
and it
possesses a “divergence form”divergence form structure, since
\begin{equation}\label{DITRO} \Delta u={\rm div}(\nabla u).\end{equation}
As a consequence of this and of the Divergence Theorem, if $\Omega$
has a boundary of class $C^1$,
the average of the Laplacian
can be reconstructed by the normal flow through the boundary of the domain, namely
\begin{equation}\label{1111DIV0934} \int_\Omega \Delta u(x)\,dx=\int_{\partial\Omega } \nabla u(x)\cdot \nu(x)\,d{\mathcal{H}}^{n-1}_x,\end{equation}
where $\nu$ denotes the external unit normal of $\Omega$ and ${\mathcal{H}}^{n-1}$
is the $(n-1)$-dimensional Hausdorff measure.
It is also useful to remark that the Laplacian measures the “infinitesimal distance”
between the value of a function at a given point and the average nearby, namely:
Let $x_0\in\R^n$ and $r>0$. Suppose that $u\in C^2(B_r(x_0))$.
\begin{equation}\label{LI} \lim_{\rho\searrow0} \frac1{\rho^2}\left(
\fint_{B_\rho(x_0)} u(x)\,dx-u(x_0)\right)=\frac{1}{2(n+2)}\,\Delta u(x_0).\end{equation}
We start by recalling a standard relation between the Lebesgue measure of
the unit ball and the $(n-1)$-dimensional Hausdorff measure of the unit sphere. Namely,
using polar coordinates, we see that
\begin{equation}\label{B1}
|B_1|={\mathcal{H}}^{n-1}(\partial B_1)\,\int_0^1 t^{n-1}\,dt=\frac{ {\mathcal{H}}^{n-1}(\partial B_1)}{n}.\end{equation}
We now point out a useful cancellation property. If $T_j$ is the reflection across the $j$th coordinate, i.e.
$$ T_j (y_1,\dots,y_n)=(y_1,\dots,y_{j-1},-y_j,y_{j+1},\dots,y_n),$$
and $g: B_\rho\to \R$ is such that $g(T_j(y))=-g(y)$ for all $y\in B_\rho$, using the change of variable $Y:=T_j(y)$
it follows that
\begin{equation}\label{22345XXse5q6u5at3i332o2n} \int_{B_\rho} g(y)\,dy=-\int_{B_\rho} g(T_j(y))\,dy=-\int_{B_\rho} g(Y)\,dY,\end{equation}
and therefore
$$ \int_{B_\rho} g(y)\,dy=0.$$
Applying this observation to the function $y_j$ for every $j\in\{1,\dots,n\}$,
we find that
\begin{equation}\label{CAN1} \int_{B_\rho} y_j\,dy=0.\end{equation}
Similarly, applying the previous observation to the function $y_j y_k$
for every $j\ne k\in\{1,\dots,n\}$,
\begin{equation}\label{CAN2} \int_{B_\rho} y_j y_k\,dy=0.\end{equation}
Now we perform an explicit computation in the ball by taking advantage
of the symmetry between coordinate exchanges.
Namely, for every $j\in\{1,\dots,n\}$, using polar coordinates we have that
\begin{equation}\label{H66}
n\int_{B_\rho} y_{j}^2\,dy=
\sum_{k=1}^n\int_{B_\rho} y_k^2\,dy=
\int_{B_\rho} |y|^2\,dy={\mathcal{H}}^{n-1}(\partial B_1)\,\int_0^\rho t^{n+1}\,dt=
\frac{ {\mathcal{H}}^{n-1}(\partial B_1)\, \rho^{n+2} }{n+2}.
\end{equation}
Now, we focus on the proof
of the desired result in (<ref>). For this,
given $\rho>0$ and $x\in B_\rho(x_0)$, we use the Taylor expansion
\begin{equation}\label{TAYTAY}
u(x)=u(x_0)+\nabla u(x_0)\cdot (x-x_0)+\frac12 D^2 u(x_\star)(x-x_0)\cdot(x-x_0),\end{equation}
for a suitable $x_\star \in B_\rho(x_0)$ and we find that
\begin{eqnarray*}
\fint_{B_\rho(x_0)} u(x)\,dx-u(x_0)&=&\fint_{B_\rho(x_0)}\big( u(x)-u(x_0)\big)\,dx\\&=&
\fint_{B_\rho(x_0)} \left(
\nabla u(x_0)\cdot (x-x_0)+\frac12 D^2 u(x_\star)(x-x_0)\cdot(x-x_0)\right)
\,dx\\&=&
\sum_{j=1}^n
\fint_{B_\rho} \frac{\partial u}{\partial x_j}(x_0)\,y_j\,dy+\frac12\sum_{j,k=1}^n
\fint_{B_\rho} \frac{\partial^2 u}{\partial x_j\partial x_k}(x_\star)\,y_j y_k\,dy\\&=&\frac12
\sum_{j,k=1}^n
\fint_{B_\rho} \frac{\partial^2 u}{\partial x_j\partial x_k}(x_\star)\,y_j y_k\,dy,
\end{eqnarray*}
where the latter identity is a consequence of (<ref>)
and (<ref>).
Consequently, by (<ref>),
\begin{equation}\label{ETA0}\begin{split} \fint_{B_\rho(x_0)} u(x)\,dx-u(x_0)\,&=\,\frac12
\sum_{j,k=1}^n
\fint_{B_\rho} \frac{\partial^2 u}{\partial x_j\partial x_k}(x_\star)\,y_j y_k\,dy\\&=\,\frac12
\sum_{j=1}^n
\fint_{B_\rho} \frac{\partial^2 u}{\partial x_j^2 }(x_0)\,y_j^2\,dy+\eta(\rho)
\begin{equation}\label{ETA} \eta(\rho):=
\frac12 \sum_{j,k=1}^n\fint_{B_\rho}\left(\frac{\partial^2 u}{\partial x_j\partial x_k}(x_\star)-
\frac{\partial^2 u}{\partial x_j\partial x_k}(x_0)\right)\,y_j y_k\,dy.\end{equation}
Now we observe that
\begin{equation}\label{H9}
\lim_{\rho\searrow0}\frac{\eta(\rho)}{\rho^{2}}=0.
\end{equation}
Indeed, since $x_\star$ approaches $x_0$ as $\rho\searrow0$,
given any $\e>0$, if $\rho$ is small enough we have that
$$ \left|\frac{\partial^2 u}{\partial x_j\partial x_k}(x_\star)-
\frac{\partial^2 u}{\partial x_j\partial x_k}(x_0)\right|\le\e$$
and consequently, using again (<ref>),
$$ |\eta(\rho)|\le
\frac{\e}2 \sum_{j=1}^n\fint_{B_\rho} y_j^2\,dy=
\frac{ {\mathcal{H}}^{n-1}(\partial B_1)\, \rho^{2}\,\e }{2(n+2)\,|B_1|}
From this, the claim in (<ref>) plainly follows.
The desired result in (<ref>) is now a direct consequence of (<ref>),
(<ref>), (<ref>)
and (<ref>), since
\begin{eqnarray*}&&
\lim_{\rho\searrow0} \frac1{\rho^2}\left(
\fint_{B_\rho(x_0)} u(x)\,dx-u(x_0)\right)=
\lim_{\rho\searrow0} \left(
\frac1{2\rho^2}
\sum_{j=1}^n
\fint_{B_\rho} \frac{\partial^2 u}{\partial x_j^2}(x_0)\,y_j^2\,dy+\frac{\eta(\rho)}{\rho^2}\right)
\\&&\qquad=
\frac{ {\mathcal{H}}^{n-1}(\partial B_1) }{2n(n+2)\,|B_1|}
\sum_{j=1}^n \frac{\partial^2 u}{\partial x_j^2}(x_0)=\frac{1}{2(n+2)}\,\Delta u(x_0),
\end{eqnarray*}
as desired.
For completeness, as a variant of Theorem <ref>,
we also point out a similar result for the limit of the spherical
Let $x_0\in\R^n$ and $r>0$. Suppose that $u\in C^2(B_r(x_0))$.
\begin{equation*} \lim_{\rho\searrow0} \frac1{\rho^2}\left(
\fint_{\partial
B_\rho(x_0)} u(x)\,d{\mathcal{H}}^{n-1}_x-u(x_0)\right)=
\frac{1}{2n}\,\Delta u(x_0).\end{equation*}
By the
Taylor expansion in (<ref>) and two odd cancellation
arguments, as $\rho\searrow0$ we have
\begin{eqnarray*}
&& \fint_{\partial
B_\rho(x_0)} u(x)\,d{\mathcal{H}}^{n-1}_x-u(x_0)\\&=&
\fint_{\partial
B_\rho(x_0)} \left(
\nabla u(x_0)\cdot (x-x_0)+\frac12 D^2 u(x_0)(x-x_0)\cdot(x-x_0)\right)
\,d{\mathcal{H}}^{n-1}_x
\frac12\,\fint_{\partial
B_\rho(x_0)} D^2 u(x_0)(x-x_0)\cdot(x-x_0)
\,d{\mathcal{H}}^{n-1}_x
\frac12\,\sum_{i,j=1}^n\fint_{\partial
B_\rho} \partial_{ij} u(x_0)y_iy_j
\,d{\mathcal{H}}^{n-1}_y
\frac12\,\sum_{i=1}^n\fint_{\partial
B_\rho} \partial_{ii} u(x_0)y_i^2
\,d{\mathcal{H}}^{n-1}_y
\end{eqnarray*}
Now, for every $i\in\{1,\dots,n\}$, we have that
$$ n\,\int_{\partial B_\rho} y_i^2
\,d{\mathcal{H}}^{n-1}_y=\sum_{k=1}^n
\int_{\partial B_\rho} y_k^2
\,d{\mathcal{H}}^{n-1}_y=\int_{\partial B_\rho} |y|^2
\,d{\mathcal{H}}^{n-1}_y=
\int_{\partial B_\rho} \rho^2
\,d{\mathcal{H}}^{n-1}_y=\rho^{2}\,{\mathcal{H}}^{n-1}(\partial B_\rho)
and consequently
\begin{eqnarray*}
&& \fint_{\partial
B_\rho(x_0)} u(x)\,d{\mathcal{H}}^{n-1}_x-u(x_0)=
\frac1{2n}\,\sum_{i=1}^n \partial_{ii} u(x_0)\rho^2
+o(\rho^2)=\frac{\rho^2}{2n}\,\Delta u(x_0)+o(\rho^2),
\end{eqnarray*}
that plainly leads to the desired result.
We point out that
Theorems <ref> and <ref>
are special cases of a more general type of result, known in the literature
as Pizzetti's FormulaPizzetti's Formula, involving the
higher order Laplace operator
$$ \Delta^k:=\underbrace{\Delta\dots\Delta}_{\footnotesize{\mbox{$k$ times}}},$$
with $k\in\N$ and the convention that $\Delta^0$ is the identity operator.
We recall this setting for the sake of completeness:
Let $x_0\in\R^n$ and $r>0$. Suppose that $u\in C^{2N}(B_r(x_0))$. Then, as $\rho\searrow0$,
\begin{eqnarray}&&
\label{PALLAPIZZE1} \fint_{B_\rho(x_0)}u(x)\,dx
n\, \Gamma\left(\frac{n}2\right)\sum_{k=0}^N
\frac{\rho^{2k}}{2^{2k+1}\,
k!\,\Gamma\left(\frac{n}2+k+1\right)}\Delta^k u(x_0)+o(\rho^{2N})
\\
{\mbox{and }}&&
\label{PALLAPIZZE2} \fint_{\partial B_\rho(x_0)}u(x)\,d{\mathcal{H}}^{n-1}_x=
\Gamma\left(\frac{n}2\right)\sum_{k=0}^N
\frac{\rho^{2k}}{2^{2k}\,k!\,\Gamma\left(\frac{n}2+k\right)}\Delta^k u(x_0)+o(\rho^{2N}).\end{eqnarray}
Here above, $\Gamma$ stands for the Euler Gamma Function, defined,
for every $z\in\C$ with $\Re z>0$, by
\begin{equation}\label{EULEROGA}
\Gamma (z):=\int _{0}^{+\infty }t^{z-1}e^{-t}\,dt.\end{equation}
We start with an observation that can be considered
as a version of the Multinomial Theorem for the Laplace operator. We claim that,
for every $k\in\N$,
\begin{equation}\label{MULTOLPALPA}
\Delta^k u=\sum_{{\alpha\in\N^n}\atop{|\alpha|=k}}\frac{k!}{\alpha!}\,D^{2\alpha}u.
\end{equation}
Here, we are using the multi-index notation for which $\alpha=(\alpha_1,\dots,\alpha_n)$,
$|\alpha|=\alpha_1+\dots+\alpha_n$, $\alpha!=\alpha_1!\dots\alpha_n!$
and $2\alpha=(2\alpha_1,\dots,2\alpha_n)$.
We prove (<ref>) by induction over $k$.
Indeed, when $k=0$,
the claim in (<ref>) reduces to the true identity $u=u$.
Then, we suppose that (<ref>) holds true for the index $k$
and observe that, being $e_i$ the $i$th element of the Euclidean basis,
\begin{eqnarray*}&&
\Delta^{k+1} u=\Delta\left(
\sum_{{\alpha\in\N^n}\atop{|\alpha|=k}}\frac{k!}{\alpha!}\,D^{2\alpha}u\right)=
\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}
\left(
\sum_{{\alpha\in\N^n}\atop{|\alpha|=k}}\frac{k!}{\alpha!}\,D^{2\alpha}u\right)
\sum_{{{\alpha\in\N^n}\atop{|\alpha|=k}}\atop{1\le i\le n}}
\frac{k!}{\alpha!}\,D^{2(\alpha+e_i)}u\\&&\qquad\qquad\qquad=
\sum_{{{\alpha\in\N^n}\atop{|\alpha|=k}}\atop{1\le i\le n}}
\frac{k!\,(\alpha_i+1)}{(\alpha+e_i)!}\,D^{2(\alpha+e_i)}u=
\sum_{{{\beta\in\N^n}\atop{|\beta|=k+1}}\atop{1\le i\le n}}
\frac{k!\,\beta_i}{\beta !}\,D^{2\beta}u=
\sum_{{{\beta\in\N^n}\atop{|\beta|=k+1}}}
\frac{k!\,|\beta|}{\beta!}\,D^{2\beta}u\\&&\qquad\qquad\qquad=
\sum_{{{\beta\in\N^n}\atop{|\beta|=k+1}}}
\frac{k!\,(k+1)}{\beta!}\,D^{2\beta}u=
\sum_{{{\beta\in\N^n}\atop{|\beta|=k+1}}}
\frac{(k+1)!}{\beta!}\,D^{2\beta}u,
\end{eqnarray*}
which completes the inductive step and provides the proof of (<ref>).
Now we point out that, by odd symmetry
of the function $x_i\mapsto x_i^{\alpha_i}$,
if $\alpha_i$ is an odd integer then $ \int_{B_\rho} x^{\alpha}\,dx=0$
and consequently
\begin{equation}\label{MULTOLPALPA2} \int_{B_\rho} x^{\alpha}\,dx=0\qquad{\mbox{unless $\alpha=2\beta$ for some~$\beta\in\N^n$.}}
\end{equation}
In the same way,
\begin{equation}\label{MULTOLPALPA3} \int_{\partial B_\rho} x^{\alpha}\,d{\mathcal{H}}^{n-1}_x=0\qquad{\mbox{unless $\alpha=2\beta$ for some~$\beta\in\N^n$.}}
\end{equation}
It is now convenient to perform some specific calculation
related to spherical integrals.
So as to achieve this goal,
recalling the definition (<ref>) of
Euler Gamma Function
using the substitution $\tau:=t^2$,
we point out that, for every $b>-1/2$,
\begin{equation}\label{KSM-ONSN Dcakgau934} \int_0^{+\infty} t^{2b} e^{-t^2}\,dt=
\frac12\,\int_0^{+\infty} \tau^{\frac{2b-1}2} e^{-\tau} \,d\tau
Consequently, for every $\alpha\in\N^n$,
\begin{equation}\label{KSM-ONSN Dcakgau9342}
\int_{\R^n} x^{2\alpha}e^{-|x|^2}\,dx=\prod_{i=1}^n
\int_{\R} x_i^{2\alpha_i}e^{-x_i^2}\,dx_i=2^n\,
\prod_{i=1}^n
\int_{0}^{+\infty} x_i^{2\alpha_i}e^{-x_i^2}\,dx_i=
\prod_{i=1}^n\Gamma\left({\frac{2\alpha_i+1}2}\right).
\end{equation}
On the other hand, using polar coordinates and once again (<ref>),
\begin{equation*}
\int_{\R^n} x^{2\alpha}e^{-|x|^2}\,dx=\int_{\partial B_1}
\left(\int_0^{+\infty} r^{2|\alpha|+n-1}
\omega^{2\alpha}e^{-r^2}\,dr
\right)\,d{\mathcal{H}}^{n-1}_\omega=
\frac{1}2\,\Gamma\left({\frac{2|\alpha|+n}2}\right)\,
\int_{\partial B_1}
\omega^{2\alpha}\,d{\mathcal{H}}^{n-1}_\omega.
\end{equation*}
Comparing this with (<ref>),
we deduce that
\begin{equation}\label{KSMX:att89i45tgsbad2glifdf0athbnergr}
\int_{\partial B_1}
\omega^{2\alpha}\,d{\mathcal{H}}^{n-1}_\omega
\displaystyle\Gamma\left({\frac{2|\alpha|+n}2}\right)
\end{equation}
It is also instructive to recall that, integrating by parts,
\begin{equation}\label{GAMMAPIU1} \Gamma (z+1)=\int _{0}^{+\infty }t^{z}e^{-t}\,dt=-
\int _{0}^{+\infty }t^{z}\,\frac{d}{dt}e^{-t}\,dt
=\int _{0}^{+\infty }zt^{z-1}e^{-t}\,dt=z\,\Gamma(z),\end{equation}
from which it follows by induction that
$$ \Gamma(j+1)=j!\qquad{\mbox{for every }}j\in\N.$$
Furthermore, making use of (<ref>) with $b:=0$,
\begin{equation}\label{ZKMdgamzo34}
\Gamma\left(\frac12\right)=2\int_0^{+\infty}e^{-t^2}\,dt=
\int_{-\infty}^{+\infty}e^{-t^2}\,dt=\sqrt{\pi}.
\end{equation}
Now we claim that, for every $j\in\N$,
\begin{equation}\label{9ijn8uh9ijGSDNlsodgue0olg:SL467A}
\Gamma\left(j+1+\frac12\right)=\frac{\sqrt\pi\,(2j+1)!}{2^{2j+1} \,j!}
\end{equation}
This statement is actually a particular case
of the Legendre Duplication Formula,
or of the
Gauß Multiplication Formula, but we provide a direct proof of (<ref>)
for the facility of the reader. For this, we argue by induction over $j$.
We remark that
\begin{equation}\label{GAMMA3mez} \Gamma\left(1+\frac12\right)=\frac12\,\Gamma\left(\frac12\right)
\frac{\sqrt\pi}{2}
\end{equation}
thanks to (<ref>) and (<ref>),
and this gives the claim in (<ref>)
when $j=0$.
Suppose now that (<ref>)
holds true for $j$. Then, we use again (<ref>) to obtain that
\begin{eqnarray*}
\Gamma\left(j+1+\frac12\right)=
\left(j+1+\frac12\right)\,\frac{\sqrt\pi\,(2j+1)!}{2^{2j+1} \,j!}\\&&\qquad=
\left(2j+3\right)\,\frac{\sqrt\pi\,(2j+1)!}{2^{2j+2} \,j!}=
\left(2j+3\right)\left(2j+2\right)\,\frac{\sqrt\pi\,(2j+1)!}{2^{2j+3}\,(j+1) \,j!}
=\frac{\sqrt\pi\,(2j+3)!}{2^{2j+3} \,(j+1)!},
\end{eqnarray*}
thus completing the inductive step and establishing (<ref>).
Hence, in light of (<ref>)
and (<ref>),
\begin{equation}\label{KSMX:att89i45tgsbad2glifdf0athbnergr-2}
\int_{\partial B_1}
\omega^{2\alpha}\,d{\mathcal{H}}^{n-1}_\omega
\displaystyle\Gamma\left({\frac{2|\alpha|+n}2}\right)
\frac{\sqrt\pi\,(2\alpha_i-1)!}{2^{2\alpha_i-1} \,(\alpha_i-1)!}
\displaystyle\prod_{i=1}^n\prod_{j=0}^{\alpha_i-1}(\alpha_i
\begin{equation}\label{8-9-46-249-2394-84rjf-238rikf223-4yci-239rij7n-12era}
\begin{split} &\int_{B_1} x^{2\alpha}\,dx=
\int_{\partial B_1}\left(\int_0^1 r^{2|\alpha|+n-1}\omega^{2\alpha}\,dr\right)\,
\frac{1}{2|\alpha|+n}\int_{\partial B_1}
\omega^{2\alpha}\,
d{\mathcal{H}}^{n-1}_\omega \\&\qquad\qquad\qquad\qquad
\frac{2^{n+1-2|\alpha|}\,\pi^{\frac{n}{2}}
\displaystyle\prod_{i=1}^n\prod_{j=0}^{\alpha_i-1}(\alpha_i
\end{equation}
We now prove (<ref>). For this, up to a translation,
we can reduce to the case $x_0=0$. We exploit the Taylor expansion
\begin{equation}\label{MULTOLPALPA4} u(x)=\sum_{{\alpha\in\N^n}\atop{|\alpha|\le 2N}}
\frac{D^\alpha u(0)}{\alpha!}\,x^\alpha+o(x^{2N})\end{equation}
and we average over $B_\rho$, thus finding that, for small $\rho$,
\begin{eqnarray*}
\fint_{B_\rho}u(x)\,dx&=&\sum_{{\alpha\in\N^n}\atop{|\alpha|\le 2N}}
\frac{D^\alpha u(0)}{\alpha!}\,\fint_{B_\rho}x^\alpha\,dx+o(\rho^{2N})\\
&=&\sum_{{\beta\in\N^n}\atop{|\beta|\le N}}
\frac{D^{2\beta} u(0)}{(2\beta)!}\,\fint_{B_\rho}x^{2\beta}\,dx+o(\rho^{2N})\\
&=&\sum_{{\beta\in\N^n}\atop{|\beta|\le N}}
\frac{D^{2\beta} u(0)\,\rho^{2|\beta|}}{(2\beta)!}\,\fint_{B_1}y^{2\beta}\,dy
\\&=&\sum_{k=0}^N\sum_{{\beta\in\N^n}\atop{|\beta|=k}}
\frac{D^{2\beta} u(0)\,\rho^{2k}}{(2\beta)!}\,\fint_{B_1}y^{2\beta}\,dy+o(\rho^{2N})
\\&=&
\sum_{k=0}^N\sum_{{\beta\in\N^n}\atop{|\beta|=k}}
\frac{D^{2\beta} u(0)\,\rho^{2k}}{(2\beta)!\,|B_1|}\,\frac{2^{n+1-2k}\,
\pi^{\frac{n}{2}}\displaystyle\prod_{i=1}^n\prod_{j=0}^{\beta_i-1}(\beta_i
\sum_{k=0}^N\sum_{{\beta\in\N^n}\atop{|\beta|=k}}
\frac{D^{2\beta} u(0)\,\rho^{2k}}{ \beta!\,|B_1|}\,
\frac{2^{1-2k}\,\pi^{\frac{n}{2}}
\\&=&
\sum_{k=0}^N
\frac{\Delta^k u(0)\,\rho^{2k}}{ |B_1|}\,
\frac{2^{1-2k}\,\pi^{\frac{n }{2}}
thanks to (<ref>), (<ref>) and (<ref>).
Hence, since, owing to (<ref>),
(used here with $\alpha:=0$) and (<ref>),
\begin{equation}\label{MULTOLPALPA280kdsmc8724yrth0923}
\Gamma\left({\frac{n}2}\right)
=\frac{2 }{{\mathcal{H}}^{n-1}(\partial B_1)}\,\left(
\Gamma\left({\frac{1}2}\right)\right)^n=\frac{2\pi^{\frac{n}2} }{{\mathcal{H}}^{n-1}(\partial B_1)}
=\frac{2\pi^{\frac{n}2}}{n\,| B_1|},\end{equation}
we obtain that
\begin{equation*}
\begin{split}
\fint_{B_\rho}u(x)\,dx\,&=\,n\,
\Gamma\left({\frac{n}2}\right)
\sum_{k=0}^N
\frac{
\Delta^k u(0)\,\rho^{2k}}{2^{2k}\,k!\,\displaystyle(2k+n)\,\Gamma\left({\frac{2k+n}2}\right)}
\Gamma\left({\frac{n}2}\right)
\sum_{k=0}^N
\frac{\Delta^k u(0)\,\rho^{2k}
\end{split}
\end{equation*}
where (<ref>) has been used again in the last line.
This completes the proof of (<ref>).
Now, to establish (<ref>),
we exploit (<ref>), (<ref>), (<ref>)
and (<ref>), and we see that
\begin{eqnarray*}
\fint_{\partial B_\rho}u(x)\,d{\mathcal{H}}^{n-1}_x
&=&\sum_{{\alpha\in\N^n}\atop{|\alpha|\le 2N}}
\frac{D^\alpha u(0)}{\alpha!}\,\fint_{\partial B_\rho}x^\alpha\,d{\mathcal{H}}^{n-1}_x+o(\rho^{2N})\\
&=&\sum_{{\beta\in\N^n}\atop{|\beta|\le N}}
\frac{D^{2\beta} u(0)}{(2\beta)!}\,\fint_{\partial B_\rho}x^{2\beta}\,
&=&\sum_{{\beta\in\N^n}\atop{|\beta|\le N}}
\frac{D^{2\beta} u(0)\,\rho^{2|\beta|}}{(2\beta)!}\,\fint_{\partial
\\&=&\sum_{k=0}^N
\sum_{{\beta\in\N^n}\atop{|\beta|=k}}
\frac{D^{2\beta} u(0)\,\rho^{2k}}{(2\beta)!}\,\fint_{\partial
\sum_{{\beta\in\N^n}\atop{|\beta|=k}}
\frac{D^{2\beta} u(0)\,\rho^{2k}}{(2\beta)!\,{\mathcal{H}}^{n-1}(\partial B_1)}\,
\frac{2^{n+1-2k}\,\pi^{\frac{n}{2}}
\displaystyle\prod_{i=1}^n\prod_{j=0}^{\beta_i-1}(\beta_i
\sum_{{\beta\in\N^n}\atop{|\beta|=k}}
\frac{D^{2\beta} u(0)\,\rho^{2k}}{ \beta!\,{\mathcal{H}}^{n-1}(\partial B_1)}\,
\frac{2^{1-2k}\,\pi^{\frac{n}{2}}
\sum_{k=0}^N
\frac{\Delta^k u(0)\,\rho^{2k}}{k!\,{\mathcal{H}}^{n-1}(\partial B_1)}\,
\frac{2^{1-2k}\,\pi^{\frac{n}{2}}
This, together with (<ref>), proves (<ref>), as desired
(alternatively, one could have proved one between (<ref>)
and (<ref>) and obtained the other by either integration or differentiation
in $\rho$).
We stress that Theorems <ref> and <ref>
are particular cases of (<ref>)
and (<ref>)
respectively, just corresponding to the case $N:=1$.
In spite of its simple flavor, Theorem <ref> reveals one of the fundamental
features of the Laplacian, which will also play an important role in the Mean Value Formula
that will be described in Theorem <ref>. Also, it immediately leads to
the fact that the Laplace operator is invariant under rotation, namely:
Let ${\mathcal{R}}:\R^n\to\R^n$ be a rotation
and $u\in C^2(\R^n)$. Let $u_{\mathcal{R}}(x):=u({\mathcal{R}}x)$.
$$ \Delta u_{\mathcal{R}}(x)=\Delta u({\mathcal{R}}x).$$
Let $x_0\in\R^n$.
By Theorem <ref>, using the change of variable $y:={\mathcal{R}}(x-x_0)$
and the notation $v(x):=u(x+{\mathcal{R}}x_0)$, and exploiting the translation invariance in (<ref>),
we have
\begin{eqnarray*}
&& \frac{1}{2(n+2)}\,\Delta u_{\mathcal{R}}(x_0)=
\lim_{\rho\searrow0} \frac1{\rho^2}\left(
\fint_{B_\rho(x_0)} u_{\mathcal{R}}(x)\,dx-u_{\mathcal{R}}(x_0)\right)=
\lim_{\rho\searrow0} \frac1{\rho^2}\left(
\fint_{B_\rho(x_0)} u({\mathcal{R}}x)\,dx-u({\mathcal{R}}x_0)\right)\\&&\qquad=
\lim_{\rho\searrow0} \frac1{\rho^2}\left(
\fint_{B_\rho} u(y+{\mathcal{R}}x_0)\,dy-u({\mathcal{R}}x_0)\right)=
\lim_{\rho\searrow0} \frac1{\rho^2}\left(
\fint_{B_\rho} v(y)\,dy-v(0)\right)\\&&\qquad=\frac{1}{2(n+2)}\,\Delta v(0)=
\frac{1}{2(n+2)}\,\Delta u({\mathcal{R}}x_0)
This proves the desired result (for a proof not relying on
Theorem <ref> but rather on a direct computation in matrix form
see e.g. <cit.>).
Of course, while the integral characterization of the Laplacian presented in Theorem <ref>
is conceptually very useful and reveals a deep geometric structure of the operator,
the explicit differential structure in (<ref>) is often
simpler to exploit for explicit calculations.
As an example, we recall the following Bochner IdentityBochner Identity:
For a given $C^3$ function $u$,
$$ \Delta \left( \frac {|\nabla u|^{2}}{2}\right)=
\nabla( \Delta u)\cdot\nabla u
\begin{equation}\label{D22H}
|D^2u|^2:=\sum_{i,j=1}^n (\partial_{ij} u)^2
By a direct computation,
\begin{equation*}\begin{split}&
\Delta \left( \frac {|\nabla u|^{2}}{2} \right)=\frac12\,\sum_{i,j=1}^n
\partial_{ii} (\partial_j u)^2=
\sum_{i,j=1}^n
\partial_i (\partial_j u\,\partial_{ij}u)\\&\qquad
+\partial_j u\,\partial_{iij}u)
\Big)=|D^2u|^2+\sum_{j=1}^n\partial_ju\,\partial_j(\Delta u)
CHAPTER: THE LAPLACE OPERATOR AND HARMONIC FUNCTIONS
This chapter is devoted to the analysis of the Laplace operator
and of the functions that lie in its kernel.
§ THE LAPLACIAN AND THE MEAN VALUE FORMULA
Among the several properties that a given function may possess,Mean Value Formula
a very relevant one is “harmonicity”, corresponding to the vanishing of
the trace of the Hessian matrix (in particular, these functions
are “saddle-looking” with respect to their tangent planes at every point).
The precise setting that we consider is the following:
Given an open set $\Omega\subseteq\R^n$ and
a function $u\in C^2(\Omega)$, we say that $u$ is harmonic in $\Omega$
if $\Delta u(x)=0$ for every $x\in\Omega$.
For example, constant and linear functions are harmonic in all $\R^n$.
Also, the functions $u:\R^2\to\R$ given by $u(x_1,x_2)=x_1 x_2$,
$u(x_1,x_2)=x_1^2- x_2^2$ and $u(x_1,x_2)=e^{x_1}\sin x_2$ are harmonic.
Other examples of harmonic functions in domains of $\R^2$ can be obtained via complex analysis,
identifying $(x,y)\in\R^2$ with $z=x+iy\in \C$,
since the real and imaginary parts of holomorphic functions are harmonic, see <cit.>.
In particular, for every $j\in\N$, using the notation $r=|z|=\sqrt{x^2+y^2}$
and $z=|z| e^{i\vartheta}=re^{i\vartheta}$,
the functions
\begin{equation}\label{REj} \Re z^j =r^j \cos(j\vartheta)\qquad{\mbox{and}}\qquad
\Im z^j =r^j \sin(j\vartheta)\end{equation}
are harmonic in all $\R^2$, and so are the functions
$$ \Re e^z =e^x \cos y\qquad{\mbox{and}}\qquad
\Im e^z =e^x \sin y.$$
In addition, given $\alpha>0$, the function $z\mapsto z^\alpha:=r^\alpha e^{i\alpha\vartheta}$ is
well defined and holomorphic in $\vartheta\in(-\pi,\pi)$, hence
$$ \Re z^\alpha =r^\alpha \cos(\alpha\vartheta)\qquad{\mbox{and}}\qquad
\Im z^\alpha =r^\alpha \sin(\alpha\vartheta)$$
are harmonic in $\R^2\setminus\ell$, being $\ell:=(-\infty,0]\times\{0\}$.
One of the most striking properties of harmonic functions
is that their value at any point is precisely equal to the average
of the values around such point.
In this sense,
the values attained by harmonic functions happen to be “perfectly balanced”,
according to the following result:
Given an open set $\Omega\subseteq\R^n$ and
a function $u\in L^1_{\rm loc}(\Omega)$,
the following conditions are equivalent:
(i). The function $u$
belongs to $C^2(\Omega)$ and
is harmonic in $\Omega$.
(ii). For almost
every $x_0\in\Omega$ and almost
every $r>0$ such that $B_r(x_0)\Subset\Omega$,
we have that
$$ u(x_0)=\fint_{\partial B_r(x_0)} u(x)\,d{\mathcal{H}}^{n-1}_x.$$
(iii). For almost
every $x_0\in\Omega$ and almost every $r>0$ such that $B_r(x_0)\Subset\Omega$,
we have that
$$ u(x_0)=\fint_{B_r(x_0)} u(x)\,dx.$$
Additionally, if $u$ satisfies any of the equivalent
conditions (i), (ii) or (iii), then[The equivalence
between conditions (i), (ii) and (iii)
highlights an interesting regularizing property of the Laplace
operator, since locally integrable functions
satisfying either (ii) or (iii) turn out to belong to $C^2(\Omega)$
hence their Laplacian can be computed pointwise and it is equal to zero.
A similar regularizing effect will be highlighted by the forthcoming
Lemma <ref>.
Also, the last statement
in Theorem <ref> concerning the smoothness of $u$
gives that harmonic functions are automatically $C^\infty(\Omega)$:
this statement will be strengthened in Theorem <ref>,
where we will show in fact that harmonic functions
are real analytic.]
it belongs to $C^\infty(\Omega)$.
We start by showing that
\begin{equation}\label{MEG9ifjliio0oO1}
{\mbox{if~$u$ satisfies either~(ii) or~(iii),
then~$u\in C^\infty(\Omega)$,}}
\end{equation}
up to redefining $u$ in a set of null Lebesgue measure.
To this end, we use a mollification argument.
We[As customary, here and in the following,
the subscript $0$ in $C^\infty_0$ means “with compact support in”.] take $\tau\in C^\infty_0(B_1,\,[0,+\infty))$ to
be radially symmetric and such that $\int_{B_1}\tau(x)\,dx=1$.
Given $\eta>0$, we let $\tau_\eta(x):=\frac1{\eta^n}
\tau\left(\frac{x}{\eta}\right)$
and define $u_\eta:=u*\tau_\eta$.
We pick a point $\overline{x}\in\Omega$
and $R>0$
such that $B_{2R}(\overline{x})\Subset\Omega$, and we show
that, when $\eta\in(0,R)$,
\begin{equation}\label{MEG9ifjliio0oO2}
{\mbox{if~$u$ satisfies either~(ii) or~(iii),
then~$u=u_\eta$ a.e. in~$B_{R}(\overline{x})$.}}
\end{equation}
Indeed, if $u$ satisfies (ii),
for each $x\in B_{R}(\overline{x})$ and $r\in(0,\eta]$
we have that $B_r(x)\subseteq
B_\eta(x)\subseteq B_{R+\eta}(\overline{x})\Subset\Omega$.
using polar coordinates (see e.g. <cit.>),
for almost any $x\in B_{R}(\overline{x})$,
\begin{eqnarray*}
u_\eta(x)&=&\int_{B_\eta(x)} \tau_\eta(x-y)\,u(y)\,dy
\\&=&\int_0^\eta \left[\int_{\partial B_r(x)}
\tau_\eta(x-\omega)\,u(\omega)\,d{\mathcal{H}}^{n-1}_\omega
\right]\,dr\\&=&\int_0^\eta \left[\int_{\partial B_r(x)}
\tau_\eta(re_1)\,u(\omega)\,d{\mathcal{H}}^{n-1}_\omega
\right]\,dr\\&=&\int_0^\eta \left[{\tau_\eta(re_1)}\,{{\mathcal{H}}^{n-1}(\partial B_r)}\,
\fint_{\partial B_r(x)}
\right]\,dr\\&=&u(x)\,
\int_0^\eta {\tau_\eta(re_1)}\,{{\mathcal{H}}^{n-1}(\partial B_r)}\,\,dr.
\end{eqnarray*}
Accordingly, since
\begin{eqnarray*}&& 1=\int_{B_\eta}\tau_\eta(y)\,dy
=\int_0^\eta \left[\int_{\partial B_r}
\tau_\eta(\omega)\,d{\mathcal{H}}^{n-1}_\omega
\right]\,dr\\&&\qquad\qquad=
\int_0^\eta \left[\int_{\partial B_r}
\tau_\eta(re_1)\,d{\mathcal{H}}^{n-1}_\omega
\right]\,dr=\int_0^\eta {\tau_\eta(re_1)}\,{{\mathcal{H}}^{n-1}(\partial B_r)}\,\,dr
we gather that $u_\eta(x)=u(x)$ for almost all $x\in
and, as a result, (<ref>)
holds true when condition (ii) is satisfied.
Also, if $u$ fulfills condition (iii), then by polar coordinates
and (<ref>),
for almost every $x_0\in\Omega$ and almost
every $r>0$ such that $B_r(x_0)\Subset\Omega$,
we have that
\begin{eqnarray*}&&
\fint_{\partial B_r(x_0)} u(y)\,dy=\frac1{r^{n-1}\mathcal{H}^{n-1}(\partial B_1)}
\,\frac{d}{dr}\int_{ B_r(x_0)} u(y)\,dy
=\frac1{n r^{n-1}}
\,\frac{d}{dr}\left( r^n\fint_{ B_r(x_0)} u(y)\,dy\right)\\&&\qquad=
\frac{u(x_0)}{n r^{n-1}}
\,\frac{d}{dr}\left( r^n\right)=u(x_0),
\end{eqnarray*}
which gives (ii).
This reduces us to the previous case, and therefore the proof
of (<ref>) is complete.
In turn, we have that (<ref>)
entails (<ref>), as desired.
Now we will show that (i) implies (ii) which implies (iii) which implies (i).
Let us assume that (i) holds true and let $B_r(x_0)\Subset\Omega$.
Then, $u$ is harmonic in $B_\rho(x_0)$ for every $\rho\in(0,r]$. Accordingly, by
the application of the Divergence Theorem given in equation (<ref>),
$$ 0=
\int_{B_\rho(x_0)} \Delta u(x)\,dx=\int_{\partial B_\rho(x_0) } \nabla u(x)\cdot \nu(x)\,d{\mathcal{H}}^{n-1}_x
=\frac1\rho\,\int_{\partial B_\rho(x_0) } \nabla u(x)\cdot (x-x_0)\,d{\mathcal{H}}^{n-1}_x.$$
On the other hand,
\begin{eqnarray*}&& \frac{d}{d\rho} \left(\fint_{\partial B_\rho(x_0)} u(x)\,d{\mathcal{H}}^{n-1}_x\right)
=\frac{d}{d\rho} \left(\fint_{\partial B_1} u(x_0+\rho\omega)\,d{\mathcal{H}}^{n-1}_\omega\right)\\&&\qquad=
\fint_{\partial B_1} \nabla u(x_0+\rho\omega)\cdot\omega\,d{\mathcal{H}}^{n-1}_\omega=\frac1\rho\,\fint_{\partial B_\rho(x_0) } \nabla u(x)\cdot (x-x_0)\,d{\mathcal{H}}^{n-1}_x.\end{eqnarray*}
These observations entail that, for every $\rho\in(0,r]$,
$$ \frac{d}{d\rho} \left(\fint_{\partial B_\rho(x_0)} u(x)\,d{\mathcal{H}}^{n-1}_x\right)=0,$$
$$ \fint_{\partial B_\rho(x_0)} u(x)\,d{\mathcal{H}}^{n-1}_x {\mbox{ is constant for all }}\rho\in(0,r].$$
In particular,
$$ \fint_{\partial B_r(x_0)} u(x)\,d{\mathcal{H}}^{n-1}_x=\lim_{\rho\searrow0}
\fint_{\partial B_\rho(x_0)} u(x)\,d{\mathcal{H}}^{n-1}_x=u(x_0)$$
and this shows that (ii) holds true.
The fact that (ii) implies (iii) is a consequence of polar coordinates. Indeed, if (ii) is satisfied, then
\begin{eqnarray*}&&
\fint_{B_r(x_0)} u(x)\,dx=\frac{1}{|B_r|} \int_0^r \left(
\int_{\partial B_\rho(x_0)} u(x)\,d{\mathcal{H}}^{n-1}_x\right)\,d\rho
\\&&\qquad\qquad=\frac{{\mathcal{H}}^{n-1}(\partial B_1)}{|B_1| \,r^n} \int_0^r \left(
\rho^{n-1}
\fint_{\partial B_\rho(x_0)} u(x)\,d{\mathcal{H}}^{n-1}_x\right)\,d\rho\\&&
\qquad\qquad=
\frac{{\mathcal{H}}^{n-1}(\partial B_1)\,u(x_0)}{|B_1| \,r^n} \int_0^r
\rho^{n-1}\,d\rho=\frac{{\mathcal{H}}^{n-1}(\partial B_1)\,u(x_0)}{|B_1| \,n}.
\end{eqnarray*}
This and (<ref>) yield that
$$ \fint_{B_r(x_0)} u(x)\,dx=u(x_0),$$
that is (iii).
Let us now suppose that (iii) holds true. Then, by (<ref>)
Theorem <ref>, for every $x_0\in\Omega$,
\lim_{r\searrow0} \frac1{r^2}\left(
\fint_{B_r(x_0)} u(x)\,dx-u(x_0)\right)=\frac{1}{2(n+2)}\,\Delta u(x_0),
thus showing the validity of (i).
For a comprehensive survey on
mean value properties of harmonic and closely related functions, see <cit.>.
A simple byproduct of the Mean Value Formula in Theorem <ref>
is the following interesting geometric observation:
Let $n\ge2$.
A harmonic function does not possess isolated zeroes.
Let $u$ be harmonic in some open set $\Omega\subseteq\R^n$
and suppose that $u(x_0)=0$. Arguing for a contradiction,
we suppose that there exists $r>0$ such that $B_r(x_0)\Subset\Omega$
and $u\ne0$ in $B_r(x_0)\setminus\{x_0\}$. Thus, by continuity and the fact that $B_r(x_0)\setminus \{x_0\}$
is a connected set when $n\ge2$,
by possibly replacing $u$ with $-u$,
we can suppose that $u>0$ in $B_r(x_0)\setminus\{x_0\}$. This and
Theorem <ref>(iii) yield that
$$ 0=u(x_0)=\fint_{B_r(x_0)} u(x)\,dx>0,$$
which is a contradiction.
We remark that Corollary <ref> does not hold true
when $n=1$, since the function $u(x)=x$ for all $x\in\R$ is harmonic but
possesses an isolated zero.
It can be useful to stress that the
“almost every $x_0\in\Omega$” and “almost
every $r>0$” in Theorem <ref>(ii)-(iii)
can be replaced by the simpler
“every $x_0\in\Omega$” and “every $r>0$” thanks to a continuity argument:
the details go as follows.
Given an open set $\Omega\subseteq\R^n$ and
a function $u\in C(\Omega)$,
the following conditions are equivalent:
(i). The function $u$
belongs to $C^2(\Omega)$ and
is harmonic in $\Omega$.
(ii). For every $x_0\in\Omega$ and
every $r>0$ such that $B_r(x_0)\Subset\Omega$,
we have that
$$ u(x_0)=\fint_{\partial B_r(x_0)} u(x)\,d{\mathcal{H}}^{n-1}_x.$$
(iii). For
every $x_0\in\Omega$ and every $r>0$ such that $B_r(x_0)\Subset\Omega$,
we have that
$$ u(x_0)=\fint_{B_r(x_0)} u(x)\,dx.$$
We prove that condition (i) is equivalent to condition (ii)
(similarly, one can prove the equivalence between conditions (i) and (iii)).
Assume (i) here. Then, condition (i) in Theorem <ref> holds true,
which entails condition (ii) in Theorem <ref>.
This gives that condition (ii) here is satisfied for almost every $x_0\in\Omega$ and almost
every $r>0$. Let now $\bar x\in\Omega$ and $\bar r>0$ such that $B_{\bar r}(\bar x)\Subset\Omega$.
Let $x^{(j)}$ be a sequence converging to $\bar{x}$ as $j\to+\infty$,
with $x^{(j)}$ belonging to the above mentioned set of full measure for which (ii) holds true.
Let also $r\in (\bar{r}-3|\bar{x}-x^{(j)}|,\bar{r}-2|\bar{x}-x^{(j)}|)$
and notice that $B_r(x^{(j)})\subseteq B_{\bar r}(\bar{x})\Subset\Omega$. Accordingly,
we can find $r_j\in(\bar{r}-3|\bar{x}-x^{(j)}|,\bar{r}-2|\bar{x}-x^{(j)}|)$ such that
$$ u(x^{(j)})=\fint_{\partial B_{r_j}(x^{(j)})} u(x)\,d{\mathcal{H}}^{n-1}_x.$$
Passing to the limit as $j\to+\infty$ and using the continuity of $u$, we obtain (ii) here, as desired.
Suppose now that (ii) here holds true.
Then, condition (ii) in Theorem <ref> holds true,
which entails condition (i) in Theorem <ref>, that is condition (i) here.
It is instructive to emphasize that the notion of harmonicity is “local”: since it relies
on the value of the derivative of a function at points, if ${\mathcal{I}}$
is a sets of indexes, $\Omega_i\subseteq\R^n$
are open sets for every $i\in {\mathcal{I}}$
and $u$ is harmonic in each of the $\Omega_i$, then
\begin{equation}\label{2432rf4g-2rfjv-1k2ermfAKSd56yu-124rktgmMqe22rtS-23t4yhtj0}{\mbox{$u$ is harmonic in }}\bigcup_{i\in{\mathcal{I}}}\Omega_i.\end{equation}
As a result, the Mean Value Formula in Theorem <ref>
(or its modification in Corollary <ref>)
can be also localized, according to this observation:
Given an open set $\Omega\subseteq\R^n$ and
a function $u\in C(\Omega)$,
the following conditions are equivalent:
(i). The function $u$
belongs to $C^2(\Omega)$ and
is harmonic in $\Omega$.
(ii). For
every $x_0\in\Omega$ there exists $r_0>0$ such that $B_{r_0}(x_0)\Subset\Omega$ and for
every $r\in(0,r_0)$
we have that
$$ u(x_0)=\fint_{\partial B_r(x_0)} u(x)\,d{\mathcal{H}}^{n-1}_x.$$
(iii). For
every $x_0\in\Omega$ there exists $r_0>0$ such that $B_{r_0}(x_0)\Subset\Omega$ and for
every $r\in(0,r_0)$
we have that
$$ u(x_0)=\fint_{B_r(x_0)} u(x)\,dx.$$
We prove that condition (i) is equivalent to condition (ii)
(similarly, one can prove the equivalence between conditions (i) and (iii)).
On the one hand, condition (i) here coincides with condition (i) in Corollary <ref>
which entails condition (ii) in Corollary <ref>, which in turn entails
condition (ii) here.
On the other hand, if condition (ii) here holds,
by Corollary <ref>(i) we infer that
every $x_0\in\Omega$ there exists $r_0(x_0)>0$ such that $B_{r_0(x_0)}(x_0)\Subset\Omega$
and $u$ is harmonic in $B_{r_0(x_0)}(x_0)$.
That is, in view of (<ref>), $u$ is harmonic in the set
$$ \widetilde\Omega:=\bigcup_{x_0\in\Omega} B_{r_0(x_0)}(x_0)\supseteq\Omega,$$
which establishes (i).
A useful consequence of Corollary <ref> is the so-called
Schwarz reflection principle, which allows to extend a harmonic function vanishing
on a hyperplane by odd reflection:
Let $\Omega\subseteq\R^n$ be an open set.
\begin{eqnarray*}&& \Omega_+:=\Big\{x=(x',x_n)\in\Omega{\mbox{ s.t. }}x_n>0\Big\},\\
&& \Omega_0:=\Big\{x=(x',x_n)\in\Omega{\mbox{ s.t. }}x_n=0\Big\},\\
&& \Omega_-:=
\Big\{x=(x',x_n){\mbox{ s.t. $x_n<0$ and }}(x',-x_n)\in\Omega_+\Big\}
\\{\mbox{and }}&&
\Omega_\star:=\Omega_+\cup\Omega_0\cup\Omega_-.\end{eqnarray*}
Let $u\in C^2(\Omega_+)\cap C(\Omega_+\cup \Omega_0)$ be harmonic
and such that $u(x)=0$ along $\Omega_0$.
Then, the function
$$ \Omega_\star\ni x\mapsto u_\star(x)=u_\star(x',x_n)
:=\begin{dcases} u(x)& {\mbox{ if }}x\in\Omega_+\cup\Omega_0,\\-
u(x',-x_n)& {\mbox{ if }}x\in\Omega_-
\end{dcases}$$
is harmonic in $\Omega_\star$.
We observe that
\begin{equation}\label{09io29203yoiewfgh7328ryfu237o8trgfoy2fr73twefg83246tyg89ergfwu-1203ury}
{\mbox{$u_\star$ is continuous and~$u_\star(x',0)=0$
for all~$x=(x',0)\in\Omega_0$.}}\end{equation}
Furthermore, if $x\in\Omega_-$ then $\Delta u_\star(x)=-\Delta u(x',-x_n)=0$
and accordingly $u_\star$ is harmonic in $\Omega_+\cup\Omega_-$, due
to (<ref>).
This and Corollary <ref>(iii) give that
\begin{equation}\label{09io29203yoiewfgh7328ryfu237o8trgfoy2fr73twefg83246tyg89ergfwu-1203ury2}
\begin{split}&{\mbox{for every~$\bar x\in\Omega_+\cup\Omega_-$
there exists~$\bar{r}>0$ such that for all~$r\in(0,\bar{r})$ we have that
}} \\&\fint_{B_r(\bar x)}u_\star(x)\,dx=u_\star(\bar x).\end{split}\end{equation}
Now we take $x_0\in\Omega_0$ and $r>0$ such that $B_r(x_0)\Subset\Omega_\star$
and we observe that
\begin{equation*}
\begin{split}&
\fint_{B_r(x_0)}u_\star(x)\,dx=
\frac{1}{|B_r|}\left(
\int_{B_r(x_0)\cap \Omega_+}u_\star(x)\,dx+
\int_{B_r(x_0)\cap \Omega_-}u_\star(x)\,dx
\right)
\\&\qquad=
\frac{1}{|B_r|}\left(
\int_{B_r(x_0)\cap \Omega_+}u(x)\,dx-
\int_{B_r(x_0)\cap \Omega_-}u(x',-x_n)\,dx
\right)\\&\qquad=
\frac{1}{|B_r|}\left(
\int_{B_r(x_0)\cap \Omega_+}u(x)\,dx-
\int_{B_r(x_0)\cap \Omega_+}u(x',x_n)\,dx
\right)=0=u_\star(x_0),
\end{split}
\end{equation*}
thanks to (<ref>).
From this and (<ref>),
it follows that for every $\bar x\in\Omega_\star$
there exists $\bar{r}>0$ such that for all $r\in(0,\bar{r})$ we have that $\fint_{B_r(\bar x)}u_\star(x)\,dx=u_\star(\bar x)$.
This and Corollary <ref> give that $u_\star$ is harmonic in $\Omega_\star$,
as desired.
The following classical identities, which are useful variations
of the Divergence Theorem, are often very helpful
and they are named[One of the cornerstones
of the foundation of potential theory
was indeed George Green's 1828 essay <cit.>,
see Figure <ref>,
which founded the mathematical theory of electricity and magnetism. The first edition of the essay
was printed for the author and sold on a subscription basis to only 51 people.
In 1850–1854, the essay was transcribed in <cit.>, with several
typographical corrections and a reference section added. Interestingly,
George Green was almost entirely self-taught, having
received only about one year of formal schooling, between the ages of 8 and 9.
By the way, the story of George Green is truly amazing. George Green's
father, also called George Green, was a baker in Nottingham, the UK town linked to the legend of Robin Hood.
Actually this legend does reflect long-lasting social problems
in the area which also affected the Green family.
In particular, at some point bakers were blamed for the incessant rise of the price of bread
and crowds of people broke into bakers to steal food and, in this circumstance,
the Green family's bakery was also attacked.
The bakery was however probably doing well from the financial point of view, since, the year after these riots,
the little George was sent to allegedly the best and most expensive school in Nottingham, where he was taught
for four terms, which, as mentioned above, amount to his whole formal training:
at nine, the boy starts working in his father's bakery business.
And this business kept being profitable, allowing the Green family to buy a land
and build a brick wind corn-mill
(the mill was renovated in 1986 and is now a science center, see Figure <ref>).
George Green Jr. then fell in love with the daughter
of the manager of the mill, named Jane Smith. They never married but they had together seven children.
Green also joined the Nottingham Subscription Library, thus finding access to a few scientific books and articles,
and also some works published in other countries.
Green used to study and do mathematics on the top floor of the mill and it is probably here
that the famous essay by him was conceived and written.
The 51 subscribers who bought the first edition of the book (at the price of 7 pounds and
6 pence) where probably for a vast majority members of the Nottingham Subscription Library and likely
their mathematical proficiency was insufficient to fully appreciate the content of the essay.
However, among these subscribers there was also Sir Edward Thomas Bromhead, 2nd Baronet, wealthy landowner, mathematician and founder of the Analytical Society, a precursor of the Cambridge Philosophical Society. Bromhead
realized that the essay was the production of a brilliant scientist and
invited Green to send any further papers to the Royal Society of London, the Royal Society of Edinburgh and the Cambridge Philosophical Society. Green took Bromhead's offer as mere politeness and did not respond for two years.
Then, following Bromhead's encouragement, Green wrote three further papers and, having
accumulated considerable wealth and land owned,
was able to abandon his miller duties, pursue mathematical studies
and, aged nearly forty, enroll as an undergraduate at the University of Cambridge.
There are rumors that, at Cambridge, Green
may have succumbed to alcohol, possibly losing the endorsement of
his earlier supporters (we forgot to mention that Bromhead was also approached by local ministers
to help to establish a Temperance Society).
George Green died age 48 and
the Nottingham Review published the following short obituary:
“we believe he was the son of a miller, residing near Nottingham, but having a taste for study, he applied his gifted mind to the science of mathematics [...]. Had his life been prolonged, he might have stood eminently high as a mathematician”.
They obviously did not understand that he had already stood most eminently among his contemporaries
and left an essay that would have revolutionized the history of science.
Historians of science are unsure how Green managed to acquire his formidable mathematical knowledge
with so little formal training. One possibility however, is that the Leibniz-Newton calculus controversy
(the silly dispute about who had first invented calculus) resounded
to a great disadvantage of the English school, which remained locked into the Newtonian notation of calculus,
stubbornly rejecting the notation introduced by Leibnitz and adopted
by continental mathematicians, which ultimately proved to be more flexible and effective.
It is possible that, being self-taught, Green had the possibility of getting in contact with Leibnitz's notation
(or possibly develop his own approach to calculus and analysis) without a rigid bias of an academia
influenced by dummy politics and sterile nationalism (well, anyway, a good notation
is always helpful, but in Green's case personal talent and inventiveness certainly made the difference).
For further readings on the figure of George Green see <cit.>.] “Green's Identities”Green's Identities:
The title page to Green's original essay (Public Domain image from
Let $\Omega\subseteq\R^n$ be a bounded open set of class $C^1$, with exterior normal $\nu$,
and let $\varphi$, $\psi\in C^2(\Omega)\cap C^1(\overline\Omega)$.
\begin{equation}\label{GRr1}
\int_\Omega \Big( \varphi(x)\Delta \psi(x)+\nabla \varphi(x)\cdot\nabla \psi(x)\Big)\,dx=
\int_{\partial\Omega} \varphi(x)\frac{\partial \psi}{\partial\nu}(x)\,d{\mathcal{H}}^{n-1}_x
\end{equation}
\begin{equation}\label{GRr2}
\int_\Omega \Big( \varphi(x)\Delta \psi(x)-\psi(x)\Delta \varphi(x)\Big)\,dx=
\int_{\partial\Omega}\left(\varphi(x)\frac{\partial \psi}{\partial\nu}(x)-
\psi(x)\frac{\partial \varphi}{\partial\nu}(x)\right)\,d{\mathcal{H}}^{n-1}_x
By the Divergence Theorem,
\begin{eqnarray*}&&
\int_\Omega \Big( \varphi(x)\Delta \psi(x)+\nabla \varphi(x)\cdot\nabla \psi(x)\Big)\,dx=
\int_\Omega \div\big( \varphi(x)\nabla \psi(x)\big)\,dx
\int_{\partial\Omega} \varphi(x)\frac{\partial \psi}{\partial\nu}(x)\,d{\mathcal{H}}^{n-1}_x
that is (<ref>).
Also, exchanging the roles of $\varphi$ and $\psi$ in (<ref>),
\[
\int_\Omega \Big( \psi(x)\Delta \varphi(x)+\nabla \psi(x)\cdot\nabla \varphi(x)\Big)\,dx=
\int_{\partial\Omega} \psi(x)\frac{\partial \varphi}{\partial\nu}(x)\,d{\mathcal{H}}^{n-1}_x
Subtracting this from (<ref>) we obtain (<ref>).
We observe that identity (<ref>) can now be considered as a special
case of (<ref>). If creatively exploited,
Green's Identities are very useful to deduce important integral formulas,
which in turn entail structural information on several relevant equations.
As a prototype of this idea, we recall the classical Pohožaev IdentityPohožaev Identity
(see <cit.>):
Let $\Omega$ be a bounded open set in $\R^n$ with $C^1$ boundary
and $u\in C^2(\Omega)\cap C^1(\overline\Omega)$ be a solution of
\begin{equation}\label{EQU-peru} \begin{dcases}
\Delta u=f (u)& {\mbox{ in }}\Omega,\\
u=0&{\mbox{ on }}\partial\Omega,
\end{dcases}\end{equation}
for some $f\in L^\infty_{\rm loc}(\R)$.
Green's mill (photo by Kev747, image from
Wikipedia, licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license).
Let also
$$ F(r):=\int_0^r f(t)\,dt.$$
$$ \frac12\,\int_{\partial\Omega} (\partial_\nu u(x))^2\,(x\cdot\nu(x))\,d{\mathcal{H}}^{n-1}_x=
\frac{n-2}2\,\int_\Omega u(x) \,f(u(x))\,dx-n\int_\Omega F(u(x))\,dx.
The idea of the proof is to test the equation against
the radial derivative $\nabla u(x)\cdot x$ using suitable integration by parts.
Namely, from (<ref>),
\begin{equation*}
\begin{split}&
\int_\Omega \Delta u(x)\,(\nabla u(x)\cdot x)\,dx
=\int_\Omega f(u(x))(\nabla u(x)\cdot x)\,dx\\&\qquad
=\int_\Omega \nabla \big( F(u(x))\big)\cdot x\,dx
=\int_\Omega\Big( \div \big( F(u(x))\, x\big) -nF(u(x))\Big)\,dx.\end{split}
\end{equation*}
This and the Divergence Theorem,
recalling the boundary condition in (<ref>),
give that
\begin{equation}\label{SPL8ygbT2Ter}
\begin{split}&
\int_\Omega \Delta u(x)\,(\nabla u(x)\cdot x)\,dx+
n\int_\Omega F(u(x))\,dx=
\int_\Omega \div \big( F(u(x))\, x\big)\,dx\\&\qquad
=\int_{\partial\Omega} F(u(x))\, (x\cdot\nu(x))\,d{\mathcal{H}}^{n-1}_x
=\int_{\partial\Omega} F(0)\, (x\cdot\nu(x))\,d{\mathcal{H}}^{n-1}_x
\begin{eqnarray*}&&
\Delta u\,(\nabla u\cdot x)=\div\Big((\nabla u\cdot x)\nabla u\Big)-
\nabla(\nabla u\cdot x)\cdot\nabla u
=\div\Big((\nabla u\cdot x)\nabla u\Big)-\sum_{i,j=1}^n
\partial_{ij}u\,\partial_i u\,x_j-|\nabla u|^2\\&&\qquad
=\div\Big((\nabla u\cdot x)\nabla u\Big)-\frac12\,\sum_{j=1}^n
\partial_{j}|\nabla u|^2\,x_j-|\nabla u|^2
=\div\Big((\nabla u\cdot x)\nabla u\Big)-\frac12\,\nabla (|\nabla u|^2)\cdot
x-|\nabla u|^2\\&&\qquad=
\div\Big((\nabla u\cdot x)\nabla u\Big)-\frac12\,\left[
\div\Big(|\nabla u|^2\,x\Big)
-n|\nabla u|^2
\right]-|\nabla u|^2\\&&\qquad=
\div\Big((\nabla u\cdot x)\nabla u\Big)-\frac12\,
\div\Big(|\nabla u|^2\,x\Big)
+\frac{n-2}2|\nabla u|^2.
\end{eqnarray*}
Thus, making use again
of the Divergence Theorem and of the first Green's Identity (<ref>)
we find that
\begin{equation}\begin{split}\label{dtv654574v6} &\int_\Omega \Delta u(x)\,(\nabla u(x)\cdot x)\,dx\\
(\nabla u(x)\cdot x)\partial_\nu u(x)\,d{\mathcal{H}}^{n-1}_x
-\frac12\,\int_{\partial\Omega}|\nabla u(x)|^2\,x\cdot\nu(x)\,d{\mathcal{H}}^{n-1}_x
+\frac{n-2}2\,\int_\Omega|\nabla u(x)|^2\,dx\\
(\nabla u(x)\cdot x)\partial_\nu u(x)\,d{\mathcal{H}}^{n-1}_x
-\frac12\,\int_{\partial\Omega}|\nabla u(x)|^2\,x\cdot\nu(x)\,d{\mathcal{H}}^{n-1}_x
-\frac{n-2}2\,\int_\Omega\Delta u(x)\,u(x)\,dx
Now we observe that $\nabla u=\pm|\nabla u|\nu$ on $\partial\Omega$, and therefore, for every $x\in\Omega$,
$$(\nabla u(x)\cdot x)\partial_\nu u(x)=|\nabla u(x)|^2(\nu(x)\cdot x)(\nu(x)\cdot\nu(x))=|\nabla u(x)|^2(\nu(x)\cdot x).
Plugging this information into (<ref>), we find that
$$\int_\Omega \Delta u(x)\,(\nabla u(x)\cdot x)\,dx=
\frac12\,\int_{\partial\Omega}|\nabla u(x)|^2\,x\cdot\nu(x)\,d{\mathcal{H}}^{n-1}_x
-\frac{n-2}2\,\int_\Omega\Delta u(x)\,u(x)\,dx.$$
Combining this and (<ref>), we obtain the desired result.
Equations as in (<ref>)
are often called “semilinear” semilinear equation
since they are not linear in $u$ (unless the source term $f$
is linear) but they are linear in the second derivative of $u$.
solutions of semilinear equations[We refer to the footnotes
on pages BIS Pohozaev Identity
and TRIS Pohozaev Identity for motivational comments about semilinear equations.] enjoy the special
feature of having constant Laplacian along their level sets,
namely if $u$ solves (<ref>),
given any $c\in\R$, we have that $\Delta u=f(c)$
on $\{u=c\}$.
As a consequence of the Pohožaev Identity
in Theorem <ref>, one obtains
nonexistence results, as the one in the forthcoming Corollary <ref>.
For this, we give the following definition:
Let $\Omega\subseteq \R^n$. Given $x_0\in\Omega$, we say that $\Omega$ is Aped
with respect to $x_0$ if for every $x\in\Omega$ we have that $tx+(1-t)x_0\in\Omega$ for all $t\in[0,1]$.
Furthermore, we say that $\Omega$ is starshaped if there exists $x_0\in\Omega$
such that $\Omega$ is starshaped with respect to $x_0$.
With this, we give the following nonexistence result:
Let $n\ge 3$ and $p > \frac{n+ 2}{n - 2}$.
Let $\Omega$ be a bounded starshaped
open set in $\R^n$ with $C^1$ boundary.
Let $u\in C^2(\Omega)\cap C^1(\overline\Omega)$ be a solution of
\begin{equation*} \begin{dcases}
\Delta u=-|u|^{p-1}u& {\mbox{ in }}\Omega,\\
u=0&{\mbox{ on }}\partial\Omega.
\end{dcases}\end{equation*}
Then, $u$ vanishes identically.
Up to a translation, we suppose that
\begin{equation}\label{ISpapkrfeppa}
{\mbox{$\Omega$ is
starshaped with respect to the origin.}}\end{equation}
We claim that
\begin{equation}\label{PDMASNDc}
{\mbox{$x\cdot\nu(x)\geq0$ for every~$x\in\partial\Omega$.}}\end{equation}
To check this, given $x_0\in\partial\Omega$, we write $\Omega$
in the vicinity of $x_0$ as the superlevels of some function $\Phi\in C^1(\R^n)$
with $\nabla\Phi(x_0)\ne0$,
that is we take $\rho>0$ such that $\Omega\cap B_\rho(x_0)=\{\Phi>0\}\cap
B_\rho(x_0)$. In this way, we have that $\nu=-\frac{\nabla\Phi}{|\nabla\Phi|}$
on $\partial\Omega$.
Also, by (<ref>),
we have that $t x_0\in\overline\Omega$ for every $t\in[0,1]$.
As a result, for $t\in[0,1]$ sufficiently close to $1$,
we have $\Phi(tx_0)\ge0$. Therefore,
$$ 0\ge
\lim_{t\nearrow1}\frac{\Phi(tx_0)}{t-1}=
\lim_{t\nearrow1}\frac{\Phi(tx_0)-\Phi(x_0)}{t-1}=\nabla\Phi(x_0)\cdot x_0=
-|\nabla\Phi(x_0)|\,\nu(x_0)\cdot x_0.
This proves (<ref>).
We now exploit the
Pohožaev Identity in Theorem <ref> with $f(u):=-|u|^{p-1}u$,
and hence $F(r):=-\frac{|r|^{p+1}}{p+1}$. In this way, using (<ref>),
we find that
\begin{eqnarray*}&& 0\le \frac12\,\int_{\partial\Omega}
(\partial_\nu u(x))^2\,(x\cdot\nu(x))\,d{\mathcal{H}}^{n-1}_x=-
\frac{n-2}2\,\int_\Omega |u(x)|^{p+1} \,dx+
\frac{n}{p+1}\,\int_\Omega |u(x)|^{p+1}\,dx\\&&\qquad\qquad=
\frac{p(2-n)+n+2}{2(p+1)}\,
\int_\Omega |u(x)|^{p+1} \,dx\le0.
\end{eqnarray*}
In particular,
$$ \frac{p(2-n)+n+2}{2(p+1)}\,\int_\Omega |u(x)|^{p+1} \,dx=0,$$
from which the desired result follows.
A natural question is whether or not
the average over balls and spheres in the Mean Value Formulas of Theorem <ref>
can be substituted with averages on different sets.
As we will see in Section <ref>,
this is not the case and in fact the geometry of the balls and spheres
play a decisive
role in the Mean Value Formula (this classical problem
was pioneered in <cit.>.
§ WEAK SOLUTIONS
We present here a classical resultweak solution
often referred to with the name of Weyl's LemmaWeyl's Lemma:
Let $\Omega\subseteq\R^n$ be an open set
and let $u\in L^1_{\rm loc}(\Omega)$. Assume that
\begin{equation}\label{GRAVSSEFORC2}
\int_\Omega u(x)\,\Delta\varphi(x)\,dx=0\qquad{\mbox{for every }}\varphi\in
Then, $u$ is harmonic in $\Omega$.
We stress that the desired claim
follows directly from (<ref>) and the
second Green's Identity (<ref>) when $u\in C^2(\Omega)$.
If instead $u$ is merely locally integrable in $\Omega$,
we use a mollification argument.
To this end,
we take $\tau\in C^\infty_0(B_1,\,[0,+\infty))$
with $\int_{B_1}\tau(x)\,dx=1$.
Given $\eta>0$, we let $\tau_\eta(x):=\frac1{\eta^n}
\tau\left(\frac{x}{\eta}\right)$
and define $u_\eta:=u*\tau_\eta$.
Then, given $x_0\in\Omega$ and $\rho>0$ such that $B_{2\rho}(x_0)\Subset\Omega$,
for all $\varphi\in C^\infty_0(B_\rho(x_0))$ and all $\eta\in(0,\rho)$ we have that
\begin{equation*}\begin{split}&
\int_\Omega u_\eta(x)\,\Delta\varphi(x)\,dx=
\iint_{B_{2\rho}(x_0)\times\Omega} u(y)\,\tau_\eta(x-y)\Delta\varphi(x)\,dx\,dy\\&\qquad=
\int_\Omega u(x)\,(\tau_\eta*\Delta\varphi)(x)\,dx
=\int_\Omega u(x)\,\Delta\varphi_\eta(x)\,dx,\end{split}
\end{equation*}
where $\varphi_\eta:=\varphi*\tau_\eta\in C^\infty_0(B_{2\rho}(x_0))\subseteq
As a result, from (<ref>), we deduce that
$$ \int_\Omega u_\eta(x)\,\Delta\varphi(x)\,dx=0\qquad{\mbox{ for every }}
\varphi\in C^\infty_0(B_\rho(x_0)),$$
as long as $\eta\in(0,\rho)$.
Since $u_\eta\in C^2(B_\rho(x_0))$,
this gives that $u_\eta$ is harmonic in $B_\rho(x_0)$.
Owing to this and to
Theorem <ref>(iii),
for every ball $B_r(\overline{x})\Subset B_\rho(x_0)$,
we have that
$$ u_\eta(\overline{x})=\fint_{B_r(\overline{x})} u_\eta(x)\,dx.$$
We now send $\eta\searrow0$ and (see e.g. Theorems 9.6 and 9.13
in <cit.>) we conclude that,
whenever $B_r(\overline{x})\Subset B_\rho(x_0)$
and $\overline{x}$ is a Lebesgue
density point for $u$,
\begin{equation} \label{3LS2345Dscdfoj74586vvyjf}
u(\overline{x})=\fint_{B_r(\overline{x})} u(x)\,dx.\end{equation}
Furthermore, by the Dominated Convergence Theorem,
for every $\widetilde{x}\in\Omega$ and $r>0$ such that $B_r(
\widetilde{x})\subset\Omega$,
$$ \lim_{p\to\widetilde{x}}\int_{B_r(p)} u(x)\,dx=
\int_{B_r(\widetilde{x})} u(x)\,dx.$$
This and (<ref>) give that, up
to continuously extend $u$ in a set of null Lebesgue measure
in $B_\rho(x_0)$, we have that
\[ u(\overline{x})=\fint_{B_r(\overline{x})} u(x)\,dx\qquad{\mbox{for every }}
\overline{x}\in B_\rho(x_0),\]
as long as $B_r(\overline{x})\Subset B_\rho(x_0)$.
Using again Theorem <ref>, we thereby conclude that $u$
is harmonic in $B_\rho(x_0)$. This gives that $\Delta u(x_0)=0$
for every $x_0\in\Omega$, as desired.
Let $\Omega\subseteq\R^n$ be an open set
and $u_k$ be a sequence of harmonic functions in $\Omega$.
Suppose that $u_k\to u$ in $L^1_{\rm loc}(\Omega)$. Then, $u$
is harmonic in $\Omega$.
Let $\Omega'\Subset\Omega$ and $\varphi\in
By the
second Green's Identity (<ref>) we know that
\begin{eqnarray*}&&
\left|\int_\Omega u(x)\,\Delta\varphi(x)\,dx\right|
\le
\int_{\Omega'} |u(x)-u_k(x)|\,|\Delta\varphi(x)|\,dx
+\left|\int_\Omega u_k(x)\,\Delta\varphi(x)\,dx\right|\\&&\qquad
\le \|\varphi\|_{C^2(\Omega')}\,\|u-u_k\|_{L^1(\Omega')}
+\left|\int_\Omega \Delta u_k(x)\,\varphi(x)\,dx\right|=\|\varphi\|_{C^2(\Omega')}\,\|u-u_k\|_{L^1(\Omega')}.
\end{eqnarray*}
Hence, sending $k\to+\infty$,
$$ \int_\Omega u(x)\,\Delta\varphi(x)\,dx=0.$$
The desired result thus follows from
Lemma <ref>.
An alternative proof of Corollary <ref>
can be obtained also using directly the
Mean Value Formula in Theorem <ref>(iii).
See also <cit.>
and the references therein for a more complete discussion
on the role played by “weak” or “distributional”
formulations of partial differential equations and a careful discussion
of the functional analysis methods involved in such a theory.
A classical application of the weak setting of partial differential
equations is provided by Kato's InequalityKato's Inequality, see <cit.>,
as presented in the following result.
For this, we use the standard notation, for every $r\in\R$,
$$\sign(r):=\begin{dcases} \frac{r}{|r|}
&{\mbox{ if }}r\ne0,\\0&{\mbox{ if }}r=0.\end{dcases}$$
\begin{equation}\label{L1co} u\in L^1_{\rm loc}(\R^n)\end{equation}
be such that there exists $f\in L^1_{\rm loc}(\R^n)$ satisfying
\begin{equation}\label{L1co2} \int_{\R^n} u(x)\,\Delta\psi(x)\,dx
=\int_{\R^n} f(x)\,\psi(x)\,dx\qquad{\mbox{ for all }}\psi\in C^\infty_0(\R^n). \end{equation}
Then, for[In jargon,
condition <ref> can be rewritten by stating that the Laplacian
of $u$, as defined in the weak sense, is actually a locally integrable function
that is denoted by $f$ (and
this is of course the case for smooth functions $u$).
Similarly, equation (<ref>) can be written as
$$\Delta|u|\ge\sign(u)\,\Delta u$$
in the weak sense.
This can be also considered as a “limit case” of the following
observation: if $\Phi\in C^1(\R)$ is a convex function, then
(u(x\pm he_i))-\Phi(u(x))\ge\Phi'(u(x))
(u(x\pm he_i)-u(x))$$
and accordingly
$$ \Phi(u(x+ he_i))+\Phi(u(x- he_i))-2\Phi(u(x))\ge
\Phi'(u(x))
(u(x+he_i)+u(x- he_i)-2u(x)),$$
which leads to
$$ \Delta\Big(\Phi(u(x))\Big)\ge\Phi'(u(x))\,\Delta u(x).$$
With this respect, Kato's Inequality (<ref>)
corresponds, formally, to the limit case in which $\Phi(t)=|t|$,
that produces $\Phi'(t)=\sign(t)$ (at least when $t\ne0$,
and the corresponding inequality holding true in the weak sense).] every $\varphi\in C^\infty_0(\R^n,\,[0,+\infty))$,
\begin{equation}\label{MSN:LASsmmS2}\int_{\R^n} |u(x)|\,\Delta\varphi(x)\,dx\geq\int_{\R^n}\sign(u(x))\,\varphi(x)\, f(x)\,dx.\end{equation}
Given $\eta>0$, we let $\tau_\eta(x):=\frac1{\eta^n}
\tau\left(\frac{x}{\eta}\right)$
and define $u_\eta:=u*\tau_\eta$. In this setting, by (<ref>),
possibly up to subsequences, we have that $u_\eta$ converges
to $u$ in $L^1_{\rm loc}(\R^n)$
(see <cit.>) and almost everywhere,
with additionally $|u_\eta|\le h$ for a suitable $h\in L^1_{\rm loc}(\R^n)$
(see <cit.>).
For this reason,
\begin{equation}\label{NOSMDBERPPLSBE} \int_{\R^n} |u(x)|\,
\Delta\varphi(x)\,dx
=\lim_{\eta\searrow0}\int_{\R^n} |u_\eta(x)|\,\Delta\varphi(x)\,dx.\end{equation}
for every $\psi\in C^\infty_0(\R^n)$,
\begin{eqnarray*}&& \int_{\R^n} u_\eta(x)\,\Delta\psi(x)\,dx
\int_{\R^n} u(y)\,\tau_\eta(x-y)\,\Delta\psi(x)\,dx\right]\,dy
=\int_{\R^n} u(y)\,\Delta\psi_\eta(y)\,dy\\&&\qquad
=\int_{\R^n} f(y)\,\psi_\eta(y)\,dy
\int_{\R^n} f(y)\,\psi(x)\,\tau_\eta(x-y)\,dy
\right]\,dx=\int_{\R^n} f_\eta(x)\,\psi(x)\,dx,
\end{eqnarray*}
and accordingly $\Delta u_\eta=f_\eta$.
In this way, possibly extracting a subsequence,
we deduce that $\Delta u_\eta$ converges
to $f$ in $L^1_{\rm loc}(\R^n)$
(see <cit.>) and almost everywhere,
with additionally $|\Delta u_\eta|\le H$ for a suitable $H\in L^1_{\rm loc}(\R^n)$
(see <cit.>).
Now we let $\e>0$ and set $v_{\e,\eta}(x):=\sqrt{(u_\eta(x))^2+\e^2}$.
In this way, the function $v_{\e,\eta}$ belongs to $C^\infty(\R^n)$
and $\sign(v_{\e,\eta}(x))=1$ for all $x\in\R^n$. Furthermore,
$$ 2v_{\e,\eta}
\nabla v_{\e,\eta}=\nabla v_{\e,\eta}^2=\nabla(u_\eta^2+\e^2)=2u_\eta
\nabla u_\eta.$$
$$ |u_\eta|\,|\nabla v_{\e,\eta}|\le v_{\e,\eta}\,|\nabla v_{\e,\eta}|= |u_\eta
|\,|\nabla u_\eta|$$
$$ |\nabla v_{\e,\eta}|^2+v_{\e,\eta}\Delta v_{\e,\eta}
= \div(v_{\e,\eta}\nabla v_{\e,\eta})=\div(u_\eta
\nabla u_\eta)=|\nabla u_\eta|^2+u_\eta
\Delta u_\eta.$$
As a result,
$$ v_{\e,\eta}\Delta v_{\e,\eta}
=|\nabla u_\eta|^2-|\nabla v_{\e,\eta}|^2+u_\eta
\Delta u_\eta\geq u_\eta\Delta u_\eta.$$
We thus define $\sigma_{\e,\eta}(x):=\frac{u_\eta(x)}{v_{\e,\eta}(x)}$
and find that $\Delta v_{\e,\eta}\geq\sigma_{\e,\eta}\Delta u_\eta
$, and then
\begin{equation}\label{KA78:01} \int_{\R^n} v_{\e,\eta}(x)\,\Delta\varphi(x)
\,dx\geq\int_{\R^n}\sigma_{\e,\eta}
(x)\,\varphi(x)\, \Delta u_\eta(x)\,dx,\end{equation}
for every $\varphi\in C^\infty_0(\R^n,\,[0,+\infty))$.
It is also helpful to observe that $|v_{\e,\eta}|=v_{\e,\eta}\le
|u_\eta|+\e\le h+1$ and that, for a.e. $x\in\R^n$,
$$ \lim_{\eta\searrow0}v_{\e,\eta}(x)=\sqrt{(u(x))^2+\e}.$$
We can therefore exploit the Dominated
Convergence Theorem to find that
\begin{equation}\label{CO6789SEDJSMBIMserD9ofkvRASHJD}
\lim_{\eta\searrow0}\int_{\R^n} v_{\e,\eta}(x)\,\Delta\varphi(x)\,dx
=\int_{\R^n} \sqrt{(u(x))^2+\e}
\;\Delta\varphi(x)\,dx.
\end{equation}
Additionally, for a.e. $x\in\R^n$,
$$ \lim_{\eta\searrow0}\sigma_{\e,\eta}(x)=
\frac{u(x)}{\sqrt{(u(x))^2+\e}} ,$$
and $|\sigma_{\e,\eta}|\le1$. Hence, by the Dominated Convergence Theorem,
\begin{equation*}
\lim_{\eta\searrow0}
\int_{\R^n}\sigma_{\e,\eta}
(x)\,\varphi(x)\, \Delta u_\eta(x)\,dx=
\int_{\R^n}\frac{u(x)}{\sqrt{(u(x))^2+\e}}\,\varphi(x)\, f(x)\,dx.\end{equation*}
Combining this fact with (<ref>),
we can pass (<ref>) to the limit as $\eta\searrow0$
and see that,
for every $\varphi\in C^\infty_0(\R^n,\,[0,+\infty))$,
\begin{equation*} \int_{\R^n} \sqrt{(u(x))^2+\e}
\;\Delta\varphi(x)\,dx\geq\int_{\R^n}\frac{u(x)}{\sqrt{(u(x))^2+\e}}
\,\varphi(x)\, f(x)\,dx.\end{equation*}
By sending $\e\searrow0$, we thereby obtain
the desired result in (<ref>).
As a consequence of
Kato's Inequality in Theorem <ref>, we present a classification
result for global weak solutions of the equation $\Delta u=Vu+cu$,
see <cit.> for additional details.
Let $V\in L^2_{\rm loc}(\R^n,\,[0,+\infty))$.
Let $u\in L^2(\R^n)$ and assume that, for
every $\varphi\in C^\infty_0(\R^n)$,
\begin{equation}\label{LSKMD-SKD}\int_{\R^n} u(x)\,\Delta\varphi(x)\,dx
=\int_{\R^n} V(x)\,u(x)\,\varphi(x)\,dx.\end{equation}
Then, $u$ vanishes identically.
To exploit Theorem <ref>,
we notice that condition (<ref>) is fulfilled
with $f(x):= V(x)\,u(x)\in L^1_{\rm loc}(\R^n)$, thanks to (<ref>).
Therefore, in light of (<ref>),
for every $\varphi\in C^\infty_0(\R^n,\,[0,+\infty))$,
\begin{equation}\label{GDBFNIqrtJSMFFIDIRENCEXNCKFNEJFINFNIFG}
\int_{\R^n} |u(x)|\,\Delta\varphi(x)\,dx
\geq\int_{\R^n}\sign(u(x))\,\varphi(x)\,
we take $\tau\in C^\infty_0(B_1,\,[0,+\infty))$
with $\int_{B_1}\tau(x)\,dx=1$.
Given $\eta>0$, we let $\tau_\eta(x):=\frac1{\eta^n}
\tau\left(\frac{x}{\eta}\right)$
and define $w_\eta:=|u|*\tau_\eta$. Notice that $|u|\in L^2(\R^n)$
and therefore $w_\eta\to|u|$
in $L^2(\R^n)$
(see e.g. <cit.>).
As a result, possibly up to a subsequence,
there exists $h\in L^2(\R^n)$ such that
\begin{equation}\label{KS-DUEMD-L2ANSAVSAILA}
|w_\eta(x)|\le h(x)\end{equation}
a.e. $x\in\R^n$ (see e.g. <cit.>).
for every $\varphi\in C^\infty_0(\R^n,\,[0,+\infty))$,
\begin{eqnarray*}
&& \int_{\R^n} w_\eta(x)\,\Delta\varphi(x)\,dx
=\int_{\R^n} \left[\int_{\R^n}
\\&&\qquad
=\int_{\R^n} |u(y)|\,\Delta\varphi_\eta(y)\,dy
\geq\int_{\R^n}\sign(u(y))\,\varphi_\eta(y)\,
V(y) \,|u(y)|\,dy\ge0
thanks to (<ref>).
As a result, we find that $\Delta w_\eta\ge0$.
Thus, if $R>1$ and $\xi_R
\in C^\infty_0(B_R,\,[0,1])$ with $\xi_R=1$ in $B_{R-1}$
and $|\nabla\xi_R|\le 2$, letting $\zeta_R:=\xi_R^2$
we find that
\begin{eqnarray*}&&
\int_{\R^n} \zeta_R(x)\,|\nabla w_\eta(x)|^2\,dx
=-\int_{\R^n} \div\big(\zeta_R(x)\,\nabla w_\eta(x)\big)\,w_\eta(x)\,dx\\
-\int_{\R^n} \nabla\zeta_R(x)\cdot\nabla w_\eta(x) \,w_\eta(x)\,dx
-\int_{\R^n} \zeta_R(x)\,\Delta w_\eta(x)\,w_\eta(x)\,dx\\&&\qquad\le
-\int_{\R^n} \nabla\zeta_R(x)\cdot\nabla w_\eta(x) \,w_\eta(x)\,dx=
-2\int_{\R^n} \xi_R(x)\,\nabla\xi_R(x)\cdot\nabla w_\eta(x) \,w_\eta(x)\,dx
\\&&\qquad\le\frac12\int_{\R^n} \zeta_R(x)\,|\nabla w_\eta(x)|^2\,dx+
2\int_{\R^n}|\nabla\xi_R(x)|^2 \,(w_\eta(x))^2\,dx. \end{eqnarray*}
For this reason, and recalling (<ref>),
\begin{eqnarray*}&&\frac12
\int_{\R^n} \zeta_R(x)\,|\nabla w_\eta(x)|^2\,dx\le
8\int_{\R^n\setminus B_{R-1}}(w_\eta(x))^2\,dx\le 8\int_{\R^n\setminus B_{R-1}}(h(x))^2\,dx
\end{eqnarray*}
and therefore
\begin{eqnarray*}&&\int_{\R^n} |\nabla w_\eta(x)|^2\,dx=\lim_{R\to+\infty}
\int_{B_{R-1}} |\nabla w_\eta(x)|^2\,dx
\le\lim_{R\to+\infty}
\int_{\R^n} \zeta_R(x)\,|\nabla w_\eta(x)|^2\,dx\\&&\qquad\le16
\lim_{R\to+\infty}
\int_{\R^n\setminus B_{R-1}}(h(x))^2\,dx=0.
\end{eqnarray*}
This leads to $w_\eta$ being constant, and thus constantly equal to
zero, due to (<ref>). From this,
taking the limit as $\eta\searrow0$, we find that $u$ is
constantly equal to
zero as well.
§ THE LAPLACE-BELTRAMI OPERATOR
The Laplace operator in $\R^n$ is actually a “special case”
of a more general operator acting
on functions defined on manifolds embedded in the
Euclidean space (or, even more generally, on Riemannian and pseudo-Riemannian manifolds).
For concreteness, though more general settings
can be taken into account (see also the comments on page DAC3aC22), we consider here the case of
a hypersurface $\Sigma=\partial E$ of class $C^3$,
for a bounded and open set $E\subseteq\R^n$,
and we denote by $\nu$ its unit exterior normal and by $d_\Sigma$ the
signed distance functionsigned distance function
to $\Sigma$ (say, with the convention that $d_\Sigma\ge0$ in $E$ and $d_\Sigma\le0$ in $\R^n\setminus E$). We point out that $d_\Sigma$ is also of class $C^3$
in a suitably small neighborhood ${\mathcal{N}}$ of $\Sigma$,
and for every $x\in{\mathcal{N}}$ there exists a unique point $\pi_\Sigma(x)\in\Sigma$ (often called the “projectionprojection of $x$
onto $\Sigma$”) such that
\begin{equation}\label{PROIE}
moreover $\pi_\Sigma$ is of class $C^2({\mathcal{N}})$ and
\begin{equation}\label{PROIE2} \nabla d_\Sigma(x)=-\nu(\pi_\Sigma(x)),\end{equation}
see[The intuition behind (<ref>)
is sketched in Figure <ref>. Roughly speaking,
one can consider a point $p$
and measure its distance from $\Sigma$ by considering the ball
centered at $p$ and tangent to $\Sigma$. Taking derivatives
of the distance function with respect to “tangential directions”
corresponds to moving $p$ infinitesimally
to the point $p'$ and considering the ball
centered at $p'$ and tangent to $\Sigma$: since $\Sigma$
detaches “quadratically” from its tangent hyperplane at $p$,
this new ball is a small perturbation of the translation of the original ball
(thus producing a zero tangential derivative).
taking normal derivatives
of the distance function
corresponds to moving $p$ infinitesimally
to the point $p''$ and considering the ball
centered at $p''$ and tangent to $\Sigma$:
in this case, the new ball has a radius
equal to the one of the old ball, plus the distance between $p$ and $p''$,
up to small perturbations (and this produces a unit normal derivative).
The minus sign in (<ref>) is due to the fact that the outer normal of $E$
points towards the region in which the sign distance is negative.]
e.g. Lemma 14.16 in <cit.> or Appendix B
in <cit.>. For our purposes ${\mathcal{N}}$ will be always supposed to be a conveniently
small neighborhood of $\Sigma$.
Taking derivatives of the distance function.
Given a function $u:\Sigma\to\R$, this framework allows us to
define the
“normal extensionnormal extension of $u$ outside $\Sigma$” for all $x\in{\mathcal{N}}$ as
\begin{equation}\label{DEEST} u_{\mbox{\scriptsize{ext}}}(x):= u(\pi_\Sigma(x)).\end{equation}
Notice that if $p\in\Sigma$ and $|t|$ is sufficiently small such that $p+t\nu(p)\in{\mathcal{N}}$,
then $
As a result,
\begin{equation}\label{CE}\begin{split}&
0=\frac{d}{dt} u(p)=\frac{d}{dt} u_{\mbox{\scriptsize{ext}}}(p+t\nu(p))=\nabla
\nabla
\end{equation}
The Laplace-Beltrami operatorLaplace-Beltrami operator of a function $u\in C^2(\Sigma)$ is then defined, for each $p\in\Sigma$, by
\begin{equation}\label{BEL} \Delta_\Sigma u(p):=\Delta u_{\mbox{\scriptsize{ext}}}(p),\end{equation}
where $\Delta$ represents here the standard Laplacian acting on functions in $C^2({\mathcal{N}})$.
Interestingly, the
Laplace-Beltrami operator is compatible with the
gradient structure intrinsic to $\Sigma$. To this end, one defines
the “tangential gradient”tangential gradient as the projection onto the tangent plane, namely,
for every $f\in C^1({\mathcal{N}})$ and any $p\in{\mathcal{N}}$,
\begin{equation}\label{IGRA}
\nabla_\Sigma f(p):=\nabla f(p)-\big(\nabla f(p)\cdot\nu_{\mbox{\scriptsize{ext}}}(p)\big)\,\nu_{\mbox{\scriptsize{ext}}}(p).\end{equation}
Also, if $f\in C^1(\Sigma)$, we define its tangential gradient via the tangential
gradient of the normal extension, namely
\begin{equation}\label{INvjkds-3}
\nabla_\Sigma f:=\nabla_\Sigma f_{\mbox{\scriptsize{ext}}}.\end{equation}
We observe that, in view of (<ref>), on $\Sigma$ we have that
\begin{equation}\label{INvjkds-1}
\nabla f_{\mbox{\scriptsize{ext}}}\cdot\nu=0
\end{equation}
and thus, by (<ref>),
\begin{equation}\label{INvjkds-2} \nabla_\Sigma f_{\mbox{\scriptsize{ext}}}
=\nabla f_{\mbox{\scriptsize{ext}}}.\end{equation}
In analogy with the tangential gradient defined in (<ref>),
one can introduce the “tangential divergence”tangential divergence
of a vector field $F\in C^1({\mathcal{N}},\R^n)$ at points of ${\mathcal{N}}$ as
\begin{equation}\label{VB-d1} \div_\Sigma F:=\div F-\nabla(F\cdot\nu_{\mbox{\scriptsize{ext}}})\cdot\nu_{\mbox{\scriptsize{ext}}}.\end{equation}
Roughly speaking, one is “removing” here the normal contribution
of the full divergence.
Also, if $F\in C^1(\Sigma,\R^n)$, one defines its tangential
divergence as the tangential divergence of its
normal extension, namely
\begin{equation}\label{VB-d2} \div_\Sigma F:=\div_\Sigma F_{\mbox{\scriptsize{ext}}},\end{equation}
where $F_{\mbox{\scriptsize{ext}}}$ is the vector field obtained
by the normal extension of all the components of $F$. In this
situation, we point out that, for each $p\in{\mathcal{N}}$,
$$ F_{\mbox{\scriptsize{ext}}}(p)\cdot\nu_{\mbox{\scriptsize{ext}}}(p)=
whence, in light of (<ref>), $\nabla
\big( F_{\mbox{\scriptsize{ext}}}\cdot\nu_{\mbox{\scriptsize{ext}}}\big)
\cdot\nu=0$ on $\Sigma$.
Combining this with (<ref>) and (<ref>), it follows that
\begin{equation}\label{0909} \div_\Sigma F=\div F_{\mbox{\scriptsize{ext}}}\qquad{\mbox{on }}\;\Sigma.\end{equation}
Concerning the definition of tangential gradient and divergence,
a caveat should be taken into account: namely,
given a function $u\in C^1({\mathcal{N}})$
(or a vector field $F\in C^1({\mathcal{N}},\R^n)$)
one can consider the restriction $u\big|_\Sigma
\in C^1(\Sigma)$
(or $F\big|_\Sigma\in C^1(\Sigma,\R^n)$)
and then compute the tangential gradient
of $u\big|_\Sigma$ (or the tangential divergence of $F\big|_\Sigma$) on $\Sigma$, according to
definition (<ref>) (or definition (<ref>)),
that is using the normal extension
defined in (<ref>). The value obtained
in this way coincides with the tangential gradient of $u$
computed via definition (<ref>) (or the tangential divergence of $F$ computed via definition (<ref>)) evaluated at $\Sigma$. As a matter of fact, the values of a function
on $\Sigma$ suffice to compute its tangential gradient
(as well as the values of a vector field
on $\Sigma$ suffice to compute its tangential divergence), according to the following observation:
Let $u$, $\widetilde u\in C^1({\mathcal{N}})$ be
such that $u=\widetilde u$ on $\Sigma$
and let $\nabla_\Sigma u$ and $\nabla_\Sigma \widetilde u$ be computed as in (<ref>). Then, on $\Sigma$
we have that
\begin{equation}\label{FEU1}
\nabla_\Sigma u=
\nabla_\Sigma \widetilde u.\end{equation}
let $F$, $\widetilde F\in C^1({\mathcal{N}},\R^n)$ be
such that $F=\widetilde F$ on $\Sigma$
and let $\div_\Sigma F$ and $\div_\Sigma \widetilde F$ be computed as in (<ref>). Then, on $\Sigma$
we have that
\begin{equation}\label{FEU2}
\div_\Sigma F=
\div_\Sigma \widetilde F.\end{equation}
First of all, we observe that, if $F=(F_1,\dots,F_n)$, then
\begin{equation}\label{FEU}
\div_\Sigma F=\sum_{j=1}^n\nabla_\Sigma F_j\cdot e_j.
\end{equation}
Indeed, using (<ref>) and (<ref>), and then also (<ref>),
we see that
\begin{eqnarray*}&&
\div_\Sigma F-\sum_{j=1}^n\nabla_\Sigma F_j\cdot e_j\\&=&
\sum_{j=1}^n\partial_j F_j
\left(\nabla F_j-\big(\nabla F_j\cdot\nu_{\mbox{\scriptsize{ext}}}\big)\,\nu_{\mbox{\scriptsize{ext}}}
\right)\cdot e_j\\&=&
\big(\nabla F_j\cdot\nu_{\mbox{\scriptsize{ext}}}\big)\big(\nu_{\mbox{\scriptsize{ext}}}
\cdot e_j\big)\\&=&
\end{eqnarray*}
thus proving (<ref>).
Now we prove (<ref>). For this, we set $w:=u-\widetilde u$
and we remark that $w=0$ on $\Sigma$. Given a point $p$ of $\Sigma$, we suppose
that in a neighborhood of $p$ the hypersurface $\Sigma$ is parameterized by the graph
of a function $\psi:\R^{n-1}\to\R$ (up to renumbering the variables, we also assume that this
graph occurs in the $n$th coordinate direction, with the set $E$
lying above the graph), namely there exists $r>0$ such that
\begin{equation} \label{uojw-S29-32jfewnb}
B_r(p)\cap E=\{x_n>\psi(x')\}\cap B_r(p).\end{equation}
Notice that, on $\Sigma$,
\begin{equation}\label{NBORM} \nu=\frac{\left( \nabla'\psi,-1\right)}{\sqrt{1+|\nabla' \psi|^2}},\end{equation}
where the notation
\begin{equation}\label{NABLEP}
\nabla':=(\partial_1,\dots,\partial_{n-1})\end{equation}
has been used.
In this way, in the vicinity of $p$ we can write that $w(x',\psi(x'))=0$ and
$$ 0=\nabla' \Big(w(x',\psi(x'))\Big)=\nabla'w(x',\psi(x'))+\partial_nw(x',\psi(x'))\,\nabla'\psi(x').
Consequently, using (<ref>), we find that, on $\Sigma$,
in the vicinity of $p$,
\begin{eqnarray*}
\nabla_\Sigma u-\nabla_\Sigma\widetilde u&=&
\nabla w-\big(\nabla w\cdot\nu\big)\,\nu\\&=&
\Big( -\partial_nw\,\nabla'\psi,\partial_nw\Big)-\left(
\Big( -\partial_nw\,\nabla'\psi,\partial_nw\Big)\cdot
\frac{\left( \nabla'\psi,-1\right)}{\sqrt{1+|\nabla' \psi|^2}}\right)\frac{\left( \nabla'\psi,-1\right)}{\sqrt{1+|\nabla' \psi|^2}}\\
&=&\Big( -\partial_nw\,\nabla'\psi,\partial_nw\Big)+
\frac{\partial_nw\left( |\nabla'\psi|^2+1\right)}{\sqrt{1+|\nabla' \psi|^2}}\frac{\left( \nabla'\psi,-1\right)}{\sqrt{1+|\nabla' \psi|^2}}\\
&=&\Big( -\partial_nw\,\nabla'\psi,\partial_nw\Big)+
\Big(\partial_nw\, \nabla'\psi,-\partial_nw\Big)\\
\end{eqnarray*}
which establishes (<ref>).
To prove (<ref>), we exploit (<ref>) (applied to the scalar component
functions $F_j$ and $\widetilde F_j$) and (<ref>)
to compute that
\begin{eqnarray*}
&&\div_\Sigma F-
\div_\Sigma \widetilde F
=\sum_{j=1}^n\nabla_\Sigma F_j\cdot e_j-
\sum_{j=1}^n\nabla_\Sigma \widetilde F_j\cdot e_j
\end{eqnarray*}
This completes the proof of (<ref>).
In a nutshell, the content of Lemma <ref>
is that different extensions of a smooth object defined only on $\Sigma$ do not alter the tangential “first order”
operators, since the tangent hyperplane of $\Sigma$ “detaches quadratically”
from $\Sigma$ (we will find however that
“second order” operators are sensitive to different
types of extensions, see (<ref>) below).
We observe that the Laplace-Beltrami operator possesses
a “tangential divergence form structure”divergence form, to be compared with
the classical one in (<ref>), namely:
For every $u\in C^2(\Sigma)$, on $\Sigma$ we have that
$$ \Delta_\Sigma u=\div_\Sigma(\nabla_\Sigma u).$$
Using in order (<ref>), (<ref>), (<ref>),
(<ref>) and (<ref>),
we see that, on $\Sigma$,
\begin{equation*}\begin{split}&
\Delta_\Sigma u-\div_\Sigma(\nabla_\Sigma u)=
\Delta u_{\mbox{\scriptsize{ext}}}-\div_\Sigma
(\nabla_\Sigma u_{\mbox{\scriptsize{ext}}})=\Delta u_{\mbox{\scriptsize{ext}}}-\div_\Sigma
(\nabla u_{\mbox{\scriptsize{ext}}})\\&\qquad=
\Delta u_{\mbox{\scriptsize{ext}}}-
\div (\nabla u_{\mbox{\scriptsize{ext}}})-\nabla(\nabla u_{\mbox{\scriptsize{ext}}}
\cdot\nu_{\mbox{\scriptsize{ext}}})\cdot\nu
\qedhere\end{split}\end{equation*}
For further use, it is now useful to recall an asymptotic
result about the sets obtained by “thickening” $\Sigma$.
For general and precise formulas
computing tubular neighborhoods of hypersurfaces
see e.g. <cit.>.
Let $\alpha$ be a continuous function on $\Sigma$
and $\beta$ be a continuous function on ${\mathcal{N}}$.
Let $\e>0$ and
\begin{equation}\label{KMSD-fpblf} \Sigma_\e(\alpha):=\big\{
p+t\nu(p),\;p\in\Sigma,\;0\le t\le \e\alpha(p)
\big\}.\end{equation}
Then, as $\e\searrow0$,
\begin{equation}\label{dxs} \int_{\Sigma_\e(\alpha)} \beta(y)\,dy= \e
\int_\Sigma \alpha_+(p)\,\beta(p)\,d{\mathcal{H}}^{n-1}_p+o(\e),
\end{equation}
where $\alpha_+(p):=\max\{\alpha(p),\,0\}$.
Local charts for $\Sigma$ and a partition of unity.
In local coordinates, we write a surface element of $\Sigma$
as a graph of a function
\begin{equation}\label{psi9itg}\psi:U\subseteq\R^{n-1}\to\R,\end{equation}
say in the $n$th direction, with normal as in (<ref>).
In this way, points $y$ in this element of $\Sigma_\e(\alpha)$ are of the form
\begin{equation}\label{psi9itg2} y=(x',\psi(x'))+
\frac{t\,\left( \nabla'\psi(x'),-1\right)}{\sqrt{1+|\nabla' \psi(x')|^2}}
with~$x'\in U$ and~$0<t<\e\alpha(x',\psi(x'))$.}}\end{equation}
That is, one can consider a partition of unity
(see e.g. <cit.>) made of functions $\phi_i\in C^\infty_0({\mathcal{N}},\,[0,1])$
with ${i\in\N}$ and finite overlapping supports, each compactly contained
in a local chart of $\Sigma$,
such that $\sum_{i\in\N}\phi_i=1$ in a given neighborhood ${\mathcal{N}}'$
of $\Sigma$ (with $\Sigma\subseteq{\mathcal{N}}'\Subset{\mathcal{N}}$, see Figure <ref>).
Then, letting $\beta_i:=\beta\phi_i$, it suffices to prove (<ref>) with $\beta$ replaced by $\beta_i$,
\begin{equation*} \int_{\Sigma_\e(\alpha)} \beta(y)\,dy=
\sum_{i\in\N}\int_{\Sigma_\e(\alpha)} \beta_i(y)\,dy
\qquad{\mbox{and}}\qquad
\int_\Sigma \alpha_+(p)\,\beta(p)\,d{\mathcal{H}}^{n-1}_p=
\sum_{i\in\N}\int_\Sigma \alpha_+(p)\,\beta_i(p)\,d{\mathcal{H}}^{n-1}_p.
\end{equation*}
Therefore, from now on, to prove (<ref>), up to replacing $\beta$
with $\beta_i$, we can suppose that, in the support of $\beta$,
the hypersurface $\Sigma$ is a graph of a function $\psi$ as in (<ref>),
say in the $n$th direction, and, for small $\e$, the tubular neighborhood $ {\Sigma_\e(\alpha)}$
can be written as the set of points $y$ in (<ref>).
For convenience, one can denote $x_n:=t$ and $x:=(x',x_n)$
in (<ref>), and thus describe $ {\Sigma_\e(\alpha)}$ in the support of $\beta$
as the collection of points
$$ y=(x',\psi(x'))+
\frac{x_n\,\left( \nabla'\psi(x'),-1\right)}{\sqrt{1+|\nabla' \psi(x')|^2}}
with $x\in U\times[0,\e\alpha(x',\psi(x'))]$
(if the latter quantity is well defined, i.e. if $\alpha(x',\psi(x'))\ge0$).
Notice in particular that $x_n=O(\e)$ and we thus consider,
for small $\e$, the change of variable relating $y$
and $x$, with
$$ \frac{\partial y}{\partial x}=\left(
\begin{matrix}
\partial_{x_1} y_1&\;\dots\;&\partial_{x_{n-1}} y_1&\;\,\partial_{x_n} y_1
\\ &\ddots\\
\partial_{x_1} y_{n-1}&
\;\dots\;&\partial_{x_{n-1}} y_{n-1}&\;\,\partial_{x_n} y_{n-1}\\
\partial_{x_1} y_{n}&\;\dots\;&\partial_{x_{n-1}} y_n&\;\,\partial_{x_n} y_n
\end{matrix}
\right)
\begin{matrix}
1&\;\dots\;0&\;\,\partial_{x_1} \psi/R
\\ &\ddots\\
0&\;\dots\;1&\;\,\partial_{x_{n-1}} \psi/R\\
\partial_{x_1} \psi&\;\dots\;\partial_{x_{n-1}} \psi&\;\,-1/R
\end{matrix}
\right)+O(\e),$$
with $R:=\sqrt{1+|\nabla' \psi(x')|^2}$.
As a result,
$$ \left|\det\frac{\partial y}{\partial x}
\right|=\frac{
(\partial_{x_1} \psi)^2+\dots(\partial_{x_{n-1}} \psi)^2+1}{R}
We remark that $R\,dx'$ is the surface element on $\Sigma$
(see e.g. <cit.>), hence we write
the volume element of (<ref>) in the form
$$ dy=\left|\det\frac{\partial y}{\partial x}
\right|\,dx
That is,
\begin{eqnarray*}
&&\int_{\Sigma_\e(\alpha)} \beta(y)\,dy=
\int_{\Sigma_\e(\alpha)} \big(\beta(\pi_\Sigma(y))+o(1)\big)\,dy\\&&\qquad=
\int_\Sigma\left[ \int_0^{\e\alpha_+(p)}
\big(\beta(p)+o(1)\big)\,dx_n\right]\,d{\mathcal{H}}^{n-1}_p
\int_\Sigma \alpha_+(p)\,\beta(p)\,d{\mathcal{H}}^{n-1}_p+o(\e),
\end{eqnarray*}
giving (<ref>)
as desired.
The tangential differential setting provides another useful form of “integration by parts formula”
according to Theorem <ref> below. Differently from the Euclidean case,
this result takes into account an additional term coming from the geometry of $\Sigma$.
For this,
it is useful to introduce the mean curvaturemean curvature
at a point $x\in\Sigma$, defined as
\begin{equation}\label{MC}
H(x):=\div_\Sigma \nu(x).
\end{equation}
See e.g. Section 1.2 in <cit.> for a geometric
description of the mean curvature.
Then, we have the following result, sometimes called the “Tangential Divergence Theorem”Tangential Divergence Theorem:
For every $F\in C^1(\Sigma,\R^n)$ and $\varphi\in C^1(\Sigma)$,
$$ \int_\Sigma \div_\Sigma F(x)\;\varphi(x)\,d{\mathcal{H}}^{n-1}_x=
\int_\Sigma F(x)\cdot\Big(H(x)\nu(x)\varphi(x)-\nabla_\Sigma\varphi(x)\Big)\,d{\mathcal{H}}^{n-1}_x.$$
Given $\e>0$, to be taken conveniently small,
we consider a tubular neighborhood of $\Sigma$ of radius $\e$, namely we set $
\Sigma_\e$ as in (<ref>) with $\alpha:=1$.
The hypersurface $\Sigma$ and the “parallel hypersurface”
at distance $\varepsilon$.
Let us now analyze the exterior normal $\nu_{\Sigma_\e}$
along $\partial \Sigma_\e$.
We stress that $\partial \Sigma_\e=\{d_\Sigma=\e\}\cup\{d_\Sigma=-\e\}$
and thus we denote by $\nu^{(\pm)}_\e$ the exterior normal
of $\Sigma_\e$ along $\{d_\Sigma=\pm\e\}$, and, for clarity,
by $\nu_\Sigma$ the exterior normal of $E$ along $\Sigma$, see Figure <ref>.
In this way, if $x\in\{d_\Sigma=\e\}$ the exterior normal $\nu^{(+)}_\e$
at $x\in\{d_\Sigma=\e\}$ is minus
the exterior normal $\nu_\Sigma$ of $E$ at $\pi_\Sigma(x)$,
while the
exterior normal $\nu^{(-)}_\e$ at $x\in\{d_\Sigma=-\e\}$ is plus
the exterior normal $\nu_\Sigma$ of $E$ at $\pi_\Sigma(x)$, that is
\begin{equation}\label{221}
\begin{split}&
\nu^{(+)}_\e(x)=-\nu_\Sigma(\pi_\Sigma(x))=-\nu_{\mbox{\scriptsize{ext}}}(x)\qquad{\mbox{if }}x\in\{d_\Sigma=\e\}\\
{\mbox{and }}\qquad&
\nu^{(-)}_\e(x)=\nu_\Sigma(\pi_\Sigma(x))=\nu_{\mbox{\scriptsize{ext}}}(x)\qquad{\mbox{if }}x\in\{d_\Sigma=-\e\}.
\end{split}\end{equation}
Moreover, by (<ref>),
used here with $\alpha:=1$
and $\beta:=
\div F_{\mbox{\scriptsize{ext}}}\;\varphi
+ F\cdot\nabla\varphi_{\mbox{\scriptsize{ext}}}$,
\begin{equation}\label{BGio}
\begin{split}
&\int_\Sigma \div_\Sigma F(x)\;\varphi(x)\,d{\mathcal{H}}^{n-1}_x+\int_\Sigma F(x)\cdot\nabla_\Sigma\varphi(x)\,d{\mathcal{H}}^{n-1}_x\\=\,&
\int_\Sigma \Big( \div F_{\mbox{\scriptsize{ext}}}(x)\;\varphi(x)
+ F(x)\cdot\nabla\varphi_{\mbox{\scriptsize{ext}}}(x)\Big)\,d{\mathcal{H}}^{n-1}_x
\\=\,&
\lim_{\e\searrow0}\frac1{\e}\,
\int_{\Sigma_\e} \Big( \div F_{\mbox{\scriptsize{ext}}}(x)\;\varphi_{\mbox{\scriptsize{ext}}}
(x)+ F_{\mbox{\scriptsize{ext}}}(x)\cdot
\nabla\varphi_{\mbox{\scriptsize{ext}}}(x)\Big)\,dx\\=\,&
\lim_{\e\searrow0}\frac1{\e}\,
\int_{\Sigma_\e} \div \Big(
\varphi_{\mbox{\scriptsize{ext}}}
\int_{\Sigma_\e} \div G_{\mbox{\scriptsize{ext}}}(x)\,dx
where $G:=\varphi F$.
We also point out that, if $\widetilde G:=G\cdot\nu$,
\begin{eqnarray*}&&
\div\Big( (G_{\mbox{\scriptsize{ext}}}\cdot\nu_{\mbox{\scriptsize{ext}}})\nu_{\mbox{\scriptsize{ext}}}\Big)-
\nabla(G_{\mbox{\scriptsize{ext}}}\cdot\nu_{\mbox{\scriptsize{ext}}})\cdot\nu_{\mbox{\scriptsize{ext}}}=\nabla \widetilde G_{\mbox{\scriptsize{ext}}}\cdot\nu_{\mbox{\scriptsize{ext}}}=0,
\end{eqnarray*}
thanks to (<ref>). Consequently,
\begin{eqnarray*}&&
\int_\Sigma H(x) F(x)\cdot\nu(x)\varphi(x)\,d{\mathcal{H}}^{n-1}_x=
\int_\Sigma (G(x)\cdot\nu(x)) \div\nu_{\mbox{\scriptsize{ext}}}(x) \,d{\mathcal{H}}^{n-1}_x\\&&\qquad
=\int_\Sigma(G_{\mbox{\scriptsize{ext}}}(x)\cdot\nu_{\mbox{\scriptsize{ext}}}(x)) \div\nu_{\mbox{\scriptsize{ext}}}(x) \,d{\mathcal{H}}^{n-1}_x=\lim_{\e\searrow0}\frac1{
\e}\,
\int_{\Sigma_\e}(G_{\mbox{\scriptsize{ext}}}(x)\cdot\nu_{\mbox{\scriptsize{ext}}}(x))\div\nu_{\mbox{\scriptsize{ext}}}(x)\,dx\\&&\qquad=\lim_{\e\searrow0}\frac1{
\e}\,
\int_{\Sigma_\e}
\div\Big( (G_{\mbox{\scriptsize{ext}}}(x)\cdot\nu_{\mbox{\scriptsize{ext}}}(x)
\end{eqnarray*}
This and (<ref>), together with (<ref>), lead to
\begin{equation*}\begin{split}&
\int_\Sigma \div_\Sigma F(x)\;\varphi(x)\,d{\mathcal{H}}^{n-1}_x+
\int_\Sigma F(x)\cdot\nabla_\Sigma\varphi(x)\,d{\mathcal{H}}^{n-1}_x
-\int_\Sigma H(x) F(x)\cdot\nu(x)\varphi(x)\,d{\mathcal{H}}^{n-1}_x\\=\,&
\lim_{\e\searrow0}\frac1{\e}\,
\int_{\Sigma_\e} \div\Big( G_{\mbox{\scriptsize{ext}}}(x)-
\cdot\nu_{\mbox{\scriptsize{ext}}}(x))\nu_{\mbox{\scriptsize{ext}}}(x)\Big)\,dx\\=\,&
\lim_{\e\searrow0}\frac1{\e}\,
\int_{\partial\Sigma_\e}
\Big( G_{\mbox{\scriptsize{ext}}}(x)-
\cdot\nu_{\mbox{\scriptsize{ext}}}(x))\nu_{\mbox{\scriptsize{ext}}}(x)\Big)
\cdot\nu_{\Sigma_\e}(x)
\,d{\mathcal{H}}^{n-1}_x\\=\,&
\lim_{\e\searrow0}\frac1{\e}\,
\Bigg(-\int_{\{d_\Sigma=\e\}}
\Big( G_{\mbox{\scriptsize{ext}}}(x)-
\cdot\nu_{\mbox{\scriptsize{ext}}}(x))\nu_{\mbox{\scriptsize{ext}}}(x)\Big)
\cdot\nu_{\mbox{\scriptsize{ext}}}(x)
\,d{\mathcal{H}}^{n-1}_x\\&\qquad+
\int_{\{d_\Sigma=-\e\}}
\Big( G_{\mbox{\scriptsize{ext}}}(x)-
\cdot\nu_{\mbox{\scriptsize{ext}}}(x))\nu_{\mbox{\scriptsize{ext}}}(x)\Big)
\cdot\nu_{\mbox{\scriptsize{ext}}}(x)
\,d{\mathcal{H}}^{n-1}_x
\Bigg)
\\=\,&
\lim_{\e\searrow0}\frac1{\e}\,
\Bigg(-\int_{\{d_\Sigma=\e\}}
\Big( (G_{\mbox{\scriptsize{ext}}}(x)
\cdot\nu_{\mbox{\scriptsize{ext}}}(x))-(G_{\mbox{\scriptsize{ext}}}(x)
\cdot\nu_{\mbox{\scriptsize{ext}}}(x))\Big)
\,d{\mathcal{H}}^{n-1}_x\\&\qquad+
\int_{\{d_\Sigma=-\e\}}
\Big( (G_{\mbox{\scriptsize{ext}}}(x)
\cdot\nu_{\mbox{\scriptsize{ext}}}(x))-(G_{\mbox{\scriptsize{ext}}}(x)
\cdot\nu_{\mbox{\scriptsize{ext}}}(x))\Big)
\,d{\mathcal{H}}^{n-1}_x
\Bigg)
\\=\,&\lim_{\e\searrow0}\frac1{\e}\,\big(0-0\big)
\\=\,&0.\qedhere\end{split}
\end{equation*}
The content of Theorem <ref> happens to be a special case
of some “Minkowski Integral Formulas”Minkowski Integral Formula, see e.g. <cit.>.
For instance, a classical identity is the following one:
We have that
$$ {\mathcal{H}}^{n-1}(\Sigma)=\frac1{n-1}
\int_\Sigma H(x)\,x\cdot\nu(x)\,d{\mathcal{H}}^{n-1}_x.$$
For every $x\in{\mathcal{N}}$,
let $F(x):=x$.
Then, by Lemma <ref> and (<ref>), recalling also (<ref>), for $x\in\Sigma$ we have that
\begin{eqnarray*}&&
\div_\Sigma F(x)=\div F(x)-\nabla(F(x)\cdot\nu_{\mbox{\scriptsize{ext}}}(x))\cdot\nu(x)=
x_j\nabla(\nu_{\mbox{\scriptsize{ext}}}(x)\cdot e_j)\cdot\nu(x)=n-1.
\end{eqnarray*}
The desired result follows by exploiting the Tangential Divergence Theorem
(see Theorem <ref>)
with $\varphi:=1$.
From the Tangential Divergence Theorem <ref>, one also obtains:
For every $u\in C^2(\Sigma)$ and $\varphi\in C^1(\Sigma)$,
\begin{equation}\label{PAR-091} \int_\Sigma \Delta_\Sigma u(x)\,\varphi(x)\,d{\mathcal{H}}^{n-1}_x=
-\int_\Sigma \nabla_\Sigma u(x)\cdot
\nabla_\Sigma\varphi(x)\,d{\mathcal{H}}^{n-1}_x.\end{equation}
Moreover, for every $u\in C^2(\Sigma)$ and $\varphi\in C^2(\Sigma)$,
\begin{equation}\label{PAR-092}
\int_\Sigma \Delta_\Sigma u(x)\,\varphi(x)\,d{\mathcal{H}}^{n-1}_x=
\int_\Sigma u(x)\,\Delta_\Sigma\varphi(x)\,d{\mathcal{H}}^{n-1}_x.\end{equation}
Let $u\in C^2(\Sigma)$
and $F:=\nabla_\Sigma u$. Then, by (<ref>), (<ref>)
and (<ref>), on $\Sigma$ we have that
$$ F\cdot\nu=\nabla_\Sigma u_{\mbox{\scriptsize{ext}}}\cdot\nu=\nabla
if $\varphi\in C^1(\Sigma)$,
by Lemma <ref> and Theorem <ref>,
\begin{eqnarray*}&&
\int_\Sigma \Delta_\Sigma u(x)\,\varphi(x)\,d{\mathcal{H}}^{n-1}_x+
\int_\Sigma \nabla_\Sigma u(x)\cdot
\nabla_\Sigma\varphi(x)\,d{\mathcal{H}}^{n-1}_x\\&=&
\int_\Sigma \div_\Sigma F(x)\,\varphi(x)\,d{\mathcal{H}}^{n-1}_x+
\int_\Sigma F(x)\cdot
\nabla_\Sigma\varphi(x)\,d{\mathcal{H}}^{n-1}_x\\&=&
\int_\Sigma H(x)\varphi(x)F(x)\cdot\nu(x)
\,d{\mathcal{H}}^{n-1}_x\\&=&0.
\end{eqnarray*}
This establishes (<ref>).
Now we suppose that also $\varphi\in C^2(\Sigma)$,
and we write (<ref>) exchanging the roles of $u$ and $\varphi$,
$$ \int_\Sigma \Delta_\Sigma \varphi(x)\,u(x)\,d{\mathcal{H}}^{n-1}_x=
-\int_\Sigma \nabla_\Sigma \varphi(x)\cdot
\nabla_\Sigma u(x)\,d{\mathcal{H}}^{n-1}_x.$$
From this and (<ref>), the desired result in (<ref>)
plainly follows.
Now we give an explicit formula for the mean curvaturemean curvature
with respect to the function that describes locally the hypersurface $\Sigma$.
We use the notation in (<ref>)
and $\div':=\partial_1+\dots+\partial_{n-1}$. With this, we have:
Let $p=(p',p_n)\in \Sigma$.
Assume that there exists $\rho>0$
such that $E$ in $B_\rho(p)$ can be written as a supergraph
of a function $\psi:\R^{n-1}\to\R$, namely
$$ E\cap B_\rho(p)=\{ x_n>\psi(x')\}\cap B_\rho(p).$$
Then, at $p$,
\begin{equation}\label{LAMEANCURVA} H=\div'\left(
\frac{\nabla' \psi}{\sqrt{1+|\nabla' \psi|^2}}
\right).\end{equation}
Given $x=(x',x_n)\in\R^n$, we define
$$ \nu_\star(x):=\frac{\left( \nabla'\psi(x'),-1\right)}{\sqrt{1+|\nabla' \psi(x')|^2}}.$$
By (<ref>), we know that $\nu_\star$ coincides with $\nu$
(and thus with $ \nu_{\mbox{\scriptsize{ext}}}$) on $\Sigma$ in the vicinity of $p$
and therefore,
using (<ref>)
and successively (<ref>), (<ref>)
and (<ref>), we have that,
for all $x\in\Sigma$
in the vicinity of $p$,
\begin{equation}\label{kPmsi}
H(x)=\div_\Sigma \nu(x)
=\div_\Sigma \nu_{\mbox{\scriptsize{ext}}}(x)=\div_\Sigma \nu_\star=
\div \nu_\star-\nabla(\nu_\star\cdot\nu_{\mbox{\scriptsize{ext}}})\cdot\nu.
\end{equation}
We also observe that $\nu_\star(x)$ does not depend on $x_n$ and therefore
\begin{equation}\label{LKJNnbvdceij84} \div \nu_\star=
\div'\left(
\frac{\nabla' \psi}{\sqrt{1+|\nabla' \psi|^2}}
\right).\end{equation}
Furthermore, for every $j\in\{1,\dots,n\}$,
$$ 0=\partial_j \frac12=\partial_j\frac{ |\nu_\star|^2}2=
\frac12\sum_{m=1}^n \partial_j (\nu_\star\cdot e_m)^2=
\sum_{m=1}^n (\nu_\star\cdot e_m)\,
\partial_j(\nu_\star\cdot e_m).
That is, on $\Sigma$,
$$ \sum_{m=1}^n (\nu\cdot e_m)\,
\partial_j(\nu_\star\cdot e_m)=0.$$
Similarly, for every $j\in\{1,\dots,n\}$, on $\Sigma$,
$$ \sum_{m=1}^n (\nu\cdot e_m)\,
\partial_j(\nu_{\mbox{\scriptsize{ext}}}\cdot e_m)=0.$$
Hence, on $\Sigma$,
\begin{eqnarray*}
\nabla(\nu_\star\cdot\nu_{\mbox{\scriptsize{ext}}})\cdot\nu&=&\sum_{j=1}^n
\partial_j(\nu_\star\cdot\nu_{\mbox{\scriptsize{ext}}})\,(\nu\cdot e_j)
\\&=&
\sum_{j,m=1}^n
\partial_j\Big(
(\nu_\star\cdot e_m)(\nu_{\mbox{\scriptsize{ext}}}\cdot e_m)\Big)\,(\nu\cdot e_j)
\\&=&
\sum_{j,m=1}^n\left[
(\nu_{\mbox{\scriptsize{ext}}}\cdot e_m)(\nu\cdot e_j)\,
\partial_j
(\nu_\star\cdot e_m)
+(\nu_\star\cdot e_m)(\nu\cdot e_j)\,\partial_j(\nu_{\mbox{\scriptsize{ext}}}\cdot e_m)\right]\\&=&
\sum_{j,m=1}^n\left[
(\nu\cdot e_m)(\nu\cdot e_j)\,
\partial_j
(\nu_\star\cdot e_m)
+(\nu\cdot e_m)(\nu\cdot e_j)\,\partial_j(\nu_{\mbox{\scriptsize{ext}}}\cdot e_m)\right]\\&=&0.
\end{eqnarray*}
This, (<ref>)
and (<ref>) yield the desired result.
As a consequence of Theorem <ref>, we also point out that
the mean curvature is monotone with respect to set inclusions:
Let $E$, $F\subseteq\R^n$ be bounded and open sets
with $C^3$ boundary, with mean curvature $H_{\partial E}$ and $H_{\partial F}$, respectively.
Assume that $E\subseteq F$ and that $p\in (\partial E)\cap(\partial F)$. Then, $H_{\partial E}(p)\ge
H_{\partial F}(p)$.
We remark that the external normal $\nu_{\partial E}$ of $E$
coincides with the external normal $\nu_{\partial F}$ of $F$ at $p$,
hence we can write both $E$ and $F$ as graphs with respect to a common coordinate direction.
That is, without loss of generality, we may assume that there exist $\rho>0$ and two functions $\psi_E$,
$\psi_F:\R^{n-1}\to\R$ such that $p=(p',p_n)=(p',\psi_E(p'))=(p',\psi_F(p'))$,
$$ E\cap B_\rho(p)=\{ x_n>\psi_E(x')\}\cap B_\rho(p)\qquad{\mbox{ and }}\qquad
F\cap B_\rho(p)=\{ x_n>\psi_F(x')\}\cap B_\rho(p).$$
Also, since $E\subseteq F$, we have that $\psi_E\ge\psi_F$ in the vicinity of $p'$.
Therefore, the function $\phi:=\psi_E-\psi_F$ has a local minimum at $p'$, entailing that $\nabla\phi(p')=0$
and the Hessian of $\phi$ at $p$, that we denote by $D^2_{x'}\phi(p)$,
is a nonnegative definite matrix. From this and Theorem <ref> we deduce that
at $p'$
\begin{equation}\label{TBO-PLA-S0}\begin{split}H_{\partial E}
-H_{\partial F}\,&=\div'\left(
\frac{\nabla' \psi_E}{\sqrt{1+|\nabla' \psi_E|^2}}
\right)-\div'\left(
\frac{\nabla' \psi_F}{\sqrt{1+|\nabla' \psi_F|^2}}
\right)\\&=\frac{\Delta' \psi_E}{\sqrt{1+|\nabla' \psi_E|^2}}-\frac{\Delta' \psi_F}{\sqrt{1+|\nabla' \psi_F|^2}}
-\sum_{i,j=1}^{n-1}\left(\frac{\partial_{ij}\psi_E\partial_i\psi_E\,\partial_j\,\psi_E}{(1+|\nabla' \psi_E|^2)^{3/2}}
-\frac{\partial_{ij}\psi_F\,\partial_i\psi_F\,\partial_j\psi_F}{(1+|\nabla' \psi_F|^2)^{3/2}}\right)\\&=
\frac{1}{({1+|\nabla' \psi_E|^2})^{3/2}}\left(
\big({1+|\nabla' \psi_E|^2}\big)\sum_{i=1}^n(D^2_{x'}\phi(p) \,e_i)\cdot e_i-{
(D^2_{x'}\phi(p) \,\nabla'\psi_E)\cdot \nabla'\psi_E}
\right).\end{split}\end{equation}
We now observe that if $M\in{\rm Mat}(m\times m)$
is symmetric and nonnegative definite, and $v\in\R^m$, then
\begin{equation}\label{TBO-PLA-S}
|v|^2\sum_{i=1}^m Me_i\cdot e_i\ge Mv\cdot v.
\end{equation}
To prove this,
we use the Spectral Theorem to find
an orthonormal basis $\{\eta_1,\dots,\eta_m\}$ of $\R^m$
consisting of eigenvectors of $M$. Namely,
$$ M\eta_i=\lambda \eta_i\qquad{\mbox{for all }}i\in\{1,\dots,m\},$$
with $\lambda_1,\dots,\lambda_m\ge0$.
Thus, we write $v$ in terms of this new basis as
$$ v=\sum_{i=1}^m (v\cdot \eta_i)\eta_i$$
and we have that
\begin{eqnarray*}
Mv\cdot v=\sum_{i,j=1}^m (v\cdot \eta_i)
(v\cdot \eta_j)M\eta_i\cdot\eta_j=
\sum_{i,j=1}^m \lambda_i (v\cdot \eta_i)
(v\cdot \eta_j)(\eta_i\cdot\eta_j)
\sum_{i=1}^m \lambda_i (v\cdot \eta_i)^2.\end{eqnarray*}
Applying this identity with $e_k$ instead of $v$,
we also obtain that, for each $k\in\{1,\dots,m\}$,
\begin{eqnarray*}
Me_k\cdot e_k=
\sum_{i=1}^m \lambda_i (e_k\cdot \eta_i)^2.\end{eqnarray*}
As a result,
\begin{equation}\label{KND st2T2}\begin{split}&
|v|^2\sum_{k=1}^m Me_k\cdot e_k-Mv\cdot v
\sum_{i=1}^m \lambda_i \left(\sum_{k=1}^m|v|^2(e_k\cdot \eta_i)^2
(v\cdot \eta_i)^2
\right)\\&\qquad\qquad\ge|v|^2\sum_{i=1}^m \lambda_i \left(\sum_{k=1}^m(e_k\cdot \eta_i)^2
\right).\end{split}
\end{equation}
Hence, since
$$ 1=|\eta_i|^2=\left|
\sum_{k=1}^m(\eta_i\cdot e_k)e_k
\right|^2=\sum_{k=1}^m(\eta_i\cdot e_k)^2,$$
we obtain (<ref>) as a consequence of (<ref>).
From (<ref>)
and (<ref>)
we conclude that $H_{\partial E}-H_{\partial F}\ge0$, as desired.
There exists $p\in\Sigma$ such that $H(p)>0$.
Since $E$ is bounded, we have that $E\subseteq B_R$ for some $R>0$.
By decreasing $R$ till the external ball touches $\partial E$, we can thus suppose that $E\subseteq B_R$
and there exists $p\in(\partial E)\cap(\partial B_R)$. Then, by
Corollary <ref>, $H(p)\ge\frac{n-1}{R}>0$.
To further analyze the interplay between the Laplace-Beltrami operator and the mean curvature of $\Sigma$,
one can also introduce the “tangential Laplacian”, defined, for every $p\in\Sigma$
and every $u\in C^2({\mathcal{N}})$, as
\begin{equation}\label{TANLAP}
\Delta_T u(p):=\Delta u(p)-\big( D^2 u(p)\,\nu(p)\big)\cdot\nu(p).\end{equation}
That is, the tangential Laplacian is the “$(n-1)$-dimensional Laplacian”
on the tangent plane to $\Sigma$ at $p$.
We stress that the tangential Laplacian does not coincide with the Laplace-Beltrami operator
and in fact their difference is related to the mean curvature of $\Sigma$, as pointed out by the following result:
Let $u\in C^2({\mathcal{N}})$. Then, on $\Sigma$,
$$ \Delta_\Sigma u = \Delta_T u-H\nabla u\cdot \nu.$$
Using Lemma <ref>, (<ref>)
and (<ref>), we see that, on $\Sigma$,
\begin{eqnarray*}
\Delta_\Sigma u&=&\div_\Sigma(\nabla_\Sigma u)\\
&=&\div_\Sigma\Big(\nabla u-\big(\nabla u\cdot\nu\big)\,\nu\Big)\\
&=&\div\Big(\nabla u-\big(\nabla u\cdot\nu\big)\,\nu\Big)_{\mbox{\scriptsize{ext}}}\\
&=&\div\Big((\nabla u)_{\mbox{\scriptsize{ext}}}\Big)-\nabla
\Big( \big(\nabla u\cdot\nu\big)_{\mbox{\scriptsize{ext}}}\Big)\cdot\nu -\big(\nabla u\cdot\nu\big)\div\nu_{\mbox{\scriptsize{ext}}}.
\end{eqnarray*}
Thus, by (<ref>), (<ref>) and (<ref>),
\begin{eqnarray*}
\Delta_\Sigma u&=&
\div_\Sigma(\nabla u)-0-\big(\nabla u\cdot\nu\big)H.
\end{eqnarray*}
As a consequence, by (<ref>), and using again (<ref>), on $\Sigma$,
\begin{eqnarray*}
\Delta_\Sigma u&=&
\div( \nabla u)-\nabla\big( \nabla u\cdot\nu_{\mbox{\scriptsize{ext}}}\big)\cdot\nu
-\big(\nabla u\cdot\nu\big)H\\&=&
\Delta u-(D^2 u\,\nu)\cdot\nu-\sum_{j=1}^n\nabla (\nu_{\mbox{\scriptsize{ext}}}
\cdot e_j)\cdot\nu\, \partial_j u
-\big(\nabla u\cdot\nu\big)H\\&=&
\Delta_T u-0-\big(\nabla u\cdot\nu\big)H,
\end{eqnarray*}
as desired.
The role of the curvature in second order operators.
An intuitive explanation for the role played by the curvature of $\Sigma$ in the computation
of second order operator naturally arises from Figure <ref>. Namely, near a given point $p\in\Sigma$,
the second order behavior of the hypersurface is well approximated by its osculating sphere,
hence, for simplicity, we presented in Figure <ref> the case $n=2$, $\Sigma=\partial B_r$
and, up to a rotation, $p=re_1$. We consider a “tangential infinitesimal increment” $he_2$.
Notice from Figure <ref> that $h=r\tan\vartheta$, and accordingly $\cos\vartheta=\frac{r}{\sqrt{r^2+h^2}}=1-\frac{h^2}{2r^2}+o(h^2)$ and $\sin\vartheta=\frac{h\cos\vartheta}{r}=\frac{h}{r}+o(h^2)$.
Then, the second order incremental quotient of $u_{\mbox{\scriptsize{ext}}}$ in the tangential direction $e_2$ is therefore
\begin{eqnarray*}&& \frac{u_{\mbox{\scriptsize{ext}}}(p+he_2)+
\frac{u(r\cos\vartheta,r\sin\vartheta)+
\frac{u\left(r-\frac{h^2}{2r}+o(h^2),h+o(h^2)\right)+
\frac{1}{h^2}\Bigg(\left(-\partial_1 u(r,0)\frac{h^2}{2r}
+\partial_2 u(r,0)h+\frac12\partial_{22} u(r,0)h^2\right)\\&&\qquad\qquad\qquad\qquad+
\left(-\partial_1 u(r,0)\frac{h^2}{2r}
-\partial_2 u(r,0)h+\frac12\partial_{22} u(r,0)h^2\right)
-\frac{1}{r}\,\partial_1 u(r,0)+\partial_{22} u(r,0)+o(1),
\end{eqnarray*}
which, as $h\searrow0$, recalling that the curvature of the circle is $\frac1r$,
converges to $-H\partial_\nu u(p)+\partial_{22} u(p)$, and this corresponds to the identity found in
Theorem <ref>.
A simple consequence of
Theorem <ref> is also the expression of the Laplacian along level setsLaplacian along level sets
of the function $u$, according to the following observation:
Let $u\in C^2({\mathcal{N}})$. Suppose that $u$ is constant along $\Sigma$. Then, on $\Sigma$,
$$ \Delta u= H\,\partial_\nu u+\partial_{\nu\nu} u
Let $c\in\R$ be such that $u=c$ on $\Sigma$.
Then, $u_{\mbox{\scriptsize{ext}}}(x)=u(\pi_\Sigma(x))=c$ for
all $x\in{\mathcal{N}}$, whence $\Delta_\Sigma u=0$.
one combines Theorem <ref> and the definition
of tangential Laplacian in (<ref>) to see that, on $\Sigma$,
\begin{equation*}
H\nabla u\cdot \nu=
\Delta_\Sigma u +H\nabla u\cdot \nu=
\Delta_T u=\Delta u-( D^2 u\,\nu\big)\cdot\nu
For completeness, we now give another formula to compute the mean curvature of
the boundary of a smooth set. This formula is perhaps not very handy for explicit calculations, but it has the
conceptual advantage of expressing the mean curvature as a local average of a set against its complement
(thus entailing that a set whose boundary has zero mean curvature has the property that the volume density of
the set is well compensated at each boundary point by the volume density of its complement). The precise result
goes as follows:
For every $p\in\Sigma=\partial E$,
\begin{equation}\label{IniZibnepbu} H(p)=\lim_{r\searrow0} \frac{1}{c r^{n+1}}\int_{B_r(p)}\big(\chi_{\R^n\setminus E}(x)-\chi_E(x)\big)\,dx,\end{equation}
for a suitable positive constant $c$ depending only on $n$.
Up to a translation and a rotation, we assume that the point $p$ is the origin and that $E$ in the vicinity
of the origin is the superlevel set of a function $\psi$, as in (<ref>), with $\psi(0)=0$ and $\nabla\psi(0)=0$.
Consequently, given $\e>0$ there exists $r_\e>0$ such that if $r\in(0,r_\e)$ then
\begin{equation}\label{0okS2890io3rhnfwe-4PA} \begin{split}&U:=B_r\cap\left\{
x_n\ge \frac12 D^2\psi(0)x'\cdot x'+\e|x'|^2\right\}\subseteq E\cap B_r\\
&\qquad\subseteq B_r\cap\left\{
x_n\ge \frac12 D^2\psi(0)x'\cdot x'-\e|x'|^2\right\}=:V,\end{split}\end{equation}
see Figure <ref>.
The sets in (<ref>).
Accordingly, if $y\in B_r\setminus V$ then $\chi_{\R^n\setminus E}(y)-\chi_E(y)=1$
and if $y\in U$ then $\chi_{\R^n\setminus E}(y)-\chi_E(y)=-1$. It is therefore convenient to define
$$ S:=B_r\cap\left\{
x_n- \frac12 D^2\psi(0)x'\cdot x'\in\big(-\e|x'|^2,\,\e|x'|^2\big)\right\}$$
and note that
$$ B_r= U\cup S\cup( B_r\setminus V).$$
These observations give that
\begin{equation}\label{APCOSMFe}
\int_{B_r}\big(\chi_{\R^n\setminus E}(x)-\chi_E(x)\big)\,dx
=|B_r\setminus V|-|U|+\int_S\big(\chi_{\R^n\setminus E}(x)-\chi_E(x)\big)\,dx.
\end{equation}
\begin{eqnarray*}
|S|\le2\e\int_{|x'|< r}|x'|^2\,dx' =O(\e r^{n+1}),
\end{eqnarray*}
whence we infer from (<ref>) that
\begin{equation}\label{APCOSMFe-32}
\int_{B_r}\big(\chi_{\R^n\setminus E}(x)-\chi_E(x)\big)\,dx
=|B_r\setminus V|-|U|+O(\e r^{n+1}).
\end{equation}
\begin{eqnarray*}
|U|=\left| B_r\cap\left\{x_n\ge \frac12 D^2\psi(0)x'\cdot x'\right\}\right|+O(\e r^{n+1})
\end{eqnarray*}
\begin{eqnarray*}
|B_r\setminus V|&=&\left| B_r\cap\left\{x_n< \frac12 D^2\psi(0)x'\cdot x'\right\}\right|+O(\e r^{n+1})\\&=&
\left| B_r\cap\left\{x_n> -\frac12 D^2\psi(0)x'\cdot x'\right\}\right|+O(\e r^{n+1}),
\end{eqnarray*}
where the change of variable $x_n\mapsto -x_n$ has been used in the last equality.
From these observations, using the short notation $M(x'):=\frac12 D^2\psi(0)x'\cdot x'$ and (<ref>) we find that
\begin{equation}\label{APCOSMFe-32090sd}
\begin{split}&
\int_{B_r}\big(\chi_{\R^n\setminus E}(x)-\chi_E(x)\big)\,dx\\
\left| B_r\cap\big\{x_n> -M(x')\big\}\right|-\left| B_r\cap\big\{x_n\ge M( x')\big\}\right|+O(\e r^{n+1}).
\end{split}\end{equation}
Now, letting $C:=\frac12 |D^2\psi(0)|$,
we have that
\begin{equation}\label{GAVBuTGBStnao0os-1}\begin{split}
\left| B_r\cap\big\{x_n\ge M( x')\big\}\right|&=\left| B_r\cap\big\{M(x')\le x_n\le C|x'|^2\big\}\right|+
\left| B_r\cap\big\{x_n\ge C|x'|^2\big\}\right|\\&=
\left| B_r\cap\big\{M(x')\le x_n\le C|x'|^2\big\}\right|+A_r\\
\end{split}\end{equation}
\begin{eqnarray*}&&A_r:=\left| B_r\cap\big\{x_n\ge C|x'|^2\big\}\right|,\\&&
E_r:=\left| \big\{|x'|<r\big\}\cap\big\{M(x')\le x_n\le C|x'|^2\big\}\right|\\{\mbox{and}}\qquad&&
F_r:=\left| \big\{|x'|<r<|x|\big\}\cap\big\{M(x')\le x_n\le C|x'|^2\big\}\right|.
\end{eqnarray*}
Additionally, if $x\in F_r$ then
$$r^2\le|x|^2= |x'|^2+x_n^2\le|x'|^2+(C|x'|^2)^2=|x'|^2+C^2|x'|^4\le (1+C^2r^2)|x'|^2
\le (1+\e)|x'|^2,$$
as long as $r$ is small enough. This entails that
$$ F_r\subseteq \left\{ |x'|\in\left[ \frac{r}{\sqrt{1+\e}},r\right]\right\}
\cap\big\{| x_n|\le C|x'|^2\big\},$$
whence $|F_r|=O(\e r^{n+1})$.
From this observation and (<ref>) we arrive at
\begin{eqnarray*}
\left| B_r\cap\big\{x_n\ge M( x')\big\}\right|&=&E_r+A_r+O(\e r^{n+1})\\
&=& \left| \big\{|x'|<r\big\}\cap\big\{M(x')\le x_n\le C|x'|^2\big\}\right|+A_r+O(\e r^{n+1}).\end{eqnarray*}
Thus, replacing $M$ with $-M$ and noticing that $A_r$ remains unchanged under this modification,
\begin{eqnarray*}
\left| B_r\cap\big\{x_n> -M( x')\big\}\right|&=&
\left| \big\{|x'|<r\big\}\cap\big\{-M(x')\le x_n\le C|x'|^2\big\}\right|+A_r+O(\e r^{n+1}).\end{eqnarray*}
Using these remarks and (<ref>) we thereby conclude that
\begin{eqnarray*}
\int_{B_r}\big(\chi_{\R^n\setminus E}(x)-\chi_E(x)\big)\,dx\\
\left(
\left| \big\{|x'|<r\big\}\cap\big\{-M(x')\le x_n\le C|x'|^2\big\}\right|+A_r\right)\\&&\qquad\qquad\quad
-\left( \left| \big\{|x'|<r\big\}\cap\big\{M(x')\le x_n\le C|x'|^2\big\}\right|+A_r\right)+O(\e r^{n+1})\\
\left| \big\{|x'|<r\big\}\cap\big\{-M(x')\le x_n\le C|x'|^2\big\}\right|\\&&\qquad\qquad\quad
- \left| \big\{|x'|<r\big\}\cap\big\{M(x')\le x_n\le C|x'|^2\big\}\right|+O(\e r^{n+1})\\
&&\qquad\quad=\,\int_{\{|x'|<r\}} \left(\int_{-M(x')}^{M(x')} dx_n\right)\,dx'
+O(\e r^{n+1})\\&&\qquad\quad=\,
D^2\psi(0)\int_{\{|x'|<r\}} x'\cdot x'\,dx'
+O(\e r^{n+1})\\&&\qquad\quad=\,
\sum_{i=1}^{n-1} \partial^2_i \psi(0)\int_{\{|x'|<r\}} x_i^2\,dx'
+O(\e r^{n+1})\\&&\qquad\quad=\,
\frac1{n-1}\sum_{i=1}^{n-1} \partial^2_i \psi(0)\int_{\{|x'|<r\}} |x'|^2\,dx'
+O(\e r^{n+1})\\&&\qquad\quad=\,
cr^{n+1} \Delta\psi(0)
+O(\e r^{n+1}),
\end{eqnarray*}
for a suitable $c>0$, from which (<ref>) plainly follows in view of (<ref>) .
§ THE LAPLACE-BELTRAMI OPERATOR IN LOCAL COORDINATES
Regarding the setting in Section <ref>,
it is sometimes useful to express geometric
differential operators (such as tangential gradients and divergences,
as well as the
Laplace-Beltrami operator) in “local coordinates”
with respect to the manifold $\Sigma$.
That is, one supposes that an element of $\Sigma$
is locally parameterized by a diffeomorphism $f:D\to\Sigma$,
for some domain $D\subset\R^{n-1}$, see Figure <ref>.
One considers the metrics ${\mbox{\Large{\calligra{g}}}}$,
which can be identified with a symmetric $(n-1)\times(n-1)$ matrix with elements
\begin{equation}\label{METRIX}
g_{ij}:=\frac{\partial f}{\partial\eta_i}\cdot\frac{\partial f}{\partial\eta_j}.\end{equation}
We observe that the matrix ${\mbox{\Large{\calligra{g}}}}\,$
is the product between the transpose of $D_\eta f$ and $D_\eta f$ itself, being $(D_\eta f)_{kj}=
\frac{\partial f}{\partial\eta_j}\cdot e_k$, for $j\in\{1,\dots,n-1\}$ and $k\in\{1,\dots,n\}$. Therefore,
by the Binet-Cauchy Formula (see e.g. <cit.>), the square of
the determinant of ${\mbox{\Large{\calligra{g}}}}\,$
is the sum of the squares
of the determinants of each $ (n-1)\times(n-1)$-submatrix of $D_\eta f$, namely
\begin{equation}\label{DETTE} \det {\mbox{\Large{\calligra{g}}}}=\sqrt{\sum_{k=1}^n\left[\;
\det\left( \begin{matrix}\frac{\partial f}{\partial\eta}\cdot e_1
\\
\vdots\\
\frac{\partial f}{\partial\eta}\cdot e_{k-1}\\
\frac{\partial f}{\partial\eta}\cdot e_{k+1}\\ \vdots\\
\frac{\partial f}{\partial\eta}\cdot e_n\end{matrix}
\right)\;\right]^2
For instance, if $\Sigma$ is locally the graph (say, in the $n$th direction)
of a function $\psi$, one can take $f(\eta):=(\eta,\psi(\eta))$,
notice that in this case $\frac{\partial f}{\partial\eta_i}=\left(e_i',\,\frac{\partial \psi}{\partial\eta_i}\right)$,
which are linearly independent,
being $\{e'_1,\dots,e'_{n-1}\}$ the Euclidean basis of $\R^{n-1}$,
obtain that
$$g_{ij}=\delta_{ij}+\frac{\partial\psi}{\partial\eta_i}\,\frac{\partial \psi}{\partial\eta_j}$$
and deduce from (<ref>) that
\begin{equation*} \det {\mbox{\Large{\calligra{g}}}}=\sqrt{1+|\partial_\eta\psi|^2
We also remark that, for every $v=(v_1,\dots,v_{n-1})\in\R^{n-1}$,
$$ ({\mbox{\Large{\calligra{g}}}}\, v)\cdot v=\sum_{i,j=1}^{n-1}\left(
\delta_{ij}+\frac{\partial\psi}{\partial\eta_i}\,\frac{\partial \psi}{\partial\eta_j}\right)v_i\,v_j=
|v|^2+\left(\frac{\partial\psi}{\partial\eta}\cdot v\right)^2\ge|v|^2,
whence ${\mbox{\Large{\calligra{g}}}}\,$ is positive definite.
Local coordinates for $\Sigma$.
In particular, we have that ${\mbox{\Large{\calligra{g}}}}\,$ is invertible
and we denote the corresponding inverse matrix by $g^{ij}$.
We also set $g:=\det{\mbox{\Large{\calligra{g}}}}\,$.
In this setting, given $u:\Sigma\to\R$, we can consider $
U:D\to\R$ defined by $U(\eta):=u(f(\eta))$.
it is convenient to identify vectors $\tau\in\R^n$ which are tangent to $\Sigma$
at a given point with suitable vectors in $\R^{n-1}$. This identification
should also reconstruct the Euclidean
scalar products of tangent vectors in $\R^n$
from a suitable product between the associated vectors in $\R^{n-1}$. To this end,
given $p\in\Sigma$, we can assume that $0\in D$ and $p=f(0)$ and,
for all $i\in\{1,\dots,n\}$, we consider the tangent vectors
$$ \tau_i:=\frac{d}{dt} f(te_i)\Big|_{t=0}
= \frac{\partial f}{\partial\eta_i}(0)
a tangent vector $V\in\R^n$ to $\Sigma$ at $p$
we write it in the form
\begin{equation}\label{0oLSPpSLkmdc} V=\sum_{i=1}^{n-1} V^i\,\tau_i
Thus, we can associate to the tangent vector $V$
its $(n-1)$-dimensional representation
\begin{equation}\label{THETAVE}\Theta(V)=(\Theta^1(V),\dots,\Theta^{n-1}(V))=(V^1,\dots,V^{n-1})\in\R^{n-1}.\end{equation}
In this way, considering the scalar product induced by
the metrics between $a=(a^1,\dots,a^{n-1})$, $b=(b^1,\dots,b^{n-1})\in\R^{n-1}$ defined as
$$ g(a,b):=\sum_{i,j=1}^{n-1} g_{ij} \,a^i\, b^j,$$
it follows that, for every tangent vector $V$, $W\in\R^n$,
\begin{equation}\label{MANMGBmhsl04599} V\cdot W=
\sum_{i,j=1}^n V^i\,W^j\,{\tau_i\cdot\tau_j}
=\sum_{i,j=1}^n \Theta^i(V)\,\Theta^j(W)\,g_{ij}=
\end{equation}
We also recall a handy way to use the metric to reconstruct the coordinates of a tangent
vector with respect to the (not necessarily
orthonomal) basis of the tangent space given by $\{\tau_1,\dots,\tau_{n-1}\}$.
Let $V$ be a tangent vector field on $\Sigma$ as in (<ref>),
and recall the notation in (<ref>).
Then, for every $j\in\{1,\dots,n-1\}$,
\begin{equation}\label{LAMNjjma834RSIi2} \Theta^j(V) = \sum_{i=1}^{n-1} g^{ij}\,V\cdot\frac{\partial f}{\partial\eta_i}.\end{equation}
\begin{equation}\label{LAMNjjma834RSIi} V=
\sum_{i,j=1}^{n-1}
g^{ij}\,V\cdot\frac{\partial f}{\partial\eta_i}\,\frac{\partial f}{\partial\eta_j}
It follows from (<ref>), that, for every $j\in\{1,\dots,n-1\}$,
$$ V\cdot\frac{\partial f}{\partial\eta_j}=
\sum_{i=1}^{n-1} V^i\,\tau_i\cdot
\frac{\partial f}{\partial\eta_j}
=\sum_{i=1}^{n-1} V^i\frac{\partial f}{\partial\eta_i}\cdot
\frac{\partial f}{\partial\eta_j}=
\sum_{i=1}^{n-1} g_{ij}V^i.$$
That is, if we consider the vectors $W:=\left(V\cdot\frac{\partial f}{\partial\eta_1},\dots,
V\cdot\frac{\partial f}{\partial\eta_{n-1}}
\right)$ and $\Theta:=\Theta(V)=(V^1,\dots,V^{n-1})$, in matrix notation we have that $W={{\mbox{\Large{\calligra{g}}}}}\,\Theta $
and thus $\Theta ={{\mbox{\Large{\calligra{g}}}}}^{-1}W$, which gives, for each $j\in\{1,\dots,n-1\}$,
$$\Theta^j=\sum_{i=1}^{n-1}g^{ij} W_i,$$
from which (<ref>) plainly follows.
In view of (<ref>), we see that (<ref>) reduces to
\begin{equation*} V=\sum_{j=1}^{n-1} V^j\frac{\partial f}{\partial\eta_j}=
\sum_{i,j=1}^{n-1}
g^{ij}\,V\cdot\frac{\partial f}{\partial\eta_i}\,\frac{\partial f}{\partial\eta_j}
which is (<ref>).
Formula (<ref>) can be straightforwardly extended to all (not necessarily tangent) vector fields:
If $V$ is a vector field on $\Sigma$,
\begin{equation*} V=(V\cdot\nu)\nu+
\sum_{i,j=1}^{n-1}
g^{ij}\,V\cdot\frac{\partial f}{\partial\eta_i}\,\frac{\partial f}{\partial\eta_j}
From (<ref>),
considering the tangent vector field $\widetilde{V}:=V-(V\cdot\nu)\nu$, we see that
\begin{equation*}\begin{split}& V-(V\cdot\nu)\nu=\widetilde{V}=
\sum_{i,j=1}^{n-1}
g^{ij}\,\widetilde{V}\cdot\frac{\partial f}{\partial\eta_i}\,\frac{\partial f}{\partial\eta_j}\\&\qquad\qquad=
\sum_{i,j=1}^{n-1}
g^{ij}\,\Big(V-(V\cdot\nu)\nu\Big)\cdot\frac{\partial f}{\partial\eta_i}\,\frac{\partial f}{\partial\eta_j}=\sum_{i,j=1}^{n-1}
g^{ij}\,{V}\cdot\frac{\partial f}{\partial\eta_i}\,\frac{\partial f}{\partial\eta_j}
The above setting allows us to write the tangential gradient in local coordinates, according
to the following result:
Let $u\in C^1({\mathcal{N}})$.
For every $i\in\{1,\dots,n-1\}$, on $\Sigma$,
$$ \Theta^i(\nabla_\Sigma u) = \sum_{j=1}^{n-1} g^{ij}\frac{\partial U}{\partial\eta_j}.$$
We stress that $\nabla_\Sigma u$ is a tangent vector since,
by (<ref>),
$$ \nabla_\Sigma u\cdot\nu=
\Big(\nabla u-\big(\nabla u\cdot\nu\big)\,\nu\Big)\cdot\nu=\big(\nabla u\cdot\nu\big)-
\big(\nabla u\cdot\nu\big)\nu\cdot\nu=0.$$
Also, by the Chain Rule, for all $i\in\{1,\dots,n-1\}$,
$$\frac{\partial U}{\partial\eta_i}(\eta)=\frac{\partial }{\partial\eta_i}(u(f(\eta))=\nabla u(f(\eta))\cdot\frac{\partial U}{\partial\eta_i}(f(\eta))$$
and thus, in light of (<ref>),
\begin{equation*}
\begin{split}& \Theta^j(\nabla_\Sigma u) = \sum_{i=1}^{n-1} g^{ij}\,\nabla_\Sigma u\cdot\frac{\partial f}{\partial\eta_i}=
\sum_{i=1}^{n-1} g^{ij}\,\Big(\nabla u-\big(\nabla u\cdot\nu\big)\,\nu\Big)\cdot\frac{\partial f}{\partial\eta_i}\\&\qquad\quad\qquad=
\sum_{i=1}^{n-1} g^{ij}\, \nabla u\cdot\frac{\partial f}{\partial\eta_i}=\sum_{i=1}^{n-1} g^{ij}\frac{\partial U}{\partial\eta_i}.
\qedhere\end{split}\end{equation*}
Correspondingly to Lemma <ref>, one can write the tangential divergence
in local coordinates as follows:
Let $F\in C^1(\Sigma,\R^n)$ be a tangent vector field. Then, on $\Sigma$,
$$ \div_\Sigma F=
\sum_{i=1}^{n-1}\frac1{\sqrt{g}}\frac{\partial}{\partial\eta_i}\Big(
\sqrt{g}\,\Theta^i(F)
\Big).$$
We consider a smooth function $\varphi$ supported in a local chart and
let $\Phi(\eta):=\varphi(f(\eta))$.
We recall (see <cit.>)
that the surface element of $\Sigma$ in local coordinates
can be written as $\sqrt{g}\,d\eta$, therefore, by Theorem <ref>,
and using that $F$ is tangential,
\begin{eqnarray*}&& \div_\Sigma F\,\Phi\,\sqrt{g}\,d\eta=\div_\Sigma F\,\varphi\,d{\mathcal{H}}^{n-1}=
H F\cdot \nu\,\varphi\,d{\mathcal{H}}^{n-1}-F\cdot
\nabla_\Sigma\varphi\,d{\mathcal{H}}^{n-1}=
\nabla_\Sigma\varphi\,d{\mathcal{H}}^{n-1}
This and (<ref>) lead to
$$ \div_\Sigma F\,\Phi\,\sqrt{g}\,d\eta=
-\sum_{i,j=1}^n g_{ij}\, \Theta^i(F)\,\Theta^j(\nabla_\Sigma\varphi)\,d{\mathcal{H}}^{n-1}.$$
Since, in light of Lemma <ref>,
$$ \Theta^j(\nabla_\Sigma \varphi) = \sum_{k=1}^{n-1} g^{jk}\frac{\partial \Phi}{\partial\eta_k},$$
we thereby obtain that
$$ \div_\Sigma F\,\Phi\,\sqrt{g}\,d\eta=
-\sum_{i,j,k=1}^n g_{ij}\, g^{jk}\,\Theta^i(F)\,
\frac{\partial \Phi}{\partial\eta_k}\,\sqrt{g}\,d\eta
=-\sum_{i=1}^n \Theta^i(F)\,
\frac{\partial \Phi}{\partial\eta_i}\,\sqrt{g}\,d\eta.$$
The latter term can be integrated by parts, giving
$$ \div_\Sigma F\,\Phi\,\sqrt{g}\,d\eta=
\sum_{i=1}^n
\frac{\partial }{\partial\eta_i}\Big(
\Theta^i(F)\,
\,\sqrt{g}\Big)\Phi\,d\eta.$$
Consequently, since $\varphi$ (and hence $\Phi$) is an arbitrary
test function, we conclude that
\begin{equation*}
\div_\Sigma F\, \sqrt{g}=
\sum_{i=1}^n
\frac{\partial }{\partial\eta_i}\Big(
\Theta^i(F)\,
\,\sqrt{g}\Big).
\qedhere\end{equation*}
Let $F\in C^1(\Sigma,\R^n)$. Then, on $\Sigma$,
$$ \div_\Sigma F=
\sum_{i=1}^{n-1}\frac1{\sqrt{g}}\frac{\partial}{\partial\eta_i}\Big(
\sqrt{g}\,\Theta^i\big(\widetilde F\big)
\Big)+H(F\cdot\nu),$$
where $\widetilde{F}:=F-(F\cdot\nu)\nu$.
By (<ref>), we know that the tangential divergence is
linear with respect to the vector fields.
\begin{equation}\label{0OKMS8138OMS}
\div_\Sigma F=
\div_\Sigma \widetilde{F}+\div_\Sigma \big((F\cdot\nu)\nu\big)
We also remark that if $\varphi\in C^1(\Sigma)$, on $\Sigma$ we have that
$$ \div_\Sigma(\varphi\nu)=\div((\varphi\nu)_{\mbox{\scriptsize{ext}}})
-\nabla\big( (\varphi\nu)_{\mbox{\scriptsize{ext}}}\cdot\nu_{\mbox{\scriptsize{ext}}}\big)\cdot\nu_{\mbox{\scriptsize{ext}}}
\varphi\div\nu_{\mbox{\scriptsize{ext}}}
thanks to (<ref>)
and (<ref>). This and (<ref>) entail that
\begin{equation*}
\div_\Sigma F=
\div_\Sigma \widetilde{F}+H(F\cdot\nu)
Since $\widetilde{F}$ is a tangential vector field, we can employ Lemma <ref>
and obtain the desired result.
As a variant of
Corollary <ref>, we also have the
following useful expression in coordinates
of the tangential divergence of a vector field:
Let $F\in C^1({\mathcal{N}},\R^n)$. Then, on $\Sigma$,
$$ \div_\Sigma F=
\sum_{i,j=1}^{n-1}g^{ij}\left(DF\,\frac{\partial f}{\partial\eta_i}\right)\cdot
\frac{\partial f}{\partial\eta_j}.
For every $k\in\{1,\dots,n\}$,
by Corollary <ref>, applied here to $V:=\frac{\partial F}{\partial x_k}$, we see that,
\begin{equation*} \frac{\partial F}{\partial x_k}=
\left(\frac{\partial F}{\partial x_k}\cdot\nu\right)\nu+
\sum_{i,j=1}^{n-1}
g^{ij}\,\frac{\partial F}{\partial x_k}\cdot\frac{\partial f}{\partial\eta_i}\,\frac{\partial f}{\partial\eta_j}
As a result,
\begin{equation*} \div F=\sum_{k=1}^n
\frac{\partial F}{\partial x_k}\cdot e_k=\sum_{k=1}^n
\left(\frac{\partial F}{\partial x_k}\cdot\nu\right)\nu\cdot e_k+
\sum_{{1\le i, j\le n-1}\atop{1\le k\le n}}
g^{ij}\,\frac{\partial F}{\partial x_k}\cdot\frac{\partial f}{\partial\eta_i}\,\frac{\partial f}{\partial\eta_j}\cdot e_k.
\end{equation*}
From this, we deduce that
\begin{eqnarray*}
&&\sum_{i,j=1}^{n-1}g^{ij}\left(DF\,\frac{\partial f}{\partial\eta_j}\right)\cdot
\frac{\partial f}{\partial\eta_i}
\sum_{{1\le i, j\le n-1}\atop{1\le k\le n}}g^{ij} \frac{\partial F}{\partial x_k}\cdot
\frac{\partial{f}}{\partial\eta_i}\;\frac{\partial f}{\partial\eta_j}\cdot e_k\\&&\qquad=\div F-\sum_{k=1}^n
\left(\frac{\partial F}{\partial x_k}\cdot\nu\right)\nu\cdot e_k=
\div F-(DF\,\nu)\cdot\nu.
\end{eqnarray*}
That is, recalling the definition of tangential divergence in (<ref>), and exploiting also (<ref>),
\begin{equation*}\begin{split}
&\sum_{i,j=1}^{n-1}g^{ij}\left(DF\,\frac{\partial f}{\partial\eta_j}\right)\cdot
\frac{\partial f}{\partial\eta_i}-\div_\Sigma F=\sum_{i,j=1}^{n-1}g^{ij}\left(DF\,\frac{\partial f}{\partial\eta_j}\right)\cdot
\frac{\partial f}{\partial\eta_i}-\div F+\nabla(F\cdot\nu_{\mbox{\scriptsize{ext}}})\cdot\nu_{\mbox{\scriptsize{ext}}}
\\&\qquad
=\nabla(F\cdot\nu_{\mbox{\scriptsize{ext}}})\cdot\nu_{\mbox{\scriptsize{ext}}}-(DF\,\nu)\cdot\nu=\sum_{k,m=1}^n (F\cdot e_m)\,\left(\frac{\partial\nu_{\mbox{\scriptsize{ext}}}}{\partial x_k}\cdot e_m\right)
\,(\nu_{\mbox{\scriptsize{ext}}}\cdot e_k)\\&\qquad
=\sum_{m=1}^n (F\cdot e_m)\,\nabla(\nu_{\mbox{\scriptsize{ext}}}\cdot e_m)\cdot\nu_{\mbox{\scriptsize{ext}}}=0
\end{split}\end{equation*}
As a direct consequence of
Lemmata <ref>, <ref> and <ref>,
we obtain that
the Laplace-Beltrami operator can be written in terms of these local coordinates, according to the following result:
Let $u\in C^2({\mathcal{N}})$. Then,
\begin{equation}\label{Jns89ijgm5jjgfjgjg2}
\Delta_\Sigma u =\sum_{i,j=1}^{n-1}\frac1{\sqrt{g}}\frac{\partial}{\partial\eta_i}\left(
\sqrt{g}\,g^{ij}
\frac{\partial U}{\partial\eta_j}
\right).
\end{equation}
As a technical remark, we observe that, for the purpose of defining the Laplace-Beltrami operator, the regularity of $\Sigma$
can be reduced from $C^3$ to $C^2$.
Indeed, on page DAC3aC2 we assumed $\Sigma$ of class $C^3$, which was used
to deduce that $\nu$ was of class $C^2$, and accordingly, by (<ref>), that $\pi_\Sigma$
was of class $C^2$. From this, we deduced that $u_{\mbox{\scriptsize{ext}}}$ was of class $C^2$
if so was $u:\Sigma\to\R$, thanks to (<ref>), and the $C^2$ regularity of $u_{\mbox{\scriptsize{ext}}}$
was utilized to give the definition of the
Laplace-Beltrami operator in (<ref>).
On the other hand, thanks to Corollary <ref>,
we can now point out that a $C^2$ regularity assumption on $\Sigma$ would suffice
for formula (<ref>). Hence, taking (<ref>) instead of (<ref>)
definition of the
Laplace-Beltrami operator would allow us to work with $\Sigma$ of class $C^2$
(alternatively, one can define the Laplace-Beltrami operator as in (<ref>) for $\Sigma$ of class $C^3$
and then extend it by approximation when $\Sigma$ is of class $C^2$, using (<ref>)
to pass to the limit).
For additional information on the Laplace-Beltrami operator see e.g. <cit.>
and the references therein.
§ THE LAPLACIAN IN SPHERICAL COORDINATES
Let $u:\R^n\setminus\{0\}\to\R$.
For every $r>0$ and $\vartheta\in\partial B_1$, we define
\begin{equation*}
u_0(r,\vartheta):= u(r\vartheta).
\end{equation*}
In this setting, one can compute the Laplace operator in spherical coordinates accordingLaplacian in spherical coordinates
to the following formula:
Let $u\in C^2(\R^n\setminus\{0\})$. Then,
$$ \Delta u(x)=\partial_{rr}u_0(r,\vartheta)+\frac{n-1}{r}\,\partial_{r}u_0(r,\vartheta)+\Delta_{\partial B_r}u(r\vartheta),$$
for every $x\in\R^n\setminus\{0\}$, where $r:=|x|$ and $\vartheta:=\frac{x}{|x|}$.
We take $E:=B_r$ and $\Sigma:=\partial B_r$. In this way, we have that $\nu(x)=\frac{x}{|x|}$
for every $x\in\partial B_r$.
Then, by (<ref>), $\nu_{\mbox{\scriptsize{ext}}}(x)= \frac{x}{|x|}$.
Hence, we deduce from (<ref>) and (<ref>)
that, for every $x\in\partial B_r$,
\begin{equation} \label{8yujbggd}
H(x)=\div_{\partial B_r} \nu(x)=\div \nu_{\mbox{\scriptsize{ext}}}(x)=\div\frac{x}{|x|}=
\frac{n-1}{|x|}=\frac{n-1}r.\end{equation}
In addition, if $x\in\partial B_r$,
\begin{eqnarray*}
\nabla u(x)\cdot \nu(x)=\lim_{h\to0}\frac{u(x+h\nu(x))-u(x)}{h}=\lim_{h\to0}\frac{u_0(r+h,\vartheta)-u_0(r,\vartheta)}{h}=\partial_ru_0(r,\vartheta)
\end{eqnarray*}
\begin{eqnarray*}&& (D^2 u(x)\,\nu(x))\cdot\nu(x)=\lim_{h\to0}\frac{u(x+h\nu(x))+u(x-h\nu(x))-2u(x)}{h^2}\\&&\qquad=
\lim_{h\to0}\frac{u_0(r+h,\vartheta)+u_0(r-h,\vartheta)-2u_0(r,\vartheta)}{h^2}=\partial_{rr}u_0(r,\vartheta).
\end{eqnarray*}
Using these identities and (<ref>) in combination with
Theorem <ref>, we conclude that, at $x\in\partial B_r$,
\begin{equation*} \Delta_{\partial B_r} u = \Delta_T u-H\nabla u\cdot \nu=
\Delta u-(D^2 u\,\nu)\cdot\nu-\frac{n-1}r \,\partial_r u_0=
\Delta u-\partial_{rr}u_0-\frac{n-1}r \,\partial_r u_0
Given $r>0$, an equivalent formulation
of Theorem <ref> can be given by replacing
the Laplace-Beltrami of $u$ along $\partial B_r$
with the Laplace-Beltrami
of the map $\partial B_1\ni\vartheta\mapsto u_0(r,\vartheta)$
along $\partial B_1$, as follows:
Let $u\in C^2(\R^n\setminus\{0\})$. Then,
$$ \Delta u(x)=\partial_{rr}u_0(r,\vartheta)+
\frac{n-1}{r}\,\partial_{r}u_0(r,\vartheta)+\frac1{r^2}\,
\Delta_{\partial B_1}u_0(r,\vartheta),$$
for every $x\in\R^n\setminus\{0\}$, where $r:=|x|$ and $\vartheta:=\frac{x}{|x|}$.
In light of Theorem <ref>, it suffices to show that,
given $r>0$,
\begin{equation}\label{TRGSATTHNBELBRTYUTT9}
\Delta_{\partial B_r}u(r\vartheta)=\frac1{r^2}\Delta_{\partial B_1}u_0(r,\vartheta)
\qquad{\mbox{for all }}\vartheta\in\partial B_1.
\end{equation}
To this end, we perform a careful scaling argument.
Given $r>0$, for every $\vartheta\in\partial B_1$
we let $v^{(r)}(\vartheta):=u_0(r,\vartheta)$.
Thus, the
normal extension of $v^{(r)}$ outside $\partial B_1$,
as introduced in (<ref>), is
the function
\begin{equation}\label{WRBSY0perIKSMaismoBBeJ}
Notice that $w^{(r)}(\lambda x)=
u_0\left(\frac{r\lambda x}{|\lambda x|}\right)=u_0\left(\frac{rx}{|x|}\right)=w^{(r)}(x)$,
whence $w^{(r)}$ is positively homogeneous of degree zero.
Therefore, the function $\phi^{(r)}:=\Delta w^{(r)}$ is homogeneous
of degree $-2$.
Thus, recalling the Laplace-Beltrami definition
in (<ref>), for every $y\in\partial B_1$,
\begin{equation}\label{CQWSXRF3O5M4P4AS5S6784P234}
\Delta_{\partial B_1} u_0(r,y)=\Delta_{\partial B_1} v^{(r)}(y)=
\Delta w^{(r)}(y)=\phi^{(r)}(y)=r^2\,\phi^{(r)}(ry).\end{equation}
Additionally, we have that
normal extension of $u$ outside $\partial B_r$
is the function $\R^n\setminus\{0\}\ni x\mapsto u\left(\frac{rx}{|x|}\right)$,
which coincides with $w^{(r)}(x)$, thanks to (<ref>).
Therefore, by the Laplace-Beltrami definition
in (<ref>) we have that for every $x\in\partial B_r$
$$ \Delta_{\partial B_r} u(x)=\Delta w^{(r)}(x)=\phi^{(r)}(x)
where $y:=\frac{x}r$. Hence, comparing with (<ref>),
we conclude that $\Delta_{\partial B_r} u(ry)=r^{-2}
\Delta_{\partial B_1} u_0(r,y)$ for every $y\in\partial B_1$,
and this completes the proof of (<ref>).
As a special case of
Theorem <ref>, one obtains that if $u\in C^2(\R^n\setminus\{0\})$ is radially symmetric,
i.e. $u(x)=u_0(|x|)$ for some $u_0:(0,+\infty)\to\R$,
\begin{equation}\label{ROTSE} \Delta u(x)=u_0''(r)+\frac{n-1}{r}\,u_0'(r),\end{equation}
with $r=|x|$. Of course, this formula can also be obtained
by direct computations, see e.g. <cit.>.
We also point out that if $\alpha\in\R$ and $u_\alpha(x):=|x|^\alpha$, then, taking $\Sigma:=\partial B_1$,
we have that $u_\alpha=1$ on $\partial B_1$,
\begin{equation}\label{MN:ikmc}
\Delta u_\alpha=\alpha\,(n+\alpha-2),\qquad\Delta_T u_\alpha=\alpha\,(n-1)\qquad{\mbox{and}}\qquad
\Delta_{\partial B_1} u_\alpha=0\qquad{\mbox{on }}\,\partial B_1.
\end{equation}
In particular, notice that the normal
extension of the function identically
equal to $1$ on $\partial B_1$ according to definition (<ref>)
is the function identically
equal to $1$ on $\R^n$, corresponding to $u_\alpha$ with $\alpha:=0$.
The other values of $\alpha$ provide different extensions
of the function identically
equal to $1$ on $\partial B_1$ and, in this case,
the Laplace-Beltrami operator is not equal to the full Laplacian
of these extensions. Namely, geometric second order
operators can be reconstructed by full operators by extension,
but they are sensitive to the different type of extension chosen
(differently from first order operators,
as discussed in Lemma <ref>).
§ THE KELVIN TRANSFORM
The Kelvin TransformKelvin Transform
is a useful tool[The Kelvin Transform is named after
William Thomson, 1st Baron Kelvin. Actually, Thomson himself
was named Baron Kelvin after the River Kelvin that flows past
the University of Glasgow where Thomson used to work.
See Figure <ref>
for a photochrom print of the river and the university dating back to the end of nineteenth century.
See also Figure <ref>
for a caricature of Lord Kelvin (by caricaturist
Sir Leslie Matthew Ward, a.k.a. Spy),
published in the magazine Vanity Fair in 1897.]
to reduce the analysis of partial differential
equations in exterior domains to that of interior ones
(and vice versa).
Meander of the River Kelvin with the Gilmorehill campus of the University of Glasgow (Public Domain image from
Moreover, this transformation
possesses a number of geometric and analytic properties
that make it handy in several occasions.
To introduce[For the sake of clarity,
we presented the Kelvin Transform with the aim
of highlighting its remarkable analytic, algebraic and geometric
properties. On the other hand,
the Kelvin Transform is naturally motivated by some
important physical considerations inspired by the method of image
charges: the reader who wishes to go straight to this motivation
can look at Section <ref>.
Also, for simplicity, we take $\partial B_1$ as the reference set which
remains invariant for the Kelvin Transform, but we observe that a similar theory
can be carried out by leaving invariant $\partial B_R$ (for this, one can either proceed by
scaling or change (<ref>)
into ${\mathcal{K}}(x):=\frac{R^2x}{|x|^2}$.] it, we define, for all $x\in\R^n\setminus\{0\}$,
\begin{equation}\label{KSM:J:KELVIN1}
We list here some interesting properties of the Kelvin Transform:
The Kelvin Transform is an
involution, meaning that ${\mathcal{K}}({\mathcal{K}}(x))=x$
for all $x\in\R^n\setminus\{0\}$.
Also, for all $x$, $y\in\R^n\setminus\{0\}$,
\begin{eqnarray}
\label{KSM:J:KELVIN2}&& |{\mathcal{K}}(x)|\,|x|=1,\\
\label{KSM:J:KELVIN3}&& \frac{|x|^2}{1-|x|^2}=\frac{1}{|{\mathcal{K}}(x)|^2-1},\\
\label{KSM:J:KELVIN4}
&& \frac{{\mathcal{K}}(x)\cdot {\mathcal{K}}(y)}{|{\mathcal{K}}(x)|\,|{\mathcal{K}}(y)|}=\frac{x\cdot y}{|x|\,|y|},\\
\label{KSM:J:KELVIN5}&& |{\mathcal{K}}(x)-{\mathcal{K}}(y)|=\frac{|x-y|}{|x|\,|y|}.
\end{eqnarray}
Moreover[We point out that (<ref>)
states that the Kelvin Transform is
angle preserving. Furthermore (<ref>)
gives that the Kelvin Transform is conformal, since its Jacobian matrix
is a scalar function times an orthogonal matrix: indeed,
$$ \sum_{k=1}^n
\left(
\delta_{ik}-\frac{2x_ix_k}{|x|^2}
\right)\left(
\delta_{kj}-\frac{2x_kx_j}{|x|^2}
\right)=\delta_{ij}.$$
for each $e\in\partial B_1$,
\begin{equation}\label{KSM:J:KELVIN9}
and, for each $i$, $j\in\{1,\dots,n\}$,
\begin{equation}\label{KSM:J:KELVIN8}
\partial_{x_j}{\mathcal{K}}(x)\cdot e_i=\frac{1}{|x|^2}\left(
\delta_{ij}-\frac{2x_ix_j}{|x|^2}
\right).
\end{equation}
We remark that (<ref>) is a direct consequence of the definition
in (<ref>).
$$ {\mathcal{K}}({\mathcal{K}}(x))=\frac{{\mathcal{K}}(x)}{|{\mathcal{K}}(x)|^2}=
that proves the involution property.
From (<ref>) we also deduce that
$$ \frac{1}{|{\mathcal{K}}(x)|^2-1}=
\frac{1}{|x|^{-2}-1}=\frac{|x|^2}{1-|x|^{2}},$$
that is (<ref>).
Caricature of Lord Kelvin (Public Domain image from
The claim in (<ref>)
is also a straightforward byproduct of the definition
in (<ref>) and (<ref>). In addition,
using (<ref>) in combination with (<ref>),
\begin{eqnarray*}&&
|x|^2|y|^2\big( |{\mathcal{K}}(x)|^2+|{\mathcal{K}}(y)|^2-2{\mathcal{K}}(x)\cdot {\mathcal{K}}(y)\big)\\&&\qquad
=|x|^2|y|^2\left( |{\mathcal{K}}(x)|^2+|{\mathcal{K}}(y)|^2-2|{\mathcal{K}}(x)|\,| {\mathcal{K}}(y)|
\frac{x\cdot y}{|x|\,|y|}
\right)=
|x|^2|y|^2\left( \frac{1}{|x|^2}+\frac1{|y|^2}-
\frac{2x\cdot y}{|x|^2\,|y|^2}
\right)\\&&\qquad
=\big( |y|^2+|x|^2-
2x\cdot y
\big)=
\end{eqnarray*}
which establishes (<ref>).
Additionally, using (<ref>) once again, if $e\in\partial B_1$ then
\begin{eqnarray*}&& |x|^2\,|{\mathcal{K}}(x)-e|^2-|x-e|^2=
|x|^2\Big(|{\mathcal{K}}(x)|^2+1-2{\mathcal{K}}(x)\cdot e\Big)-
\Big(|x|^2+1-2x\cdot e\Big)\\&&\qquad=
1+|x|^2-2|x|^2 {\mathcal{K}}(x)\cdot e-\Big(|x|^2+1-2x\cdot e\Big)=
and thus[Of course, many of the identities
in Lemma <ref> can be proved
using different strategies. For instance, one
could also deduce (<ref>)
directly from (<ref>)
by noticing that, when $|e|=1$,
$$ |{\mathcal{K}}(x)-e|=
] we have proved (<ref>).
$$ \partial_{x_j}{\mathcal{K}}(x)\cdot e_i=\partial_{x_j}
\frac{x\cdot e_i}{|x|^2}
that is (<ref>).
We remark that the Kelvin Transform
is an “inversion of the sphere”, namely, in view of (<ref>),
we have that ${\mathcal{K}}(B_1\setminus\{0\})=\R^n\setminus B_1$,
${\mathcal{K}}(\R^n\setminus B_1)=B_1\setminus\{0\}$
and ${\mathcal{K}}(\partial B_1)=\partial B_1$.
Also, the Kelvin Transform acts naturally on functions,
in a nicely compatible way with respect to the Laplace operator. Namely,
\begin{equation}\label{SOLLE:0} u_{\mathcal{K}}(x):= |x|^{2-n}u({\mathcal{K}}(x)),\end{equation}
we have:
If $v:=u_{\mathcal{K}}$, then $v_{\mathcal{K}}=u$.
Moreover[In the second part of Theorem <ref>, it is convenient to exclude
the case $n=1$ to avoid integrability issues (from the technical point of view,
this condition is used in (<ref>)
to avoid contributions coming from infinity). As an illustrative example
of the loss of integrability in (<ref>)
when $n=1$, one can consider a function $v\in C^\infty_0((-1,1))$ such that $v(x)=1$ for each $x\in\left[-\frac12,\frac12\right]$
and notice that, in view of (<ref>), $v_{\mathcal{K}}(x)= |x| v\left(\frac{x}{|x|^2}\right)=|x|=x$
for each $x\ge2$. Accordingly, in this case $\nabla v_{\mathcal{K}}=1$ in $(2,+\infty)$, whence
$$ \int_{\R^n} |\nabla
v_{\mathcal{K}}(x)|^2\,dx\ge\int_2^{+\infty} dx=+\infty.$$
However, the claim in (<ref>)
is valid in dimension $1$ as well, as it can be checked
by differentiation (though it is arguably not very useful in this case).]
if $n\ge2$ and $u\in C^1(\overline{B_1})$ (respectively, if $u\in C^1(\R^n\setminus B_1)$),
then, for every $v\in C^\infty_0(B_1)$
(respectively, $v\in C^\infty_0(\R^n\setminus B_1)$,
\begin{equation}\label{KS:09876543209876543lkjhgfduerhfnv3ue-0}
\int_{\R^n}\nabla u_{\mathcal{K}}(x)\cdot\nabla
\int_{\R^n}\nabla u(x)\cdot\nabla
In addition to that,
if $u\in C^2(\overline{B_1})$ (respectively, if $u\in C^2(\R^n\setminus B_1)$), for every $x\in\R^n\setminus B_1$ (respectively,
for every $x\in B_1\setminus\{0\}$) it holds that
\begin{equation}\label{KS:09876543209876543lkjhgfduerhfnv3ue}
\Delta u_{\mathcal{K}}(x)=
\frac {1}{|x|^{n+2}}\,
\Delta u\big({\mathcal{K}}(x)\big).\end{equation}
We observe that, if $v:=u_{\mathcal{K}}$, then
$$ v({\mathcal{K}}(x))=u_{\mathcal{K}}({\mathcal{K}}(x))=
thanks to (<ref>) and Lemma <ref>,
and, for this reason,
$$ v_{\mathcal{K}}(x)=
This proves the desired involution property.
Now we establish (<ref>)
and (<ref>).
For this, we consider $v\in C^\infty_0(B_1)$
(respectively, $v\in C^\infty_0(\R^n\setminus B_1)$.
We let $M(x)$ be the matrix with entries $
\delta_{ij}-\frac{2x_ix_j}{|x|^2}$ for all $i$, $j\in\{1,\dots,n\}$
and we recall that $M(x)$ is orthogonal (see the footnote
on page VEDHNDOCOSIJNDFSUCJFM).
As a result, we have that $\det M(x) =1$.
Using (<ref>), we see that the change
of variable $y={\mathcal{K}}(x)$ leads to $dy=\left|
\det D{\mathcal{K}}(x)\right|\,dx=\left|
\det\frac{M(x)}{|x|^2}\right|\,dx
Moreover, by (<ref>),
$$ \nabla u_{\mathcal{K}}(x)=\nabla\Big( |x|^{2-n}u({\mathcal{K}}(x))\Big)=
(2-n)|x|^{-n}xu({\mathcal{K}}(x))+|x|^{-n}M(x)\nabla u({\mathcal{K}}(x)),$$
and a similar formula holds for $v$ replacing $u$.
\begin{equation}\label{JHSNDsikcmsdSfgaisdwedsagfdsgf39450ksdxc}\begin{split}&
\int_{\R^n}\nabla u_{\mathcal{K}}(x)\cdot\nabla
\int_{\R^n}
\Big[
+(2-n) |x|^{-2n}u({\mathcal{K}}(x))\big(M(x)\nabla v({\mathcal{K}}(x))\big)\cdot x
\\&\qquad+(2-n) |x|^{-2n}v({\mathcal{K}}(x))\big(M(x)\nabla u({\mathcal{K}}(x))\big)\cdot x
\\&\qquad+|x|^{-2n}\big(M(x)\nabla u({\mathcal{K}}(x))\big)\cdot
\big(M(x)\nabla v({\mathcal{K}}(x))\big)
\Big]\,dx.
\end{split}\end{equation}
We also remark that
\begin{eqnarray*}&&
\div\Big(|x|^{2-2n}x u({\mathcal{K}}(x))v({\mathcal{K}}(x))\Big)\\&=&
\div\Big(|x|^{2-2n}x\Big) u({\mathcal{K}}(x))v({\mathcal{K}}(x))\\&&\qquad
+|x|^{2-2n}x\cdot\nabla\Big( u({\mathcal{K}}(x))\Big)
|x|^{2-2n}x u({\mathcal{K}}(x))\cdot\nabla\Big(v({\mathcal{K}}(x))\Big)\\&=&
|x|^{-2n}x\cdot\big(M(x)\nabla u({\mathcal{K}}(x))\big)
+|x|^{-2n}x\cdot\big(M(x)\nabla v({\mathcal{K}}(x))\big)
\end{eqnarray*}
This and the Divergence Theorem give that
\begin{equation}\label{ATTn1}
\begin{split}&
\int_{\R^n}
\Big[
+(2-n) |x|^{-2n}u({\mathcal{K}}(x))\big(M(x)\nabla v({\mathcal{K}}(x))\big)\cdot x
\\&\qquad+(2-n) |x|^{-2n}v({\mathcal{K}}(x))\big(M(x)\nabla u({\mathcal{K}}(x))\big)\cdot x
\Big]\,dx=0.\end{split}
\end{equation}
Plugging this information into (<ref>),
we thus conclude that
$$ \int_{\R^n}\nabla u_{\mathcal{K}}(x)\cdot\nabla
\int_{\R^n}|x|^{-2n}\big(M(x)\nabla u({\mathcal{K}}(x))\big)\cdot
\big(M(x)\nabla v({\mathcal{K}}(x))\big)\,dx.$$
Therefore, using the orthogonality of $M(x)$
and changing variable,
\begin{eqnarray*}
\int_{\R^n}\nabla u_{\mathcal{K}}(x)\cdot\nabla
v_{\mathcal{K}}(x)\,dx&=&\int_{\R^n}|x|^{-2n}\nabla u({\mathcal{K}}(x))\cdot\nabla
&=&\int_{\R^n}\nabla u(y)\cdot\nabla
\end{eqnarray*}
This proves (<ref>).
Now we prove (<ref>).
For this, we utilize (<ref>)
to see that
\begin{eqnarray*}
-\int_{\R^n}|{\mathcal{K}}(y)|^{n+2}\,\Delta u_{\mathcal{K}}({\mathcal{K}}(y))\,
-\int_{\R^n}|y|^{-n-2}\,\Delta u_{\mathcal{K}}({\mathcal{K}}(y))\,
-\int_{\R^n}|x|^{2-n}\,\Delta u_{\mathcal{K}}(x)\,
-\int_{\R^n}\Delta u_{\mathcal{K}}(x)\,
\int_{\R^n}\nabla u_{\mathcal{K}}(x)\cdot\nabla
v_{\mathcal{K}}(x)\,dx=\int_{\R^n}\nabla u(y)\cdot\nabla
\int_{\R^n}\Delta u(y)\,
\end{eqnarray*}
This yields that $|{\mathcal{K}}(y)|^{n+2}\,\Delta u_{\mathcal{K}}({\mathcal{K}}(y))=
\Delta u(y)$, from which we obtain (<ref>).
Another interesting geometric property of the Kelvin Transform is that it
carries spheres and hyperplanes into spheres and hyperplanes:
The Kelvin Transform carries
* spheres not passing through the origin
into spheres not passing through the origin,
* spheres passing through the origin into hyperplanes not passing through the origin,
* and hyperplanes not passing through the origin into
spheres passing through the origin.
Also, the Kelvin Transform leaves invariant all the
hyperplanes passing through the origin.
More explicitly, if $p\in\R^n$, $r\in(0,+\infty)$, $\omega\in\partial B_1$ and $c\in\R$ we have that
\begin{equation}\label{4567k-0olk-45rt-HSMgAolfLIkdgnN89kAm-01}
\begin{split}
&{\mathcal{K}}\Big( \{ x\in\R^n\setminus\{0\} {\mbox{ s.t. }} |x-p|^2=r^2\}\Big)\\
\displaystyle\left\{ x\in\R^n\setminus\{0\} {\mbox{ s.t. }} \left|x-\frac{p}{|p|^2-r^2}\right|^2=\frac{r^2}{(|p|^2-r^2)^2}\right\}&{\mbox{ if }}r\ne|p|
\\
\displaystyle\left\{ x\in\R^n\setminus\{0\} {\mbox{ s.t. }} p\cdot x=\frac12\right\}&{\mbox{ if }}r=|p|,
\end{dcases}\end{split}\end{equation}
\begin{equation}\label{4567k-0olk-45rt-HSMgAolfLIkdgnN89kAm-02}
\begin{split}
&{\mathcal{K}}\Big( \{ x\in\R^n\setminus\{0\} {\mbox{ s.t. }} \omega\cdot x=c\}\Big)\\&\qquad\qquad=\begin{dcases}
\displaystyle\left\{ x\in\R^n\setminus\{0\} {\mbox{ s.t. }} \left|x-\frac{\omega}{2c}\right|^2=\frac{1}{4c^2}\right\}&{\mbox{ if }}c\ne0
\\
\displaystyle\left\{ x\in\R^n\setminus\{0\} {\mbox{ s.t. }} \omega\cdot x=0\right\}&{\mbox{ if }}c=0.
\end{dcases}\end{split}\end{equation}
Let $x\ne0$ and $y:={\mathcal{K}}(x)$.
Since, by Lemma <ref>,
the Kelvin Transform is an involution, we have that $x={\mathcal{K}}(y)=\frac{y}{|y|^2}$. Therefore, if $|x-p|^2=r^2$ then
\begin{equation*}
\begin{split}&
0=|y|^2\left( |x-p|^2-r^2\right)=|y|^2\left( \left|\frac{y}{|y|^2}-p\right|^2-r^2\right)\\&\qquad\qquad
1-2 p\cdot y+(|p|^2-r^2)|y|^2.\end{split}
\end{equation*}
Now, if $r=|p|$ this boils down to $p\cdot y=\frac12$. If instead $r\ne|p|$, then
\begin{eqnarray*}&&
0=\frac{1}{|p|^2-r^2}-\frac{2 p\cdot y}{|p|^2-r^2}+|y|^2=
\frac{1}{|p|^2-r^2}-\frac{|p|^2}{(|p|^2-r^2)^2}+\left|y-\frac{p}{|p|^2-r^2}\right|^2\\&&\qquad\qquad=-
\frac{r^2}{(|p|^2-r^2)^2}+\left|y-\frac{p}{|p|^2-r^2}\right|^2.
\end{eqnarray*}
Furthermore, we notice that, since $y:={\mathcal{K}}(x)=\frac{x}{|x|^2}$,
\begin{eqnarray*}
|y-p|^2=|y|^2-2p\cdot y +|p|^2=\frac1{|x|^2}-\frac{2p\cdot x}{|x|^2}+|p|^2.
\end{eqnarray*}
Thus, if $r=|p|$ and $p\cdot x=\frac12$, then
\frac1{|x|^2}-\frac{1}{|x|^2}+|p|^2=|p|^2=r^2.$$
If instead $r\ne|p|$ and $\left|x-\frac{p}{|p|^2-r^2}\right|^2=\frac{r^2}{(|p|^2-r^2)^2}$, then
\begin{eqnarray*}
\end{eqnarray*}
These considerations establish (<ref>).
Now, if $\omega\cdot x=c$, we find that $\omega\cdot y=c|y|^2$.
This reduces to $\omega\cdot y=0$ if $c=0$. Instead, if $c\ne0$,
$$ 0=|y|^2-\frac{\omega\cdot y}{c}=
\left| y-\frac{\omega}{2c}\right|^2-\frac{1}{4c^2}.$$
Viceversa, if $\omega\cdot x=0$, then $\omega\cdot y=0=c$.
If instead $\left| x-\frac{\omega}{2c}\right|^2=\frac{1}{4c^2}$, then
$$\omega\cdot y=\frac{\omega\cdot x}{|x|^2}=\frac{c|x|^2}{|x|^2}=c,
thus completing the proof of (<ref>).
We refer to Figure <ref> for a graphical representation
of the situation discussed in detail in Proposition <ref>.
The Kelvin Transform carries the sphere $\partial B_1(e_1)$ into the hyperplane $\left\{x_1=\frac12\right\}$.
A small neighborhood of $2e_1$ in $B_1(e_1)$ is carried into a small neighborhood of $\frac{e_1}2$
in $\left\{x_1>\frac12\right\}$.
§ THE FUNDAMENTAL SOLUTION
Here we will describe the notion of fundamental solutionfundamental solution of the Laplace equation of the Laplace[Usually,
though the notation is certainly not uniform across the literature, the equation $\Delta u(x)=0$
for all $x$ in $\Omega$ is often referred to with the name of “Laplace equation”,
and correspondingly $\Delta u(x)=f(x)$ with the name of “Poisson equation”. When the
Laplace equation (or, sometimes, the Poisson equation) is complemented
with a boundary datum $u=g$ along $\partial\Omega$ people speak about the “Dirichlet problem”.]
which is physically motivated by the electrostatic (or gravitational) potential produced
by a point charge (or a point mass). For this, the isotropy and homogeneity of the ambient space
play a crucial role by inducing rotational and translational symmetries.
To start with, in view of the polar coordinates representation of the Laplacian in (<ref>),
one can explicitly find all the rotationally symmetric harmonic functions in $\R^n\setminus\{0\}$,
according to the following observation:
If $v\in C^2(\R^n\setminus\{0\})$ (respectively, if $v\in C^2(B_R\setminus\{0\})$
for some ball $B_R\subset\R^n$)
is rotationally symmetric and harmonic in $\R^n\setminus\{0\}$
(respectively, in $B_R\setminus\{0\}$),
then there exist $a$, $b\in\R$ such that, for every $x\in\R^n\setminus\{0\}$
(respectively, for every $x\in B_R\setminus\{0\}$), we have that
$$ v(x)=\begin{dcases}
\frac{a}{|x|^{n-2}}+b&{\mbox{ if }}n\ne2,\\
a\ln |x|+b&{\mbox{ if }}n=2.
\end{dcases}$$
We argue in $\R^n\setminus\{0\}$, the case in $B_R\setminus\{0\}$ being similar.
We use the notation $r:=|x|$ and let $v_0:(0,+\infty)\to\R$ be such that $v(x)=v_0(|x|)$.
Then, by (<ref>), in $\R^n\setminus\{0\}$,
\begin{equation*}
0=\Delta v=v_0''+\frac{n-1}r\,v_0'=r^{1-n}\,\frac{d}{dr} \big( r^{n-1} v_0'\big),
\end{equation*}
therefore there exists $a\in\R$ such that $r^{n-1}v_0'(r)=a$ for every $r>0$. Integrating this expression,
we obtain that there exists $b\in\R$ such that
$$ v_0(r)=\begin{dcases}
\frac{a r^{2-n}}{2-n}+b&{\mbox{ if }}n\ne2,\\
a\ln r+b&{\mbox{ if }}n=2.
\end{dcases}$$
Hence, by renaming the constant $a$,
\begin{equation*} v_0(r)=\begin{dcases}
\frac{a }{r^{n-2}}+b&{\mbox{ if }}n\ne2,\\
a\ln r+b&{\mbox{ if }}n=2.
\end{dcases}\end{equation*}
and the desired result plainly follows.
The function $\Gamma_\rho$ (if $n\ge3$).
The functions introduced in Lemma <ref>
play a pivotal role in the development of the theory of elliptic
partial differential equations, since not only, as observed in Lemma <ref>,
they satisfy pointwise outside the origin the equation $\Delta v=0$,
but also, as we will see now, up to normalization constants,
they provide the “fundamental solutions” of the Laplace operators,
that is their Laplacian (in a suitable distributional sense,
as we will precise in Theorem <ref>) is (minus)
the Dirac Delta Function
at the origin.
To clarify this concept, in light of Lemma <ref>,
it is useful to define
\begin{equation}\label{OVERLINEV} \overline v(r)=\begin{dcases}
\frac{1}{r^{n-2}}&{\mbox{ if }}n\ne2,\\
-\ln r&{\mbox{ if }}n=2
\end{dcases}\end{equation}
and choose the normalizing constant
\begin{equation}\label{COIENEE} c_n:=\begin{dcases}\frac1{n(n-2)\,| B_1|}
& {\mbox{ if }} n\ne2,\\
\frac1{2\pi} & {\mbox{ if }} n=2.\end{dcases}\end{equation}
The reason for choosing $c_n$ in this way is that, for every $\rho>0$, the function
\begin{equation}\label{COIENEE-0986ytufgkbv-0-rjfeonvnb2GDB} \overline\Gamma_\rho(r):=\begin{dcases}
\frac{\rho^2-r^2}{2n |B_\rho|}+c_n \,\overline v(\rho) & {\mbox{ if }}r\in(0,\rho),\\ c_n\,
\overline v(r) & {\mbox{ if }}r\in[\rho,+\infty)
\end{dcases}\end{equation}
is such that $\overline\Gamma_\rho'\in C^{0,1}((0,+\infty))$. Let also
\begin{equation}\label{GAROGRA}
\Gamma_\rho(x):=\overline\Gamma_\rho(|x|).\end{equation}
We remark that $\Gamma_\rho$ is obtained, roughly speaking, by “gluing a paraboloid near the origin”
to the harmonic function outside the origin that was introduced in
Lemma <ref>, see Figure <ref>. Moreover, this paraboloid is normalized to
have Laplacian that integrates to $-1$ in $B_\rho$, namely
\begin{equation}\label{DEGAM} \Delta \Gamma_\rho=\begin{dcases}
-\frac{1}{|B_\rho|}&{\mbox{ in }}B_\rho,\\
0&{\mbox{ in }}\R^n\setminus\overline{B_\rho}.
\end{dcases}\end{equation}
We then consider the formal limit as $\rho\searrow0$ of $\Gamma_\rho$, namely we define
\begin{equation}\label{GAMMAFU} \Gamma(x):=c_n\overline v(|x|)=
\begin{dcases}
\frac{c_n}{|x|^{n-2}}&{\mbox{ if }}n\ne2,\\
-c_n\ln |x|&{\mbox{ if }}n=2
We stress that when $n=3$ this function represents, up to a
normalizing constant, the electrostatic potential
generated by a point charge (as well as the gravitational potential generated by a point mass,
or the equilibrium temperature produced by a concentrated heat source),
and we can exploit this physical motivation in any dimension as well,
at least[It is
usually a challenging task to understand higher dimensions, and the change of the fundamental
solution when $n=2$ is a deep feature to keep in mind. To ease the intuition and possibly recover, at least
at a heuristic level, lower dimensional cases from higher dimensional ones, we propose some
reflections about how electrostatic potentials of a line of uniformly distributed charges
in $\R^n$ produces the fundamental solutions in $\R^{n-1}$ (up to
some renormalization that plays a role in low dimension). For this,
we consider a uniform distribution of charges
on the line $L:=\{x=(x',x_n)\in\R^{n-1}\times\R$
s.t. $x'=0\}$, which (up to physical constant) produces an electrostatic
potential at the point $p=(p',0)\in\R^{n-1}\times\{0\}$ of the type
$$ U(r):=\frac1{c_n}\int_L \Gamma_n(p-x)\,d{\mathcal{H}}^{n-1}_x
=\int_L \overline{v}(|p-x|)\,d{\mathcal{H}}^{n-1}_x
where $r:=|p'|$, and $\overline{v}_n$ and $\Gamma_n$
are as in (<ref>) and (<ref>),
with the subscript $n$ exploited to underline the dependence of these functions upon the dimension.
Strangely enough, the easiest case to understand is the high dimensional one: namely,
when $n\ge4$ we obtain
$$ U(r)=\int_{-\infty}^{+\infty}\left( r^2+x_n^2\right)^{\frac{2-n}2}\,dx_n=
\int_{-\infty}^{+\infty}\left( 1+t^2\right)^{\frac{2-n}2}\,dt=cr^{3-n}=c\overline{v}_{n-1}(r),
where $c>0$, showing that the potential of the charged line produces
the fundamental solution in one dimension less, up to a normalizing constant.
The lower dimensional cases $n\in\{2,3\}$ are instead more tricky,
since the corresponding constant $c$ would diverge. To make the argument
work (at least at a heuristic level) one needs to perform a “renormalization”
procedure, formally “subtracting infinity” to the potential (as a matter of fact,
potentials are always defined “up to additive constants”, since the physical forces
come from their derivatives). Concretely, when $n\in\{2,3\}$ one has to replace the previous definition of $U$
by the following renormalized one:
$$ U(r):=\lim_{R\to+\infty}\left(
\int_{-R}^{+R}\overline{v}_n\left(\sqrt{r^2+x_n^2}\right)\,dx_n-\phi_n(R)\right),$$
for a suitable renormalization $\phi_n(R)$. The previous computations
give that one can choose $\phi_n$ to be identically zero when $n\ge3$,
but, as we will see now, the cases $n\in\{2,3\}$ do require a more specific choice.
If $n=3$, we choose $\phi(R):=2\ln(2R)$, and we thereby obtain
\begin{eqnarray*}
\int_{-R}^{+R}\frac1{\sqrt{r^2+x_n^2}}\,dx_n-2\ln(2R)\right)\\&=&
\lim_{R\to+\infty}\left(\ln\left(1+\frac{2 R (\sqrt{r^2 + R^2} + R)}{r^2} \right)
\\&=&\lim_{R\to+\infty}
\ln\frac{r^2+2 R (\sqrt{r^2 + R^2} + R)}{4r^2 R^2} \\&=&
\ln\frac{1}{r^2}\\&=&2\overline{v}_2 (r).
\end{eqnarray*}
If instead $n=2$, we choose $\phi(R):=-2R(\ln R+1)$, and we have
\begin{eqnarray*}
-\frac12\int_{-R}^{+R}\ln\big(r^2+x_n^2\big)\,dx_n+2R(\ln R+1)\right)\\
-2 r \arctan\frac{R}r -R \ln (r^2 + R^2)+2R\ln R\right)\\
-2 r \arctan\frac{R}r -R \ln \frac{r^2 + R^2}{R^2}\right)\\
-2 r \arctan\frac{R}r -R \ln \left(1+\frac{r^2}{R^2}\right)\right)\\
&=&-\pi r\\&=&-\pi \overline{v}_1(r).
\end{eqnarray*}
The negative sign here above is consistent with the fact that the constant
in (<ref>) is negative when $n=1$.
Related approaches for constructing lower dimensional effective potentials
deal with the interaction of a charged line with a “test line” (rather than a “test point”).
Another approach to recover dimension $1$ directly from dimension $3$ is to consider the renormalized
electrostatic potential of a charged plate (say, $\R^2\times\{0\}$) at the point $(0,0,r)$: in this case
the computation, up to dimensional constants, would involve a corrector of the type $2\pi R$ and go as follows:
\begin{eqnarray*}
U(r)&=&\lim_{R\to+\infty} \left[\int_0^R \left(\int_{\{|x'|=\rho,\;x_3=0\}}\frac{d{\mathcal{H}}^{2}_x}{|x-(0,0,r)|}\right)\,d\rho-2\pi R\right]\\
&=&\lim_{R\to+\infty} \left[\int_0^R \frac{2\pi\rho\,d\rho}{\sqrt{\rho^2+r^2}}-2\pi R\right]\\&=&
\lim_{R\to+\infty}2\pi\sqrt{R^2 + r^2}-2\pi r-2\pi R\\
&=&-2\pi r\\&=&-2\pi \overline{v}_1(r).
\end{eqnarray*}
to facilitate our mathematical intuition.
In our setting, the function $\Gamma$ is the “fundamental solution”
of the Laplace operator, in the sense made precise by the following result:
For every $\varphi\in C^\infty_0(\R^n)$,
$$ \int_{\R^n}\Gamma(x)\, \Delta\varphi(x) \,dx=-\varphi(0).$$
Since $\overline\Gamma_\rho'\in C^{0,1}((0,+\infty))$,
there exists $C>0$ such that $|\overline\Gamma_\rho'|+|\overline\Gamma_\rho''|\le C$ a.e. in $\R^n$.
As a result, for every $i$, $j\in\{1,\dots,n\}$, and a.e. $x\in\R^n\setminus B_{\rho/2}$,
$$ |\partial_{ij}\Gamma_\rho(x)|=\left|
\overline\Gamma_\rho''(|x|)\,\frac{x_i\,x_j}{|x|^2}
\overline\Gamma_\rho'(|x|)\,\frac{|x|^2\delta_{ij}-x_i\,x_j}{|x|^3}
\right|\le 4C\left(1+\frac1\rho\right).
As a result, letting $A_{\rho,\e}:=B_{\rho+\e}\setminus B_{\rho-\e}$,
\begin{equation}\label{OK-plkdm-1}\lim_{\e\searrow0}
\left|
\int_{A_{\rho,\e}}\Delta\Gamma_\rho(x)\,\varphi(x) \,dx
\right|\le \lim_{\e\searrow0}
\end{equation}
\begin{equation}\label{OK-plkdm-2}\lim_{\e\searrow0}
\left|
\int_{A_{\rho,\e}}\Gamma_\rho(x)\,\Delta\varphi(x) \,dx
\right|\le \lim_{\e\searrow0}n\,\|\Gamma_\rho\|_{L^\infty(S)}
\,\|D^2\varphi\|_{L^\infty(\R^n)}\,\big|A_{\rho,\e}\big|=0,
\end{equation}
where $S$ is the support of $\varphi$.
Now we point out that if $F:\R^n\to\R$ is a continuous vector field, then
\begin{equation}\label{CVF}\lim_{\e\searrow0}
\int_{\partial A_{\rho,\e}} F(x)\cdot\nu(x)\,d{\mathcal{H}}^{n-1}_x=0.
\end{equation}
For this, given $\eta>0$, we consider a smooth vector field $F_\eta$ such that $\|F-F_\eta\|_{L^\infty(B_{2\rho},\R^n)}\le\eta$ and we use the Divergence Theorem to see that
\begin{eqnarray*}&&
\lim_{\e\searrow0}\left|
\int_{\partial A_{\rho,\e}} F(x)\cdot\nu(x)\,d{\mathcal{H}}^{n-1}_x
\right|\\&\le&
\lim_{\e\searrow0}
\|F-F_\eta\|_{L^\infty(\partial A_{\rho,\e},\R^n)}\,{\mathcal{H}}^{n-1}(\partial A_{\rho,\e})+
\left|
\int_{\partial A_{\rho,\e}} F_\eta(x)\cdot\nu(x)\,d{\mathcal{H}}^{n-1}_x
\right|
\\&\le&2\eta{\mathcal{H}}^{n-1}(\partial B_\rho)+\lim_{\e\searrow0}
\left|
\int_{ A_{\rho,\e}}\div F_\eta(x)\,dx
\right|\\&\le&
2\eta{\mathcal{H}}^{n-1}(\partial B_\rho)+\lim_{\e\searrow0}
n\,\|F_\eta\|_{C^1(B_{2\rho},\R^n)}\,| A_{\rho,\e}|
\\&=&2\eta{\mathcal{H}}^{n-1}(\partial B_\rho).
\end{eqnarray*}
We now send $\eta\searrow0$ and we conclude the proof of (<ref>).
As a consequence of (<ref>), we find that
$$ \lim_{\e\searrow0}
\int_{\partial A_{\rho,\e}}
\Gamma_\rho(x)\frac{\partial \varphi}{\partial\nu}(x) \,d{\mathcal{H}}^{n-1}_x
=0\qquad{\mbox{ and }}\qquad
\lim_{\e\searrow0}
\int_{\partial A_{\rho,\e}}
\varphi(x)\frac{\partial \Gamma_\rho}{\partial\nu}(x)\,d{\mathcal{H}}^{n-1}_x
Hence, by (<ref>), (<ref>)
the second Green's Identity (recall (<ref>)),
\begin{equation}\label{IJSfabnxcstukcIDjmcopiougf087}
\begin{split}& \int_{\R^n}\Gamma_\rho(x)\, \Delta\varphi(x) \,dx=\lim_{\e\searrow0}
\int_{\R^n\setminus A_{\rho,\e}}\Gamma_\rho(x)\, \Delta\varphi(x) \,dx
\\&\qquad=\lim_{\e\searrow0}\left[
\int_{\R^n\setminus A_{\rho,\e}}\Delta\Gamma_\rho(x)\, \varphi(x) \,dx-
\int_{\partial A_{\rho,\e}}\left(\Gamma_\rho(x)\frac{\partial \varphi}{\partial\nu}(x)-
\varphi(x)\frac{\partial \Gamma_\rho}{\partial\nu}(x)\right)\,d{\mathcal{H}}^{n-1}_x\right]\\&\qquad=
\int_{\R^n }\Delta\Gamma_\rho(x)\, \varphi(x) \,dx.
\end{split}\end{equation}
Thus, using (<ref>),
$$ \int_{\R^n}\Gamma_\rho(x)\, \Delta\varphi(x) \,dx=
-\fint_{B_\rho} \varphi(x) \,dx.$$
The desired result now follows be sending $\rho\searrow0$
(notice that $|\Gamma_\rho|\le\Gamma$ which is locally integrable,
hence the Dominated Convergence Theorem can be utilized here).
A variant of Theorem <ref> goes as follows:
Let $\Omega$ be a bounded open set in $\R^n$
with $C^1$ boundary. Let $x_0\in\Omega$.
Then, for every $\varphi\in C^2(\Omega)\cap C^1(\overline\Omega)$,
\begin{equation}\label{300G} \int_{\Omega}
\Gamma(x-x_0)\, \Delta\varphi(x) \,dx-
\int_{\partial\Omega}
\left(\Gamma (x-x_0)\frac{\partial \varphi}{\partial\nu}(x)-
\varphi(x)\frac{\partial \Gamma}{\partial\nu}(x-x_0)\right)\,d{\mathcal{H}}^{n-1}_x
Up to replacing $\varphi(x)$ with $\widetilde\varphi(x):=\varphi(x+x_0)$,
we can assume that $x_0=0$.
We take $\rho>0$ so small such that $B_\rho\Subset\Omega$,
and thus $\Gamma_\rho=\Gamma$ in a neighborhood of $\partial\Omega$.
Then, we replace (<ref>)
in this framework by
\begin{equation*}
\begin{split}& \int_{\Omega}
\Gamma_\rho(x)\, \Delta\varphi(x) \,dx=\lim_{\e\searrow0}
\int_{\Omega\setminus A_{\rho,\e}}\Gamma_\rho(x)\, \Delta\varphi(x) \,dx
\\&\qquad=\lim_{\e\searrow0}\left[
\int_{\Omega\setminus A_{\rho,\e}}\Delta\Gamma_\rho(x)\, \varphi(x) \,dx+
\int_{\partial\Omega}\left(\Gamma_\rho(x)\frac{\partial \varphi}{\partial\nu}(x)-
\varphi(x)\frac{\partial \Gamma_\rho}{\partial\nu}(x)\right)\,d{\mathcal{H}}^{n-1}_x
\right.\\&\qquad\qquad\qquad\left.-
\int_{\partial A_{\rho,\e}}\left(\Gamma_\rho(x)\frac{\partial \varphi}{\partial\nu}(x)-
\varphi(x)\frac{\partial \Gamma_\rho}{\partial\nu}(x)\right)\,d{\mathcal{H}}^{n-1}_x\right]\\&\qquad=
\int_{\Omega }\Delta\Gamma_\rho(x)\, \varphi(x) \,dx+
\int_{\partial \Omega}\left(\Gamma(x)\frac{\partial \varphi}{\partial\nu}(x)-
\varphi(x)\frac{\partial \Gamma }{\partial\nu}(x)\right)\,d{\mathcal{H}}^{n-1}_x
\end{split}\end{equation*}
and we conclude as in the proof of Theorem <ref>
by sending $\rho\searrow0$.
The identity (<ref>)
is sometimes called “Green's Representation Formula”Green's Representation Formula: interestingly,
it allows one to reconstruct the pointwise value of a function from
its Laplacian in a given domain and its value and the values
of its normal derivative at the boundary of the domain.
The fundamental solution
can be used to construct regular solutions of the Poisson equation
with sufficiently regular right hand side. As an example, we provide the
following result (a more precise version will follow from the
Schauder estimates in Proposition <ref>).
Let $f\in C^1_0(\R^n)$. Then, the function $v:=-\Gamma*f$ belongs to $C^2(\R^n)$
and satisfies $\Delta v=f$ in $\R^n$.
We notice that $v$ is well defined since $\Gamma$ is locally integrable
and ${f}$ is bounded and with bounded support.
In fact, $v\in L^\infty(\R^n)$.
Also, by Theorem <ref>, for every $\varphi\in C^\infty_0(\R^n)$
and every $x\in\R^n$ we know that
\int_{\R^n}\Gamma(Y-x)\, \Delta\varphi(Y) \,dY=
\int_{\R^n}\Gamma(y)\, \Delta\varphi(x+y) \,dy=-\varphi(x)$$
and therefore
\begin{equation}\label{KA-OS-ikjfmf-S-kCONd}
\begin{split}&
\int_{\R^{n}}v(Y)\,
\Delta\varphi(Y) \,dY=-
\iint_{\R^{2n}}\Gamma(Y-x)\,{f}(x)\,
\Delta\varphi(Y) \,dx\,dY=\int_{\R^n}\varphi(x)\,
\end{split}\end{equation}
We observe that if $\phi\in L^\infty(\R^n)\cap L^1(\R^n)$ then, for all $i\in\{1,\dots,n\}$,
\begin{equation}\label{PRIMA98DER}
\partial_i(\Gamma*\phi)=(\partial_i\Gamma)*\phi.
\end{equation}
To check this, given $\rho>0$ we use the regularization $\Gamma_\rho$ of the fundamental solution
introduced in (<ref>) and (<ref>).
We define $\psi:=\Gamma*\phi$ and $\psi_\rho:=\Gamma_\rho*\phi$ and we observe that, for every $x\in\R^n$,
\begin{equation}\label{IHFSHNTHSYDHMSWMDTKFDOFLL}
\begin{split}&
\frac{c_n}{2n|B_\rho|}\int_{B_\rho} (\rho^2-|y|^2)|\phi(y)|\,dy\\
\frac{C\,\|\phi\|_{L^\infty(\R^n)}}{\rho^n}
\int_0^\rho(\rho^2-r^2)\,r^{n-1}\,dr\le C\,\|\phi\|_{L^\infty(\R^n)}\,\rho^2
\end{split}\end{equation}
for some constant $C>0$ depending only on $n$ and possibly varying from line to line.
\begin{eqnarray*}
&&\left|\partial_i \psi_\rho(x)-(\partial_i\Gamma)*\phi(x)\right|=
\left|\partial_i\Gamma_\rho*\phi(x)-(\partial_i\Gamma)*\phi(x)\right|\\&&\qquad\le
\frac{C}{\rho^n}\int_{B_\rho}|y|\,|\phi(x-y)|\,dy+C\int_{B_\rho}|y|^{1-n}\,|\phi(x-y)|\,dy
\le C\,\|\phi\|_{L^\infty(\R^n)}\,\rho.
\end{eqnarray*}
From this and (<ref>) we infer that $\psi_\rho$ converges uniformly to $\psi$
and its derivative to $(\partial_i\Gamma)*\phi$, whence it follows that $\partial_i\psi=(\partial_i\Gamma)*\phi$,
thus establishing (<ref>).
We also remark that $\partial_j(\Gamma*f)=\Gamma*(\partial_jf)$
(see e.g. <cit.>) and therefore it follows from (<ref>) that $
\partial_{ij}^2 v=-(\partial_i\Gamma)*(\partial_j f)$.
Since $\partial_j f$ is continuous and compactly supported
and $\partial_i\Gamma$ is locally integrable, we have that $\partial_{ij}^2 v\in L^\infty(\R^n)$.
Moreover, we suppose that the support of $f$ is contained in a bounded set $\Omega$ and, for every $x\in\R^n$,
we denote by $\Omega_x$ the bounded set containing all the points $y$ such that $x-y$
belong to $\displaystyle\bigcup_{p\in\Omega} B_1(p)$.
Then, if $x\in\R^n$ and $x_k\to x$ as $k\to+\infty$, we have that
$$ |\partial_i\Gamma(y)\partial_j f(x_k-y)|\le\|{f}\|_{C^1(\R^n)}\,|\nabla\Gamma(y)|\,\chi_{\Omega_x}(y),$$
and the latter is an integrable function of $y\in\R^n$.
As a result, by the Dominated Convergence Theorem,
$$ \lim_{k\to+\infty}\partial_{ij}^2v(x_k)=
-\lim_{k\to+\infty}\int_{\R^n}\partial_i\Gamma(y)\partial_j f(x_k-y)\,dy
=\int_{\R^n}\partial_i\Gamma(y)\partial_j f(x-y)\,dy=\partial_{ij}^2v(x),$$
whence $
v\in C^2(\R^n)$.
This and (<ref>) yield that $\Delta v=f$.
The function $-\Gamma*f$ in Proposition <ref>
(or sometimes, up to a sign convention, the function $\Gamma*f$)
is often referred to with the name of Newtonian potentialNewtonian potential.
A neat intuition for the result in Proposition <ref> comes from
physical motivations: namely, since the fundamental solution $-\Gamma(x-y)$ (disregarding physical constants and possible sign conventions) corresponds to the
gravitygravitation field at the point $x$ generated by a pointwise mass located at the point $y$, it follows that
the quantity $-\Gamma*f(x)$ (which agrees with
the the Newtonian potentialNewtonian potential)
corresponds to the gravitational field at the point $x$ generated by a distribution $f$ of masses, since it can be
seen as the superposition of $\Gamma(x-y)f(y)\,dy$. By
Gauß' Law, the flux of the field through a given surface $\partial\Omega$ (seen as the boundary
of a bounded set $\Omega$) corresponds to the
total mass comprised inside $\partial\Omega$, namely
$$ \int_\Omega f(x)\,dx=-\int_{\partial\Omega} \nabla(\Gamma*f)(x)\cdot\nu(x)\,d{\mathcal{H}}^{n-1}_x.$$
Using the Divergence Theorem on the latest integral we thereby find that
$$ \int_\Omega f(x)\,dx=-\int_{\Omega}\div\Big( \nabla(\Gamma*f)(x)\Big)\,dx
=-\int_\Omega \Delta(\Gamma*f(x))\,dx.$$
Hence, since $\Omega$ is an arbitrary domain, we obtain that $f=-\Delta(\Gamma*f)$,
which is precisely the content of Proposition <ref>.
In this sense, Proposition <ref> is a mathematically structured statement
of Gauß' Law (with some care devoted on the regularity of the distribution of mass $f$
required for the result to hold true).
As a byproduct of our analysis, one can also deal with the gravitational potential of a homogeneous ball and show that at all external points this potential is equal
to the potential of the material point of the same mass placed in its center (see (<ref>)
for an application to physical geodesy):
Let $R>0$ and $x\in\R^n\setminus \overline{B_R}$. Then,
$$ \int_{B_R} \Gamma(x-y)\,dy=|B_R|\,\Gamma(x).$$
Given $y\in B_R$, we have that $\Delta\Gamma(x-y)=0$ for every $x\in\R^n\setminus B_R$. In particular,
by the Green's Identity (<ref>),
if $r\in(0,R)$ and $\psi(y):=\frac{|y|^2-r^2}{2n}$,
\begin{eqnarray*}&&\int_{B_r}
\Gamma(x-y)\,dy=\int_{B_r} \Gamma(x-y)\Delta \psi(y)\,dy\\&&
\qquad=\int_{B_r}\psi(y)\Delta \Gamma(x-y) \,dy+
\int_{\partial B_r}\left(\Gamma(x-y)\frac{\partial \psi}{\partial\nu}(y)-
\psi(y)\frac{\partial \Gamma}{\partial\nu}(x-y)\right)\,d{\mathcal{H}}^{n-1}_y
\\&&\qquad=0+\int_{\partial B_r}\left(\frac{r}n\,\Gamma(x-y)-0\right)\,d{\mathcal{H}}^{n-1}_y=\frac{r}n\,\int_{\partial B_r} \Gamma(x-y)\,d{\mathcal{H}}^{n-1}_y
As a result, defining
$$ \Psi(r):=r^{-n}\int_{B_r}\Gamma(x-y)\,dy,$$
and using
polar coordinates (see e.g. <cit.>), for $r\in(0,R)$ we have
\begin{eqnarray*}\Psi'(r)=-nr^{-n-1}\int_{B_r}\Gamma(x-y)\,dy
r^{-n}\int_{\partial B_r}\Gamma(x-y)\,d{\mathcal{H}}^{n-1}_y
This gives that $\Psi$ is constant in $(0,R]$ and thus
\begin{equation*} \Gamma(x)=\lim_{r\searrow0}\fint_{B_r}\Gamma(x-y)\,dy=\lim_{r\searrow0}\frac{\Psi(r)}{|B_1|}=\lim_{r\nearrow R}\frac{\Psi(r)}{|B_1|}=
\fint_{B_R}\Gamma(x-y)\,dy.\qedhere\end{equation*}
§ BACK TO THE KELVIN TRANSFORM:
THE METHOD OF IMAGE
We discuss here a simple, but very influential, technique from electrostaticselectrostatics
that naturally leads to the Kelvin Transform introduced in Section <ref>.method of image charges
This method uses the fundamental solution
showcased in Section <ref>, interpreting it as the electrostatic
potential generated by a point charge (the method can also be
considered as an inspiration for the construction
of the Green Function
of the ball in the forthcoming Theorem <ref>).
The details of this motivation go as follows.
Suppose that we have a positive unit point charge located at some point
in the ball $B_1$. Is it possible to place a negative point charge (not necessarily a
unit charge) somewhere outside of the ball in order to make $\partial B_1$
a surface with constant, say zero, potential? If so, can we determine the position and
intensity of this
auxiliary charge?
For the mathematical setting for this situation we suppose[The case $n=2$
requires some conceptual modification:
when $n=2$, rather than imposing that $\partial B_1$
is a surface with zero potential, one requires the weaker condition
that it is an equipotential surface, and the value of the potential
may depend on $x_0$. To compensate
this weaker potential condition however, in dimension $2$
one can additionally impose that the image charge has
also unit intensity (but it is negatively charged), that
is, up to a sign, both the original charge and the mirror charge have
the same intensity.
In this way, equation (<ref>)
when $n=2$ is replaced by
$$ \ln|x-x_0|-
\ln |x-T(x_0)|=\beta(x_0)\qquad{\mbox{for every }}x\in\partial B_1,$$
for a suitable $\beta(x_0)$.
That is, setting $\gamma(x_0):=e^{\beta(x_0)}$,
$$ 1+|x_0|^2-2x\cdot x_0=|x-x_0|^2=(\gamma(x_0))^2 |x-T(x_0)|^2=
(\gamma(x_0))^2(1+|T(x_0)|^2-2x\cdot T(x_0)).
This identity is the counterpart of (<ref>)
when $n=2$ (with the notation $\mu(x_0)=\frac{1}{(\gamma(x_0))^2}$),
hence the computation provided in these pages (and leading to (<ref>))
would give $\gamma(x_0)=|x_0|$ and hence $\beta(x_0)=\ln|x_0|$.]
that $n\ne2$. In this way, up to a normalizing constant,
we can suppose that the electrostatic potential at some point $x\in\R^n$
generated by
a positive unit charge located at $x_0\in B_1$ is equal to $\frac1{|x-x_0|^{n-2}}$,
recall (<ref>).
So, suppose that we place a charge of some intensity $-\alpha(x_0)\in(-\infty,0)$
at some point $T(x_0)\in\R^n\setminus B_1$. The potential
generated by this auxiliary charge is equal to $-\frac{\alpha(x_0)}{|x-T(x_0)|^{n-2}}$.
Therefore, the condition that $\partial B_1$ is a surface with zero potential
boils down to the relation
\begin{equation}\label{hifbdnHDNDNbdsfidbiwtg49673fggMS}
\frac1{|x-x_0|^{n-2}}-
\frac{\alpha(x_0)}{|x-T(x_0)|^{n-2}}=0\qquad{\mbox{for every }}x\in\partial B_1.
\end{equation}
Setting $\mu(x_0):=(\alpha(x_0))^{\frac2{n-2}}$, we therefore obtain that,
for every $x\in\partial B_1$,
\begin{equation}\label{2345tyKS:KJHGFDOIUYTROIHGFD0987654}\begin{split}&
1+|T(x_0)|^2-2x\cdot T(x_0)
=|x|^2+|T(x_0)|^2-2x\cdot T(x_0)=
\mu(x_0)\,|x-x_0|^2=\mu(x)\,\big(|x|+|x_0|^2-2x\cdot x_0\big)=
\mu(x_0)\,\big(1+|x_0|^2-2x\cdot x_0\big),\end{split}
\end{equation}
that is
\begin{equation}\label{LAN DcindfgbfdizPM-10293tu4y}
=2x\cdot \big( T(x_0)-
\mu(x_0)\, x_0\big)
\qquad{\mbox{for every }}x\in\partial B_1.
\end{equation}
Now, given $x\in\partial B_1$, we exploit (<ref>)
both for $x$ and for $-x\in\partial B_1$, finding that
$$ 2x\cdot \big( T(x_0)-
\mu(x_0)\, x_0\big)=1+|T(x_0)|^2-\mu(x_0)\,\big(1+|x_0|^2\big)=
-2x\cdot \big( T(x_0)-
\mu(x_0)\, x_0\big).$$
\begin{equation}\label{IDKMsubiadkfgbK9idfsaj9irtkgki0d-1}
x\cdot \big( T(x_0)-
\mu(x_0)\, x_0\big)=0
\qquad{\mbox{for every }}x\in\partial B_1
\end{equation}
and thus (<ref>) reduces to
\begin{equation}\label{IDKMsubiadkfgbK9idfsaj9irtkgki0d-2}
\end{equation}
As a matter of fact, by choosing $x$ as
an element of the Euclidean basis in (<ref>)
we obtain that
\begin{equation}\label{KS:rujmaldmcgab dtedyfwiosxicaqs3d}
T(x_0)\cdot e_i=
\mu(x_0)\, x_0\cdot e_i\qquad{\mbox{for every }}i\in\{1,\dots,n\}
\end{equation}
and therefore
$$ |T(x_0)|^2=\sum_{i=1}^n(T(x_0)\cdot e_i)^2
=(\mu(x_0))^2\,\sum_{i=1}^n( x_0\cdot e_i)^2
Substituting this information into (<ref>)
we find that
$$ 0=1+(\mu(x_0))^2\,|x_0|^2-\mu(x_0)\,\big(1+|x_0|^2\big)$$
and accordingly either $\mu(x_0)=1$ or $\mu(x_0)=\frac1{|x_0|^2}$.
However, the possibility $\mu(x_0)=1$ must be ruled out,
otherwise, by (<ref>), we would
have that $\R^n\setminus B_1\ni T(x_0)=x_0\in B_1$ which is
a contradiction.
In this way, we have establishes that
\begin{equation}\label{2345tyKS:KJHGFDOIUYTROIHGFD0987654BIS}
\mu(x_0)=\frac1{|x_0|^2},\end{equation}
and thus necessarily $x_0\ne0$ if we want our problem to have a solution,
\begin{equation}\label{SOLLE:1}
\alpha(x_0)=\frac1{|x_0|^{n-2}}.\end{equation}
Furthermore, we obtain from (<ref>) that
\begin{equation}\label{SOLLE:2}
\end{equation}
Thus, conditions (<ref>)
and (<ref>) provide the solution of our
image electrostatic charge problem, specifying, respectively,
the intensity of the image charge and its spatial location.
Remarkably, the spatial location of the image charge in (<ref>)
coincides with the Kelvin Transform in (<ref>)
and the charge intensity in (<ref>) coincides with
the multiplicative factor in the functional action of the
Kelvin Transform as defined in (<ref>).
§ MAXIMUM PRINCIPLES
Maximum Principles are one of the cornerstonesMaximum Principle
of the theory of elliptic partial differential equations
and they can also
be seen as the real analysis counterpart of the Maximum Modulus Principle
which is in turn one of the backbones of complex analysis (see e.g. <cit.>).
Roughly speaking, the main idea is that if $\Delta u=0$ then
each point of the graph of $u$ is necessarily a “saddle point”
and therefore $u$ cannot have local maxima or minima.
In these pages, we will obtain the simplest
possible versions of the Maximum Principle for the Laplace operator.
First of all, we present
a statement, often called
Strong Maximum Principle, that goes as follows:
Let $\Omega\subseteq\R^n$ be open and connected,
and let $u\in C^2(\Omega)$.
(i). If $\Delta u\ge0$ in $\Omega$
and there exists $\overline{x}\in\Omega$
such that $u(\overline{x})=\sup_\Omega u$, then u is constant.
(ii). If $\Delta u\le0$
in $\Omega$ and there exists $\underline{x}\in\Omega$
such that $u(\underline{x})=\inf_\Omega u$, then u is constant.
(iii). If $u$ is harmonic
in $\Omega$, then it cannot
attain an interior maximum or minimum value unless it is constant.
We observe that the claim in (ii) follows from (i)
by changing $u$ into $-u$. Also, the claim in (iii) follows from (i) and (ii).
Therefore, it suffices to prove (i). To this end,
we define
\begin{equation}\label{S:osl4995ffdssSloamed} {\mathcal{U}}:=\left\{x\in\Omega{\mbox{ s.t. }}u(x)=\sup_\Omega u\right\}
We remark that $\overline{x}\in{\mathcal{U}}$
and thus ${\mathcal{U}}\ne\varnothing$. The continuity of $u$
also gives that ${\mathcal{U}}$ is closed in $\Omega$.
Electromagnetic levitation, after
René Magritte's The Castle of the Pyrenees.
Moreover, if $r>0$ is such that $B_r(\overline{x})\Subset\Omega$,
we let $H(x):=\Gamma(x-\overline{x})-\Gamma(re_1)$
and we deduce from Green's Representation Formula (<ref>)
and the Divergence Theorem (recall also the setting of the fundamental
solution in (<ref>)
and (<ref>), as well as the measure theoretic
identity in (<ref>))
\begin{equation}\label{0987654hdhdnknn098765hjoiuytre-0987654adsdoodo}
\begin{split}&
H(x)\,\Delta u(x)\,dx=
\int_{B_r(\overline{x})}
\Gamma(x-\overline{x})\,\Delta u(x)\,dx-\Gamma(re_1)
\int_{B_r(\overline{x})}\Delta u(x)\,dx
\\&\qquad
\int_{\partial B_r(\overline{x})}
\left(\Gamma (x-\overline{x})\frac{\partial u}{\partial\nu}(x)-
u(x)\frac{\partial \Gamma}{\partial\nu}(x-\overline{x})\right)\,d{\mathcal{H}}^{n-1}_x
\\&\qquad\qquad-u(\overline{x})-\Gamma(re_1)
\int_{\partial B_r(\overline{x})}\frac{\partial u}{\partial\nu}(x)\,d{\mathcal{H}}^{n-1}_x
\\&\qquad
=\frac{r}{n\,|B_1|}\,\int_{\partial B_r(\overline{x})}
\frac{u(x)}{|
-\sup_\Omega u\\&\qquad= \fint_{\partial B_r(\overline{x})}
-\sup_\Omega u.
\end{split}
\end{equation}
Now we take $\rho>0$ such that $B_\rho(\overline{x})\Subset\Omega$
and we claim that
\begin{equation}\label{TGASBDCryuiTTskdfroppnuo}
{\mbox{for every~$p\in B_\rho(\overline{x})$ it holds that~$u(p)=\,$}}\sup_\Omega u.
\end{equation}
Indeed, suppose not. Then, we have that
$$ \fint_{B_\rho(\overline{x})}u(x)\,dx<\sup_\Omega u.$$
Thus, using polar coordinates and (<ref>),
\begin{eqnarray*}
&&\sup_\Omega u>\frac{1}{| B_\rho|}\,
\int_0^\rho\left(
\int_{\partial B_r(\overline{x})}u(x)\,d{\mathcal{H}}^{n-1}_x
\right)\,dr\ge
\frac{{\mathcal{H}}^{n-1}(\partial B_1)}{|B_1|\rho^{n}}\,
\int_0^\rho\left(r^{n-1}
\sup_\Omega u
\right)\,dr\\&&\qquad\qquad=
\frac{{\mathcal{H}}^{n-1}(\partial B_1)}{n\,|B_1|}\,\sup_\Omega u
=\sup_\Omega u.
\end{eqnarray*}
This contradiction proves (<ref>).
As a consequence
of (<ref>), we have that ${\mathcal{U}}$
is open. Then, by the connectedness of $\Omega$, it follows that ${\mathcal{U}}=\Omega$
and consequently $u(x)=\sup_\Omega u$ for all $x\in\Omega$.
A more general version of this result, without assuming the function $u$
to be smooth, will be given in Lemma <ref>.
An interesting consequence of the Strong Maximum Principle in Theorem <ref>
is the observation, originally due to Earnshaw's Theorem
Samuel Earnshaw, according to which[Of course, if we put a positive charge at $e_1$ and another at $-e_1$,
a positive charge at the origin would be in equilibrium (by symmetry), but this would be an unstable equilibrium,
since a small perturbation of the position of the charge at the origin in the $e_2$ direction would make it drift away.
Given a collection of point charges in $\R^n$, with $n\ge2$, the corresponding harmonic electrostatic potential $u$
would generate a force field $F=-\nabla u$
(recall the discussion on electromagnetic fields in Section <ref>). The equilibria correspond to the zeros of $F$,
and the notion of stability adopted here
means that if we move the particle a little bit away from the equilibrium the electrostatic force will tend to bring it back to the equilibrium.
Hence, if, say, the origin were a stable equilibrium
in the above setting, then $F(0)=0$ and, if $x$ is close enough to the origin,
then $F(x)\cdot x\le0$; that is, $\nabla u(0)=0$ and, if $x$ is close enough to the origin, $\nabla u(x)\cdot x\ge0$.
In particular, there exists $\rho>0$ such that if $x\in B_\rho\setminus\{0\}$ and $\omega:=\frac{x}{|x|}$ then
$$ u(x)-u(0)=u(|x|\omega)-u(0)=\int_0^{|x|} \frac{d}{dr} u(r\omega)\,dr=
\int_0^{|x|} \nabla u(r\omega)\cdot\omega\,dr=\frac{1}{r}\,
\int_0^{|x|} \nabla u(r\omega)\cdot(r\omega)\,dr\ge0.$$
In particular, a stable equilibrium at the origin would produce a local minimum for $u$,
which would violate Theorem <ref>(iii).
This proves Earnshaw's Theorem in (<ref>).
Actually, Theorem <ref>(iii) gives, more generally, that
the electrostatic potential cannot
have a maximum or a minimum value at any point in space not occupied
by an electric charge
(saddle points are instead possible and would correspond to unstable
equilibria, as discussed above, see Figure <ref>).
We stress that the notion of stability used here is the one formulated
by Maxwell in <cit.>]
|
# SparseDNN: Fast Sparse Deep Learning Inference on CPUs
Ziheng Wang<EMAIL_ADDRESS>Stanford University
###### Abstract.
The last few years have seen gigantic leaps in algorithms and systems to
support efficient deep learning inference. Pruning and quantization algorithms
can now consistently compress neural networks by an order of magnitude. For a
compressed neural network, a multitude of inference frameworks have been
designed to maximize the performance of the target hardware. While we find
mature support for quantized neural networks in production frameworks such as
OpenVINO and MNN, support for pruned sparse neural networks is still lacking.
To tackle this challenge, we present SparseDNN, a sparse deep learning
inference engine targeting CPUs. We present both kernel-level optimizations
with a sparse code generator to accelerate sparse operators and novel network-
level optimizations catering to sparse networks. We show that our sparse code
generator can achieve significant speedups over state-of-the-art sparse and
dense libraries. On end-to-end benchmarks such as Huggingface pruneBERT,
SparseDNN achieves up to 5x throughput improvement over dense inference with
state-of-the-art OpenVINO. Open source library at
https://github.com/marsupialtail/sparsednn
## 1\. Introduction
Neural networks have started to top benchmarks in computer vision (CV) and
natural language processing (NLP). However, recent neural network
architectures, such as deep convolutional networks and transformer networks
like BERT, suffer from high memory footprint and FLOP count due to the massive
number of weights (Devlin et al., 2018; Howard et al., 2017). For example,
while GPT-3 can accomplish an array of amazing feats, it has a whopping 175
billion parameters (Brown et al., 2020). Assuming the model is in 32-bit
floating point, it would require around 20 V100 GPUs just to store the model.
To facilitate model deployment in production systems with strict memory and
latency constraints, there has been significant research interest in how to
compress deep neural networks (DNN) (Gale et al., 2019; Han et al., 2015). The
objective is to significantly reduce the number of parameters in a DNN to
reduce the inference latency and memory footprint, while not losing too much
accuracy. Recent neural network pruning research has established that
different criteria, such as L0 norm or L1 norm (magnitude), can be used to
remove as much as 90% of the weights in modern neural networks with little
loss in accuracy (Molchanov et al., 2017; Louizos et al., 2017; Gale et al.,
2019), if there are no constraints on which weights can be removed. These
unconstrained pruning techniques typically result in unstructured sparse
weight matrices.
Unfortunately, it was soon recognized that unstructured sparsity patterns are
hard to support efficiently on modern CPUs and GPUs typically used for deep
learning inference. This spurred the developments of custom hardware (Han et
al., 2016) as well as “structured” pruning methods that prune blocks of
weights at once. The resulting weights from these structured pruning methods
often can be used directly in dense BLAS routines or have enough structure to
support highly performant implementations on current hardware (Gray et al.,
2017; Yao et al., 2019). However, these structured pruning methods often lead
to larger accuracy losses than unstructured pruning and often do no better
than a smaller dense network (Crowley et al., 2019).
In this work, we present an inference engine, SparseDNN, to efficiently
support unstructured sparse deep learning inference. We show that through both
network-level and kernel-level optimizations, we can significantly speed up
the execution of unstructured sparse neural networks on modern datacenter CPUs
where most deep learning inference takes place today (Gaurav Batra and
Santhanam, 2018). The design of SparseDNN is inspired by other state-of-the-
art inference systems for dense neural networks such as TensorRT, OpenVINO and
MNN (Nvidia, 2018; Jiang et al., 2020b; Gorbachev et al., 2019), as well as
performance engineering research targeting sparse matrix multiplications and
convolutions in Intel Libxsmm and SkimCaffe (Georganas et al., 2019; Park et
al., 2016). SparseDNN leverages both novel network-level optimizations such as
weight permutation and kernel-level optimizations through a sparse kernel
generator which achieves significantly superior performance than Libxsmm and
SkimCaffe (Georganas et al., 2019; Park et al., 2016). End-to-end, we
demonstrate up to 5x speedup over dense inference with OpenVINO on sparse
networks like Huggingface pruneBERT (Sanh et al., 2020).
## 2\. Background
This section introduces some background concepts that motivate the development
of SparseDNN.
### 2.1. Deep Learning Inference Engines
As neural network models become increasingly commonplace in production, there
have been a number of different open and closed source inference frameworks
from both academia and industry. Some notable examples include OpenVINO for
CPUs, TensorRT for Nvidia GPUs and MNN for mobile devices (Gorbachev et al.,
2019; Nvidia, 2018; Jiang et al., 2020b). Almost all deep learning inference
engines consist of two stages: preparation and execution. In the preparation
stage, optimizations are performed on the neural network to generate a binary
suitable for execution. Typically, this only has to be done once for each
network and hardware backend, which allows significant computational resources
to be spent as they can be amortized over all subsequent inference requests in
the execution stage. The preparation stage typically determines how
efficiently a neural network can be executed.
Most of the deep learning inference engines divide the preparation stage into
two phases. In the first phase, network level optimizations such as layer
fusion, removing redundant operators and data format selection, are performed.
Latest neural network inference engines such as OpenVINO and TensorRT have
become sophisticated enough to perform fusions of adjacent depthwise and
groupwise convolutions (Gorbachev et al., 2019; Nvidia, 2018). Many of these
optimizations, such as layer fusion, transfer directly to sparse neural
networks. In addition, SparseDNN leverages network-level optimizations
specific to sparse neural networks, such as weight permutation.
In the second phase, each operator in the optimized network is then mapped to
an efficient kernel implementation depending on the target hardware. The
kernels can be selected from a kernel library or generated specifically for
the operator. OpenVINO and TensorRT rely on Intel oneDNN (formerly MKL-DNN)
and Nvidia cuDNN respectively (Gorbachev et al., 2019; Nvidia, 2018). MNN
relies on a semi-automated search technique to generate the kernels from a
pre-defined number of optimization strategies (Jiang et al., 2020b). TVM takes
it a step further and performs compilation and autotuning for each kernel
(Chen et al., 2018). In SparseDNN, we adopt the last approach. For each sparse
operator, we use a sparse code generator to compile a kernel optimized for its
sparsity pattern.
Crucially, there is often a feedback loop between the first phase and the
second phase. For example, if the kernel library’s version of fused depthwise-
groupwise convolution is slower than doing them separately, the fusion should
be rolled back in the first phase. In the case of SparseDNN, there are
sparsity patterns that our sparse code generator prefers, which guides the
weight permutation optimizations in the first phase.
Once the neural network is primed for execution, there are typically two modes
of inference, synchronous and asynchronous. On CPU inference engines such as
OpenVINO, synchronous inference uses all available CPU cores to process one
input example at a time through the neural network to minimize latency,
whereas asynchronous inference allows each CPU to process a different stream
of input examples to avoid synchronization costs to maximize throughput
(Gorbachev et al., 2019).
Figure 1. The architecture of the SparseDNN system. The preparation phase is
shown on the left while the execution phase is shown on the right.
### 2.2. SpMM and Sparse Convolutions
It is evident that the quality of the kernel implementations of the operators
is a crucial factor in the performance of the inference engine. Dense deep
learning libraries typically rely on highly optimized BLAS libraries such as
oneDNN and cuDNN that builds on decades of performance engineering research
(Goto and Van De Geijn, 2008; Chetlur et al., 2014).
In a sparse neural network in CV or NLP, the core operation at inference time
is typically sparse matrix – dense matrix multiplication (abbreviated as SpMM)
and sparse convolution (Elsen et al., 2019; Howard et al., 2017; Tan and Le,
2019; Devlin et al., 2018; Wang et al., 2019). Sparse convolution can in turn
be cast as a SpMM computation either through the im2col transformation
(Chetlur et al., 2014) or direct convolution (Park et al., 2016). This
suggests that speeding up SpMM is critical in speeding up sparse inference.
Although SpMM is a highly optimized kernel in scientific computing, we find
that many existing SpMM implementations are poorly suited for accelerating
unstructured sparse DNNs (Yang et al., 2018; Hong et al., 2019, 2018). They
are often catered to scientific computing applications where the sparse matrix
is massive and highly sparse. A typical sparse matrix in a SpMM problem,
circuit5M (Davis and Hu, 2011), is 5.5 million by 5.5 million with less than
0.01% nonzeros. Unfortunately, sparse matrices encountered in deep learning
are small and only moderately sparse. A typical sparse matrix in a deep
learning problem, the sixth layer in a pruned MobileNet V1 network, is 512 by
512 with 10% nonzeros (Elsen et al., 2019).
Many recent works have attempted to cater to SpMM problems that arise in deep
learning. Intel Libxsmm and SkimCaffe provide efficient sparsity support
tailored for small matrix sizes encountered in deep learning on CPUs
(Georganas et al., 2019; Park et al., 2016). We use these as the main
benchmark for SparseDNN’s kernels in this paper. In addition, Elsen et. al
demonstrated that unstructured sparsity can be fast on Arm mobile processors
with sufficient performance engineering (Elsen et al., 2019). Sputnik and
SparseRT demonstrated similar results on Nvidia GPUs (Gale et al., 2020; Wang,
2020).
The approach of SparseRT’s GPU optimizations is particularly interesting. The
author recognizes that the sparsity patterns of the sparse weight matrices in
deep learning are statically known before inference. The sparsity pattern can
thus be leveraged to optimize kernel performance through the inspector-
executor approach (Strout et al., 2004). SparseDNN also incorporates this
technique to optimize CPU SpMM kernels. In addition, we take it one step
further through incorporating network-level optimizations that statically
reorder the sparsity patterns in the sparse weights guided by the preferences
of our code generator.
## 3\. SparseDNN Architecture
This section describes the architecture of the SparseDNN sparse deep learning
inference system. The overall architecture, illustrated in Figure 1, is
similar to other inference systems such as OpenVINO, TensorRT and MNN (Jiang
et al., 2020b; Nvidia, 2018; Gorbachev et al., 2019).
To set up a neural network for inference in SparseRT, the neural network is
first implemented using SparseDNN’s C++ API. We selected this approach instead
of parsing from a higher-level representation in Tensorflow or Pytorch because
of its flexibility. Existing high-level frameworks also only typically have
incomplete and experimental support for sparse operators.
SparseDNN adopts the aforementioned preparation-execution paradigm. During the
preparation stage, SparseDNN performs network level optimizations, and selects
the best kernel implementation for each layer. SparseDNN also allows for both
synchronous and asynchronous inference in the execution stage.
### 3.1. Network Optimizations
In SparseDNN, extensive network-level optimizations are performed in the
preparation phase. SparseDNN supports standard layer fusion strategies such as
fusing a convolution layer and a subsequent element-wise layer. SparseDNN also
finds the best kernel implementation for each operator in the neural network.
Sparse kernels are generated from a sparse code generator based on their
associated sparse weight matrix, which is assumed to be statically known. The
sparse code generator will be discussed in detail in the next subsection. For
dense kernels, SparseDNN selects between a default Intel oneDNN backend and
SparseDNN’s own handtuned kernel library, which can be more performant in
certain cases.
In addition to these standard optimizations performed in the prepration phase,
SparseDNN also performs network-level optimizations specific to sparse neural
networks. We will describe two such optimizations below.
#### 3.1.1. Activation Buffer Reuse
Memory buffers that store input activations for earlier layers of the network
can be repurposed and rewritten in later layers. This is a simple optimization
that is often done to minimize the memory footprint of the inference engine on
edge devices (Jiang et al., 2020b). However, on sparse networks, it also has a
big impact on inference performance. Sparse operators are typically memory-
bound instead of compute-bound. As a result, their performance can often be
substantially improved if their inputs and outputs physically reside in cache
rather than DRAM. On CPUs, the more frequently used a memory buffer is, the
more likely it resides in cache. As a result, cache hit rates can be improved
with activation buffer reuse.
#### 3.1.2. Weights Permutation
This is an optimization that is only relevant to sparse networks. For dense
linear algebra operations such as matrix multiplication, permuting the rows of
the dense matrix has no effect on the operator’s runtime. However, for sparse
operators, permuting the rows of the sparse weight matrix can substantially
change the operator’s performance by changing the sparsity pattern. There have
been many recent works in the scientific computing community that examine
beneficial reorderings to apply to a sparse matrix (Jiang et al., 2020a).
In the context of neural network inference, permuting the rows of a sparse
weight matrix by itself is unacceptable as it will lead to the wrong output.
However, if a sparse matrix multiplication is followed by another sparse
matrix multiplication for example, one could permute the columns of the second
sparse matrix to offset the permutation of the rows of the first sparse matrix
and preserve the same output. This observation allows us to change some of the
sparsity patterns of the weight matrices in the network in the preparation
phase.
In the results presented here, we apply a simple greedy algorithm, shown in
Algorithm 1, to permute the rows of the weight matrices to maximize the
similarity between the nonzero patterns between adjacent matrices. First, the
row with the most number of nonzeros is selected to be the new first row.
Then, the row with the most number of nonzero columns which overlapped with
the nonzero columns in the previous row is selected to be the next row. This
process is repeated until there are no rows left. We empirically found that
this simple approach performed comparably to more complicated approaches that
use locality sensitive hashing and clustering on the problems tested in this
paper (Jiang et al., 2020b). It is straightforward to experiment with
different algorithms for different neural networks with SparseDNN.
Algorithm 1 Weight Reordering Algorithm, $nnz(A[i])$ stands for the set of
nonzero column positions in the $i^{th}$ row of A.
1: Input: sparse matrix $A$ with shape $m\times n$
2: Output: permuted sparse matrix $B$ with shape $m\times n$
3: $rem=\\{0...m\\}$
4: $row=\underset{x\in rem}{\arg\max}(|nnz(A[x])|)$
5: $B[0]=A[row]$
6: $rem$.remove($row$)
7: for $i$ in $1...m$ do
8: $row=\underset{x\in rem}{\arg\max}$ $(|nnz(B[i-1])\cap nnz(A[x])|)$
9: $B[i]=A[row]$
10: $rem$.remove($row$)
11: end for
### 3.2. Sparse Code Generator
After network optimizations, SparseRT attempts to map each operator in the
network to an optimized kernel implementation. Kernels for sparse operators
are generated from a sparse code generator at this stage. We recognize that
most operators in deep learning can be represented at their core by a matrix
multiplication (Georganas et al., 2019). The sparse code generator focuses on
generating high performance SpMM routines. We support sparse convolutions by
converting them into a SpMM problem with the direct convolution method
described in (Park et al., 2016).
#### 3.2.1. High Performance Dense GeMM
The optimization strategies employed by our sparse code generator are inspired
by optimization techniques commonly used in dense linear algebra, in
particular, dense matrix multiplication (GeMM). We briefly review these
techniques here for reference.
High performance GeMM kernels typically start with a highly tuned microkernel
which performs a small matrix multiplication using the SIMD instructions of
the CPU. For example, a typical strategy for this microkernel is illustrated
in Figure 2. The microkernel computes the outer product between a column slice
in A and a row slice in B. It typically employs register blocking to make the
best use of the vector registers. By tuning the size of the A and B slices, it
can trade off the number of loads vs. the number of computes to strike the
best load/compute ratio for a particular hardware architecture.
Armed with this tunable microkernel, the GeMM problem becomes how to decompose
the computation into a sequence of microkernel calls. Typically, the
implementation then optimizes for memory locality by using tiling. Tiling
breaks down the matrix multiplication into chunks that are small enough to fit
into on-chip caches on the CPU, which often offer an order of magnitude faster
access than the off-chip main memory. Typically, a tile from each input matrix
operand is loaded, and a tile of the output is computed using the microkernel.
We will now proceed to describe our SpMM strategy, which applies the key
aforementioned optimization strategies of vectorization, register blocking and
tiling.
Figure 2. Strategy for GeMM. An output tile in C is produced by multiplying
two input tiles from A and B. The microkernel computes the outer product
between a column in the A tile and a row in the B tile. This outer product is
accumulated to the output tile. The program then calls the microkernel again
on the next column in A and the next row in B.
#### 3.2.2. SpMM Strategy
Figure 3. SpMM microkernel. Only the nonzero elements in an A column slice are
processed. We iterate through the nonzeros in the A column slice, broadcasting
each element into vector registers. Then the corresponding B row slice is
loaded into vector registers. The vector registers holding A and B values are
then multiplied and accumulated to the vector registers holding the output
tile.
In SpMM, we adopt the same general strategy as dense matrix multiplication, as
illustrated in Figure 3. Similar to Libxsmm and SkimCaffe, we focus on
optimizing a sparse microkernel (Georganas et al., 2019; Park et al., 2016).
Our sparse microkernel is based on the dense microkernel described above.
Since A is sparse, only the nonzero elements in each A column slice needs to
be processed. Since B is dense, the vector registers that load the B row slice
are fully packed. For each A column slice, we iterate over the nonzeros. We
broadcast each nonzero to a vector register, multiply it with the vector
registers holding the B row slice, and accumulate the result to the output
tile. Although the accumulation position depends on the random nonzero
position, we can use registers for the output tile by statically encoding the
accumulation position in the vector register name. This strategy is inspired
by the code unrolling approach first described in (Wang, 2020), and represents
one crucial difference between our code generator and Intel’s Libxsmm and
SkimCaffe (Georganas et al., 2019; Park et al., 2016).
Similar to dense GeMM, we also employ tiling in SpMM. Since the nonzero
locations in the A column slice can be random due to unstructured sparsity, we
can expect random memory accesses into the dense B matrix. We tile the dense B
matrix to optimize for CPU cache behavior along the reduction dimension at the
cost of reloading the output tile from memory. This is commonly referred to as
“split-K” in dense GeMM optimization literature (Kerr et al., 2017). This
optimization significantly improves performance on SpMM problems with large
reduction dimensions.
Since the sparsity pattern is statically known, we can perform autotuning on
the sparse weight matrix to find the best settings for the microkernel and the
tiling factor. Our sparse code generator directly generates x86 assembly in
contrast to the approaches in Libxsmm and SkimCaffe (Georganas et al., 2019;
Park et al., 2016). This was mainly done to accelerate the autotuning process
by bypassing the compiler and improve register placement in the generated
code.
Finally, we notice that our generated kernel traverses the sparse matrix in a
Z-curve whose precise shape depends on the dimensions of the microkernel
output tile and split-K tiling strategy. Typical sparse data formats such as
CSR and CSC do not support efficient traversal of the sparse matrix in this
manner. Since we statically know the exact order in which the sparse matrix
values will be accessed depending on our microkernel parameters and tiling
strategy, we reorganize the sparse matrix values such that they will be
accessed sequentially. In effect, each sparse matrix is stored in its own
custom format, optimized for the SpMM routine. This represents another
difference between our approach and Libxsmm and SkimCaffe.
#### 3.2.3. Sparse Convolution
We apply the direct convolution approach to implement sparse convolutions from
our SpMM code generator (Park et al., 2016). Briefly, the sparse convolution
is treated as an SpMM where the sparse matrix is the filters treated as an
$OC$ by $IC\times F_{x}\times F_{y}$ matrix. $OC$ and $IC$ are output and
input channel counts and $F_{x}$ and $F_{y}$ are filter dimensions. The dense
matrix is a virtual matrix that is realized on the fly from the input dense
activations with the proper offsets (Park et al., 2016). The SpMM is then
optimized using the techniques described in the last subsection.
### 3.3. Optimized Dense Kernels
Dense operations in the neural network, such as pooling, self attention, and
softmax, are mostly handled by the Intel oneDNN library. However, SparseDNN
allows users to use other custom dense kernels. In this work, we use hand-
optimized dense kernels for grouped convolutions. This is because the
optimized kernels in oneDNN use a custom memory format for the input
activations that is different from what our sparse kernels expect.
In addition, the user could generate dense kernels from deep learning compiler
frameworks such as TVM or Triton to use as plug-ins instead of oneDNN kernels
if they provide better performance (Tillet et al., 2019; Chen et al., 2018).
We do not explore this option in this paper.
## 4\. Results
We evaluate SparseDNN in two different ways. We first evaluate our sparse code
generator via kernel-level benchmarks on single operators. We then evaluate
SparseDNN as a system with two end-to-end neural network inference benchmarks
in CV and NLP.
We consider two different CPU backends, both available via Amazon AWS: AMD
Epyc 7R32 processors and Intel Xeon(R) Platinum 8124M processors. One key
difference between the AMD processor and the Intel processor is that AMD
hardware only supports AVX2 vector intrinsics while the Intel processor
supports AVX512 instructions. We generate AVX2 code for AMD and AVX512 code
for Intel processors.
### 4.1. Kernel Benchmarks
Figure 4. SpMM benchmark results for SparseDNN demonstrating speedup against
Intel Libxsmm on a) one thread b) four threads. The plot shows the cumulative
distribution function of speedups over the benchmark set of problems.
We evaluate our sparse code generator on two different operations, SpMM and
sparse convolution, on a suite of typical problems encountered in deep
learning. We consider both single-threaded performance and multi-threaded
performance on 4 physical cores. As aforementioned, both scenarios are useful
for end-to-end network inference. Single threaded performance is important for
asynchronous inference, where each CPU handles a separate stream of inference
requests, while multi-threaded performance is important for synchronous
inference where all CPUs handle the same stream of inference requests.
We compare our SpMM routines against dense GeMM kernels from Intel MKL on
Intel CPUs and OpenBLAS on AMD CPUs. We also compare against the optimized
SpMM implementation from Libxsmm on both Intel and AMD CPUs (Georganas et al.,
2019; Park et al., 2016) . We compare our sparse convolution routines against
dense convolution kernels provided by Intel oneDNN on both Intel and AMD CPUs.
We also compare against the optimized sparse convolution implementation in
Intel SkimCaffe (Park et al., 2016).
To test our SpMM routines, we first examine a benchmark suite of synthetic
sparse matrices with sizes ranging from 256 by 256 to 2048 by 2048, with
sparsity ratios at 70%, 80% and 90% and 95%. The sparsity pattern is generated
by populating the matrices with random numbers drawn from a unit Gaussian
distribution and then setting values below a certain threshold to zero. We
find that the sparsity pattern generated closely resembles the ones obtained
via magnitude pruning. The dense matrix has either 128, 512 or 2048 columns.
This resulted in a total of 768 test matrices.
The benchmarking results are presented in Figure 4. For single-threaded
comparisons, we achieve a geometric mean of 2.5x speedup over the optimized
SpMM routine in Libxsmm on Intel CPUs and 1.3x on AMD CPUs. We can see from
Figure 4a that for single-threaded benchmark, SparseDNN is faster than Libxsmm
on most problems in the benchmark suite, reaching a speedup factor of 10x on
the best performing problems on Intel CPU.
The speedup achieved for multi-threaded performance shown in Figure 4b was
lower given the added cost of synchronization, which we do not improve. We
achieve a geometric mean of 1.7x speedup over Libxsmm on Intel CPUs and
breakeven on AMD CPUs. The speedup against dense BLAS libraries is dependent
on the sparsity level, which will be elaborated in the next section.
Figure 5. Single-threaded sparse convolution benchmark. Multi-threaded
performance is similar.
To test our sparse convolution routines, we use a benchmark suite of sparse
convolution problems with random sparsity generated in the same way as our
SpMM benchmark. The number of input and output channels range from 32 to 256,
while the image dimension is one of 7, 14, 28 and 56. The filter size is 3 by
3 with a padding size of 1. The problems are at either 90 or 95 percent
sparsity. Compared to the sparse convolution kernels in SkimCaffe (Park et
al., 2016), we obtain a geometric mean 22% speedup on Intel and 9% slowdown on
AMD single thread, as shown in Figure 5. Compared to the optimized dense
convolution routines in oneDNN, we obtain a geometric mean speedup of 3.4x at
90% sparsity and 6.3x at 95% sparsity for AMD CPU. We obtain a geometric mean
speedup of 3.0x at 90% sparsity and 4.8x at 95% sparsity for Intel CPU. The
results for multi-threaded are similar.
### 4.2. Performance Analysis
Figure 6. Geometric mean speedup obtained vs a) Libxsmm and b) dense BLAS
kernel as a function of sparsity ratio. The matrices in the benchmark set had
70%, 80%, 90% and 95% sparsity. The break-even point is around 70% in
comparison to dense BLAS kernels, suggesting that sparse networks with more
than 70% zeros can be sped up using SparseDNN. Figure 7. The speedup obtained
against Libxsmm is strongly inversely correlated with the size of the dense
matrix (r = -0.51).
This benchmark suite is aimed at answering two questions: 1) At what sparsity
level can our sparse code generator provide advantages over Libxsmm and dense
GeMM? 2) How does the performance change depending on the dimensions of the
sparse and dense matrix operands?
Figure 6 shows how the speedups achieved by SparseDNN with respect to Libxsmm
SpMM routines and dense BLAS libraries scale with the sparsity level. From
Figure 6a we can see that SparseDNN performs better with respect to Libxsmm
with increasing sparsity, with a break-even point of around 70% sparsity on
Intel CPUs. SparseDNN performs worse on AMD CPUs, with a break-even point of
around 80% sparsity.
Compared to dense BLAS libraries MKL and OpenBLAS in Figure 6b, we achieve a
break-even point of around 70% sparsity for both CPU architectures. This
observation suggests that unstructured sparsity can translate into real
speedups as long as the neural network has 70% sparsity. At 95% sparsity, we
can achieve around 4x speedup over dense BLAS routines.
Crucially, SparseDNN’s break-even point with dense BLAS routines roughly
coincides with the SparseDNN’s break-even point with Libxsmm, which suggests
that SparseDNN performs better for the vast majority of sparse problems in the
regime where sparse computation is faster than dense computation. This shows
the efficacy of incorporating the known sparsity pattern of the sparse matrix
into static optimizations, which is the key difference between our approach
and Libxsmm and SkimCaffe.
We notice that the speedups we obtain relative to the dense BLAS libraries are
consistent across both architectures while we perform worse against Libxsmm on
AMD CPUs. This is attributed to the observation that the Intel Libxsmm library
actually performs slightly better on the AMD CPU than on the Intel CPU tested,
compared to the dense BLAS baselines.
The most important factor in determining SparseDNN’s performance is the dense
matrix size. In Figure 7, we plot the single threaded speedup achieved by
SparseDNN compared to Libxsmm on Intel CPUs against the size of the dense
matrix. The results for AMD CPUs and multi-thread are similar. We see that
while SparseDNN can achieve $>$ 10x speedups for small matrices, it is often
on par with Libxsmm for very large matrices.
As for sparse convolution, we find that we outperform SkimCaffe on problems
where the number of output channels exceed the number of input channels, while
we perform worse on problems where the number of output channels is small. We
also perform relatively worse on small images since we do not register block
across rows of the image as in SkimCaffe.
In summary, our sparse code generation technique outperforms state-of-the-art
sparse libraries, and achieves several times speedup over dense BLAS
libraries. This suggests that unstructured sparsity can be used efficiently to
speed up neural network inference.
### 4.3. End-to-end Benchmarks
Figure 8. We show the latency/throughput tradeoff achieved by the three
inference systems under synchronous and asynchronous inference. The dot
corresponding to synchronous inference is always to the left of the dot
corresponding to asynchronous inference. Ideally, we want low latency and high
throughput (upper left corner of graph). a) Intel pruneBERT. b) AMD pruneBERT.
c) Intel MobileNet V1. d) AMD MobileNet V1.
To evaluate the end-to-end inference performance of SparseDNN, we choose two
different neural networks from CV and NLP that are representative of different
operations popular in deep learning. The first benchmark is MobileNet V1,
which is composed solely of grouped convolutions and 1x1 convolutions. We use
a 90% unstructured sparse model, where only the 1x1 convolutions are pruned
(Elsen et al., 2019). The grouped convolutions are dense. This model achieves
68.4% top-1 accuracy on ImageNet, vs 70.9% from an unpruned model.
The second benchmark is pruneBERT by Huggingface (Sanh et al., 2020). It is a
95% unstructured sparse BERT model, consisting mostly of fully connected
layers and self attention units. Only the fully connected weights are pruned
in this model. The self attention unit is dense. This model achieves an F1
score of 80 on the SQuAD v1.1 benchmark, vs. the unpruned model’s score of 88.
We benchmark only the encoder stack. We use 32-bit floating point for both
models.
We compare our end-to-end inference performance with two different inference
frameworks. The first is the network implemented in Pytorch with dense
operators, backed by the Intel MKL-DNN kernel library. Pytorch only provides
experimental support for sparse operators. We found that the sparse operators
are only faster than the dense equivalent under very high ($>$98%) sparsity
ratio, making them unsuitable for our benchmarks.
The second is the Intel OpenVINO inference framework (Gorbachev et al., 2019).
Intel OpenVINO is a highly optimized closed source framework for CPUs that
perform complex optimizations such as layer fusion and intermediate activation
format selection. It is typically faster than a naive Pytorch or Tensorflow
implementation, even on AMD CPUs. However, it is less flexible than Pytorch
due to the lack of an API to construct the network layer by layer. Instead,
models must be converted to a supported IR format using the provided
conversion scripts for different frontends such as Tensorflow 1.x and ONNX.
While we were not able to convert the Huggingface pruneBERT model to OpenVINO
IR, we were able to benchmark a supported BERT architecture that has the same
encoder stack.
Since this paper was first written, Neuralmagic Inc. has open-sourced a sparse
neural network inference engine Deepsparse that is able to achieve impressive
performance on a set of neural networks sparsified with another library
SparseML. Deepsparse also optionally supports direct conversion from an ONNX
model. Unfortunately the ONNX support falls short for both of the custom-
pruned models used in this paper. Deepsparse also does not offer an operator-
level API to support custom neural networks. Due to these difficulties, we
don’t include Deepsparse as a comparison point.
For the three inference systems, SparseDNN, Pytorch and OpenVINO, we test the
achieved latency and throughput in both synchronous and asynchronous
execution. For both Intel and AMD CPUs, we assume we have access to four
physical cores. We present our end-to-end results in Figure 8. The result from
synchronous inference (dot on the left) always has lower latency and lower
throughput than the result from asynchronous inference (dot on the right).
Results in the upper left corner of the graphs are more desirable as they
correspond to higher throughput and lower latency.
In all cases, Intel OpenVINO outperforms Pytorch. The performance gap is
especially large for MobileNet V1, where OpenVINO employs data layout
transformations and fuses some depthwise and groupwise convolution layers.
We see that SparseDNN is able to outperform OpenVINO significantly in all
cases except synchronous inference for MobileNet V1 on Intel CPUs. SparseDNN
is particularly efficient in asynchronous inference, achieving a 5x
improvement in throughput for pruneBERT on Intel CPUs with lower latency than
synchronous inference with OpenVINO. This suggests that pruneBERT inference
with SparseDNN on a single core is faster than dense OpenVINO inference on
four cores.
Although we can achieve even greater reductions in latency with synchronous
inference, SparseDNN incurs a much steeper throughput penalty than OpenVINO or
Pytorch. This reflects the earlier kernel-level benchmark results where
SparseDNN does comparatively poorly in a multi-threaded scenario compared to
single-thread. We are currently working on improving our OpenMP based
threading implementation in SparseDNN to use an optimized custom thread pool
implementation, as done in Amazon Sagemaker Neo (Liu et al., 2019).
The performance gap between OpenVINO and SparseDNN decreases for MobileNet V1.
This is due to the lower sparsity ratio (90% vs. 95%) and better performance
of OpenVINO. While the sparse 1x1 contractions are greatly accelerated in
SparseDNN, the dense depthwise convolution layers are handled better in
OpenVINO. OpenVINO uses a custom data format that allows vectorization over
the channels, and fuses some depthwise layers into the following 1x1
convolutions. We were not able to achieve end-to-end speedups using the oneDNN
API for the depthwise layers. This prompted us to use hand-optimized depthwise
convolution kernels, which resulted in a 50% gain in throughput on Intel CPUs
and a 58% gain in throughput on AMD CPUs over OpenVINO.
While the efficient sparse kernels are responsible for the bulk of the
performance improvement, the network-level optimizations were also beneficial.
Although on pruneBERT, these optimization only improved inference throughput
by six and three percent respectively on Intel and AMD CPUs, they had a much
greater effect on MobileNet V1. On Intel CPUs, they improved throughput by 27%
while on AMD CPUs, they improved throughput by 21%.
## 5\. Conclusion and Future Work
In this work, we present SparseDNN, a deep learning inference engine that
supports unstructured sparsity. At its core, SparseDNN relies on a sparse code
generator that leverages the static sparsity patterns of the sparse weights to
generate optimized kernels that are significantly faster than current state-
of-the-art libraries. Network-level optimizations catering to sparsity such as
buffer reuse and weights permutation improve the end-to-end inference results
further, resulting in large latency and throughput gains over highly optimized
dense inference engines.
It is important to note that SparseDNN currently does not handle activation
sparsity typically induced by activation functions such as relu. Activation
sparsity is typically dynamic, which does not allow for the static
optimization techniques leveraged by our code generator. In addition, its
presence heavily depends on the activation function used, with newer
activation functions such as gelu (Hendrycks and Gimpel, 2016) inducing much
less sparsity than relu.
Although currently applied to sparse inference, SparseDNN can also be applied
to sparse training. This is because current sparse training algorithms
typically use a fixed sparse network architecture or a fixed sparsity pattern
for a number of iterations (Frankle and Carbin, 2018; Evci et al., 2019). As a
result, the sparsity pattern is in effect statically known, and the sparse
kernels can be generated by SparseDNN before training or just-in-time every
time the architecture changes.
We note that in addition to network pruning, quantization is also a popular
approach to compress neural networks. Recently, pruning has been successfully
combined with quantization to produce sparse quantized networks (Kozlov et
al., 2020). We are currently working on adding support for those sparse low-
precision operators.
This work adds to a growing corpus of literature that suggests unstructured
sparsity is a viable option to obtain tangible speedups on commodity hardware
(Gale et al., 2020; Elsen et al., 2019; Park et al., 2016; Wang, 2020; Chen,
2019). We hope that by efficiently supporting unstructured sparsity in end-to-
end network inference, SparseDNN can make it easier to productionize novel
pruning algorithms and inspire future research into neural network
compression.
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# On the Mean First Arrival Time of Brownian Particles on Riemannian Manifolds
M. Nursultanov School of Mathematics and Statistics, University of Sydney
<EMAIL_ADDRESS>, L. Tzou School of Mathematics and Statistics,
University of Sydney<EMAIL_ADDRESS>and J.C. Tzou Department of
Mathematics and Statistics, Macquarie University<EMAIL_ADDRESS>
###### Abstract.
We use geometric microlocal methods to compute an asymptotic expansion of mean
first arrival time for Brownian particles on Riemannian manifolds. This
approach provides a robust way to treat this problem, which has thus far been
limited to very special geometries. This paper can be seen as the Riemannian
3-manifold version of the planar result of [1] and thus enable us to see the
full effect of the local extrinsic boundary geometry on the mean arrival time
of the Brownian particles. Our approach also connects this question to some of
the recent progress on boundary rigidity and integral geometry [23, 20].
###### 2010 Mathematics Subject Classification:
Primary: 58J65 Secondary: 60J65, 58G15, 92C37
M. Nursultanov and L. Tzou are partially supported by ARC DP190103302 and ARC
DP190103451 during this work. We thank Ben Goldys for the helpful discussion.
## 1\. Introduction
Let $(M,g,\partial M)$ be a compact connected orientable Riemannian manifold
with non-empty smooth boundary and without loss of generality we may assume
that it is an open subset of an orientable Riemannian manifold $(\tilde{M},g)$
without boundary oriented by the Riemannian volume form ${\rm dvol}_{g}$. Let
also $(X_{t},\mathbb{P}_{x})$ be the Brownian motion on $M$ with initial
condition at $x$, that is, the stochastic process generated by the Laplace-
Beltrami operator $\Delta_{g}$ (this article uses the convention
$\Delta_{g}=-d^{*}d$ with negative spectrum, where $d$ is the exterior
derivative). For any $\Gamma\subset\partial M$ open we denote by
$\tau_{\Gamma}$ the first time the Brownian motion $X_{t}$ hits $\Gamma$, that
is
$\tau_{\Gamma}:=\inf\\{t\geq 0:X_{t}\in\Gamma\\}.$
In the case when $\Gamma=\Gamma_{\epsilon,a}$ is a small elliptic window of
eccentricity $\sqrt{1-a^{2}}$ and size $\epsilon\to 0^{+}$ (to be made precise
later), the narrow escape/mean first arrival time problem wishes to derive an
asymptotic expansion as $\epsilon\to 0$ for the expected value
$\mathbb{E}[\tau_{\Gamma_{\epsilon,a}}|X_{0}=x]$ of the first arrival time
$\tau_{\Gamma_{\epsilon,a}}$ amongst all Brownian particles starting at $x$.
Another quantity of interest is the average expected value over $M$:
$|M|^{-1}\int_{M}\mathbb{E}[\tau_{\Gamma_{\epsilon,a}}|X_{0}=x]{\rm
dvol}_{g}(x).$
Here $|M|$ denotes the Riemannian volume of $M$ with respect to the metric
$g$.
Many problems in cellular biology may be formulated as mean first arrival time
problems; a collection of analysis methods, results, applications, and
references may be found in [10]. For example, cells have been modelled as
simply connected two-dimensional domains with small absorbing windows on the
boundary representing ion channels or target binding sites; the quantity
sought is then the mean time for a diffusing ion or receptor to exit through
an ion channel or reach a binding site [28, 8, 24].
There has been much progress for this problem in the setting of planar
domains, and we refer the readers to [8, 24, 31, 1] and references therein for
a complete bibliography. An important contribution was made in the planar case
by [1] to introduce rigor into the computation of [24]. The use of layered
potential in [1] also cast this problem in the mainstream language of elliptic
PDE and facilitates some of the approach we use in this article.
Few results exists for three dimensional domains in $\mathbb{R}^{n}$ or
Riemannian manifolds; see [3, 27, 30, 5] and references therein. The
additional difficulties introduced by higher dimension are highlighted in the
introduction of [1] and the challenges in geometry are outlined in [30]. In
the case when $M$ is a domain in $\mathbb{R}^{3}$ with Euclidean metric and
$\Gamma_{\epsilon,a}$ is a single small disk absorbing window, [27, 30] gave
an expansion for the average of the expected first arrival time, averaged over
$M$, up to an unspecified $O(1)$ term:
(1.1)
$\displaystyle|M|^{-1}\int_{M}\mathbb{E}[\tau_{\Gamma_{\epsilon,a}}|X_{0}=x]{\rm
dvol}_{g}(x)\sim\frac{|M|}{4\epsilon}\left[1-\frac{\epsilon}{\pi}H\log\epsilon+\mathcal{O}(\epsilon)\right].$
Here, $H$ is the mean curvature of the boundary at the center of the absorbing
window. The case when $\Gamma_{\epsilon,a}$ is a small elliptic window was
also addressed in [27, 30].
When $M$ is a three dimensional ball with multiple circular absorbing windows
on the boundary, an expansion capturing the explicit form of the
$\mathcal{O}(1)$ correction in equation (1.1) in terms of the Neumann Green’s
function and its regular part was done in [3]. The method of matched
asymptotic used there required the explicit computation of the Neumann Green’s
function, which is only possible in special geometries with high degrees of
symmetry/homogeneity. In these results one does not see the full effects of
local geometry. This result was also rigorously proved in [2] but with a
better estimate for the error term.
In this paper we outline an approach which allows one to derive all the main
terms of $\mathbb{E}[\tau_{\Gamma_{\epsilon,a}}|X_{0}=x]$ (up to a remainder
vanishing as $\epsilon\to 0$) for Riemannian manifolds of dimension three with
a multiple number of small absorbing windows which are boundary geodesic balls
or ellipses. We will only demonstrate this approach for one absorbing window
so as to not obscure the main idea. In the case when the window is a geodesic
ball our approach also adapts naturally to Riemannian manifolds of any
dimension as the proof of Proposition 1.1 as well as the analysis for
inverting a key integral equation on the ball in Section 4 both carry through
to higher dimensions.
We discuss briefly here on how to obtain a comprehensive singularity expansion
at the boundary for the Neumann Green’s function on a Riemannian manifold as
the Euclidean case was of interest in [30] and [3]. We will define in Section
3 the Neumann Green’s function $G(x,z)$ on $(M,g,\partial M)$ which satisfies
$\Delta_{g}G(x,z)=-\delta_{x}\ ,\partial_{\nu_{z}}G(x,z)\mid_{z\in\partial
M}=\frac{-1}{|\partial M|},\ \ \int_{\partial M}G(x,z){\rm dvol}_{\partial
M}(z)=0,$
where $z\in\partial M\mapsto\nu_{z}$ is a outward pointing normal vector field
and $|\partial M|$ is the area of the boundary.
Singer-Schuss-Holcman in [30] highlighted the difficulty in obtaining a
comprehensive singularity expansion of $G(x,z)\mid_{x,z\in\partial M,x\neq z}$
in a neighbourhood of the diagonal $\\{x=z\\}$ when $M$ is a bounded domain in
$\mathbb{R}^{n}$, but it turns out that even when $M$ is a general Riemannian
manifold this question can be treated by the standard pseudodifferential
operators approach. We only carry out this calculation in three dimensions as
it pertains to our application. Readers who are interested in the higher
dimensional analogue can follow our treatment to carry out the (cumbersome)
calculations for themselves:
###### Proposition 1.1.
For $x,y\in\partial M$, set $H(x)$ to be the mean curvature of $\partial M$ at
$x$, $d_{h}(x,y)$ the geodesic distance on the boundary given by metric
$h:=\iota_{\partial M}^{*}g$, $d_{g}(x,y)$ the geodesic distance given by the
metric $g$, and
${\rm{II}}_{x}(V):={\rm{II}}_{x}(V,V),\ \ V\in T_{x}\partial M$
the scalar second fundamental quadratic form (see pages 235 and 381 of [16]
for definitions).
i) The map
$f\in C^{\infty}(\partial M)\mapsto\left.\left(\int_{\partial
M}G(\cdot,y)f(y)d{\rm vol}_{h}(y)\right)\right|_{\partial M}$
is well defined and extends to a map from $H^{k}(\partial M)\to
H^{k+1}(\partial M)$ for all $k\in\mathbb{R}$ whose Schwartz kernel we will
denote by $G_{\partial M}(x,y)\in{\mathcal{D}}^{\prime}(\partial
M\times\partial M)$. Here the map $u\in H^{1}(M)\mapsto u\mid_{\partial M}\in
H^{1/2}(\partial M)$ is the trace map.
ii) There exists an open neighbourhood of the diagonal
${\rm Diag}:=\\{(x,y)\in\partial M\times\partial M\mid x=y\\}$
such that in this neighbourhood, the singularity structure of $G_{\partial
M}(x,y)$ is given by:
$\displaystyle\quad\quad G_{\partial M}(x,y)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}d_{g}(x,y)^{-1}-\frac{1}{4\pi}{H(x)}\log
d_{h}(y,x)$
$\displaystyle+\frac{1}{16\pi}\left({\rm{II}}_{x}\left(\frac{\exp_{x;h}^{-1}(y)}{|\exp_{x;h}^{-1}(y)|_{h}}\right)-{\rm{II}}_{x}\left(\frac{*\exp_{x;h}^{-1}(y)}{|\exp_{x;h}^{-1}(y)|_{h}}\right)\right)+R(x,y),$
where $R(\cdot,\cdot)\in C^{0,\mu}(\partial M\times\partial M)$, for all
$\mu<1$, is called the regular part of the Green’s function and $*$ is the
Hodge-star operator (i.e. rotation by $\pi/2$ on the surface $\partial M$).
We recall the definition of the exponential map. Let $(X,g_{0})$ be a
geodesically complete manifold. For any $x\in X$ and $V\in T_{x}X$ there
exists a unique geodesic $\gamma_{g_{0}}(t)=\gamma_{g_{0}}(t;V)$, defined on
$[0,1]$, such that $\gamma_{g_{0}}(0)=x$, $\gamma_{g}^{\prime}(0)=V$. The
exponential map based at $x$ is then a map taking $T_{x}X\to X$ defined by
$\exp_{x;g_{0}}:V\mapsto\gamma_{g_{0}}(1;V).$
Observe that when $M$ is a Euclidean ball the singular term involving the
second fundamental forms vanishes due to homogeneity and therefore does not
show up. This is consistent with the explicit formula derived in [3].
An explicit formula for the regular part is only possible in special
geometries such as the one considered in [3]. However, our approach in
arriving at (1.1) also provides a way to numerically compute $R(x,y)$ via a
Fredholm integral equation. See Remark 3.3.
We will use the formula in Proposition 1.1 to derive the mean first arrival
time of a Brownian particle on a Riemannian manifold with a single absorbing
window which is a small geodesic ellipse. As mentioned earlier, our method
extends to multiple windows but we present the single window case to simplify
notations. We first state the result when the window is a geodesic disk of the
boundary $\partial M$ around a fixed point since the statement is cleaner:
###### Theorem 1.2.
Let $(M,g,\partial M)$ be a smooth Riemannian manifold of dimension three with
boundary and let $|M|$ be its volume.
i) Fix $x^{*}\in\partial M$ and let $\Gamma_{\epsilon}$ be a boundary geodesic
ball centered at $x^{*}$ of geodesic radius $\epsilon>0$. For each
$x\notin\Gamma_{\epsilon}$,
$\mathbb{E}[\tau_{\Gamma_{\epsilon}}|X_{0}=x]=F(x)+C_{\epsilon}-|M|G(x,x^{*})+r_{\epsilon}(x),$
with $\|r_{\epsilon}\|_{C^{k}(K)}\leq C_{k,K}\epsilon$ for any integer $k$ and
compact set $K\subset\overline{M}$ which does not contain $x^{*}$. The
function $F$ is the unique solution to the boundary value problem
$\Delta_{g}F=-1,\ \ \partial_{\nu}F=-|M|/|\partial M|,\ \ \int_{\partial
M}F=0.$
The constant $C_{\epsilon}$ is, modulo an error of $O(\epsilon\log\epsilon)$,
given by
$\displaystyle
C_{\epsilon}=\frac{|M|}{4\epsilon}-\frac{1}{4\pi}H(x^{*})|M|\log\epsilon+R(x^{*},x^{*})|M|-F(x^{*})-\frac{|M|H(x^{*})}{4\pi}\left(2\log
2-\frac{3}{2}\right),$
where $R(x^{*},x^{*})$ is the evaluation at $(x,y)=(x^{*},x^{*})$ of the
kernel $R(x,y)$ in (1.1).
ii) One has that the integral of
$\mathbb{E}[\tau_{\Gamma_{\epsilon,a}}|X_{0}=x]$ over $M$ satisfies
$\int_{M}\mathbb{E}[\tau_{\Gamma_{\epsilon,a}}|X_{0}=x]{\rm
dvol}_{g}(x)=\int_{M}F(x){\rm
dvol}_{g}(x)+C_{\epsilon}|M|-F(x^{*})|M|+O(\epsilon).$
Theorem 1.2 does not realize the full power of Proposition 1.1 as it does not
see the non-homogeneity of the local geometry at $x^{*}$ (only the mean
curvature $H(x^{*})$ shows up). This is due to the fact that we are looking at
windows which are geodesic balls. If we replace geodesic balls with geodesic
ellipses, we see that the second fundamental form term in (1.1) contributes to
a term in $\mathbb{E}[\tau_{\Gamma_{\epsilon,a}}|X_{0}=x]$ which is the
difference of principal curvatures.
To this end let $E_{1}(x^{*}),E_{2}(x^{*})\in T_{x^{*}}\partial M$ be the unit
eigenvectors of the shape operator at $x^{*}$ corresponding respectively to
the principal curvatures $\lambda_{1}(x^{*}),\ \lambda_{2}(x^{*})$. For $1\geq
a>0$ fixed, let
$\Gamma_{\epsilon,a}:=\\{\exp_{x^{*};h}(\epsilon t_{1}E_{1}(x^{*})+\epsilon
t_{2}E_{2}(x^{*}))\mid t_{1}^{2}+a^{-2}t_{2}^{2}\leq 1\\}$
be a small geodesic ellipse.
###### Theorem 1.3.
Let $(M,g,\partial M)$ be a smooth Riemannian manifold of dimension three with
boundary.
i) For each $x\in M\backslash\Gamma_{\epsilon,a}$,
$\mathbb{E}[\tau_{\Gamma_{\epsilon,a}}|X_{0}=x]=F(x)+C_{\epsilon,a}-|M|G(x,x^{*})+r_{\epsilon}(x)$
with $\|r_{\epsilon}\|_{C^{k}(K)}\leq C_{k,K}\epsilon$ for any integer $k$ and
compact set $K\subset\overline{M}$ which does not contain $x^{*}$. The
function $F$ is the unique solution to the boundary value problem
$\Delta_{g}F=-1,\ \ \partial_{\nu}F=-|M|/|\partial M|,\ \ \int_{\partial
M}F=0.$
The constant $C_{\epsilon,a}$ is given by
(1.3) $\displaystyle C_{\epsilon,a}=$
$\displaystyle\frac{|M|K_{a}}{4a\epsilon\pi^{2}}-\frac{1}{4\pi}H(x^{*})|M|\log\epsilon+aR(x^{*},x^{*})|M|-F(x^{*})$
$\displaystyle-\frac{|M|H(x^{*})}{16\pi^{3}}\int_{\mathbb{D}}\frac{1}{(1-|s^{\prime}|^{2})^{1/2}}\int_{\mathbb{D}}\frac{\log\left((t_{1}-s_{1})^{2}+a^{2}(t_{2}-s_{2})^{2}\right)^{1/2}}{(1-|t^{\prime}|^{2})^{1/2}}dt^{\prime}ds^{\prime}$
$\displaystyle+\frac{|M|(\lambda_{1}-\lambda_{2})}{64\pi^{3}}\int_{\mathbb{D}}\frac{1}{(1-|s^{\prime}|^{2})^{1/2}}\int_{\mathbb{D}}\frac{(t_{1}-s_{1})^{2}-a^{2}(t_{2}-s_{2})^{2}}{(t_{1}-s_{1})^{2}+a^{2}(t_{2}-s_{2})^{2}}\frac{1}{(1-|t^{\prime}|^{2})^{1/2}}dt^{\prime}ds^{\prime}$
$\displaystyle+O(\epsilon\log\epsilon),$
where
$K_{a}=\frac{\pi}{2}\int_{0}^{2\pi}{\left(\cos^{2}\theta+\frac{\sin^{2}\theta}{a^{2}}\right)^{-1/2}}d\theta$
and $\mathbb{D}$ is the two dimensional unit disk centered at the origin.
ii) One has that the integral of
$\mathbb{E}[\tau_{\Gamma_{\epsilon,a}}|X_{0}=x]$ over $M$ satisfies
$\int_{M}\mathbb{E}[\tau_{\Gamma_{\epsilon,a}}|X_{0}=x]{\rm
dvol}_{g}(x)=\int_{M}F(x){\rm
dvol}_{g}+C_{\epsilon,a}|M|-F(x^{*})|M|+O(\epsilon).$
Note that while the dependence on the eccentricity of the ellipse is hidden in
the integrals, the dependence on the difference of the principal curvatures
$(\lambda_{1}-\lambda_{2})$ is easy to see in this formula. The integral which
multiplies $(\lambda_{1}-\lambda_{2})$ turns out to vanish when $a=1$ which
makes the above result consistent with Theorem 1.2.
The fact that our result is valid on general Riemannian three manifolds allows
for the incorporation of spatial heterogeneity such as anisotropic diffusion.
In contrast to [30], the fact that we are able to obtain explicitly an
expression for the $O(1)$ term in (1.1) is due to the fact that in Proposition
1.1 we have the expansion of $G_{\partial M}(x,z)$ all the way to a remainder
$R(x,y)$, which is Hölder continuous at the diagonal. We also appeal to some
recent advances in integral geometry [29, 20, 23, 22, 12] to address the
comment in [1] on the difficulty of treating this problem in higher
dimensions.
The strategy and organization of this paper will be as follows. In Section 2
we will give a brief overview of pseudodifferential operators and their
associated Schwartz kernels. The machinery of pseudodifferential operators
serve as a bridge between the geometric and analytic objects appearing in
Proposition 1.1 and we will compute their coordinate expression. In Section 3
we will use the tools we developed in Section 2 to prove Propostion 1.1. A
singularity expansion for the Green’s function such as Proposition 1.1 is the
gateway for obtaining the asymptotic expansions of Theorems 1.2 and 1.3.
However, there is an additional hurdle of inverting an integral transform as
mentioned in [1]. Here we make use of some recent advancements in integral
geometry and geometric rigidity [23, 12, 20] to overcome these difficulties.
This approach is described in Section 4. Finally, in Section 5 we carry out
the asymptotic calculation using the tools we have developed. The appendices
characterizes the expected first arrival time
$\mathbb{E}[\tau_{\Gamma_{\epsilon,a}}|X_{0}=x]$ as the solution of an
elliptic mixed boundary value problem. This is classical in the Euclidean case
(see [26]) but we could not find a reference for the general case of a
Riemannian manifold with boundary.
## 2\. Overview of Pseudodifferential Operators
### 2.1. Basic Definitions
We give some basic definitions and properties of pseudodifferential operators.
For a comprehensive treatment we refer the reader to Chapt 7 of [34] or the
book [32]. Readers who are already familiar with microlocal analysis can skip
this section.
As usual, $C^{\infty}$ denotes the space of smooth functions. We use notation
$C_{c}^{\infty}$ for compactly suported smooth functions and $D^{\prime}$ for
its dual. By $C^{k}$, we denote the space of $k$ time continuously
differentiable functions. The spce of functions from $C^{k}$, whose $k$th
derivatives are Hölder continuous with exponent $\mu\in(0,1]$, is denoted by
$C^{k,\mu}$
Let $a(x,\xi)$ be a smooth function on $T^{*}\mathbb{R}^{n}$ and for all
$l\in\mathbb{R}$ we say that $a\in S^{l}_{1}(T^{*}\mathbb{R}^{n})$ (or simply
$S^{l}_{1}$) if for all multi-indices $\alpha,\beta$ there are constants
$C_{\alpha,\beta}$ such that
(2.1) $\displaystyle|D_{\xi}^{\alpha}D_{x}^{\beta}a(x,\xi)|\leq
C_{\alpha,\beta}\langle\xi\rangle^{l-|\alpha|}$
where $D_{\xi}^{\alpha}=(-i)^{|\alpha|}\partial_{\xi}^{\alpha}$,
$D_{x}^{\beta}=(-i)^{|\beta|}\partial_{x}^{\beta}$ , and
$\langle\xi\rangle:=(1+|\xi|^{2})^{1/2}$. These are the Kohn-Nirenberg
symbols. This class of symbols contain the classical symbols, denoted by
$S_{cl}^{l}(T^{*}\mathbb{R}^{n})$, which are defined by those $a(x,\xi)\in
S^{l}_{1}(T^{*}\mathbb{R}^{n})$ satisfying
(2.2) $\displaystyle a(x,\xi)\sim\sum_{m=0}^{\infty}a_{l-m}(x,\xi),$
where each $a_{l-m}$ are homogeneous in the sense that
$a_{l-m}(x,\tau\xi)=\tau^{l-m}a(x,\xi)$ for all $x\in\mathbb{R}^{n}$, $\tau>1$
and $|\xi|>1$. The expression (2.2) means that for all $N$,
$a(x,\xi)-\sum_{m=0}^{N}a_{l-m}(x,\xi)\in S^{l-N-1}_{1}(T^{*}\mathbb{R}^{n}).$
If $a(x,\xi)\in S^{l}_{1}$ we can define an operator
$a(x,D):C^{\infty}_{c}(\mathbb{R}^{n})\to{\mathcal{D}}^{\prime}(\mathbb{R}^{n})$
by
(2.3) $\displaystyle a(x,D)u:=\int_{\mathbb{R}^{n}}e^{i\xi\cdot
x}a(x,\xi)\hat{u}(\xi)d\xi,$
where
$\hat{u}(\xi):={\mathcal{F}}u:=(2\pi)^{n}\int_{\mathbb{R}^{n}}e^{-ix\cdot\xi}u(x)dx$
is the Fourier transform. Recall that the absolutely convergent integral
representation of the Fourier transform is well defined as an automorphism of
the Schwartz class functions $S(\mathbb{R}^{n})$ but extends to an
automorphism of the tempered distributions $S^{\prime}(\mathbb{R}^{n})$. (See
[4] for a comprehensive guide to distribution theory and definition of these
spaces).
Operators taking
$C^{\infty}_{c}(\mathbb{R}^{n})\to{\mathcal{D}}^{\prime}(\mathbb{R}^{n})$
which have the above representation are said to be in
$\Psi^{l}_{1}(\mathbb{R}^{n})$ and are called pseudodifferential operators.
For the symbol class $S_{1}^{l}(T^{*}\mathbb{R}^{n})$, Lemma 1.1 in Chapter
7.1 of [34] extends $a(x,D)$ to map $S^{\prime}(\mathbb{R}^{n})\to
S^{\prime}(\mathbb{R}^{n})$.
The classical pseudodifferential operators $\Psi^{l}_{cl}(\mathbb{R}^{n})$ are
defined analogously by requiring that $a(x,\xi)$ belongs to $S^{l}_{cl}$. Note
that knowing the operator $a(x,D)\in\Psi^{l}_{1}(\mathbb{R}^{n})$ we can
recover $a(x,\xi)\in S^{l}_{1}$ by the formula
(2.4) $\displaystyle a(x,\xi)=e_{-\xi}(x)a(x,D)e_{\xi},$
where $e_{\xi}(x):=e^{i\xi\cdot x}$. Note that if $A(x,y)$ is the Schwartz
kernel of the operator $a(x,D)$ then
(2.5) $\displaystyle
a(0,\xi)={\mathcal{F}}_{y}^{-1}(A(0,y))(\xi)=\int_{\mathbb{R}^{n}}e^{i\xi\cdot
y}A(0,y)dy.$
Let $X$ be a compact manifold without boundary. An operator
$\mathcal{A}:C^{\infty}(X)\to{\mathcal{D}}^{\prime}(X)$111We shall avoid a
discussion of distributional sections of density bundles by noting that in our
setting of $X$ is always prescribed with a Riemannian volume form which
provides a natural trivialization of density bundles. As such all
distributional sections of density bundles are identified with sections of the
trivial line bundle in this way. is said to be in $\Psi^{l}_{1}(X)$ if there
exists coordinate covers $\\{(O_{j},\Phi_{j})\mid\Phi_{j}:O_{j}\to
U_{j}\subset\mathbb{R}^{n}\\}$ and a partition of unity $\\{\chi_{j}\\}$
subordinate to $\\{O_{j}\\}$ such that the map
(2.6) $\displaystyle
u\mapsto\left(\chi_{k}\mathcal{A}\chi_{j}\Phi_{j}^{*}u\right)\circ\Phi_{k}^{-1}$
from $C^{\infty}(U_{j})\to{\mathcal{E}}^{\prime}(U_{k})$ belongs to
$\Psi^{l}_{1}(\mathbb{R}^{n})$.
If $a\in C^{\infty}(T^{*}X)$ we say that it belongs to the symbol class
$S^{l}_{1}(T^{*}X)$ if
$\chi_{j}\circ\Phi_{j}^{-1}a(\Phi_{j}^{-1}(\cdot),\Phi_{j}^{*}\cdot)\in
S^{l}_{1}(T^{*}\mathbb{R}^{n})$
for all $j$. The classical pseudodifferential operators $\Psi^{l}_{cl}(X)$ and
classical symbols $S^{l}_{cl}(T^{*}X)$ are defined analogously. These
definitions depend a-priori on the choice of coordinate systems but turn out
to be invariant (see Chapt 7 [34]).
There exists a linear isomorphism
(2.7) $\displaystyle\sigma_{l}:\Psi^{l}_{cl}(X)/\Psi^{l-1}_{cl}(X)\to
S^{l}_{cl}(T^{*}X)/S^{l-1}_{cl}(T^{*}X)$
called the principal symbol map. For each $\mathcal{A}\in\Psi^{l}_{cl}(X)$ it
can be defined at each $x\in X$ by taking a coordinate neighborhood $O$
containing $x$ and a $\chi\in C^{\infty}_{c}(O)$ which is identically $1$ near
$x$ then considering the operator given in (2.6) for $\chi_{j}=\chi_{k}=\chi$
and $\Phi_{j}=\Phi_{k}=\Phi$. As the resulting operator in (2.6) is in
$\Psi^{l}_{cl}(\mathbb{R}^{n})$ with symbol $a\in
S^{l}_{cl}(T^{*}\mathbb{R}^{n})$, we may set
$\sigma_{l}(\mathcal{A})(x,\Phi^{*}\xi):=a_{l}(\Phi(x),\xi)$
for all $x\in X$ and $\xi\in T^{*}_{\Phi_{j}(x)}\mathbb{R}^{n}$. This
definition depends a-priori on the choice of coordinate systems but turns out
to be invariant (see Chapt 5 of [32]). In practice these computations are
often done in normal coordinates centered at the point of interest $x\in X$
then computing the inverse Fourier transform as in (2.5).
One important property of $\sigma_{l}$ which we will use is that it respects
the product structure of $\Psi^{l}_{cl}$ and $S^{l}_{cl}$:
(2.8)
$\displaystyle\sigma_{l}(\mathcal{A})\sigma_{m}({\mathcal{B}})=\sigma_{l+m}(\mathcal{A}{\mathcal{B}})$
for $\mathcal{A}\in\Psi^{l}_{cl}(X)$ and ${\mathcal{B}}\in\Psi^{m}_{cl}(X)$.
### 2.2. Coordinate Calculations
We make some calculations for some geometric objects which will naturally
appear in the singularity expansion for $G_{\partial M}$. These identities
will be useful in proving Proposition 1.1.
Let $(M,g,\partial M)$ be a three dimensional Riemannian manifold with non-
empty smooth boundary which inherits the metric $h:=\iota_{\partial M}^{*}g$.
Denote by ${\rm{II}}$ the scalar second fundamental form on the surface
$\partial M$ and $H(x)$ be the mean curvature at $x\in\partial M$. Let
$S\partial M$ denote the unit-sphere bundle over $\partial M$,
$S\partial M=\\{v\in T\partial M\mid\|v\|_{h}=1\\}.$
For any $x\in\partial M$, let $E_{1}(x),E_{2}(x)\in S_{x}\partial M$ be
principal directions (i.e. unit eigenvectors) of the induced shape operator
with eigenvalues $\lambda_{1}(x)$ and $\lambda_{2}(x)$. We will drop the
dependence in $x$ from our notation when there is no ambiguity.
We choose $E_{1}$ and $E_{2}$ such that $E_{1}^{\flat}\wedge
E_{2}^{\flat}\wedge\nu^{\flat}$ is a positive multiple of the volume form
${\rm dvol}_{g}$ (see p.26 of [16] for the “musical isomorphism” notation of ♭
and ♯). Here we use $\nu$ to denote the outward pointing normal vector field
so that it is consistent with most PDE literature. However, in defining
${\rm{II}}$ and the shape operator we will follow geometry literature
(e.g.[16]) and use the inward pointing normal so that the sphere embedded in
$\mathbb{R}^{3}$ would have positive mean curvature in our convention.
For two points $x,y\in\partial M$ there are two distances to consider. The
first is the shortest path amongst those that stay on the boundary which we
denote by $d_{h}(x,y)$ and the other is the distance measured by paths
allowing to enter $M$, which we denote by $d_{g}(x,y)$. Clearly,
$d_{h}(x,y)\geq d_{g}(x,y)$.
For a fixed $x_{0}\in\partial M$, we will denote by
$B_{h}(\rho;x_{0})\subset\partial M$ the geodesic disk of radius $\rho>0$
(with respect to the metric $h$) centered at $x_{0}$ and $\mathbb{D}_{\rho}$
to be the Euclidean disk in $\mathbb{R}^{2}$ of radius $\rho$ centered at the
origin. In what follows $\rho$ will always be smaller than the injectivity
radius of $(\partial M,h)$. Letting $t=(t_{1},t_{2},t_{3})\in\mathbb{R}^{3}$,
we will construct a coordinate system $x(t;x_{0})$ by the following procedure:
Write $t\in\mathbb{R}^{3}$ near the origin as $t=(t^{\prime},t_{3})$ for
$t^{\prime}=(t_{1},t_{2})\in\mathbb{D}_{\rho}$. Define first 222for example it
is obvious below that $E_{1}$ and $E_{2}$ are elements of the tangent space
over $x_{0}$ as they are inserted into the argument of
$\exp_{x_{0};h}(\cdot)$.
$x((t^{\prime},0);x_{0}):={\rm{exp}}_{x_{0};h}(t_{1}E_{1}+t_{2}E_{2}),$
where ${\rm{exp}}_{x_{0};h}(V)$ denotes the time $1$ map of $h$-geodesics with
initial point $x_{0}$ and initial velocity $V\in T_{x_{0}}\partial M$. The
coordinate $t^{\prime}\in\mathbb{D}_{\rho}\mapsto x((t^{\prime},0);x_{0})$ is
then an $h$-geodesic coordinate system for a neighborhood of $x_{0}$ on the
boundary surface $\partial M$. We can extend this to become a coordinate
system for points in $M$ near $x_{0}$ so that $t\mapsto x(t;x_{0})$ is a
boundary normal coordinate system with $t_{3}>0$ in $M$ as the boundary
defining function. Readers wishing to know more about boundary normal
coordinates can refer to [17] for a brief recollection of the basic properties
we use here and Prop 5.26 of [16] for a detailed construction.
For convenience we will write $x(t^{\prime};x_{0})$ in place of
$x((t^{\prime},0);x_{0})$. The boundary coordinate system $t\mapsto
x(t;x_{0})$ has the advantage that the metric tensor $g$ can be expressed as
(2.9)
$\displaystyle\sum_{j,k=1}^{3}g_{j,k}(t)dt_{j}dt_{k}=\sum_{\alpha,\beta=1}^{2}h_{\alpha,\beta}(t^{\prime},t_{3})dt_{\alpha}dt_{\beta}+dt_{3}^{2},$
where $h_{\alpha,\beta}(t^{\prime},0)=h_{\alpha,\beta}(t^{\prime})$ is the
expression of the boundary metric $h$ in the $h$-geodesic coordinate system
$x(t^{\prime};x_{0})$. Note that $(g_{j,k}(t))_{j,k=1}^{3}$ and
$(h_{\alpha,\beta}(t^{\prime},t_{3}))_{\alpha,\beta=1}^{2}$ are symmetric
positive definite $3\times 3$ and $2\times 2$ matrices varying smoothly with
respect to the variable $t=(t^{\prime},t_{3})=(t_{1},t_{2},t_{3})$.
For $\epsilon>0$ sufficiently small we define the (rescaled) $h$-geodesic
coordinate by the following map
$x^{\epsilon}(\cdot;x_{0}):t^{\prime}=(t_{1},t_{2})\in\mathbb{D}\mapsto
x(\epsilon t^{\prime};x_{0})\in B_{h}(\epsilon;x_{0}),$
where $\mathbb{D}$ is the unit disk in $\mathbb{R}^{2}$. We derive some
coordinate expressions for some of the geometric objects we will consider
later.
###### Lemma 2.1.
Let
$h(s^{\prime})=\sum_{\alpha,\beta=1}^{2}h_{\alpha,\beta}(s^{\prime})ds_{\alpha}ds_{\beta}$
be the pullback metric of $h$ by $s^{\prime}\mapsto x(s^{\prime};x_{0})$ on
$\mathbb{D}_{\rho}$. Denote by
$r=|s^{\prime}-t^{\prime}|_{h(s^{\prime})}:=\left(\sum_{\alpha,\beta=1}^{2}h_{\alpha,\beta}(s^{\prime})(s_{\alpha}-t_{\alpha})(s_{\beta}-t_{\beta})\right)^{1/2}$
so that $t^{\prime}=s^{\prime}+r\omega$ where $\omega\in
S_{s^{\prime}}\mathbb{D}_{\rho}$. We have that
$d_{h}(x(s^{\prime};x_{0}),x(t^{\prime};x_{0}))^{2}=r^{2}\sum\limits_{j,k=1}^{2}H_{j,k}(s^{\prime},r,\omega)\omega_{j}\omega_{k}$
for matrix $H_{j,k}(s^{\prime},r,\omega)$ jointly smooth in
$(s^{\prime},r,\omega)$. It also satisfies
$H_{j,k}(s^{\prime},0,\omega)=h_{j,k}(s^{\prime})$ and
$\sum_{j,k}\partial_{r}H_{j,k}(s^{\prime},0,\omega)\omega_{k}\omega_{j}=O(s^{\prime}).$
###### Proof.
Expressing $t^{\prime}$ using $(s^{\prime},r,\omega)$ we have that (see e.g.
[35] Lemma 4.8)
$d_{h}(x(s^{\prime};x_{0}),x(t^{\prime};x_{0}))=r\left(\sum_{\alpha,\beta}H_{\alpha,\beta}(s^{\prime},r,\omega)\omega_{\alpha}\omega_{\beta}\right)^{1/2}$
with $H_{\alpha,\beta}$ symmetric, even under the map
$(r,\omega)\mapsto(-r,-\omega)$, and
$H_{\alpha,\beta}(s^{\prime},0,\omega)=h_{\alpha,\beta}(s^{\prime})$. Setting
$s^{\prime}=0$ and using the fact that we are using normal coordinates we
obtain
$r^{2}=r^{2}\left(\sum_{\alpha,\beta}H_{\alpha,\beta}(0,r,\omega)\omega_{\alpha}\omega_{\beta}\right)$.
Now Taylor expanding $H_{\alpha,\beta}(0,r,\omega)$ in $r$, we see that
$r^{2}=r^{2}\left(1+\sum_{\alpha,\beta}\partial_{r}H_{\alpha,\beta}(0,0,\omega)r\omega_{\alpha}\omega_{\beta}+O(r^{2})\right).$
The $r^{2}$ terms on left-hand and right-hand sides cancel, leaving
$0=\partial_{r}H_{\alpha,\beta}(0,0,\omega)r^{3}\omega_{\alpha}\omega_{\beta}+O(r^{4}).$
Divide through by $r^{3}$ and take the limit as $r\to 0$ we see that
$\sum_{\alpha,\beta}\partial_{r}H_{\alpha,\beta}(0,0,\omega)\omega_{\alpha}\omega_{\beta}=0$
for any $\omega\in S_{0}\mathbb{D}$. ∎
###### Lemma 2.2.
We use the same notation for $\omega$ and $r$ as in Lemma 2.1. One has that
$\displaystyle
d_{h}(x(s^{\prime};x_{0}),x(t^{\prime};x_{0}))^{-1}=r^{-1}+O(s^{\prime})+O(r),$
where $O(r)$ (respectively $O(s^{\prime})$) denotes smooth functions of
$(s^{\prime},r,\omega)\in\mathbb{D}_{\rho}\times\mathbb{R}\times
S_{s^{\prime}}\mathbb{D}_{\rho}$ which vanish to first order as $r\to 0$
(respectively $s^{\prime}\to 0$).
###### Proof.
From Lemma 2.1 we have that
$d_{h}(x(s^{\prime};x_{0}),x(t^{\prime};x_{0}))^{-1}=r^{-1}\left(\sum_{\alpha,\beta}h_{\alpha,\beta}(s^{\prime})\omega_{\alpha}\omega_{\beta}+r\partial_{r}H_{\alpha,\beta}(s^{\prime},0,\omega)\omega_{\alpha}\omega_{\beta}+O(r^{2})\right)^{-1/2}.$
Using the fact that $\omega\in S_{s^{\prime}}\mathbb{D}$ with respect to the
metric given by $h_{\alpha,\beta}$ we have that
$d_{h}(x(s^{\prime};x_{0}),x(t^{\prime};x_{0}))^{-1}=r^{-1}\left(\sum_{\alpha,\beta}1+r\partial_{r}H_{\alpha,\beta}(s^{\prime},0,\omega)\omega_{\alpha}\omega_{\beta}+O(r^{2})\right)^{-1/2}.$
For $r$ and $s^{\prime}$ sufficiently small we may use Taylor’s expansion to
obtain the desired property. The fact that
$\sum_{\alpha,\beta}\partial_{r}H_{\alpha,\beta}(s^{\prime},0,\omega)\omega_{\alpha}\omega_{\beta}=O(s^{\prime})$
is stated in Lemma 2.1 ∎
###### Corollary 2.3.
For $\epsilon>0$ sufficiently small we have that
$\displaystyle
d_{h}(x^{\epsilon}(s^{\prime};x_{0}),x^{\epsilon}(t^{\prime};x_{0}))^{-1}=\epsilon^{-1}r^{-1}+\epsilon
r^{-1}A(\epsilon,s^{\prime},r,\omega)$
for some smooth function $A$ in the variables
$(\epsilon,s^{\prime},r,\omega)\in[0,\epsilon_{0}]\times\mathbb{D}\times\mathbb{R}\times
S^{1}$. Here we use $r=|s^{\prime}-t^{\prime}|$ and $t^{\prime}=s+r\omega$.
###### Lemma 2.4.
Let $x(\cdot;x_{0})$ be the coordinate system at the beginning of this section
centered at $x_{0}$. For $s^{\prime},t^{\prime}\in\mathbb{R}^{2}$ sufficiently
small, we have that
$d_{g}(x(s^{\prime};x_{0}),x(t^{\prime};x_{0}))^{2}=r^{2}\left(1+r{\bf\tilde{G}}(s^{\prime},\omega)+O(r^{2})\right),$
where $r=|s^{\prime}-t^{\prime}|_{h(s^{\prime})}$ and
$t^{\prime}=s^{\prime}+r\omega$. Here ${\bf\tilde{G}}(s^{\prime},\omega)$ is a
smooth function of $(s^{\prime},\omega)$ which vanishes at $s^{\prime}=0$. The
$O(r^{2})$ term is a smooth function in
$(s^{\prime},r,\omega)\in\mathbb{D}_{\rho}\times\mathbb{R}\times
S_{s^{\prime}}\mathbb{D}_{\rho}$ which vanishes to second order as $r\to 0$.
###### Proof.
We begin with the identity that for any $s$ and $t$,
$d_{g}(x(s;x_{0}),x(t;x_{0}))^{2}=\sum_{j,k=1}^{3}{\bf
G}_{j,k}(s,t)(s_{j}-t_{j})(s_{k}-t_{k}),$
with ${\bf G}_{j,k}(s,s)=g_{j,k}(s)$ given by (2.9). Now set
$s=(s^{\prime},0)$ and $t=(t^{\prime},0)$ we have
$d_{g}(x(s^{\prime};x_{0}),x(t^{\prime};x_{0}))^{2}=r^{2}\left(1+\sum_{j,k=1}^{2}\partial_{r}{\bf
G}^{0}_{j,k}(s^{\prime},0,\omega)r\omega_{j}\omega_{k}+O(r^{2})\right)$
where ${\bf G}^{0}_{j,k}(s^{\prime},r,\omega):={\bf
G}_{j,k}(s^{\prime},s^{\prime}+r\omega),$ is even in
$(r,\omega)\mapsto-(r,\omega)$ and $O(r^{2})$ is a smooth function of
$(s^{\prime},r,\omega)$ which is even and vanishes to second order as $r\to
0$. Observe that $\omega\mapsto\partial_{r}{\bf
G}^{0}_{j,k}(s^{\prime},0,\omega)$ is odd in $\omega$.
We now need to argue that $\sum_{\alpha,\beta}\partial_{r}{\bf
G}^{0}_{j,k}(0,0,\omega)\omega_{\alpha}\omega_{\beta}=0$. Setting
$s^{\prime}=0$ in the above identity and using the fact that we are using
boundary normal coordinates we have
$r^{2}=d_{h}(x(0;x_{0}),x(t^{\prime};x_{0}))^{2}\geq
d_{g}(x(0;x_{0}),x(t^{\prime};x_{0}))^{2}=r^{2}\left(1+\sum_{j,k=1}^{2}\partial_{r}{\bf
G}^{0}_{j,k}(0,0,\omega)r\omega_{j}\omega_{k}+O(r^{2})\right).$
Subtracting off the $r^{2}$ terms and dividing by $r^{3}$ we see that as $r\to
0$,
$\sum_{j,k=1}^{2}\partial_{r}{\bf
G}^{0}_{j,k}(0,0,\omega)\omega_{j}\omega_{k}\leq 0$
for all $\omega\in S^{1}$. We now use the fact that $\partial_{r}{\bf
G}_{j,k}(0,0,\omega)$ is odd to see that
$\sum_{j,k=1}^{2}\partial_{r}{\bf
G}^{0}_{j,k}(0,0,\omega)\omega_{j}\omega_{k}=0.$
∎
Just as how Lemma 2.2 and Corollary 2.3 followed from Lemma 2.1, we have the
following:
###### Corollary 2.5.
Using the expression $r=|s^{\prime}-t^{\prime}|_{h(s^{\prime})}$ and
$t^{\prime}=s^{\prime}+r\omega$, we have that
$\displaystyle
d_{g}(x(s^{\prime};x_{0}),x(t^{\prime};x_{0}))^{-1}=r^{-1}+O(s^{\prime})+O(r),$
where $O(r)$ (respectively $O(s^{\prime})$) denotes smooth function of
$(s^{\prime},r,\omega)\in\mathbb{D}_{\rho}\times\mathbb{R}\times
S_{s^{\prime}}\mathbb{D}_{\rho}$ which vanishes to first order as $r\to 0$
(respectively $s^{\prime}\to 0$).
###### Corollary 2.6.
For $\epsilon>0$ sufficiently small we have that
$\displaystyle
d_{g}(x^{\epsilon}(s^{\prime};x_{0}),x^{\epsilon}(t^{\prime};x_{0}))^{-1}=\epsilon^{-1}r^{-1}+\epsilon
r^{-1}A(\epsilon,s^{\prime},r,\omega)$
for some smooth function $A$ in the variables
$(\epsilon,s^{\prime},r,\omega)\in[0,\epsilon_{0}]\times\mathbb{D}\times\mathbb{R}\times
S^{1}$. Here we use $r=|s^{\prime}-t^{\prime}|$ and $t^{\prime}=s+r\omega$.
###### Lemma 2.7.
In the coordinate given by $y=x(s^{\prime};x_{0})$,
$\partial_{\nu_{y}}d_{g}(x_{0},y)=\frac{\lambda_{1}(x_{0})s_{1}^{2}+\lambda_{2}(x_{0})s_{2}^{2}}{2|s^{\prime}|}+O(|s^{\prime}|^{2})$
where $O(|s^{\prime}|^{2})$ denotes a smooth function of the variables
$(|s^{\prime}|,\frac{s^{\prime}}{|s^{\prime}|})$ which vanishes to order 2 at
the origin.
###### Proof.
By Gauss Lemma, for $y$ near $x_{0}$, ${\rm
grad}_{y}d_{g}(x_{0},y)=\dot{\gamma}(d_{g}(x_{0},y))$ where $\gamma(\cdot)$ is
the unit speed geodesic in $(M,g)$ from $x_{0}$ to $y$. Therefore we have that
(2.10)
$\displaystyle\partial_{\nu_{y}}d_{g}(x_{0},y)=-\left\langle\nu_{y},\frac{{\rm{exp}}_{y;g}^{-1}(x_{0})}{|{\rm{exp}}_{y;g}^{-1}(x_{0})|_{g}}\right\rangle_{g}.$
In the coordinates given by $s\mapsto x(s;x_{0})$ the Christoffel symbols are
(2.11)
$\displaystyle\Gamma_{3,3}^{3}=\Gamma_{\alpha,3}^{3}=\Gamma_{3,\alpha}^{3}=0,\
\ \Gamma_{\alpha,\beta}^{3}=-\frac{1}{2}\partial_{3}h_{\alpha,\beta}.$
Choose $(\hat{s}^{\prime},0)\in\mathbb{R}^{3}$ so that
$x(\hat{s}^{\prime},0;x_{0})=y$. By Lemma 2.4, $d_{g}(y,x_{0})$ is a smooth
function of $(|\hat{s}^{\prime}|,\frac{\hat{s}^{\prime}}{|\hat{s}^{\prime}|})$
when we write $y=x(\hat{s}^{\prime};x_{0})$ in these coordinates. Let
$V(y):=\frac{{\rm{exp}}_{y;g}^{-1}(x_{0})}{|{\rm{exp}}_{y;g}^{-1}(x_{0})|_{g}}=\sum_{j=1}^{3}V_{j}(\hat{s}^{\prime})\partial_{j}$
be the unique unit vector over $y$ so that the $(M,g)$ unit velocity geodesic
in these coordinates starting at $y$ with initial direction $V(y)$ reaches
$x_{0}$ in time $d_{g}(y,x_{0})$. In the coordinates given by
$s^{\prime}\mapsto x(s^{\prime};x_{0})$, we want to argue that
$V_{j}(\hat{s}^{\prime})$ are smooth functions of
$(|\hat{s}^{\prime}|,\frac{\hat{s}^{\prime}}{|\hat{s}^{\prime}|})$. To do so,
observe that in $g$-geodesic coordinates centered around $x_{0}$ this is of
course the case. The result for any other coordinate system can then be
obtained via a change of variable.
Note that the outward pointing normal is given by $-\partial_{3}$ since
$s_{3}>0$ in $M$. Using (2.10) and the expression for the metric (2.9) we have
that in the chosen boundary normal coordinate system
(2.12) $\displaystyle\partial_{\nu_{y}}d_{g}(x_{0},y)=V_{3}.$
After time $\tau$, the geodesic with initial position $(\hat{s}^{\prime},0)$
and initial unit velocity $V$ can be written in the $s=(s^{\prime},s_{3})$
coordinate as
$s(\tau)=(\hat{s}^{\prime},0)+\tau V+r(\tau;V)$
for some remainder $r=(r_{1},r_{2},r_{3})$ which has initial condition
$r(0)=\dot{r}(0)=0$. Taylor expanding $r(\cdot;V)$ we have that
(2.13) $\displaystyle s(\tau)=(\hat{s}^{\prime},0)+\tau
V+\frac{\tau^{2}}{2}\ddot{r}(0;V)+\tau^{3}r^{\prime}(\tau;V)$
for some $r^{\prime}(\tau;V)$ depending smoothly on $\tau$ and $V$.
Due to (2.12), we are particularly interested in the evolution of the $r_{3}$
component which solves the ODE
$\displaystyle\ddot{r}_{3}(\tau;V)$ $\displaystyle=$
$\displaystyle-\sum_{j,k=1}^{3}\Gamma^{3}_{j,k}(s(\tau))(V_{j}+\dot{r}_{j})(V_{k}+\dot{r}_{k})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\alpha,\beta=1}^{2}\partial_{3}h_{\alpha,\beta}(s(\tau))(V_{\alpha}+\dot{r}_{\alpha})(V_{\beta}+\dot{r}_{\beta}).$
The last equality comes from (2.11).
Now set $\tau=d_{g}(y,x_{0})$ so that $s(\tau)=0$, we have from (2.13) that
for $\alpha=1,2$,
$V_{\alpha}=-\frac{\hat{s}_{\alpha}}{d_{g}(y,x_{0})}+d_{g}(y,x_{0})f_{\alpha}$
for functions $f_{\alpha}$ which is smooth in $V$ and $d_{g}(y,x_{0})$ and
thus smooth functions of the
$(\hat{s}^{\prime},\hat{s}^{\prime}/|\hat{s}^{\prime}|)$ variable. Inserting
this into (2.2) yields
$\frac{d_{g}(x,y)^{2}}{2}\ddot{r}_{3}(0;V)=\frac{1}{4}\sum_{\alpha,\beta}\partial_{3}h_{\alpha,\beta}(0)\hat{s}_{\alpha}\hat{s}_{\beta}+d_{g}(y,x)^{3}f$
for some function $f$ which is smooth in the variable
$(|\hat{s}^{\prime}|,\hat{s}^{\prime}/|\hat{s}^{\prime}|)$. Inserting this
expression into (2.13) for $\tau=d_{g}(x_{0},y)$ we have that
$\displaystyle V_{3}(\hat{s}^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{d_{g}(x,y)}{2}\ddot{r}_{3}(0;V)+d_{g}(x_{0},y)^{2}r^{\prime}_{3}(d_{g}(x_{0},y),V(s^{\prime}))$
$\displaystyle=$
$\displaystyle-\frac{1}{4}\sum_{\alpha,\beta}\frac{\partial_{3}h_{\alpha,\beta}(0)}{d_{g}(x_{0},y)}\hat{s}_{\alpha}\hat{s}_{\beta}+d_{g}(y,x_{0})^{2}(f+r^{\prime})$
Using Lemma 2.4 we have that
$d_{g}(x_{0},y)^{-1}=|\hat{s}^{\prime}|^{-1}+F(\frac{\hat{s}^{\prime}}{|\hat{s}^{\prime}|},s^{\prime})$
for some smooth function $F(\cdot,\cdot)$. We may thus write
$V_{3}(\hat{s}^{\prime})=-\frac{1}{4}\sum_{\alpha,\beta}\frac{\partial_{3}h_{\alpha,\beta}(0)}{|\hat{s}^{\prime}|}\hat{s}_{\alpha}\hat{s}_{\beta}+O(|\hat{s}^{\prime}|^{2}),$
where $O(|\hat{s}^{\prime}|^{2})$ is a smooth function of
$(\frac{\hat{s}^{\prime}}{|\hat{s}^{\prime}|},\hat{s}^{\prime})$ which
vanishes to order $2$ near the origin. Now use the fact that
$-\frac{1}{2}\partial_{3}h_{\alpha,\beta}(0)$ is the coordinate expression for
the scalar second fundamental form with respect to the normal given by
$\partial_{3}$ 333recall that we are using the convention where ${\rm{II}}$
and shape operator are defined with respect to the inward pointing normal at
the point $x_{0}$ (see Proposition 8.17 of [16]) and our coordinate system
$x(s^{\prime};x_{0})$ is chosen so that the shape operator is diagonalized at
$x_{0}$. Therefore,
$V_{3}(\hat{s}^{\prime})=\frac{1}{2}\sum_{\alpha,\beta}\frac{\lambda_{1}\hat{s}_{1}^{2}+\lambda_{2}\hat{s}_{2}^{2}}{|\hat{s}^{\prime}|}+O(|\hat{s}^{\prime}|^{2}).$
In view of (2.12) we have proven the required identity. ∎
Let $x_{0}\in\partial M$ and define
$R_{{\rm{II}}}(\cdot,\cdot),R_{{\rm{II}}*}(\cdot,\cdot)\in
L^{\infty}(B_{h}(\rho;x_{0})B_{h}(\rho;x_{0}))$ by
(2.15) $\displaystyle
R_{{\rm{II}}}(x,y):={\rm{II}}_{x}\left(\frac{{\rm{exp}}_{x;h}^{-1}y}{|{\rm{exp}}_{x;h}^{-1}y|_{h}},\frac{{\rm{exp}}_{x;h}^{-1}y}{|{\rm{exp}}_{x,h}^{-1}y|_{h}}\right),$
$\displaystyle
R_{{\rm{II}}*}(x,y):={\rm{II}}_{x}\left(*\frac{{\rm{exp}}_{x;h}^{-1}y}{|{\rm{exp}}_{x;h}^{-1}y|_{h}},*\frac{{\rm{exp}}_{x;h}^{-1}y}{|{\rm{exp}}_{x,h}^{-1}y|_{h}}\right).$
Here $*$ is the Hodge star operator associated to the metric $h$.
###### Lemma 2.8.
Let $x(\cdot;x_{0}):\mathbb{D}_{\rho}\to\partial M$ be a normal coordinate
system for $(\partial M,h)$ centred around $x_{0}\in\partial M$ and let $h$
denote the pullback metric tensor on on $\mathbb{D}_{\rho}$ under this
coordinate. Then for all $s^{\prime},t^{\prime}\in\mathbb{D}_{\rho}$
sufficiently close to the origin,
$\frac{\exp^{-1}_{t^{\prime};h}(s^{\prime})}{|\exp^{-1}_{t^{\prime};h}(s^{\prime})|_{h(t^{\prime})}}=\sum_{j=1}^{2}\omega_{j}\partial_{j}+O(t^{\prime})+O(r),$
where $\omega:=\frac{s^{\prime}-t^{\prime}}{r}\in S^{1}$ with
$r:=|s^{\prime}-t^{\prime}|$ being the Euclidean distance between $s^{\prime}$
and $t^{\prime}$. The $O(t^{\prime})$ (resp $O(r)$) term denotes a smooth map
of $(t^{\prime},\omega,r)\in\mathbb{D}_{\rho}\times S^{2}\times[0,\rho]$ which
vanishes to order $1$ as $t^{\prime}\to 0$ (resp $r\to 0$).
###### Proof.
This comes from the fact that for some matrix $H_{j,k}(s^{\prime},t^{\prime})$
smooth in $(s^{\prime},t^{\prime})$
$d_{h}(s^{\prime},t^{\prime})^{2}=\sum_{j=1}^{2}H_{j,k}(s^{\prime},t^{\prime})(s_{j}-t_{j})(s_{k}-t_{k}),$
where
$H_{j,k}(t^{\prime},t^{\prime})=h_{j,k}(t^{\prime})=\delta_{j,k}+O(|t^{\prime}|^{2})$.
Therefore
$d_{h(s^{\prime},t^{\prime})}(s^{\prime},t^{\prime})=|s^{\prime}-t^{\prime}|+|s^{\prime}-t^{\prime}|F\left(t^{\prime},\frac{s^{\prime}-t^{\prime}}{|s^{\prime}-t^{\prime}|},|s^{\prime}-t^{\prime}|\right)$
for some smooth function $F\in C^{\infty}(\mathbb{D}_{\rho}\times
S^{1}\times[0,r_{0}])$ which is $O(t^{\prime})+O(r)$.
A coordinate calculation yields that
$\frac{\exp^{-1}_{t^{\prime},h}(s^{\prime})}{|\exp^{-1}_{t^{\prime},h}(s^{\prime})|_{h(t^{\prime})}}={\rm
grad}_{t^{\prime}}d_{h}(s^{\prime},t^{\prime})$ so
$\frac{\exp^{-1}_{t^{\prime},h}(s^{\prime})}{|\exp^{-1}_{t^{\prime},h}(s^{\prime})|_{h(t^{\prime})}}=\sum_{j,k=1}^{2}h_{j,k}(t^{\prime})\partial_{t_{j}}d_{h^{\theta}(s^{\prime},t^{\prime})}(s^{\prime},t^{\prime})\partial_{k}=\sum_{j}\omega_{j}\partial_{j}+O(t^{\prime})+O(r).$
∎
###### Corollary 2.9.
Let $x_{0}\in\partial M$ and $\lambda_{1}(x_{0})$ and $\lambda_{2}(x_{0})$ be
the eigenvalues of the shape operator at $x_{0}$. Then,
i) in the $t^{\prime}\mapsto x(t^{\prime};x_{0})$ coordinate system prescribed
at the beginning of this section,
$R_{\rm{II}}(x(t^{\prime};x_{0}),x(s^{\prime};x_{0}))-\left(\lambda_{1}(x_{0})\frac{(s_{1}-t_{1})^{2}}{|s^{\prime}-t^{\prime}|^{2}}+\lambda_{2}(x_{0})\frac{(s_{2}-t_{2})^{2}}{|s^{\prime}-t^{\prime}|^{2}}\right)=O(r)+O(t^{\prime})$
$R_{{\rm{II}}*}(x(t^{\prime};x_{0}),x(s^{\prime};x_{0}))-\left(\lambda_{2}(x_{0})\frac{(s_{1}-t_{1})^{2}}{|s^{\prime}-t^{\prime}|^{2}}+\lambda_{1}(x_{0})\frac{(s_{2}-t_{2})^{2}}{|s^{\prime}-t^{\prime}|^{2}}\right)=O(r)+O(t^{\prime})$
The $O(t^{\prime})$ (resp $O(r)$) term denotes a smooth function of
$(t^{\prime},\omega,r)\in\mathbb{D}_{\rho}\times S^{2}\times[0,\rho]$ which
vanishes to order $1$ as $t^{\prime}\to 0$ (resp $r\to 0$).
ii)In the $t^{\prime}\mapsto x^{\epsilon}(t^{\prime};x_{0})$ coordinate
systems prescribed at the beginning of this section,
$R_{\rm{II}}(x^{\epsilon}(t^{\prime};x_{0}),x^{\epsilon}(s^{\prime};x_{0}))-\left(\lambda_{1}(x_{0})\frac{(s_{1}-t_{1})^{2}}{|s^{\prime}-t^{\prime}|^{2}}+\lambda_{2}(x_{0})\frac{(s_{2}-t_{2})^{2}}{|s^{\prime}-t^{\prime}|^{2}}\right)=\epsilon
R_{\epsilon}(t^{\prime},\omega,r),$
$R_{{\rm{II}}*}(x^{\epsilon}(t^{\prime};x_{0}),x^{\epsilon}(s^{\prime};x_{0}))-\left(\lambda_{2}(x_{0})\frac{(s_{1}-t_{1})^{2}}{|s^{\prime}-t^{\prime}|^{2}}+\lambda_{1}(x_{0})\frac{(s_{2}-t_{2})^{2}}{|s^{\prime}-t^{\prime}|^{2}}\right)=\epsilon
R_{\epsilon}(t^{\prime},\omega,r),$
where $R_{\epsilon}(t^{\prime},\omega,r)$ is smooth with derivatives of all
orders uniformly bounded in $\epsilon$.
###### Proof.
Since $x^{\epsilon}(t^{\prime};x_{0})=x(\epsilon t^{\prime};x_{0})$ we have
that ii) is a consequence of i). For i), we will only prove this for
$R_{\rm{II}}$ since the statement for $R_{{\rm{II}}*}$ can be obtained via a
rotation.
Recall that the normal coordinate system $t^{\prime}\mapsto
x(t^{\prime};x_{0})$ is chosen so that at $x_{0}$ the coordinate vectors
$\\{\partial_{t_{1}},\partial_{t_{2}}\\}$ pushes forward under $x(\cdot;x_{0}$
to become of eigenvectors of the shape operator. Because of this we have that
the pull-back of ${\rm{II}}_{x}(\cdot,\cdot)$ under this coordinate system is
given by
${\rm{II}}_{x(t^{\prime};x_{0})}=\sum_{j,k=1}^{2}{\rm{II}}_{j,k}dt_{j}dt_{k}$
where ${\rm{II}}_{j,k}=\lambda_{j}(x_{0})\delta_{j,k}+O(t^{\prime})$. Here
$O(t^{\prime})$ denotes a smooth function of $t^{\prime}$ which vanishes at
the origin. Using the expression derived in Lemma 2.8 for the vector
$\frac{{\rm{exp}}_{x;h}^{-1}y}{|{\rm{exp}}_{x;h}^{-1}y|_{h}}$ in the
coordinate given by $x=x(t^{\prime};x_{0})$ and $y=x(s^{\prime};x_{0})$, we
have the desired expression for
$R_{{\rm{II}}}(x(t^{\prime};x_{0}),x(s^{\prime};x_{0}))$. ∎
### 2.3. Operator Estimates
In this section we derive Sobolev estimates for some integral kernels we will
encounter when obtaining the asymptotic expansions of Theorems 1.2 and 1.3. As
these depend on a parameter $\epsilon>0$ and do not immediately fit into the
framework of semiclassical calculus, we need to keep track of the bounds by
hand.
It is useful to take the Fourier transform with respect to only some
variables. Let $u(s^{\prime},t^{\prime})$ be a family of tempered
distributions in $t^{\prime}\in\mathbb{R}^{2}$ depending smoothly on the
parameter $s^{\prime}\in\mathbb{R}^{2}$. That is, it is the distribution
defined by
$\phi\mapsto\int_{\mathbb{R}^{2}}u(s^{\prime},t^{\prime})\phi(t^{\prime})dt^{\prime}$
for all $\phi\in S(\mathbb{R}^{2})$. We denote by
$\mathcal{F}_{t^{\prime}}(u(s^{\prime},t^{\prime}))(\xi^{\prime})$ to be the
Fourier transform with respect to the $t^{\prime}$ variable only.
###### Lemma 2.10.
Let $A(s^{\prime},\omega)$ be a smooth function on
$(s^{\prime},\omega)\in\mathbb{R}^{2}\times S^{1}$. For $j\geq 0$ and
$\xi^{\prime}\neq 0$ we have that for any multi-index $\alpha$,
$D^{\alpha}_{s^{\prime}}\mathcal{F}_{t^{\prime}}\left(A\left(s^{\prime},\frac{t^{\prime}}{|t^{\prime}|}\right)|t^{\prime}|^{j-1}\right)\left(\xi^{\prime}\right)=\mathcal{F}_{t^{\prime}}\left(D^{\alpha}_{s^{\prime}}A\left(s^{\prime},\frac{t^{\prime}}{|t^{\prime}|}\right)|t^{\prime}|^{j-1}\right)\left(\xi^{\prime}\right).$
Furthermore,
$\mathcal{F}_{t^{\prime}}\left(A\left(s^{\prime},\frac{t^{\prime}}{|t^{\prime}|}\right)|t^{\prime}|^{j-1}\right)\left(\xi^{\prime}\right)$
is jointly smooth in
$(s^{\prime},\xi^{\prime})\in\mathbb{R}^{2}\times\mathbb{R}^{2}\backslash\\{0\\}$.
###### Proof.
Let $\chi\in C^{\infty}_{c}(\mathbb{R}^{2})$ be identically $1$ near the
origin. We can write for any positive integer $k$ and $\xi^{\prime}\neq 0$,
$\displaystyle\mathcal{F}_{t^{\prime}}\left(A\left(s^{\prime},\frac{t^{\prime}}{|t^{\prime}|}\right)|t^{\prime}|^{j-1}\right)\left(\xi^{\prime}\right)$
$\displaystyle=$
$\displaystyle\mathcal{F}_{t^{\prime}}\left(A\left(s^{\prime},\frac{t^{\prime}}{|t^{\prime}|}\right)|t^{\prime}|^{j-1}\chi(t^{\prime})\right)\left(\xi^{\prime}\right)$
$\displaystyle+$
$\displaystyle|\xi^{\prime}|^{-2k}\mathcal{F}_{t^{\prime}}\left(\Delta_{t^{\prime}}^{k}\left(A\left(s^{\prime},\frac{t^{\prime}}{|t^{\prime}|}\right)|t^{\prime}|^{j-1}(1-\chi(t^{\prime}))\right)\right)\left(\xi^{\prime}\right).$
The first integral is absolutely convergent by the $\chi(t^{\prime})$ cut-off.
The second Fourier transform is also absolutely convergent owing to the fact
that the integrand is smooth and $\Delta_{t^{\prime}}^{k}$ makes the integrand
decay quickly as $t^{\prime}\to\infty$ provided $k$ is chosen large enough. ∎
###### Lemma 2.11.
Let $A_{\epsilon}(s^{\prime},r,\omega)$ be a family of
$C^{\infty}_{c}(\mathbb{R}^{2}\times\mathbb{R}\times S^{1})$ whose support is
uniformly bounded in $\epsilon\in[0,\epsilon_{0}]$ and whose derivatives are
also uniformly bounded in $\epsilon$. Then for all $l\geq 0$,
$\mathcal{A}_{\epsilon}f:=\int_{\mathbb{R}^{2}}A_{\epsilon}(s^{\prime},r,\omega)r^{l}f(s^{\prime}+r\omega)drd\omega$
is a map bounded uniformly in $\epsilon$ from $H^{m}(\mathbb{R}^{2})\to
H^{m+1+l}(\mathbb{R}^{2})$.
###### Proof.
We prove the estimates only for Schwartz functions $f\in S(\mathbb{R}^{2})$.
We first expand
(2.16) $\displaystyle
A_{\epsilon}=\sum_{j=1}^{N}\partial_{r}^{j}A_{\epsilon}(s^{\prime},0,\omega)r^{j}+r^{N+1}R_{\epsilon}(s^{\prime},r,\omega).$
All terms are smooth in its variables with derivatives uniformly bounded in
$\epsilon$. Estimating the remainder term in (2.16) is easy:
$\mathcal{A}_{R}f:=\int_{S^{1}}\int_{0}^{\infty}r^{N+1+l}R_{\epsilon}(s^{\prime},r,\omega)f(s^{\prime}+r\omega)=\int_{\mathbb{R}^{2}}|s^{\prime}-t^{\prime}|^{N+l}R_{\epsilon}\left(s^{\prime},|s^{\prime}-t^{\prime}|,\frac{s^{\prime}-t^{\prime}}{|s^{\prime}-t^{\prime}|}\right)f(t^{\prime})dt^{\prime}.$
For any positive integer $m$ we write $f=\langle D\rangle^{2m}\langle
D\rangle^{-2m}f$. Choose $N>>m$ so that there is sufficient smoothness in the
integral kernel to integrate by parts the formula
$\mathcal{A}_{R}f=\int_{\mathbb{R}^{2}}|s^{\prime}-t^{\prime}|^{N+l}R_{\epsilon}\left(s^{\prime},|s^{\prime}-t^{\prime}|,\frac{s^{\prime}-t^{\prime}}{|s^{\prime}-t^{\prime}|}\right)\langle
D_{t^{\prime}}\rangle^{2m}\langle
D_{t^{\prime}}\rangle^{-2m}f(t^{\prime})dt^{\prime}.$
We see from this that for a fixed positive integer $m$ we may choose $N$ large
enough so that $\mathcal{A}_{R}:H^{-2m}(\mathbb{R}^{2})\to
H^{2m}(\mathbb{R}^{2})$ is uniformly bounded in $\epsilon$.
For the integral involving the main term of (2.16), we write
$\mathcal{A}_{j}f:=\int_{S^{1}}\int_{0}^{\infty}\partial_{r}^{j}A_{\epsilon}(s^{\prime},0,\omega)r^{j+l}f(s^{\prime}+r\omega).$
Let $\chi\in C^{\infty}_{c}(\mathbb{R}^{2})$ be $1$ near the origin and write
(2.17)
$\displaystyle\mathcal{A}_{j}f=\mathcal{A}_{j}\chi(D)f+\mathcal{A}_{j}(1-\chi(D))f.$
To see the mapping property of the first term of (2.17) we write it out in
Cartesian coordinates
$\mathcal{A}_{j}\chi(D)f=\int_{\mathbb{R}^{2}}\partial_{r}^{j}A_{\epsilon}\left(s^{\prime},0,\frac{t^{\prime}}{|t^{\prime}|}\right)|t^{\prime}|^{j-1+l}(\check{\chi}*f)(s^{\prime}-t^{\prime})dt^{\prime}.$
Since
$\partial_{r}^{j}A_{\epsilon}\left(s^{\prime},0,\frac{t^{\prime}}{|t^{\prime}|}\right)$
is smooth in $s^{\prime}$ with derivatives bounded uniformly in $\epsilon$ and
$\chi(D)$ is smoothing, we have that
$\mathcal{A}_{j}\chi(D):H^{-m}(\mathbb{R}^{2})\to H^{m}(\mathbb{R}^{2})$ for
any positive integer $m$ with bound uniform in $\epsilon$.
The second term of (2.17) is a pseudodifferential operator with full symbol
$a_{j}(s^{\prime},\xi^{\prime};\epsilon):=(1-\chi(\xi^{\prime}))\mathcal{F}_{t^{\prime}}\left(\partial_{r}^{j}A_{\epsilon}\left(s^{\prime},0,\frac{t^{\prime}}{|t^{\prime}|}\right)|t^{\prime}|^{j-1+l}\right)$
and away from $\xi=0$ we can deduce from Lemma 2.10 that
$\mathcal{F}_{t^{\prime}}\left(\partial_{r}^{j}A_{\epsilon}\left(s^{\prime},0,\frac{t^{\prime}}{|t^{\prime}|}\right)|t^{\prime}|^{j-1+l}\right)=|\xi^{\prime}|^{-j-l-1}\tilde{a}_{j}(s^{\prime},\xi^{\prime}/|\xi^{\prime}|;\epsilon)$
for some $\tilde{a}_{j}(s^{\prime},\omega;\epsilon)\in
C_{c}^{\infty}(\mathbb{R}^{2}\times S^{1})$ with derivatives uniformly bounded
in $\epsilon$. The operator $\langle
D\rangle^{m+l+1}\mathcal{A}_{j}(1-\chi(D))\langle D\rangle^{-m}$ then has full
symbol in $S^{0}$ with symbol seminorms bounded uniformly in $\epsilon$. We
can now apply Calderón-Vailancourt Theorem to deduce that
$\mathcal{A}_{j}(1-\chi(D))$ is bounded uniformly in $\epsilon$ from
$H^{m}(\mathbb{R}^{2})\to H^{m+l+1}(\mathbb{R}^{2})$. ∎
### 2.4. Symbol Computations
Compute symbol by taking the Fourier transform and multiply by $2\pi$. We
compute the principal symbols of some of the main operators which we will
encounter. The following list of inverse Fourier transforms will be useful for
later computations and we will leave its proof to the reader:
###### Lemma 2.12.
In $\mathbb{R}^{2}$ with $\xi=(\xi_{1},\xi_{2})$ and $x=(x_{1},x_{2})$ one has
that for $|\xi|\geq 1$,
i) ${\mathcal{F}}^{-1}(\log|x|)(\xi)=-2\pi|\xi|^{-2}$,
${\mathcal{F}}^{-1}(|x|^{-1})(\xi)=2\pi|\xi|^{-1}$
ii) ${\mathcal{F}}^{-1}(x_{j}|x|^{-1})(\xi)=2\pi i\xi_{j}|\xi|^{-3}$
iii) ${\mathcal{F}}^{-1}(x_{j}^{2}|x|^{-3})(\xi)=2\pi\xi_{k}^{2}|\xi|^{-3}$,
$k\neq j$
iv)
${\mathcal{F}}^{-1}(x_{j}^{2}|x|^{-2})(\xi)=2\pi(\xi_{k}^{2}-\xi_{j}^{2})|\xi|^{-4}$,
$k\neq j$.
###### Remark 2.13.
Note that we ignore the behaviour of $\mathcal{F}^{-1}(\cdot)$ near $\xi=0$ as
they are irrelevant to the principal symbol computations we are interested in.
###### Lemma 2.14.
Let $A$ be a pseudodifferential operator on $\partial M$ whose singularity
along the diagonal is given by $d_{g}(x,y)^{-1}$. Then $\sigma(A)(x,\xi)\equiv
2\pi|\xi|^{-1}_{h(x)}$.
###### Proof.
We compute the symbol at $x_{0}\in\partial M$ using the normal coordinate
$t^{\prime}\mapsto x(t^{\prime};x_{0})$. By Corollary 2.5
$d(x(t^{\prime};x_{0}),x_{0})^{-1}=|t^{\prime}|^{-1}+O(|t^{\prime}|)$. By
Lemma 2.12, the inverse Fourier transform of the leading order singularity is
$2\pi|\xi|^{-1}$ for $\xi$ large and therefore $\sigma(A)(x,\xi)\equiv
2\pi|\xi|^{-1}_{h(x)}$. ∎
###### Lemma 2.15.
Let $A$ be a pseudodifferential operator on $\partial M$ whose singularity
near the diagonal is given by
$\partial_{\nu_{y}}d_{g}(x,y)^{-1}\mid_{x,y\in\partial M\times\partial M}$.
Then
$\sigma(A)(x_{0},\xi)\equiv-\pi\frac{{\rm{II}}_{x_{0}}(*\xi^{\sharp})}{|\xi|_{h(x_{0})}^{3}}$
where $\xi\mapsto\xi^{\sharp}$ denotes the musical isomorphism from
$T^{*}\partial M$ to $T\partial M$ induced by the boundary metric $h$ and $*$
is the Hodge star operator in this metric. Here we use ${\rm{II}}_{x}(V)$ to
denote the quadratic form ${\rm{II}}_{x}(V,V)$ for $V\in T_{x}\partial M$.
###### Proof.
Using Lemma 2.7 we see that in normal coordinates given by
$x(s^{\prime};x_{0})$ the leading order singularity of
$\partial_{\nu_{y}}d_{g}(x_{0},y=x(s^{\prime};x_{0}))$ is given by
$-\frac{\lambda_{1}s_{1}^{2}+\lambda_{2}s_{2}^{2}}{2|s^{\prime}|^{3}}$. By
Lemma 2.12 we have that
${\mathcal{F}}^{-1}\left(-\frac{\lambda_{1}s_{1}^{2}+\lambda_{2}s_{2}^{2}}{2|s^{\prime}|^{3}}\right)=-\pi\frac{\lambda_{1}\xi_{2}^{2}+\lambda_{2}\xi_{1}^{2}}{|\xi|^{3}}.$
This is precisely the normal coordinate expression for
$-\pi\frac{{\rm{II}}_{x_{0}}(*\xi^{\sharp})}{|\xi|_{h(x_{0})}^{3}}$. ∎
###### Proposition 2.16.
Let $H(x)$ denote the mean curvature of $\partial M$ at $x$, ${\rm{II}}_{x}$
the second fundamental form of $\partial M$ at $x\in\partial M$, and
${\rm{II}}_{x}(V):={\rm{II}}_{x}(V,V)$ for $V\in T_{x}\partial M$. Define
(2.18) $\displaystyle\quad\quad K(x,y):={H(x)\pi}\log
d_{h}(y,x)-\frac{\pi}{4}\left({\rm{II}}_{x}\left(\frac{\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)-{\rm{II}}_{x}\left(\frac{*\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)\right),$
where $*$ is the Hodge star operator for the metric $h$. Let
$A:C^{\infty}(\partial M)\to{\mathcal{D}}^{\prime}(\partial M)$ be the
operator defined by
$A:u\mapsto\int_{\partial M}K(x,y)u(y){\rm dvol}_{h}.$
Then $A\in\Psi_{cl}^{-2}(\partial M)$ with principal symbol $a(x,\xi)\in
C^{\infty}(T^{*}\partial M\backslash\\{0\\})$ given by
$a(x,\xi)\equiv-\frac{2\pi^{2}}{|\xi|_{h}^{4}}{\rm{II}}_{x}(*\xi^{\sharp}),$
where $\xi\mapsto\xi^{\sharp}\in T\partial M$ denotes the raising of index
with respect to the metric $h$ on $\partial M$.
###### Proof.
To see that $A$ is a classical $\Psi$DO, we use Lemma 2.1 and Corollary 2.9 to
see that the coordinate expression for the integral kernel $K(x,y)$ satisfies
the polyhomogeneous conditions of Prop 2.8 in Chapter 7 of [34]. Therefore
$A\in\Psi^{-2}_{cl}(\partial M)$.
The principal symbol computation is done using normal coordinates. Fix
$x_{0}\in\partial M$ and denote by
$t^{\prime}\mapsto x(t^{\prime};x_{0}):=\exp_{x_{0};h}(t^{\prime})$
the normal coordinate system around $x_{0}$. By a rotation we can choose the
coordinates so that $\partial_{t_{j}}x(0;x_{0})\in T_{x_{0}}\partial M$ is an
eigenvector of the shape operator at $x_{0}$ with eigenvalue $\lambda_{j}$.
According to Lemma 2.1 and Corollary 2.9, in these coordinates the terms of
$K(x_{0},y)$ can be expressed as
(2.19) $\displaystyle
K(x_{0},x(t^{\prime};x_{0}))=\frac{\lambda_{1}+\lambda_{2}}{2}\pi\log|t^{\prime}|-\frac{\pi}{4}\left(\frac{\lambda_{1}t_{1}^{2}+\lambda_{2}t_{2}^{2}}{|t^{\prime}|^{2}}-\frac{\lambda_{1}t_{2}^{2}+\lambda_{2}t_{1}^{2}}{|t^{\prime}|^{2}}\right)$
for $t^{\prime}$ close to the origin. Computing the principal symbol
$a(x_{0},\xi)$ amounts to taking the inverse Fourier transform of the above
expression, and observe the behaviour as $|\xi|\to\infty$. Use the formula in
Lemma 2.12 , we obtain
$a(x_{0},\xi)\equiv-\frac{2\pi^{2}}{|\xi|^{4}}(\lambda_{1}\xi_{2}^{2}+\lambda_{2}\xi_{1}^{2})=-\frac{2\pi^{2}}{|\xi|_{h}^{4}}{\rm{II}}_{x}(*\xi^{\sharp}).$
The last equality holds due to the fact that we are using normal coordinates.
∎
## 3\. Proof of Proposition 1.1
In this section we use layer potential methods to pick out the singularity
structure of the Neumann Green’s function at the boundary. Assume without loss
of generality that $M$ is an open subset of a compact Riemannian manifold
$(\tilde{M},g)$ without boundary. Choose $M^{\prime}\subset\tilde{M}$ a
manifold with boundary which compactly contains $M$. For all $F\in
C^{\infty}_{0}(M^{\prime})$, standard elliptic theory shows that there exists
a unique solution $U_{F}\in C^{\infty}(M^{\prime})$ to
$\Delta_{g}U_{F}=F,\ \ U_{F}\mid_{\partial M^{\prime}}=0.$
The map $F\mapsto U_{F}$ is a continuous linear operator from
$C^{\infty}_{0}(M^{\prime})\to{\mathcal{D}}^{\prime}(M^{\prime})$ and is
therefore given by a Schwartz kernel
$E(x,y)\in{\mathcal{D}}^{\prime}(M^{\prime}\times M^{\prime})$ which we call
the Green’s function. Note that for any $u\in C_{0}^{\infty}(M^{\prime})$, if
we fix $x\in M^{\prime}$ then by definition
$u(x)=\int_{M^{\prime}}\Delta_{g}u(y)E(x,y)d{\rm vol}_{g}(y)=\langle
u(\cdot),\Delta_{g}E(x,\cdot)\rangle.$
We formally write
(3.1) $\displaystyle\Delta_{g}E(x,\cdot)=\delta_{x}(\cdot)\ \ {\rm on}\ \
M^{\prime}$
Note that if $M$ is a bounded domain in $\mathbb{R}^{3}$ then $\tilde{M}$ can
be chosen to be the flat torus and $E(x,y)$ can be chosen to be
$\frac{-1}{4\pi|x-y|}$ for the appropriate constant $c\in\mathbb{R}$.
Using standard elliptic parametrix construction in normal coordinates we
express $E(x,y)$ in the following way.
###### Lemma 3.1.
For all $x,y\in M^{\prime}$
(3.2) $\displaystyle
E(x,y)=-\frac{1}{4\pi}d_{g}(x,y)^{-1}+\Psi_{cl}^{-4}(\tilde{M}).$
Here $\Psi_{cl}^{-4}(\tilde{M})$ denotes the Schwartz kernel of an operator in
$\Psi_{cl}^{-4}(\tilde{M})$.
###### Proof.
Let $P\in\Psi_{cl}^{-2}(\tilde{M})$ be a parametrix for the $\Delta_{g}$ on
the closed compact manifold $\tilde{M}$ without boundary meaning that
$\Delta_{g}P=I+\Psi^{-\infty}(\tilde{M}).$
By ellipticity, for any $\chi_{0}\in C^{\infty}_{c}(M^{\prime})$ we have
$\chi_{0}(x)\chi_{0}(y)(E(x,y)-P(x,y))\in\Psi^{-\infty}(\tilde{M}).$
Therefore it suffices to show that
(3.3) $\displaystyle P-P_{-2}\in\Psi_{cl}^{-4}(\tilde{M})$
where $P_{-2}\in\Psi^{-2}(\tilde{M})$ is defined by
$(P_{-2}u)(x):=\int_{\tilde{M}}\frac{-\chi(x,y)}{4\pi}d_{g}(x,y)^{-1}u(y)d{\rm
vol}_{g}(y)$
for some smooth function $\chi(\cdot,\cdot)\in
C^{\infty}(\tilde{M}\times\tilde{M})$ satisfying $\chi(x,y)=\chi(y,x)$,
$\chi(x,y)=1$ if $d_{g}(x,y)<{\rm Inj}_{\tilde{M}}/4$ and
${\rm supp}(\chi(\cdot,\cdot))\subset\subset\\{(x,y)\mid d_{g}(x,y)<{\rm
Inj}_{\tilde{M}}/2\\}.$
Here ${\rm Inj}_{\tilde{M}}$ is the injectivity radius of the closed compact
Riemannian manifold $(\tilde{M},g)$.
By elliptic regularity (3.3) is equivalent to showing that
(3.4) $\displaystyle\Delta_{g}P_{-2}-I\in\Psi_{cl}^{-2}(\tilde{M}).$
Taking the adjoint and use the self-adjointness of both $\Delta_{g}$ and
$P_{-2}$, this is the same as
(3.5) $\displaystyle P_{-2}\Delta_{g}-I\in\Psi_{cl}^{-2}(\tilde{M}).$
Using the principal symbol map defined in (2.7) it amounts to showing that
(3.6) $\displaystyle\sigma_{-1}(P_{-2}\Delta_{g}-I)=0$
as an element of the quotient space $S^{-1}_{cl}/S^{-2}_{cl}$. In fact, since
the symbol is classical, we now choose $\sigma_{-1}(P_{-2}\Delta_{g}-I)$ to be
the representative in the equivalence class which is positively homogeneous of
degree $-1$.
For each $y_{0}\in\tilde{M}$ and covector $\eta_{0}\in S^{*}_{y_{0}}\tilde{M}$
in the unit cosphere bundle we will show that
(3.7) $\displaystyle|\sigma_{-1}(P_{-2}\Delta_{g}-I)(y_{0},\tau\eta_{0})|\leq
C_{y_{0},\eta_{0}}\tau^{-2}$
as $\tau\to\infty$. Homogeneity would then ensure that
$\sigma_{-1}(P_{-2}\Delta_{g}-I)(y,\eta)=0$ for all $(y,\eta)\in
T^{*}\tilde{M}$ which would then ensure (3.6).
To this end let $V_{1},V_{2},V_{3}\in S_{y_{0}}\tilde{M}$ be three orthonormal
vectors and choose normal coordinate given by
$\Phi(t):=\exp_{y_{0}}(\sum_{j=1}^{3}t_{j}V_{j})$ for
$|t|_{\mathbb{R}^{3}}<{\rm Inj}_{\tilde{M}}/2$. Let $\chi_{\mathbb{R}^{3}}\in
C_{c}^{\infty}(\mathbb{R}^{3})$ take the value $1$ in an open set containing
the origin but ${\rm
supp}(\chi_{\mathbb{R}^{3}})\subset\subset\\{t\in\mathbb{R}^{3}\mid|t|_{\mathbb{R}^{3}}<{\rm
Inj}_{\tilde{M}}/2\\}$. Similarly let $\chi_{\tilde{M}}\in
C^{\infty}(\mathbb{R}^{3})$ take the value $1$ in an open set containing
$y_{0}$ but ${\rm supp}(\chi_{\tilde{M}})\subset\subset\\{x\mid
d_{g}(x,y_{0})<{\rm Inj}_{\tilde{M}}/2\\}$.
Define the pullback operators $A,B:C^{\infty}(\mathbb{R}^{3})\to
C^{\infty}(\mathbb{R}^{3})$ by
$A:u\mapsto\Phi_{*}\left(\chi_{\tilde{M}}P_{-2}\Phi^{*}\left(\chi_{\mathbb{R}^{3}}u\right)\right),\
B:u\mapsto\Phi_{*}\left(\chi_{\tilde{M}}\Delta_{g}\Phi^{*}\left(\chi_{\mathbb{R}^{3}}u\right)\right)$
where $\Phi^{*}$ and $\Phi_{*}$ are pullback by $\Phi$ and $\Phi^{-1}$
respectively.
Thanks to the invariance of the principal symbol map under symplectomorphism,
we have
(3.8)
$\displaystyle\sigma_{-1}(P_{-2}\Delta_{g}-I)(y_{0},\Phi^{*}\xi)=\sigma_{-1}(AB-I)(0,\xi)$
for all $\xi\in T^{*}_{0}\mathbb{R}^{3}$. We see then that (3.6) amounts to
showing that $AB-I$ satisfies
(3.9) $\displaystyle\sigma_{-1}(AB-I)(0,\xi)=0.$
Let $a(t,\xi)$ and $b(t,\xi)$ be the full symbol of $A$ and $B$ respectively.
The full symbol of $A$ can be computed by the formula
$a(t,\xi)=e_{-\xi}(t)\left(Ae_{\xi}\right)(t)$
where $e_{\xi}(t):=e^{-it\cdot\xi}$. Since we are using the normal coordinate
around $y_{0}$, $d_{g}(y_{0},\Phi(t))=|t|_{\mathbb{R}^{3}}$ and
$\Phi_{*}d{\rm Vol}_{g}=\sqrt{|g|}dt=dt+H_{2}(t)dt$
where $H_{2}(t)$ is a smooth function vanishing to order $2$ at $t=0$. So
$(Ae_{\xi})(0)=\int_{\mathbb{R}^{3}}\frac{-4\pi}{|t|}e^{-it\cdot\xi}dt+\int_{\mathbb{R}^{3}}\frac{c}{|t|}e^{-it\cdot\xi}\tilde{H}_{2}(t)dt$
for some smooth and compactly supported function $\tilde{H}_{2}(t)$ vanishing
to order $2$ at $t=0$. Computing the first term directly and treat the second
term by expanding $H_{2}(t)$ using Taylor expansion we see that
(3.10) $\displaystyle
a(0,\xi)=(Ae_{\xi})(0)=|\xi|^{-2}+S_{cl}^{-4}(T^{*}\mathbb{R}^{3}).$
Since $B$ is the Laplace operator in the coordinate given by $\Phi$ we have
that
(3.11) $\displaystyle
b(t,\xi)=\sum_{j,k=1}^{3}g^{j,k}(t)\xi_{j}\xi_{k}+\frac{1}{\sqrt{|g|}}\sum_{j,k=1}^{3}\partial_{t_{j}}(\sqrt{|g|}g^{j,k})(t)\xi_{k}$
Composition calculus gives that if $c(x,\xi)$ is the full symbol of the
operator $AB$ then
(3.12) $\displaystyle
c(0,\xi)=a(0,\xi)b(0,\xi)+-i\sum_{j=1}^{3}\partial_{\xi_{j}}a(0,\xi)\partial_{t_{j}}b(t,\xi)\mid_{t=0}+S_{cl}^{-2}.$
Substituting into (3.12) the expression we have in (3.10), (3.11), and the
fact that in normal coordinates $g^{j,k}(t)=\delta^{j,k}+O(|t|^{2})$ for $t$
in a neighbourhood of the origin we have that the second term in (3.12) drops
out. So the full symbol of $AB-I$ at the point $(0,\xi)\in
T^{*}\mathbb{R}^{3}$ is
$c(0,\xi)-1\in S_{cl}^{-2}.$
In light of (3.8), for each fixed $\xi\in T^{*}_{y_{0}}\tilde{M}$,
$|\sigma_{-1}(P_{-2}\Delta_{g}-I)(y_{0},\tau\Phi^{*}\xi)|<C_{\xi}\tau^{-2}$
as $\tau\to\infty$. Therefore (3.7) is verified.
∎
For all $f\in C^{\infty}(\partial M)$ we define as in [34] the operators
$S,N\in\Psi_{cl}^{-1}(\partial M)$ by the following
(3.13) $\displaystyle Sf(x):=\int_{\partial M}E(x,y)f(y){\rm vol}_{h},\ \
Nf(x):=2\int_{\partial M}\partial_{\nu_{y}}E(x,y)f(y){\rm vol}_{h}$
for $x\in\partial M$. Note that $Nf(x)$ is different from (see [34] Chapt 7
Sect 11)
$\lim_{x\to\partial M,x\in M}2\int_{\partial
M}\partial_{\nu_{y}}E(x,y)f(y)dy=f(x)+Nf(x).$
Modulo lower order pseudodifferential operator, $S$ and $N$ are given by the
integral kernels $d_{g}(x,y)^{-1}$ and $\partial_{\nu_{y}}d_{g}(x,y)^{-1}$
respectively. Indeed, using (3.2) and equation (11.14) on page 38 of [34], we
see that for $(x,y)\in\partial M\times\partial M$ in a neighbourhood of the
diagonal,
(3.14) $\displaystyle S=-\frac{1}{4\pi}d_{g}(x,y)^{-1}+\Psi_{cl}^{-3}(\partial
M),\ \
N=-\frac{1}{2\pi}\partial_{\nu_{y}}d_{g}(x,y)^{-1}+\Psi_{cl}^{-2}(\partial
M).$
Using (3.1) we can construct the so called Neumann Green’s function on $M$ via
the following procedure. For each fixed $x\in M^{o}$ we can solve the
following Neumann boundary value problem to obtain the correction term
$C(x,y)$ as a function of $y\in M$
$\Delta_{g}C(x,y)=0,\ ,\
\partial_{\nu_{y}}C(x,y)=\partial_{\nu_{y}}E(x,y)-\frac{1}{|\partial M|},\ \
\int_{\partial M}C(x,y)d{\rm vol}_{h}=\int_{\partial M}E(x,y)d{\rm vol}_{h}.$
Setting $G(x,y)=-E(x,y)+C(x,y)$ we get, for each fixed $x\in M$ the unique
solution (as a distribution in $z$) $G(x,z)$ to
(3.15) $\displaystyle\Delta_{g}G(x,z)=-\delta_{x}(z)\
,\partial_{\nu_{z}}G(x,z)\mid_{z\in\partial M}=\frac{-1}{|\partial M|},\ \
\int_{\partial M}G(x,z)d{\rm vol}_{h}=0.$
Fix for the time being $y\neq z$ in the interior of $M$ and observe that
$x\mapsto G(z,x)$ is smooth in a neighbourhood of the singularity of the map
$x\mapsto G(x,y)$ and vice versa. Therefore we can integrate by parts the the
expression $G(z,y)=-\int_{M}G(z,x)\Delta_{x}G(y,x)dx$ to obtain
$G(z,y)-G(y,z)=\int_{\partial
M}G(y,x)\partial_{\nu_{x}}G(z,x)-G(z,x)\partial_{\nu_{x}}G(y,x){\rm
vol}_{h}(x).$
The boundary and orthogonality conditions in (3.15) ensures that the right
side vanishes so we have
(3.16) $\displaystyle G(z,y)=G(y,z).$
Let $\Lambda:H^{k}(\partial M)\to H^{k-1}(\partial M)$ denote the Dirichlet-
to-Neumann map (see [17] for definition) whose range is precisely a
codimension one subspace of $H^{k-1}$ which annihilates the constant function.
By the orthogonality condition in (3.15), the behaviour of $G(x,y)$ is
uniquely characterized by its action on the range of $\Lambda$. To this end,
for $f\in C^{\infty}(\partial M)$, denote its harmonic extension by $u_{f}$.
Integrating by parts the expression $0=\int_{M}\Delta_{g}u_{f}(x)G(x,y)dx$ for
$z$ in the interior of $M$ we have
$\displaystyle u(y)$ $\displaystyle=$ $\displaystyle\int_{\partial
M}\partial_{\nu}u_{f}(x)G(x,y){\rm dvol}_{h}(x)+\frac{1}{|\partial
M|}\int_{\partial M}f{\rm dvol}_{h}$ $\displaystyle=$
$\displaystyle\int_{\partial M}(\Lambda f)(x)G(x,y){\rm
dvol}_{h}(x)+\frac{1}{|\partial M|}\int_{\partial M}f{\rm dvol}_{h}$
Observe that any $\tilde{f}\in C^{\infty}(\partial M)$ has a unique
decomposition $\tilde{f}=c+\Lambda f$ for some constant function $c$ and $f\in
C^{\infty}(\partial M)$ satisfying $\int_{\partial M}f=0$. Therefore, using
the orthogonality condition of (3.15) and taking the trace of (3) we see that
the map
$\tilde{f}\mapsto\left.\left(\int_{\partial M}\tilde{f}(x)G(x,y){\rm
dvol}_{h}(x)\right)\right|_{y\in\partial M}$
is well defined and takes $C^{\infty}(\partial M)\to C^{\infty}(\partial M)$.
We denote this operator by $G_{\partial M}$ and its Schwartz kernel by
$G_{\partial M}(x,y)$. Going back to (3) we see that
$f=G_{\partial M}\Lambda f+Pf,$
where
(3.18) $\displaystyle Pf:=|\partial M|^{-1}\int_{\partial M}f$
is smoothing. In operator form this is
(3.19) $\displaystyle I=G_{\partial M}\Lambda+P.$
Since $\Lambda\in\Psi^{1}_{cl}(\partial M)$ is elliptic (see [17]) we can
conclude that $G_{\partial M}\in\Psi_{cl}^{-1}(\partial M)$ which maps
$H^{k}(\partial M)\to H^{k+1}(\partial M)$ for all $k\in\mathbb{R}$. This
completes the proof of Proposition 1.1 part i).
###### Remark 3.2.
A quick way to prove part ii) of Proposition 1.1 would be to observe that
(3.19) implies $G_{\partial M}$ is a parametrix for $\Lambda$. The symbol
expansion for $\Lambda$ has already been computed in [17] so constructing its
parametrix follows from standard pseudodifferential calculus. However, we will
choose instead to take the layer potential approach since it iwill be more
conducive for future numerical implementations. See Remark 3.3 below.
Applying on the right the single-layered potential $S$ defined in(3.13) and
using identity (11.58) of [34] we have
(3.20) $\displaystyle G_{\partial M}=-2S+G_{\partial M}N^{*}+2PS.$
Iterating this equation and using intertwining property (11.59) of [34] we get
(3.21) $\displaystyle G_{\partial M}=-2S-2NS+\Psi_{cl}^{-3}(\partial M).$
By (2.8) the principal symbol of the operator $NS$ is simply the product of
the principal symbols of $S$ with the principal symbol of $N$. The leading
singularities of the operators $S$ and $N$ are given in (3.14) and the
principal symbols of these kernels are computed in Lemmas 2.14 and 2.15.
Therefore, using Proposition 2.16, we see that modulo $\Psi_{cl}^{-3}(\partial
M)$, the integral kernel of $NS$ is given by
$\frac{1}{8\pi}{H(x)}\log
d_{h}(y,x)-\frac{1}{32\pi}\left({\rm{II}}_{x}\left(\frac{\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)-{\rm{II}}_{x}\left(\frac{*\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)\right)$
when $x,y\in\partial M$ are close to each other.
Inserting this into (3.21) we get that when $x,y\in\partial M$ are close to
each other,
$\displaystyle\quad\quad G_{\partial M}(x,y)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}d_{g}(x,y)^{-1}-\frac{1}{4\pi}{H(x)}\log
d_{h}(y,x)$
$\displaystyle+\frac{1}{16\pi}\left({\rm{II}}_{x}\left(\frac{\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)-{\rm{II}}_{x}\left(\frac{*\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)\right)+R(x,y),$
where $R(x,y)$ is the Schwartz kernel of an operator in
$\Psi_{cl}^{-3}(\partial M)$ which we call the regular part of $G(x,y)$.
Observe that since the principal symbol of $R$ is in
$S_{cl}^{-3}(T^{*}\partial M)$ and $\partial M$ is dimension $2$, Sobolev
embedding yields that
(3.23) $\displaystyle R(\cdot,\cdot)\in C^{0,\alpha}(\partial M\times\partial
M)$
for all $\alpha<1$. The proof of Proposition 1.1 is now complete.
###### Remark 3.3.
Note that (1.1) peels off the "singular part" of the distribution $G_{\partial
M}(x,y)$ and gives us the representation
$G_{\partial M}(x,y)=G_{sing}(x,y)+R(x,y)$
with the singularity structure of $G_{sing}$ explicitly given by (1.1).
Inserting this representation of $G_{\partial M}$ into (3.20) gives the
following integral equation for the regular part $R(x,y)$:
$R(I-N^{*})=-G_{sing}-2S+G_{sing}N^{*}+2PS$
where the operators $P$, $S$, and $N$ are given by (3.18) and (3.13).
Since $N^{*}\in\Psi^{-1}_{cl}(\partial M)$, it is a compact operator which
makes $I-N^{*}$ Fredholm with index zero. Therefore, numerically computing for
$R(x,y)$ amounts to solving a Fredholm boundary integral equation subject to
the orthogonality condition
$\int_{\partial M}G_{\partial M}(x,z){\rm dvol}_{h}(z)=0.$
## 4\. Inverting the Normal Operator
Let $\Omega\subset\mathbb{R}^{n}$ be a bounded convex domain with smooth
boundary. We will analyze the mapping properties of the operator
(4.1) $\displaystyle
L:f\mapsto\int_{\Omega}\frac{f(s)}{|s-t|^{n-1}}ds_{1}\dots ds_{n}$
and its inverse. Methods do exist [18],[7] for the explicit expression of the
inverse of $L$ when $\Omega=\mathbb{D}$ (which is sufficient for our setting).
When $\Omega$ is a two dimensional ellipse [27] computed explicitly the
inverse of $L$ acting on the constant function.
The purpose of Section 4.1 is to provide a geometric perspective to the
operator $L$ one of the advantages of which is that it provides an explicit
formula for $L^{-1}(1)$ when $\Omega$ is a ball of any dimension. Our
perspective is based on some of the recent progress on integral geometry (in
particular [23], [20], [12]). Since the explicit formulas and estimates will
be valid in all dimensions, this will provide the key ingredient in proving
Theorem 1.3 in all dimensions. When $\Omega$ is not necessarily $\mathbb{B}$,
this geometric point of view may also potentially provide ways to relate some
of the quantities of interest to the geometry of $\Omega$.
Section 4.2 will provide some explicit formulas for the composition of
$L^{-1}$ with other operators in the case when $\Omega=\mathbb{D}$, the two
dimensional disk. Section 4.3 will do the same for when $\Omega$ is the two
dimensional ellipse although the formulas will not be as explicit.
### 4.1. Mapping Properties of L
Denote by
$\partial_{+}S\Omega:=\\{(x,v)\in\partial\Omega\times S^{n-1}\mid
v\cdot\nu(x)\leq 0\\}$
to be the set of inward pointing unit vectors on $\partial\Omega$. Note that
this is a closed submanifold of the sphere bundle $S\Omega$ and thus inherits
its smooth structure. Define the X-Ray transform
$I:C^{\infty}(\overline{\Omega})\to C^{\infty}(\partial_{+}S\Omega)$ by
$If(x,v):=\int_{0}^{\tau(x,v)}f(x+tv)dt$ where $\tau(x,v)$ is the time it
takes for a ray of unit velocity $v$ starting at $x\in\overline{\Omega}$ to
reach the boundary $\partial\Omega$. Note that because $\Omega$ is assumed to
be convex, $\tau(x,v)$ is a smooth function on $\partial_{+}SM$. Furthermore,
$I$ is injective by [23].
By [29] Theorem 4.2.1 this operator extends to an operator $I:L^{2}(\Omega)\to
L^{2}_{\mu}(\partial_{+}S\Omega)$ where $\mu$ is the measure given by
$\mu=|\nu(x)\cdot v|{\rm dvol}_{\partial\Omega}{\rm dvol}_{S^{n-1}}$. This
$L^{2}$ space mapping property allows us to define the adjoint operator
$I^{*}$ given by (see [23])
(4.2) $\displaystyle I^{*}\omega(x)=\int_{S^{n-1}}\omega(x+\tau(x,v)v){\rm
dvol}_{S^{n-1}}(v)$
when acting on smooth functions $\omega$. This allows us to define a self-
adjoint normal operator $I^{*}I:L^{2}(\Omega)\to L^{2}(\Omega)$. It turns out
that by [23] the Schwartz kernel of $I^{*}I$ is precisely $2|s-t|^{-n+1}$ and
therefore $I^{*}I=2L$. Let $d_{\Omega}(\cdot)$ be any smooth positive function
on $\Omega$ which is equal to $dist(\cdot,\partial\Omega)$ near the boundary.
By Theorem 2.2 and Theorem 4.4 of [20] respectively, we have that
(4.3) $\displaystyle I^{*}I:d_{\Omega}^{-1/2}C^{\infty}(\overline{\Omega})\to
C^{\infty}(\overline{\Omega})$
is a bijection and
(4.4) $\displaystyle 2L=I^{*}I:\\{u\in H^{-1/2}(\mathbb{R}^{n})\mid{\rm
supp}(u)\subset\Omega\\}\simeq H^{1/2}(\Omega)^{*}\to H^{1/2}(\Omega)$
is a self-adjoint homeomorphism. Thus there exists a unique function $u_{0}\in
d_{\Omega}^{-1/2}C^{\infty}(\overline{\Omega})$ such that $Lu_{0}=1$ which is
equivalent to $I^{*}Iu_{0}=2$. To compute $u_{0}$, observe that if we find
$u_{0}$ such that
(4.5) $\displaystyle Iu_{0}(x,v)=\frac{2}{{\rm Vol}(S^{n-1})}$
for all $(x,v)\in\partial_{+}S\Omega$ then by (4.2) we would have
$I^{*}Iu_{0}=2$. The solution of (4.5) is easy to compute explicitly when
$\Omega=\mathbb{B}$. Indeed, direct computation shows that choosing
(4.6) $\displaystyle u_{0}(x)=\frac{2}{\pi{\rm Vol}(S^{n-1})\sqrt{1-|x|^{2}}}$
one satisfies (4.5). In particular if $n=2$ (which is the case we are
interested in) we have that
(4.7) $\displaystyle L^{-1}(1)=u_{0}(x)=\frac{1}{\pi^{2}\sqrt{1-|x|^{2}}},\ \
{\rm in\ dimension\ 2}$
###### Remark 4.1.
This process of computing solution to $I^{*}Iu_{0}=const$ by solving for
$Iu_{0}=const$ unfortunately only works for $\Omega=\mathbb{B}$. In fact,
thanks to the rigidity result of [12], we know that $Iu_{0}=1$ is solvable iff
$\Omega=\mathbb{B}$. However, for more general domains the geometric view
presented here could potentially allow one to apply the reconstruction formula
for inverting $I$ [22] to solve $I^{*}Iu_{0}=1$ explicitly. To do so one must
first invert $I^{*}$ (which has a large kernel but is surjective) into the
range of $I$ and it is not clear how to do this when $\Omega\neq\mathbb{B}$.
### 4.2. Integrals Involving L inverse of 1.
We also define $R_{\log}$ and $R_{\infty}$ to be operators with kernels
$\log|s^{\prime}-t^{\prime}|$ and
$\frac{(s_{1}-t_{1})^{2}-(s_{2}-t_{2})^{2}}{|s^{\prime}-t^{\prime}|^{2}}$
respectively.
###### Lemma 4.2.
The operators $R_{\infty}$ and $R_{\log}$ are bounded maps from
$H^{1/2}(\mathbb{D})^{*}$ to $H^{3/2}(\mathbb{D})$.
###### Proof.
Observe that the integral kernels of both $R_{\log}$ and $R_{\infty}$ extends
naturally to kernels representing operators in $\Psi^{\infty}(\mathbb{R}^{2})$
which we denote by $\tilde{R}_{\log}$ and $\tilde{R}_{\infty}$ respectively.
We denote by $E:H^{1/2}(\mathbb{D})^{*}\to\\{u\in
H^{-1/2}(\mathbb{R}^{2})\mid{\rm supp}(u)\subset\Omega\\}$ to be the
isomorphism obtained by the trivial extension. Let $\chi\in
C^{\infty}_{0}(\mathbb{R}^{2})$ be identically $1$ on $\mathbb{D}$. Then we
have that
(4.8)
$\displaystyle\left(R_{\log}u\right)\mid_{\mathbb{D}}=\left(\chi\tilde{R}_{\log}\chi
Eu\right)\mid_{\mathbb{D}}$
and the same holds for $R_{\infty}$.
By Proposition 7.2.8 of [34] we have that both $\chi\tilde{R}_{\log}\chi$ and
$\chi\tilde{R}_{\infty}\chi$ are pseudodifferential operators of order $-2$.
Therefore by (4.8) both $R_{\log}$ and $R_{\infty}$ are bounded operators from
$H^{1/2}(\mathbb{D})^{*}$ to $H^{3/2}(\mathbb{D})$.
∎
The following lemma was proved in [7, Theorem 4.2].
###### Lemma 4.3.
For $u\in H^{\frac{1}{2}}(\mathbb{D})$, it follows
(4.9)
$\displaystyle\left\langle{L}^{-1}u,1\right\rangle=\frac{1}{\pi^{2}}\int_{\mathbb{D}}\frac{u(t^{\prime})}{(1-|t^{\prime}|^{2})^{\frac{1}{2}}}dt_{1}dt_{2}.$
###### Proof.
By (4.4) $L:H^{\frac{1}{2}}(\mathbb{D})^{*}\to H^{\frac{1}{2}}(\mathbb{D})$ is
a self-adjoint homeomorphism. The result of this Lemma is a direct
consequence. ∎
###### Lemma 4.4.
Let
$\displaystyle
f(s^{\prime})=R_{log}L^{-1}1=\int_{\mathbb{D}}\log|t^{\prime}-s^{\prime}|[L^{-1}1](t^{\prime})dt_{1}dt_{2},$
then
$\displaystyle
f(s^{\prime})=\frac{2}{\pi}\log|s^{\prime}|+\frac{2}{\pi}\left(\frac{1}{2}\log\left|(1-|s^{\prime}|^{2})^{\frac{1}{2}}+1\right|-\frac{1}{2}\log\left|(1-|s^{\prime}|^{2})^{\frac{1}{2}}-1\right|\right)-\frac{2}{\pi}(1-|s^{\prime}|^{2})^{\frac{1}{2}}.$
###### Proof.
Note that $\frac{1}{2\pi}\log|t^{\prime}-s^{\prime}|$ is the fundamental
solution for the Laplace operator in $\mathbb{R}^{2}$, therefore,
$\displaystyle\Delta
f=2\pi[L^{-1}1]=\frac{2}{\pi}\frac{1}{(1-|t^{\prime}|^{2})^{\frac{1}{2}}}$
Since $[{L}^{-1}1](t^{\prime})$ is radially symmetric,
$f(t^{\prime})=\tilde{f}(r)$ where $r=|t^{\prime}|$. Writing the Laplace
operator in polar coordinates, we get
$\displaystyle(r\tilde{f}_{r})_{r}=\frac{2}{\pi}\frac{r}{(1-r^{2})^{\frac{1}{2}}}.$
Integration gives
$\displaystyle\tilde{f}_{r}(r)=\frac{2}{\pi}\left(\frac{C_{1}}{r}-\frac{(1-r^{2})^{\frac{1}{2}}}{r}\right)$
and
$\displaystyle\quad\tilde{f}(r)=\frac{2}{\pi}\left(C_{1}\log
r-\frac{1}{2}\log\left|(1-r^{2})^{\frac{1}{2}}-1\right|+\frac{1}{2}\log\left|(1-r^{2})^{\frac{1}{2}}+1\right|-(1-r^{2})^{\frac{1}{2}}+C_{2}\right).$
Let us find $C_{1}$ and $C_{2}$. Note that $\tilde{f}(r)$ does not have
singularity at $r=0$, namely,
$\displaystyle\tilde{f}(0)=f(0)=\frac{1}{\pi^{2}}\int_{\mathbb{D}}\frac{\log|t^{\prime}|}{(1-|t^{\prime}|^{2})^{\frac{1}{2}}}=\frac{2}{\pi}\int_{0}^{1}\frac{r\log
r}{(1-r^{2})^{\frac{1}{2}}}dr=\frac{2}{\pi}(\log 2-1).$
Therefore, the identities
$\displaystyle C_{1}\log
r-\frac{1}{2}\log\left|(1-r^{2})^{\frac{1}{2}}-1\right|=\frac{1}{2}\log\left|\frac{r^{2C_{1}}}{(1-r^{2})^{\frac{1}{2}}-1}\right|=\frac{1}{2}\log\left|\frac{r^{2C_{1}}}{-\frac{1}{2}r^{2}+O(r^{4})}\right|,$
as $r\rightarrow 0$, implies that $C_{1}=1$. Hence, putting $r=0$ into (4.4),
gives
$\displaystyle\frac{2}{\pi}(\log 2-1)=\frac{2}{\pi}\left(\frac{1}{2}\log
2+\frac{1}{2}\log 2-1+C_{2}\right),$
so that $C_{2}=0$. ∎
Due to Lemmas 4.4 and 4.3, the following identity is a direct computation:
###### Lemma 4.5.
The following identity holds
$\displaystyle\left\langle
L^{-1}R_{log}L^{-1}1,1\right\rangle=\frac{8}{\pi^{2}}\log
2-\frac{6}{\pi^{2}}.$
###### Lemma 4.6.
The following idetity holds
$\displaystyle\left\langle L^{-1}R_{\infty}L^{-1}1,1\right\rangle=0.$
###### Proof.
By Lemmas (4.7) and 4.3, we know that
$\displaystyle\left\langle
L^{-1}R_{\infty}L^{-1}1,1\right\rangle=\frac{1}{\pi^{2}}\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{(s_{1}-t_{1})^{2}-(s_{2}-t_{2})^{2}}{|s^{\prime}-t^{\prime}|^{2}}\frac{1}{(1-|s^{\prime}|^{2})^{\frac{1}{2}}}ds^{\prime}\frac{1}{(1-|t^{\prime}|^{2})^{\frac{1}{2}}}dt^{\prime}.$
Consider the following two changes of variables for the right-hand side
$\displaystyle(s_{1},s_{2},t_{1},t_{2})=(r\cos\phi,r\sin\phi,\rho\cos\theta,\rho\sin\theta),$
$\displaystyle(s_{1},s_{2},t_{1},t_{2})=(r\sin\phi,r\cos\phi,\rho\sin\theta,\rho\cos\theta).$
The results differ by multiplying by $-1$, which means that the right-hand
side is 0. ∎
### 4.3. Explicit Formulas in 2 Dimensional Ellipse
We now compute the inverse of the map
$f\mapsto\int_{{\mathcal{E}}_{a}}\frac{f(s^{\prime})}{|s^{\prime}-t^{\prime}|}ds^{\prime}$
where the domain of integration is the two dimensional ellipse
${\mathcal{E}}_{a}:=\\{s_{1}^{2}+\frac{s_{2}^{2}}{a^{2}}=1\\}$ instead of a
ball. A change of variable leads us to consider the operator
(4.11)
$L_{a}f=a\int_{\mathbb{D}}\frac{f(s^{\prime})}{\left((t_{1}-s_{1})^{2}+a^{2}(t_{2}-s_{2})^{2}\right)^{1/2}}ds^{\prime}$
acting on functions of the disk $\mathbb{D}$. By [27] we have that
(4.12) $L_{a}\left({K_{a}}^{-1}{(1-|t^{\prime}|^{2})^{-1/2}}\right)=1,$
on $\mathbb{D}$ where
$K_{a}=\frac{\pi}{2}\int_{0}^{2\pi}\frac{1}{\left(\cos^{2}\theta+\frac{\sin^{2}\theta}{a^{2}}\right)^{1/2}}d\theta.$
By (4.4) this is the unique solution in $H^{1/2}(\mathbb{D})^{*}$ to
$L_{a}u=1$.
Next we denote
$R_{\log,a}f(t^{\prime}):=a\int_{\mathbb{D}}\log\left((t_{1}-s_{1})^{2}+a^{2}(t_{2}-s_{2})^{2}\right)^{1/2}f(s^{\prime})ds^{\prime},$
$R_{\infty,a}f(t^{\prime}):=a\int_{\mathbb{D}}\frac{(t_{1}-s_{1})^{2}-a^{2}(t_{2}-s_{2})^{2}}{(t_{1}-s_{1})^{2}+a^{2}(t_{2}-s_{2})^{2}}f(s^{\prime})ds^{\prime}.$
For general $a$, the quantities $\langle
L^{-1}_{a}(1),\mathbb{R}_{\log,a}L_{a}^{-1}(a)\rangle$ and $\langle
L^{-1}_{a}(1),\mathbb{R}_{\infty,a}L_{a}^{-1}(a)\rangle$ cannot be computed as
explicitly as in the case when $a=1$ in Section 4.2.
## 5\. Asymptotic Expansion of the Singularly Perturbed Problems
### 5.1. Asymptotic Expansion of Mixed Boundary Value Problems
We are now ready to compute the asymptotic expansion for the mixed boundary
value problem $u_{\epsilon}\in H^{1}(M)$,
(5.1) $\displaystyle\Delta_{g}u_{\epsilon}=-1,\ \
u_{\epsilon}\mid_{\Gamma_{\epsilon,a}}=0,\ \
\partial_{\nu}u_{\epsilon}\mid_{\partial M\backslash\Gamma_{\epsilon,a}}=0$
which gives the compatibility condition
(5.2) $\displaystyle\int_{\partial M}\partial_{\nu}u_{\epsilon}{\rm
dvol}_{h}=-|M|.$
All integrals on open subsets of $\partial M$ are with respect to the volume
form given by the metric $h$.
Using Green’s formula as in [9], also in [1], we can deduce that for points
$x\in M^{o}$, $u_{\epsilon}(x)$ satisfies the integral equation
(5.3) $\displaystyle u_{\epsilon}(x)=F(x)+C_{\epsilon,a}+\int_{\partial
M}G(x,y)\partial_{\nu}u_{\epsilon}(y){\rm dvol}_{h}(y),$
where $C_{\epsilon,a}=|\partial M|^{-1}\int_{\partial M}u_{\epsilon}$ and
$F(x)=\int_{M}G(x,y)$ solves the boundary value problem
(5.4) $\displaystyle\Delta F=-1,\ \ \partial_{\nu}F=-|M|/|\partial M|,\ \
\int_{\partial M}F{\rm dvol}_{h}=0.$
By Proposition 1.1 we can take the trace of (5.3) to the boundary and restrict
to the open subset $\Gamma_{\epsilon,a}^{o}\subset\partial M$. Using (5.1) we
see that
$0=F(x)+C_{\epsilon,a}+\int_{\Gamma_{\epsilon,a}}G_{\partial
M}(x,y)\partial_{\nu}u_{\epsilon}(y){\rm dvol}_{h}(y)$
for $x\in\Gamma_{\epsilon,a}^{o}$. We now replace $G_{\partial M}(x,y)$ for
$x,y\in\Gamma_{\epsilon,a}^{o}$ with the expression in (1.1) to obtain
$\displaystyle-F(x)-C_{\epsilon,a}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{\Gamma_{\epsilon,a}}d_{g}(x,y)^{-1}\partial_{\nu}u_{\epsilon}(y){\rm
dvol}_{h}(y)-\frac{H(x)}{4\pi}\int_{\Gamma_{\epsilon,a}}\log
d_{h}(x,y)\partial_{\nu}u_{\epsilon}(y){\rm dvol}_{h}(y)$
$\displaystyle+\frac{1}{16\pi}\int_{\Gamma_{\epsilon,a}}\left({\rm{II}}_{x}\left(\frac{\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)-{\rm{II}}_{x}\left(\frac{*\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)\right)\partial_{\nu}u_{\epsilon}(y){\rm
dvol}_{h}(y)$
$\displaystyle+\int_{\Gamma_{\epsilon,a}}R(x,y)\partial_{\nu}u_{\epsilon}(y){\rm
dvol}_{h}(y).$
We will write this integral equation in the coordinate system given by
(5.6) $\displaystyle\mathbb{D}\ni(s_{1},s_{2})\mapsto
x^{\epsilon}(s_{1},as_{2};x^{*})\in\Gamma_{\epsilon,a},$
where $x^{\epsilon}(\cdot;x^{*}):{\mathcal{E}}_{a}\to\Gamma_{\epsilon,a}$ is
the coordinate defined in Section 2.2. To simplify notation we will drop the
$x^{*}$ in the notation and denote $x^{\epsilon}(\cdot;x^{*})$ by simply
$x^{\epsilon}(\cdot)$.
Note that in these coordinates the volume form for $\partial M$ is given by
(5.7) $\displaystyle{\rm
dvol}_{h}=a\epsilon^{2}(1+\epsilon^{2}Q_{\epsilon}(s^{\prime}))ds_{1}\wedge
ds_{2},\ s^{\prime}\in\mathbb{D}$
for some smooth function $Q_{\epsilon}(s^{\prime})$ whose derivatives of all
orders are bounded uniformly in $\epsilon$. We denote
(5.8)
$\displaystyle\psi_{\epsilon}(s^{\prime}):=\partial_{\nu}u_{\epsilon}(x^{\epsilon}(s_{1},as_{2})).$
The compatibility condition (5.2) written using the expression for the volume
form (5.7) is then
(5.9)
$\displaystyle\int_{\mathbb{D}}\psi_{\epsilon}(s^{\prime})(1+\epsilon^{2}Q_{\epsilon}(s^{\prime}))ds_{1}ds_{2}=-\frac{|M|}{a\epsilon^{2}}.$
Let us unwrap the right hand side of (5.1) term by term in the coordinate
given by $x^{\epsilon}(\cdot)$. Write out the integral of the first term using
the expression of the volume form (5.7) and the expression for
$d_{g}(x,y)^{-1}$ in Corollary 2.6 and taking into account that the coordinate
system is scaled by a factor $a$ as in (5.6) gives
(5.10)
$\int_{\Gamma_{\epsilon,a}}d_{g}(x,y)^{-1}\partial_{\nu}u_{\epsilon}(y){\rm
dvol}_{h}(y)=a\epsilon\int_{\mathbb{D}}\frac{1}{\left((t_{1}-s_{1})^{2}+a^{2}(t_{2}-s_{2})^{2}\right)^{1/2}}\psi_{\epsilon}(s^{\prime})ds+\epsilon^{3}\mathcal{A}_{\epsilon}\psi_{\epsilon},$
for some operator $\mathcal{A}_{\epsilon}$ whose Schwartz kernel is given by
the second term of the expansion in Corollary 2.6. Here due to Lemma 2.11 we
have that
$\mathcal{A}_{\epsilon}:H^{1/2}(\mathbb{D};ds^{\prime})^{*}\to
H^{1/2}(\mathbb{D};ds^{\prime})$
with operator norm bounded uniformly in $\epsilon$. From here on we will
denote by $\mathcal{A}_{\epsilon}$ any operator which takes
$H^{1/2}(\mathbb{D};ds^{\prime})^{*}\to H^{1/2}(\mathbb{D};ds^{\prime})$ whose
operator norm is bounded uniformly in $\epsilon$.
Doing the same thing for the second term of (5.1) while using Lemma 2.2, Lemma
2.11, and (5.9) gives
(5.11) $\displaystyle H(x)\int_{\Gamma_{\epsilon,a}}\log
d_{h}(x,y)\partial_{\nu}u_{\epsilon}(y){\rm dvol}_{h}(y)$ $\displaystyle=$
$\displaystyle-H(x^{*})|M|\log\epsilon+\epsilon^{2}H(x^{*})R_{\log,a}\psi_{\epsilon}+\epsilon^{3}\mathcal{A}_{\epsilon}\psi_{\epsilon}$
$\displaystyle+O_{H^{1/2}(\mathbb{D})}(\epsilon\log\epsilon)$
where $R_{\log,a}$ is defined at the very beginning of Section 4.2. Here
$O_{H^{1/2}(\mathbb{D})}(\epsilon\log\epsilon)$ denotes a function with
$H^{1/2}(\mathbb{D};ds^{\prime})$ norm vanishing to order
$\epsilon\log\epsilon$. Note the volume for we use here is now the Euclidean
one rather than ${\rm dvol}_{h}$ given by (5.7).
Finally, for the third term of (5.1) we get by using the coordinate expression
derived in Corollary 2.9 and the estimate of Lemma 2.11:
(5.12)
$\displaystyle\int_{\Gamma_{\epsilon,a}}\left({\rm{II}}_{x}\left(\frac{\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)-{\rm{II}}_{x}\left(\frac{*\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)\right)\partial_{\nu}u_{\epsilon}(y)$
$\displaystyle=$
$\displaystyle\epsilon^{2}(\lambda_{1}-\lambda_{2})R_{\infty,a}\psi_{\epsilon}$
$\displaystyle+\epsilon^{3}\mathcal{A}_{\epsilon}\psi_{\epsilon},$
where $R_{\infty,a}$ is defined in Section 4. Inserting into (5.1) the
identities (5.10), (5.11), and (5.12) we have
$\displaystyle\frac{-F(x^{*})-C_{\epsilon,a}-H(x^{*})|M|(4\pi)^{-1}\log\epsilon}{\epsilon}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}L_{a}\psi_{\epsilon}-\epsilon\frac{H(x^{*})}{4\pi}R_{\log,a}\psi_{\epsilon}+\epsilon\frac{\lambda_{1}-\lambda_{2}}{16\pi}R_{\infty,a}\psi_{\epsilon}$
$\displaystyle+a\epsilon\int_{\mathbb{D}}R(x^{\epsilon}(t^{\prime}),x^{\epsilon}(s^{\prime}))\psi_{\epsilon}(s^{\prime})+\epsilon^{2}\mathcal{A}_{\epsilon}\psi_{\epsilon}$
$\displaystyle+O_{H^{1/2}(\mathbb{D})}(\log\epsilon).$
We would like to approximate the integral kernel
$R(x^{\epsilon}(t^{\prime}),x^{\epsilon}(s^{\prime}))$ by the constant
$R(x^{*},x^{*})$ and this is the content of
###### Lemma 5.1.
Let
$T_{\epsilon}:C_{c}^{\infty}(\mathbb{D})\to{\mathcal{D}}^{\prime}(\mathbb{D})$
be the operator defined by the integral kernel
$R(x^{\epsilon}(t^{\prime};x^{*}),x^{\epsilon}(s^{\prime};x^{*}))-R(x^{*},x^{*}).$
Then
$\|T_{\epsilon}\|_{(H^{1/2}(\mathbb{D}))^{*}\to H^{1/2}(\mathbb{D})}\leq
C\epsilon\log\epsilon.$
###### Proof.
Set
$T(t^{\prime},s^{\prime}):=R(x(t^{\prime};x^{*}),x(s^{\prime};x^{*}))-R(x^{*},x^{*})$
for $t^{\prime}$ and $s^{\prime}$ small and extend it to be a smooth compactly
supported kernel otherwise. The kernel for $T_{\epsilon}$ is then $T(\epsilon
t^{\prime},\epsilon s^{\prime})$. Note that
(5.14) $\displaystyle T(0,0)=0.$
Observe that if $\chi\in C_{c}^{\infty}(\mathbb{R}^{2})$ which is identically
$1$ on $\mathbb{D}$ then the operator $T_{\epsilon}$ acting on distributions
supported in $\mathbb{D}$ is given by the Schwartz kernel
$\chi(s^{\prime})\chi(t^{\prime})T(\epsilon t^{\prime},\epsilon s^{\prime})$
for $t^{\prime},s^{\prime}\in\mathbb{D}$ and $\epsilon>0$ small.
Observe that $T(t^{\prime},s^{\prime})$ is the integral kernel for an operator
in $\Psi^{-3}_{cl}(\mathbb{R}^{2})$. Applying Prop 2.8 in Chap 7 of [34] we
can deduce that for all $k$ we may choose $N$ sufficiently large such that
$T(t^{\prime},s^{\prime})=\sum_{l=0}^{N}\left(q_{l}(t^{\prime},t^{\prime}-s^{\prime})+p_{l}(t^{\prime},t^{\prime}-s^{\prime})\log|t^{\prime}-s^{\prime}|\right)+R_{k}(t^{\prime},s^{\prime}),$
where for each integer $l$ and multi-index $\gamma$,
$D^{\gamma}_{t^{\prime}}q_{l}(t^{\prime},\cdot)$ is a bounded continuous
function of $t^{\prime}$ with value in the space of smooth (away from the
origin) homogeneous distributions of degree $l+1$, $p_{l}(t^{\prime},\cdot)$
are homogenous polynomials of degree $l+1$ with coefficients which are smooth
functions of $t^{\prime}$, and for all multi-indices $\gamma$,
$D^{\gamma}_{t^{\prime}}R_{k}(t^{\prime},\cdot)$ bounded continuous function
of $t^{\prime}$ with value in $C^{k}(\mathbb{R}^{2})$.
Using (5.14) along with the homogenenity degree of $q_{l}$ and $p_{l}$ we see
that $R_{k}(0,0)=0$. Therefore, for $s^{\prime},t^{\prime}\in\mathbb{D}$ the
integral kernel of $T_{\epsilon}$ is
(5.15) $\displaystyle T_{\epsilon}(t^{\prime},s^{\prime})$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{N}\left(\epsilon^{l+1}q_{l}(\epsilon
t^{\prime},t^{\prime}-s^{\prime})+\epsilon^{l+1}p_{l}(\epsilon
t^{\prime},t^{\prime}-s^{\prime})\log\epsilon+\epsilon^{l+1}p_{l}(\epsilon
t^{\prime},t^{\prime}-s^{\prime})\log|t^{\prime}-s^{\prime}|\right)$
$\displaystyle+$ $\displaystyle R_{k}(\epsilon t^{\prime},\epsilon
s^{\prime}).$
The kernel $R_{k}(s^{\prime},t^{\prime})$ is sufficiently smooth. Therefore,
by doing a Taylor expansion and using $R_{k}(0,0)=0$ we see that the integral
kernel $\chi(s^{\prime})\chi(t^{\prime})R_{k}(\epsilon t^{\prime},\epsilon
s^{\prime})$ takes $H^{1/2}(\mathbb{D})^{*}\to H^{1/2}(\mathbb{D})$ with norm
$\epsilon$. The worst term in the polyhomogeneous expansion part of (5.15)
happens when $l=0$ and this term is given by
$\epsilon q_{0}(\epsilon t^{\prime},z^{\prime})+p_{0}(\epsilon
t^{\prime},z^{\prime})\log|z^{\prime}|+\epsilon\log\epsilon p_{0}(\epsilon
t^{\prime},z^{\prime}),$
where $z^{\prime}=t^{\prime}-s^{\prime}$. Recall that both $q_{0}$ and $p_{0}$
are homogeneous of degree $1$ in $z^{\prime}$ so writing $z^{\prime}=r\omega$
then applying Lemma 2.11 we have uniform estimates in $\epsilon$ for the
kernels $\chi(s^{\prime})\chi(t^{\prime})q_{0}(\epsilon
t^{\prime},t^{\prime}-s^{\prime})$ and
$\chi(s^{\prime})\chi(t^{\prime})p_{0}(\epsilon
t^{\prime},t^{\prime}-s^{\prime})$. For the term involving
$\log|s^{\prime}-t^{\prime}|$, we use the fact that
$p_{0}(t^{\prime},z^{\prime})$ is a linear function in $z^{\prime}$ whose
coefficients are smooth functions of $t^{\prime}$. Therefore, if $u\in
C^{\infty}_{c}(\mathbb{R}^{2})$,
$\displaystyle\int_{\mathbb{R}^{2}}\chi(t^{\prime})p_{0}(\epsilon
t^{\prime},t^{\prime}-s^{\prime})\log|t^{\prime}-s^{\prime}|u(s^{\prime})ds^{\prime}$
$\displaystyle=$
$\displaystyle\chi(t^{\prime})\int_{\mathbb{R}^{2}}\sum_{j}c_{j}(\epsilon
t^{\prime})(t_{j}-s_{j})\log|s^{\prime}-t^{\prime}|u(s^{\prime})ds^{\prime}$
$\displaystyle=$ $\displaystyle\chi(t^{\prime})\sum_{j}c_{j}(\epsilon
t^{\prime})\int_{\mathbb{R}^{2}}a_{j}(\xi^{\prime})\hat{u}(\xi^{\prime})e^{it^{\prime}\cdot\xi^{\prime}}d\xi^{\prime},$
where $a_{j}(\cdot)\in{\mathcal{S}}^{\prime}(\mathbb{R}^{2})$ for $j=1,2$ are
derivatives of the distribution ${\rm PF}|\xi|^{-2}$ with respect to
$\partial_{\xi_{j}}$. We refer the reader to (8.31) in Chapt 3 of [33] for the
definition of the the distribution ${\rm PF}|\xi|^{-2}$. From this we see that
the integral kernel
$\chi(t^{\prime})\chi(s^{\prime})p_{0}(\epsilon
t^{\prime},t^{\prime}-s^{\prime})\log|t^{\prime}-s^{\prime}|$
also maps $H^{1/2}(\mathbb{D})^{*}\to H^{1/2}(\mathbb{D})$ with uniform bound
in $\epsilon$. ∎
Due to Lemma 5.1 we can write (5.1) as
(5.16)
$\displaystyle\frac{2\pi}{\epsilon}\left(aR(x^{*},x^{*})|M|-F(x^{*})-C_{\epsilon,a}-\frac{H(x^{*})|M|\log\epsilon}{4\pi}\right)=$
$\displaystyle\left(L_{a}-\frac{\epsilon
H(x^{*})}{2}R_{\log,a}+\frac{\epsilon(\lambda_{1}-\lambda_{2})}{8}R_{\infty,a}\right)\psi_{\epsilon}$
$\displaystyle+\epsilon
T_{\epsilon}\psi_{\epsilon}++\epsilon^{2}\mathcal{A}_{\epsilon}\psi_{\epsilon}+O_{H^{1/2}(\mathbb{D})}(\log\epsilon).$
We hit both sides with $L_{a}^{-1}$ and use (4.12) and (4.4) to get the
identity
(5.17)
$\displaystyle\frac{2\pi}{\epsilon}\left(aR(x^{*},x^{*})|M|-F(x^{*})-C_{\epsilon,a}-\frac{H(x^{*})|M|\log\epsilon}{4\pi}\right)\frac{1}{K_{a}(1-|t^{\prime}|^{2})^{1/2}}=$
$\displaystyle\left(I-\frac{\epsilon
H(x^{*})}{2}L_{a}^{-1}R_{\log,a}+\frac{\epsilon(\lambda_{1}-\lambda_{2})}{8}L_{a}^{-1}R_{\infty,a}+T^{\prime}_{\epsilon}\right)\psi_{\epsilon}+O_{H^{1/2}(\mathbb{D})^{*}}(\log\epsilon).$
for some $T^{\prime}_{\epsilon}:H^{1/2}(\mathbb{D})^{*}\to
H^{1/2}(\mathbb{D})^{*}$ with operator norm $O(\epsilon^{2}\log\epsilon)$. Use
the mapping properties from Lemma 4.2 we see that the right side can be
inverted by Neumann series to deduce
(5.18) $\displaystyle\psi_{\epsilon}=-\frac{2\pi C_{\epsilon,a}}{\epsilon
K_{a}(1-|t^{\prime}|^{2})^{1/2}}+C_{\epsilon,a}O_{H^{1/2}(\mathbb{D})^{*}}(1)+O_{H^{1/2}(\mathbb{D})^{*}}(\epsilon^{-1}\log\epsilon).$
Insert (5.18) into (5.9) we get that
(5.19) $\displaystyle
C_{\epsilon,a}=\frac{|M|K_{a}}{4a\epsilon\pi^{2}}+C^{\prime}_{\epsilon,a}$
with $C^{\prime}_{\epsilon,a}=O(\log\epsilon)$. Insert (5.19) into (5.18)
(5.20)
$\displaystyle\psi_{\epsilon}=\frac{-|M|}{2a\pi\epsilon^{2}(1-|t^{\prime}|^{2})^{1/2}}+\psi_{\epsilon}^{\prime}=-\frac{|M|K_{a}}{2a\pi\epsilon^{2}}L_{a}^{-1}(1)+\psi^{\prime}_{\epsilon}$
with $\|\psi_{\epsilon}^{\prime}\|_{H^{1/2}(\mathbb{D};ds^{\prime})^{*}}\leq
C\epsilon^{-1}\log\epsilon$. Insert (5.19) and (5.20) into (5.17) we get
(5.21)
$\displaystyle\frac{2\pi}{\epsilon}\left(aR(x^{*},x^{*})|M|-F(x^{*})-C^{\prime}_{\epsilon,a}-\frac{H(x^{*})|M|\log\epsilon}{4\pi}\right)\frac{1}{K_{a}(1-|t^{\prime}|^{2})^{1/2}}=$
$\displaystyle\psi^{\prime}_{\epsilon}+\frac{|M|K_{a}}{2\pi\epsilon}\left(\frac{H(x^{*})}{2}L_{a}^{-1}R_{\log,a}-\frac{(\lambda_{1}-\lambda_{2})}{8}L_{a}^{-1}R_{\infty,a}\right)L_{a}^{-1}(1)+O_{H^{1/2}(\mathbb{D})^{*}}(\log\epsilon).$
Inserting the expression (5.20) into (5.9) we get that
(5.22)
$\displaystyle\int_{\mathbb{D}}\psi^{\prime}_{\epsilon}(s^{\prime})ds_{1}ds_{2}=O(1).$
Multiply (5.21) by $\epsilon$ then integrate over $\mathbb{D}$ . Then (5.22)
implies
$\displaystyle C^{\prime}_{\epsilon,a}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}H(x^{*})|M|\log\epsilon+aR(x^{*},x^{*})|M|-F(x^{*})$
$\displaystyle-\frac{|M|H(x^{*})K_{a}^{2}}{16\pi^{3}}\int_{\mathbb{D}}L_{a}^{-1}R_{\log,a}L_{a}^{-1}1(s^{\prime})ds^{\prime}$
$\displaystyle+\frac{|M|(\lambda_{1}-\lambda_{2})K_{a}^{2}}{64\pi^{3}}\int_{\mathbb{D}}L_{a}^{-1}R_{\infty,a}L_{a}^{-1}1(s^{\prime})ds^{\prime}+O(\epsilon\log\epsilon).$
Since $L_{a}^{-1}$ is self-adjoint, we can express the last two integrals more
explicitly:
$\displaystyle\int_{\mathbb{D}}L_{a}^{-1}R_{\log,a}L_{a}^{-1}1(s^{\prime})ds^{\prime}=K_{a}^{-2}\langle(1-|s^{\prime}|^{2})^{-1/2},R_{\log,a}(1-|s^{\prime}|^{2})^{-1/2}\rangle,$
$\displaystyle\int_{\mathbb{D}}L_{a}^{-1}R_{\infty,a}L_{a}^{-1}1(s^{\prime})ds^{\prime}=K_{a}^{-2}\langle(1-|s^{\prime}|^{2})^{-1/2},R_{\infty,a}(1-|s^{\prime}|^{2})^{-1/2}\rangle.$
We summarize this calculation into the following:
###### Proposition 5.2.
We have that
$\psi_{\epsilon}=\frac{-|M|}{2a\pi\epsilon^{2}(1-|t^{\prime}|^{2})^{1/2}}+\psi_{\epsilon}^{\prime}$
with
$\psi_{\epsilon}^{\prime}=O_{H^{1/2}(\mathbb{D})^{*}}(\epsilon^{-1}\log\epsilon)$.
Furthermore
(5.24) $\displaystyle C_{\epsilon,a}=$
$\displaystyle\frac{|M|K_{a}}{4a\epsilon\pi^{2}}-\frac{1}{4\pi}H(x^{*})|M|\log\epsilon+aR(x^{*},x^{*})|M|-F(x^{*})$
$\displaystyle-\frac{|M|H(x^{*})}{16\pi^{3}}\langle(1-|s^{\prime}|^{2})^{-1/2},R_{\log,a}(1-|s^{\prime}|^{2})^{-1/2}\rangle$
$\displaystyle+\frac{|M|(\lambda_{1}-\lambda_{2})}{64\pi^{3}}\langle(1-|s^{\prime}|^{2})^{-1/2},R_{\infty,a}(1-|s^{\prime}|^{2})^{-1/2}\rangle$
$\displaystyle+O(\epsilon\log\epsilon),$
where $F$ is the unique solution to (5.4) and $R(x^{*},x^{*})$ is the
evaluation at $(x,y)=(x^{*},x^{*})$ of the kernel $R(x,y)$ in (1.1).
Observe that in the case of the disc (i.e. $a=1$) we have that
$\langle(1-|s^{\prime}|^{2})^{-1/2},R_{\log,a}(1-|s^{\prime}|^{2})^{-1/2}\rangle=\pi^{2}\left({8}\log
2-{6}\right)$
by Lemma 4.5 and
$\langle(1-|s^{\prime}|^{2})^{-1/2},R_{\infty,a}(1-|s^{\prime}|^{2})^{-1/2}\rangle=0$
by Lemma 4.6. Thus the formula (5.24) simplifies to
$\displaystyle C_{\epsilon}:=C_{\epsilon,1}$ $\displaystyle=$
$\displaystyle\frac{|M|K_{a}}{4a\epsilon\pi^{2}}-\frac{1}{4\pi}H(x^{*})|M|\log\epsilon+aR(x^{*},x^{*})|M|-F(x^{*})$
$\displaystyle-$ $\displaystyle\frac{|M|H(x^{*})}{16\pi}\left(8\log
2-6\right)+O(\epsilon\log\epsilon),$
when $a=1$.
### 5.2. Proof of Theorems 1.2 and 1.3
We now prove Theorem 1.3. Theorem 1.2 follows from the explicit expression for
$C_{\epsilon}$ in (5.1).
By the result of Appendix A we have that
$u_{\epsilon}=\mathbb{E}[\tau_{\Gamma_{\epsilon,a}}|X_{0}=x]$ solves the mixed
boundary value problem (5.1) so using Proposition 5.2, (5.3), and (5.7), the
expansion for $\mathbb{E}[\tau_{\Gamma_{\epsilon,a}}|X_{0}=x]$ is given by
$u_{\epsilon}(x)=\mathbb{E}[\tau_{\Gamma_{\epsilon,a}}|X_{0}=x]=F(x)+C_{\epsilon,a}-|M|G(x,x^{*})+r_{\epsilon}(x)$
for each $x\in M\backslash\Gamma_{\epsilon,a}$. Here $F$ is the unique
solution to (5.4) and the remainder $r_{\epsilon}$ is given by
(5.26) $\displaystyle r_{\epsilon}(x)=\int_{\partial
M}(G(x,y)-G(x,x^{*}))\partial_{\nu}u_{\epsilon}(y){\rm dvol}_{h}(y).$
Let $K\subset\overline{M}$ be a compact subset of $\overline{M}$ which has
positive distance from $x^{*}$ and consider $x\in K$. Writing out this
integral in the $x^{\epsilon}(\cdot;x^{*})$ coordinate system and use (5.8),
(5.7), and the expression of $\psi_{\epsilon}$ derived in Proposition 5.2 we
get
$\displaystyle r_{\epsilon}(x)$ $\displaystyle=$
$\displaystyle\epsilon\int_{\mathbb{D}}\frac{-|M|}{2\pi(1-|s^{\prime}|^{2})^{1/2}}L(x,\epsilon
s^{\prime})(1+\epsilon^{2}Q_{\epsilon}(s^{\prime}))ds^{\prime}$
$\displaystyle+$ $\displaystyle
a\epsilon^{3}\int_{\mathbb{D}}\psi^{\prime}_{\epsilon}(s^{\prime})L(x,\epsilon
s^{\prime})(1+\epsilon^{2}Q_{\epsilon}(s^{\prime}))ds^{\prime}$
for some function $L(x,s^{\prime})$ jointly smooth in $(x,s^{\prime})\in
K\times\mathbb{D}$. The second integral formally denotes the duality between
$H^{1/2}(\mathbb{D})^{*}$ and $H^{1/2}(\mathbb{D})$. The estimate for
$\psi^{\prime}_{\epsilon}$ derived in Proposition 5.2 now gives for any
integer $k$ and any compact set $K$ not containing $x^{*}$,
$\|r_{\epsilon}\|_{C^{k}(K)}\leq C_{k,K}\epsilon$.
Our pseudodifferential characterization of $G_{\partial M}$ also allows us to
compute the asymptotic of the average $\int_{M}u_{\epsilon}$. Indeed,
integrating (5.3) over $M$ we get
(5.28) $\int_{M}u_{\epsilon}{\rm dvol}_{g}=\int_{M}F(x){\rm
dvol}_{g}+C_{\epsilon,a}|M|+\int_{M}\int_{\Gamma_{\epsilon,a}}G(x,y)\partial_{\nu}u_{\epsilon}(y){\rm
dvol}_{h}(y){\rm dvol}_{g}(x).$
We compute the last integral by noting that
$v(x):=\int_{\Gamma_{\epsilon,a}}G(x,y)\partial_{\nu}u_{\epsilon}(y){\rm
dvol}_{h}(y)$
is the unique solution to the Dirichlet boundary value problem:
(5.29) $\displaystyle\Delta_{g}v=0,\ \ v(x)\mid_{\partial
M}=\int_{\Gamma_{\epsilon,a}}G_{\partial M}(x,y)u_{\epsilon}(y){\rm
dvol}_{h}(y)\in H^{1/2}(\partial M).$
We concluded the boundary value is in $H^{1/2}$ because $G_{\partial
M}\in\Psi^{-1}_{cl}(\partial M)$ by (1.1).
Let a sequence of smooth functions $f_{j}\to\partial_{\nu}u_{\epsilon}$ in
$H^{-1/2}(\partial M)$ and let $v_{j}$ solve
$\Delta_{g}v_{j}=0,\ \ v_{j}(x)\mid_{\partial M}=\int_{\partial M}G_{\partial
M}(x,y)f_{j}(y){\rm dvol}_{h}(y).$
Standard elliptic theory shows that $v_{j}\to v$ in $H^{1}(M)$. Therefore
$\displaystyle\int_{M}\int_{\Gamma_{\epsilon,a}}G(x,y)\partial_{\nu}u_{\epsilon}(y)$
$\displaystyle=$ $\displaystyle\lim_{j}\int_{M}\int_{\partial
M}G(x,y)f_{j}(y)$ $\displaystyle=$ $\displaystyle\lim_{j}\int_{\partial
M}f_{j}(y)\int_{M}G(x,y)$ $\displaystyle=$
$\displaystyle\lim_{j}\int_{\partial M}f_{j}(y)F(y)$ $\displaystyle=$
$\displaystyle\langle\partial_{\nu}u_{\epsilon},F\rangle=-F(x^{*})|M|+O(\epsilon),$
where $F$ is the solution to the boundary value problem (5.4) and
$\langle\cdot,\cdot\rangle$ denotes the pairing between $H^{-1/2}(\partial M)$
and $H^{1/2}(\partial M)$. The last equality comes from (5.2), smoothness of
$F$, and ${\rm supp}(\partial_{\nu}u_{\epsilon})\subset\Gamma_{\epsilon,a}$.
Inserting this into (5.28) we have
$\int_{M}u_{\epsilon}{\rm dvol}_{g}=\int_{M}F(x){\rm
dvol}_{g}+C_{\epsilon,a}|M|-F(x^{*})|M|+O(\epsilon).$
The constant $C_{\epsilon,a}$ is given by Proposition 5.2.
## 6\. Appendix A -Elliptic Equation for the first passage time
In this appendix we show that $u(x):=\mathbb{E}[\tau_{\Gamma}|X_{0}=x]$
satisfies the boundary value problem (5.1). This is standard material but we
could not find a suitable reference which precisely addresses our setting. As
such we are including this appendix for the convenience of the reader.
Let $(M,g,\partial M)$ be an orientable compact connected Riemannian manifold
with non-empty smooth boundary oriented by ${\rm dvol}_{g}$. Let also
$(X_{t},\mathbb{P}_{x})$ be the Brownian motion on $M$ starting at $x$, that
is, the stochastic process generated by the Laplace-Beltrami operator
$\Delta_{g}$. Let $\Gamma$ be a geodesic ball on $\partial M$ with radius
$\varepsilon>0$. We denote by $\tau_{\Gamma}$ the first time the Brownian
motion $X_{t}$ hits $\Gamma$, that is
$\tau_{\Gamma}:=\inf\\{t\geq 0:X_{t}\in\Gamma\\}.$
We set
$\mathcal{P}_{\Gamma}(t,x):=\mathbb{P}[\tau_{\Gamma}\leq t|X_{0}=x].$
Let us note that $\mathcal{P}_{\Gamma}(t,x)$ is the probability that the
Brownian motion hits $\Gamma$ before or at time $t$, and therefore, satisfies
(6.1) $\mathcal{P}_{\Gamma}(0,x)=0,\quad x\in M\setminus\Gamma,$ (6.2)
$\mathcal{P}_{\Gamma}(t,x)=1,\quad(t,x)\in[0,\infty)\times\Gamma.$
Note that, for any compact subset $\Gamma\subset M$, it follows444Note that in
[6] and [12], the authors consider the manifold together with its boundary,
and $C_{c}^{\infty}(M)$, $C_{0}^{\infty}(M)$ denote the set of smooth (up to
the boundary) functions with compact support. In case of compact manifold,
these sets coincide with $C^{\infty}(\overline{M})$.
$\text{Cap}(\Gamma,M):=\inf_{u\in
C^{\infty}(\overline{M}),\left.u\right|_{\Gamma}=1}\int_{M}\|\nabla
u\|^{2}{\rm dvol}_{g}=0.$
Then, [13, Theorem 1.5] implies that $(M,g,\partial M)$ is parabolic, that is,
the probability that the Brownian motion ever hits any compact set $F$ with
non-empty interior is $1$. Since $\Gamma\subset\partial M$ is connected with
non-empty interior on $\partial M$, we can extend $M$ to a compact connected
Riemannian manifold $\tilde{M}$ such that $\overline{\tilde{M}\setminus M}$ is
compact with non-empty interior and $\overline{\tilde{M}\setminus M}\cap
M=\Gamma$. Note that, the Brownian motion, starting at any point
$M\setminus\Gamma$, hits $\overline{\tilde{M}\setminus M}$ if and only if it
hits $\Gamma$. Therefore, the parabolicity condition of $(M,g)$ gives
(6.3) $\lim_{t\rightarrow\infty}\mathcal{P}_{\Gamma}(t,x)=1,\quad x\in M.$
Further, let us define the mean first arrival time $u$, as
(6.4)
$u(x):=\mathbb{E}[\tau_{\Gamma}|X_{0}=x]:=\int_{0}^{\infty}td\mathcal{P}_{\Gamma}(t,x),$
where the integral is a Riemann-Stieltjes integral. To investigate $u$, let us
recall some properties of $\mathcal{P}_{\Gamma}$. By Remmark 2.1 in [6], it
follows that
$1-\mathcal{P}_{\Gamma}(t,x)=\left(e^{t\Delta_{mix}}1\right)(x),$
where $e^{t\Delta_{mix}}$ is the semigroup with infinitesimal generator
$\Delta_{mix}$, and $\Delta_{mix}$ is the Laplace operator $\Delta_{g}$
corresponding to the Dirichlet boundary condition on $\Gamma$ and Neumann
boundary condition on $\partial M\setminus\Gamma$, which is defined as follows
(6.5) $\displaystyle\mathrm{D}(\Delta_{mix}):=\\{u\in
H^{1}(M):\;\Delta_{g}u\in
L^{2}\;\left.u\right|_{\Gamma}=0,\;\left.\partial_{\nu}u\right|_{\Gamma^{c}}=0\\}$
(6.6) $\displaystyle\Delta_{mix}u=\Delta_{g}u\quad
u\in\mathrm{D}(\Delta_{mix}).$
In (6.5) we define $\partial_{\nu}u\in H^{-1/2}(\partial M)$ using the same
method for defining the Dirichlet to Neumann map. That is, for $u\in H^{1}(M)$
such that $\Delta_{g}u\in L^{2}(M)$, the distribution
$\partial_{\nu}u\left.\right|_{\partial M}\in H^{-1/2}(\partial M)$ acts on
$f\in H^{1/2}(\partial M)$ via
$\langle\partial_{\nu}u\left.\right|_{\partial M},f\rangle:=\int_{M}\Delta
u_{g}\overline{v_{f}}\;dvol_{g}+\int_{M}g(du,dv_{f})\;dvol_{g},$
where $v_{f}\in H^{1}(M)$ is the harmonic extension of $f$. We say that
$\partial_{\nu}u\left.\right|_{\overline{\omega}}=0$, for non-empty open set
$\omega\subset\partial M$, if $\langle\partial_{\nu}u\left.\right|_{\partial
M},f\left.\right|_{\partial M}\rangle=0$ for all $f\in H^{1/2}(\partial M)$
such that $f|_{\partial M\setminus\overline{\omega}}=0$.
Note that if $u$ sufficiently regular, for instance $u\in H^{2}(M)$, then
$\langle\partial_{\nu}u\left.\right|_{\partial M},f\rangle$ is equal to the
boundary integral of $\partial_{\nu}u\left.\right|_{\partial M}$ and $f$.
In fact, $\Delta_{mix}$ can be equivalently defined by quadratic form; see
Proposition 7.1 in Appendix B. Moreover, $\Delta_{mix}$ is the non-positive
self-adjoint operator with the discrete spectrum, consisting of negative
eigenvalues accumulating at $-\infty$; see Proposition 7.1 in Appendix. Hence,
$\Delta_{mix}$ satisfies the quadratic estimate
$\int_{0}^{\infty}\|t\Delta_{mix}(1+t^{2}\Delta_{mix}^{2})^{-1}u\|^{2}_{L^{2}}\frac{dt}{t}\leq
C\|u\|^{2},$
for some $C>0$ and all $u\in L^{2}(M)$; see for instance [19, p. 221].
Therefore, $\Delta_{mix}$ admits the functional calculus defined in [21].
###### Remark 6.1.
The functional calculus in [21] is defined for a concrete operator, which is
denoted by $T$ in the notation used in that article. However, $\Delta_{mix}$
satisfy all necessary conditions to admit this functional calculus.
Therefore, the semigroup $e^{t\Delta_{mix}}$, which is contracting by Hille-
Yosida theorem [14, Theorem 8.2.3], can be defined as follows
$e^{t\Delta_{mix}}u=\frac{1}{2\pi
i}\int_{\gamma_{a,\alpha}}e^{t\zeta}(\zeta-\Delta_{mix})^{-1}ud\zeta,\qquad
u\in L^{2}(M),$
where $a\in(\tau,0)$, $\alpha\in(0,\frac{\pi}{2})$, and $\gamma_{a,\alpha}$ is
the anti-clockwise oriented curve:
$\gamma_{a,\alpha}:=\\{\zeta\in\mathbb{C}:\textrm{Re}\zeta\leq a,\text{
}|\textrm{Im}\zeta|=|\textrm{Re}\zeta-a|\tan\alpha\\}.$
Let $\varepsilon>0$ such that $a+\varepsilon<0$. Then
$\Delta_{mix}+\varepsilon$ is also a negative self-adjoint operator, and hence
generates contracting semigroup, $e^{t(\Delta_{mix}+\varepsilon)}$, as above.
By definition, we obtain, for $u\in L^{2}(M)$,
(6.7) $\displaystyle e^{t\Delta_{mix}}u$ $\displaystyle=\frac{1}{2\pi
i}\int_{\gamma_{a,\alpha}}e^{t\zeta}(\zeta-\Delta_{mix})^{-1}ud\zeta$
$\displaystyle=\frac{e^{-t\varepsilon}}{2\pi
i}\int_{\gamma_{a,\alpha}}e^{t(\zeta+\varepsilon)}(\zeta+\varepsilon-(\Delta_{mix}+\varepsilon))^{-1}ud\zeta$
$\displaystyle=\frac{e^{-t\varepsilon}}{2\pi
i}\int_{\gamma_{a+\varepsilon,\alpha}}e^{t\xi}(\xi-(\Delta_{mix}+\varepsilon))^{-1}ud\xi=e^{-t\varepsilon}e^{t(\Delta_{mix}+\varepsilon)}u,$
where
$\gamma_{a+\varepsilon,\alpha}=\gamma_{a,\alpha}+\varepsilon\subset\\{\textrm{Re}\xi<0\\}$.
Let $f_{1}$ the constant function on $M$ equals $1$. By Theorem 8.2.2 in [14],
we know, for $\lambda>0$,
$(\lambda-(\Delta_{mix}+\varepsilon))^{-1}f_{1}=\int_{0}^{\infty}e^{-\lambda
t}e^{t(\Delta_{mix}+\varepsilon)}f_{1}dt.$
Let us choose $\lambda=\varepsilon$, then, by using (6.7), we obtain
$-\Delta_{mix}^{-1}f_{1}=\int_{0}^{\infty}e^{-\varepsilon
t}e^{t(\Delta_{mix}+\varepsilon)}f_{1}dt=\int_{0}^{\infty}e^{t\Delta_{mix}}f_{1}dt$
and hence,
(6.8)
$\int_{0}^{\infty}1-\mathcal{P}_{\Gamma}(t,x)dt=-(\Delta_{mix}^{-1}f_{1})(x)<\infty.$
Therefore, the dominated convergence theorem implies
$\lim_{b\rightarrow\infty}\int_{0}^{b}\left(\mathcal{P}_{\Gamma}(b,x)-\mathcal{P}_{\Gamma}(t,x)\right)dt=\int_{0}^{\infty}1-\mathcal{P}_{\Gamma}(t,x)dt<\infty.$
Hence, by using (6.4) and integration by parts, we obtain
$\displaystyle u(x)$
$\displaystyle=\lim_{b\rightarrow\infty}\left(\mathcal{P}_{\Gamma}(b,x)b-\int_{0}^{b}\mathcal{P}_{\Gamma}(t,x)dt\right)=\lim_{b\rightarrow\infty}\int_{0}^{b}\left(\mathcal{P}_{\Gamma}(b,x)-\mathcal{P}_{\Gamma}(t,x)\right)dt<\infty$
$\displaystyle=\int_{0}^{\infty}1-\mathcal{P}_{\Gamma}(t,x)dt.$
Therefore, by (6.8), we obtain
$\Delta_{mix}u=-f_{1}=-1.$
In particular, $u\in\mathrm{D}(\Delta_{mix})$, and hence,
$u\mid_{\Gamma}=0,\qquad\partial_{\nu}u\mid_{\partial M\backslash\Gamma}=0.$
We see that (5.1) is satisfied.
## 7\. Appendix B - Quadratic Form
Let $(M,g,\partial M)$ be a compact connected Riemannian manifold with non-
empty smooth boundary. Let $\Gamma$ be a closed subset of $\partial M$ such
that $\partial M\setminus\overline{\Gamma^{c}}$ is a non-empty open set.
Consider the quadratic form
(7.1) $\displaystyle a[u,v]:=\int_{M}g(du,d\overline{v}){\rm dvol}_{g},\qquad
u,v\in\mathrm{D}(a):=\\{u\in H^{1}(M):\;\left.u\right|_{\Gamma}=0\\}.$
Note that $\mathrm{D}(a)$ is closed subspace of $H^{1}(M)$ containing
$H_{0}^{1}(M)$ and $a[\cdot,\cdot]$ is a non-negative, closed, densely defined
form. Therefore, by Friedrichs Theorem 2.23 in [15], it generates a non-
negative self-adjoint operator $-\Delta_{a}$ in $L^{2}(M)$ whose domain is
contained in $D(a)$ such that $(-\Delta_{a}u,u)_{L^{2}(M)}=a[u,u]$ for
$u\in\mathrm{D}(-\Delta_{a})$.
Let us show that the resolvents of $-\Delta_{a}$ are compact. Assume that $s$
belongs to the resolvent set of $-\Delta_{a}$. Since
$-\Delta_{a}(-\Delta_{a}-s)^{-1}:L^{2}(M)\rightarrow L^{2}(M)$ is bounded, it
is suffices to show that $D(-\Delta_{a})$, endowed with the graph norm,
compactly embedded into $L^{2}(M)$. Since, for $u\in D(-\Delta_{a})$,
$\|du\|^{2}_{L^{2}(M)}=(du,du)_{L^{2}(M)}=a[u,u]=(-\Delta_{a}u,u)_{L^{2}(M)}\leq\frac{1}{4}(\|-\Delta_{a}u\|_{L^{2}(M)}+\|u\|_{L^{2}(M)})^{2},$
we see that any bounded sequence in $D(-\Delta_{a})$, endowed with the graph
norm, is bounded in $H^{1}(M)$, and hence, it contains a Cauchy subsequence in
$L^{2}(M)$ by Rellich-Kondrachov theorem. This implies that the resolvents of
$-\Delta_{a}$ are compact, and hence, the spectrum of $-\Delta_{a}$ is
discrete, consisting of non-negative eigenvalues accumulating at $+\infty$.
Assume that $\lambda=0$ is an eigenvalue, and let $u_{0}\in\mathrm{D}(a)$ be a
corresponding eigenfunction.
Then the Poincaré-Wirtinger inequality gives, for some $C>0$,
$\left\|u-\frac{1}{|M|}\int_{M}u{\rm dvol}_{g}\right\|_{L^{2}}\leq
C\|du\|_{L^{2}}=C(-\Delta_{a}u,u)_{L^{2}}=0,$
so that $u_{0}$ is a constant in $L_{2}(M)$. Since $u_{0}\in H^{1}(M)$, we
conclude that $u_{0}=const$ in $L^{2}(\partial M)$, and hence, $u_{0}=0$ by
choice of $\Gamma$. Therefrore $\lambda=0$ is not an eigenvalue, and
consequently, the spectrum of $-\Delta_{a}$ is positive. For sake of
completeness, we prove the following well known result.
###### Proposition 7.1.
Let $-\Delta_{a}$ be the operator defined above and $-\Delta_{mix}$ be the
operator defined in Section 6, then $-\Delta_{a}=-\Delta_{mix}$. In
particular, $-\Delta_{mix}$ is a self-adjoint operator with the positive
discrete spectrum accumulating at infinity.
###### Proof.
Assume that $u$, $v\in H^{1}(M)$ and $\Delta_{g}u$, $\Delta_{g}u\in L^{2}(M)$.
Let $V$ be the harmonic extension of $v\left.\right|_{\partial M}$, then
$\omega:=V-v\in H^{1}_{0}(M)$ and $\Delta_{g}\omega\in L^{2}(M)$. By
$H^{2}$-regularity, $\omega\in H^{2}(M)$. The generalized Green’s identity
gives
$-\int_{M}u\Delta_{g}\omega\;dvol_{g}+\int_{M}\Delta_{g}u\omega\;dvol_{g}=\langle\partial_{\nu}u\left.\right|_{\partial
M},\omega\left.\right|_{\partial M}\rangle-\langle u\left.\right|_{\partial
M},\partial_{\nu}\omega\left.\right|_{\partial M}\rangle,$
where the first term of the right hand side vanishes since $\omega\in
H_{0}^{1}(M)$. Therefore we get
$\displaystyle 0$ $\displaystyle=-\int_{M}u\Delta_{g}\omega\;dvol_{g}+\langle
u\left.\right|_{\partial M},\partial_{\nu}\omega\left.\right|_{\partial
M}\rangle+\int_{M}\Delta_{g}u\omega\;dvol_{g}$
$\displaystyle=\int_{M}g(du,d\overline{\omega})\;dvol_{g}+\int_{M}\Delta_{g}u\omega\;dvol_{g}.$
Hence, we obtain
$\langle\partial_{\nu}u\left.\right|_{\partial M},v\left.\right|_{\partial
M}\rangle=\int_{M}\Delta_{g}u\overline{v}\;dvol_{g}+\int_{M}g(du,d\overline{v})\;dvol_{g}$
for $u$, $v\in H^{1}(M)$ and $\Delta_{g}u$, $\Delta_{g}u\in L^{2}(M)$.
Assume that $u\in\mathrm{D}(\Delta_{mix})\subset\mathrm{D}(a)$ and
$v\in\mathrm{D}(\Delta_{a})$, then, by above formula,
$\displaystyle
a[u,v]=\int_{M}g(du,d\overline{v})\;dvol_{g}=-\int_{M}\Delta_{g}u\overline{v}\;dvol_{g}+\langle\partial_{\nu}u\left.\right|_{\partial
M},v\left.\right|_{\partial M}\rangle$
Note that the last term vanishes since $u\in\mathrm{D}(\Delta_{mix})$ and
$v\in\mathrm{D}(a)$, so that
$a[u,v]=-\int_{M}\Delta_{g}u\overline{v}\;dvol_{g}=-\int_{M}\Delta_{mix}u\overline{v}\;dvol_{g}.$
Since this holds for all $v\in\mathrm{D}(\Delta_{a})$, it follows from Theorem
2.1., in [15], that $u\in\mathrm{D}(\Delta_{a})$ and
$\Delta_{a}u=\Delta_{mix}u$.
Conversely, assume that $u\in\mathrm{D}(\Delta_{a})$, then
$-\Delta_{g}u=-\Delta_{a}u\in L^{2}(M)$. Then, it follows
$\displaystyle\langle\partial_{\nu}u\left.\right|_{\partial
M},f\rangle=\int_{M}\Delta
u_{g}\overline{V_{f}}\;dvol_{g}+\int_{M}g(du,dV_{f})\;dvol_{g}=a[u,V_{f}]-a[u,V_{f}]=0.$
for any $f\in H^{1/2}(\partial M)$ such taht $f\left.\right|_{\Gamma}=0$. This
means that $\partial_{\nu}\left.u\right|_{\Gamma^{c}}=0$, so that $u\in
D(\Delta_{mix})$ and $\Delta_{a}u=\Delta_{mix}u$.
∎
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|
11institutetext: Metadata Research Center, Drexel University, Philadelphia PA
19104, USA
11email<EMAIL_ADDRESS>
# HIVE-4-MAT: Advancing the Ontology Infrastructure for Materials
Science††thanks: Supported by NSF Office of Advanced Cyberinfrastructure
(OAC): #1940239.
Jane Greenberg 0000-0001-7819-5360 Xintong Zhao 0000-0001-8401-356X Joseph
Adair 0000-0001-5646-9041 Joan Boone 0000-0001-5646-9041 Xiaohua Tony Hu
0000-0002-4777-3022
###### Abstract
This paper introduces Helping Interdisciplinary Vocabulary Engineering for
Materials Science (HIVE-4-MAT), an automatic linked data ontology application.
The paper provides contextual background for materials science, shared
ontology infrastructures, and knowledge extraction applications. HIVE-4-MAT’s
three key features are reviewed: 1) Vocabulary browsing, 2) Term search and
selection, and 3)Knowledge Extraction/Indexing, as well as the basics of named
entity recognition (NER). The discussion elaborates on the importance of
ontology infrastructures and steps taken to enhance knowledge extraction. The
conclusion highlights next steps surveying the ontology landscape, including
NER work as a step toward relation extraction (RE), and support for better
ontologies.
###### Keywords:
Materials Science Ontology Ontology Infrastructure Helping Interdisciplinary
Vocabulary Engineering Named Entity Recognition Knowledge Extraction
## 1 Introduction
A major challenge in materials science research today is that the artifactual
embodiment is primarily textual, even if it is in digital form. Researchers
analyze materials through experiments and record their findings in textual
documents such as academic literature and patents. The most common way to
extract knowledge from these artifacts is to read all the relevant documents,
and manually extract knowledge. However, reading is time-consuming, and it is
generally unfeasible to read and mentally synthesize all the relevant
knowledge[26, 28]. Hence, effectively extracting knowledge and data becomes a
problem. One way to address this challenge is through knowledge extraction
using domain-specific ontologies [18]. Unfortunately, materials science work
in this area is currently hindered by limited access to and use of relevant
ontologies. This situation underscores the need to improve the state of
ontology access and use for materials science research, which is the key goal
of the work presented here.
This paper introduces Helping Interdisciplinary Vocabulary Engineering for
Materials Science (HIVE-4-MAT), an automatic linked data ontology application.
The contextual background covers materials science, shared ontology
infrastructures, and knowledge extraction applications. HIVE-4-MAT’s basic
features are reviewed, followed by a brief discussion and conclusion
identifying next steps.
## 2 Background
### 2.1 Materials Science
Materials science is an interdisciplinary field that draws upon chemistry,
physics, engineering and interconnected disciplines. The broad aim is to
advance the application of materials for scientific and technical endeavors.
Accordingly, materials science researchers seek to discover new materials or
alter existing ones; with the overall aim of offering more robust, less
costly, and/or less environmentally harmful materials.
Materials science researchers primarily target solid matter, which retains its
shape and character compared to liquid or gas. There are four key classes of
solid materials: metals, polymers, ceramics, and composites. Researchers
essentially process (mix, melt, etc.) elements in a controlled way, and
measure performance by examining a set of properties. Table 1 provides two
high-level examples of materials classes, types, processes, and properties.
| MATERIAL
---
CLASS & TYPE
MANUFACTURING PROCESS | | PROPERTIES
---
(examples)
| Class:
---
Polymer
Type:
Polyethylene[21]
| Polymerization (distillation of
---
ethane into fractions, some of
which are combined with catalysts)
| Melt temperature
---
Tensile strength
Flexurile strength
(or bend strength)
Shrink Rate
| Class: Metal
---
Type: Steel
| Iron ore is heated and forged in
---
blast furnaces, where the impurities
are altered and carbon is added.
| Yield strength
---
Tensile strength
Thermal conductivity
Resistance to
wear/corrosion
Formability
Table 1: Examples of Materials classes and types, processes, and properties
The terms in Table 1 have multiple levels (sub-types or classes) and variants.
For example, there is stainless steel and surgical steel. Moreover, the
universe of properties, which is large, extends even further when considering
nano and kinetic materials. This table illustrates the language, hence the
ontological underpinnings, of materials science, which is invaluable for
knowledge extraction. Unfortunately, the availability of computationally ready
ontologies applicable to materials science is severely limited, particularly
compared to biomedicine and biology.
### 2.2 Ontologies: Shared Infrastructure and Knowledge Extraction
Applications
Ontologies have provided a philosophical foundation and motivation for
scientific inquiry since ancient times[15]. Today, computationally ready
ontologies conforming to linked data standards[9] offer a new potential for
data driven discovery[14]. Here, the biomedical and biology communities have
taken the lead in developing a shared infrastructure, through developments
such as the National Center for Biological Ontologies (NCBO) Bioportal[29, 4]
and the OBO foundry[25, 6]. Another effort is the FAIRsharing portal[23, 1],
providing access to a myraid of standards, databases, and other resources[31].
Shared ontology infrastructures help standardize language and support data
interoperability across communities. Additionally, the ontological resources
can aid knowledge extraction and discovery. Among one of the best known
applications in this area is Aronson’s [8] $MetaMap$, introduced in 2001. This
application extracts key information from textual documents, and maps the
indexing to the metathesaurus ontology. The $MetaMap$ application is widely-
used for extraction of biomedical information. The HIVE application[16],
developed by the Metadata Research Center, Drexel University, also supports
knowledge extraction in a same way, although results are limited by the depth
of the ontologies applied. For example, biomedicine ontologies, which often
have a rich and deep network of terms, will produce better results compared to
more simplistic ontologies targeting materials science[33, 32].
Overall, existing ontology infrastructure and knowledge extraction approaches
are applicable to materials science. In fact, biology and biomedical
ontologies are useful for materials science research, and researchers have
been inspired by these developments to develop materials science ontologies[7,
11, 17]. Related are nascent efforts developing shared metadata and ontology
infrastructures for materials science. Examples include the NIST Materials
Registry[5] and the Industrial Ontology Foundry[2]. These developments and the
potential to leverage ontologies for materials science knowledge extraction
motivate our work to advance HIVE-4-MAT. They have also had a direct impact on
exploring the use of NER to assist in the development richer ontologies for
materials science [33].
### 2.3 Named Entity Recognition
The expanse and depth of materials science ontologies is drastically limited,
pointing to a need for richer ontologies; however, ontology development via
manual processes is a costly undertaking. One way to address this challenge is
to through relation extraction and using computational approaches to develop
ontologies. To this end, named entity recognition (NER) can serve as an
invaluable first step, as explained here.
The goal of Named Entity Recognition (NER) is to recognize key information
that are related to predefined semantic types from input textual
documents[20]. As an important component of information extraction (IE), it is
widely applied in tasks such as information retrieval, text summarization,
question answering and knowledge extraction.
The semantic types can vary depending on specific task types. For example,
when extracting general information, the predefined semantic types can be
location, person, or organization. NER approaches have been also proven
effective to biomedical information extraction; an example from SemEval2013
task 9[24] about NER for drug-drug interaction is shown in Figure 1 below.
Figure 1: An NER Example from a SemEval Task
As shown in the Figure 1, the NER pharmaceutical model receives the textual
input (e.g. sentences), and returns whether there are important information
entities that belong to any predefined labels, such as brand name and drug
name.
A similar undertaking has been pursued by Weston et al.[28], with their NER
model designed for inorganic materials information extraction. Their model
includes seven entity labels and testing has resulted in an overall f1-score
of 0.87[28]. This work has inspired the HIVE team to use NER, as a step toward
relation extraction, and the development of richer ontologies for materials
science.
## 3 Purpose and Goals
Goals of this paper are to:
1. 1.
Introduce HIVE
2. 2.
Demonstrate HIVE’s three key features Vocabulary browsing, term search and
selection, and knowledge extraction/indexing
3. 3.
Provide an example of our NER work, as a foundation for relation extraction.
## 4 HIVE-4-MAT: Prototype Development and Features
Hive is a linked data automatic metadata generator tool developed initially as
a demonstration for the Dryad repository[16, 30], and incorporated into the
DataNet Federation Consortium’s iRODS system[12]. Ontologies encoded in the
Simple Knowledge Organization System (SKOS) format are shared through a HIVE-
server. Currently, HIVE 2.0 uses Rapid Automatic Keyword Extraction (RAKE), an
unsupervised algorithm that processes and parses text into a set of candidate
keywords based on co-occurrence[22]. Once the list of candidate keywords is
selected from the SKOS encoded ontologies, the HIVE system matches candidate
keywords to terms in the selected ontologies. Figure 2 provides an overview of
the HIVE model.
HIVE-4-MAT builds on the HIVE foundation, and available ontologies have been
selected for either broad or targeted applicability to materials science. The
prototype includes the following ten ontologies: 1)Bio-Assay Ontology
(BioAssay), 2) Chemical Information Ontology (CHEMINF), 3) Chemical Process
Ontology (prochemical), (4) Library of Congress Subject Headings (LCSH), 5)
Metals Ontology, 6) National Cancer Institute Thesaurus (NCIT), 7) Physico-
Chemical Institute and Properties (FIX), 8) Physico-chemical process (REX), 9)
Smart Appliances REFerence Ontology (SAREF), and 10) US Geological Survey
(USGS).
Figure 2: Overview of HIVE Structure
Currently, HIVE-4-MAT has three main features:
* $\bullet$
Vocabulary browsing (Figure 3 and Figure 4)
* $\bullet$
Term search and selection (Figure 5)
* $\bullet$
Knowledge Extraction/Indexing (Figure 6)
Figure 3: Lists of Vocabularies/Ontologies
The vocabulary browsing feature allows a user to view and explore the
ontologies registered in HIVE-4-MAT. Figure 3 presents the full list of
currently available ontologies, and Figure 4 provides an example navigating
through the hierarchy of the Metals ontology. The left-hand column (Figure 4)
displays the hierarchical levels of this ontology; the definition, and the
right-hand side displays the alternative name, broader concepts and narrow
concepts.
### 4.1 Mapping Input Text to Ontologies
The term search and selection feature in Figure 5 allows a user to select a
set of ontologies and enter a search term. In this example, eight of the 10
ontologies are selected, and the term thermoelectric is entered as a search
concept. Thermoelectrics is an area of research that focuses on materials
conductivity of temperature (heat or cooling) for energy production. In this
example, the term was only found in the LCSH, which is a general domain
ontology. The lower-half of Figure 5 shows the term relationships. There are
other tabs accessible to see the JSON-LD, SKOS-RDF/XML and other encoding.
This feature also allows a user to select an encoded term for a structure
database system, such as a catalog, or for inclusion in a knowledge graph.
Figure 4: Vocabularies/Ontologies Structure Figure 5: Term Search
Figure 6 illustrates the Knowledge Extraction/Indexing Feature. To reiterate,
reading research literature is time-consuming. Moreover, it is impossible for
a researcher to fully examine and synthesize all of the knowledge from
existing work. HIVE-4-MAT’s indexing functionality allows a researcher or a
digital content curator to upload a batch of textual resources, or simply
input a uniform resource locator (URL) for a web resource, and automatically
index the textual content using the selected ontologies. Figure 6 provides an
example using the Wikipedia content for Wikipedia page on Metal[3]. The
visualization of the HIVE-4-MAT’s results helps a user to gain an
understanding of the knowledge contained within the resource, and they can
further navigate the hypertext to confirm the meaning of a term within the
larger ontological structure.
Figure 6: Keyword Extraction
### 4.2 Building NER for Information Extraction
Inspired by the work of Weston et al.[28], the HIVE team is also exploring the
performance and applications of NER as part of knowledge extraction in
materials science. Research in this area may also serve to enhance HIVE.
Weston et al.[28] focus on inorganic materials, and appear to be one of the
only advanced initiative’s in this area. Our current effort focuses on
building a test dataset for organic materials discovery, with the larger aim
of expanding research across materials science.
To build our corpus, we used Scopus API[27] to collect a sample of abstracts
from a set of journals published by Elsevier that cover organic materials. The
research team has identified and defined a set of seven key entities to assist
with the next step of of training our model. These entities have the following
semantic labels: (1) Molecules/fragments, (2) Polymers/organic materials, (3)
Descriptors, (4) Property, (5) Application, (6) Reaction and (7)
Characterization method. Members of our larger research team are actively
annotating the abstracts using these semantic labels as shown in Figure 7. The
development a test dataset is an important research step, and will help our
team move forward testing our NER model and advancing knowledge extraction
options for materials science in our future work.
Figure 7: Example from our In-Progress Organic Dataset
## 5 Discussion
The demonstration of HIVE and reporting of initial work with NER is motivated
by the significant challenge materials science researchers face gleaning
knowledge from textual artifacts. Although this challenge pervades all areas
of scientific research, disciplines such as biology, biomedicine, astronomy,
and other earth sciences have a much longer history of open data and ontology
development, which drives knowledge discovery. Materials science has been slow
to embrace these developments, most likely due to the disciplines connection
with competitive industries. Regardless of the reasons impacting timing, there
is clearly increased interest and acceptance of a more open ethos across
materials science, as demonstrated by initiatives outlined by Himanen et al.
in 2019 [19]. Two key examples include NOMADCoE [13] and the Materials Data
Facility [10], which are inspired by the FAIR principles [13, 31]. These
developments provide access to structured data, although, still the majority
of materials science knowledge remains hidden in textually dense artifacts.
More importantly, these efforts recognize the value of access to robust and
disciplinary relevant ontologies. HIVE-4-MAT complements these developments
and enables materials science researchers not only to gather, register, and
browse ontologies; but, also the ability to automatically apply both general
and targeted ontologies for knowledge extraction. Finally, the HIVE-4-MAT
output provides researchers with a structured display of knowledge that was
previously hidden within unstructured text.
## 6 Conclusion
This paper introduced the HIVE-4-MAT application, demonstrated HIVE’s three
key features, and reported on innovative work underway exploring NER. The
progress has been encouraging, and plans are underway to further assess the
strengths and limitation of existing ontologies for materials science.
Research here will help our team target areas where richer ontological
structures are needed. Another goal is to test additional algorithms with the
HIVE-4-MAT application, as reported by White, et al[30]. Finally, as the team
moves forward, it is critical to recognize that ontologies, alone, are not
sufficient for extracting knowledge, and it is important to consider other
approaches for knowledge extraction, such as Named Entity Recognition (NER)
and Relation Extraction (RE) can complement and enrich current apporaches. As
reported above, the HIVE team is also pursuing research in this area as
reported by Zhao[33], which we plan to integrate with the overall HIVE-4-MAT.
## 7 Acknowledgment
The research reported on in this paper is supported, in part, by the U.S.
National Science Foundation, Office of Advanced Cyberinfrastructure (OAC):
Grant: #1940239. Thank you also to researchers in Professor Steven Lopez’s
lab, Northeastern University, and Semion Saiki, Kebotix for assistance in
developing the entity set for organic materials.
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|
# Following up TESS Single Transits With Archival Photometry and Radial
Velocities
Xinyu Yao Department of Physics, Lehigh University, 16 Memorial Drive East, Bethlehem, PA 18015, USA Joshua Pepper Department of Physics, Lehigh University, 16 Memorial Drive East, Bethlehem, PA 18015, USA B. Scott Gaudi Department of Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus, OH 43210, USA Paul A. Dalba NSF Astronomy & Astrophysics Postdoctoral Fellow Department of Earth & Planetary Sciences, University of California Riverside, 900 University Ave, Riverside, CA 92521, USA Jennifer A. Burt Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena CA 91109, USA Robert A. Wittenmyer University of Southern Queensland, Centre for Astrophysics, West Street, Toowoomba, QLD 4350, Australia Diana Dragomir Department of Physics and Astronomy, University of New Mexico, 1919 Lomas Blvd NE, Albuquerque, NM 87131, USA Joseph E. Rodriguez Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA Steven Villanueva, Jr. Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Daniel J. Stevens Eberly Fellow Center for Exoplanets and Habitable Worlds, The Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA Department of Astronomy & Astrophysics, The Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA Keivan G. Stassun Vanderbilt University, Department of Physics & Astronomy, 6301 Stevenson Center Lane, Nashville, TN 37235, USA David J. James ASTRAVEO LLC, PO Box 1668, MA 01931
###### Abstract
NASA’s Transiting Exoplanet Survey Satellite (TESS) mission is expected to
discover hundreds of planets via single transits first identified in their
light curves. Determining the orbital period of these single transit
candidates typically requires a significant amount of follow-up work to
observe a second transit or measure a radial velocity orbit. In Yao et al.
(2019), we developed simulations that demonstrated the ability to use archival
photometric data in combination with TESS to “precover” the orbital period for
these candidates with a precision of several minutes, assuming circular
orbits. In this work, we incorporate updated models for TESS single transits,
allowing for eccentric orbits, along with an updated methodology to improve
the reliability of the results. Additionally, we explore how radial velocity
(RV) observations can be used to follow up single transit events, using
strategies distinct from those employed when the orbital period is known. We
find that the use of an estimated period based on a circular orbit to schedule
reconnaissance RV observations can efficiently distinguish eclipsing binaries
from planets. For candidates that pass reconnaissance RV observations, we
simulate RV monitoring campaigns that enable one to obtain an approximate
orbital solution. We find this method can regularly determine the orbital
periods for planets more massive than $0.5M_{\rm J}$ with orbital periods as
long as 100 days.
planets and satellites: detection — planets and satellites: general — methods:
data analysis
## 1 Introduction
Follow-up observations of transiting exoplanets generally require precise
ephemerides. Any attempt to conduct transmission or emission spectroscopy,
whether from the ground or from space, requires the ability to accurately
predict future transits so as not to waste valuable telescope time or miss
part of the event. When candidate transiting planets have large uncertainties
in their ephemerides, it becomes difficult to schedule and thus successfully
perform follow-up observations. That is most often the case for long-period
transit candidates, which we define here as candidates with orbital periods
comparable to or greater than the time baseline of their originating transit
survey. This situation was vividly demonstrated by Benneke et al. (2017), who
found that the initially calculated ephemeris of the planet K2-18b ($P=33$
days with two transits shown in K2 data) was off by 2 hours before they
recovered the correct ephemeris using the Spitzer telescope. More recently,
Ikwut-Ukwa et al. (2020) combined observations from Kepler and TESS for known
K2 planets to reduce the uncertainty on future transit times from hours to
minutes through 2030. Ephemeris errors of order hours would preclude the
possibility of future observations from facilities such as the James Webb
Space Telescope (JWST).
Reliable ephemerides are also crucial to further dynamical studies of
transiting planets. Wang et al. (2015) found that half of the ten long period
exoplanets (periods between 430 days and 670 days) discovered by Kepler show
transit timing variations (TTVs) ranging from $\sim$2 to 40 hours. For planet
discoveries in which the initial ephemerides are poorly constrained,
additional transit observations might be needed to fix the ephemerides to
permit later TTV analysis (Dalba & Muirhead, 2016; Dalba & Tamburo, 2019).
Obtaining such ephemerides from archival data saves valuable observing time,
which can be helpful even in the cases of shorter-period planets that TESS is
likely to detect.
The Transiting Exoplanet Survey Satellite (TESS) is designed to detect
transiting planets orbiting bright stars across the whole sky. Launched in
April 2018, TESS has observed 26 sectors and has discovered $\sim$ 2000 planet
candidates so far. Among those, 51 have been confirmed and published. However,
90% of the TESS planet discoveries have orbital periods shorter than 20
days111https://exoplanetarchive.ipac.caltech.edu/ due to the short observing
time for most of the sky ($\sim 27$ days). Cooke et al. (2018) and Villanueva
et al. (2019) estimated that TESS will detect hundreds of planets with long
orbital periods via single transits in their TESS light curves.
In Yao et al. (2019) (henceforth Y19), we investigated the ability to recover
the ephemerides of TESS single transit candidates using archival data from the
Kilodegree Extremely Little Telescope (KELT) ground-based transit survey
(Pepper et al., 2007, 2012). The process of using archival data to detect a
signal originally revealed in later observations is sometimes called
“precovery”. In that work, we inserted simulated transit signals into KELT
light curves, and explored the recoverability of the signals when combined
with the information from TESS observations. We found that a significant
subset of large planets in long orbits that show single transits in TESS could
be detected in KELT light curves, enabling precise measurements of their
ephemerides. This type of approach was successfully carried out by Gill et al.
(2020c) to recover the ephemeris of a single-eclipse TESS eclipsing binary
using archival photometry from the WASP survey (Pollacco et al., 2006).
There are many ways to consider how to best follow up and confirm single-
transit candidates. These approaches are different from those used to follow
up transit candidates where the orbital period is known, since in those cases,
photometric follow-up can be scheduled at specific times to catch future
transits, and radial velocity (RV) observations can be timed to properly
sample the orbital phase. In one approach to follow up single transits, Cooke
& Pollacco (2020) explored the use of photometric versus spectroscopic follow-
up observations to confirm single transits. That analysis considered the use
of three specific instruments: photometric observations using the Next
Generation Transit survey (NGTS; Wheatley et al., 2018), and radial velocity
observations using the HARPS (Mayor et al., 2003) and CORALIE (Queloz et al.,
2000) spectrographs. To compare cases, Cooke & Pollacco (2020) considered the
observing time required for a given instrument to achieve a detection of the
planet past a given signal-to-noise ratio (SNR) threshold. In §5 below, we
compare our approach to that work.
In this paper, we improve upon the simulations in Y19 by incorporating more
realistic transit models and orbital configurations. We also explore the use
of RV observations of single-transit candidates to both eliminate certain
types of false positives, and to confirm their planetary nature by measuring
their orbits. We do not compare single-transit follow-up strategies to those
used for multi-transit candidates, since the unavailability of a known period
means that the efficiency or expense of the efforts cannot be directly
compared. The paper is structured as follows: §2 updates the earlier analysis
by deriving recovery rates of TESS single transit candidates with KELT
photometry using realistic eccentricities for the simulated sample, and
applying a more sophisticated calculation for the recovery rate. §3 discusses
the use of RV observations to distinguish planetary systems from high-mass-
ratio eclipsing binaries. §4 presents simulations of RV observations to
confirm planet candidates by constraining their orbital periods. We explore
the results and implications of the result in §5, comparing this approach to
other techniques, and in §6 we summarize our findings.
## 2 Updates to the Recovery Analysis
In Y19, we explored transit recovery of TESS single transits using KELT
photometry to pre-cover the signals. We simulated TESS single transits with
known periods ranging from 13.5 to 300 days in circular and centrally
transiting (i.e. equatorial) orbits, and the transit depth was assigned
randomly from 3 mmag to 20 mmag in log space. Then we inserted the periodic
transit signal into detrended KELT photometry, using a subset of 130,000 KELT
light curves with RMS scatter below 30 mmag, out of all 5.8 million KELT light
curves. The selection of only low-noise KELT light curves effectively
eliminates cases with stellar variability larger than the KELT photometric
scatter. We then used the Box-fitting Least Squares (BLS) algorithm (Kovács et
al., 2002) to try to recover the signal. We use the version of BLS with a
fixed transit duration and a fixed $T_{C}$, since the parameters $T_{C}$ and
duration for a given transit signal will be known with high fidelity from the
TESS observations. The period was free to vary and we searched from 13.5 days
to 300 days, with a frequency resolution of 300,000 (the number of trial
frequencies scanned, evenly spaced in frequency). The recovered period was
identified as the period corresponding to the strongest peak in the BLS
periodogram. As the number of the light curves for the simulations is large,
no visual inspection was conducted to check the recovered period. For a
successful recovery of an inserted transit signal, we require that the percent
difference between the recovered period and the inserted period be within
0.01%. In that approach, we calculated the recovery rate for the KELT
photometry for a range of transit durations and transit depths. That is
essentially an injection/recovery test of long-period transits with the KELT
photometry under the assumption that the transit time and duration are known
before searching for the signal in the KELT data. Since planets with orbital
periods longer than 10 days actually have a broad distribution of orbital
eccentricities (Marcy & Butler, 2000; Winn & Fabrycky, 2015), the duration and
window coverage of transits is affected by the orbital eccentricity (Barnes,
2007; Burke, 2008; Kane et al., 2012; Kipping, 2014). Transiting planets with
eccentric orbits will not have uniformly distributed arguments of periastron
$\omega$; in eccentric orbits, $\omega$ is more likely to be close to $\pi/2$
(as shown in Figure 4 from Burke, 2008) and the planet is more likely to
transit, which means the typical transit duration is shorter, for a fixed
orbital period. Therefore, compared with circular orbits, planets in eccentric
orbits will on average have fewer data points during the transit, which causes
the SNR of the transit signals to decrease, and thus recovery rate for planets
with more realistic (eccentric) orbits will be lower than cases with circular
orbits. Therefore the recovery results from Y19 represent an overestimate of
the recovery rates.
We address that issue by repeating the Y19 simulations but now consider
eccentric orbits. To determine the orbital eccentricities in the updated
simulations, we adopt the beta distribution with parameters $\alpha=0.867$ and
$\beta=3.03$ from Kipping (2013), and use the algorithm ECCSAMPLES (Kipping,
2014) to generate the orbital eccentricity and argument of periastron for each
planet. Other parameter distributions such as orbital period, planetary
radius, and orbital inclination are kept the same as in the circular orbital
case. Using the same criteria from Y19 to calculate the fraction of
successfully recovered orbital periods (a period precision of better than
0.01%), the average recovery rate declines as expected, and we find 5% fewer
recovered planets, where the overall recovery rate across all ranges of
orbital period and planet radius drops from 33% to 28%.
We now explore ways to improve the utility of the recovery process. We
continue to operate under the assumption that the transit signal in the TESS
light curve has a high enough signal-to-noise ratio for the transit duration
and $T_{\rm C}$ to be calculated precisely, but without enough signal-to-noise
ratio to strongly constrain the eccentricity just from the light curve.
Along with updating the Y19 analysis by incorporating eccentric orbits, we
utilize the transit duration as observed by TESS in an additional way. In Y19
we required the recovered transit signal to have a similar duration as in the
TESS light curve, but we did not constrain the range of possible periods to
search based on the transit duration. Here, we consider the fact that the
orbital period of a transiting planet can be estimated from the observed
parameters of single transit assuming a circular orbit with a central transit
(Seager & Mallén-Ornelas, 2003; Yee & Gaudi, 2008; Winn, 2010). We do not
assume that the orbits are necessarily circular, but rather make an educated
guess that the orbit is not extremely eccentric, which allows us to more
efficiently and reliably search a more limited range of possible periods, as
described below.
An alternate approach to our assumption of central transits is to use the
transit shape as measured by TESS to estimate the impact parameter based on
the transit shape in the TESS light curve. However, that method relies on high
SNR of the TESS detection, along with assumptions for the stellar limb-
darkening. Since we want to apply this method to as many of the TESS single
transits as possible, we instead take the alternate approach of assuming
central transits throughout this analysis.
Equation (7) in Yee & Gaudi (2008) expresses the relation between the orbital
period and transit duration for a transiting planet. The way we incorporate
this constraint is by calculating the orbital period from the transit duration
under the assumption of circular and centrally transiting orbits ($P_{\rm
cal}$) (Winn, 2010),
$P_{cal}=365\;\mathrm{d}\left(\frac{T_{dur}}{13\;\mathrm{hr}}\right)^{3}\left(\frac{\rho_{\star}}{\rho_{\odot}}\right).$
(1)
and requiring that the recovered orbital period from the BLS search ($P_{\rm
BLS}$) be within the range of $P_{\rm cal}$ described below. Even though
$P_{\rm cal}$ is based on the assumption of a circular orbit, we can still use
it to constrain the search for eccentric transits. The transit duration in Eq.
1 is the full-width half-maximum of the transit in hours.
Figure 1 shows the distribution of the ratio $P_{\rm cal}/P_{\rm BLS}$, along
with the recovery rates. Since the BLS search is performed across period space
for all light curves, we expect some spurious signals at all period ranges. By
requiring $P_{\rm BLS}$ to be sufficiently close to $P_{\rm cal}$, we can
eliminate many spurious cases from the analysis and improve the reliability of
the surviving signals. We thus implemented an additional cut in the analysis
by requiring that $P_{\rm cal}/P_{\rm BLS}$ to be between 0.25 and 1.96, which
effectively removes cases where the expected recovery rate is below 20%. We
selected that particular cutoff to reflect a subjective judgement regarding
the balance between improving the overall recovery rate, and retaining a large
fraction of the total sample. We then calculated the overall recovery rate for
the remaining simulated light curves.
Using this constraint, the overall recovery rate improves by $\sim$12%, from
28% to 40%, as a result of excluding cases where the period calculated from
BLS is likely inconsistent with the transit duration. In this case, that
restriction excluded $\sim$41% of the original set of simulated light curves,
finding the BLS-derived period to be unlikely to be consistent with the
transit duration. This recovery rate is also higher than the overall recovery
rate found in Y19, even though we now account for eccentric orbits. The
improvement in the recovery rate from 28% to 40% does not mean that we can
detect more long-period planets from TESS, but that the reliability of the
results from searching the KELT data is more trustworthy.
Figure 1: Distributions of the ratios between $P_{\rm cal}$ and $P_{\rm BLS}$
for eccentric orbits (blue) and the associated recovery rate as a function of
the period ratio (orange). The green horizontal dotted line marks the average
recovery rate in Y19 (restricted to circular orbits). The green horizontal
dashed line marks the 20% recovery rate. The vertical dashed grey lines
indicate the period ratio corresponding to the 20% recovery rate.
To show the result in a more useful form for follow-up observers, we adopt the
signal-to-pink noise ratio (SPN) as a signal strength criterion to estimate
the confidence of the recovery result. The SPN ratio is a variant of the S/N
computed using a “pink” total noise that includes both uncorrelated (“white”)
and correlated (“red”) noise sources. More details of the definition of SPN
can be found in Hartman & Bakos (2016), and details of how we utilize this
statistic are in section 4.2 of Y19. In short, we calculate the empirical SPN
threshold where a given percentage of all test cases are successfully
recovered, and use that as the ”recovery confidence”. For instance, in a given
range of transit depth and duration, if we find that 90% of all test signals
are recovered in the KELT light curves, then that SPN value is our 90%
confidence threshold. Similarly to Figure 13 in Y19, Figure 2 shows the SPN
thresholds for 10%, 50% and 90% recovery confidence with the fraction of KELT
light curves that are above those thresholds across transit depth versus
transit duration, using the same parameter ranges as in Y19. In that figure,
we show the results separately for the KELT-South and KELT-North data sets,
since the KELT-North dataset generally includes more epochs spread over a
longer time baseline as compared to the KELT-South dataset. For example,
consider the center bin in the upper panel. The lower right row within that
bin refers to the SPN required for a transit to be recovered 90% of the time.
We find that SPN value to be 7.3, so that if a transit signal with an SPN
value greater than 7.3 is seen, an observer can have 90% confidence the signal
is real. The percentage in that row indicates that of the KELT light curves
with inserted transit durations and depth in that corresponding range, 26%
yielded SPN values greater than 7.3.
Figure 2: The fractions of KELT-North (top) and KELT-South (bottom) light
curves that pass SPN thresholds in transit depth / transit duration bins, and
the corresponding SPN values in parentheses. In each box are three percentage
values and corresponding SPN values. The percentages reflect the fraction of
KELT light curves in that bin of depth/duration space such that if the SPN
value is greater than the indicated value, there is a given likelihood that
the recovered period is correct. Those likelihoods are 10%, 50%, or 90%, from
upper left to lower right in each bin. The color bar indicates the fractional
change in the fraction of KELT light curves that pass the 50% confidence
threshold between this analysis and the results from Y19, (discussion in
text). The items marked ‘nan’ represent cases where none of the inserted
signals were recovered in the KELT light curves at the corresponding
confidence level for the depth and duration ranges for that bin.
The color of the cells in Figure 2 indicate whether the updated analysis
yields better (blue) or worse (red) recovery rates compared to Figure 13 in
Y19. That is, the colors indicate the differential improvement (or decline) in
recovery rates compared to the assumption of a circular orbit. The figure
conveys two sets of information - the absolute recovery rates, using the
methodology described above, as well as the relative increase or decrease in
recovery rates across parameters space compared to Y19.
The colors in the figure reflect the impact of the two key changes we have
made. The use of eccentric orbits decreases the recovery rate throughout the
analysis. That is because for the same transit duration, the real orbital
periods for eccentric transiting planets are generally longer than the ones
with circular orbits. Therefore the SNR of the transit signal in the light
curves is lower (the converse of the effect mentioned above) when we are
observing multiple transits. On the other hand, the use of the additional
information coming from the transit duration using $P_{\rm cal}$ improves the
reliability of the recovery (and thus the overall recovery rates) due to
constraining the period search range. Figure 2 shows that for transit
durations shorter than 8 hours, the former effect dominates, while at longer
transit durations, the latter effect dominates.
## 3 Distinguishing Eclipsing Binaries From Planets
Although in this analysis we have inserted transit-like signals into the KELT
light curves as a phenomenological feature, agnostic as to the physical cause
of the signal, we now explore not just how to detect those photometric
signals, but also how to distinguish their physical cause.
One feature of the mass-radius relationship for stars and planets is that the
onset of electron degeneracy in the core leads to a flat mass-radius curve
from the massive planet regime to the small star regime (Zapolsky & Salpeter,
1969; Chabrier & Baraffe, 2000; Lynden-Bell & O’Dwyer, 2001; Fortney et al.,
2007). That is, compact objects with masses between Jovian planets and late-
type stars all have radii of roughly 1 $R_{\rm J}$, despite an almost two
orders of magnitude range in masses. Among our simulation samples, $\sim$65%
of planetary candidates have radii larger than $1R_{\rm J}$. That means a
large number of transit signals in our simulations may be caused by eclipsing
binaries instead of planets. They can include a larger star being eclipsed by
a later-type dwarf star, yielding a primary eclipse depth comparable to that
of a transiting giant planet, or an eclipsing binary with a large primary
transit that is diluted by blending with a nearby bright star. In this section
we will concentrate on ways to identify the first of these scenarios. We will
ignore transit-like signals caused by blended eclipsing binaries. Those types
of false-positive scenarios can be addressed through careful analysis of the
TESS pixel data by looking for centroid shifts (Bryson et al., 2013, Guerrero,
et al. submitted), or via photometric monitoring with higher angular
resolution or other spectroscopic techniques (see Collins et al. (2018) for a
discussion on vetting different false positive types).
One tool for distinguishing transiting planets from eclipsing binaries is the
use of radial velocity (RV) observations. Here we explore how to conduct RV
observations in the case where the ephemeris of the transit signal is unknown
or poorly constrained, as in the case of single transits. This process often
involves two stages of follow-up observations. The first stage, often referred
to as reconnaissance spectroscopy, involves two spectroscopic observations.
These observations are taken at the predicted orbit’s quadrature phases, and
can identify SB1s, and visual inspection of the spectra can identify SB2s.
Once those scenarios are ruled out, further RV observations are obtained over
the full phase of the orbit to characterize the orbital elements, e.g., the RV
semiamplitude, argument of periastron, and eccentricity.
In the case of an unknown orbital period, a single spectroscopic observation
can still identify SB2-type cases, but without knowing the predicted
quadrature times, it is not obvious how to perform reconnaissance spectroscopy
so as to rule out EBs. However, if the system has a large semi-amplitude, two
random observations could show a large offset indicating the presence of an
EB. There has been extensive work conducting both RV and photometric follow-up
observations of single-transit candidates from the NGTS team, recovering the
ephemerides of two EBs that showed single transits in TESS photometry (Lendl
et al., 2020; Gill et al., 2020a), along with the discovery of a long-period
planet from what was initially thought to be a single transit in TESS data,
which was later revealed to contain an additional transit initially obscured
by scattered light in the photometry (Dalba et al., 2020; Gill et al., 2020b).
Here we investigate a comprehensive approach to RV follow-up.
The dataset used in Y19 and in §2 consists of KELT light curves in which we
inserted a set of simulated transit signals. In this section, we do not use
any light curves, but we apply the properties of the simulated planetary
systems from that analysis, namely the orbital periods, radii of eclipsing
objects, eccentricities, and argument of periastron, and the masses and radii
of the associated stars, with the stellar parameters taken from the TESS Input
Catalog (TIC-8).
We first introduce EB cases to the simulated signals. Note that the signals
were drawn from a range of empirical properties (depth, duration, period,
etc.), without reference to physical properties (companion mass, radius,
semimajor axis, etc.). We selected $\sim$48,000 targets from the KELT-North
sample of stars described in §2 that have radii of the transiting body that
are larger than $1R_{\rm J}$, as inferred from the simulated transit depths
and the estimated stellar radii from the TIC-8. This subset of the full light
curve simulation includes the candidates for which there is ambiguity about
the physical nature of the eclipsing body. In the process of assigning masses,
we now consider two cases: if the transiting objects are all planets, the mass
($M_{p}$) was randomly assigned between $0.2M_{\rm J}$ and $3M_{\rm J}$ in log
space; and if the transiting objects are stars, the mass ($\rm M_{\sec}$) was
calculated based on the mass-radius relation in the stellar region derived
from Chen & Kipping (2017). We are essentially assuming that objects with
masses between $3M_{\rm J}$ and $80M_{\rm J}$ are rarer than planets or stars,
i.e. that there exists a “Brown-dwarf desert” (e.g., Grether & Lineweaver,
2006). The semi-amplitude ($K$) of the radial velocity curve for planets was
calculated using the above parameters:
$K=28.4\;\mathrm{m/s}\left(\frac{P}{1\;\mathrm{yr}}\right)^{-1/3}\left(\frac{M_{\rm
p}}{M_{\rm J}}\right)\left(\frac{M_{\star}}{\rm
M_{\odot}}\right)^{-2/3}\left({1-e^{2}}\right)^{-1/2}.$ (2)
For the case of binary stars, we replace $M_{\rm p}$ with $M_{\sec}$ and
$M_{\star}$ with $(M_{\star}+M_{\sec})$ in Equation 2. Note that, since we
know these systems are seen nearly edge-on because they exhibit transits or
eclipses, Equation 2 assumes $\sin{i}$ $\simeq$ 1\. We therefore ignore the
$\sin{i}$ dependence on $K$ in Equation 2. We used the python package RadVel
(Fulton et al., 2018) to synthesize RV curves based on those parameters.
While the observational strategies of RV follow-up can vary significantly, we
simulated a plausible follow-up approach that might be undertaken by a mid-
size RV survey facility such as TRES (Szentgyorgyi & Furész, 2007), MINERVA
(Swift et al., 2015), MINERVA-Australis (Addison et al., 2019), the APF (Vogt
et al., 2014), or a similar facility. We assume that the time interval between
the TESS observations of the single transit and the first RV observation is
three months, which is consistent with the TESS data release procedure, and
the time required to identify candidates and begin scheduling RV observations.
We also tested lengths of time intervals of 1 month and 12 months,
respectively, and found very little effect on the results.
The RV precision of an observation depends on the telescope, the instrument
and various stellar properties such as brightness, effective temperature, and
rotational velocity, among other factors. Although we explored the possibility
of simulating the RV precision for each observation based on the stellar
properties and assumed noise model for a given instrument, we ultimately found
such an effort to be unfeasible. Even for a specific telescope and noise
model, there is a spread in RV precision for stars of a given brightness, with
many ways that the stellar properties can influence the RVs, such as the
effects of stellar rotation, spot coverage, and chromospheric activity.
Therefore, we decided to adopt a uniform RV precision of 20 m/s in the
simulation, assuming the stellar rotation velocities are lower than 10 km/s.
These assumptions were based on the statistics of real observational data from
the CHIRON spectrograph (Tokovinin et al., 2013).
Since the true orbital period is not precisely known from TESS single transit
light curves particularly when assuming eccentric orbits, we modeled the
approach of an observer making an educated guess about the quadrature times
based on $P_{\rm cal}$. We then determined the RV values at those times
according to the model RV, and then added offsets to the calculated RV using
values drawn from a Gaussian distribution with a width of 20 m/s. We then
approximate the RV semi-amplitude as half the difference in the simulated RV
measurements at the estimated times of quadrature. This makes the inherent
assumption that the orbit is approximately circular, and thus the quadrature
times are approximately known, and that the semiamplitude is just one half the
difference between the measurements at quadrature (ignoring measurement
errors).
We do not consider whether the calculated quadrature time is during local
night, but since the potential orbital periods are known to be much longer
than 24 hours, we assume that observations will take place within 12 hours of
the calculated quadrature, and that issue does not have a major impact on
these results. Another potential issue with this approach is that we assume
that all planet candidates included are observable at some point in the night
through the campaign. If targets are randomly distributed in right ascension,
then some targets will inevitably be unobservable for significant lengths of
time. But with a significant number of total candidates, the observers can
select a subset that should be observable through the whole campaign.
In the upper panel of Figure 3, we show the distribution of estimated $\rm K$
for the planet and star samples, respectively. The lower panel indicates the
fraction of the samples at a given estimated $\rm K$ in which the candidates
are planets. If the difference between the two measured RVs is smaller than 50
m/s, the transit signal has a less than 10% probability of being a stellar
companion, and more RV follow-up is merited to determine the full dynamical
orbit. If the RV difference is larger than 200 m/s, it is likely to be an EB
with greater than 90% probability. For RV differences between those ranges,
additional observations are required to identify potential false positives.
Figure 3: The upper panel shows the distributions of half the difference
between two RV measurements at estimated quadrature times for the case of
planetary companions (orange) and a portion of the distribution for stellar
companions (blue). The difference between the RV observations is a rough proxy
for the semiamplitude. The lower panel indicates the fraction of the samples
at a given calculated semi-amplitude in which the candidates are planets.
## 4 Radial Velocity Detection of Planets
RV observations to dynamically detect planets are generally conducted in one
of two ways. If the orbital period of a system is known from multiple
transits, RV observations can be obtained across a full orbital phase, to
measure the orbital reflex motion of the star due to the planet (e.g. Burt et
al., 2018; Medina et al., 2018). If the orbital period is not known, as in a
blind RV search, one star or a set of stars can be monitored by (semi)regular
RV observations until orbital motion is seen in a phase-folded search of the
RV data. In these situations, RV monitoring, which can detect the
gravitational reflex motion that a planet induces on its host star throughout
the entire orbit, can be used to determine the orbital parameters.
In this section we conduct a simulation of what such an RV campaign might look
like, and we examine the distribution of stellar and planet parameters for the
systems for which that campaign could successfully detect the orbit. It should
be noted that RV confirmation of a single-transit candidate must operate quite
differently than for a multiple-transit candidate. Since the photometry
provides no direct measure of the orbital period, the RV observations must
effectively operate as a blind RV search, as in the second case noted above.
The approach described here aims to provide a more efficient way to conduct
that sort of search.
To create the simulation, we randomly selected 10% of the stars from the
sample in §2 that have $T_{\rm eff}$ between 4000K and 7400K (i.e., FGK type
stars) which are best suited for RV follow-up. This results in a sample of
$\sim$5600 stars in the KELT-North set and $\sim$5400 stars in the KELT-South
set. Note again that the data set involved in this analysis consists of the
simulated transit signals, not the KELT light curves themselves. We start with
the empirical signal properties, and translate those into associated physical
properties of the transiting companions, as we did in §3. In this case, if the
transiting objects have radii larger than $1R_{\rm J}$, the mass was randomly
assigned between $0.2M_{\rm J}$ and $3M_{\rm J}$ in log space; if planets have
radii smaller than $1R_{\rm J}$, the mass was calculated based on the mass-
radius relation in the Neptunian region derived from Chen & Kipping (2017).
The RV observations are simulated over a 3-month time span. That duration was
chosen as a plausible length of time for a dedicated RV campaign of this type.
A natural consequence of this is that systems with true periods longer than 3
months will be incompletely sampled, which we discuss below. The parameters
$P$, $e$, $\omega$ and $T_{\rm C}$ used to generate the RV curves are the same
as those from §2, in which we have included eccentric orbits, and we again
assume an RV precision of 20 m/s. We randomly selected 7 days each month in
which one observation of the target star is obtained each night, for a total
of 21 observations per star over that time frame. We assume that the
stipulation of 7 RV observations per month can be met even in the presence of
weather, which we account for by the random placement of the observations
within each month. We assigned RV values according to the theoretical RV
curve, and then added Gaussian noise with a distribution width of 20 m/s to
the RV points.
It should be noted that this approach does not account for dynamic scheduling
of upcoming RV observations during a campaign based on an evaluation of the RV
results up to that point. That topic has been explored extensively before
(e.g. Kane et al., 2007; Ford, 2008; Loredo et al., 2011), but not in the
specific case of a singly-transiting planet where a date of conjunction and
the planet size are known, but not the orbital period. In a related analysis,
Cabona et al. (2020) examined the efficacy of different scheduling strategies
when conducting follow-up RV observations of small TESS targets using the
ESPRESSO instrument (Pepe et al., 2014) on the VLT. However, that work
considers only transit candidates with known periods, and so addresses a
different scientific question than considered here.
There are various software tools that perform RV fitting, such as RadVel
(Fulton et al., 2018) and the Joker (Price-Whelan et al., 2017) and EXOFASTv2
(Eastman et al., 2019). In the next step, we conducted a maximum likelihood
Keplerian model fit as implemented in the RadVel package for the synthesized
RV data for each target. In the fitting process, the transit time $T_{\rm C}$
was fixed to the simulated value as would be measured from the TESS single
transit, and we set boundaries from 13.5 days to 300 days for the orbital
period. Other parameters ($K$, $e\sin\omega$, $e\cos\omega$) were free to
vary. For the initial guess of the orbital period, we used $P_{\rm L-S}$ from
a Lomb-Scargle period search of the simulated RV data. The initial guess of
$K$ was calculated based on the assumption of a circular and centrally
transiting orbit with $P_{\rm L-S}$, known stellar mass $M_{\star}$, and
$M_{\rm p}$ calculated based on the $M_{p}-R_{p}$ relation from Chen & Kipping
(2017) as described above.
There are a number of ways to consider the precision on the period of an
exoplanet derived from RV observations. For the purposes of confirming a
planet, a fractional period precision of tens of percent may be sufficient.
For conducting intensive transit follow-up however, such as atmospheric
characterization, an absolute precision of better than 30 minutes would
typically be required. For the analysis here, we consider an RV-fitted period
to be correct if the fractional difference between the RV-fitted period
$P_{\rm RV,fit}$ and $P_{\rm real}$ is smaller than 5%. While that precision
would not be sufficient for atmospheric observations, or even for photometric
ephemeris confirmation in some cases, it would be more than sufficient to
identify good candidates for long-period transiting planets from the TESS
sample of single-transit candidates. At that point, an observer can conduct
additional RV observations beyond the campaign envisioned here, with higher
cadence, or with another facility with greater RV precision to bring the
measured period from the new RV observations to a level sufficient for
photometric ephemeris confirmation to obtain the higher absolute precision on
the transit time needed for intensive transit observations. For similar
reasons, we consider an RV-fitted $K$ to be correct if the fractional
difference between $K_{\rm fit}$ and $K_{\rm real}$ is smaller than 50%. Our
goals at this stage are to obtain approximate orbital solutions and to
differentiate likely planets from false positives, rather than obtaining a
complete system solution at the end of the campaign.
It is the case that while the simulated RV campaign lasts for only 90 days, we
search for orbital periods out to 300 days. We made this choice for two
reasons. First, it is possible to obtain an RV detection of the orbital signal
with only partial phase coverage, although the fractional period precision of
the resulting fit is typically quite poor. Since a real campaign, regardless
of its duration, will end up observing some systems with long orbital periods,
we wanted to test the ability to extract such signals. In most cases of
periods much longer than the campaign duration, only a linear trend will be
identified. We discuss such cases in more detail below.
Figure 4 shows the distributions of the real and fitted parameters $P_{\rm
real}$, $P_{\rm RV,fit}$, $K_{\rm real}$, $K_{\rm fit}$ and $e_{\rm real}$.
There is clearly a significant population along the diagonal in each panel
which represents the correctly fitted samples, along with clusters running
along the diagonals in the period plots which indicate cases off by factors of
2 or 1/2. As expected, systems with high eccentricity and small semi-
amplitudes tend to be recovered least well. Systems with long orbital periods
are also poorly recovered, although that is partly due to the fact that the
90-day span of the RV observations is shorter than the orbital period for some
of the systems. By comparing the fitted parameters with real values, we found
the median absolute deviation (MAD) of ($P_{\rm RV,fit}$ versus $P_{\rm
real}$) is 7.2 days and the MAD of ($K_{\rm fit}$ versus $K_{\rm real}$) is
8.2 m/s. Although these intermediate results provide a rough estimate of the
ephemerides of the candidates, these ephemerides are not sufficiently precise
to schedule follow up observations during transits. We improve upon these
results below by identifying the more reliable fitted periods.
Figure 5 shows two examples that were successfully fitted and two examples
where the fitting failed. We indicate the locations of these four examples on
the plots in Figure 4. The two successful fits (cases A and B) show good
agreement between the true periods and the fitted periods for both shorter and
longer orbital periods. Example C shows a case where the true semiamplitude is
smaller than the scatter in the RV points, and example D shows a case where
the true period is significantly longer than the time baseline of the RV data.
Figure 4: Scatter plots for the simulated samples, comparing the real orbital
periods and semi-amplitudes with their fitted values, colored by different
real parameter values. We indicate the real and fitted values for the 4
examples in Figure 5 with filled black points. In the upper plots, the
populations that run along side the diagonal represent cases where the fitted
period is off by a factor of 2 or 1/2.
Figure 5: Two successfully fitted RV samples (top panel) and two failed fitted
samples (bottom panel). All these four examples are identified in Figure 4 and
Figure 6.
We compute the reduced chi square ($\chi_{\rm dof}^{2}$) between the best-fit
model and the simulated data. By inspecting the fitting results, we found
nearly all samples have $\chi_{\rm dof}^{2}$<2, which indicates that
$\chi_{\rm dof}^{2}$ is not an efficient tool to measure the confidence of the
fitted RV period. That situation arises because the amplitudes of the RV
signals in our sample are often not that much larger than the per-point errors
that we are modeling. As $\chi_{\rm dof}^{2}$ is not an efficient tool to
measure the confidence of the fitted RV period, we instead conduct a False
Alarm Probability (FAP) test of the RV fits. For each star in the simulation,
we ran 100 iterations of the above analysis after randomizing the RV
observations and the observing times. We selected the smallest reduced chi
square $\chi_{\rm dof{FAP<1\%}}^{2}$ to indicate the 1% FAP value for that
star. Figure 6 shows the distribution of period recovery accuracy compared to
the ratio between $\chi_{\rm dof}^{2}$ and $\chi_{\rm dof{FAP<1\%}}^{2}$,
which we refer to as the 1% FAP ratio. The plots in that figure include
horizontal lines indicating the 5% accuracy threshold on period recovery, and
vertical lines at the FAP ratio of unity indicating whether the $\chi_{\rm
dof}^{2}$ fit was larger or smaller than the 1% FAP threshold. The region to
the right of the FAP threshold indicates systems where the RV fit is not very
reliable, and tends to include long-period and low-amplitude systems. The
region to the left of the FAP threshold indicates reliable results, and we
find that 85% of those results yield orbital periods that are within 5% of the
true period. Of the 15% cases that are considered reliable from the FAP cut
but do not match the orbital period (the upper left region in the first
panel), 82% of those have true periods longer than the observing campaign,
indicating that we see a broad trend without the leverage to accurately
measure the period. We conclude that the 1% FAP ratio can be used as an
efficient tool to measure the reliability of the fit. Similar to Figure 4,
systems with a relatively short period and large semi-amplitude are most
likely to be correctly fitted, as expected.
Figure 6: Scatter plots of the percentage difference between the fitted period
$P_{\rm RV,fit}$ and $P_{\rm real}$ versus the 1% FAP ratio $\chi_{\rm
dof}^{2}$/$\chi_{\rm dof{FAP<1\%}}^{2}$. The color bars represent the real
value of period, eccentricity, and semi-amplitude.
Figure 7 provides a different way to view the utility of the 1% FAP ratio. The
figure shows the distribution of the 1% FAP ratios for all test cases, with
one set of values for the period fit, and another for the semi-amplitude fit,
separated according to passing the fit threshold. At a 1% FAP ratio around
unity, half of the cases were correctly fitted, and at a 1% FAP ratio around
0.3, nearly all samples are correctly fit. These distributions provide
confidence metrics for the fitted period and semi-amplitude.
As discussed above, the precision of a fitted period has different
implications depending on the goals of an investigation. While a given fitted
period might have a low fractional uncertainty, if an observer wants to
establish the true transit ephemeris, they will often want to obtain a fitted
period with a specific absolute uncertainty, corresponding to the time span
over which the second transit is expected at a confidence of, say, 1$\sigma$
or 68%, so as to schedule follow-up photometry to measure the transit. Figure
8 shows another version of Figure 7, by using a fixed absolute uncertainty on
the period of 8 hours rather than a fixed fractional uncertainty on the
period. This version provides a more useful reference for observers to plan
photometric follow-up observations, assuming an 8-hr night for observing.
Specifically, if an observer wants to obtain follow-up photometry for targets
with an RV-fitted period with precision smaller than 8 hours, with at least
50% confidence, then they should select targets with a 1% FAP ratio smaller
than 0.3 (see §5 below for details on implementing this process.)
Figure 7: Distribution of the 1% FAP ratio ($\chi_{\rm dof}^{2}$/$\chi_{\rm
dof{FAP<1\%}}^{2}$) for tested RV fits, colored by whether the fitted period
was close to the correct period based on a period precision of 5%, with the
grey color indicating the overlap of the two. The dot-dashed histogram
represents the cases where the semi-amplitude was near the real value. The
lower panel indicates the fraction of the tested RV fittings at a given bin in
1% FAP ratio in which the period and semi-amplitude were correctly fitted. At
a 1% FAP ratio of about 1, half of the test samples have well-fitted periods
and semi-amplitudes.
Figure 8: Distribution of the 1% FAP ratio ($\chi_{\rm dof}^{2}$/$\chi_{\rm
dof{FAP<1\%}}^{2}$) for tested RV fits, colored by whether the fitted period
was close to the correct period based on an 8-hour absolute accuracy. The dot-
dashed histogram represents the cases where the semi-amplitude was within 50%
of the correct value.
Figure 9 reproduces Figure 4, showing the distributions of the real and fitted
parameters, but this time only for the samples with 1% FAP ratio smaller than
1. In this case, the median absolute deviation (MAD) of the difference between
$P_{\rm RV,fit}$ and $P_{\rm real}$ is 0.3 days, and the MAD of the offset
between $K_{\rm fit}$ and $K_{\rm real}$ is 6.5 m/s. There remain some
clusters of fits that are off by a factor of 2 or 1/2. This result shows the
clear improvement in the accuracy of the RV results after excluding poor-
quality fits. We find that the systems most amenable to successful recovery
are those with an orbital period shorter than 100 days and planetary mass
greater than half a Jupiter mass (see Figure 10).
We found that orbital periods as long as $\sim$180 days can be determined to a
fractional accuracy of 10-20%, in which case around half of the orbital phase
was observed, and the RV data usually show some linear trends. These systems
usually contain a planet with a high mass yielding a large semi-amplitude.
Detection of such systems in a campaign as mapped out here would prompt
observers to take additional RV observations to refine the period.
Figure 9: Scatter plots for the simulated samples, similar to Figure 4. In
this figure, only cases where the 1% FAP ratio is smaller than 1 are
displayed.
Figure 10: Distribution of the planet masses for the successfully and
unsuccessfully recovered simulated systems.
## 5 Discussion
In this work, we have updated the simulation process introduced in Y19 by
incorporating orbits with realistic eccentricity distributions, and improved
the recovery rates by using $P_{\rm cal}$ to refine the searched periods. The
results from this work use more realistic descriptions of planetary orbits,
and should be more useful for the observers to schedule photometric follow-up.
If an observer has a particular single transit TESS candidate, they can
calculate $P_{\rm cal}$, use the KELT light curve to compute $P_{\rm BLS}$ and
the associated SPN, and check the confidence of $P_{\rm BLS}$ from Figure 2.
They can use that information to prioritize and schedule photometric
observations during the predicted transit window for that candidate to confirm
the ephemeris. A similar procedure can be used with any archival photometry,
so long as a sensitivity analysis along the lines of Y19 and 2 is conducted to
determine the SPN threshold associated with a given confidence level for a
particular range of transit depth and duration.
That analysis can still be improved. We assume central transits in our
simulations, but real transits, with a range of impact parameters, will be
shorter for the same stellar properties and orbital period, leading to
additional errors in the value of $P_{\rm cal}$. However, the overall size of
that effect is small (Seager & Mallén-Ornelas, 2003). It is the case that one
could estimate the impact parameter and other orbital properties from the TESS
light curve, but we have assumed here that while the TESS light curve is
likely to have high enough photometric precision to obtain fairly reliable
measurements of the transit depth and duration, it should not be assumed that
the transit shape, along with ingress/egress times, can always be measured to
high precision. In cases with extremely high photometric precision, the
transit shape itself can be used to identify or discount false positive
scenarios, but that typically can only be done for very bright stars, such as
the case of HR 858, which has multiple planets orbiting a $T=5.9$ star
(Vanderburg et al., 2019). Another issue we did not consider is whether the
presence of additional planets or stellar variability due to rotational
modulation from spots/plages in the system could create RV signatures that
would interfere with the ability to recover the orbit of the transiting
planet.
We also explored in §3 how RV observations can be used for initial vetting of
single-transit candidates, by simulating the use of two RV observations tied
to the predicted quadrature times based on simplified orbital assumptions. We
showed that for most system configurations, such observations should be
readily able to distinguish stellar binaries from planetary systems.
We then examined an example campaign of RV observations to confirm the single-
transit candidates by measuring the orbital motion of the stellar host due to
the planet. We posited a particular observing campaign that is within the
capability of multiple current facilities. For a TESS single transit, even
after an initial pair of RV observations discounts the presence of an
unblended stellar binary, there is a broad range of possible periods (Figure
1) even after accounting for the transit duration and stellar mass. Our
simulations show that RV campaigns of the type we considered can successfully
measure the orbital motion due to the planet in $\sim$ 30% of cases we
simulated. The results of such an RV campaign, making use of modest telescope
and spectroscopic facilities, can constrain orbital period to within 5% with
85% confidence and the mass of the companion to within 50% with 88%
confidence. That information can then be used to plan a photometric campaign
over one or several nights to catch another transit and precisely determine
the ephemeris for future atmospheric study, or conduct additional, targeted RV
observations, potentially with higher RV precision to further constrain the
period before conducting photometric observations. A flow chart in Figure 11
provides a schematic illustration of this RV follow-up process we have
outlined. While this approach requires a number of steps to refine the precise
orbital period, it will typically be more feasible for most observers than
blind photometric follow-up of a candidate, as in the case of NGTS-11b
described below.
Figure 11: A flow chart illustrating the process to determine the ephemerides
of TESS single transits described in this paper.
The achievable precision of an actual RV follow-up campaign depends on many
factors, such as weather, observational windows, the number of RV observations
that can be acquired, the properties of the target stars (e.g., magnitude,
rotational velocity, effective temperature), the resolution of the
spectrograph ($\frac{\lambda}{\Delta\lambda}$), the exposure time, etc.
Therefore, we find it impractical to incorporate all these factors into a
generalized simulation, but these results can be used as a set of guidelines
for real-world follow-up efforts. In future work, we intend to explore how the
fitting of the RV signals can be improved with other software tools such as
EXOFASTv2 (Eastman et al., 2019) or Juliet (Espinoza et al., 2019), as opposed
to RadVel.
The TESS Single Transit Planet Candidates (TSTPC) working group has identified
a sample of more than 100 single transit candidates in TESS FFI data during
Sectors 1 through 13 (Villanueva et al in prep). Assuming circular orbits to
calculate the orbital period, and estimating the planetary mass using the
calculated planetary radius and the mass-radius relation from Chen & Kipping
(2017) we find that about 70% of the TSTPC-identified candidates would be
amenable to confirmation using these techniques.
Cooke & Pollacco (2020) have approached follow-up observations of TESS single
transits in a different way, as mentioned in section 1. That work asks a
narrower question than we deal with here. They investigate how one should
conduct follow-up observations when deciding between a blind photometric
search or RV search, for two (or three) specific observing facilities. They
calculate the effectiveness of using photometry or spectroscopy for exoplanet
follow-up as a function of $\rm R_{\star}$, $\rm R_{p}$ and $\rm P$. They
assume circular orbits for the transit candidates, and expect full, dedicated
access the the observing facilities, which in that case consist of the NGTS
photometric telescopes, and the CORALIE and HARPS spectrographs. For
photometric follow-up observations, they model regular, nightly observations,
in a mode exemplified by the successful confirmation of the ephemeris of the
planet NGTS-11b. In that case, Gill et al. (2020b) describe how the follow-up
procedure dedicated 79 nights of photometric monitoring in a blind search for
the detection of a second transit of a TESS single-transit candidate. Such an
approach is complementary to the process we describe in this paper and in Y19.
When archival photometry is available, we can restrict the likely ephemerides
to certain time ranges or to exact values, but if such photometry is not
available, the approach used by Cooke & Pollacco (2020) provides an alternate
path, when sufficient resources are available. The RV follow-up approach they
describe accounts for a more exact calculation of the RV precision as a
function of target magnitude for NGTS, HARPS, and CORALIE and the associated
SNR of the RV detection, and the assumption of circular orbits rather than a
range of eccentricities represents a different set of assumptions than we use.
We believe that the approach described in this work should be useful for
observers working with a range of follow-up observing facilities and various
levels of access, while the Cooke & Pollacco (2020) analysis is most
appropriate for those with dedicated access to high-performance instruments.
## 6 Summary
TESS will discover hundreds of planet candidates with long orbital periods via
single transits in their TESS light curves during the prime mission (Cooke et
al., 2018; Villanueva et al., 2019). Y19 demonstrated that KELT data can
recover the ephemerides of some single-transit candidates to a fractional
precision of 0.01%. In this work, we have incorporated more realistic models
for TESS single transits with eccentric orbits instead of circular orbits,
which is common for long-period exoplanets. We also improved the reliability
of precovery using transit duration constraints.
In addition to improving the precovery simulations, we explored the use of RV
follow-up observations in the case of single transit events. We found that the
use of the estimated period based on a circular orbit to schedule
reconnaissance RV observations around quadrature can efficiently distinguish
EBs from planets. For candidates that pass reconnaissance RV observations, we
simulated RV monitoring campaigns to obtain an orbital solution. We found that
the use of a $\chi_{\rm dof}^{2}$ fit to a Keplerian model, combined with an
FAP analysis, is sufficient to obtain an approximate orbital solution for
planets more massive than $0.5M_{\rm J}$ with orbital periods as long as 100
days, providing sufficient constraints for additional detailed orbital
refinement and photometric ephemeris determination.
JP and XY acknowledge support from NASA grant 80NSSC19K0387 under the TESS
Guest Investigator program G011229. PD acknowledges support from a National
Science Foundation (NSF) Astronomy & Astrophysics Postdoctoral Fellowship
under award AST-1903811. Part of this research was carried out at the Jet
Propulsion Laboratory, California Institute of Technology, under a contract
with the National Aeronautics and Space Administration (NASA). KGS
acknowledges partial support from NASA grant 17-XRP17 2-0024. BSG was
supported by a Thomas Jefferson Chair for Space Exploration endowment at the
Ohio State University. DJS is supported as an Eberly Research Fellow by the
Eberly College of Science at the Pennsylvania State University. The Center for
Exoplanets and Habitable Worlds is supported by the Pennsylvania State
University, the Eberly College of Science, and the Pennsylvania Space Grant
Consortium. KELT was partially supported by NSF CAREER Grant AST-1056524.
Funding for the TESS mission is provided by the NASA Explorer Program. This
work has made use of NASA’s Astrophysics Data System. This research made use
of Astropy,222http://www.astropy.org a community-developed core Python package
for Astronomy (Astropy Collaboration et al., 2013, 2018). We acknowledge
support for the KELT project through the Vanderbilt Initiative in Data-
intensive Astrophysics, Ohio State University, and Lehigh University.
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# Variational Quantum Support Vector Machine based on $\Gamma$ matrix
expansion
and Variational Universal-Quantum-State Generator
Motohiko Ezawa Department of Applied Physics, University of Tokyo, Hongo
7-3-1, 113-8656, Japan
###### Abstract
We analyze a binary classification problem by using a support vector machine
based on variational quantum-circuit model. We propose to solve a linear
equation of the support vector machine by using a $\Gamma$ matrix expansion.
In addition, it is shown that an arbitrary quantum state is prepared by
optimizing a universal quantum circuit representing an arbitrary $U(2^{N})$
based on the steepest descent method. It may be a quantum generalization of
Field-Programmable-Gate Array (FPGA).
Introduction
Quantum computation is a hottest topic in contemporary physicsFeynman ; DiVi ;
Nielsen . An efficient application of quantum computations is machine
learning, which is called quantum machine learningLloyd ; Schuld ; Biamonte ;
Wittek ; Harrow ; Wiebe ; Reben ; ZLi ; SchuldB ; Hav ; Lamata ; Cong . A
support vector machine is one of the most fundamental algorithms for machine
learningVap ; Noble ; Suy , which classifies data into two classes by a
hyperplane. The optimal hyperplane is determined by an associated linear
equation $F|\psi_{\text{in}}\rangle=|\psi_{\text{out}}\rangle$, where $F$ and
$|\psi_{\text{out}}\rangle$ are given. A quantum support vector machine solves
this linear equation by a quantum computerZhaokai ; Hav ; Reben . Usually, the
linear equation is solved by the Harrow-Hassidim-Lloyd (HHL) algorithmHHL .
However, this algorithm requires many quantum gates. Thus, the HHL algorithm
is hard to be executed by using a near-term quantum computer. Actually, this
algorithm has experimentally been verified only for two and three qubitsXCai ;
Barz ; JPan . In addition, it requires a unitary operator to execute
$e^{iFt}$, which is quite hard to be implemented.
The number of qubits in current quantum computers is restricted. Variational
quantum algorithms are appropriate for these small-qubit quantum computers,
which use both quantum computers and classical computers. Various methods have
been proposed such as Quantum Approximate Optimization Algorithm (QAOA)QAOA ,
variational eigenvalue solverPeru , quantum circuit learningMitarai and
quantum linear solverPrie ; Xxu . We use wave functions with variational
parameters in QAOA, which are optimized by minimizing the expectation value of
the Hamiltonian. A quantum circuit has variational parameters in quantum
circuit learningMitarai , which are optimized by minimizing a certain cost
function. A quantum linear solver solves a linear equation by variational
ansatzPrie ; Xxu . The simplest method of the optimization is a steepest-
descent method.
In this paper, we present a variational method for a quantum support vector
machine by solving an associated linear equation based on variational quantum
circuit learning. We propose a method to expand the matrix $F$ by the $\Gamma$
matrices, which gives simple quantum circuits. We also propose a variational
method to construct an arbitrary state by using a universal quantum circuit to
represent an arbitrary unitary matrix $U(2^{N})$. We prepare various internal
parameters for a universal quantum circuit, which we optimize by minimizing a
certain cost function. Our circuit is capable to determine the unitary
transformation $U$ satisfying
$U|\psi_{\text{initial}}\rangle=|\psi_{\text{final}}\rangle$ with arbitrary
given states $|\psi_{\text{initial}}\rangle$ and
$|\psi_{\text{final}}\rangle$. It will be a quantum generalization of field-
programmable-gate array (FPGA), which may execute arbitrary outputs with
arbitrary inputs.
Risults
Support vector machine.
A support vector machine (SVM) is a computer algorithm that learns by examples
to assign labels to objects. It is a typical method to solve a binary-
classification problemVap . A simplest example reads as follows. Suppose that
there are red and blue points whose distributions are almost separated into
two dimensions. We classify these data points into two classes by a line, as
illustrated in Fig.1.
In general, $M$ data points are spattered in $D$ dimensions, which we denote
$\boldsymbol{x}_{j}$, where $1\leq j\leq M$. The problem is to determine a
hyperplane,
$\boldsymbol{\omega}\cdot\boldsymbol{x}+\omega_{0}=0,$ (1)
separating data into two classes with the use of a support vector machine. We
set
$\boldsymbol{\omega}\cdot\boldsymbol{x}+\omega_{0}>0$ (2)
for red points and
$\boldsymbol{\omega}\cdot\boldsymbol{x}+\omega_{0}<0$ (3)
for blue points. These conditions are implemented by introducing a function
$f\left(\boldsymbol{x}\right)=\text{sgn}\left(\boldsymbol{\omega}\cdot\boldsymbol{x}+\omega_{0}\right),$
(4)
which assigns $f\left(\boldsymbol{x}\right)=1$ to red points and
$f\left(\boldsymbol{x}\right)=-1$ to blue points. In order to determine
$\omega_{0}$ and $\boldsymbol{\omega}$ for a given set of data
$\boldsymbol{x}_{j}$, we introduce real numbers $\alpha_{j}$ by
$\boldsymbol{\omega}=\sum_{j=1}^{M}\alpha_{j}\boldsymbol{x}_{j}.$ (5)
A support vector machine enables us to determine $\omega_{0}$ and $\alpha_{j}$
by solving the linear equation
$F\left(\begin{array}[]{c}\omega_{0}\\\ \alpha_{1}\\\ \vdots\\\
\alpha_{M}\end{array}\right)=\left(\begin{array}[]{c}0\\\ y_{1}\\\ \vdots\\\
y_{M}\end{array}\right),$ (6)
where $y_{i}=f(x_{i})=\pm 1$, and $F$ is a $(M+1)\times(M+1)$ matrix given by
$F=\left(\begin{array}[]{cccc}0&1&\cdots&1\\\ 1&&&\\\
\vdots&&K+I_{M}/\gamma&\\\ 1&&&\end{array}\right).$ (7)
Here,
$K_{ij}=\boldsymbol{x}_{i}\cdot\boldsymbol{x}_{j},$ (8)
is a Kernel matrix, and $\gamma$ is a certain fixed constant which assures the
existence of the solution of the linear equation (6) even when the red and
blue points are slightly inseparable. Note that $\gamma\rightarrow\infty$
corresponds to the hard margin condition. Details of the derivation of Eq.(6)
are given in Method A.
Figure 1: (a) Binary classification of red and blue points based on a quantum
support vector machine with soft margin. A magenta (cyan) line obtained by an
exact solution (variational method). (b) Evolution of the cost function. The
vertical axis is the log${}_{10}E_{\text{cost}}$. The horizontal axis is the
variational step number. We have used $r=2$, $\xi_{1}=0.001$ and
$\xi_{2}=0.0005$ and $\gamma=1$. We have runed simulations ten times.
Quantum linear solver based on $\Gamma$ matrix expansion.
We solve the linear equation (6) by a quantum computer. In general, we solve a
linear equation
$F\left|\psi_{\text{in}}\right\rangle=c\left|\psi_{\text{out}}\right\rangle,$
(9)
for an arbitrary given non-unitary matrix $F$ and an arbitrary given state
$\left|\psi_{\text{out}}\right\rangle$. Here, the coefficient $c$ is
introduced to preserve the norm of the state, and it is given by
$c=\sqrt{\left\langle\psi_{\text{in}}\right|F^{\dagger}F\left|\psi_{\text{in}}\right\rangle}.$
(10)
The HHL algorithmHHL is a most famous algorithm to solve this linear equation
by a quantum computer. We first construct a Hermitian matrix by
$H=\left(\begin{array}[]{cc}0&F\\\ F^{\dagger}&0\end{array}\right).$ (11)
Then, a unitary matrix associated with $F$ is uniquely obtained by $e^{iHt}$.
Nevertheless, it requires many quantum gates. In addition, it is a nontrivial
problem to implement $e^{iHt}$.
Recently, variational methods have been proposedPrie to solve the linear
equation (9). In one of the methods, the matrix $F$ is expanded in terms of
some unitary matrices $U_{j}$ as
$F=\sum_{j=0}^{2^{N}-1}c_{j}U_{j}.$ (12)
In general, a complicated quantum circuit is necessary to determine the
coefficient $c_{j}$.
We start with a trial state $|\tilde{\psi}_{\text{in}}\rangle$ to determine
the state $|\psi_{\text{in}}\rangle$. Application of each unitary matrix to
this state is efficiently done by a quantum computer,
$U_{j}|\tilde{\psi}_{\text{in}}\rangle=|\tilde{\psi}_{\text{out}}^{\left(j\right)}\rangle$,
and we obtain
$F|\tilde{\psi}_{\text{in}}\rangle=\sum_{j=0}^{2^{N}-1}c_{j}U_{j}|\tilde{\psi}_{\text{in}}\rangle=\sum_{j=0}^{2^{N}-1}c_{j}|\tilde{\psi}_{\text{out}}^{\left(j\right)}\rangle\equiv
c|\tilde{\psi}_{\text{out}}\rangle,$ (13)
where $|\tilde{\psi}_{\text{out}}\rangle$ is an approximation of the given
state $\left|\psi_{\text{out}}\right\rangle$. We tune a trial state
$|\tilde{\psi}_{\text{in}}\rangle$ by a variational method so as to minimize
the cost functionPrie
$E_{\text{cost}}\equiv
1-\left|\langle\tilde{\psi}_{\text{out}}|\psi_{\text{out}}\rangle\right|^{2},$
(14)
which measures the similarity between the approximate state
$|\tilde{\psi}_{\text{out}}\rangle$ and the state
$\left|\psi_{\text{out}}\right\rangle$ in (9). We have $0\leq
E_{\text{cost}}\leq 1$, where $E_{\text{cost}}=0$ for the exact solution. The
merit of this cost function is that the inner product is naturally calculated
by a quantum computer.
Let the dimension of the matrix $F$ be $2^{N}$. It is enough to use $N$
satisfying $2^{N-1}<D\leq 2^{N}$ without loss of generality by adding trivial
$2^{N}-D$ components to the linear equation. We propose to expand the matrix
$F$ by the gamma matrices $\Gamma_{j}$ as
$F=\sum_{j=0}^{2^{N}-1}c_{j}\Gamma_{j},$ (15)
with
$\Gamma_{j}=\bigotimes_{\beta=1}^{N}\sigma_{\alpha}^{\left(\beta\right)},$
(16)
where $\alpha$ $=0,x,y$ and $z$.
The merit of our method is that it is straightforward to determine $c_{j}$ by
the well-known formula
$c_{j}=\text{Tr}\left[\Gamma_{j}F\right].$ (17)
In order to construct a quantum circuit to calculate $c_{j}$, we express the
matrix $F$ by column vectors as
$F=\left\\{\left|f_{0}\right\rangle,\cdots,\left|f_{2^{N}-1}\right\rangle\right\\}.$
(18)
We have $\left(\left|f_{q-1}\right\rangle\right)_{p}=F_{pq}$, where subscript
$p$ denotes the $p$-th component of $\left|f_{q-1}\right\rangle$. Then $c_{j}$
is given by
$c_{j}=\sum_{q=0}^{2^{N}-1}\left(\Gamma_{j}\left|f_{q}\right\rangle\right)_{q}=\sum_{q=0}^{2^{N}-1}\left\langle\\!\left\langle
q\right|\right.\Gamma_{j}\left|f_{q}\right\rangle,$ (19)
where the subscript $q$ denotes the ($q+1$)-th component of
$\Gamma_{j}\left|f_{q}\right\rangle$. We have introduced a notation
$\left|q\right\rangle\\!\rangle\equiv|n_{1}n_{2}\cdots n_{N}\rangle$ with
$n_{i}=0,1$, where $q$ is the decimal representation of the binary number
$n_{1}n_{2}\cdots n_{N}$. See explicit examples for one and two qubits in
Method B.
The state $\left.\left|q\right\rangle\\!\right\rangle\equiv|n_{1}n_{2}\cdots
n_{N}\rangle$ is generated as follows. We prepare the NOT gates
$\sigma_{x}^{\left(i\right)}$ for the $i$-th qubit if $n_{i}=1$. Using all
these NOT gates we define
$U_{X}^{\left(q\right)}=\bigotimes\limits_{n_{i}=1}\sigma_{x}^{\left(i\right)}.$
(20)
We act it on the initial state $\left|0\right\rangle\\!\rangle$ and obtain
$U_{X}^{\left(q\right)}\left|0\right\rangle\\!\rangle=\left|q\right\rangle\\!\rangle.$
(21)
Next, we construct a unitary gate $U_{f_{q}}$ generating
$\left|f_{q}\right\rangle$,
$U_{f_{q}}\left|0\right\rangle\\!\rangle=\left|f_{q}\right\rangle.$ (22)
We will discuss how to prepare $U_{f_{q}}$ by a quantum circuit soon later;
See Eq.(28). By using these operators, $c_{j}$ is expressed as
$c_{j}=\sum_{q=0}^{2^{N}-1}\left\langle\\!\left\langle
0\right|\right.U_{X}^{\left(q\right)}\Gamma_{j}U_{f_{q}}\left|0\right\rangle\\!\rangle,$
(23)
which can be executed by a quantum computer. We show explicit examples in
Fig.2.
Once we have $c_{j}$, the final state is obtained by applying $\Gamma_{j}$ to
$|\tilde{\psi}_{\text{in}}\rangle$ and taking sum over $j$, which leads to
$|\tilde{\psi}_{\text{out}}\rangle=F|\tilde{\psi}_{\text{in}}\rangle=\sum_{j=0}^{2^{N}-1}c_{j}\Gamma_{j}|\tilde{\psi}_{\text{in}}\rangle.$
(24)
The implementation of the $\Gamma$ matrix is straightforward in quantum
circuit, because the $\Gamma$ matrix is composed of the Pauli sigma matrices,
as shown in Fig.2.
Figure 2: Quantum circuits determining $c_{j}$. We show an example with (a)
$\Gamma_{yx0}=\sigma_{y}\otimes\sigma_{x}\otimes\sigma_{0}$.
$U_{X}^{\left(6\right)}\left.\left|0\right\rangle\\!\right\rangle=\sigma_{x}^{\left(1\right)}\sigma_{x}^{\left(2\right)}\left|000\right\rangle=\left|110\right\rangle=\left.\left|6\right\rangle\\!\right\rangle$
and (b) $\Gamma_{xyz}=\sigma_{x}\otimes\sigma_{y}\otimes\sigma_{z}$.
$U_{X}^{\left(5\right)}\left|0\right\rangle\\!\rangle=\sigma_{x}^{\left(1\right)}\sigma_{x}^{\left(3\right)}\left|000\right\rangle=\left|101\right\rangle=\left.\left|5\right\rangle\\!\right\rangle$.
We may use the steepest descent method to find an optimal trial state
$|\tilde{\psi}_{\text{in}}\rangle$ closest to the state
$|\psi_{\text{in}}\rangle$. We calculate the difference of the cost function
$\Delta E_{\text{cost}}$ when we slightly change the trial state
$|\tilde{\psi}_{\text{in}}(t)\rangle$ at step $t$ by the amount of
$\Delta|\tilde{\psi}_{\text{in}}(t)\rangle$ as
$\Delta E_{\text{cost}}\equiv
E_{\text{cost}}\left(|\tilde{\psi}_{\text{in}}(t)\rangle+\Delta|\tilde{\psi}_{\text{in}}(t)\rangle\right)-E_{\text{cost}}\left(|\tilde{\psi}_{\text{in}}(t)\rangle\right)\simeq\frac{\Delta
E_{\text{cost}}}{\Delta|\tilde{\psi}_{\text{in}}(t)\rangle}\Delta|\tilde{\psi}_{\text{in}}(t)\rangle.$
(25)
We explain how to construct $|\tilde{\psi}_{\text{in}}(t)\rangle$ by a quantum
circuit soon later; See Eq.(28). Then, we renew the state as
$|\tilde{\psi}_{\text{in}}(t)\rangle\rightarrow|\tilde{\psi}_{\text{in}}(t)\rangle-\eta_{t}\frac{\Delta
E_{\text{cost}}}{\Delta|\tilde{\psi}_{\text{in}}(t)\rangle}\Delta|\tilde{\psi}_{\text{in}}(t)\rangle,$
(26)
where we use an exponential function for $\eta_{t}$,
$\eta_{t}=\xi_{1}e^{-\xi_{2}t}.$ (27)
We choose appropriate constants $\xi_{1}$ and $\xi_{2}$ for an efficient
search of the optimal solution, whose explicit examples are given in the
caption of Fig.2. We stop the renewal of the variational step when the
difference $\Delta|\tilde{\psi}_{\text{in}}(t)\rangle$ becomes sufficiently
small, which gives the optimal state of the linear equation (9).
Variational universal-quantum-state generator.
In order to construct the trial state $|\tilde{\psi}_{\text{in}}(t)\rangle$,
it is necessary to prepare an arbitrary state $\left|\psi\right\rangle$ by a
quantum circuit. Alternatively, we need such a unitary transformation $U$ that
$U\left|0\right\rangle\\!\rangle=\left|\psi\right\rangle.$ (28)
It is known that any unitary transformation is done by a sequential
application of the Hadamard, the $\pi/4$ phase-shift and the CNOT gatesDeutsch
; Dawson . Indeed, an arbitrary unitary matrix is decomposable into a
sequential application of quantum gatesDeutsch ; Dawson , each of which is
constructed as a universal quantum circuit systematicallyKraus ; Vidal ; Motto
; Shende ; Vatan ; Sousa . Universal quantum circuits have so far been
demonstrated experimentally for two and three qubitsHanne ; DiCarlo ; Qiang ;
Roy .
We may use a variational method to construct $U$ satisfying Eq.(28). Quantum
circuit learning is a variational methodMitarai , where angle variables
$\theta_{i}$ are used as variational parameters in a quantum circuit $U$, and
the cost function is optimized by tuning $\theta_{i}$. We propose to use a
quantum circuit learning for a universal quantum circuit. We show that an
arbitrary state $\left|\psi\left(\theta_{i}\right)\right\rangle$ can be
generated by tuning $U\left(\theta_{i}\right)$ starting from the initial state
$\left|0\right\rangle\\!\rangle$ as
$U\left(\theta_{i}\right)\left|0\right\rangle\\!\rangle=\left|\psi\left(\theta_{i}\right)\right\rangle.$
(29)
We adjust $\theta_{i}$ by minimizing the cost function
$E_{\text{cost}}\left(\theta_{i}\right)\equiv
1-\left|\left\langle\psi\left(\theta_{i}\right)\left|\psi\right\rangle\right.\right|^{2},$
(30)
which is the same as that of the variational quantum support vector machine.
We present explicit examples of universal quantum circuits for one, two and
three qubits in Method C.
Figure 3: Evolution of the cost function for (a) two qubits and (b) three
qubits. The vertical axis is the log${}_{10}E_{\text{cost}}$. The horizontal
axis is the number of variational steps. We use $c_{1}=0.005$ and
$c_{2}=0.005$ for both the two- and three-qubit universal quantum circuits. We
prepare random initial and final states, where we have runed simulations ten
times.
Quantum Field-Programmable-Gate Array.
We next consider a problem to find a unitary transformation $U_{\text{ini-
fin}}$ which maps an arbitrary initial state
$\left|\psi_{\text{initial}}\right\rangle$ to an arbitrary final state
$\left|\psi_{\text{final}}\right\rangle$,
$U_{\text{ini-
fin}}\left|\psi_{\text{initial}}\right\rangle=\left|\psi_{\text{final}}\right\rangle.$
(31)
Since we can generate an arbitrary unitary matrix as in Eq.(28), it is
possible to generate such matrices $U_{\text{ini}}$ and $U_{\text{fin}}$ that
$U_{\text{ini}}\left|0\right\rangle\\!\rangle=\left|\psi_{\text{initial}}\right\rangle,\qquad
U_{\text{fin}}\left|0\right\rangle\\!\rangle=\left|\psi_{\text{final}}\right\rangle.$
(32)
Then, Eq.(31) is solved as
$U_{\text{fin}}=U_{\text{ini-fin}}U_{\text{ini}},$ (33)
since $U_{\text{ini-
fin}}\left|\psi_{\text{initial}}\right\rangle=U_{\text{ini-
fin}}U_{\text{ini}}\left|0\right\rangle\\!\rangle=\left|\psi_{\text{final}}\right\rangle=U_{\text{fin}}\left|0\right\rangle\\!\rangle$.
An FPGA is a classical integrated circuit, which can be programmable by a
customer or a designer after manufacturing in a factory. An FPGA executes any
classical algorithms. On the other hand, our variational universal quantum-
state generator creates an arbitrary quantum state. We program by using the
variational parameters $\theta_{i}$. In this sense, the above quantum circuit
may be considered as a quantum generalization of FPGA, which is a quantum FPGA
(q-FPGA).
We show explicitly how the cost function is renewed for each variational step
in the case of two- and three-qubit universal quantum circuits in Fig.3, where
we have generated the initial and the final states randomly. We optimize 15
parameters $\theta_{i}$ for two-qubit universal quantum circuits and 82
parameters $\theta_{i}$ for three-qubit universal quantum circuits. We find
that $U_{\text{ini-fin}}$ is well determined by variational method as in
Fig.3.
Variational quantum support vector machine.
We demonstrate a binary classification problem in two dimensions based on the
support vector machine. We prepare a data set, where red points have a
distribution around $\left(r\cos\Theta,r\sin\Theta\right)$ with variance $r$,
while blue points have a distribution around
$\left(-r\cos\Theta,-r\sin\Theta\right)$ with variance $r$. We assume the
Gaussian normal distribution. We choose $\Theta$ randomly. We note that there
are some overlaps between the red and blue points, which is the soft margin
model.
As an example, we show the distribution of red and blue points and the lines
obtained by the variational method marked in cyan and by the direct solution
of (6) marked in magenta in Fig.1. They agrees well with one another, where
both of the lines well separate red and blue points. We have prepared 31 red
points and 32 blue points, and used six qubits.
Discussion
We have proposed that the matrix $F$ is efficiently inputted into a quantum
computer by using the $\Gamma$-matrix expansion method. There are many ways to
use a matrix in a quantum computer such as linear regression and principal
component analysis. Our method will be applicable to these cases.
Although it is possible to obtain the exact solution for the linear equation
by the HHL algorithm, it requires many gates. On the other hand, it is often
hard to obtain the exact solution by variational methods since trial functions
may be trapped to a local minimum. However, this problem is not serious for
the machine learning problem because it is more important to obtain an
approximate solution efficiently rather than an exact solution by using many
gates. Indeed, our optimized hyperplane also well separates red and blue
points as shown in Fig.1(a).
In order to classify $M$ data, we need to prepare $\log_{2}M$ qubits. It is
hard to execute a large number of data points by current quantum computers.
Recently, it is shown that electric circuits may simulate universal quantum
gatesEzawaUniv ; EzawaDirac ; LCBit based on the fact that the Kirchhoff law
is rewritten in the form of the Schrödinger equationEzawaSch . Our variational
algorithm will be simulated by using them.
Methods
A: Support vector machine.
A support vector machine is an algorithm for supervised learningVap ; Noble ;
Suy . We first prepare a set of training data, where each point is marked
either in red or blue. Then, we determine a hyperplane separating red and blue
points. After learning, input data are classified into red or blue by
comparing the input data with the hyperplane. The support vector machine
maximizes a margin, which is a distance between the hyperplane and data
points. If red and blue points are perfectly separated by the hyperplane, it
is called a hard margin problem [Fig.4(a)]. Otherwise, it is called a soft
margin problem [Fig.4(b)].
We minimize the distance $d_{j}$ between a data point $\boldsymbol{x}_{j}$ and
the hyperplane given by
$d_{j}=\frac{\left|\boldsymbol{\omega}\cdot\boldsymbol{x}_{j}+\omega_{0}\right|}{\left|\boldsymbol{\omega}\right|}.$
(34)
We define support vectors $\boldsymbol{x}$ as the closest points to the
hyperplane. There is such a vector in each side of the hyperplane, as shown in
Fig.4(a). This is the origin of the name of the support vector machine.
Without loss of generality, we set
$\left|\boldsymbol{\omega}\cdot\boldsymbol{x}+\omega_{0}\right|=1$ (35)
for the support vectors, because the hyperplane is present at the equidistance
of two closest data points and because it is possible to set the magnitude of
$\left|\boldsymbol{\omega}\cdot\boldsymbol{x}+\omega_{0}\right|$ to be $1$ by
scaling $\boldsymbol{\omega}$ and $\omega_{0}$. Then, we maximize the distance
$d=\frac{\left|\boldsymbol{\omega}\cdot\boldsymbol{x}+\omega_{0}\right|}{\left|\boldsymbol{\omega}\right|}=\frac{1}{\left|\boldsymbol{\omega}\right|},$
(36)
which is identical to minimize $\left|\boldsymbol{\omega}\right|$.
Figure 4: Illustration of the hyperplane and the support vector. Two support
vectors are marked by red and blue squares. (a) Hard margin where red and blue
points are separated perfectly, and (b) soft margin where they are separated
imperfectly.
First, we consider the hard margin problem, where red and blue points are
perfectly separable. All red points satisfy
$\boldsymbol{\omega}\cdot\boldsymbol{x}_{j}+\omega_{0}>1$ and all blue points
satisfy $\boldsymbol{\omega}\cdot\boldsymbol{x}_{j}+\omega_{0}<-1$. We
introduce variables $y_{j}$, where $y_{j}=1$ for red points and $y_{j}=-1$ for
blue points. Using them, the condition is rewritten as
$\left(\boldsymbol{\omega}\cdot\boldsymbol{x}_{j}+\omega_{0}\right)y_{j}\geq
1$ (37)
for each $j$. The problem is reduced to find the minimum of
$\left|\boldsymbol{\omega}\right|^{2}$ under the above inequalities. The
optimization under inequality conditions is done by the Lagrange multiplier
method with the Karush-Kuhn-Tucker conditionKKT . It is expressed in terms of
the Lagrangian as
$L\left(\boldsymbol{\omega},\omega_{0},\boldsymbol{\alpha}\right)=\frac{1}{2}\left|\boldsymbol{\omega}\right|^{2}-\sum_{j}\beta_{j}[\left(\boldsymbol{\omega}\cdot\boldsymbol{x}_{j}+\omega_{0}\right)y_{j}-1],$
(38)
where $\beta_{j}$ are Lagrange multipliers to ensure the constraints.
For the soft margin case, we cannot separate two classes exactly. In order to
treat this case, we introduce slack variables $\xi_{j}$ satisfying
$\left(\boldsymbol{\omega}\cdot\boldsymbol{x}_{j}+\omega_{0}\right)y_{j}\geq
1-\xi_{j},\qquad\xi_{j}\geq 0$ (39)
and redefine the cost function as
$E_{\text{cost}}=\frac{1}{2}\left|\boldsymbol{\omega}\right|^{2}+\gamma\sum_{j=1}^{M}\xi_{j}^{2}.$
(40)
Here, $\gamma=\infty$ corresponds to the hard margin. The second term
represents the penalty for some of data points to have crossed over the
hyperplane. The Lagrangian is modified as
$L\left(\boldsymbol{\omega},\omega_{0},\xi_{i},\boldsymbol{\beta}\right)=\frac{1}{2}\left|\boldsymbol{\omega}\right|^{2}+\gamma\sum_{j=1}^{M}\xi_{j}^{2}-\sum_{j=1}^{M}\left[\left(\boldsymbol{\omega}\cdot\boldsymbol{x}_{j}+\omega_{0}\right)\beta_{j}y_{j}-\left(1-\xi_{i}\right)\right].$
(41)
The stationary points are determined by
$\displaystyle\frac{\partial L}{\partial\boldsymbol{\omega}}=$
$\displaystyle\boldsymbol{\omega}-\sum_{j=1}^{M}\beta_{j}y_{j}\boldsymbol{x}_{j}=0,$
(42) $\displaystyle\frac{\partial L}{\partial\omega_{0}}=$
$\displaystyle-\sum_{j=1}^{M}\beta_{j}y_{j}=0,$ (43)
$\displaystyle\frac{\partial L}{\partial\xi_{j}}=$
$\displaystyle\gamma\xi_{j}-\beta_{j}=0,$ (44) $\displaystyle\frac{\partial
L}{\partial\beta_{j}}=$
$\displaystyle\left(\boldsymbol{\omega}\cdot\boldsymbol{x}_{j}+\omega_{0}\right)y_{j}-\left(1-\xi_{i}\right)=0.$
(45)
We may solve these equations to determine $\boldsymbol{\omega}$ and $\nu_{j}$
as
$\boldsymbol{\omega}=\sum_{j=1}^{M}\beta_{j}y_{j}\boldsymbol{x}_{j},$ (46)
from (42), and
$\xi_{j}=\beta_{j}/\gamma$ (47)
from (44). Inserting them into (45), we find
$y_{j}\sum_{i=1}^{M}\left(\beta_{i}y_{i}\boldsymbol{x}_{i}\cdot\boldsymbol{x}_{j}+\omega_{0}\right)-\left(1-\beta_{j}/\gamma\right)=0.$
(48)
Since $y_{j}^{2}=1$, it is rewritten as
$\omega_{0}+\sum_{i=1}^{M}\left(\boldsymbol{x}_{i}\cdot\boldsymbol{x}_{j}+\delta_{ij}/\gamma\right)\beta_{i}y_{i}=y_{j}.$
(49)
Since $\beta_{j}$ appears always in a pair with $y_{j}$, we introduce a new
variable defined by
$\alpha_{j}=\beta_{j}y_{j},$ (50)
and we define the Kernel matrix $K_{ij}$ as
$K_{ij}=\boldsymbol{x}_{i}\cdot\boldsymbol{x}_{j}.$ (51)
Then, $\omega_{0}$ and $\alpha_{j}$ are obtained by solving linear equations
$\displaystyle\sum_{i=1}^{M}\alpha_{j}=$ $\displaystyle 0,$ (52)
$\displaystyle\omega_{0}+\sum_{i=1}^{M}\left(\boldsymbol{x}_{i}\cdot\boldsymbol{x}_{j}+\delta_{ij}/\gamma\right)\alpha_{i}=$
$\displaystyle y_{j},$ (53)
which are summarized as
$\left(\begin{array}[]{cccc}0&1&\cdots&1\\\ 1&&&\\\ \vdots&&K+I_{M}/\gamma&\\\
1&&&\end{array}\right)\left(\begin{array}[]{c}\omega_{0}\\\ \alpha_{1}\\\
\vdots\\\ \alpha_{M}\end{array}\right)=\left(\begin{array}[]{c}0\\\ y_{1}\\\
\vdots\\\ y_{M}\end{array}\right),$ (54)
which is Eq.(6) in the main text. Finally, $\boldsymbol{\omega}$ is determined
by
$\boldsymbol{\omega}=\sum_{j=1}^{M}\alpha_{j}\boldsymbol{x}_{j}.$ (55)
Once the hyperplane is determined, we can classify new input data into red if
$\boldsymbol{\omega}\cdot\boldsymbol{x}_{j}+\omega_{0}>0$ (56)
and blue if
$\boldsymbol{\omega}\cdot\boldsymbol{x}_{j}+\omega_{0}<0.$ (57)
Thus, we obtain the hyperplane for binary classification.
B: $\Gamma$ matrix expansion.
We explicitly show how to calculate $c_{j}$ in (17) based on the $\Gamma$
matrix expansion for the one and two qubits.
One qubit:
We show an explicit example of the $\Gamma$-matrix expansion for one qubit.
Ome qubit is represented by a $2\times 2$ matrix,
$F=\left(\begin{array}[]{cc}F_{11}&F_{12}\\\ F_{21}&F_{22}\end{array}\right).$
(58)
The column vectors are explicitly given by
$\displaystyle\left|f_{1}\right\rangle=$
$\displaystyle\left(\begin{array}[]{c}F_{11}\\\
F_{21}\end{array}\right)=F_{11}\left|0\right\rangle+F_{21}\left|1\right\rangle,\qquad$
(61) $\displaystyle\left|f_{2}\right\rangle=$
$\displaystyle\left(\begin{array}[]{c}F_{12}\\\
F_{22}\end{array}\right)=F_{12}\left|0\right\rangle+F_{22}\left|1\right\rangle.$
(64)
The coefficient $c_{j}$ in (17) is calculated as
$c_{j}=\text{Tr}\left[\sigma_{j}F\right]=\langle
0|\sigma_{j}\left|f_{1}\right\rangle+\left\langle
1\right|\sigma_{j}\left|f_{2}\right\rangle=\sum_{p=0,1}\langle
p|\sigma_{j}\left|f_{p}\right\rangle=\sum_{p=0,1}\langle
0|U_{X}^{\left(p\right)}\sigma_{j}U_{f_{p}}\left|0\right\rangle.$ (65)
Two qubits:
Next, we show an explicit example of the $\Gamma$-matrix expansion for two
qubits. Two qubits are represented by a $4\times 4$ matrix,
$F=\left(\begin{array}[]{cccc}F_{11}&F_{12}&F_{13}&F_{14}\\\
F_{21}&F_{22}&F_{23}&F_{24}\\\ F_{31}&F_{32}&F_{33}&F_{34}\\\
F_{41}&F_{42}&F_{43}&F_{44}\end{array}\right).$ (66)
The column vectors are explicitly given by
$\displaystyle\left|f_{1}\right\rangle=$
$\displaystyle\left(\begin{array}[]{c}F_{11}\\\ F_{21}\\\ F_{31}\\\
F_{41}\end{array}\right)=F_{11}\left|00\right\rangle+F_{21}\left|01\right\rangle+F_{31}\left|10\right\rangle+F_{41}\left|11\right\rangle,$
(71) $\displaystyle\left|f_{2}\right\rangle=$
$\displaystyle\left(\begin{array}[]{c}F_{12}\\\ F_{22}\\\ F_{32}\\\
F_{42}\end{array}\right)=F_{12}\left|00\right\rangle+F_{22}\left|01\right\rangle+F_{32}\left|10\right\rangle+F_{42}\left|11\right\rangle,$
(76) $\displaystyle\left|f_{3}\right\rangle=$
$\displaystyle\left(\begin{array}[]{c}F_{13}\\\ F_{23}\\\ F_{33}\\\
F_{43}\end{array}\right)=F_{13}\left|00\right\rangle+F_{23}\left|01\right\rangle+F_{33}\left|10\right\rangle+F_{43}\left|11\right\rangle,$
(81) $\displaystyle\left|f_{4}\right\rangle=$
$\displaystyle\left(\begin{array}[]{c}F_{14}\\\ F_{24}\\\ F_{34}\\\
F_{44}\end{array}\right)=F_{14}\left|00\right\rangle+F_{24}\left|01\right\rangle+F_{34}\left|10\right\rangle+F_{44}\left|11\right\rangle.$
(86)
The coefficient $c_{j}$ in (17) is calculated as
$c_{j}=\text{Tr}\left[\Gamma_{j}F\right]=\left\langle
00\right|\Gamma_{j}\left|f_{1}\right\rangle+\left\langle
01\right|\Gamma_{j}\left|f_{2}\right\rangle+\left\langle
10\right|\Gamma_{j}\left|f_{3}\right\rangle+\left\langle
11\right|\Gamma_{j}\left|f_{4}\right\rangle=\sum_{p=0}^{3}\langle\\!\left\langle
p\right|\Gamma_{j}\left|f_{p}\right\rangle=\sum_{p=0}^{3}\langle\\!\left\langle
0\right|U_{X}^{\left(p\right)}\Gamma_{j}U_{f_{p}}\left|0\right\rangle\\!\rangle.$
(87)
C: Universal quantum circuits.
Angle variables are used as variational parameters in a universal quantum
circuit learning. We present examples for one, two and three qubits.
One-qubit universal quantum circuit:
The single-qubit rotation gates are defined by
$\displaystyle R\left(\theta,\phi\right)=$
$\displaystyle\exp\left[-i\theta\left(\sigma_{x}\cos\phi+\sigma_{y}\sin\phi\right)/2\right],$
(88) $\displaystyle R_{z}\left(\phi_{z}\right)=$
$\displaystyle\exp\left[-i\sigma_{z}\phi_{z}/2\right].$ (89)
The one-qubit universal quantum circuit is constructed as
$U^{\left(1\right)}\left(\theta,\phi,\phi_{z}\right)=R\left(\theta,\phi\right)R_{z}\left(\phi_{z}\right)=\left(\begin{array}[]{cc}e^{-i\phi_{z}/2}\cos\frac{\theta}{2}&-ie^{i\left(\phi_{z}/2-\phi\right)}\sin\frac{\theta}{2}\\\
-ie^{-i\left(\phi_{z}/2-\phi\right)}\sin\frac{\theta}{2}&e^{i\phi_{z}/2}\cos\frac{\theta}{2}\end{array}\right).$
(90)
We show a quantum circuit in Fig.5(a). There are three variational parameters.
It is obvious that an arbitrary state is realized starting from the state
$\left|0\right\rangle$ as
$U\left(1\right)\left(\begin{array}[]{c}1\\\
0\end{array}\right)=\left(\begin{array}[]{c}e^{-i\phi_{z}/2}\cos\frac{\theta}{2}\\\
-ie^{-i\left(\phi_{z}/2-\phi\right)}\sin\frac{\theta}{2}\end{array}\right).$
(91)
Figure 5: Universal quantum circuits for (a) one, (b) two and (c) three
qubits.
Two-qubit universal quantum circuit:
The two-qubit universal quantum circuit is constructed asHanne
$\displaystyle U\left(2\right)\equiv$
$\displaystyle\left[U^{\left(1\right)}\left(\theta_{A},\phi_{A},\phi_{z,A}\right)\otimes
U^{\left(1\right)}\left(\theta_{B},\phi_{B},\phi_{z,B}\right)\right]U_{G}\left[R\left(\theta_{E},0\right)\otimes
R\left(\frac{3\pi}{2},0\right)\right]U_{G}\left[R\left(\theta_{F},\frac{\pi}{2}\right)\otimes
R\left(\frac{3\pi}{2},\theta_{G}\right)\right]$ $\displaystyle
U_{G}\left[U^{\left(1\right)}\left(\theta_{C},\phi_{C},\phi_{z,C}\right)\otimes
U^{\left(1\right)}\left(\theta_{D},\phi_{D},\phi_{z,D}\right)\right],$ (92)
where the entangling two-qubit gate is defined byHanne
$U_{G}=e^{-i\pi/4}\exp\left[\frac{i\pi}{4}\sigma_{z}\otimes\sigma_{z}\right].$
(93)
The two-qubits universal quantum circuit contains 15 variational parameters.
We show a quantum circuit in Fig.5(b).
Three-qubit universal quantum circuit:
The three-qubit universal quantum circuit is constructed as
$U\left(3\right)\equiv\left[U_{A}^{\left(2\right)}\otimes
U_{A}^{\left(1\right)}\right]U_{A}\left(3\right)\left[U_{B}^{\left(2\right)}\otimes
U_{B}^{\left(1\right)}\right]U_{C}\left(3\right)\left[U_{C}^{\left(2\right)}\otimes
U_{C}^{\left(1\right)}\right]U_{B}\left(3\right)\left[U_{D}^{\left(2\right)}\otimes
U_{D}^{\left(1\right)}\right],$ (94)
where $U_{A}^{\left(1\right)}$, $U_{B}^{\left(1\right)}$,
$U_{C}^{\left(1\right)}$, and $U_{D}^{\left(1\right)}$ are one-qubit universal
quantum circuits, while $U_{A}^{\left(2\right)}$, $U_{B}^{\left(2\right)}$,
$U_{C}^{\left(2\right)}$, and $U_{D}^{\left(2\right)}$ are two-qubit universal
quantum circuit and
$\displaystyle U_{A}\left(3\right)=$
$\displaystyle\exp\left[i\left(\theta_{xxz}^{A}\sigma_{x}\otimes\sigma_{x}\otimes\sigma_{z}+\theta_{yyz}^{A}\sigma_{x}\otimes\sigma_{x}\otimes\sigma_{z}+\theta_{zzz}^{A}\sigma_{z}\otimes\sigma_{z}\otimes\sigma_{z}\right)\right],$
(95) $\displaystyle U_{B}\left(3\right)=$
$\displaystyle\exp\left[i\left(\theta_{xxz}^{B}\sigma_{x}\otimes\sigma_{x}\otimes\sigma_{z}+\theta_{yyz}^{B}\sigma_{x}\otimes\sigma_{x}\otimes\sigma_{z}+\theta_{zzz}^{B}\sigma_{z}\otimes\sigma_{z}\otimes\sigma_{z}\right)\right],$
(96) $\displaystyle U_{C}\left(3\right)=$
$\displaystyle\exp\left[i\left(\theta_{xxx}^{C}\sigma_{x}\otimes\sigma_{x}\otimes\sigma_{x}+\theta_{yyx}^{C}\sigma_{y}\otimes\sigma_{y}\otimes\sigma_{x}+\theta_{zzx}^{C}\sigma_{z}\otimes\sigma_{z}\otimes\sigma_{x}+\theta_{00x}^{C}\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{x}\right)\right].$
(97)
Eplicit quantum circuits for $U_{A}\left(3\right)$, $U_{B}\left(3\right)$ and
$U_{C}\left(3\right)$ are shown in Ref.Sousa . The three-qubits universal
quantum circuit contains 82 variational parameters. We show a quantum circuit
in Fig.5(c).
Multi-qubit universal quantum circuit:
General multi-qubit universal quantum circuit is constructed in Ref.Motto .
The minimum numbers of variational parameters are $4^{N}-1$ for $N$-qubit
unicersal quantum circuits. However, we need more variational parameters in
the currently known algorithm for $N\geq 3$.
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Acknowledgements
The author is very much grateful to E. Saito and N. Nagaosa for helpful
discussions on the subject. This work is supported by the Grants-in-Aid for
Scientific Research from MEXT KAKENHI (Grants No. JP17K05490 and No.
JP18H03676). This work is also supported by CREST, JST (JPMJCR16F1 and
JPMJCR20T2).
Author contributions
M.E. conceived the idea, performed the analysis, and wrote the manuscript.
Additional information
Competing financial and non-financial interests: The author declares no
competing financial and non-financial interests.
|
# Experimental demonstration of indefinite causal order induced quantum heat
extraction
Huan Cao1,2 These two authors contributed equally to this work. Ning-ning
Wang1,2 These two authors contributed equally to this work. Zhih-Ahn Jia1,2
Chao Zhang1,2<EMAIL_ADDRESS>Yu Guo1,2 Bi-Heng Liu1,2 Yun-Feng
Huang1,2<EMAIL_ADDRESS>Chuan-Feng Li1,2<EMAIL_ADDRESS>Guang-Can Guo1,2
1CAS Key Laboratory of Quantum Information, University of Science and
Technology of China, Hefei, 230026, China
2CAS Center For Excellence in Quantum Information and Quantum Physics, Hefei,
230026, China
###### Abstract
In the classical world, physical events always happen in a fixed causal order.
However, it was recently revealed that quantum mechanics allows events to
occur with indefinite causal order (ICO). In this study, we use an optical
quantum switch to experimentally investigate the application of ICO in
thermodynamic tasks. Specifically, we demonstrate that when a working system
interacts with two thermal reservoirs in an ICO, non-classical heat transfer
can be observed, even through they share the same temperature. Using such a
process, we simulate an ICO refrigeration cycle and investigate its
properties. We also show that by passing through the ICO channel multiple
times, one can extract more heat per cycle and thus obtain a higher
refrigeration efficiency. Our results provide inspirations for further
improving the efficiency of quantum thermodynamic tasks and shed new light on
the development of a new class of thermodynamic resource theories without
presumed causal order.
It is a deeply rooted concept that in a physical theory, there is a well-
defined pre-existing classical causal structure for which physical events
happen in a definite causal order. However, from the Bell-Kochen-Specker
theorem Bell (1964); Kochen and Specker (1975), we known that quantum
mechanics is incompatible with the viewpoint that physical observables have
pre-existing values independent of the measurement context. Inspired by this,
many recent studies on causal structure of quantum mechanics have shown that
if we assume the causal relation to obey the laws of quantum mechanics, it is
possible for two events to occur with a superposed causal order. Thus, there
is no pre-existing causal relation Oreshkov et al. (2012); Brukner (2014). The
quantum causal structure becomes especially crucial when quantum physics and
general relativity become relevant Oriti (2009); Hossenfelder (2017); Marletto
and Vedral (2017); Bose et al. (2017); Peres and Terno (2004). A typical
example is the quantum spacetime causal structure in the study of quantum
gravity Christodoulou and Rovelli (2019); DeWitt (1967); Rovelli (1990);
Gambini et al. (2004); Hardy (2009). For instance, when a massive object is in
a superposition of two or more distinct spatial positions, the states of the
gravitational field produced by the object will also be in a superposition,
implying that the spacetime causal order that originates from the spacetime
geometry will also be in a superposition.
In addition to the fundamental properties of indefinite causal order (ICO) and
causal structure, the applications of ICO as an operational resource in
quantum information and computation also attract considerable research
interests Taddei et al. (2019); Jia et al. (2019). It provides remarkable
enhancements ranging from channel discrimination Chiribella (2012),
communication and computation complexity Guérin et al. (2016); Feix et al.
(2015); Araújo et al. (2014) to quantum metrology Zhao et al. (2020);
Mukhopadhyay et al. (2018) , quantum information transmission Ebler et al.
(2018); Salek et al. (2018); Chiribella et al. (2018) etc. Recently, some of
these predictions have also been studied in the experiment, in which optical
quantum switch is utilized to simulate the ICO process Procopio et al. (2015);
Rubino et al. (2017); Goswami et al. (2018); Guo et al. (2020); Goswami et al.
(2020); Wei et al. (2019).
In thermodynamics, entropy in closed systems always tends to increase
definitely. An interesting question is what will happen when apply ICO in
thermodynamic tasks. One of such example is the recent discovery of ICO-based
quantum refrigeration Felce and Vedral (2020). There are several different
ways for quantum refrigeration: the standard one is powered by energy injected
by a time dependent driving force Campisi et al. (2015); Campisi and Fazio
(2016); the Maxwell demon can also be used to steer the heat by means of a
feedback control loop Maruyama et al. (2009); Elouard et al. (2017); while
another method is to use invasive quantum measurements as a resource to power
the refrigeration Buffoni et al. (2019). All the above refrigeration protocols
works in a world with pre-existing causal structures. The ICO-based protocol
provides a good supplement for which no pre-existing causal relation is
assumed Felce and Vedral (2020). This sheds new light on both of the quantum
causal structure and quantum thermodynamics.
In this paper, we experimentally demonstrate the advantages of ICO in
performing quantum thermodynamic tasks Mukhopadhyay et al. (2018); Guha et al.
(2020); Felce and Vedral (2020); Markes and Hardy (2011). In particular, by
using an optical quantum simulator, we demonstrate the ICO induced heat
extraction and investigate its feasibility to construct a quantum
refrigerator. We also show that by interacting with reservoirs in an ICO
multiple times, one can extract more heat from the reservoirs. The high
accuracy achieved in our experiment is expected to lead to more operational
protocols and thus contribute to a broader research into the use of ICO as a
resource.
Protocol outline.— Before detailing the experimental realization of the ICO-
driven quantum heat extraction, let us first recall how the protocol works.
Consider a system with Hamiltonian $\mathcal{H}$ and energy eigenstates
$\left|n\right\rangle$ for energy level $E_{n}$. After thermocontact with a
thermal reservoir with inverse temperature $\beta$, the resulting equilibrium
state of the system is always $T=e^{-\beta\mathcal{H}/Z}=\sum_{n}e^{-\beta
E_{n}}/Z\left|n\right\rangle\left\langle n\right|$ regardless of the initial
system state $\rho$, where
$Z=\mathrm{Tr}\left(e^{-\beta\mathcal{H}}\right)=\sum_{n}e^{-\beta E_{n}}$ is
the partition function. This thermodynamics operation can be characterized by
a quantum channel, namely, a completely positive trace preserving (CPTP) map
$\mathcal{T}\colon\mathcal{L}\left(\mathcal{H}\right)\rightarrow\mathcal{L}\left(\mathcal{H}\right)$
for which $\mathcal{T}\left(\rho\right)=T$ for all density operators $\rho$.
We denote the Kraus operators for $\mathcal{T}$ as $K_{i}$. The Kraus
decomposition is thus $\mathcal{T}\left(\rho\right)=\sum_{i}K_{i}\rho
K_{i}^{\dagger}$, where the Kraus operators $\left\\{K_{i}\right\\}$ satisfy
$\sum_{i}K_{i}^{\dagger}K_{i}=I$.
Now consider the situation where the system in state $\rho$ undergoes
thermocontact sequentially with two identical thermal reservoires with the
same temperature. If we assume the definite causal order, then the process is
given either by $\mathcal{T}^{1}\circ\mathcal{T}^{2}\left(\rho\right)$ or
$\mathcal{T}^{2}\circ\mathcal{T}^{1}\left(\rho\right)$, or potentially a
classical probabilistic mixture of these two processes, and the same thermal
state $T$ is obtained. However, when in a world with ICO, the two events
‘thermocontact with thermal reservoir 1 firstly’ and ‘thermocontact with
thermal reservoir 2 firstly’ can occur in a superposed causal order. This will
yield an interesting result—the resulting state is different from $T$. Such an
operation can be optically simulated using the quantum switch Chiribella
(2012); Chiribella et al. (2013); Procopio et al. (2015). The action of ICO is
achieved by routing photons through two channels with the visiting order being
tailored by the control qubit Procopio et al. (2015); Rubino et al. (2017);
Goswami et al. (2018); Guo et al. (2020); Goswami et al. (2020); Wei et al.
(2019). when control qubit $\left|\phi_{c}\right\rangle=\left|1\right\rangle$
or $\left|\phi_{c}\right\rangle=\left|0\right\rangle$ , the operations
$\mathcal{T}^{2}\circ\mathcal{T}^{1}$ or $\mathcal{T}^{1}\circ\mathcal{T}^{2}$
is carried out respectively. We denote the corresponding channel as
$\mathcal{S}^{T}$. In terms of Kraus operators, we have
$\mathcal{S}^{T}\left(\rho_{c}\otimes\rho\right)=\sum_{ij}M_{ij}\left(\rho_{c}\otimes\rho\right)M_{ij}^{\dagger}$
and
$M_{ij}=\left|0\right\rangle\left\langle
0\right|_{c}K_{i}^{1}K_{j}^{2}+\left|1\right\rangle\left\langle
1\right|_{c}K_{j}^{2}K_{i}^{1}$ (1)
where $K_{i}^{1}$ ($K_{j}^{2}$) represents the $i-$th ($j-$th) Kraus operator
of the thermalizing channels $\mathcal{T}^{1}$ ($\mathcal{T}^{2}$).
Considering the most simple non-trivial case, a two-level system, the ground
state is $\left|0\right\rangle$ (the excited state is $\left|1\right\rangle$)
with energy $E_{0}=0$ ($E_{1}=\Omega$), thus, the Hamiltonian for the system
is $\mathcal{H}=\Omega\left|1\right\rangle\left\langle 1\right|$. The thermal
state at a given temperature is $\rho=diag\left(1,e^{-\beta\Omega}\right)/Z$,
where $Z=1+e^{-\beta\Omega}$. In the following we set $\Omega=1$ for
simplicity. If the ancillary control qubit is initialized as
$\left|\phi_{c}\right\rangle=\left(\left|0\right\rangle+\left|1\right\rangle\right)_{c}/\sqrt{2}$.
the resultant output state undergoing ICO with two identical thermalizing
channels will be
$\displaystyle\mathcal{S}\left(\rho_{c}\otimes\rho\right)$ $\displaystyle=$
$\displaystyle\sum_{ij}M_{ij}\left(\rho_{c}\otimes\rho\right)M_{ij}^{\dagger}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[\left(\left|0\right\rangle\left\langle
0\right|_{c}+\left|1\right\rangle\left\langle 1\right|_{c}\right)\otimes
T\right.$ $\displaystyle\left.+\left(\left|0\right\rangle\left\langle
1\right|_{c}+\left|1\right\rangle\left\langle 0\right|_{c}\right)\otimes T\rho
T\right]$
Where the control qubit is
$\rho_{c}=\left|\phi_{c}\right\rangle\left\langle\phi_{c}\right|$. Note that
the ICO evolution inside the quantum switch entangles the control qubit with
the resultant system state; if the control qubit is projected into
$\left|\pm\right\rangle=\left(\left|0\right\rangle\pm\left|1\right\rangle\right)_{c}/\sqrt{2}$,
the system state will collapse into
$\mathrm{Tr}_{c}\left[\left|\pm\right\rangle\left\langle\pm\right|_{c}\mathcal{S}\left(\rho_{c}\otimes\rho\right)\right]=\frac{1}{2}\left(T\pm
T\rho T\right)$ (3)
with the corresponding probability $p_{\pm}=\frac{1}{2}\mathrm{Tr}\left[T\pm
T\rho T\right]$. We note that the temperature of output system state could be
different from the thermal state $T$, with the temperature depending on the
measurement outcome in $\left|\pm\right\rangle_{c}$ basis and initial system
state $\rho$. This intriguing phenomenon suggests that the ICO can either
extract heat from or dump heat into the reservior. Hence ICO is potentially
useful for realizing thermodynamic tasks, such as refrigerating or cooling the
input system.
Figure 1: Schematic of the experimental apparatus. The quantum switch contains
two identical thermalizing channels (with pink planes underneath) in a
indefinite causal order. One of the causal order is presented by a bluish
optical path, the other is presented by a red optical path. The BS2 concludes
the superposition. BS: beam splitter, PBS: polarization beam spliter, HWP:
half wave plate, IF: interferometer filter.
Experimental implementation.— We build a high-performance tabletop photonic
quantum switch to realize the ICO process. A spontaneous parametric down
conversion (SPDC) produces heralded single photons at 780 nm. The heralded
photons, which act as working substances, are then fed into a Mach-Zenhder
interferometer, as shown in Fig 1. Here we use photonic polarization to
simulate the two energy levels of the working system, where horizontal
(vertical) polarization state H (V) represents the ground (excited) state. The
population of the excited state is related to the temperature of the working
system. A system state at an arbitrary temperature is prepared by randomly
rotating the polarization of the photons into H or V with a probability
proportional to its temperature. Energy detection can be realized by
measurements in Pauli $\sigma_{z}$ basis; thus, it is possible to detect the
temperature. To realize the ICO quantum channel (see eq.(Experimental
demonstration of indefinite causal order induced quantum heat extraction)),
the beam splitter 1 (BS1) introduces two spatial modes as the control qubit.
The polarization qubit undergoes the causal order
$\mathcal{T}^{1}\circ\mathcal{T}^{2}$ in one spatial mode, while
$\mathcal{T}^{2}\circ\mathcal{T}^{1}$ in the other one. BS2 then coherently
combines these spatial modes and projects the control qubit onto
$\left|\pm\right\rangle$. The ICO is valid when the spatial modes inside
quantum switch admit the interferometer. A phase-locking system is adopted to
ensure the stability of the path interferometer, with an average
interferometric visibility of more than 99.7% for four hours, which
sufficiently guarantees the ICO performance (See Section II in the
Supplementary Material for details). The action of the thermalizing channel
can be simulated using a probabilistic mixture of two fully amplitude damping
(AD) channels Fisher et al. (2012); Lu et al. (2017); Mancino et al. (2017),
$\mathcal{T}\left(\rho\right)=\sum_{i=0}^{3}p_{i}K_{i}\rho K_{i}^{\dagger}$,
where the $\left\\{K_{0},K_{1}\right\\}$ ($\left\\{K_{2},K_{3}\right\\}$)
constitute a channel damping to ground (excited) state, and
$\overrightarrow{p}=\left\\{p_{0},p_{1},p_{2},p_{3}\right\\}$ denote the
probability vectors, which depend on the temperature of the thermalizing
channel. The AD channel is realized using polarization M-Z interferometer-like
setup with half-wave plates (HWPs) and polarization beam splitters (PBSs), as
shown in Fig.1. We have performed the process tomography of the thermalizing
channel at several different temperatures. The average process fidelity
exceeded 99.9%, verifying the credible simulation of the channel (see section
II in Supplementary Material for details).
Figure 2: (a) Energy change $\Delta
E_{\pm}=\mathrm{Tr}\left(\rho_{\pm}\mathcal{H}\right)-\mathrm{Tr}\left(\rho\mathcal{H}\right)$
where $\rho_{\pm}=\left[T\pm T\rho T\right]/\mathrm{Tr}\left[T\pm T\rho
T\right]$ is the normalized output system state with the control qubit
measured in $\left|\pm\right\rangle$. The weighted energy change is presented
as $\Delta\widetilde{E}_{\pm}$. We account for the deviation of experimental
data to theoretical prediction when the $E_{C}\rightarrow 0$ that at ultra
cool temperature, the success probability $P_{-}$ approaches to zero, thus the
collecting signal counting is low but the noise due to experimental
imperfection never decreases, which causes the signal-to-noise level to
gradually deteriorate. (b) The energy of the working system
$E_{\rho}=\mathrm{Tr}\left(\rho\mathcal{H}\right)$ when the control qubit is
measured at
$\left|+\right\rangle,\left|-\right\rangle,\left|0\right\rangle,\left|1\right\rangle$.
The histograms represent the theoretical prediction, while the plots represent
the experimental result. The dotted line shows the corresponding quantity of
the input state. The quoted error in the experimental data points in plots
reflects the impact of Poissonian statistics on the collecting count numbers.
Results of the ICO process.— We first verify the non-classical heat transfer
driven by the ICO. We implement numbers of experiments by traversing the
temperatures of thermalizing channel which belongs to the reservoir in the
following discussion. The system state is initialized into the thermal state
with the same temperature ($1/\beta_{C}$) as the reservoir, $\rho=T$. Fig. 2
(a) shows the measured energy change $\Delta E$ of the system qubit (as the
working substance) after passing through the ICO channel. The horizontal
coordinate $E_{C}=e^{-\beta_{C}}/\left(1+e^{-\beta_{C}}\right)$ ranges from 0
to 0.5, corresponding to the temperatures from zero to infinity (here we use
the energy of the thermal state of the working system to represent the
reservoir’s temperature for simplicity). By extrapolating our experimental
data ( red and blue dots in Fig. 2 (a)) to theoretical prediction (red and
blue lines in Fig. 2 (a)), we find that the working substance can extract the
heat flow from the reservoir when the control qubit is measured in
$\left|-\right\rangle$, even though they initially share the same temperature.
This intriguing phenomenon is applicable in arbitrary temperature case with
the exceptions of zero and infinity. The heat transfer decreases when the
temperature increases, whereas the successful probability
$p_{-}=\mathrm{Tr}\left[\left|-\right\rangle\left\langle-\right|S\left(\rho_{c}\otimes\rho\right)\right]$
increases (inset in Fig. 2 (a)). The weighted energy change
$\Delta\widetilde{E}_{\pm}=p_{\pm}\Delta E_{\pm}$ is also presented (orange
and green dots and lines 2 (a)). For comparison, we also measure the energy
transfer when the control qubit is measured in computational basis
$\left\\{\left|0\right\rangle,\left|1\right\rangle\right\\}$, by exemplifying
the reservoir’s temperature to be $E_{C}=0.25$ (Fig.2(b)). Such a case yields
the classical outcome in which the output working substance has the same
temperature as the reservoir, and no heat extraction could achieve.
Obviously, the non-classical heat transfer driven by ICO can be used for
thermodynamic tasks. For example, when the working substance appears at the
heating branch, we can interact it with an external hot reservoir to release
the extracted heat, thus refrigerating the reservoir; when the working
substance appears at the cooling branch, we can send it back to the reservoir
to erase the unwanted heat exchange. For the opposite purpose (i.e., to
construct a heat engine or cool the working substance), one can instead
discard the heating branch and reserve the cooling branch. An interesting
question is whether the working substance can become colder or hotter after
passing through the ICO channel multiple times.
Figure 3: (a) Energy of the working system after passing through the ICO
channel multiple times. Each step will generate a two-branch outcome. The
latter step is performed only when the former step yields a failed outcome
$\left|+\right\rangle$ (multi-pass condition, as shown by the arrows in each
step, with intermediate ones are abbreviated with ellipsis). The saturated
energy in both outcomes are indicated by red and blue lines. In this
experiment, the initial working substance had the same temperature as the
reservoir with $E_{C}=0.25$. (b) Coefficients of the quantum refrigerator
based on the classical strategy (blue) and multi-pass strategy (red). Here the
coefficient is calculated in steady-state solution. Tht lines show the
theoretical predictions, while the dots are the experimental data.
Results of multi-pass ICO.— Our second result is to investigate this multi-
pass strategy. We start with the working substance at the same temperature as
the reservoirs. As an example, we still adopt the initial temperature to be
$E_{C}=0.25$, as shown by the first point in Fig.3(a). At each step, a single
run of quantum switch is carried out. Hence one initial state will generate a
two-branch outcome (indicated by the arrows in Fig.3(a)). Here we only
consider the case in which the working substance becomes colder, and then send
it into the next step as initial state.In the experiment, since photons will
be annihilated after being measured in each step, we use the measurement
results (classical information) to determine the state preparation in the next
step to simulate this iterative process (as the loop depicted in Fig.1). The
experimental results for the 10-steps ICO are summarized in Fig.3(a), in which
the iteration process is indicated by the arrows. We observe that when the
multi-pass ($N\geq 2$) ICO is implemented , a colder working substance in the
unwanted branch could jump into a higher temperature compared to in the
single-step process ($N=1$). This means that the multi-pass ICO may release
the restriction for external reservoir for heat dumping. Interestingly,
Fig.3(a) shows that the working substance will quickly saturates to a specific
temperature, which means the output working substance remains unchanged in its
input state when the control qubit is measured to be $\left|+\right\rangle$.
We theoretically calculate this steady-state solution for all temperatures of
the reservoir and also experimentally sample five points
$E_{C}=\left\\{0.05,0.15,0.25,0.35,0.45\right\\}$, finding that the working
substance always tends toward this steady-state solution after several
iterations (see section I of Supplementary Material for details).
Figure 4: Schematic diagram of a single cycle of quantum refrigerator. The
cold and hot reservoir is denoted by the cyan and moccasin squares
respectively, and the working substance is denoted by the ball with the color
corresponding to its temperature. The working substance is initially in
thermal equilibrium with the cold reservoir and then undergoes the ICO. The
cycle continues (going up into the hot reservoir) by control qubit on
$\left|-\right\rangle$, or recycle (going down). Two recycling strategies are
presented by the dotted arrow (classical strategy) and solid arrow (multi-pass
strategy). Before entering the ICO quantum switch, erasing the register of
control qubit consumes the quantum work $\Delta W$. The total heat extraction
per cycle $\Delta E$ is related to the temperature of particle entering
($\beta_{-}$) and leaving the external hot reservoir ($\beta_{H}$) (depicted
by the particles on the centreline).
Construction of a quantum refrigerator—In the following we consider the
refrigeration task driven by ICO. We construct an operational cycle to realize
a refrigerator. The schematic diagram of a single cycle of quantum
refrigerator is shown in Fig.4, of which the complete process is described as
follows. In the first stroke (i), the working substance is initialized by
classically interacting with the cold reservoir (Preparing a colder working
substance required additional work cost, thus is excluded in our discuss).
Then interacts with the two cold reservoirs superposed in ICO. In the second
stroke (ii), the control qubit is measured. If the control qubit is collapsed
into $\left|-\right\rangle$, the working system successfully extracts heat
from the cold reservoir, followed by proceeding to next stroke. If the control
qubit collapses into $\left|+\right\rangle$, two alternative strategies are
available as listed above: (a) the working system classically contacts the
cold reservoir to recover its initialized state, thereby undoing the unwanted
heat change, and a new cycle is implemented. (termed classical strategy); and
(b) the ICO quantum switch is repeatedly passed through until the desired
outcome is obtained. (termed multi-pass strategy). In the third stroke (iii),
The working substance makes classical thermal contact with external hot
reservoir for heat release, followed by a classical thermal contact with cold
reservoir once again for initialization; subsequently, a new cycle is started.
To evaluate the performance of the quantum refrigerator, we introduce the
coefficient of performance, which is calculated by dividing the heat change
from the cold reservoir by the work cost of measurement Abdelkhalek et al.
(2016). To calculate the heat flow, we adopt the assumption that all the
thermalizations are isochoric, which assures that the internal energy change
of working system refers to the heat exchange with reserviors. The heat flow
out of cold reservoirs per cycle lies in two aspects: one results from
themalization inside quantum switch in stroke (i) when a desired outcome
$\left|-\right\rangle$ of the control qubit is measured. We calculate the heat
flow extractied from cold reservior $\Delta E_{1}=E_{-}-E_{C}$, where
$E_{-}=\mathrm{Tr}\left(\rho_{-}\mathcal{H}\right)$. The other aspect results
from the classical thermal contact of working system with cold reservoir in
stroke (iii), which is given by $\Delta E_{2}=-\left(E_{H}-E_{C}\right)$,
where $E_{H}$ is the energy of the thermal state of the external hot
reservoir. The total heat change in cold reservoir in a single cycle is
$\Delta E=\Delta E_{1}+\Delta E_{2}$, related to the temperature of hot
reservoir and cold reservoir. Here, we fix the hot reservoir
$\beta_{H}=\beta_{C}$ to maximize the heat change. In this case, the total
heat change $\Delta E=E_{-}-E_{C}$.
The magic of the ICO refrigerator can also be explained by the Maxwell’s
demon-like cooling mechanism. In each cycle of the quantum refrigerator, the
control qubit need to be measured and the result is stored in a register. A
following operation is determined by the information in the register, as shown
by the loop in Fig.1. Since we assume the control qubit is degenerate in
energy, the measurement itself does not cost energy. Rather, the energy cost
comes from resetting the register to its initial state in order to proceed to
the next cycle, which refers to Landauer’s erasure Landauer (1961). The work
cost is given by $\Delta W=-\frac{1}{\beta_{R}}S$, where $S=\left(p_{+}\ln
p_{+}+p_{-}\ln p_{-}\right)$ is the Shannon entropy of the register and
$\beta_{R}$ is the inverse temperature of the resetting reservoir. Therefore
the coefficient of quantum refrigerator is given by
$\eta=\frac{\Delta E}{\bar{n}\Delta W}$ (4)
where $\bar{n}=\frac{1}{p_{-}}$ is the average number of measurements consumed
to realize one cycle of the quantum refrigerator.
For the classical strategy, the coefficient of quantum refrigerator can be
directly calculated by Eq. (4). When it comes to multi-pass strategy, it
potentially contains many steps per cycle. The heat flow extraction or work
cost in each step is not identical. However, we can rationally approximate the
coefficient for multi-pass strategy by assuming that the quantum refrigerator
working at the steady-state point (see section I of Supplementary Material for
details). That is because single cycle is much likely to undergo multiple
steps and evolve into equilibrium, especially for a low temperature of cold
reservoir in practice, where the probability of mesurement outcome
$\left|-\right\rangle$ approaches to zero with the temperature decreasing. we
comparatively present the coefficients of quantum refrigerator under both the
classical and multi-pass strategy in Fig. 3(d), in which the solid lines show
the theoretical prediction and dots show the experimental results for several
sample temperatures $E_{C}=\left\\{0.05,0.15,0.25,0.35,0.45\right\\}$. The
results show that the multi-pass strategy is always superior to classical
strategy. The results also provide a deep insights into how to improve the
efficiency of quantum thermodynamic tasks by replacing irreversible isochoric
thermalizations with a reversible process to the extent possible.
We provide the strategies of quantum refrigerator that involve both quantum
and classical operation. Dispite a measurement inside strokes, both the
outcomes are all taken into account, hence the refrigerator cycle does not
depend on postselection to gain its advantages. In addition, instead of
feedback control loop in which the control qubit is measured, the quantum
refrigerator can also be realized without projection but rather consumption of
purity of control qubit Felce and Vedral (2020). In the present work, we have
demonstrated ICO-driven nonclassical heat transfer between system and
reservoirs even though they share the same initial temperature, for which heat
change via direct thermalization would be inaccessible. Our findings are
beneficial for advancing a new class of resource theories in quantum
thermodynamics based ICO. Especially in the case where the control qubit is
degenerate in energy, which means it can not be used for direct
thermalization, the ICO provide an effective way to ultilize its free energy.
Our results confirm that the ICO can be a useful resource in quantum
thermodynamics, which provide a new paradigm of work extraction alternative to
other non-classical features Perarnau-Llobet et al. (2015); Francica et al.
(2017). Therefore we expect that our work will advance further investigations
on the exotic properties of indefinite causal order, as well as its
superiority in quantum tasks.
We thank Yong-Xiang Zheng, Xue Li, Xiao Liu for beneficial discussions. This
work was supported by National Natural Science Foundation of China (11734015,
62075208, 11774335, 11874345, 11821404, 11904357), Anhui Initiative in Quantum
Information Technologies (AHY070000, AHY020100, AHY060300), National Key
Research and Development Program of China (2017YFA0304100, 2016YFA0301300,
2016YFA0301700), Key Research Program of Frontier Sciences, CAS (QYZDY-SSW-
SLH003), Science Foundation of the CAS (ZDRW-XH-2019-1), the Fundamental
Research Funds for the Central Universities, Science and Technological Fund of
Anhui Province for Outstanding Youth (2008085J02).
Noted—— Recently, we become aware of a related work by Nie et al. which
experimentally study quantum thermodynamics driven by ICO on nuclear spins
using the nuclear magnetic resonance system Nie et al. (2020).
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|
[2010]35A01, 35K57, 35K58, 35Q92
We study the global existence of classical solutions to volume-surface reaction-diffusion systems with control of mass. Such systems appear naturally from modeling evolution of concentrations or densities appearing both in a volume domain and its surface, and therefore have attracted considerable attention. Due to the characteristic volume-surface coupling, global existence of solutions to general systems is a challenging issue. In particular, the duality method, which is powerful in dealing with mass conserved systems in domains, is not applicable on its own. In this paper, we introduce a new family of $L^p$-energy functions and combine them with a suitable duality method for volume-surface systems, to ultimately obtain global existence of classical solutions under a general assumption called the intermediate sum condition. For systems that conserve mass, but do not satisfy this condition, global solutions are shown under a quasi-uniform condition, that is, under the assumption that the diffusion coefficients are close to each other. In the case of mass dissipation, we also show that the solution is bounded uniformly in time by studying the system on each time-space cylinder of unit size, and showing that the solution is sup-norm bounded independently of the cylinder. Applications of our results include global existence and boundedness for systems arising from membrane protein clustering or activation of Cdc42 in cell polarization.
§ INTRODUCTION AND MAIN RESULTS
§.§ Problem setting
Let $\Omega\subset\R^n$, $n\geq 1$, be a bounded domain with smooth boundary $M:= \partial\Omega$. Let $m_1\geq 0$ and $m_2\geq 0$. We consider the following volume-surface reaction-diffusion system
\begin{equation}\left\{
\begin{aligned}
\frac{\partial u_{i}}{\partial t}&=d_{i}\Delta u_{i}+F_{i}(u), &&\text{ on \ensuremath{\Omega\times(0,T)\text{ for \ensuremath{i=1,...,m_1}}}}\\
d_{i}\frac{\partial u_{i}}{\partial\eta}&=G_{i}(u,v), &&\text{ on \ensuremath{M\times(0,T)\text{ for \ensuremath{i=1,...,m_1}}}}\\
\frac{\partial v_{j}}{\partial t}&=\delta_{j}\Delta_M v_{j}+H_{j}(u,v), && \text{ on \ensuremath{M\times(0,T)\text{ for \ensuremath{j=1,...,m_2}}}}\\
u&=u_0 && \text{ on \ensuremath{\overline{\Omega}\times{\{0\}}}}\\
v&=v_0 && \text{ on \ensuremath{M\times{\{0\}}}}
\end{aligned}\right.\label{eq:mainsys}
\end{equation}
where $\pa/\pa{\eta}$ denotes the outward normal flux on the boundary $M$, $u = (u_1,u_2,\ldots, u_{m_1})$, $v = (v_1,v_2, \ldots, v_{m_2})$ are vectors of concentrations, $F_{i}:\R^{m_1}\to \R^{m_1}$ and $G_{i},H_{j}:\R^{m_1+m_2}\to \R^{m_1+m_2}$ satisfy assumptions which will be specified later. The operators $\Delta$ and $\Delta_M$ represent the traditional Laplacian in $\Omega$ and Laplace-Beltrami operator on $M$. The initial
data $u_0=(u_{0,{i}})_{i=1,\ldots, m_1}$ and $v_0=(v_{0,{j}})_{j=1,\ldots, m_2}$ are assumed to be smooth, bounded and component-wise non-negative on $\overline{\Omega}$ and $M$, respectively. Also, when $n=1$, we assume $m_2 = 0$. Throughout this paper, we will assume the following conditions on the domain
* $\Omega\subset\mathbb R^n$, $n\geq 1$, is bounded domain with smooth boundary $M = \pa\Omega$ such that $\Omega$ lies locally on one side of $M$;
on the diffusion coefficients
* $d_i > 0$ and $\delta_j > 0$ for all $i=1,\ldots,\; m_1, j=1,\ldots, m_2$;
and on the nonlinearities
* (local Lipschitz continuity) $F_i: \R_{+}^{m_1} \to \R$, $G_i: \R_+^{m_1+m_2}\to \R_+$, $H_j: \R_+^{m_1+m_2}\to \R$ are locally Lipschitz continuous for $i\in \{1,\ldots, m_1\}$ and $j\in \{1, \ldots, m_2\}$;
* (quasi-positivity)
$$F_{i}(u),G_{i}(u,v)\ge0\text{ for all }u\in \R_{+}^{m_1}, v\in \R_{+}^{m_2}\text{ with }u_{i}=0, \; \forall i=1,\ldots, m_1,$$
\begin{equation*}
H_j(u,v) \geq 0\text{ for all } u\in \R_+^{m_1}, v\in \R_+^{m_2} \text{ with } v_{j} = 0, \; \forall j=1,\ldots, m_2.
\end{equation*}
* (mass control) there exist $a_i,b_j>0$ and $L\in \R$, $K\geq 0$ such that
\begin{equation}\label{eq:balsysmain}
\begin{aligned}\sum_{i=1}^{m_1}a_i F_i(u)&\le L\left(\sum_{i=1}^{m_1}u_i\right) + K,\\ \sum_{i=1}^{m_1}a_i G_i(u,v)+\sum_{j=1}^{m_2}b_j H_j(u,v)&\le L\left(\sumi u_i + \sumj v_j\right) + K,
\end{aligned}
\end{equation}
for all $u\in \R_+^{m_1}$ and $v\in \R_+^{m_2}$;
* (polynomial growth) there are constants $K_1, r > 0$ such that, for all $i=1,\ldots, m_1$, $j=1,\ldots, m_2$,
\begin{equation}
F_i(u),G_i(u,v),H_j(u,v)\le K_1\left(\sum_{i=1}^{m_1}u_{i}^{r}+\sum_{j=1}^{m_2}v_{j}^{r}+1\right),\label{eq:polybound}
\end{equation}
for all $u\in \R_{+}^{m_1}$ and $v\in \R_{+}^{m_2}$.
With (<ref>), it can be shown that (<ref>) possesses a unique local strong solution on a maximal interval $(0,T_{\max})$, cf. [36]. Moreover,
\begin{equation}\label{blowup-criterion}
T_{\max}<+\infty \quad \Longrightarrow \quad \limsup_{t\nearrow T_{\max}}\bra{\sumi \|u_i(t)\|_{\LO{\infty}} +\sumj \|v_j(t)\|_{\LM{\infty}}} = +\infty.
\end{equation}
The quasi-positivity assumption (<ref>) assures that the solution to (<ref>) is (component-wise) non-negative (as long as it exists) if the initial data is non-negative. This assumption also has a simple physical interpretation: if a concentration is zero, it cannot be consumed in a reaction. The condition (<ref>) implies that the total mass of the system is finite for all time, which, together with the non-negativity of solutions, implies that
\begin{equation*}
u_i \in L^\infty(0,T;\LO{1}),\; v_j\in L^\infty(0,T;\LM{1}) \quad \forall i=1,\ldots, m_1, \; j=1,\ldots, m_2, \; T>0.
\end{equation*}
Comparing to (<ref>), these a-priori estimates are far from enough to conclude the global existence of solutions to (<ref>). In fact, it was shown in [32] that the above assumptions are very likely not enough to prevent blow-up in finite time even for reaction-diffusion systems in domains. The global well-posedness question for (<ref>) is, therefore, challenging and generally open.
In this paper, we will show that under the above natural assumptions, and a so-called intermediate sum condition, there exists a unique global strong solution to (<ref>). Moreover, if the total mass is bounded by a constant independent of time, the solution is sup-norm bounded uniformly for all time.
§.§ State of the art
Volume-surface (or bulk-surface) systems of the form (<ref>) have recently attracted a lot of attention. On the one hand, such systems arise naturally from many applications. For example, in asymmetric stem cell division [15, 13], in modeling receptor-ligand dynamics in cell biology [17, 2], in crystal growth [20, 40], in population modeling [8, 6], in chemistry with fast sorption [38, 1], or in fluid mechanics [25, 16], and so on.
On the other hand, the volume-surface coupling yields new and highly non-trivial challenges in the analysis of such systems. Of interest and importance is the question of global existence of solutions, and most of the existing works rely heavily on special structures of the considered systems. For instance, when the system is linear, or the reactions have at most linear growth, global (weak, strong) solutions were shown in [15, 19]. For nonlinear coupling on the boundary, [13] shows global bounded solutions for the reversible reaction $\alpha \mathcal U \leftrightharpoons \beta \mathcal V$, where $\mathcal U$ and $\mathcal V$ are volumic- and surface-concentrations, respectively. For general systems, [11] showed that if the $L^\infty$-bound is preserved, then one can get global classical solutions. The works [36, 37] showed global existence, and boundedness of solutions to volume-surface systems assuming a linear upper bound for $F_i$, $G_i$, and the sum of $G_i + H_j$, i.e. for any $i\in \{1,\ldots, m_1\}$ and any $j\in \{1,\ldots, m_2\}$,
\begin{equation}\label{linear_growth}
F_i(u) \leq L\bra{1+\sum_{k=1}^{m_1} u_k}, \quad G_i(u,v) + H_j(u,v) \leq L\bra{1+\sum_{k=1}^{m_1}u_k + \sum_{l=1}^{m_2}v_l}
\end{equation}
for some $L>0$. Due to these restrictions, the results therein are not applicable in many systems arising from cell biology (see e.g. [34, 7]).
It is also remarked that significant progress has been made lately concerning mass controlled reaction-diffusion systems in domains, i.e. when $m_2 = 0$ and $G\equiv 0$. For instance, a major problem concerning the global existence of strong solutions for systems with quadratic nonlinearities in all dimensions has been settled in three recent works [39, 10, 14]. If the nonlinearities have $L^1(\Omega\times(0,T))$-bound a priori, global weak solutions can be shown [30]. In particular, by assuming only an entropy inequality, [12] shows global existence of renormalized solutions without any restriction on the growth of nonlinearities. We emphasize that none of these works seem to readily extend to the case of volume-surface systems.
In this paper, by introducing a new family of $L^p$-energy function and combining it with duality methods for volume-surface systems, we show the global existence of solutions to a large class of systems of type (<ref>), which include the systems in [34, 7] as special cases.
§.§ Main results and Key ideas
To study the global existence of solutions to (<ref>), we assume the so-called intermediate sum condition (see e.g. [27]) for the nonlinearities of (<ref>), i.e. there exists an $(m_1+m_2)\times (m_1+m_2)$ lower
triangular matrix $A$ with non-negative elements everywhere and positive entries on the main diagonal,
and constants $L_2\ge0$, $\mM >0$, such that
\begin{equation}\label{eq:intsum1}
A\begin{bmatrix}F(u)\\ \vec{0}_{m_2}\end{bmatrix}
\le L_2\bra{\sumi u_i^{\pO}+1}\vec{1}_{m_1+m_2},
\end{equation}
\begin{equation}\label{eq:intsum2}
A\begin{bmatrix}G(u,v)\\H(u,v)\end{bmatrix}\le L_2\bra{\sumi u_i^{\pM} + \sumj v_j^{\pM}+1}\begin{bmatrix}\vec{1}_{m_1}\\ \vec{0}_{m_2}\end{bmatrix} + L_2\bra{\sumi u_i^{\mM} + \sumj v_j^{\mM}+1}\begin{bmatrix}\vec{0}_{m_1}\\ \vec{1}_{m_2}\end{bmatrix},
\end{equation}
for all $u\in \R_{+}^{m_1}$ and $v\in \R_{+}^{m_2}$, where the exponent $\pO, \pM$ and $\mM$ will be specified later,
$F(u) = (F_1(u), \ldots, F_{m_1}(u))$ and $G(u,v) = (G_1(u,v), \ldots, G_{m_2}(u,v))$. We denote by $\vec{1}_m$ (rsp. $\vec{0}_m$) the vector of all $1s$ elements (rsp. $0s$ elements) in $\mathbb R^m$. By dividing each row of $A$ by the diagonal element, we can assume w.l.o.g. that $a_{ii} = 1$ for all $i=1,\ldots, m_1+m_2$, and we will do so for the rest of this paper.
It's emphasized that assumptions (<ref>) and (<ref>) do not require the components of the reaction vector fields $F$, $G$ and $H$ to be at most of order $\pO$, $\pM$ or $\mM$. It simply requires a "trade-off" of higher order terms between components of $F$, $G$ and $H$, and additionally requires that $F_1(u)$ be bounded above by a polynomial in $u$ of order $p_\Omega$ and $G_1(u,v)$ be bounded above by a polynomial in $u$ and $v$ of order $\pM$.
The first main result of this paper is the following theorem.
\begin{equation}\label{pOpM}
1\leq \pO < 1+\frac{2}{n}, \quad \text{ and } \quad 1\leq \pM < 1+\frac{1}{n},
\end{equation}
\begin{equation}\label{mM}
1\leq \mM \leq 1+\frac{4}{n+1}.
\end{equation}
Then for any nonnegative, bounded initial data $(u_0,v_0)\in (W^{2-2/p}(\Omega))^{m_1}\times (W^{2-2/p}(M))^{m_2}$ for some $p>n$ satisfying the compatibility condition
\begin{equation}\label{compatibility}
d_i\pa_{\eta}u_{i,0} = G_i(u_0,v_0) \quad \text{on M} \quad \text{ for all } i=1,\ldots, m_1,
\end{equation}
the system (<ref>) has a unique global classical solution. Moreover, if $L < 0$ or $L=K=0$ in (<ref>), then
\begin{equation*}
\sup_{i=1,\ldots, m_1}\sup_{j=1,\ldots, m_2}\sup_{t\geq 0}\bra{\|u_i(t)\|_{L^\infty(\Omega)} + \|v_j(t)\|_{L^\infty(M)}} <+\infty.
\end{equation*}
* The first remark is that the smoothness of initial data as well as the compatibility condition (<ref>) are only used
to obtain a local classical solution (see e.g. [36] or [19]). We believe that this local existence can in fact be proved for merely bounded initial data, though, due to the volume-surface coupling, the proof is more delicate comparing to the case of systems in domains (see e.g. [5]). The details are left for the interested reader.
* The intermediate sum conditions involving $F$ and $G$ in (<ref>) and (<ref>) are used to construct $L^p$-energy functions, which is essential in the proof of global existence. In fact, one can construct such functions under more general (but technical) assumptions (see Remark <ref>). We choose to present Theorem <ref> under the intermediate sum conditions (<ref>) and (<ref>) as they are more constructive, and appear naturally in many applications (see Section <ref>).
* The condition of $\mM$ in (<ref>) can be slightly improved as
\begin{equation*}
\mM < 1 + \frac{2(2+\eps)}{n+1}
\end{equation*}
for a sufficiently small $\eps>0$.
* The assumption $L<0$ or $L=K=0$ is used obtain uniform-in-time $L^1$-bounds (see Lemma <ref>), i.e.
\begin{equation*}
\sup_{t\geq 0}\bra{\|u_i(t)\|_{\LO{1}} + \|u_i\|_{L^1(M\times{(t,t+1)})} + \|v_j(t)\|_{\LM{1}}} < +\infty,
\end{equation*}
which eventually leads to the uniform-in-time $L^\infty$-bound. If this $L^1$-bound can be obtain differently, for instance, by using special structures of a specific model, the assumption $L <0 $ or $L=K=0$ can be removed.
* If $L<0$ and $K=0$ in (<ref>), we can show that the global solution decays exponentially to zero as $t\to \infty$. The details are left for the interested reader.
* The sub-critical order of $\pO$ (and $\pM$) in (<ref>) is typical for reaction-diffusion systems (in domains) once an $L^1$-bound is known (see e.g. [4]). It is again emphasized that we do not assume the nonlinearities, but only an intermediate sum of them, to have the growth of order $\pO$.
* It's finally remarked that the results of Theorem <ref> are new even in the case without surface concentrations, i.e. $m_2 = 0$.
Let us recall that for reaction-diffusion systems in domains, i.e. $m_2=0$ and homogeneous boundary conditions $G_i \equiv 0$, $\forall i=1,\ldots, m_1$, the global existence with control of mass and intermediate sum conditions has been successfully relied on the famous duality method, cf. [30, 27]. Thanks to this method, the control of mass condition implies an $L^{2+\eps}$-estimate a-priori. This, together with the intermediate sum condition, allows the use of a bootstrap procedure to ultimately obtain $L^{\infty}$-estimate, which is sufficient for the global existence. It is remarked that the duality is not only crucial to get the initial $L^{2+\eps}$-bound, but also important in the bootstrapping procedure. Such a strategy, unfortunately, fails to apply to volume-surface systems of the form (<ref>), see [26]. In this paper, our first key idea to overcome this difficulty is to introduce a new family of $L^p$-energy functions. Some preliminary ideas of such $L^p$-energy functions have been carried out also for reaction-diffusion systems, but have been less noticed comparing to the duality method[28, 21, 3][Aside from this method's technicality, another reason for this ignorance might be that the elegant duality method, cf. [32], has proved to be very efficient in studying reaction-diffusion systems with control of mass, see also the survey [30]. A recent work [35] studied reaction-diffusion systems with nonlinear boundary conditions to which this $L^p$-energy method is well adapted while the duality method seems not. It is also remarked that the assumption therein do not allow for the use of intermediate sums, and restrict to primary application of the results to two component systems.]. The main aim is to provide a generalization of the usual method of constructing $L^p$-energy functions by (traditionally) multiplying both sides of a parabolic equation for a function $w(x,t)$ by $w(x,t)^{p-1}$. Instead, we consider $L^p$-energy functions consisting of all (mixed) multi-variable polynomials of order $p$ with carefully chosen coefficients. Certainly, the essential difficulty is to choose an $L^p$-energy function so that it is “compatible" with both diffusion and reaction parts of the system, i.e. its evolution in time should lead to useful a-priori estimates. We show in this paper that the intermediate sum conditions (<ref>) and (<ref>) allow us to find such an energy function. Yet, this strategy alone is not enough to obtain sufficient estimates for global existence due to the surface concentrations. Our second key idea is, therefore, to combine the constructed $L^p$-energy functions with a duality method for volume-surface reaction-diffusion systems. This method has been used in previous work, see e.g. [36, 26], but due to the lack of $L^p$-estimates obtained in the current work, it has only applied to systems with restrictive conditions, for instance under assumptions (<ref>). We stress, as it will become apparent in our paper, that only the combination of these two ideas makes it possible to obtain global existence of (<ref>) under the general assumptions (<ref>) and (<ref>).
Let us briefly sketch the main ideas of the proof of Theorem <ref>, which can be roughly divided in several steps.
* Step 1. Firstly, by (<ref>) one has
\begin{equation*}
\|u_i\|_{L^\infty(0,T;\LO{1})}, \|u_i\|_{L^1(M\times(0,T))}, \|v_j\|_{L^\infty(0,T;\LM{1})} \leq C_T
\end{equation*}
where $C_T$ is a constant depending on the time horizon $T>0$.
* Step 2. The intermediate sum condition (<ref>) allows us to construct for any $2\leq p\in \mathbb N$ an $L^p$-energy function of the form
\begin{equation*}
\L_p[u](t) = \intO \sum_{|\beta|=p}\begin{pmatrix}
p\\ \beta \end{pmatrix}\theta^{\beta^2}u^{\beta}(x,t)
\end{equation*}
with the convention $u^{\beta} = \prod\limits_{i=1}^{m_1}u_i^{\beta_i}$, $\theta^{\beta^2} = \prod\limits_{i=1}^{m_1}\theta_i^{\beta_i^2}$, and $\begin{pmatrix}p\\\beta\end{pmatrix} = \dfrac{p!}{\beta_{1}!\beta_2!\cdots\beta_{m_1}!}$, where $\theta\in (0,\infty)^{m_1}$ are chosen suitably[It is remarked that such a function $\L_p[u]$ can be constructed under a more general but less constructive condition than (<ref>) (see Lemma <ref> and Remark <ref>).]. Since all $u_i$ are non-negative, $\bra{\L_p[u]}^{1/p}$ is an equivalent norm to the usual $L^p$-norm of $u$.
Due to the volume-surface coupling, it does not seem possible to show that $\L_p[u]$ is non-increasing in time, or even bounded. Instead, one obtains for any $\eps>0$ a constant $C_\eps>0$ such that
\begin{equation}\label{intro1}
\quad \bra{\L_p[u]}' + C\sumi\bra{\intO u_i^{p-1+\pO} + \intM u_i^{p-1+\pM}} \leq C_{\eps} + \eps\sumj\intM v_j^{p-1+\pM}.
\end{equation}
This estimate is crucial in the analysis of this paper. The “left over" $L^p$-integrals of surface concentrations $v_j$ in (<ref>) are treated using a duality method in the next steps.
* Step 3. We derive an improved duality estimate using the dual problem suited for volume-surface systems
\begin{equation*}
\begin{cases}
\pa_t \phi + \Delta \phi = 0, & (x,t)\in \Omega\times(0,T),\\
\pa_t \phi + \delta \Delta_M \phi = -\psi, &(x,t)\in M\times (0,T),\\
\phi(x,T) = 0, &x\in\overline{\Omega}
\end{cases}
\end{equation*}
and (<ref>) to obtain
\begin{equation}\label{twoplus}
\sumi\bra{\|u_i\|_{L^{2+\eps}(\Omega\times(0,T))}+\|u_i\|_{L^{2+\eps}(M\times(0,T))}} + \sumj\|v_j\|_{L^{2+\eps}(M\times(0,T))} \leq C_T
\end{equation}
for some $\eps>0$.
* Step 4. By using the estimates in (<ref>) and the duality method, we are able to show: for any $p>1$, any $k\in \{1,\ldots, m_2\}$, and any $\eps>0$, there exists $C_{T,\eps}>0$ such that
\begin{equation*}
\|v_k\|_{L^p(M\times(0,T))} \leq C_{T,\eps} + C_T\sum_{j=1}^{k-1}\|v_j\|_{L^p(M\times(0,T))} + \eps\sumj \|v_j\|_{L^p(M\times(0,T))}.
\end{equation*}
This ultimately leads to the boundedness of $v_j$ in $L^p(M\times(0,T))$, and consequently, thanks to (<ref>), the boundedness of $u_i$ in $L^p(\Omega\times(0,T))$ as well as in $L^p(M\times(0,T))$.
From $u_i\in L^p(\Omega\times(0,T))$, $u_i\in L^p(M\times(0,T))$, and $v_j\in L^p(M\times(0,T))$ for all $p>1$, by using the polynomial growth of the nonlinearities in (<ref>) and regularization of the heat operator with inhomogeneous boundary conditions, we obtain finally $u_i\in L^\infty(\Omega\times(0,T))$ and $v_j\in L^\infty(M\times(0,T))$, which concludes the global existence of bounded solutions.
* Step 5. To obtain uniform-in-time bounds of the global solution, we use for each $\tau\in \mathbb N$, a smooth cut-off function $\vat: \R \to [0,1]$ with $\vat|_{(-\infty,\tau]} = 0$ and $\vat|_{[\tau,\infty)} = 1$ to study (<ref>) on the cylinder $\Omega\times(\tau,\tau+2)$. By repeating the previous steps on this cylinder, we obtain that $u_i$ and $v_j$ are bounded in $L^\infty(\Omega\times(\tau,\tau+2))$ and $L^\infty(M\times(\tau,\tau+2))$, respectively, uniformly in $\tau\in \mathbb N$. This concludes the uniform-in-time bounds of the global solution, and consequently completes the proof of Theorem <ref>.
It is noticed from the proof of Theorem <ref> that the $L^p$-energy method is used for the volume concentrations, while the duality method is well adapted for the surface concentration. An improved duality method, see e.g. [9], shows that one can deal with higher order nonlinearities in the intermediate sums by assuming quasi-uniform diffusion coefficients, i.e. when the diffusion coefficients are not too different from each other. Following this idea, the second main result of this paper shows global existence of strong solutions to (<ref>) with large order $\mM$ of the nonlinearities on the boundary, assuming that the diffusion coefficients $\delta_j$ are quasi-uniform. To state our second main theorem, we denote by
\begin{equation}\label{dmaxmin}
\delta_{\max} = \max\{\delta_1,\ldots, \delta_{m_2}\}, \quad \delta_{\min} = \min\{\delta_1,\ldots, \delta_{m_2}\}.
\end{equation}
Let $\mM > 0$ be fixed. Assume that there exists a constant $\Lam>0$ such that
\begin{equation}\label{mM_general}
\Lam\geq \frac{(\mM-1)(n+1)}{2}, \quad \text{ or equivalently } \quad \mM \leq 1+\frac{2\Lam}{n+1}
\end{equation}
\begin{equation}\label{quasi-uniform}
\frac{\delta_{\max}-\delta_{\min}}{\delta_{\max}+\delta_{\min}}C_{\mr,\Lambda'}^M < 1,
\end{equation}
where $\Lambda' = \Lambda/(\Lambda - 1)$, and $C_{\mr,\Lambda'}^M$, which depends only on $\Lam$ and $M$, is the maximal regularity constant in Lemma <ref>. Then for any nonnegative, initial data $(u_0,v_0)\in (W^{2-2/p}(\Omega))^{m_1}\times (W^{2-2/p}(M))^{m_2}$ for some $p>n$ satisfying the compatibility condition (<ref>), the system (<ref>) has a unique nonnegative, global strong solution. Moreover, if $L<0$ or $L=K=0$ in (<ref>), the solution is bounded uniformly in time, i.e.
\begin{equation*}
\sup_{i=1,\ldots, m_1}\sup_{j=1,\ldots, m_2}\sup_{t\geq 0}\bra{\|u_i(t)\|_{L^\infty(\Omega)} + \|v_j(t)\|_{L^\infty(M)}} <+\infty.
\end{equation*}
* We will show in this paper that for any fixed $\delta_{\max}$ and $\delta_{\min}$, condition (<ref>) is always satisfied with $\Lam = 2$. This implies that Theorem <ref> is in fact a consequence of Theorem <ref>.
* We see that for fixed $\mM>0$, condition (<ref>) is fulfilled if we fix $\Lam = (\mM-1)(n+1)/2$ and require the surface diffusion coefficients $\delta_j$ to be close enough to each other (relatively to their sums). Since $C_{\mr,q}^M$ is increasing as $p\searrow 1$ and $\lim_{q\searrow 1}C_{\mr,q}^M = +\infty$, (<ref>) means that $\delta_j$ are required to get closer to each other as the nonlinearity order $\mM$ in the intermediate sum increases.
* Similarly to Remark <ref>, we believe that the results of Theorem <ref> still hold true for non-negative, and bounded initial data $(u_0,v_0)\in L^\infty(\Omega)^{m_1}\times L^\infty(M)^{m_2}$.
Thanks to the quasi-uniform condition (<ref>), one can use the duality method to obtain some $L^{\Lam}$-estimates on $u_i$ and $v_j$, which should form the starting point of a bootstrap argument. Unfortunately, $L^\Lam$-estimates just fall short in case $\Lam$ satisfies (<ref>) with an equality. To overcome this difficulty, an important observation is that the condition (<ref>) is “open" in the sense that if (<ref>) is true for some $\Lam$, then it also holds for $\Lam+\eps$ with $\eps>0$ small enough[This was observed and utilized in many recent works, see e.g. [9, 33].]. This observation allows us to use an improved duality argument to prove that $u_i$ are bounded in $L^{\Lam+\eps}(\Omega\times(0,T))$ and $L^{\Lam+\eps}(M\times(0,T))$, and $v_j$ are bounded in $L^{\Lam+\eps}(M\times(0,T))$, for some small $\eps>0$. Starting from these estimates, we can use the duality method as in Step 4 above to get $L^p$-estimates for the solution for all $p\geq 1$. This is enough to conclude that the solution is global. To show the uniform-in-time bound, we again use the smooth cut-off function $\vat$ and repeat the arguments for global existence on each cylinder $\Omega\times(\tau,\tau+2)$, to obtain $L^\infty$-bound which are independent of $\tau\in \mathbb N$.
One notices in Theorem <ref> that by imposing the quasi-uniform condition on diffusion coefficients we are able to improve only the order of the nonlinearities in intermediate sums for surface concentrations. The reason is that with the $L^1$-estimates implied from (<ref>), the upper bound $1+\frac 2n$ of $\pO$ seems to be the critical exponent to obtain $L^p$-energy estimates for all $p\geq 1$. For a specific system, it might be possible to obtain better a-priori estimates (see Section <ref>), which consequently allows a larger range of $\pO$, $\pM$ and $\mM$. More precisely, we have the following conditional result.
Assume there exist $a, b\ge 1$ such that, for any $T>0$, and for all $i=1,\ldots, m_1$, $j=1,\ldots, m_2$
\begin{equation}\label{gg8}
\|u_i\|_{L^\infty(0,T;\LO{a})} + \|u_i\|_{L^b(M\times(0,T))} + \|v_j\|_{L^{b}(M\times(0,T))} \leq \F(T)
\end{equation}
where $\F\in C([0,\infty))$. Assume that
\begin{equation}\label{pOpM_ab}
1\leq \pO < 1 + a\cdot\min\left\{\frac{2}{n}; \frac{3}{n+2}\right\}, \quad \text{and} \quad 1\leq \pM < 1 + \frac{a}{n}
\end{equation}
\begin{equation}\label{mM_ab}
1\leq \mM < 1 + \frac{2b}{n+1}.
\end{equation}
Then for any nonnegative initial data $(u_0,v_0)\in (W^{2-2/p}(\Omega))^{m_1}\times (W^{2-2/p}(M))^{m_2}$ for some $p>n$ satisfying the compatibility condition
\begin{equation*}
d_i\pa_{\eta}u_{i,0} = G_i(u_0,v_0) \quad \text{on M} \quad \text{ for all } i=1,\ldots, m_1,
\end{equation*}
the system (<ref>) has a unique global classical solution. Moreover, if $\sup_{t\geq 0}\F(t) <+\infty$, then
\begin{equation*}
\sup_{i=1,\ldots, m_1}\sup_{j=1,\ldots, m_2}\sup_{t\geq 0}\bra{\|u_i(t)\|_{L^\infty(\Omega)} + \|v_j(t)\|_{L^\infty(M)}} <+\infty.
\end{equation*}
It is remarked that Theorem <ref> does not impose the mass control assumption (<ref>), since the condition is only used to obtain a-priori estimates of type (<ref>), which are now given.
Organization of the paper. In the next section, we present the construction of $L^p$-energy functions, and show its relation to the intermediate sum condition (<ref>). Section <ref> is devoted to an improved duality method for volume-surface systems, where we show that assumption (<ref>), in combination with the previously constructed $L^p$-energy functions, gives $L^{\Lam+\eps}$-estimates of the solutions. In Section <ref>, we start with the proof of Theorem <ref> in subsections <ref> and <ref>, where the first subsection shows the global existence while the second one proves the uniform-in-time bound of the solutions. As pointed out in Remark <ref>, we prove Theorem <ref> in subsection <ref> by showing that (<ref>) is always satisfied for $\Lam=2$. The last subsection <ref> presents the proof of Theorem <ref>. The last section is devoted to applications of our results to some recent models arising from cell biology. It is noted that previous works are unlikely to be applicable to these systems. Finally, we give in the Appendix <ref> two technical lemmas concerning the
construction of $L^p$-energy functions.
Notation. For the rest of this paper, we will use the following notation:
* As some of our intermediate lemmas are of independent interest, we use the convention
“Theorem X. ((A)-(B)-(C))"
to indicate that this theorem assumes only conditions (A), (B), (C) (besides the assumptions stated explicitly therein). It's also useful to verify which condition is applicable to which lemmas or theorems.
* For $0\leq \tau < T$,
\begin{equation*}
Q_{\tau,T}:= \Omega\times (\tau, T) \quad \text{and}\quad M_{\tau,T}:= M\times(\tau,T).
\end{equation*}
When $\tau = 0$, we simply write $Q_T$ and $M_T$.
* For $1\leq p < \infty$,
\begin{equation*}
\|f\|_{L^p(Q_{\tau,T})}:= \bra{\int_\tau^T\intO |f(x,t)|^p}^{\frac 1p}
\end{equation*}
and for $p = \infty$,
\begin{equation*}
\|f\|_{L^\infty(Q_{\tau,T})}:= \mathrm{ess\,sup}_{Q_T}|f(x,t)|.
\end{equation*}
The spaces $L^p(M_{\tau,T})$ with $1\leq p\leq \infty$ are defined in the similar way.
* For $1\leq p \leq \infty$,
\begin{equation*}
W^{2,1}_p(Q_{\tau,T}):= \left\{f\in L^p(Q_{\tau,T}): \pa_t^r\pa_x^sf\in L^p(Q_{\tau,T}) \; \forall r,s\in \mathbb N, \, 2r+s\leq 2\right\}
\end{equation*}
with the norm
\begin{equation*}
\|f\|_{W^{2,1}_p(Q_{\tau,T})}:= \sum_{2r+s\leq 2}\|\pa_t^r\pa_x^s f\|_{L^p(Q_{\tau,T})}.
\end{equation*}
§ CONSTRUCTION OF $L^P$-ENERGY FUNCTIONS
In this section, we firstly state local existence of (<ref>), and provide the blow-up criterion, which were proved in [36]. The main part concerns the construction of an $L^p$-like energy function, and its relation to the intermediate sum condition (<ref>). Estimates derived from this energy function are crucial in the sequel analysis of this paper.
For any smooth initial data $(u_0,v_0)\in (W^{2-2/p}(\Omega))^{m_1}\times (W^{2-2/p}(M))^{m_2}$ for some $p>n$ satisfying the compatibility condition (<ref>), there exists a unique classical solution to (<ref>) on a maximal interval $(0,T_{\max})$, i.e. for any $0<T<T_{\max}$,
\begin{equation*}
(u,v)\in C([0,T];L^p(\Omega)^{m_1}\times L^p(M)^{m_2}) \cap L^\infty(0,T;L^\infty(\Omega)^{m_1}\times \LM{\infty}^{m_2}),
\end{equation*}
for any $p>n$,
\begin{equation*}
u\in (C^{2,1}(\overline{\Omega}\times(\tau,T)))^{m_1}, \quad v \in (C^{2,1}(M\times(\tau,T)))^{m_2} \quad \text{ for all } \quad 0<\tau<T,
\end{equation*}
and the equations (<ref>) satisfy pointwise.
The following blow-up criterion holds
\begin{equation}\label{blowup}
T_{\max}<+\infty \quad \Longrightarrow \quad \limsup_{t\nearrow T_{\max}}\bra{\sumi\|u_i(t)\|_{\LO{\infty}} + \sumj\|v_j(t)\|_{\LM{\infty}}} = +\infty.
\end{equation}
Moreover, if (<ref>) holds, then $(u(t),v(t))$ is (component-wise) non-negative provided the initial data $(u_0,v_0)$ is non-negative.
Thanks to Theorem <ref>, the global existence of (<ref>) follows if we can show that
\begin{equation}\label{eee}
\sumi\|u_i\|_{\LQ{\infty}} + \sumj\|v_j\|_{\LS{\infty}} \leq C_T
\end{equation}
where $C_T$ depends continuously on $T>0$, and $C_T$ is finite for all $T>0$. For simplicity, we consider for the rest of this paper $0<T<T_{\max}$, and ultimately prove the estimate (<ref>).
We first show that, under the mass control condition (<ref>), the solution is bounded in $L^\infty(0,T;\LO{1})$ and $L^\infty(0,T;\LM{1})$.
There exists a constant $C_T$ depending on $T$ such that
\begin{equation}\label{L1_bound}
\sumi\|u_i\|_{L^\infty(0,T;\LO{1})} + \sumj\|v_j\|_{L^\infty(0,T;\LM{1})} \leq C_T.
\end{equation}
Moreover, there exists a constant $C_T$ depending on $T$ such that
\begin{equation}\label{L1_MT_bound}
\sumi\|u_i\|_{\LS{1}} \leq C_T.
\end{equation}
Thanks to (<ref>), we have
\begin{equation}\label{ee1}
\pa_t\bra{\sumi \intO a_iu_i + \sumj \intM b_jv_j} \leq L\intO\bra{\sumi u_i + 1} + L\intM\bra{\sumi u_i + \sumj v_j + 1}.
\end{equation}
Thus, for some constant $C>0$,
\begin{equation}\label{ee2}
\sumi \intO a_iu_i(t) + \sumj \intM b_jv_j(t) \leq C\int_0^t\bra{\sumi \intO a_iu_i + \sumj\intM b_jv_j} + C\sumi\int_0^t\intM u_i + C.
\end{equation}
We need to deal with the boundary integral of $u_i$ on the right hand side. Let $K>0$ be a constant to be determined later and $\phi_0$ is a non-negative, smooth function in $\overline{\Omega}$ satisfying $\pa_{\eta}\phi_0 = 1 + K\phi_0$ on $M$. Suppose $\phi\in C^{2,1}(\bar{\Omega}\times[0,t])$ be a nonnegative function such that $\phi_t+\Delta\phi=0$ on $\Omega\times(0,t)$, $\frac{\partial \phi}{\partial\eta}=1 + K\phi$ on $M\times(0,t)$, and $\phi(\cdot,t) = \phi_0$ on $\overline{\Omega}$. It follows from the comparison principle that $0\leq \phi$. Moreover, if we set $\theta = -\pa_t\phi - \Delta_M\phi$ on $M\times(0,t)$.
By integration by parts we have for $i=1,...,m_1$
\begin{equation}\label{eq:L1esta}
\begin{aligned}
\int_0^t\intM a_id_iu_i(1+K\phi)&=\int_0^t\intM a_id_i u_i \frac{\partial \phi}{\partial\eta}\\
&\le \int_0^t\intM \phi \cdot a_i G_i(u,v)+\int_0^t\intO \phi\cdot a_i F_i(u) \\
&\quad +\int_0^t\intO a_iu_i\left(d_i-1\right)\Delta \phi+a_i\intO\phi(\cdot,0)u_i(\cdot,0).
\end{aligned}
\end{equation}
Furthermore, $j=1,...,m_2$
\begin{equation}\label{eq:L1estb}
\begin{aligned}
\int_0^t\intM b_jv_j \theta &=\int_0^t\intM b_jv_j(-\phi_t-\Delta_M\phi)\\
&\leq \int_0^t\intM \left( \phi \cdot b_j H_j(u,v) + b_jv_j(\delta_j-1)\Delta_M\phi \right)+b_j\intM v_j(\cdot,0)\phi(\cdot,0)
\end{aligned}
\end{equation}
Now sum (<ref>) from $i=1,...,m_1$, with (<ref>) from $j=1,...,m_2$, and apply (<ref>), it follows that
\begin{equation*}
\begin{aligned}
&\sumi \int_0^t\intM a_id_iu_i(1+K\phi) + \sumj \int_0^t\intM b_jv_j\theta\\
&\leq L\int_0^t\intO \phi\bra{\sumi u_i + 1} + L\int_0^t\intM\phi\bra{\sumi u_i + \sumj v_j + 1}\\
&\quad + \sumi \int_0^t\intO a_iu_i(d_i-1)\Delta \phi + \sumi a_i\intO \phi(\cdot,0)u_{i,0}\\
&\quad + \sumj \int_0^t\intM b_jv_j(\delta_j-1)\Delta_M\phi + \sumj \intM \phi(\cdot,0)v_{j,0}\\
&\leq L\sumi\int_0^t\intM \phi u_i + C\sumi\int_0^t\intO a_i u_i + C\sumj \int_0^t\intM b_j v_j + C(1+t)
\end{aligned}
\end{equation*}
where the last step uses $\phi\in C^{2,1}(\overline \Omega\times[0,t])$. By choosing $K$ large enough such that $Kd_ia_i \geq L$ for all $i=1,\ldots, m_1$, and using $v_j\geq 0$ and $\theta\in \LS{\infty}$, we obtain
\begin{equation}\label{ee3}
\sumi \int_0^t\intM a_id_iu_i \leq C\sumi \int_0^t\intO a_iu_i + C\sumj \int_0^t\intM b_jv_j + C(1+t).
\end{equation}
Inserting this into (<ref>) yields
\begin{equation*}
\sumi \intO a_iu_i(t) + \sumj \intM b_jv_j(t) \leq C\int_0^t\bra{\sumi \intO a_iu_i + \sumj\intM b_jv_j} + C(1+t).
\end{equation*}
A direct application of Gronwall's inequality gives the estimates (<ref>). The bound (<ref>) follows directly from (<ref>).
In the following lemma, we show that (<ref>) implies the existence of functions $g_j, j=m_1-1,\ldots, 1$, which allow us to construct a desired $L^p$-energy function.
There exist componentwise increasing functions $g_j:\R^{m_1-j}\to \R$ for $j=1,...,m_1-1$, such that if $\ell\in (0,\infty)^{m_1}$ with $\ell_j> g_j(\ell_{j+1},...,\ell_{m_1})$ for $j=1,...,m_1-1$ then there exists $L_\ell>0$ such that
\begin{equation}\label{f1}
\sumi \ell_i F_i(u)\le L_\ell \left(\sumi u_i^{\pO}+1\right)\text{ for all }u\in \mathbb{R}_+^{m_1},
\end{equation}
\begin{equation}\label{f2}
\sumi \ell_i G_i(u,v)\le L_\ell \left(\sumi u_i^{\pM} + \sumj v_j^{\pM}+1\right)\text{ for all }u\in \mathbb{R}_+^{m_1}, v\in \R_+^{m_2}.
\end{equation}
Without loss of generality we assume that and $a_{i,j}>0$ for $i>j$ with $i,j\in{1,...,m_1}$. We construct two sequences of functions $g_j: \R^{m_1-j}\to \R$, $j=m_1-1,m_1-2,\ldots, 1$ and $\alpha_j: \R^{m_1-j+1} \to \R$, $j=m_1,\ldots, 1$ inductively as follows:
* $g_{m_1-1}(x_{m_1}):= a_{m_1,m_1-1}x_{m_1}$. We also define the function $\alpha_{m_1}(x_{m_1}):= x_{m_1}$. Note that $\wh{\alpha}_{m_1}:= \alpha_{m_1}(\ell_{m_1}) = \ell_{m_1}>0$.
* For $i=m_1-2,m_1-3,\ldots, 2, 1$, we constructed the function $g_i$ using established functions $g_j$ and $\alpha_j$ for $j\ge i+1$. More precisely, we define
\begin{equation*}
g_{i}(x_{i+1},x_{i+2},\ldots, x_{m_1}):= \sum_{j=i+1}^{m_1}a_{j,i}\alpha_j(x_j,x_{j+1}, \ldots, x_{m_1}),
\end{equation*}
\begin{equation*}
\alpha_i(x_i,\ldots, x_{m_1}):= x_i - g_i(x_{i+1},x_{i+2},\ldots, x_{m_1}).
\end{equation*}
Due to the assumptions of $(\ell_i)_{i=1,\ldots, m_1}$,
\begin{equation*}
\wh{\alpha}_i:= \alpha_i(\ell_i,\ell_{i+1},\ldots, \ell_{m_1})=\ell_i - g_i(\ell_{i+1},\ldots, \ell_{m_1})>0.
\end{equation*}
Therefore, we have
\begin{align*}
\sumi \ell_iF_i(u) &= \sumi\sbra{\wh{\alpha}_i + g_i(\ell_{i+1},\ldots, \ell_{m_1})}F_i(u)\\
&=\sumi\sbra{\wh{\alpha}_i + \sum_{j=i+1}^{m_1}\wh{\alpha}_ja_{j,i}}F_i(u)\\
&=\sumi \sum_{j=i}^{m_1}\wh{\alpha}_ja_{j,i}F_i(u) \quad (\text{since } a_{i,i}=1)\\
&=\sumi \wh{\alpha}_i\sum_{j=1}^{i}a_{i,j}F_j(u)\\
&\leq \sumi \wh{\alpha}_iL\sbra{\sumi u_i^{\pO}+1} \quad (\text{due to } (\ref{eq:intsum1}))
\end{align*}
which implies directly (<ref>). The inequality (<ref>) can be verified in the same way, so we omit it here.
The above lemma shows that the intermediate sum conditions involving $F$ and $G$ in (<ref>) and (<ref>) imply the existence of functions $g_j$, $j=1,\ldots, m_1-1$. It will be shown later on that once we have these functions $g_j$, an $L^p$-energy can be constructed. This means that the conclusion of Theorem <ref> still holds true if we assume the existence of $g_j$, $j=1,\ldots, m_1-1$, instead of the intermediate sum conditions (involving $F$ and $G$) (<ref>) and (<ref>). We choose to present Theorem <ref> under (<ref>) and (<ref>) as they are more constructive, and they appear naturally in many applications (see Section <ref>).
It is important to remark that the existence of functions $g_j$ is more general than the intermediate sum conditions (<ref>)–(<ref>). For example, the nonlinearities
\begin{align*}
\end{align*}
clearly do not satisfy (<ref>) for high dimensions $n\geq 3$. However, if we define $g_1(x) = x$, then for any $\ell = (\ell_1,\ell_2)$ with $\ell_1 > g_1(\ell_2) = \ell_2$ we have
\begin{align*}
\ell_1F_1(u_1,u_2) + \ell_2F_2(u_1,u_2) &= u_1\sbra{\ell_1 u_2^3 - \ell_2 u_2^4 + (\ell_2-\ell_1)u_1^3}\\
&\leq u_1\sbra{\ell_1u_2^3 - \ell_2 u_2^4} \leq L_\ell (1+u_1)
\end{align*}
for all $u_1,u_2\ge 0$, and a constant $L_\ell$ depending only on $\ell$.
The following interpolation inequality might be of independent interest.
Assume that $w:\Omega \to [0,\infty)$ with $\|w\|_{\LO{a}} \leq K$. Then for any $p\geq 2a$, $\eps>0$, and any $\rO, \rM$ satisfying
\begin{equation*}
1\leq \rO < 1+ \frac{2a}{n} \quad \text{ and } 1\leq \rM < 1 + \frac{a}{n},
\end{equation*}
there exists a constant $C_{p,\eps,K}$ depending on $p, \eps$ and $K$, but not on $w$, such that
\begin{equation}\label{c1_1}
\intO w^{p-1+\rO} + \intM w^{p-1+\rM} \leq \eps\bra{\intO w^{p-2}|\na w|^2 + \intO w^{p}} + C_{p,\eps,K}.
\end{equation}
We will estimate the domain term on the left hand side of (<ref>), while the boundary term will follow as a consequence. First, by using Sobolev's embedding we have
\begin{equation}\label{c2_1}
\begin{aligned}
\intO w^{p-2}|\na w|^2 + \intO w^p = \frac{4}{p^2}\intO|\na(w^{p/2})|^2 + \intO(w^{p/2})^2\geq \frac{4}{p^2}\|w^{p/2}\|_{H^1(\Omega)}^2.
\end{aligned}
\end{equation}
Define $\rO = 1+\eta$ and $\beta = \frac{2\eta}{p}$. Then we have $\eta\in [0,2a/n)$ and $\beta \in [0,\frac{4a}{np})$, and
\begin{equation*}
\intO w^{p-1+\rO} = \intO w^{p+\eta} = \intO|w^{p/2}|^{2+\beta} = \|y\|_{\LO{2+\beta}}^{2+\beta},
\end{equation*}
where $y:= w^{p/2}$.
Thanks to the Gagliardo-Nirenberg inequality, we have
\begin{equation*}
\|y\|_{\LO{2+\beta}}^{2+\beta} \leq C_{\rm{GN}}\|y\|_{H^1(\Omega)}^{\alpha(2+\beta)}\|y\|_{\LO{1}}^{(1-\alpha)(2+\beta)}
\end{equation*}
where $\alpha \in (0,1)$ satisfies
\begin{equation*}
\frac{1}{2+\beta} = \bra{\frac 12 - \frac 1n}\alpha + \frac{1-\alpha}{1}.
\end{equation*}
It follows that
\begin{equation*}
\alpha(2+\beta) = \frac{2n(\beta+1)}{n+2}\quad \text{and}\quad (1-\alpha)(2+\beta) = \frac{4+2\beta - \beta n}{n+2}.
\end{equation*}
\begin{equation*}
\|y\|_{\LO{2+\beta}}^{2+\beta} \leq C_{\rm{GN}}\|y\|_{H^1(\Omega)}^{\frac{2n(\beta+1)}{n+2}}\|y\|_{\LO{1}}^{\frac{4+2\beta-\beta n}{n+2}}.
\end{equation*}
Since $\beta<4a/(np)$ and $p\geq 2a$, it holds $\frac{2n(\beta+1)}{n+2}<2$. We can then use Young's inequality to estimate
\begin{equation}\label{c0-1}
\|y\|_{\LO{2+\beta}}^{2+\beta} \leq \eps \|y\|_{H^1(\Omega)}^2 + C_{\eps}\|y\|_{\LO{1}}^{\frac{4+2\beta-\beta n}{2-n\beta}}.
\end{equation}
Changing the variable $y = w^{p/2}$ we have
\begin{equation}\label{c0}
\|y\|_{\LO{1}}^{\frac{4+2\beta-\beta n}{2-n\beta}} = \|w\|_{\LO{\frac p2}}^{\frac{p(4+2\beta-\beta n)}{2(2-n\beta)}}.
\end{equation}
If $p = 2a$, then this term is bounded by a constant depending on $K$, since $\|w\|_{\LO{a}} \leq K$. If $p>2a$, we use interpolation inequality to have
\begin{equation}\label{c0_1}
\|w\|_{\LO{\frac p2}} \leq \|w\|_{\LO{p+\eta}}^{\theta}\|w\|_{\LO{a}}^{1-\theta} \leq K^{1-\theta}\|w\|_{\LO{p+\eta}}^{\theta}
\end{equation}
where $\theta \in (0,1)$ satisfies
\begin{equation*}
\frac{2}{p} = \frac{\theta}{p+\eta} + \frac{1-\theta}{a}.
\end{equation*}
Note that
\begin{equation}\label{c0_2}
\theta {\frac{p(4+2\beta-\beta n)}{2(2-n\beta)}} = \frac{(p+\eta)(p-2a)(4+2\beta-\beta n)}{2(2-n\beta)(p+\eta-a)} < p+\eta
\end{equation}
due to $\beta=2\eta/p$ and $\eta < 2a/n$. From (<ref>)–(<ref>) and Young's inequality, it follows that
\begin{equation*}
C_\eps\|y\|_{\LO{1}}^{\frac{4+2\beta-\beta n}{2-n\beta}} \leq C_\eps C_K\|w\|_{\LO{p+\eta}}^{\frac{(p+\eta)(p-2a)(4+2\beta-\beta n)}{2(2-n\beta)(p+\eta-a)}} \leq \frac 12 \|w\|_{\LO{p+\eta}}^{p+\eta} + C_{p,\eps,K}.
\end{equation*}
Inserting this into (<ref>), and noting that $\|y\|_{\LO{2+\beta}}^{2+\beta} = \|w\|_{\LO{p+\eta}}^{p+\eta}$, we get
\begin{equation*}
\|w\|_{\LO{p+\eta}}^{p+\eta} \leq 2\eps\|w^{p/2}\|_{H^1(\Omega)}^2 + C_{p,\eps,K}.
\end{equation*}
Combining this with (<ref>) leads to the desired estimate for the domain term, i.e.
\begin{equation}\label{c2}
\intO w^{p-1+\rO} \leq \eps\bra{\intO w^{p-2}|\na w|^2 + \intO w^p} + C_{p,\eps,K}.
\end{equation}
To treat the boundary term $\intM w^{p-1+\pM}$, we first define $\pM = 1+\xi$ for $\xi \in [0,a/n)$. We then use the following interpolation trace inequality, see e.g. <cit.>,
\begin{equation*}
\begin{aligned}
\intM w^{p-1+\pM} = \intM w^{p+\xi} &\leq C\intO w^{p+\xi - 1}|\na w| + C\intO w^{p+\xi}\\
&\leq \frac{\eps}{2}\intO w^{p-2}|\na w|^2 + C_{\eps}\intO w^{p+2\xi} + C\intO w^{p+\xi}\\
&\leq \frac{\eps}{2}\intO w^{p-2}|\na w|^2 + C_{\eps}\intO w^{p+2\xi} + C.
\end{aligned}
\end{equation*}
Since $\xi \in [0,a/n)$, $2\xi \in [0,2a/n)$. Therefore, we can use (<ref>) to show
\begin{equation*}
C_\eps\intO w^{p+2\xi} \leq \frac{\eps}{2}\bra{\intO w^{p-2}|\na w|^2 + \intO w^{p}} + C_{p,\eps,K}.
\end{equation*}
Therefore, we obtain the estimate for the boundary term in (<ref>), and thus finish the proof of Lemma <ref>.
To construct our $L^p$-energy function, we write $\mathbb{Z}_{+}^{k}$ for the set of all $k$-tuples of non negative integers. Addition and
scalar multiplication by non negative integers of elements in $\mathbb{Z}_{+}^{k}$
is understood in the usual manner. If $\beta=(\beta_{1},...,\beta_{k})\in \mathbb{Z}_{+}^{k}$ and $p\in \mathbb N$,
then we define $\beta^{p}=((\beta_{1})^{p},...,(\beta_{k})^{p})$.
Also, if $\alpha=(\alpha_{1},...,\alpha_{k})\in \mathbb{Z}_{+}^{k}$, then
we define $|\alpha|=\sum_{i=1}^{k}\alpha_{i}$. Finally, if $z=(z_{1},...,z_{k})\in \mathbb{R}_{+}^{k}$
and $\alpha=(\alpha_{1},...,\alpha_{k})\in \mathbb{Z}_{+}^{k}$, then we define
$z^{\alpha}=z_{1}^{\alpha_{1}}\cdot...\cdot z_{k}^{\alpha_{k}}$,
where we interpret $0^{0}$ to be $1$. For $2\leq p\in \mathbb N$, we build our $L^p$-energy function of the form
\begin{equation}\label{Lp}
\L_p[u](t) = \intO \H_p[u](t),
\end{equation}
\begin{equation}\label{Hp}
\H_p[u](t) = \sum_{\beta\in \mathbb Z_+^{m_1}, |\beta| = p}\begin{pmatrix}
p\\ \beta\end{pmatrix}\theta^{\beta^2}u(t)^{\beta},
\end{equation}
and the positive constants $\theta= (\theta_1,\ldots, \theta_{m_1})$ are to be chosen. For convenience, hereafter we drop the subscript $\beta\in \mathbb Z_+^{m_1}$ in the sum as it should be clear.
The main result of this section is the following lemma.
For any positive integer $p\geq 2$ and any constant $\eps>0$, there exists $K_{p,\eps}>0$ such that
\begin{equation}\label{eps_critical}
\sumi\bra{\norm{u_i}_{\LQ{p-1+\pO}}^{p-1+\pO} + \norm{u_i}_{\LS{p-1+\pM}}^{p-1+\pM}} \leq K_{p,\eps}(1+T) + \eps\sumj\norm{v_j}_{\LS{p-1+\pM}}^{p-1+\pM}
\end{equation}
for a possibly different constant $K_{p,\eps}$.
Consequently, for any $1<p<\infty$ and any $\eps>0$, there exists a constant $K_{p,\eps}>0$ such that
\begin{equation}\label{ff4}
\sumi\bra{\|u_i\|_{\LQ{p}}+\|u_i\|_{\LS{p}}} \leq K_{p,\eps}(1+T) + \eps\sumj\|v_j\|_{\LS{p}}.
\end{equation}
First, we choose $\theta=(\theta_{1},...,\theta_{m_1})$
with $\theta_{i}\ge1$ for all $i=1,...,m_1$ such that
* The matrix $\mathscr{M}=\left(\mathscr{M}_{i,j}\right)$ is positive
definite, where
\begin{equation}\label{M}
\mathscr{M}_{i,j}=\begin{cases}
\begin{array}{cc}
d_{i}\theta_{i}^{2}, & \text{ if \ensuremath{i=j}}\\
\frac{d_{i}+d_{j}}{2}, & \text{ if \ensuremath{i\ne j}}
\end{array}\end{cases}
\end{equation}
* $\theta_j> \underset{i_1,...,i_{m_1-j}\in\{1,...,2p-1\}}{\max}g_j\left(\theta_{j+1}^{i_1},...,\theta_{m_1}^{i_{m_1-j}}\right)$ for all $j=1,...,m_1-1$, where the functions $g_j$ are given by Lemma <ref>.
Such a choice of $\theta$ is possible since the off-diagonal elements of $\mathscr{M}$ are fixed, therefore we choose successively $\theta_{m_1}>0$, $\theta_{m_1-1}$, $\ldots$, $\theta_1$ such that (<ref>) is fulfilled, and $\theta_{j}$ is large enough such that $\mathscr{M}$ is diagonally dominant, which implies its positive definiteness. With this chosen $\theta$, we define for $2\leq p\in \mathbb Z$ our $L^p$-energy function $\L_p[u]$ as in (<ref>). Then, thanks to Lemma <ref>
\begin{equation}\label{f5}
\begin{aligned}
\bra{\L_p[u]}'(t)&=\int_{\Omega}\sum_{|\beta|=p-1}\left(\begin{array}{c}
\beta
\end{array}\right)\theta^{\beta^{2}}u^{\beta}\sum_{i=1}^{m_1}\theta_{i}^{2\beta_{i}+1}\frac{\partial}{\partial t}u_{i}\\
\beta
\end{array}\right)\theta^{\beta^{2}}u^{\beta}\sum_{i=1}^{m_1}\theta_{i}^{2\beta_{i}+1}\left(d_{i}\Delta u_{i}+F_{i}(u)\right).
\end{aligned}
\end{equation}
Integration by parts gives
\[
\int_{\Omega}\sum_{|\beta|=p-1}\left(\begin{array}{c}
\beta
\end{array}\right)\theta^{\beta^{2}}u^{\beta}\sum_{i=1}^{m_1}d_{i}\Delta u_{i}=:(I)+(II),
\]
\[
\beta
\end{array}\right)\theta^{\beta^{2}}u^{\beta}\sum_{i=1}^{m_1}\theta_{i}^{2\beta_{i}+1}G_{i}(u,v),
\]
and, thanks to Lemma <ref>,
\begin{equation}\label{f3}
\beta
\end{array}\right)\theta^{\beta^{2}}u^{\beta}\sum_{l=1}^{n}\sum_{i,j=1}^{m_1}a_{i,j}\frac{\partial}{\partial x_{l}}u_{i}\frac{\partial}{\partial x_{l}}u_{j}
\end{equation}
\begin{equation}\label{a_ij}
\begin{array}{cc}
d_{j}\theta_{i}^{2\beta_{i}+1}\theta_{j}^{2\beta_{j}+1}, & \text{ if \ensuremath{i\ne j}},\\
d_{i}\theta_{i}^{4\beta_{i}+4}, & \text{ if \ensuremath{i=j}}.
\end{array}\end{cases}
\end{equation}
Note that
\begin{equation}\label{f4}
\sum_{i,j=1}^{m_1}a_{i,j}\frac{\partial}{\partial x_{l}}u_{i}\frac{\partial}{\partial x_{l}}u_{j}=\sum_{i,j=1}^{m_1}b_{i,j}\frac{\partial}{\partial x_{l}}u_{i}\frac{\partial}{\partial x_{l}}u_{j}
\end{equation}
\[
\begin{array}{cc}
\frac{d_{i}+d_{j}}{2}\theta_{i}^{2\beta_{i}+1}\theta_{j}^{2\beta_{j}+1}, & \text{ if \ensuremath{i\ne j}}\\
d_{i}\theta_{i}^{4\beta_{i}+4}, & \text{ if \ensuremath{i=j}}
\end{array}\end{cases}
\]
Furthermore, if we define $\mathscr B = \left(b_{i,j}\right)$, and the $m_1\times m_1$ diagonal matrix
\[
\mathscr C=\text{diag}\left(\theta_{i}^{-2\beta_{i}-1}\right)
\]
\[
\mathscr C^{-1}\mathscr M\mathscr C^{-1}=\mathscr{B}
\]
where $\mathscr{M}$ is defined in (<ref>). Consequently, from the choice of $\theta$, the matrix $\mathscr B$
is positive definite. Therefore, there exists $\lambda>0$ such that
\begin{equation*}
\sum_{i,j=1}^{m_1}b_{i,j}\frac{\pa}{\pa x_l}u_i\frac{\pa}{\pa x_l}u_j \geq \lambda \sum_{i=1}^{m_1}\abs{\frac{\pa}{\pa x_l}u_i}^2.
\end{equation*}
Inserting this into (<ref>) and (<ref>) leads to, for some $c_p>0$,
\begin{equation}\label{eq:gradpstuff}
\begin{aligned}
(II) &\leq -\lambda \int_{\Omega}\sum_{|\beta|=p-2}\left(\begin{array}{c}
\beta
\end{array}\right)\theta^{\beta^{2}}u^{\beta}\sum_{l=1}^{n}\sum_{i=1}^{m_1}\abs{\frac{\partial}{\partial x_{l}}u_{i}}^2\\
&\leq -c_p\sum_{i=1}^{m_1}\intO u_i^{p-2}|\na u_i|^2.
\end{aligned}
\end{equation}
Also, $\theta_i\le\theta_i^{2\beta_i+1}\le\theta_i^{2p-1}$ for all $i=1,...,m_1$. Therefore, from Lemma <ref>, (<ref>)–(<ref>) and the choice of $\theta$, there is a value $L_\theta>0$ such that
\[
\sum_{i=1}^{m_1}\theta_{i}^{2\beta_{i}+1}G_{i}(u,v)\le L_\theta \left(\sum_{i=1}^{m_1}u_{i}^{\pM}+\sum_{j=1}^{m_2}v_{j}^{\pM}+1\right)
\]
\[
\sum_{i=1}^{m_1}\theta_{i}^{2\beta_{i}+1}F_{i}(u)\le L_\theta \left(\sum_{i=1}^{m_1}u_{i}^{\pO}+1\right).
\]
As a result, there exist $c_{p},K_{p,\theta}>0$ so that
\begin{equation}
\sum_{|\beta|=p-1}\left(\begin{array}{c}
\beta
\end{array}\right)\theta^{\beta^{2}}u^{\beta}\sum_{i=1}^{m_1}\theta_{i}^{2\beta_{i}+1}G_{i}(u,v)\le K_{p,\theta}\sum_{j=1}^{m_1}u_{j}^{p-1}\left(\sum_{i=1}^{m_1}u_{i}^{\pM}+\sum_{j=1}^{m_2}v_{j}^{\pM}+1\right),\label{eq:Gpstuff}
\end{equation}
\begin{equation}
\sum_{|\beta|=p-1}\left(\begin{array}{c}
\beta
\end{array}\right)\theta^{\beta^{2}}u^{\beta}\sum_{i=1}^{m_1}\theta_{i}^{2\beta_{i}+1}F_{i}(u)\le K_{p,\theta}\left(\sum_{i=1}^{m_1}u_{i}^{p-1+\pO}+1\right)\label{eq:Fpstuff}.
\end{equation}
By applying (<ref>), (<ref>) and (<ref>) into (<ref>), there exists $K_{p,\theta}>0$ such that
\begin{equation}\label{eq:LprimeEqn}
\begin{aligned}
(\L_p[u])'(t)&+c_p\sum_{i=1}^{m_1}\int_\Omega u_i^{p-2}|\nabla u_i|^2\\
&\le K_{p,\theta}\left[1+\int_\Omega \sum_{j=1}^{m_1}u_j^{p-1+\pO}+ \sumi\intM u_i^{p-1+\pM}+ \sumi\sumj\int_M u_i^{p-1}v_j^{\pM}\right]\\
&\leq K_{p,\theta,\eps}\sbra{1+\sumi\intO u_i^{p-1+\pO} + \sumi\intM u_i^{p-1+\pM}} + \eps\sumj\intM v_j^{p-1+\pM},
%&\leq K_{p,\theta,\eps}\sbra{1+\sumi\intO u_i^{p-1+\pO}} + \eps\sumj\intM v_j^{p}
\end{aligned}
\end{equation}
where we used Young's inequality at the last step. Adding
$$\sumi\bra{\frac{c_p}{2}\intO u_i^p + \intO u_i^{p-1+\pO} + \intM u_i^{p-1+\pM}}$$
to both sides gives
\begin{equation}\label{ff5}
\begin{aligned}
\bra{\L_p[u]}'(t) + \frac{c_p}{2}\sumi \bra{\intO u_i^{p-2}|\na u_i|^2 + \intO u^p} + \frac 12\sumi\bra{\intO u_i^{p-1+\pO}+\intM u_i^{p-1+\pM}}\\
\leq \sumi\bra{\frac{c_p}{2}\intO u_i^p + \intO u_i^{p-1+\pO} + \intM u_i^{p-1+\pM}} + K_{p,\theta,\eps} + \eps\sumj\intM v_j^{p-1+\pM}\\
\leq K_{p,\theta,\eps} + \eps\sumi\bra{\intO u_i^{p-2}|\na u_i|^2 + \intO u_i^p} + \eps\sumj\intM v_j^{p-1+\pM}
\end{aligned}
\end{equation}
where we used Lemma <ref> at the last step. Integrating the resultant on $(0,T)$ finishes the proof of (<ref>).
To prove (<ref>), we first show that for each $p\in \mathbb N$, $\eps>0$, there exists $K_{p,\eps}$ such that
\begin{equation}\label{ff13}
\sumi \|u_i\|_{\LQ{p-1+\pM}}^{p-1+\pM} \leq K_{p,\eps,T} + \eps\sumj\|v_j\|_{\LS{p-1+\pM}}^{p-1+\pM}.
\end{equation}
Indeed, from the $L^1$-bound in Lemma <ref> and interpolation inequality we have, for $\gamma\in (0,1)$ with $\frac{1}{p-1+\pM} = \frac{\gamma}{1} + \frac{1-\gamma}{p-1+\pO}$,
\begin{align*}
\sumi\|u_i\|_{\LQ{p-1+\pM}}^{p-1+\pM} &\leq \sumi\|u_i\|_{\LQ{1}}^{\gamma(p-1+\pM)}\sumi\|u_i\|_{\LQ{p-1+\pO}}^{(1-\gamma)(p-1+\pM)}\\
&\leq K_T\sumi\|u_i\|_{\LQ{p-1+\pO}}^{(1-\gamma)(p-1+\pM)}\\
&\leq K_T\bra{1+\sumi\|u_i\|_{\LQ{p-1+\pO}}^{p-1+\pO}}\\
&\leq K_{p,\eps,T} + \eps\sumj\|v_j\|_{\LS{p-1+\pM}}^{p-1+\pM}.
\end{align*}
The estimate (<ref>) now follows from (<ref>), (<ref>), and interpolation.
§ DUALITY METHOD
We first collect some useful results which will be used in the following sections. The next lemma is about the $L^p$-maximal regularity for heat equation in a smooth manifold without boundary.
Let $1<p<\infty$, $0\leq \tau < T < \infty$. There exists a constant $C_{\mr,p}^M$ ($\mr$ stands for “maximal regularity"), which depends only on $p$, $M$, if the dimension $n$, such that, for any $\ff\in L^p(M_{\tau,T})$, and $\uu$ is the solution to
\begin{equation}\label{parabolic_M}
\begin{cases}
\pa_t\uu - \Delta_{M} \uu = \ff, &(x,t)\in M_{\tau,T},\\
\uu(x,0) = 0, &x\in M,
\end{cases}
\end{equation}
we have the following estimate
\begin{equation*}
\|\Delta_{M} \uu\|_{L^p(M_{\tau,T})} \leq C_{\mr,p}^M\|\ff\|_{L^p(M_{\tau,T})}.
\end{equation*}
The proof of this lemma can be found in <cit.>[A similar result was shown in <cit.> where the constant $C_{\mr,p}^M$ might depend on the time horizon $T - \tau$]. We emphasize that fact that the constant $C_{\mr,p}^M$ is independent of $T$ and $\tau$.
The following crucial lemma provides the regularity of the duality problem suited for volume-surface systems.
Assume that $0<\tau<T$, $1<q<\infty$ and $\psi\in L^{q}(M_{\tau,T})$. Let $\phi$ be the solution to
\begin{equation}\label{dual_eq}
\left\{
\begin{aligned}
\pa_t\phi + \Delta \phi = 0, &\text{ on }Q_{\tau,T},\\
\pa_t\phi + \delta \Delta_M \phi = -\psi, &\text{ on } M_{\tau,T},\\
\phi(x,T) = 0 &\text{ on } \overline{\Omega}.
\end{aligned}
\right.
\end{equation}
Then, we have the following estimate
\begin{equation}\label{scaled_ineq}
\|\Delta_M \phi\|_{L^q(M_{\tau,T})} \leq \frac{C_{\mr,q}^M}{\delta}\|\psi\|_{L^q(M_{\tau,T})},
\end{equation}
where $C_{\mr,q}^M$ is the maximal regularity constant in Lemma <ref>. Moreover, with $\xi = \frac{q}{n+1}$ we have
\begin{equation}\label{W21_estimate}
\|\phi\|_{W^{2,1}_{q+\xi}(Q_{\tau,T})} + \|\phi\|_{W^{2,1}_q(M_{\tau,T})} + \|\phi(\tau)\|_{\LO{q+\xi}}+ \|\phi(\tau)\|_{\LM{q}} \leq C_{T-\tau} \|\psi\|_{L^q(M_{\tau,T})},
\end{equation}
\begin{equation}\label{flux_estimate}
\norm{\pa_{\eta}\phi}_{L^{q+\xi}(M_{\tau,T})} \leq C_{T-\tau}\|\psi\|_{L^{q}(M_{\tau,T})}.
\end{equation}
\begin{equation}\label{Lq^*}
\|\phi\|_{L^{q^{\dag}}(Q_{\tau,T})} + \|\phi\|_{L^{q^*}(M_{\tau,T})} \leq C_{T-\tau}\|\psi\|_{L^{q}(M_{\tau,T})}
\end{equation}
\begin{equation}\label{crit_exp}
q^{\dag} =
\begin{cases}
\frac{(n+2)q}{n+1-2q} &\text{ if } q < \frac{n+1}{2},\\
<+\infty \text{ abitrary } &\text{ if } q = \frac{n+1}{2},\\
+\infty &\text{ if } q > \frac{n+1}{2}.
\end{cases}\quad
\text{ and }\quad
q^* = \begin{cases}
\frac{(n+1)q}{n+1-2q} &\text{ if } q < \frac{n+1}{2},\\
<+\infty \text{ abitrary } &\text{ if } q = \frac{n+1}{2},\\
+\infty &\text{ if } q > \frac{n+1}{2}.
\end{cases}
\end{equation}
Moreover, if $\psi \geq 0$ a.e. in $M_{\tau,T}$, then $\phi\geq 0$.
* At the first glance, the dual problem (<ref>) looks like a backward parabolic equation. However, since the “initial data" is considered at $t = T$, by the change of variable $s = T-\tau-t$, (<ref>) transforms into the usual forward parabolic equation.
* To solve the dual problem, we first solve the boundary equation $\pa_t\phi + \delta\Delta_M\phi = -\psi$ and define $\Psi:= \phi|_{M_{\tau,T}}$. Then, we consider the heat operator with inhomogeneous Dirichlet boundary condition: $\pa_t \phi + \Delta \phi = 0$ in $Q_{\tau,T}$ and $\phi|_{M_{\tau,T}} = \Psi$. See more details in [36].
The bound
\begin{equation*}
\norm{\phi}_{W_q^{2,1}(M_{\tau,T})} \leq C_{T-\tau}\norm{\psi}_{L^q(M_{\tau,T})}
\end{equation*}
can be found in [36]. In particular,
\begin{equation*}
\|\pa_t\phi\|_{L^q(M_{\tau,T})} \leq C_{T-\tau}\|\psi\|_{L^q(M_{\tau,T})}.
\end{equation*}
Therefore, by Hölder's inequality
\begin{equation*}
\|\phi(\tau)\|_{\LM{q}}^q = \intM\abs{\int_\tau^T\pa_t\phi}^q \leq (T-\tau)^{q-1}\intM\int_{\tau}^{T}|\pa_t\phi|^q \leq C_{T-\tau}\|\psi\|_{L^{q}(M_{\tau,T})}^q
\end{equation*}
which implies
\begin{equation*}
\|\phi(\tau)\|_{\LM{q}} \leq C_{T-\tau}\|\psi\|_{L^q(M_{\tau,T})}.
\end{equation*}
To show the improved bound in $Q_{\tau,T}$ and for the normal derivative, we use the following embedding, cf. <cit.>,
\begin{equation*}
W_q^{2,1}(M_{\tau,T})\hookrightarrow L^{r}(\tau,T;W^{2-\frac{1}{r}, r}(M))\cap W^{1-\frac{1}{2r}, r}(0,T;L^r(M))
\end{equation*}
for all
\begin{equation*}
r \leq \frac{(n+2)q}{n+1} = q + \frac{q}{n+1}.
\end{equation*}
Therefore, by choosing $\xi=\frac{q}{n+1}$ and defining $r= q + \xi$, we have $\Psi:= \phi|_{M_T}\in L^{r}(\tau,T;W^{2-\frac{1}{r}, r}(M))\cap W^{1-\frac{1}{2r}, r}(0,T;L^r(M))$. Now we can apply the maximal regularity for equation with inhomogeneous Dirichlet boundary condition, see e.g. [31],
\begin{equation*}
\begin{cases}
\pa_t\phi + \Delta \phi = 0, &\text{ on } Q_{\tau,T},\\
\phi = \Psi, &\text{ on } M_{\tau,T},\\
\phi(x,T) = 0, &\text{ on } \overline{\Omega},
\end{cases}
\end{equation*}
to obtain
\begin{align*}
\norm{\phi}_{W_{q+\xi}^{2,1}(Q_{\tau,T})} \leq C_{T-\tau}\norm{\Psi}_{L^{r}(\tau,T;W^{2-\frac{1}{r}, r}(M))\cap W^{1-\frac{1}{2r}, r}(0,T;L^r(M))}\\
\leq C_{T-\tau}\norm{\phi}_{W_q^{2,1}(M_{\tau,T})} \leq C_{T-\tau}\norm{\psi}_{L^q(M\times(\tau,T))}.
\end{align*}
The estimate of the normal derivative (<ref>) follows from the bound of $\phi$ in $W_{q+\xi}^{2,1}(Q_{\tau,T})$ and Lemma <cit.>.
The estimate (<ref>) then follows from (<ref>) and the embedding theory (<cit.>).
It remains to show (<ref>). We define scaled functions $\wh{\phi}(x,t) = \phi(x,t/\delta)$ and $\wh{\psi}(x,t) = \psi(x,t/\delta)$. From the equation of $\phi$, it follows that
\begin{equation*}
\begin{cases}
\partial_t \wh{\phi} - \Delta_M\wh{\phi} = \frac{1}{\delta}\wh{\psi}, &(x,t)\in M_{\delta\tau,\delta T},\\
\wh{\phi}(x,\delta \tau) = 0, &x\in M.
\end{cases}
\end{equation*}
From Lemma <ref>,
\begin{equation*}
\|\Delta_M\wh{\phi}\|_{L^q(M_{\delta \tau, \delta T})} \leq \frac{C_{\mr,q}^M}{\delta}\|\wh \psi\|_{L^q(M_{\delta \tau, \delta T})}.
\end{equation*}
By switching back to the original variables we obtain easily
\begin{equation*}
\delta\|\Delta_M\phi\|_{L^q(M_{\tau,T})}^q = \int_{\delta \tau}^{\delta T}\intM |\Delta_M\wh\phi|^q \leq \bra{\frac{C_{\mr,q}^M}{\delta}}^q\int_{\delta \tau}^{\delta T}\intM |\wh \psi|^q = \frac{\bra{C_{\mr,q}^M}^q}{\delta^{q-1}}\|\psi\|_{L^q(M_{\tau,T})}^q,
\end{equation*}
which yields the desired inequality (<ref>).
We show that if (<ref>) is true for some $\Lam$, then it's also true for $\Lam+\varkappa$ for small enough $\varkappa>0$ in the following sense.
Assume that (<ref>) holds for some $\Lam$. Then there exists $\varkappa_0>0$ such that
\begin{equation*}
\frac{\delta_{\max} - \delta_{\min}}{\delta_{\max} + \delta_{\min}}C_{\mr,(\Lam+\varkappa)'}^M < 1 \quad \text{ for all } \quad 0<\varkappa<\varkappa_0
\end{equation*}
\begin{equation*}
(\Lam+\vk)' = \frac{\Lam+\vk}{\Lam + \vk - 1}
\end{equation*}
is the Hölder conjugate exponent of $\Lam+\vk$.
It's sufficient to show that
\begin{equation}\label{claim2}
\bra{C_{\mr,\Lam'}^M}^-:= \liminf_{\eta\to 0^+}C_{\mr,\Lam' - \eta}^M \leq C_{\mr,\Lam'}^M.
\end{equation}
Let $\Lam_\eta'$ satisfy
\begin{equation*}
\frac{1}{\Lam_\eta'} = \frac{1}{2}\sbra{\frac{1}{\Lam'} + \frac{1}{\Lam'-\eta}} \quad \text{ or equivalently } \quad \Lam_\eta' = \Lam' - \frac{\Lam'\eta}{2\Lam' - \eta}.
\end{equation*}
By the Riesz-Thorin interpolation theorem (cf. <cit.>),
\begin{equation*}
C_{\mr,\Lam_\eta'}^M \leq C_{\mr,\Lam'}^{1/2}C_{\mr,\Lam'-\eta}^{1/2}.
\end{equation*}
By letting $\eta \to 0$,
\begin{equation*}
\bra{C_{\mr,\Lam'}^M}^{-} \leq C_{\mr,\Lam'}^{1/2}\sbra{\bra{C_{\mr,\Lam'}^M}^{-}}^{1/2}
\end{equation*}
which yields the desired claim (<ref>) and thus finishes the proof of Lemma <ref>.
There exist constants $C_T$ and $\gamma>0$ such that
\begin{equation*}
\sumi\bra{\norm{u_i}_{\LQ{\Lambda+\gamma}} + \norm{u_i}_{\LS{\Lambda+\gamma}}} +\sumj \norm{v_j}_{\LS{\Lambda+\gamma}} \leq C_T.
\end{equation*}
Let $\vk_0$ be given in Lemma <ref>, and choose $\vk \in (0,\vk_0)$ small enough such that
\begin{equation*}
\frac{\Lam+\vk}{n+1} > \frac{\vk}{\Lam-1}.
\end{equation*}
Note that this is equivalent to
\begin{equation}\label{l1}
\Lam' = \frac{\Lam}{\Lam-1}< (\Lam+\varkappa)' + \frac{(\Lam+\varkappa)'}{n+1}
\end{equation}
where $(\Lam+\vk)'$ is the Hölder conjugate exponent of $\Lam+\vk$. Since we only need (<ref>) for $\vk$ sufficiently small and positive, note that (<ref>) is true when $\vk =0$, and therefore it is true for small positive $\vk$.
Let $0\leq \psi\in \LS{(\Lam+\vk)'}$ with $ \|\psi\|_{\LS{(\Lam+\vk)'}} = 1$, and let $\phi$ be the solution to the dual problem (<ref>) with
\begin{equation*}
\delta = \frac{\delta_{\max}+\delta_{\min}}{2},
\end{equation*}
where $\delta_{\max}$ and $\delta_{\min}$ are defined in (<ref>).
Thanks to (<ref>) in Lemma <ref> and (<ref>), we have
\begin{equation}\label{eq:W12p}
\|\phi\|_{W^{2,1}_{\Lam'}(Q_T)} + \norm{\pa_{\eta}\phi}_{\LS{\Lam'}} + \|\phi\|_{W^{2,1}_{(\Lam+\vk)'}(M_T)} + \norm{\phi(0)}_{\LO{\Lam'}} + \norm{\phi(0)}_{\LM{(\Lam+\vk)'}}\le C_{\Lam',T}.
\end{equation}
In particular, thanks to Lemma <ref>,
\begin{equation*}
\norm{\Delta_M \phi}_{\LS{(\Lam+\vk)'}} \leq \frac{C_{\mr,(\Lam+\vk)'}^M}{\delta} = \frac{2C_{\mr,(\Lam+\vk)'}^M}{\delta_{\max}+\delta_{\min}}.
\end{equation*}
By integration by parts, we have
\begin{align}
0 &= -\sumi\intQT a_iu_i(\pa_t\phi + \Delta\phi)\nonumber\\
&=\sumi\intO a_iu_{i,0}\phi(0) - \sumi \intMT a_id_iu_i\pa_{\eta}\phi + \sumi \intMT a_i\phi d_i\pa_{\eta}u_i\nonumber\\
&\quad + \sumi \intQT a_i\phi(\pa_t u_i - d_i\Delta u_i) + \sumi \intQT(d_i-1)a_i u_i\Delta \phi\nonumber\\
&\leq C\sumi \|u_{i,0}\|_{\LO{\Lam}}\norm{\phi(0)}_{\LO{\Lam'}} + C\sumi \norm{u_i}_{\LS{\Lam}}\norm{\pa_{\eta}\phi}_{\LS{\Lam'}}\nonumber\\
&\quad + \intMT \phi \bra{\sumi a_iG_i(u,v)} + \intQT \phi \sumi a_iF_i(u) + C\sumi\norm{u_i}_{\LQ{\Lam}}\norm{\Delta\phi}_{\LQ{\Lam'}}\nonumber\\
&\leq C_T\sumi\bra{ \norm{u_{i,0}}_{\LO{\Lam}} +\norm{u_i}_{\LQ{\Lam}}+ \norm{u_i}_{\LS{\Lam}}}\nonumber\\
&\quad + L\intQT \phi\bra{\sumi u_i+1} + \intMT \phi\bra{\sumi a_iG_i(u,v)}\nonumber\\
&\leq C_T\sumi\bra{1+ \norm{u_{i,0}}_{\LO{\Lam}} +\norm{u_i}_{\LQ{\Lam}}+ \norm{u_i}_{\LS{\Lam}}} + \intMT \phi\bra{\sumi a_iG_i(u,v)}.\label{est1}
\end{align}
On the other hand, we have
\begin{align}
&\intMT \bra{\sumj b_jv_j}\psi\nonumber\\
&= -\sumj \intMT b_jv_j(\pa_t\phi + \delta\Delta_M \phi)\nonumber\\
&= \sumj \intM v_{j,0}\phi(0) +\sumj\intMT b_j\phi(\pa_tv_j - \delta_j\Delta_Mv_j) + \sumj\intMT b_j(\delta_j - \delta)v_j\Delta_M\phi\nonumber\\
&\leq \sumj\norm{v_{j,0}}_{\LM{\Lam+\vk}}\norm{\phi(0)}_{\LM{(\Lam+\vk)'}} + \sumj\intMT \phi b_jH_j(u,v)\nonumber\\
&\quad + \frac{\delta_{\max}-\delta_{\min}}{2}\norm{\sumj b_jv_j}_{\LS{\Lam+\vk}}\norm{\Delta_M\phi}_{\LS{(\Lam+\vk)'}}\nonumber\\
&\leq C_{(\Lam+\vk)',T}\sumj\norm{v_{j,0}}_{L^{\Lam+\vk}(M)} + \intMT \phi \bra{ \sumj b_jH_j(u,v)}\nonumber\\
&\quad + \frac{\delta_{\max}-\delta_{\min}}{\delta_{\max}+\delta_{\min}}C_{\mr,(\Lam+\vk)'}^M\norm{\sumj b_jv_j}_{\LS{\Lam+\vk}}.\label{est2}
\end{align}
By summing (<ref>) and (<ref>) and using the second inequality (<ref>), we have
\begin{equation}\label{est3}
\begin{aligned}
&\intMT \bra{\sumj b_jv_j}\psi\\
&\leq C_T\sumi\bra{1+\norm{u_{i,0}}_{\LO{\Lam}}+\norm{u_i}_{\LQ{\Lam}}+\norm{u_i}_{\LS{\Lam}}} + C_T\sumj\|v_{j,0}\|_{\LM{\Lam+\vk}}\\
&\quad + \intMT\phi\bra{\sumi u_i + \sumj v_j+1}+ \frac{\delta_{\max}-\delta_{\min}}{\delta_{\max}+\delta_{\min}}C_{\mr,(\Lam+\vk)'}^M\norm{\sumj b_jv_j}_{\LS{\Lam+\vk}}.
\end{aligned}
\end{equation}
We show now that by choosing $\vk$ small enough, we have $\norm{\phi}_{\LS{\Lam'}} \leq C_T$. Indeed, from the boundary bound in (<ref>) we can use the embedding theorem to have
\begin{equation*}
\norm{\phi}_{\LS{h(\vk)}} \leq C_{T}\norm{\phi}_{W^{2,1}_{(\Lam+\vk)'}(M_T)} \leq C_T
\end{equation*}
where $h(\vk) = \frac{(n+1)(\Lam+\vk)'}{n+1-2(\Lam+\vk)'}$ when $(\Lam+\vk)' < \frac{n+1}{2}$, and $h(\vk)$ can be chosen arbitrarily large if $(\Lam+\vk)' \geq \frac{n+1}{2}$. Since $h(\vk)$ is strictly increasing in $\vk$ and $h(0) > \Lam'$, we can choose $\vk \in (0,\vk_0)$ small enough such that
\begin{equation}\label{vk_small}
\Lam' < h(\vk) = \frac{(n+1)(\Lam+\vk)'}{n+1-2(\Lam+\vk)'},
\end{equation}
and consequently,
\begin{equation*}
\norm{\phi}_{\LS{\Lam'}} \leq C_T\norm{\phi}_{\LS{h(\vk)}} \leq C_{T}.
\end{equation*}
Thus, we can estimate
\begin{equation}\label{est4}
\begin{aligned}
&\intMT\phi\bra{\sumi u_i + \sumj v_j + 1}\\
&\leq \bra{C_T+\sumi \norm{u_i}_{\LS{\Lam}}}\norm{\phi}_{\LS{\Lam'}} + \frac{1}{\min_{j=1,\ldots, m_2}b_j}\norm{\phi}_{\LS{\Lam'}}\norm{\sumj b_jv_j}_{\LS{\Lam}}\\
&\leq C_T\bra{1+\sumi \norm{u_i}_{\LS{\Lam}} + \norm{\sumj b_jv_j}_{\LS{\Lam}}}.
\end{aligned}
\end{equation}
Thanks to the interpolation inequality and the $L^1$-bound in Lemma <ref> we have
\begin{equation}\label{est5}
\norm{\sumj b_jv_j}_{\LS{\Lam}} \leq \norm{\sumj b_jv_j}_{\LS{1}}^{\beta}\norm{\sumj b_jv_j}_{\LS{\Lam+\vk}}^{1-\beta} \leq C_{T,\beta}\norm{\sumj b_jv_j}_{\LS{\Lam+\vk}}^{1-\beta}
\end{equation}
where $\beta\in (0,1)$ satisfies $\frac{1}{\Lam} = \frac{\beta}{1} + \frac{1-\beta}{\Lam+\vk}$. Inserting (<ref>) and (<ref>) into (<ref>), and applying (<ref>) we have for any $\eps >0$
\begin{equation*}
\begin{aligned}
\intMT \bra{\sumj b_jv_j}\psi &\leq C_{T,\eps}\sumi\bra{1+\norm{u_{i,0}}_{\LO{\Lam}}} + C_T\sumj\|v_{j,0}\|_{\LM{\Lam+\vk}} +\eps\norm{\sumj b_jv_j}_{\LS{\Lam+\vk}}\\
&\quad + \frac{\delta_{\max}-\delta_{\min}}{\delta_{\max}+\delta_{\min}}C_{\mr,(\Lam+\vk)'}^M\norm{\sumj b_jv_j}_{\LM{\Lam+\vk}}+ C_{T,\beta}\norm{\sumj b_jv_j}_{\LM{\Lam+\vk}}^{1-\beta}.
\end{aligned}
\end{equation*}
By Young's inequality we have
\begin{equation*}
C_{T,\beta}\norm{\sumj b_jv_j}_{\LM{\Lam+\vk}}^{1-\beta} \leq C_{T,\eps,\beta} + \eps\norm{\sumj b_jv_j}_{\LS{\Lam+\vk}},
\end{equation*}
and consequently,
\begin{equation*}
\begin{aligned}
\intMT\bra{\sumj b_jv_j}\psi\leq C_{T,\eps,\beta} + \bra{2\eps+ \frac{\delta_{\max}-\delta_{\min}}{\delta_{\max}+\delta_{\min}}C_{\mr,(\Lam+\vk)'}^M}\norm{\sumj b_jv_j}_{\LS{\Lam+\vk}}.
\end{aligned}
\end{equation*}
Since $0\leq \psi\in \LS{(\Lam+\vk)'}$ with $\norm{\psi}_{\LS{(\Lam+\vk)'}} = 1$ arbitrary, we obtain by duality
\begin{equation*}
\begin{aligned}
\norm{\sumj b_jv_j}_{\LS{\Lam+\vk}} \leq C_{T,\eps,\beta} + \bra{2\eps+ \frac{\delta_{\max}-\delta_{\min}}{\delta_{\max}+\delta_{\min}}C_{\mr,(\Lam+\vk)'}^M}\norm{\sumj b_jv_j}_{\LS{\Lam+\vk}}.
\end{aligned}
\end{equation*}
Now thanks to Lemma <ref> we can choose $\eps$ small enough such that, for all $\vk\in (0,\vk_0)$ verifying (<ref>),
\begin{equation*}
2\eps+ \frac{\delta_{\max}-\delta_{\min}}{\delta_{\max}+\delta_{\min}}C_{\mr,(\Lam+\vk)'}^M < 1,
\end{equation*}
and finally obtain
\begin{equation*}
\norm{\sumj b_jv_j}_{\LS{\Lam+\vk}} \leq \bra{1-2\eps- \frac{\delta_{\max}-\delta_{\min}}{\delta_{\max}+\delta_{\min}}C_{\mr,(\Lam+\vk)'}^M}^{-1}C_{T,\eps,\beta}.
\end{equation*}
Combining this with (<ref>) we can finish the proof of Proposition <ref>.
§ PROOF OF MAIN RESULTS
§.§ Theorem <ref>: Global existence
Let $\{y_j\}_{j=1,\ldots, m_2}$ be a sequence of non-negative numbers. Assume that there is a constant $K>0$ such that, for any $\eps>0$, there exists $C_\eps>0$ independent of $\{y_i\}$ such that if $k\in \{1,\ldots, m_2\}$ then
\begin{equation*}
y_k \leq C_\eps+ K\sum_{j=1}^{k-1}y_j + \eps\sumj y_j,
\end{equation*}
where if $k=1$, the sum $\sum_{j=1}^{k-1}y_j$ is neglected. Then there exists a constant $C$ independent of the sequence $\{y_j\}$ such that
\begin{equation}\label{bound_elementary}
\sumj y_j \leq C.
\end{equation}
We prove by induction that for each $k\in \{1,\ldots, m_2\}$, there exists a constant $B_k, R_k>0$ independent of $\eps$ such that
\begin{equation}\label{induction}
y_k \leq B_k + R_k\eps \sumj y_j.
\end{equation}
With $k=1$, (<ref>) follows for $B_1 = C_\eps$ and $R_1 = 1$. Assume that this is true for all $j=1,\ldots, k-1$. Then
\begin{align*}
y_k \leq C_\eps+K\sum_{j=1}^{k-1}y_j + \eps \sumj y_j \leq C_\eps + K\sum_{j=1}^{k-1}\bra{B_j + R_j\eps\sumj y_j} + \eps\sumj y_j \\
\leq \underbrace{\sbra{C_\eps+K\sum_{j=1}^{k-1}B_j}}_{=: B_k} + \underbrace{\sbra{K\sum_{j=1}^{k-1}R_j + 1}}_{=:R_k}\eps\sumj y_j
\end{align*}
which proves (<ref>). Now summing (<ref>) for $k=1,\ldots, m_2$ gives
\begin{equation*}
\sum_{k=1}^{m_2}y_k \leq \sum_{k=1}^{m_2}B_k + \bra{\sum_{k=1}^{m_2}R_k}\eps\sumj y_j,
\end{equation*}
and consequently, by choosing $\eps = \frac 12\bra{\sum_{k=1}^{m_2}R_k}^{-1}$, we obtain the desired bound (<ref>).
For any $p>\Lam+\vk$, any $\eps>0$ and any $k\in \{1,\ldots, m_2\}$, there exists a constant $C_{T,\eps}$ depending on $T$ and $\eps$ such that
\begin{equation}\label{inductive_estimate}
\|v_k\|_{\LS{p}} \leq C_{T,\eps} + C_T\sum_{j=1}^{k-1}\|v_j\|_{\LS{p}} + \eps \sumj \|v_j\|_{\LS{p}}.
\end{equation}
Naturally, when $k = 1$, the first sum on the right-hand side is neglected.
It's enough to show (<ref>) for $p$ large enough. Let $0\le \psi \in \LS{p'}$ with $\|\psi\|_{\LS{p'}} = 1$, and $\phi$ be the solution to (<ref>) with $\delta = \delta_k$. Recall that from the assumption (<ref>) we have for any $k = 1,\ldots, m_2$
\begin{equation*}
\sumi a_{(k+m_1)i}G_i(u,v) + \sum_{j=1}^k a_{(k+m_1)(j+m_1)}H_j(u,v) \leq L_2\sbra{\sumi u_i^{\mM} + \sumj v_j^{\mM} + 1}
\end{equation*}
which implies, recalling $a_{kk} = 1$ for all $k=1,\ldots, m_1+m_2$,
\begin{equation}\label{a1}
\begin{aligned}
& \leq -\sumi a_{(k+m_1)i}G_i(u,v) - \sum_{j=1}^{k-1} a_{(k+m_1)(j+m_1)}H_j(u,v)\\
&\quad + L_2\sbra{\sumi u_i^{\mM} + \sumj v_j^{\mM} + 1}.
\end{aligned}
\end{equation}
By integration by parts we have
\begin{equation}\label{a2}
\begin{aligned}
\intMT v_k\psi &= -\intMT v_k(\partial_t \phi + \delta_k \Delta_M \phi)\\
&= \intM v_{k,0}\phi(0) + \intMT \phi (\partial_t v_k - \delta_k \Delta_Mv_k)\\
&= \intM v_{k,0}\phi(0) + \intMT \phi H_k(u,v)\\
&=: (A) + (B).
\end{aligned}
\end{equation}
From Lemma <ref> we have
\begin{equation}\label{A}
|(A)| \leq C\|v_{k,0}\|_{\LM{p}}\|\phi(0)\|_{\LM{p'}} \leq C\|v_{k,0}\|_{\LM{p}}.
\end{equation}
To estimate $(B)$ we use $\phi\geq 0$ and (<ref>) to have
\begin{equation}\label{B}
\begin{aligned}
(B) &\leq -\sumi \intMT a_{(k+m_1)i}G_i(u,v)\phi - \sum_{j=1}^{k-1}\intMT a_{(k+m_1)(j+m_1)}H_j(u,v)\phi\\
&\quad + L_2\intMT \phi\sbra{\sumi u_i^{\mM} + \sumj v_j^{\mM} + 1 }\\
&=: (B1) + (B2) + (B3).
\end{aligned}
\end{equation}
Estimate of $(B1)$. From the equation (<ref>) we have
\begin{align*}
% \begin{aligned}
(B1) & = -\sumi \intMT a_{(k+m_1)i}G_i(u,v)\phi\\
& = -\sumi \intMT a_{(k+m_1)i}(d_i\partial_\eta u_i) \phi\\
&= -\sumi a_{(k+m_1)i}\sbra{\intQT d_i\Delta u_i \phi + \intMT d_iu_i\partial_{\eta}\phi - \intQT d_iu_i\Delta \phi}\\
&= -\sumi a_{(k+m_1)i}\sbra{\intQT \sbra{\partial_t u_i - F_i(u)} \phi + \intMT d_iu_i\partial_{\eta}\phi - \intQT d_iu_i\Delta \phi}\\
&= \sumi a_{(k+m_1)i}\intO u_{i,0}\phi(0) + \intQT \phi\sbra{\sumi a_{(k+m_1)i}F_i(u)}\\
&\quad + \sumi a_{(k+m_1)i}d_i\intMT u_i\partial_{\eta}\phi + \sumi a_{(k+m_1)i}\intQT u_i\sbra{\partial_t \phi + d_i\Delta \phi}\\
&=: (B11) + (B12) + (B13) + (B14).
% \end{aligned}
\end{align*}
By using Lemma <ref> we have thanks to Hölder's inequality
\begin{equation}\label{B11}
|(B11)| \leq C\sumi\|u_{i,0}\|_{\LO{p}}\|\phi(0)\|_{\LO{p'}} \leq C_T\sumi\|u_{i,0}\|_{\LO{p}}.
\end{equation}
From (<ref>) we have
\begin{equation*}
\sumi a_{(k+m_1)i}F_i(u) \leq L_2\sbra{\sumi u_i^{\pO} + 1}.
\end{equation*}
Therefore, by Hölder's inequality,
\begin{equation}\label{B12r}
\begin{aligned}
|(B12)| \leq L_2\intQT \phi \sbra{\sumi u_i^{\pO} + 1} \leq L_2\sumi \intQT \phi u_i^{\pO} + C_T\|\phi\|_{\LQ{p'}}.
\end{aligned}
\end{equation}
From Lemma <ref> we have
\begin{equation}\label{ff3}
\|\phi\|_{\LQ{(p')^{\dag}}} \leq C_T
\end{equation}
with $(p')^{\dag}$ defined similarly to $q^{\dag}$ in (<ref>). For any $\beta\in (0,1)$, it follows from Hölder's inequality that
\begin{equation}\label{c5}
\intQT \phi u_i^{\pO} \leq \bra{\intQT \phi ^{(p')^{\dag}}}^{\frac{1}{(p')^\dag}}\bra{\intQT u_i^{(p_\Omega-\beta)s}}^{\frac 1s}\bra{\intQT u_i^p}^{\frac{\beta}{p}}
\end{equation}
\begin{equation}\label{Sigma}
\frac{1}{(p')^\dag} + \frac 1s + \frac{\beta}{p} = 1.
\end{equation}
This implies
\begin{equation*}
s = \frac{(n+2)p}{(n+2)(p-\beta) + 2p - (n+1)(p-1)}.
\end{equation*}
Therefore, from (<ref>) and the fact that $\Lam\geq 2$, we can always choose $\beta\in (0,1)$ and $p$ large enough such that
\begin{equation*}
(\pO - \beta)s < \Lam+\vk,
\end{equation*}
and consequently
\begin{equation*}
\|u_i\|_{\LQ{(\pO-\beta)s}} \leq C_T\|u_i\|_{\LQ{\Lam+\vk}} \leq C_T.
\end{equation*}
Thus it follows from (<ref>) and (<ref>) that
\begin{equation}\label{c6}
\intQT \phi u_i^{\pO} \leq \|\phi\|_{\LQ{(p')^\dag}}\|u_i\|_{\LQ{(p_\Omega-\beta)s}}^{\pO-\beta}\|u_i\|_{\LQ{p}}^{\beta} \leq C_T\|u_i\|_{\LQ{p}}^{\beta}.
\end{equation}
Therefore, from (<ref>) we get the estimate for $(B12)$,
\begin{equation}\label{B12}
|(B12)| \leq C_{T} + \sumi\|u_i\|_{\LQ{p}}^{\beta} \leq C_T + \sumi \|u_i\|_{\LQ{p}},
\end{equation}
since $\beta\in (0,1)$. Due to Hölder's inequality, Lemmas <ref> and <ref> we have
\begin{equation}\label{B13}
\begin{aligned}
|(B13)| \leq C\sumi\|u_i\|_{\LS{p}}\|\partial_{\eta}\phi\|_{\LS{p'}}\leq C_T\sumi\|u_i\|_{\LS{p}}.
\end{aligned}
\end{equation}
Finally, since $\partial_t \phi + \Delta \phi = 0$ in $Q_T$ we estimate $(B14)$ as
\begin{equation}\label{B14}
\begin{aligned}
|(B14)| \leq \sumi a_{(k+m_1)i}|d_i -1|\intQT |u_i||\Delta \phi| &\leq C\sumi \|u_i\|_{\LQ{p}}\|\Delta\phi\|_{\LQ{p'}}\\
& \leq C\sumi \|u_i\|_{\LQ{p}}.
\end{aligned}
\end{equation}
From (<ref>), (<ref>), (<ref>), and (<ref>) we obtain the estimate for $(B1)$ as
\begin{equation}\label{B1}
|(B1)| \leq C_{T} + \sumi\bra{\|u_i\|_{\LQ{p}}+\|u_i\|_{\LS{p}}}.
\end{equation}
Estimate of $(B2)$. Since $\partial_tv_j - \delta_j\Delta_Mv_j = H_j(u,v)$, we have
\begin{equation}\label{B2}
\begin{aligned}
|(B2)| &= \abs{\sum_{j=1}^{k-1}a_{(k+m_1)(j+m_1)}\intMT(\partial_tv_j - \delta_j\Delta_Mv_j)\phi}\\
&= \abs{\sum_{j=1}^{k-1}a_{(k+m_1)(j+m_1)}\sbra{\intM v_{j,0}\phi(0) - \intMT v_j(\partial_t\phi + \delta_j \Delta_M\phi)}}\\
&\leq C\sum_{j=1}^{k-1}\biggl[\|v_{j,0}\|_{\LM{p}}\|\phi(0)\|_{\LM{p'}}\\
&\qquad\qquad\quad + \|v_j\|_{\LS{p}}\bra{|\delta_j-\delta_k|\|\Delta_M\phi\|_{\LS{p'}} + \|\psi\|_{\LS{p'}}}\biggr]\\
&\leq C\sum_{j=1}^{k-1}\bra{\|v_{j,0}\|_{\LM{p}} + \|v_j\|_{\LS{p}}}
\end{aligned}
\end{equation}
where we used Lemma <ref> at the last step.
Estimate of $(B3)$.
We split $(B3)$ into three parts, as
\begin{equation*}
(B3) = L_2\sumi \intMT \phi u_i^{\mM} + L_2\sumj \intMT \phi v_j^{\mM} + L_2\intMT \phi =: (B31) + (B32) + (B33).
\end{equation*}
The term $(B33)$ can be estimated directly as
\begin{equation}\label{B33}
|(B33)| \leq C_T\|\phi\|_{\LS{p'}} \leq C_T.
\end{equation}
For any $\alpha \in (0,1)$ we use Hölder's inequality to estimate
\begin{equation}\label{ff1}
\begin{aligned}
\intMT \phi u_i^{\mM} = \intMT \phi u_i^{\mM-\alpha}u_i^{\alpha}\leq \bra{\intMT \phi^{(p')^*}}^{\frac{1}{(p')^*}}\bra{\intMT u_i^{(\mM-\alpha)r}}^{\frac{1}{r}}\bra{\intMT u_i^{p}}^{\frac{\alpha}{p}}
\end{aligned}
\end{equation}
where $(p')^*$ is defined similarly as $q^*$ in (<ref>), and
\begin{equation}\label{ff11}
\frac{1}{(p')^*} + \frac{1}{r} + \frac{\alpha}{p} = 1.
\end{equation}
It follows that
\begin{equation}\label{ff12}
r = \frac{(n+1)p}{(n+1)(1-\alpha)+2p}.
\end{equation}
We now choose $\alpha \in (0,1)$ such that
\begin{equation}\label{alpha}
1-\alpha < \frac{2\vk}{(n+1)\sbra{1-\frac{\Lam+\vk}{p}}}.
\end{equation}
Combining this with (<ref>) gives
\begin{equation*}
\mM - \alpha < \frac{(\Lam+\vk)\sbra{(n+1)(1-\alpha) + 2p}}{(n+1)p},
\end{equation*}
and consequently
\begin{equation*}
(\mM - \alpha)r < \Lam+\vk.
\end{equation*}
Therefore, it follows from (<ref>) that
\begin{equation*}
\intMT \phi u_i^{\mM} \leq \|\phi\|_{\LS{(p')^*}}\|u_i\|_{\LS{(\mM-\alpha)r}}^{\mM-\alpha}\|u_i\|_{\LS{p}}^{\alpha} \leq C_T\|u_i\|_{\LS{p}}^{\alpha}
\end{equation*}
thanks to (<ref>) and $\|u_i\|_{\LS{(\mM-\alpha)r}} \leq C_T\|u_i\|_{\LS{\Lam+\vk}} \leq C_T$. Therefore, we have
\begin{equation}\label{B31}
|(B31)| \leq C_T\sumi\|u_i\|_{\LS{p}}^{\alpha} \leq C_T + \sumi \|u_i\|_{\LS{p}}.
\end{equation}
In the same way we can estimate
\begin{equation}\label{B32}
|(B32)| \leq C_T\sumj\|v_j\|_{\LS{p}}^{\alpha}.
\end{equation}
From (<ref>), (<ref>), and (<ref>) we obtain
\begin{equation}\label{B3}
|(B3)| \leq C_{T} + \sumi \|u_i\|_{\LS{p}}+ \sumj\|v_j\|_{\LS{p}}^{\alpha},
\end{equation}
with $\alpha \in (0,1)$ satisfying (<ref>).
From the estimates of $(B1), (B2), (B3)$ in (<ref>), (<ref>), (<ref>), we get
\begin{equation*}
|(B)| \leq C_T + C_T\bra{\sum_{j=1}^{k-1}\|v_j\|_{\LS{p}} + \sumj\|v_j\|_{\LS{p}}^{\alpha} + \sumi\bra{\|u_i\|_{\LQ{p}} + \|u_i\|_{\LS{p}}}}.
\end{equation*}
By using estimate (<ref>) and Young's inequality, we find
\begin{equation*}
|(B)| \leq C_T + C_T\bra{\sum_{j=1}^{k-1}\|v_j\|_{\LS{p}} + \eps\sumj\|v_j\|_{\LS{p}}}.
\end{equation*}
Together with (<ref>), we get finally from (<ref>),
\begin{equation*}
\intMT v_k\psi \leq C_{T} + C_T\bra{\sum_{j=1}^{k-1}\|v_j\|_{\LS{p}} + \eps\sumj\|v_j\|_{\LS{p}}},
\end{equation*}
which yields the desired estimate (<ref>) thanks to duality.
By combining Lemmas <ref>, <ref>, and <ref> we get
For any $2\leq p \in \mathbb Z$, there exists a constant $C_{T,p}$ such that
\begin{equation*}
\sumi\bra{\|u_i\|_{\LQ{p}} + \|u_i\|_{\LS{p}}} + \sumj\|v_j\|_{\LS{p}} \leq C_{T,p}.
\end{equation*}
From Lemmas <ref> and <ref> we obtain
\begin{equation*}
\sumj\|v_j\|_{\LS{p}} \leq C_{T,p}.
\end{equation*}
This and Lemma <ref> imply the desired estimate.
As a consequence, we obtain the global existence of (<ref>) in Theorem <ref> part (i).
Let $p>n$.
For any non-negative initial data $(u_0,v_0) \in (W^{2-2/p}(\Omega))^{m_1}\times (W^{2-2/p}(M))^{m_2}$ satisfying the compatibility condition (<ref>), the system (<ref>) has a unique global strong solution in all dimensions.
From Lemma <ref> and (<ref>), the nonlinearities $F_i(u)$ are bounded in $\LQ{p}$, and $G_i(u,v), H_j(u,v)$ are bounded in $\LS{p}$, for all $p\geq 1$. It follows that
\begin{equation*}
\begin{cases}
\pa_t u_i - d_i\Delta u_i = F_i(u)\in \LQ{p},\\
d_i\pa_{\eta}u_i = G_i(u,v)\in \LS{p},
\end{cases}
\end{equation*}
for any $p\geq 1$. By the regularizing effect of linear parabolic equations with inhomogeneous boundary conditions, cf. <cit.>, it follows that $\|u_i\|_{\LQ{\infty}}$ is bounded. Similarly, $\|v_j\|_{\LS{\infty}}$ is bounded. This implies that the solution is bounded in $L^{\infty}$, which implies the desired global existence.
§.§ Theorem <ref>: Uniform-in-time bounds
To complete the proof of Theorem <ref>, it remains to show the uniform-in-time bound of the solution. To this end, we study the system (<ref>) on each cylinder $Q_{\tau,\tau+1} = \Omega\times(\tau,\tau+1)$, $\tau\in \mathbb N$.
For the rest of this section, all constants are independent of $\tau$ unless otherwise stated. We also consider an increasing, smooth function $\varphi\in C^\infty(\R;[0,1])$ such that $\varphi(s) = 0$ for $s\in (-\infty,0]$, $\varphi(s) = 1$ for $s\geq 1$, and its shifted version $\varphi_\tau(\cdot) = \varphi(\cdot - \tau)$. By multiplying the system (<ref>) by $\vat$ we have the truncated system
\begin{equation}\label{shifted_sys}
\begin{cases}
\pa_t(\vat u_i) = d_i\Delta(\vat u_i) + \vat' u_i + \vat F_i(u), &(x,t)\in Q_{\tau,\tau+2},\; i=1,\ldots, m_1,\\
d_i\pa_\eta(\vat u_i) = \vat G_i(u,v), &(x,t)\in M_{\tau,\tau+2},\; i=1,\ldots, m_1,\\
\pa_t(\vat v_j) = \delta_j\Delta_M(\vat v_j) + \vat' v_j + \vat H_j(u,v), &(x,t)\in M_{\tau,\tau+2}, \; j = 1,\ldots, m_2,
\end{cases}
\end{equation}
with zero initial data
\begin{equation}\label{zero_initial}
\begin{cases}
(\vat u_i)(x,\tau) = 0, & x\in\Omega,\; i=1,\ldots, m_1,\\
(\vat v_j)(x,\tau) = 0, & x\in M, \; j=1,\ldots, m_2.
\end{cases}
\end{equation}
If $L< 0$ or $L=K=0$ in (<ref>), then
\begin{equation*}
\sup_{t\ge 0}\sumi\|u_i(t)\|_{\LO{1}} + \sup_{t\ge 0}\sumj\|v_j(t)\|_{\LM{1}} + \sup_{\tau\in\mathbb N}\sumi\|u_i\|_{\LStau{1}} \leq C.
\end{equation*}
From (<ref>) it follows that
\begin{equation*}
\frac{d}{dt}\bra{\sumi\intO a_iu_i + \sumj\intM b_jv_j} \leq L\bra{\sumi \intO u_i + \sumj \intM v_j} + K.
\end{equation*}
If $L = K = 0$, it yields directly by integrating on $(0,t)$ that
\begin{equation}\label{b1}
\sup_{t\ge 0}\sumi\|u_i(t)\|_{\LO{1}} + \sup_{t\ge 0}\sumj\|v_j(t)\|_{\LM{1}} \leq C.
\end{equation}
If $L<0$, there exists $\sigma>0$ such that
\begin{equation*}
\frac{d}{dt}\bra{\sumi\intO a_iu_i + \sumj\intM b_jv_j} \leq -\sigma\bra{\sumi\intO a_iu_i + \sumj\intM b_jv_j} + K,
\end{equation*}
which also implies (<ref>) thanks to Gronwall's lemma. Let $\phi\in C^{2,1}(\bar{\Omega}\times[\tau,\tau+2])$ be a nonnegative function such that $\phi_t +\Delta \phi = 0$ on $Q_{\tau,\tau+2}$, $\pa_{\eta}\phi = 1$ on $M_{\tau,\tau+2}$ and $\phi(\cdot,\tau+2) = 0$ in $\bar \Omega$. Define $\theta = -\pa_t\phi - \Delta_M\phi$ on $M\times(\tau,\tau+2)$. By integration by parts,
\begin{equation}\label{d1}
\begin{aligned}
\intMtautwo a_id_i(\vat u_i) &= \intMtautwo a_id_i(\vat u_i)\pa_{\eta}\phi\\
&= \intMtautwo \vat \phi \cdot a_iG_i(u,v) + a_i\intQtautwo \phi\vat' u_i\\
&\quad + \intQtautwo \vat \phi \cdot a_iF_i(u) + \intQtautwo a_i(\vat u_i)(d_i-1)\Delta \phi,
\end{aligned}
\end{equation}
\begin{equation}\label{d2}
\begin{aligned}
\intMtautwo b_j(\vat v_j)\theta = \intMtautwo\sbra{\vat\phi\cdot b_jH_j(u,v) + \vat' v_j\phi + b_jv_j(\delta_j - 1)\Delta_M\phi}.
\end{aligned}
\end{equation}
By summing (<ref>) in $i=1,\ldots, m_1$, summing (<ref>) in $j=1,\ldots, m_2$, and adding the resultants we can apply (<ref>) (recalling $L=0$) to get
\begin{equation*}
\begin{aligned}
& \sumi \intMtautwo a_id_i(\vat u_i) + \sumj \intMtautwo b_j(\vat v_j)\theta\\
& \leq \sumi \sbra{\intQtautwo \phi \vat' u_i + \intQtautwo a_i(\vat u_i)(d_i-1)\Delta \phi} + K\bra{\intQtautwo \vat \phi + \intMtautwo \vat \phi}\\
&\quad + \sumj \sbra{\intMtautwo \phi \vat' u_i + \intMtautwo b_j(\vat v_j)(\delta_j-1)\Delta_M\phi}.
\end{aligned}
\end{equation*}
Thanks to the fact that $\theta \in \LStaut{\infty}$, $\sup_{t\geq 0}\|u_i(t)\|_{\LO{1}} \leq C$ and $\sup_{t\geq 0}\|v_j(t)\|_{\LM{1}} \leq C$, $\phi \in C^{2,1}(\bar \Omega\times[\tau,\tau+2])$, we conclude that
\begin{equation*}
\sumi \intMtautwo \vat u_i \leq C \quad \text{ for all } \quad \tau\in \mathbb N.
\end{equation*}
Since $\vat \ge 0$ and $\vat|_{[\tau+1,\tau+2]} = 1$, we get finally
\begin{equation*}
\sup_{\tau\in \mathbb N}\sumi\|u_i\|_{\LStau{1}} \leq C.
\end{equation*}
If $L<0$ or $L=K=0$ in (<ref>), then for any positive integer $p\ge 2$, and any $\eps>0$, there exists $K_{p,\eps}>0$ such that
\begin{equation}\label{d3_1}
\begin{aligned}
\sumi \bra{\intQtautwo (\vat u_i)^{p-1+\pO} + \intMtautwo (\vat u_i)^{p-1+\pM}}\\
\leq K_{p,\eps} + \eps\sbra{\sumi\bra{\intQtautwo u_i^{p-1+\pM} + \intMtautwo u_i^{p-1+\pM}}+ \sumj \intMtautwo v_j^{p-1+\pM}}.
\end{aligned}
\end{equation}
As a consequence, for any $1<p<\infty$ and any $\eps>0$, there exists $K_{p,\eps}>0$ such that
\begin{equation}\label{ff6}
\begin{aligned}
\sumi\bra{\|\varphi_\tau u_i\|_{\LQtaut{p}} + \|\varphi_\tau u_i\|_{\LStaut{p}}}\\
\leq K_{p,\eps} + \eps\sbra{\sumi\bra{\|u_i\|_{\LQtaut{p}} + \|u_i\|_{\LStaut{p}}} + \sumj\|v_j\|_{\LStaut{p}}}.
\end{aligned}
\end{equation}
Recall the Lyapunov-like function $\L_p[u]$ in (<ref>), with $\theta$ is chosen in (<ref>) and (<ref>). Thanks to (<ref>),
\begin{equation*}
\begin{aligned}
\bra{\L_p[u]}'(t) + C\sumi \bra{\intO u_i^{p-1+\pO} + \intM u_i^{p-1+\pM}}\leq K_{p,\theta}\sbra{1 + \eps\sumj\intM v_j^{p-1+\pM}}.
\end{aligned}
\end{equation*}
Therefore, we have
\begin{equation}\label{d4}
\begin{aligned}
\bra{\vat\L_p[u]}' + C\sumi \bra{\intO \vat u_i^{p-1+\pO} + \intM \vat u_i^{p-1+\pM}}\\
\leq \vat' \L_p(t) + K_{p,\theta,\eps}\vat + \eps\sumj\intM \vat v_j^{p-1+\pM}.
\end{aligned}
\end{equation}
Since $0\leq \vat \leq 1$,
\begin{equation*}
\vat u_i^{p-1+\pO} \ge (\vat u_i)^{p-1+\pO}, \quad \vat u_i^{p-1+\pM} \ge (\vat u_i)^{p-1+\pM}.
\end{equation*}
From (<ref>) and $0\leq \vat' \leq C$, it follows that
\begin{equation*}
\abs{\vat' \L_p[u](t)} \leq C\sumi \intO u_i^p \leq C_\eps + \eps\sumi \intO u_i^{p-1+\pM}.
\end{equation*}
Putting all these into (<ref>) and integrating the resultant on $(\tau,\tau+2)$, noticing that $\vat(\tau) = 0$, we obtain
\begin{equation*}
\begin{aligned}
\intQtautwo (\vat u_i)^{p-1+\pO} + \intMtautwo (\vat u_i)^{p-1+\pM}\\
\leq K_{p,\theta,\eps} + \eps\sbra{\sumi\bra{\intQtautwo u_i^{p-1+\pM} + \intMtautwo u_i^{p-1+\pM}} + \sumj\intMtautwo v_j^{p-1+\pM}},
\end{aligned}
\end{equation*}
which is the desired estimate (<ref>). From this, (<ref>) can be obtained similarly to the last step of Lemma <ref>.
We need the following elementary result whose proof is straightforward.
Let $\{y_n\}_{n\ge 0}$ be a nonnegative sequence and $\mathscr N = \{n\in \mathbb N: y_{n-1}\leq y_n \}$. If there exists $K>0$ (independent of $n$) such that
\begin{equation*}
y_n \leq K \quad \text{ for all } \quad n\in \mathscr N,
\end{equation*}
\begin{equation*}
y_n \leq \max\{y_0, K\} \quad \text{ for all } \quad n\in \mathbb N.
\end{equation*}
There exist constants $C>0$ and $\gamma>0$ such that for all $\tau\in \mathbb N$,
\begin{equation}\label{desired1}
\sumi\bra{\|u_i\|_{\LQtau{\Lam+\gamma}}+\|u_i\|_{\LStau{\Lam+\gamma}}} + \sumj\|v_j\|_{\LStau{\Lam+\gamma}} \leq C.
\end{equation}
As in Lemma <ref>, we choose $\vk>0$ small enough such that (<ref>) holds. Let $0\leq \psi \in \LStaut{(\Lam+\vk)'}$ with $\|\psi\|_{\LStaut{(\Lam+\vk)'}} = 1$, and let $\phi$ be the solution to (<ref>) with $T = \tau+2$ and
\begin{equation*}
\delta = \frac{\delta_{\max}+\delta_{\min}}{2}
\end{equation*}
with $\delta_{\max}$ and $\delta_{\min}$ are in (<ref>). From Propositions <ref> and (<ref>), we have
\begin{equation}\label{e1}
\|\phi\|_{W^{2,1}_{\Lam'}(Q_{\tau,\tau+2})} + \|\pa_\eta\phi\|_{\LStaut{\Lam'}} + \|\phi\|_{W^{2,1}_{(\Lam+\vk)'}(M_{\tau,\tau+2})} \leq C.
\end{equation}
In particular,
\begin{equation}\label{ff7}
\|\Delta_M\phi\|_{\LStaut{(\Lam+\vk)'}} \leq \frac{2C_{\mr,(\Lam+\vk)'}^M}{\delta_{\max}+\delta_{\min}}.
\end{equation}
By integration by parts (see the proof of Lemma <ref>) we have
\begin{equation}\label{d11}
\begin{aligned}
0&= -\sumi \intQtautwo a_i(\vat u_i)(\pa_t\phi + \Delta \phi)\\
&= \sumi\intQtautwo a_i\phi(\vat' u_i + \vat F_i(u)) - \sumi \intMtautwo d_ia_i\vat u_i\pa_{\eta}\phi\\
&\quad + \sumi \intMtautwo \phi \vat a_iG_i(u,v) + \sumi \intQtautwo a_i(d_i-1)\vat u_i\Delta\phi
\end{aligned}
\end{equation}
\begin{equation}\label{d12}
\begin{aligned}
\intMtautwo\bra{\sumj b_j\vat v_j}\psi &=\sumj \intMtautwo b_j\phi \vat' v_j + \sumj \intMtautwo \phi \vat b_jH_j(u,v) \\
&\quad + \sumj \intMtautwo b_j(\delta_j-\delta)(\vat v_j)\Delta_M\phi.
\end{aligned}
\end{equation}
Sum (<ref>) and (<ref>), and note that either $L < 0$ or $L=K=0$ in (<ref>) and (<ref>). We provide the argument in the case when $L=K=0$, and leave the similar case when $L<0$ for the reader. We calculate
\begin{equation}\label{d13}
\begin{aligned}
\intMtautwo\bra{\sumj b_j\vat v_j}\psi &\leq \sumi\intQtautwo \phi a_i\vat' u_i + \sumi \intQtautwo a_i(d_i-1)\vat u_i\Delta \phi\\
&\quad-\sumi\intMtautwo a_id_i\vat u_i\pa_{\eta}\phi + \sumj \intMtautwo b_j\phi \vat' v_j \\
&\quad + \sumj \intMtautwo b_j(\delta_j-\delta)(\vat v_j)\Delta_M\phi\\
&\quad=: (I)+(II)+(III)+(IV)+(V).
\end{aligned}
\end{equation}
We estimate five terms on the right hand side of (<ref>) as follows. We use (<ref>) to estimate, for $\gamma\in (0,1)$ such that $\frac{1}{\Lam}=\frac{\gamma}{1}+\frac{1-\gamma}{\Lam+\vk}$ and any $\eps>0$,
\begin{equation}\label{e3}
\begin{aligned}
|(I)| &\leq C\sumi\|\phi\|_{\LQtaut{\Lam'}}\|u_i\|_{\LQtaut{\Lam}} \leq C\sumi\|u_i\|_{\LQtaut{1}}^{\gamma}\|u_i\|_{\LQtaut{\Lam+\vk}}^{1-\gamma}\\
&\leq C\sumi\|u_i\|_{\LQtaut{\Lam+\vk}}^{1-\gamma} \leq C_\eps + \eps\sumi\|u_i\|_{\LQtaut{\Lam+\vk}}.
\end{aligned}
\end{equation}
\begin{equation}\label{e4}
|(II)| \leq C\sumi\|u_i\|_{\LQtaut{\Lam}}\|\Delta\phi\|_{\LQtaut{\Lam'}} \leq C_\eps + \eps\sumi\|u_i\|_{\LQtaut{\Lam+\vk}},
\end{equation}
\begin{equation}\label{e5}
|(III)| \leq C\sumi \|u_i\|_{\LStaut{\Lam}}\|\pa_\eta\phi\|_{\LStaut{\Lam'}} \leq C_\eps + \eps\sumi \|u_i\|_{\LStaut{\Lam+\vk}},
\end{equation}
\begin{equation}\label{e6}
\begin{aligned}
|(IV)| \leq C\sumj\|\phi\|_{\LStaut{\Lam'}}\|v_j\|_{\LStaut{\Lam}}\leq C_\eps +\eps\sumj \|v_j\|_{\LStaut{\Lam+\vk}}.
\end{aligned}
\end{equation}
For $(V)$ we estimate using (<ref>)
\begin{equation}\label{e7}
\begin{aligned}
|(V)| &\leq \frac{\delta_{\max}-\delta_{\min}}{\delta_{\max}+\delta_{\min}}\intMtautwo\abs{\sumj b_j\vat v_j}|\Delta_M\phi|\\
&\leq \frac{\delta_{\max}-\delta_{\min}}{\delta_{\max}+\delta_{\min}}\norm{\sumj b_j\vat v_j}_{\LStaut{\Lam+\vk}}\|\Delta_M\|_{\LStaut{(\Lam+\vk)'}}\\
&\leq \frac{\delta_{\max}-\delta_{\min}}{\delta_{\max}+\delta_{\min}}C_{\mr,(\Lam+\vk)'}^M\norm{\sumj b_j\vat v_j}_{\LStaut{\Lam+\vk}}.
\end{aligned}
\end{equation}
Using (<ref>)–(<ref>) into (<ref>), it follows from duality, (<ref>) and Lemma <ref>, that
\begin{equation}\label{e9}
\begin{aligned}
&\norm{\sumj b_j\vat v_j}_{\LStaut{\Lam+\vk}}\\
&\leq C_\eps + \eps \sbra{\sumi\bra{\|u_i\|_{\LQtaut{\Lam+\vk}} + \|u_i\|_{\LStaut{\Lam+\vk}}} + \sumj\|v_j\|_{\LStaut{\Lam+\vk}}}.
\end{aligned}
\end{equation}
Combining (<ref>) with with (<ref>) in Lemma <ref> (choosing $p = \Lam+\vk$) we have
\begin{equation}\label{ff8}
\begin{aligned}
&\sumi \bra{\|\vat u_i\|_{\LQtaut{\Lam+\vk}} + \|\vat u_i\|_{\LStaut{\Lam+\vk}}} + \norm{\vat \sumj b_j v_j}_{\LStaut{\Lam+\vk}}\\
&\leq C_\eps +\eps \sbra{\sumi\bra{\|u_i\|_{\LQtaut{\Lam+\vk}} + \|u_i\|_{\LStaut{\Lam+\vk}}} + \sumj\|v_j\|_{\LStaut{\Lam+\vk}}}.
\end{aligned}
\end{equation}
Recall that $\vat\geq 0$ and $\vat|_{[\tau,\tau+1]}\equiv 1$, it follows from (<ref>) that
\begin{equation}\label{ff9}
\begin{aligned}
&\sumi \bra{\|u_i\|_{\LQtau{\Lam+\vk}} + \|u_i\|_{\LStau{\Lam+\vk}}} + \norm{\sumj b_jv_j}_{\LStau{\Lam+\vk}}\\
&\leq C_\eps +C\eps \sbra{\sumi\bra{\|u_i\|_{\LQtaut{\Lam+\vk}} + \|u_i\|_{\LStaut{\Lam+\vk}}} + \norm{\sumj b_jv_j}_{\LStaut{\Lam+\vk}}}.
\end{aligned}
\end{equation}
For $\tau \in \mathbb N$, we define
\begin{equation*}
y_{\tau}:= \sumi\bra{\|u_i\|_{\LQtau{\Lam+\vk}} + \|u_i\|_{\LStau{\Lam+\vk}}} + \norm{\sumj b_jv_j}_{\LStau{\Lam+\vk}}.
\end{equation*}
Inequality (<ref>) implies
\begin{equation}\label{ff10}
y_\tau \leq C + C\eps(y_\tau + y_{\tau+1}).
\end{equation}
Define $\mathscr N = \{\tau \in \mathbb N: y_{\tau}\leq y_{\tau+1}\}$. Then for any $\tau\in \mathscr N$, by choosing $\eps$ sufficiently small, we obtain from (<ref>)
\begin{equation*}
y_\tau \leq C,
\end{equation*}
where $C$ is independent of $\tau$. From Lemma <ref>, we have
\begin{equation*}
y_\tau \leq C \quad \text{ for all } \quad \tau\in \mathbb N,
\end{equation*}
which proves the desired estimate (<ref>).
Assume that $L<0$ or $L=K=0$ in (<ref>).
For any $\tau\in \mathbb N$, $2\leq p$, any $k\in \{1,\ldots, m_2\}$, and any $\eps>0$, there exists a constant $C_\eps>0$ such that
\begin{equation}\label{d6}
\begin{aligned}
\|\vat v_k\|_{\LStaut{p}} \leq C_\eps &+ C_\eps\sum_{j=1}^{k-1}\|\vat v_j\|_{\LStaut{p}}+ \eps\sumi\bra{\|u_i\|_{\LQtaut{p}} + \|u_i\|_{\LStaut{p}}}\\
&+ \eps\sumj\|v_j\|_{\LStaut{p}} .
\end{aligned}
\end{equation}
Consequently, for any $\eps>0$, there exists $C_\eps>0$ such that
\begin{equation}\label{d6_1}
\|\vat v_k\|_{\LStaut{p}} \leq C_\eps + \eps\sumi\bra{\|u_i\|_{\LQtaut{p}} + \|u_i\|_{\LStaut{p}}} + \eps\sumj\|v_j\|_{\LStaut{p}}
\end{equation}
for all $k=1,\ldots, m_2$.
Let $0\leq \psi \in \LStaut{p'}$ with $\|\psi\|_{\LStaut{p'}} = 1$, and $\phi$ be the solution to (<ref>) with $T = \tau+2$. Thanks to the calculations of Lemma <ref>, we can write
\begin{equation}\label{d5}
\begin{aligned}
&\intMtautwo (\vat v_k)\psi\\
&\leq \intMtautwo \phi \vat' v_k + \sumi \intQtautwo a_{(k+m_1)i}\phi \vat' u_i + \sumi \intQtautwo a_{(k+m_1)i}\phi \vat F_i(u)\\
&\quad + \sumi \intQtautwo a_{(m+k_1)i}(\vat u_i)(d_i-1)\Delta \phi + \sumi \intMtautwo a_{(k+m_1)i}d_i(\vat u_i)\pa_{\eta}\phi\\
&\quad - \sum_{j=1}^{k-1}\intMtautwo a_{(k+m_1)(k+j)}\phi \vat (\pa_t v_j - \delta_j\Delta_M v_j) + L_2\intMtautwo \phi \vat \sbra{\sumi u_i^{\mM} + \sumj v_j^{\mM} +1}\\
&\quad =: (I) + (II) + (III) + (IV) + (V) + (VI) + (VII).
\end{aligned}
\end{equation}
We estimate the terms on the right hand side of (<ref>) separately.
* Estimate $(I)$. From Lemma <ref>, there exists $q>p'$ such that
\begin{equation*}
\|\phi\|_{\LStaut{q}} \leq C\|\psi\|_{\LStaut{p'}} = C.
\end{equation*}
Therefore, by Young's inequality and $q' = \frac{q}{q-1}$ is the Hölder conjugate exponent of $q$, we have
\begin{equation*}
|(I)| \leq C\|\phi\|_{\LStaut{q}}\|v_k\|_{\LStaut{q'}} \leq C\|v_k\|_{\LStaut{q'}}.
\end{equation*}
Since $q>p' = \frac{p}{p-1}$, we have $q' < p$. Therefore, it follows from Hölder's inequality that
\begin{align*}
\|v_k\|_{\LStaut{q'}} &\leq \|v_k\|_{\LStaut{1}}^{\theta_0}\|v_k\|_{\LStaut{p}}^{1-\theta_0} \\
&\leq C\|v_k\|_{\LStaut{p}}^{1-\theta_0}\leq C_\eps + \eps \|v_k\|_{\LStaut{p}},
\end{align*}
with $\theta_0\in (0,1)$ satisfying $\frac{1}{q'} = \frac{\theta_0}{1} + \frac{1-\theta_0}{p}$. Therefore,
\begin{equation}\label{estI}
|(I)| \leq C_\eps + \eps \|v_k\|_{\LStaut{p}}.
\end{equation}
* Estimate $(II)$. Similarly to the estimate of $(I)$, we can use Lemma <ref> to estimate
\begin{equation}\label{estII}
|(II)| \leq C\sumi\|u_i\|_{\LQtaut{p}}^{1-\theta_1} \leq C_\eps + \eps\sumi\|u_i\|_{\LQtaut{p}}
\end{equation}
for some $\theta_1\in (0,1)$.
* Estimate $(III)$. We use the condition (<ref>) to find
\begin{equation*}
(III) \leq L_2 \intQtautwo \phi \vat \sbra{\sumi u_i^{\pO}+1}.
\end{equation*}
Looking at the estimate of $(B12)$ in Lemma <ref>, and using $|\vat'| \leq C$, we have
\begin{equation*}
|(III)| \leq C\sumi \intQtautwo \phi u_i^{\pO} + C\|\phi\|_{\LQtaut{p'}} \leq C\sumi \intQtautwo \phi u_i^{\pO} + C.
\end{equation*}
Similarly to (<ref>)–(<ref>),
\begin{equation*}
\begin{aligned}
\intQtautwo \phi u_i^{\pO} &\leq \|\phi\|_{\LQtaut{(p')^{\dag}}}\|u_i\|_{\LQtaut{(\pO-\beta)s}}^{\pO-\beta}\|u_i\|_{\LQtaut{p}}^{\beta}\\
&\leq C\|u_i\|_{\LQtaut{p}}^{\beta} \leq C_\eps+ \eps\|u_i\|_{\LQtaut{p}}
\end{aligned}
\end{equation*}
where $\beta$ and $s$ are in (<ref>). Therefore,
\begin{equation}\label{estIII}
|(III)| \leq C_\eps + \eps\sumi \|u_i\|_{\LQtaut{p}}.
\end{equation}
* Estimate $(IV)$. Thanks to Lemma <ref> we have for some $s>p'$,
\begin{equation*}
\|\Delta \phi\|_{\LQtaut{s}} \leq C\|\psi\|_{\LStaut{p'}} \leq C.
\end{equation*}
Therefore, for $s' = \frac{s}{s-1}$,
\begin{equation*}
|(IV)| \leq C\|\Delta \phi\|_{\LQtaut{s}}\sumi\|u_i\|_{\LQtaut{s'}} \leq C\sumi \| u_i\|_{\LQtaut{s'}}.
\end{equation*}
Since $s>p'$, $s' < p = \frac{p'}{p'-1}$. Therefore, by interpolation inequality
\begin{align*}
\|u_i\|_{\LQtaut{s'}} \leq \|u_i\|_{\LQtaut{1}}^{\theta_3}\|u_i\|_{\LQtaut{p}}^{1-\theta_3}\leq C\|u_i\|_{\LQtaut{p}}^{1-\theta_3}
\end{align*}
with $\theta_3\in (0,1)$ satisfying $\frac{1}{s'} = \frac{\theta_3}{1} + \frac{1-\theta_3}{p}$. From that we obtain
\begin{equation}\label{estIV}
|(IV)| \leq C\sumi \|u_i\|_{\LQtaut{p-1+\pO}}^{1-\theta_3}\leq C_\eps + \eps\sumi \|u_i\|_{\LQtaut{p}}.
\end{equation}
* Estimate $(V)$. From (<ref>) in Lemma <ref> we have for $\xi = \frac{p'}{n+1}$
\begin{equation*}
\|\pa_{\eta}\phi\|_{\LStaut{p'+\xi}} \leq C\|\psi\|_{\LStaut{p'}} \leq C.
\end{equation*}
Therefore, with $s = \frac{p'+\xi}{p'+\xi - 1} = \frac{p+\xi(p-1)}{p+(p-1)(\xi-1)} < p$, we can estimate
\begin{equation}\label{estV}
\begin{aligned}
|(V)| &\leq C\sumi \|u_i\|_{\LStaut{\frac{p'+\xi}{p'+\xi - 1}}}\|\pa_{\eta}\phi\|_{\LStaut{p'+\xi}}\\
&\leq C\sumi \|u_i\|_{\LStaut{\frac{p'+\xi}{p'+\xi - 1}}}\\
&\leq C\sumi\|u_i\|_{\LStaut{1}}^{\theta_4}\|u_i\|_{\LStaut{p}}^{1-\theta_4} \quad \bra{\text{ with } \frac{1}{\frac{p'+\xi}{p'+\xi - 1}} = \frac{\theta_4}{1} + \frac{1-\theta_4}{p}}\\
&\leq C_\eps + \eps \sumi\|u_i\|_{\LStaut{p}}.
\end{aligned}
\end{equation}
* Estimate $(VI)$. By integration by parts,
\begin{equation*}
(VI) = \sum_{j=1}^{k-1}\intMtautwo \sbra{a_{(k+m_1)(k+j)}v_j\phi \vat' + v_j\vat \bra{(\delta_j-\delta_k)\Delta_M\phi + \psi}}.
\end{equation*}
Similar to the estimate of $(I)$ above
\begin{align*}
\abs{\sum_{j=1}^{k-1}\intMtautwo a_{(k+m_1)(k+j)}v_j\phi \vat'} &\leq C\sum_{j=1}^{k-1}\|v_j\|_{\LStaut{p}}^{1-\theta_5}\\
&\leq C_\eps + \eps \sumj \|v_j\|_{\LStaut{p}}
\end{align*}
for some $\theta_5\in (0,1)$. By Hölder's inequality and
$$\|\Delta_{M} \phi\|_{\LStaut{p'}} \leq C\|\psi\|_{\LStaut{p'}} \leq C,$$
we get
\begin{equation*}
\begin{aligned}
&\abs{\sum_{j=1}^{k-1}\intMtautwo v_j\vat \sbra{(\delta_j-\delta_k)\Delta_M\phi + \psi}}\\
&\leq C\sum_{j=1}^{k-1}\|\vat v_j\|_{\LStaut{p}}\bra{\|\Delta_M\phi\|_{\LStaut{p'}} + \|\psi\|_{\LStaut{p'}}}\\
&\leq C\sum_{j=1}^{k-1}\|\vat v_j\|_{\LStaut{p}}.
\end{aligned}
\end{equation*}
Therefore, we have
\begin{equation}\label{estVI}
|(VI)|\leq C_\eps + \eps\sumj\|v_j\|_{\LStaut{p}}+ C\sum_{j=1}^{k-1}\|\vat v_j\|_{\LStaut{p}}.
\end{equation}
* Estimate $(VII)$. We use similar estimates to that of $(B3)$ in (<ref>)–(<ref>). More precisely, with $\alpha$ and $r$ are in (<ref>)–(<ref>) we have
\begin{equation*}
\|u_i\|_{\LStaut{(\mM-\alpha)r}}^{\mM -\alpha} \leq C\|u_i\|_{\LStaut{\Lam+\vk}}^{\mM - \alpha} \leq C.
\end{equation*}
From Lemma <ref>,
\begin{equation*}
\|\phi\|_{\LStaut{(p')^*}} \leq C\|\psi\|_{\LStaut{p'}} \leq C.
\end{equation*}
\begin{align*}
\sumi\intMtautwo \phi u_i^{\mM} &\leq \sumi\|\phi\|_{\LStaut{(p')^*}}\|u_i\|_{\LStaut{(\mM-\alpha)r}}^{\mM-\alpha}\|u_i\|_{\LStaut{p}}^{\alpha}\\
&\leq C\sumi\|u_i\|_{\LStaut{p}}^{\alpha}\\
&\leq C_\eps + \eps\sumi \|u_i\|_{\LStaut{p}}.
\end{align*}
\begin{equation*}
\sumi \intMtautwo \phi v_j^{\mM} \leq C\sumj\|v_j\|_{\LStaut{p}}^{\alpha} \leq C_\eps + \eps\sumj\|v_j\|_{\LStaut{p}}.
\end{equation*}
\begin{equation*}
L_2\intMtautwo \phi \vat \leq C\|\phi\|_{\LStaut{p'}} \leq C.
\end{equation*}
\begin{equation}\label{estVII}
|(VII)| \le C_\eps + \eps\sumi \|u_i\|_{\LStaut{p}} + \eps\sumj\|v_j\|_{\LStaut{p}}.
\end{equation}
Applying all the estimates of $(I)$ to $(VII)$ in (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) into (<ref>) we obtain
\begin{equation*}
\begin{aligned}
\intMtautwo (\vat v_k)\psi&\leq C_\eps + C_\eps\sum_{j=1}^{k-1}\|\vat v_j\|_{\LStaut{p}}\\ &+\eps\sumi\bra{\|u_i\|_{\LQtaut{p}} + \sumi \|u_i\|_{\LStaut{p}}} + \eps\sumj\| v_j\|_{\LStaut{p}}.
\end{aligned}
\end{equation*}
From this we get the estimate (<ref>) due to duality. Finally (<ref>) follows from (<ref>) by induction.
We are now ready to show the uniform-in-time bound in Theorem <ref>.
Assume that $L<0$ or $L=K=0$ in (<ref>).
The global solution to (<ref>) is bounded uniformly in time, i.e.
\begin{equation*}
\sup_{i=1,\ldots, m_1}\sup_{j=1,\ldots, m_2}\sup_{t\geq 0}\sbra{\|u_i(t)\|_{L^\infty(\Omega)} + \|v_j(t)\|_{L^\infty(M)}} <+\infty.
\end{equation*}
We claim that, for any $2\leq p$, there exists a constant $C_p>0$ such that
\begin{equation}\label{claim}
\sumi \bra{\|u_i\|_{\LQtau{p}} + \|u_i\|_{\LStau{p}}} + \sumj\|v_j\|_{\LStau{p}} \leq C_p \quad \text{ for all } \quad \tau\in \mathbb N.
\end{equation}
Indeed, from (<ref>) in Lemma <ref> and (<ref>) in Lemma <ref>, we get for any $\eps>0$ a constant $C_\eps>0$ such that
\begin{align*}
&\sumi\bra{\|\vat u_i\|_{\LQtaut{p}} + \|\vat u_i\|_{\LStaut{p}}} + \sumj\|\vat v_j\|_{\LStaut{p}} \\
&\leq C_\eps + \eps\sbra{\sumi\bra{\|u_i\|_{\LQtaut{p}} + \|u_i\|_{\LStaut{p}}} + \sumj\|v_j\|_{\LStaut{p}}}.
\end{align*}
Using the same arguments as at the end of the proof of Lemma <ref>, we obtain (<ref>).
Now we can use (<ref>) in the truncated system (<ref>), with $p$ is large enough, to conclude that there exists $C_\infty>0$ independent of $\tau\in \mathbb N$ such that
\begin{equation*}
\sumi\|u_i\|_{\LQtau{\infty}} + \sumj\|v_j\|_{\LStau{\infty}} \leq C_\infty \quad \text{ for all } \quad \tau\in \mathbb N,
\end{equation*}
which finishes the proof of Theorem <ref>.
§.§ Proof of Theorem <ref>
Thanks to Theorem <ref>, it's sufficient to show that (<ref>) always holds for $\Lam = 2$. Indeed, in this case $\Lam' = 2$. We only need to show that
\begin{equation}\label{claim3}
C_{\mr,2}^M \leq 1.
\end{equation}
To do that, we multiply the equation (<ref>) by $-\Delta_M\uu$ in $L^2(\Omega)$, to get
\begin{equation*}
\frac{1}{2}\frac{d}{dt}\|\na_M\uu\|_{\LM{2}}^2 + \|\Delta_M \uu\|_{L^2(\Omega)}^2 = -\intM \ff \Delta_M\uu \leq \frac 12\|\ff\|_{\LM{2}}^2 + \frac 12\|\Delta_M \uu\|_{\LM{2}}^2,
\end{equation*}
\begin{equation*}
\frac{d}{dt}\|\na_M\uu\|_{\LM{2}}^2 + \|\Delta_M \uu\|_{\LM{2}}^2 \leq \|\ff\|_{\LM{2}}.
\end{equation*}
Integrating this on $(0,T)$ and using $\uu(\cdot,0) = 0$ we obtain
\begin{equation*}
\|\na_M\uu(\cdot,T)\|_{\LM{2}}^2 + \|\Delta_M\uu\|_{\LS{2}}^2 \leq \|\ff\|_{\LS{2}}^2,
\end{equation*}
which implies
\begin{equation*}
\|\Delta_M\uu\|_{\LS{2}} \leq \|\ff\|_{\LS{2}},
\end{equation*}
and hence, the desired estimate (<ref>).
§.§ Proof of Theorem <ref>
First, thanks to Lemmas <ref> and <ref>, we have, similarly to (<ref>) and (<ref>), for any $p>1$ and $\eps>0$ a constant $K_{p,\eps>0}$ such that
\begin{equation*}
\sumi\bra{\|u_i\|_{\LQ{p}} + \|u_i\|_{\LS{p}}} \leq K_{p,\eps}(1+T) + \eps\sumj\|v_j\|_{\LS{p}}.
\end{equation*}
Following Lemma <ref>, we will show that for any $\eps>0$, and any $k\in \{1,\ldots, m_2\}$, there exists $C_{T,\eps}>0$ such that
\begin{equation}\label{gg4}
\|v_k\|_{\LS{p}} \leq C_{T,\eps} + C_T\sum_{j=1}^{k-1}\|v_j\|_{\LS{p}} + \eps\sumj \|v_j\|_{\LS{p}}.
\end{equation}
We consider the duality equation (<ref>) with $0\leq \psi \in \LS{p'}$ satisfying $\|\psi\|_{\LS{p'}} = 1$. The same integration by parts arguments in Lemma <ref> gives
\begin{equation}\label{gg5}
\begin{aligned}
\intMT v_k\psi &\leq \intM v_{k,0}\phi(0) + \sumi a_{(k+m_1)i}\intO u_{i,0}\phi(0) + L_2\intQT \phi\sbra{\sumi u_i^{\pO}+1}\\
&\quad + \sumi a_{(k+m_1)i}d_i\intMT u_i\pa_{\eta}\phi + \sumi a_{(k+m_1)i}\intQT u_i\sbra{\pa_t\phi + d_i\Delta \phi}\\
&\quad +\sum_{j=1}^{k-1}a_{(k+m_1)(k+j)}\intM v_{j,0}\phi(0) - \sum_{j=1}^{k-1}\intMT v_j\sbra{\pa_t\phi + \delta_j\Delta_M\phi}\\
&\quad + L_2\intMT \phi\sbra{\sumi u_i^{\mM} + \sumj v_j^{\mM}+1}.
\end{aligned}
\end{equation}
All the terms on the right hand side of (<ref>) can be estimated similarly to Lemma <ref> except for two sums
\begin{equation*}
(\mathsf A) = \sumi\intQT \phi u_i^{\pO} \quad \text{ and } \quad (\mathsf B) = \sumi\intMT \phi u_i^{\mM} + \sumj\intMT \phi v_j^{\mM}.
\end{equation*}
To estimate ($\mathsf A$) we first use Lemma <ref> to have
\begin{equation*}
\|\phi\|_{\LQ{(p')^\dag}} \leq C_T\|\psi\|_{\LQ{p'}} = C_T.
\end{equation*}
By Hölder's inequality
\begin{align}\label{gg6}
\int_0^T\intO \phi u_i^{\pO - \alpha} u_i^{\alpha} \leq \|\phi\|_{\LQ{(p')^{\dag}}}\|u_i\|_{\LQ{(\pO-\alpha)s}}^{\pO - \alpha}\|u_i\|_{\LQ{p}}^{\alpha} \leq C_T\|u_i\|_{\LQ{(\pO-\alpha)s}}^{\pO - \alpha}\|u_i\|_{\LQ{p}}^{\alpha}
\end{align}
\begin{equation*}
\frac{1}{(p')^{\dag}} + \frac 1s + \frac{\alpha}{p}=1.
\end{equation*}
From this
\begin{align*}
\frac 1s = 1 - \frac \alpha p - \frac{1}{(p')^{\dag}}
= 1 - \frac \alpha p - \frac{n+1-2p'}{(n+2)p'}
= \frac{n(1-\alpha) + 3p + 1 - 2\alpha}{(n+2)p}
\end{align*}
or equivalently
\begin{equation*}
s = \frac{(n+2)p}{n(1-\alpha) + 3p + 1 - 2\alpha}.
\end{equation*}
Note that as $p\to\infty$, $s \to \frac{n+2}{3}$. Since
\begin{equation*}
\pO < 1 + \frac{3a}{n+2}
\end{equation*}
we can choose $\alpha \in (0,1)$ and $p$ large enough such that
\begin{equation*}
(\pO - \alpha)s \leq \frac{3a}{n+2}\frac{n+2}{3} = a.
\end{equation*}
We then use this and the assumption $\|u_i\|_{L^\infty(0,T;\LO{a})} \leq \F(T)$ in (<ref>) to get
\begin{equation*}
\intQT \phi u_i^{\pO} \leq C_T\F(T)^{\pO-\alpha}\|u_i\|_{\LQ{p}}^{\alpha} \leq C_{T,\eps} + \eps\|u_i\|_{\LQ{p}},
\end{equation*}
which gives the estimate
\begin{equation*}
(\mathsf A) \leq C_{T,\eps} + \eps\sumi\|u_i\|_{\LQ{p}}.
\end{equation*}
The estimate of ($\mathsf B$) can be done in the same way as for $(B3)$ in the proof of Lemma <ref>, where the integrability $b$ plays the role of $\Lam+\vk$ therein, so we omit it here. Ultimately, we have for any $\eps>0$ a constant $C_{T,\eps}>0$ such that
\begin{equation*}
(\mathsf B) \leq C_{T,\eps} + \sumi \|u_i\|_{\LS{p}} + \sumj\|v_j\|_{\LS{p}}.
\end{equation*}
With the estimates of $(\mathsf A)$ and $(\mathsf B)$, one can proceed from (<ref>) the same way in Lemma <ref> to finally obtain (<ref>). Lemma <ref> then gives
\begin{equation*}
\sumj \|v_j\|_{\LS{p}} \leq C_T
\end{equation*}
and consequently
\begin{equation*}
\sumi\bra{\|u_i\|_{\LQ{p}} + \|v_j\|_{\LS{p}}} \leq C_T
\end{equation*}
for all $p\geq 1$. This is enough to conclude the global existence of solutions to (<ref>). The uniform-in-time bound is obtained by using the truncated function $\varphi_\tau$ and working on each cylinder $Q_{\tau,\tau+2}$ for $\tau\in \mathbb N$. We omit the details since they are very similar to that of Section <ref>.
§ APPLICATIONS
In this section, we show the application of our theorems to some models arising from cell biology. It's worth emphasizing that, all known results for volume-surface systems seem not applicable to these systems.
§.§ Membrane protein clustering
In a recent work of Lucas M. Stolerman et al [34], the authors focus on stability analysis of a bulk-surface reaction-diffusion model of membrane protein clustering. In their model, $U$ represents a volume component which diffuses in the cytoplasm. It binds with the membrane and forms a surface monomer $A_1$ via a reaction flux (see below). Then subsequent oligomerization at the membrane is given by $A_{j-1}+A_1 \rightleftharpoons A_j$ for $j=2,3,...,N$. If we model the cytoplasm as a smooth bounded region $\Omega \subset \R^n$, and the membrane (boundary of $\Omega$) by $M$, then the model in [34] has the form
\begin{equation}
\begin{cases}
\frac{\partial u}{\partial t}=d\Delta u, & \text{ on }\Omega\times(0,T)\\
d\frac{\partial u}{\partial\eta}=F(u,v_1,...,v_N), & \text{ on }M\times(0,T)\\
\frac{\partial v_1}{\partial t}=\delta_{1}\Delta_M v_{1}+G_{1}(u,v_1,...,v_N), & \text{ on }M\times(0,T)\\
\vdots&\\
\frac{\partial v_N}{\partial t}=\delta_{N}\Delta_M v_{N}+G_{N}(u,v_1,...,v_N), & \text{ on }M\times(0,T)\\
u=u_0 & \text{ on }\overline{\Omega}\times{0}\\
v=v_0 & \text{ on }M\times{0}
\end{cases}\label{eq:example1}
\end{equation}
Here, $d$ and $\delta_i$ are positive diffusion coefficients, for $i=1,...,N$, $u$ represents the cytoplasm concentration density of $U$, and $v_1,...,v_N$ represent the membrane concentration densities of $A_1,...,A_N$, $u_0$ and the vector $v_0=(v_{0_i})$ represent bounded nonnegative initial data, and $k_0$, $k_b$, $k_d$ and $k_2,...,k_N$ are positive constants. Also,
\begin{align*}
G_1(u,v_1,...,v_N) & =(K_0+k_bv_N)u-k_dv_1-2k_mv_1^2+2k_2v_2
G_j(u,v_1,...,v_N)&=k_gv_1v_{j-1}-k_gv_1v_j-k_jv_j+k_{j+1}v_{j+1},\text{ for }j=3,...,N-1\\
\end{align*}
Note that the nonlinearities $F$ and $G_j$, $j=1,\ldots, N$, are locally Lipschitz continuous,
\begin{equation}\label{eq:qpf}
F(0,v_1,...,v_N)\ge 0\text{ for all }(v_1,...,v_N) \in \R_+^N
\end{equation}
\begin{equation}\label{eq:qpG}
G_i(u,v_1,...,v_N)\ge 0\text { for all }u,v_1,...,v_N\ge 0\text{ with }v_i=0\text{ for }i=1,...,N,
\end{equation}
and they are polynomial. That means the assumptions (<ref>), (<ref>), (<ref>) are fulfilled.
It's also simple to check that
\begin{equation}\label{eq:minibal}
F(u,v_1,...,v_N)+\sum_{j=1}^NjG_j(u,v_1,...,v_N)=0,\text{ for all }u,v_1,...,v_N\ge 0,
\end{equation}
and thus (<ref>) is satisfied with $L = K = 0$. It remains to check the intermediate sum conditions (<ref>) and (<ref>), i.e. there is $(N+1)\times(N+1)$ lower triangular matrix $A=(a_{i,j})$ with positive entries on the diagonal, and nonnegative entries below the diagonal, and constants $K_1,K_2\ge 0$ so that
\begin{equation}\label{eq:miniintsum}
A\begin{bmatrix}F(u,v_1,...,v_N)\\G_1(u,v_1,...,v_N)\\\vdots\\G_N(u,v_1,...,v_N)\end{bmatrix}\le K_1\vec{1}\left(u+\sum_{j=1}^n v_j+K_2\right),\text{ for all }u,v_1,...,v_N\ge 0.
\end{equation}
Indeed, this is done by choosing $K_1=2\max_{{i=2,...,m}}k_i$, $K_2=0$, and
\[A=\begin{bmatrix}
\vdots&\vdots&\ddots&\vdots\\
\end{bmatrix}.
\]
Since all the assumptions in Theorem <ref> are satisfied, we have the following
For any non-negative initial data $(u_0,v_0)\in W^{2-2/p}(\Omega)\times (W^{2-2/p}(M))^N$ for $p>n$ satisfying the compatibility condition
\begin{equation*}
d\pa_{\eta}u_0 = F(u_0, v_{1,0}, \ldots, v_{N,0}) \quad \text{ on }\quad M,
\end{equation*}
there exists a unique global classical solution to (<ref>) which is uniformly bounded in time, i.e.
\begin{equation*}
\sup_{t\geq 0}\bra{\|u(t)\|_{\LO{\infty}} + \sum_{j=1}^N\|v_j(t)\|_{\LM{\infty}}} < +\infty.
\end{equation*}
§.§ Activation of Cdc42 in cell polarization
This second example is from the recent paper [7] where the authors derive a mathematical model for the activation of Cdc42 in cell polarization. The system reads as[The model considered herein is slightly different from that of [7], where $Q(A,I,G) = k_1G(k_{\max}-(A+I))$, since we take into account the saturation. It's noted that their choice of nonlinearity might lead to negative concentrations, for instance, when $I_0 \equiv 0$, $\beta \geq A_0(x) > k_{\max}+1$ for $x\in\Omega$, and $G_0(x)\geq \alpha>0$ for a large enough constant $\alpha$.]
\begin{equation}\label{eq:example2}
\begin{cases}
\pa_t G = D_G\Delta G, &(x,t)\in Q_T,\\
D_G\pa_\eta G = Q(A,I,G), &(x,t)\in M_T,\\
\pa_tA = D_A\Delta_MA + F(A,I), &(x,t)\in M_T,\\
\pa_t I = D_I\Delta_M I + H(A,I,G), &(x,t)\in M_T,\\
G(x,0) = G_0(x), &x\in\Omega,\\
A(x,0) = A_0(x), \; I(x,0) = I_0(x), &x\in M,
\end{cases}
\end{equation}
\begin{equation*}
Q(A,I,G) = -k_1G(k_{\max} - (A+I))_+ + k_{-1}I, F(A,I) = k_2I - k_{-2}A + k_3A^2I
\end{equation*}
\begin{equation*}
H(A,I,G) = -F(A,I)-Q(A,I,G),
\end{equation*}
with positive constant rates $k_1, k_{-1}, k_2, k_{-2}, k_{\max}$. Here $G$, $A$, $I$ are concentrations of the GTP-, GDP-, and GDI-bound form respectively. The interested reader is referred to [7] for more details. It's easy to check that all the assumptions (<ref>), (<ref>), (<ref>) (with $L=K=0$) and (<ref>) are fulfilled. It's also clear that
\begin{equation*}
\begin{bmatrix}
1 & 0 & 0\\
1 & 1 & 0\\
1 & 1 & 1
\end{bmatrix}
\begin{pmatrix} Q\\ H \\ F \end{pmatrix} \le \begin{pmatrix} k_{-1}I\\ k_{-2}A\\ 0\end{pmatrix},
\end{equation*}
which means that the intermediate sum condition (<ref>) is satisfied (condition (<ref>) is trivially fulfilled since we have no nonlinearities for the equation of $G$). Therefore Theorem <ref> applies and we have the following result.
For any non-negative initial data $(G_0, A_0, I_0)\in W^{2-2/p}(\Omega)\times(W^{2-2/p}(M))^{2}$ for $p>n$ satisfying the compatibility condition
\begin{equation*}
D_G\pa_{\eta}G_0 = Q(A_0,I_0,G_0) \quad \text{ on } \quad M,
\end{equation*}
there exists a unique global classical solution to (<ref>) which is bounded uniformly in time,
\begin{equation*}
\sup_{t\ge 0}\bra{\|G(t)\|_{\LO{\infty}} + \|A(t)\|_{\LM{\infty}} + \|I(t)\|_{\LM{\infty}}} < +\infty.
\end{equation*}
§.§ A system with better a-priori estimates
In this section we consider a system where better a-priori estimates can be derived using the system's special structures, which in turn allows us to obtain global existence for higher order nonlinearities. More precisely, the system reads
\begin{equation}\label{gg1}
\begin{cases}
\pa_t u_1 - d_1\Delta u_1 = f_1(u) = u_1^3 - u_2^3, &x\in\Omega,\\
\pa_t u_2 - d_2\Delta u_2 = f_2(u) = -u_1^3 + u_2^3, &x\in\Omega,\\
d_1\pa_{\eta}u_1 = g_1(u,v) = u_1 - u_2 - 2u_1^3 + u_2^2 - v_2^2, &x\in M,\\
d_2\pa_{\eta}u_2 = g_2(u,v) = -u_1 + u_2 - u_2^3 - v_1^2, &x\in M,\\
\pa_t v_1 - \delta_1\Delta_M v_1 = h_1(u,v) = 2u_1^3 + u_1u_2^2 - v_2^6, &x\in M,\\
\pa_t v_2 - \delta_2 \Delta_M v_2 = h_2(u,v) = 2u_2^3 + u_1^3 - v_1^6, &x\in M,\\
u(x,0) = u_0(x), &x\in\Omega,\\
v(x,0) = v_0(x), w(x,0) = w_0(x), &x\in M.
\end{cases}
\end{equation}
It's obvious that
\begin{equation*}
f_1(u) + f_2(u) \leq 0.
\end{equation*}
By using the Young's inequality
\begin{equation}\label{gg2}
u_1u_2^2 \leq \frac{u_1^3}{3} + \frac{2u_2^3}{3}
\end{equation}
we can check that
\begin{equation*}
2g_1(u,v) + 2g_2(u,v) + h_1(u,v) + h_2(u,v) \leq -u_1^3 + u_1u_2^2 - 2u_3^3 \leq -\frac{2}{3}u_1^3 - \frac 43 u_2^3 \leq 0.
\end{equation*}
Therefore, assumption (<ref>) is satisfied. It's easy to check that by choosing the matrix
\begin{equation*}
A = \begin{pmatrix}
\end{pmatrix},
\end{equation*}
the assumptions (<ref>) and (<ref>) are satisfied with
\begin{equation*}
\pO = 3, \quad \pM = 2, \quad \text{ and } \quad \mM = 3.
\end{equation*}
Therefore, the results of Theorem <ref> are not applicable to obtain global existence to (<ref>). We show here that by utilizing the special structure of (<ref>), one can use Theorem <ref> to still get global solutions. Indeed, direct computations give
\begin{equation}\label{gg3}
\begin{aligned}
\pa_t\bra{\intO \bra{u_1^4+u_2^4} + \intM (v_1 +v_2)} + 12d_1\|u_1\na u_1\|_{\LO{2}}^2 + 12d_2\|u_2\na u_2\|_{\LO{2}}^2\\
= 4\intO (f_1(u)u_1^3 + f_2(u)u_2^3) + 4\intM \bra{g_1(u,v)u_1^3 + g_2(u,v)u_2^3} + \intM\bra{h_1(u,v) + h_2(u,v)}.
\end{aligned}
\end{equation}
We have
\begin{equation*}
f_1(u)u_1^3 + f_2(u)u_2^3 = -(u_1^3 - u_2^3)^2 \leq 0,
\end{equation*}
\begin{align*}
&4\bra{g_1(u,v)u_1^3 + g_2(u,v)u_2^3}+h_1(u,v) + h_2(u,v)\\
&\leq 4u_1^4 -8u_1^6 + 4u_1^3u_2^2 + 4u_2^4 - 4u_2^6 + 2u_1^3 + u_1u_2^2 - v_2^6 + 2u_2^3 + u_1^3 - v_1^6 \\
&\leq -C(u_1^6 + u_2^6+v_1^6 + v_2^6)+ C
\end{align*}
where we used (<ref>) and
\begin{equation*}
4u_1^4 + 4u_1^3u_2^2 + 4u_2^4 + 2u_1^3 + u_1u_2^2 + 2u_2^3 + u_1^3 \leq \eps\bra{u_1^6 + u_2^6} + C_\eps
\end{equation*}
for any $\eps>0$. Thus we get from (<ref>) that
\begin{align*}
\pa_t\bra{\intO (u_1^4 + u_2^4) + \intM\bra{v_1 + v_2}}+ C\intM\bra{u_1^6 + u_2^6 + v_1^6 + v_2^6} \leq C.
\end{align*}
By integrating in time we get
\begin{equation*}
\sup_{i=1,2}\|u_i\|_{L^\infty(0,T;\LO{4})}^4 + \sup_{i=1,2}\bra{\|u_i\|_{\LS{6}}^6 + \|v_i\|_{\LS{6}}^6} \leq CT.
\end{equation*}
Thus, Theorem <ref> is applicable with $a = 4$ and $b = 6$, which allows us to get global existence of (<ref>) in the physical dimension $n= 3$. The uniform-in-time bound of solutions remains unclear.
Let $n\leq 3$. For any non-negative initial data $(u_0,v_0)\in W^{2-2/p}(\Omega)^2\times W^{2-2/p}(M)^2$ satisfying the compatibility condition
\begin{equation*}
d_1\pa_{\eta}u_{1,0} = g_1(u_0,v_0), \quad d_2\pa_{\eta}u_{2,0} = g_2(u_0,v_0), \quad \text{ on } \quad M,
\end{equation*}
there exists a unique global classical solution to (<ref>).
§ TECHNICAL LEMMAS
In this appendix, we prove two technical lemmas that are used in Lemma <ref>, more precisely the time derivative of the function $\H_p[u]$ and the integration by parts in (<ref>)–(<ref>).
Suppose $m_1\in \mathbb N$, $\theta= (\theta_1,\ldots, \theta_{m_1})$, where $\theta_1,...,\theta_{m_1}$ are positive real numbers, $\beta\in \mathbb Z_+^{m_1}$, and $\H_p[u]$ is defined in (<ref>). Then
$$\frac{\partial}{\partial t}\H_0[u](t)=0,\text{ }\frac{\partial}{\partial t}\H_1[u](t)=\sum_{j=1}^{m_1}\theta_j\frac{\partial}{\partial t}u_j(t),$$
and for $p\in\mathbb N$ such that $p\ge 2$,
\begin{equation*}
\frac{\partial}{\partial t}\H_p[u](t) = \sum_{|\beta| = p-1}\begin{pmatrix} p\\ \beta \end{pmatrix} \theta^{\beta^2}u(t)^{\beta}\sum_{j=1}^{m_1}\theta_j^{2\beta_j+1}\frac{\partial}{\partial t}u_j(t).
\end{equation*}
The results for $\H_0[u](t)$ and $\H_1[u](t)$ are trivial. The same is true for the case when $m_1=1$. Suppose $p\ge 2$ and $m_1\ge 2$. We proceed by induction on the value $m_1$, and assume $k\in \mathbb N$ such that the result is is true for $m_1=k$. Suppose $m_1=k+1$ and denote
$$\tilde\beta=(\beta_2,...,\beta_{m_1})\text{ and }\tilde u=(u_2,...,u_{m_1}).$$
Then we can rewrite $\H_p[u]$ as
\begin{equation}\label{Hp-eq1}
\H_p[u] = \sum_{\beta_1=0}^p\frac{1}{\beta_1!}\theta_1^{\beta_1^2}u_1^{\beta_1}\sum_{|\tilde\beta|=p-\beta_1}\begin{pmatrix}
p\\ \tilde\beta\end{pmatrix}\tilde\theta^{\tilde\beta^2}\tilde u^{\tilde\beta}.
\end{equation}
\begin{align}\label{Hp-eq2}
\frac{\partial}{\partial t}\H_p[u] &= \sum_{\beta_1=1}^p\frac{1}{\beta_1!}\theta_1^{\beta_1^2}\beta_1u_1^{\beta_1-1}\frac{\partial}{\partial t}u_1\sum_{|\tilde\beta|=p-\beta_1}\begin{pmatrix}
p\\ \tilde\beta\end{pmatrix}\tilde\theta^{\tilde\beta^2}\tilde u^{\tilde\beta}\nonumber\\
&+\sum_{\beta_1=0}^p\frac{1}{\beta_1!}\theta_1^{\beta_1^2}u_1^{\beta_1}\frac{\partial}{\partial t}\left(\sum_{|\tilde\beta|=p-\beta_1}\begin{pmatrix}
p\\ \tilde\beta\end{pmatrix}\tilde\theta^{\tilde\beta^2}\tilde u^{\tilde\beta}\right)\nonumber\\
&=\sum_{\beta_1=1}^p\frac{1}{\beta_1!}\theta_1^{\beta_1^2}\beta_1u_1^{\beta_1-1}\frac{\partial}{\partial t}u_1\sum_{|\tilde\beta|=p-\beta_1}\begin{pmatrix}
p\\ \tilde\beta\end{pmatrix}\tilde\theta^{\tilde\beta^2}\tilde u^{\tilde\beta}\nonumber\\
&+\sum_{\beta_1=0}^{p-1}\frac{1}{\beta_1!}\theta_1^{\beta_1^2}u_1^{\beta_1}\frac{p!}{(p-\beta_1)!}\frac{\partial}{\partial t}H_{p-\beta_1}[\tilde u].
\end{align}
Now, from our induction hypothesis,
\begin{equation}\label{Hp-eq3}
\frac{\partial}{\partial t}\H_{p-\beta_1}[\tilde u] = \sum_{|\tilde\beta| = p-\beta_1-1}\begin{pmatrix} p-\beta_1\\ \tilde\beta \end{pmatrix} \tilde\theta^{\tilde\beta^2}\tilde u^{\tilde\beta}\sum_{j=1}^{m_1-1}\tilde\theta_j^{2\tilde\beta_j+1}\frac{\partial}{\partial t}\tilde u_j.
\end{equation}
Therefore, substituting (<ref>) into (<ref>), and noting that $\tilde u_j=u_{j+1}$ and $\tilde\theta_j=\theta_{j+1}$, gives
\begin{align*}\label{Hp-eq4}
\frac{\partial}{\partial t}\H_p[u] &=\sum_{\beta_1=1}^p\frac{1}{\beta_1!}\theta_1^{\beta_1^2}\beta_1u_1^{\beta_1-1}\frac{\partial}{\partial t}u_1\sum_{|\tilde\beta|=p-\beta_1}\begin{pmatrix}
p\\ \tilde\beta\end{pmatrix}\tilde\theta^{\tilde\beta^2}\tilde u^{\tilde\beta}\nonumber\\
&+\sum_{\beta_1=0}^{p-1}\frac{1}{\beta_1!}\theta_1^{\beta_1^2}u_1^{\beta_1}\frac{p!}{(p-\beta_1)!}\sum_{|\tilde\beta| = p-\beta_1-1}\begin{pmatrix} p-\beta_1\\ \tilde\beta \end{pmatrix} \tilde\theta^{\tilde\beta^2}\tilde u^{\tilde\beta}\sum_{j=1}^{m_1-1}\tilde\theta_j^{2\tilde\beta_j+1}\frac{\partial}{\partial t}\tilde u_j\nonumber\\
&=\sum_{\beta_1=0}^{p-1}\frac{1}{\beta_1!}\theta_1^{(\beta_1+1)^2}u_1^{\beta_1}\frac{\partial}{\partial t}u_1\sum_{|\tilde\beta|=p-\beta_1-1}\begin{pmatrix}
p\\ \tilde\beta\end{pmatrix}\tilde\theta^{\tilde\beta^2}\tilde u^{\tilde\beta}\nonumber\\
&+\sum_{\beta_1=0}^{p-1}\frac{1}{\beta_1!}\theta_1^{\beta_1^2}u_1^{\beta_1}\frac{p!}{(p-\beta_1)!}\sum_{|\tilde\beta| = p-\beta_1-1}\begin{pmatrix} p-\beta_1\\ \tilde\beta \end{pmatrix} \tilde\theta^{\tilde\beta^2}\tilde u^{\tilde\beta}\sum_{j=1}^{m_1-1}\tilde\theta_j^{2\tilde\beta_j+1}\frac{\partial}{\partial t}\tilde u_j\nonumber\\
&=\sum_{|\beta|=p-1}\begin{pmatrix}p\\\beta\end{pmatrix}\theta^{\beta^2}u^{\beta}\theta_1^{2\beta_1+1}\frac{\partial}{\partial t}u_1
+\sum_{|\beta|=p-1}\begin{pmatrix}p\\\beta\end{pmatrix}\theta^{\beta^2}u^{\beta}\sum_{j=2}^{m_1}\theta_j^{2\beta_j+1}\frac{\partial}{\partial t}u_j\nonumber\\
&=\sum_{|\beta|=p-1}\begin{pmatrix}p\\\beta\end{pmatrix}\theta^{\beta^2}u^{\beta}\sum_{j=1}^{m_1}\theta_j^{2\beta_j+1}\frac{\partial}{\partial t}u_j.
\end{align*}
Therefore, the result follows from induction.
Suppose $m_1\in \mathbb N$, $\theta= (\theta_1,\ldots, \theta_{m_1})$, where $\theta_1,...,\theta_{m_1}$ are positive real numbers. If $p\in\mathbb N$ such that $p\ge 2$, then
$$\sum_{|\beta|=p-1}\begin{pmatrix}p\\ \beta\end{pmatrix}\theta^{\beta^2}\sum_{i=1}^{m_1}\theta_i^{2\beta_i+1}d_i\nabla u_i\cdot\nabla u^\beta =\sum_{|\beta|=p-2}\begin{pmatrix}p\\ \beta\end{pmatrix}\theta^{\beta^2}u^\beta\sum_{l=1}^n\sum_{i,j=1}^{m_1}a_{i,j}\frac{\partial u_i}{\partial x_l}\frac{\partial u_j}{\partial x_l},$$
where $(a_{i,j})$ is the $m_1\times m_1$ symmetric matrix with entries
\begin{align}
\begin{matrix}
\frac{d_i+d_j}{2}\theta_i^{2\beta_i+1}\theta_j^{2\beta_j+1},&\text{if }i\ne j\\
d_i\theta_i^{4\beta_i+4},&\text{if }i=j
\end{matrix}
\right..
\end{align}
The result is easily verified when $m_1=1$, regardless of the choice of $p$, and for $p=2$, regardless of the choice of $m_1$. Suppose $p\ge 2$ and $m_1\ge 2$. We proceed by induction on the value $m_1$, and assume $k\in \mathbb N$ such that the result is is true for $m_1=k$. Suppose $m_1=k+1$ and (as in the proof of Lemma <ref>) denote
$$\tilde\beta=(\beta_2,...,\beta_{m_1})\text{ and }\tilde u=(u_2,...,u_{m_1}).$$
\begin{align}\label{Hp-eq5}
\sum_{|\beta|=p-1}\begin{pmatrix}p\\ \beta\end{pmatrix}\theta^{\beta^2}\sum_{i=1}^{m_1}\theta_i^{2\beta_i+1}d_i\nabla u_i\cdot\nabla u^\beta &=\sum_{\beta_1=0}^{p-1}\sum_{|\tilde\beta|=p-\beta_1-1}\begin{pmatrix}p\\ \beta\end{pmatrix}\theta_1^{\beta_1^2}\tilde\theta^{\tilde\beta^2}\biggl[\theta_1^{2\beta_1+1}d_1\nabla u_1\cdot\nabla u^\beta\nonumber\\
&+\sum_{i=1}^{m_1-1}\tilde\theta_i^{2\tilde\beta_i+1}d_{i+1}\nabla\tilde u_i\cdot\nabla\left(u_1^{\beta_1}\tilde u^{\tilde\beta}\right)\biggr].
\end{align}
Note that for $1\le \beta_1\le p-2$ and $|\tilde\beta|=p-\beta_1-1$
\begin{align}\label{Hp-eq6}
\nabla\left(u_1^{\beta_1}\tilde u^{\tilde\beta}\right)=\beta_1 u_1^{\beta_1-1}\tilde u^{\tilde\beta}\nabla u_1+\sum_{j=1,\tilde\beta_j\ne 0}^{m_1-1}\tilde\beta_j u_1^{\beta_1} \tilde u^{\tilde\beta-e_j}\nabla \tilde u_j.
\end{align}
Therefore, from (<ref>) and (<ref>), we have
\begin{align}\label{Hp-eq7}
\sum_{|\beta|=p-1}\begin{pmatrix}p\\ \beta\end{pmatrix}\theta^{\beta^2}\sum_{i=1}^{m_1}\theta_i^{2\beta_i+1}d_i\nabla u_i\cdot\nabla u^\beta=\text{I}+\text{II},
\end{align}
\begin{align}\label{Hp-eq8}
\text{I}=&\sum_{\beta_1=0}^{p-1}\sum_{|\tilde\beta|=p-\beta_1-1}\begin{pmatrix}p\\\beta\end{pmatrix}\theta_1^{\beta_1^2}\tilde\theta^{\tilde\beta^2}\biggl[\theta_1^{2\beta_1+1}d_1\nabla u_1\cdot \nabla u^{\beta}\nonumber\\
&+\sum_{i=1,\beta_1\ne 0}^{m_1-1}\tilde\theta_i^{2\tilde\beta_i+1}d_{i+1}\nabla \tilde u_i\cdot\beta_1 u_1^{\beta_1-1}\tilde u^{\tilde\beta}\nabla u_1\biggr]
\end{align}
\begin{align}\label{Hp-eq9}
\text{II}=\sum_{\beta_1=0}^{p-2}\sum_{|\tilde\beta|=p-\beta_1-1}\begin{pmatrix}p\\\beta\end{pmatrix}\theta_1^{\beta_1^2}\tilde\theta^{\tilde\beta^2}\sum_{i,j=1,\tilde\beta_j\ne 0}^{m_1-1}\tilde\theta_i^{2\tilde\beta_i+1}d_{i+1}\nabla \tilde u_i\cdot\tilde\beta_ju_1^{\beta_1}\tilde u^{\tilde\beta-e_j}\nabla \tilde u_j.
\end{align}
Above, $e_j$ denotes row $j$ of the $(m_1-1)\times (m_1-1)$ identity matrix. We start with the analysis of II. We can rewrite
\begin{align}\label{Hp-eq10}
\text{II}&=\sum_{\beta_1=0}^{p-2}\frac{1}{\beta_1!}\theta_1^{\beta_1^2}u_1^{\beta_1}\sum_{|\tilde\beta|=p-\beta_1-1}\begin{pmatrix}p\\\tilde\beta\end{pmatrix}\tilde\theta^{\tilde\beta^2}\sum_{i,j=1,\tilde\beta_j\ne 0}^{m_1-1}\tilde\theta_i^{2\tilde\beta_i+1}d_{i+1}\nabla \tilde u_i\cdot\tilde\beta_j\tilde u^{\tilde\beta-e_j}\nabla \tilde u_j\nonumber\\
&=\sum_{\beta_1=0}^{p-2}\frac{1}{\beta_1!}\theta_1^{\beta_1^2}u_1^{\beta_1}\sum_{|\tilde\beta|=p-\beta_1-1}\begin{pmatrix}p\\\tilde\beta\end{pmatrix}\tilde\theta^{\tilde\beta^2}\sum_{i,j=1,\tilde\beta_j\ne 0}^{m_1-1}\tilde\theta_i^{2\tilde\beta_i+1}d_{i+1}\nabla \tilde u_i\cdot \nabla\tilde u^{\tilde\beta}\nonumber\\
&=\sum_{\beta_1=0}^{p-2}\frac{1}{\beta_1!}\theta_1^{\beta_1^2}u_1^{\beta_1}\frac{p!}{(p-\beta_1)!}\sum_{|\tilde\beta|=p-\beta_1-1}\begin{pmatrix}p-\beta_1\\\tilde\beta\end{pmatrix}\tilde\theta^{\tilde\beta^2}\sum_{i,j=1,\tilde\beta_j\ne 0}^{m_1-1}\tilde\theta_i^{2\tilde\beta_i+1}d_{i+1}\nabla \tilde u_i\cdot \nabla\tilde u^{\tilde\beta}\nonumber\\
&=\sum_{\beta_1=0}^{p-2}\frac{1}{\beta_1!}\theta_1^{\beta_1^2}u_1^{\beta_1}\frac{p!}{(p-\beta_1)!}\sum_{|\tilde\beta|=p-\beta_1-2}\begin{pmatrix}p-\beta_1\\ \tilde\beta\end{pmatrix}\tilde\theta^{\tilde\beta^2}\tilde u^{\tilde\beta}\sum_{l=1}^n\sum_{i,j=1}^{m_1-1} a_{i+1,j+1}\frac{\partial \tilde u_i}{\partial x_l}\frac{\partial \tilde u_j}{\partial x_l}\nonumber\\
&=\sum_{|\beta|=p-2}\begin{pmatrix}p\\\beta\end{pmatrix}\theta^{\beta^2}u^\beta\sum_{l=1}^n\sum_{i,j=2}^{m_1}a_{i,j}\frac{\partial u_i}{\partial x_l}\frac{\partial u_j}{\partial x_l}
\end{align}
where the last step follows from the induction hypothesis. Now let's investigate I. We begin by expanding the $\nabla u^\beta$ term to find
\begin{align}\label{Hp-eq11}
\text{I}=&\sum_{\beta_1=0}^{p-1}\sum_{|\tilde\beta|=p-\beta_1-1}\begin{pmatrix}p\\\beta\end{pmatrix}\theta_1^{\beta_1^2}\tilde\theta^{\tilde\beta^2}\biggl[\theta_1^{2\beta_1+1}d_1\nabla u_1\cdot \biggl(\beta_1u_1^{\beta_1-1}\tilde u^{\tilde\beta}\nabla u_1\nonumber\\
&+\sum_{i=1}^{m_1-1}\tilde\beta_iu_1^{\beta_1}\tilde u^{\tilde\beta-e_i}\nabla \tilde u_i\biggr)+\sum_{i=1,\beta_1\ne 0}^{m_1-1}\tilde\theta_i^{2\tilde\beta_i+1}d_{i+1}\nabla \tilde u_i\cdot\beta_1 u_1^{\beta_1-1}\tilde u^{\tilde\beta}\nabla u_1\biggr]\nonumber\\
\end{align}
\begin{align}\label{Hp-eq12}
\text{I}_{1,1}&=\sum_{\beta_1=1}^{p-1}\sum_{|\tilde\beta|=p-\beta_1-1}\begin{pmatrix}p\\\beta\end{pmatrix}\theta_1^{\beta_1^2}\tilde\theta^{\tilde\beta^2}\sum_{l=1}^n \theta_1^{2\beta_1+1}d_1\beta_1u_1^{\beta_1-1}\tilde u^{\tilde\beta}\left(\frac{\partial u_1}{\partial x_l}\right)^2\nonumber\\
&=\sum_{\beta_1=1}^{p-1}\sum_{|\tilde\beta|=p-(\beta_1-1)-2}\begin{pmatrix}p\\ (\beta_1-1,\tilde\beta) \end{pmatrix}\theta^{(\beta_1-1,\tilde\beta)^2} u^{(\beta_1-1,\tilde\beta)} \sum_{l=1}^n \theta_1^{4(\beta_1-1)+4}d_1\left(\frac{\partial u_1}{\partial x_l}\right)^2\nonumber\\
&=\sum_{\beta_1=0}^{p-2}\sum_{|\tilde\beta|=p-\beta_1-2}\begin{pmatrix}p\\ \beta \end{pmatrix}\theta^{\beta^2} u^{\beta} \sum_{l=1}^n \theta_1^{4\beta_1+4}d_1\left(\frac{\partial u_1}{\partial x_l}\right)^2\nonumber\\
&=\sum_{|\beta|=p-2}\begin{pmatrix}p\\ \beta \end{pmatrix}\theta^{\beta^2} u^{\beta} \sum_{l=1}^n \theta_1^{4\beta_1+4}d_1\left(\frac{\partial u_1}{\partial x_l}\right)^2,
\end{align}
and for $i\in\{2,...,m_1\}$,
\begin{align}\label{Hp-eq13}
\text{I}_{i,1}&=\sum_{\beta_1=0}^{p-1}\sum_{|\tilde\beta|=p-\beta_1-1}\begin{pmatrix}p\\\beta\end{pmatrix}\theta^{\beta^2}\sum_{l=1}^n \left[ \theta_1^{2\beta_1+1}d_1\tilde\beta_{i-1}u_1^{\beta_1}\tilde u^{\tilde\beta-e_{i-1}}+\tilde\theta_{i-1}^{2\tilde\beta_{i-1}+1}d_i\beta_1u_1^{\beta_1-1}\tilde u^{\tilde\beta} \right]\frac{\partial u_1}{\partial x_l}\frac{\partial u_i}{\partial x_l}\nonumber\\
&=\sum_{\beta_1=0}^{p-2}\sum_{|\tilde\beta|=p-\beta_1-1,\tilde\beta_{i-1}\ne 0}\begin{pmatrix}p\\(\beta_1,\tilde\beta-e_{i-1})\end{pmatrix}\theta^{\beta^2}\sum_{l=1}^n\theta_1^{2\beta_1+1}d_1u_1^{\beta_1}\tilde u^{\tilde\beta-e_{i-1}}\frac{\partial u_1}{\partial x_l}\frac{\partial u_i}{\partial x_l}\nonumber\\
&+\sum_{\beta_1=1}^{p-1}\sum_{|\tilde\beta|=p-\beta_1-1}\begin{pmatrix}p\\(\beta_1-1,\tilde\beta)\end{pmatrix}\theta^{\beta^2}\sum_{l=1}^n\theta_i^{2\beta_i+1}d_iu_1^{\beta_1-1}\tilde u^{\tilde\beta}\frac{\partial u_1}{\partial x_l}\frac{\partial u_i}{\partial x_l}\nonumber\\
&=\sum_{|\beta|=p-2}\begin{pmatrix}p\\\beta\end{pmatrix}\theta^{\beta^2}u^\beta\sum_{l=1}^n\theta_1^{2\beta_1+1}\theta_i^{2\beta_i+1}d_1\frac{\partial u_1}{\partial x_l}\frac{\partial u_i}{\partial x_l}\nonumber\\
&+\sum_{|\beta|=p-2}\begin{pmatrix}p\\\beta\end{pmatrix}\theta^{\beta^2}u^\beta\sum_{l=1}^n\theta_1^{2\beta_1+1}\theta_i^{2\beta_i+1}d_i\frac{\partial u_1}{\partial x_l}\frac{\partial u_i}{\partial x_l}\nonumber\\
&=\sum_{|\beta|=p-2}\begin{pmatrix}p\\\beta\end{pmatrix}\theta^{\beta^2}u^\beta\sum_{l=1}^n\theta_1^{2\beta_1+1}\theta_i^{2\beta_i+1}(d_1+d_i)\frac{\partial u_1}{\partial x_l}\frac{\partial u_i}{\partial x_l}.
\end{align}
The result follows by combining (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>).
Acknowledgement. This work is partially supported by NAWI Graz, and the International Research Training Group IGDK 1754 “Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures”, funded by the German Research Council (DFG) project number 188264188/GRK1754 and the Austrian Science Fund (FWF) under grant number W 1244-N18.
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|
# Semi-supervised Keypoint Localization
Olga Moskvyak, Frederic Maire, Feras Dayoub
School of Electrical Engineering and Robotics
Queensland University of Technology, Australia
<EMAIL_ADDRESS>
&Mahsa Baktashmotlagh
School of Information Technology and Electrical Engineering
The University of Queensland, Australia
<EMAIL_ADDRESS>
###### Abstract
Knowledge about the locations of keypoints of an object in an image can assist
in fine-grained classification and identification tasks, particularly for the
case of objects that exhibit large variations in poses that greatly influence
their visual appearance, such as wild animals. However, supervised training of
a keypoint detection network requires annotating a large image dataset for
each animal species, which is a labor-intensive task. To reduce the need for
labeled data, we propose to learn simultaneously keypoint heatmaps and pose
invariant keypoint representations in a semi-supervised manner using a small
set of labeled images along with a larger set of unlabeled images. Keypoint
representations are learnt with a semantic keypoint consistency constraint
that forces the keypoint detection network to learn similar features for the
same keypoint across the dataset. Pose invariance is achieved by making
keypoint representations for the image and its augmented copies closer
together in feature space. Our semi-supervised approach significantly
outperforms previous methods on several benchmarks for human and animal body
landmark localization.
## 1 Introduction
Detecting keypoints helps with fine-grained classification (Guo & Farrell,
2019) and re-identification (Zhu et al., 2020; Sarfraz et al., 2018). In the
domain of wild animals (Mathis et al., 2018; Moskvyak et al., 2020; Liu et
al., 2019a; b), annotating data is especially challenging due to large pose
variations and the need for domain experts to annotate. Moreover, there is
less commercial interest in keypoint estimation for animals compared to
humans, and little effort is invested in collecting and annotating public
datasets.
Unsupervised detection of landmarks111We use terms keypoints or landmarks
interchangeably in our work. These terms are more generic than body joints
(used in human pose estimation) because our method is applicable to a variety
of categories. (Jakab et al., 2018; Thewlis et al., 2017; 2019) can extract
useful features, but are not able to detect perceptible landmarks without
supervision. On the other hand, supervised learning has the risk of
overfitting if trained only on a limited number of labeled examples. Semi-
supervised learning combines a small amount of labeled data with a large
amount of unlabeled data during training. It is mostly studied for
classification task (van Engelen & Hoos, 2019) but it is also important for
keypoint localization problem because annotating multiple keypoints per image
is a time-consuming manual work, for which precision is the most important
factor. Pseudo-labeling (Lee, 2013) is a common semi-supervised approach where
unlabeled examples are assigned labels (called pseudo-labels) predicted by a
model trained on a labeled subset. A heuristic unsupervised criterion is
adopted to select the pseudo-labeled data for a retraining procedure. More
recently, the works of (Dong & Yang, 2019; Radosavovic et al., 2018) apply
variations to selection criteria in pseudo-labeling for semi-supervised facial
landmark detection. However, there are less variations in facial landmark
positions than in human or animal body joints, where there is a high risk of
transferring inaccurate pseudo-labeled examples to the retraining stage that
is harmful for the model.
Previous work of (Honari et al., 2018) in semi-supervised landmark detection
utilizes additional class attributes and test only on datasets that provide
these attribute annotations. Our work focuses on keypoint localization task in
a common real-world scenario where annotations are provided for a small subset
of data from a large unlabeled dataset. More specifically, we propose a method
for semi-supervised keypoint localization that learns a list of heatmaps and a
list of semantic keypoint representations for each image (Figure 1). A
semantic keypoint representation is a vector of real numbers in a low-
dimensional space relative to the image size, and the same keypoints in
different images have similar representations. We leverage properties that are
specific to the landmark localization problem to design constraints for
jointly optimizing both representations.
We extend a transformation consistency constraint of (Honari et al., 2018) to
be able to apply it on each representation differently (i.e. transformation
equivariant constraint for heatmaps and transformation invariant constraint
for semantic representations). Moreover, we formulate a semantic consistency
constraint that encourages detecting similar features across images for the
same landmark independent of the pose of the object (e.g. an eye in all images
should look similar). Learning both representations simultaneously allows us
to use the power of both supervised and unsupervised learning.
Figure 1: Our semi-supervised keypoint localization system learns a list of
heatmaps and a list of semantic keypoint representations for each image. In
addition to a supervised loss optimized on the labeled subset of the data, we
propose several unsupervised constraints of transformation equivariance,
transformation invariance, and semantic consistency.
Our work is motivated by data scarcity in the domain of wild animals, but is
not limited to animals, and as well, it is applicable to human body landmarks
detection. The contribution of our work is three-fold:
* •
We propose a technique for semi-supervised keypoint localization that jointly
learns keypoint heatmaps and semantic representations optimised with
supervised and unsupervised constraints;
* •
Our method can be easily added to any existing keypoint localization networks
with no structural and with minimal computational overhead;
* •
We evaluate the proposed method on annotated image datasets for both humans
and animals. As demonstrated by our results, our method significantly
outperforms previously proposed supervised and unsupervised methods on several
benchmarks, using only limited labeled data.
The paper is organised as follows. Related work on semi-supervised learning
and keypoint localization is reviewed in Section 2. Our proposed method is
described in Section 3. Experimental settings, datasets and results are
discussed in Section 4.
## 2 Related Work
Keypoint localization. Supervised keypoint localization research is driven by
a few large datasets with labeled keypoints that span across several common
research domains including human pose estimation (Andriluka et al., 2014) and
facial keypoints (Sagonas et al., 2016). Challenges in obtaining keypoint
annotations have led to the rise in unsupervised landmark localization
research. Several unsupervised methods leverage the concept of equivariance
which means that landmark coordinates stay consistent after synthetic
transformations or in subsequent video frames. Thewlis et al. (2017) propose
to learn viewpoint-independent representations that are equivariant to
different transformations and Dong et al. (2018) exploit the coherence of
optical flow as a source of supervision. Zhang et al. (2018) learn landmark
encodings by enforcing constraints that reflect the necessary properties for
landmarks such as separability and concentration. Jakab et al. (2018) propose
a generative approach where the predicted heatmaps are used to reconstruct the
input image from a transformed copy. Recent work (Thewlis et al., 2019)
enforce the consistency between instances of the same object by exchanging
descriptor vectors. These methods are mostly evaluated on faces of people that
have less degrees of freedom during movements and transformations than human
or animal body joints. We compare our method to the combination of supervised
and aforementioned unsupervised methods in Section 4.
Semi-supervised learning is the most studied for the classification task.
Pseudo-labeling (Lee, 2013) is a method that uses the model’s class
predictions as artificial labels for unlabeled examples and then trains the
model to predict these labels. Another technique is a consistency
regularization which states that realistic perturbations of input examples
from unlabeled dataset should not significantly change the output of a neural
network. Consistency regularization is used in $\Pi$-model (Laine & Aila,
2017) and further improved by Temporal Ensembling (Laine & Aila, 2017) which
maintains an exponential moving average prediction for each training example
and Mean Teacher (Tarvainen & Valpola, 2017) that averages model weights
instead of model predictions. Recent methods UDA (Xie et al., 2019),
ReMixMatch (Berthelot et al., 2020) and FixMatch (Sohn et al., 2020) use a
combination of consistency loss, pseudo-labeling and advanced augmentation
techniques in addition to color perturbations and spatial transformations. In
this work, we investigate adjustments required to apply consistency loss to
keypoint localization which we discuss in Section 3.2.
Semi-supervised learning for keypoint localization. To the best of our
knowledge, there are a few works in semi-supervised keypoint localization.
Dong & Yang (2019) build on the pseudo-labeling technique and propose a
teacher model and two students to generate more reliable pseudo-labels for
unlabeled images. However, the method is evaluated on face landmarks and in
cases with high variations of poses, there is a high possibility of inaccurate
pseudo-labels that cannot be filtered out and be harmful during the retraining
stage. Honari et al. (2018); Ukita & Uematsu (2018) learn keypoints in a semi-
supervised manner but utilise extra annotations to guide landmark learning
such as action labels (running, jumping) for juman joints or emotion labels
(smiling, yawning) for facial keypoint localization. Different from previous
work our approach does not use any class labels and learns directly from
unlabeled data with high pose variations.
## 3 Semi-supervised learning for keypoint localization
In this work, we propose a semi-supervised technique for keypoint localization
that learns from an image set where ground truth annotations are provided only
for a small subset of the dataset. The overall architecture consists of two
components: a keypoint localization network (KLN) that outputs keypoint
heatmaps of the image, and a keypoint classification network (KCN) that
classifies keypoints given a semantic keypoint representation as input. Our
method does not pose any constraints on the architecture of the KLN and it can
be added to any existing keypoint localization network with minimal
modifications.
We optimize heatmaps with the supervised loss and the transformation
equivariance constraint. Simultaneously, keypoint representations are
optimized with transformation invariance and semantic consistency constraints
(Figure 1). We discuss each constraint and related components of the
architecture in the next sections.
### 3.1 Semantic keypoint representations
Figure 2: Semantic consistency criteria. Keypoint representation is defined
for each keypoint by multiplying a corresponding predicted heatmap
${\bm{h}}_{i}$ with intermediate features $F$. Keypoint representations are
classified with a shared network $\phi$ and the feedback is added to the total
loss.
Keypoint heatmaps are optimized to estimate locations of keypoints in the
image. However, heatmaps do not carry any information about a semantic type of
the keypoint (e.g, a beak or an eye for a bird). In semi-supervised regime,
the feedback provided by unlabeled examples are not as effective as the ones
coming from labeled examples. To extract useful information from unlabeled
images, we propose learning a semantic keypoint representation. In particular,
keypoint localization network is encouraged to detect similar features for the
same semantic keypoint across the dataset by incorporating the feedback from a
keypoint representation classifier in the objective function.
Motivation for our approach is that the same keypoints should activate the
same feature maps. Let us consider KLN as a function $\displaystyle
f({\bm{x}};{\bm{\theta}})$ with an input image ${\bm{x}}$ and trainable
parameters ${\bm{\theta}}$ that outputs heatmaps ${\bm{h}}=\displaystyle
f({\bm{x}};{\bm{\theta}})$. We collect intermediate feature maps from KLN,
upscale them to the spatial dimension of output heatmaps, concatenate by
channels, and pass through a convolutional layer with $C$ filters of size one
(Figure 2). The resulting feature map $F$ has the shape $(C,H,W)$. Then,
feature maps $F$ are element-wise multiplied with each keypoint heatmap
${\bm{h}}_{i},i\in\\{1,...,K\\}$ seperately to mask out activations
corresponding to the detected keypoint. The output of this operation is $K$
feature maps of size $(C,H,W)$. Global Max Pooling (GMP) is applied over
feature maps to keep the highest value for each channel. We call the produced
vector ${\bm{z}}_{i}=\text{GMP}(F\odot{\bm{h}}_{i})$ for each keypoint
$i\in\\{1,...,K\\}$ a semantic keypoint representation.
Finally, we pass keypoint representations to a simple KCN ($\phi$) which is a
fully connected network with an input and an output layer for classification
with cross-entropy loss. The feedback from the cross-entropy loss makes up a
semantic consistency (SC) loss:
$\displaystyle{\mathcal{L}}_{\text{sc}}({\bm{x}})=-\frac{1}{K}\sum_{i=1}^{K}\hat{y}_{i}\log(\phi({\bm{z}}_{i}))$
(1)
where $\hat{y}$ is a vector of ground truth semantic labels for keypoints
because the order of keypoints in a heatmap is fixed.
One advantage of our method is its efficiency as it only adds a small number
of parameters to the network to address the task of keypoint representation
classification. Specifically, KCN is a small fully connected network shared
between keypoints and it has less than a thousand of parameters depending on
the number of keypoints. Our approach is related to attention modules (Vaswani
et al., 2017; Hu et al., 2020) as our network has the ability to focus on a
subset of features using element-wise multiplication with heatmaps. However,
our model uses this attention-based mechanism to learn additional keypoint
representations from unlabeled data by optimizing a set of unsupervised
losses.
### 3.2 Transformation consistency constraint
The difference between keypoint heatmaps and semantic keypoint representations
is that the former is transformation equivariant and the latter is
transformation invariant. In other words, the output heatmaps should be
consistent with viewpoint variations of the image while keypoint
representations should be preserved for all different transformations of the
image. We call this property a transformation consistency constraint.
Transformation equivariance (TE) enforces a commutative property on the
landmark localization and augmentation operations that include spatial
transformation (e.g, rotations and translations), meaning that the order of
applying these two operations does not matter. Let $g(\cdot,{\bm{s}})$ be an
augmentation function with augmentation parameters ${\bm{s}}$ which are not
trainable and sampled randomly each time. Transformation equivariance
constraint is formulated as: $\displaystyle f\circ
g({\bm{x}})=g\circ\displaystyle f({\bm{x}})$. We measure a transformation
equivariance loss $\displaystyle{\mathcal{L}}_{\text{te}}$ over predicted
heatmaps by squared Euclidean distance:
$\displaystyle{\mathcal{L}}_{\text{te}}({\bm{x}};{\bm{\theta}})=\displaystyle\mathbb{E}_{{\bm{x}}}\Big{[}\displaystyle||\displaystyle
f(g({\bm{x}},{\bm{s}});{\bm{\theta}})-g(\displaystyle
f({\bm{x}};{\bm{\theta}}),{\bm{s}})||^{2}\Big{]}$ (2)
Note that, after applying a transformation, some landmarks may go outside of
the image boundary, and cause the visibility issue. This problem is alleviated
in our formulation by applying the same transformation to an image. This is
different from equivariant landmark transformation (ELT) loss proposed by
Honari et al. (2018) which computes an inverse transformation instead. In
essence, inverse transformation cannot bring these landmarks back meaning that
inverse transformation does not output the original image. Our approach avoids
this issue.
Transformation invariance (TI) of keypoint representations is enforced by
pulling corresponding vectors for the image and its augmented copy closer
together. First, we concatenate keypoint representations in one vector to get
a holistic representation ${\bm{z}}$ of the image ${\bm{x}}$:
${\bm{z}}=[{\bm{z}}_{1},{\bm{z}}_{2},...,{\bm{z}}_{K}].$ (3)
We apply a random spatial transformation to the input image to get image
${\bm{x}}^{\prime}$, compute keypoint representations
${\bm{z}}_{1}^{\prime},{\bm{z}}_{2}^{\prime},...,{\bm{z}}_{K}^{\prime}$, and
concatenate them to get a vector ${\bm{z}}^{\prime}$. Finally, we enforce pose
invariance by penalizing a distance between representations of original and
transformed images and formulate a transformation invariance loss
$\displaystyle{\mathcal{L}}_{\text{ti}}$:
$\displaystyle{\mathcal{L}}_{\text{ti}}({\bm{x}},{\bm{x}}^{\prime})=\displaystyle\mathbb{E}_{{\bm{x}},{\bm{x}}^{\prime}}\Big{[}\displaystyle||{\bm{z}}-{\bm{z}}^{\prime}||^{2}\Big{]}$
(4)
The overall objective is the weighted sum of losses:
$\displaystyle{\mathcal{L}}=\lambda_{1}\displaystyle{\mathcal{L}}_{\text{sup}}+\lambda_{2}\displaystyle{\mathcal{L}}_{\text{sc}}+\lambda_{3}\displaystyle{\mathcal{L}}_{\text{te}}+\lambda_{4}\displaystyle{\mathcal{L}}_{\text{ti}}$
(5)
where $\displaystyle{\mathcal{L}}_{\text{sup}}$ is a supervised mean squared
error between predicted and ground truth heatmaps for the labeled subset.
Parameters $\lambda_{i}$ are defined experimentally.
## 4 Experiments
### 4.1 Datasets
We evaluate our method on two datasets with annotated human body joints and
two datasets of wild animals.
MPII Human Pose dataset (Andriluka et al., 2014) is a collection of images
showing people doing real-world activities with annotations for the full body.
Due to the fact that test annotations are not released publicly, we use
training and validation splits of MPII in our experiments. We use 10,000
images for training to speed up experiments as we run multiple training runs
for each subset of labeled examples. Our validation and test sets consist of
3,311 and 2,958 images respectively. Annotations contain coordinates for 16
body joints with a visibility flag.
LSP (Leeds Sports Pose) (Johnson & Everingham, 2010; 2011) dataset is a
collection of annotated images with people doing sports such as athletics,
badminton or soccer. Each image has been annotated with 14 joint locations. We
use 10,000 images from extended (Johnson & Everingham, 2011) version for
training and 2,000 images from original (Johnson & Everingham, 2010) dataset
for testing and validation.
CUB-200-2011 (Welinder et al., 2010) is a dataset of 200 fine-grained classes
of bird species. We split dataset into training, validation and testing with
disjoint classes so test classes does not appear during training. First 100
classes are used for training (5,864 images), 50 classes for validation (2,958
images) and the last 50 classes (2,966 images) for testing. Each image is
annotated with 15 body keypoints such as beak, left eye and throat. We use
class label only for splitting the dataset and do not use it anywhere in out
method.
ATRW (Li et al., 2019) is a dataset of Amur tigers images captured in multiple
wild zoos in unconstrained settings. Professionals annotated 15 skeleton
keypoints for each tiger. We use 3,610 images for training, 516 for validation
and 1,033 for testing with annotations provided by authors. This dataset is
more challenging than birds as four-legged animals exhibit more pose
variations.
Training set for each dataset is split into labeled and unlabeled subsets by
randomly picking 5%, 10%, 20% or 50% of the training examples and discarding
the labels for the rest of the data. The procedure is repeated three times so
all experiments are run three times to obtain the mean and standard deviation
of the results. Validation and test sets are fixed for all experiments.
Validation set is used to tune hyperparameters and test set is used to report
the final results. The order of the keypoints is explicitly defined in
annotations and is fixed for the training and inference.
The evaluation metric is PCK (probability of correct keypoint) from (Yang &
Ramanan, 2013) where a keypoint is considered correctly localized if it falls
within $\alpha l$ pixels of the ground truth position ($\alpha$ is a constant
and $l$ is the maximum side of the bounding box). The<EMAIL_ADDRESS>$(\alpha=0.1)$
score is reported for LSP, CUB-200-2011 and ATRW datasets. For MPII we use an
adaptation (Andriluka et al., 2014) which is PCKh (head-normalized probability
of correct keypoint) where $l$ is the head size that corresponds to 60% of the
diagonal length of the ground truth head bounding box (provided in the MPII
annotations).
### 4.2 Implementation details
Images are resized to the input size $256\times 256$ and heatmaps are
predicted at size $64\times 64$. We adapt HRNet-32 (Sun et al., 2019)
architecture as KLN because it is originally designed for keypoint
localization and retains features at high spatial dimension (e.g. $64\times
64$ for the input of size $256\times 256$). We collect intermediate features
at the output of each multi-scale subnetwork, after concatenation we get 352
channels and then apply 64 convolutional filters of size one. GMP results in
representations of length 64 for each keypoint. We also experimented with
collecting more features from different layers but it did not improve the
performance. KCN is a fully connected network that accepts keypoint
representation of size 64 and classifies keypoints based on their semantic
labels (from 10 to 17 depending on the dataset).
We use perspective transformations as an augmentation function $g$ where
parameters ${\bm{s}}$ of the transformation are sampled randomly using a
method from (Moskvyak & Maire, 2017) to avoid extreme warping. We also
experimented with simple affine transformations but perspective gave better
results most likely due to higher variability of transformations.
Unsupervised losses may hurt the learning at the beginning because output
heatmaps and intermediate feature maps are random during first epochs. A
possible solution is to vary the contribution of unsupervised losses according
to a predefined strategy. To avoid tuning many hyperparameters, our semi-
supervised approach uses ground truth heatmaps in unsupervised losses for the
labeled samples in a batch. This approach has only one hyperparameter -
percentage of the labeled samples in a batch. We found that there is enough
feedback from labeled examples when the batch has 50% of labeled and 50% of
unlabeled examples.
We adopt Adam (Kingma & Ba, 2015) optimizer with learning rate $10^{-4}$ for
all experiments. Models are trained until the accuracy on the validation set
has stopped improving. The weights of loss components were determined
experimentally
$(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})=(10^{3},0.5,10^{2},10^{2})$.
We provide the sensitivity analysis in Section 4.
### 4.3 Results
Table 1<EMAIL_ADDRESS>score for keypoint localization with different percentage of labeled images. We report mean and standard deviation from three runs for different randomly sampled labeled subsets. Pseudo-labeled (PL) baseline is not evaluated for 100% of labeled data because there is no unlabeled data to generate pseudo-labels for. | Percentage of labeled images
---|---
Method | 5% | 10% | 20% | 50% | 100%
Dataset 1: MPII
HRNet (Sun et al., 2019) | 66.22$\pm$1.60 | 69.18$\pm$1.03 | 71.83$\pm$0.87 | 75.73$\pm$0.35 | 81.11$\pm$0.15
PL (Radosavovic et al., 2018) | 62.44$\pm$1.75 | 64.78$\pm$1.44 | 69.35$\pm$1.11 | 77.43$\pm$0.48 | -
ELT (Honari et al., 2018) | 68.27$\pm$0.64 | 71.03$\pm$0.46 | 72.37$\pm$0.58 | 77.75$\pm$0.31 | 81.01$\pm$0.15
Gen (Jakab et al., 2018) | 71.59$\pm$1.12 | 72.63$\pm$0.62 | 74.95$\pm$0.32 | 79.86$\pm$0.19 | 80.92$\pm$0.32
Ours | 74.15$\pm$0.83 | 76.56$\pm$0.48 | 78.46$\pm$0.36 | 80.75$\pm$0.32 | 82.12$\pm$0.14
Dataset 2: LSP
HRNet (Sun et al., 2019) | 40.19$\pm$1.46 | 45.17$\pm$1.15 | 55.22$\pm$1.41 | 62.61$\pm$1.25 | 72.12$\pm$0.30
PL (Radosavovic et al., 2018) | 37.36$\pm$1.89 | 42.05$\pm$1.68 | 48.86$\pm$1.23 | 64.45$\pm$0.96 | -
ELT (Honari et al., 2018) | 41.77$\pm$1.56 | 47.22$\pm$0.91 | 57.34$\pm$0.94 | 66.81$\pm$0.62 | 72.22$\pm$0.13
Gen (Jakab et al., 2018) | 61.01$\pm$1.41 | 67.75$\pm$1.00 | 68.80$\pm$0.91 | 69.70$\pm$0.77 | 72.25$\pm$0.55
Ours | 66.98$\pm$0.94 | 69.56$\pm$0.66 | 71.85$\pm$0.33 | 72.59$\pm$0.56 | 74.29$\pm$0.21
Dataset 3: CUB-200-2011
HRNet (Sun et al., 2019) | 85.77$\pm$0.38 | 88.62$\pm$0.14 | 90.18$\pm$0.22 | 92.60$\pm$0.28 | 93.62$\pm$0.13
PL (Radosavovic et al., 2018) | 86.31$\pm$0.45 | 89.51$\pm$0.32 | 90.88$\pm$0.28 | 92.78$\pm$0.27 | -
ELT (Honari et al., 2018) | 86.54$\pm$0.34 | 89.48$\pm$0.25 | 90.86$\pm$0.13 | 92.26$\pm$0.06 | 93.77$\pm$0.18
Gen (Jakab et al., 2018) | 88.37$\pm$0.40 | 90.38$\pm$0.22 | 91.31$\pm$0.21 | 92.79$\pm$0.14 | 93.62$\pm$0.25
Ours | 91.11$\pm$0.33 | 91.47$\pm$0.36 | 92.36$\pm$0.30 | 92.80$\pm$0.24 | 93.81$\pm$0.13
Dataset 4: ATRW
HRNet (Sun et al., 2019) | 69.22$\pm$0.87 | 77.55$\pm$0.84 | 86.41$\pm$0.45 | 92.17$\pm$0.18 | 94.44$\pm$0.10
PL (Radosavovic et al., 2018) | 67.97$\pm$1.07 | 75.26$\pm$0.74 | 84.69$\pm$0.57 | 92.15$\pm$0.24 | -
ELT (Honari et al., 2018) | 74.53$\pm$1.24 | 80.35$\pm$0.96 | 87.98$\pm$0.47 | 92.80$\pm$0.21 | 94.75$\pm$0.14
Gen (Jakab et al., 2018) | 89.54$\pm$0.57 | 90.48$\pm$0.49 | 91.16$\pm$0.13 | 92.27$\pm$0.24 | 94.80$\pm$0.13
Ours | 92.57$\pm$0.64 | 94.29$\pm$0.66 | 94.49$\pm$0.36 | 94.63$\pm$0.18 | 95.31$\pm$0.12
Comparison with the supervised baseline. We train HRNet-32 (Sun et al., 2019)
with the supervised loss as a baseline from the official implementation on the
labeled subsets with 5%, 10%, 20%, 50% and 100% of the dataset. The baseline
performance decreases significantly when the amount of training data is
reduced on human poses and tigers datasets (Table 1). On birds dataset, we
observe only a small decrease in the baseline score (Table 1). We explain it
by the fact that there are more variations in poses of four-legged animals and
human body joints than of birds. Supervised results on MPII are lower than the
official ones because the training set is smaller and we do not include
additional tricks during training (e.g. half body transforms) and testing
(post-processing and averaging over flipped images).
Our method significantly improves the baseline on all datasets (Table 1). Our
proposed unsupervised constraints are the most beneficial for low data regimes
with 5%, 10% and 20% labeled images. For example, our method increases the
score from 40% to 66% on LSP dataset with 5% of labeled data. On the
challenging tigers dataset, our approach reaches the score of 92% trained with
only 5% labeled examples when the supervised model shows the score 69% while
trained on the same labeled data. Experiments show that the influence of
additional unsupervised losses decreases when more labeled examples are added
to the training. Experiments show that our method on 100% labeled data
outperforms the supervised baseline by a small margin because by learning
supplementary semantic keypoint representations with unsupervised losses the
model learns to generalize better.
Table 2: Ablation study on LSP. We isolate gains from each unsupervised loss to evaluate the contribution of each constraint. SC - semantic consistency TE - transformation equivariance and TI - transformation invariance. TE is not evaluated separately because it does not optimize both representations. Results are reported on one run. | Percentage of labeled images
---|---
Unsupervised losses | 5% | 10% | 20% | 50% | 100%
TE + TI + SC | 66.32 | 69.09 | 71.62 | 72.19 | 74.44
TE + TI | 46.76 | 55.18 | 64.01 | 67.54 | 72.11
SC | 64.74 | 67.43 | 69.65 | 70.61 | 72.85
TI + SC | 65.23 | 68.11 | 70.12 | 71.28 | 73.56
TE + SC | 65.78 | 68.51 | 70.56 | 71.77 | 73.89
TI | 43.62 | 53.74 | 61.12 | 65.32 | 71.80
Supervised baseline | 39.16 | 44.36 | 54.23 | 61.73 | 71.91
Table 3: Influence of the amount of unlabeled data. We train our model with the fixed 5% of labeled data and vary the amount of unlabeled data. Results are reported on one run. The results are compared with the supervised baseline (SB) trained with the same labeled data. | Percentage of unlabeled images |
---|---|---
Dataset | 10% | 20% | 50% | 100% | SB
CUB-200-2011 | 87.01 | 88.33 | 89.44 | 91.34 | 85.33
ATRW | 72.04 | 76.65 | 86.56 | 93.02 | 69.84
Comparison with the pseudo-labeled baseline. We apply pseudo-labeled (PL)
method from Radosavovic et al. (2018) on our datasets (Table 1). We use the
same model HRNet-32 as in all our experiments for a fair comparison. Overall,
the pseudo-labeled baseline is inferior to our method on all datasets used in
our study. We explain it by the fact that Radosavovic et al. (2018) trained on
datasets that are by order of magnitude larger than our data so models
pretrained on the labeled subset are already good enough to generate reliable
pseudo-labels.
Comparison with related methods. We compare our approach with previously
proposed semi-supervised and unsupervised methods for landmark detection
(Table 1). The equivariant landmark transformation (ELT) loss from (Honari et
al., 2018) forces a model to predict equivariant landmarks with respect to
transformations applied to an image. ELT loss gives a small improvement over
the baseline model and is inferior to our method on all datasets. Jakab et al.
(2018) learn keypoints without supervision by encouraging the keypoints to
capture the geometry of the object by learning to generate the input image
given its predicted keypoints and an augmented copy. For a fair comparison we
inject the models from Jakab et al. (2018) into our training pipeline and add
the supervised loss for the labeled examples in each batch. All other
parameters are kept the same including augmentation, subsets of data and
training schedule. We observe that the generation approach improves over ELT
loss and the baseline however it is inferior to our method. The generation
approach also introduces more parameters (in the reconstruction part of the
network) than our approach that adds only a small keypoint classifier network.
Ablation study. We investigate the influence of different loss components of
our methods on LSP dataset (Table 2). At first, we remove semantic consistency
loss component (Eq. 1) and observe the significant drop in the score
especially in low labeled data regime. For example, with 5% of labeled data
the score drops from 66% when trained with the combination TE + TI + SC to 46%
for the combination TE + TI. When we return semantic consistency and remove
transformation consistency losses (Eq. 2, 4), the results are reduced
slightly. The results of ablation study shows that the semantic consistency
loss component is more influential than the transformation consistency. Both
TE and TI losses contribute to the performance gain and their combination
achieves better results than each loss separately. We argue that our TE loss
is an improvement over ELT loss (Honari et al., 2018). We replaced our TE loss
with an inverse transformation loss of Honari et al. (2018) in our framework,
and applied it on ATRW and CUB-200-2011 datasets with 20% of labeled data. We
observed that the score decreased by 1% on both datasets.
We also analyse the influence of the amount of unlabeled data in our method
(Table 3). We conduct experiments where the amount of labeled examples is
fixed at 5% and the number of unlabeled examples is reduced to 50%, 20% and
10% of the number of original unlabeled samples. We observe that the score
goes down as the amount of unlabeled data is reduced. Our method outperforms
the supervised score only by a small margin with 10% of unlabeled data. We
conclude that the number of unlabeled examples plays an important role in
training with our unsupervised losses.
We conduct an ablation study to get an insight on using ground truth heatmaps
in unsupervised losses. Experiments on the LSP dataset show a decrease of 1-2%
in the score for all cases when ground truth heatmaps are not used (Table 4).
The results prove the benefit of using the signal from available ground truth
heatmaps.
Table 4: An ablation study of using ground truth heatmaps in unsupervised losses on LSP dataset. Results are reported on one run. | Percentage of labeled images
---|---
Method | 5% | 10% | 20% | 50% | 100%
With g/t heatmaps | 66.32 | 69.09 | 71.62 | 72.19 | 74.44
Without g/t heatmaps | 64.75 | 67.27 | 69.91 | 70.55 | 73.65
Sensitivity analysis of weight loss components. We fixed the weight
$\lambda_{1}=10^{3}$ and tested weights: $\lambda_{2}=(0.1,0.5,1.)$,
$\lambda_{3}=(10^{1},10^{2},10^{3})$ and $\lambda_{4}=(10^{1},10^{2},10^{3})$.
The ranges of weight values are different due to differences in scales for
mean squared error and cross-entropy loss. Experiments on LSP dataset show
that our method is not sensitive to variations of TE ($\lambda_{3}$) and TI
($\lambda_{4}$) losses (Figure 3). The most notable drop in accuracy is
observed when the weight of SC loss ($\lambda_{2}$) is reduced to 0.1 and the
accuracy is at the same level when $\lambda_{2}$ equals 0.5 and 1.0. We select
the combination of $(\lambda_{2},\lambda_{3},\lambda_{4})=(0.5,10^{2},10^{2})$
that achieves the highest score.
Figure 3: Sensitivity analysis of the weights of loss components.
Analysis of keypoint representations. We analyze the learned keypoint
representation with t-SNE (van der Maaten & Hinton, 2008). The t-SNE algorithm
maps a high dimensional space (64 dimensions in our case) into a two-
dimensional while preserving the similarity between points. The t-SNE plot for
the keypoint representations of LSP test set (Figure 4) shows that
representations for the same keypoints are clustered together.
Figure 4: tSNE visualization of keypoint embeddings for human body landmarks
on LSP test set.
## 5 Conclusion
We presented a new method for semi-supervised keypoint localization. We show
that reliable keypoints can be obtained with a limited number of labeled
examples. This is achieved by learning semantic keypoint representations
simultaneously with keypoint heatmaps using a set of unsupervised constraints
tailored for the keypoint localization task. We applied our method to predict
human body joints and animal body keypoints and demonstrated that it
outperforms current supervised and unsupervised methods. Moreover, it reaches
the same performance as the model trained on the whole labeled dataset with
only 10% of labeled images on tigers ATRW dataset and with 50% labeled images
on challenging human poses LSP dataset. We plan to investigate the
applicability of our method to domain adaptation for keypoint localization in
the future work.
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# Eigenvalue Inequalities for the
Clamped Plate Problem of $\mathfrak{L}^{2}_{\nu}$ Operator
Lingzhong Zeng Lingzhong Zeng
College of Mathematics and Statistics
Jiangxi Normal University, Nanchang 330022, China<EMAIL_ADDRESS>
###### Abstract.
$\mathfrak{L}_{II}$ operator is introduced by Y.-L. Xin (_Calculus of
Variations and Partial Differential Equations. 2015, 54(2):1995-2016)_, which
is an important extrinsic elliptic differential operator of divergence type
and has profound geometric meaning. In this paper, we extend
$\mathfrak{L}_{II}$ operator to more general elliptic differential operator
$\mathfrak{L}_{\nu}$, and investigate the clamped plate problem of
bi-$\mathfrak{L}_{\nu}$ operator, which is denoted by
$\mathfrak{L}_{\nu}^{2}$, on the complete Riemannian manifolds. A general
formula of eigenvalues for the $\mathfrak{L}_{\nu}^{2}$ operator is
established. Applying this formula, we estimate the eigenvalues with lower
order on the Riemannian manifolds. As some further applications, we establish
some eigenvalue inequalities for this operator on the translating solitons
with respect to the mean curvature flows, submanifolds of the Euclidean
spaces, unit spheres and projective spaces. In particular, for the case of
translating solitons, all of the eigenvalue inequalities are universal.
††footnotetext: Key words and phrases: mean curvature flows;
$\mathfrak{L}_{\nu}^{2}$ operator; clamped plate problem; eigenvalues;
Riemannian manifolds; translating solitons.††footnotetext: 2010 Mathematics
Subject Classification: 35P15, 53C40.
## 1\. Introduction
Suppose that $(\mathcal{M}^{n},g)$ is an $n$-dimensional, complete, noncompact
Riemannian manifold with smooth metric $g$ and $\Omega$ is a bounded domain
with piecewise smooth boundary $\partial\Omega$ on $\mathcal{M}^{n}$. We
consider the following fixed membrane problem of Laplacian on
$\mathcal{M}^{n}$:
(1.1) ${\begin{cases}\ \Delta u=-\lambda u,\ \ &{\rm in}\ \ \ \ \Omega,\\\ \
u=0,\ \ &{\rm on}\ \ \partial\Omega,\end{cases}}$
where $\Delta$ denotes the Laplacian on the Riemannian manifold
$\mathcal{M}^{n}$. Let $\lambda_{k}$ denote the $k^{th}$ eigenvalue, and then
the spectrum of the eigenvalue problem (1.1) is discrete and satisfies
$0<\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{k}\leq\cdots\rightarrow+\infty,$
where each eigenvalue is repeated according to its multiplicity. Supposing
that $\Omega$ is a bounded domain on the two dimensional plane
$\mathbb{R}^{2}$, eigenvalue problem (1.1) describes an interesting physical
phenomenon of two-dimensional membrane vibration. For this case, Payne, Pólya
and Weinberger [24] investigated Dirichlet problem (1.1) of Laplacian and
proved that
(1.2) $\lambda_{2}+\lambda_{3}\leq 6\lambda_{1}$
in 1956. Furthermore, they proposed a famous conjecture for
$\Omega\subset\mathbb{R}^{n}$ as follows:
(1.3) $\frac{\lambda_{2}+\lambda_{3}+\cdots+\lambda_{n+1}}{\lambda_{1}}\leq
n\frac{\lambda_{2}(\mathbb{B}^{n})}{\lambda_{1}(\mathbb{B}^{n})},$
where $\lambda_{i}(\mathbb{B}^{n})(i=1,2)$ denotes the $i^{th}$ eigenvalue of
Laplacian on $\mathbb{B}^{n}\subset\mathbb{R}^{n}$, and $\mathbb{B}^{n}$ is an
$n$-dimensional ball with the same volume as $\Omega$, i.e.,
$Vol(\Omega)=Vol(\Omega^{\ast})$. Attacking this conjecture, Brands [6]
improved (1.2) to the following:
$\lambda_{2}+\lambda_{3}\leq\lambda_{1}(3+\sqrt{7}),$ when $n=2$. Soon
afterwards, Hile and Protter [18] obtained $\lambda_{2}+\lambda_{3}\leq
5.622\lambda_{1}.$ In 1980, Marcellini [23] obtained
$\lambda_{2}+\lambda_{3}\leq(15+\sqrt{345})/6\lambda_{1}.$ In 2011, Chen and
Zheng [10] proved $\lambda_{2}+\lambda_{3}\leq 5.3507\lambda_{1}.$ For the
general dimension, Ashbaugh and Benguria [1] established a universal
inequality as follows:
(1.4) $\frac{\lambda_{2}+\lambda_{3}+\cdots+\lambda_{n+1}}{\lambda_{1}}\leq
n+4,$
for $\Omega\subset\mathbb{R}^{n}$ in 1993. As for further references on the
solution of this conjecture, we refer the readers to [2, 3, 16, 18, 25]. In
2008, Sun, Cheng and Yang [26] studied Dirichlet problem (1.1) on the bounded
domains in a complex projective space and a unit sphere, and they derived some
universal eigenvalue inequalities with lower order. In 2008, Chen and Cheng
[8] showed that inequality (1.4) remains true when $\Omega$ is a bounded
domain in a complete Riemannian manifold isometrically minimally immersed in
$\mathbb{R}^{n+p}$. Furthermore, Ashbaugh and Benguria [1] (cf. Hile and
Protter [18] ) improved (1.4) to
(1.5) $\frac{\lambda_{2}+\lambda_{3}+\cdots+\lambda_{n+1}}{\lambda_{1}}\leq
n+3+\frac{\lambda_{1}}{\lambda_{2}}.$
In 2012, Cheng and Qi [14] proved that, for any positive integer $j$, where
$1\leq j\leq n+2$, eigenvalues satisfy at least one of the following universal
inequalities:
$\displaystyle{\rm(1)}\ \
\frac{\lambda_{2}}{\lambda_{1}}<2-\frac{\lambda_{1}}{\lambda_{j}},\ \ \
{\rm(2)}\ \
\frac{\lambda_{2}+\lambda_{3}+\cdots+\lambda_{n+1}}{\lambda_{1}}\leq
n+3+\frac{\lambda_{1}}{\lambda_{j}}.$
In 2002, Levitin and Parnovski [22] proved an abstract algebraic inequality.
Applying this algebraic inequality, they generalized (1.4) to the following
eigenvalue inequality:
(1.6)
$\frac{\lambda_{j+1}+\lambda_{j+2}+\cdots+\lambda_{j+n}}{\lambda_{j}}\leq
n+4,$
where $j$ is any positive integer.
To describe vibrations of a clamped plate in elastic mechanics, one usually
consider the following Dirichlet problem of biharmonic operator :
(1.7) $\left\\{\begin{array}[]{l}\Delta^{2}u=\Lambda u,\quad\text{ in
}\Omega,\\\ u=\frac{\partial u}{\partial\textbf{n}}=0,\quad\text{ on
}\partial\Omega,\end{array}\right.$
where $\Delta$ is the Laplacian in $\mathbb{R}^{n}$ and $\Delta^{2}$ is the
biharmonic operator in $\mathbb{R}^{n}$, and this eigenvalue problem is called
a clamped plate problem. In 1956, Payne, Pólya and Weinberger [24] also
established a universal inequality for eigenvalue problem (1.7). They obtained
(1.8)
$\Lambda_{k+1}-\Lambda_{k}\leq\frac{8(n+2)}{n^{2}}\frac{1}{k}\sum_{i=1}^{k}\Lambda_{i}.$
In 1984, by improving Hile and Protter’s method in [18], Hile and Yeh [19]
obtained
(1.9)
$\sum_{i=1}^{k}\frac{\Lambda_{i}^{\frac{1}{2}}}{\Lambda_{k+1}-\Lambda_{i}}\geq\frac{n^{2}k^{3/2}}{8(n+2)}\left(\sum_{i=1}^{k}\Lambda_{i}\right)^{-1/2},$
which generalizes the above result obtained by Payne, Pólya and Weinberger.
Furthermore, in $1990,$ Hook [20], Chen and Qian [12] proved, independently,
the following inequality:
(1.10)
$\frac{n^{2}k^{2}}{8(n+2)}\leq\left[\sum_{i=1}\frac{\Lambda_{i}^{\frac{1}{2}}}{\Lambda_{k+1}-\Lambda_{i}}\right]\sum_{i=1}^{k}\Lambda_{i}^{\frac{1}{2}}.$
In 1999, Ashbaugh pointed out “ _whether one can establish inequalities for
eigenvalues of the vibrating clamped plate problem which are analogous
inequalities of Yang in the case of the eigenvalue problem of the Laplacian
with Dirichlet boundary condition_ ” in his survey paper [4]. In 2006, Cheng
and Yang [15] gave an affirmative answer to the problem posed by Ashbaugh.
Specifically, they obtained the following:
(1.11)
$\Lambda_{k+1}-\frac{1}{k}\sum_{i=1}^{k}\Lambda_{i}\leq\left[\frac{8(n+2)}{n^{2}}\right]^{\frac{1}{2}}\frac{1}{k}\sum_{i=1}^{k}\left[\Lambda_{i}\left(\Lambda_{k+1}-\Lambda_{i}\right)\right]^{\frac{1}{2}},$
which is sharper than
(1.12)
$\Lambda_{k+1}\leq\left[1+\frac{8(n+2)}{n^{2}}\right]\frac{1}{k}\sum_{i=1}^{k}\Lambda_{i}.$
It is easy to see that inequality (1.12) is better than inequality (1.8) given
by Payne, Pólya and Weinberger. In 1999, Ashbaugh [4] announced two universal
eigenvalue inequalities which are analogous to (1.4) for any dimension $n$ as
follows:
(1.13)
$\sum_{\alpha=1}^{n}\left(\Lambda_{\alpha+1}^{\frac{1}{2}}-\Lambda_{1}^{\frac{1}{2}}\right)\leq
4\Lambda_{1}^{\frac{1}{2}},$
and
(1.14) $\sum_{\alpha=1}^{n}\left(\Lambda_{\alpha+1}-\Lambda_{1}\right)\leq
24\Lambda_{1}.$
Next, we consider that $X:\mathcal{M}^{n}\rightarrow\mathbb{R}^{n+p}$ is an
isometric immersion from an $n$-dimensional, oriented, complete Riemannian
manifold $\mathcal{M}^{n}$ to the Euclidean space $\mathbb{R}^{n+p}$, and let
$\Omega$ be a bounded domain with smooth boundary $\partial\Omega$ in
$\mathcal{M}^{n}$. Assume that$\left\\{e_{1},\ldots,e_{n}\right\\}$ is a local
orthonormal basis of $\mathcal{M}^{n}$ with respect to the induced Riemannian
metric $g$, and $\\{e_{n+1},\ldots,e_{n+p}\\}$ is the corresponding local unit
orthonormal normal vector fields. Assume that
$\textbf{H}=\frac{1}{n}\sum_{\alpha=n+1}^{n+p}H^{\alpha}e_{\alpha}=\frac{1}{n}\sum_{\alpha=n+1}^{n+p}\left(\sum_{i=1}^{n}h_{ii}^{\alpha}\right)e_{\alpha},$
is the mean curvature vector field, and
$H=|\textbf{H}|=\frac{1}{n}\sqrt{\sum_{\alpha=n+1}^{n+p}\left(\sum_{i=1}^{n}h_{ii}^{\alpha}\right)^{2}},$
is the mean curvature of $\mathcal{M}^{n}$ in this paper. Let $\Pi$ denote the
set of all isometric immersions from $\mathcal{M}^{n}$ into a Euclidean space.
In 2010, Cheng, Huang and Wei [13] proved
(1.15) $\sum^{n}_{\alpha=1}(\Lambda_{\alpha+1}-\Lambda_{1})^{\frac{1}{2}}\leq
4\left[\left(\frac{n}{2}+1\right)\Lambda^{\frac{1}{2}}_{1}+C_{0}\right]^{\frac{1}{2}}\left(\Lambda^{\frac{1}{2}}_{1}+C_{0}\right)^{\frac{1}{2}},$
where
$C_{0}=\frac{1}{4}\inf_{\sigma\in\Pi}\max_{\Omega}\left(n^{2}H^{2}\right).$
In particular, when $\mathcal{M}^{n}$ is an $n$-dimensional complete minimal
submanifold in a Euclidean space, (1.15) implies
(1.16)
$\sum^{n}_{\alpha=1}(\Lambda_{\alpha+1}-\Lambda_{1})^{\frac{1}{2}}\leq[8(n+2)\Lambda_{1}]^{\frac{1}{2}}.$
In 2011, Wang and Xia [29] investigated the eigenvalues with higher order of
bi-harmonic operator on the complete Riemannian manifolds and proved the
following inequality:
(1.17)
$\displaystyle\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)^{2}\leq$
$\displaystyle\frac{4}{n}\left\\{\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)^{2}\left[\left(\frac{n}{2}+1\right)\Lambda_{i}^{\frac{1}{2}}+C_{0}\right]\right\\}^{\frac{1}{2}}$
$\displaystyle\times\left\\{\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)\left(\Lambda_{i}^{\frac{1}{2}}+C_{0}\right)\right\\}^{\frac{1}{2}},$
where
$C_{0}=\frac{1}{4}\inf_{\sigma\in\Pi}\max_{\Omega}\left(n^{2}H^{2}\right).$
Let $\nu$ be a constant vector field defined on $\mathcal{M}^{n}$. Throughout
this paper, we use $\langle\cdot,\cdot\rangle_{g}$, $|\cdot|_{g}^{2}$, ${\rm
div}$, $\Delta$, $\nabla$ and $\nu^{\top}$ to denote the Riemannian inner
product with respect to the induced metric $g$, norm associated with the inner
product $\langle\cdot,\cdot\rangle_{g}$, divergence, Laplacian, the gradient
operator on $\mathcal{M}^{n}$ and the projective of the vector $\nu$ on the
tangent bundle $T\mathcal{M}^{n}$, respectively. Next, we define an elliptic
operator on $\mathcal{M}^{n}$ as follows:
(1.18)
$\mathfrak{L}_{\nu}=\Delta+\langle\nu,\nabla(\cdot)\rangle_{g_{0}}=e^{-\langle\nu,X\rangle_{g_{0}}}{\rm
div}(e^{\langle\nu,X\rangle_{g_{0}}}\nabla(\cdot)),$
where $\langle\cdot,\cdot\rangle_{g_{0}}$ denotes the standard inner product
of $\mathbb{R}^{n+p}$. Correspondingly, we use $|\cdot|_{g_{0}}$ to denote the
norm on $\mathbb{R}^{n+p}$ associated with the standard inner product
$\langle\cdot,\cdot\rangle_{g_{0}}$. In particular, we assume that $\nu$ is a
unit constant vector defined on a translating soliton in the sense of the
means curvature flows and denote it by $\nu_{0}$. Then, the differential
operator will be denoted by $\mathfrak{L}_{II}$, which is introduced by Xin in
[30] and of important geometric meaning. We refer the readers to section 5 for
details.
###### Remark 1.1.
It can be shown that the elliptic differential operator $\mathfrak{L}_{\nu}$
is a self-adjoint operator with respect to the weighted measure
$e^{\langle\nu,X\rangle_{g_{0}}}dv$. Namely, for any $u,w\in
C_{1}^{2}(\Omega)$, the following formula holds:
(1.19) $\displaystyle-\int_{\Omega}\langle\nabla u,\nabla
w\rangle_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv=\int_{\Omega}(\mathfrak{L}_{\nu}w)ue^{\langle\nu,X\rangle_{g_{0}}}dv=\int_{\Omega}(\mathfrak{L}_{\nu}u)we^{\langle\nu,X\rangle_{g_{0}}}dv.$
Just like the other weighted Laplacian, for example, $\mathcal{L}$ operator
and Witten-Laplacian, $\mathcal{L}_{\nu}$ operator is also very important in
geometric analysis. However, the eigenvalues of such an operator are rarely
studied as far as we know. Therefore, it is very urgent for us to exploit the
eigenvalue problem of $\mathcal{L}_{\nu}$ operator. Next, let us consider an
eigenvalue problem of $\mathfrak{L}_{\nu}^{2}$ operator on the bounded domain
$\Omega\subset\mathcal{M}^{n}$ with Dirichlet boundary condition:
(1.20) ${\begin{cases}\ \mathfrak{L}_{\nu}^{2}u=\Lambda u,\ \ &{\rm in}\ \ \ \
\Omega,\\\ \ u=\frac{\partial u}{\partial\textbf{n}}=0,\ \ &{\rm on}\ \
\partial\Omega,\end{cases}}$
where n denotes the normal vector to the boundary $\partial\Omega$. The main
goal of this paper is to establish some eigenvalue inequalities with lower
order for clamped plate problem (1.20) of $\mathfrak{L}_{\nu}^{2}$ operator on
$\mathcal{M}^{n}$. However, for the eigenvalues with higher order of
$\mathcal{L}_{\nu}$ operator and $\mathcal{L}^{2}_{\nu}$ operator, we also
obtain some eigenvalue inequalities in some separated papers, elsewhere. Now,
let us state the main result as follows.
###### Theorem 1.1.
Let $\mathcal{M}^{n}$ be an $n$-dimensional complete Riemannian manifold
isometrically embedded into the Euclidean space $\mathbb{R}^{n+p}$ with mean
curvature $H$. Then, for any $j$, where $j=1,2,\cdots$, eigenvalues of clamped
plate problem (1.20) satisfy
(1.21)
$\displaystyle\sum_{i=1}^{n}\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}$
$\displaystyle\leq
4\left\\{\left(\Lambda_{1}^{\frac{1}{2}}+4\widetilde{C}_{1}\Lambda_{1}^{\frac{1}{4}}+4\widetilde{C}_{1}^{2}+C_{1}\right)\left[\left(\frac{n}{2}+1\right)\Lambda_{1}^{\frac{1}{2}}+4\widetilde{C}_{1}\Lambda_{1}^{\frac{1}{4}}+4\widetilde{C}_{1}^{2}+C_{1}\right]\right\\}^{\frac{1}{2}},$
where $C_{1}$ is given by
$C_{1}=\frac{1}{4}\inf_{\sigma\in\Pi}\max_{\Omega}\left(n^{2}H^{2}\right),$
and $\widetilde{C}_{1}$ is given by
$\widetilde{C}_{1}=\frac{1}{4}\max_{\Omega}|\nu^{\top}|_{g_{0}}.$
###### Remark 1.2.
In theorem 1.1, if $|\nu|_{g_{0}}=0$, then one can deduce (1.15) from
(LABEL:z-ineq-1).
###### Corollary 1.1.
Under the assumption of theorem 1.1, we have
$\sum_{i=1}^{n}\left\\{\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}-\Lambda_{1}^{\frac{1}{2}}\right\\}\leq
4\left(\Lambda_{1}^{\frac{1}{2}}+4\widetilde{C}_{1}\Lambda_{1}^{\frac{1}{4}}+4\widetilde{C}_{1}^{2}+C_{1}\right),$
where $C_{1}$ is given by
$C_{1}=\frac{1}{4}\inf_{\sigma\in\Pi}\max_{\Omega}\left(n^{2}H^{2}\right),$
and $\widetilde{C}_{1}$ is given by
$\widetilde{C}_{1}=\frac{1}{4}\max_{\Omega}|\nu^{\top}|_{g_{0}}.$
###### Corollary 1.2.
Under the assumption of theorem 1.1, we have
(1.22)
$\displaystyle\sum_{i=1}^{n}\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}\leq
6\left\\{\left(\Lambda_{1}^{\frac{1}{2}}+C_{2}\right)\left[\left(\frac{n}{3}+1\right)\Lambda_{1}^{\frac{1}{2}}+C_{2}\right]\right\\}^{\frac{1}{2}},$
where $C_{2}$ is given by
$C_{2}=\frac{1}{6}\inf_{\sigma\in\Pi}\max_{\Omega}\left(n^{2}H^{2}+3|\nu^{\top}|_{g_{0}}^{2}\right).$
###### Corollary 1.3.
Under the assumption of theorem 1.1, we have
$\sum_{i=1}^{n}\left\\{\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}-\Lambda_{1}^{\frac{1}{2}}\right\\}\leq
6\left(\Lambda_{1}^{\frac{1}{2}}+C_{2}\right),$
where $C_{2}$ is given by
$C_{2}=\frac{1}{6}\inf_{\sigma\in\Pi}\max_{\Omega}\left(n^{2}H^{2}+3|\nu^{\top}|_{g_{0}}^{2}\right).$
This paper is organized as follows. In Section 2, we prove a general formula
for eigenvalues of the clamped plate problem of $\mathfrak{L}_{\nu}^{2}$
operator on the complete Riemannian manifolds. In section 3, we prove some
results of Chen and Cheng type, which will be very useful in the proof of our
main reuslts. By making use of the general formula and some results of Chen-
Cheng type, we give the proofs of theorem 1.1, corollary 1.1, corollary 1.2
and corollary 1.3 in section 4. Applying theorem 1.1, we obtain several
universal inequalities for the eigenvalues of $\mathfrak{L}_{II}^{2}$ operator
on the translating solitons with respect to the mean curvature flows in
section 5. We note that all of eigenvalue inequalities of
$\mathfrak{L}_{II}^{2}$ operator on the translating solitons are universal.
Finally, as some further applications of theorem 1.1, we obtain some
eigenvalue inequalities of $\mathfrak{L}_{\nu}^{2}$ operator on the minimal
submanifolds isometrically embedded into the Euclidean spaces, submanifolds
isometrically embedded (or immersed) into the unit spheres and projective
spaces in section 6.
###### Acknowledgment.
The research was partially supported by the National Natural Science
Foundation of China (Grant Nos. 11861036 and 11826213) and the Natural Science
Foundation of Jiangxi Province (Grant No. 20171ACB21023). The author shall
express his sincere gratitude to the anonymous referees for their helpful
comments and suggestions.
## 2\. Some Lemmas and their Proofs
In this section, we establish a general formula, which will play an important
role in the proof of theorem 1.1.
Assume that $\zeta\in C^{2}_{0}(\Omega)$, and we define $\xi$ as follows:
(2.1) $\displaystyle\xi:$
$\displaystyle=u_{1}\mathfrak{L}_{\nu}^{2}\zeta+2\langle\nabla
u_{1},\nabla(\mathfrak{L}_{\nu}\zeta)\rangle_{g}+2\mathfrak{L}_{\nu}\zeta\mathfrak{L}_{\nu}u_{1}$
$\displaystyle\quad+2\mathfrak{L}_{\nu}\langle\nabla\zeta,\nabla
u_{1}\rangle_{g}+2\langle\nabla\zeta,\nabla(\mathfrak{L}_{\nu}u_{1})\rangle_{g}.$
Then, we have the following lemma.
###### Lemma 2.1.
Let $(\mathcal{M}^{n},g)$ be an $n$-dimensional complete Riemannian manifold
and $u_{1}$ the first eigenfunction of the eigenvalue problem (1.20). Then, we
have
(2.2) $\int_{\Omega}\xi u_{1}e^{\langle\nu,X\rangle_{g_{0}}}dv=0.$
###### Proof.
By the definition of $\xi$ given by (2.1), we know that
(2.3) $\displaystyle\int_{\Omega}\xi u_{1}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\int_{\Omega}u_{1}^{2}\mathfrak{L}_{\nu}^{2}\zeta
e^{\langle\nu,X\rangle_{g_{0}}}dv+2\int_{\Omega}u_{1}\langle\nabla
u_{1},\nabla(\mathfrak{L}_{\nu}\zeta)\rangle_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad+2\int_{\Omega}u_{1}\mathfrak{L}_{\nu}\zeta\mathfrak{L}_{\nu}u_{1}e^{\langle\nu,X\rangle_{g_{0}}}dv+2\int_{\Omega}u_{1}\mathfrak{L}_{\nu}\langle\nabla\zeta,\nabla
u_{1}\rangle_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad+2\int_{\Omega}u_{1}\langle\nabla\zeta,\nabla(\mathfrak{L}_{\nu}u_{1})\rangle_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
By Stokes’s theorem, we have
(2.4) $\displaystyle\int_{\Omega}u_{1}^{2}\mathfrak{L}_{\nu}^{2}\zeta
e^{\langle\nu,X\rangle_{g_{0}}}dv+2\int_{\Omega}u_{1}\langle\nabla
u_{1},\nabla(\mathfrak{L}_{\nu}\zeta)\rangle_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\int_{\Omega}u_{1}^{2}\mathfrak{L}_{\nu}^{2}\zeta
e^{\langle\nu,X\rangle_{g_{0}}}dv+\int_{\Omega}\langle\nabla
u_{1}^{2},\nabla(\mathfrak{L}_{\nu}\zeta)\rangle_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\int_{\Omega}u_{1}^{2}\mathfrak{L}_{\nu}^{2}\zeta
e^{\langle\nu,X\rangle_{g_{0}}}dv-\int_{\Omega}u_{1}^{2}\mathfrak{L}_{\nu}^{2}\zeta
e^{\langle\nu,X\rangle_{g_{0}}}dv$ $\displaystyle=0.$
Applying Stokes’s theorem again, we infer that
(2.5) $\displaystyle 0$ $\displaystyle=\int_{\Omega}{\rm
div}\left(u_{1}\mathfrak{L}_{\nu}u_{1}e^{\langle\nu,X\rangle_{g_{0}}}\nabla\zeta\right)dv$
$\displaystyle=\int_{\Omega}\langle\nabla
u_{1},\nabla\zeta\rangle_{g}\mathfrak{L}_{\nu}u_{1}e^{\langle\nu,X\rangle_{g_{0}}}dv+\int_{\Omega}u_{1}\langle\nabla(\mathfrak{L}_{\nu}u_{1}),\nabla\zeta\rangle_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad+\int_{\Omega}u_{1}\mathfrak{L}_{\nu}u_{1}\langle\nu,\nabla\zeta\rangle_{g_{0}}e^{\langle\nu,X\rangle_{g_{0}}}dv+\int_{\Omega}u_{1}\mathfrak{L}_{\nu}u_{1}\Delta\zeta
e^{\langle\nu,X\rangle_{g_{0}}}dv$ $\displaystyle=\int_{\Omega}\langle\nabla
u_{1},\nabla\zeta\rangle_{g}\mathfrak{L}_{\nu}u_{1}e^{\langle\nu,X\rangle_{g_{0}}}dv+\int_{\Omega}u_{1}\langle\nabla(\mathfrak{L}_{\nu}u_{1}),\nabla\zeta\rangle_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad\quad+\int_{\Omega}u_{1}\mathfrak{L}_{\nu}u_{1}\mathfrak{L}_{\nu}\zeta
e^{\langle\nu,X\rangle_{g_{0}}}dv.$
From (2.3), (LABEL:le-eq2) and (2.5), we derive (2.2). Hence, we finish the
proof of this lemma.
∎
###### Lemma 2.2.
_(General formula)_ Let $\Omega$ be a bounded domain on an $n$-dimensional
complete Riemannian submanifold $(\mathcal{M}^{n},g)$ isometrically immersed
into the Euclidean space $\mathbb{R}^{n+p}$, and $\Lambda_{i}$ be the
$i^{\text{th}}$ eigenvalue of the eigenvalue problem (1.20) and $u_{i}$ be the
orthonormal eigenfunction corresponding to $\Lambda_{i},$ that is,
$\left\\{\begin{array}[]{ll}\mathfrak{L}_{\nu}^{2}u_{i}=\Lambda_{i}u_{i},&\text{
in }\Omega\\\ u=\frac{\partial u}{\partial\textbf{n}}=0,&\text{ on
}\partial\Omega\\\
\int_{\Omega}u_{i}u_{j}e^{\langle\nu,X\rangle_{g_{0}}}dv=\delta_{ij},&\forall
i,j=1,2,\ldots\end{array}\right.$
where n is an outward normal vector field of $\partial\Omega$. If
$\phi_{i}(i\geq 2)\in C^{4}(\Omega)\cap C^{3}(\partial\Omega)$ satisfies
$\int_{\Omega}\phi_{i}u_{1}u_{j+1}e^{\langle\nu,X\rangle_{g_{0}}}dv=0$ for
$1\leq j<i,$ then for any positive integer $i,$ we have
(2.6)
$\displaystyle\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}\int_{\Omega}\left|u_{1}\nabla\phi_{i}\right|^{2}_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\leq\left(\frac{\delta}{2}+\frac{1}{2\delta}\right)\int_{\Omega}\Upsilon(\phi_{i})e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad\quad-\delta\int_{\Omega}\Phi(\phi_{i})e^{\langle\nu,X\rangle_{g_{0}}}dv,$
where
(2.7)
$\Upsilon(\phi_{i})=\left(u_{1}\mathfrak{L}_{\nu}\phi_{i}+2\langle\nabla\phi_{i},\nabla
u_{1}\rangle_{g}\right)^{2},$
(2.8)
$\Phi(\phi_{i})=\left|\nabla\phi_{i}\right|^{2}_{g}u_{1}\mathfrak{L}_{\nu}u_{1},$
and $\delta$ is any positive constant.
###### Proof.
In order to prove (2.6), let us define
(2.9) $\psi_{i}:=\left(\phi_{i}-a_{i}\right)u_{1},$
where $i\geq 2$ and
$a_{i}=\int_{\Omega}\phi_{i}u_{1}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
It is not difficult to check that
$\int_{\Omega}\psi_{i}u_{1}e^{\langle\nu,X\rangle_{g_{0}}}dv=0.$
Noticing
$\int_{\Omega}\phi_{i}u_{1}u_{j+1}e^{\langle\nu,X\rangle_{g_{0}}}dv=0\ \ {\rm
for}\ \ 1\leq j<i,$
we infer
$\int_{\Omega}\psi_{i}u_{j+1}e^{\langle\nu,X\rangle_{g_{0}}}dv=0,\text{ for
}1\leq j<i,$
and
$\psi_{i}\bigg{|}_{\partial\Omega}=\frac{\partial\psi_{i}}{\partial\nu}\bigg{|}_{\partial\Omega}=0.$
From the Rayleigh-Ritz inequality, we have
(2.10)
$\Lambda_{i+1}\int_{\Omega}\psi_{i}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv\leq\int_{\Omega}\psi_{i}\mathfrak{L}_{\nu}^{2}\psi_{i}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
According to the definition of the function $\psi_{i},$ one has
$\displaystyle\mathfrak{L}_{\nu}\left(\psi_{i}\right)=\mathfrak{L}_{\nu}\left(\phi_{i}u_{1}\right)-a_{i}\mathfrak{L}_{\nu}u_{1}=u_{1}\mathfrak{L}_{\nu}\phi_{i}+2\left\langle\nabla\phi_{i},\nabla
u_{1}\right\rangle_{g}+\phi_{i}\mathfrak{L}_{\nu}u_{1}-a_{i}\mathfrak{L}_{\nu}u_{1},$
and
$\displaystyle\mathfrak{L}_{\nu}^{2}\left(\psi_{i}\right)$
$\displaystyle=\mathfrak{L}_{\nu}\left(\mathfrak{L}_{\nu}\left(\psi_{i}\right)\right)$
$\displaystyle=\mathfrak{L}_{\nu}\left(u_{1}\mathfrak{L}_{\nu}\phi_{i}+2\left\langle\nabla\phi_{i},\nabla
u_{1}\right\rangle_{g}+\phi_{i}\mathfrak{L}_{\nu}u_{1}-a_{i}\mathfrak{L}_{\nu}u_{1}\right)$
$\displaystyle=u_{1}\mathfrak{L}_{\nu}^{2}\phi_{i}+2\left\langle\nabla
u_{1},\nabla\left(\mathfrak{L}_{\nu}\phi_{i}\right)\right\rangle_{g}+2\mathfrak{L}_{\nu}\phi_{i}\mathfrak{L}_{\nu}u_{1}+2\mathfrak{L}_{\nu}\left\langle\nabla\phi_{i},\nabla
u_{1}\right\rangle_{g}$
$\displaystyle+2\left\langle\nabla\phi_{i},\nabla\left(\mathfrak{L}_{\nu}u_{1}\right)\right\rangle_{g}+\phi_{i}\mathfrak{L}_{\nu}^{2}u_{1}-a_{i}\mathfrak{L}_{\nu}^{2}u_{1}$
$\displaystyle=\tau_{i}+\Lambda_{1}\psi_{i},$
where
$\displaystyle\epsilon_{i}=u_{1}\mathfrak{L}_{\nu}^{2}\phi_{i}+2\left\langle\nabla
u_{1},\nabla\left(\mathfrak{L}_{\nu}\phi_{i}\right)\right\rangle_{g}+2\mathfrak{L}_{\nu}\phi_{i}\mathfrak{L}_{\nu}u_{1}$
$\displaystyle+2\mathfrak{L}_{\nu}\left\langle\nabla\phi_{i},\nabla
u_{1}\right\rangle_{g}+2\left\langle\nabla\phi_{i},\nabla\left(\mathfrak{L}_{\nu}u_{1}\right)\right\rangle_{g}.$
From (2.10), we conclude that
(2.11)
$\displaystyle\left(\Lambda_{i+1}-\Lambda_{1}\right)\int_{\Omega}\psi_{i}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\leq\int_{\Omega}\psi_{i}\epsilon_{i}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\int_{\Omega}\epsilon_{i}\phi_{i}u_{1}e^{\langle\nu,X\rangle_{g_{0}}}dv-
a_{i}\int_{\Omega}\epsilon_{i}u_{1}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
By lemma 2.1, we know that
(2.12) $\int_{\Omega}\epsilon_{i}u_{1}e^{\langle\nu,X\rangle_{g_{0}}}dv=0.$
Applying Stokes’ theorem, we have the following equalities:
(2.13) $\displaystyle 2\int_{\Omega}\phi_{i}u_{1}\left\langle\nabla
u_{1},\nabla\left(\mathfrak{L}_{\nu}\phi_{i}\right)\right\rangle_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad=\int_{\Omega}\left(2u_{1}\mathfrak{L}_{\nu}\phi_{i}\left\langle\nabla
u_{1},\nabla\phi_{i}\right\rangle_{g}+u_{1}^{2}\left(\mathfrak{L}_{\nu}\phi_{i}\right)^{2}-\phi_{i}u_{1}^{2}\mathfrak{L}_{\nu}^{2}\phi_{i}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv,$
(2.14) $\displaystyle
2\int_{\Omega}\phi_{i}u_{1}\mathfrak{L}_{\nu}\left\langle\nabla\phi_{i},\nabla
u_{1}\right\rangle_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad=\int_{\Omega}\left(2\mathfrak{L}_{\nu}\phi_{i}u_{1}\left\langle\nabla\phi_{i},\nabla
u_{1}\right\rangle_{g}+4\left\langle\nabla\phi_{i},\nabla
u_{1}\right\rangle_{g}^{2}+2\phi_{i}\mathfrak{L}_{\nu}u_{1}\left\langle\nabla\phi_{i},\nabla
u_{1}\right\rangle_{g}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv,$
and
(2.15) $\displaystyle
2\int_{\Omega}\phi_{i}u_{1}\left\langle\nabla\phi_{i},\nabla\left(\mathfrak{L}_{\nu}u_{1}\right)\right\rangle_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad=-2\int_{\Omega}\left(\left|\nabla\phi_{i}\right|^{2}_{g}u_{1}\mathfrak{L}_{\nu}u_{1}+\phi_{i}\mathfrak{L}_{\nu}u_{1}\left\langle\nabla\phi_{i},\nabla
u_{1}\right\rangle_{g}+\phi_{i}u_{1}\mathfrak{L}_{\nu}\phi_{i}\mathfrak{L}_{\nu}u_{1}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv.$
Combining (2.13)-(2.15), we infer that
(2.16)
$\displaystyle\int_{\Omega}\epsilon_{i}\phi_{i}u_{1}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\int_{\Omega}\left(\left(\mathfrak{L}_{\nu}\phi_{i}\right)^{2}u_{1}^{2}+4\left\langle\nabla\phi_{i},\nabla
u_{1}\right\rangle_{g}^{2}+4u_{1}\mathfrak{L}_{\nu}\phi_{i}\left\langle\nabla\phi_{i},\nabla
u_{1}\right\rangle_{g}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad\quad-\int_{\Omega}2\left|\nabla\phi_{i}\right|^{2}_{g}u_{1}\mathfrak{L}_{\nu}u_{1}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\int_{\Omega}\left(u_{1}\mathfrak{L}_{\nu}\phi_{i}+2\left\langle\nabla\phi_{i},\nabla
u_{1}\right\rangle_{g}\right)^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv-\int_{\Omega}2\left|\nabla\phi_{i}\right|_{g}^{2}u_{1}\mathfrak{L}_{\nu}u_{1}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
Substituting (2.12) and (LABEL:th-i-ineq) into (2.11), one can conclude that
$\displaystyle\left(\Lambda_{i+1}-\Lambda_{1}\right)\int_{\Omega}\psi_{i}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\leq\int_{\Omega}\left(u_{1}\mathfrak{L}_{\nu}\phi_{i}+2\left\langle\nabla\phi_{i},\nabla
u_{1}\right\rangle_{g}\right)^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad-\int_{\Omega}2\left|\nabla\phi_{i}\right|_{g}^{2}u_{1}\mathfrak{L}_{\nu}u_{1}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
On the other hand, we have
$\displaystyle\int_{\Omega}$
$\displaystyle\psi_{i}\left(u_{1}\mathfrak{L}_{\nu}\phi_{i}+2\langle\nabla\phi_{i},\nabla
u_{1}\rangle_{g}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\int_{\Omega}\left(\phi_{i}-a_{i}\right)u_{1}\left(u_{1}\mathfrak{L}_{\nu}\phi_{i}+2\langle\nabla\phi_{i},\nabla
u_{1}\rangle_{g}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\int_{\Omega}\phi_{i}u_{1}\left(u_{1}\mathfrak{L}_{\nu}\phi_{i}+2\langle\nabla\phi_{i},\nabla
u_{1}\rangle_{g}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\int_{\Omega}\left(\phi_{i}u_{1}^{2}\mathfrak{L}_{\nu}\phi_{i}+2\phi_{i}u_{1}\left\langle\nabla\phi_{i},\nabla
u_{1}\right\rangle_{g}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=-\int_{\Omega}\left|u_{1}\nabla\phi_{i}\right|^{2}_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
So, for any $\delta>0,$ we have
$\displaystyle\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}\int_{\Omega}\left|u_{1}\nabla\phi_{i}\right|_{g}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad=\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}\int_{\Omega}-\psi_{i}\left(u_{1}\mathfrak{L}_{\nu}\phi_{i}+2\langle\nabla\phi_{i},\nabla
u_{1}\rangle_{g}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad\leq\frac{\delta}{2}\left(\Lambda_{i+1}-\Lambda_{1}\right)\int_{\Omega}\psi_{i}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv+\frac{1}{2\delta}\int_{\Omega}\left(u_{1}\mathfrak{L}_{\nu}\phi_{i}+2\langle\nabla\phi_{i},\nabla
u_{1}\rangle_{g}\right)^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad\leq\left(\frac{\delta}{2}+\frac{1}{2\delta}\right)\int_{\Omega}\left(u_{1}\mathfrak{L}_{\nu}\phi_{i}+2\langle\nabla\phi_{i},\nabla
u_{1}\rangle_{g}\right)^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv-\delta\int_{\Omega}\left|\nabla\phi_{i}\right|_{g}^{2}u_{1}\mathfrak{L}_{\nu}u_{1}e^{\langle\nu,X\rangle_{g_{0}}}dv,$
which means that (2.6) is true. This completes the proof of Lemma 2.2.
∎
## 3\. Some Results of Chen-Cheng Type
In order to prove our main results, the following lemmas will play very
important roles. The first lemma reads as follows:
###### Lemma 3.1.
Let $\mathcal{M}^{n}$ be an $n$-dimensional submanifold in Euclidean space
$\mathbb{R}^{n+p}$, and $y=(y^{1},y^{2},\cdots,y^{n+p})$ be the position
vector of a point $p\in\mathcal{M}^{n}$ with
$y^{\alpha}=y^{\alpha}(x^{1},\cdots,x^{n})$, $1\leq\alpha\leq n+p$, where
$(x^{1},\cdots,x^{n})$ denotes a local coordinate system of $\mathcal{M}^{n}$.
Then, we have
(3.1) $\sum^{n+p}_{\alpha=1}\langle\nabla y^{\alpha},\nabla
y^{\alpha}\rangle_{g}=n,$ (3.2)
$\displaystyle\sum^{n+p}_{\alpha=1}\langle\nabla y^{\alpha},\nabla
u\rangle_{g}\langle\nabla y^{\alpha},\nabla w\rangle_{g}=\langle\nabla
u,\nabla w\rangle_{g},$
for any functions $u,w\in C^{1}(\mathcal{M}^{n})$,
(3.3) $\displaystyle\sum^{n+p}_{\alpha=1}(\Delta y^{\alpha})^{2}=n^{2}H^{2},$
(3.4) $\displaystyle\sum^{n+p}_{\alpha=1}\Delta y^{\alpha}\nabla
y^{\alpha}=\textbf{0},$
where $H$ is the mean curvature of $\mathcal{M}^{n}$.
A proof of lemma 3.1 can be found in [8]. Or see [13]. Similarly, we have the
following lemma.
###### Lemma 3.2.
Let $\left(x^{1},\cdots,x^{n}\right)$ be an arbitrary coordinate system in a
neighborhood $U$ of $P$ in $\mathcal{M}^{n}.$ Assume that $y$ with components
$y^{\alpha}$ defined by
$y^{\alpha}=y^{\alpha}\left(x^{1},\cdots,x^{n}\right),1\leq\alpha\leq n+p,$ is
the position vector of $P$ in $\mathbb{R}^{n+p}$. Then, we have
(3.5) $\sum_{\alpha=1}^{n+p}\left\langle\nabla
y^{\alpha},\nu\right\rangle_{g_{0}}^{2}=|\nu^{\top}|_{g_{0}}^{2},$
where $\nabla$ is the gradient operator on $\mathcal{M}^{n}$.
###### Proof.
Equality (3.5) can be proved as follows:
$\displaystyle\sum_{\alpha=1}^{n+p}\langle\nabla
y^{\alpha},v\rangle_{g_{0}}^{2}$
$\displaystyle=\sum_{\alpha=1}^{n+p}\langle\nabla
y^{\alpha},\nu^{\top}\rangle_{g_{0}}^{2}=\sum_{\alpha=1}^{n+p}\left(\nu^{\top}y^{\alpha}\right)^{2}=\left|\nu^{\top}\right|_{g_{0}}^{2}.$
Therefore, it finishes the proof of lemma 3.2.∎
###### Lemma 3.3.
Let $\left(x^{1},\cdots,x^{n}\right)$ be an arbitrary coordinate system in a
neighborhood $U$ of $P$ in $\mathcal{M}^{n}.$ Assume that $y$ with components
$y^{\alpha}$ defined by
$y^{\alpha}=y^{\alpha}\left(x^{1},\cdots,x^{n}\right),1\leq\alpha\leq n+p,$ is
the position vector of $P$ in $\mathbb{R}^{n+p}$. Then, we have
(3.6) $\sum_{\alpha=1}^{n+p}\left\langle\nabla y^{\alpha},\nabla
u\right\rangle_{g}\left\langle\nabla
y^{\alpha},\nu\right\rangle_{g_{0}}\leq|\nabla u|_{g}|\nu^{\top}|_{g_{0}},$
where $\nabla$ is the gradient operator on $\mathcal{M}^{n}$.
###### Proof.
By the Cauchy-Schwarz inequality, we have
(3.7) $\displaystyle\sum_{\alpha=1}^{n+p}\left\langle\nabla y^{\alpha},\nabla
u\right\rangle_{g}\left\langle\nabla y^{\alpha},\nu\right\rangle_{g_{0}}$
$\displaystyle\leq\left(\sum_{\alpha=1}^{n+p}\left\langle\nabla
y^{\alpha},\nabla
u\right\rangle_{g}^{2}\right)^{\frac{1}{2}}\left(\sum_{\alpha=1}^{n+p}\left\langle\nabla
y^{\alpha},\nu\right\rangle_{g_{0}}^{2}\right)^{\frac{1}{2}}.$
It follows from (3.2) that,
(3.8) $\sum_{\alpha=1}^{n+p}\left\langle\nabla y^{\alpha},\nabla
u\right\rangle_{g}^{2}=|\nabla u|_{g}^{2}.$
From (3.5), (3.7) and (3.8), we get (3.6). Therefore, we finish the proof of
this lemma. ∎
From (3.1), we have
(3.9) $\int_{\Omega}u_{i}^{2}\sum_{\alpha=1}^{n+p}\left|\nabla
y^{\alpha}\right|_{g}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv=n.$
According to (3.2), one has
(3.10) $\sum_{\alpha=1}^{n+p}\left\langle\nabla y^{\alpha},\nabla
u_{i}\right\rangle_{g}^{2}=\left|\nabla u_{i}\right|_{g}^{2}.$
It follows from (3.4) that,
(3.11) $\sum_{\alpha=1}^{n+p}\Delta y^{\alpha}\left\langle\nabla
y^{\alpha},\nabla
u_{i}\right\rangle_{g}=\sum_{\alpha=1}^{n+p}\left\langle\Delta
y^{\alpha}\nabla y^{\alpha},\nabla u_{i}\right\rangle_{g}=0,$
and
(3.12) $\sum_{\alpha=1}^{n+p}\Delta y^{\alpha}\left\langle\nabla
y^{\alpha},\nu\right\rangle_{g_{0}}=\sum_{\alpha=1}^{n+p}\left\langle\Delta
y^{\alpha}\nabla y^{\alpha},\nu\right\rangle_{g_{0}}=0.$
From (3.6) and (3.11), we obtain
(3.13)
$\displaystyle\sum_{\alpha=1}^{n+p}\mathfrak{L}_{\nu}y^{\alpha}\left\langle\nabla
y^{\alpha},\nabla u_{i}\right\rangle_{g}=\sum_{\alpha=1}^{n+p}\left(\Delta
y^{\alpha}+\left\langle\nabla
y^{\alpha},\nu\right\rangle_{g_{0}}\right)\left\langle\nabla y^{\alpha},\nabla
u_{i}\right\rangle_{g}\leq|\nabla u_{1}|_{g}|\nu^{\top}|_{g_{0}}.$
Let $y^{1},y^{2},\ldots,y^{n+p}$ be the standard coordinate functions of
$\mathbb{R}^{n+p}$ and define an $((n+p)\times(n+p))$-matrix $D$ by
$D:=\left(d_{\alpha\beta}\right),$where
$d_{\alpha\beta}=\int_{\Omega}y^{\alpha}u_{1}u_{\beta+1}.$ Using the
orthogonalization of Gram and Schmidt, we know that there exist an upper
triangle matrix $R=\left(R_{\alpha\beta}\right)$ and an orthogonal matrix
$Q=\left(\tau_{\alpha\beta}\right)$ such that
$R=QD,$
i.e.,
$\displaystyle
R_{\alpha\beta}=\sum_{\gamma=1}^{n+p}\tau_{\alpha\gamma}d_{\gamma\beta}=\int_{\Omega}\sum_{\gamma=1}^{n+p}\tau_{\alpha\gamma}y^{\gamma}u_{1}u_{\beta+1}=0,$
for $1\leq\beta<\alpha\leq n+p$. Defining
(3.14) $h_{\alpha}=\sum_{\gamma=1}^{n+p}\tau_{\alpha\gamma}y^{\gamma},$
we have
$\int_{\Omega}h_{\alpha}u_{1}u_{\beta+1}=0,$
where $1\leq\beta<\alpha\leq n+p.$ Since $Q$ is an orthogonal matrix, by lemma
3.1 and lemma 3.3, we have the following lemma.
###### Lemma 3.4.
Under the above convention, we have
(3.15) $\sum_{\alpha=1}^{n+p}\left|\nabla h_{\alpha}\right|_{g}^{2}=n,$
(3.16) $\sum_{\alpha=1}^{n+p}\left(\Delta h_{\alpha}\right)^{2}=n^{2}H^{2},$
(3.17) $\sum_{\alpha=1}^{n+p}\Delta h_{\alpha}\left\langle\nabla
h_{\alpha},\nabla u_{1}\right\rangle_{g}=0,$
(3.18) $\sum_{\alpha=1}^{n+p}\Delta h_{\alpha}\left\langle\nabla
h_{\alpha},\nu\right\rangle_{g_{0}}=0,$
(3.19) $\sum_{\alpha=1}^{n+p}\left\langle\nabla h_{\alpha},\nabla
u_{1}\right\rangle_{g}\left\langle\nabla
h_{\alpha},\nu\right\rangle_{g_{0}}\leq|\nabla
u_{1}|_{g}|\nu^{\top}|_{g_{0}},$
(3.20) $\sum_{\alpha=1}^{n+p}\left\langle\nabla
h_{\alpha},\nu\right\rangle_{g_{0}}^{2}=\left|\nu^{\top}\right|_{g_{0}}^{2},$
and
(3.21) $\sum_{\alpha=1}^{n+p}\left\langle\nabla h_{\alpha},\nabla
u_{1}\right\rangle_{g}^{2}=\left|\nabla u_{1}\right|_{g}^{2}.$
## 4\. Proofs of Main Results
In this section, we would like to give the proofs of the main results.
From (3.16), (3.18) and (3.20), we obtain
(4.1)
$\sum_{\alpha=1}^{n+p}\left(\mathfrak{L}_{\nu}h_{\alpha}\right)^{2}=n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2}.$
Utilizing (3.17) and (3.19), one has
(4.2) $\sum_{\alpha=1}^{n+p}\mathfrak{L}_{\nu}h_{\alpha}\left\langle\nabla
h_{\alpha},\nabla u_{1}\right\rangle_{g}\leq|\nabla
u_{1}|_{g}|\nu^{\top}|_{g_{0}}.$
###### Lemma 4.1.
For any $i=1,2,\cdots k$ and $\alpha=1,2,\cdots,n+p$, let
(4.3)
$\widehat{\Upsilon}=\sum^{n+p}_{\alpha=1}\int_{\Omega}\Upsilon(h_{\alpha})e^{\langle\nu,X\rangle_{g_{0}}}dv,$
where function $\Upsilon$ is given by (2.7) and $h_{\alpha}$ is given by
(3.14). Then, we have
(4.4) $\displaystyle\widehat{\Upsilon}$
$\displaystyle\leq\int_{\Omega}\left[4\left|\nabla
u_{1}\right|_{g}^{2}+u_{1}^{2}\left(n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2}\right)\right]e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\ \ \ \ \ \
+4\left(\int_{\Omega}u_{1}^{2}|\nu^{\top}|_{g_{0}}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv\right)^{\frac{1}{2}}\left(\int_{\Omega}|\nabla
u_{1}|_{g}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv\right)^{\frac{1}{2}},$
and
(4.5) $\displaystyle\widehat{\Upsilon}\leq\int_{\Omega}\left[6\left|\nabla
u_{1}\right|_{g}^{2}+u_{1}^{2}\left(\left(n^{2}H^{2}+3|\nu^{\top}|_{g_{0}}^{2}\right)\right)\right]e^{\langle\nu,X\rangle_{g_{0}}}dv.$
###### Proof.
By (3.14) and (4.3), we have
(4.6) $\displaystyle\widehat{\Upsilon}$
$\displaystyle=\sum^{n+p}_{\alpha=1}\int_{\Omega}\left(u_{1}\mathfrak{L}_{\nu}h_{\alpha}+2\langle\nabla
h_{\alpha},\nabla
u_{1}\rangle_{g}\right)^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\int_{\Omega}\sum^{n+p}_{\alpha=1}\left(u^{2}_{1}(\mathfrak{L}_{\nu}h_{\alpha})^{2}+4u_{1}\mathfrak{L}_{\nu}h_{\alpha}\langle\nabla
h_{\alpha},\nabla u_{1}\rangle_{g}+4\langle\nabla h_{\alpha},\nabla
u_{1}\rangle_{g}^{2}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv.$
From (3.17), (3.21), (4.1) and (4.2), we infer that
(4.7)
$\displaystyle\widehat{\Upsilon}\leq\int_{\Omega}\left[u^{2}_{1}(n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2})+4|\nabla
u_{1}|_{g}^{2}\right]e^{\langle\nu,X\rangle_{g_{0}}}dv+4\int_{\Omega}(u_{1}|\nu^{\top}|_{g_{0}})|\nabla
u_{1}|_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
Furthermore, by Cauchy-Schwarz inequality, we have
(4.8) $\displaystyle 4\int_{\Omega}(u_{1}|\nu^{\top}$
$\displaystyle|_{g_{0}})|\nabla u_{1}|_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\leq
4\left(\int_{\Omega}(u_{1}|\nu^{\top}|_{g_{0}})^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv\right)^{\frac{1}{2}}\left(\int_{\Omega}|\nabla
u_{1}|_{g}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv\right)^{\frac{1}{2}}.$
From (4.7) and (4.8), we yield
$\displaystyle\widehat{\Upsilon}$
$\displaystyle\leq\int_{\Omega}\left[4\left|\nabla
u_{1}\right|_{g}^{2}+u_{1}^{2}\left(n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2}\right)\right]e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\ \ \ \ \ \
+4\left(\int_{\Omega}(u_{1}|\nu^{\top}|_{g_{0}})^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv\right)^{\frac{1}{2}}\left(\int_{\Omega}|\nabla
u_{1}|_{g}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv\right)^{\frac{1}{2}}.$
By mean inequality, we obtain
(4.9) $4\int_{\Omega}(u_{1}|\nu^{\top}|_{g_{0}})|\nabla
u_{1}|_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv\leq
2\int_{\Omega}(u_{1}|\nu^{\top}|_{g_{0}})^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv+2\int_{\Omega}|\nabla
u_{1}|_{g}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
Therefore, by (4.7) and (4.9), we derive that
$\displaystyle\widehat{\Upsilon}$ $\displaystyle\leq\int_{\Omega}4\left|\nabla
u_{1}\right|_{g}^{2}+u_{1}^{2}\left(n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\ \ \ \ \ \
+2\int_{\Omega}(u_{1}|\nu^{\top}|_{g_{0}})^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv+2\int_{\Omega}|\nabla
u_{1}|_{g}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv,$
which gives (4.5). Therefore, we finish the proof of this lemma.
∎
By (3.15), we have the following lemma.
###### Lemma 4.2.
For any $i=1,2,\cdots k$ and $\alpha=1,2,\cdots,n+p$, let
$\widehat{\Phi}=\sum^{n+p}_{\alpha=1}\int_{\Omega}\Phi(h_{\alpha})e^{\langle\nu,X\rangle_{g_{0}}}dv,$
where function $\Phi$ is given by (2.8) and $h_{\alpha}$ is given by (3.14).
Then, we have
(4.10)
$\widehat{\Phi}=n\int_{\Omega}u_{1}\mathfrak{L}_{\nu}u_{1}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
Now, we give the proof of theorem 1.1.
_Proof of theorem_ 1.1. From (2.6), noticing the definitions of
$\widehat{\Phi}$ and $\widehat{\Upsilon}$, we have
(4.11)
$\displaystyle\sum^{n+p}_{\alpha=1}\left(\Lambda_{\alpha+1}-\Lambda_{1}\right)^{\frac{1}{2}}\int_{\Omega}\left|u_{1}\nabla
h_{\alpha}\right|_{g}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv\leq\left(\frac{\delta}{2}+\frac{1}{2\delta}\right)\widehat{\Upsilon}-\delta\widehat{\Phi}.$
By divergence theorem and Cauchy-Schwarz inequality, we conclude that
$\int_{\Omega}\left|\nabla
u_{1}\right|_{g}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv\leq\Lambda_{1}^{\frac{1}{2}},$
which gives
(4.12) $\widehat{\Phi}\geq-n\Lambda_{1}^{\frac{1}{2}}.$
Since eigenvalues are invariant under isometries, letting
$C_{1}=\frac{1}{4}\inf_{\sigma\in\Pi}\max_{\Omega}\left(n^{2}H^{2}\right),$
and
$\widetilde{C}_{1}=\frac{1}{4}\max_{\Omega}|\nu^{\top}|_{g_{0}},$
where $\Pi$ denotes the set of all isometric immersions from $\mathcal{M}^{n}$
into the Euclidean space $\mathbb{R}^{n+p}$, by inequality (4.12), we infer
that
(4.13) $\displaystyle\widehat{\Upsilon}$
$\displaystyle\leq\int_{\Omega}\left[4\left|\nabla
u_{1}\right|_{g}^{2}+u_{1}^{2}\left(n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2}\right)\right]e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad+4\Lambda^{\frac{1}{4}}_{1}\left(\int_{\Omega}\left(u_{1}|\nu^{\top}|_{g_{0}}\right)^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv\right)^{\frac{1}{2}}$
$\displaystyle\leq
4\left(\Lambda_{1}^{\frac{1}{2}}+4\widetilde{C}_{1}\Lambda_{1}^{\frac{1}{4}}+4\widetilde{C}_{1}^{2}+C_{1}\right).$
For $\forall x\in\mathcal{M}^{n},$ by a transformation of orthonormal frame if
necessary, it is not difficult to prove that, for any $\alpha$,
(4.14) $\left|\nabla h_{\alpha}\right|_{g}^{2}\leq 1,$
where $\alpha=1,2,\cdots,n+p$. It is clear that
(4.15)
$\displaystyle\sum_{\alpha=1}^{n+p}\left(\Lambda_{\alpha+1}-\Lambda_{1}\right)^{\frac{1}{2}}\int_{\Omega}\left|u_{1}\nabla
h_{\alpha}\right|_{g}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\geq\sum_{i=1}^{n}\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}\int_{\Omega}\left|u_{1}\nabla
h_{i}\right|^{2}_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad\quad+\left(\Lambda_{n+1}-\Lambda_{1}\right)^{\frac{1}{2}}\sum_{j=n+1}^{n+p}\int_{\Omega}\left|u_{1}\nabla
h_{j}\right|_{g}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
Hence, from (3.15), (4.14) and (LABEL:Sum-3.21), we infer that,
$\displaystyle\sum_{\alpha=1}^{n+p}\left(\Lambda_{\alpha+1}-\Lambda_{1}\right)^{\frac{1}{2}}\int_{\Omega}\left|u_{1}\nabla
h_{\alpha}\right|_{g}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\geq\sum_{i=1}^{n}\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}\int_{\Omega}\left|u_{1}\nabla
h_{i}\right|^{2}_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad\quad+\left(\Lambda_{n+1}-\Lambda_{1}\right)^{\frac{1}{2}}\left(n-\sum_{j=1}^{n}\int_{\Omega}\left|u_{1}\nabla
h_{j}\right|_{g}^{2}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\sum_{i=1}^{n}\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}\int_{\Omega}\left|u_{1}\nabla
h_{i}\right|^{2}_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad\quad+\left(\Lambda_{n+1}-\Lambda_{1}\right)^{\frac{1}{2}}\sum_{j=1}^{n}\left(1-\int_{\Omega}\left|u_{1}\nabla
h_{j}\right|_{g}^{2}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\geq\sum_{i=1}^{n}\int_{\Omega}\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}\int_{\Omega}\left|u_{1}\nabla
h_{i}\right|^{2}_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad\quad+\sum_{j=1}^{n}\left(\Lambda_{j+1}-\Lambda_{1}\right)^{\frac{1}{2}}\int_{\Omega}\left(u_{1}^{2}-\left|u_{1}\nabla
h_{j}\right|_{g}^{2}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv,$
which implies that
(4.16)
$\displaystyle\sum_{\alpha=1}^{n+p}\left(\Lambda_{\alpha+1}-\Lambda_{1}\right)^{\frac{1}{2}}\int_{\Omega}\left|u_{1}\nabla
h_{\alpha}\right|_{g}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv\geq\sum_{j=1}^{n}\left(\Lambda_{j+1}-\Lambda_{1}\right)^{\frac{1}{2}}.$
Using (4.11), (4.12), (4.13) and (4.16), we have
$\sum_{j=1}^{n}\left(\Lambda_{j+1}-\Lambda_{1}\right)^{\frac{1}{2}}\leq
4\left(\frac{\delta}{2}+\frac{1}{2\delta}\right)\left(\Lambda_{1}^{\frac{1}{2}}+4\widetilde{C}_{1}\Lambda_{1}^{\frac{1}{4}}+4\widetilde{C}_{1}^{2}+C_{1}\right)+n\delta\Lambda_{1}^{\frac{1}{2}}.$
Taking
$\delta=\frac{\sqrt{\Lambda_{1}^{\frac{1}{2}}+4\widetilde{C}_{1}\Lambda_{1}^{\frac{1}{4}}+4\widetilde{C}_{1}^{2}+C_{1}}}{\sqrt{\left(\frac{n}{2}+1\right)\Lambda_{1}^{\frac{1}{2}}+4\widetilde{C}_{1}\Lambda_{1}^{\frac{1}{4}}+4\widetilde{C}_{1}^{2}+C_{1}}},$
we get (LABEL:z-ineq-1). Therefore, it completes the proof of theorem 1.1.
$None$
According to theorem 1.1, we would like to give the proof of corollary 1.1.
_Proof of Corollary_ 1.1 Since
$\displaystyle
4\left\\{\left(\Lambda_{1}^{\frac{1}{2}}+4\widetilde{C}_{1}\Lambda_{1}^{\frac{1}{4}}+4\widetilde{C}_{1}^{2}+C_{1}\right)\left[\left(\frac{n}{2}+1\right)\Lambda_{1}^{\frac{1}{2}}+4\widetilde{C}_{1}\Lambda_{1}^{\frac{1}{4}}+4\widetilde{C}_{1}^{2}+C_{1}\right]\right\\}^{\frac{1}{2}}$
$\displaystyle\leq
n\Lambda_{1}^{\frac{1}{2}}+4\left(\Lambda_{1}^{\frac{1}{2}}+4\widetilde{C}_{1}\Lambda_{1}^{\frac{1}{4}}+4\widetilde{C}_{1}^{2}+C_{1}\right),$
where $C_{1}$ is given by
$C_{1}=\frac{1}{4}\inf_{\sigma\in\Pi}\max_{\Omega}\left(n^{2}H^{2}\right),$
and $\widetilde{C}_{1}$ is given by
$\widetilde{C}_{1}=\frac{1}{4}\max_{\Omega}|\nu^{\top}|_{g_{0}},$
from (LABEL:z-ineq-1), we then obtain
$\sum_{i=1}^{n}\left\\{\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}-\Lambda_{1}^{\frac{1}{2}}\right\\}\leq
4\left(\Lambda_{1}^{\frac{1}{2}}+4\widetilde{C}_{1}\Lambda_{1}^{\frac{1}{4}}+4\widetilde{C}_{1}^{2}+C_{1}\right).$
This finishes the proof of corollary 1.1.
$None$
_Proof of corollary_ 1.2. From (4.5), we have
(4.17) $\displaystyle\widehat{\Upsilon}$
$\displaystyle\leq\int_{\Omega}\left[6\left|\nabla
u_{1}\right|_{g}^{2}+u_{1}^{2}\left(n^{2}H^{2}+3|\nu^{\top}|_{g_{0}}^{2}\right)\right]e^{\langle\nu,X\rangle_{g_{0}}}dv.$
According to (2.6) and the definitions of $\widehat{\Phi}$ and
$\widehat{\Upsilon}$, we derive that
(4.18)
$\displaystyle\sum^{n+p}_{\alpha=1}\left(\Lambda_{\alpha+1}-\Lambda_{1}\right)^{\frac{1}{2}}\int_{\Omega}\left|u_{1}\nabla
h_{\alpha}\right|_{g}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv\leq\left(\frac{\delta}{2}+\frac{1}{2\delta}\right)\widehat{\Upsilon}-\delta\widehat{\Phi}.$
Since eigenvalues are invariant under isometries, defining
(4.19)
$C_{2}=\frac{1}{6}\inf_{\sigma\in\Pi}\max_{\mathcal{M}^{n}}\left(n^{2}H^{2}+3|\nu^{\top}|_{g_{0}}^{2}\right),$
where $\Pi$ denotes the set of all isometric immersions from $\mathcal{M}^{n}$
into the Euclidean space $\mathbb{R}^{n+p}$, by divergence theorem and Cauchy-
Schwarz inequality, we infer that
(4.20) $\displaystyle\widehat{\Upsilon}\leq\int_{\Omega}\left[6\left|\nabla
u_{1}\right|_{g}^{2}+u_{1}^{2}\left(n^{2}H^{2}+3|\nu^{\top}|_{g_{0}}^{2}\right)\right]e^{\langle\nu,X\rangle_{g_{0}}}dv\leq
6\Lambda_{1}^{\frac{1}{2}}+6C_{2}.$
Using (4.12), (4.16), (4.18) and (4.20), we have
$\sum_{j=1}^{n}\left(\Lambda_{j+1}-\Lambda_{1}\right)^{\frac{1}{2}}\leq\left(\frac{\delta}{2}+\frac{1}{2\delta}\right)\left(6\Lambda_{1}^{\frac{1}{2}}+6C_{2}\right)+n\delta\Lambda_{1}^{\frac{1}{2}}.$
Taking
$\delta=\frac{\sqrt{6\Lambda_{1}^{\frac{1}{2}}+6C_{2}}}{\sqrt{(2n+6)\Lambda_{1}^{\frac{1}{2}}+6C_{2}}},$
we have (LABEL:z-ineq-1). Therefore, it completes the proof of corollary 1.2.
$None$
_Proof of corollary_ 1.3. The method of the proof is the same as corollary
1.1. Hence, we omit it.
$None$
## 5\. Eigenvalue Inequalities on the Translating Solitons
In this section, we would like to discuss the eigenvalues of
$\mathcal{L}_{II}^{2}$ on the complete translating solitons.
Firstly, let us consider a smooth family of immersions
$X_{t}=X(\cdot,t):\mathcal{M}^{n}\rightarrow\mathbb{R}^{n+p}$ with
corresponding images $\mathcal{M}^{n}_{t}=X_{t}(\mathcal{M}^{n})$ such that
the following mean curvature equation system [21]:
(5.1)
${\begin{cases}&\frac{d}{dt}X(x,t)=\textbf{H}(x,t),x\in\mathcal{M}^{n},\\\
&X(\cdot,0)=X(\cdot),\end{cases}}$
is satisfied, where $\textbf{H}(x,t)$ is the mean curvature vector of
$\mathcal{M}_{t}^{n}$ at $X(x,t)$ in $\mathbb{R}^{n+p}$. We assume that
$\nu_{0}$ is a constant vector with unit length and denote $\nu_{0}^{N}$ the
normal projection of $\nu_{0}$ to the normal bundle of $\mathcal{M}^{n}$ in
$\mathbb{R}^{n+p}$. A submanifold
$X:\mathcal{M}^{n}\rightarrow\mathbb{R}^{n+p}$ is said to be a translating
soliton of the mean curvature flow (5.1), if it satisfies
(5.2) $\textbf{H}=\nu_{0}^{N},$
which is a special solution of the mean curvature flow equations (5.1).
Translating solitons also occur as Type-II singularity of the mean curvature
flow equations (5.1), which play an important role in the study of the mean
curvature flow [5]. In [30], Xin studied some basic properties of translating
solitons: the volume growth, generalized maximum principle, Gauss maps and
certain functions related to the Gauss maps. In addition, he carried out
point-wise estimates and integral estimates for the squared norm of the second
fundamental form. By utilizing these estimates, Xin proved some rigidity
theorems for translating solitons in the Euclidean space in higher
codimension. Recently, Chen and Qiu [11] proved a nonexistence theorem for
spacelike translating solitons. These results are established by using a new
Omori-Yau maximal principle.
When $\nu_{0}$ is a unit vector field satisfying (5.2),
$\mathfrak{L}_{\nu_{0}}$ exactly is an $\mathfrak{L}_{II}$ operator, which is
introduced by Xin in [30] and similar to the $\mathfrak{L}$ operator
introduced by Colding and Minicozzi in [17]. Therefore, $\mathfrak{L}_{\nu}$
operator can be viewed as a extension of $\mathfrak{L}_{II}$ operator. As an
application of theorem 1.1, we study the eigenvalues of bi-$\mathfrak{L}_{II}$
operator, which is denoted by $\mathfrak{L}_{II}^{2}$, on the complete
translating solitons. In other words, we prove the following theorem.
###### Theorem 5.1.
Let $\mathcal{M}^{n}$ be an $n$-dimensional complete translating soliton
isometrically embedded into the Euclidean space $\mathbb{R}^{n+p}$ with mean
curvature $H$. Then, eigenvalues of clamped plate problem (1.20) of the
$\mathfrak{L}_{II}^{2}$ operator satisfy
(5.3)
$\displaystyle\sum_{i=1}^{n}\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}\leq
4\left\\{\left(\Lambda_{1}^{\frac{1}{2}}+\Lambda^{\frac{1}{4}}_{1}+\frac{n^{2}}{4}\right)\left[\left(\frac{n}{2}+1\right)\Lambda_{1}^{\frac{1}{2}}+\Lambda^{\frac{1}{4}}_{1}+\frac{n^{2}}{4}\right]\right\\}^{\frac{1}{2}}.$
###### Proof.
Since $\mathcal{M}^{n}$ is an $n$-dimensional complete translator
isometrically embedded into the Euclidean space $\mathbb{R}^{n+p}$, we have
(5.4) $\textbf{H}=\nu_{0}^{\perp},$
and
(5.5) $|\nu_{0}^{\top}|_{g_{0}}^{2}\leq|\nu_{0}|_{g_{0}}^{2}=1,$
which implies that
(5.6)
$n^{2}H^{2}+|\nu_{0}^{\top}|_{g_{0}}^{2}=n^{2}|\nu_{0}^{\perp}|_{g_{0}}^{2}+|\nu_{0}^{\top}|_{g_{0}}^{2}\leq
n^{2}.$
Uniting (5.4), (5.5) and (5.6), we yield
(5.7)
$\frac{1}{4}\int_{\Omega}u_{i}^{2}\left(n^{2}H^{2}+|\nu_{0}^{\top}|_{g_{0}}^{2}\right)e^{\langle\nu_{0},X\rangle_{g_{0}}}dv\leq\frac{n^{2}}{4}.$
Substituting (5.7) into (LABEL:z-ineq-1), we obtain
$\displaystyle\sum_{i=1}^{n}\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}\leq
4\left\\{\left(\Lambda_{1}^{\frac{1}{2}}+\frac{n^{2}}{4}+\Lambda^{\frac{1}{4}}_{1}\right)\left[\left(\frac{n}{2}+1\right)\Lambda_{1}^{\frac{1}{2}}+\frac{n^{2}}{4}+\Lambda^{\frac{1}{4}}_{1}\right]\right\\}^{\frac{1}{2}}.$
Therefore, we finish the proof of this theorem.
∎
###### Corollary 5.1.
Under the same assumption as theorem 5.1, eigenvalues of eigenvalue problem
(1.20) of $\mathfrak{L}_{II}^{2}$ operator satisfy
(5.8)
$\sum_{i=1}^{n}\left\\{\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}-\Lambda_{1}^{\frac{1}{2}}\right\\}\leq
4\left(\Lambda_{1}^{\frac{1}{2}}+\Lambda^{\frac{1}{4}}_{1}+\frac{n^{2}}{4}\right).$
###### Proof.
The method of proof is similar to corollary 1.1. Thus, we omit it. ∎
###### Corollary 5.2.
Under the same assumption as theorem 5.1, for any $n\geq 2$, eigenvalues of
clamped plate problem (1.20) of the $\mathfrak{L}_{II}^{2}$ operator satisfy
(5.9)
$\displaystyle\sum_{i=1}^{n}\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}\leq
6\left\\{\left(\Lambda_{1}^{\frac{1}{2}}+\frac{n^{2}}{6}\right)\left[\left(\frac{n}{3}+1\right)\Lambda_{1}^{\frac{1}{2}}+\frac{n^{2}}{6}\right]\right\\}^{\frac{1}{2}}.$
###### Proof.
The method of the proof is similar to the proof of corollary 1.2. Thus, we
omit it here.∎
According to corollary 5.3, we can prove the following corollary.
###### Corollary 5.3.
Under the same assumption as theorem 5.1, for any $n\geq 2$, eigenvalues of
clamped plate problem (1.20) of the $\mathfrak{L}_{II}^{2}$ operator satisfy
(5.10)
$\sum_{i=1}^{n}\left\\{\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}-\Lambda_{1}^{\frac{1}{2}}\right\\}\leq
6\left(\Lambda_{1}^{\frac{1}{2}}+\frac{n^{2}}{6}\right).$
###### Remark 5.1.
Since inequality (5.3), (5.8), (5.9) and (5.10) are not dependent on the
domain $\Omega$, they are universal.
## 6\. Further Applications
In this section, we would like to give some further applications of theorem
1.1. Specially, we establish some eigenvalue inequalities on the minimal
submanifolds of the Euclidean spaces, unit spheres and projective spaces.
Firstly, we consider that $(\mathcal{M}^{n},g)$ is an $n$-dimensional complete
minimal submanifold isometrically embedded into the $(n+p)$-dimensional
Euclidean space $\mathbb{R}^{n+p}$. Then, we know that the mean curvature
vanishes. Therefore, one can deduce the following corollary from theorem 1.1.
###### Corollary 6.1.
Let $(\mathcal{M}^{n},g)$ be an $n$-dimensional complete minimal submanifold
isometrically embedded into the Euclidean space $\mathbb{R}^{n+p}$. Then,
eigenvalues of eigenvalue problem (1.20) of $\mathfrak{L}_{\nu}^{2}$ operator
satisfy
(6.1)
$\displaystyle\sum_{i=1}^{n}\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}\leq
4\left\\{\left(\Lambda_{1}^{\frac{1}{2}}+4\Lambda^{\frac{1}{4}}_{1}C_{3}+4C_{3}^{2}\right)\left[\left(\frac{n}{2}+1\right)\Lambda_{1}^{\frac{1}{2}}+4\Lambda^{\frac{1}{4}}_{1}C_{3}+4C_{3}^{2}\right]\right\\}^{\frac{1}{2}},$
where $C_{3}$ is given by
$C_{3}=\frac{1}{4}\max_{\Omega}|\nu^{\top}|_{g_{0}}.$
Next, we consider that $(\mathcal{M}^{n},g)$ is an $n$-dimensional submanifold
isometrically immersed in the unit sphere
$\mathbb{S}^{n+p-1}(1)\subset\mathbb{R}^{n+p}$ with mean curvature vector
$\overline{\textbf{H}}$. We use $\overline{\Pi}$ to denote the set of all
isometric immersions from $\mathcal{M}^{n}$ into the unit sphere
$\mathbb{S}^{n+p-1}(1)$. By theorem 1.1, we have the following corollary.
###### Corollary 6.2.
If $(\mathcal{M}^{n},g)$ be an $n$-dimensional submanifold isometrically
immersed in the unit sphere $\mathbb{S}^{n+p-1}(1)\subset\mathbb{R}^{n+p}$
with mean curvature vector $\overline{\textbf{H}}$. Then, eigenvalues of
eigenvalue problem (1.20) of $\mathfrak{L}_{\nu}^{2}$ operator satisfy
(6.2)
$\displaystyle\sum_{i=1}^{n}\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}\leq$
$\displaystyle
4\left\\{\Lambda_{1}^{\frac{1}{2}}+4\widetilde{C}_{3}\Lambda_{1}^{\frac{1}{4}}+4\widetilde{C}_{4}^{2}+C_{4}\right\\}^{\frac{1}{2}}$
$\displaystyle\times\left\\{\left(\frac{n}{2}+1\right)\Lambda_{1}^{\frac{1}{2}}+4\widetilde{C}_{4}\Lambda_{1}^{\frac{1}{4}}+4\widetilde{C}_{4}^{2}+C_{4}\right\\}^{\frac{1}{2}},$
where
$C_{4}=\frac{1}{4}\inf_{\overline{\sigma}\in\overline{\Pi}}\max_{\Omega}n^{2}(|\overline{\textbf{H}}|^{2}+1),$
and
$\widetilde{C}_{4}=\frac{1}{4}\max_{\Omega}|\nu^{\top}|_{g_{0}}.$
###### Proof.
Since the unit sphere can be canonically imbedded into Euclidean space, we
have the following diagram:
$\textstyle{\mathcal{M}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j\circ
f}$$\scriptstyle{f}$$\textstyle{\mathbb{S}^{n+p-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{\mathbb{R}^{n+p}}$
where $j:\mathbb{S}^{n+p-1}(1)\rightarrow\mathbb{R}^{n+p}$ is the canonical
imbedding from the unit sphere $S^{n+p-1}(1)$ into $\mathbb{R}^{n+p},$ and
$f:\mathcal{M}^{n}\rightarrow\mathbb{S}^{n+p-1}(1)$ is an isometrical
immersion. Then, $j\circ f:\mathcal{M}^{n}\rightarrow\mathbb{R}^{n+p}$ is an
isometric immersion from $\mathcal{M}^{n}$ to $\mathbb{R}^{n+p}.$ Let
$\overline{\textbf{H}}$ and H be the mean curvature vector fields of $f$ and
$j\circ f,$ respectively; then
$\left|\textbf{H}\right|^{2}=|\overline{\textbf{H}}|^{2}+1.$
Applying theorem 1.1 directly, we can get (6.2). Therefore, we finish the
proof of corollary 6.2.∎
In particular, we assume that $(\mathcal{M}^{n},g)$ is an $n$-dimensional unit
sphere $\mathbb{S}^{n}(1)$, and then, the mean curvature equals to $1$. This
is, $\left|\overline{\textbf{H}}\right|=0$, and thus we have
$\left|\textbf{H}\right|=1$. Furthermore, by theorem 1.1, we obtain the
following corollary.
###### Corollary 6.3.
Let $(\mathcal{M}^{n},g)$ be an $n$-dimensional unit sphere
$\mathbb{S}^{n}(1)$ and $\Omega$ is a bounded domain on $\mathbb{S}^{n}(1)$.
Then, eigenvalues of eigenvalue problem (1.20) of $\mathfrak{L}_{\nu}^{2}$
operator satisfy
$\displaystyle\sum_{i=1}^{n}\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}\leq
4\left\\{\left(\Lambda_{1}^{\frac{1}{2}}+4C_{5}\Lambda^{\frac{1}{4}}_{1}+\frac{n^{2}}{4}+4C_{5}^{2}\right)\left[\left(\frac{n}{2}+1\right)\Lambda_{1}^{\frac{1}{2}}+4C_{5}\Lambda^{\frac{1}{4}}_{1}+\frac{n^{2}}{4}+4C_{5}^{2}\right]\right\\}^{\frac{1}{2}},$
where $C_{5}$ is given by
$C_{5}=\frac{1}{4}\max_{\Omega}|\nu^{\top}|_{g_{0}}.$
Next, let us recall some results for submanifolds on the projective spaces.
For more details, we refer the readers to [7, 9]. Let $\mathbb{F}$ denote the
field $\mathbb{R}$ of real numbers, the field $\mathbb{C}$ of complex numbers
or the field $\mathbb{Q}$ of quaternions. For convenience, we introduce the
integers
(6.3)
$d_{\mathbb{F}}=\operatorname{dim}_{\mathbb{R}}\mathbb{F}=\left\\{\begin{array}[]{ll}1,&\text{
if }\mathbb{F}=\mathbb{R};\\\ 2,&\text{ if }\mathbb{F}=\mathbb{C};\\\
4,&\text{ if }\mathbb{F}=\mathbb{Q}.\end{array}\right.$
Let us denote by $\mathbb{F}P^{m}$ the $m$-dimensional real projective space
if $\mathbb{F}=\mathbb{R}$, the complex projective space with real dimension
$2m$ if $\mathbb{F}=\mathbb{C}$, and the quaternionic projective space with
real dimension $4m$ if $\mathbb{F}=\mathbb{Q}$, respectively. Here, the
manifold $\mathbb{F}P^{m}$ carries a canonical metric so that the Hopf
fibration
$\pi:\mathbb{S}^{d_{\mathbb{F}}\cdot(m+1)-1}\subset\mathbb{F}^{m+1}\rightarrow\mathbb{F}P^{m}$
is a Riemannian submersion. Hence, the sectional curvature of
$\mathbb{R}P^{m}$ is $1$, the holomorphic sectional curvature is $4$ and the
quaternion sectional curvature is $4$. Let
$\mathcal{H}_{m+1}(\mathbb{F})=\left\\{A\in\mathcal{A}_{m+1}(\mathbb{F})\mid
A^{*}:=\overline{{}^{t}A}=A\right\\}$
be the vector space of $(m+1)\times(m+1)$ Hermitian matrices with coefficients
in the field $\mathbb{F}$, where $\mathcal{A}$ denotes the space of all
$(m+1)\times(m+1)$ matrices over $\mathbb{F}$. We can endow
$\mathcal{H}_{m+1}(\mathbb{F})$ with the inner product
$\langle A,B\rangle=\frac{1}{2}\operatorname{tr}(AB),$
where tr $(\cdot)$ denotes the trace for the given $(m+1)\times(m+1)$ matrix.
Clearly, the map
$\psi:\mathbb{S}^{d_{\mathbb{F}}\cdot(m+1)-1}\subset\mathbb{F}^{m+1}\rightarrow$
$\mathcal{H}_{m+1}(\mathbb{F})$ given by
$\psi=\left(\begin{array}[]{llll}\left|z_{0}\right|^{2}&z_{0}\overline{z_{1}}&\cdots&z_{0}\overline{z_{m}}\\\
z_{1}\overline{z_{0}}&\left|z_{1}\right|^{2}&\cdots&z_{1}\overline{z_{m}}\\\
\cdots&\cdots&\cdots&\cdots\\\
z_{m}\overline{z_{0}}&z_{m}\overline{z_{1}}&\cdots&\left|z_{m}\right|^{2}\end{array}\right)$
induces through the Hopf fibration an isometric embedding $\psi$ from
$\mathbb{F}P^{m}$ into $\mathcal{H}_{m+1}(\mathbb{F}).$ Moreover,
$\psi\left(\mathbb{F}P^{m}\right)$ is a minimal submanifold of the hypersphere
$\mathbb{S}\left(\frac{I}{m+1},\sqrt{\frac{m}{2(m+1)}}\right)$ of
$\mathcal{H}_{m+1}(\mathbb{F})$ with radius $\sqrt{\frac{m}{2(m+1)}}$ and
center $\frac{I}{m+1}$, where $I$ is the identity matrix. In addition, we need
a result as follows (cf. lemma 6.3 in Chapter 4 in [7]):
###### Lemma 6.1.
Let $f:\mathcal{M}^{n}\rightarrow\mathbb{F}P^{\text{m }}$ be an isometric
immersion, and let $\widehat{\textbf{H}}$ and H be the mean curvature vector
fields of the immersions $f$ and $\psi\circ f,$ respectively (here $\psi$ is
the induced isometric embedding $\psi$ from $\mathbb{F}P^{m}$ into
$\mathcal{H}_{m+1}(\mathbb{F})$ explained above). Then, we have
$\left|\textbf{H}\right|^{2}=|\widehat{\textbf{H}}|^{2}+\frac{4(n+2)}{3n}+\frac{2}{3n^{2}}\sum_{i\neq
j}K\left(e_{i},e_{j}\right),$
where $\left\\{e_{i}\right\\}_{i=1}^{n}$ is a local orthonormal basis of
$\Gamma(T\mathcal{M}^{n})$ and $K$ is the sectional curvature of
$\mathbb{F}P^{m}$ expressed $by$
$K\left(e_{i},e_{j}\right)=\left\\{\begin{array}[]{ll}1,&\text{ if
}\mathbb{F}=\mathbb{R};\\\ 1+3\left(e_{i}\cdot Je_{j}\right)^{2},&\text{ if
}\mathbb{F}=\mathbb{C};\\\ 1+\sum_{r=1}^{3}3\left(e_{i}\cdot
J_{r}e_{j}\right)^{2},&\text{ if }\mathbb{F}=\mathbb{Q},\end{array}\right.$
where $J$ is the complex structure of $\mathbb{C}P^{m}$ and $J_{r}$ is the
quaternionic structure of $\mathbb{Q}P^{m}$.
Therefore, one can infer from lemma 6.1 that
(6.4)
$\left|\textbf{H}\right|^{2}=\left\\{\begin{array}[]{ll}|\widehat{\textbf{H}}|^{2}+\frac{2(n+1)}{2n},&\text{
for }\mathbb{R}P^{m};\\\
|\widehat{\textbf{H}}|^{2}+\frac{2(n+1)}{2n}+\frac{2}{n^{2}}\sum_{i,j=1}^{n}\left(e_{i}\cdot
Je_{j}\right)^{2}\leq|\widehat{\textbf{H}}|^{2}+\frac{2(n+2)}{n},&\text{ for
}\mathbb{C}P^{m};\\\
|\widehat{\textbf{H}}|^{2}+\frac{2(n+1)}{2n}+\frac{2}{n^{2}}\sum_{i,j=1}^{n}\sum_{r=1}^{3}\left(e_{i}\cdot
J_{r}e_{j}\right)^{2}\leq|\widehat{\textbf{H}}|^{2}+\frac{2(n+4)}{n},&\text{
for }\mathbb{Q}P^{m}.\end{array}\right.$
Hence, it follows from (6.4) that,
(6.5)
$\left|\textbf{H}\right|^{2}\leq\widehat{H}^{2}+\frac{2\left(n+d_{\mathbb{F}}\right)}{n},$
where $\widehat{H}$ denotes the mean curvature of $\mathcal{M}^{n}$
isometrically immersed into the projective space $\mathbb{F}P^{m}$, this is to
say that,
$\widehat{H}=|\widehat{\textbf{H}}|.$
We note that the equality in (6.5) holds if and only if $\mathcal{M}^{n}$ is a
complex submanifold of $\mathbb{C}P^{m}$ (for the case $\mathbb{C}P^{m}$ )
while $n\equiv 0(\bmod 4)$ and $\mathcal{M}^{n}$ is an invariant submanifold
of $\mathbb{Q}P^{m}\left(\text{ for the case }\mathbb{Q}P^{m}\right)$. We use
$\widehat{\Pi}$ to denote the set of all isometric immersions from
$\mathcal{M}^{n}$ into a projective space $\mathbb{F}P^{m}$. Then, from
theorem 1.1, we can prove the following corollary.
###### Corollary 6.4.
If $\mathcal{M}^{n}$ is isometrically immersed in a projective space
$\mathbb{F}P^{m}$ with mean curvature vector $\widehat{\textbf{H}}$, Then,
eigenvalues of eigenvalue problem (1.20) of $\mathfrak{L}_{\nu}^{2}$ operator
satisfy
(6.6)
$\displaystyle\sum_{i=1}^{n}\left(\Lambda_{i+1}-\Lambda_{1}\right)^{\frac{1}{2}}$
$\displaystyle\leq
4\left\\{\left(\Lambda_{1}^{\frac{1}{2}}+4\widetilde{C}_{5}\Lambda_{1}^{\frac{1}{4}}+4\widetilde{C}_{6}^{2}+C_{6}\right)\left[\left(\frac{n}{2}+1\right)\Lambda_{1}^{\frac{1}{2}}+4\widetilde{C}_{6}\Lambda_{1}^{\frac{1}{4}}+4\widetilde{C}_{6}^{2}+C_{6}\right]\right\\}^{\frac{1}{2}},$
where $C_{6}$ is given by
$C_{6}=\frac{1}{4}\inf_{\widehat{\sigma}\in\widehat{\Pi}}\max_{\Omega}\left(n^{2}|\widehat{\textbf{H}}|^{2}+2n\left(n+d_{\mathbb{F}}\right)\right),$
and $\widetilde{C}_{6}$ is given by
$\widetilde{C}_{6}=\frac{1}{4}\max_{\Omega}|\nu^{\top}|_{g_{0}},$
and $d_{\mathbb{F}}=\operatorname{dim}_{\mathbb{R}}\mathbb{F}$ defined by
(6.3).
###### Proof.
Since there is a canonical imbedding from
$\mathbb{F}P^{m}(\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{Q})$ to Euclidean
space $\mathcal{H}_{m+1}(\mathbb{F})$, then for compact manifold
$\mathcal{M}^{n}$ isometrically immersed into the projective space
$\mathbb{F}P^{m},$ we have the following diagram:
$\textstyle{\mathcal{M}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi\circ
f}$$\scriptstyle{f}$$\textstyle{\mathbb{F}P^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\mathcal{H}_{m+1}(\mathbb{F})}$
where $\psi:\mathbb{F}P^{m}\rightarrow\mathcal{H}_{m+1}(\mathbb{F})$ denotes
the canonical imbedding from $\mathbb{F}P^{m}$ into
$\mathcal{H}_{m+1}(\mathbb{F}),$ and $f:\mathcal{M}^{n}\rightarrow$
$\mathbb{F}P^{m}$ denotes an isometric immersion from $\mathcal{M}^{n}$ to
$\mathbb{F}P^{m}$. Then, $\psi\circ
f:\mathcal{M}^{n}\rightarrow\mathcal{H}_{m+1}(\mathbb{F})$ is an isometric
immersion from $\mathcal{M}^{n}$ to $\mathcal{H}_{m+1}(\mathbb{F})$. Applying
(6.5) and theorem 1.1, one can get (6.6). Thus, it completes the proof of
corollary 6.4.
∎
## References
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* [6] J. J. A. M. Brands, Bounds for the ratios of the first three membrane eigenvalues. Arch. Rational Mech. Anal., 1964, 16 (4): 265-268.
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* [18] F. N. Hile and M. H. Protter, Inequalities for eigenvalues of the Laplacian. Indiana Univ. Math. J., 1980, 29(4): 523-538.
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* [22] M. Levitin and L. Parnovski, Commutators, spectral trace identities, and universal estimates for eigenvalues, J. Funct. Anal., 2002, 192: 425-445.
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|
# Eigenvalues of Xin-Laplacian on
Complete Riemannian manifolds
Lingzhong Zeng and Zhouyuan Zeng Lingzhong Zeng
School of Mathematics and Statistics
Jiangxi Normal University, Nanchang 330022, China<EMAIL_ADDRESS>
Zhouyuan Zeng
School of Mathematics and Statistics
Jiangxi Normal University, Nanchang 330022, China<EMAIL_ADDRESS>
###### Abstract.
In this paper, we firstly consider Dirichlet eigenvalue problem which is
related to Xin-Laplacian on the bounded domain of complete Riemannian
manifolds. By establishing the general formulas, combining with some results
of Chen and Cheng type, we prove some eigenvalue inequalities. As some
applications, we consider the eigenvalues on some Riemannian manifolds
admitting with special functions, the translating solitons, minimal
submanifolds on the Euclidean spaces, submanifolds on the unit spheres,
projective spaces and so on. In particular, for the case of translating
solitons, some eigenvalue inequalities are universal. Moreover, we investigate
the closed eigenvalue problem for the Xin-Laplacian and generalize the
Reilly’s result on the first eigenvalue of the Laplace-Beltrami operator. As
some remarkable applications, we obtain a very sharp estimate for the upper
bound of the second nonzero eigenvalue(without counting multiplicities of
eigenvalues) of the Laplace-Beltrami operator on the minimal isoparametric
hypersurfaces and focal submanifolds in the unit sphere, which leads to a
conjecture and is the most fascinating part of this paper.
††footnotetext: Key words and phrases: isoparametric hypersurface,
$\mathfrak{L}_{\nu}$ operator; eigenvalues; Riemannian manifolds; universal
inequality; translating solitons.††footnotetext: 2010 Mathematics Subject
Classification: 35P15, 53C40,53C30.
## 1\. Introduction
Let $\mathcal{M}^{n}$ be an $n$-dimensional, complete Riemannian submanifold
isometrically immersed into the $(n+p)$-dimensional Euclidean space
$\mathbb{R}^{n+p}$. Suppose that $g_{0}$ is the standard metric on the
Euclidean space $\mathbb{R}^{n+p}$ and $g$ is a Riemannian metric on
$\mathcal{M}^{n}$ induced from the Euclidean space $\mathbb{R}^{n+p}$. Assume
that $\left\\{e_{1},\cdots,e_{n}\right\\}$ is a local orthonormal basis of
$\mathcal{M}^{n}$ with respect to the induced metric $g$, and
$\\{e_{n+1},\cdots,e_{n+p}\\}$ is the local unit orthonormal normal vector
fields. Assume that
$\textbf{H}=\frac{1}{n}\sum_{\alpha=n+1}^{n+p}H^{\alpha}e_{\alpha}=\frac{1}{n}\sum_{\alpha=n+1}^{n+p}\left(\sum_{i=1}^{n}h_{ii}^{\alpha}\right)e_{\alpha},\
\ {\rm and}\ \
H=\frac{1}{n}\sqrt{\sum_{\alpha=n+1}^{n+p}\left(\sum_{i=1}^{n}h_{ii}^{\alpha}\right)^{2}}$
are the mean curvature vector field and the mean curvature of
$\mathcal{M}^{n}$, respectively. Throughout this paper, we use
$\langle\cdot,\cdot\rangle_{g}$, $|\cdot|_{g}^{2}$, ${\rm div}$, $\Delta$,
$\nabla$ and $\nu^{\top}$ to denote the Riemannian inner product associated
with the induced metric $g$, norm with respect to the inner product
$\langle\cdot,\cdot\rangle_{g}$, divergence, Laplacian, the gradient operator
on Riemannian manifolds $\mathcal{M}^{n}$ and the projection of the vector
$\nu$ onto the tangent bundle of $\mathcal{M}^{n}$, respectively. Moreover, we
use $\langle\cdot,\cdot\rangle_{g_{0}}$, $|\cdot|_{g_{0}}^{2}$ and
$\nu^{\bot}$ to represent the standard Euclidean inner product, norm on
$\mathbb{R}^{n+p}$ and the projection of $\nu$ onto the normal bundle of
$\mathcal{M}^{n}$, respectively. Next, we define Xin-Laplacian (or call it
$\mathfrak{L}_{\nu}$ operator) as follows:
(1.1)
$\mathfrak{L}_{\nu}(\cdot)=\Delta(\cdot)+\langle\nu,\nabla(\cdot)\rangle_{g_{0}}=e^{-\langle\nu,X\rangle_{g_{0}}}{\rm
div}(e^{\langle\nu,X\rangle_{g_{0}}}\nabla(\cdot)).$
Xin-Laplacian is an elliptic differential operator and introduced by Xin in
[66]. From the viewpoint of geometry, Xin-Laplacian plays an important role
for the geometric understanding of the translating solitons, see [19, 66]. We
remark that this operator is similar to the $\mathfrak{L}$ operator introduced
by Colding and Minicozzi in [25] and Witten-Laplacian given by
$\Delta_{f}(\cdot)=\Delta(\cdot)-\langle\nabla f,\nabla(\cdot)\rangle_{g}$,
where $f$ is a potential function defined on $\mathcal{M}^{n}$(cf. [28, 29,
64]). It can be shown that the elliptic differential operator
$\mathfrak{L}_{\nu}$ is a self-adjoint operator with respect to the weighted
measure $e^{\langle\nu,X\rangle_{g_{0}}}dv$, namely, for any $u,w\in
C_{0}^{2}(\Omega)$,
(1.2) $\displaystyle-\int_{\Omega}\langle\nabla u,\nabla w\rangle
e^{\langle\nu,X\rangle_{g_{0}}}dv=\int_{\Omega}(\mathfrak{L}_{\nu}w)ue^{\langle\nu,X\rangle_{g_{0}}}dv=\int_{\Omega}(\mathfrak{L}_{\nu}u)we^{\langle\nu,X\rangle_{g_{0}}}dv.$
From more analytic viewpoint, just like $\mathfrak{L}$ operator and Witten-
Laplacian, it is of great importance to explore some analytic properties of
Xin-Laplacian. For example, one can consider Liouville property, spectrum of
Xin-Laplacian, mean value inequality, Gauss maps, heat kernel associated with
the Xin-Laplacian and so on. In particular, the first eigenvalue will lead to
a lot of very profound results in understanding some geometric structure of
translating solitons although we does not cover this aspect of the research in
this paper.
Let $\Omega$ be a bounded domain on an $n$-dimensional Riemannian manifold
$\mathcal{M}^{n}$ with piecewise smooth boundary $\partial\Omega$. We consider
Dirichlet eigenvalue problem of Xin-Laplacian on complete Riemannian manifolds
as follows:
(1.3) $\mathfrak{L}_{\nu}u+\Lambda u=0,\ \ {\rm in}\ \ \ \ \Omega,\ \ {\rm
and}\ \ \ u=0,\ \ {\rm on}\ \ \partial\Omega.$
Assume that $\Lambda_{k}$ denotes the $k^{th}$ eigenvalue corresponding to the
eigenfunction $u_{k}$. Then, the eigenvalue problem (1.3) has real and
discrete spectrum satisfying the following inequalities:
$0<\Lambda_{1}<\Lambda_{2}\leq\Lambda_{3}\leq\cdots\leq\Lambda_{k}\leq\cdots\uparrow+\infty,$
where each eigenvalue is repeated according to its multiplicity.
On one hand, suppose that $M^{n}$ is an $n$-dimensional Euclidean space
$\mathbb{R}^{n}$, Payne, Pólya and Weinberger [50] investigated the
eigenvalues for Dirichlet eigenvalue problem (1.3) of Laplacian and obtained a
universal inequality as follows:
(1.4) $\Lambda_{k+1}-\Lambda_{k}\leq\frac{4}{nk}\sum^{k}_{i=1}\Lambda_{i}.$
Here, the words “universal inequality” means that the spectrum is subject to
universal bounds by which certain expressions involving eigenvalues dominate
others with no reference to the geometry of bounded domain $\Omega$ but
reference to the dimension $n$. The study of the universal inequalities are
stemmed from Payne, Pólya and Weinberger’s important work in 1956 (cf. [50]).
Furthermore, in various settings, many mathematicians extended the universal
inequality of Payne, Pólya and Weinberger. In particular, Hile and Protter
[35] proved the following universal inequality of eigenvalues:
(1.5)
$\sum^{k}_{i=1}\frac{\Lambda_{i}}{\Lambda_{k+1}-\Lambda_{i}}\geq\frac{nk}{4},$
which is sharper than inequality (1.4) given by Payne, Pólya and Weinberger.
Furthermore, an amazing contribution to eigenvalue inequality is that Yang
[67] (cf. [21]) obtained a very sharp universal inequality:
(1.6)
$\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{4}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\Lambda_{i}.$
From (1.6), one can obtain
(1.7) $\Lambda_{k+1}\leq\frac{1}{k}(1+\frac{4}{n})\sum^{k}_{i=1}\Lambda_{i}.$
The inequalities (1.6) and (1.7) are called by Ashbaugh Yang’s first
inequality and second inequality, respectively (cf. [5], [6]). In fact,
Chebyshev’s inequality implies following connections
$\eqref{y1-ineq}\Rightarrow\eqref{y2-ineq}\Rightarrow\eqref{hp-
ineq}\Rightarrow\eqref{ppw-ineq}.$
Let $\Psi$ denote the set of all isometric immersions from $\mathcal{M}^{n}$
into the Euclidean space $\mathbb{R}^{n+p}$. In an important literature [14],
Chen and Cheng investigated Dirichlet problem of Laplacian on the Riemannian
manifolds in 2008. In details, based on an extrinsic method on the mean
curvature of the immersion, they proved
(1.8)
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{4}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{1}{4}C_{1}\right),$
where
$C_{1}=\inf_{\psi\in\Psi}\max_{\Omega}n^{2}H^{2}.$
On the other hand, letting $\Omega$ be a bounded domain on the plane
$\mathbb{R}^{2}$, Payne, Pólya and Weinberger [50] proved that its lower order
eigenvalues satisfy
(1.9) $\Lambda_{2}+\Lambda_{3}\leq 6\Lambda_{1},$
which leads to a famous conjecture for $\Omega\subset\mathbb{R}^{n}$ as
follows (see [1]):
###### Genernal Payne-Pólya-Weinberger Conjecture.
Let $\Omega$ be a bounded domain on an $n$-dimensional Euclidean space
$\mathbb{R}^{n}$. Assume that $\Lambda_{i}$ is the $i$-th eigenvalue of
Dirichlet problem (1.3) for the Laplace operator on $\mathbb{R}^{n}$. Then,
inequality
(1.10) $\frac{\Lambda_{2}+\Lambda_{3}+\cdots+\Lambda_{n+1}}{\Lambda_{1}}\leq
n\frac{\Lambda_{2}(\mathbb{B}^{n})}{\Lambda_{1}(\mathbb{B}^{n})}$
holds, where $\Lambda_{i}(\mathbb{B}^{n})(i=1,2)$ denotes the $i^{th}$
eigenvalue of Laplacian on the ball $\mathbb{B}^{n}$ with the same volume as
the bounded domain $\Omega$, i.e., $Vol(\Omega)=Vol(\Omega^{\ast})$.
Attacking this conjecture, Brands [10] improved Payne, Pólya and Weinberger’s
inequality (1.9) to the following:
$\Lambda_{2}+\Lambda_{3}\leq\Lambda_{1}(3+\sqrt{7}),$ when $n=2$. Furthermore,
Hile and Protter [35] obtained $\Lambda_{2}+\Lambda_{3}\leq 5.622\Lambda_{1}.$
In 1980, Marcellini [45] proved
$\Lambda_{2}+\Lambda_{3}\leq(15+\sqrt{345})/6\Lambda_{1}.$ In 2011, by a new
approach, Chen and Zheng [15] proved $\Lambda_{2}+\Lambda_{3}\leq
5.3507\Lambda_{1}.$ For general case, Ashbaugh and Benguria [4] made a
fundamental contribution for establishing a surprising universal inequality as
follows:
(1.11) $\frac{\Lambda_{2}+\Lambda_{3}+\cdots+\Lambda_{n+1}}{\Lambda_{1}}\leq
n+4.$
for $\Omega\subset\mathbb{R}^{n}$, in 1993. For more references on the
solution of this conjecture, we refer the readers to [2, 3, 24, 35] and
references therein. In particular, an amazing breakthrough was made by
Ashbaugh and Benguria in [2](or see[3]). They affirmatively settled the
general Payne, Pólya and Weinberger’s Conjecture under certain special case.
More specifically, by dealing with some good properties of Bessel functions,
Ashbaugh and Benguria proved a famous conjecture listed in problem collection
of Yau [68](or cf. [1]) as follows:
###### Therorem (Payne-Pólya-Weinberger Conjecture).
Let $\Omega$ be a bounded domain on an $n$-dimensional Euclidean space
$\mathbb{R}^{n}$. Assume that $\Lambda_{i}$ is the $i$-th eigenvalue of the
Dirichlet problem (1.3) for the Laplace operator on $\mathbb{R}^{n}$. Then,
the following eigenvalue inequality
$\frac{\Lambda_{2}}{\Lambda_{1}}\leq\frac{\Lambda_{2}(\mathbb{B}^{n})}{\Lambda_{1}(\mathbb{B}^{n})}$
holds, where $\Lambda_{i}(\mathbb{B}^{n})(i=1,2)$ denotes the $i^{th}$
eigenvalue of Laplacian on the ball $\mathbb{B}^{n}$ with the same volume as
the bounded domain $\Omega$, i.e., $Vol(\Omega)=Vol(\Omega^{\ast})$.
In 2008, Chen and Cheng [14] proved (1.11) still holds when $\Omega$ is a
bounded domain in a complete Riemannian manifold isometrically minimally
immersed in $\mathbb{R}^{n+p}$ . Furthermore, Ashbaugh and Benguria [4] (cf.
Hile and Protter [35] ) improved the above result to the following interesting
universal inequality:
(1.12) $\frac{\Lambda_{2}+\Lambda_{3}+\cdots+\Lambda_{n+1}}{\Lambda_{1}}\leq
n+3+\frac{\Lambda_{1}}{\Lambda_{2}}.$
Very recently, Cheng and Qi [19] have proved that, for any $1\leq j\leq n+2$,
eigenvalues satisfy at least one of the following:
$\displaystyle{\rm(1)}\ \
\frac{\Lambda_{2}}{\Lambda_{1}}<2-\frac{\Lambda_{1}}{\Lambda_{j}},\ \ \
{\rm(2)}\ \
\frac{\Lambda_{2}+\Lambda_{3}+\cdots+\Lambda_{n+1}}{\Lambda_{1}}\leq
n+3+\frac{\Lambda_{1}}{\Lambda_{j}}.$
In 2002, Levitin and Parnovski [41] proved an algebraic inequality, and by
using this algebraic inequality, they generalized (1.11) to
(1.13)
$\frac{\Lambda_{j+1}+\Lambda_{j+2}+\cdots+\Lambda_{j+n}}{\Lambda_{j}}\leq
n+4,$
where $j$ is any positive integer. For general Riemannian manifold
$\mathcal{M}^{n}$ isometrically immersed into the $(n+p)$-dimensional
Euclidean space $\mathbb{R}^{n+p}$, in [14], Chen and Cheng obtained
(1.14)
$\displaystyle\frac{\lambda_{2}+\lambda_{3}+\cdots+\lambda_{n+1}}{\lambda_{1}}\leq
n+4$
where
$\lambda_{i}=\Lambda_{i}+\frac{1}{4}\inf_{\psi\in\Psi}\max_{\Omega}n^{2}H^{2}.$
Also, Soufi, Harrell, Ilias and other mathematicians made many very important
contributions to eigenvalue problem of some self-adjoint elliptic differential
operators. In particular, Soufi, Harrell, Ilias studied the eigenvalues of
Schrödinger operator, and by the same algebraic argument, they established
some interesting inequalities of Payne-Pólya-Weinberger type in [56], which
generalizes inequality (1.14).
Next, let $\mathcal{M}^{n}$ be an $n$-dimensional compact Riemannian manifolds
without boundary. We consider the following closed eigenvalue problem of the
differential operator $\mathfrak{L}_{\nu}$ on the Riemannian manifolds
$\mathcal{M}^{n}$:
(1.15) $\mathfrak{L}_{\nu}\overline{u}+\overline{\Lambda}\overline{u}=0,\ \
{\rm in}\ \ \ \ \mathcal{M}^{n}.$
Let $\overline{\Lambda}_{k}$ denote the $k$-th eigenvalue of the closed
eigenvalue problem (1.15), which is corresponding to the eigenfunction
$\overline{u}_{k}$. Similarly, the spectrum of the eigenvalue problem (1.15)
is discrete and satisfies
$0=\overline{\Lambda}_{0}<\overline{\Lambda}_{1}\leq\overline{\Lambda}_{2}\leq\cdots\leq\overline{\Lambda}_{k}\leq\cdots\rightarrow+\infty,$
where each eigenvalue is repeated according to its multiplicity. Clearly, when
$\nu$ vanishes, closed eigenvalue problem (1.15) becomes a classical closed
eigenvalue problem of Laplace-Beltrami operator:
(1.16) $\Delta\overline{u}+\overline{\Lambda}\overline{u}=0,\ \ {\rm in}\ \ \
\ \mathcal{M}^{n}.$
Assume that $\overline{\Lambda}_{k}$ denotes the $k^{th}$ eigenvalue
corresponding to the eigenfunction $\overline{u}_{k}$. Let
$\overline{\Gamma}_{i}$ be the $i$-th distinct eigenvalue of the closed
eigenvalue problem (1.16) of Beltrami Laplacian on a compact Riemannian
manifold without boundary, where $i=0,1,2,\cdots$. In other words, without
counting multiplicity of each eigenvalue, one has the following strict
inequalities:
$0=\overline{\Gamma}_{0}<\overline{\Gamma}_{1}<\overline{\Gamma}_{2}<\overline{\Gamma}_{3}<\cdots\uparrow+\infty.$
Motivation. It is a very fundamental problem to investigate the eigenvalues of
some elliptic operators on the Riemannian manifolds. Usually, there are two
important problems to be considered in spectral geometry: From an analytic
perspective, given some geometric and topological structures on the manifolds,
one tents to determine the dates or demonstrate certain behaviors of spectrum
of elliptic operators; Conversely, from a geometric viewpoint, one always
wants to obtain some information on the topology and geometry of manifolds
when some spectrum dates are given. The motivation of this paper focuses on
the former part. Although we do does not address the latter part, a conjecture
is proposed deriving from some estimates for upper bounds of the closed
eigenvalue problem (1.16) of Laplace-Beltrami operator. As the authors know,
there is few of investigation for the spectrum of Xin-Laplacian. Thus, it is
very urgent for us to consider the eigenvalue problem of Xin-Laplacian.
Inspired by the previous work and the above statements, it is natural for us
to discuss the following problem.
###### Problem A.
Can we establish some inequalities for the lower and higher order eigenvalues
of Dirichlet problem (1.3)? Furthermore, for translating solitons, whether the
spectral behavior of Xin-Laplacian has a similar rigidity just like the
Dirichlet Laplacian on the domain of the Euclidean space or not?
In 1982, Yau posed a famous conjecture as follows:
###### Yau’s Conjecture.
(cf. [68]) The first nontrivial(non-zero) eigenvalue of Laplace-Beltrami
operator for every closed embedding minimal hypersurface in the unit sphere
equals to the dimension of the hypersurface.
Yau’s conjecture attracted the attention of many mathematicians, and we will
briefly describe its progress in Subsection 6.3. Up to now, it was far from
settled. Yau’s conjecture is concerned with the first eigenvalues, which is
solved by Tang and Yan in [61] when the hypersurface is assumed to be
isoparametric. The authors think that the second eigenvalue is also important
subject, and thus it is natural to ask the following question.
###### Problem B.
Assume that $\mathcal{M}^{n}$ is an $n$-dimensional minimal hypersurface
embedded into the unit sphere $\mathbb{S}^{n+1}(1)$. How can we estimate
accurately the second nontrivial eigenvalue (without counting the
multiplicities of eigenvalues) of eigenvalue problem (1.16)? Furthermore,
suppose that $\mathcal{M}^{n}$ is a isoparametric hypersurface or focal
submanifold of the unit sphere, can we directly calculate the date of the
second nontrivial eigenvalue?
In this paper, we make an affirmative answer to Problem A and partially answer
Problem B. Based on those arguments, we propose some conjectures. This paper
is organized as follows.
In Section 2, we prove several auxiliary lemmas. Applying those auxiliary
lemmas, we establish some general formulas for Dirichlet eigenvalue problem
(1.3).
Furthermore, applying those general formulas, we prove the following
eigenvalue inequalities in Section 3:
(1.17)
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{4}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+D_{1}\Lambda_{i}^{\frac{1}{2}}+\frac{1}{4}D_{1}^{2}+\frac{1}{4}C_{1}\right),$
and
(1.18)
$\displaystyle\sum_{l=1}^{n}\Lambda_{j+l}\leq(4+n)\Lambda_{i}+4D_{1}\Lambda_{i}^{\frac{1}{2}}+D_{1}^{2}+C_{1},$
where
$C_{1}=\inf_{\psi\in\Psi}\max_{\Omega}n^{2}H^{2}\ \ {\rm and}\ \
D_{1}=\max_{\Omega}|\nu^{\top}|_{g_{0}}.$
See Theorem 3.1 and Theorem 3.3 for details. Observing the right hand of
(1.1), we know that the Xin-Laplacian not only depends on the metric $g$ on
the Riemannian manifold $\mathcal{M}^{n}$ but also depends on the standard
metric $g_{0}$ on the Euclidean space. Therefore, it is different from the
Witten-Laplacian, which only depends on the Riemannian metric on
$\mathcal{M}^{n}$. It is well know that the Witten-Laplacian is unitarily
equivalent to the Schrödinger operator, which means that one can estimate the
eigenvalues of Witten-Laplacian by Schrödinger operator techniques. See [54]
for details. However, Xin-Laplacian is not unitarily equivalent to the
Schrödinger operator. As a consequence, some methods associated with unitarily
equivalent no longer works in our situations. Therefore, we remark that our
method is different from the method due to Levitin and Parnovski [41], where
they utilized some algebraic techniques to prove some desired results.
In Section 4, we discuss the eigenvalues of $\mathfrak{L}_{II}$ operator on
the translating solitons. To be special, we obtain the following universal
inequalities:
(1.19)
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{4}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\Lambda_{i}^{\frac{1}{2}}+\frac{n^{2}}{4}\right),$
and
(1.20)
$\displaystyle\sum^{n}_{k=1}\Lambda_{j+k}\leq(n+4)\Lambda_{j}+n^{2}+4\Lambda_{i}^{\frac{1}{2}}.$
See Theorem 4.1 and Theorem 4.2. One could hope that eigenvalue inequalities
are universal for the Dirichlet problem of some elliptic operators on
Riemannian manifolds, but, unfortunately, this is not always possible. In
general, it is not easy to obtain universal inequalities for weighted
Laplacian and even Laplacian on the complete Riemannian manifolds. Therefore,
our work can be regarded as a new contribution to universal inequality.
Furthermore, by using (1.19), we give some estimates for the upper bounds of
the $k$-th eigenvalue and gaps of the consecutive eigenvalues of
$\mathfrak{L}_{II}$ operator on the translating solitons.
As some further applications, we discuss the eigenvalues on the minimal
submanifolds on the Euclidean spaces, submanifolds on the unit spheres,
projective spaces in Section 5. In addition, we also consider the eigenvalues
on some manifolds admitting with special functions such as Cartan-Hadamard
manifolds, product manifolds and homogeneous manifolds and so on in this
section. We refer the readers to Corollary 5.1-5.5 for details.
Before starting Problem B, we motivate the study of closed eigenvalue problem
(1.15) and establish some eigenvalue inequalities in Section 6. Furthermore,
as some remarkable applications, we prove some eigenvalue inequalities of Xin-
Laplacian on the minimal submanifolds in the unit sphere and generalize the
Reilly’s result on the first eigenvalue of the Laplace-Beltrami operator. More
importantly, we suppose that $\mathcal{M}^{n}$ is an $n$-dimensional compact
minimal isoparametric hypersurface in the unit sphere $\mathbb{S}^{n+1}(1)$,
and prove that eigenvalues of the closed eigenvalue problem (1.16) of the
Laplace-Beltrami operator satisfy
(1.21) $\frac{1}{n}\sum_{k=1}^{n}\overline{\Lambda}_{n_{0}+k}\leq 2n+4,$
where $n_{0}$ denotes the value of the multiplicity of the first eigenvalue,
which gets a very sharp estimation for the upper bound of the second
eigenvalues as follows:
$\overline{\Gamma}_{2}\leq 2n+4.$
Here, we do not count the multiplicity of eigenvalues. As a byproduct, our
result further hints that $2n$ could be the second non-zero eigenvalue in term
of a class of isoparametric hypersurfaces of OT-FKM type. For further details,
we refer the readers to Remark 6.6. Clearly, focal submanifolds are some
important minimal submanifolds of the unit spheres. In the remainder part of
this section, we also discuss the eigenvalues of the Laplacian on them and
some upper bounds are obtained.
Based on some arguments for the eigenvalues of Xin-Laplacian on the complete
Riemannian manifolds, several conjectures are posed in Section 7. In
particular, the last conjecture, i.e., Conjecture 7.3, is closely related to
two famous conjecture proposed by Yau and Chern. Here, it is necessary to
emphasize that conjecture 7.3 is presented entirely based on a very sharp
estimates for upper bound in Section 6. In addition, we note that many
important examples hint that this conjecture is true and the second non-zero
eigenvalue ( without counting the multiplicity of eigenvalues) will perfectly
characterize the isoparametric hypersurfaces if it is true. To solve this
conjecture, we must have an in-depth understanding for the topology of minimal
hypersurfaces on the unit spheres. Therefore, the authors think that the study
of this conjecture maybe have an important and far-reaching impact on the
isoparametric theory.
## 2\. Several Auxiliary Lemmas and General Formulas
### 2.1. General Formulas
In this subsection, we would like to establish two general formulas, which
will play critical roles in the proofs of main results. Our first general
formula says the following.
###### Proposition 2.1.
Let $\phi_{l}$, $l=1,2,\cdots,m$, be smooth functions on an $n$-dimensional
complete Riemannian manifold $\mathcal{M}^{n}$ and $\Lambda_{k}$ the
$k^{\text{th}}$ eigenvalue of (1.3). Then, for any $j=1,2,\cdots$, there
exists an orthogonal matrix $A=(a_{ls})_{m\times m}$ such that
$\Phi_{l}=\sum_{s=1}^{m}a_{ls}\phi_{s}$
satisfy
(2.1)
$\sum^{m}_{l=1}(\Lambda_{j+l}-\Lambda_{j})\|u_{j}\nabla\Phi_{l}\|^{2}_{\Omega}\leq\sum^{m}_{l=1}\int_{\Omega}\big{(}u_{j}\mathfrak{L}_{\nu}\Phi_{l}+2\langle\nabla\Phi_{l},\nabla
u_{j}\rangle_{g}\big{)}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv,$
where $u_{j}$ is an orthonormal eigenfunction corresponding to eigenvalue
$\Lambda_{j}$ and
$\|f(x)\|_{\Omega}^{2}=\int_{\Omega}f(x)^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
In order to prove Proposition 2.1, we need the following auxiliary lemmas.
###### Lemma 2.2.
Let $\phi$ be a smooth function on an $n$-dimensional complete Riemannian
manifold $\Omega$. Assume that $\Lambda_{i}$ is the $i^{\text{th}}$ eigenvalue
of the Dirichlet eigenvalue problem (1.3) and $u_{i}$ is an orthonormal
eigenfunction corresponding to $\Lambda_{i}$ such that
$\mathfrak{L}_{\nu}u_{i}=-\Lambda_{i}u_{i},$
and
$\int_{\Omega}u_{i}u_{j}e^{\langle\nu,X\rangle_{g_{0}}}dv=\delta_{ij},$
where $i,j=1,2,\cdots$. Then, for any $j=1,2,\cdots,$ the following equation
(2.2)
$\|u_{j}\nabla\phi\|_{\Omega}^{2}=\sum^{\infty}_{k=1}(\Lambda_{k}-\Lambda_{j})\sigma_{jk}^{2},$
holds, where
$\sigma_{jk}=\int_{\Omega}\phi u_{j}u_{k}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
###### Proof.
Since $\\{u_{k}\\}^{\infty}_{k=1}$ is an orthonormal basis of the weighted
$L^{2}(\Omega)$, then, for any $j$, where $j=1,2,\cdots$, we know that
(2.3) $\phi u_{j}=\sum^{\infty}_{k=1}\sigma_{jk}u_{k}.$
According to Parseval’s identity, it is not difficult to check that
$\|\phi u_{j}\|^{2}_{\Omega}=\sum^{\infty}_{k=1}\sigma_{jk}^{2}.$
By a simple computation, we immediately derive
$\displaystyle\int_{\Omega}(\mathfrak{L}_{\nu}\left(\phi
u_{j})-\phi\mathfrak{L}_{\nu}u_{j}\right)u_{k}e^{\langle\nu,X\rangle_{g_{0}}}dv=(\Lambda_{j}-\Lambda_{k})\int_{\Omega}\phi
u_{j}u_{k}e^{\langle\nu,X\rangle_{g_{0}}}dv,$
and
$\mathfrak{L}_{\nu}(\phi
u_{j})=\phi\mathfrak{L}_{\nu}u_{j}+u_{j}\mathfrak{L}_{\nu}\phi+2\langle\nabla\phi,\nabla
u_{j}\rangle_{g}.$
Therefore, we have
(2.4) $\int_{\Omega}(u_{j}\mathfrak{L}_{\nu}\phi+2\langle\nabla\phi,\nabla
u_{j}\rangle_{g})u_{k}e^{\langle\nu,X\rangle_{g_{0}}}dv=(\Lambda_{j}-\Lambda_{k})\int_{\Omega}\phi
u_{j}u_{k}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
Furthermore, from (2.3), we deduce
(2.5)
$\displaystyle\sum^{\infty}_{k=1}(\Lambda_{k}-\Lambda_{j})\sigma_{jk}^{2}$
$\displaystyle=\sum^{\infty}_{k=1}(\Lambda_{k}-\Lambda_{j})\left(\int_{\Omega}\phi
u_{j}u_{k}e^{\langle\nu,X\rangle_{g_{0}}}dv\right)^{2}$
$\displaystyle=\sum^{\infty}_{k=1}\Lambda_{k}\left(\int_{\Omega}\phi
u_{j}u_{k}e^{\langle\nu,X\rangle_{g_{0}}}dv\right)^{2}-\Lambda_{j}\|\phi
u_{j}\|_{\Omega}^{2}.$
By using (2.3), we derive
$\mathfrak{L}_{\nu}(\phi
u_{j})=\sum^{\infty}_{k=1}\sigma_{jk}\mathfrak{L}_{\nu}u_{k}=-\sum^{\infty}_{k=1}\sigma_{jk}\Lambda_{k}u_{k},$
which implies
(2.6) $\phi u_{j}\mathfrak{L}_{\nu}(\phi
u_{j})=-\sum^{\infty}_{k=1}\sigma_{jk}\Lambda_{k}u_{k}\phi u_{j}.$
Hence, it follows from (2.6) that
(2.7) $\displaystyle\int_{\Omega}\phi u_{j}\mathfrak{L}_{\nu}(\phi
u_{j})e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=-\sum^{\infty}_{k=1}\int_{\Omega}\sigma_{jk}\Lambda_{k}u_{k}\phi
u_{j}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=-\sum^{\infty}_{k=1}\Lambda_{k}\int_{\Omega}\phi
u_{j}u_{k}e^{\langle\nu,X\rangle_{g_{0}}}dv\int_{\Omega}u_{k}\phi
u_{j}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=-\sum^{\infty}_{k=1}\Lambda_{k}\left(\int_{\Omega}\phi
u_{j}u_{k}e^{\langle\nu,X\rangle_{g_{0}}}dv\right)^{2}.$
Combining (2.5) with (2.7), we have
$\displaystyle\sum^{\infty}_{k=1}(\Lambda_{k}-\Lambda_{j})\sigma_{jk}^{2}$
$\displaystyle=\int_{\Omega}(-\phi u_{j}\mathfrak{L}_{\nu}(\phi
u_{j})-\Lambda_{j}\phi^{2}u_{j}^{2})e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\int_{\Omega}(-\phi
u_{j}^{2}\mathfrak{L}_{\nu}\phi-2\langle\nabla\phi,\nabla u_{j}\rangle_{g}\phi
u_{j})e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\int_{\Omega}\left(|\nabla\phi|_{g}^{2}u_{j}^{2}+\frac{1}{2}\langle\nabla\phi^{2},\nabla
u_{j}^{2}\rangle_{g}-\frac{1}{2}\langle\nabla\phi^{2},\nabla
u_{j}^{2}\rangle_{g}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\|u_{j}\nabla\phi\|^{2}_{\Omega}.$
This completes the proof of Lemma 2.2. ∎
By making use of the Lemma 2.2, we give the proof of Proposition 2.1.
_Proof of Proposition_ 2.1. For any $j=1,2,\cdots$, we consider the following
$m\times m$-matrix:
$C:=\left(\int_{\Omega}\big{(}u_{j}\mathfrak{L}_{\nu}\phi_{l}+2\langle\nabla\phi_{l},\nabla
u_{j}\rangle_{g}\big{)}u_{j+s}e^{\langle\nu,X\rangle_{g_{0}}}dv\right)_{m\times
m}.$
According to the Gram-Schmidt orthogonalization, we know that there exists an
orthogonal matrix $A=(a_{ls})$ such that
$Q=AC=(q_{ls})_{m\times m}=\begin{aligned}
\left(\begin{array}[]{cccc}q_{11}&q_{12}&\cdots&q_{1m}\\\
0&q_{22}&\cdots&q_{2m}\\\ \vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&q_{mm}\\\
\end{array}\right),\end{aligned}$
that is,
$\displaystyle q_{ls}$
$\displaystyle=\sum_{i=1}^{m}a_{li}\int_{\Omega}\biggl{(}u_{j}\mathfrak{L}_{\nu}\phi_{i}+2\langle\nabla\phi_{i},\nabla
u_{j}\rangle_{g}\biggl{)}u_{j+s}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\int_{\Omega}\left(u_{j}\mathfrak{L}_{\nu}\left(\sum_{i=1}^{m}a_{li}\phi_{i}\right)+2\langle\nabla\left(\sum_{i=1}^{m}a_{li}\phi_{i}\right),\nabla
u_{j}\rangle_{g}\right)u_{j+s}e^{\langle\nu,X\rangle_{g_{0}}}dv,$
with $q_{ls}=0$ for $l>s$. For
$\Phi_{l}=\sum_{i=1}^{m}a_{li}\phi_{i},$
we have
$\displaystyle q_{ls}$
$\displaystyle=\int_{\Omega}\left(u_{j}\mathfrak{L}_{\nu}\Phi_{l}+2\left\langle\nabla\Phi_{l},\nabla
u_{j}\right\rangle_{g}\right)u_{j+s}e^{\langle\nu,X\rangle_{g_{0}}}dv=0,\
\text{for $l>s$}.$
Applying the Lemma 2.2 to functions $\Phi_{l}$, we yield
(2.8) $\displaystyle\|u_{j}\nabla\Phi_{l}\|^{2}_{\Omega}$
$\displaystyle=\sum^{\infty}_{k=1}(\Lambda_{k}-\Lambda_{j})\beta^{2}_{ljk}$
$\displaystyle=\sum^{j-1}_{k=1}(\Lambda_{k}-\Lambda_{j})\beta^{2}_{ljk}+\sum^{j+l-1}_{k=j}(\Lambda_{k}-\Lambda_{j})\beta^{2}_{ljk}+\sum^{\infty}_{k=j+l}(\Lambda_{k}-\Lambda_{j})\beta^{2}_{ljk},$
where
$\beta_{ljk}:=\int_{\Omega}\Phi_{l}u_{j}u_{k}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
According to (2.4) in place of $\phi$ with $\Phi_{l}$, it is easy to verify
that
(2.9)
$\int_{\Omega}(u_{j}\mathfrak{L}_{\nu}\Phi_{l}+2\langle\nabla\Phi_{l},\nabla
u_{j}\rangle_{g})u_{k}e^{\langle\nu,X\rangle_{g_{0}}}dv=(\Lambda_{j}-\Lambda_{k})\int_{\Omega}\Phi_{l}u_{j}u_{k}e^{\langle\nu,X\rangle_{g_{0}}}dv,$
which implies
$(\Lambda_{k}-\Lambda_{j})\int_{\Omega}\Phi_{l}u_{j}u_{k}e^{\langle\nu,X\rangle_{g_{0}}}dv=(\Lambda_{k}-\Lambda_{j})\beta_{ljk}=0,\
\text{for $k=j,j+1,\cdots,j+l-1$}.$
From (LABEL:2.8), we conclude
(2.10) $\displaystyle\|u_{j}\nabla\Phi_{l}\|^{2}_{\Omega}$
$\displaystyle\leq\sum^{\infty}_{k=j+l}(\Lambda_{k}-\Lambda_{j})\beta^{2}_{ljk}.$
Hence, from (2.9), (2.10) and Parseval’s identity, we infer that,
$\displaystyle\sum^{m}_{l=1}(\Lambda_{j+l}-\Lambda_{j})\|u_{j}\nabla\Phi_{l}\|^{2}_{\Omega}$
$\displaystyle\leq\sum^{m}_{l=1}\sum^{\infty}_{k=j+l}(\Lambda_{k}-\Lambda_{j})^{2}\beta^{2}_{ljk}$
$\displaystyle\leq\sum^{m}_{l=1}\int_{\Omega}\big{(}u_{j}\mathfrak{L}_{\nu}\Phi_{l}+2\langle\nabla\Phi_{l},\nabla
u_{j}\rangle_{g}\big{)}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
The proof ends.
$None$
In what follows, we would like to prove the second general formula for
eigenvalues, which generalizes a formula established by Cheng and Yang in [21]
for the eigenvalue problem of the Laplacian. We remark that the original
method of this proof is due to Cheng and Yang in [21]. However, for the
convenience of readers, we shall give a self contained proof.
###### Proposition 2.3.
Let $(\mathcal{M}^{n},g)$ be an $n$-dimensional complete noncompact Riemannian
manifold. Assume that $\Lambda_{i}$ is the $i^{\text{th}}$ eigenvalue of the
Dirichlet eigenvalue problem (1.3) and $u_{i}$ is an orthonormal eigenfunction
corresponding to $\Lambda_{i}$ such that
$\mathfrak{L}_{\nu}u_{i}=-\Lambda_{i}u_{i},$
and
$\int_{\Omega}u_{i}u_{j}e^{\langle\nu,X\rangle_{g_{0}}}dv=\delta_{ij},$
where $i,j=1,2,\cdots$. Then, for any function $\varphi(x)\in C^{2}(\Omega)$
and any positive integer $k$, eigenvalues of the Dirichlet eigenvalue problem
(1.3) satisfy
(2.11)
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\|u_{i}\nabla\varphi\|_{\Omega}^{2}\leq\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\|2\langle\nabla\varphi,\nabla
u_{i}\rangle_{g}+u_{i}\mathfrak{L}_{\nu}\varphi\|_{\Omega}^{2}.$
Before giving the proof of Proposition 2.3, we shall introduce several
notations. Set
$\sigma_{ij}:=\int_{\Omega}\varphi
u_{i}u_{j}e^{\langle\nu,X\rangle_{g_{0}}}dv,$
and
$\displaystyle\Theta_{i}:=-\int_{\Omega}\zeta_{i}(u_{i}\mathfrak{L}_{\nu}\varphi+2\langle\nabla\varphi,\nabla
u_{i}\rangle_{g})e^{\langle\nu,X\rangle_{g_{0}}}dv.$
where $\varphi(x)\in C^{2}(\Omega)$, and
$\zeta_{i}:=\varphi u_{i}-\sum^{k}_{j=1}\sigma_{ij}u_{j}.$
Define
$\tau_{ij}:=-\int_{\Omega}(u_{j}\mathfrak{L}_{\nu}\varphi+2\langle\nabla\varphi,\nabla
u_{j}\rangle_{g})u_{i}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
In order to prove Proposition 2.3, we need the following several auxiliary
lemmas. The first auxiliary lemma is show that $\tau_{ji}$ is skew-symmetric.
###### Lemma 2.4.
Under the assumption of Proposition 2.3, we have
(2.12) $\tau_{ij}=(\Lambda_{i}-\Lambda_{j})\sigma_{ij},$
and
(2.13) $\tau_{ij}=-\tau_{ji}.$
###### Proof.
By utilizing (1.2), it is easy to verify that
$\tau_{ij}=(\Lambda_{i}-\Lambda_{j})\sigma_{ij}.$ By the definition of
$\sigma_{ij}$, we know that $\sigma_{ij}=\sigma_{ji}.$ Therefore, we have
$\tau_{ij}=-\tau_{ji}.$ This finishes the proof of this Lemma.
∎
Next, we shall give an estimate for the lower bound of $\Theta_{i}$.
###### Lemma 2.5.
Under the assumption of Proposition 2.3, we have
(2.14)
$\displaystyle(\Lambda_{k+1}-\Lambda_{i})\|\zeta_{i}\|_{\Omega}^{2}\leq\Theta_{i}.$
###### Proof.
Since $u_{j}$ is an orthonormal eigenfunction corresponding to the eigenvalue
$\Lambda_{j}$, $\\{u_{j}\\}^{\infty}_{j=1}$ forms an orthonormal basis of the
weighted $L^{2}(\Omega)$. Furthermore, by the Rayleigh-Ritz inequality, we
have
(2.15)
$\displaystyle\Lambda_{k+1}\leq-\frac{\displaystyle\int_{\Omega}\varphi\mathfrak{L}_{\nu}\varphi
e^{\langle\nu,X\rangle_{g_{0}}}dv}{\displaystyle\int_{\Omega}\varphi^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv},$
for any function $\varphi$ satisfing
$\int_{\Omega}\varphi u_{j}e^{\langle\nu,X\rangle_{g_{0}}}dv=0,\ \ 1\leq j\leq
k.$
By a direct calculation, it is not difficult to verify that
(2.16)
$\displaystyle\int_{\Omega}\zeta_{i}u_{l}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=0,$
for $1\leq i,l\leq k.$ Clearly, (2.15) implies
$\Lambda_{k+1}\leq-\frac{\displaystyle\int_{\Omega}\zeta_{i}\mathfrak{L}_{\nu}\zeta_{i}e^{\langle\nu,X\rangle_{g_{0}}}dv}{\displaystyle\int_{\Omega}\zeta^{2}_{i}e^{\langle\nu,X\rangle_{g_{0}}}dv}.$
Since
$\displaystyle\mathfrak{L}_{\nu}\zeta_{i}=u_{i}\mathfrak{L}_{\nu}\varphi-\Lambda_{i}\varphi
u_{i}+2\langle\nabla\varphi,\nabla
u_{j}\rangle_{g}+\sum^{k}_{j=1}\Lambda_{j}\sigma_{ij}u_{j},$
from (2.16), we have
$\displaystyle(\Lambda_{k+1}-\Lambda_{i})\|\zeta_{i}\|_{\Omega}^{2}\leq-\int_{\Omega}\zeta_{i}(u_{i}\mathfrak{L}_{\nu}\varphi+2\langle\nabla\varphi,\nabla
u_{i}\rangle_{g})e^{\langle\nu,X\rangle_{g_{0}}}dv=\Theta_{i}.$
Thus, we finish the proof of this Lemma.
∎
From Lemma 2.5, we have the following lemma.
###### Lemma 2.6.
Under the assumption of Proposition 2.3, we have
(2.17)
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\Theta_{i}\leq\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\|u_{i}\mathfrak{L}_{\nu}\varphi+2\langle\nabla\varphi,\nabla
u_{i}\rangle_{g}-\sum_{j=1}^{k}\tau_{ij}u_{j}\|_{\Omega}^{2}.$
###### Proof.
Firstly, we give an estimate for the upper bound of $\Theta_{i}$. From (2.14),
(2.16) and the Cauchy-Schwarz inequality, we infer
(2.18) $\displaystyle\Theta_{i}$
$\displaystyle=-\int_{\Omega}\zeta_{i}\left(u_{i}\mathfrak{L}_{\nu}\varphi+2\langle\nabla\varphi,\nabla
u_{i}\rangle_{g}-\sum_{j=1}^{k}\tau_{ij}u_{j}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\leq\left\\{\|\zeta_{i}\|_{\Omega}^{2}\|u_{i}\mathfrak{L}_{\nu}\varphi+2\langle\nabla\varphi,\nabla
u_{i}\rangle_{g}-\sum_{j=1}^{k}\tau_{ij}u_{j}\|_{\Omega}^{2}\right\\}^{\frac{1}{2}}.$
Uniting (2.14) and (2.18), we obtain
$\displaystyle(\Lambda_{k+1}-\Lambda_{i})\Theta^{2}_{i}$
$\displaystyle\leq(\Lambda_{k+1}-\Lambda_{i})\|\zeta_{i}\|_{\Omega}^{2}\|u_{i}\mathfrak{L}_{\nu}\varphi+2\langle\nabla\varphi,\nabla
u_{i}\rangle_{g}-\sum_{j=1}^{k}\tau_{ij}u_{j}\|_{\Omega}^{2}$
$\displaystyle\leq\Theta_{i}\|u_{i}\mathfrak{L}_{\nu}\varphi+2\langle\nabla\varphi,\nabla
u_{i}\rangle_{g}-\sum_{j=1}^{k}\tau_{ij}u_{j}\|_{\Omega}^{2}.$
Therefore, we have
(2.19)
$\displaystyle(\Lambda_{k+1}-\Lambda_{i})^{2}\Theta_{i}\leq(\Lambda_{k+1}-\Lambda_{i})\|u_{i}\mathfrak{L}_{\nu}\varphi+2\langle\nabla\varphi,\nabla
u_{i}\rangle_{g}-\sum_{j=1}^{k}\tau_{ij}u_{j}\|_{\Omega}^{2}.$
Summing on $i$ from $1$ to $k$ for (2.19), we derive
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\Theta_{i}\leq\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\|u_{i}\mathfrak{L}_{\nu}\varphi+2\langle\nabla\varphi,\nabla
u_{i}\rangle_{g}-\sum_{j=1}^{k}\tau_{ij}u_{j}\|_{\Omega}^{2},$
as claimed. Thus, the proof of this lemma ends.
∎
Applying Lemma 2.4, Lemma 2.5 and Lemma 2.6, we give the proof of Proposition
2.3.
_Proof of Proposition_ 2.3. By the definition of $\tau_{ij}$ and (2.12), it is
not difficult to infer that
(2.20)
$\displaystyle\|u_{i}\mathfrak{L}_{\nu}\varphi+2\langle\nabla\varphi,\nabla
u_{i}\rangle_{g}-\sum^{n}_{j=1}\tau_{ij}u_{j}\|_{\Omega}^{2}$
$\displaystyle=\|u_{i}\mathfrak{L}_{\nu}\varphi+2\langle\nabla\varphi,\nabla
u_{i}\rangle_{g}\|_{\Omega}^{2}-\sum^{n}_{j=1}\tau_{ij}^{2}$
$\displaystyle=\|u_{i}\mathfrak{L}_{\nu}\varphi+2\langle\nabla\varphi,\nabla
u_{i}\rangle_{g}\|_{\Omega}^{2}-\sum^{n}_{j=1}(\Lambda_{i}-\Lambda_{j})^{2}\sigma_{ij}^{2},$
According to the definitions of $\Theta_{i}$ and $\zeta_{i}$, it follows from
(2.12) that,
(2.21) $\displaystyle\Theta_{i}$ $\displaystyle=-\int_{\Omega}\left(\varphi
u_{i}-\sum^{k}_{j=0}\sigma_{ij}u_{j}\right)\Bigg{(}u_{i}\mathfrak{L}_{\nu}\varphi+2\langle\nabla\varphi,\nabla
u_{i}\rangle_{g}\Bigg{)}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=-\int_{\Omega}(\varphi
u^{2}_{i}\mathfrak{L}_{\nu}\varphi+2\varphi u_{i}\langle\nabla\varphi,\nabla
u_{i}\rangle_{g})e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad~{}+\sum^{k}_{j=1}a_{ij}\int_{\Omega}u_{j}(u_{i}\mathfrak{L}_{\nu}\varphi+2\langle\nabla\varphi,\nabla
u_{i}\rangle_{g})e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=-\int_{\Omega}\left(\varphi\mathfrak{L}_{\nu}\varphi-\frac{1}{2}\mathfrak{L}_{\nu}\varphi^{2}\right)u^{2}_{i}e^{\langle\nu,X\rangle_{g_{0}}}dv+\sum^{k}_{j=1}\sigma_{ij}\tau_{ij}$
$\displaystyle=\int_{\Omega}\langle\nabla\varphi,\nabla\varphi\rangle
u^{2}_{i}e^{\langle\nu,X\rangle_{g_{0}}}dv+\sum^{k}_{j=1}(\Lambda_{i}-\Lambda_{j})\sigma_{ij}^{2}.$
A simple calculation shows that
(2.22)
$\displaystyle\sum_{i,j=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}(\Lambda_{i}-\Lambda_{j})\sigma^{2}_{ij}=-\sum_{i,j=1}^{k}(\Lambda_{k+1}-\Lambda_{i})(\Lambda_{i}-\Lambda_{j})^{2}\sigma^{2}_{ij}.$
Furthermore, uniting (2.19), (LABEL:2.18), (2.21) and (2.22), we get
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\|u_{i}\nabla\varphi\|_{\Omega}^{2}\leq\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\|2\langle\nabla\varphi,~{}\nabla
u_{i}\rangle_{g}+u_{i}\mathfrak{L}_{\nu}\varphi\|_{\Omega}^{2}.$
Therefore, we finish the proof of this proposition.
$None$
### 2.2. Extrinsic Formulas
In order to prove our main results, we need the following lemma. A proof of it
can be found in [14].
###### Lemma 2.7.
For an $n$-dimensional submanifold $\mathcal{M}^{n}$ in Euclidean space
$\mathbb{R}^{n+p}$, let $x=(x_{1},x_{2},\cdots,x_{n+p})$ is the position
vector of a point $p\in\mathcal{M}^{n}$ with
$x_{\alpha}=x_{\alpha}(y_{1},\cdots,y_{n})$, $1\leq\alpha\leq n+p$, where
$(y_{1},\cdots,y_{n})$ denotes a local coordinate system of $\mathcal{M}^{n}$.
Then, we have
$\sum^{n+p}_{\alpha=1}\langle\nabla x_{\alpha},\nabla
x_{\alpha}\rangle_{g}=n,$ $\displaystyle\sum^{n+p}_{\alpha=1}\langle\nabla
x_{\alpha},\nabla u\rangle_{g}\langle\nabla x_{\alpha},\nabla
w\rangle_{g}=\langle\nabla u,\nabla w\rangle_{g},$
for any functions $u,w\in C^{1}(\mathcal{M}^{n})$,
$\displaystyle\sum^{n+p}_{\alpha=1}(\Delta x_{\alpha})^{2}=n^{2}H^{2},$
and
$\displaystyle\sum^{n+p}_{\alpha=1}\Delta x_{\alpha}\nabla x_{\alpha}=0,$
where $H$ is the mean curvature of $\mathcal{M}^{n}$.
We choose a new coordinate system
$\bar{y}=\left(\bar{y}^{1},\cdots,\bar{y}^{n+p}\right)$ of $\mathbb{R}^{n+p}$
given by
$y-y(P)=\bar{y}A,$
such that
$\left(\frac{\partial}{\partial\bar{y}^{1}}\right)_{P},\cdots,\left(\frac{\partial}{\partial\bar{y}^{n}}\right)_{P}\operatorname{span}T_{P}\mathcal{M}^{n}$,
and at $P,$
$\langle\frac{\partial}{\partial\bar{y}^{i}},\frac{\partial}{\partial\bar{y}^{j}}\rangle_{g}=\delta_{ij},$
where $A=\left(a_{\beta}^{\alpha}\right)\in O(n+p)$ is an $(n+p)\times(n+p)$
orthogonal matrix. Let
(2.23)
$\nu=\sum^{n+p}_{\theta=1}\nu_{\theta}\frac{\partial}{\partial\overline{y}^{\theta}}\in\mathbb{R}^{n+p},$
and
$g_{0\alpha\beta}=\langle\frac{\partial}{\partial\overline{y}^{\alpha}},\frac{\partial}{\partial\overline{y}^{\beta}}\rangle_{g_{0}}.$
Let $w$ be a smooth function defined on the Riemannian manifold
$\mathcal{M}^{n}$. Under the local coordinate system
$\bar{y}=\left(\bar{y}^{1},\cdots,\bar{y}^{n}\right)$, by an easy exercise,
one can show that
(2.24)
$\nu^{\top}=\sum^{n}_{\theta=1}\nu_{\theta}\frac{\partial}{\partial\overline{y}^{\theta}},$
and
(2.25) $\displaystyle\langle\nu,\nabla
w\rangle_{g_{0}}=\sum^{n}_{i=1}\nu_{i}\frac{\partial
w}{\partial\overline{y}^{i}}.$
By Cauchy-Schwarz inequality, we have
(2.26) $\displaystyle\left(\sum^{n}_{\theta=1}\nu_{\theta}\frac{\partial
w}{\partial\overline{y}^{\theta}}\right)^{2}\leq\left(\sum^{n}_{\theta=1}\nu_{\theta}^{2}\right)\cdot\sum^{n}_{\theta=1}\left(\frac{\partial
w}{\partial\overline{y}^{\theta}}\right)^{2}.$
Therefore, combining (2.24), (2.25) and (2.26), we can prove the following
lemma.
###### Lemma 2.8.
Let $w$ be a smooth function defined on the Riemannian manifold
$\mathcal{M}^{n}$, then we have
(2.27) $\langle\nu,\nabla w\rangle_{g_{0}}\leq|\nu^{\top}|_{g_{0}}|\nabla
w|_{g}.$
By a direct computation, one can show the following results of Chen and Cheng
type.
###### Lemma 2.9.
_(Result of Cheng and Chen Type)_ Let $\left(y_{1},\cdots,y_{n}\right)$ be an
arbitrary coordinate system in a neighborhood $U$ of $P$ in $\mathcal{M}^{n}.$
Assume that $x$ with components $x_{\alpha}$ defined by
$x_{\alpha}=x_{\alpha}\left(y_{1},\cdots,y_{n}\right)$, where $1\leq\alpha\leq
n+p$ is the position vector of $P$ in $\mathbb{R}^{n+p}$. Then, we have
(2.28) $\sum_{\alpha=1}^{n+p}\left\langle\nabla
x_{\alpha},\nu\right\rangle_{g_{0}}^{2}=|\nu^{\top}|_{g_{0}}^{2},$
where $\nabla$ is the gradient operator on $\mathcal{M}^{n}$.
From Cauchy-Schwarz inequality, Lemma 2.7 and Lemma 2.9, we have the following
lemma.
###### Lemma 2.10.
_(Result of Cheng and Chen Type)_ Let $\left(y_{1},\cdots,y_{n}\right)$ be an
arbitrary coordinate system in a neighborhood $U$ of $P$ in $\mathcal{M}^{n}.$
Assume that $x$ with components $x_{\alpha}$ defined by
$x_{\alpha}=x_{\alpha}\left(y_{1},\cdots,y^{n}\right)$, where $1\leq\alpha\leq
n+p$, is the position vector of $P$ in $\mathbb{R}^{n+p}$. Then, we have
(2.29) $\sum_{\alpha=1}^{n+p}\left\langle\nabla x_{\alpha},\nabla
u\right\rangle_{g}\left\langle\nabla
x_{\alpha},\nu\right\rangle_{g_{0}}\leq|\nabla u|_{g}|\nu^{\top}|_{g_{0}},$
where $\nabla$ is the gradient operator on $\mathcal{M}^{n}$.
Let $x_{1},x_{2},\cdots,x_{n+p}$ be the standard coordinate functions of
$\mathbb{R}^{n+p}$ and define an $((n+p)\times(n+p))$-matrix $D$ by
$D:=\left(d_{\alpha\beta}\right),~{}{\rm
where}~{}d_{\alpha\beta}=\int_{\Omega}x_{\alpha}u_{1}u_{\beta+1}.$ Using the
orthogonalization of Gram and Schmidt, it is easy to see that there exist an
upper triangle matrix $R=\left(R_{\alpha\beta}\right)$ and an orthogonal
matrix $Q=\left(\tau_{\alpha\beta}\right)$ such that $R=QB,$ i.e.,
$\displaystyle
R_{\alpha\beta}=\sum_{\gamma=1}^{n+p}\tau_{\alpha\gamma}d_{\gamma\beta}=\int_{\Omega}\sum_{\gamma=1}^{n+p}\tau_{\alpha\gamma}x_{\gamma}u_{1}u_{\beta+1}=0,$
for $1\leq\beta<\alpha\leq n+p$. Defining
(2.30) $h_{\alpha}=\sum_{\gamma=1}^{n+p}\tau_{\alpha\gamma}x_{\gamma},$
we have
$\int_{\Omega}h_{\alpha}u_{1}u_{\beta+1}=0,$
where $1\leq\beta<\alpha\leq n+p.$ Since
$h_{\alpha}=\sum_{\gamma=1}^{n+p}\tau_{\alpha\gamma}x_{\gamma}$
and $Q$ is an orthogonal matrix, by Lemma (2.7), Lemma (2.9) and Lemma (2.10),
we can show the following lemma.
###### Lemma 2.11.
Under the above convention, we have
(2.31) $\sum_{\alpha=1}^{n+p}\left|\nabla h_{\alpha}\right|_{g}^{2}=n,$
(2.32) $\sum_{\alpha=1}^{n+p}\left(\Delta h_{\alpha}\right)^{2}=n^{2}H^{2},$
(2.33) $\sum_{\alpha=1}^{n+p}\Delta h_{\alpha}\left\langle\nabla
h_{\alpha},\nabla u_{1}\right\rangle_{g}=0,$
(2.34) $\sum_{\alpha=1}^{n+p}\Delta h_{\alpha}\left\langle\nabla
h_{\alpha},\nu\right\rangle_{g_{0}}=0,$
(2.35) $\sum_{\alpha=1}^{n+p}\left\langle\nabla
h_{\alpha},\nu\right\rangle_{g_{0}}^{2}=\big{|}\nu^{\top}\big{|}_{g_{0}}^{2},$
(2.36) $\sum_{\alpha=1}^{n+p}\left\langle\nabla h_{\alpha},\nabla
u_{1}\right\rangle_{g}\left\langle\nabla
h_{\alpha},\nu\right\rangle_{g_{0}}\leq|\nabla
u_{1}|_{g}|\nu^{\top}|_{g_{0}},$
and
(2.37) $\sum_{\alpha=1}^{n+p}\left\langle\nabla h_{\alpha},\nabla
u_{1}\right\rangle_{g}^{2}=\left|\nabla u_{1}\right|_{g}^{2}.$
###### Proof.
The proof is direct and one only need to notice that $Q$ is a orthogonal
matrix. Here, we omit the details. ∎
## 3\. Two Bounds for the Eigenvalues
In this section, applying general formulas, we give some bounds of the
eigenvalues.
### 3.1. Bound of Yang Type
In this paper, we investigate the eigenvalues of Dirichlet problem (1.3) of
Xin-Laplacian on the complete Riemannian manifolds. The first purpose of this
paper is to prove an inequality of eigenvalues with higher order as follows.
###### Theorem 3.1.
Let $(\mathcal{M}^{n},g)$ be an $n$-dimensional complete Riemannian manifold
isometrically embedded into the Euclidean space $\mathbb{R}^{n+p}$ with mean
curvature $H$. Assume that $\Lambda_{i}$ denotes the $i$-th eigenvalue of the
Dirichlet problem (1.3) of the Xin-Laplacian. Then, we have
(3.1)
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{4}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+D_{1}\Lambda_{i}^{\frac{1}{2}}+\frac{1}{4}D_{1}^{2}+\frac{1}{4}C_{1}\right),$
and
(3.2)
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{6}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{1}{2}D_{1}^{2}+\frac{1}{6}C_{1}\right),$
where
$C_{1}=\inf_{\psi\in\Psi}\max_{\Omega}n^{2}H^{2}\ \ and\ \
D_{1}=\max_{\Omega}|\nu^{\top}|_{g_{0}}.$
###### Remark 3.1.
In Theorem 3.1, one can show that (3.2) is sharper than (3.1), when
(3.3) $\displaystyle
2\Lambda_{i}^{\frac{1}{2}}\left(\int_{\Omega}u_{i}^{2}|\nu^{\top}|^{2}_{g_{0}}e^{\langle\nu,X\rangle_{g_{0}}}dv\right)^{\frac{1}{2}}\geq\Lambda_{i}+\int_{\Omega}u_{i}^{2}|\nu^{\top}|_{g_{0}}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv;$
while (3.1) is sharper than (3.2), when inequality (3.3) goes the other way.
###### Remark 3.2.
In Theorem 3.1, assuming that $|\nu^{\top}|_{g_{0}}=0$, one can deduce the
following inequality:
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{4}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{1}{4}C_{1}\right),$
where $C_{1}=\inf_{\psi\in\Psi}\max_{\Omega}n^{2}H^{2},$ which is given by
Chen and Cheng in [14].
###### Remark 3.3.
For Theorem 3.1, an analogous version with respect to the $\mathfrak{L}$
operator is obtained by Chen and Peng in [18] and a similar result for the
drifting Laplacian is obtained by Xia and Xu in [65].
In this subsection, we give the proof of Theorem 3.1.Proof of Theorem 3.1.
From Nash’s Theorem, there exists an isometric immersion from
$\mathcal{M}^{n}$ into the $(n+p)$-dimensional Euclidean space
$\mathbb{R}^{n+p}$. Assume $x_{1},\cdots,x_{n+p}$ are $(n+p)$ coordinate
functions of $\mathbb{R}^{n+p}$, then $x_{1},\cdots,x_{n+p}$ are defined on
$\mathcal{M}^{n}$ globally. Taking $\varphi=x_{\alpha}$, for $1\leq\alpha\leq
n+p$, we have, from the Proposition 2.3,
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\|u_{i}\nabla
x_{\alpha}\|_{\Omega}^{2}\leq\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\|2\langle\nabla
x_{\alpha},\nabla
u_{i}\rangle_{g}+u_{i}\mathfrak{L}_{\nu}x_{\alpha}\|_{\Omega}^{2}.$
Taking sum on $\alpha$ from 1 to $n+p$, we have
(3.4)
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\sum^{n+p}_{\alpha=1}\|u_{i}\nabla
x_{\alpha}\|_{\Omega}^{2}$
$\displaystyle\leq\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\sum^{n+p}_{\alpha=1}\|2\langle\nabla
x_{\alpha},\nabla
u_{i}\rangle_{g}+u_{i}\mathfrak{L}_{\nu}x_{\alpha}\|_{\Omega}^{2}.$
Applying Lemma 2.7, Lemma 2.9 and Lemma 2.10, we have
$\displaystyle\sum^{n+p}_{\alpha=1}\|2\langle\nabla x_{\alpha},\nabla
u_{i}\rangle_{g}+u_{i}\mathfrak{L}_{\nu}x_{\alpha}\|_{\Omega}^{2}$
$\displaystyle\leq
4\Lambda_{i}+\int_{\Omega}u_{i}^{2}(n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2})e^{\langle\nu,X\rangle_{g_{0}}}dv+4\int_{\Omega}u_{i}|\nabla
u_{i}|_{g}|\nu|_{g_{0}}e^{\langle\nu^{\top},X\rangle_{g_{0}}}dv.$
Furthermore, by Cauchy-Schwarz inequality, we infer that
(3.5) $\displaystyle\sum^{n+p}_{\alpha=1}\|2\langle\nabla x_{\alpha},\nabla
u_{i}\rangle_{g}+u_{i}\mathfrak{L}_{\nu}x_{\alpha}\|_{\Omega}^{2}$
$\displaystyle\leq
4\Lambda_{i}+\int_{\Omega}u_{i}^{2}(n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2})e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad+4\left(\int_{\Omega}|\nabla
u_{i}|^{2}_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv\right)^{\frac{1}{2}}\left(\int_{\Omega}u_{i}^{2}|\nu^{\top}|^{2}_{g_{0}}e^{\langle\nu,X\rangle_{g_{0}}}dv\right)^{\frac{1}{2}}$
$\displaystyle=4\Lambda_{i}+\int_{\Omega}u_{i}^{2}(n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2})e^{\langle\nu,X\rangle_{g_{0}}}dv+4\Lambda_{i}^{\frac{1}{2}}\left(\int_{\Omega}u_{i}^{2}|\nu^{\top}|^{2}_{g_{0}}e^{\langle\nu,X\rangle_{g_{0}}}dv\right)^{\frac{1}{2}}.$
By the mean value inequality, we have
(3.6) $\displaystyle\sum^{n+p}_{\alpha=1}\|2\langle\nabla x_{\alpha},\nabla
u_{i}\rangle_{g}+u_{i}\mathfrak{L}_{\nu}x_{\alpha}\|_{\Omega}^{2}$
$\displaystyle\leq
4\Lambda_{i}+\int_{\Omega}u_{i}^{2}(n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2})e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\quad+2\int_{\Omega}|\nabla
u_{i}|^{2}_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv+2\int_{\Omega}u_{i}^{2}|\nu^{\top}|^{2}_{g_{0}}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=6\Lambda_{i}+\int_{\Omega}u_{i}^{2}(n^{2}H^{2}+3|\nu^{\top}|_{g_{0}}^{2})e^{\langle\nu,X\rangle_{g_{0}}}dv.$
Therefore, from (LABEL:3.2), (LABEL:3.3), (LABEL:3.3-1) and Lemma 2.7, we
infer that
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}$
$\displaystyle\leq\frac{4}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})$
$\displaystyle\times\left(\Lambda_{i}+\frac{1}{4}\int_{\Omega}u_{i}^{2}\left(n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv+\Lambda_{i}^{\frac{1}{2}}\left(\int_{\Omega}u_{i}^{2}|\nu^{\top}|^{2}_{g_{0}}e^{\langle\nu,X\rangle_{g_{0}}}dv\right)^{\frac{1}{2}}\right),$
and
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{6}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{1}{6}\int_{\Omega}u_{i}^{2}\left(n^{2}H^{2}+3|\nu^{\top}|_{g_{0}}^{2}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv\right).$
Since eigenvalues are invariant in the sense of isometries, defining
$C_{1}=\inf_{\psi\in\Psi}\max_{\Omega}n^{2}H^{2}$, and
$D_{1}=\max_{\Omega}|\nu^{\top}|_{g_{0}}$, where $\Psi$ denotes the set of all
isometric immersions from $\mathcal{M}^{n}$ into a Euclidean space, we have
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{4}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+D_{1}\Lambda_{i}^{\frac{1}{2}}+\frac{1}{4}D_{1}^{2}+\frac{1}{4}C_{1}\right),$
and
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{6}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{1}{2}D_{1}^{2}+\frac{1}{6}C_{1}\right),$
as claimed. It finishes the proof of Theorem 3.1.
$\square$
By observing the proof of the Theorem3.1, one has the following corollary.
###### Corollary 3.2.
For an $n$-dimensional complete Riemannian manifold $\mathcal{M}^{n}$, there
exists a function $H$ such that eigenvalues $\Lambda_{i}$ of the Dirichlet
eigenvalue problem (1.3) of the differential operator $\mathfrak{L}_{\nu}$
satisfy
(3.7) $\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}$
$\displaystyle\leq\frac{4}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})$
$\displaystyle\times\left(\Lambda_{i}+\frac{1}{4}\int_{\Omega}u_{i}^{2}\left(n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2}+4\Lambda_{i}^{\frac{1}{2}}|\nu^{\top}|_{g_{0}}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv\right),$
and
(3.8) $\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}$
$\displaystyle\leq\frac{6}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})$
$\displaystyle\times\left(\Lambda_{i}+\frac{1}{4}\int_{\Omega}u_{i}^{2}\left(n^{2}H^{2}+3|\nu^{\top}|_{g_{0}}^{2}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv\right).$
### 3.2. Bound of Payne-Pólya-Weinberger Type
In this subsection, we shall give a bound of Payne-Pólya-Weinberger type for
the Xin-Laplacian as follows.
###### Theorem 3.3.
Let $\mathcal{M}^{n}$ be an $n$-dimensional complete Riemannian manifold
isometrically embedded into the Euclidean space $\mathbb{R}^{n+p}$ with mean
curvature $H$. Then, for any $j(j=1,2,\cdots)$, Dirichlet problem (1.3) of the
Xin-Laplacian satisfy
(3.9)
$\displaystyle\sum_{l=1}^{n}\Lambda_{j+l}\leq(4+n)\Lambda_{i}+4D_{1}\Lambda_{i}^{\frac{1}{2}}+D_{1}^{2}+C_{1},$
and
(3.10)
$\displaystyle\sum_{l=1}^{n}\Lambda_{j+l}\leq(6+n)\Lambda_{i}+3D_{1}^{2}+C_{1},$
where
$C_{1}=\inf_{\psi\in\Psi}\max_{\Omega}n^{2}H^{2}\ \ and\ \
D_{1}=\max_{\Omega}|\nu^{\top}|_{g_{0}}.$
###### Remark 3.4.
In Theorem 3.3, when $|\nu^{\top}|_{g_{0}}=0$, for all $j=1,2,\cdots,$ we have
$\frac{\Lambda_{j+1}+\Lambda_{j+2}+\cdots+\Lambda_{j+n}}{\Lambda_{j}}\leq
n+4,$
which generalizes Ashbaugh and Benguria’s universal inequality (1.11).
Proof of Theorem 3.3. Nash’s Theorem implies that there exists an isometric
immersion from $\mathcal{M}^{n}$ into $\mathbb{R}^{n+p}$. Let
$x_{1},\cdots,x_{n+p}$ be coordinate functions of $\mathbb{R}^{n+p}$. Then
$x_{1},\cdots,x_{n+p}$ are defined on $\mathcal{M}^{n}$ globally. Applying the
Proposition 2.1 to functions $\phi_{l}=x_{l}$, we obtain
(3.11)
$\displaystyle\sum^{n+p}_{l=1}(\Lambda_{j+l}-\Lambda_{j})\|u_{j}\nabla\Phi_{l}\|^{2}_{\Omega}\leq\sum^{n+p}_{l=1}\int_{\Omega}\big{(}u_{j}\mathfrak{L}_{\nu}\Phi_{l}+2\langle\nabla\Phi_{l},\nabla
u_{j}\rangle_{g}\big{)}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv,$
with
$\Phi_{l}=\sum_{s=1}^{n+p}a_{ls}x_{s},$
where $A=(a_{ij})_{(n+p)\times(n+p)}$ is an orthogonal matrix. Furthermore, we
know that $\Phi_{l}$ satisfies Proposition 2.1 since $A=(a_{lt})$ is an
orthogonal matrix. By an orthogonal transformation, it is not hard to prove,
for any $l$, $|\nabla\Phi_{l}|_{g}^{2}\leq 1.$ Furthermore, with a simple
calculation, we derive
$\displaystyle\quad~{}\sum^{n+p}_{l=1}(\Lambda_{j+l}-\Lambda_{j})\|\nabla\Phi_{l}u_{j}\|^{2}_{\Omega}$
$\displaystyle\geq\sum^{n}_{l=1}(\Lambda_{j+l}-\Lambda_{j})\|\nabla\Phi_{l}u_{j}\|^{2}_{\Omega}+(\Lambda_{j+n+1}-\Lambda_{j})\sum^{n+p}_{l=n+1}\|\nabla\Phi_{l}u_{j}\|^{2}_{\Omega}$
$\displaystyle=\sum^{n}_{l=1}(\Lambda_{j+l}-\Lambda_{j})\|\nabla\Phi_{l}u_{j}\|^{2}_{\Omega}+(\Lambda_{j+n+1}-\Lambda_{j})\int_{\Omega}\left(\sum_{l=1}^{n}(1-|\nabla\Phi_{l}|_{g}^{2})\right)u_{j}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\geq\sum^{n}_{l=1}(\Lambda_{j+l}-\Lambda_{j})\|\nabla\Phi_{l}u_{j}\|^{2}_{\Omega}+\sum^{n}_{l=1}(\Lambda_{j+l}-\Lambda_{j})\int_{\Omega}\left(u_{1}^{2}-|\nabla\Phi_{l}|_{g}^{2}\right)u_{j}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv,$
which tells us
(3.12)
$\displaystyle\sum_{l=1}^{n}(\Lambda_{j+l}-\Lambda_{j})\leq\int_{\Omega}\sum^{n+p}_{l=1}(u_{j}\mathfrak{L}_{\nu}\Phi_{l}+2\langle\nabla\Phi_{l},\nabla
u_{j}\rangle_{g})^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
From Lemma 2.7, we have
(3.13)
$\displaystyle\sum^{n+p}_{l=1}\left(\Delta\Phi_{l}\right)^{2}=n^{2}H^{2},$
(3.14) $\displaystyle\sum^{n+p}_{l=1}\Delta\Phi_{l}\nabla\Phi_{l}=0,$
and
(3.15) $\displaystyle\sum^{n+p}_{l=1}\langle\nabla\Phi_{l},\nabla
w\rangle_{g}\langle\nabla\Phi_{l},\nabla v\rangle_{g}=\langle\nabla w,\nabla
v\rangle_{g},$
since $A=(a_{lt})$ is an $(n+p)\times(n+p)$-orthogonal matrix. Substituting
(3.13), (3.14), (3.15) into (3.12), we infer that
$\displaystyle\sum_{l=1}^{n}(\Lambda_{j+l}-\Lambda_{j})$
$\displaystyle\leq\int_{\Omega}\sum^{n+p}_{l=1}\left\\{u_{j}\mathfrak{L}_{\nu}\Phi_{l}+2\langle\nabla\Phi_{l},\nabla
u_{j}\rangle_{g}\right\\}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\leq
4\Lambda_{i}+\int_{\Omega}\Big{\\{}u_{j}^{2}(n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2})\Big{\\}}e^{\langle\nu,X\rangle_{g_{0}}}dv+4\int_{\Omega}u_{i}|\nabla
u_{j}|_{g}|\nu|_{g_{0}}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
Therefore, by Cauchy-Schwarz inequality and the mean value inequality, we have
$\displaystyle\sum_{l=1}^{n}(\Lambda_{j+l}-\Lambda_{j})\leq
4\Lambda_{i}+\int_{\Omega}u_{j}^{2}\left(n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2}+4\Lambda_{i}^{\frac{1}{2}}|\nu^{\top}|_{g_{0}}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv$
and
$\displaystyle\sum_{l=1}^{n}(\Lambda_{j+l}-\Lambda_{j})\leq
6\Lambda_{i}+\int_{\Omega}u_{j}^{2}\left(n^{2}H^{2}+3|\nu^{\top}|_{g_{0}}^{2}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv,$
as desired. The proof of the Theorem 3.3 is finished.
$\square$
###### Corollary 3.4.
For an $n$-dimensional complete Riemannian manifold $\mathcal{M}^{n}$, there
exists a function $H$ such that, for any $j=1,2,\cdots$, eigenvalues of the
Dirichlet eigenvalue problem (1.3) of the differential operator
$\mathfrak{L}_{\nu}$ satisfy
$\displaystyle\sum_{l=1}^{n}\Lambda_{j+l}\leq(4+n)\Lambda_{i}+\int_{\Omega}u_{j}^{2}\left(n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2}+4\Lambda_{i}^{\frac{1}{2}}|\nu^{\top}|_{g_{0}}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv,$
and
$\displaystyle\sum_{l=1}^{n}\Lambda_{j+l}\leq(6+n)\Lambda_{i}+\int_{\Omega}u_{j}^{2}\left(n^{2}H^{2}+3|\nu^{\top}|_{g_{0}}^{2}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv.$
###### Remark 3.5.
In some of the most important early literatures [31, 32, 33], Harrell II,
Michel, Stubbe developed an algebraic technique to discuss the eigenvalue
problems and this technique enable them to explore the universal inequality in
various of settings. Recently, based on an algebraic technique, some similar
eigenvalue inequalities of Schrödinger operator are established by Soufi,
Harrell and Ilias in their cerebrated paper [56]. However, the method
estimating the eigenvalues in this paper is different from their one.
### 3.3. A Remark on Theorem 3.1 and Theorem 3.3
According to Colin de Verdière’s construction [26] and the celebrated
isometric embedding theorem due to Nash and Moser, there exist no universal
inequalities for the eigenvalues of the Laplace operator on the bounded domain
of a Riemannian submanifold isometrically embedded into the Euclidean space,
unless it is a submanifold with constant mean curvature. Likewise, for some
submanifolds of the Euclidean spaces, eigenvalues of the Xin-Laplacian do not
satisfy universal inequalities. Therefore, in the absence of any other
condition of intrinsic geometry, the spectrum of drifting Laplacian naturally
contains information about the extrinsic geometry of a submanifold when it is
embedded into certain Euclidean space. However, for translating solitons, one
can obtain some universal inequalities. For example, see Section 4.
## 4\. Eigenvalues on the Translating Solitons
In this section, we would like to exploit the eigenvalues of Xin-Laplacian on
the translating solitons.
### 4.1. Translating Solitons Associated with MCF
Let $X:\mathcal{M}^{n}\rightarrow\mathbb{R}^{n+p}$ be an isometric immersion
from an $n$-dimensional, oriented, complete Riemannian manifold
$\mathcal{M}^{n}$ to the Euclidean space $\mathbb{R}^{n+p}$. We consider a
smooth family of immersions
$X_{t}=X(\cdot,t):\mathcal{M}^{n}\rightarrow\mathbb{R}^{n+p}$ with
corresponding images $\mathcal{M}^{n}_{t}=X_{t}(\mathcal{M}^{n})$ such that
the following mean curvature equation system [37, 40]:
(4.1)
${\begin{cases}&\frac{d}{dt}X(x,t)=\textbf{H}(x,t),x\in\mathcal{M}^{n}\\\
&X(\cdot,0)=X(\cdot),\end{cases}}$
is satisfied, where $\textbf{H}(x,t)$ is the mean curvature vector of
$\mathcal{M}_{t}$ at $X(x,t)$ in $\mathbb{R}^{n+p}$. We let $\nu_{0}$ be a
constant vector with is a constant vector with unit length in
$\mathbb{R}^{n+p}$. We denote $\nu_{0}^{\bot}$ the normal projection of
$\nu_{0}$ to the normal bundle of $\mathcal{M}^{n}$ in $\mathbb{R}^{n+p}$. A
submanifold $X:\mathcal{M}^{n}\rightarrow\mathbb{R}^{n+p}$ is said to be a
translating soliton of the mean curvature flow (4.1), if it satisfies:
$\textbf{H}=\nu_{0}^{\bot},$ which is a special solution of the mean curvature
flow equation (4.1). They are not only special solutions to the mean curvature
flow equations, but they often occur as Type-II singularity of a mean
curvature flow, which play an important role in the study of the mean
curvature flow [7]. In [66], Xin studied some basic properties of translating
solitons: the volume growth, generalized maximum principle, Gauss maps and
certain functions related to the Gauss maps. In addition, he carried out
point-wise estimates and integral estimates for the squared norm of the second
fundamental form. By utilizing these estimates, Xin proved some rigidity
theorems for translating solitons in the Euclidean space in higher codimension
in [66]. Recently, Chen and Qiu [17] proved established a nonexistence theorem
for the spacelike translating solitons. These results are proved by using a
new Omori-Yau maximal principle. To agree with the notation appearing in [66],
we denote $\mathfrak{L}_{\nu_{0}}$ by $\mathfrak{L}_{II}$ henceforth in the
section.
### 4.2. Eigenvalue Inequality of Yang Type
As an application of Theorem 3.1, we study the eigenvalues of
$\mathfrak{L}_{II}$ operator on the complete translating solitons in this
section. More precisely, we prove the following theorem.
###### Theorem 4.1.
Let $\mathcal{M}^{n}$ be an $n$-dimensional complete translating soliton
isometrically embedded into the Euclidean space $\mathbb{R}^{n+p}$. Then, the
eigenvalues $\Lambda_{i}(1\leq i\leq k)$ of Dirichlet eigenvalue problem (1.3)
of the differential operator $\mathfrak{L}_{II}$ satisfy
(4.2)
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{4}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\Lambda_{i}^{\frac{1}{2}}+\frac{n^{2}}{4}\right),$
and, for any $n\geq 2$,
(4.3)
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{6}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{n^{2}}{6}\right).$
###### Proof.
Since $\mathcal{M}^{n}$ is an $n$-dimensional complete translating soliton
isometrically embedded into the $(n+p)$ dimensional Euclidean space
$\mathbb{R}^{n+p}$, we have
(4.4) $H=\nu_{0}^{\perp}\ \ {\rm and}\ \ |\nu_{0}|_{g_{0}}^{2}=1,$
which implies that
(4.5)
$\int_{\Omega}n^{2}H^{2}e^{\langle\nu_{0},X\rangle_{g_{0}}}dv=\int_{\Omega}n^{2}|\nu_{0}^{\perp}|^{2}e^{\langle\nu_{0},X\rangle_{g_{0}}}dv\leq
n^{2}.$
Combining with (4.4) and (4.5) yields
(4.6)
$\frac{1}{4}\int_{\Omega}u_{i}^{2}\left(n^{2}H^{2}+|\nu_{0}^{\top}|_{g_{0}}^{2}\right)e^{\langle\nu_{0},X\rangle_{g_{0}}}dv\leq\frac{n^{2}}{4}.$
Substituting (4.6) into (3.7), we obtain (4.2). The proof of (4.3) is
similar.∎
### 4.3. Eigenvalue Inequality of Levitin-Parnovski Type
Applying Theorem 3.3, we can show the following theorem.
###### Theorem 4.2.
Let $\mathcal{M}^{n}$ be an $n$-dimensional complete translating soliton
isometrically embedded into the Euclidean space $\mathbb{R}^{n+p}$. Then, for
any $j=1,2,\cdots$, the eigenvaluesof Dirichlet eigenvalue problem (1.3) of
the differential operator $\mathfrak{L}_{II}$ satisfy
$\displaystyle\sum^{n}_{k=1}\Lambda_{j+k}\leq(n+4)\Lambda_{j}+n^{2}+4\Lambda_{i}^{\frac{1}{2}},$
and, for any $n\geq 2$,
$\displaystyle\sum^{n}_{k=1}\Lambda_{j+k}\leq(n+6)\Lambda_{j}+n^{2}.$
###### Proof.
The proof is similar to Theorem 4.1. Therefore, we omit it here.
∎
###### Remark 4.1.
Roughly speaking, it is very difficult to obtain universal inequalities of
Witten-Laplacian on the Ricci solitons in the sense of Ricci flows or the
self-shrinkers in the sense of the mean curvature flows unless it is a trivial
Ricci soliton and there are some special assumption for the potential function
$f$. For example, see [65]. However, it is surprising that, in Theorem 4.2 and
Theorem 4.1, the eigenvalue inequalities are universal.
### 4.4. Estimates for the Upper Bounds and Gaps of Consecutive Eigenvalues
In what follows, we will give several applications of Theorem 4.2 and Theorem
4.1. First of all, by (4.2), we have
(4.7)
$n\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)^{2}\leq\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)\left(4\Lambda_{i}+4\Lambda^{\frac{1}{2}}_{i}+n^{2}\right).$
Since the formula (4.7) is a quadratic inequality of $\Lambda_{k+1}$,
according to the direct but somewhat tedious calculation, one can get
$\displaystyle\Lambda_{k+1}$
$\displaystyle\leq\frac{1}{k}\sum_{i=1}^{k}\left[\frac{2}{n}\left(\Lambda_{i}+\Lambda^{\frac{1}{2}}_{i}\right)+\Lambda_{i}\right]+\frac{n}{2}$
$\displaystyle+\Bigg{\\{}\left[\frac{1}{k}\frac{2}{n}\sum_{i=1}^{k}\left(\Lambda_{i}+\Lambda^{\frac{1}{2}}_{i}\right)+\frac{n}{2}\right]^{2}-\left(1+\frac{4}{n}\right)\frac{1}{k}\sum_{i=1}^{k}\left(\Lambda_{i}-\frac{1}{k}\sum_{j=1}^{k}\Lambda_{j}\right)^{2}$
$\displaystyle+\frac{1}{k}\frac{4}{n}\left[\sum_{i=1}^{k}\Lambda^{\frac{1}{2}}_{i}\left(\frac{1}{k}\sum_{j=1}^{k}\Lambda_{j}-\Lambda_{i}\right)\right]\Bigg{\\}}^{\frac{1}{2}}.$
Thus, we have the following estimates for the upper bound of the eigenvalues
of Xin-Laplacian on the translating solitons.
###### Corollary 4.3.
For an $n$-dimensional complete translating soliton $(\mathcal{M}^{n},g)$, the
$k^{\text{th}}$ eigenvalue $\Lambda_{k}$ of the Dirichlet eigenvalue problem
(1.3) of the differential operator $\mathfrak{L}_{II}$ satisfy,
(4.8) $\displaystyle\Lambda_{k+1}$
$\displaystyle\leq\frac{1}{k}\sum_{i=1}^{k}\left[\frac{2}{n}\left(\Lambda_{i}+\Lambda^{\frac{1}{2}}_{i}\right)+\Lambda_{i}\right]+\frac{n}{2}+\Bigg{\\{}\left[\frac{1}{k}\frac{2}{n}\sum_{i=1}^{k}\left(\Lambda_{i}+\Lambda^{\frac{1}{2}}_{i}\right)+\frac{n}{2}\right]^{2}$
$\displaystyle-\left(1+\frac{4}{n}\right)\frac{1}{k}\sum_{i=1}^{k}\left(\Lambda_{i}-\frac{1}{k}\sum_{j=1}^{k}\Lambda_{j}\right)^{2}+\frac{1}{k^{2}}\frac{4}{n}\sum_{i,j=1}^{k}\left(\Lambda_{i}^{\frac{1}{2}}\Lambda_{j}-\Lambda_{i}^{\frac{3}{2}}\right)\Bigg{\\}}^{\frac{1}{2}}.$
If we use a positive real-valued function $f(n)$ to replace $n$ in the proofs
of Theorem 2.1 in [22], then we can extend Cheng and Yang’s recursion formula
to the following general case.
###### Theorem 4.4.
( A recursion formula of Cheng and Yang Type). Let
$\mu_{1}\leq\mu_{2}\leq\dots,\leq\mu_{k+1}$ be any positive real numbers
satisfying
$\sum_{i=1}^{k}(\mu_{k+1}-\mu_{i})^{2}\leq\frac{4}{f(n)}\sum_{i=1}^{k}\mu_{i}(\mu_{k+1}-\mu_{i}).$
Define
$\Lambda_{k}=\frac{1}{k}\sum_{i=1}^{k}\mu_{i},\qquad
T_{k}=\frac{1}{k}\sum_{i=1}^{k}\mu_{i}^{2},\ \ \
F_{k}=\left(1+\frac{2}{f(n)}\right)\Lambda_{k}^{2}-T_{k}.$
Then, we have
$F_{k+1}\leq C(n,k)\left(\frac{k+1}{k}\right)^{\frac{4}{f(n)}}F_{k},$
where
$C(n,k)=1-\frac{1}{3f(n)}\left(\frac{k}{k+1}\right)^{\frac{4}{f(n)}}\frac{\left(1+\frac{2}{f(n)}\right)\left(1+\frac{4}{f(n)}\right)}{(k+1)^{3}}<1.$
By Theorem 4.4, we can show the following proposition. See [16].
###### Proposition 4.5.
Let $\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{k+1}$ be any positive
real numbers satisfying the following inequality
(4.9)
$\sum_{i=1}^{k}\left(\lambda_{k+1}-\lambda_{i}\right)^{2}\leq\frac{4}{f(n)}\sum_{i=1}^{k}\lambda_{i}\left(\lambda_{k+1}-\lambda_{i}\right),$
with $f(n)>0$. Then we have
(4.10)
$\lambda_{k+1}\leq\left(1+\frac{4}{f(n)}\right)k^{\frac{2}{f(n)}}\lambda_{1}.$
###### Proof.
By using the same approach in [22], we can give the proof of this proposition.
Here, we leave out the details of this proof.∎
Taking $f(n)=\frac{2n}{3}$ and $\lambda_{i}=\Lambda_{i}+\frac{n^{2}}{6}$ in
(4.9), we can give an estimate for the upper bound of the eigenvalues of
$\mathfrak{L}_{II}$ on the translating solitons.
###### Corollary 4.6.
For an $n$-dimensional complete translating soliton $(\mathcal{M}^{n},g)$, the
$k^{\text{th}}$ eigenvalue $\Lambda_{k}$ of the Dirichlet eigenvalue problem
(1.3) of the differential operator $\mathfrak{L}_{II}$ satisfy, for any $k\geq
1$,
(4.11)
$\Lambda_{k+1}+\frac{n^{2}}{6}\leq\left(1+\frac{6}{n}\right)\left(\Lambda_{1}+\frac{n^{2}}{6}\right)\
k^{\frac{3}{n}}.$
###### Remark 4.2.
The upper bound (4.11) is not sharp in the sense of order of eigenvalues.
Recall that, in [22], by establishing a recursion formula, Cheng and Yang gave
a sharp upper bound in the sense of order of eigenvalues. Unfortunately, Cheng
and Yang’s recursion formula dose not work in our situation. In other words,
Proposition 4.5 can not apply directly to inequality (4.2). Therefore, we are
fail to get a sharp upper bound of Cheng and Yang type. To get a sharp upper
bound, it seems that a similar recursion formula needs to be proved.
Next, we give an estimate for the upper bound of the gap of consecutive
eigenvalues of the Dirichlet problem (1.3) of $\mathfrak{L}_{II}$ operator on
the translating solitons.
###### Corollary 4.7.
Under the same condition as Theorem 4.1, we have
(4.12) $\displaystyle\Lambda_{k+1}-\Lambda_{k}\leq
2\left\\{\left[\frac{1}{k}\frac{2}{n}\sum_{i=1}^{k}\left(\Lambda_{i}+\Lambda^{\frac{1}{2}}_{i}\right)+\frac{n}{2}\right]^{2}\right.$
$\displaystyle\left.-\left(1+\frac{4}{n}\right)\frac{1}{k}\sum_{i=1}^{k}\left(\Lambda_{i}-\frac{1}{k}\sum_{j=1}^{k}\Lambda_{j}\right)^{2}+\frac{1}{k^{2}}\frac{4}{n}\sum_{i,j=1}^{k}\left(\Lambda_{i}^{\frac{1}{2}}\Lambda_{j}-\Lambda_{i}^{\frac{3}{2}}\right)\right\\}^{\frac{1}{2}},$
and
(4.13) $\displaystyle\Lambda_{k+1}$ $\displaystyle-\Lambda_{k}\leq
2\left[\left(\frac{3}{n}\frac{1}{k}\sum_{i=1}^{k}\Lambda_{i}+\frac{n}{3}\right)^{2}-\left(1+\frac{6}{n}\right)\frac{1}{k}\sum_{j=1}^{k}\left(\Lambda_{j}-\frac{1}{k}\sum_{i=1}^{k}\Lambda_{i}\right)^{2}\right]^{\frac{1}{2}}.$
###### Proof.
The strategy of this proof is similar to Corollary 1 in [20]. For
completeness, we give the proof of this corollary. Since $k$ is an any
integer, we know that (4.8) is also true if we replace $k+1$ with $k$.
Equivalently, we have
$n\sum_{i=1}^{k-1}\left(\Lambda_{k}-\Lambda_{i}\right)^{2}\leq\sum_{i=1}^{k-1}\left(\Lambda_{k}-\Lambda_{i}\right)\left(4\Lambda_{i}+4\Lambda_{i}^{\frac{1}{2}}+n^{2}\right).$
Namely, $\Lambda_{k}$ also satisfies the same quadratic inequality. Therefore,
we infer
(4.14) $\displaystyle\Lambda_{k+1}$
$\displaystyle\geq\frac{1}{k}\sum_{i=1}^{k}\left[\frac{2}{n}\left(\Lambda_{i}+\Lambda^{\frac{1}{2}}_{i}\right)+\Lambda_{i}\right]+\frac{n}{2}+\Bigg{\\{}\left[\frac{1}{k}\frac{2}{n}\sum_{i=1}^{k}\left(\Lambda_{i}+\Lambda^{\frac{1}{2}}_{i}\right)+\frac{n}{2}\right]^{2}$
$\displaystyle-\left(1+\frac{4}{n}\right)\frac{1}{k}\sum_{i=1}^{k}\left(\Lambda_{i}-\frac{1}{k}\sum_{j=1}^{k}\Lambda_{j}\right)^{2}+\frac{1}{k^{2}}\frac{4}{n}\sum_{i,j=1}^{k}\left(\Lambda_{i}^{\frac{1}{2}}\Lambda_{j}-\Lambda_{i}^{\frac{3}{2}}\right)\Bigg{\\}}^{\frac{1}{2}}.$
From (4.8) and (4.14), we get (LABEL:gap-product-1). By modifying Cheng and
Yang’s proof presented in [22], one can get (4.13). Thus, this completes the
proof of Corollary 4.7.
∎
## 5\. Further Applications
In this section, we would like to give some applications of Theorem 3.1 and
Theorem 3.3. Specially, we obtain some eigenvalue inequalities on the
submanifolds of the Euclidean spaces, unit spheres and projective spaces.
### 5.1. Eigenvalues on the Manifolds of Euclidean Space and Unit Sphere
Firstly, we suppose that $\mathcal{M}^{n}$ is an $n$-dimensional complete
submanifold isometrically embedded into the $(n+p)$-dimensional Euclidean
space $\mathbb{R}^{n+p}$ with the mean curvature $H\equiv 0$. Then, according
to Theorem 3.1, one can deduce the following corollary.
###### Corollary 5.1.
Let $(\mathcal{M}^{n},g)$ be an $n$-dimensional complete minimal submanifold
isometrically embedded into the Euclidean space $\mathbb{R}^{n+p}$. Then, for
any $j=1,2,\cdots$, eigenvalues of the Dirichlet problem (1.3) of the Xin-
Laplacian satisfy
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}$
$\displaystyle\leq\frac{4}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+D_{4}\Lambda_{i}^{\frac{1}{2}}+\frac{1}{4}D_{4}^{2}\right),$
and
$\displaystyle\sum_{l=1}^{n}(\Lambda_{j+l}-\Lambda_{j})\leq
4\Lambda_{j}+4D_{4}\Lambda_{j}^{\frac{1}{2}}+D_{4}^{2},$
where $D_{4}$ is given by $D_{4}=\max_{\Omega}|\nu^{\top}|_{g_{0}}.$
Next, we consider that $(\mathcal{M}^{n},g)$ is an $n$-dimensional submanifold
isometrically immersed in the unit sphere
$\mathbb{S}^{n+p-1}(1)\subset\mathbb{R}^{n+p}$ with mean curvature vector
$\overline{\textbf{H}}$. We use $\overline{\Psi}$ to denote the set of all
isometric immersions from $\mathcal{M}^{n}$ into the unit sphere
$\mathbb{S}^{n+p-1}(1)$. By Theorem 3.1, we have the following corollary.
###### Corollary 5.2.
If $(\mathcal{M}^{n},g)$ be an $n$-dimensional submanifold isometrically
immersed in the unit sphere $\mathbb{S}^{n+p-1}(1)\subset\mathbb{R}^{n+p}$
with mean curvature vector $\overline{\emph{{H}}}$. Then, eigenvalues of the
Dirichlet problem (1.3) of the Xin-Laplacian satisfy
(5.1) $\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}$
$\displaystyle\leq\frac{4}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\left[\Lambda_{i}+D_{5}\Lambda_{i}^{\frac{1}{2}}+\frac{1}{4}\left(D_{5}^{2}+C_{5}\right)\right],$
and, for any $j=1,2,\cdots$,
(5.2) $\displaystyle\sum_{l=1}^{n}(\Lambda_{j+l}-\Lambda_{j})\leq
4\Lambda_{j}+4D_{5}\Lambda_{j}^{\frac{1}{2}}+D_{5}^{2}+C_{5},$
where
$C_{5}=\inf_{\overline{\sigma}\in\overline{\Psi}}\max_{\Omega}n^{2}(|\overline{\emph{{H}}}|^{2}+1)\
\ and\ \ D_{5}=\max_{\Omega}|\nu^{\top}|_{g_{0}}.$
###### Proof.
Since the unit sphere can be canonically imbedded into Euclidean space, we
have the following diagram:
$\textstyle{\mathcal{M}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi\circ\sigma}$$\scriptstyle{\sigma}$$\textstyle{\mathbb{S}^{n+p-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\mathbb{R}^{n+p}}$
where $\psi:\mathbb{S}^{n+p-1}(1)\rightarrow\mathbb{R}^{n+p}$ is the canonical
imbedding from the unit sphere $S^{n+p-1}(1)$ into $\mathbb{R}^{n+p},$ and
$\sigma:\mathcal{M}^{n}\rightarrow\mathbb{S}^{n+p-1}(1)$ is an isometrical
immersion. Then, the composite map
$\psi\circ\sigma:\mathcal{M}^{n}\rightarrow\mathbb{R}^{n+p}$ is an isometric
immersion from $\mathcal{M}^{n}$ to $\mathbb{R}^{n+p}.$ Let
$\overline{\textbf{H}}$ and H be the mean curvature vector fields of $\sigma$
and $\psi\circ\sigma,$ respectively. Then, we have
$\left|\textbf{H}\right|^{2}=|\overline{\textbf{H}}|^{2}+1.$ Applying Theorem
3.1 directly, we can get (5.1) and (5.2). Therefore, we finish the proof of
this corollary.∎
In particular, we assume that $(\mathcal{M}^{n},g)$ is an $n$-dimensional unit
sphere $\mathbb{S}^{n}(1)$, and then, the mean curvature equals to $1$. This
is, $\left|\overline{\textbf{H}}\right|=0$, and thus, we have
$\left|\textbf{H}\right|=1$. Therefore, by Corollary 5.2, we obtain the
following corollary.
###### Corollary 5.3.
Let $(\mathcal{M}^{n},g)$ be an $n$-dimensional unit sphere
$\mathbb{S}^{n}(1)$ and $\Omega$ is a bounded domain on $\mathbb{S}^{n}(1)$.
Then, eigenvalues of the Dirichlet problem (1.3) of the Xin-Laplacian satisfy
$\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}$
$\displaystyle\leq\frac{4}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\left[\Lambda_{i}+D_{5}\Lambda_{i}^{\frac{1}{2}}+\frac{1}{4}\left(D_{5}^{2}+n^{2}\right)\right],$
and, for any $j=1,2,\cdots$,
$\displaystyle\sum_{l=1}^{n}(\Lambda_{j+l}-\Lambda_{j})\leq
4\Lambda_{j}+4D_{5}\Lambda_{j}^{\frac{1}{2}}+D_{5}^{2}+n^{2},$
where $D_{5}$ is given by
$D_{5}=\frac{1}{4}\max_{\Omega}|\nu^{\top}|_{g_{0}}.$
### 5.2. Eigenvalues on the Submanifolds of the Projective Spaces
Next, let us recall some results for submanifolds on the projective spaces.
For more details, we refer the readers to [13, 56]. Let $\mathbb{F}$ denote
the field $\mathbb{R}$ of real numbers, the field $\mathbb{C}$ of complex
numbers or the field $\mathbb{Q}$ of quaternions. In a natural way,
$\mathbb{R}\subset\mathbb{C}\subset\mathbb{Q}$. For each element $z$ of
$\mathbb{F}$, we define the conjugate of $z$ as follows: If
$z=z_{0}+z_{1}i+z_{2}j+z_{3}k\in\mathbb{Q}$ with
$z_{0},z_{1},z_{2},z_{3}\in\mathbb{R},$ then
$\bar{z}=z_{0}-z_{1}i-z_{3}j-z_{3}k.$ If $z$ is in $\mathbb{C},\bar{z}$
coincides with the ordinary complex conjugate. Let us denote by
$\mathbb{F}P^{m}$ the $m$-dimensional real projective space if
$\mathbb{F}=\mathbb{R}$, the complex projective space with real dimension $2m$
if $\mathbb{F}=\mathbb{C}$, and the quaternionic projective space with real
dimension $4m$ if $\mathbb{F}=\mathbb{Q}$, respectively. For convenience, we
introduce the integers
(5.3)
$d_{\mathbb{F}}=\operatorname{dim}_{\mathbb{R}}\mathbb{F}=\left\\{\begin{array}[]{ll}1,&\text{
if }\mathbb{F}=\mathbb{R};\\\ 2,&\text{ if }\mathbb{F}=\mathbb{C};\\\
4,&\text{ if }\mathbb{F}=\mathbb{Q}.\end{array}\right.$
It is well known that the manifold $\mathbb{F}P^{m}$ carries a canonical
metric so that the Hopf fibration
$\rho:\mathbb{S}^{d_{\mathbb{F}}\cdot(m+1)-1}\subset\mathbb{F}^{m+1}\rightarrow\mathbb{F}P^{m}$
is a Riemannian submersion. Hence, the sectional curvature of
$\mathbb{R}P^{m}$ is $1$, the holomorphic sectional curvature is $4$ and the
quaternion sectional curvature is $4$. We use $\mathcal{A}$ to denote the
space of all $(m+1)\times(m+1)$ matrices over $\mathbb{F}$ and let
$\mathcal{H}_{m+1}(\mathbb{F})=\left\\{A\in\mathcal{A}_{m+1}(\mathbb{F})\mid
A^{*}:=\overline{{}^{t}A}=A\right\\}$
be the vector space of $(m+1)\times(m+1)$ Hermitian matrices with coefficients
in the field $\mathbb{F}$. Then, one can endow $\mathcal{H}_{m+1}(\mathbb{F})$
with an inner product as follows: $\langle
A,B\rangle=\frac{1}{2}\operatorname{tr}(AB),$ where tr $(\cdot)$ denotes the
trace for the given $(m+1)\times(m+1)$ matrix. It is clear that the map
$\rho:\mathbb{S}^{d_{\mathbb{F}}\cdot(m+1)-1}\subset\mathbb{F}^{m+1}\rightarrow\mathcal{H}_{m+1}(\mathbb{F})$
given by
$\rho(\textbf{z})=\textbf{z}\textbf{z}^{\ast}=\left(\begin{array}[]{llll}\left|z_{0}\right|^{2}&z_{0}\overline{z_{1}}&\cdots&z_{0}\overline{z_{m}}\\\
z_{1}\overline{z_{0}}&\left|z_{1}\right|^{2}&\cdots&z_{1}\overline{z_{m}}\\\
\cdots&\cdots&\cdots&\cdots\\\
z_{m}\overline{z_{0}}&z_{m}\overline{z_{1}}&\cdots&\left|z_{m}\right|^{2}\end{array}\right)$
induces an isometric embedding $\rho$ from $\mathbb{F}P^{m}$ into
$\mathcal{H}_{m+1}(\mathbb{F})$ through the Hopf fibration, where
$\textbf{z}=(z_{0},z_{1},\cdots,z_{m})\in\mathbb{S}^{d_{\mathbb{F}}\cdot(m+1)-1}.$
Moreover, $\rho\left(\mathbb{F}P^{m}\right)$ is a minimal submanifold of the
hypersphere $\mathbb{S}\left(\frac{I}{m+1},\sqrt{\frac{m}{2(m+1)}}\right)$ of
$\mathcal{H}_{m+1}(\mathbb{F})$ with radius $\sqrt{\frac{m}{2(m+1)}}$ and
center $\frac{I}{m+1}$, where $I$ is the identity matrix. In addition, we need
the following lemma (see Lemma 6.3 in Chapter 4 in [13], [53] and a proof of
this lemma in [60]):
###### Lemma 5.4.
Let $f:\mathcal{M}^{n}\rightarrow\mathbb{F}P^{\text{m }}$ be an isometric
immersion, and let $\widehat{\textbf{H}}$ and H be the mean curvature vector
fields of the immersions $f$ and $\rho\circ f,$ respectively (here $\rho$ is
the induced isometric embedding $\rho$ from $\mathbb{F}P^{m}$ into
$\mathcal{H}_{m+1}(\mathbb{F})$ explained above). Then, we have
$\left|\textbf{H}\right|^{2}=|\widehat{\textbf{H}}|^{2}+\frac{4(n+2)}{3n}+\frac{2}{3n^{2}}\sum_{i\neq
j}K\left(e_{i},e_{j}\right),$
where $\left\\{e_{i}\right\\}_{i=1}^{n}$ is a local orthonormal basis of
$\overline{\Gamma}(T\mathcal{M}^{n})$ and $K$ is the sectional curvature of
$\mathbb{F}P^{m}$ expressed $by$
$K\left(e_{i},e_{j}\right)=\left\\{\begin{array}[]{ll}1,&\text{ if
}\mathbb{F}=\mathbb{R};\\\ 1+3\left(e_{i}\cdot Je_{j}\right)^{2},&\text{ if
}\mathbb{F}=\mathbb{C};\\\ 1+\sum_{r=1}^{3}3\left(e_{i}\cdot
J_{r}e_{j}\right)^{2},&\text{ if }\mathbb{F}=\mathbb{Q},\end{array}\right.$
where $J$ is the complex structure of $\mathbb{C}P^{m}$ and $J_{r}$ is the
quaternionic structure of $\mathbb{Q}P^{m}$.
$None$
Therefore, one can infer from Lemma 5.4 that
$\left|\textbf{H}\right|^{2}=\left\\{\begin{array}[]{ll}|\widehat{\textbf{H}}|^{2}+\frac{2(n+1)}{2n},&\text{
for }\mathbb{R}P^{m};\\\
|\widehat{\textbf{H}}|^{2}+\frac{2(n+1)}{2n}+\frac{2}{n^{2}}\sum_{i,j=1}^{n}\left(e_{i}\cdot
Je_{j}\right)^{2}\leq|\widehat{\textbf{H}}|^{2}+\frac{2(n+2)}{n},&\text{ for
}\mathbb{C}P^{m};\\\
|\widehat{\textbf{H}}|^{2}+\frac{2(n+1)}{2n}+\frac{2}{n^{2}}\sum_{i,j=1}^{n}\sum_{r=1}^{3}\left(e_{i}\cdot
J_{r}e_{j}\right)^{2}\leq|\widehat{\textbf{H}}|^{2}+\frac{2(n+4)}{n},&\text{
for }\mathbb{Q}P^{m}.\end{array}\right.$
Hence, from the above equation, one can verify the following inequality:
(5.4)
$\left|\textbf{H}\right|^{2}\leq|\widehat{\textbf{H}}|^{2}+\frac{2\left(n+d_{\mathbb{F}}\right)}{n}.$
We note that the equality in (5.4) holds if and only if $\mathcal{M}^{n}$ is a
complex submanifold of $\mathbb{C}P^{m}$ (for the case $\mathbb{C}P^{m}$ )
while $n\equiv 0(\bmod 4)$ and $\mathcal{M}^{n}$ is an invariant submanifold
of $\mathbb{Q}P^{m}\left(\text{ for the case }\mathbb{Q}P^{m}\right)$. By
Theorem 3.1 and 3.3, we can show the following corollary.
###### Corollary 5.5.
If $\mathcal{M}^{n}$ is isometrically immersed in a projective space
$\mathbb{F}P^{m}$ with mean curvature vector $\widehat{\textbf{H}}$, Then,
eigenvalues of the Dirichlet problem (1.3) of the Xin-Laplacian satisfy
(5.5) $\displaystyle\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}$
$\displaystyle\leq\frac{4}{n}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\left[\Lambda_{i}+D_{6}\Lambda_{i}^{\frac{1}{2}}+\frac{1}{4}\left(D_{6}^{2}+C_{6}\right)\right],$
and, for any $j=1,2,\cdots$,
(5.6) $\displaystyle\sum_{l=1}^{n}(\Lambda_{j+l}-\Lambda_{j})\leq
4\Lambda_{j}+4D_{6}\Lambda_{j}^{\frac{1}{2}}+D_{6}^{2}+C_{6},$
where
$C_{6}=\frac{1}{4}\inf_{\psi\in\Psi}\max_{\Omega}\left(n^{2}H^{2}+2n\left(n+d_{\mathbb{F}}\right)\right),$
and $D_{6}=\frac{1}{4}\max_{\Omega}|\nu^{\top}|_{g_{0}},$ and
$d_{\mathbb{F}}=\operatorname{dim}_{\mathbb{R}}\mathbb{F}$ defined by (5.3).
###### Proof.
As we know, there exists a canonical imbedding map
$\rho:\mathbb{F}P^{m}\rightarrow\mathcal{H}_{m+1}(\mathbb{F})$ from
$\mathbb{F}P^{m}(\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{Q})$ to Euclidean
space $\mathcal{H}_{m+1}(\mathbb{F})$. Therefore, for compact manifold
$\mathcal{M}^{n}$ isometrically immersed into the projective space
$\mathbb{F}P^{m},$ one has the following diagram:
$\textstyle{\mathcal{M}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho\circ
f}$$\scriptstyle{f}$$\textstyle{\mathbb{F}P^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\mathcal{H}_{m+1}(\mathbb{F})}$
$f:\mathcal{M}^{n}\rightarrow$ $\mathbb{F}P^{m}$ denotes an isometric
immersion from $\mathcal{M}^{n}$ to $\mathbb{F}P^{m}$. Then, the composite map
$\rho\circ f:\mathcal{M}^{n}\rightarrow\mathcal{H}_{m+1}(\mathbb{F})$ is an
isometric immersion from $\mathcal{M}^{n}$ to $\mathcal{H}_{m+1}(\mathbb{F})$.
According to inequality (5.4) and Theorem 3.1, we can conclude (5.5) and
(5.6). Hence, it completes the proof of corollary 5.5.
∎
### 5.3. Manifolds Admitting Some Special Functions
In this section, we would like to discuss the eigenvalue of Xin-Laplacian on
the manifolds admitting some special functions.
###### Theorem 5.6.
Let $\mathcal{M}^{n}$ be an $n$-dimensional complete Riemannian manifold and
let $\Omega$ be a bounded domain with smooth boundary in $\mathcal{M}^{n}$.
Denote by $\Lambda_{i}$ the $i$ -th eigenvalue of the problem (1.3) of the
differential operator $\mathfrak{L}_{\nu}$. If there exist a function
$\mathcal{W}:\Omega\rightarrow\mathbb{R}$ and a positive constant $C_{3}$ such
that $|\nabla\mathcal{W}|_{g}=1$, and $|\Delta\mathcal{W}|_{a}\leq C_{3}$,
where $|w|_{a}$ denotes the absolute value of $w$, then
(5.7)
$\displaystyle\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)^{2}\leq\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)\left(4\Lambda_{i}+4\left(C_{3}+D_{3}\right)\Lambda_{i}^{\frac{1}{2}}+\left(C_{3}+D_{3}\right)^{2}\right),$
where $D_{3}=\max_{\Omega}|\nu^{\top}|_{g_{0}}$.
###### Remark 5.1.
Let $\mathcal{M}^{n}$ be an $n$-dimensional connected complete Riemannian
manifold whose Ricci curvature satisfies
$\operatorname{Ric}_{\mathcal{M}^{n}}\geq-(n-1)\kappa^{2},\kappa\geq 0.$
Suppose that there is a smooth function $\mathcal{W}$ on $\mathcal{M}^{n}$
with $|\nabla\mathcal{W}|_{g}=1$. Then, we have
$|\Delta\mathcal{W}|_{a}\leq(n-1)\kappa^{2}.$ See Remark 3.6 in [52].
Furthermore, we consider that $\xi:[0,+\infty)\rightarrow M$ is a geodesic
ray, namely a unit speed geodesic with $d(\xi(s),\xi(t))=t-s$ for any $t>s>0.$
Then the Busemann function $b_{\xi}$ corresponding to $\xi$ is defined as
$b_{\xi}(q):=\lim_{t\rightarrow+\infty}(d(q,\xi(t))-t).$
If $\mathcal{M}^{n}$ is an Hadamard manifold, then it is known that $b_{\xi}$
is a convex function of class $C^{2}$ with $|\nabla b_{\xi}|_{g}\equiv 1$ and
these conditions characterize Busemann functions (see [8, 34]). Thus, the
Bussemann functions on the Cartan-Hadamard manifolds $\mathcal{M}^{n}$ satisfy
the conditions in Theorem 5.6. Also, if $\mathcal{N}^{n-1}$ is complete
Riemannian manifold with Ricci curvature bounded below and if
$\mathcal{M}^{n}=\mathcal{N}^{n-1}\times\mathbb{R}$ is the product of
$\mathcal{N}$ and $\mathbb{R}$ with the product metric, then the function
$f:\mathcal{M}^{n}\rightarrow\mathbb{R}$ given by $f(p,t)=t$ satisfies the
conditions of Theorem 5.6.
###### Remark 5.2.
Let $\mathcal{M}^{n}=\mathbb{R}\times\mathcal{N}^{n-1}$ be the complete
manifold with the warped product metric $ds_{M}^{2}=$
$dt^{2}+\exp(2t)ds_{N}^{2},$ where $\mathcal{N}^{n-1}$ is a complete manifold.
If the Ricci curvature of $\mathcal{N}$ is non-negative, then
$\operatorname{Ric}_{\mathcal{M}^{n}}\geq-(n-1)$, which means that the
function $f:\mathcal{M}^{n}\rightarrow\mathbb{R}$ given by $f(p,t)=t$
satisfies $|\nabla f|_{g}=1\ \ and\ \ |\Delta f|_{a}\leq n-1.$ See [53] for
details. Consequently, the product Riemannian manifold $\mathcal{M}^{n}$
satisfies the condition in Theorem 5.6.
_Proof of Theorem_ 5.6. Substituting $\varphi=\mathcal{W}$ into (2.11), and
utilizing (2.8) and Cauchy-Schwarz inequality, we infer that
$\displaystyle\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)^{2}$
$\displaystyle\leq\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)\int_{\Omega}\left(u_{i}\left(\Delta\mathcal{W}+\langle\nu,\nabla\mathcal{W}\rangle_{g_{0}}\right)+2\left\langle\nabla\mathcal{W},\nabla
u_{i}\right\rangle_{g}\right)^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\leq\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)\int_{\Omega}\left(\left|u_{i}\right|(|\Delta\mathcal{W}|_{a}+|\nu^{\top}|_{g_{0}}|\nabla\mathcal{W}|_{g})+2|\nabla\mathcal{W}|_{g}\left|\nabla
u_{i}\right|_{g}\right)^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv,$
since $|\nabla\mathcal{W}|_{g}=1$. Furthermore, we have
(5.8) $\displaystyle\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)^{2}$
$\displaystyle\leq\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)\int_{\Omega}\left[\left(C_{3}+|\nu^{\top}|_{g_{0}}\right)\left|u_{i}\right|_{a}+2\left|\nabla
u_{i}\right|_{g}\right]^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\leq\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)\Bigg{[}\left(C_{3}+D_{3}\right)^{2}\left\|u_{i}\right\|_{\Omega}^{2}+4\left(C_{3}+D_{3}\right)\left\|u_{i}\left|\nabla
u_{i}\right|_{g}\right\|_{\Omega}+4\left\|\left|\nabla
u_{i}\right|_{g}\right\|_{\Omega}^{2}\Bigg{]},$
since $|\nabla\mathcal{W}|_{g}=1$, and $|\Delta\mathcal{W}|_{a}\leq C_{3}$,
where $D_{3}=|\nu^{\top}|_{g_{0}}.$ By Cauchy-Schwarz inequality, we derive
(5.9) $\left\|u_{i}\left|\nabla
u_{i}\right|_{g}\right\|_{\Omega}\leq\left(\left\|u_{i}\right\|_{\Omega}\right)^{\frac{1}{2}}\left(\left\|\left|\nabla
u_{i}\right|_{g}\right\|_{\Omega}\right)^{\frac{1}{2}}=\Lambda_{i}^{\frac{1}{2}}.$
From (5.8) and (5.9), we yield
$\displaystyle\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)^{2}\leq\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)\left(\left(C_{3}+D_{3}\right)+2\Lambda_{i}^{1/2}\right)^{2},$
as we desired. Hence, this completes the proof of this theorem.
$None$
###### Theorem 5.7.
Let $\mathcal{M}^{n}$ be an $n$-dimensional complete Riemannian manifold and
let $\Omega$ be a bounded domain with smooth boundary in $\mathcal{M}^{n}$.
Denote by $\Lambda_{i}$ the $i$ -th eigenvalue of the problem (1.3) of the
differential operator $\mathfrak{L}_{\nu}$. If $\Omega$ admits an eigenmap
$f=\left(f_{1},f_{2},\cdots,f_{m+1}\right):\Omega\rightarrow\mathbb{S}^{m}(1)$
corresponding to an eigenvalue $\eta,$ that is,
$\Delta f_{\alpha}=-\eta f_{\alpha},\ \ where\ \ \alpha=1,\cdots,m+1,$
and
$\sum_{\alpha=1}^{m+1}f_{\alpha}^{2}=1,$
then
(5.10)
$\displaystyle\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)^{2}\leq\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)\left(4\Lambda_{i}+4D_{3}\Lambda_{i}^{1/2}+D_{3}^{2}+\eta\right),$
where $\mathbb{S}^{m}(1)$ is the unit sphere of dimension $m$ and
$D_{3}=\max_{\Omega}|\nu^{\top}|_{g_{0}}.$
###### Remark 5.3.
Let $\mathcal{M}^{n}$ be a compact homogeneous Riemannian manifold. Then,
Riemannian manifold $\mathcal{M}^{n}$ admits eigenmaps to some unit sphere for
the first positive eigenvalue of the Laplacian. See [42]. Therefore, it
satisfies the condition of Theorem 5.7.
_Proof of Theorem_ 5.7. Taking the Laplacian for the following equation
(5.11) $\sum_{\alpha=1}^{m+1}f_{\alpha}^{2}=1$
and using the fact that $\Delta f_{\alpha}=-\eta f_{\alpha},\ \ {\rm where}\ \
\alpha=1,\cdots,m+1,$ we have
(5.12) $\sum_{\alpha=1}^{m+1}\left|\nabla f_{\alpha}\right|_{g}^{2}=\eta.$
Taking the gradient for the equation (5.11), we have
(5.13) $\sum_{\alpha=1}^{m+1}f_{\alpha}\nabla f_{\alpha}=\textbf{0}.$
By Cauchy-Schwarz inequality, (2.27) and (5.12), we have
(5.14)
$\displaystyle\int_{\Omega}\sum_{\alpha=1}^{m+1}\left(u_{i}\left\langle\nu,\nabla
f_{\alpha}\right\rangle_{g_{0}}\right)^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\int_{\Omega}u_{i}^{2}\sum_{\alpha=1}^{m+1}\left\langle\nu,\nabla
f_{\alpha}\right\rangle_{g_{0}}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle\leq\eta\int_{\Omega}u_{i}^{2}|\nu^{\top}|^{2}_{g_{0}}e^{\langle\nu,X\rangle_{g_{0}}}dv,$
(5.15)
$\displaystyle\int_{\Omega}\sum_{\alpha=1}^{m+1}\left(4u_{i}\left\langle\nu,\nabla
f_{\alpha}\right\rangle_{g_{0}}\left\langle\nabla u_{i},\nabla
f_{\alpha}\right\rangle_{g}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv\leq\eta\int_{\Omega}\left(4u_{i}|\nu^{\top}|_{g_{0}}|\nabla
u_{i}|_{g}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv,$
and
(5.16) $\displaystyle 4\int_{\Omega}\sum_{\alpha=1}^{m+1}\left\langle\nabla
u_{i},\nabla
f_{\alpha}\right\rangle_{g}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv\leq
4\eta\int_{\Omega}|\nabla u_{i}|^{2}_{g}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
From (5.12), we have
(5.17)
$\sum_{\alpha=1}^{m+1}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\|u_{i}\nabla
f_{\alpha}\|_{\Omega}^{2}=\eta\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)^{2}.$
Taking $\varphi=f_{\alpha}$ in (2.11) and summing over $\alpha$, we infer that
(5.18)
$\displaystyle\sum_{\alpha=1}^{m+1}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})^{2}\|u_{i}\nabla
f_{\alpha}\|_{\Omega}^{2}$
$\displaystyle\leq\sum_{\alpha=1}^{m+1}\sum^{k}_{i=1}(\Lambda_{k+1}-\Lambda_{i})\|2\langle\nabla
f_{\alpha},\nabla
u_{i}\rangle_{g}+u_{i}\mathfrak{L}_{\nu}f_{\alpha}\|_{\Omega}^{2}.$
Substituting (2.27), (5.12)-(5.17) into (5.17), we conclude that,
$\displaystyle\eta$
$\displaystyle\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)^{2}$
$\displaystyle\leq\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)\int\sum_{\Omega}^{m+1}\left(u_{i}\left(\Delta
f_{\alpha}+\left\langle\nu,\nabla
f_{\alpha}\right\rangle_{g_{0}}\right)+2\left\langle\nabla f_{\alpha},\nabla
u_{i}\right\rangle_{g}\right)^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv$
$\displaystyle=\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)\int_{\Omega}\sum_{\alpha=1}^{m+1}\left(-\eta
u_{i}f_{\alpha}+u_{i}\left\langle\nu,\nabla
f_{\alpha}\right\rangle_{g_{0}}+2\left\langle\nabla u_{i},\nabla
f_{\alpha}\right\rangle_{g}\right)^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
$\displaystyle\leq\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)\left(\eta^{2}+\int_{\Omega}\left(4\left|\nabla
u_{i}\right|_{g}^{2}+4\left|u_{i}\right|_{g}\left|\nabla
u_{i}\right|_{g}|\nu^{\top}|_{g_{0}}+u_{i}^{2}|\nu^{\top}|_{g_{0}}^{2}\right)\eta
e^{\langle\nu,X\rangle_{g_{0}}}dv\right)$
$\displaystyle\leq\sum_{i=1}^{k}\left(\Lambda_{k+1}-\Lambda_{i}\right)\left(\eta^{2}+\left(4\Lambda_{i}+4D_{3}\Lambda_{i}^{1/2}+D_{3}^{2}\right)\eta\right),$
where
$D_{3}=\max_{\Omega}|\nu^{\top}|.$
Thus, we can obtain (5.10).
$None$
## 6\. The Closed Eigenvalue Problem
In this section, we investigate eigenvalue inequalities for the closed
eigenvalue problem on the compact Riemannian manifolds.
### 6.1. Estimates for the Eigenvalue of Closed Eigenvalue Problem
Let $\mathfrak{L}_{\nu}$ be an $n$-dimensional compact Riemannian manifolds
without boundary. In this subsection, we would like to study closed eigenvalue
problem (1.15) and establish some eigenvalue inequalities. By the same method
as the proof of Proposition 2.1, one can prove the following propsition.
###### Proposition 6.1.
Let $\overline{\phi}_{l}$, $l=1,2,\cdots,m$, be smooth functions on an
$n$-dimensional closed Riemannian manifold $\mathcal{M}^{n}$. Assume that
$\overline{\Lambda}_{i}$ is the $i^{\text{th}}$ eigenvalue of the closed
eigenvalue problem (1.15) and $\overline{u}_{i}$ is an orthonormal
eigenfunction corresponding to $\overline{\Lambda}_{i}$, where
$i=0,1,2,\cdots$, such that
$\mathfrak{L}_{\nu}\overline{u}_{i}=-\overline{\Lambda}_{i}\overline{u}_{i},$
and
$\int_{\mathcal{M}^{n}}\overline{u}_{i}\overline{u}_{j}e^{\langle\nu,X\rangle_{g_{0}}}dv=\delta_{ij},$
for any $i,j=0,1,2,\cdots$. Then, for any $j=0,1,2,\cdots$, there exists an
orthogonal matrix $A=(a_{lt})_{m\times m}$ such that
$\overline{\Phi}_{l}=\sum_{s=1}^{m}a_{ls}\overline{\phi}_{s}$ satisfy
(6.1)
$\sum^{m}_{l=1}(\overline{\Lambda}_{j+l}-\overline{\Lambda}_{j})\|\overline{u}_{j}\nabla\overline{\Phi}_{l}\|^{2}_{\mathcal{M}^{n}}\leq\sum^{m}_{l=1}\int_{\mathcal{M}^{n}}\big{(}\overline{u}_{j}\mathfrak{L}_{\nu}\overline{\Phi}_{l}+2\langle\nabla\overline{\Phi}_{l},\nabla\overline{u}_{j}\rangle_{g}\big{)}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv.$
Synthesizing Proposition 6.1, Lemma 2.7, Lemma 2.9 and Lemma 2.10, we can
prove the following theorem.
###### Theorem 6.2.
Let $\mathcal{M}^{n}$ be an $n$-dimensional compact Riemannian manifold
without boundary isometrically embedded into the Euclidean space
$\mathbb{R}^{n+p}$. Then, for any $j=0,1,2,\cdots$, the eigenvalues of closed
eigenvalue problem (1.15) of Xin-Laplacian satisfy
(6.2)
$\displaystyle\sum^{n}_{k=1}\overline{\Lambda}_{j+k}\leq(n+4)\overline{\Lambda}_{j}+4\overline{D}_{1}\overline{\Lambda}_{j}^{\frac{1}{2}}+\overline{D}^{2}_{1}+\overline{C}_{1},$
and
(6.3)
$\displaystyle\sum^{n}_{k=1}\overline{\Lambda}_{j+k}\leq(n+6)\overline{\Lambda}_{j}+3\overline{D}^{2}_{1}+\overline{C}_{1},$
where
$\overline{C}_{1}=\inf_{\psi\in\Psi}\max_{\mathcal{M}^{n}}n^{2}H^{2}\ \ and\ \
\overline{D}_{1}=\max_{\mathcal{M}^{n}}|\nu^{\top}|_{g_{0}}.$
###### Proof.
By making use of the same proof as in the proof of Theorem 3.3, we can prove
this theorem if one notices to count the number of eigenvalues from $1$. ∎
For the sake of the appearance of the mean curvature, it is very natural to
generalize an important result obtained by Rielly in [51]. Indeed, by Theorem
6.2, we have following corollary.
###### Corollary 6.3.
Let $\mathcal{M}^{n}$ be an $n$-dimensional compact Riemannian manifold
without boundary. Then, for any $j=0,1,2,\cdots$, the eigenvalues of closed
eigenvalue problem (1.15) of Xin-Laplacian satisfy
(6.4)
$\displaystyle\sum^{n}_{k=1}\overline{\Lambda}_{j+k}\leq(n+4)\overline{\Lambda}_{j}+\int_{\mathcal{M}^{n}}\overline{u}_{j}^{2}\left(n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2}+4\overline{\Lambda}_{j}^{\frac{1}{2}}|\nu^{\top}|_{g_{0}}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv,$
and
(6.5)
$\displaystyle\sum^{n}_{k=1}\overline{\Lambda}_{j+k}\leq(n+6)\overline{\Lambda}_{j}+\int_{\mathcal{M}^{n}}\overline{u}_{j}^{2}\left(n^{2}H^{2}+3|\nu^{\top}|_{g_{0}}^{2}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv.$
###### Corollary 6.4.
For an $n$-dimensional complete submanifold $\mathcal{M}^{n}$ in the Euclidean
space $\mathbb{R}^{n+p}$, eigenvalues of the closed eigenvalue problem (1.15)
of the differential operator $\mathfrak{L}_{\nu}$ satisfy
(6.6)
$\displaystyle\sum^{n}_{k=1}\overline{\Lambda}_{k}\leq\dfrac{\int_{\mathcal{M}^{n}}\left(n^{2}H^{2}+3|\nu^{\top}|_{g_{0}}^{2}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv}{\int_{\mathcal{M}^{n}}e^{\langle\nu,X\rangle_{g_{0}}}dv}.$
###### Proof.
Since $\overline{\Lambda}_{0}=0$ and $\overline{u}_{0}$ is constant, by taking
$j=0$ in the Theorem 6.2, we can infer that,
$\displaystyle\sum^{n}_{k=1}\overline{\Lambda}_{k}\leq\dfrac{\int_{\mathcal{M}^{n}}\left(n^{2}H^{2}+3|\nu^{\top}|_{g_{0}}^{2}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv}{\int_{\mathcal{M}^{n}}e^{\langle\nu,X\rangle_{g_{0}}}dv},$
where we have used a fact as follows:
$\int_{\mathcal{M}^{n}}\overline{u}_{0}^{2}e^{\langle\nu,X\rangle_{g_{0}}}dv=1.$
Therefore, we finish the proof of this corollary. ∎
###### Remark 6.1.
If we take $\nu=0$, the operator $\mathfrak{L}_{\nu}$ is the Laplace-Beltrami
operator and we have
$\displaystyle\sum^{n}_{k=1}\overline{\Lambda}_{k}\leq\dfrac{n^{2}\int_{\mathcal{M}^{n}}{H}^{2}dv}{\int_{\mathcal{M}^{n}}dv},$
which is a remarkable result obtained by Ilias and Makhoul in [38]. Also, see
[56]. In particular, when $M^{n}$ is an $n$-dimensional unit sphere
$\mathbb{S}^{n}(1)$ and $\nu$ is a zero vector, the identity holds. Hence, our
result is a generalization of Reilly’s result in [51] on the first eigenvalue
$\overline{\Lambda}_{1}\leq\dfrac{n\int_{\mathcal{M}^{n}}{H}^{2}dv}{\int_{\mathcal{M}^{n}}dv}.$
Next, we consider that $\mathcal{M}^{n}$ is an $n$-dimensional compact minimal
submanifold in the unit sphere $\mathbb{S}^{n+p}(1)$. For this case, we have
the following result.
###### Theorem 6.5.
Let $\mathcal{M}^{n}$ be an $n$-dimensional compact minimal submanifold in the
unit sphere $\mathbb{S}^{n+p}(1)$. Then, for any $j$, where $j=0,1,2,\cdots$,
eigenvalues of the closed eigenvalue problem (1.15) of the differential
operator $\mathfrak{L}_{\nu}$ satisfy
(6.7)
$\displaystyle\sum^{n}_{k=1}\overline{\Lambda}_{j+k}\leq(n+4)\overline{\Lambda}_{j}+4\overline{D}_{2}\overline{\Lambda}_{j}^{\frac{1}{2}}+\overline{D}^{2}_{2}+n^{2},$
and
(6.8)
$\displaystyle\sum^{n}_{k=1}\overline{\Lambda}_{j+k}\leq(n+6)\overline{\Lambda}_{j}+3\overline{D}_{2}^{2}+n^{2},$
where
$\overline{D}_{2}=\inf_{\psi\in\Psi}\max_{\mathcal{M}^{n}}|\nu^{\top}|_{g_{0}}.$
###### Proof.
Since $\mathcal{M}^{n}$ is an $n$-dimensional minimal submanifold in the unit
sphere $\mathbb{S}^{n+p}(1)$, then $\mathcal{M}^{n}$ can be seen as a compact
submanifold in $\mathbb{R}^{n+p+1}$ with mean curvature $H\equiv 1$.
Therefore, by Corollary 6.3, we know that both inequalities (6.7) and (6.8)
hold. ∎
By the same strategy as the proof of Proposition 2.3, one also can prove the
following proposition.
###### Proposition 6.6.
Let $(\mathcal{M}^{n},g)$ be an $n$-dimensional compact Riemannian manifold
without boundary. Assume that $\overline{\Lambda}_{i}$ is the $i^{\text{th}}$
eigenvalue of the closed eigenvalue problem (1.15) and $\overline{u}_{i}$ is
an orthonormal eigenfunction corresponding to $\overline{\Lambda}_{i}$,
$i=0,1,2,\cdots$, such that
$\mathfrak{L}_{\nu}\overline{u}_{i}=-\overline{\Lambda}_{i}\overline{u}_{i},$
and
$\int_{\mathcal{M}^{n}}\overline{u}_{i}\overline{u}_{j}e^{\langle\nu,X\rangle_{g_{0}}}dv=\delta_{ij},$
for any $i,j=0,1,2,\cdots$. Then, for any function $\overline{\varphi}(x)\in
C^{2}(\mathcal{M}^{n})$ and any positive integer $k$, eigenvalues of the close
eigenvalue problem (1.15) satisfy
$\displaystyle\sum^{k}_{i=0}(\overline{\Lambda}_{k+1}-\overline{\Lambda}_{i})^{2}\|\overline{u}_{i}\nabla\overline{\varphi}\|_{\mathcal{M}^{n}}^{2}\leq\sum^{k}_{i=0}(\overline{\Lambda}_{k+1}-\overline{\Lambda}_{i})\|2\langle\nabla\overline{\varphi},\nabla\overline{u}_{i}\rangle_{g}+\overline{u}_{i}\mathfrak{L}_{\nu}\overline{\varphi}\|_{\mathcal{M}^{n}}^{2},$
where
$\|\overline{\varphi}(x)\|_{\mathcal{M}^{n}}^{2}=\int_{\mathcal{M}^{n}}\overline{\varphi}^{2}(x)e^{\langle\nu,X\rangle_{g_{0}}}dv.$
By using Proposition 6.6 and Lemma 2.7, we can establish the following
eigenvalue inequality of Yang type.
###### Theorem 6.7.
Let $(\mathcal{M}^{n},g)$ be an $n$-dimensional closed Riemannian manifold
isometrically embedded into the Euclidean space $\mathbb{R}^{n+p}$. Assume
that $\overline{\Lambda}_{i}$ is the $i^{th}$ eigenvalue of eigenvalue problem
(1.15) of the Xin-Laplacian. Then, we have
(6.9)
$\displaystyle\sum^{k}_{i=0}(\overline{\Lambda}_{k+1}-\overline{\Lambda}_{i})^{2}\leq\frac{4}{n}$
$\displaystyle\sum^{k}_{i=0}(\overline{\Lambda}_{k+1}-\overline{\Lambda}_{i})\left(\overline{\Lambda}_{i}+\frac{1}{4}\overline{D}_{1}\overline{\Lambda}_{j}^{\frac{1}{2}}+\overline{D}_{1}^{2}+\frac{1}{4}\overline{C}_{1}\right),$
and
(6.10)
$\displaystyle\sum^{k}_{i=0}(\overline{\Lambda}_{k+1}-\overline{\Lambda}_{i})^{2}\leq\frac{6}{n}$
$\displaystyle\sum^{k}_{i=0}(\overline{\Lambda}_{k+1}-\overline{\Lambda}_{i})\left(\overline{\Lambda}_{i}+3\overline{D}_{1}^{2}+\frac{1}{6}\overline{C}_{1}\right),$
where
$\overline{C}_{1}=\inf_{\psi\in\Psi}\max_{\mathcal{M}^{n}}n^{2}H^{2}\ \ and\ \
\overline{D}_{1}=\max_{\mathcal{M}^{n}}|\nu^{\top}|_{g_{0}}.$
###### Proof.
The proof almost is a copy of the proof of Theorem 3.1 word by word, and the
only thing needs to be done is to notices to count the number of eigenvalues
from $0$. ∎
By Theorem 6.7, we have following corollary.
###### Corollary 6.8.
Let $(\mathcal{M}^{n},g)$ be an $n$-dimensional closed Riemannian manifold
isometrically embedded into the Euclidean space $\mathbb{R}^{n+p}$. Assume
that $\overline{\Lambda}_{i}$ is the $i^{th}$ eigenvalue of eigenvalue problem
(1.15) of the Xin-Laplacian. Then, we have
$\displaystyle\sum^{k}_{i=0}(\overline{\Lambda}_{k+1}-\overline{\Lambda}_{i})^{2}\leq\frac{4}{n}$
$\displaystyle\sum^{k}_{i=0}(\overline{\Lambda}_{k+1}-\overline{\Lambda}_{i})$
$\displaystyle\times\left[\overline{\Lambda}_{i}+\frac{1}{4}\int_{\mathcal{M}^{n}}\overline{u}_{i}^{2}\left(n^{2}H^{2}+|\nu^{\top}|_{g_{0}}^{2}+4\overline{\Lambda}_{j}^{\frac{1}{2}}|\nu^{\top}|_{g_{0}}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv\right],$
and
$\displaystyle\sum^{k}_{i=0}(\overline{\Lambda}_{k+1}-\overline{\Lambda}_{i})^{2}\leq\frac{6}{n}$
$\displaystyle\sum^{k}_{i=0}(\overline{\Lambda}_{k+1}-\overline{\Lambda}_{i})\left[\overline{\Lambda}_{i}+\frac{1}{6}\int_{\mathcal{M}^{n}}\overline{u}_{i}^{2}\left(n^{2}H^{2}+3|\nu^{\top}|_{g_{0}}^{2}\right)e^{\langle\nu,X\rangle_{g_{0}}}dv\right].$
Finally, we assume that $\mathcal{M}^{n}$ is an $n$-dimensional compact
minimal submanifold in the unit sphere $\mathbb{S}^{n+p}(1)$. For this case,
we have the following theorem.
###### Theorem 6.9.
Let $\mathcal{M}^{n}$ be an $n$-dimensional compact minimal submanifold in the
unit sphere $\mathbb{S}^{n+p}(1)$. Then, for any $j$, where $j=0,1,2,\cdots$,
eigenvalues of the closed eigenvalue problem (1.15) of differential operator
$\mathfrak{L}_{\nu}$ satisfy
(6.11)
$\displaystyle\sum^{k}_{i=0}(\overline{\Lambda}_{k+1}-\overline{\Lambda}_{i})^{2}\leq\frac{4}{n}$
$\displaystyle\sum^{k}_{i=0}(\overline{\Lambda}_{k+1}-\overline{\Lambda}_{i})\left(\overline{\Lambda}_{i}+\frac{1}{4}\overline{D}_{2}\overline{\Lambda}_{j}^{\frac{1}{2}}+\overline{D}_{2}^{2}+\frac{1}{4}n^{2}\right),$
and
(6.12)
$\displaystyle\sum^{k}_{i=0}(\overline{\Lambda}_{k+1}-\overline{\Lambda}_{i})^{2}\leq\frac{6}{n}$
$\displaystyle\sum^{k}_{i=0}(\overline{\Lambda}_{k+1}-\overline{\Lambda}_{i})\left(\overline{\Lambda}_{i}+3\overline{D}_{2}^{2}+\frac{1}{6}n^{2}\right),$
where
$\overline{D}_{2}=\inf_{\psi\in\Psi}\max_{\mathcal{M}^{n}}|\nu^{\top}|_{g_{0}}.$
###### Proof.
Since $\mathcal{M}^{n}$ is an $n$-dimensional minimal submanifold in the unit
sphere $\mathbb{S}^{n+p}(1)$, then $\mathcal{M}^{n}$ can be viewed as a
compact submanifold in $\mathbb{R}^{n+p+1}$ with mean curvature $H=1$.
Therefore, by the Corollary 6.8, it is easy to see that both inequalities
(6.11) and (6.12) hold. This completes the proof of this theorem. ∎
###### Remark 6.2.
In Theorem 6.9, we assume that $\nu=0$, and then, (6.11) implies that
(6.13)
$\displaystyle\sum^{k}_{i=0}(\overline{\Lambda}_{k+1}-\overline{\Lambda}_{i})^{2}\leq\frac{4}{n}\sum^{k}_{i=0}(\overline{\Lambda}_{k+1}-\overline{\Lambda}_{i})\left(\overline{\Lambda}_{i}+\frac{n^{2}}{4}\right),$
which is given by Cheng and Yang in [20].
### 6.2. Geometry of Isoparametric Foliations
In recent years, isoparametric theory has remarkable applications in the
research of geometry of submanifolds and spectrum analysis. For the sake of
reader’s convenience, we recall some fundamental facts about the isoparametric
hypersurfaces and focal submanifolds. For more information on isoparametric
hypersurfaces and focal submanifolds, we refer the readers to the good
articles [11, 62]. Firstly, let us introduce the definition of isoparametric
functions. Let $b$ and $a$ be a smooth function and a continuous function
defined on $\mathbb{R}$, respectively. For this purpose, let $f$ be a smooth
function defined on $\mathbb{S}^{n+1}(1)$. If $f$ satisfies
(6.14) $|\nabla f|_{g}^{2}=b(f),$
and
(6.15) $\Delta f=a(f),$
then it is said to be isoparametric. A function satisfying only (6.14) is
called transnormal. The geometric meaning of (6.14) and (6.15) is that the
regular level hypersurfaces of $f$ are parallel with each other and have
constant mean curvatures. In this sense, the regular level hypersurfaces of
$f$ are called _isoparametric hypersurfaces_ , and the two singular level sets
of $f$ are called _focal submanifolds_. Of course, one can also define
isoparametric hypersurfaces via an extrinsic geometric viewpoint as follows: A
hypersurface $\mathcal{M}^{n}$ in the $(n+1)$-dimensional unit sphere
$\mathbb{S}^{n+1}(1)$ is said to be isoparametric, if all of the principle
curvatures are constant functions. A well-known result of Cartan states that
isoparametric hypersurfaces come as a family of parallel hypersurfaces. To be
more specific, given an isoparametric hypersurface $\mathcal{M}^{n}$ in
$\mathbb{S}^{n+1}(1)$ and a smooth field $\xi$ with unit normals to
$\mathcal{M}^{n}$, for each $x\in\mathcal{M}^{n}$ and $\theta\in\mathbb{R},$
we can define $\phi_{\theta}:\mathcal{M}^{n}\rightarrow$ $\mathbb{S}^{n+1}(1)$
by
$\phi_{\theta}(x)=\cos\theta x+\sin\theta\xi(x).$
Here, $\phi_{\theta}(x)$ is the point at an oriented distance $\theta$ to
$\mathcal{M}$ along the normal geodesic through $x$. If
$\theta\neq\theta_{\epsilon}$ for any $\epsilon=1,\cdots,\ell$, where $\ell$
denotes the number of distinct constant principal curvatures, $\phi_{\theta}$
is a parallel hypersurface to $M$ at an oriented distance $\theta$. If
$\theta=\theta_{\epsilon}$ for some $\epsilon=1,\cdots,\ell,$ it is easy to
find that for any vector $Y$ in the principal distributions
$E_{\epsilon}(x)=\left\\{Y\in
T_{x}M\mid\mathcal{A}_{\xi}Y=\cot\theta_{\epsilon}Y\right\\},$
where $\mathcal{A}_{\xi}$ is a shape operator with respect to $\xi$,
$\left(\phi_{\theta}\right)_{*}Y=\left(\cos\theta-\sin\theta\cot\theta_{\epsilon}\right)Y=\frac{\sin\left(\theta_{\epsilon}-\theta\right)}{\sin\theta_{\epsilon}}Y=0.$
In other words, if $\cot\theta=\cot\theta_{\epsilon}$ is a principal curvature
of $\mathcal{M}^{n}$, $\phi_{\theta}$ is not an immersion, but is actually a
focal submanifold of codimension $m_{\epsilon}+1$ in $\mathbb{S}^{n+1}(1)$.
Using an elegant topological method, Münzner proved the remarkable result that
the number $\ell$ must be $1,2,3,4,$ or 6 $m_{\epsilon}=m_{\epsilon+2}($
indices mod
$\ell);\theta_{\epsilon}=\theta_{1}+\frac{\epsilon-1}{\ell}\pi(\epsilon=1,\cdots,\ell);$
and when $\ell$ is odd, $m_{1}=m_{2}$(cf. [49]). Münzner asserted that
regardless of the number of distinct principal curvatures of $M,$ there are
only two distinct focal submanifolds in a parallel family of isoparametric
hypersurfaces, and every isoparametric hypersurface is a tube of constant
radius over each focal submanifold. We denote the distinct focal submanifolds
by $\mathcal{M}_{1},\mathcal{M}_{2}$ according to the inverse images of
maximum or minimum values of $f$ satisfy the equations system (6.14) and
(6.15), respectively. It is well known that $\mathcal{M}_{i}$, where $i=1,2$,
are minimal submanifolds in $\mathbb{S}^{n+1}(1)$. Assuming that
$\left\\{\textbf{P}_{0},\cdots,\textbf{P}_{m}\right\\}$ is a symmetric
Clifford system on $\mathbb{R}^{2l}$, this is, $\textbf{P}_{i}$ ’s are
symmetric matrices satisfying
$\textbf{P}_{i}\textbf{P}_{j}+\textbf{P}_{j}\textbf{P}_{i}=2\delta_{ij}\textbf{I}_{2l},$
in [27], Ferus, Karcher and Münzner constructed a polynomial function $\Re$ on
$\mathbb{R}^{2l}$ as follows:
$\begin{array}[]{c}\Re:\mathbb{R}^{2l}\rightarrow\mathbb{R},\\\
\Re(x)=|x|^{4}-2\sum_{i=0}^{m}\left\langle\textbf{P}_{i}x,x\right\rangle^{2}.\end{array}$
Then, each level hypersurface of $f=\left.\Re\right|_{S^{2l-1}}$, i.e., the
preimage of some regular value of $f,$ is an isoparametric hypersurface with
four distinct constant principal curvatures. We choose $\xi=\frac{\nabla
f}{|\nabla f|},$ and it can be asserted that
$\mathcal{M}_{1}=f^{-1}(1),\mathcal{M}_{2}=f^{-1}(-1)$, with codimensions
$m_{1}+1$ and $m_{2}+1$ in $\mathbb{S}^{n+1}(1),$ respectively. The
multiplicity pairs $\left(m_{1},m_{2}\right)=(m,l-m-1),$ provided $m>0$ and
$l-m-1>0,$ where $l=k\delta(m)$ $(k=1,2,3,\cdots)$ and $\delta(m)$ is the
dimension of an irreducible module of the Clifford algebra $C_{m-1}$ on
$\mathbb{R}^{l}$. See [30].
### 6.3. Eigenvalues on the Isoparametric Hypersurfaces of Laplacian
Attacking Yau’s conjecture presented in the introduction, a significant
breakthrough to it was made by Choi and Wang [23]. They proved that the first
eigenvalue of every (embedded ) closed minimal hypersurface in
$\mathbb{S}^{n+1}(1)$ is not smaller than $\frac{n}{2}$. Furthermore, Brendle
pointed out that the first eigenvalue is larger than $\frac{n}{2}$ in his
survey paper [9]. Usually, the calculation of the eigenvalues of the Laplace-
Beltrami operator, even of the first eigenvalue, is rather complicated and
difficult. Up to now, Yau’s conjecture remains unsolved. In 2013, Tang and Yan
made an extremely important contribution to this conjecture in [61], where
they made an affirmative answer to this conjecture under the condition that
$\mathcal{M}^{n}$ is a closed embedding isoparametric hypersurfaces in
$\mathbb{S}^{n+1}(1)$. For more progress on this conjecture, we refer the
readers to [39, 48, 47, 57, 58, 59] and references therein. As a fascinating
application of Theorem 6.2, we can show the following result.
###### Theorem 6.10.
Let $\mathcal{M}^{n}$ be an $n$-dimensional compact minimal isoparametric
hypersurface in the unit sphere $\mathbb{S}^{n+1}(1)$. Then, eigenvalues of
the closed eigenvalue problem (1.15) of the Laplace-Beltrami operator satisfy
(6.16) $\frac{1}{n}\sum_{k=1}^{n}\overline{\Lambda}_{n_{0}+k}\leq 2n+4,$
where $n_{0}$ denotes the value of the multiplicity of the first eigenvalue.
###### Proof.
Assume that $\mathcal{M}^{n}$ is a unit sphere $\mathbb{S}^{n+1}(1)$, the
assertion is obvious. Now, we consider that $M^{n}$ is a minimal isoparametric
hypersurface other than $\mathbb{S}^{n}(1),$ we know that
$\overline{\Lambda}_{1}=\overline{\Lambda}_{2}=\cdots=\overline{\Lambda}_{n_{0}}=n$
according to some results showed by Tang and Yan in [61]. From (6.7), we
directly get (6.16). ∎
###### Remark 6.3.
Let $\mathcal{M}^{n}$ be an $n$-dimensional compact minimal isoparametric
hypersurface in the unit sphere $\mathbb{S}^{n+1}(1)$ and $|\nu|_{g_{0}}=0$.
Then, according to Theorem 6.10, we get an estimate for the upper bound of the
second non-zero eigenvalue without counting the multiplicities of eigenvalues
as follows:
(6.17) $\overline{\Gamma}_{2}\leq 2n+4.$
###### Remark 6.4.
Let $\mathcal{M}^{n}$ be an $n$-dimensional unit sphere $\mathbb{S}^{n}(1)$
and $|\nu|_{g_{0}}=0$. Then, we have
$\overline{\Gamma}_{2}=2n+2,$
which means that eigenvalue inequality given in Theorem 6.16 is very sharp.
###### Remark 6.5.
Let $\mathcal{M}^{n}$ be an $n$-dimensional compact minimal isoparametric
hypersurface in the unit sphere $\mathbb{S}^{n+1}(1)$ and $|\nu|_{g_{0}}=0$.
Then, from (6.13), we can obtain a weaker inequality than (6.16). To be more
specific, we have
$\overline{\Gamma}_{2}\leq 2n+4.$
###### Remark 6.6.
In [59], Solomon constructed an eigenfunction on a so-called quartic
isoparametric hypersurface $\mathcal{M}^{n}$ of OT-FKM-type, to conclude that
$\mathcal{M}^{n}$ has $2n$ as an eigenvalue to fill the gap of eigenvalue
sequence $0,n,3n,4n,\cdots$, which contain in the spectrum of Laplacian on the
quartic isoparametric hypersurface $\mathcal{M}^{n}$. Therefore, Theorem 6.10
further hints that $2n$ could be the second non-zero eigenvalue, although we
still don’t know whether $2n$ is the second non-zero eigenvalue or not.
### 6.4. Eigenvalues on the Focal Submanifolds of Laplacian
In this subsection, we are concerned with the focal submanifolds. It is
remarkable that the focal submanifolds of isoparametric hypersurfaces provide
infinitely many spherical submanifolds with abundant intrinsic and extrinsic
geometric properties. For instance, they are both minimal in a unit sphere.
Moreover, two focal submanifolds of an isoparametric hypersurface with four
distinct principal curvatures are both Willmore in a unit sphere. See [43].
Let $\mathcal{M}_{1}$ be the focal submanifold of an isoparametric
hypersurface with four distinct principal curvatures in the unit sphere
$\mathbb{S}^{n+1}(1)$ with codimension $m_{1}+1$. Tang and Yan [61]
investigated the eigenvalue of Laplacian on the focal submanifold of an
isoparametric hypersurface with four distinct principal curvatures and
obtained an estimates for the lower bound as follows:
$\Lambda_{n+3}\left(\mathcal{M}_{1}\right)\geq\frac{4(n+2)\left(m_{2}-1\right)}{n},$
which implies that
$\overline{\Gamma}_{2}\left(\mathcal{M}_{1}\right)\geq\frac{4(n+2)\left(m_{2}-1\right)}{n}.$
Applying Theorem 6.2, we can get an estimates for the upper bound of the
eigenvalues Laplacian on the focal submanifold of an isoparametric
hypersurface with four distinct principal curvatures. This is what the
following theorem states.
###### Theorem 6.11.
Let $\mathcal{M}_{1}$ be the focal submanifold of an isoparametric
hypersurface with four distinct principal curvatures with dimension
$\operatorname{dim}\mathcal{M}_{1}\geq\frac{2n}{3}+1$
in the unit sphere $\mathbb{S}^{n+1}(1)$. Then, for the eigenvalues of
Laplace-Beltrami operator, we have
(6.18)
$\displaystyle\frac{1}{m_{1}+2m_{2}}\sum^{m_{1}+2m_{2}}_{k=1}\overline{\Lambda}_{n+2+k}\leq
2(n+m_{2}+2).$
In particular, we have
(6.19) $\overline{\Gamma}_{2}\leq 2(n+m_{2}+2).$
A similar conclusion holds for $\mathcal{M}_{2}$ under an analogous condition.
###### Proof.
If $\operatorname{dim}\mathcal{M}_{1}\geq\frac{2}{3}n+1,$ Tang and Yan [61]
proved that,
$\overline{\Lambda}_{1}\left(\mathcal{M}_{1}\right)=m_{1}+2m_{2}$
with multiplicity $n+2.$ Therefore, it follows from (6.7) that,
(6.20)
$\displaystyle\sum^{m_{1}+2m_{2}}_{k=1}\overline{\Lambda}_{n+2+k}\leq\left[\left(m_{1}+2m_{2}\right)+4\right]\left(m_{1}+2m_{2}\right)+\left(m_{1}+2m_{2}\right)^{2},$
which gives (6.18), since $n=2(m_{1}+m_{2})$. From (6.18), it is not difficult
to conclude (6.19). This completes the proof of Theorem 6.11.
∎
###### Remark 6.7.
Both $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ are fully embedded in
$\mathbb{S}^{n+1}(1)$ if $\ell\geq 3$, namely, they cannot be embedded into a
hypersphere, which means that, the dimension $n-m_{1}$ (resp. $n-m_{2}$) of
$\mathcal{M}_{1}$ is an eigenvalue of $\mathcal{M}_{1}$ (resp.
$\mathcal{M}_{2}$) with multiplicity at least $n+2$ (cf. [61]).
For the focal submanifold $\mathcal{\mathcal{M}}_{1}$ of OT-FKM type in
$\mathbb{S}^{5}(1)$ with $\left(\mathcal{M}_{1},\mathcal{M}_{2}\right)=$
(1,1), Tang, Xie and Yan [63] proved that
$\overline{\Lambda}_{1}\left(\mathcal{M}_{1}\right)=\operatorname{dim}M_{1}=3$
with multiplicity $6$. Furthermore, for the focal submanifold
$\mathcal{M}_{1}$ of homogeneous OT-FKM type in $\mathbb{S}^{15}(1)$ with
$\left(m_{1},m_{2}\right)=(4,3)$, they claimed that
$\overline{\Lambda}_{1}\left(\mathcal{M}_{1}\right)=\operatorname{dim}\mathcal{M}_{1}=10$
with multiplicity $16$. Thus, we can prove the following theorems in the light
of the idea of the proof of Theorem 6.11.
###### Theorem 6.12.
For the focal submanifold $\mathcal{M}_{1}$ of OT-FKM type in
$\mathbb{S}^{5}(1)$ with $\left(m_{1},m_{2}\right)=$ (1,1), we have
(6.21) $\displaystyle\sum^{3}_{k=1}\overline{\Lambda}_{6+k}\leq 30.$
In particular, for the second eigenvalue (without considering the
multiplicity) of the Laplace-Beltrami operator, we have
(6.22) $\overline{\Gamma}_{2}\leq 10.$
###### Theorem 6.13.
For the focal submanifold $\mathcal{M}_{1}$ of homogeneous OT-FKM type in
$\mathbb{S}^{15}(1)$ with $\left(m_{1},m_{2}\right)=(4,3)$, we have
$\displaystyle\sum^{10}_{k=1}\overline{\Lambda}_{16+k}\leq 240.$
In particular, we have
$\overline{\Gamma}_{2}\leq 24.$
###### Remark 6.8.
For the focal submanifold $\mathcal{M}_{1}$ of homogeneous OT-FKM type with
$\left(m_{1},m_{2}\right)=(1,k)$, according to the Proposition 1.1 in [61] and
Theorem 6.2, we can similarly give an upper estimate for the non-zero
eigenvalue without counting the multiplicity of eigenvalues.
###### Remark 6.9.
It is well known that, when $\ell=2$, the focal submanifolds are isometric to
$\mathbb{S}^{p}(1)$ and $\mathbb{S}^{q}(1)$. Thus, their second non-zero
eigenvalues (without counting the multiplications) equal to two times of their
dimensions, respectively.
###### Remark 6.10.
When $\ell=3,$ Cartan showed that $m_{1}=m_{2}=1,2,4$ or $8.$ In the unit
spheres $\mathbb{S}^{4}(1),\mathbb{S}^{7}(1),\mathbb{S}^{13}(1)$ and
$\mathbb{S}^{25}(1)$, the focal submanifolds of them are the Veronese
embedding of $\mathbb{R}P^{2},\mathbb{C}P^{2},\mathbb{H}P^{2}$ and
$\mathbb{Q}P^{2},$ respectively. The induced metric of this $\mathbb{R}P^{2}$
minimally embedded in $\mathbb{S}^{4}(1)$ differs from the standard metric of
constant Gaussian curvature $K=1$ by a constant factor such that
$K=\frac{1}{3}$. As for $\mathbb{C}P^{2},\mathbb{H}P^{2}$ and
$\mathbb{Q}P^{2},$ these are minimally embedded in the unit spheres
$\mathbb{S}^{7}(1),\mathbb{S}^{13}(1)$ and $\mathbb{S}^{25}(1)$ respectively,
while the induced metric differs from the symmetric space metric by a constant
factor such that $\frac{1}{3}\leq\operatorname{Sec}\leq\frac{4}{3}$. According
to [55, 46], one knows that the first eigenvalues of the focal submanifolds
$\mathbb{C}P^{2},\mathbb{H}P^{2}$ and $\mathbb{Q}P^{2}$ are equal to their
dimensions, respectively. In conclusion, when $\ell=3$, one can assert that
$\displaystyle\frac{1}{n_{0}}\sum^{n_{0}}_{k=1}\overline{\Lambda}_{m_{0}+k}\leq
2n_{0}+4,$
which implies that
$\displaystyle\overline{\Gamma}_{2}\leq 2n_{0}+4,$
where $n_{0}$ denotes the dimension of focal submanifolds and $m_{0}$ denotes
the multiplicity of the first non-zero eigenvalue.
## 7\. Some Conjectures and Further Remarks
In this section, we raise some conjectures and give some further remarks to
end this paper.
Let $\Omega$ be a bounded domain on an $n$-dimensional Riemannian manifold
$\mathcal{M}^{n}$ with piecewise smooth boundary $\partial\Omega$. We consider
Dirichlet eigenvalue problem of Laplacian on complete Riemannian manifolds as
follows:
(7.1) ${\begin{cases}\ \Delta u+\Lambda u=0,\ \ &{\rm in}\ \ \ \ \Omega,\\\ \
u=0,\ \ &{\rm on}\ \ \partial\Omega.\end{cases}}$
We suppose that $\Lambda_{k}$ is the $k^{th}$ eigenvalue corresponding to the
eigenfunction $u_{k}$. It is well known that the spectrum of this eigenvalue
problem (7.1) is real and discrete. Furthermore, the following Weyl s
asymptotic formula holds (cf. [12]):
(7.2)
$\Lambda_{k}\sim\frac{4\pi^{2}}{\left(\omega_{n}\operatorname{vol}\Omega\right)^{\frac{2}{n}}}k^{\frac{2}{n}},\quad
k\rightarrow\infty.$
From this asymptotic formula (7.2), it is not difficult to infer that
$\sum_{i=1}^{k}\Lambda_{i}\sim\frac{n}{n+2}\frac{4\pi^{2}}{\left(\omega_{n}\operatorname{vol}\Omega\right)^{\frac{2}{n}}}k^{\frac{n+2}{n}},\quad
k\rightarrow\infty.$
In addition, for any positive integer $n_{1}$, it is easy to show that the
eigenvalues of the eigenvalue problem (7.1) of Laplacian satisfy:
$\displaystyle\lim_{j\rightarrow+\infty}\frac{\Lambda_{j+1}+\Lambda_{j+2}+\cdots+\Lambda_{j+n_{1}}}{\Lambda_{j}}=n_{1}.$
In particular, when $n_{1}=n$, we have
(7.3)
$\displaystyle\lim_{j\rightarrow+\infty}\frac{\Lambda_{j+1}+\Lambda_{j+2}+\cdots+\Lambda_{j+n}}{\Lambda_{j}}=n.$
From (7.5), we know that (1.11) can be improved and thus the first author
propose the following conjecture.
###### Conjecture 7.1.
Let $\Omega$ be a bounded domain with piecewise smooth boundary on an
$n$-dimensional Euclidean space $\mathbb{R}^{n}$. Then, the eigenvalues of the
eigenvalue problem (7.1) of the Laplace operator satisfy the following
universal inequality:
(7.4)
$\displaystyle\frac{\Lambda_{j+1}+\Lambda_{j+2}+\cdots+\Lambda_{j+n}}{\Lambda_{j}}\leq\frac{\Lambda_{2}+\Lambda_{3}+\cdots+\Lambda_{n+1}}{\Lambda_{2}},$
for any $j=1,2,\cdots.$
###### Remark 7.1.
If the conjecture above is true, it is natural for us to ask the same problem
for the case of general Riemannian manifolds, too.
Let $\mathfrak{L}_{\nu}$ be an $n$-dimensional compact Riemannian manifolds
without boundary. In this section, we shall investigate eigenvalues of the
closed eigenvalue problem of Laplacian on the Riemannian manifolds
$\mathcal{M}^{n}$ as follows:
$\Delta\overline{u}+\overline{\Lambda}\overline{u}=0,\ \ {\rm in}\ \ \ \
\mathcal{M}^{n}.$
###### Conjecture 7.2.
Let $\mathcal{M}^{n}$ be an $n$-dimensional compact minimal submanifold in the
unit sphere $\mathbb{S}^{n+p}(1)$. Then, the eigenvalues of the closed
eigenvalue problem (1.16) of the Laplace-Beltrami operator satisfy:
(7.5)
$\displaystyle\sum_{k=1}^{n}\overline{\Lambda}_{j+k}\leq(n+3)\overline{\Lambda}_{j}+\frac{\overline{\Lambda}_{j}^{2}}{\overline{\Lambda}_{j+1}}+n^{2}.$
###### Remark 7.2.
Provided that (7.5) is true and $\mathcal{M}^{n}$ is a compact minimal
isoparametric hypersurface in the unit sphere $\mathbb{S}^{n+1}(1)$, it is
easy to verify the following inequality
(7.6) $\overline{\Gamma}_{2}\leq\frac{2n+3+\sqrt{4n^{2}+16n+9}}{2}.$
Clearly, inequality (7.6) is sharper than inequality (6.17).
If $\mathcal{M}^{n}$ isoparametric hypersurfaces embedded in the unit sphere
$\mathbb{S}^{n+1}(1)$ with $\ell=1$, then $\overline{\Gamma}_{2}=2n+2$. If
$\mathcal{M}^{n}$ is the generalized Clifford torus
$\mathbb{S}^{p}\left(\sqrt{\frac{p}{n}}\right)\times\mathbb{S}^{q}\left(\sqrt{\frac{q}{n}}\right)$
$(p+q=n)$, by a straightforward calculation, we can show that the second
eigenvalue $\overline{\Gamma}_{2}=2n$. When $\ell=2$, as is well known, the
isoparametric hypersurface in $\mathbb{S}^{n+1}(1)$ is isometric to the
Clifford torus. Thus, $\overline{\Gamma}_{2}=2n$ when $\ell=2$. As a further
interest, based on the argument in the previous section, the first author
propose the following conjecture, which is closely related to Yau’s
Conjecture:
###### Conjecture 7.3.
Let $\mathcal{M}^{n}$ be an $n$-dimensional closed minimal hypersurface
embedded into the $(n+1)$-dimensional unit sphere $\mathbb{S}^{n+1}(1)$. Then,
we have
$2n\leq\overline{\Gamma}_{2}\leq 2n+2.$
In particular, let $\mathcal{M}^{n}$ be an $n$-dimensional closed minimal
isoparametric hypersurface embedded into $\mathbb{S}^{n+1}(1)$. Then, when
$\ell=3,4,6$, we have
$\overline{\Gamma}_{2}=2n.$
###### Remark 7.3.
Yau’s Conjecture is to consider the first non-zero eigenvalue, while
Conjecture 7.3 is to explore the second non-zero eigenvalue. However, we
consider the lower and upper bounds for the second eigenvalue without counting
the multiplications of eigenvalues in Conjecture 7.3.
###### Remark 7.4.
If the last part of Conjecture 7.3 holds, the second eigenvalue of Laplacian
will give a perfect and new character for the isoparametric hypersurfaces
embedding into the unit sphere $\mathbb{S}^{n+1}(1)$. We also note that
Conjecture 7.3 is closely related to the famous Chern’s conjecture: A closed,
minimally immersed hypersurface in $\mathbb{S}^{n+1}(1)$, whose second
fundamental form has constant length, is isoparametric. Therefore, it is to be
a fantabulous understanding for the isoparametric hypersurfaces if this
conjecture is settled.
Hsiang and Lawson [36] showed that every homogenous hypersurfaces in the unit
sphere $\mathbb{S}^{n+1}(1)$ is represented as an orbit of a linear isotropy
group of a Riemannian symmetric space of rank $2$. We refer the readers to
[44] for the list of the complete examples of homogenous hypersurfaces in the
unit sphere $\mathbb{S}^{n+1}(1)$. In what follows, there are some further
remarks on the eigenvalues of Laplace-Beltrami operator on the isoparametric
hypersurfaces.
###### Remark 7.5.
It is well known that, both $SO(3)/(\mathbb{Z}_{2}+\mathbb{Z}_{2})$ and
$SU(3)/\mathbb{T}^{2}$ are two isoparametric hypersurfaces embedded in the
unit sphere $\mathbb{S}^{n+1}(1)$ with $\ell=3$, and from [48], we know that
Conjecture 7.3 holds for the cases of $SO(3)/(\mathbb{Z}_{2}+\mathbb{Z}_{2})$
and $SU(3)/\mathbb{T}^{2}$.
###### Remark 7.6.
Assume that $\mathcal{M}^{n}$ are the cubic isoparametric minimal
hypersurfaces with $n=3,6,12,24$, the conjecture 7.3 is correct. Indeed, under
the assumption above, Solomon proved that [57, 58], without considering the
multiplicity of eigenvalues, the second non-zero eigenvalue satisfies
identity: $\overline{\Gamma}_{2}=2n$.
###### Acknowledgment.
The first author expresses his gratitude to professor Q.-M. Cheng for his
continuous encouragement and useful discussion in early years. The authors
also are debt to professor Mark S. Ashbaugh for sharing his literature [1].
The research was partially supported by the National Natural Science
Foundation of China (Grant Nos. 11861036 and 11826213) and the Natural Science
Foundation of Jiangxi Province (Grant No. 20171ACB21023).
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|
# Viral Visualizations: How Coronavirus Skeptics Use Orthodox Data Practices
to Promote Unorthodox Science Online
Crystal Lee<EMAIL_ADDRESS>0000-0001-6672-9118 Massachusetts Institute of
TechnologyCambridgeMAUSA , Tanya Yang<EMAIL_ADDRESS>Massachusetts
Institute of TechnologyCambridgeMAUSA , Gabrielle Inchoco
<EMAIL_ADDRESS>Wellesley CollegeWellesleyMAUSA , Graham M. Jones
<EMAIL_ADDRESS>Massachusetts Institute of TechnologyCambridgeMAUSA and Arvind
Satyanarayan<EMAIL_ADDRESS>Massachusetts Institute of
TechnologyCambridgeMAUSA
(2021)
###### Abstract.
Controversial understandings of the coronavirus pandemic have turned data
visualizations into a battleground. Defying public health officials,
coronavirus skeptics on US social media spent much of 2020 creating data
visualizations showing that the government’s pandemic response was excessive
and that the crisis was over. This paper investigates how pandemic
visualizations circulated on social media, and shows that people who mistrust
the scientific establishment often deploy the same rhetorics of data-driven
decision-making used by experts, but to advocate for radical policy changes.
Using a quantitative analysis of how visualizations spread on Twitter and an
ethnographic approach to analyzing conversations about COVID data on Facebook,
we document an epistemological gap that leads pro- and anti-mask groups to
draw drastically different inferences from similar data. Ultimately, we argue
that the deployment of COVID data visualizations reflect a deeper
sociopolitical rift regarding the place of science in public life.
digital ethnography, network analysis, Twitter, Facebook, data literacy, data
visualization
††journalyear: 2021††copyright: rightsretained††conference: CHI Conference on
Human Factors in Computing Systems; May 8–13, 2021; Yokohama,
Japan††booktitle: CHI Conference on Human Factors in Computing Systems (CHI
’21), May 8–13, 2021, Yokohama, Japan††doi: 10.1145/3411764.3445211††isbn:
978-1-4503-8096-6/21/05††ccs: Human-centered computing Empirical studies in
visualization††ccs: Human-centered computing Visualization theory, concepts
and paradigms††ccs: Human-centered computing Social media††ccs: Human-centered
computing Ethnographic studies
## 1\. Introduction
Throughout the coronavirus pandemic, researchers have held up the crisis as a
“breakthrough moment” for data visualization research (Shneiderman, 2020):
John Burn-Murdoch’s line chart comparing infection rates across countries
helped millions of people make sense of the pandemic’s scale in the United
States (Forrest, 2020), and even top Trump administration officials seemed to
rely heavily on the Johns Hopkins University COVID data dashboard (Mazza,
2020). Almost every US state now hosts a data dashboard on their health
department website to show how the pandemic is unfolding. However, despite a
preponderance of evidence that masks are crucial to reducing viral
transmission (Yi-Fong Su et al., 2020; CDC, 2020; Chu et al., 2020),
protestors across the United States have argued for local governments to
overturn their mask mandates and begin reopening schools and businesses. A
pandemic that affects a few, they reason, should not impinge on the liberties
of a majority to go about life as usual. To support their arguments, these
protestors and activists have created thousands of their own visualizations,
often using the same datasets as health officials.
This paper investigates how these activist networks use rhetorics of
scientific rigor to oppose these public health measures. Far from ignoring
scientific evidence to argue for individual freedom, anti-maskers often engage
deeply with public datasets and make what we call “counter-
visualizations”—visualizations using orthodox methods to make unorthodox
arguments—to challenge mainstream narratives that the pandemic is urgent and
ongoing. By asking community members to “follow the data,” these groups
mobilize data visualizations to support significant local changes.
We examine the circulation of COVID-related data visualizations through both
quantitative and qualitative methods. First, we conduct a quantitative
analysis of close to half a million tweets that use data visualizations to
talk about the pandemic. We use network analysis to identify communities of
users who retweet the same content or otherwise engage with one another (e.g.,
maskers and anti-maskers). We process over 41,000 images through a computer
vision model trained by Poco and Heer (Poco and Heer, 2017), and extract
feature embeddings to identify clusters and patterns in visualization designs.
The academic visualization research community has traditionally focused on
mitigating chartjunk and creating more intuitive visualization tools for use
by non-experts; _better_ visualizations, researchers argue, would aid public
understanding of data-driven phenomena. However, we find that anti-mask groups
on Twitter often create polished counter-visualizations that would not be out
of place in scientific papers, health department reports, and publications
like the _Financial Times_.
Second, we supplement this quantitative work with a six month-long
observational study of anti-mask groups on Facebook. The period of this study,
March to September 2020, was critical as it spanned the formation and
consolidation of these groups at the pandemic’s start. Quantitative analysis
gives us an overview of what online discourse about data and its visual
representation looks like on Twitter both within and outside anti-mask
communities. Qualitative analysis of anti-mask groups gives us an
interactional view of how these groups leverage the language of scientific
rigor—being critical about data sources, explicitly stating analytical
limitations of specific models, and more—in order to support ending public
health restrictions despite the consensus of the scientific establishment. Our
data analysis evolved as these communities did, and our methods reflect how
these users reacted in real time to the kaleidoscopic nature of pandemic life.
As of this writing, Facebook has banned some of the groups we studied, who
have since moved to more unregulated platforms (Parler and MeWe).
While previous literature in visualization and science communication has
emphasized the need for data and media literacy as a way to combat
misinformation (Guess et al., 2020; Scheufele and Krause, 2019; Fontichiaro
and Oehrli, 2016), this study finds that anti-mask groups practice a form of
data literacy in spades. Within this constituency, unorthodox viewpoints do
not result from a deficiency of data literacy; sophisticated practices of data
literacy are a means of consolidating and promulgating views that fly in the
face of scientific orthodoxy. Not only are these groups prolific in their
creation of counter-visualizations, but they leverage data and their visual
representations to advocate for and enact policy changes on the city, county,
and state levels.
As we shall see throughout this case study, anti-mask communities on social
media have defined themselves _in opposition_ to the discursive and
interpretive norms of the mainstream public sphere (e.g., against the
“lamestream media”). In media studies, the term “counterpublic” describes
constituencies that organize themselves in opposition to mainstream civic
discourse, often by agentively using communications media (Downey and Fenton,
2003). In approaching anti-maskers as a counterpublic (a group shaped by its
hostile stance toward mainstream science), we focus particular attention on
one form of agentive media production central to their movement: data
visualization. We define this counterpublic’s visualization practices as
“counter-visualizations” that use orthodox scientific methods to make
unorthodox arguments, beyond the pale of the scientific establishment. Data
visualizations are not a neutral window onto an observer-independent reality;
during a pandemic, they are an arena of political struggle.
Among other initiatives, these groups argue for open access to government data
(claiming that CDC and local health departments are not releasing enough data
for citizens to make informed decisions), and they use the language of data-
driven decision-making to show that social distancing mandates are both ill-
advised and unnecessary. In these discussions, we find that anti-maskers think
carefully about the grammar of graphics by decomposing visualizations into
layered components (e.g., raw data, statistical transformations, mappings,
marks, colors). Additionally, they debate how each component changes the
narrative that the visualization tells, and they brainstorm alternate
visualizations that would better enhance public understanding of the data.
This paper empirically shows how COVID anti-mask groups use data
visualizations to argue that the US government’s response (broadly construed)
to the pandemic is overblown, and that the crisis has long been over.
These findings suggest that the ability for the scientific community and
public health departments to better convey the urgency of the US coronavirus
pandemic may not be strengthened by introducing more downloadable datasets, by
producing “better visualizations” (e.g., graphics that are more intuitive or
efficient), or by educating people on how to better interpret them. This study
shows that there is a fundamental epistemological conflict between maskers and
anti-maskers, who use the same data but come to such different conclusions. As
science and technology studies (STS) scholars have shown, data is not a
neutral substrate that can be used for good or for ill (Bowker and Star, 2000;
Gitelman, 2013; Porter, 1995). Indeed, anti-maskers often reveal themselves to
be more sophisticated in their understanding of how scientific knowledge is
socially constructed than their ideological adversaries, who espouse naive
realism about the “objective” truth of public health data. Quantitative data
is culturally and historically situated; the manner in which it is collected,
analyzed, and interpreted reflects a deeper narrative that is bolstered by the
collective effervescence found within social media communities. Put
differently, there is no such thing as dispassionate or objective data
analysis. Instead, there are stories: stories shaped by cultural logics,
animated by personal experience, and entrenched by collective action. This
story is about how a public health crisis—refracted through seemingly
objective numbers and data visualizations—is part of a broader battleground
about scientific epistemology and democracy in modern American life.
## 2\. Related Work
### 2.1. Data and visualization literacies
There is a robust literature in computer science on data and visualization
literacy, where the latter often refers to the ability of a person “to
comprehend and interpret graphs” (Lee et al., 2017) as well as the ability to
create visualizations from scratch (Börner et al., 2019). Research in this
area often includes devising methods to assess this form of literacy (Lengler,
2006; Boy et al., 2014; Alper et al., 2017), to investigate how people
(mis)understand visualizations (Börner et al., 2016; Parsons, 2018; Calero
Valdez et al., 2018), or to create systems that help a user improve their
understanding of unfamiliar visualizations (Stoiber et al., 2019; Ruchikachorn
and Mueller, 2015; Tanahashi et al., 2016; Alberda, 2020; Adar and Lee, 2020;
Börner et al., 2019). Evan Peck et al. (Peck et al., 2019) have responded to
this literature by showing how a “complex tapestry of motivations,
preferences, and beliefs [impact] the way that participants [prioritize] data
visualizations,” suggesting that researchers need to better understand users’
social and political context in order to design visualizations that speak
powerfully to their personal experience.
Linguistic anthropologists have shown that “literacy” is not just the ability
to encode and decode written messages. The skills related to reading and
writing are historically embedded, and take on different meanings in a given
social context depending on who has access to them, and what people think they
should and shouldn’t be used for. As a consequence of such contingencies,
these scholars view literacy as multiple rather than singular, attending to
the impact of local circumstances on they way members of any given community
practice literacy and construe its value (Ahearn, 2004). Thus, media literacy,
is not simply about understanding information, but about being able to
actively leverage it in locally relevant social interactions (Hobbs, 2016).
Building on these traditions, we do not normatively assess anti-maskers’
visualization practices against a prescriptivist model of what literacy
_should_ be (according to, say, experts in human-computer interaction), but
rather seek to describe what those practices actually look like in locally
relevant contexts.
As David Buckingham (Buckingham, 2017) has noted, calls for increased literacy
have often become a form of wrong-headed solutionism that posits education as
the fix to all media-related problems. danah boyd (boyd, 2018) has documented,
too, that calling for increased media literacy can often backfire: the
instruction to “question more” can lead to a weaponization of critical
thinking and increased distrust of media and government institutions. She
argues that calls for media literacy can often frame problems like fake news
as ones of personal responsibility rather than a crisis of collective action.
Similarly, Francesca Tripodi (Tripodi, [n.d.]) has shown how evangelical
voters do not vote for Trump because they have been “fooled” by fake news, but
because they privilege the personal study of primary sources and have found
logical inconsistencies not in Trump’s words, but in mainstream media
portrayals of the president. As such, Tripodi argues, media literacy is not a
means of fighting “alternative facts.” Christopher Bail et al. (Bail et al.,
2018) have also shown how being exposed to more opposing views can actually
increase political polarization.
Finally, in his study of how climate skeptics interpret scientific studies,
Frank Fischer (Fischer, 2019) argues that increasing fact-checking or levels
of scientific literacy is insufficient for fighting alternative facts. “While
fact checking is a worthy activity,” he says, “we need to look deeper into
this phenomenon to find out what it is about, what is behind it.” Further
qualitative studies that investigate how ideas are culturally and historically
situated, as the discussion around COVID datasets and visualizations are
manifestations of deeper political questions about the role of science in
public life.
### 2.2. Critical approaches to visualization
Historians, anthropologists, and geographers have long shown how
visualizations—far from an objective representation of knowledge—are often, in
fact, representations of power (Harley, 1989; Teng, 2004; Kitchin et al.,
2011; Mueggler, 2011). To address this in practice, feminist cartographers
have developed quantitative GIS methods to describe and analyze differences
across race, gender, class, and space, and these insights are then used to
inform policymaking and political advocacy (Hanson, 1992; McLafferty, 1995;
Kwan, 2002). Best practices in data visualization have often emphasized
reflexivity as a way to counter the power dynamics and systemic inequalities
that are often inscribed in data science and design (D’Ignazio and Klein,
2020; Costanza-Chock, 2020; Correll, 2019). Central to this practice is
articulating what is excluded from the data (Onuoha, 2016; Buolamwini and
Gebru, 2018; Noble, 2018), understanding how data reflect situated knowledges
rather than objective truth (Haraway, 1988; Christin, 2016; So and Duarte,
2020), and creating alternative methods of analyzing and presenting data based
on anti-oppressive practices (Kelly, 2019; Anti-Eviction Mapping Project,
2019; Data for Black Lives, 2020). Researchers have also shown how
interpreting data visualizations is a fundamentally social, narrative-driven
endeavor (Hullman and Diakopoulos, 2011; Peck et al., 2019; Dumit and Burri,
2008). By focusing on a user’s contextual experience and the communicative
dimensions of a visualization, computer scientists have destabilized a more
traditional focus on improving the technical components of data visualization
towards understanding how users interpret and use them (Viegas and Wattenberg,
2006; Maltese et al., 2015; Lee et al., 2016).
Critical, reflexive studies of data visualization are undoubtedly crucial for
dismantling computational systems that exacerbate existing social
inequalities. Research on COVID visualizations is already underway: Emily Bowe
et al. (Bowe et al., 2020) have shown how visualizations reflect the unfolding
of the pandemic at different scales; Alexander Campolo (Campolo, 2020) has
documented how the pandemic has produced new forms of visual knowledge, and
Yixuan Zhang et al. (Zhang et al., 2020) have mapped the landscape of COVID-
related crisis visualizations. This paper builds on these approaches to
investigate the epistemological crisis that leads some people to conclude that
mask-wearing is a crucial public health measure and for others to reject it
completely.
Like data feminists, anti-mask groups similarly identify problems of political
power within datasets that are released (or otherwise withheld) by the US
government. Indeed, they contend that the way COVID data is currently being
collected is non-neutral, and they seek liberation from what they see as an
increasingly authoritarian state that weaponizes science to exacerbate
persistent and asymmetric power relations. This paper shows that more critical
approaches to visualization are necessary, and that the frameworks used by
these researchers (e.g., critical race theory, gender analysis, and social
studies of science) are crucial to disentangling how anti-mask groups mobilize
visualizations politically to achieve powerful and often horrifying ends.
## 3\. Methods
This paper pairs a quantitative approach to analyzing Twitter data (computer
vision and network analysis) with a qualitative approach to examining Facebook
Groups (digital ethnography). Drawing from social media scholarship that uses
mixed-methods approaches to examine how users interact with one another
(Burgess and Matamoros-Fernández, 2016; Moats and Borra, 2018; Kiesling et
al., 2018; Berriche and Altay, 2020; Arif et al., 2018), this paper engages
with work critical of quantitative social media methods (Baym, 2013; Wu and
Taneja, 2020) by demonstrating how interpretive analyses of social media
discussions and computational techniques can be mutually re-enforcing. In
particular, we leverage quantitative studies of social media that use network
analysis to understand political polarization (Arif et al., 2018), qualitative
analysis of comments to identify changes in online dialogue over time (Yardi
and boyd, 2010), and visualization research that reverse-engineers and
classifies chart images (Yardi and boyd, 2010; Savva et al., 2011).
### 3.1. Twitter data and quantitative analysis
#### 3.1.1. Dataset
This analysis is conducted using a dataset of tweet IDs that Emily Chen et al.
(Chen et al., 2020) assembled by monitoring the Twitter streaming API for
certain keywords associated with COVID-19 (e.g., “coronavirus”, “pandemic”,
“lockdown”, etc.) as well as by following the accounts of public health
institutions (e.g., @CDCGov, @WHO, etc). We used version 2.7 of this dataset,
which included over 390M tweets spanning January 21, 2020–July 31, 2020. This
dataset consists of tweet IDs which we “hydrated” using twarc (Documenting the
Now, 2016) into full tweets with associated metadata (e.g., hashtags,
mentions, replies, favorites, retweets, etc.).
To identify tweets that primarily discuss data visualizations about the
pandemic, we initially adopted a strategy that filtered for tweets that
contained at least one image and keyword associated with data analysis (e.g.,
“bar,” “line,” but also “trend,” “data”). Unfortunately, this strategy yielded
more noise than signal as most images in the resultant dataset were memes and
photographs. We therefore adopted a more conservative approach, filtering for
tweets that explicitly mentioned chart-related keywords (i.e., “chart(s)”,
“plot(s)”, “map(s)”, “dashboard(s)”, “vis”, “viz”, or “visualization(s)”).
This process yielded a dataset of almost 500,000 tweets that incorporated over
41,000 images. We loaded the tweets and their associated metadata into a
SQLite database, and the images were downloaded and stored on the file system.
#### 3.1.2. Image classification
To analyze the types of visualization found in the dataset, we began by
classifying every image in our corpus using the mark classification computer
vision model trained by Poco and Heer (Poco and Heer, 2017). Unfortunately,
this model was only able to classify 30% of the images. As a result, we
extracted a 4096-dimensional feature embedding for every image, and ran
k-means clustering on a 100-dimensionally reduced space of these embeddings,
for steps of k from 5–40. Two authors manually inspected the outputs of these
runs, independently identified the most salient clusters, and then cross
validated their analysis to assemble a final list of 8 relevant clusters: line
charts, area charts, bar charts, pie charts, tables, maps, dashboards, and
images. For dimensionality reduction and visualization, we used the UMAP
algorithm (McInnes et al., 2018) and iteratively arrived at the following
parameter settings: 20 neighbors, a minimum distance of 0.01, and using the
cosine distance metric. To account for UMAP’s stochasticity, we executed 10
runs and qualitatively examined the output to ensure our analyses were not
based on any resultant artifacts.
#### 3.1.3. Network analysis
Finally, to analyze the users participating in these discussions, we
constructed a network graph: nodes were users who appeared in our dataset, and
edges were drawn between pairs of nodes if a user mentioned, replied to,
retweeted, or quote-tweeted another user. The resultant graph consisted of
almost 400,000 nodes and over 583,000 edges, with an average degree of 2.9. To
produce a denser network structure, we calculated a histogram of node degrees
and identified that two-thirds of the nodes were degree 1. We then computed
subgraphs, filtering for nodes with a minimum degree of 2 through 10 and found
that degree 5 offered us a good balance between focusing on the most
influential actors in the network, without losing smaller yet salient
communities. This step yielded a subgraph of over 28,000 nodes and 104,000
edges, with an average degree of 7.3. We detected communities on this network
using the Louvain method (Blondel et al., 2008). Of the 2,573 different
communities detected by this algorithm, we primarily focus on the top 10
largest communities which account for 72% of nodes, 80% of edges, and 30% of
visualizations.
### 3.2. Facebook data and qualitative analysis
#### 3.2.1. Digital ethnography
While qualitative research can involve clinical protocols like interviews or
surveys, Clifford Geertz (Geertz, 1998) argues that the most substantial
ethnographic insights into the cultural life of a community come from “deep
hanging out,” i.e., long-term, participant observation alongside its members.
Using “lurking,” a mode of participating by observing specific to digital
platforms, we propose “deep lurking” as a way of systematically documenting
the cultural practices of online communities. Our methods here rely on robust
methodological literature in digital ethnography (Markham, 2013; Coleman,
2014), and we employ a case study approach (Small, 2009) to analyze these
Facebook groups. To that end, we followed five Facebook groups (each with a
wide range of followers, 10K-300K) over the first six months of the
coronavirus pandemic, and we collected posts throughout the platform that
included terms for “coronavirus” and “visualization” with Facebook’s
CrowdTangle tool (CrowdTangle Team, 2020). In our deep lurking, we archived
web pages and took field notes on the following: posts (regardless of whether
or not they included “coronavirus” and “data”), subsequent comments, Facebook
Live streams, and photos of in-person events. We collected and analyzed posts
from these groups from their earliest date to September 2020.
Taking a case study approach to the interactional Facebook data yields an
analysis that ultimately complements the quantitative analysis. While the
objective with analyzing Twitter data is statistical representativeness—we
investigate which visualizations are the most popular, and in which
communities—the objective of analyzing granular Facebook data is to accurately
understand social dynamics within a singular community (Small, 2009). As such,
the Twitter and Facebook analyses are foils of one another: we have the
ability to quantitatively analyze large-scale interactions on Twitter, whereas
we analyze the Facebook data by close reading and attending to specific
context. Twitter communities are loosely formed by users retweeting, liking,
or mentioning one another; Facebook groups create clearly bounded
relationships between specific communities. By matching the affordances of
each data source with the most ecologically appropriate method (network
analysis and digital ethnography), this paper meaningfully combines
qualitative and quantitative methods to understand data visualizations about
the pandemic on a deeply contextual level and at scale.
#### 3.2.2. Data collection & analysis
Concretely, we printed out posts as PDFs, tagged them with qualitative
analysis software, and synthesized themes across these comments using grounded
theory (Kozinets, 2019). Grounded theory is an inductive method where
researchers collect data and tag it by identifying analytically pertinent
themes. Researchers then group these codes into higher-level concepts. As
Kathy Charmaz (Charmaz, 2006) writes: these “methods consist of systematic,
yet flexible guidelines for collecting and analyzing qualitative data to
construct theories ‘grounded’ in the data themselves,” and these methods have
since been adapted for social media analysis (Postill and Pink, 2012). While
this flexibility allows this method to respond dynamically to changing
empirical phenomena, it can also lead to ambiguity about how new data fit with
previously identified patterns. Digital ethnography also requires a longer
time horizon than quantitative work in order to generate meaningful insights
and, on its own, does not lead immediately to quantifiable results. These
limitations are a major reason to use both qualitative and quantitative
approaches. Following Emerson et al. (Emerson et al., 2011), we employ an
integrative strategy that weaves together “exemplars” from qualitative data
alongside our interpretations. We have redacted the names of individual users
and the Facebook groups we have studied, but we have preserved the dates and
other metadata of each post within the article where possible.
#### 3.2.3. Note on terminology
Throughout this study, we use the term “anti-mask” as a synecdoche for a broad
spectrum of beliefs: that the pandemic is exaggerated, schools should be
reopening, etc. While groups who hold these beliefs are certainly
heterogeneous, the mask is a common flashpoint throughout the ethnographic
data, and they use the term “maskers” to describe people who are driven by
fear. They are “anti-mask” by juxtaposition. This study therefore takes an
emic (i.e. “insider”) approach to analyzing how members of these groups think,
talk, and interact with one another, which starts by using terms that these
community members would use to describe themselves. There is a temptation in
studies of this nature to describe these groups as “anti-science,” but this
would make it completely impossible for us to meaningfully investigate this
article’s central question: understanding what these groups mean when they say
“science.”
## 4\. Case Study
In the Twitter analysis, we quantitatively examine a corpus of tweets that use
data visualizations to discuss the pandemic, and we create a UMAP
visualization (figure 1) that identifies the types of visualizations that
proliferate on Twitter. Then, we create a network graph (figure 2) of the
users who share and interact with these data visualizations; the edges that
link users in a network together are retweets, likes, mentions. We discover
that the fourth largest network in our data consists of users promulgating
heterodox scientific positions about the pandemic (i.e., anti-maskers). By
comparing the visualizations shared within anti-mask and mainstream networks,
we discover that there is no significant difference in the kinds of
visualizations that the communities on Twitter are using to make drastically
different arguments about coronavirus (figure 3). Anti-maskers (the community
with the highest percentage of verified users) also share the second-highest
number of charts across the top six communities (table 1), are the most
prolific producers of area/line charts, and share the fewest number of photos
(memes and images of politicians; see figure 3). Anti-maskers are also the
most likely to amplify messages from their own community. We then examine the
kinds of visualizations that anti-maskers discuss (figure 4).
This leads us to an interpretive question that animates the Facebook analysis:
how can opposing groups of people use similar methods of visualization and
reach such different interpretations of the data? We approach this problem by
ethnographically studying interactions within a community of anti-maskers on
Facebook to better understand their practices of knowledge-making and data
analysis, and we show how these discussions exemplify a fundamental
epistemological rift about how knowledge about the coronavirus pandemic should
be made, interpreted, and shared.
### 4.1. Visualization design and network analysis
Figure 1. A UMAP visualization of feature embeddings of media found in our
Twitter corpus. Color encodes labeled clusters, and size encodes the amount of
engagement the media received (i.e., the sum of replies, favorites, retweets,
and quote tweets).
Scatterplot of the different types of visualizations that are present within
the corpus labeled with their types (dashboards, images, choropleth/symbol
maps, tables/screenshots, bar charts, area charts, line charts). Each cluster
has a callout with a tweet and attached visualization. For bar charts: tweet
from July 12, 2020 which includes a bar chart about coronavirus cases per 10k
people separated by race: “Breaking: This chart. Devastating. Do everything
you can to prevent the spread. From @nytimes who sued @cdc to get this data.”
For line charts, we have two tweets. The first line chart from May 2, 2020
includes a Financial Times line chart with multiple countries displaying that
daily death tolls are at their peak or falling in many Western countries. The
second line chart from June 26, 2020 is on daily confirmed COVID-19 deaths per
million, rolling 7-day average: “The COVID-19 death rate is steadily in
decline, as you see in this chart! Do not be taken by fear and paranoia.“ For
dashboards: tweet from July 17, 2020 with two choropleth maps of Georgia side-
by-side, one from July 2 and the other from July 17 with cases per 100k. “In
just 15 days the total number of #COVID19 cases in Georgia is up 49%, but you
wouldn’t know it from looking at the state’s data visualization map of cases.”
For images: a tweet from February 29, 2020 with an image of White House
officials with their heads bowed and looking solemn in a room: “I think I’d
rather see the coronavirus team in a room with a lot of screens and maps and
what not. This looks like the Black Plague team meet up.” For choropleth and
symbol maps: a tweet from July 1, 2020 with a map of central and northern
England COVID-19 infection rates: “This map (from @FT) shows the progress
we’ve made in Scotland against COVID. But we mustn’t drop our guard.” There
are no callout tweets for the visualization types of area charts and
tables/screenshots.
#### 4.1.1. Visualization types
What kinds of visualizations are Twitter users sharing about the pandemic?
Figure 1 visualizes the feature embeddings of images in our corpus, with color
encoding clusters revealed and manually curated through k-means. Each circle
is sized by the engagement the associated tweet received calculated as the sum
of the number of favorites, replies, retweets, and quote tweets. Our analysis
revealed eight major clusters: line charts (8908 visualizations, 21% of the
corpus), area charts (2212, 5%), bar charts (3939, 9%), pie charts (1120, 3%),
tables (4496, 11%), maps (5182, 13%), dashboards (2472, 6%), and images
(7,128, 17%). The remaining 6,248 media (15% of the corpus) did not cluster in
thematically coherent ways. Here, we characterize salient elements and trends
in these clusters.
Line charts represent the largest cluster of visualizations in our corpus.
There are three major substructures: the first comprises line charts depicting
the exponential growth of cases in the early stages of the pandemic, and
predominantly use log-scales rather than linear scales. Charts from John Burn-
Murdoch at the Financial Times and charts from the nonprofit Our World in Data
are particularly prominent here. A second substructure consists of line charts
comparing cases in the United States and the European Union when the US was
experiencing its second wave of cases, and the third consists of line charts
that visualize economic information. This substructure includes line charts of
housing prices, jobs and unemployment, and stock prices (the latter appear to
be taken from financial applications and terminals, and often feature
additional candlestick marks). Across this cluster, these charts typically
depict national or supranational data, include multiple series, and very
rarely feature legends or textual annotations (other than labels for each
line). Where they do occur, it is to label every point along the lines.
Features of the graph are visually highlighted by giving some lines a heavier
weight or graying other ones out.
Maps are the second largest cluster of visualizations in our corpus. The
overwhelming majority of charts here are choropleths (shaded maps where a
geographic region with high COVID rates might be darker, while low-rate
regions are lighter). Other visualizations in this cluster include cartograms
(the size of a geographic region is proportional its number of COVID
infections as a method of comparison) and symbol maps (the size of a circle
placed on a geographic region is proportional to COVID infections). The data
for these charts span several geographic scales—global trends, country-level
data (the US, China, and the UK being particularly salient), and municipal
data (states and counties). These maps generally feature heavy annotation
including direct labeling of geographic regions with the name and associated
data value; arrows and callout boxes also better contextualize the data. For
instance, in a widely shared map of the United Kingdom from the Financial
Times, annotations described how “[t]hree Welsh areas had outbreaks in
meatpacking plants in June” and that “Leicester, which is currently in an
enforced local lockdown, has the second-highest rate…” These maps depict a
wide range of data values including numbers of cases/deaths, metrics
normalized per capita, rate of change for cases and/or deaths, mask adherence
rates, and the effect of the pandemic on greenhouse gas emissions.
Interestingly, choropleth maps of the United States electoral college at both
the state- and district-level also appear in the corpus, with the associated
tweets comparing the winner of particular regions with the type of pandemic
response.
Area charts feature much heavier annotation than line charts (though fewer
than maps). Peaks, troughs, and key events (e.g., when lockdowns occurred or
when states reopened) are often shaded or labeled with arrows, and line marks
are layered to highlight the overall trend or depict the rolling average. When
these charts reflect data with a geographic correspondence, this data is often
at a more local scale; line charts typically depict national or supranational
data, and area charts more often visualized data at the state or county level.
Notable subclusters in this group include the viral “Flatten the Curve”
graphic, stacked area/streamgraphs, and “skinny bar” charts (charts of
temporal data that closely resemble area charts, but use bar marks with narrow
widths. Charts from the New York Times are especially prominent examples of
the latter category—particularly screenshots of a red chart that was featured
on the mobile front page.
Bar charts are predominantly encode categorical data and are more consistently
and more heavily annotated than area charts. In addition to the annotations
described for area charts (direct labeling of the tops of bars, labeled lines
and arrows), charts in this cluster often include concise explainer texts.
These texts include some form of extended subtitles, more descriptive axis
tick labels, or short passages before or after the bar chart that
contextualize the data. Visually, the cluster is equally split between
horizontal and vertical charts, and both styles feature a mix of layered,
grouped, and stacked bars. Bar chart “races” (e.g., those developed with the
Flourish visualization package) are one of the more frequently recurring
idioms in this cluster. These are horizontal bar charts depicting the total
number of cases per country, and animated over time.
Dashboards and images. While the remaining clusters are thematically coherent,
we did not observe as rich a substructure within them. The dashboard cluster
is overwhelmingly dominated by screenshots of the Johns Hopkins dashboard, and
the image cluster is primarily comprised of reaction memes featuring the
photos or caricatures of heads of state.
Figure 2. A network visualization of Twitter users appearing in our corpus.
Color encodes community as detected by the Louvain method (Blondel et al.,
2008), and nodes are sized by their degree of connectedness (i.e., the number
of other users they are connected to).
A network visualization of the Twitter corpus, with colorings for clusters and
labels for the most prominent Twitter users. The teal cluster has users
@EthicalSkeptic and @AlexBerenson. The orange cluster has @POTUS, @CNN, and
@realDonaldTrump. The blue cluster has @JoeBiden, @GeoRebekah, @JohnsHopkins,
@bopinion, @Rshooley, @SethAbramson, @stengel, and @alexnazaryan. The green
cluster has @nytimes, @ASlavitt, @cdc, and @DrEricDing. The red cluster has
@FT, @jburnmurdoch,@ NicolaSturgeon, @devisridhar, @MaxCRoser, @guardian. The
purple cluster has @WHO.
Table 1. Descriptive Statistics of Communities Community # | Verified Users as % of Total Users | In-Network Retweets as % of Total Retweets | Original Tweets as % of Total Tweets
---|---|---|---
1 | 8.39 | 73.30 | 22.12
2 | 14.36 | 75.45 | 44.75
3 | 22.92 | 89.32 | 34.00
4 | 10.56 | 82.17 | 37.12
5 | 12.33 | 58.29 | 21.57
6 | 8.94 | 70.97 | 37.46
Figure 3. Visualizing the distribution of chart types by network community
(with top accounts listed). While every community has produced at least one
viral tweet, anti-mask users (group 6) receive higher engagement on average.
A display of six scatter plots, each one representing a specific cluster of
users and the visualizations that they employed. There are two rows and three
columns. For the first row: the first one has a title of American politics and
media (blue) with highlighted users Johns Hopkins, Joe Biden, Rebekah Jones.
In the bottom left, there are metrics of the network. Nodes: 3,828 (13.47%),
charts: 648 (5.31%), average engagement: 131. The second one has a title of
American politics and right-wing media (red) with highlighted users Donald
Trump, CNN, and Washington Post. In the bottom left, there are metrics of the
network. Nodes: 2,896 (10.19%), charts: 1,916 (15.17%), average engagement:
18. The third one has a title of British news media (orange) with highlighted
users John Murdoch, Financial Times, and Nicola Sturgeon. In the bottom left,
there are metrics of the network. Nodes: 277 (9.5%), charts: 1,385 (11.36%),
average engagement: 94. For the second row: the fourth one has a title of
Anti-mask network (teal) with highlighted users Alex Berenson and Ethical
Skeptic. In the bottom left, there are metrics of the network. Nodes: 2,596
(9.04%), charts: 1,799 (14.75%), average engagement: 65. The fifth one has a
title of New York Times-centric network (green) with highlighted users Andy
Slavitt, New York Times, and CDC. In the bottom left, there are metrics of the
network. Nodes: 1,885 (6.63%), charts: 1,119 (9.17%), average engagement: 41.
The sixth one has a title of World Health Organization and health-related news
(purple) with highlighted users WHO, BNO News, and Helen Branswell. In the
bottom left, there are metrics of the network. Nodes: 1,484 (5.22%), charts:
1,474 (12.08%), average engagement: 34. To the right of all of the scatter
plots, there’s a legend indicating engagement sizes and visualization colors.
Figure 4. Sample counter-visualizations from the anti-mask user network. While
there are meme-based visualizations, anti-maskers on Twitter adopt the same
visual vocabulary as visualization experts and the mainstream media.
Scatterplot of the different types of visualization within the anti-mask
network, each labeled by type (area charts, bar charts, images, line charts,
table/screenshots, choropleth/symbol maps). Each cluster has a callout of a
tweet and attached visualization. For bar chart: tweet from July 12, 2020 that
compares COVID-19 deaths and non COVID deaths by age. For area charts, we have
two tweets. The first tweet is from July 7, 2020 and displays daily COVID-19
deaths, COVID-19 hospitalizations, and deaths. The second tweet from July 31,
2020 has a chart of deaths per day from March to July: “COVID-19 update: Check
out Sweden’s actual day of death chart. No lockdowns. No masks. We are all
being taken for an absolute ride. There is precisely zero evidence that masks
and/or lockdowns have had any benefit worldwide.” For images: tweet from July
19, 2020 with chart displaying COVID fatalities per million per state uses the
faces of the governors of the respective states and their relative scaled
sizes to emphasize fatalities. For choropleth/symbol maps, there are two
tweets. The first one from July 22, 2020 displays column charts throughout the
months of quarantine. The second tweet from July 27, 2020 has a chart of USA
COVID numbers as of July 19, 2020 with a highlight on the Pacific Northwest.
For tables/screenshots: tweet from June 20, 2020 with a table breaking down
COVID-related deaths and total deaths: “Another great chart that puts covid
death risk by age…proportion to other causes. College kids are more likely to
die driving to campus for workouts than they are from the coronavirus.” For
line chart: tweet from June 26, 2020 with a screenshot of a line graph of
daily confirmed COVID-19 deaths per million, rolling 7-day average: “The
COVID-19 death rate is steadily in decline, as you see in this chart! Do not
be taken by fear and paranoia.”
#### 4.1.2. User networks
What are the different networks of Twitter users who share COVID-related data
visualizations, and how do they interact with one another? Figure 2 depicts a
network graph of Twitter users who discuss (or are discussed) in conversation
with the visualizations in Figure 1. This network graph only shows users who
are connected to at least five other users (i.e., by replying to them,
mentioning them in a tweet, or re-tweeting or quote-tweeting them). The color
of each network encodes a specific community as detected by the Louvain method
(Blondel et al., 2008), and the graph accounts for the top 10 communities
(20,500 users or 72% of the overall graph). We describe the top six networks
below listed in order of size (i.e., number of users within each network).
While we have designated many of these communities with political orientation
(e.g., left- or right-wing), these are only approximations; we recognize that
these terms are fundamentally fluid and use them primarily as shorthand to
make cross-network comparisons (e.g., mainstream political/media organizations
vs. anti-mask protestors).
1\. American politics and media (blue). This community features the American
center-left, left, mainstream media, and popular or high profile figures
(inside and outside of the scientific community). Accounts include politicians
(@JoeBiden, @SenWarren), reporters (@joshtpm, @stengel), and public figures
like Floridian data scientist @GeoRebekah and actor @GeorgeTakei. The user
with the most followers in this community is @neiltyson.
2\. American politics and right-wing media (red). This community includes
members of the Trump administration, Congress, and right-wing personalities
(e.g., @TuckerCarlson). Several accounts of mainstream media organizations
also lie within this community (@CNN, @NBCNews), which reflects how often they
mention the President (and other government accounts) in their coverage.
Notably, these are official organizational accounts rather than those of
individual reporters (which mostly show up in the previous group). Several
mainstream media organizations are placed equally between these two clusters
(@NPR, @washingtonpost). The user with the most followers in this community is
@BarackObama.
3\. British news media (orange). The largest non-Americentric network roughly
corresponds to news media in the UK, with a significant proportion of
engagement targeted at the _Financial Times’_ successful visualizations by
reporter John Burn-Murdoch, as well as coverage of politician Nicola
Sturgeon’s coronavirus policies. The user with the most followers in this
community is @TheEconomist.
4\. Anti-mask network (teal). The anti-mask network comprises over 2,500 users
(9% of our network graph) and is anchored by former _New York Times_ reporter
@AlexBerenson, blogger @EthicalSkeptic, and @justin_hart. _The Atlantic_ ’s
@Covid19Tracking project (which collates COVID-19 testing rates and patient
outcomes across the United States) and @GovMikeDeWine are also classified as
part of this community. Governor DeWine of Ohio is not an anti-masker, but is
often the target of anti-mask protest given his public health policies. Anti-
mask users also lampoon _The Atlantic_ ’s project as another example of
mainstream misinformation. These dynamics of intertextuality and citation
within these networks are especially important here, as anti-mask groups often
post screenshots of graphs from “lamestream media” organizations (e.g., _New
York Times_) for the purpose of critique and analysis. The user with the most
followers in this community is COVID skeptic and billionaire @elonmusk.
5\. New York Times-centric network (green). This community is largely an
artifact of a single visualization: Andy Slavitt (@ASlavitt), the former
acting Administrator of the Centers for Medicare and Medicaid Services under
the Obama administration, posted a viral tweet announcing the _New York Times_
had sued the CDC (tagged with the incorrect handle @cdc instead of @CDCGov).
The attached bar chart showing the racial disparity in COVID cases was shared
widely with commentary directly annotated onto the graph itself, or users
analyzed the graph through quote-tweets and comments. The user with the most
followers in this community is @NYTimes.
6\. World Health Organization and health-related news organizations (purple).
This community consists of global health organizations, particularly the @WHO
and its subsidiary accounts (e.g., @WHOSEARO for Southeast Asia). The user
with the most followers in this community is @YouTube.
#### 4.1.3. Descriptive statistics of communities
Table 1 lists summary statistics for the six largest communities in our
dataset. There are three statistics of interest: the percentage of verified
users (based on the total number of users within a community), the percentage
of in-network retweets (based on a community’s total number of retweets), and
the percentage of original tweets (based on a community’s total number of
tweets). Twitter verification can often indicate that the account of public
interest is authentic (subject to Twitter’s black-boxed evaluation standards);
it can be a reasonable indication that the account is not a bot. Secondly, a
high percentage of in-network retweets can be an indicator of how insular a
particular network can be, as it shows how often a user amplifies messages
from other in-network members. Finally, the percentage of original tweets
shows how much of the content in that particular community is organic (i.e.,
they write their own content rather than simply amplifying existing work).
Communities that have users who use the platform more passively (i.e., they
prefer to lurk rather than comment) will have fewer original tweets;
communities that have higher levels of active participation will have a higher
number of original tweets as a percentage of total tweets.
The networks with the highest number of in-network retweets (which can be one
proxy for insularity) are the British media (89.32%) and the anti-mask
networks (82.17%), and the network with the highest percentage of original
tweets is the American politics and right-wing media network (44.75%).
Notably, the British news media network has both the highest percentage of
verified users (22.92%), the highest percentage of in-network retweets
(89.32%), and the fourth-highest percentage of original tweets (34.00%). As
the third largest community in our dataset, we attribute this largely to the
popularity of the graphs from the Financial Times from a few sources and the
constellation of accounts that discussed those visualizations. While other
communities (anti-mask, American politics/right-wing media, and WHO/health-
related news) shared _more_ visualizations, this network shared fewer graphs
(1,385) that showed the _second-highest_ level of engagement across the six
communities (averaging 94 likes, retweets, or mentions per visualization). The
network whose visualizations garner the highest level of engagement is the
American politics and media network (131 likes, retweets, or mentions per
visualization), but they only shared about half of the visualizations (648)
compared to their British counterparts.
Through the descriptive statistics, we find that the anti-mask community
exhibits very similar patterns to the rest of the networks in our dataset (it
has about the same number of users with the same proportion of verified
accounts). However, this community has the second highest percentage of in-
network retweets (82.17% of all retweets) across the communities, and has the
third-highest percentage of original tweets (37.12%, only trailing the World
Health Organization network, at 37.46%).
#### 4.1.4. Cross-network comparison of visualization types
Figure 3 depicts the distribution of visualization types by each community,
along with descriptive statistics on the numbers of users, charts, and average
engagement per tweet. These scatterplots show that there is little variance
between the types of visualizations that users in each network share: almost
all groups equally use maps or line, area, and bar charts. However, each group
usually has one viral visualization—in group 3 (British media), the large
yellow circle represents a map from the _Financial Times_ describing COVID-19
infections in Scotland; in group 5 (_New York Times_), the large purple circle
in the center of the chart represents the viral bar chart from Andy Slavitt
describing the racial disparities in COVID cases. The visualizations with the
highest number of engagements in each of the six communities is depicted in
figure 1.
Overall, we see that each group usually has one viral hit, but that the anti-
mask users (group 4) tend to share a wide range of visualizations that garner
medium levels of engagement (they have the third-highest number of average
engagements in the six communities; an average of 65 likes, shares, and
retweets). As a percentage of total tweets, anti-maskers have shared the
second-highest number of charts across the top six communities (1,799 charts
or 14.75%). They also use the _most_ area/line charts and the _least_ images
across the six communities (images in this dataset usually include memes or
photos of politicians). These statistics suggest that anti-maskers tend to be
among the most prolific sharers of data visualizations on Twitter, and that
they overwhelmingly amplify these visualizations to other users within their
network (88.97% of all retweets are in-network).
#### 4.1.5. Anti-mask visualizations
Figure 4 depicts the data visualizations that are shared by members of the
anti-mask network accompanied by a select tweets from each category. While
there are certainly visualizations that tend to use a meme-based approach to
make their point (e.g., “Hey Fauci…childproof chart!” with the heads of
governors used to show the rate of COVID fatalities), many of the
visualizations shared by anti-mask Twitter users employ visual forms that are
relatively similar to charts that one might encounter at a scientific
conference. Many of these tweets use area and line charts to show the
discrepancy between the number of projected deaths in previous epidemiological
and the numbers of actual fatalities. Others use unit visualizations, tables,
and bar charts to compare the severity of coronavirus to the flu. In total,
this figure shows the breadth of visualization types that anti-mask users
employ to illustrate that the pandemic is exaggerated.
### 4.2. Anti-mask discourse analysis
The Twitter analysis establishes that anti-maskers are prolific producers and
consumers of data visualizations, and that the graphs that they employ are
similar to those found in orthodox narratives about the pandemic. Put
differently, anti-maskers use “data-driven” narratives to justify their
heterodox beliefs. However, a quantitative overview of the visualizations they
share and amplify does not in itself help us understand _how_ anti-maskers
invoke data and scientific reasoning to support policies like re-opening
schools and businesses. Anti-maskers are acutely aware that mainstream
narratives use data to underscore the pandemic’s urgency; they believe that
these data sources and visualizations are fundamentally flawed and seek to
counteract these biases. This section showcases the different ways that anti-
mask groups talk about COVID-related data in discussion forums. What kinds of
concerns do they have about the data used to formulate public policies? How do
they talk about the limitations of data or create visualizations to convince
other members in their physical communities that the pandemic is a hoax?
#### 4.2.1. Emphasis on original content
Many anti-mask users express mistrust for academic and journalistic accounts
of the pandemic, proposing to rectify alleged bias by “following the data” and
creating their own data visualizations. Indeed, one Facebook group within this
study has very strict moderation guidelines that prohibit the sharing of non-
original content so that discussions can be “guided solely by the data.” Some
group administrators even impose news consumption bans on themselves so that
“mainstream” models do not “cloud their analysis.” In other words, anti-
maskers value unmediated access to information and privilege personal research
and direct reading over “expert” interpretations. While outside content is
generally prohibited, Facebook group moderators encourage followers to make
their own graphs, which are often shared by prominent members of the group to
larger audiences (e.g., on their personal timelines or on other public facing
Pages). Particularly in cases where a group or page is led by a few prominent
users, follower-generated graphs tend to be highly popular because they often
encourage other followers to begin their own data analysis projects, and
comments on these posts often deal directly with how to reverse-engineer (or
otherwise adjust) the visualization for another locality.
Since some of these groups are focused on a single state (e.g. “Reopen
Nevada”), they can fill an information gap: not every county or locality is
represented on data dashboards made by local newspapers, health departments,
or city governments—if these government entities have dashboards or open data
portals at all. In such cases, the emphasis on original content primarily
reflects a grassroots effort to ensure access to pandemic-related data where
there are no alternatives, and only secondarily serves to constitute an
alternative to ideologically charged mainstream narratives. In the rare
instances where mainstream visualizations are shared in such a group, it is
usually to highlight the ways that mainstream analysis finally matches anti-
mask projections, or to show how a journalist, government official, or
academic can manipulate the same data source to purposefully mislead readers.
In order to create these original visualizations, users provide numerous
tutorials on how to access government health data. These tutorials come either
as written posts or as live screencasts, where a user (often a group
administrator or moderator) demonstrates the process of downloading
information from an open data portal. During the livestream, they editorialize
to show which data are the most useful (e.g., “the data you can download [from
the Georgia Health Department website] is completely worthless, but the
dashboard—which has data that everyday citizens cannot access—actually shows
that there are no deaths whatsoever,” July 13, 2020). Since many local health
departments do not have the resources to stand up a new data system
specifically for COVID-19, some redirect constituents to the state health
department, which may not have granular data for a specific township available
for public use. In the absence of data-sharing between states and local
governments, users often take it upon themselves to share data with one
another (e.g., “[redacted] brings us this set of data from Minnesota. […] Here
it is in raw form, just superimposed on the model,” May 17, 2020) and they
work together to troubleshoot problems with each dataset (e.g., “thanks.
plugging [sic] in new .csv file to death dates is frustrating but worth it,”
May 2, 2020).
#### 4.2.2. Critically assessing data sources
Even as these users learn from each other to collect more data, they remain
critical about the circumstances under which the data are collected and
distributed. Many of the users believe that the most important metrics are
missing from government-released data. They express their concerns in four
major ways. First, there is an ongoing animated debate within these groups
about which metrics matter. Some users contend that deaths, not cases, should
be the ultimate arbiter in policy decisions, since case rates are easily
“manipulated” (e.g., with increased testing) and do not necessarily signal
severe health problems (people can be asymptomatic). The shift in focus is
important, as these groups believe that the emphasis on cases and testing
often means that rates of COVID deaths by county or township are not reported
to the same extent or seriously used for policy making. As one user noted,
“The Alabama public health department doesn’t provide deaths per day data
(that I can tell—you can get it elsewhere). I sent a message asking about
that. Crickets so far,” (July 13, 2020).
Second, users also believe that state and local governments are deliberately
withholding data so that they can unilaterally make decisions about whether or
not lockdowns are effective. During a Facebook livestream with a Congressional
candidate who wanted to “use data for reopening,” for example, both the
candidate and an anti-mask group administrator discussed the extent to which
state executives were willing to obscure the underlying data that were used to
justify lockdown procedures (August 30, 2020). To illustrate this, the
candidate emphasized a press conference in which journalists asked the state
executive whether they would consider making the entire contact tracing
process public, which would include releasing the name of the bar where the
outbreak started. In response, the governor argued that while transparency
about the numbers were important, the state would not release the name of the
bar, citing the possibility of stigmatization and an erosion of privacy. This
soundbite—“we have the data, but we won’t give it to you”—later became a
rallying cry for anti-mask groups in this state following this livestream. “I
hate that they’re not being transparent in their numbers and information
they’re giving out,” another user wrote. “They need to be honest and admit
they messed up if it isn’t as bad as they’re making it out to be. […] We need
honesty and transparency.”
This plays into a third problem that users identify with the existing data:
that datasets are constructed in fundamentally subjective ways. They are
coded, cleaned, and aggregated either by government data analysts with
nefarious intentions or by organizations who may not have the resources to
provide extensive documentation. “Researchers can define their data set anyhow
[sic] they like in absence of generally accepted (preferably specified)
definitions,” one user wrote on June 23, 2020. “Coding data is a big deal—and
those definitions should be offered transparently by every state. Without a
national guideline—we are left with this mess.” The lack of transparency
within these data collection systems—which many of these users infer as a lack
of honesty—erodes these users’ trust within both government institutions and
the datasets they release.
Even when local governments do provide data, however, these users also contend
that the data requires context in order for any interpretation to be
meaningful. For example, metrics like hospitalization and infection rates are
still “vulnerable to all sorts of issues that make [these] data less reliable
than deaths data” (June 23, 2020), and require additional inquiry before the
user considers making a visualization. In fact, there are multiple threads
every week where users debate how representative the data are of the
population given the increased rate of testing across many states. For some
users, random sampling is the only way to really know the true infection rate,
as (1) testing only those who show symptoms gives us an artificially high
infection rate, and (2) testing asymptomatic people tells us what we already
know—that the virus is not a threat. These groups argue that the conflation of
asymptomatic and symptomatic cases therefore makes it difficult for anyone to
actually determine the severity of the pandemic. “We are counting ‘cases’ in
ways we never did for any other virus,” a user writes, “and we changed how we
counted in the middle of the game. It’s classic garbage in, garbage out at
this point. If it could be clawed back to ONLY symptomatic and/or contacts, it
could be a useful guide [for comparison], but I don’t see that happening”
(August 1, 2020).
Similarly, these groups often question the context behind measures like
“excess deaths.” While the CDC has provided visualizations that estimate the
number excess deaths by week (CDC, 2020), users take screenshots of the
websites and debate whether or not they can be attributed to the coronavirus.
“You can’t simply subtract the current death tally from the typical value for
this time of year and attribute the difference to Covid,” a user wrote.
“Because of the actions of our governments, we are actually causing excess
deaths. Want to kill an old person quickly? Take away their human interaction
and contact. Or force them into a rest home with other infected people. Want
people to die from preventable diseases? Scare them away from the hospitals,
and encourage them to postpone their medical screenings, checkups, and
treatments […] The numbers are clear. By trying to mitigate one problem, we
are creating too many others, at too high a price” (September 5, 2020).
#### 4.2.3. Critically assessing data representations
Even beyond downloading datasets from local health departments, users in these
groups are especially attuned to the ways that specific types of
visualizations can obscure or highlight information. In response to a
visualization where the original poster (OP) created a bar chart of death
counts by county, a user commented: “the way data is presented can also show
bias. For example in the state charts, counties with hugely different
populations can be next to each other. The smaller counties are always going
to look calm even if per capita they are doing the same or worse. Perhaps you
could do a version of the charts where the hardest hit county is normalized
per capita to 1 and compare counties that way,” to which the OP responded, “it
is never biased to show data in its entirety, full scale” (August 14, 2020).
An ongoing topic of discussion is whether to visualize absolute death counts
as opposed to deaths per capita, and it is illustrative of a broader mistrust
of mediation. For some, “raw data” (e.g., counts) provides more accurate
information than any data transformation (e.g., death rate per capita, or even
visualizations themselves). For others, screenshots of tables are the most
faithful way to represent the data, so that people can see and interpret it
for themselves. “No official graphs,” said one user. “Raw data only. Why give
them an opportunity to spin?” (June 14, 2020). These users want to understand
and analyze the information for themselves, free from biased, external
intervention.
#### 4.2.4. Identifying bias and politics in data
While users contend that their data visualizations objectively illustrate how
the pandemic is no worse than the flu, they are similarly mindful to note that
these analyses only represent partial perspectives that are subject to
individual context and interpretation. “I’ve never claimed to have no bias. Of
course I am biased, I’m human,” says one prolific producer of anti-mask data
visualizations. “That’s why scientists use controls… to protect ourselves from
our own biases. And this is one of the reasons why I disclose my biases to
you. That way you can evaluate my conclusions in context. Hopefully, by
staying close to the data, we keep the effect of bias to a minimum” (August
14, 2020). They are ultimately mindful of the subjectivity of human
interpretation, which leads them to analyzing the data for themselves.
More tangibly, however, these groups seek to identify bias by being critical
about specific profit motives that come from releasing (or suppressing)
specific kinds of information. Many of the users within these groups are
skeptical about the potential benefits of a coronavirus vaccine, and as a
point of comparison, they often reference how the tobacco industry has
historically manipulated science to mislead consumers. These groups believe
that pharmaceutical companies have similarly villainous profit motives, which
leads the industry to inflate data about the pandemic in order to stoke demand
for a vaccine. As one user lamented, “I wish more of the public would do some
research into them and see how much of a risk they are but sadly most wont
[sic]—because once you do and you see the truth on them, you get labeled as an
‘antivaxxer’ which equates to fool. In the next few years, the vaccine
industry is set to be a nearly 105 billion dollar industry. People should
really consider who profits off of our ignorance” (August 24, 2020).
#### 4.2.5. Appeals to scientific authority
Paradoxically, these groups also seek ways to validate their findings through
the scientific establishment. Many users prominently display their scientific
credentials (e.g., referring to their doctoral degrees or prominent
publications in venues like Nature) which uniquely qualify them as insiders
who are most well-equipped to criticize the scientific community. Members who
perform this kind of expertise often point to 2013 Nobel Laureate Michael
Levitt’s assertion that lockdowns do nothing to save lives (Lloyd, 2020) as
another indicator of scientific legitimacy. Both Levitt and these anti-mask
groups identify the dangerous convergence of science and politics as one of
the main barriers to a more reasonable and successful pandemic response, and
they construct their own data visualizations as a way to combat what they see
as health misinformation. “To be clear. I am not downplaying the COVID
epidemic,” said one user. “I have never denied it was real. Instead, I’ve been
modeling it since it began in Wuhan, then in Europe, etc. […] What I have done
is follow the data. I’ve learned that governments, that work for us, are too
often deliberately less than transparent when it comes to reporting about the
epidemic” (July 17, 2020). For these anti-mask users, their approach to the
pandemic is grounded in a _more_ scientific rigor, not less.
#### 4.2.6. Developing expertise and processes of critical engagement
The goal of many of these groups is ultimately to develop a network of well-
informed citizens engaged in analyzing data in order to make measured
decisions during a global pandemic. “The other side says that they use
evidence-based medicine to make decisions,” one user wrote, “but the data and
the science do not support current actions” (August 30, 2020). The discussion-
based nature of these Facebook groups also give these followers a space to
learn and adapt from others, and to develop processes of critical engagement.
Long-time followers of the group often give small tutorials to new users on
how to read and interpret specific visualizations, and users often give each
other constructive feedback on how to adjust their graphic to make it more
legible or intuitive. Some questions and comments would not be out of place at
all at a visualization research poster session: “This doesn’t make sense. What
do the colors mean? How does this demonstrate any useful information?” (July
21, 2020) These communities use data analysis as a way to socialize and
enculturate their users; they promulgate data literacy practices as a way of
inculcating heterodox ideology. The transmission of data literacy, then,
becomes a method of political radicalization.
These individuals as a whole are extremely willing to help others who have
trouble interpreting graphs with multiple forms of clarification: by helping
people find the original sources so that they can replicate the analysis
themselves, by referencing other reputable studies that come to the same
conclusions, by reminding others to remain vigilant about the limitations of
the data, and by answering questions about the implications of a specific
graph. The last point is especially salient, as it surfaces both what these
groups see as a reliable measure of how the pandemic is unfolding and what
they believe they should do with the data. These online communities therefore
act as a sounding board for thinking about how best to effectively mobilize
the data towards more measured policies like slowly reopening schools. “You
can tell which places are actually having flare-ups and which ones aren’t,”
one user writes. “Data makes us calm.” (July 21, 2020)
Additionally, followers in these groups also use data analysis as a way of
bolstering social unity and creating a community of practice. While these
groups highly value scientific expertise, they also see collective analysis of
data as a way to bring communities together within a time of crisis, and being
able to transparently and dispassionately analyze the data is crucial for
democratic governance. In fact, the explicit motivation for many of these
followers is to find information so that they can make the best decisions for
their families—and by extension, for the communities around them. “Regardless
of your political party, it is incumbent on all of us to ask our elected
officials for the data they use to make decisions,” one user said during a
live streamed discussion. “I’m speaking to you as a neighbor: request the
data. […] As a Mama Bear, I don’t care if Trump says that it’s okay, I want to
make a decision that protects my kids the most. This data is especially
important for the moms and dads who are concerned about their babies” (August
30, 2020). As Kate Starbird et al. have demonstrated, strategic information
operations require the participation of online communities to consolidate and
amplify these messages: these messages become powerful when emergent, organic
crowds (rather than hired trolls and bots) iteratively contribute to a larger
community with shared values and epistemologies (Starbird et al., 2019).
Group members repost these analyses onto their personal timelines to start
conversations with friends and family in hopes that they might be able to
congregate in person. However, many of these conversations result in
frustration. “I posted virus data from the CDC, got into discussion with
people and in the end several straight out voiced they had no interest in the
data,” one user sighed. “My post said ‘Just the facts.’ [screenshot from the
CDC] People are emotionally invested in their beliefs and won’t be swayed by
data. It’s disturbing” (August 14, 2020). Especially when these conversations
go poorly, followers solicit advice from each other about how to move forward
when their children’s schools close or when family members do not “follow the
data.” One group even organized an unmasked get-together at a local restaurant
where they passed out t-shirts promoting their Facebook group, took selfies,
and discussed a lawsuit that sought to remove their state’s emergency health
order (September 12, 2020). The lunch was organized such that the members who
wanted to first attend a Trump rolling rally could do so and “drop in
afterward for some yummy food and fellowship” (September 8, 2020).
#### 4.2.7. Applying data to real-world situations
Ultimately, anti-mask users emphasize that they need to apply this data to
real-world situations. The same group that organized the get-together also
regularly hosts live-streams with guest speakers like local politicians,
congressional candidates, and community organizers, all of whom instruct users
on how to best agitate for change armed with the data visualizations shared in
the group. “You’re a mom up the street, but you’re not powerless,” emphasized
one of the guest speakers. “Numbers matter! What is just and what is true
matters. […] Go up and down the ladder—start real local. Start with the lesser
magistrates, who are more accessible, easier to reach, who will make time for
you.” (July 23, 2020)
These groups have been incredibly effective at galvanizing a network of
engaged citizens towards concrete political action. Local officials have
relied on data narratives generated in these groups to call for a lawsuit
against the Ohio Department of Health (July 20, 2020). In Texas, a coalition
of mayors, school board members, and city council people investigated the
state’s COVID-19 statistics and discovered that a backlog of unaudited tests
was distorting the data, prompting Texas officials to employ a forensic data
team to investigate the surge in positive test rates (Carroll, 2020). “There
were over a million pending assignments [that were distorting the state’s
infection rate],” the city councilperson said to the group’s 40,000+
followers. “We just want to make sure that the information that is getting out
there is giving us the full picture.” (August 17, 2020) Another Facebook group
solicited suggestions from its followers on how to support other political
groups who need data to support lawsuits against governors and state health
departments. “If you were suddenly given access to all the government records
and could interrogate any official,” a group administrator asked, “what piece
of data or documentation would you like to inspect?” (September 11, 2020) The
message that runs through these threads is unequivocal: that data is the only
way to set fear-bound politicians straight, and using better data is a
surefire way towards creating a safer community.
## 5\. Discussion
Anti-maskers have deftly used social media to constitute a cultural and
discursive arena devoted to addressing the pandemic and its fallout through
practices of data literacy. Data literacy is a quintessential criterion for
membership within the community they have created. The prestige of both
individual anti-maskers and the larger Facebook groups to which they belong is
tied to displays of skill in accessing, interpreting, critiquing, and
visualizing data, as well as the pro-social willingness to share those skills
with other interested parties. This is a community of practice (Wenger, 1998;
Lave and Wenger, 1991) focused on acquiring and transmitting expertise, and on
translating that expertise into concrete political action. Moreover, this is a
subculture shaped by mistrust of established authorities and orthodox
scientific viewpoints. Its members value individual initiative and ingenuity,
trusting scientific analysis only insofar as they can replicate it themselves
by accessing and manipulating the data firsthand. They are highly reflexive
about the inherently biased nature of any analysis, and resent what they view
as the arrogant self-righteousness of scientific elites.
As a subculture, anti-masking amplifies anti-establishment currents pervasive
in U.S. political culture. Data literacy, for anti-maskers, exemplifies
distinctly American ideals of intellectual self-reliance, which historically
takes the form of rejecting experts and other elites (Hofstadter, 1966). The
counter-visualizations that they produce and circulate not only challenge
scientific consensus, but they also assert the value of independence in a
society that they believe promotes an overall de-skilling and dumbing-down of
the population for the sake of more effective social control (Tripodi, [n.d.];
Hochschild, 2016; Elisha, 2011). As they see it, to counter-visualize is to
engage in an act of resistance against the stifling influence of central
government, big business, and liberal academia. Moreover, their simultaneous
appropriation of scientific rhetoric and rejection of scientific authority
also reflects longstanding strategies of Christian fundamentalists seeking to
challenge the secularist threat of evolutionary biology (Bielo, 2019).
So how do these groups diverge from scientific orthodoxy if they are using the
same data? We have identified a few sleights of hand that contribute to the
broader epistemological crisis we identify between these groups and the
majority of scientific researchers. For instance, they argue that there is an
outsized emphasis on _deaths_ versus cases: if the current datasets are
fundamentally subjective and prone to manipulation (e.g., increased levels of
faulty testing, asymptomatic vs. symptomatic cases), then deaths are the only
reliable markers of the pandemic’s severity. Even then, these groups believe
that deaths are an additionally problematic category because doctors are using
a COVID diagnosis as the main cause of death (i.e., people who die because of
COVID) when in reality there are other factors at play (i.e., dying with but
not because of COVID). Since these categories are fundamentally subject to
human interpretation, especially by those who have a vested interest in
reporting as many COVID deaths as possible, these numbers are vastly over-
reported, unreliable, and no more significant than the flu.
Another point of contention is that of lived experience: in many of these
cases, users do not themselves know a person who has experienced COVID, and
the statistics they see on the news show the severity of the pandemic in
vastly different parts of the country. Since they do not see their experience
reflected in the narratives they consume, they look for hyperlocal data to
help guide their decision-making. But since many of these datasets do not
always exist on such a granular level, this information gap feeds into a
larger social narrative about the government’s suppression of critical data
and the media’s unwillingness to substantively engage with the subjectivity of
coronavirus data reporting.
Most fundamentally, the groups we studied believe that science is a process,
and not an institution. As we have outlined in the case study, these groups
mistrust the scientific establishment (“Science”) because they believe that
the institution has been corrupted by profit motives and politics. The
knowledge that the CDC and academics have created cannot be trusted because
they need to be subject to increased doubt, and not accepted as consensus. In
the same way that climate change skeptics have appealed to Karl Popper’s
theory of falsification to show why climate science needs to be subjected to
continuous scrutiny in order to be valid (Fischer, 2019), we have found that
anti-mask groups point to Thomas Kuhn’s The Structure of Scientific
Revolutions to show how their anomalous evidence—once dismissed by the
scientific establishment—will pave the way to a new paradigm (“As I’ve
recently described, I’m no stranger to presenting data that are inconsistent
with the narrative. It can get ugly. People do not give up their paradigms
easily. […] Thomas Kuhn wrote about this phenomenon, which occurs repeatedly
throughout history. Now is the time to hunker down. Stand with the data,”
August 5, 2020). For anti-maskers, valid science must be a process they can
critically engage for themselves in an unmediated way. _Increased_ doubt, not
consensus, is the marker of scientific certitude.
Arguing that anti-maskers simply need more scientific literacy is to
characterize their approach as uninformed and inexplicably extreme. This study
shows the opposite: users in these communities are deeply invested in forms of
critique and knowledge production that they recognize as markers of scientific
expertise. If anything, anti-mask science has extended the traditional tools
of data analysis by taking up the theoretical mantle of recent critical
studies of visualization (D’Ignazio and Klein, 2020; Correll, 2019). Anti-mask
approaches acknowledge the subjectivity of how datasets are constructed,
attempt to reconcile the data with lived experience, and these groups seek to
make the process of understanding data as transparent as possible in order to
challenge the powers that be. For example, one of the most popular
visualizations within the Facebook groups we studied were unit visualizations,
which are popular among anti-maskers and computer scientists for the same
reasons: they provide more information, better match a reader’s mental model,
and they allow users to interact with them in new and more interesting ways
(Park et al., 2018). Barring tables, they are the most unmediated way to
interact with data: one dot represents one person.
Similarly, these groups’ impulse to mitigate bias and increase transparency
(often by dropping the use of data they see as “biased”) echoes the organizing
ethos of computer science research that seeks to develop “technological
solutions regarding potential bias” or “ground research on fairness,
accountability, and transparency” (Association for Computing Machinery, 2020).
In other words, these groups see themselves as engaging deeply within multiple
aspects of the scientific process—interrogating the datasets, analysis, and
conclusions—and still university researchers might dismiss them in leading
journals as “scientifically illiterate” (Miller, 2020). In an interview with
the Department of Health and Human Services podcast, even Anthony Fauci (Chief
Medical Advisor to the US President) noted: “one of the problems we face in
the United States is that unfortunately, there is a combination of an anti-
science bias […] people are, for reasons that sometimes are, you know,
inconceivable and not understandable, they just don’t believe science” (Fauci,
2020).
We use Dr. Fauci’s provocation to illustrate how understanding the way that
anti-mask groups think about science is crucial to grappling with the
contested state of expertise in American democracy. In a study of Tea Party
supporters in Louisiana, Arlie Russell Hochschild (Hochschild, 2016) explains
the intractable partisan rift in American politics by emphasizing the
importance of a “deep story”: a subjective prism that people use in order to
make sense of the world and guide the way they vote. For Tea Party activists,
this deep story revolved around anger towards a federal system ruled by
liberal elites who pander to the interests of ethnic and religious minorities,
while curtailing the advantages that White, Christian traditionalists view as
their American birthright. We argue that the anti-maskers’ deep story draws
from similar wells of resentment, but adds a particular emphasis on the
usurpation of scientific knowledge by a paternalistic, condescending elite
that expects intellectual subservience rather than critical thinking from the
lay public.
To be clear, we are not promoting these views. Instead, we seek to better
understand how data literacy, as a both a set of skills and a moral virtue
championed within academic computer science, can take on distinct valences in
different cultural contexts. A more nuanced view of data literacy, one that
recognizes multiplicity rather than uniformity, offers a more robust account
of how data visualization circulates in the world. This culturally and
socially situated analysis demonstrates why increasing access to raw data or
improving the informational quality of data visualizations is not sufficient
to bolster public consensus about scientific findings. Projects that examine
the cognitive basis of visualization or seek to make “better” or “more
intuitive” visualizations (Kosara, 2016) will not meaningfully change this
phenomenon: anti-mask protestors already use visualizations, and do so
extremely effectively. Moreover, in emphasizing the politicization of pandemic
data, our account helps to explain the striking correlation between practices
of counter-visualization and the politics of anti-masking. For members of this
social movement, counter-visualization and anti-masking are complementary
aspects of resisting the tyranny of institutions that threaten to usurp
individual liberties to think freely and act accordingly.
## 6\. Implications and conclusion
This paper has investigated anti-mask counter-visualizations on social media
in two ways: quantitatively, we identify the main types of visualizations that
are present within different networks (e.g., pro- and anti-mask users), and we
show that anti-mask users are prolific and skilled purveyors of data
visualizations. These visualizations are popular, use orthodox visualization
methods, and are promulgated as a way to convince others that public health
measures are unnecessary. In our qualitative analysis, we use an ethnographic
approach to illustrate how COVID counter-visualizations actually reflect a
deeper epistemological rift about the role of data in public life, and that
the practice of making counter-visualizations reflects a participatory,
heterodox approach to information sharing. Convincing anti-maskers to support
public health measures in the age of COVID-19 will require more than “better”
visualizations, data literacy campaigns, or increased public access to data.
Rather, it requires a sustained engagement with the social world of
visualizations and the people who make or interpret them.
While academic science is traditionally a system for producing knowledge
within a laboratory, validating it through peer review, and sharing results
within subsidiary communities, anti-maskers reject this hierarchical social
model. They espouse a vision of science that is radically egalitarian and
individualist. This study forces us to see that coronavirus skeptics champion
science as a personal practice that prizes rationality and autonomy; for them,
it is _not_ a body of knowledge certified by an institution of experts. Calls
for data or scientific literacy therefore risk recapitulating narratives that
anti-mask views are the product of individual ignorance rather than
coordinated information campaigns that rely heavily on networked
participation. Recognizing the _systemic_ dynamics that contribute to this
epistemological rift is the first step towards grappling with this phenomenon,
and the findings presented in this paper corroborate similar studies about the
impact of fake news on American evangelical voters (Tripodi, [n.d.]) and about
the limitations of fact-checking climate change denialism (Fischer, 2019).
Calls for media literacy—especially as an ethics smokescreen to avoid talking
about larger structural problems like white supremacy—are problematic when
these approaches are deficit-focused and trained primarily on individual
responsibility. Powerful research and media organizations paid for by the
tobacco or fossil fuel industries (Proctor, 2011; Oreskes and Conway, 2010)
have historically capitalized on the skeptical impulse that the “science
simply isn’t settled,” prompting people to simply “think for themselves” to
horrifying ends. The attempted coup on January 6, 2021 has similarly
illustrated that well-calibrated, well-funded systems of coordinated
disinformation can be particularly dangerous when they are _designed_ to
appeal to skeptical people. While individual insurrectionists are no doubt to
blame for their own acts of violence, the coup relied on a collective effort
fanned by people questioning, interacting, and sharing these ideas with other
people. These skeptical narratives are powerful because they resonate with
these these people’s lived experience and—crucially—because they are posted by
influential accounts across influential platforms.
Broadly, the findings presented in this paper also challenge conventional
assumptions in human-computer interaction research about who imagined users
might be: visualization experts traditionally design systems for scientists,
business analysts, or journalists. Researchers create systems intended to
democratize processes of data analysis and inform a broader public about how
to use data, often in the clean, sand-boxed environment of an academic lab.
However, this literature often focuses narrowly on promoting expressivity
(either of current or new visualization techniques), assuming that improving
visualization tools will lead to improving public understanding of data. This
paper presents a community of users that researchers might not consider in the
systems building process (i.e., supposedly “data illiterate” anti-maskers),
and we show how the binary opposition of literacy/illiteracy is insufficient
for describing how orthodox visualizations can be used to promote unorthodox
science. Understanding how these groups skillfully manipulate data to
undermine mainstream science requires us to adjust the theoretical assumptions
in HCI research about how data can be leveraged in public discourse.
What, then, are visualization researchers and social scientists to do? One
step might be to grapple with the social and political dimensions of
visualizations at the _beginning_ , rather than the end, of projects (Correll,
2019). This involves in part a shift from positivist to interpretivist
frameworks in visualization research, where we recognize that knowledge we
produce in visualization systems is fundamentally “multiple, subjective, and
socially constructed” (Meyer and Dykes, 2019). A secondary issue is one of
uncertainty: Jessica Hullman and Zeynep Tufekci (among others) have both
showed how _not_ communicating the uncertainty inherent in scientific writing
has contributed to the erosion of public trust in science (Hullman et al.,
2019; Tufekci, 2020b). As Tufekci demonstrates (and our data corroborates),
the CDC’s initial public messaging that masks were ineffective—followed by a
quick public reversal—seriously hindered the organization’s ability to
effectively communicate as the pandemic progressed. As we have seen, people
are not simply passive consumers of media: anti-mask users _in particular_
were predisposed to digging through the scientific literature and highlighting
the uncertainty in academic publications that media organizations elide. When
these uncertainties did not surface within public-facing versions of these
studies, people began to assume that there was a broader cover-up (Tufekci,
2020a).
But as Hullman shows, there are at least two major reasons why uncertainty
_hasn’t_ traditionally been communicated to the public (Hullman, 2020).
Researchers often do not believe that people will understand and be able to
interpret results that communicate uncertainty (which, as we have shown, is a
problematic assumption at best). However, visualization researchers also do
not have a robust body of understanding about _how_ , and when, to communicate
uncertainty (let alone how to do so effectively). There are exciting threads
of visualization research that investigate how users’ interpretive frameworks
can change the overarching narratives they glean from the data (Hullman and
Diakopoulos, 2011; Peck et al., 2019; Segel and Heer, 2010). Instead of
championing absolute certitude or objectivity, this research pushes us to ask
how scientists and visualization researchers alike might express uncertainty
in the data so as to recognize its socially and historically situated nature.
In other words, our paper introduces new ways of thinking about
“democratizing” data analysis and visualization. Instead of treating increased
adoption of data-driven storytelling as an unqualified good, we show that data
visualizations are not simply tools that people use to understand the
epidemiological events around them. They are a battleground that highlight the
contested role of expertise in modern American life.
###### Acknowledgements.
The authors thank Stephan Risi, Maeva Fincker, and Mateo Monterde for their
assistance with quantitative methods and supplemental material. We also thank
the members of the Visualization Group (especially Jonathan Zong, Alan
Lundgard, Harini Suresh, EJ Sefah, and the Fall 2020 UROP cohort: Anna Arpaci-
Dusseau, Anna Meurer, Ethan Nevidomsky, Kat Huang, and Soomin Chun). This
manuscript benefited from the insights of Rodrigo Ochigame, Hannah LeBlanc,
Meghan Kelly, Will Deringer, Blakeley H. Payne, Mariel García-Montes, and the
comments of anonymous reviewers. This project was supported by NSF Award
1900991, NSF Dissertation Improvement Grant 1941577, an SSRC Social Data
Dissertation Fellowship, and the MIT Programs for Digital Humanities.
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|
# The fast radio burst population evolves, consistent with the star-formation
rate
C.W. James,1 J.X. Prochaska2,3 J.-P. Macquart,1 F.O. North-Hickey1 K. W.
Bannister4and A. Dunning4
1International Centre for Radio Astronomy Research, Curtin University,
Bentley, WA 6102, Australia
2Kavli Institute for the Physics and Mathematics of the Universe, 5-1-5
Kashiwanoha, Kashiwa 277-8583, Japan.
3Astronomy Department, University of Washington, Seattle, WA 98195, USA.
4CSIRO Astronomy and Space Science, PO Box 76, Epping, NSW 1710, Australia
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
Fast radio bursts (FRBs) are extremely powerful sources of radio waves
observed at cosmological distances. We use a sophisticated model of FRB
observations — presented in detail in a companion paper — to fit FRB
population parameters using large samples of FRBs detected by ASKAP and
Parkes, including seven sources with confirmed host galaxies. Our fitted
parameters demonstrate that the FRB population evolves with redshift in a
manner consistent with, or faster than, the star-formation rate (SFR), ruling
out a non-evolving population at 99.9% C.L. Our estimated maximum FRB energy
is $\log_{10}E_{\rm max}[{\rm erg}]=41.84_{-0.18}^{+0.49}$ (68% C.L.) assuming
a 1 GHz emission bandwidth, with slope of the cumulative luminosity
distribution $\gamma=-1.16_{-0.12}^{+0.11}$. We find a log-mean host DM
contribution of $145_{-60}^{+64}$ pc cm-3 on top of a typical local (ISM and
halo) contribution of $\sim 80$ pc cm-3, which is higher than most literature
values. These results are consistent with the model of FRBs arising as the
high-energy limit of magnetar bursts, but allow for FRB progenitors that
evolve faster than the SFR.
###### keywords:
radio continuum: transients – methods: statistical
††pubyear: 2020††pagerange: The fast radio burst population evolves,
consistent with the star-formation rate–2
## 1 Introduction
Fast radio bursts (FRBs) are extragalactic transient radio sources of
millisecond duration (Lorimer et al., 2007; Thornton et al., 2013). Some
repeat, while most have not been observed to do so (Spitler et al., 2016;
CHIME/FRB Collaboration et al., 2019b; Fonseca et al., 2020a, b; Kumar et al.,
2019; Shannon et al., 2018; James et al., 2020), and the question of whether
or not there are one, two, or more FRB populations remains open. The recent
observation of a Galactic magnetar flare with FRB-like properties strongly
suggests such objects as an FRB progenitor class (Mereghetti et al., 2020; The
Chime/Frb Collaboration et al., 2020; Bochenek et al., 2020), though many more
have been proposed (Platts et al., 2019). Yet, this flare was three orders of
magnitude less powerful than the weakest FRBs, which in turn are orders of
magnitude weaker than the most powerful FRBs (Shannon et al., 2018). FRBs may
therefore have an unrelated origin.
If the FRB population does originate from young magnetars, they would be
expected to be closely correlated with star-forming activities, as observed
for two rapid repeaters (Tendulkar et al., 2017; Marcote et al., 2020).
However, the single largest sample of localised FRBs comes from the Australian
Square Kilometre Array Pathfinder (ASKAP; Bannister et al., 2019; Prochaska et
al., 2019; Bhandari et al., 2020b). The host galaxies of these FRBs — which
due to ASKAP’s large field of view (FOV) and higher detection threshold tend
to be the intrinsically most powerful bursts — do not show evidence for
unusual star-forming activity (Bhandari et al., 2020a; Heintz et al., 2020).
This allows for the possibility of much of this population to arise from other
sources, e.g. compact binary mergers (see Caleb et al., 2018, and references
contained therein).
A useful method to distinguish between these models comes from the evolution
of the FRB population on cosmological timescales. If FRBs originate from young
magnetars, they will closely follow the star-formation rate (SFR) (Metzger et
al., 2017), and peak in the redshift range 1–3. A binary merger scenario
however would likely lag the SFR, and possibly result in an FRB rate that is
increasing with cosmological time (Cao et al., 2018). As yet, FRB population
analysis has not been able to distinguish between these scenarios (Arcus et
al., 2021; Luo et al., 2020). Other methods yield mixed results: Hashimoto et
al. (2020) find evidence against the redshift evolution of once-off FRBs, and
some evidence for redshift evolution of the event rate for repeating FRBs;
while Locatelli et al. (2019) find evidence for an evolving FRB population for
once-off FRBs. However, neither work follows a comprehensive approach
advocated by Connor (2019), by modelling observational biases, and allowing
for the confounding effects of the FRB luminosity function.
The FRB luminosity function is interesting in and of itself. Comparisons of
the luminosity function of individual repeaters (e.g. Law et al., 2017) to the
population as a whole tests the credibility of the one-population model, while
evidence for a minimum burst energy above that produced by Galactic magnetars
would require a separate progenitor class, or at least a separate emission
mechanism. Models requiring rare events to explain FRBs can be challenged by
measurements of the absolute volumetric rate (Ravi, 2019). Estimates of the
maximum FRB energy not only challenges theoretical models and pushes up
against theoretical limits (Lu & Kumar, 2018), but affects the ability to use
FRBs as cosmological probes. Estimates of the host contribution to dispersion
measure (DM) inform us of the environment surrounding FRB progenitors.
Consequently, several groups have begun modelling the FRB population in an
attempt to derive these parameters, although the results and methods have been
inconsistent (Caleb et al., 2016; Luo et al., 2018; Lu & Piro, 2019; Luo et
al., 2020; Arcus et al., 2021; Gardenier et al., 2020).
In a companion paper (James et al., 2021), we present our method to model the
FRB population. It uses the the methodology advocated by Connor (2019), and
first implemented by Luo et al. (2020), while making several significant
advances in accuracy and precision, and taking advantage of recent FRB
localisations, and fitting for the measured signal-to-noise ratio. This models
all known observational biases in detail, allowing us to make accurate and
precise estimates of FRB population parameters, and model its cosmological
source evolution. Here, we present maximum-likelihood estimates of FRB
population parameters using FRBs observed by the Australian Square Kilometre
Array Pathfinder (ASKAP) and Parkes, and discuss the implications for the FRB
population.
## 2 Review of the model
In modelling FRB observations, it is critically important to account for a
range of observational biases. Our full treatment is contained in a (much
lengthier) companion paper, James et al. (2021). To briefly summarise, we
account for telescope beamshape, and reduced observational sensitivity to
high-DM, high-width FRBs, as recommended by Connor (2019); and fluctuations in
cosmological dispersion measure according to best-fit cosmological parameters,
local contributions from the Milky Way’s interstellar medium (ISM) and halo,
and a log-normal distribution $p({\rm DM}_{\rm host}^{\prime})$ of the host DM
contribution, as per Macquart et al. (2020). This latter contribution, defined
by
$\displaystyle p({\rm DM}_{\rm host}^{\prime})=\frac{1}{{\rm DM_{\rm
host}^{\prime}}}\frac{1}{\sigma_{\rm host}\sqrt{2\pi}}e^{-\frac{(\log{\rm
DM}^{\prime}_{\rm host}-\mu_{\rm host})^{2}}{2\sigma_{\rm host}^{2}}},$ (1)
is fit using the parameters $\mu_{\rm host}$ and $\sigma_{\rm host}$. The
effective host DM, DMhost, corrects the host DM for redshift: DM${}_{\rm
host}={\rm DM}_{\rm host}^{\prime}/(1+z)$.
Our model for the FRB population uses a power-law with cumulative slope
$\gamma$ and maximum energy $E_{\rm max}$, such that the probability of
observing an FRB above an energy threshold $E_{\rm th}$ is given by
$\displaystyle p(E>E_{\rm th})$ $\displaystyle=$
$\displaystyle\frac{\left(\frac{E_{\rm th}}{E_{\rm
min}}\right)^{\gamma}-\left(\frac{E_{\rm max}}{E_{\rm
min}}\right)^{\gamma}}{1-\left(\frac{E_{\rm max}}{E_{\rm
min}}\right)^{\gamma}}.$ (2)
The minimum FRB energy is not well-constrained by current observations, and is
set to a very low value of $10^{30}$ erg. We scale the FRB energy $E$
according to $E\sim\nu^{\alpha}$; for data taken exclusively at L-band ($\sim
1.4$ GHz), the model is almost degenerate to $\alpha$ (a conclusion also
reached by Lu & Piro, 2019; Arcus et al., 2021), and so we use a symmetric
Gaussian prior of $\alpha=-1.5\pm 0.3$ (Macquart et al., 2019).
We model the evolution of the FRB population $\Phi(z)$ (bursts per proper time
per comoving volume) by smoothly scaling the SFR with the parameter $n$,
$\displaystyle\Phi(z)$ $\displaystyle=$
$\displaystyle\frac{\Phi_{0}}{1+z}\left(\frac{{\rm SFR}(z)}{{\rm
SFR}(0)}\right)^{n}.$ (3)
We take SFR$(z)$ from Madau & Dickinson (2014),
$\displaystyle{\rm SFR}(z)$ $\displaystyle=$ $\displaystyle
1.0025738\frac{(1+z)^{2.7}}{(1+\left(\frac{1+z}{2.9}\right)^{5.6}}.$ (4)
Thus our full model treats $E_{\rm max}$, $\gamma$, $\alpha$, $n$, $\mu_{\rm
host}$ and $\sigma_{\rm host}$ as free parameters.
There is some ambiguity in the interpretation of $\alpha$, which can instead
be interpreted as a frequency-dependent rate. This is motivated by the many
FRBs with a limited band occupancy, as originally noted by Law et al. (2017)
for FRB 121102. The interpretation of $\alpha$ has slight effects on the
modelling: in the rate interpretation, the FRB rate at high $z$ is modified
directly through a further factor $\Phi(z)\sim(1+z)^{\alpha}$, while in the
spectral-index interpretation, this occurs indirectly, through the
k-correction affecting the calculation of threshold energy $E_{\rm th}$ from a
fluence threshold $F_{\rm th}$, and hence the rate via Eq. 2. In the absence
of an obvious correct treatment, we by default present results for the
spectral index interpretation, and show results for the rate interpretation in
Appendix A.
We use a sample of 24 non-localised, and seven localised, FRBs detected by
ASKAP, and 20 FRBs detected by the Parkes multibeam system. These have been
selected due to them occurring at high Galactic latitudes where the reduced
sensitivity due to high Galactic DM is unimportant. The full telescope
beamshape of each of these instruments is modelled in detail in our companion
paper, based off the methods of James et al. (2019a), while the reduction in
sensitivity to high DMs and widths is modelled using the time- and frequency
resolutions of the instruments according to Cordes & McLaughlin (2003).
## 3 Results
Our single-parameter constraints are given in Figure 1, showing results both
with and without a prior on $\alpha$. Best-fit values and confidence limits,
calculated using Wilks’ theorem (Wilks, 1962), are given in Table 1. Two-
parameter plots are given in Figure 3. We discuss the implications for each
parameter individually below.
Figure 1: Maximum likelihoods as a function of each considered variable
($E_{\rm max},\alpha,\gamma,n,\mu_{\rm host},\sigma_{\rm host}$) when
marginalised over the other five, both with (orange, lower) and without (blue,
upper) a prior on the spectral index $\alpha$. Calculation results are given
by points, with lines drawn using cubic spline smoothing. Vertical lines are
single-parameter intervals at the labelled degree of confidence calculated
using Wilks’ theorem with one degree of freedom. In the case of
$\log_{10}E_{\rm max}$, 90% and 95% lower limits are at 41.4.
### 3.1 Maximum burst energy $E_{\rm max}$
We find the maximum FRB to be energy is $\log_{10}E_{\rm max}$
(erg)$=41.84_{-0.18}^{+0.49}$ (68% C.L.). According to our method, $E_{\rm
max}$ is normalised to a bandwidth of 1 GHz at the mean frequency of the data
used as inputs (about 1350 MHz), and applies to all burst widths. A strict
lower limit on $E_{\rm max}$ is set by the intrinsically brightest localised
FRB, 190711, which — using a fluence of 34 Jy ms (Macquart et al., 2020), 1
GHz bandwidth, and $\alpha=-1.5$ — had an energy of $E_{190711}=10^{41.5}$
erg. $E_{\rm max}=10^{41.6}$ erg is however consistent with observations at
all quoted levels of confidence.
The preferred value of $E_{\rm max}$ is most strongly correlated with
$\alpha$, which effectively attenuates FRBs as a function of redshift. Upper
limits on $E_{\rm max}$ are also strongly correlated with $\gamma$, since a
large negative value of this parameter makes it unlikely to observe FRBs near
$E_{\rm max}$.
Our value of $E_{\rm max}$ lies in the middle of the values found by other
authors. From Figure 3, fixing $n=0$ as per Luo et al. (2020) would lead to a
lower value of $E_{\rm max}$, and greater consistency with that work. The
higher values of $E_{\rm max}$ found by Lu & Piro (2019), and used by Arcus et
al. (2021), arise in models that assume a 1–1 DM–z relation, which will tend
to over-estimate $E_{\rm max}$ when an FRB with a significant excess DM —
either due to its host or intervening matter — is detected.
A key implication of $E_{\rm max}$ is the distance out to which an FRB is
observable by a given telescope. For $\alpha=-1.5$, our value of
$\log_{10}E_{\rm max}$ leads to a maximum observable redshift of
$z=4^{+3}_{-0.85}$ for an instrument with 1 Jy ms threshold.
### 3.2 Intrinsic luminosity index $\gamma$
Our best-fit power-law index for the FRB population is
$\gamma=-1.16_{-0.12}^{+0.11}$ (68% C.L.). As discussed by Macquart & Ekers
(2018), this parameter primarily governs the degree to which FRBs are viewed
from the near or far Universe, with steep values of $\gamma$ (i.e. below -1.5)
leading to observations being dominated by nearby events and the event rate
being governed by $E_{\rm min}$. Our result is definitely above this value,
which is in agreement with all other calculations. It is however somewhat
steeper than the values found by other authors.
Why? Luo et al. (2020) assume no cosmological source evolution, which this
parameter is strongly correlated with. An increase of high-redshift FRBs can
be due to either a lower $\gamma$, leading to more bursts visible near $E_{\rm
max}$ in the larger volume of the distant Universe; or due to an evolving
population, as determined by $n$. This anti-correlation is clearly visible in
Figure 3. Both Arcus et al. (2021) and Lu & Piro (2019) allow source
evolution, but assume a 1–1 DM–z relation, implying a large distance for the
highest-DM FRBs. In order to fit such bursts without over-predicting a large
number of lower-energy bursts requires a flat luminosity function.
In the case that all FRBs repeat, with each FRB having the same $E_{\rm max}$
and $\gamma$ but a distribution of intrinsic rates, the intrinsic luminosity
function for the entire population will match that of each FRB. This index has
been well-measured for FRB 121102, with data giving a range
$\gamma_{121102}\approx-0.9\pm 0.2$ (Law et al., 2017; Gajjar et al., 2018;
James, 2019). This is consistent with our value for the population. However,
should $E_{\rm max}$ vary over FRBs (which is quite likely), then the value of
$\gamma$ for the population might be steeper. FRB observations by CHIME, which
have detected several FRBs with many repeat bursts (CHIME/FRB Collaboration et
al., 2019b; Fonseca et al., 2020a, b), should be able to answer the question
definitively.
### 3.3 Redshift evolution $n$
Our best-fit value of the redshift evolution scaling parameter is
$n=1.77^{+0.25}_{-0.45}$ (68% C.L.). Under the ‘rate interpretation’ of
$\alpha$, we find $n=1.26_{-0.35}^{+0.51}$. In both cases, $n=0$ is excluded
at better than 99.9%, which holds when no prior on $\alpha$ is considered.
Our detection of evolution in the FRB population supports conclusions based on
FRB localisations, which locate most FRBs within normal host galaxies (Heintz
et al., 2020); evidence associating FRBs with magnetars, such as the recent
Galactic magnetar outburst (The Chime/Frb Collaboration et al., 2020; Bochenek
et al., 2020); and observations of the host environment of FRB 121102
(Michilli et al., 2018), as well as predictions from several classes of
progenitor models (Platts et al., 2019).
This does not mean that we have confirmed that FRBs exhibit cosmological
evolution identical to the star-formation rate however. A more-general model
of source evolution, as used by Lu & Piro (2019), simply assumes a
$(1+z)^{n^{\prime}}$ dependence, i.e. it removes the denominator and
normalising constant in Eq. 4. Near $z=0$, $n^{\prime}\equiv 2.7n$ — however,
the models will diverge above $z=1$. A true detection of scaling with the
star-formation rate would require observations to be consistent with a
downturn relative to the $(1+z)^{n^{\prime}}$ model at and beyond the peak of
star-forming activity.
Figure 2: Maximum likelihood fits for source evolution parameter $n$, for
three different cases: interpreting $\alpha$ as a spectral index, and source
evolution $n$ scaling the star-formation rate as per Eq. 4 (blue);
interpreting $\alpha$ as a frequency dependent rate, again using Eq. 4 for
$n$-scaling (orange); and with $\alpha$ as a frequency-dependent rate, but $n$
scaling source evolution simply as $(1+z)^{2.7n}$ (green). The vertical lines
show $90$ C.L. intervals calculated using Wilks’ theorem.
Figure 2 plots the likelihood for both interpretations of $\alpha$, and the
$(1+z)^{2.7n}$ model. While the spectral index interpretation of $\alpha$
clearly gives a better fit, the difference in maximum likelihoods between the
two source evolution models under the rate assumption is negligible, with the
preferred value of $n$ being slightly higher under SFR-scaling to compensate
for the denominator in Eq. 4. We therefore cannot claim evidence for a down-
turn in the source evolution function due to the peak in the SFR, only that
the FRB population is evolving with cosmological time, with the rate per
comoving volume greater at higher $z$.
Our result is still a significant improvement on prior works. Previous
calculations have either had to assume a value for FRB source evolution (of
$n=0$ or $1$), due to complete degeneracy with $\alpha$ (Lu & Piro, 2019), or
otherwise could not distinguish between models (Caleb et al., 2016; Lu & Piro,
2019; Arcus et al., 2021). As previously noted, and explained in detail in our
companion paper, this degeneracy also affects this work. However, the
degeneracy is not complete — it is partially broken by the ASKAP/ICS sample of
localised FRBs, and by the Parkes sample, which probes to sufficient $z$ to be
sensitive to the non-Cartesian nature of the Universe. Of similar works, only
Caleb et al. (2016); Luo et al. (2020) model a telescope beamshape. We show in
our companion paper that these authors’ assumption of a Gaussian ($\sim$Airy)
beamshape for Parkes observations is sufficient, but doing so for ASKAP data —
as considered by Luo et al. (2020) — is inappropriate. The inability of Arcus
et al. (2021) and Lu & Piro (2019) to exclude $n=0$ may be due to their lack
of beamshape modelling. Including beamshape reveals that a larger fraction of
the sky is probed at lower sensitivity, thus increasing sensitivity to FRBs in
the local Universe relative to that in the distant Universe. Without this
effect, $n$ must be artificially decreased to model the observed number of
near-Universe bursts.
### 3.4 Excess DM distribution
Our model fits a log-normal distribution to DMhost, which nominally covers the
host galaxy and the immediate FRB environs. The fit will naturally include
deviations from the NE2001 DM model of the Milky Way and the assumed halo DM
of 50 pc cm-3. We find best-fit values of $\log_{10}\mu_{\rm
host}=2.16^{+0.20}_{-0.23}$ and $\log_{10}\sigma_{\rm
host}=0.51_{-0.10}^{+0.15}$, with both parameters being relatively independent
of the other four. Figure 3 shows that high values of $\mu_{\rm host}$ and low
values of $\sigma_{\rm host}$ are most strongly excluded. The only other
authors to fit these parameters are Macquart et al. (2020), who use a sub-set
of the data analysed in this work; our fitted value for the mean DM is greater
than theirs, but not significantly. Partially, this is because Macquart et al.
(2020) do not account for reduced sensitivity to high-DM bursts. Our inclusion
of this effect requires a greater intrinsic high-DM population to fit the same
observations. Combined with local contributions from the Milky Way’s ISM and
halo, we estimate a mean non-cosmological DM of ${\rm DM}_{\rm ISM}+{\rm
DM}_{\rm halo}+\mu_{\rm host}=50+35+145=230$ pc cm-3 at $z=0$. However, this
still allows for low values of DM observed by ASKAP (Shannon et al., 2018) and
CHIME (CHIME/FRB Collaboration et al., 2019a), since both DMhost and the
cosmological contribution can vary. This large value of mean non-cosmological
contribution helps to explain the observation by Shannon et al. (2018) that
the mean DM of the Parkes FRB sample is not as large relative to the ASKAP/ICS
sample as would be expected from the relative telescope sensitivities alone.
### 3.5 The prevalence of FRBs
We estimate the best-fit absolute rate of FRBs above $E_{\rm min}$,
$\Phi_{0}$, by maximising the product of $p_{n}$ between the ASKAP/FE and
Parkes/Mb samples with well-constrained $T_{\rm obs}$. We quote $\Phi_{39}$,
defined as the estimated rate of bursts above $10^{39}$ erg (above the maximum
allowed value of $E_{\rm min}$ James et al., 2021) per year at $z=0$. In the
case of our best-fit model, we find $\Phi_{39}=9_{-3.8}^{+2.2}\cdot 10^{4}$
bursts Gpc-3 yr-1 (90% C.L.).
This value is broadly consistent with that estimated by other authors (Ravi et
al., 2019; Lu & Piro, 2019; Luo et al., 2020), and supports the conclusion
that the majority of FRBs must either be repeaters, or cannot be due to known
populations of once-off events.
Interestingly, the best-fit parameter set under-predicts the number of FRBs
observed by ASKAP/FE (12.9 vs. 20 in 1274.6 days), and over-predicts the
number found by Parkes (17.0 vs 12 in 164.4 days). There are several possible
causes of this discrepancy. One possibility is a minimum FRB energy — or at
least a flattening of the distribution at low energies — which would reduce
the number of bursts seen by the more-sensitive Parkes telescope. Another is
the low number of FRBs detected by Parkes with SNR below 16, as noted by James
et al. (2019b), which could be an indicator of a reduced detection efficiency
to low-fluence bursts. Both are investigated and deemed unlikely in our
companion paper. A third option is that the observation times reported here
are raw observation times, and do not account for lost effective observation
time due to e.g. radio-frequency interference, which is likely more prevalent
at Parkes than ASKAP. Finally, this could be the result of simple statistical
fluctuations — if the true rates are 12.9 and 17.0 FRBs respectively, then the
product of the ASKAP and Parkes likelihoods will be this or less unlikely 7.7%
of the time.
## 4 Conclusion
We have used a precise and accurate method of modelling the results of FRB
surveys to fit the measured DM, $z$, and signal-to-noise ratios of FRBs
detected by ASKAP and Parkes. We have carefully selected our data to ensure it
is not biased due to under-reporting of observation time, or due to large
local DM contributions reducing sensitivity. Crucially, we have included a
sample of localized FRBs from ASKAP for which the redshift of the host
galaxies is measured.
These modelled observations are tested against a six-parameter model of the
FRB population. Using a maximum-likelihood approach, we have derived the
tightest constraints on FRB population parameters to date. Our value of the
maximum FRB energy of $41.84_{-0.18}^{+0.49}$ erg (68% C.L.) is mid-way
between previous estimates.The intrinsic slope of the cumulative luminosity
distribution, $\gamma$, is found to be $-1.16_{-0.12}^{+0.11}$ (68% C.L.),
consistent with, but slightly steeper than, the slope found for FRB 121102.
Importantly, we find that the FRB population evolves with redshift, scaling
with the star-formation rate (SFR) to the power of $1.77_{-0.45}^{+0.25}$ or
$1.26_{-0.35}^{+0.51}$, depending on the interpretation of FRB spectral
properties. While we cannot distinguish between SFR-scaling and a model where
the FRB population increases as a simple power of $(1+z)^{2.7n}$, in all
scenarios we exclude a non-evolving population at better than 99.9% C.L.
Our best-fit log-mean host contribution to DM of $145$ pc cm-3 is also
somewhat higher than the standard value of $100$ pc cm-3.
Such large excess dispersion measures, and a population evolution consistent
with star formation, strongly aligns with the hypothesis of FRBs originating
from young magnetars. We caution that these results apply to the total FRB
population (which may or may not consist of multiple sub-populations), and
only to that part of the population to which the ASKAP and Parkes observations
are sensitive.
## Acknowledgements
This research has made use of NASA’s Astrophysics Data System Bibliographic
Services. This research made use of Python libraries Matplotlib (Hunter,
2007), NumPy (van der Walt et al., 2011), and SciPy (Virtanen et al., 2020).
This work was performed on the gSTAR national facility at Swinburne University
of Technology. gSTAR is funded by Swinburne and the Australian Government’s
Education Investment Fund. This work was supported by resources provided by
the Pawsey Supercomputing Centre with funding from the Australian Government
and the Government of Western Australia. This research was partially supported
by the Australian Government through the Australian Research Council’s
Discovery Projects funding scheme (project DP180100857).
## Data Availability
The data underlying this article will be shared on reasonable request to the
corresponding author.
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## Appendix A Further data
Figure 3 shows the correlation plots between all two-parameter combinations,
excluding $\alpha$.
Figure 3: Two-parameter maximum likelihood results, showing 68%, 90%, and 95% confidence intervals, calculated using Wilks’ theorem and a $\chi^{2}_{2}$ distribution. | No prior | Prior on $\alpha$
---|---|---
Parameter | Best Fit | 68% C.L. | 90% C.L. | 95% C.L. | Best Fit | 68% C.L. | 90% C.L. | 95% C.L.
$\log_{10}E_{\rm max}$ | 42.16 | ${}_{-0.38}^{+0.29}$ | ${}_{-0.56}^{+0.49}$ | ${}_{-0.56}^{+0.62}$ | 41.84 | ${}_{-0.18}^{+0.49}$ | ${}_{-0.24}^{+0.67}$ | ${}_{-0.24}^{+0.82}$
$\alpha$ | -2.50 | N/A | N/A | N/A | -1.55 | ${}_{-0.21}^{+0.21}$ | ${}_{-0.33}^{+0.33}$ | ${}_{-0.41}^{+0.41}$
$\gamma$ | -1.15 | ${}_{-0.13}^{+0.12}$ | ${}_{-0.19}^{+0.19}$ | ${}_{-0.22}^{+0.24}$ | -1.16 | ${}_{-0.12}^{+0.11}$ | ${}_{-0.18}^{+0.20}$ | ${}_{-0.21}^{+0.26}$
$n$ | 2.22 | ${}_{-0.35}^{+0.56}$ | ${}_{-0.96}^{+0.71}$ | ${}_{-1.16}^{+0.86}$ | 1.77 | ${}_{-0.45}^{+0.25}$ | ${}_{-0.66}^{+0.51}$ | ${}_{-0.81}^{+0.66}$
$\mu_{\rm host}$ | 2.14 | ${}_{-0.25}^{+0.16}$ | ${}_{-0.41}^{+0.25}$ | ${}_{-0.55}^{+0.30}$ | 2.16 | ${}_{-0.23}^{+0.16}$ | ${}_{-0.39}^{+0.25}$ | ${}_{-0.50}^{+0.30}$
$\sigma_{\rm host}$ | 0.54 | ${}_{-0.13}^{+0.17}$ | ${}_{-0.18}^{+0.35}$ | ${}_{-0.22}^{+0.47}$ | 0.51 | ${}_{-0.10}^{+0.15}$ | ${}_{-0.15}^{+0.30}$ | ${}_{-0.19}^{+0.41}$
Table 1: Confidence limits on single parameters, both with (left) and without
(right) a prior on $\alpha$.
In the main body of this work, we present parameter limits at 90% C.L.
obtained when using $\alpha$ as a prior. Table 1 presents parameter limits are
different confidence levels, both with and without a prior on $\alpha$.
Furthermore, in Section 2, we note that the spectral index $\alpha$ can also
be interpreted as a frequency-dependent rate. Our standard set of results use
the spectral index interpretation. For completeness, in Table 2, we give the
single-parameter limits under the rate interpretation.
| No prior | Prior on $\alpha$
---|---|---
Parameter | Best Fit | 68% C.L. | 90% C.L. | 95% C.L. | Best Fit | 68% C.L. | 90% C.L. | 95% C.L.
$\log_{10}E_{\rm max}$ | 41.93 | ${}_{-0.55}^{+0.22}$ | ${}_{-0.55}^{+0.48}$ | ${}_{-0.55}^{+0.63}$ | 41.86 | ${}_{-0.48}^{+0.28}$ | ${}_{-0.48}^{+0.51}$ | ${}_{-0.48}^{+0.69}$
$\alpha$ | -2.50 | N/A | N/A | N/A | -1.50 | ${}_{-0.20}^{+0.21}$ | ${}_{-0.33}^{+0.35}$ | ${}_{-0.41}^{+0.41}$
$\gamma$ | -1.19 | ${}_{-0.11}^{+0.17}$ | ${}_{-0.18}^{+0.26}$ | ${}_{-0.21}^{+0.30}$ | -1.15 | ${}_{-0.16}^{+0.15}$ | ${}_{-0.22}^{+0.23}$ | ${}_{-0.25}^{+0.27}$
$n$ | 1.97 | ${}_{-1.16}^{+0.25}$ | ${}_{-1.41}^{+0.51}$ | ${}_{-1.57}^{+0.61}$ | 1.26 | ${}_{-0.35}^{+0.51}$ | ${}_{-0.56}^{+0.76}$ | ${}_{-0.71}^{+0.86}$
$\mu_{x}$ | 2.11 | ${}_{-0.23}^{+0.23}$ | ${}_{-0.36}^{+0.32}$ | ${}_{-0.50}^{+0.39}$ | 2.14 | ${}_{-0.23}^{+0.20}$ | ${}_{-0.36}^{+0.32}$ | ${}_{-0.48}^{+0.36}$
$\sigma_{x}$ | 0.48 | ${}_{-0.10}^{+0.15}$ | ${}_{-0.15}^{+0.29}$ | ${}_{-0.18}^{+0.39}$ | 0.48 | ${}_{-0.10}^{+0.14}$ | ${}_{-0.15}^{+0.27}$ | ${}_{-0.18}^{+0.36}$
Table 2: Equivalent of Table 1, calculated assuming the rate interpretation of
$\alpha$ (see Section 2).
|
# Surface plasmon resonance for in-plane birefringence measurement of
anisotropic thin organic film
Amrit Kumar Department of Physics, Birla Institute of Technology and Science,
Pilani (BITS Pilani), 333031, India. Raj Kumar Gupta<EMAIL_ADDRESS>pilani.ac.in Department of Physics, Birla Institute of Technology and Science,
Pilani (BITS Pilani), 333031, India. Manjuladevi V Department of Physics,
Birla Institute of Technology and Science, Pilani (BITS Pilani), 333031,
India. Ashutosh Joshi Department of Physics, Birla Institute of Technology
and Science, Pilani (BITS Pilani), 333031, India.
###### Abstract
The measurement of in-plane birefringence ($\Delta{n}$) of ultrathin film is
challenging due to a significant deviation of physical properties of materials
in ultrathin regime as compared to that in bulk state. Surface plasmon
resonance (SPR) phenomenon can be employed to measure change in refractive
index of ultrathin film at a very high resolution. This article discusses
simulation of SPR phenomenon in Kretschmann configuration for the measurement
of $\Delta{n}$ in organic thin film exhibiting nematic-like ordering on the
two dimensional gold surface. The distribution of plasmonic field on the gold
surface was found to be anisotropic. This suggested that the coupling
plasmonic field with that of organic thin film exhibiting nematic-like
ordering on the gold surface will be non-isotropic. Therefore, a non-zero
difference in resonance angle (RA) was obtained from SPR measurement performed
along the optic-axis (OA) and orthogonal to OA of the in-plane nematic
ordering ($\Delta\theta$). A calibration surface showing the variation of
($\Delta\theta$) as a function of $\Delta{n}$ and thickness of thin organic
film consisting of shape anisotropic tilted molecules exhibiting nematic-like
ordering on gold surface was obtained. This calibration surface was employed
for the measurement of $\Delta{n}$ of single layer of Langmuir-Blodgett films
of cadmium stearate (CdSA) and 4’-octyl-4-biphenylcarbonitrile (8CB) deposited
on SPR chips. The thickness of the LB films was estimated using X-ray
reflectivity measurement and $\Delta\theta$ was measured using a home built
SPR instrument. The $\Delta{n}$ values were found to be 0.012 and 0.022 for
ultrathin films of CdSA and 8CB molecules, respectively.
Surface plasmon resonance; Kretschmann configuration; In-plane birefringence;
Langmuir-Blodgett film; FDTD simulation
## Introduction
The optical phenomenon surface plasmon resonance (SPR) is very popular owing
to its remarkable application in the field of sensors. The phenomenon
facilitates a highly sensitive and label free sensing for a variety of
biological and chemical analytes Homola2008 ; Wu2019 ; Gupta2017 . The
underlying principle for a SPR sensor is based on measurement of changes in
refractive index (RI) at a very high resolution due to molecular interactions.
The surface plasmon polaritons (SPP) can be excited at a metal-dielectric
interface by an incident electromagnetic wave traveling via a coupling medium
with RI greater than 1.0. The resonance can be established by matching the
wavevectors of the incident and the SPP waves. At the resonance, a maximum
energy will transfer from the incident wave to the SPP wave leading to
extinction of the characteristic incident electromagnetic wave from the
spectrum Homolabook2006 ; Abdulhalim2008 ; Prabowo2018 . In the widely
utilized Kretschmann configuration of SPR, a p-polarized monochromatic
electromagnetic wave is allowed to incident on the metal surface via a
coupling prism Krets1968 ; Krets1971 . In order to establish the SPR, the
angle of incidence is varied and the reflected intensity is recorded. At
resonance, the reflected intensity diminishes to minimum. The resonance angle
is unique for the given metal-dielectric interface. Therefore, any adsorption
of analytes at the metal-dielectric interface during sensing can alter the
dielectric nature and hence resonance angle (RA) shifts. The shift in RA can
be measured very precisely and the corresponding change in RI can be
calculated theoretically using the Fresnel’s relations devSNB . In addition to
the traditional sensing applications, the SPR phenomenon can also be used for
the measurement of optical anisotropy in thin films devJMS , temperature
measurement SPR_temp ; SPRtemp2 ; Lu2016 and optical filter wangAPL . A
typical resolution of the Kretschmann configuration based SPR instrument lies
in the range of $10^{-5}$ to $10^{-7}$ RIU devSNB ; resolution1 ; Homola_rev1
. Such a high resolution in the measurement of RI using SPR was successfully
utilized for quantification of optical anisotropy in ultrathin films.
Anisotropy in thin film arises due to tilt of shape anisotropic molecules
(e.g. rod shaped calamitic liquid crystal molecules) with respect to surface
normal which may yield in-plane nematic ordering. In an earlier report by our
group, the optical anisotropy in ultrathin films was estimated experimentally
using the SPR phenomenon by measuring shift in the RA in orthogonal directions
of the films exhibiting different degree of optical anisotropy devJMS . The
reported anisotropy in the ultrathin films was estimated from SPR angle
measurements in randomly chosen orthogonal directions. In order to estimate
the in-plane birefringence ($\Delta{n}=n_{e}-n_{o}$), the SPR measurement has
to be performed along the optic axis (OA) of the thin film and orthogonal to
it. The measured values of RI along OA and orthogonal to it can be treated as
extraordinary ($n_{e}$) and ordinary ($n_{o}$) components, respectively bire .
The shift in RA along OA and orthogonal to OA of a given anisotropic thin film
exhibiting nematic ordering in two dimensional plane can be defined as
$\Delta\theta$. In the present work, we have modified our experimental setup
by integrating a rotating platform (rotation axis along X-axis, Fig. 1 ) with
a resolution of 0.1∘ to rotate the film deposited substrate and measure the
SPR response in-situ as a function of angle of rotation of the film. This
modification ensures alignment of optics for the measurement of $n_{e}$ and
$n_{o}$ and hence $\Delta{n}$ of the ultrathin film. The RI of ultrathin film
will be dependent on several factors including the surface density,
orientation of molecules, surface morphology and the thickness of the film.
Thus the RA measured using SPR phenomenon will be dependent on such factors.
Therefore, a systematic study is needed for the estimation of important
optical parameter related to thin film viz. in-plane birefringence
($\Delta{n}$).
The reports in literature in general provide the value of birefringence of the
bulk material however, due to reduction of dimension of the material, the
physical properties deviate largely from that of bulk. Therefore, measurement
of physical properties of a material at the lower dimension is essential for
material engineering followed by device fabrication. The physical properties
of the low dimensional materials like two dimensional thin film depend on its
thickness. Hence, a calibration curve is essential for quantifying the
dependencies of a physical property on any such parameters. Since, the SPR
phenomenon can be potentially employed for the measurement of RI at a very
high resolution, a small in-plane birefringence due to tilt of shape
anisotropic organic molecules even in a single layer can be measured. Such
film with tilted molecules may exhibit nematic ordering on the surface. In
this article, we present a calibration surface showing the dependency of
$\Delta\theta$ on $\Delta{n}$ and thickness of the thin organic film. The
calibration surface was obtained through simulation and it was utilized for
the estimation of $\Delta{n}$ of single layer of Langmuir-Blodgett (LB) films
of cadmium stearate (CdSA) and 4’-octyl-4-biphenylcarbonitrile (8CB)
molecules. The values of thickness and $\Delta\theta$ of the LB films of CdSA
and 8CB was obtained from X-ray reflectivity and a home built SPR instrument,
respectively and these values were used in the calibration surface for the
estimation of the respective $\Delta{n}$.
## Simulation Setup
A finite difference time domain (FDTD) method was employed for the simulation
of SPR phenomenon in the Kretschmann configuration using a commercial package
of Lumerical lum1 ; lum2 . The FDTD method is highly reliable and advantageous
over other techniques in solving Maxwell’s equations for complex geometries of
materials. The simulation setup is shown in the Figure 1(a).
Figure 1: A schematic of (a) simulation setup showing the major components as
depicted. The plane of polarization is XY. The angle of incidence of the
monochromatic light (L) is $\theta_{i}$, thickness of each material and
detector (D) are shown and (b) a single layer of shape anisotropic molecules
(rod shaped) tilted with respect to X-axis along Y-axis on the YZ plane. The
projection of the molecules is shown in black. Such projection resembles
nematic ordering on 2D plane with optic axis along Y-axis.
The simulation was carried out using a monochromatic plane wave source (L)
having a wavelength of 635 nm. The perfectly matched layer (PML) boundary
condition with steep angle profile of 12 layers was used in order to minimise
reflection from the boundary as the wave enters into the layer. Linear
discrete Fourier transform monitors were used to capture reflected and
transmitted electric field at 350 nm away from the interface. The source was
made to incident on the gold layer via glass medium at an angle of incidence
of $\theta_{i}$. In order to obtain the resonance angle, the incident angle
sweep was generated from 40o\- 48o with 251 iterations. The mesh override was
selected in the propagation direction of the plane wave to get more precise
results. The optical anisotropy was seen in case of a single layer of
materials exhibiting geometrical anisotropy at the molecular level. A common
example of such system is shown schematically in Figure 1(b). A single layer
of rod shaped molecule (calamitic liquid crystal) tilted with respect to
X-axis can have a projection on the YZ plane. If all the molecules are more or
less tilted in the same direction (here it is along Y-axis), they exhibit a
nematic-like ordering with optic axis (OA) parallel to the Y-axis. Another set
of examples are single layer of self-assembled monolayer of rod shaped
octadecanethiol or Langmuir-Blodgett film of fatty acids LBSA . To simulate
such system of anisotropic material, a thin layer of organic material was
added onto the gold layer whose in-plane birefringence $(\Delta{n})$ was
varied systematically to observe the change in the resonance angle for the
same system but measured along the OA (i.e. Y-axis) and orthogonal to it (i.e.
Z-axis) in the SPR simulation model. Since the material is organic, only the
real part of RI is considered in the simulation.
## Experimental
The Kretschmann configured SPR instrument was developed in the laboratory
devSNB . The equipment utilizes 5 mW laser of wavelength 635 nm, coupling
prism (RI=1.51) and a segmented photodiode as detector. The resolution and
sensitivity of the equipment are 1.92 $\mu{RIU}$ and 53${}^{\circ}/RIU$,
respectively. The SPR chip consists of 0.5 mm glass plate (RI=1.51) deposited
with 50 nm thick gold film through sputtering technique. The chemicals,
stearic acid and 4’-octyl-4-biphenylcarbonitrile (8CB) were procured from
Sigma-Aldrich. Both the molecules yield a very stable Langmuir monolayer at
the air-water interface and are ideal systems for utilizing them for
fundamental studies LBSA ; 8CB . A single layer of LB film of CdSA deposited
at 30 mN/m can yield an average molecular tilt of $\sim$10∘ with respect of
surface normal cdsatilt and similarly, that of 8CB deposited at 4 mN/m yields
an average molecular tilt of $\sim$60∘ with respect to the surface normal
8cbtilt . A single layer of LB films of CdSA and 8CB were deposited onto SPR
chips at target surface pressure of 30 and 4 mN/m, respectively using a LB
trough (KSV-NIMA). The thickness of the LB films were measured by X-ray
reflectivity (XRR) technique using a X-ray diffractometer equipped with thin
film analysis unit (SmartLab, Rigaku).
## Results and Discussion
Figure 2: (a) SPR spectrum of gold-air interface (b) the electric field
profile on the two dimensional gold surface (Y-Z plane) obtained from
simulation.
A p-polarized electromagnetic wave was allowed to incident at the glass-gold
interface as shown in the Figure 1. The evanescent wave generated in the gold
film can excite the surface plasmon polaritons (SPP). Figure 2(a) shows the
SPR curve for the gold-air interface. It exhibits the RA value of 44∘. The SPR
curve and hence the RA value obtained through the FDTD calculation for the
gold-air interface is in agreement with the literature Gupta2017 . The two
dimensional (2D) electric field profile due to the surface plasmon polaritons
at the resonance angle was obtained and is shown in Figure 2(b). According to
the chosen geometry, the YZ plane corresponds to the gold-air interface and
the plane of polarization is XY. The SPP are excited by the incident
p-polarized electromagnetic wave. Therefore, the electric field of the
incident electromagnetic wave is restricted in the XY plane and has zero
component along the Z-axis. This may lead to surface distribution of the
surface plasmon field to be anisotropic in nature. For a chosen 1000
nm$\times$1000 nm mesh size, the anisotropic nature of the plasmonic field can
be clearly seen in the image. This indicates that the excitation of SPP is
non-isotropic and hence there is an immense possibility that coupling of such
anisotropic field with optically anisotropic material will be direction
dependent. Therefore, the SPR measurement of such anisotropic materials in
different direction with reference to the plane of incidence can yield
different resonance angle. The materials with optical anisotropy can be
obtained either in bulk state or as a single layers of organic molecules
exhibiting some shape anisotropy. The rod shaped calamitic liquid crystal
molecules exhibit a birefringence of $\sim$0.2 in the bulk nematic phase
LCbire ; 5CBbire . The liquid crystal molecules have great technological
importance where such optical anisotropy play significant role in display
device applications. When such shape anisotropic molecules are aligned onto
solid substrate through self-assembly or a controlled Langmuir-Blodgett
deposition technique roberts , the deposited single layer can induce a degree
of optical anisotropy due to a collective tilt of the molecules with respect
to the surface normal. Hence the projections of such tilted molecules can
yield a nematic ordering on the two dimensional surface. In our simulation
setup, we created an organic layer of a given thickness whose RI is chosen to
be anisotropic by assigning different values along X, Y and Z axes. The SPR
spectra were obtained through simulation when the plane of incidence is
parallel and perpendicular to the OA of the in-plane nematic ordering in thin
film of organic material. The difference in RA was noted as $\Delta\theta$
from the SPR spectra obtained in these two geometries.
Figure 3 shows the SPR curves obtained for an anisotropic thin film of 2 nm
thickness having $\Delta{n}$ as 0.1. The corresponding RAs were obtained as
$44.45^{\circ}$ and $44.80^{\circ}$ yielding $\Delta\theta$ to be
$0.35^{\circ}$.
Figure 3: Simulated SPR spectra of a 2 nm thick organic film consisting of
shape anisotropic organic molecules exhibiting an in-plane birefringence of
0.1.
In the simulation, the SPR curves are obtained for different values of
$\Delta{n}$ and thickness of organic film and the corresponding $\Delta\theta$
were obtained. A calibration surface displaying the variation of
$\Delta\theta$ as a function of $\Delta{n}$ and film thickness ($t$) is
plotted in Figure 4.
Figure 4: Calibration surface plot showing the variation of $\Delta\theta$ as
a function of in-plane birefringence ($\Delta{n}$) and thickness of organic
film. The simulated points are shown as filled circle. The surface is
polynomially fitted.
The simulated data are fitted with a surface polynomial curve
$\Delta\theta=P_{1}+P_{2}t+P_{3}\Delta{n}+P_{4}t^{2}+P_{5}\Delta{n}^{2}+P_{6}t\Delta{n}+P_{7}t^{2}\Delta{n}+P_{8}t\Delta{n}^{2}+P_{9}\Delta{n}^{3}$
(1)
where $P_{i},i=1,2,3...9$ are the fit parameters. The fit indicator R-square
was 0.993 which suggests a good fitting. The fitted calibration surface as
represented by the Eq. 1 can be useful for the determination of $\Delta{n}$ of
thin films using SPR phenomenon in the very simple prescribed methodology as
discussed here.
We have utilized the calibration surface (Eqn. 1) for the estimation of in-
plane birefringence of ultrathin films fabricated using the standard Langmuir-
Blodgett (LB) technique. We fabricated a single layer of LB films of cadmium
stearate (CdSA) and 8CB molecules on the SPR chips at the target surface
pressure of 30 and 4 mN/m, respectively LBSA ; 8CB . The molecules in a single
layer of LB films of CdSA and 8CB were tilted by $\sim$10 and 60∘ with respect
to the substrate normal cdsatilt ; 8cbtilt . Hence, they can offer anisotropy
in the refractive indices and therefore can exhibit non-zero values of
$\Delta{n}$. The thickness of the LB films were obtained from X-ray
reflectivity measurement (Figure 5). The experimental curve was fitted using
Parrat’s formulation Parratt and the thickness of the film was estimated
therefrom. The thickness of gold film deposited over the glass plate, LB films
of CdSA and 8CB deposited over such gold substrates were estimated as 49, 2.4
and 2.0 nm, respectively.
Figure 5: X-ray reflectivity curves obtained from (a) thin films of gold, (b)
LB films of cadmium stearate (CdSA) and (c) 4’-octyl-4-biphenylcarbonitrile
(8CB). The theoretical fitting yields the thickness of gold, CdSA and 8CB
films to be 49, 2.4 and 2.0 nm, respectively.
The LB films of CdSA and 8CB were scanned using the SPR instrument. The change
in RA along the such orthogonal directions ($\Delta\theta$) were found to be
24 and 71 millidegree, respectively. Such non-zero values suggest the
anisotropy in the ultrathin films. The values of thickness and $\Delta\theta$
were substituted in the calibration surface and $\Delta{n}$ of the ultrathin
films of CdSA and 8CB were estimated as 0.012 and 0.022, respectively.
Our analysis give a strong foundation for the measurement of in-plane
birefringence of ultrathin films of organic molecules. Such information are
essential for the development of optical devices.
## Conclusion
The measurement of physical properties at a lower dimension is challenging due
to large dependencies of the properties on other parameters e.g. thickness of
the thin film, aspect ratio of nanomaterials, morphology etc. In this article,
we simulated the SPR phenomenon in Kretschmann configuration to measure the
in-plane birefringence of thin organic film. The thin film consists of rod
shaped organic molecules tilted on the gold surface and thus exhibited in-
plane nematic ordering. We performed simulation to obtain a calibration
surface showing the variation of $\Delta\theta$ as a function of $\Delta{n}$
and thickness of the film. Such calibration surface was employed for the
estimation of $\Delta{n}$ in single layer of LB films of CdSA and 8CB. This
study provides a vital methodology for the measurement of very small value of
$\Delta{n}$ even in case of a single layer of ultrathin organic film. Further
studies involve the role of percolation in quasi-two dimensional film on the
optical properties.
## Acknowledgements
We are thankful to BITS Pilani for providing Lumerical software. We are
thankful to Department of Science and Technology, India for providing the XRD
facility through FIST programme. Thanks are also due to DST India for
supporting SPR instrument from project (IDP/SEN/06/2015) and LB trough from
(CRG/2018/000755). This is a post-peer-review, pre-copyedit version of an
article published in Plasmonics. The final authenticated version is available
online at: https://doi.org/10.1007/s11468-021-01373-1.
## Funding
Not applicable.
## Conflicts of interest/Competing interests
There are no conflicts of interest/competing interests to declare.
## Availability of data and material
The datasets generated during and/or analysed during the current study are
available from the corresponding author on reasonable request.
## Code availability
Lumerical is GUI based commercial simulation package. As such code
availability is not applicable. However, some scripts can be made available on
reasonable request to corresponding author.
## Authors’ contributions
Simulation and part of experiments were done by Amrit Kumar.
Conceptualization, data analysis, manuscript preparation were done by Raj
Kumar Gupta. Data analysis and manuscript preparation were done by
Manjuladevi. SPR measurements and part of experiments were done by Ashutosh
Joshi.
## References
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|
# The $z$–DM distribution of fast radio bursts
C.W. James,1 J.X. Prochaska2,3 J.-P. Macquart,1 F.O. North-Hickey1 K. W.
Bannister4and A. Dunning4
1International Centre for Radio Astronomy Research, Curtin University,
Bentley, WA 6102, Australia
2Department of Astronomy and Astrophysics, University of California, Santa
Cruz, CA 95064, USA
3Kavli Institute for the Physics and Mathematics of the Universe, 5-1-5
Kashiwanoha, Kashiwa 277-8583, Japan.
4CSIRO Astronomy and Space Science, PO Box 76, Epping, NSW 1710, Australia
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
We develop a sophisticated model of FRB observations, accounting for the
intrinsic cosmological gas distribution and host galaxy contributions, and
give the most detailed account yet of observational biases due to burst width,
dispersion measure, and the exact telescope beamshape. Our results offer a
significant increase in both accuracy and precision beyond those previously
obtained. Using results from ASKAP and Parkes, we present our best-fit FRB
population parameters in a companion paper. Here, we consider in detail the
expected and fitted distributions in redshift, dispersion measure, and signal-
to-noise. We estimate that the unlocalised ASKAP FRBs arise from $z<0.5$, with
between a third and a half within $z<0.1$. Our predicted source-counts
(“logN–logS”) distribution confirms previous indications of a steepening index
near the Parkes detection threshold of $1$ Jy ms. We find no evidence for a
minimum FRB energy, and rule out $E_{\rm min}>10^{38.5}$ erg at 90% C.L.
Importantly, we find that above a certain DM, observational biases cause the
Macquart (DM–z) relation to become inverted, implying that the highest-DM
events detected in the unlocalised Parkes and ASKAP samples are unlikely to be
the most distant. We do not expect our quantitative estimates in this region
to be accurate until it is directly probed with localised FRBs. Since the
cause of this effect is a well-understood observational bias however, it is
guaranteed to be present to some degree. Works assuming a 1–1 DM–z relation
may therefore derive erroneous results.
###### keywords:
radio continuum: transients – methods: statistical
††pubyear: 2020††pagerange: The $z$–DM distribution of fast radio bursts–A.5
## 1 Introduction
Fast radio bursts (FRBs) are radio transients of millisecond duration and
extragalactic origin (Lorimer et al., 2007; Thornton et al., 2013). Their
progenitors are unknown, with very many production mechanisms propoed (Platts
et al., 2019). FRB surveys are providing increasingly large statistics with
which to study the FRB population (Bhandari et al., 2018; Shannon et al.,
2018; CHIME/FRB Collaboration et al., 2019a), including a handful of localised
FRBs (Tendulkar et al., 2017; Bannister et al., 2019; Prochaska et al., 2019;
Ravi et al., 2019; Marcote et al., 2020). Furthermore, Macquart et al. (2020)
have used localised FRBs as probes of the cosmological distribution of ionised
gas, illustrating their utility for cosmological studies (McQuinn, 2014; Masui
& Sigurdson, 2015; Madhavacheril et al., 2019; Caleb et al., 2019). Of the key
questions surrounding FRBs, this work focuses on FRB population statistics.
Studies of the FRB population are important both for understanding the nature
of FRBs themselves, and their use as cosmological probes. Typical fitted
parameters include the FRB luminosity function, e.g. minimum and maximum
energies, and its shape; FRB spectral properties; and the source evolution.
Studies including repeating FRBs must also fit the distribution of repetition
rates and allow for time-dependence between bursts from a single object.
Linking FRB observations to the underlying true FRB population however is
difficult. Connor (2019) review previous methods of studying the FRB
population, and emphasise that accurate estimates require accounting for the
sensitivity effects of telescope beamshape, intrinsic burst width, and the
dispersion measure distribution $p({\rm DM}|z)$ for a given redshift. In
short, one must integrate over known or hypothesised intrinsic distributions
of these variables, model observational biases, and then attempt to match
observations. Doing so improperly will produce biased results.
In what is usually seen as a different line of inquiry, cosmological studies
using FRBs take advantage of their dispersion measure (DM), which integrates
the column density of ionised gas along their line of sight. This encodes
information on the diffuse gas in voids and galactic halos which is otherwise
difficult to study. Macquart et al. (2020) have recently used the localised
FRB population to constrain the total baryon density of the Universe and the
degree of feedback. In that work, the authors analyse the probability
distribution of observed dispersion measures, DM, given the redshift $z$ of
identified FRB host galaxies, $p(z,{\rm DM})$. This controls somewhat for the
effects of the population of FRBs, which primarily affects the redshift
distribution $p(z)$. The authors do note however the potentially biasing
effects of the FRB population as observed by FRB surveys, although a
comprehensive treatement of such biasing effects is not performed due to the
intrinsic error in using a small sample size (5–7). Many cosmological studies
(such as helium reionisation) require many FRBs to both exist and be
detectable at a redshift of $z\sim 4$ (Caleb et al., 2019), which is well
beyond the most distant localized FRB to date at $z\sim 0.6$ (Law et al.,
2020).
Fundamentally, both lines of inquiry aim to study the intrinsic distribution
of FRBs in $z$–DM space, $p({\rm DM},z)$. The only difference is the aspect of
interest: population studies try to resolve a redshift distribution $p(z)$ and
treat the distribution $p({\rm DM}|z)$ as a nuissance distribution, while
cosmological studies aim to resolve $p({\rm DM}|z)$ and attempt to remove the
$p(z)$ factor. They are thus fundamentally coupled problems. Unbiased
estimates of the cosmological distribution of ionised gas require knowing the
FRB population and the consequent biasing effects on measured dispersion
measures; and understanding the FRB population requires knowing the
dispersion-measure distribution and its biasing effects on the measured
luminosity function.
Caleb et al. (2016) provide the first comprehensive model of observational
biases on a simulated burst population, and FRBPOPPY (Gardenier et al., 2019)
is being developed to include such effects. To date however, only Luo et al.
(2020) have used this approach to fit population parameters. The authors study
a sample of FRBs detected in the $\sim$1 GHz band from Parkes, the Upgraded
Molonglo Synthesis Telescope (UTMOST), the Australian Square Kilometre Array
Pathfinder (ASKAP), Arecibo, and the Greenbank Telescope (GBT). The authors
evaluate the validity of their model using MULTINEST (Feroz et al., 2009),
which applies a Bayesian framework, and find a peak FRB luminosity $L^{*}$ of
$2.9_{-1.7}^{+11.9}10^{44}$ erg s-1, a differential power-law index of
$-1.79_{-0.35}^{0.31}$, and a volumetric rate of $3.7_{-2.4}^{+5.7}\,10^{4}$
Gpc-3 yr-1 above $10^{42}$ erg s-1.
In this work, we significantly improve upon FRB population models in the
following manner:
* •
using an unbiased sample of FRBs from ASKAP and Parkes;
* •
using seven localised FRBs detected by ASKAP;
* •
correctly accounting for the full telescope beamshape;
* •
using the measured signal-to-noise ratio in probability estimates;
* •
including the intrinsic spread in the cosmological DM contribution due to
large-scale structure and galaxy halos;
* •
and allowing for redshift evolution of the FRB event rate per comoving volume.
As with other population models, to make the problem tractable, we assume that
the cosmological rate evolution, host galaxy DM contribution, burst width, and
burst energy distributions are all independent; and that FRB observations are
random and uncorrelated, i.e. we do not model rapidly repeating FRBs.
We begin by describing the ingredients to our model: a model of the DM
distribution of FRBs as a function of redshift (Section 2), based on the model
of Macquart et al. (2020); the intrinsic properties of FRBs, such as their
burst width and the luminosity function, using a standard power-law
description (Section 3); and the influence of observational effects such as
beamshape and search sensitivity (Section 4). The method to combine these to
calculate the expected “z-DM” distribution for observed FRB surveys is given
in Section 5. In Section 6, we describe the data from ASKAP and Parkes to
which we fit our model using maximum-likelihood methods. The best-fit FRB
population parameters, and their uncertainties, are given in a companion paper
(James et al., 2021). In this work, we present detailed comparisons to the
observed DM, redshift, and signal-to-noise ratio distributions in Section 8,
where we test for goodness-of-fit and search for deviations from expectations.
Section 9 shows our estimates for the expected z–DM distribution of FRBs
detected by ASKAP and Parkes. We summarize our results in Section 10. We
attach in appendices a discussion of neglected effects in our modelling, and
extra data for alternative source evolution scenarios.
## 2 Dispersion measure distribution
The distribution of dispersion measure, DM, of FRBs from a given redshift $z$,
$p({\rm DM}|z)$, is of both intrinsic interest, and is a nuissance factor in
calculating the properties of the FRB population itself. Here, we use the
method and parameters of Macquart et al. (2020). We model the DM of an FRB as
$\displaystyle{\rm DM}$ $\displaystyle=$ $\displaystyle{\rm DM}_{\rm ISM}+{\rm
DM}_{\rm halo}+{\rm DM}_{\rm cosmic}+{\rm DM}_{\rm host},$ (1)
with respective contributions from the Milky Way’s interstellar medium (ISM),
it’s halo, the cosmological distribution of ionised gas, and the FRB host. In
this work, we divide this into an ‘extragalactic’ contribution,
$\displaystyle{\rm DM}_{\rm EG}$ $\displaystyle\equiv$ $\displaystyle{\rm
DM}_{\rm cosmic}+{\rm DM}_{\rm host},$ (2)
and a ‘local’ contribution,
$\displaystyle{\rm DM}_{\rm local}$ $\displaystyle\equiv$ $\displaystyle{\rm
DM}_{\rm ISM}+{\rm DM}_{\rm halo}.$ (3)
The ‘local’ contribution is subtracted from FRB observations, and thus all
comparisons between expectations and measurements are made in terns of ${\rm
DM}_{\rm EG}$. This model slightly differs from that in Macquart et al.
(2020), who model both ${\rm DM}_{\rm host}$ and ${\rm DM}_{\rm halo}$ using
the same nuisance term, $DM_{X}$. The distinction becomes important at large
redshifts.
### 2.1 DMISM
We use the NE2001 model (Cordes & Lazio, 2002),111Ben Bar-Or, J. Prochaska,
available at https://readthedocs.org/projects/ne2001/ to estimate the Galactic
contribution to dispersion measure. Since DM is an ingredient in the
calculation of detection efficiency (see Section 4.2), the full integral for
the FRB rate extends over the pointing direction as a function of Galactic
coordinates, as discussed in Section A.3 and Eq. 44. Since most FRBs and FRB
searches have been at high Galactic latitudes however, we use the mean value
$\overline{{\rm DM}}_{\rm ISM}$ to calculate the sensitivity for each survey,
while using the individual values ${\rm DM}_{\rm ISM}$ when calculating FRB
likelihoods.
### 2.2 DMhalo
The exact contribution of the Milky Way halo to DM is uncertain, with
estimates of order 10–80 pc cm-3 (Prochaska & Zheng, 2019; Keating & Pen,
2020). FRBs have been observed down to a DM of little more than 110 pc cm-3
(CHIME/FRB Collaboration et al., 2019a) and 114 pc cm-3 (Shannon et al.,
2018), favouring the middle of this range and consistent with current
estimations based on the full set of observed DMs (Platts et al., 2020). We
therefore use a value of DM${}_{\rm halo}=50$ pc cm-3 in our default model.
Deviations between our assumed values for DMISM and DMhalo will be absorbed
into our model for the host galaxy contribution.
### 2.3 Cosmological DM
We caution that symbols $E$, $F$, and $\alpha$ have different definitions in
this section than in the remainder of this work. The notation regarding
DMcosmic is derived from Macquart et al. (2020), and we preserve it for ease
of reference to that work.
The ‘cosmological’ contribution to DM, DMcosmic, can be understood as the DM
incurred when an FRB is emitted at $z$ at a random point in the Universe and
propagates until the current epoch, $z=0$. A parameterisation based on
detailed simulations (McQuinn, 2014) is given in Macquart et al. (2020), as a
function of burst redshift $z$, as
$\displaystyle p({\rm DM_{\rm cosmic}}|z)$ $\displaystyle=$
$\displaystyle\left<{\rm DM}_{\rm cosmic}\right>p(\Delta_{\rm DM}).$ (4)
The expected value $\left<{\rm DM}_{\rm cosmic}\right>$ is calculated as per
Ioka (2003); Inoue (2004):
$\displaystyle\left<{\rm DM}_{\rm cosmic}\right>$ $\displaystyle=$
$\displaystyle\int_{0}^{z}\frac{c\bar{n}_{e}(z^{\prime})dz^{\prime}}{H_{0}(1+z^{\prime})^{2}E(z)},$
(5) $\displaystyle E(z)$ $\displaystyle=$
$\displaystyle\sqrt{\Omega_{m}(1+z^{\prime})^{3}+\Omega_{\Lambda}},$ (6)
using the mean density of ions, $\bar{n}_{e}$, and cosmological parameters
relevant for the range $0\leq z\leq 5$: $H_{0}=67.4$ km s-1 Mpc-1, and matter
and dark energy densities $\Omega_{m}=0.315$ and $\Omega_{\Lambda}=0.685$ for
a critical density Universe. See Macquart et al. (2020) and references
contained therein for further details — cosmological parameters are taken from
Planck Collaboration et al. (2018).
The probability of deviations from the mean, $p(\Delta_{\rm DM})$, is given by
$\displaystyle p(\Delta_{\rm DM})$ $\displaystyle=$ $\displaystyle
A\Delta^{-\beta}_{\rm DM}\exp\left[-\frac{(\Delta^{-\alpha}_{\rm
DM}-C_{0})^{2}}{2\alpha^{2}\sigma_{\rm DM}^{2}}\right],$ (7)
with $\alpha=3$, $\beta=3$, and $C_{0}$ being numerically tuned such that the
expectation value of the distribution is unity. The degree of feedback $F$ is
reflected in the choice of $\sigma_{\rm DM}=Fz^{-0.5}$. In this work, we fix
$F=0.32$ based on results of (Macquart et al., 2020).
The resulting distribution of DMcosmic, $p({\rm DM}_{\rm cosmic}|z)$ is shown
in Figure 1.
### 2.4 DMhost
The contribution of the FRB host galaxy (including the local environs of the
FRB itself) to DM is highly uncertain. Some FRBs, most notably FRB 121102 and
FRB 190608, show a large excess DM beyond what is expected from cosmological
and MW contributions, which cannot be explained by passage through an
intervening galaxy along the line-of-sight (Spitler et al., 2014; Chatterjee
et al., 2017; Tendulkar et al., 2017; Hardy et al., 2017; Chittidi et al.,
2020). Yet as noted in Section 2.2, many FRBs do not allow for a great deal of
excess DM. Macquart et al. (2020) generically model this large spread using a
log-normal distribution
$\displaystyle p({\rm DM}_{\rm host}^{\prime})=\frac{1}{{\rm DM_{\rm
host}^{\prime}}}\frac{1}{\sigma_{\rm
host}\sqrt{2\pi}}\exp\left[-\frac{(\log{\rm DM}^{\prime}_{\rm host}-\mu_{\rm
host})^{2}}{2\sigma_{\rm host}^{2}}\right].$ (8)
We also correct the host contribution for redshift via
$\displaystyle{\rm DM}_{\rm host}$ $\displaystyle=$ $\displaystyle\frac{{\rm
DM}_{\rm host}^{\prime}}{1+z}.$ (9)
In this work, we use $\mu_{\rm host}$ and $\sigma_{\rm host}$ as free
parameters. Thus uncertainties in other DM contributions — including from our
assumed value of feedback $F$ — will be absorbed into these quantities.
### 2.5 The intrinsic z–DM grid
The probability distribution of observation-independent factors, ${\rm
DM}_{\rm cosmic}+{\rm DM}_{\rm host}$, is given in Figure 1. In this work, a
linear grid in both $z$ and DM space is used, with 1200 DM points spaced from
0–7000 in intervals of 5 pc cm-3, and 500 in redshift from 0.01–5. FRBs have
their nominal local contributions, ${\rm DM}_{\rm local}$ subtracted from
their observed values of DM prior to evaluating their likelihood on this grid.
In this model, only a small fraction of FRBs will have a DM very much larger
than the mean. In particular, for $z<1.5$, the majority of the spread in DM
comes from the host galaxy, rather than the cosmological contribution.
Figure 1: Distribution $p({\rm DM}|z)$ of observation-independent DM
parameters, $DM_{\rm cosmic}$ only (top), and also including the best-fit
distribution for $DM_{\rm host}$ derived in James et al. (2021) (bottom), as a
function of redshift $z$, showing contours.
## 3 FRB population
### 3.1 Energetics
Our model of the FRB population $\Phi$ is consistent with that used in the
literature. We adopt a power-law distribution of burst energies $E$ between
$E_{\rm min}$ and $E_{\rm max}$ with integral slope $\gamma$. We use ‘burst
energy’ as the isotropic equivalent energy at $1.3$ GHz, and assume an
effective bandwidth of 1 GHz when converting between ‘per Hz’ and total
quantities.
In this model, the probability of a burst occurring above a threshold $E_{\rm
th}$ is a piecewise function,
$\displaystyle p(E>E_{\rm th})$ $\displaystyle=$ $\displaystyle
1~{}~{}~{}(E<E_{\rm min})$ $\displaystyle p(E>E_{\rm th})$ $\displaystyle=$
$\displaystyle 0~{}~{}~{}(E>E_{\rm max})$ $\displaystyle p(E>E_{\rm th})$
$\displaystyle=$ $\displaystyle\frac{\left(\frac{E_{\rm th}}{E_{\rm
min}}\right)^{\gamma}-\left(\frac{E_{\rm max}}{E_{\rm
min}}\right)^{\gamma}}{1-\left(\frac{E_{\rm max}}{E_{\rm
min}}\right)^{\gamma}}~{}{\rm otherwise}.$ (10)
Observations show no evidence for a minimum FRB energy, and indeed the event
rate is generally insensitive to $E_{\rm min}$ for $\gamma>-1.5$ (Macquart &
Ekers, 2018b). Thus we use a very low value of $E_{\rm min}=10^{30}$ erg,
which is several orders of magnitude below the minimum detected burst energy
of all known FRBs — including SGR 1935+2154 at $10^{34}$–$10^{35}$ erg (The
Chime/Frb Collaboration et al., 2020; Bochenek et al., 2020) — and treat this
as a fixed parameter. We re-examine this assumption in Section 8.3.
Several other authors have used a Schechter function to model the FRB
luminosity function (Lu & Piro, 2019; Luo et al., 2020), which adds an
exponential cut-off of the form $\exp(-E/E_{\rm max})$ to the luminosity
function of Eq. 37. This is neither observationally nor theoretically
motivated — the form of the Schechter function is used to model galaxy
luminosities — and adds computational complexity, but avoids the unphysicality
of a sharp cut-off. A preliminary investigation showed that using the
Schechter function provided effectively identical fits to the data. We
therefore advocate for the simple power-law model out of simplicity.
To calculate $E_{\rm th}$, we convert between FRB energy $E$ and observable
fluence $F$ using
$\displaystyle E(F)$ $\displaystyle=$ $\displaystyle\frac{4\pi
D_{L}^{2}(z)}{(1+z)^{2+\alpha}}\Delta\nu F,$ (11)
where $\alpha$ is the spectral index ($F\propto\nu^{\alpha}$), and $\Delta\nu$
the bandwidth (here we use 1 GHz). Macquart et al. (2019) fit $\alpha$ to 23
FRBs detected by ASKAP in Fly’s Eye mode, finding $\alpha=-1.5_{-0.3}^{+0.2}$.
Thus we use a default value of $\alpha=-1.5$. We return to the interpretation
of $\alpha$ shortly.
### 3.2 Population evolution
The rate of FRBs per comoving volume will likely be a function redshift. While
FRB host galaxies do not appear to be drawn from a population sampled
proportionally to their star-forming activity (Safarzadeh et al., 2020), they
certainly are not exclusively associated with very old galaxies in which star-
forming activity has ceased (Bhandari et al., 2020a; Heintz et al., 2020). We
therefore adopt the approach of Macquart & Ekers (2018b) and generically model
the population evolution of FRBs as being to some power of the star-formation
rate, i.e.
$\displaystyle\Phi(z)$ $\displaystyle=$
$\displaystyle\frac{\Phi_{0}}{1+z}\left(\frac{{\rm SFR}(z)}{{\rm
SFR}(0)}\right)^{n},$ (12)
with $\Phi_{0}$ — and hence $\Phi(z)$ — taking the units of bursts per proper
time per comoving volume, i.e. bursts yr-1 Mpc-3. The factor of $(1+z)^{-1}$
converts between proper time in the emission and observer frames. We take
SFR$(z)$ from Madau & Dickinson (2014),
$\displaystyle{\rm SFR}(z)$ $\displaystyle=$ $\displaystyle
1.0025738\frac{(1+z)^{2.7}}{1+\left(\frac{1+z}{2.9}\right)^{5.6}}.$ (13)
This model is useful in that $n$ can be scaled as a smooth parameter. However
it does not accurately model the source evolution should FRB progenitors
originate from e.g. binary mergers with long delay times, as investigated by
Cao et al. (2018).
The total FRB rate in a given redshift interval $dz$ and sky area $d\Omega$
will also be proportional to the total comoving volume $dV$,
$\displaystyle\frac{dV}{d\Omega dz}$ $\displaystyle=$ $\displaystyle
D_{H}\frac{(1+z)^{2}D_{A}^{2}(z)}{E(z)},$ (14)
for angular diameter distance $D_{A}$, Hubble distance $D_{H}=c/H_{0}$, and
scale factor $E(z)$ from Eq. 6.
Figure 2: Relative rate of FRB detections in $z$–DM space when ignoring
beamshape and burst width effects for the best-fit FRB population parameters
presented in James et al. (2021). All observational biases from Section 4 have
been ignored, and a constant detection threshold of 1 Jy ms is used. The
Macquart relation, approximated by DM=$\left<{\rm DM}_{\rm
cosmic}\right>$+$\exp(\mu_{\rm host})$, is also shown. The contours represent
50% (dotted), 90% (dash-dot), and 99% (dashed) of the FRB population.
Applying this model with the best-fit FRB population parameters derived in our
accompanying work to the DM-distribution of Section 2 with a nominal threshold
of 1 Jy ms produces the distribution of FRBs shown in Figure 2. This ignores
the important observational biases to be introduced in Section 4, and hence a
quantitative analysis of the implications are left to Section 8. However, it
is clear that while at least 90% of FRBs will follow a 1–1 DM-z relation (the
Macquart relation Macquart et al., 2020), a significant minority will lie well
above the DM–z curve. Indeed the highest DM events in a large sample are not
likely to be the most distant. Consider DM$\geq 3000$ pc cm-3 in Figure 2.
Such events can be produced by $z\sim$3.2 lying on the Macquart relation.
However, they must be the most intrinsically luminous FRBs to be detectable.
At $z\sim$1.6, observations probe a factor of $\sim 4$ further down the energy
distribution, allowing a greater number of events to be visible, and its high-
DM tail may dominate the DM$\geq 3000$ pc cm-3 event rate. This effect becomes
more important for steeper luminosity distributions (large negative $\gamma$)
— this plot uses $\gamma=-1.2$.
### 3.3 Interpretation of $\alpha$
Many FRBs have a limited band occupancy (originally noted for FRB 121102 by
Law et al., 2017), in which case the notion of a spectral index for an
individual FRB has little meaning. In this case, the results of Macquart et
al. (2019) can be interpreted as meaning either that there are more low-
frequency FRBs, or that low-frequency FRBs are stronger. For an experiment
with bandwidth similar to or less than that of the FRB bandwidth (which is the
case with the data used here — see Section 6), the latter interpretation
behaves identically to that of broadband bursts defined by a spectral index.
However, the interpretation of an FRB population with a frequency-dependent
rate does not. We denote this interpretation as the ‘rate interpretation’ of
$\alpha$, and that of Macquart et al. (2019) as the ‘spectral index’
interpretation.
Under the spectral index interpretation of $F(\nu)\sim\nu^{\alpha}$, a
negative $\alpha$ increases the detection threshold $E_{\rm th}$ at high $z$
due to the k-correction factor of $(1+z)^{-\alpha}$ through Eq. 11. This in
turn decreases the rate by a factor $(1+z)^{\gamma\alpha}$ when $E_{\rm
th}<<E_{\rm max}$ through Eq. 10. Under the rate interpretation, the FRB
population itself behaves as $\Phi\sim\Phi(z,\nu)=\Phi(z)\nu^{\alpha}$, and
the k-correction therefore directly changes the rate, adding an additional
factor of $(1+z)^{\alpha}$ to Eq. 12. Therefore, when $\gamma=-1$, and $E_{\rm
th}<<E_{\rm max}$, the two interpretations are identical. The situation
becomes less simple near $E_{\rm max}$, which is frequency dependent under the
spectral index interpretation, and constant under the rate interpretation —
and the true behaviour may be more complicated than either result.
Ultimately, we expect further observational data to be required to
discriminate between the two scenarios, and consider both equally plausible
for the time being. In this work, we present results using the spectral-index
interpretation, but give additional data for the rate interpretation when
constraining FRB population parameters.
## 4 Detection threshold — observational biases
FRB surveys usually calculate the fluence threshold above which FRBs would be
detected using the radiometer equation, referenced to a 1 ms duration burst,
using the sensitivity of the telescope at beam centre. This readily calculable
value represents an unrealistic ideal. Bursts of longer duration will be
harder to detect due to increased noise, while those viewed away from beam
centre will be seen at less sensitivity. Furthermore, incoherent dedispersion
searches will not perfectly match the shape of an FRB to the time–frequency
resolution of the search, resulting in a lower detection efficiency.
In this work, we model the effective fluence threshold $F_{\rm th}$ as a
function of nominal fluence threshold at 1 ms $F_{1}$, beam sensitivity $B$
(normalized to a maximum of 1), and an efficiency factor due to burst
duration, $\eta$, as
$\displaystyle F_{\rm th}$ $\displaystyle=$ $\displaystyle\frac{F_{1}}{\eta
B}.$ (15)
This results in a theoretical distribution of bursts in z–DM space, $p(z,{\rm
DM})$, as
$\displaystyle p(z,{\rm DM})=\int dB\int d\eta p(z,{\rm DM}|F_{\rm
th}(\eta,B))\Omega(B)p(\eta),$ (16)
where $p(z,{\rm DM}|F_{\rm th})$ is the distribution at a fixed threshold
(Figure 2), $\Omega(B)$ is the region of sky at which the beam sensitivity is
$B$, and $p(\eta)$ is the probability that burst properties lead to a total
detection efficiency $\eta$. The effects of these two factors are investigated
in the following sections.
### 4.1 Beamshape
A telescope’s beamshape is usually represented as the relative sensitivity $B$
as a function of the sky position $\Omega$ relative to boresight, $B(\Omega)$,
such that $B(0)=1$. The beamshape is often approximated as a Gaussian or Airy
function, although precise measurements of $B$ can become important when
attempting to localise FRBs detected in multiple beams, or estimating the
relative rate of single- vs multiple-beam detections (Vedantham et al., 2016;
Macquart & Ekers, 2018b).
For the purpose of estimating the number of FRBs detected however, the
‘inverse beamshape’, $\Omega(B)$, which describes the amount of sky $\Omega$
viewed at any given sensitivity $B$, becomes more relevant (James et al.,
2019a). Most calculations of FRB rates have characterised a telescope beam as
viewing out to the FWHM at full sensitivity, i.e. $\Omega(B)=\Omega_{\rm
FWHM}\delta(B-1)$ (e.g. Thornton et al., 2013; Bhandari et al., 2018). Others
have used a Gaussian approximation for the beamshape (e.g. Lawrence et al.,
2017), which is equivalent to $\Omega(B)d\log B={\rm const}$. We here analyse
the sufficiency of these approximations, using for the Gaussian approximation
$\sigma=({\rm FWHM}/2)(2\log 2)^{0.5}$, where the full width at half maximum
(FWHM) assumes an Airy disk, i.e. HPBW=$1.22\lambda/D$ for wavelength at
central frequency $\lambda$ and dish diameter $D$.
ASKAP FRB observations have varied the observation frequency and configuration
of beams formed from ASKAP’s phased array feeds (PAFs). However, the majority
of both fly’s eye and incoherent sum observations have used the ‘closepack36’
configuration at a central frequency of 1.296 GHz. We therefore use the
beamshape derived in James et al. (2019a). In the case of the Parkes
multibeam, we use a central frequency of 1.382 GHz, and the simulations of K.
Bannister (published as Vedantham et al., 2016) and A. Dunning (referenced as
‘private communication’ by Macquart & Ekers, 2018a)), which produce equivalent
results for $\Omega(B)$. This also allows us to conclude that while the
ability to localise FRBs detected in multiple beams may be limited by
systematic uncertainties in the beamshape (Macquart & Ekers, 2018a), the
inverse beamshape $B(\Omega)$ is robust against such certainties, since it
does not care about where on the sky any given patch of sensitivity is
located.
Figure 3: Inverse beamshape $\Omega(B)$ as a function of beam sensitivity $B$
for ASKAP (closepack36 configuration at 1.296 GHz) and the Parkes Multibeam
(1.382 GHz). Shown are the best measurements (lines), FWHM approximation
(crosses), the numerical approximations used in this work (circles), and the
Gaussian beamshape (flat horizontal lines). The points for the FWHM and
numerical approximations represent $\delta$ functions — in the FWHM case, the
true amplitude is shown, while in the numerical approximations, the amplitudes
are renormalised by their spacing in $\log B$ for comparability with the best
measurements. Figure 4: Top: dependence of expected FRB DM distributions (pc
cm-3) on beamshape models for three different FRB surveys. The beamshape
models considered are shown in Figure 3: the‘FWHM’ approximation, the ‘Full’
beamshape $\Omega(B)$, a numerical approximation used in ‘this work’, and a
Gaussian beamshape. All curves for each telescope are normalised such that the
distribution for the full beamshapes peaks at unity. Bottom: the difference
between the ‘Full’ beamshape and that found when using the ‘FWHM’, ‘Approx.’,
and Gaussian approximations.
Figure 3 shows the resulting ‘inverse beamshape’ function $\Omega(B)$. This is
compared to the equivalent $\Omega(B)$ when using the Gaussian and
$\Omega_{\rm FWHM}$ approximations. Since implementing the full function
$\Omega(B)$ in the calculation of the z–DM distribution is numerically
expensive, we investigate the accuracy of reducing $\Omega(B)$ to a small
number of values. A set of such values is also shown in Figure 3. The accuracy
of all approximations is assessed against the full beamshape, by comparing the
total predicted number of events and the mean value of DM to those calculated
for the full beamshape, and also assessing the maximum difference between the
$p({\rm DM})$ curves, as shown in Figure 4. Table 1 lists the resulting
errors. This is evaluated for the best-fit set of parameters found in our
accompanying paper — however, a brief investigation has shown that results are
not sensitive to the assumed parameters within a reasonable range.
Table 1: Percentage errors in the total FRB rate and mean DM value, $\overline{\rm DM}$, and maximum deviation $|\delta{\rm DM}|_{\rm max}$, when using different beam approximations when compared to that found for the full beamshape function $\Omega(B)$, for each of three FRB surveys considered. Survey | Approximation | Rate | $\overline{\rm DM}$ | $|\delta{\rm DM}|_{\rm max}$
---|---|---|---|---
ASKAP/FE | FWHM | +76 | +6 | 0.1
This work | +17 | +0.2 | 0.009
Gauss | +513 | -3 | 0.07
ASKAP/ICS | FWHM | +73 | +8 | 0.11
This work | +5.8 | +0.2 | 0.005
Gauss | +440 | -4 | 0.08
Parkes/Mb | FWHM | +20 | +14 | 0.15
This work | +2.8 | +0.2 | 0.005
Gauss | +160 | +1 | 0.011
We find that using five values of $B$ for ASKAP, and ten for Parkes, achieves
an approximate DM distribution with an error in $p({\rm DM})$ of less than 1%,
and an error in the mean value $\overline{\rm DM}$ of only 0.2%. In contrast,
using the FWHM approximation, which is the standard in the current literature,
results in $\mathcal{O}\sim$10% deviations in the DM distribution and its mean
value, and pushes the mean towards higher values. The total expected detection
rate found when using the FWHM approximation is almost double that when found
when using the full ASKAP beam, but there is only a 20% excess for Parkes.
When assuming Gaussian beams, a huge excess in the total rate of ASKAP bursts
is predicted, since this does not account for the closely packed, and thus
overlapping, beams. We note that uncertainties in the true ASKAP beamshape due
to the calibration procedure (see James et al., 2019a) are less than the
errors introduced by our numerical approximation.
In the case of Parkes/Mb, the excess rate when using a Gaussian beam is due to
outer beams being less sensitive than the central beam at which the
sensitivity is usually calculated. However, the Gaussian beam approximation
accurately estimates $\overline{\rm DM}$ and the shape of $p({\rm DM})$
distribution. This suggests that even very complex beamshapes, such as that of
the Canadian Hydrogen Intensity Mapping Experiment (CHIME), could be included
in our model in a relatively simplified manner.
### 4.2 Detection efficiency
We model the threshold at which an FRB of fixed fluence $F$ can be detected as
scaling with the square root of its effective width, $w_{\rm eff}$, relative
to the nominal width of 1 ms, using an efficiency factor $\eta$:
$\displaystyle\eta$ $\displaystyle\equiv$ $\displaystyle\frac{F_{1}}{F_{\rm
th}(w_{\rm eff})}$ (17) $\displaystyle=$ $\displaystyle\left(\frac{w_{\rm
eff}}{1\,{\rm ms}}\right)^{-0.5}.$ (18)
The effective width is modelled as per Cordes & McLaughlin (2003), being a
function of its intrinsic duration $w_{\rm int}$, scattered width $w_{\rm
scat}$, DM smearing within each frequency channel $w_{\rm DM}$, and the time-
resolution of the search $w_{\rm samp}$:
$\displaystyle w_{\rm eff}$ $\displaystyle=$ $\displaystyle\sqrt{w_{\rm
int}^{2}+w_{\rm scat}^{2}+w_{\rm DM}^{2}+w_{\rm samp}^{2}}.$ (19)
Often, the scattered width and intrinsic width are indistinguishable, and
their separation only becomes important for telescopes observing at different
frequencies. We therefore define the ‘incident’ width, $w_{\rm inc}$, as
$\displaystyle w_{\rm inc}^{2}$ $\displaystyle=$ $\displaystyle w_{\rm
int}^{2}+w_{\rm scat}^{2}.$ (20)
An alternative model is presented by Arcus et al. (2021), which is based on
fits to simulated ASKAP and Parkes FRBs. Since it is not clear how the fit
parameters translate to the general case, and because we wish to present a
broadly scalable model, we do not use their formulation. We remark rather that
the widely used model of Eq. 19 should be investigated further.
In order to model the distribution of $w_{\rm inc}$, $p(w_{\rm inc})$, we use
a log-normal distribution,
$\displaystyle p(w_{\rm inc})dw_{\rm inc}=\frac{w_{\rm
inc}}{\log\sigma_{w}(2\pi)^{0.5}}e^{-\frac{1}{2}\left(\frac{\log w_{\rm
inc}-\log\mu_{w}}{\log\sigma_{w}}\right)^{2}}.$ (21)
We do not include any DM or $z$ dependence in the width distribution — see
Appendix A.4 for further discussion on this topic.
Unlike Luo et al. (2020), we do not include fits of the model parameters
$\mu_{w}$ and $\sigma_{w}$ as part of our general fitting process. Rather, we
use the low correlation between $\mu_{w}$, $\sigma_{w}$ and other parameters
to first fit for these values using a preliminary parameter set, and then
check that the fit is still valid for the final parameter set presented in our
companion paper (James et al., 2021).
Figure 5: The effect of intrinsic burst width on survey sensitivity. Blue: intrinsic distribution of burst widths $w_{\rm int}$ (peak set to unity). Orange/green/red: the simulated width distributions expected from the three FRB surveys after accounting for observational bias. Two normalisations are shown: the upper curves have been normalised to a peak probability of unity, for comparison with the fit to observed values by Arcus et al. (2021) (shown in black); while the lower curves are normalised relative to the rate for $w=0$, and thus illustrate the relative reduction in event rate as a function of intrinsic width. The upper curves for ASKAP/FE and ASKAP/ICS coincide almost identically with the results of Arcus et al. (2021). Table 2: The effects of different assumptions on the intrinsic burst width distribution at $1.3$ GHz for three FRB surveys. Given are the total detection rate (normalised to a rate of unity at $w=0$), and mean DM $\overline{\rm DM}$. The width distributions are parameterised via Eq. 21, with given values of $\mu_{w}$ and $\sigma_{w}$ corresponding to no intrinsic width, the observed width distribution of Arcus et al. (2021), when accounting for observational bias to find the intrinsic distribution, and when numerically approximating ($\sim$) the intrinsic distribution with a few characteristic values. Parameter | $\mu_{w}$ | $\sigma_{w}$ | ASKAP/FE | ASKAP/ICS | Parkes
---|---|---|---|---|---
Rate | 0 | 0 | 1 | 1 | 1
2.67 | 2.07 | 0.46 | 0.51 | 0.20
5.49 | 2.46 | 0.27 | 0.30 | 0.11
$\sim$5.49 | $\sim$2.46 | 0.27 | 0.30 | 0.11
$\overline{\rm DM}$ | 0 | 0 | 263 | 371 | 488
2.67 | 2.07 | 286 | 397 | 724
5.49 | 2.46 | 293 | 401 | 726
$\sim$5.49 | $\sim$2.46 | 292 | 400 | 724
Figure 6: Effect of using different distributions of the intrinsic width
$w_{\rm int}$ on the expected DM distribution of FRBs, $p({\rm DM})$, from the
FRB surveys considered here: ASKAP/FE (left-most), ASKAP/ICS (centre), and
Parkes/Mb (right-most). Assuming no intrinsic width (blue solid), using a log-
normal distribution of widths with parameters from Arcus et al. (2021) (green
dotted), using a log-normal distribution with parameters derived in this work
(red dashed), and numerically approximating the latter (black dot-dashed). The
distributions are normalised to a peak of unity for illustrative purposes.
Arcus et al. (2021) use Eq. 21 to model the observed width distribution
$p(w_{\rm inc})$ of ASKAP and Parkes FRBs, finding $\mu_{w}=2.67$ ms and
$\sigma_{w}=2.07$. We instead use Eq. 21 to model the intrinsic width
distribution, and vary $\mu_{w},\sigma_{w}$ until the simulated width
distribution of the ASKAP/FE survey matches the parameterisation of observed
widths by Arcus et al. (2021). We obtain $\mu_{w}=5.49$ and $\sigma_{w}=2.46$,
and then proceed to use these values in further calculations to optimise FRB
population parameters. This is shown in Figure 5, with total rates, and mean
estimated DM values, given in Table 2. Finally, we re-evaluate the fits using
the final optimal set of parameters we present in our companion paper (James
et al., 2021).
Comparing the intrinsic (blue) and observed (black, coloured) distributions in
Figure 5, modelling the intrinsic FRB rate, and accounting for observational
bias, correctly reproduces the observed FRB width distribution as estimated by
Arcus et al. (2021). The effect of observational bias is clearly reflected in
the total expected FRB rate, with high-width bursts much less likely to be
detected. The magnitude of this effect, shown in Table 2, depends on the
significance of the other terms in Eq. 19. When the time resolution is poor
($w_{\rm samp}$ large), the effect of a large intrinsic width is less — thus
the reduction in rate for ASKAP is less than that for Parkes surveys. A
second-order effect is that more-sensitive surveys, which probe further into
the Universe and see FRBs with on-average higher DMs, are also less sensitive
to $w_{\rm int}$, since $w_{\rm DM}$ is larger. Hence the reduction in rate
for ASKAP/ICS is slightly lower than for ASKAP/FE, while the greatest effect
is for the Parkes/Mb survey, where the intrinsic FRB width reduces the number
of detected FRBs by a factor of 10.
A consequence of this is that the true number of high-width FRBs will be very
difficult to estimate, so that our log-normal model is effectively untested
beyond $w=10$ ms. Thus while we estimate that ASKAP/FE and ASKAP/ICS miss
$\sim 70$% of FRBs due to their intrinsic width, and Parkes/Mb 90%, we do not
consider this quantitatively reliable — there may be virtually any number of
high-width events remaining to be detected.
Whatever the lost rate, losses will preferentially arise from the nearby
Universe where $w_{\rm DM}$ is low. Including the width distribution therefore
increases the mean expected DM, $\overline{\rm DM}$. This effect is small
($\sim 10$%) for ASKAP observations, but more significant for Parkes ($\sim
30$%).
Finally, we note that while including the width distribution is clearly very
important, the details matter less. Using the parameters of Arcus et al.
(2021), or approximating the true distribution with a small number of points
for computational efficiency, produces an almost identical value of
$\overline{\rm DM}$. This also means that the loss of efficiency to high-width
bursts for the HEIMDALL222 https://sourceforge.net/p/heimdall-astro/wiki/Home/
search pipeline found by Gupta et al. (2021) — which is commonly used in
Parkes FRB searches — is insignificant to our modelling.
### 4.3 Numerical implementation
The integrals over $B$ and $\eta$ in Eq. 16 are numerically expensive.
Furthermore, we have shown above that approximating the beamshape with 5–10
values, and the width distribution with five, allows for a very good
approximation to the continuous distributions. Therefore, we approximate these
continuous distributions with these discrete distributions, i.e.
$\displaystyle\Omega(B)$ $\displaystyle\sim$
$\displaystyle\sum_{i}^{N_{B}}\Omega_{i}\delta(B-B_{i})$ (22) $\displaystyle
p(\eta(DM,w))$ $\displaystyle\sim$
$\displaystyle\sum_{i}^{N_{w}}p_{i}\delta(\eta(DM,w_{w})-\eta(DM,w_{i})),$
(23)
recalling that $\eta$ is a function of both DM and $w$ through Eq. 18 and 19.
Therefore, the z–DM distribution of Eq. 16 becomes a weighted sum over
individual distributions at fixed observational thresholds,
$p(z,{\rm DM})=\sum_{i=1}^{N_{B}}\sum_{j=1}^{N_{w}}p\left(z,{\rm DM},F_{\rm
th}(B_{i},\eta({\rm DM},w_{j}))\right)\Omega_{i}p_{j}.$ (24)
## 5 Methodology
The ingredients described above are implemented in Python. Here, we describe
its effective implementation, which is identical in approach to the method
proposed by Connor (2019) and implemented by Luo et al. (2020), even if the
exact details differ.
We consider FRB data from some number of independent FRB surveys. The total
likelihood ${\mathcal{L}}$ of the outcome of multiple independent FRB surveys
is simply a product of their individual likelihoods, ${\mathcal{L}}_{i}$,
$\displaystyle{\mathcal{L}}=\prod_{i}^{N_{\rm surveys}}{\mathcal{L}}_{i},$
(25)
for surveys $i=1\ldots N_{\rm surveys}$. We describe ${\mathcal{L}}_{i}$ as
the product of three independent terms:
$\displaystyle{\mathcal{L}}_{i}=p_{n}(N_{i})\prod_{j=1}^{N_{i}}p_{\rm
dmz}({\rm DM}_{j},z_{j})p_{s}(s_{j}|{\rm DM}_{j},z_{j}).$ (26)
Here, $p_{n}$ is the probability of detecting $N_{i}$ FRBs in survey $i$,
while $p_{\rm dmz}$ is the likelihood of the $j^{\rm th}$ FRB from survey $i$
being detected with dispersion measure DMj and, when applicable, at redshift
$z_{j}$. We also include the probability $p_{s}$ of an FRB being detected with
fluence $F$ a factor $s$ above the fluence threshold $F_{\rm th}$ given it was
observed with properties ${\rm DM}_{j},z_{j}$. Each of these terms is
described independently below.
It is also possible to separate further terms in Eq. 26. As noted by Vedantham
et al. (2016), for a multibeam instrument, the relative likelihood of a
single- vs multi-beam detection, and the relative likelihood of detection in
different beams of varying sensitivity, are functions of the FRB fluence
distribution. Such measures are only relevant when the true FRB fluence $F$
cannot be resolved, as is common with FRB detections by the Parkes multibeam.
When $F$ can be reconstructed, as is the case for CRAFT FRB detections with
ASKAP (Shannon et al., 2018), then the survey acceptance to that particular
FRB can be calculated exactly, and $p(s)$ in Eq. 26 can be written in terms of
$p(F)$. Similarly, we will not include the observed value of an FRB’s width
when evaluating $p_{\rm dmz}$ and $p_{s}$, with only the overall width
distribution being accounted for. We consider that adding such terms will
yield only a small increase in analytic power for resolving the FRB
population, at the cost of a large increase in complexity. Thus they are
ignored — however we do acknowledge that we are discarding a small amount of
information by doing so.
### 5.1 Probability of $N$ detections, $p_{n}(N)$
Ignoring the correlations caused by repeating FRBs (see the discussion in
Section 5.4), the total number of observed FRBs in survey $i$, $N_{i}$ comes
from a Poisson distribution,
$\displaystyle p_{n}(N_{i})$ $\displaystyle=$
$\displaystyle\frac{\left<N_{i}\right>^{N_{i}}\exp(-\left<N_{i}\right>)}{N_{i}!},$
(27)
where $\left<N_{i}\right>$ is the expectation value of $N_{i}$. The
calculation of $\left<N_{i}\right>$ is the heart of the problem that we
address in this work, since it must necessarily incorporate all relevant
properties that affect the detection rate.
Combining the dependencies in Sections 2–4, $\left<N_{i}\right>$ is calculated
as
$\displaystyle\left<N_{i}\right>$ $\displaystyle=$ $\displaystyle T_{i}R_{i},$
(28) $\displaystyle R_{i}$ $\displaystyle=$ $\displaystyle\int
dz\,\Phi(z)\frac{dV(z)}{d\Omega dz}\int d{\rm DM}p({\rm DM}|z)$
$\displaystyle\int dB\,\Omega(B)\int dwp(w)p(E>E_{\rm th}(B,w,z,{\rm DM})).$
Here, $T_{i}$ is the survey duration, which multiplies a rate $R_{i}$ to
produce the total expected number of bursts. $\Phi(z)$ is the FRB source
evolution function (Eq. 12), $dV(z)/d\Omega/dz$ is the comoving volume per
steradian per redshift interval from Eq. 14, $p({\rm DM}|z)$ is the
extragalactic DM distribution found by convolving Eq. 4 and 8 (shown in Figure
1); $\Omega(B)$ is the beamshape discussed in Section 4.1 and approximated as
per Eq. 22; $p(w)$ is the width distribution of Eq. 21, discretised as per Eq.
23; and $p(E>E_{\rm th})$ is the cumulative energy function of Eq. 10. The
dependency of this threshold $E_{\rm th}$ on the parameters $(B,w,z,{\rm DM})$
is encapsulated in Eq. 11, Eq. 15, and Eq. 18.
For some surveys, no controlled survey time $T_{i}$ is available, and this
term is simply set to unity in Eq. 26. However, $R_{i}$ can be calculated
regardless of knowledge of $T_{i}$. For those surveys with known $T_{i}$, the
most likely value of the lead constant in the population function of Eq. 13,
$\Phi_{0}$, can be estimated without recalculating the integral of Eq. 5.1.
### 5.2 Calculating $p_{\rm dmz}$
The probability of an FRB being observed with a given dispersion measure DM
and redshift $z$ is given by the appropriate integrand of Eq. 5.1,
$\displaystyle p_{\rm dmz}$ $\displaystyle=$ $\displaystyle
R_{i}^{-1}\Phi(z)\frac{dV(z)}{d\Omega dz}p({\rm DM}|z)\int dB\Omega(B)$ (31)
$\displaystyle\int dwp(w)p(E>E_{\rm th}(B,w,z,{\rm DM})).$
For FRBs with no measured host redshift, the relevant quantity is
$\displaystyle p_{\rm dm}$ $\displaystyle=$ $\displaystyle\int p_{\rm dmz}dz,$
(32)
and $p_{\rm dm}$ replaces $p_{\rm dmz}$ in Eq. 26. The rate $R_{i}$ is used as
a normalising factor in Eq. 31, so that
$\displaystyle\int\int p_{\rm dmz}d{\rm DM}dz=\int p_{\rm dm}d{\rm DM}=1.$
(33)
The shape of $p_{\rm dmz}$ in $z$–DM space is a primary quantity of interest
in this work.
### 5.3 Calculating $p_{s}({\rm s}$)
The measured fluence $F$ of an FRB also holds information on the FRB
population. However, in many telescope systems — and notably for Parkes
(Macquart & Ekers, 2018a) — $F$ is not directly measured, since the location
of the FRB in the beam is not known. Furthermore, we are interested in
$p(F|F_{\rm th})$, i.e. the probability of measuring $F$ given an FRB has been
detected at threshold $F_{\rm th}$, which itself has complex dependency
through Eq. 15.
This difficulty can be overcome by noting that the signal-to-noise ratio, SNR,
is a readily observable parameter for an FRB, and most FRB-hunting systems use
a well-defined threshold SNR, SNRth, to distinguish FRBs from noise. As per
James et al. (2019b), who base their work on Crawford et al. (1970), we define
$\displaystyle s$ $\displaystyle\equiv$ $\displaystyle\frac{\rm SNR}{\rm
SNR_{\rm th}}$ (34) $\displaystyle=$ $\displaystyle\frac{F}{F_{\rm th}}$ (35)
where $F_{\rm th}$ is the fluence threshold to that FRB. As detailed in
Section 4, $F_{\rm th}$ is a function of the burst DM, width, and the location
in which it is observed by the telescope’s beam, so that neither term in Eq.
35 is known. However, the ratio is preserved in the measurable quantities of
Eq. 34, making $s$ a very useful observable.
The probability $p(s)$ of observing $s$ in the range $s$ to $s+ds$ given that
an FRB has already been observed is
$\displaystyle p_{s}(s|{\rm DM},z)$ $\displaystyle=$
$\displaystyle\frac{1}{p_{\rm dmz}}\int dB\Omega(B)\int dwp(w)\frac{dp(sF_{\rm
th})}{ds},$ $\displaystyle p_{s}(s|{\rm DM})$ $\displaystyle=$
$\displaystyle\frac{1}{p_{\rm dm}}\int\Phi(z)\frac{dV(z)}{d\Omega dz}p({\rm
DM}|z)dz$ (36) $\displaystyle\int dB\Omega(B)\int dwp(w)\frac{dp(sF_{\rm
th})}{ds}$
for localised and unlocalised FRBs respectively. In the integrands, $p(sF_{\rm
th})$ is the probability of detecting a fluence $F=sF_{\rm th}$ in an interval
between $s$ and $s+ds$, where $F_{\rm th}$ depends on $B$, $w$, and ${\rm DM}$
as per Eq. 15.
The probability $p(sF_{\rm th})$ can be found from Eq. 10. Given that such an
FRB has been observed at all, the integral distribution of $E$ given that an
FRB has been detected can be found by replacing $E_{\rm th}$ with $E$ and
$E_{\rm min}$ with $E_{\th}$. Differentiating by $E$ produces the probability
amplitude
$\displaystyle\frac{dp(E_{\rm obs}|E_{\rm th})}{dE}$ $\displaystyle=$
$\displaystyle\frac{\gamma}{E_{\rm th}}\frac{\left(\frac{E}{E_{\rm
th}}\right)^{\gamma-1}}{1-\left(\frac{E_{\rm max}}{E_{\rm
th}}\right)^{\gamma}}$ (37) $\displaystyle=$
$\displaystyle\frac{\gamma}{E_{\rm
th}}\frac{s^{\gamma-1}}{1-\left(\frac{E_{\rm max}}{E_{\rm
th}}\right)^{\gamma}}.$
The value of $E_{\rm th}$ can be found as a function of $z$ by inserting
$F_{\rm th}$ into Eq. 11; thus $p(sF_{\rm th})$ is equivalent to $p(sE_{\rm
th}(z,F_{\rm th}))$. Relating $dp/dE$ from Eq. 37 to the required $dp/ds$ of
Eq. 36 produces
$\displaystyle\frac{dp(E_{\rm obs}|E_{\rm th})}{ds}$ $\displaystyle=$
$\displaystyle\frac{dp(E_{\rm obs}|E_{\rm th})}{dE}\frac{dE}{dF}\frac{dF}{ds}$
(38) $\displaystyle=$ $\displaystyle\frac{dp(E_{\rm obs}|E_{\rm
th})}{dE}\frac{E_{\rm th}}{F_{\rm th}}F_{\rm th}$ (39) $\displaystyle=$
$\displaystyle\gamma\frac{s^{\gamma-1}}{1-\left(\frac{E_{\rm max}}{E_{\rm
th}}\right)^{\gamma}}.$ (40)
Inserting Eq. 40 into Eq. 36, and integrating over the appropriate dimensions,
produces the desired $p_{s}$.
### 5.4 What about repeating FRBs?
By writing the individual burst probabilities as being independent in Eq. 26,
and assuming that the number of detected bursts follows a Poisson distribution
in Eq. 27, we ignore the potential of FRBs to repeat. While the fraction of
the FRB population which is observed to repeat is a current topic of debate,
it is certain that many do. Formally, the FRB population described in Section
3 represents all _bursts_ , rather than all FRB emitters, and the summation of
Eq. 26 runs over all detected bursts. The distinction becomes irrelevant for
distant, rarely repeating sources for which only ever zero or one bursts will
be observed. For bright, nearby repeaters, the probability of having such an
object in a survey’s field of view is rare, especially when burstiness and/or
periodicity is included (Oppermann et al., 2018; Chime/Frb Collaboration et
al., 2020; Rajwade et al., 2020; Cruces et al., 2020), and an observation of
zero bursts will be more likely than that estimated by Eq. 27. Conversely, the
probability of observing many bursts will also be high, with observations of
single bursts being much rarer than otherwise expected.
We note that the only FRB survey to ever observe an FRB repeat in an unbiased
way are the observations by CHIME (CHIME/FRB Collaboration et al., 2019b;
Fonseca et al., 2020a, b) — all other repeating FRBs have been discovered in
targeted follow-up observations. This suggests that the majority of FRB
observations can safely be classified as being in a ‘one burst per progenitor’
regime, regardless of the true fraction which are actually repeating objects.
For these, our approach should be valid. We revisit this assumption in Section
8.4.
## 6 Surveys
Estimates of the FRB population have been made using data from many
telescopes, which are often drawn from FRBCAT (Petroff et al., 2016). Due to
the large number of FRBs they have detected and published, results from Parkes
and ASKAP remain the most important, and we focus on these instruments here.
Other important instruments we wish to examine in future works include the
Upgraded Molonglo Observatory Synthesis Telescope (UTMOST), and the Canadian
Hydrogen Intensity Mapping Experiment (CHIME).
The sensitivity of an FRB survey — and hence the functions $p_{N}$, $p_{\rm
zdm}$, and $p_{s}$ from Section 5 — depends on the local contribution to DM,
and hence varies with the value of ${\rm DM}_{\rm ISM}$. Since this fluctuates
pointing-by-pointing, in theory these functions must be recalculated for every
single pointing direction, which becomes computationally prohibitive.
Evaluating $p_{\rm zdm}$ and $p_{s}$ however for the measured DMEG and $s$ of
an FRB is much quicker. This motivates grouping FRB observations not just by
telescope, but also by other observational properties, such as Galactic
latitude. Here, we use five groups, as described below.
### 6.1 Parkes
All Parkes FRBs published so-far have used the multibeam (‘Mb’) receiver
(Staveley-Smith et al., 1996). However, a new ultra-wideband receiver is now
in place (Hobbs et al., 2020), which is being used for FRB searches and
follow-up observations. We therefore refer to results from Parkes as
“Parkes/Mb” to distinguish this from future works.
Of the many Parkes FRB discoveries, we consider only those by the High Time
Resolution Universe (HTRU; Keith et al., 2010; Thornton et al., 2013; Champion
et al., 2016; Petroff et al., 2014) and Survey for Pulsars and Extragalactic
Radio Bursts (SUPERB; Keane et al., 2017; Bhandari et al., 2018)
collaborations to have an unbiased estimate of observation time, $T_{i}$. This
is because their observation time and pointing directions were pre-determined,
and the results published regardless of outcome. Other results suffer from
publication bias whereby non-detections are less likely to be published. Thus,
while their discovery can contribute to individual FRB likelihoods via $p_{\rm
dmz}$ and $p_{s}$, no well-defined observation time $T_{i}$ exists for use in
Eq. 5.1, and the term $p_{i}(N_{i})$ must be neglected in Eq. 26. Thus they
cannot contribute to estimates of the total FRB rate.
Since the distribution of DMISM affects telescope sensitivity, surveys at low
Galactic latitudes have significantly reduced sensitivity compared to those at
high latitudes, and the full distribution of time spent at each ${\rm DM}_{\rm
ISM}$ must be accounted for. In particular, ${\rm DM}_{\rm ISM}$ will vary
significantly for each pointing at low latitudes, making estimates numerically
taxing. We therefore include only FRBs detected at mid
($19.5^{\circ}<|b|<42^{\circ}$) and high ($42^{\circ}le|b|\leq 90^{\circ}$)
Galactic latitudes. This criteria leaves 12 FRBs detected in a total of 164.4
days by HTRU and SUPERB (Bhandari et al., 2018), and another 8 FRBs by other
groups with no reliable observation time. A full list is given in Table 4.
Early searches for FRBs with Parkes used a sparse grid of DMs and arrival
times, resulting in sensitivities that would fluctuate by $\pm 15$% (Keane &
Petroff, 2015b). This was corrected with the use of
Heimdall333http://sourceforge.net/projects/heimdall-astro/, which has been
used to (re)process the data from HTRU and SUPERB. While early HTRU searches
extended only to DM=2000 pc cm-3 (Thornton et al., 2013), latter searches
extended this to 5000 pc cm-3; and while the SUPERB ‘F’ pipeline looks for
FRBs with DM$\leq 2000$ pc cm-3, the SUPERB ‘T’ pipeline extends the search to
10,000 pc cm-3. Thus we treat all Parkes FRB searches as fully covering DM
space.
For the Parkes multibeam, nominal sensitivity to a burst at beam centre is
$F_{1}=0.5$ Jy ms to a 1 ms duration burst (Keane et al., 2017) — this is an
approximation, since different FRB searches used slightly different values of
SNRth. We neglect the effect of 1-bit sampling with early searches for FRBs
with Parkes, which would have slightly degraded the sensitivity of these
observations. These and other properties are summarised in Table 4.
Table 3: Observational properties of follow-up observations, for ASKAP Fly’s Eye (FE) and incoherent sum (ICS) modes (Shannon et al., 2018; Bannister et al., 2019); the Parkes multibeam (MB) receiver (Keane et al., 2017); and the Greenbank Telescope’s (GBT’s) 820 MHz primary focus and L-band receivers (Kumar et al., 2019). From left to right: the telescope and receiver names, the central frequency $\bar{\nu}$ and total bandwidth $\Delta\nu$, time- and frequency- resolutions $\delta t$ and $\delta\nu$, and nominal sensitivity to a 1 ms duration burst. Telescope | $N_{\rm FRB}$ | $T_{\rm obs}$ [hr] | $\bar{\nu}$ | $\Delta\nu$ | $\delta t$ | $\delta\nu$ | $F_{1}$
---|---|---|---|---|---|---|---
| | Mode | [MHz] | [MHz] | [ms] | [MHz] | [Jy ms]
ASKAP/FE | 20 | 26,616 | 1315 | 336 | 1.2565 | 1 | 21.9
ASKAP/ICS | 7 | N/A | 1315 | 336 | 1.2565 | 1 | 4.4
Parkes/Mb | 13 | 3,946 | 1382 | 337.1 | 0.064 | 0.39 | 0.5
### 6.2 ASKAP
The Commensal Real-time ASKAP Fast Transients (CRAFT) group have performed
several FRB surveys with ASKAP. The majority of ASKAP FRBs have been observed
in single-antenna (“Flye’s Eye”, or “FE”) mode during the ‘lat50’ survey, i.e.
while observing Galactic latitudes of $|b|=50^{\circ}\pm 5^{\circ}$ (Bannister
et al., 2017; Shannon et al., 2018). Twenty FRBs have been initially reported
(Macquart et al., 2019; Bhandari et al., 2019), with a total recorded data
time of $1274.6$ antenna-days duration (James et al., 2019a). A further six
FRBs have been detected in a variety of surveys (Macquart et al., 2019;
Bhandari et al., 2019; Qiu et al., 2019), of which four satisfy the Galactic
latitude requirement. These are listed in Table 5. We err on the side of
caution and do not assume a known observation time for this last category,
since several other FRB searches outside the lat50 survey have been performed
and were not reported. As with Parkes, $p_{i}(N_{i})$ in Eq. 26 is only
evaluated for the former category.
All ASKAP/FE searches have used the same frequency range and time/spectral
resolutions, as given in Table 3. The beamshape and threshold for this survey
are given in James et al. (2019a).
ASKAP has recently been observing in incoherent sum mode (ICS), with voltage
buffers used in offline analysis to localise FRBs to sub-arcsecond precision
(Bannister et al., 2019). Follow-up observations with radio and optical
instruments have determined the redshifts of the host galaxies of each FRB,
allowing the DM–$z$ grid to be directly probed for the first time. This mode
has undergone an extended period of commissioning, with the number of
telescopes, observation frequency, and time resolution of the search all
varying. The total observation time is difficult to estimate, again precluding
the use of observation time in this survey’s likelihood calculation. A
comprehensive analysis would involve recalculating the $z$–DM grid for each
and every observed burst. For reasons of computational efficiency, we instead
use mean observation parameters to evaluate the likelihood on this grid. This
precludes the use of FRB 191001, which was detected at a lower frequency
during commensal observations (Bhandari et al., 2020b). The remaining seven
FRBs used are given in Table 6.
Table 4: Properties of fast radio bursts detected by the Parkes multibeam
system, and used in this work. Given is the original FRB designation; measured
total DM and DMISM estimated by the NE2001 (Cordes & Lazio, 2002) in pc cm-3,
and ratio of measured to threshold SNR. Entries with a ‘∗’ indicates that this
value is approximate. Parkes: total $T_{\rm obs}=164.4$ days
---
FRB | DM | DMISM | s | Ref.
110214 | 168.9 | 32 | 1.44 | Petroff et al. (2019)
110220 | 944.4 | 36 | 5.44 | Thornton et al. (2013)
110627 | 723 | 48 | 1.22
110703 | 1103.6 | 33 | 1.78
120127 | 553.3 | 33 | 1.22
090625 | 899.55 | 32 | 2.8 | Champion et al. (2016)
121002 | 1629.18 | 72 | 1.6
130626 | 952.4 | 65 | 2
130628 | 469.88 | 52 | 2.9
130729 | 861 | 32 | 1.4
151230 | 960.4 | 48 | 1.7 | Bhandari et al. (2018)
160102 | 2596.1 | 36 | 1.6
Parkes: unnormalised observation time
FRB | DM | DMISM | s | Ref.
010305 | 350 | 44 | 1.02∗ | Zhang et al. (2020a)
010312 | 1187 | 51 | 1.1∗ | Zhang et al. (2019)
010724 | 375 | 45 | 2.3∗ | Lorimer et al. (2007)
131104 | 779 | 71 | 3.06 | Ravi et al. (2015)
140514 | 562.7 | 36 | 1.6 | Petroff et al. (2015)
150807 | 266.5 | 38 | 5∗ | Ravi et al. (2016)
180309 | 263.52 | 46 | 41.1 | Osłowski et al. (2019)
180311 | 1570.9 | 46 | 1.15
Table 5: As per Table 4, for ASKAP/FE observations. ASKAP/FE: $T_{\rm
obs}=1274.6$ days
---
FRB | DM | DMISM | s | Ref.
170107 | 609.5 | 37 | 1.68 | Bannister et al. (2017)
170416 | 523.2 | 40 | 1.38 | Shannon et al. (2018)
170428 | 991.7 | 40 | 1.11
170707 | 235.2 | 38.5 | 1.00
170712 | 312.8 | 35.9 | 1.34
170906 | 390.3 | 38.9 | 1.79
171003 | 463.2 | 40.5 | 1.45
171004 | 304.0 | 38.5 | 1.15
171019 | 460.8 | 37 | 2.46
171020 | 114.1 | 38.4 | 2.05
171116 | 618.5 | 35.8 | 1.24
171213 | 158.6 | 36.8 | 2.64
171216 | 203.1 | 37.2 | 1∗
180110 | 715.7 | 38.8 | 3.75
180119 | 402.7 | 35.6 | 1.67
180128.0 | 441.4 | 32.3 | 1.31
180128.2 | 495.9 | 40.5 | 1.01
180130 | 343.5 | 38.7 | 1.08
180131 | 657.7 | 39.5 | 1.45
180212 | 167.5 | 30.5 | 1.93
ASKAP/FE: unnormalised time
FRB | DM | DMISM | s | Ref.
180417 | 474.8 | 26.1 | 1.84 | Agarwal et al. (2019)
180515 | 355.2 | 32.6 | 1.27 | Bhandari et al. (2019)
180324 | 431 | 64 | 1.03 | Macquart et al. (2019)
180525 | 388.1 | 30.8 | 2.88
Table 6: As per Table 4, for ASKAP/ICS observations, with redshift $z$ included. ASKAP/ICS: unnormalised time |
---|---
FRB | DM | DMISM | s | z | Ref.
180924 | 362.4 | 40.5 | 2.34 | 0.3214 | Bannister et al. (2019)
181112 | 589.0 | 40.2 | 2.14 | 0.4755 | Prochaska et al. (2019)
190102 | 364.5 | 57.3 | 1.38 | 0.291 | Macquart et al. (2020)
190608 | 339.5 | 37.2 | 1.79 | 0.1178
190611.2 | 322.2 | 57.6 | 1.03 | 0.378
190711 | 594.6 | 56.6 | 2.64 | 0.522
190714 | 504.7 | 38.5 | 1.19 | 0.209 | Heintz et al. (2020)
## 7 Initial results
### 7.1 Calculations
We use a brute-force approach to find the best-fit values of $E_{\rm
max},\alpha,\gamma,n,\mu_{\rm host},\sigma_{\rm host}$, and evaluate Eq. 25
over a multi-demensional cube of parameter values. The resulting likelihood
dependence on each is calculated for single (pairs of) parameters by
marginalising over the remaining five (four) parameters. That is, the value
taken is the maximum likelihood found when the remaining five (four) values
are varied over their full range.
In this work, we use a frequentist approach to setting confidence limits.
Confidence intervals are determined using Wilks’ theorem,
$\displaystyle 2\left(\log L_{\rm max}-\log L\right)$ $\displaystyle\sim$
$\displaystyle\chi^{2}_{\rm ndf}$ (41)
where $\chi^{2}_{\rm ndf}$ is a chi-square distribution with ndf degrees of
freedom, here equal to the number of parameters which have not been
marginalised over (either one or two throughout this work; Wilks, 1962). The
51 FRBs used in this work should satisfy the large-N limit required for Eq. 41
to be valid.
### 7.2 Degeneracy
Figure 7: Degeneracy of the marginalised likelihood with $\alpha$. Top:
dependence of $E_{\rm max}$, bottom: dependence on $n$. The contours give the
68%, 90%, and 95% confidence limits. The step-like behaviour in $E_{\rm max}$
is due to coarse gridding.
Initial calculations revealed a degeneracy in the fitting parameters between
$E_{\rm max}$, $\alpha$, and $n$. This is shown in Figure 7, which plots the
variation of the marginalised likelihood over these parameters. In each case,
there is a broad maximum extending over the full range of $\alpha$ ($-5$ –
$-0.5$). The reason for this degeneracy is that these three parameters are
strongly related to the high-DM, high-z cut-off in the observed FRB
distribution. This is restricted by the lack of ASKAP/ICS-localised FRBs at
high redshifts, and by the number of high-DM FRBs detected by ASKAP/FE and
Parkes multibeam observations. Having a steep spectral index (very negative
value of $\alpha$) provides a mechanism to reduce the expected number of FRBs
via the k-correction, and is both consistent with, and requires, a higher
value of $E_{\rm max}$. Similarly, it also allows for a rapidly evolving
population with redshift (high $n$); a very large number of high-$z$ events is
otherwise excluded when $\alpha$ is near zero. This degeneracy was also noted
by Lu & Piro (2019), who by default use $\alpha=-1.5$.
Without further data, or a prior on any of these three parameters, this
degeneracy cannot be broken. All results from the literature on $E_{\rm max}$
and $n$ derive from similar analyses as in this work, albeit with simpler
methods, and are therefore not independent. However, the work of Macquart et
al. (2019), who analyse the instrument-corrected spectral structure of bursts
detected in ASKAP/FE observations, provides an independent constraint on this
parameter. We therefore use a Gaussian prior on $\alpha$, with mean at
$\alpha=-1.5$, and $\sigma_{\alpha}=0.3$, reflecting their result of
$\alpha=-1.5_{-0.3}^{+0.2}$. In doing so, we note that the spectral index
model may be interpreted as a frequency-dependent rate as discussed in Section
3, and that while a spectral break below GHz frequencies is expected
(Sokolowski et al., 2018), the FRBs used in this analysis were all observed at
GHz frequencies, and thus the results are only sensitive to spectral behaviour
at — and due to redshift, above — 1 GHz.
### 7.3 Comparison of results
The limits on single FRB population parameters presented in James et al.
(2021) are obtained with this approach. We observe that while our prior on
$\alpha$ shifts the preferred values of the other parameters by small amounts,
it is not a large influence, and primarily acts to limit very strong source
evolution models with $n>2$. Table 7 compares these results with a prior on
$\alpha$ to those of other authors, as well as a brief summary of which
effects are included.
Author | $E_{\rm max}$ | $\alpha$ | $\gamma$ | $n$ | $\mu_{\rm host}$ | $\sigma_{\rm host}$ | Beam | DM|z | $\eta({\rm DM})$ | $f(w)$
---|---|---|---|---|---|---|---|---|---|---
Lu & Piro (2019) | $43.1^{+1.1}_{-0.7}{}^{b}$ | $-1.5^{a}$ | $-0.6\pm 0.3$ | $0.3^{+1.0}_{-1.1}{}^{d}$ | N | N | N | N | N | N
Luo et al. (2020) | $41.16^{+0.51}_{-0.19}{}^{b}$ | $0^{a}$ | $-0.79_{-0.18}^{+0.16}$ | $0^{a}$ | $30^{a,c,e}$ | $0.17^{a,c,e}$ | Y | Y | Y | Y
Arcus et al. (2021) | $45.1^{a}$ | $-2.1^{1.1}_{-1.4}$ | $-0.7_{-0.2}^{+0.2}$ | $1\pm 1$ | N | N | N | N | Y | Y
Caleb et al. (2016) | 41.2g | 0a | N/A | 0,1a | 100a | 0a | Y | Y | Y | Y
Macquart et al. (2020) | N/A | N/A | N/A | N/A | $65^{+95}_{-25}{}^{e}$ | $0.9_{-0.6}^{+0.9}{}^{e}$ | N | Y | N | N
This work | $41.84_{-0.18}^{+0.49}$ | $-1.55^{+0.21}_{-0.21}{}^{f}$ | $-1.16_{-0.12}^{+0.11}$ | $1.77_{-0.45}^{+0.25}$ | $145_{-60}^{+64}$ | $0.51_{-0.10}^{+0.15}$ | Y | Y | Y | Y
Table 7: Summary of results and methods used by other authors in estimating
FRB population parameters. Shown in columns 2–7 are the best-fit parameter
values and corresponding uncertainties, converted to the units and notation
used in this work. For the sake of comparability, all limits have been
converted to 1$\sigma$ standard deviations assuming that uncertainties are
Gaussian distributed. Columns 8–11 summarise the different effects that were
(Y) or were not (N) included in the modelling: accounting for instrumental
beamshape, a distribution in dispersion measure for a given redshift, a DM-
dependent telescope threshold, and a distribution of burst widths $f(w)$ and
its corresponding effects on sensitivity. Note that a ‘Y’ does not necessarily
mean a comparable treatment to this work. a Parameter held fixed by the
authors. b indicates the use of a Schechter function, with exponential decay
after $E_{\rm max}$. c The host galaxy contribution is modelled according to
the “ALGs(YMW16)” model from Luo et al. (2018), with approximate mean $\sim
0.8$ and standard deviation $0.2$ in $\log_{10}$ DM, to which is added a
‘local’ contribution with a uniform distribution from 0–50, for a total mean
of $30$ and quadrature-added deviation $15$, i.e. $0.17$ in $\log_{10}$. d The
$(1+z)^{\beta}$ scaling used in this work is converted to $n$ via $n=2.7\beta$
from Eq. 13. e These values are not explicitly quoted, and were approximately
derived from plots shown in the text. f This constraint primarily arises from
the prior of Macquart et al. (2019). g A log-normal luminosity function was
used; here we quote the mean of $10^{38.2}$ erg plus three standard deviations
of $1$ in $\log_{10}$.
## 8 Results 2: redshift and dispersion measure
The best fitting parameter set (James et al., 2021) allows a comparison
between the expected and observed distributions of FRBs in DM, z, and SNR
space. However, each combination of allowed parameters produces a unique map
of the z–DM distribution of FRBs for each telescope. Rather than present a
very large number of plots, we use the following approach to identify a finite
number of reasonable possibilities.
Table 8: Parameter sets used in this work: the best-fit values found in our companion paper (James et al., 2021); sets allowing the parameter in the left-most column to take its minimum and maximum value within 90% C.L; the best-fitting parameter set when forcing $n=0$, i.e. no source evolution; and a set when setting $E_{\rm min}=10^{38.5}$ erg, but otherwise identical to the best fit. Set | $\log_{10}E_{\rm min}$ | $\log_{10}E_{\rm max}$ | $\alpha$ | $\gamma$ | $n$ | $\mu_{x}$ | $\sigma_{x}$
---|---|---|---|---|---|---|---
Best fit | 30 | 41.84 | -1.55 | -1.16 | 1.77 | 2.16 | 0.51
$\log_{10}E_{\rm max}$ | 30.0 | 41.6 | -1.5 | -0.8 | 1.0 | 2.0 | 0.9
| 30.0 | 42.51 | -1.5 | -1.25 | 1.91 | 2.0 | 0.45
$\alpha$ | 30.0 | 41.9 | -1.88 | -1.15 | 1.9 | 2.25 | 0.5
| 30.0 | 41.64 | -1.2 | -1.12 | 1.4 | 2.25 | 0.5
$\gamma$ | 30.0 | 42.08 | -1.5 | -1.34 | 2.08 | 2.25 | 0.5
| 30.0 | 41.8 | -1.5 | -0.96 | 1.52 | 2.0 | 0.6
$n$ | 30.0 | 41.8 | -1.5 | -1.1 | 1.11 | 2.25 | 0.5
| 30.0 | 41.88 | -1.75 | -1.2 | 2.28 | 2.15 | 0.5
$\mu_{\rm host}$ | 30.0 | 42.18 | -1.5 | -1.1 | 1.78 | 1.77 | 0.59
| 30.0 | 41.8 | -1.5 | -1.2 | 1.67 | 2.41 | 0.56
$\sigma_{\rm host}$ | 30.0 | 42.16 | -1.5 | -1.2 | 1.8 | 2.08 | 0.36
| 30.0 | 42.0 | -1.5 | -1.1 | 1.6 | 2.0 | 0.81
$n=0$ | 30. | 41.6 | -1.25 | -1.1 | 0. | 2.5 | 0.6
$E_{\rm min}$ | 38.5 | 41.84 | -1.55 | -1.16 | 1.77 | 2.16 | 0.51
For each parameter, we take its value at both the upper and lower limits of
its one-dimensional 90% C.L., and choose the corresponding values of the other
parameters. This results in 12 further parameter sets, which are listed in
Table 8. For comparison, we also include the best-fit parameter set assuming
no source evolution.
For each of these parameter sets — including the best-fit set — we generate
the expected FRB distribution in DM–z space, and compare this to the observed
values. Finally, we also consider the best-fit set of parameters when setting
$n=0$, in order to illustrate how well this case fits the data. All the
results shown in this section are calculated using the spectral index
interpretation of $\alpha$.
In interpreting the results in this section, we caution that the data to which
we compare expectations have been used to determine the FRB population
parameters, so the two are not independent. However, given that only eight
free parameters (the standard six, plus two for the FRB width distribution)
are used to fit multiple observables from three different FRB surveys, good
agreement should not be the result of over-fitting, but rather indicate a
genuine correspondence between the models used in this work and reality.
### 8.1 Observed and predicted distributions
\begin{overpic}[width=195.12767pt]{Figures/DMZ_fitting/DMerr.pdf}
\put(30.0,65.0){\Large All}
\end{overpic}\begin{overpic}[width=195.12767pt]{Figures/DMZ_fitting/DMerr_pks.pdf}
\put(15.0,65.0){\Large Parkes/Mb}
\end{overpic}\begin{overpic}[width=195.12767pt]{Figures/DMZ_fitting/DMerr_fe.pdf}
\put(30.0,65.0){\Large ASKAP/FE}
\end{overpic}\begin{overpic}[width=195.12767pt]{Figures/DMZ_fitting/DMerr_ics.pdf}
\put(30.0,65.0){\Large ASKAP/ICS} \end{overpic}
Figure 8: Observed (histogram) and predicted (lines) distributions of DM for
all surveys considered (upper left), and the individual Parkes multibeam
(upper right), ASKAP Fly’s Eye (lower left), and ASKAP ICS (lower right)
surveys. Predictions show the best-fit over the entire parameter space, when
constrained to no source evolution, when allowing $E_{\rm min}$ to vary, and
when varying parameters within their 90% C.L.
\begin{overpic}[width=195.12767pt]{Figures/DMZ_fitting/Zerr_pks.pdf}
\put(20.0,65.0){\Large Parkes/Mb}
\end{overpic}\begin{overpic}[width=195.12767pt]{Figures/DMZ_fitting/Zerr_fe.pdf}
\put(20.0,65.0){\Large ASKAP/FE}
\end{overpic}\begin{overpic}[width=195.12767pt]{Figures/DMZ_fitting/Zerr_ics.pdf}
\put(20.0,65.0){\Large ASKAP/ICS} \end{overpic}
Figure 9: Predicted distributions (lines) of $z$ for Parkes multibeam (top)
and ASKAP Fly’s Eye (middle) surveys, with the ASKAP ICS results (bottom) also
showing the observed values (histogram). Predictions show the best-fit over
the entire parameter space, when constrained to no source evolution, when
setting $E_{\rm min}=10^{39}$ erg, and when varying parameters within their
90% C.L.
The predicted DM and $z$ distributions from all fourteen cases — best-fit, no
source evolution, and twelve sets reflecting parameter uncertainty — are shown
in Figures 8 and 9 respectively. In the case of the DM distribution, data and
predictions are shown individually from each survey described in Section 6,
and stacked together to allow a better comparison. For the $z$ distribution,
the only available data comes from ASKAP/ICS observations — however,
predictions from each individual survey are also shown.
The first question to ask is — are the best fits indeed good fits? Our best-
fitting parameter estimates (James et al., 2021) do not necessarily indicate
that the modelled DM, z, and SNR distributions are good fits to the data —
merely that they are the best fits given the form of the model used. A by-eye
analysis shows that the predictions from the best-fit parameter set are indeed
a good match to the data. However, there appears to be an over-prediction of
FRBs at low DM and particularly at low redshift. This is a common feature over
all the parameter sets within the 90% error margins, although the degree of
peakedness near $z=0$ varies greatly.
We consider four possible explanations for this below.
### 8.2 Random fluctuations
Figure 10: Observed and predicted cumulative distribution functions for the DM
from all considered surveys (above) and $z$ from the ASKAP ICS survey (below).
Predictions show the best-fit over the entire parameter space, when
constrained to no source evolution, and when allowing $E_{\rm min}$ to vary.
Figure 11: Observed value (red line) and simulated distribution (blue
histogram) of the KS statistics for the DM (top) and redshift (bottom)
distributions under the best-fit hypothesis. Shown are the corresponding
p-values to reject this hypothesis.
The observed deficit of low-DM and low-z FRBs may simply be a statistical
fluctuation. To evaluate this, we perform Kolmogorov-Smirnov tests on the
ASKAP ICS z distribution and total DM distribution (Kolmogorov, 1933; Smirnov,
1948). Predicted and measured cumulative distributions of both DM and z are
shown in Figure 10 — the KS-statistic is the maximum absolute difference
between the two curves. To evaluate the significance of this statistic, we
take the best-fit curves as the truth, and randomly generate 10,000 samples
from each. Histogramming the results produces the expected distributions of
the KS statistic under the null hypothesis that the best-fit prediction is
true. Comparing this distribution to the observed value of the KS statistic in
Figure 11 shows that our observed values of DM and $z$ are completely
consistent with predictions, with larger values of the KS statistics observed
in 17% and 41% of cases for the DM and z distributions respectively.
Performing a similar analysis using the $n=0$ distribution gives p-values of
22% and 2% for the DM and z distributions respectively. We therefore conclude
that the apparent deficit of FRBs at low DM and redshift compared with
predictions is consistent within $1\,\sigma$ statistical fluctuations of
expectations. Nonetheless, we proceed with further analysis since the presence
or otherwise of a minimum energy, or effects due to a large fraction of the
population being repeaters, is of great interest.
### 8.3 Evidence for a minimum energy $E_{\rm min}$
In our standard modelling, we have set the minimum FRB energy $E_{\rm min}$ to
an extremely low value of $10^{30}$ erg — well below the characteristic
energies of observed FRBs, effectively making it zero. This is because values
of $\gamma>-1.5$ render FRB observations primarily sensitive to $E_{\rm max}$
(Macquart & Ekers, 2018b), while bursts from FRB 121102 have been observed at
much lower energies than are likely to be probed by Parkes and ASKAP
observations (e.g. Law et al., 2017).
However, a clear possible explanation for the apparent deficit of FRBs at low
DM-z is a minimum FRB energy. Without such a cut-off, as telescopes probe ever
lower values of $E$ in the nearby Universe, the predicted number of FRB
observations per comoving volume will increase without limit when
$\gamma\leq-1$. If $-1\geq\gamma>-1.5$, the reducing volume will somewhat
compensate, and the total rate will remain finite. This gives rise to the
sharp increase in the best-fit expected redshift distributions near $z=0$ in
Figure 9, although such a peak may not be present within 90% error margins.
Figure 12: Evolution of likelihoods as a function of minimum energy $E_{\rm
min}$, relative to the value at $E_{\rm min}=10^{30}$ erg. Shown is the total
likelihood; the likelihood over all surveys, split into contributions from the
z–DM distribution p(z,DM), the number of events p(N), and the signal-to-noise
probability p(SNR). The p(z,DM) contribution is further split into components
from the ASKAP/FE, ASKAP/ICS, and Parkes/Multibeam surveys.
To investigate the effect of $E_{\rm min}$, we fix the best-fit parameters,
and vary $E_{\rm min}$. The evolution of the likelihoods is shown in Figure
12. Interestingly, the total likelihood decreases with increasing $E_{\rm
min}$, with $E_{\rm min}=10^{38.5}$ erg the 90% C.L. upper limit. Why? As
expected from Figures 8 and 9, the $p(z,DM)$ contribution increases with
$E_{\rm min}$, peaking near $10^{38.5}$ erg. This peak is a combination of the
ASKAP/ICS observations strongly favouring a large $E_{\rm min}\approx
10^{40.5}$ erg, and the Parkes multibeam observations strongly favouring
$E_{\rm min}<10^{39}$ erg, with the ASKAP/FE observations being relatively
neutral until $E_{\rm min}>10^{40}$ erg.
The reason why the Parkes observations are strongly against high values of
$E_{\rm min}$ is that sensitive telescopes with small fields of view are
highly unlikely to observe low-DM bursts, unless the small volume of the near
Universe corresponding to such a low DM is populated by many (necessarily low-
energy) FRBs. The lowest-DM burst detected by Parkes during the included
surveys, FRB 110214, had a DM of $168.5$ pc cm-3 (Petroff et al., 2019).
Setting $E_{\rm min}=10^{38.5}$ implies reduced FRB rates for redshifts closer
than $z=0.194$, assuming a limiting fluence of $0.3$ Jy ms, with those bursts
that are detected likely to have a SNR significantly greater than threshold. A
redshift of $z=0.194$ implies a most likely dispersion measure of
approximately 281 pc cm-3, being composed of a cosmological contribution of
DM${}_{\rm cosmic}=83$ pc cm-3, local contribution of 82 pc cm-3, and our
model best-fit value of $\mu_{\rm host}/(1+z)=116$ pc cm-3. The observation of
FRBs by Parkes with DMs below this value therefore disfavour a significant
minimum energy $E_{\rm min}$.
This is illustrated in Figure 13, where we show model predictions of $p_{s}$,
and the observed values of $s$, for each Parkes-detected FRB. This is done for
the best-fit model, and when using $E_{\rm min}=10^{38.5}$ erg. While the
$E_{\rm min}$ model predicts that the high fluence of FRB 180309 with a DM of
$\sim$263 pc cm-3 and $s=41$ is slightly more likely, it predicts that FRB
110214, with a DM of $\sim$169 pc cm-3, is much less likely to have its
observed value of $s=1.44$.
In this work, we have not included $E_{\rm min}$ as a global minimisation
parameter due to computational constraints. Might there be some other
combination of parameters for which a significant $E_{\rm min}$ is found? To
investigate this, we have repeated the $E_{\rm min}$ optimisation for all
parameter sets in Table 8. Only for the parameter set minimising $\gamma$ do
we find a significant value of $E_{\rm min}$ to be preferred, with a best-fit
value of $10^{38.0}$ erg, and 90% upper limit of $10^{38.8}$ erg. This makes
sense, since otherwise a steep energy function would over-predict the number
of near-Universe FRBs. The resulting gain in likelihood acts to slightly
weaken our confidence in the lower limits on $\gamma$, e.g. the 90% lower
limit shifts to 81% confidence.
Our findings clearly do not indicate that there is no minimum FRB energy.
Within the confines of our power-law model, the data used appear insensitive
to $E_{\rm min}<10^{37}$ erg, and we can only rule out $E_{\rm min}>10^{38.5}$
erg at 90% C.L. — but it does run counter to the findings of Luo et al.
(2020). These authors find a most likely minimum luminosity of $10^{42}$ erg
s-1, which is approximately $10^{39}$ erg assuming a standard 1 ms burst,
although they conclude rather that this finding is due to the limit of their
sample.
Figure 13: Likelihoods of observing FRBs with a given $s$, $p_{s}(s)$, as a
function of their dispersion measure, DM, for the Parkes multibeam
observations, weighted by $s$ for clarity. This is calculated for best-fit
parameters using $E_{\rm min}=10^{30}$ erg (top), and $E_{\rm min}=10^{38.5}$
erg (bottom).
### 8.4 Influence of repetition
In this study, we have ignored repeating FRBs on the basis that none of the
FRBs detected by ASKAP and Parkes have been observed more than once — even
though some are known to repeat (Kumar et al., 2019; Patek & Chime/Frb
Collaboration, 2019; Kumar et al., 2020), the surveys were not sensitive
enough to probe this. However, the method of Section 5 covers the entirety of
z–DM space, regardless of whether or not an FRB has been detected at that
point. Therefore, for any hypothesised repeating FRB, there will always be a
sufficiently nearby volume of the Universe where any survey would be expected
to detect more than one burst if the FRB landed in its field of view. If such
a repeating FRB happens to be located in that volume, the observed burst rate
will be greater than expected — but if one does not, it will be less.
This effect is analysed in the context of Canadian Hydrogen Intensity Mapping
Experiment (CHIME) observations by Gardenier et al. (2020), who note that the
DM distribution of repeating FRBs should be lower than that of repeaters
observed only once. While this has not yet been observed (CHIME/FRB
Collaboration et al., 2019a, b; Fonseca et al., 2020a, b), this may be due to
a very broad distribution of intrinsic repetition rates (James et al., 2020b).
As discussed by James (2019), the ASKAP/FE observations have deeply probed
some regions of sky with over 50 antenna-days spent on individual fields. This
has allowed strong limits to be placed on FRB repetitions — but also made the
observations more susceptible to whether or not a strong repeater at $z<0.2$
was present in these fields. We observe that the low-DM deficit is greatest in
ASKAP/FE observations, and lowest for Parkes observations, consistent with
this prediction.
Until more is known about the population of repeating FRBs, we cannot further
quantify this effect, except to add that this effect is guaranteed to be
present to some degree (some FRBs definitely do repeat), with its importance
increasing if all FRBs are explained by strong repeaters, and lessening as a
lower fraction of all FRBs are due to repeaters, and those repeaters are weak.
### 8.5 Minimum search DM
Many FRB surveys use a minimum searched DM, either to exclude Galactic FRB
candidates, or due to local RFI with intrinsic DM of $0$ that nonetheless
contaminates searches. For example, Thornton et al. (2013) reject bursts with
DM$<100$ pc cm-3, although in our model nearby FRBs could have a DM of as
little as DM$\approx 80$ pc cm-3. This effect will sometimes exclude FRBs in
the very nearby Universe, potentially resulting in the observed deficit. Only
two bursts however — 171020 (Shannon et al., 2018) and 180729.J0558+56
(CHIME/FRB Collaboration et al., 2019a) — have been detected near this limit,
and the deficit extends well beyond 100 pc cm-3, so we consider this an
unlikely cause. Analysis of historical Parkes data by Zhang et al. (2020b) has
found one new FRB at 350 pc cm-3 (which was too recent to include in this
analysis), but none at lower DM.
### 8.6 Summary: redshift and dispersion measure distributions
Having established the robustness of these results, we summarise the
predictions for the redshift distributions from Figure 9. In all scenarios,
the redshift distribution of the ASKAP/FE detections lie in the range $z_{\rm
max}<0.7$, and most fits find $z_{\rm max}<0.5$. Over all parameter sets,
between 32% and 49% of ASKAP/FE bursts should originate from within $z<0.1$,
confirming that these bursts are ideal follow-up targets due to their likely
proximity. This suggests that the limits set on the repeatability of
individual FRBs by James et al. (2020a) are significantly stronger than
published, since those authors conservatively assume maximal redshifts. It
also lends additional weight to the possible association of FRB 171020 with a
galaxy at 40 Mpc by Mahony et al. (2018).
The predicted $z$-distribution of Parkes bursts is significantly broader than
that of ASKAP/FE observations, although the no-evolution scenario still
suggests a large fraction (35–60% at 90% confidence over all parameter sets)
in the $z<0.5$ Universe, with 6–20% (over all parameter sets) being within
$z<0.1$.
A key test of our prediction of a large number of near-Universe FRBs will be
future ASKAP/ICS detections. So-far, all bursts detected by ASKAP’s ICS mode
have been in a limited range of both DM and $z$, which seem to be from the
central part of all predicted distributions. Our best-fit model predicts that
24% of ASKAP/ICS localisations should lie within $z<0.1$, with a range over
all sets of 14–32%. To date, none have been observed — even amongst the
unpublished ones not included here.
### 8.7 Source counts distribution
Figure 14: Predicted source counts distribution, $p(s)$, multiplied by
$s^{1.5}$ for clarity, for top: ASKAP/lat50, and bottom: Parkes/Mb, according
to the best-fit parameter set (black solid), when setting $E_{\rm
min}=10^{39}$ erg (red dashed), and when varying parameters within their 90%
C.L. (thin grey). Also shown are lines of constant power-law slopes to guide
the eye. The fluctuations above $s=10$ are due to finite gridding in $z$ and
DM.
The slope of the source counts (“logN–logS”) distribution was one of the first
FRB observables to be analysed. Adapted from its original use in the study of
radio galaxies characterised by their flux S (Ryle, 1968), applied to FRBs,
the source counts distribution is the number N of observed FRBs as a function
of fluence threshold $F_{\rm th}$. In an infinite Euclidean Universe, the
distribution is expected to have a form
$\displaystyle N(F_{\rm th})$ $\displaystyle=$ $\displaystyle
C\left(\frac{F_{\rm th}}{F_{0}}\right)^{a},$ (42)
with $a=-1.5$, and $C$ and $F_{0}$ normalising constants. Studies using a
variety of methods, with different treatments — or neglect of — observational
biases in $F_{\rm th}$, and using data from telescopes with different
detection thresholds, found inconsistent values of $a$ in the range of
$-0.8\geq a\geq-2.6$ (Vedantham et al., 2016; Oppermann et al., 2018; Lawrence
et al., 2017; Macquart & Ekers, 2018a; Bhandari et al., 2018). James et al.
(2019b), reviving methods applied to radio galaxy studies by Crawford et al.
(1970), argue that $s$ — defined in Eq. 34 — is a bias-independent measure of
the source-counts slope, and find $a=-1.18\pm 0.24$ (68% C.L.) for Parkes
FRBs, and $a=-2.2\pm 0.47$ for ASKAP/FE FRBs, equating to a $2.6\sigma$
tension. This was qualitatively consistent with Macquart & Ekers (2018b), who
argue that at high values of $F_{\rm th}$, the slope should be Euclidean
($a=-1.5$), while the parameters of the FRB population will lead to a
flattening at lower thresholds.
Our model can be used to derive the expected distribution of $s$ using Eq. 36,
by integrating over all values of DM and $z$, then converting this
differential distribution to a cumulative distribution. The results for
ASKAP/FE and Parkes/Mb observations are given in Figure 14.
We immediately see that in all scenarios, ASKAP/FE observations, with a higher
base threshold of $26$ Jy ms, are expected to follow a Euclidean ($a\sim-1.5$)
distribution, while near-threshold ($1\leq s\leq 10$) Parkes/Mb observations
exhibit a flatter source-counts index near $a=-1.3$. Different scenarios
predict source-counts indices in the range $-1.4\leq a\leq-1.6$ above $s=10$
for Parkes/Mb observations. The existence or otherwise of a minimum energy
does not significantly affect the distribution.
Our results in all scenarios are consistent with the findings of James et al.
(2019b), with the greatest tension — about $1.5\sigma$ — being between the
source-counts slope that those authors find for ASKAP of $-2.2\pm 0.47$ (68%
C.L.), and the approximate range of $-1.4$ to $-1.6$ found here. In
particular, we also find that the source-counts index for Parkes/Mb is
expected to be flatter than for ASKAP/lat50, and furthermore, the apparent
‘deficit’ of FRBs with low values of $s$ is potentially attributable to the
true behaviour of the FRB population, rather than statistical fluctuations or
a measurement bias. This also suggests that the lower values of $a$ found by
previous authors — Vedantham et al. (2016); Oppermann et al. (2018); Lawrence
et al. (2017) — incorporating data from telescopes more sensitive than Parkes
may have been correct.
## 9 The $z$–DM distribution
We argue in this work that the best representation of the FRB population
observable by a telescope is a two-dimensional function of extragalactic
(cosmological plus host) dispersion measure, and red shift. Necessarily, the
observable fraction of this distribution is a function of survey parameters,
and also local DM contribution, which reduces sensitivity as it increases at
lower Galactic latitudes. Our best-fit z–DM distributions for the three
surveys considered are plotted in Figure 15.
Figure 15: Normalised probability distribution $p(z,{\rm DM}_{\rm EG})$ of the
redshift $z$ and extragalactic dispersion measure ${\rm DM}_{\rm EG}$ of FRBs
detected in ASKAP Fly’s Eye (top), ICS (middle) and Parkes multibeam (bottom)
surveys. Contours indicate the intervals in which 50% (dotted), 90% (dot-
dash), and 99% (dashed) of all FRBs would be detected. Red dots indicate the
values of DMEG and host galaxy redshift $z$ for ASKAP ICS localized FRBs.
We describe the general features of these plots. Most FRBs are expected to lie
near to, or below, the Macquart relation, being the 1–1 correspondence of FRB
redshfit with DM. This relation, with a slope of approximately 100 pc cm-3 per
0.1 in redshift, continues up to some maximum detectable distance, being
$z$=0.5, 0.9, and 2.4 for ASKAP/FE, ASKAP/ICS, and Parkes/Mb respectively.
However, the majority of FRBs will not be found near the maximum redshift,
simply because there are very many more FRBs with lower energy, and much more
of the sky covered at lower sensitivity. This is most evident with Parkes/Mb
observations, where the most likely half of all FRB observations will arise on
the Macquart relation with $z<1$.
Off the Macquart relation, there is a significant fraction of FRBs expected to
be found with higher than expected DMs, due to a combination of their host and
cosmological contributions. Only our ASKAP/ICS (localised) FRB sample has
provable examples, being FRB 190714 (504.7 pc cm-3 from $z=0.209$) and FRB
190608 (339.5 pc cm-3 from $z=0.1178$), although a third (FRB 191001, with
506.92 pc cm-3 from $z=0.23$) was excluded due to being detected at a lower
frequency. This effect is less pronounced for more-sensitive surveys, since
excess host contributions are dominated by cosmological ones. We emphasise
that much of the structure above the Macquart relation — i.e. in the high-DM
region — is poorly constrained, since our adopted log-normal distributions may
not reflect reality.
At very high DMs, only near-Universe FRBs are observable, since a burst must
be observed with very high fluence to overcome the detection bias against high
DM. The upper bound of the 99% region (dashed lines) slopes backward, against
the Macquart relation, since more low-energy FRBs with large excess DM are
predicted than high-energy FRBs lying on the Macquart relation. We do not
expect our quantitative estimates in this region to be accurate until it is
directly probed with localised FRBs. Since the cause of this effect is a well-
understood observational bias however, it will clearly be present to some
degree.
### 9.1 The Macquart Relation
The ‘Macquart Relation’ is the general one-to-one-ness of the relationship
between redshift and DM of FRBs. It is predicted from the distribution of
baryonic matter in the Universe (Inoue, 2004), and first evidence for it was
given in Shannon et al. (2018), by comparing the DM distributions of ASKAP/FE
and Parkes/Mb populations, where the higher sensitivity of Parkes allowed it
to probe more-distant FRBs with higher DMs. The relation was first observed
directly by Macquart et al. (2020), who showed that the redshifts of localized
FRBs were consistent with the baryonic content of the Universe.
For the purpose of comparing survey results, we propose a useful distinction:
the ‘weak’ Macquart relation (which might more accurately be titled the
‘Shannon’ relation), where telescopes with higher sensitivity on average
observe more-distant FRBs with higher DM; and the ‘strong’ version (or true
Macquart relation), where the DMs of FRBs _within_ a survey are a good proxy
for their redshift. Several authors use a 1–1 z–DM relation in performing
estimates of the FRB population from the DMs of un-localised FRBs (Shannon et
al., 2018; Deng et al., 2019; Lu & Piro, 2019; James et al., 2020a).
Figure 16: Probability $p(z|{\rm DM})$ of an FRB originating from redshift $z$
given it has been observed with the specified DM in the ASKAP/FE (top),
ASKAP/ICS (middle), and Parkes/Mb (bottom) surveys.
Clearly, the weak version of the relation is well-established, and an obvious
consequence of the cosmological nature of FRBs. Here, we test the strong
version, which can be readily tested by calculating
$\displaystyle p(z|{\rm DM})$ $\displaystyle=$ $\displaystyle\frac{p(z,{\rm
DM})}{p({\rm DM})}$ (43)
for a given survey. This is shown for each survey in Figure 16.
In each survey, the Macquart relation applies up to a maximum redshift $z_{\rm
max}$, beyond which it reverses. The reversal can be intuitively understood by
realising that at the maximum redshift at which an FRB can be detected, FRBs
with any excess of DM can not be detected, due to DM smearing reducing
sensitivity. Therefore, the only way to detect FRBs with a DM lying above that
expected from a burst originating at $z_{\rm max}$ is to have the burst
originate at a nearer redshift. As noted in Section 3 and Fig. 2, the
increased number of FRBs in the local Universe is related to the cumulative
luminosity index $\gamma=-1.16$, with more negative values leading to more
nearby high-DM events.
The reversal of the Macquart relation has several practical consequences.
Firstly: for surveys with a large sample of FRBs, the burst with the greatest
DM will _not_ be the most distant. An excellent example of this phenomenon is
FRB 170428 (ASKAP; Shannon et al., 2018), which is most likely to originate
below $z=0.3$, rather than the value of $z\sim 1$ expected from the Macquart
relation. FRB 160102, observed by Parkes, is another likely candidate. The
implication is that works using a 1–1 DM–z relation will vastly over-estimate
the maximum FRB energy, since they will necessarily attribute a large distance
and therefore high energy to the highest DM event, which may in fact be quite
local.
A key test of this reversal would be the detection of an FRB with DM$\gtrsim
1000$ pc cm-3 by ASKAP in ICS mode, and its subsequent localisation to a
redshift $z\lesssim 0.5$. Again, we note that this reversal of the Macquart
relation will always be present to some extent, since it is fundamentally due
to observational effects which are known and understood — it is merely the
extent of this effect which is currently uncertain.
Finally, we note that the existence of FRBs with very high DMs has raised the
possibility of using FRBs to probe for the signature of Helium reionisation
(Deng & Zhang, 2014; Caleb et al., 2019; Linder, 2020). While this is by no
means ruled out, it emphasises that doing so will require FRBs to be
localised, since simple measures of FRB properties as a function of DM will
yield a very large scatter in redshifts, and hence reduced statistical power.
## 10 Conclusion
We have developed a precise model of FRB observations, including observational
biases due to the full telescope beamshape, degradation in efficiency due to
DM, and intrinsic burst width. None of these effects are fundamentally new;
many others should take credit for highlighting their importance, and Luo et
al. (2020) should be attributed with a first analysis using these techniques.
Here, we have improved upon this method by using an unbiased data sample,
adding new observations of localized FRBs, studying the effects of source
evolution, including the likelihood of the observed signal strength, and
improving the beam model. We show that ignoring, or incorrectly modelling,
these factors leads to significant biases in the expected redshifts of
observable FRBs.
We have used our approach to model FRB observations with ASKAP in both fly’s
eye and incoherent sum mode, and Parkes multibeam observations. We have
carefully selected our data to ensure it is not biased due to under-reporting
of observation time, or due to large local DM contributions reducing
sensitivity. Crucially, we have included a sample of localized FRBs from ASKAP
for which the redshift of the host galaxies is measured.
The $z$, DM, and SNR distributions of FRBs predicted by our best-fit
population estimates, presented in James et al. (2021), are tested against
observations. We find no evidence for, and some evidence against, a lower
bound to the FRB energy distribution, although we only exclude $E_{\rm
min}\geq 10^{38.5}$ erg (90% C.L.). No such minimum energy is expected for the
magnetar-origin hypothesis, which links the observed extragalactic FRB
population to radio bursts from magnetar flares in our own Galaxy.
Our model also allows us to make inferences on the redshifts of the un-
localized samples of FRBs detected by ASKAP and Parkes. We find these to be
somewhat lower than expected from the Macquart relation, and indeed that the
highest-DM events are likely not the most distant, due to observational
effects causing an inversion of the Macquart relation, and the relatively
steep best-fit value of the FRB energy distribution. The ability of this model
to place priors on the expected redshift of FRBs given their measured DMs will
also aid FRB localisation efforts, especially for those bursts — such as FRB
171020 — which have uncertain, but promising, host associations.
For the first time, we have incorporated the measured signal-to-noise ratio
$s$ into FRB population modelling, allowing the use of this observable to
constrain population parameters, and to predict the source-counts (“logN-
logS”) distribution. In all scenarios, we find a steepening of this
distribution from Parkes to ASKAP, consistent with the predictions of Macquart
& Ekers (2018b) and the observations of James et al. (2019b).
Anomalies between observations and our model include a lack of ASKAP
incoherent sum (localised) FRBs within $z<0.1$, an over (under) prediction of
FRBs from Parkes multibeam (ASKAP fly’s eye) observations, and an over-
prediction of low-DM bursts. We note several potential explanations for these
discrepencies, although none of them have a high statistical significance.
Future observations of localised FRBs with very large excess DMs, and/or from
the local Universe, would verify our predictions. FRB surveys with very
sensitive radio telescopes such as the Five-hundred-meter Aperture Spherical
Telescope (FAST), or repeating previous Parkes and ASKAP surveys at different
frequencies, would help to further constrain FRB population models. In
particular, the application of this model to the large sample of bursts
observed by CHIME would be particularly useful, although it would then need to
be adapted to include repeating FRBs. We also aim to investigate improved
numerical/computational methods to speed up calculations to allow the
inclusion of further data.
## Acknowledgements
This research has made use of NASA’s Astrophysics Data System Bibliographic
Services. This research made use of Python libraries Matplotlib (Hunter,
2007), NumPy (van der Walt et al., 2011), and SciPy (Virtanen et al., 2020).
This work was performed on the gSTAR national facility at Swinburne University
of Technology. gSTAR is funded by Swinburne and the Australian Government’s
Education Investment Fund. This work was supported by resources provided by
the Pawsey Supercomputing Centre with funding from the Australian Government
and the Government of Western Australia. This research was partially supported
by the Australian Government through the Australian Research Council’s
Discovery Projects funding scheme (project DP180100857).
## Data Availability
The data underlying this article will be shared on reasonable request to the
corresponding author.
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## Appendix A Neglected effects
Here we discuss effects that affect search sensitivity which have been
neglected in this work.
### A.1 Fine sensitivity effects
The precise sensitivity of an incoherent FRB search to an FRB depends not only
on the ‘coarse’ effects of effective pulse width, but also the exact alignment
of a burst with the time–frequency binning of the data compared to the FRB
arrival time, and the DMs which are searched compared to the exact DM of the
FRB. This effect is analysed by Keane & Petroff (2015a) for early FRB search
methods, finding fluctuations of up to $\pm$15%, although in current FRB
search methods fluctuations are essentially zero. This was also found to be
the case for an internal investigation into the CRAFT FRB searches with ASKAP
for Shannon et al. (2018), with fluctuations of $\pm$3%. Such effects are thus
ignored in this work, and likely should be in all future works.
The presence of radio-frequency interference (RFI) during observations can
result in a loss of effective bandwidth — and hence sensitivity — via vetoed
frequency channels; loss of effective observing time if the RFI results in
FRBs being completely unobservable; and reduced sensitivity to FRBs in a
certain parameter space (particularly low-DM or high-width FRBs). The
sensitivity of ASKAP to such effects has been studied using pulsar calibration
observations by James et al. (2019a), finding a typical 10% loss of effective
observation time and $\sim$10% fluctuation in sensitivity.
Further details of search pipelines can effect burst sensitivity. The zero-DM
subtraction method — i.e. subtracting the mean detected power prior to
dedispersion — will reduce sensitivity to low-DM events and very bright
bursts, although typically only if the total dispersion sweep across the
detection bandwidth is low. For instance, we have found a few percent bias in
the estimated SNR for ASKAP searches when the expected SNR is greater than one
hundred. An analysis of the HEIMDALL software — used in most Parkes FRB
searches — using real-time injected FRBs by Gupta et al. (2021) finds a 20%
reduction in SNR compared to expectations. This mostly affected bursts with
widths above 20 ms however, and we find in Section 4 our estimates of the DM–z
distribution to be insensitive to the population of very wide bursts. For
ASKAP, the expected SNR of pulsar bursts was found by Shannon et al. (2018) to
be within a few percent of that found by the Fast Real-time Engine for
Dedispersing Amplitudes (FREDDA) algorithm used by the CRAFT collaboration.
We also note that in this work we use SNR as the true SNR of the FRB, when by
definition this has a $1\sigma$ error of unity. Murdoch et al. (1973) find
that for SNR above six, the measured SNR can be approximated as the true SNR
for statistical purposes, and so we have here ignored this effect, rather than
marginalising over it.
Our beamshape model for Parkes uses the mean observation frequency, while for
ASKAP it is uniformly weighted over all frequencies. A preference for e.g.
low-frequency bursts would result in a slight increase in the survey effective
area for both instruments, with only the difference in increase being relevant
when estimating population parameters other than the absolute FRB rate. We
consider this a third-order effect for the frequencies and bandwidths
considered here.
Finally, we have not discussed errors in $F_{0}$. The absolute scale of the
mean ASKAP threshold was calculated by James et al. (2019a) by referencing
observations to Parkes, and using Hydra A as an absolute flux calibrator. Thus
we expect errors in flux calibration for Parkes and ASKAP to be linked, and
cancel to first order. However, individual antenna sensitivity for ASKAP was
found to vary by $\sim\pm 5\%$, and this is similar to the level of
uncertainty found when performing calibration observations for Bannister et
al. (2019). Perhaps the largest source of uncertainty in $F_{0}$ comes from
the quoted threshold of 0.5 Jy ms for Parkes, which has an inherent rounding
uncertainty of $\pm 10\%$, and thus a corresponding rate uncertainty of $\pm
15\%$ for a cumulative source counts index of $-1.5$ (see Section 8.7).
### A.2 Validity of the measurement of $\alpha$
We take as given the measurement of the spectral index by Macquart et al.
(2019). However, as with all other observables used in this work, it is valid
to ask: why do we not fit the observed distribution, taking selection effects
into account? This is in-general important, since telescope beam widths are
broader at low frequencies, and hence more low-frequency FRBs will be
observed. While these authors corrected observed FRBs for frequency-dependent
sensitivity effects, they did not use such effects to model the FRBs they did
not observe, and the result is thus susceptible to biases due to selection
effects.
We first note that in the spectral index interpretation of $\alpha$, there is
no such effect — all FRBs are broadband, and every measured FRB tends to have
the same spectral index. In the rate interpretation however, where every FRB
is narrow-band, each burst is observed according to the sensitivity at its
particular frequency. For experiments with widely spaced beams, the observed
sky area scales as $f^{-2}$, so that far more low-frequency bursts would be
observed even for a frequency-independent rate. In such a case, the observed
frequency-dependence rate should be used as a fitted parameter.
The measurements of Macquart et al. (2019) however were made with ASKAP, with
closely spaced beams overlapping at the half-power points. Making a toy
calculation, using a Euclidean source-counts such that rate is proportional to
beam power1.5, we find that in the narrow-bandwidth limit, one would find an
FRB rate proportional to $f^{-1}$, and hence $\alpha=1$. We emphasise that
ASKAP FRBs are absolutely not in this narrow-bandwidth limit — while they do
show narrow-band features, these tend to be spread across the spectrum
(Shannon et al., 2018). We thus expect the bias due to this effect on the
measurement of $\alpha$ by Macquart et al. (2019) to be very small, and since
$\alpha$ is a small influence on estimates of other parameters in the study,
do not consider this further. It would be important for CHIME, for instance.
### A.3 Pointing
Here we treat a given FRB survey as having a constant local DM component,
DM${}_{\rm local}=$DM${}_{\rm MW}+$DMhost. This is because a greater local DM
reduces sensitivity to extragalatic FRBs due to DM smearing.
When this effect becomes significant, this requires extending the integral in
Eq. 5.1 to
$\displaystyle<N_{i}>$ $\displaystyle=$ $\displaystyle\sum_{k}T_{i,k}\int
dz\Phi(z)\frac{dV(z)}{d\Omega dz}\int d{\rm DM}_{\rm EG}p({\rm DM}_{\rm
EG}|z)$ (44) $\displaystyle\int dB\Omega(B)\int dwp(w)p(E>E_{\rm th})$
$\displaystyle E_{\rm th}$ $\displaystyle\sim$ $\displaystyle E_{\rm
th}(B,w,z,{\rm DM}={\rm DM}_{\rm EG}+{\rm DM}_{\rm local,k})$ (45)
where $T_{ik}$ is the total time spent observing at ${\rm DM}_{\rm local}={\rm
DM}_{\rm local,k}$.
### A.4 Scattering
In Section 4, we combine the intrinsic burst width $w_{\rm int}$ and scatter-
broadening width $w_{\rm scat}$ into the incident width $w_{\rm inc}$.
However, while $w_{\rm int}$ can reasonably be assumed to be independent of
other FRB properties, it is plausible that $w_{\rm inc}$ will be correlated
with both DM and/or $z$, and it will certainly be frequency-dependent. This is
due to interstellar scattering, which is implicitly included through modelling
of observed FRB widths, but is not explicitly accounted for as per e.g. Caleb
et al. (2016). We largely avoid this question in this work by choosing surveys
of similar frequency — however, we consider evidence for such a correlation
here.
Most FRB searches have time resolutions in the range of 100 $\mu$s–1 ms, and
resolving scattering tails from the intrinsic burst structure is difficult:
Qiu et al. (2020) are able to do this for only six of the 33 FRBs observed at
the typically 1.26 ms real-time resolution of ASKAP FRB searches. The broader
time-frequency structure exhibited by many repeating FRBs (Fonseca et al.,
2020a, b) however can often be resolved at these resolutions (e.g. Hessels et
al., 2019), while observations down to 10 $\mu$s have revealed a 1 GHz
scattering time of only 24 $\mu$s in FRB 121102, which is obtained indirectly
from the measured scintillation bandwidth (Michilli et al., 2018).
The most reliable way to resolve these two contributions is to use
time–frequency data at the Nyquist resolution. ASKAP (Cho et al., 2020; Day et
al., 2020) and UTMOST (Farah et al., 2018, 2019) have analysed such data for
six FRBs each. Bursts were found to have strong sub-burst structures down to
50 $\mu$s, with intrinsic widths as low as 10 ms. Scattering was conclusively
measured for a total of four ASKAP FRBs, being in the range 40 $\mu$s–3.3 ms
at 1.27 GHz, while for 181112, it was at most 20 $\mu$s. UTMOST FRBs had 835
MHz scattering times of 4 $\mu$s–30 ms, with one upper limit at 0.2 ms.
Clearly, there is a broad distribution of FRB intrinsic widths and scattering
times. There are general effects that this distribution can have on the DM-$z$
distribution of FRBs.
The most complicated potential interaction of scattering and the intrinsic
width is one that is dependent on the exact position of the FRB in DM–z space.
If indeed FRBs do arise from two source populations with different
cosmological evolutions or different host galaxy properties, or elsewise one
population of object with properties that age on cosmological timescales, the
intrinsic width distribution may have some DM–$z$ dependence. This possibility
should not be ignored. However, the most likely redshift dependence arises
from the scattering term.
Theoretical studies have examined expectations for scattering of FRBs during
cosmic propagation, with effects attributed to the intergalactic medium (IGM),
intra-cluster medium (ICM), and the halos and interstellar medium (ISM) of
intersected galaxies.
The general form of the redshift dependence is analysed by Macquart & Koay
(2013), finding that scattering due to the IGM will increase as $(1+z)^{2}$ to
$z\lesssim 1$, and as $(1+z)^{0.2-0.5}$ for $z\gtrsim 1$. This is in contrast
to the contribution from hosts, which scales as $(1+z)^{-3}$. Results on the
absolute magnitude of the scattering depend on the assumed minimum and maximum
length scales of the turbulence. Macquart & Koay (2013) argue that for
realistic turbulence parameters, the total amount of scattering from the IGM
and ICM is expected to be low, at $\lesssim 1$ and $\lesssim 5$ ms at 300 MHz,
respectively. Zhu et al. (2018) simulate FRBs propagating in a clumpy IGM.
Examining the dependence of the mean scattering time $\bar{\tau}$ on DM, they
find $\bar{\tau}\sim{\rm DM}^{1-2}$ for voids, clusters, and filaments over a
range of simulation parameters. They also conclude that unreasonably high
turbulence scales would be required to achieve scattering values comparable to
that observed in FRBs.
In some cases, FRBs localisations have been sufficient to identify
intersections of the line-of-sight with Galactic halos, placing upper limits
on the degree of scattering caused by such intersections (Prochaska et al.,
2019; Connor et al., 2020).
Qiu et al. (2020) find no evidence however for a DM-dependence of scattering.
Both measurements and expectation therefore suggest that the observed
distribution of scattering arises from the host galaxies. We do not however
attempt to model this in this work, since the $(1+z)^{2}$ reduction in width
at high redshift will be in any case insignificant against the DM smearing
effect. A full model of scattering will however become important when
including observations over a wide frequency range.
### A.5 Influence of specific parameters
This work uses measured FRB detection numbers, dispersion measures, redshifts,
and strengths from three different FRB searches. It is useful to probe the
influence of each of these on our overall result.
To do so, we show in Figure 17 our confidence limits for each parameter using
our prior on $\alpha$ when removing the likelihood corresponding to $p_{N}$
27, $p_{\rm DMZ}$ (eq.31), $p_{s}$ (eq.36), and when removing ASKAP/FE,
ASKAP/ICE, and Parkes/Mb observations. We do this under the spectral index
interpretation of $\alpha$ — results are similar under the rate
interpretation.
Figure 17: Resulting 68%, 90%, and 95% confidence limits (respectively: blue,
upper; green, middle; red, lower) on individual parameters under the spectral
index interpretation of $\alpha$, using in descending order: all available
information, removing information on the number of detected FRBs, their DMs
and redshifts, their measured signal-to-noise ratios, and data from ASKAP/FE,
ASKAP/ICS, and Parkes/MB, respectively.
Unsurprisingly, the most dramatic effect occurs when removing information on
measured DM and $z$. In this limit, we do not find lower bounds on $E_{\rm
max}$ in the range $\log_{10}E_{\rm max}{\rm[erg]}>40$, and limits on other
parameters are very broad. This is not a very interesting, informative, or
unexpected result, and we ignore the “No $p_{\rm DMZ}$” in the following
discussion. However, the fact that some constraints are nonetheless derived is
evidence for the source-counts distribution of FRBs containing useful
information (Macquart & Ekers, 2018b).
For all other parameters, we obtain a variety of modified limits. Clearly, the
ASKAP/ICS sample is most constraining to $E_{\rm max}$, while without the
ASKAP/FE sample, the constraints are narrower. This is natural, since the
ASKAP/FE sample probes the most intrinsically luminous bursts, without which a
fit for a narrow population distribution is possible.
The slope of the cumulative luminosity function $\gamma$ is constrained to by
flat by $p_{n}$, without which a steeper (lower) value would be obtained,
while both the Parkes/Mb sample, and $p_{\rm snr}$, prevent flat (higher)
values. This is a very interesting result. As noted in our companion paper,
our best fit under-predicts the number of ASKAP/FE FRBs (12.9 vs. 20 in 1274.6
days), and over-predicts those for Parkes/Mb (19.2 vs 12 in 164.4 days). One
would expect that removing either Parkes/Mb or $p_{n}$ would have a similar
effect by removing this tension. However, removing Parkes/Mb also removes the
Parkes DM distribution. This then suggests that the Parkes DM distribution is
indicative of a steep source-counts spectrum, whereas the number of FRBs
detected by Parkes is indicative of a flat spectrum. We do not have a full
solution to this quandry, although it is clearly related to the discussion in
Section 8. We do not elaborate further, since this discrepancy is of only
marginal statistical significance.
Perhaps our most significant result — strong evidence for source evolution
with redshift, i.e. $n>0$ — is most strongly disfavoured by $p_{n}$, such that
when this is removed, we prefer a strongly evolving population. This,
predictably, is related to the low number of bursts observed by Parkes/Mb
relative to ASKAP/FE compared to expectations discussed above, which alone
argues against source evolution. Removing any of the three FRB surveys, or
$p_{\rm snr}$, simply results in mildly less evidence for $n>0$ — there is no
single result strongly favouring $n>0$. We thus conclude that our conclusion
on $n$ robust, especially considering uncertainties discussed in Appendix A.1.
Our host galaxy parameters, $\mu_{\rm host}$ and $\sigma_{\rm host}$ — recall
these are in log DM space — are most affected by the highest- and lowest-
sensitivity surveys, ASKAP/FE and Parkes/Mb. Without these observations, very
low values of $\mu_{\rm host}$ (and hence very large values of $\sigma_{\rm
host}$) become possible. This validates the use of DM distributions only when
studying these parameters. Neither $p_{\rm snr}$ nor $p_{n}$ have a
significant effect.
|
# Two-dimensional Inflow-Wind Solution of Hot Accretion Flow. I. Hydrodynamics
Amin Mosallanezhad School of Mathematics and Statistics, Xi’an Jiaotong
University, Xi’an, Shaanxi 710049, People’s Republic of China Fatemeh Zahra
Zeraatgari School of Mathematics and Statistics, Xi’an Jiaotong University,
Xi’an, Shaanxi 710049, People’s Republic of China Liquan Mei School of
Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049,
People’s Republic of China De-Fu Bu Key Laboratory for Research in Galaxies
and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences,
80 Nandan Road, Shanghai 200030, People’s Republic of China
###### Abstract
We solve the two-dimensional hydrodynamic equations of hot accretion flow in
the presence of the thermal conduction. The flow is assumed to be in steady-
state and axisymmetric, and self-similar approximation is adopted in the
radial direction. In this hydrodynamic study, we consider the viscous stress
tensor to mimic the effects of the magnetorotational instability for driving
angular momentum. We impose the physical boundary conditions at both the
rotation axis and the equatorial plane and obtain the solutions in the full
$r-\theta$ space. We have found that thermal conduction is indispensable term
for investigating the inflow-wind structure of the hot accretion flows with
very low mass accretion rates. One of the most interesting results here is
that the disc is convectively stable in hot accretion mode and in the presence
of the thermal conduction. Furthermore, the properties of wind and also its
driving mechanisms are studied. Our analytical results are consistent with
previous numerical simulations of hot accretion flow.
Accretion; High energy astrophysics; Black hole physics; Low-luminosity active
galactic nuclei
††journal: ApJ
## 1 Introduction
Various black hole accretion models have been proposed in the past several
decades including standard thin disc (Shakura & Sunyaev 1973), super-Eddington
accretion flow (slim disc, Abramowicz et al. 1988), and also hot accretion
flow (Narayan & Yi 1994; Abramowicz et al. 1995). Based upon the temperature
of the accretion flow, these models can be divided into cold and hot modes
where the standard thin disc and super-Eddington accretion flow belong to cold
mode (Yuan & Narayan 2014).
In recent years, wind has become a fascinating subject in the study of
accretion flows in both cold and hot modes. Wind appears to carry significant
mass, angular momentum and energy away from the disc, and has the potential
for a greater impact on its surrounding. This discernible effect on its
environment is persuasive enough for interest in this topic. Further, the
elimination of mass and angular momentum from the disc might essentially alter
the accretion process (Shields et al. 1986). There have been substantial
observational evidence of wind in cold accretion mode via blue shifted
absorption lines, in luminous AGNs (Crenshaw et al. 2003; Tombesi et al. 2010,
2014; Liu et al. 2013; King & Pounds 2015) and X-ray binaries in high/soft
state (Neilsen & Homan 2012; Homan et al. 2016; Díaz Trigo & Boirin 2016).
Nevertheless, the challenging objects for detection are systems in hot
accretion mode, since the accreting gas is virially hot and fully ionized in
low-luminous AGNs (LLAGNs) (Tombesi et al. 2010, 2014; Crenshaw & Kraemer
2012; Cheung et al. 2016) and low/hard state of X-ray binaries (Homan et al.
2016; Munoz-Darias et al. 2019).
Wind can be driven by different physical mechanisms such as thermal, radiation
and magnetic pressures. Radiation pressure as well as magnetic forces would
act on smaller scales (Proga et al. 2000; Fukumura et al. 2015), while thermal
driving might work in outer regions of the disc, and could adjust the mass
accretion rate through the disc (Shakura & Sunyaev 1973; Begelman et al. 1983;
Shields et al. 1986). A large number of numerical simulations have been done
to show the wind is potentially able to transfer a significant amount of power
from the black hole accreting system in hot accretion mode (e.g., Stone et al.
1999; Igumenshchev & Abramowicz 1999, 2000; Hawley & Balbus 2002; Pang et al.
2011; Yuan et al. 2012a, b; Bu et al. 2013; Narayan et al. 2012; Li et al.
2013); however, the actual mechanism driving such winds is still a source of
much debate.
Analytical method is often invoked to investigate the existence of wind from
accretion flow in hydrodynamic (HD) and magnetohydrodynamic (MHD) studies
(e.g., Bu et al. 2009; Mosallanezhad et al. 2014; Samadi et al. 2017; Bu &
Mosallanezhad 2018; Kumar & Gu 2018; Zeraatgari et al. 2020) and calculate the
physical properties of the wind. In principle, it is very difficult and time
consuming to perform numerical simulations with the most updated physical
terms as well as different input parameters to examine the dependency of the
results to the initial set of parameters. Therefore, analytical studies are
very powerful tools for better understanding the dependency of inflow-wind
structure of the system to the physical parameters and attempt to find the
real mechanism of producing wind in such accreting systems.
The analytical study of hot accretion flow started from height-averaged,
radially self-similar solutions presented by Narayan & Yi 1994 which could not
show a clear picture of the vertical structure of such a system. Next, in
Narayan & Yi 1995 (henceforth NY95), they revisited their solutions in
$r-\theta$ plane in spherical coordinates. They adopted self-similar solutions
in radial direction and to solve the equations in the $\theta$-direction used
proper boundary conditions at both equatorial plane and rotation axis.
Unfortunately, the solutions did not show an inflow-wind structure, since they
considered $v_{\theta}=0$ and zero radial velocity at the pole, i.e.,
$v_{r}(0)=0$. By eliminating $v_{\theta}$ and adopting axisymmetric and steady
state assumptions, the radial self-similar solution of the density followed
by: $\rho\propto r^{-3/2}$ (see the first term in equation (B1)). Their
solutions also became singular when $\gamma=5/3$ albeit the numerical
simulations of hot accretion flow suggest that in the non-relativistic cases,
$\gamma$ is very close to $5/3$ (e.g. Balbus & Hawley 1998; Blandford &
Begelman 1999). However, based on their positive value of Bernoulli parameter,
they argued that bipolar outflow must exist near the rotation axis.
After that, Xu & Chen 1997 (henceforth XC97) brought up all components of the
velocity including $v_{\theta}$, adopted Fourier series and obtained accretion
and ejection solutions. Blandford & Begelman 2004 also represented self-
similar two-dimensional solutions for radiatively inefficient accretion flows
with outflow called adiabatic inflow-outflow solutions (ADIOS). They suggest
that the mass accretion rate decreases inward due to the mass loss in the
outflow. Based on their solution, the mass inflow and outflow fluxes follow
the power law of $\dot{M}\propto r^{s}$ with equal and opposite values. They
also determined the power index $s$ to be in the range $0\leqslant s\leqslant
1$.
Tanaka & Menou 2006 (henceforth TM06) followed NY95 and they mainly focused on
the effects of the thermal conduction on the global properties of hot
accretion flows. Even though they did not include $v_{\theta}$, their
solutions obtained positive radial velocity near the rotation axis
interpreting the outflow. Here, we intend to emphasize the requisite role of
$v_{\theta}$ in wind studies. As an example, XC97 considered $v_{\theta}$ but
some researchers argued that the solution would not be correct (e.g., Xue &
Wang 2005, Jiao & Wu 2011). This is mainly because, they believed the mass
flux of the outflow became exactly equal to the mass flux of the inflow at a
certain radius. To avoid this knotty issue, for instance, Xue & Wang 2005
truncated the solutions where $v_{r}=0$ by prescribing the opening angle
$\theta_{0}$ and considered this place as the surface of the disk. They set
the sound speed on the surface as
$c_{s}/\Omega_{k}\lesssim(\pi/2-\theta_{0})r$. Therefore, the second boundary
is set as an input parameter rather than being calculated in their solution.
Actually, their solutions were limited to the inflow region near the
equatorial plane and a surface from which the wind would blow out. Also, Jiao
& Wu 2011 integrated the equations from equatorial plane toward the rotation
axis and stopped integration where the density or gas pressure became
negative. Although an outflow structure could be shown, their solutions did
not reach to the pole, $\theta=0$, where some of the physical boundary
conditions must be satisfied.
Khajenabi & Shadmehri 2013 (henceforth KS13) also solved the HD equations of
hot accretion flow in the presence of the thermal conduction as well as all
three components of the velocity. Further, they imposed the same physical
constraint as Xue & Wang 2005 for the opening angle and obtained this angle
self-consistently from their numerical integration starting from equatorial
plane. Their results showed that the thermal conduction affected the opening
angle of the wind as an increase of that would shrink the size of the wind
region.
The main aim of this paper is to find inflow-wind solution of hot accretion
flow in the whole vertical direction by imposing proper boundary conditions at
both the rotation axis and the equatorial plane and then compare the results
with those above mentioned studies. In the following section, we will find the
origin of discrepancy between all those aforementioned analytical solutions by
focusing on the continuity and the energy equations. We will illustrate that
it would be possible to solve the hydrodynamic equations of hot accretion flow
in the whole $\theta$-direction using two point boundary value problem.
In this paper, to mimic the effects of magnetorotational instability (MRI) in
driving angular momentum, we will consider the viscous stress tensor. Another
aim of the present paper is to find the dependency of the results to the
initial set of parameters such as the conductivity coefficient, density index,
and advection parameter. Note here that, the gradient of the magnetic pressure
is one of the driving mechanisms for producing wind and plays an important
role in the dynamics of hot accretion flows. In the sense of on this point, we
intend to embark on a series of papers to solve hydrodynamic (HD) and
magnetohydrodynamic (MHD)111The MHD equations will be solved in the second
paper of these series and the HD and MHD results will be compared there.
equations of hot accretion flow using appropriate boundary conditions at the
rotation axis as well as the equatorial plane.
The remainder of the paper is organized as follows. In section 2, we will
investigate the origin of discrepancy between all previous analytical
solutions and find the reason why their solutions could not reach the rotation
axis. The basic HD equations, physical assumptions, self-similar solutions,
and also the boundary conditions will be introduced in Section 3. In Section
4, the detailed explanations of numerical results will be presented. Finally,
in section 5, we will provide the summary and discussion.
## 2 The origin of discrepancy between previous analytical solutions
Based on Reynolds transport theorem, the time rate of change of integrals of
physical quantities within material volumes can be calculated as,
$\frac{d\mathcal{F}}{dt}=\int_{V}\left(\frac{dF}{dt}+F\nabla\cdot\bm{v}\right)dV,$
(1)
where, $F(\bm{x},t)$ can be any scalar, vector, or tensor field, with
$\mathcal{F}=\int_{V}F(\bm{x},t)dV$ (see equation (18.2) in Mihalas & Mihalas
1984 for more details). According to the conservation law of mass for the
fluid, the mass within a material volume must be always the same, i.e.,
$\frac{d}{dt}\int_{V}\rho\,dV=0.$ (2)
By applying Reynolds transport theorem for $F=\rho$, we have:
$\int_{V}\left(\frac{\mathrm{d}\rho}{\mathrm{d}t}+\rho\nabla\cdot\bm{v}\right)\,dV=0.$
(3)
In principle, this integral will be vanished only if the integrand vanishes at
all points in the flow field. The integrand then is the continuity equation
(see equation 11). In spherical coordinates, by imposing the axisymmetric
$(\partial/\partial\phi=0)$ and steady state $(\partial/\partial t=0)$
assumptions into the equation of continuity, it will be reduced to the
equation (B1). Integrating the first term of this equation over angle will
give us,
$\dot{M}=-\int 2\pi r^{2}\sin\theta\rho v_{r}\,d\theta,$ (4)
where, $\dot{M}$ is the net mass accretion rate. In NY95 since $v_{\theta}=0$,
the second term was spontaneously eliminated from the equation (B1). As a
result, the density index of their self-similar solutions became $n=3/2$ in
$\rho\propto r^{-n}$ (see equations (2.11)-(2.15) of NY95). In the case of
$v_{\theta}\neq 0$, the sum of two terms in the equation (B1) must be always
zero at each specific radius to satisfy the continuity equation. Keeping the
second term of equation (B1) would also cause flattening of the density
profile, i.e., density index would be in the range of $0<n<3/2$.
In XC97, they included $v_{\theta}$ in their solutions and found two different
kinds of solutions, i.e., accretion-outflow and ejection-outflow for the
density index of $n<3/2$ and $n>3/2$, respectively. However, the ejection-
outflow with the large value for density index of $n>3/2$ might not be
correct. The reason would emanate from the energy equation which is defined
as,
$Q_{\mathrm{adv}}=Q_{+}-Q_{-}$ (5)
where $Q_{\mathrm{adv}}$, $Q_{+}$, and $Q_{-}$ are energy advection, total
heating and cooling terms per unit volume, respectively. In hot accretion
flows or radiative inefficient accretion flows (RIAFs), it is common to omit
$Q_{-}$ and introduce advection parameter as $f\equiv Q_{\mathrm{adv}}/Q_{+}$
with $0<f\leq 1$. Hence, the energy equation of the hot accretion flow in XC97
was written as,
$Q_{\mathrm{adv}}\equiv fQ_{+}$ (6)
with the total energy heating released by the viscosity. By assuming
axisymmetric, steady state, and radially self-similar approximation, the
advection and the viscous terms of the energy equation were reduced to the
dimensionless forms as described in equation (A14) and equation (A15),
respectively (see Appendix A for more details). Based upon the global picture
of the accretion flow, inflow happens around the equatorial plane while the
outflow does appear at high latitudes. To show that the range of the density
index, only for highly non-relativistic cases, must be $n<3/2$, in what
follows, we investigate the energy equation in inflow and wind regions in
details.
### 2.1 Energy equation in the inflow region
As proved by Mihalas & Mihalas 1984, the viscous dissipation term is always
positive everywhere, $Q_{\mathrm{vis}}>0$. Besides, the symmetry boundary
conditions dictate that the latitudinal component of the velocity should be
null at the equatorial plane, i.e., $v_{\theta}(\pi/2)=0$ (see equation (26))
which means that the first term of the equation (A14) is dominated at
midplane. Necessarily, this term must be positive to satisfy the energy
equation,
$\left[n-\frac{1}{\gamma-1}\right]p_{\mathrm{g}}v_{r}>0,$ (7)
where $\gamma$ is the adiabatic index. Since the gas pressure is always
positive and $v_{r}<0$ at the equatorial plane, then the above equation will
be satisfied there only if,
$n-\frac{1}{\gamma-1}<0.$ (8)
As mentioned in the introduction, numerical simulations of the hot accretion
flow suggest that in the non-relativistic cases, $\gamma$ is very close to
$5/3$. Consequently, to satisfy the above equation, with $\gamma=5/3$, the
density index must be222It should be emphasized here that this analysis is
very specific since we adopt self-similar approximation with constant Mach
number along the radial direction and $\gamma=5/3$ which is only applicable
for highly non-relativistic cases.,
$n<\frac{3}{2}.$ (9)
This value for n clearly proves that the bipolar ejection outflow solution of
XC97 is not physical where they set n = 2.5 (see section 3 and also panel (b)
of figure 1 in XC97).
### 2.2 Energy equation in the wind region
At high latitudes, if wind exists and launches, the radial velocity must be
positive ($v_{r}>0$). At the rotation axis, we have similar boundary condition
for the latitudinal component of the velocity, i.e., $v_{\theta}(0)=0$.
Additionally, as proved from the energy equation in the inflow region, the
density index must be $n<3/2$. Therefore, the advection term at the pole must
be negative as,
$\left[n-\frac{1}{\gamma-1}\right]p_{\mathrm{g}}v_{r}<0.$ (10)
Since the viscous heating term of the energy equation is always positive, to
satisfy the above equation, we need an additional term in the right-hand side
of the energy equation which allows for the positive radial velocity near the
pole. TM06 and KS13 showed that thermal conduction can be negative enough at
high latitudes to overcome positive sum of the viscous heating terms and
generate positive nonzero radial velocity about the rotation axis.
In essence, in hot accretion flows with very low mass accretion rates, the
electron mean-free path is much larger than electron gyro-radius. Hence,
Coulomb collisions are not exceptional. For instance, in our Galactic Center,
Sgr $\mathrm{A^{*}}$, with mass accretion rate $\dot{\mathrm{M}}\approx
10^{-7}-10^{-8}\,\mathrm{M}_{\odot}\mathrm{yr^{-1}}$, the electron mean-free
path $l\sim 1.3\times 10^{17}\mathrm{cm}$ and the gyro-radius of electrons
$R_{\mathrm{gyro}}(=m_{\mathrm{e}}v_{\mathrm{th}}c/qB)\sim 10^{5}\mathrm{cm}$
($m_{\mathrm{e}}$, $v_{\mathrm{th}}$, $c$, $q$ and $B$, are electron mass,
thermal speed of electron, speed of light, electron charge, and magnetic
field, respectively). Thermal conduction can play a striking role in such a
system as can make a heat flux from hot inner regions to the cold outermost of
the flow. The gas in outer region has the possibility to be heated to a
temperature higher than the local virial temperature. This increase in the
temperature can drive thermal outflow/wind and consequently decrease the mass
accretion rate.
In this paper, we will solve the HD equations of the hot accretion flow with
thermal conduction in the whole $\theta$ direction. We will also consider all
components of the velocity and the viscous stress tensor and impose the
boundary conditions at both the rotation axis and the equatorial plane.
## 3 Basic Equations and Assumptions
The HD equations of the hot accretion flow with thermal conduction can be
written as,
$\frac{\mathrm{d}\rho}{\mathrm{d}t}+\rho\nabla\cdot\bm{v}=0,$ (11)
$\rho\frac{\mathrm{d}\bm{v}}{\mathrm{d}t}=-\rho\nabla\psi-\nabla
p+\nabla\cdot\bm{\sigma},$ (12)
$Q_{\mathrm{adv}}=Q_{\mathrm{vis}}+Q_{\mathrm{c}},$ (13)
where, $\rho$ is the mass density, $\bm{v}$ is the velocity,
$\psi\left[=-GM/r\right]$ is the Newtonian potential (where $r$ is the
distance from central black hole, $M$ is the black hole mass, and $G$ is the
gravitational constant), $p$ is the gas pressure, $\bm{\sigma}$ is the viscous
stress tensor, and the $\mathrm{d}/\mathrm{d}t\equiv\partial/\partial
t+\bm{v}\cdot\nabla$ denotes the Lagrangian or comoving derivative. In the
equation (13), $Q_{\mathrm{adv}}$, $Q_{\mathrm{vis}}$, and $Q_{\mathrm{c}}$
are advection, viscous and thermal conduction terms of the energy equation,
respectively, which can be described as,
$Q_{\mathrm{adv}}=\rho\frac{\mathrm{d}e}{\mathrm{d}t}-\frac{p}{\rho}\frac{\mathrm{d}\rho}{\mathrm{d}t},$
(14)
$Q_{\mathrm{vis}}=f\nabla\bm{v}:\bm{\sigma},$ (15)
$Q_{\mathrm{c}}=-\nabla\cdot\bm{F}_{\mathrm{c}}.$ (16)
Here, $e$ is the internal energy of the gas, and $\bm{F}_{\mathrm{c}}$ is the
heat flux due to the thermal conduction. We adopt the adiabatic equation of
state as $p=\left(\gamma-1\right)\rho e$. In purely HD limit,
$\bm{F}_{\mathrm{c}}$ is defined as,
$\bm{F}_{\mathrm{c}}=-\chi\nabla T,$ (17)
where $\chi$ is the thermal diffusivity and $T$ is the gas temperature. In
one-temperature structure, as our case, $T$ can be written as,
$T=\frac{\mu m_{p}}{k_{\\!{}_{B}}}\frac{p}{\rho},$ (18)
where, $\mu$, $m_{p}$, and $k_{\\!{}_{B}}$ are mean molecular weight, proton
mass, and Boltzmann constant, respectively. Numerical simulations of Sharma et
al. 2008 and Bu et al. 2016), assumed $\kappa=\chi
T/p\equiv\alpha_{\mathrm{c}}\left(GMr\right)^{1/2}$ with the dimensionless
conductivity takes the value of $\alpha_{\mathrm{c}}\simeq(0.2-2)$. Here, we
follow those numerical simulations by adopting the same definition for thermal
conduction and consider the same range for $\alpha_{c}$ except in section 4.4
we consider small value of $\alpha_{c}$ to check the dependency of the results
to this parameter. The viscous stress tensor is given by,
$\sigma_{ij}=\rho\nu\left[\left(\frac{\partial v_{j}}{\partial
x_{i}}+\frac{\partial v_{i}}{\partial
x_{j}}\right)-\frac{2}{3}\left(\nabla\cdot\bm{v}\right)\delta_{ij}\right],$
(19)
where $\nu$ is called the kinematic viscosity coefficient and $\delta_{ij}$ is
the usual Kronecker delta. Note that the bulk viscosity is neglected here. It
is well known that in real accretion flow, the angular momentum is transferred
by Maxwell stress associated with MHD turbulence driven by MRI (Balbus &
Hawley 1998). In our current HD case, we approximate the effect of the
magnetic stress by adding viscous terms in momentum and energy equations (see
e.g., Yuan et al. 2012b). The kinematic viscosity coefficient is calculated
with the $\alpha$-prescription (Shakura & Sunyaev 1973) as,
$\nu=\alpha\frac{p}{\rho\Omega_{\\!{{}_{K}}}},$ (20)
where $\Omega_{\\!{{}_{K}}}\equiv(GM/r^{3})^{1/2}$ is the Keplerian angular
velocity and $\alpha$ is the viscosity parameter. We use spherical coordinates
$(r,\theta,\phi)$ to solve the full set of equations including all viscous
terms. The disc is taken to be axisymmetric and steady state. By implementing
all the above mentioned assumptions and definitions into equations (11)-(13),
we obtain the partial differential equations (PDEs) presented in Appendix B
(see equations (B1)-(B5) for more details).
The global numerical simulations of hot accretion flows show that the physical
variables of the flow can be described by power-law function of radius far
away from the radial boundaries. For instance, the radial profile of the
density follows $\rho(r)\propto r^{-n}$ with $n<3/2$ ( see, e.g., Stone et al.
1999; Yuan et al. 2012a, b, 2015). In order to solve equations (B1)-(B5) by
numerical methods, we impose self-similar solutions to remove the radial
dependency of the variables. To do so, we introduce the fiducial radius
$r_{\\!{}_{0}}$ with the self-similar solutions as a power-law form of
$(r/r_{\\!{}_{0}})$. Accordingly, the physical variables of the hot accretion
flow will be written in the following forms,
Figure 1: Latitudinal profile of the physical variables of the fiducial model.
Top row: radial velocity in the unit of Keplerian velocity, $v_{\\!{}_{K}}$;
the dash-dotted line shows the location of $v_{r}=0$ that is about
$52^{\circ}$ (left panel); latitudinal velocity, $v_{\theta}$ (middle panel);
angular velocity in the unit of Keplerian angular velocity,
$\Omega_{\\!{}_{K}}$ (right panel). Bottom row: density in the unit of density
of mid-plane at $r_{\\!{}_{0}}$, i.e., $\rho_{\\!{}_{0}}$ (left panel); gas
pressure in the unit of $\rho_{\\!{}_{0}}v_{\\!{}_{0}}^{2}$ where
$v_{\\!{}_{0}}(\equiv\sqrt{GM/r_{\\!{}_{0}}})$ is the Keplerian velocity at
$r_{0}$ (middle panel); Mach numbers (right panel).
$\rho\left(r,\theta\right)=\rho_{\\!{}_{0}}\left(\frac{r}{r_{\\!{}_{0}}}\right)^{-n}\bar{\rho}(\theta),$
(21)
$v_{r}(r,\theta)=v_{\\!{}_{0}}\left(\frac{r}{r_{\\!{}_{0}}}\right)^{-1/2}\bar{v}_{r}(\theta),$
(22)
$v_{\theta}(r,\theta)=v_{\\!{}_{0}}\left(\frac{r}{r_{\\!{}_{0}}}\right)^{-1/2}\bar{v}_{\theta}(\theta),$
(23)
$v_{\phi}(r,\theta)=v_{\\!{}_{0}}\left(\frac{r}{r_{\\!{}_{0}}}\right)^{-1/2}\bar{\Omega}(\theta)\sin(\theta),$
(24)
$p(r,\theta)=p_{\\!{}_{0}}\left(\frac{r}{r_{\\!{}_{0}}}\right)^{-n-1}\bar{p}_{\mathrm{g}}(\theta),$
(25)
where $r_{\\!{}_{0}}$, $\rho_{\\!{}_{0}}$,
$v_{\\!{}_{0}}\left[=\sqrt{GM/r_{\\!{}_{0}}}\right]$, and
$p_{\\!{}_{0}}\left[=\rho_{\\!{}_{0}}v_{\\!{}_{0}}^{2}\right]$, are the units
of length, density, velocity, and gas pressure, respectively. Substituting the
above self-similar solutions into equations (B1)-(B5), the radial dependency
will be removed and the system of PDEs will be reduced to a set of ordinary
differential equations (ODEs) presented in Appendix C. The ODE equations
(C1)-(C5) consist of five physical variables: $v_{r}(\theta)$,
$v_{\theta}(\theta)$, $\Omega(\theta)$, $\rho(\theta)$,
$p_{\mathrm{g}}(\theta)$ and also their first and second derivatives. As we
mentioned in introduction, the integration will not stop at some angle near
the rotation axis (see e.g., Jiao & Wu 2011, Mosallanezhad et al. 2016, Samadi
& Abbassi 2016). Instead the computational domain will be extended from the
rotation axis, $\theta=0$, to the equatorial plane, $\theta=\pi/2$. Following
previous analytical solutions of hot accretion flow, e.g., NY95 and TM06, all
physical variables are assumed to be even symmetric, continuous, and
differentiable at both boundaries. Since we also include the latitudinal
component of the velocity, $v_{\theta}$, its value will be null at both the
equatorial plane and the rotation axis. Thus, the following boundary
conditions at $\theta=0$ and $\theta=\pi/2$ will be imposed:
$\frac{\mathrm{d}\rho}{\mathrm{d}\theta}=\frac{\mathrm{d}p_{\mathrm{g}}}{\mathrm{d}\theta}=\frac{\mathrm{d}\Omega}{\mathrm{d}\theta}=\frac{\mathrm{d}v_{r}}{\mathrm{d}\theta}=v_{\theta}=0.$
(26)
To satisfy all boundary conditions at both ends, the relaxation method will be
adopted mainly because the set of ODEs have extraneous solutions, and also
there exists singularity at the rotation axis. To have a good resolution at
both sides where the boundary conditions set, we divide the $\theta$ direction
into 5000 grids with stretch grid as follow: From $\theta=0$ to $\theta=\pi/4$
the grid size ratio is set as
$\mathrm{d}\theta_{\mathrm{i+1}}/\mathrm{d}\theta_{\mathrm{i}}=1.003$ while
from $\theta=\pi/4$ to $\theta=\pi/2$ the grid size ratio is set as
$\mathrm{d}\theta_{\mathrm{i+1}}/\mathrm{d}\theta_{\mathrm{i}}=0.997$. We
calculate the variables at the cell center of these grids. The absolute error
tolerance is set to $10^{-15}$. The most difficult part of solving the set of
equations is providing an appropriate guess for the required solutions. In
this study, we use Fourier cosine series for the initial guess of all physical
variables except for $v_{\theta}(\theta)$ which we used Fourier sine series.
Since our solutions are satisfying the boundary conditions at both the
rotation axis as well as the equatorial plane, it does assure us that well-
behaved solutions will be derived in the whole $\theta$ direction. In the next
section we will explain the behaviors of the physical variables in detail.
## 4 Numerical Results
### 4.1 The solutions of the fiducial model
Figure 2: Two-dimensional distribution of the density (left panel) and the
temperature (right panel) based on the self-similar solutions. Both panels are
overlaid with the poloidal velocity,
$\vec{v}_{p}=v_{r}\hat{\bm{r}}+v_{\theta}\hat{\bm{\theta}}$ . In the left
panel, the arrows are plotted only at six different radii in the unit of
schwarzschild radius, i.e., $r=[100,150,200,250,300,350]\,r_{\mathrm{s}}$ to
show the strength of the outflow. The white dash-dotted lines show the
location of $v_{r}=0$. In the right panel, the poloidal velocity is normalized
with its absolute value, $|\vec{v}_{p}|=\sqrt{v_{r}^{2}+v_{\theta}^{2}}$ to
denote the direction of the vectors. Here,
$T_{\mathrm{vir}}(r_{\\!{}_{0}})=GMm_{p}/(3k_{B}r_{\\!{}_{0}})$ is the virial
temperature at $r_{\\!{}_{0}}=10\,r_{\mathrm{s}}$.
To solve the system of ODEs numerically, we integrate the equations (C1)-(C5)
from the rotation axis to the equatorial plane to get the latitudinal profiles
of all physical variables of our fiducial model shown in Figure 1. The
parameters are set as $\alpha=0.15$, $\alpha_{c}=0.2$, $n=0.85$, $f=1$, and
$\gamma=5/3$. The velocities are scaled with the Keplerian velocity at
$r=r_{\\!{}_{0}}$, i.e., $v_{\\!{}_{0}}=\sqrt{GM/r_{\\!{}_{0}}}$. The density
is normalized with $\rho_{\\!{}_{0}}$, and the pressure is also normalized
with $\rho_{\\!{}_{0}}v_{\\!{}_{0}}^{2}$ at $r_{\\!{}_{0}}$. We assumed the
radius $r_{\\!{}_{0}}$ is located at the equatorial plane where the maximum
density of the accretion flow is accumulated there. In top left panel of
Figure 1, $v_{r}=0$ shows the inclination of the inflow to the wind region
which is around $\theta=52^{\circ}$333We made a parameter study and found that
inclination of the inflow to the wind region does not change too much with
different set of input parameters. The changes are only in the range of
$\theta\sim 50^{\circ}-55^{\circ}$.. As it is seen, $v_{r}$ is negative in the
inflow region where the matter goes toward the BH while in the wind region it
becomes positive and reaches a value slightly above Keplerian velocity of that
radius. In the top middle panel, $v_{\theta}$ is negative in the vertical
direction from the equatorial plane toward the rotation axis. Furthermore, due
to the boundary conditions it is zero in both boundaries and minimum around
$\theta\simeq 20^{\circ}$. In top right panel, the angular velocity tendency,
as it increases from the inflow region toward the wind region, shows that the
angular momentum is transported away from the system by the wind. Moreover,
the angular velocity exceeds Keplerian velocity around $\theta\simeq
45^{\circ}$ and is quite super-Keplerian in the wind region. In bottom left
panel, the concentration of the density is at the equatorial plane and then
drops toward the wind region so that reaches its minimum value around the
rotation axis. The trend of the gas pressure, bottom middle panel, is also the
same for the density, maximum at the equator and the minimum at the rotation
axis. From the bottom right panel, $v_{r}$ and $v_{\theta}$ both remain
subsonic in whole $r-\theta$ domain. The main reason is that the density drops
faster than the pressure from equatorial plane toward the rotation axis.
Therefore, the sound speed, $c_{\mathrm{s}}=\sqrt{\gamma p/\rho}$, or
equivalently the gas temperature increases as $\theta$ angle decreases. This
panel clearly shows that both $|v_{r}|/c_{\mathrm{s}}$ and
$|v_{\theta}|/c_{\mathrm{s}}$ do not pass through a sonic point (where the
singularity exists) at small angles. The behavior of Mach number for $v_{r}$
is that it first drops and then increases going from $\theta=52^{\circ}$
(where the radial velocity is zero, $v_{r}=0$) toward the rotation axis. Also,
$|v_{\theta}|/c_{\mathrm{s}}$ first increases until around $\theta\simeq
35^{\circ}$ and then decreases toward the rotation axis, and this is because
we plot the absolute value of Mach numbers. The overall behavior of the
physical variables are almost the same as those in XC97, TM06, and KS13. More
precisely, the similarity can be seen in the inflow-wind structure, inflow
with negative radial velocity around the equatorial plane and wind with
positive radial velocity around the rotation axis. It is very diagnostic in
the density profile which drops from the equatorial plane to the rotation axis
in all above studies showing a disc-shape structure (see also left panel of
Figure 2). Since in the present study we include $v_{\theta}$, we can easily
compare our results with XC97 and KS13. For instance, the wind radial velocity
is much more higher than the inflow radial velocity at high latitudes which is
exactly similar to the Figure 2 of XC97 and also Figure 1 of KS13444Note here
that the main differences between our solution and the accretion outflow
solution of XC97 are; (1) we include thermal conduction, and (2) the value of
the radial velocity at the equatorial plane is self-consistently determined
here, while in XC97, it is fixed as $v_{r}(\pi/2)=-0.05v_{\\!{}_{K}}$.. The
trend of the angular velocity in the present work increasing toward the
rotation axis is similar to that in both XC97 and KS13, which helps the wind
to accelerate with high speed and cause the angular momentum is transferred
outward (see Figure 2 of XC97 and also the bottom panel of figure 2 in KS13).
However, in figure 2 of TM06, angular velocity, $\Omega$, decreases with
decreasing $\theta$. One of the differences of the present work and XC97 with
KS13 is that the angular velocity becomes super-Keplerian at small angles.
Another one is U-shape behavior of $v_{\theta}$, it first falls down and then
goes up toward the rotation axis, while it keeps increasing continuously
toward the opening angle in KS13. The reason can be interpreted due to
different assumptions which these studies have made in their calculations. In
addition, we and XC97 both put the second boundary condition at the rotation
axis where the gradient of the physical variables as well as $v_{\theta}$ must
be zero. Nevertheless, KS13 started the integration from the equator and went
to a certain inclination where the physical constraint was satisfied and
considered this inclination as the upper boundary of the accretion flow.
Overall, due to appropriately imposing physical boundary conditions at the
rotation axis as well as considering thermal conduction in this study, our
results show strong wind at high latitudes.
Figure 3: Velocity field in three dimensional space (left plot), in four
cylindrical radii, i.e., $R=[4,8,15,25]\,r_{\mathrm{s}}$ and also in four
different heights, i.e., $z=[0,10,25,40]\,r_{\mathrm{s}}$. The two dimensional
velocity fields have been plotted at four different heights (four right
panels).
The two dimensional inflow-wind structure of the flow is shown in Figure 2, in
the density and temperature contours. The left panel is overlaid with the
poloidal velocity at six different radii in the unit of Schwarzchild radius,
i.e., $r=[100,150,200,250,300,350]\,r_{\mathrm{s}}$ to show the strength of
the wind. In the right panel, the poloidal velocity is normalized with its
absolute value, $|\vec{v}_{p}|=\sqrt{v_{r}^{2}+v_{\theta}^{2}}$, which denotes
the direction of the vectors. As it can be seen from velocity field, around
mid-plane the flow is accreted toward the BH until the white dash-dotted line
($v_{r}=0$) in the left panel. From this line to the high latitudes the
velocity vector deflects outward and the flow escapes away. Moreover, it is
clear that the higher the latitudes, the stronger the wind is. It is also
noted that the density and temperature profiles are plotted based on our self-
similar solutions with density index $n=0.85$ (see equations (21)-(25)). From
this figure, the maximum amount of the density is located at the inner region
of the equatorial plane. So, as we go to outer regions and small $\theta$
angles, the density decreases rapidly and reaches its minimum in the disc. The
density contour is identical to that plotted in the top left panel of figure 2
in XC97 for the accretion outflow solution. The torus-like shape of the
density is also in agreement with the numerical simulations of hot accretion
flow (see e.g., Yuan et al. 2012b). From the right panel, the minimum
temperature belongs to the inflow region inside the disc, while the wind
region has the highest temperature which is much less dense than the disc.
Indeed, at high latitudes, thermal conduction heats up the flow then, the
temperature goes up and leads to a rapid change of the gradient of the gas
pressure, launching the thermal wind (see also top left panel of Figure 4).
Figure 4: Angular distributions of the radial forces (top left panel) and
angular forces (top right panel) in the unit of gravitational force, and the
force analysis in the inflow/wind region to show the driving mechanism of the
wind at $r=70\,r_{s}$ (bottom panel). The forces include gravity (black),
centrifugal force (green), gradient of gas pressure (blue), and their sum
(red). In the bottom panel, the length of the arrows schematically denote the
magnitude of the forces while the direction of arrows show that of the forces.
The dash–dotted line shows the location of $v_{r}=0$, and the dotted line is
for the radius where the forces are calculated. The forces are calculated at
two representative locations in wind and inflow regions, $\theta=30^{\circ}$
and $\theta=85^{\circ}$, respectively.
To show the strength of the wind, we also plot three dimensional (3D) velocity
field in Figure 3 in four cylindrical radii, i.e.,
$\mathrm{R}=[4,8,15,25]\,r_{\mathrm{s}}$, and also in four different heights,
$z=[0,10,25,40]\,r_{\mathrm{s}}$ (left panel). From the 3D figure, it is clear
that the flow is purely inflow at the equatorial plane of the disc, $z=0$. In
higher $z$, the velocity vectors are not parallel to the equatorial plane and
deflected away. In addition, arrows become stronger around the rotation axis
at small heights as well as small radii. For the better view of the velocity
field, the two dimensional (2D) plots in four different heights in $x-y$ plane
are shown in the right panel of Figure 3.
It is of interest to know which mechanism is dominant to drive wind in the hot
accretion flow in HD case. For this purpose, in Figure 4, we analyze all
forces in the unit of the gravitational force, including the gradient of gas
pressure (blue), the centrifugal force (green), the gravitational force
(black), and the sum of the forces (red). It is also worthwhile to find the
prominent radial and angular components of the forces in the inflow and wind
regions. In the top left panel, the radial components of the forces are shown.
Since the gravitational force only changes with radius, in this panel, it has
a fixed negative value, i.e., $F_{\mathrm{gravity}}=-1$, in whole $\theta$
direction. In the inflow region, the gravity is prevailing while from
$\theta\simeq 52^{\circ}$ upward, the gradient of the gas pressure becomes
dominant (wind region). As it can be seen, the sum of the radial components of
the gradient of the gas pressure and the centrifugal force cannot dominate the
gravity in the inflow region so, the total force remains negative there. This
is also clear from the bottom panel of Figure 4, where the forces are plotted
at the region near the equatorial plane, i.e., $\theta=85^{\circ}$, since the
latitudinal components of the forces are almost negligible. The top right
panel of this figure shows that the vertical component of the centrifugal
force is always positive and dominant in the inflow region. On the other hand,
in the wind region, the $\theta$ component of the gradient of the gas pressure
is dominated and the sum of these forces in the vertical direction becomes
negative. In bottom panel of Figure 4, the forces are calculated at two
representative locations in wind and inflow regions, $\theta=30^{\circ}$ and
$\theta=85^{\circ}$, respectively. The dash-dotted line represents the barrier
between inflow and wind regions corresponded to $v_{r}=0$ and the dotted line
is for the radius where the forces are evaluated, i.e.,
$r=70\,r_{\mathrm{s}}$. This panel clearly shows that in the inflow region
gravity is dominant so the matter moves inward, and in the wind region the sum
of the forces is outward due to the strong value of the gradient of the gas
pressure. These results are in agreement with those presented in numerical HD
simulations of the hot accretion flow (see Yuan et al. 2012b).
Figure 5: Latitudinal profiles of the viscous heating (dotted line), the
conduction (dashed line), and the advection (solid line) terms of energy
equation presented in equation (C5).
Figure 5 shows the latitudinal profile of three terms of energy equation
including viscous heating (dotted line), conduction (dashed line), and
advection (solid line) based on the solutions of the fiducial model. This
figure shows that in the inflow region the viscosity and advection terms have
positive values while the thermal conduction has a negative value, so the
advection can cool the flow in the disc. At the polar region, the thermal
conduction is much more negative and the viscous heating term is almost
negligible. Consequently, the sum of these two terms permits a negative
advection term near the rotation axis which means the advection acts to heat
the flow and produce wind. Note that the latitudinal profile of three terms of
the energy equation are not identical to the ones presented in TM06 because of
two main reasons; (1) the thermal conduction term defined here is not similar
to that written in TM06 (we follow the numerical simulation of Sharma et al.
2008 since the TM06 definition of the thermal conduction is not consistent
with the self-similarity adopted here555In TM06, since the density was
proportional to $\propto r^{-3/2}$, the thermal conductivity coefficient
defined as $\lambda(r)=\lambda_{0}r^{-1}$ to preserve the radial self-
similarity of the solutions. Here, to satisfy the radial self-similar
solutions, we cannot follow TM06 definition for thermal conduction since the
density is proportional to $\propto r^{-n}$.). (2) The latitudinal component
of the velocity, $v_{\theta}$, exists in both advection and viscous terms of
our equations, while this component of the velocity was ignored in TM06.
### 4.2 Bernoulli parameter
In almost all previous analytical solutions of the hot accretion flow, the
Bernoulli parameter was calculated to show the existence of the wind/outflow.
The Bernoulli parameter is defined as the sum of the kinetic energy, the
potential energy and the enthalpy of the accreting gas. In fact, the Bernoulli
parameter has been of significant concern as it shows whether wind or outflow
would probably emanate from the accretion flow (Narayan & Yi 1995). The
Bernoulli parameter can be defined as,
$Be=\frac{1}{2}\bm{v}^{2}+h+\psi,$ (27)
where, $h=\gamma p/\left[\rho(\gamma-1)\right]$ is the enthalpy.
Figure 6: Latitudinal profile of the Bernoulli parameter (solid line),
corresponding kinetic energy, $\frac{1}{2}\bm{v}^{2}$ (dashed line), enthalpy,
$h$ (dash-dotted line), and gravitational energy, $\phi$ (dotted line).
Figure 6 shows the Bernoulli parameter (solid line) and its three terms in
Equation (27), including kinetic energy (dashed line), enthalpy (dash-dotted
line), and gravitational energy (dotted line). From this Figure, the Bernoulli
parameter is negative around the equatorial plane, and its value becomes
larger and positive at the high latitudes. In the region near the equatorial
plane, the gravitational energy is the dominant one and it is followed by the
Bernoulli parameter. Additionally, in high latitudes, the Bernoulli parameter
follows the enthalpy due to this fact that the density drops faster than the
pressure from equatorial plane to the rotation axis. Therefore, the square of
the sound speed, and equivalently the enthalpy rises as $\theta$ angle
decreases. The trend of the Bernoulli parameter is similar to accretion
outflow solution in XC97. However, in that solution the Bernoulli parameter is
always positive independent of $\theta$.
### 4.3 Convective stability
In this subsection, we investigate the convective stability of hot accretion
flows based on our self-similar solutions. In this regard, we use the well-
known Solberg-Høiland criterions in cylindrical coordinates $(R,\phi,z)$. If
the disc is convectively stable, the two following Solberg-Høiland criterions
must be positive as,
$\frac{1}{R^{3}}\frac{\partial l^{2}}{\partial
R}-\frac{1}{C_{P}\rho}\bm{\nabla}P\cdot\bm{\nabla}S>0,$ (28)
$-\frac{\partial P}{\partial z}\left(\frac{\partial l^{2}}{\partial
R}\frac{\partial S}{\partial z}-\frac{\partial l^{2}}{\partial
z}\frac{\partial S}{\partial R}\right)>0,$ (29)
where $l\left[=r\sin\theta v_{\phi}\right]$ is the specific angular momentum
per unit mass, $C_{P}$ is the specific heat at constant pressure, $P$ is the
total pressure which equals to the gas pressure in current study, and $S$ is
the entropy defined as,
$dS\propto d\,\ln\left(\frac{P}{\rho^{\gamma}}\right).$ (30)
The first criterion can be reduced as,
$N_{\mathrm{eff}}=\kappa^{2}+N_{\\!{}_{R}}^{2}+N_{z}^{2}>0,$ (31)
with
$\kappa^{2}=\frac{1}{R^{3}}\frac{\partial l^{2}}{\partial R},$ (32)
$N_{\\!{}_{R}}^{2}=-\frac{1}{\gamma\rho}\frac{\partial P}{\partial
R}\frac{\partial}{\partial R}\ln\left(\frac{P}{\rho^{\gamma}}\right),$ (33)
$N^{2}_{z}=-\frac{1}{\gamma\rho}\frac{\partial P}{\partial
z}\frac{\partial}{\partial z}\ln\left(\frac{P}{\rho^{\gamma}}\right).$ (34)
In the above equations, $N_{\mathrm{eff}}$, $\kappa$, $N_{\\!{}_{R}}^{2}$ and
$N^{2}_{z}$ are the effective frequency, the epicyclic frequency, and the $R$
and $z$ components of the Brunt-Väisälä frequency, respectively. The $\partial
P/\partial z$ is always negative, therefore the second Solberg-Høiland
criterion can be reduced as,
$\Delta_{lS}\equiv\frac{\partial l^{2}}{\partial R}\frac{\partial}{\partial
z}\ln\left(\frac{P}{\rho^{\gamma}}\right)-\frac{\partial l^{2}}{\partial
z}\frac{\partial}{\partial R}\ln\left(\frac{P}{\rho^{\gamma}}\right)>0.$ (35)
The following transformations will be adopted to find the angular dependency
of two Solberg-Høiland criterions in spherical coordinates,
$\frac{\partial}{\partial R}=\sin\theta\frac{\partial}{\partial
r}+\frac{\cos\theta}{r}\frac{\partial}{\partial\theta}$ (36)
$\frac{\partial}{\partial z}=\cos\theta\frac{\partial}{\partial
r}-\frac{\sin\theta}{r}\frac{\partial}{\partial\theta}.$ (37)
Figure 7: Analysis of convective stability. The latitudinal profiles of
$\kappa^{2}$, $N_{R}^{2}$, $N_{z}^{2}$, and $N_{\mathrm{eff}}^{2}$ normalized
by $\Omega_{K}^{2}$, and $\Delta_{lS}$ normalized by $v_{K}^{2}$. Figure 8:
The dependency of the solution to the density index, $n$. Here, $\alpha=0.15$,
$\gamma=5/3$ and $\alpha_{c}=0.2$.
In Figure 7 we show the angular variations of $\kappa^{2}$,
$N^{2}_{\\!{}_{R}}$, $N^{2}_{z}$, $N^{2}_{\mathrm{eff}}$ normalized by
$\Omega^{2}_{K}$ and also $\Delta_{lS}$ normalized by
$v^{2}_{\scriptscriptstyle K}$. We can see that $\kappa^{2}$ is positive while
$N^{2}_{R}$ is negative in whole $\theta$ angles. On the other hand,
$N^{2}_{z}$ is only negative in a small area near the rotationa axis so, in
the rest of the area is positive. Inevitably, $N^{2}_{\mathrm{eff}}$ becomes
positive and the first criterion would be satisfied (the light solid line).
Moreover, as it is shown in this Figure, $\Delta_{lS}\geq 0$ in whole domain
which means the second criterion is also satisfied. Since, both Solberg-
Høiland criterions are satisfied here, we conclude that the disc is
convectively stable in the presence of thermal conduction. Note that, the
numerical MHD simulations (Narayan et al. 2012; Yuan et al. 2012a) show
magnetic field can cause the flow becomes convective stable. We believe that
our results are totally in agreement with the prediction of the numerical MHD
simulations. We adopted viscosity to mimic the effect of MRI in driving
angular momentum from the system. In addition, thermal conduction can
convectively stabilize the accretion system in hot accretion mode. In our
future MHD study, we will investigate the convective stability of the hot
accretion flow in more details.
Figure 9: The dependency of the solution to the conductivity coefficient,
$\alpha_{c}$. Here, $\alpha=0.15$, $\gamma=5/3$ and $n=0.85$. Figure 10: The
dependency of the solution to the advection parameter; $f=0.7$ (dotted line),
$f=1$ (solid line). Here, $\alpha=0.2$, $\gamma=5/3$ and $n=0.85$.
### 4.4 Dependency of the Solutions on Input Parameters
In this section, we investigate the dependency of the solutions to the input
parameters including, density index, $n$, conductivity parameter,
$\alpha_{c}$, and advection parameter, $f$. As it was mentioned before,
similar to XC97 and TM06, we obtained the solutions in the whole $\theta$
direction, from the rotation axis to the equatorial plane, showing the strong
wind at high latitudes. However, in contrast to XC97 which solved the HD
equations of hot accretion flow without thermal conduction, we evinced that
the thermal conduction should be inevitably considered in the energy equation
(see section 2.1). Moreover, as introduced in Section 3, the density index,
$n$, of the self-similar solutions shows how the density changes along the
radius. In the case of $n=3/2$, the solution is similar to TM06 where
$v_{\theta}$ was eliminated from the system of the equations. In section 2 ,
we also showed that for highly non-relativistic cases ($\gamma=5/3$) the
density index must be $n<3/2$. 666The density index of the accretion outflow
solution of XC97 is in the same range as we considered, i.e., $n<3/2$, where
they set $n=0.5$. Although, the real accretion systems around the BHs, where
this conclusion may not be satisfied there, happen in the relativistic regime,
specially when the thermal energy of the gas becomes comparable to (or
exceeds) the rest mass energy of the electron (Chattopadhyay & Ryu 2009, Kumar
et al. 2013).
We have shown the dependency of the results to the index parameter, $n$, in
Figure 8. We consider three different values of the density index, i.e.,
$n=0.55,0.85,\,\text{and}\,1.15$. From the top left panel of this figure, we
can see that the maximum amount of the radial velocity of the wind is for
$n=1.15$. The latitudinal component of the velocity is always negative and
becomes null at both boundaries for all values of $n$ (see top right panel).
In addition, from $n=0.55$ to $n=1.15$, $v_{\theta}$ decreases which shows an
opposite behavior in respect to $v_{r}$. This result can be predictable from
continuity equation, (see equation (B1)). In fact, in our current study, this
equation has two terms and the sum of these terms should be always equal to
zero in the whole $\theta$ direction. Therefore, in a fixed density index, any
increase in $v_{r}$ cause a decrease in $v_{\theta}$. The bottom right panel
of Figure 8 illustrates that the density profile drops faster with $n=0.55$,
rather than $n=1.15$. Also, the flow rotates faster for low density indices.
The numerical simulations of the hot accretion flow also studied the
dependency of the solutions to the conductivity coefficient, $\alpha_{c}$
(e.g., Bu et al. 2016). At a fixed radius, from the time averaged of the
solutions in steady state, they found that the density and pressure change
slightly with increasing $\alpha_{c}$. To compare our results with numerical
simulations, we also plot Figure 9. We pick three values of the conductivity
coefficient for this comparison, $\alpha_{c}=[0.02,0.2,2.0]$. In the top left
panel, it is shown that as the value of $\alpha_{c}$ increases, the radial
velocity at the equator becomes slightly larger which is consistent with the
results obtained in KS13 (see figure 1 of KS13). Moreover, our results show
that for the wind region unlike KS13, $v_{r}$ is still rising which is
commensurate with the numerical simulations of Bu et al. 2016. In the top
right panel, $v_{\theta}$ decreases with increasing conductivity coefficient.
This trend is consistent with the continuity equation terms discussed above.
However, KS13 found that $v_{\theta}$ had increasing tendency with an
enhancement in the thermal conduction (see top panel of figure 2 in KS13). The
behavior of $v_{\theta}$ in the present study and also in KS13 are not totally
similar since, we impose $v_{\theta}=0$ at the rotation axis as a symmetric
boundary condition. From the bottom left panel, we find that the value of
$\alpha_{c}$ affects the angular velocity of the flow, where the flow rotates
slower with the greater values of the conductivity coefficient. The changes of
the angular velocity with thermal conduction is consistent with the result of
TM06 (see top right panel of figure 2 in TM06). From the density profile, the
flow indicates a more spherical structure with a rise in thermal conduction
which is consistent with numerical simulations and also KS13 (see top panel of
figure 3 in KS13).
We further treated the dependency of the solution to the advection parameter
in Figure 10 for two different values; $f=0.7$ (dotted line) and $f=1$ (solid
line). In top left panel, the radial velocity, $v_{r}$, of the wind drops with
growing of the advection parameter. In top right panel, the minimum of the
latitudinal velocity, $v_{\theta}$, moves toward the rotation axis as
advection parameter increases. Moreover, $v_{\theta}$ in comparison with
$v_{r}$ shows inverse behavior as it increases with the growth of advection
parameter. The enhancement of the $f$ would raise the angular velocity of the
gas particles significantly as you can see in the bottom left panel of this
figure. However, from the bottom right panel, the density profile would not
extensively suffer from the changes of the advection parameter777It should be
mentioned here that at small $f$, the radiation may not be totally ignored as
in this paper..
In this work, we solved the HD equations with considering thermal conduction
as well as the viscosity to mimic the effects of magnetic field. This will
motivate us to solve the full set of MHD equations to understand the
properties and nature of the hot accretion flow in a more realistic case.
Therefore, in our future studies we mostly focus on MHD equations and
investigative the dependency of the solutions to different configurations of
magnetic field. We will also compare the HD and MHD analytical solutions to
find a unique and accurate solution for the hot accretion flow which might be
useful for numerical simulations.
## 5 Summary and Discussion
In summary, we have shown thermal conduction is crucial term for investigating
the inflow-wind structure of hot accretion flows. As argued in section 2,
thermal conduction is only significant in very low accretion rate systems
which are suitable for very low luminosity AGNs, e.g., our Galactic center Sgr
$\mathrm{A^{*}}$, super massive black holes in early-type galaxies, and a
considerable number of quiescent X-ray binaries. On this matter, we have
solved the two-dimensional HD equations of hot accretion flow with the
inclusion of the thermal conduction and all components of the viscous stress
tensor. For simplicity, we adopted steady state as well as axisymmetric
assumptions and used self-similar approximation in radial direction. Our
integration starts from the rotation axis and stops at the equatorial plane
so, the solutions will be obtained in the full $r-\theta$ space. We imposed
the physical boundary conditions at both boundaries and used relaxation method
to solve the coupled system of equations.
We have obtained an inflow-wind solution extended over full range of $\theta$
direction. The inflow region is around the equatorial plane, while the wind
region is located at high latitudes around the rotation axis, i.e.,
$0<\theta\leqslant 52^{\circ}$, (see Figure 2). The density is torus-like and
its concentration occurs in the equatorial plane and decreases in the wind
region. From our results, angular velocity behavior shows that wind is able to
transfer angular momentum outward. Moreover, the gradient of the gas pressure
is the dominant force in driving wind. Our results also show there is no sonic
point in our calculation domain. Analysis of the energy balance between
advection, thermal conduction and viscous heating (Figure 5) indicates that
while viscous dissipation heats the flow everywhere, thermal conduction cools
it at the equator and polar region. Therefore, thermal conduction conducts the
heat from inner regions to outer regions and proceeds as a mechanism for
launching thermal wind. Moreover, to balance the two terms in the right hand
side of the energy equation (viscous heating and thermal conduction),
advection cools the gas in the disc region while in the polar region, acts as
a heating mechanism and heats the flow. The Bernoulli parameter is positive in
wind region and negative in the inflow region. Therefore, it could be still an
useful value, if not be an arguable factor, to show the existence of wind in
the hot accretion flows. We compared our results with previous related
analytical studies on hot accretion flow with and without thermal conduction.
We also treated the convective stability of the hot accretion flow in HD case
and found that the disc is convectively stable in the presence of the thermal
conduction. From parameter study of the density index, $n$, the radial
velocity, $v_{r}$, increases with growth of $n$ in the wind region. Whilst,
the latitudinal velocity, $v_{\theta}$, and angular velocity, $\Omega$,
decrease with increasing of $n$. In this work, we explored the role of thermal
conduction in hot accretion flows in HD case with parameter study (Figure 9).
From our results, thermal conduction did not have effective role to change
$v_{r}$ and $v_{\theta}$ in inflow region. However, in wind region, an inverse
behavior were shown for these two components of the velocity. With an increase
in the conductivity coefficient, the radial velocity enhances while the
latitudinal component drops. Moreover, the disc rotates slower with growth of
the thermal conduction strength. Thermal conduction also slightly does enlarge
the density in the entire of the flow. These results are fully consistent with
the numerical simulations of the hot accretion flow (e.g., Bu et al. 2016). To
tackle the influence of the advection parameter on the physical variables, we
did a comparison between two values, $f=0.7$ and $f=1$. A rise in the
advection parameter would increase both the latitudinal and the angular
velocities while decrease the radial velocity. More, growth of the advection
parameter could not make substantial change in the density profile of the
accretion flow.
There are several caveats in this study which will be improved in our future
studies. The first one is that we only solved the HD equations of hot
accretion flow. In a real accretion flow, angular momentum is transferred by
Maxwell stress associated with MHD turbulence driven by MRI. In addition,
magnetic field is one of the driving mechanisms for wind production. With this
aim, we will solve the MHD equations of the accretion flow. In our next papers
we mainly focus on different magnetic field configurations on the structure of
the hot accretion flow and compare the results with HD case. A further
simplification here is that we adopted a single one temperature fluid. In hot
accretion model, it is expected the ions be much more hotter than the
electrons (see, Rees et al. 1982; Yuan & Narayan 2014). Thus, two different
energy equations for electrons and ions should be solved.
## Aknowledgments
The authors would like to thank the referee for his/her thoughtful and
constructive comments. Fatemeh Zahra Zeraatgari is supported by the National
Natural Science Foundation of China (grant No. 12003021). This work is
supported by the Science Challenge Project of China (grant No. TZ2016002), the
China Postdoctoral Science Foundation (grants No. 2019M663665, 2020M673371).
De-Fu Bu is supported by the Natural Science Foundation of China (grant No.
11773053). Amin Mosallanezhad also thanks the support of Dr. X. D. Zhang at
the Network Information Center of Xi’an Jiaotong University. The computation
has made use of the High Performance Computing (HPC) platform of Xi’an
Jiaotong University.
## Appendix A Energy Equation presented in Xu & Chen 1997
XC97 solved the HD equations of hot accretion flows which was very similar to
the equations described in this paper. The only difference is the definition
of the energy equations. The energy equation described in XC97 can be written
as,
$Q_{\mathrm{adv}}=fQ_{\mathrm{vis}},$ (A1)
with
$Q_{\mathrm{adv}}=\rho\frac{\mathrm{d}e}{\mathrm{d}t}-\frac{p}{\rho}\frac{\mathrm{d}\rho}{\mathrm{d}t},$
(A2)
$Q_{\mathrm{vis}}=\nabla\bm{v}:\bm{\sigma},$ (A3)
By imposing steady state, $\partial/\partial t=0$, and axisymmetric,
$\partial/\partial\phi=0$, assumptions, the components of stress tensor,
$\sigma$, in the spherical coordinates are given by:
$\sigma_{rr}=\rho\nu\left[2\frac{\partial v_{r}}{\partial
r}-\frac{2}{3}\left(\nabla\cdot\bm{v}\right)\right],$ (A4)
$\sigma_{\theta\theta}=\rho\nu\left[2\left(\frac{1}{r}\frac{\partial
v_{\theta}}{\partial\theta}+\frac{v_{r}}{r}\right)-\frac{2}{3}\left(\nabla\cdot\bm{v}\right)\right],$
(A5)
$\sigma_{\phi\phi}=\rho\nu\left[2\left(\frac{v_{r}}{r}+\frac{v_{\theta}\cot\theta}{r}\right)-\frac{2}{3}\left(\nabla\cdot\bm{v}\right)\right],$
(A6)
$\sigma_{r\theta}=\rho\nu\left[r\frac{\partial}{\partial
r}\left(\frac{v_{\theta}}{r}\right)+\frac{1}{r}\frac{\partial
v_{r}}{\partial\theta}\right],$ (A7)
$\sigma_{r\phi}=\rho\nu\left[r\frac{\partial}{\partial
r}\left(\frac{v_{\phi}}{r}\right)+\frac{1}{r\sin\theta}\frac{\partial
v_{r}}{\partial\phi}\right],$ (A8)
$\sigma_{\theta\phi}=\rho\nu\left[\frac{\sin\theta}{r}\frac{\partial}{\partial\theta}\left(\frac{v_{\phi}}{\sin\theta}\right)+\frac{1}{r\sin\theta}\frac{\partial
v_{\theta}}{\partial\phi}\right],$ (A9)
where $\nabla\cdot\bm{v}$ is written as
$\nabla\cdot\bm{v}=\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}v_{r}\right)+\frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}\left(v_{\theta}\sin\theta\right).$
(A10)
Substituting the above terms into the advection and the viscous terms of the
energy equations, we have,
$Q_{\mathrm{adv}}=\rho\left[v_{r}\frac{\partial e}{\partial
r}+\frac{v_{\theta}}{r}\frac{\partial
e}{\partial\theta}\right]-\frac{p}{\rho}\left[v_{r}\frac{\partial\rho}{\partial
r}+\frac{v_{\theta}}{r}\frac{\partial\rho}{\partial\theta}\right]$ (A11)
$Q_{\mathrm{vis}}=f\Bigg{[}\frac{\partial v_{r}}{\partial
r}\sigma_{rr}+\frac{\partial v_{\theta}}{\partial
r}\sigma_{r\theta}+\frac{\partial v_{\phi}}{\partial
r}\sigma_{r\phi}+\frac{1}{r}\left(\frac{\partial
v_{r}}{\partial\theta}-v_{\theta}\right)\sigma_{r\theta}+\frac{1}{r}\left(\frac{\partial
v_{\theta}}{\partial\theta}+v_{r}\right)\sigma_{\theta\theta}+\frac{1}{r}\frac{\partial
v_{\phi}}{\partial\theta}\sigma_{\theta\phi}\\\
-\frac{v_{\phi}}{r}\left(\sigma_{r\phi}+\sigma_{\theta\phi}\cot\theta\right)+\frac{\sigma_{\phi\phi}}{r}\left(v_{r}+v_{\theta}\cot\theta\right)\Bigg{]}$
(A12)
They assumed fully advection case, i.e., $f=1$. Substituting self-similar
approximation, equation (21)-(25), into the above terms, the radial dependency
will be removed. Therefore, the dimensionless form of the energy equation
described in XC97 can be written as,
$q_{\mathrm{adv}}=q_{\mathrm{vis}}$ (A13)
with
$q_{\mathrm{adv}}=\left[n-\frac{1}{\gamma-1}\right]p_{\mathrm{g}}v_{r}+\frac{v_{\theta}}{\gamma-1}\left[\frac{\mathrm{d}p_{\mathrm{g}}}{\mathrm{d}\theta}-\gamma\frac{p_{\mathrm{g}}}{\rho}\frac{\mathrm{d}\rho}{\mathrm{d}\theta}\right],$
(A14)
$q_{\mathrm{vis}}=\alpha
p_{\mathrm{g}}\Biggr{[}\frac{1}{2}v_{r}^{2}+2\left(v_{r}+\frac{\mathrm{d}v_{\theta}}{\mathrm{d}\theta}\right)^{2}+2\left(v_{r}+v_{\theta}\cot\theta\right)^{2}+\frac{1}{4}\left(2\frac{\mathrm{d}v_{r}}{\mathrm{d}\theta}-3v_{\theta}\right)^{2}+\left[\frac{9}{4}\Omega^{2}+\left(\frac{\mathrm{d}\Omega}{\mathrm{d}\theta}\right)^{2}\right]\sin^{2}\theta\\\
-\frac{2}{3}\left(\frac{3}{2}v_{r}+\frac{\mathrm{d}v_{\theta}}{\mathrm{d}\theta}+v_{\theta}\cot\theta\right)^{2}\Biggr{]}.$
(A15)
## Appendix B Partial Differential Equations in Spherical Coordinates
In this study to reach the final PDEs governing the system, we adopt spherical
coordinates $(r,\theta,\phi)$. We consider the disc to be axisymmetric and
steady state. We further assume the velocity field consists of all its
components as $\bm{v}=v_{r}\hat{r}+v_{\theta}\hat{\theta}+v_{\phi}\hat{\phi}$.
All components of the viscous stress tensor will be included as described in
equations (A4)-(A9). By imposing the assumptions and definitions introduced in
section 3, the equations (11)-(13) will be reduced to the following system of
PDEs. Thus, we can rewrite the continuity equation as,
$\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\rho
v_{r}\right)+\frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}\left(\rho
v_{\theta}\sin\theta\right)=0.$ (B1)
Three components of the equation of motion, equation (12), can be read as,
$\rho\left[v_{r}\frac{\partial v_{r}}{\partial
r}+\frac{v_{\theta}}{r}\left(\frac{\partial
v_{r}}{\partial\theta}-v_{\theta}\right)-\frac{v_{\phi}^{2}}{r}\right]=-\frac{GM\rho}{r^{2}}-\frac{\partial
p}{\partial r}+\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\sigma_{rr}\right)+\frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\sigma_{r\theta}\right)-\frac{1}{r}\left(\sigma_{\theta\theta}+\sigma_{\phi\phi}\right),$
(B2)
$\rho\left[v_{r}\frac{\partial v_{\theta}}{\partial
r}+\frac{v_{\theta}}{r}\left(\frac{\partial
v_{\theta}}{\partial\theta}+v_{r}\right)-\frac{v_{\phi}^{2}}{r}\cot\theta\right]=-\frac{1}{r}\frac{\partial
p}{\partial\theta}+\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\sigma_{r\theta}\right)+\frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\sigma_{\theta\theta}\right)+\frac{1}{r}\left(\sigma_{r\theta}-\sigma_{\phi\phi}\cot\theta\right),$
(B3)
$\rho\left[v_{r}\frac{\partial v_{\phi}}{\partial
r}+\frac{v_{\theta}}{r}\frac{\partial
v_{\phi}}{\partial\theta}+\frac{v_{\phi}}{r}\left(v_{r}+v_{\theta}\cot\theta\right)\right]=\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\sigma_{r\phi}\right)+\frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\sigma_{\theta\phi}\right)+\frac{1}{r}\left(\sigma_{r\phi}+\sigma_{\theta\phi}\cot\theta\right),$
(B4)
And finally, the energy equation can be written as,
$\rho\left[v_{r}\frac{\partial e}{\partial
r}+\frac{v_{\theta}}{r}\frac{\partial
e}{\partial\theta}\right]-\frac{p}{\rho}\left[v_{r}\frac{\partial\rho}{\partial
r}+\frac{v_{\theta}}{r}\frac{\partial\rho}{\partial\theta}\right]=f\Bigg{[}\frac{\partial
v_{r}}{\partial r}\sigma_{rr}+\frac{\partial v_{\theta}}{\partial
r}\sigma_{r\theta}+\frac{\partial v_{\phi}}{\partial
r}\sigma_{r\phi}+\frac{1}{r}\left(\frac{\partial
v_{r}}{\partial\theta}-v_{\theta}\right)\sigma_{r\theta}+\frac{1}{r}\left(\frac{\partial
v_{\theta}}{\partial\theta}+v_{r}\right)\sigma_{\theta\theta}\\\
+\frac{1}{r}\frac{\partial
v_{\phi}}{\partial\theta}\sigma_{\theta\phi}-\frac{v_{\phi}}{r}\left(\sigma_{r\phi}+\sigma_{\theta\phi}\cot\theta\right)+\frac{\sigma_{\phi\phi}}{r}\left(v_{r}+v_{\theta}\cot\theta\right)\Bigg{]}+\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\chi\frac{\partial T}{\partial
r}\right)+\frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}\left(\frac{\chi\sin\theta}{r}\frac{\partial
T}{\partial\theta}\right).$ (B5)
## Appendix C Ordinary Differential Equations
By substituting self-similar solutions presented in section 3 into the partial
differential equations (B1)–(B5), the following coupled ODEs in the $\theta$
direction will be obtained 888Note that, for simplicity, in equations
(C1)-(C5), we remove the $\theta$ dependency of the variables as well as their
derivatives.:
$\left[\left(\frac{3}{2}-n\right)\bar{v}_{r}+\frac{\mathrm{d}\bar{v}_{\theta}}{\mathrm{d}\theta}+\bar{v}_{\theta}\cot\theta\right]\bar{\rho}+\bar{v}_{\theta}\frac{\mathrm{d}\bar{\rho}}{\mathrm{d}\theta}=0$
(C1)
$\bar{\rho}\left[-\frac{1}{2}\bar{v}_{r}^{2}+\bar{v}_{\theta}\frac{\mathrm{d}\bar{v}_{r}}{\mathrm{d}\theta}-\bar{v}_{\theta}^{2}-\bar{\Omega}^{2}\sin^{2}\theta\right]=-\bar{\rho}+\left(n+1\right)\bar{p}_{\mathrm{g}}+\alpha\left(\frac{\mathrm{d}\bar{v}_{r}}{\mathrm{d}\theta}-\frac{3}{2}\bar{v}_{\theta}\right)\frac{\mathrm{d}\bar{p}_{\mathrm{g}}}{\mathrm{d}\theta}\\\
+\alpha\bar{p}_{\mathrm{g}}\left[\frac{2}{3}\left(n+1\right)\left(3\bar{v}_{r}+\bar{v}_{\theta}\cot\theta+\frac{\mathrm{d}\bar{v}_{\theta}}{\mathrm{d}\theta}\right)-\frac{7}{2}\left(\frac{\mathrm{d}\bar{v}_{\theta}}{\mathrm{d}\theta}+\bar{v}_{\theta}\cot\theta\right)+\frac{\mathrm{d}^{2}\bar{v}_{r}}{\mathrm{d}\theta^{2}}+\frac{\mathrm{d}\bar{v}_{r}}{\mathrm{d}\theta}\cot\theta-6\bar{v}_{r}\right],$
(C2)
$\bar{\rho}\left[\frac{1}{2}\bar{v}_{r}\bar{v}_{\theta}+\bar{v}_{\theta}\frac{\mathrm{d}\bar{v}_{\theta}}{\mathrm{d}\theta}-\bar{\Omega}^{2}\sin\theta\cos\theta\right]=-\frac{\mathrm{d}\bar{p}_{\mathrm{g}}}{\mathrm{d}\theta}+\alpha\left(\bar{v}_{r}+\frac{4}{3}\frac{\mathrm{d}\bar{v}_{\theta}}{\mathrm{d}\theta}-\frac{2}{3}\bar{v}_{\theta}\cot\theta\right)\frac{\mathrm{d}\bar{p}_{\mathrm{g}}}{\mathrm{d}\theta}\\\
+\alpha\bar{p}_{\mathrm{g}}\left[\left(n+1\right)\left(\frac{3}{2}\bar{v}_{\theta}-\frac{\mathrm{d}\bar{v}_{r}}{\mathrm{d}\theta}\right)+4\frac{\mathrm{d}\bar{v}_{r}}{\mathrm{d}\theta}+\frac{4}{3}\left(\frac{\mathrm{d}^{2}\bar{v}_{\theta}}{\mathrm{d}\theta^{2}}+\frac{\mathrm{d}\bar{v}_{\theta}}{\mathrm{d}\theta}\cot\theta-\bar{v}_{\theta}\csc^{2}\theta\right)-\frac{5}{2}\bar{v}_{\theta}\right],$
(C3)
$\bar{\rho}\left[\frac{1}{2}\bar{v}_{r}\bar{\Omega}+2\bar{v}_{\theta}\bar{\Omega}\cot\theta+\bar{v}_{\theta}\frac{\mathrm{d}\bar{\Omega}}{\mathrm{d}\theta}\right]=\alpha\bar{p}_{\mathrm{g}}\left[\frac{3}{2}\left(n-2\right)\bar{\Omega}+3\frac{\mathrm{d}\bar{\Omega}}{\mathrm{d}\theta}\cot\theta+\frac{\mathrm{d}^{2}\bar{\Omega}}{\mathrm{d}\theta^{2}}\right]+\alpha\frac{\mathrm{d}\bar{\Omega}}{\mathrm{d}\theta}\frac{\mathrm{d}\bar{p}_{\mathrm{g}}}{\mathrm{d}\theta},$
(C4)
$\left(n-\frac{1}{\gamma-1}\right)\bar{p}_{\mathrm{g}}\bar{v}_{r}+\frac{\bar{v}_{\theta}}{\gamma-1}\left(\frac{\mathrm{d}\bar{p}_{\mathrm{g}}}{\mathrm{d}\theta}-\gamma\frac{\bar{p}_{\mathrm{g}}}{\bar{\rho}}\frac{\mathrm{d}\bar{\rho}}{\mathrm{d}\theta}\right)=\alpha
f\bar{p}_{\mathrm{g}}\Biggr{[}\frac{1}{2}\bar{v}_{r}^{2}+2\left(\bar{v}_{r}+\frac{\mathrm{d}\bar{v}_{\theta}}{\mathrm{d}\theta}\right)^{2}+2\left(\bar{v}_{r}+\bar{v}_{\theta}\cot\theta\right)^{2}\\\
+\frac{1}{4}\left(2\frac{\mathrm{d}\bar{v}_{r}}{\mathrm{d}\theta}-3\bar{v}_{\theta}\right)^{2}+\left[\frac{9}{4}\bar{\Omega}^{2}+\left(\frac{\mathrm{d}\bar{\Omega}}{\mathrm{d}\theta}\right)^{2}\right]\sin^{2}\theta-\frac{2}{3}\left(\frac{3}{2}\bar{v}_{r}+\frac{\mathrm{d}\bar{v}_{\theta}}{\mathrm{d}\theta}+\bar{v}_{\theta}\cot\theta\right)^{2}\Biggr{]}\\\
+\alpha_{c}\left[\left(n-\frac{1}{2}\right)\bar{p}_{\mathrm{g}}+\left(\frac{\mathrm{d}\bar{p}_{\mathrm{g}}}{\mathrm{d}\theta}-\frac{\bar{p}_{\mathrm{g}}}{\bar{\rho}}\frac{\mathrm{d}\bar{\rho}}{\mathrm{d}\theta}\right)\cot\theta+\frac{\mathrm{d}^{2}\bar{p}_{\mathrm{g}}}{\mathrm{d}\theta^{2}}-\frac{1}{\bar{\rho}}\frac{\mathrm{d}\bar{\rho}}{\mathrm{d}\theta}\frac{\mathrm{d}\bar{p}_{\mathrm{g}}}{\mathrm{d}\theta}+\frac{\bar{p}_{\mathrm{g}}}{\bar{\rho}^{2}}\left(\frac{\mathrm{d}\bar{\rho}}{\mathrm{d}\theta}\right)^{2}-\frac{\bar{p}_{\mathrm{g}}}{\bar{\rho}}\frac{\mathrm{d}^{2}\bar{\rho}}{\mathrm{d}\theta^{2}}\right].$
(C5)
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|
# The Peculiar Transient AT2018cow: A Possible Origin of A Type Ibn/IIn
Supernova
Danfeng Xiang Physics Department and Tsinghua Center for Astrophysics (THCA), Tsinghua University, Beijing, 100084, China Xiaofeng Wang Physics Department and Tsinghua Center for Astrophysics (THCA), Tsinghua University, Beijing, 100084, China Beijing Planetarium, Beijing Academy of Sciences and Technology, Beijing, 100044, China Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, 210023, China Weili Lin Jun Mo Han Lin Physics Department and Tsinghua Center for Astrophysics (THCA), Tsinghua University, Beijing, 100084, China Jamison Burke Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117-5575, USA Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA Daichi Hiramatsu Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117-5575, USA Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA Griffin Hosseinzadeh Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138-1516, USA D. Andrew Howell Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117-5575, USA Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA Curtis McCully Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117-5575, USA Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA Stefan Valenti Department of Physics and Astronomy, University of California, 1 Shields Avenue, Davis, CA 95616-5270, USA József Vinkó Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly-Thege M. út 15-17, Budapest 1121, Hungary ELTE Eötvös Loránd University, Institute of Physics, Pázmany Péter sétány 1/A, Budapest, 1117, Hungary Department of Optics & Quantum Electronics, University of Szeged, Dóm tér 9, Szeged, 6720 Hungary Department of Astronomy, University of Texas at Austin, Austin, TX, 78712, USA J. Craig Wheeler Department of Astronomy, University of Texas at Austin, Austin, TX, 78712, USA Shuhrat A. Ehgamberdiev Ulugh Beg Astronomical Institute, Uzbekistan Academy of Sciences, Uzbekistan, Tashkent, 100052, Uzbekistan Davron Mirzaqulov Ulugh Beg Astronomical Institute, Uzbekistan Academy of Sciences, Uzbekistan, Tashkent, 100052, Uzbekistan Attila Bódi Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly-Thege M. út 15-17, Budapest 1121, Hungary CSFK Lendület Near-Field Cosmology Research Group ELTE Eötvös Loránd University, Institute of Physics, Pázmany Péter sétány 1/A, Budapest, 1117, Hungary Zsófia Bognár Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly-Thege M. út 15-17, Budapest 1121, Hungary CSFK Lendület Near-Field Cosmology Research Group ELTE Eötvös Loránd University, Institute of Physics, Pázmany Péter sétány 1/A, Budapest, 1117, Hungary Borbála Cseh Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly-Thege M. út 15-17, Budapest 1121, Hungary Ottó Hanyecz Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly-Thege M. út 15-17, Budapest 1121, Hungary Bernadett Ignácz Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly-Thege M. út 15-17, Budapest 1121, Hungary Csilla Kalup Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly-Thege M. út 15-17, Budapest 1121, Hungary Réka Könyves-Tóth Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly-Thege M. út 15-17, Budapest 1121, Hungary Levente Kriskovics Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly-Thege M. út 15-17, Budapest 1121, Hungary Eötvös Loránd University, Institute of Physics, Pázmany Péter sétány 1/A, Budapest, 1117, Hungary András Ordasi Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly-Thege M. út 15-17, Budapest 1121, Hungary András Pál Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly-Thege M. út 15-17, Budapest 1121, Hungary Eötvös Loránd University, Institute of Physics, Pázmany Péter sétány 1/A, Budapest, 1117, Hungary Eötvös Loránd University, Department of Astronomy, Pázmany Péter sétány 1/A, Budapest, 1117, Hungary Krisztián Sárneczky Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly-Thege M. út 15-17, Budapest 1121, Hungary Bálint Seli Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly-Thege M. út 15-17, Budapest 1121, Hungary Róbert Szakáts Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly-Thege M. út 15-17, Budapest 1121, Hungary T. Arranz-Heras Observadores de Supernovas Group (ObSN), Spain Obs. Las Pequeras, 40470 Navas de Oro, Segovia, Spain R. Benavides-Palencia Observadores de Supernovas Group (ObSN), Spain Obs. Posadas MPC J53, 14730 Posadas, Córdoba, Spain D. Cejudo-Martínez Observadores de Supernovas Group (ObSN), Spain Obs. El Gallinero, 28192 El Berrueco, Madrid, Spain P. De la Fuente-Fernández Observadores de Supernovas Group (ObSN), Spain Obs. Llanes, 33784 Llanes, Asturias, Spain A. Escartín-Pérez Observadores de Supernovas Group (ObSN), Spain Obs. Belako, 48100 Mungia, Vizcaya, Spain F. García-De la Cuesta Observadores de Supernovas Group (ObSN), Spain Obs. La Vara MPC J38, 33784 Muñas de Arriba, Asturias, Spain J.L. González-Carballo Observadores de Supernovas Group (ObSN), Spain Obs. Cerro del Viento MPC I84, 06010 Badajoz, Spain R. González-Farfán Observadores de Supernovas Group (ObSN), Spain Obs. Uraniborg MPC Z55, 41400 Écija, Sevilla, Spain F. Limón-Martínez Observadores de Supernovas Group (ObSN), Spain Obs. Mazariegos MPC I99, 34170 Mazariegos, Palencia, Spain A. Mantero Observadores de Supernovas Group (ObSN), Spain Obs. Bernezzo MPC C77, 12010 Bernezzo, Cuneo, Italy R. Naves-Nogués Observadores de Supernovas Group (ObSN), Spain Obs. Montcabrer MPC 213, 08348 Cabrils, Barcelona, Spain M. Morales-Aimar Observadores de Supernovas Group (ObSN), Spain Obs. Sencelles MPC K14, 07140 Sencelles, Islas Baleares, Spain V. R. Ruíz-Ruíz Observadores de Supernovas Group (ObSN), Spain New Mexico Skies (USA), iTelescope Siding Spring (Asutralia), AstroCamp (Nerpio, Spain) F.C. Soldán-Alfaro Observadores de Supernovas Group (ObSN), Spain Obs. Amanecer de Arrakis MPC Z74, 41500 Alcalá de Guadaira, Sevilla, Spain J. Valero-Pérez Observadores de Supernovas Group (ObSN), Spain Obs. Ponferrada MPC Z70, 24411 Ponferrada, León, Spain F. Violat-Bordonau Observadores de Supernovas Group (ObSN), Spain Obs. Norba Caesarina MPC Z71, 10195 Cáceres, Spain Tianmeng Zhang Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, 10101, Beijing School of Astronomy and Space Science, University of Chinese Academy of Sciences, 101408, Beijing Jujia Zhang Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650216, China Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing, 100012, China Xue Li Zhihao Chen Hanna Sai Wenxiong Li Physics Department and Tsinghua Center for Astrophysics (THCA), Tsinghua University, Beijing, 100084, China
###### Abstract
We present our photometric and spectroscopic observations on the peculiar
transient AT2018cow. The multi-band photometry covers from peak to $\sim$70
days and the spectroscopy ranges from 5 to $\sim$50 days. The rapid rise
($t_{\mathrm{r}}$$\lesssim$2.9 days), high luminosity
($M_{V,\mathrm{peak}}\sim-$20.8 mag) and fast decline after peak make
AT2018cow stand out of any other optical transients. While we find that its
light curves show high resemblance to those of type Ibn supernovae. Moreover,
the spectral energy distribution remains high temperature of $\sim$14,000 K
after $\sim$15 days since discovery. The spectra are featureless in the first
10 days, while some broad emission lines due to H, He, C and O emerge later,
with velocity declining from $\sim 14,000$ km s-1 to $\sim$3000 km s-1 at the
end of our observations. Narrow and weak He I emission lines emerge in the
spectra at $t>$20 days since discovery. These emission lines are reminiscent
of the features seen in interacting supernovae like type Ibn and IIn
subclasses. We fit the bolometric light curves with a model of circumstellar
interaction (CSI) and radioactive decay (RD) of 56Ni and find a good fit with
ejecta mass $M_{\mathrm{ej}}\sim 3.16$ $\mathrm{M_{\odot}}$, circumstellar
material mass $M_{\mathrm{CSM}}\sim 0.04$ $\mathrm{M_{\odot}}$, and ejected
56Ni mass $M_{{}^{56}\mathrm{Ni}}\sim 0.23$ $\mathrm{M_{\odot}}$. The CSM
shell might be formed in an eruptive mass ejection of the progenitor star.
Furthermore, host environment of AT2018cow implies connection of AT2018cow
with massive stars. Combining observational properties and the light curve
fitting results, we conclude that AT2018cow might be a peculiar interacting
supernova originated from a massive star.
CSM interaction – supernova: general – : supernova: individual (AT2018cow,
SN2006jc)
††software: ZrutyPhot (Mo et al. in prep.), IRAF (Tody, 1993, 1986), Firefly
(Wilkinson et al., 2017).
††thanks: E-mail<EMAIL_ADDRESS>
## 1 Introduction
The studies of time domain astronomy cover a variety of optical transients,
including novae, supernovae (SNe), tidal disruption events (TDE), and
kilonovae, etc. With different physical origins, these transients exhibit a
huge diversity in evolutionary properties, especially optical light curves.
The evolutionary time scales and luminosity of different transients are
directly related to their physical origins. There are a group of transients
with very high luminosity and short timescale of evolution, such as the so-
called fast evolving luminous transients (FELTs) (e.g. Rest et al., 2018).
They have much faster rise and decline in light curves than regular SNe. And
many of them have peak luminosity much higher than normal SNe, close to the
superluminous supernovae (SLSNe, Quimby et al., 2011; Howell, 2017). The
physical origins of these FELTs are still unclear. Among them, some are
characterized by very blue color, indicating high temperature, which are also
called fast-rising blue optical transients (FBOTs, e.g. Drout et al., 2014;
Arcavi et al., 2016).
A recently discovered extragalactic transient, AT2018cow (ATLAS18qqn), has
caught much attention due to its peculiar behaviour in its light curves and
spectral evolution. AT2018cow was discovered by ATLAS on MJD 58285.44 (UT Jun.
16.44, 2018, UT dates are used throughout this paper), with a magnitude of
14.76$\pm$0.10 mag in ATLAS orange-band (Smartt et al., 2018). It is located
far from the center of the host galaxy CGCG 137-068 ($z=0.0141$, $D_{L}=63$
Mpc111We assume a flat universe with
$H_{0}=67.7~{}\mathrm{km~{}s^{-1}~{}{Mpc^{-1}}}$, $\Omega_{\mathrm{M}}=0.307$
(Planck Collaboration et al., 2016).). This distance means that AT2018cow is
as luminous as the peak of SNe Ia at discovery. As soon as this transient
source was reported, astronomers from all over the world were actively
conducting its follow-up observations in all bands, including ultra-violet
(UV), optical, X-ray, radio and $\gamma$-ray. AT2018cow is found to evolve
rapidly with a rise time less than 3 days and peak magnitude <$-$20 mag. The
photospheric temperature is measured to be $\sim$30,000 K near the peak and it
still maintains high temperature of $\sim$15,000 K after $\sim$20 days after
discovery (Prentice et al., 2018; Perley et al., 2019). All of these features
suggest that AT2018cow can be put into the FBOTs.
The close distance makes AT2018cow the first FELT/FBOT which has well-
sequenced photometric and spectroscopic observations in wavebands ranging from
X-ray to radio (e.g., Prentice et al., 2018; Perley et al., 2019; Kuin et al.,
2019; Margutti et al., 2019; Ho et al., 2019), making it a rare sample for the
study of FBOT-like objects. In previous studies, several possible physical
mechanisms have been proposed for AT2018cow, e.g., tidal disruption of a star
into an intermediate mass black hole (Perley et al., 2019; Kuin et al., 2019,
Li et al. in prep.), central-engine powered supernova (Prentice et al., 2018;
Margutti et al., 2019), interaction of a condensed CSM and the supernova shock
(Margutti et al., 2019; Leung et al., 2020), electron-capture collapse of a
white dwarf (Lyutikov & Toonen, 2019). And Margutti et al. (2019) suggests
that there should be a deeply embedded X-ray source in an asymmetrical ejecta.
In this paper, we present our optical photometric and spectroscopic
observations of AT2018cow. Spectroscopic observations spanned from Jun. 21,
2018 to Aug. 14, 2018. and photometric observations lasted until September 21,
2018. In Sec. 2 we describe our spectroscopic and photometric observations as
well as data processing. In Sec. 3 we analyse the observational properties of
AT2018cow, including light-curve and spectral evolution. The analysis of the
host galaxy is presented in Sec. 4. In Sec. 5 we explore the possible physical
origins of AT2018cow. Further discussion and final summary are given in Sec. 6
and 7, respectively.
## 2 Observations and Data Reduction
### 2.1 Photometric Observations
The optical photometric observations of AT2018cow were monitored by several
observatories, including the 0.8-m Tsinghua University-NAOC telescope (TNT,
Huang et al., 2012) at Xinglong Observatory of NAOC, the AZT-22 1.5-m
telescope (hereafter AZT) at Maidanak Astronomical Observatory (Ehgamberdiev,
2018), telescopes of the Las Cumbres Observatory network (LCO), and telescope
of Konkoly Observatory in Hungary (hereafter KT). Photometric and
spectroscopic data from LCO were obtained via the Global Supernova Project
(GSP). We also collected early time photometric data from Observadores de
Supernovas Group (ObSN) in Spain. The TNT and LCO observations were obtained
in standard Johnson-Cousin $UBV$ bands and SDSS $gri$ bands. Long time and
short-cadenced observations in $UBVRI$ bands were obtained by AZT. The Konkoly
observations were obtained in $BVRI$ bands. Data from ObSN were obtained in
$BVRI$ and $gr$ bands. The entire dataset covers phases from MJD 58286.89
(Jun. 17.89, 2018) to MJD 58348.74 (Aug. 18.74, 2018). The earliest
photometric data point comes from ObSN in $V$-band on MJD 58286.89, which is
$\sim$0.27 day earlier than that presented in Prentice et al. (2018). Besides
the fast rise, the object faded very quickly. The late time photometry may be
influenced by contamination from the galaxy. Thus for AZT, LCO and KT, we
obtained reference images in each band in Mar. 2019, Oct. 2018, and Feb. 2019,
respectively. The reference images were obtained in all corresponding bands
except for the $U$-band of AZT. For TNT images, since the source is still
bright during observations, the influence of the background is negligible.
Although the observations continued after Aug. 18, 2018, the object became too
faint to be distinguished from the background.
All $UBVRI$ and $gri$ images are pre-processed using standard IRAF222IRAF is
distributed by the National Optical Astronomy Observatories, which are
operated by the Association of Universities for Research in Astronomy, Inc.,
under cooperative agreement with the National Science Foundation (NSF).
routines, which includes corrections for bias, flat field, and removal of
cosmic rays. To remove the contamination from the host galaxy, we applied
template subtraction to the AZT, LCO and KT images. Note that the $U$-band
images were not host subtracted. The instrumental magnitudes of both AT2018cow
and the reference stars were then measured using the standard point spread
function (PSF). And then the instrumental magnitudes were converted to
standard Johnson and SDSS gri-band magnitudes using the zero points and color
terms of each telescope. The resultant magnitudes are listed in Tab. 1. We
also include the early photometry from Prentice et al. (2018) for comparison.
The light curves are shown in Fig. 1.
Table 1: Portion of optical photometric observations of AT2018cow. MJD | mag. | mag. error | band | Telescope/reference
---|---|---|---|---
58285.4400 | 14.700 | 0.100 | o | Smartt et al. (2018)
58286.1950 | 14.320 | 0.010 | i | Fremling (2018)
58286.8880 | 13.695 | … | V | ObSN
58287.1130 | 13.593 | … | V | ObSN
58287.1500 | 13.400 | 0.050 | g | Prentice et al. (2018)
58287.1500 | 13.800 | 0.100 | r | Prentice et al. (2018)
58287.1500 | 14.100 | 0.100 | i | Prentice et al. (2018)
58287.4440 | 13.674 | … | V | ObSN
58287.9270 | 13.771 | … | V | ObSN
58287.9400 | 14.021 | … | I | ObSN
58287.9460 | 13.926 | … | r | ObSN
58287.9520 | 13.742 | … | V | ObSN
58287.9540 | 13.692 | … | B | ObSN
58287.9540 | 13.692 | … | g | ObSN
58287.9750 | 13.725 | … | R | ObSN
58288.0677 | 13.809 | 0.021 | B | LCO
58288.0677 | 13.939 | 0.013 | V | LCO
58288.0677 | 13.787 | 0.011 | g | LCO
58288.0677 | 14.573 | 0.016 | i | LCO
58288.0677 | 14.295 | 0.017 | r | LCO
Figure 1: Light curves obtained from various telescopes. Discovery magnitude
in $orange$-band from Smartt et al. (2018) (orange) and early follow-up
photometry from Fremling (2018) and Prentice et al. (2018) are also plotted as
empty squares. The pre-discovery detection limits are from Fremling (2018) and
Prentice et al. (2018). The magnitudes in different bands are shifted for
better display.
It can be seen that AT2018cow rises to a peak at MJD$\sim$58287.0 in $V$, $R$
and $I$ bands, where the light curves are better sampled around the peak. The
latest non-detection limit is on MJD 58284.13 in $g$-band (Prentice et al.,
2018), so the rise time of AT2018cow is less than 2.9 days. If we take the
median of the first detection (i.e. discovery by ATLAS) MJD 58285.44 and the
latest non-detection (i.e. MJD 58284.13) as the first light time, then the
rise time is $\sim$2.2 days. We apply an explosion time on
MJD=58284.79$\pm$0.66 throughout this paper. This rise time is too short
compared to supernovae, which usually have rise time of more than 10 days.
After the peak, the light curves decline as fast as 0.33 mag d-1, 0.27 mag
d-1, 0.22 mag d-1, within the first 10 days in $V$, $R$ and $I$-bands,
respectively.
### 2.2 Optical Spectroscopic Observations
Our first spectrum was taken at Jul. 21, 2018 by the 2.16-m telescope at
Xinglong Observatory of NAOC (hereafter XLT). A total of 31 spectra were
collected with different telescopes, including the XLT, the 2-m Faulkes
Telescope North (FTN) of the Las Cumbres Observatory network, and the 9.2-m
Hobby-Eberly Telescope (HET). The details of the spectroscopic observations
are listed in Tab. 2.
All spectra were reduced using the standard IRAF routines, which involves
corrections for bias, flat field, and removal of cosmic rays. The Fe/Ar and
Fe/Ne arc lamp spectra obtained during the observation nights are used to
calibrate the wavelength of the spectra, and standard stars observed on the
same night at similar airmasses as the supernova were used to calibrate the
flux of spectra. The spectra were further corrected for continuum atmospheric
extinction during flux calibration using mean extinction curves obtained at
Xinglong Observatory and Haleakala Observatory in Hawaii, respectively.
Moreover, telluric lines were removed from the spectra of XLT and FTN. We
recalibrated the fluxes of the spectra to the multi-band photometry data. The
UV data from Perley et al. (2019) are included in the recalibration process.
The recalibrated spectra are shown in Fig. 2.
On Sep. 17, 2019, when AT2018cow already faded away in the host galaxy, a
spectrum was obtained at the site of AT2018cow by HET. There are some narrow
absorption lines in the resultant spectrum, which are an artifact of data
reduction. HET LRS2 is an IFU spectrograph having 280 individual fibers packed
close together in a rectangular pattern, with a field-of-view of
12"$\times$6", which is smaller than the size of the host galaxy of AT2018cow.
Since the data reduction pipeline determines the background by combining the
fibers having the lowest flux level, the background will necessarily contain
some of the galaxy features. Thus the spectra show some fake absorption lines
resulting from subtraction of the emission lines from other faint part of the
host galaxy. These fake lines are manually removed from the spectrum. A
detailed analysis on this spectrum is presented in Sec. 4.
Table 2: Log of optical spectroscopy of AT2018cow. UT | MJD | Telescope | Wav. range (Å) | Instrument | Exposure time (s)
---|---|---|---|---|---
2018/06/21.58 | 58290.58 | XLT | 3970-8820 | BFOSC | 2400
2018/06/22.32 | 58291.32 | HET | 3640-10298 | LRS2 | 300
2018/06/23.64 | 58292.64 | XLT | 3970-8820 | BFOSC | 2400
2018/06/24.50 | 58293.50 | FTN | 3500-10000 | FLOYDS | 1200
2018/06/26.30 | 58295.30 | HET | 3640-10300 | LRS2 | 500
2018/06/26.39 | 58295.39 | FTN | 3500-10000 | FLOYDS | 1200
2018/06/26.54 | 58295.54 | XLT | 3970-8820 | BFOSC | 1200
2018/06/27.57 | 58296.57 | XLT | 3970-8820 | BFOSC | 2400
2018/06/28.35 | 58297.35 | FTN | 3500-10000 | FLOYDS | 1200
2018/06/28.55 | 58297.55 | XLT | 3970-8820 | BFOSC | 1200
2018/06/30.38 | 58299.38 | FTN | 3500-10000 | FLOYDS | 1200
2018/07/01.57 | 58300.57 | XLT | 3970-8820 | BFOSC | 1500
2018/07/04.48 | 58303.48 | FTN | 3500-10000 | FLOYDS | 1200
2018/07/06.43 | 58305.43 | FTN | 3500-10000 | FLOYDS | 2700
2018/07/08.37 | 58307.37 | FTN | 3500-10000 | FLOYDS | 2700
2018/07/10.33 | 58309.33 | FTN | 3500-10000 | FLOYDS | 2700
2018/07/11.41 | 58310.41 | FTN | 3500-10000 | FLOYDS | 2700
2018/07/12.25 | 58311.25 | HET | 6440-10300 | LRS2 | 1000
2018/07/13.32 | 58312.32 | FTN | 3500-10000 | FLOYDS | 2700
2018/07/14.35 | 58313.35 | FTN | 3500-10000 | FLOYDS | 2700
2018/07/15.25 | 58314.25 | HET | 3640-6970 | LRS2 | 800
2018/07/16.31 | 58315.31 | FTN | 3500-10000 | FLOYDS | 2700
2018/07/17.35 | 58316.35 | FTN | 3500-10000 | FLOYDS | 2700
2018/07/19.35 | 58318.35 | FTN | 3500-10000 | FLOYDS | 3600
2018/07/22.28 | 58321.28 | FTN | 3500-10000 | FLOYDS | 3600
2018/07/24.34 | 58323.34 | FTN | 3500-10000 | FLOYDS | 3600
2018/07/25.32 | 58324.32 | FTN | 3500-10000 | FLOYDS | 3600
2018/07/26.34 | 58325.34 | FTN | 3500-10000 | FLOYDS | 3600
2018/07/31.37 | 58330.37 | FTN | 4800-10000 | FLOYDS | 3600
2018/08/03.25 | 58333.25 | FTN | 3500-10000 | FLOYDS | 3600
2018/08/14.18 | 58344.18 | HET | 3640-8300 | LRS2 | 1800
2019/09/17.09 | 58743.09 | HET | 3640-10200 | LRS2 | 1800
Figure 2: Optical spectra of AT2018cow. The numbers indicate days since MJD
58285. Host galaxy emissions are not removed. The data are smoothed by a bin
of 20 Å for better display.
## 3 Observational Properties
### 3.1 Light Curves and Color Evolution
The light curves of AT2018cow show much faster evolution than other optical
transients. In Fig. 3 we compare the $V$-band light curves of AT2018cow with
other SNe of different subtypes, including the peculiar fast-evolving
transient KSN2015K (Rest et al., 2018). One can see that both the rise and
decline of AT2018cow are faster than any other known fast-evolving supernovae.
The rise time is very close to KSN2015K, while AT2018cow is about 2 mags
brighter. Most SLSNe have much slower evolution so we do not show them in the
plot. AT2018cow is close to the type Ibn SN iPTF15ul (Hosseinzadeh et al.,
2017) in peak luminosity, while it is similar to the type Ibn SN 2006jc in
terms of fast decline after the peak. It should be noted that the high
luminosity as well as the rapid evolution seen in AT 2018cow lie in the range
of SNe Ibn.
Figure 3: The V-band light curves of AT2018cow compared with other optical
transients. Different colors are used to distinguish object types. The green
shaded area shows the template R-band light curves of SNe Ibn from
Hosseinzadeh et al. (2017). Data references: SN 2011fe (Zhang et al., 2016),
KSN2015K (Rest et al., 2018), SN 1998bw (Galama et al., 1998; Sollerman et
al., 2000; McKenzie & Schaefer, 1999), SN 2002ap (Foley et al., 2003), SN
2017ein (Xiang et al., 2019; Van Dyk et al., 2018), SN 1994I (Richmond et al.,
1996; Yokoo et al., 1994), SN 2007gr (Chen et al., 2014), iPTF16asu
(Whitesides et al., 2017), SN 2008D (Mazzali et al., 2008; Modjaz et al.,
2009; Bianco et al., 2014; Brown et al., 2014), SN 2004aw (Taubenberger et
al., 2006), SN 2006jc (Bianco et al., 2014; Brown et al., 2014; Drout et al.,
2011),SN 2010al (Hicken et al., 2017; Brown et al., 2014), ASASSN-14ms (Wang,
et al. 2020, in prep.), SN 2015U (Shivvers et al., 2016; Tsvetkov et al.,
2015), SN 2011hw (Brown et al., 2014; Smith et al., 2012b), LSQ13ccw (Smartt
et al., 2015), iPTF15ul (Hosseinzadeh et al., 2017), SN 2014av (Pastorello et
al., 2016), SN 2009ip: Mauerhan et al. (2013); Smartt et al. (2015). Part of
the reference data are obtained via the Open Supernova Catalog (Guillochon et
al., 2017).
During our observations, AT2018cow maintains very blue color (i. e.
$B-V\sim-0.1$ mag, Fig. 4). Thus, it should suffer little reddening from its
host galaxy. This can also be verified by the absence of Na I D absorption
line in the spectra. We only consider the Galactic extinction of $E(B-V)$=0.08
(Schlafly & Finkbeiner, 2011) for AT2018cow, and ignore the host extinction in
this paper. As also proposed by Perley et al. (2019), the photospheric
temperature of AT2018cow is as high as $\sim$30,000 K near the maximum light,
and is still as high as $\sim$14,000 K at $\sim$50 days after discovery. This
is not seen in any other optical transients ever discovered. For supernovae,
the photospheric temperature can be high in early times but usually cools down
to $\sim$5000 K in a few weeks after explosion, since the energy source is not
strong enough to maintain a very high temperature. So the color of normal SNe
will become red in late phases. In Fig. 4 we show the $B-V$ color evolution of
AT2018cow in comparison with other SNe. The color evolution of AT2018cow
resembles that of SN 2006jc. Assuming a blackbody SED shape, the spectra of SN
2006jc also seem to present unusually high effective temperature, $\sim$15,000
K on day 8, then the temperature grows to 25,000 K on day 25 and drops to
15,000 K around day 60. The temperature decreases to $\sim$3,500 K and then
keeps flat after day 80. Nevertheless, the interaction and blending of iron
lines may indeed contribute to the high temperature.
Figure 4: Upper: $B-V$ evolution of AT2018cow, in comparison with other well
observed SNe. Symbols and references are the same as in Fig. 3. Lower:
temperature evolution of AT2018cow. The dased line shows the estimation of the
early temperature as $T\propto t^{-0.5}$, which is used to estimate the early
time bolometric luminosity (see Sec. 5).
Another interesting point is that the photospheric radius seems to be
decreasing since the very beginning, unlike that of normal SNe, which will
increase before peak and then decrease as a result of the expansion and
dilution of the ejecta. The absence of an expansion phase is the main problem
of the supernova origin for AT2018cow.
### 3.2 Spectral Evolution: Signatures of Interaction
The spectra of AT2018cow are characterized by featureless blue continuum in
the first $\sim$10 days after discovery, and then some broad emission features
emerge later, with possible contaminations from the host galaxy. Featureless
and blue spectra are common in SNe due to high photospheric temperature at
early phases. Then spectral lines appear as the temperature decreases. We
create normalized spectra of AT2018cow from the observed spectra by
subtracting and deviding the best fit single blackbody continuum of each
spectrum. In the first 10 days, the spectra are characterized by a wide
feature near 5000Å, as shown in Fig. 5. Later on, many broad emission lines
emerge, overlapped with many narrow and strong emission lines. And there is
flux excess in the red end, which is probably due to dust emission in later
phases. As proposed by Fox & Smith (2019), the spectra of AT2018cow might have
shown signatures of circurmstellar interaction (CSI) like SNe Ibn and IIn.
While the typical features of CSI are narrow emission lines of H and He. The
last spectrum taken by HET shows many narrow emission lines (FWHM $\approx$ 4
Å) which are apparently from the background host galaxy. Although other
spectra of AT2018cow do show strong and broader H$\alpha$ lines since day 8
(Fig. 2), it is quite possible that the lines of H are from the host galaxy,
not AT2018cow. The reason is that those spectra do not have such high
resolution as that in HET spectrum, so the narrow lines are broadened. To
figure out whether the narrow emission lines are from the host galaxy or
AT2018cow, we measured the FWHMs of H$\alpha$ line in each spectrum and
compared it with other lines in the same spectrum. The results show that the
width of H$\alpha$ lines are only slightly broader than (by less than 10 Å,
within the uncertainty) other narrow lines such as [NII] and [SII], indicating
that they are probably from the host galaxy. Thus we conclude that there is no
significant narrow emissions of H in AT2018cow.
To better look into the spectral features of AT2018cow at t>10 days, we
carefully subtracted the narrow emission lines of H$\alpha$, [NII]
$\lambda$6548,6583 and [SII] $\lambda$6730,6716 from the spectra. For the
spectra taken from $\sim$10 days to $\sim$59 days after discovery, we identify
shallow and broad emission lines that can be attributed to HI, HeI, HeII, OI,
OIII and CIII lines (as shown in Fig. 6). The OI, OIII, CIII and HeII lines
dissipated after around day 45. The peaks of these lines are all slightly
redshifted by up to 2000 km s-1. The emission lines of AT2018cow are much
broader than most SNe Ibn and IIn. The HeI $\lambda$5876 line has an FWHM of
$\sim$300Å ($v\sim 15,000~{}\mathrm{km~{}s^{-1}}$) at day 14, which is one
magnitude higher than that of most SNe Ibn ( $v\sim$1000
$\mathrm{km~{}s^{-1}}$). In late phases, the broad lines become narrower, with
the FWHM decreasing to $\sim$3000 km s-1 on day 59. Meanwhile, these broad
emission lines are redshifted with velocities decreasing from $\sim$1800
$\mathrm{km\ s^{-1}}$ when they first emerge, to hundreds of km s-1 in late
phases. In the region of H$\alpha$, there is a broad emission line, which
should be a blending of H$\alpha$ and HeI $\lambda$6678\. This line is seen
getting narrower over time and splitted into two lines since t$\sim$30 day,
and the peaks moves to the rest wavelength. In addition to the long existing
broad emission lines, weak and narrow (FWHM$\sim$800$-$1000 km s-1) HeI
$\lambda$6678 line emerged in the spectra since t$\sim$20 days. This narrow
line is certain to be from AT2018cow, as it does not appear in the spectrum of
the host galaxy. To conclude, the broad emission lines of highly ionized
elements (CIII, OIII) indicate that there is possible CSM interaction at very
early time (t<10 days). And the appearance of narrow He emission lines in late
times (t>20 days) implies the existence of another distant CSM formed around
the progenitor object.
It is natural to think of an interacting SN picture for AT2018cow. Fox & Smith
(2019) found the similarity between AT2018cow and some SNe Ibn and SNe IIn.
Here we argue that although AT2018cow show signitures of interaction similar
to SNe Ibn and SNe IIn, its spectral evolution is quite different from that of
SNe Ibn and IIn. In Fig. 7 we show the spectral evolution of AT2018cow
compared with some well observed SNe Ibn, SN 2006jc (Pastorello et al., 2007;
Smith et al., 2008), SN 2015U (Pastorello et al., 2015a; Shivvers et al.,
2016) and SN 2002ao (Pastorello et al., 2008a), and a typical SN IIn 2010jl
(Smith et al., 2012c; Zhang et al., 2012). From Fig. 7 we can see the
diversity of SNe Ibn. AT2018cow seems to have different spectral features from
any other interacting SNe, as it has weaker lines at all phases. At earlier
phases AT2018cow is characterized by blue featureless continum like that seen
in some CCSNe as a result of hight temperature, i. e. SN 2015U from our
comparison sample. While SN 2015U shows a narrow P-cygni absorption feature,
indicating the recombination of He in the CSM (Shivvers et al., 2016). Note
that the emission lines of AT2018cow emerged at later phases, and are much
weaker compared to SN 2006jc and SN 2002ao. Moreover, the Ca lines are very
strong in SN 2006jc and SN 2015U, but are weak in AT2018cow. At late times,
AT2018cow show similarities to SN 2002ao, both being dominated by broad lines.
While P-cygni absorption of HeI lines are present in SN 2002ao, and the lines
are stronger. The line velocity of HeI at t$\sim+$24d is $\sim$8500 km s-1 for
SN 2002ao, slightly higher than AT2018cow ($\sim$7000 km s-1). SN 2002ao is
claimed as 06jc-like, which are proposed as Wolf-Rayet (WR) stars exploded in
a He-rich CSM (Pastorello et al., 2008a). Although SNe Ibn and IIn are
distinguished by the strength of the H emission lines, there are some
transitional objects which show roughly equal strength of H and He emission
lines, for example SN 2005la (Pastorello et al., 2008b) and SN 2011hw (Smith
et al., 2012b; Pastorello et al., 2015b).
In most interacting supernovae, like SNe IIn and Ibn, the emission lines have
velocities in range of tens to a few thousand km s-1, depending on the wind
velocities of the progenitor stars. The wind velocities are related to the
type of the progenitor stars. At same metallicities, stars with larger initial
masses are expected to have stronger stellar winds therefore higher wind
velocities when they evolve to the end of life (see Smith, 2014 and references
therein). Some of the objects show intermediate width emission lines (1000 km
s-1 < $v$ < 4000 km s-1), like SN 2006jc. In spectra of SN 2006jc, the bluer
He lines show narrow P-Cygni profiles, while the redder He lines show an
intermediate width emission component (FWHM$\approx$ 3000 km s-1) (Foley et
al., 2007). The broad emission features in AT2018cow are apparently different
from the spectral features in ordinary SNe II. The lack of absorption features
implies that AT2018cow is possibly more similar to SNe IIn/Ibn, rather than
SNe IIP/IIL. While the velocities of the broad emission component in AT2018cow
($v\sim$10,000 km s-1) are much higher than normal SNe Ibn/IIn. The lack of
narrow emission lines in AT2018cow and relatively weak lines make it unique
among interacting SNe. While this is not an argument against the interacting
SN origin of AT2018cow, because spectral diversity is seen in other SNe Ibn
and SNe IIn (e.g., Hosseinzadeh et al., 2017). Absence of narrow lines might
be resulted from a closely located CSM which was immediately swept up by the
shock within a short time period.
Figure 5: Normalized spectra of AT2018cow in the first 10 days. The shaded
areas mark the region of telluric lines. Figure 6: Normalized spectra of
AT2018cow after 10 days since discovery. The spectra are normalized and the
narrow emission lines from the host galaxy in the ragion around H$\alpha$ in
spectra of AT2018cow are manually subtracted for better view. The numbers on
the right mark the time in days since discovery (MJD 58285). Figure 7:
Normalized spectra of AT2018cow compared with other supernovae. All spectra
are normalized by the fitted blackbody continum. The numbers in the brackets
are the phases relative to $V$-band maxmum, which for AT2018cow we adopt MJD
58287. The narrow H$\alpha$ lines in spectra of AT2018cow are manully
subtracted for better view.
## 4 Host galaxy environment
We notice that the spectra of AT2018cow are almost featureless at early phases
(t<10 d). Later on the spectra are some broader features overlapped with many
narrow emission lines which are most probably due to the emission from the
background galaxy. We obtained a spectrum of the host at the location of
AT2018cow with the 9.2-m HET on Sep. 17, 2019 (corresponding to $\sim$460 days
after discovery), as shown in Fig. 8.
Figure 8: Upper panel: The spectrum taken at the lcation of AT2018cow. Lower
panel: Firefly fitting of the host spectra at the site of the object and
galaxy center. The over-plotted colored lines are the best fit models of
Firefly.
The spectrum is characterized by that of a typical HII region, which implies
that this region is currently at gas phase and star-forming. One can see
strong emission lines of H, He, N, S and O, and a NaI D absorption line at the
rest wavelength of the Milky Way. With this spectrum, we are able to measure
the intensities of the emission lines and then derive the properties of the
local environment. Following Curti et al. (2017), we get a local metallicity
of 12+log(O/H)$\sim$8.65$\pm$0.07, which is solar-like and among the range of
other SNe Ibn (Pastorello et al., 2016). The star formation rate (SFR) can be
derived from the luminosity of the H$\alpha$ emission line, for which we
measured as $L$(H$\alpha$)$\approx 1.82\times 10^{39}$ erg s-1. This is
consistent with the result from Lyman et al. (2020) $L$(H$\alpha$)$\approx
1.35\times 10^{39}$ erg s-1 at the site of AT2018cow, considering that we
applied a larger distance. Using the conversion factor given in Sullivan et
al. (2001), we get SFR(H$\alpha$)${}_{local}\approx 0.015$
$\mathrm{M_{\odot}}$ yr-1. We also examine the [OII]$\lambda$3727 line, and
get $L$([OII])$\approx 8.76\times 10^{38}$ erg s-1. With the relation given in
Kennicutt (1998), we get SFR([OII])${}_{local}\approx 0.012$
$\mathrm{M_{\odot}}$ yr-1, which is consistent with that from H$\alpha$ line.
To get more information of the local environment of AT2018cow, we use Firefly
(Wilkinson et al., 2017) to fit the spectrum with stellar population models.
The input models are two M11 libs: MILES and STELIB (Maraston & Strömbäck,
2011), and initial mass function ‘Kroupa’ (Kroupa, 2001) is adopted in the
fit. Fig. 8 shows the best fit specta, from which we get a stellar mass of
$M_{\star}\sim 5\times 10^{6}$ $\mathrm{M_{\odot}}$. Combining the above SFR
and stellar mass infomation, we can get a local specific star formation rate
(sSFR) as log(sSFR)${}_{local}\sim-$8.5 (yr-1).
The Sloan Digital Sky Survey (SDSS; Abolfathi et al., 2018) has taken one
spectrum at the center of the host galaxy of AT2018cow on MJD 53566. As the
HET spectrum we obtained only provides the local information, we also use the
SDSS spectrum to measure the above corresponding parameters for the whole
galaxy. The resulting metallicity is the same as that measured from the HET
spectrum spotted at the site of AT2018cow, while the SFR is measured as
SFR(H$\alpha)_{center}\approx 0.008$ $\mathrm{M_{\odot}}$ $\mathrm{yr^{-1}}$
if we do not consider any host extinction. The Firefly fit shows that the
stellar mass of the nucleus is $M_{\star}\sim 2.6\times 10^{8}$
$\mathrm{M_{\odot}}$. We caution that the SDSS spectrum only includes the flux
from the galaxy center, thus the SFR is expected to be lower. For the whole
galaxy, we refer to the results from other studies. Perley et al. (2019) and
Lyman et al. (2020) found stellar mass and SFR in good agreement with each
other, although they adopted different distances. At $D_{L}$=63 Mpc, stellar
masses in these two studies become $M_{\star}\approx 1.56\times
10^{9}$$\mathrm{M_{\odot}}$ and $1.85\times 10^{9}$$\mathrm{M_{\odot}}$,
respectively. And the SFR from Lyman et al. (2020) becomes 0.20
$\mathrm{M_{\odot}}$ yr-1. In the following discussion, we adopt an average of
these results, i.e. $M_{\star}\approx 1.70\times 10^{9}$$\mathrm{M_{\odot}}$,
SFR$\approx$0.21 $\mathrm{M_{\odot}}$ yr-1, and log(sSFR)$\approx-$9.88
(yr-1).
The host environment may provide a clue to the physical origin of AT2018cow.
We compare the host environment parameters with other well studied transients,
including type Ia supernovae (SNe Ia, Smith et al., 2012a; Galbany et al.,
2014), core-collapse supernovae (CCSNe, Svensson et al., 2010; Galbany et al.,
2014), superluminous supernovae (SLSNe, Angus et al., 2016), and gamma-ray
bursts (GRBs, Svensson et al., 2010). As shown in Fig. 9, the host galaxy of
AT2018cow is located among SNe Ia, CCSNe and GRBs, but away from the SLSNe
group. The host galaxy of AT2018cow has stellar mass close to the median of
GRBs, but at the lower end of the SNe group, except for SLSNe. We can not say
for sure which group it should belong to, and it is likely that AT2018cow is
distinct from SLSNe, although AT2018cow has a peak luminosity comparable to
them. Meanwhile, the local high SFR of AT2018cow may imply that AT2018cow is
probably originated from a massive star.
Figure 9: The host-galaxy parameters of AT2018cow compared with other types of
transients.
## 5 Modeling the rapid evolving light curves
The physical interpretation of AT2018cow is still in debate, although there
are already several papers trying to uncover its physical origin. The
radioactive decay of 56Ni is a well known energy source for supernovae
(Arnett, 1982). The bolometric light curve of AT2018cow can not be powered by
pure 56Ni , as the peak luminosity would require an ejected 56Ni mass of
$\sim$ 6 $\mathrm{M_{\odot}}$ but a low ejecta mass < 1 $\mathrm{M_{\odot}}$.
In the above analysis, we find high resemblance of the light curves of
AT2018cow to that of SNe Ibn, and signatures of CSI are found in the spectra,
so we try to fit the light curves of AT2018cow using the CSI model. The fast-
declining and luminous bolometric light curve of SN 2006jc has been
successfully modeled by CSI models (e.g., Chugai, 2009; Tominaga et al.,
2008). The rapid declining light curves can be related to the early shock-
cooling from the progenitor envelope. Since the progenitor has lost most of
its hydrogen envelope, the shock-cooling should be weak and short for the
core-collapse of a massive star. Another reasonable interpretation is the
interaction of the supernova ejecta with the surrounding circumstellar medium
(CSM). This can be supported by the emission lines in the spectra (see Sec.
3.2).
Figure 10: Bolometric light curve of AT2018cow. Best-fit models of CSI+RD with
s=0, 2 are plotted in blue and red solid lines, respectively. The two
components of the models are shown by dashed (CSI) and dotted (RD) lines,
respectively. The best-fit magnetar model is plotted as a magenta solid line.
A pure RD model is also shown in a grey dotted line as a reference.
We construct the bolometric light curves by integrating the UV and optical
flux (the UV data are taken from Perley et al., 2019), and then apply a model
in which CSI is dominating the early time light curve. In order to constrain
the fitting better, especially to obtain data before peak, we estimate the
pre-peak bolometric luminosities based on the following assumptions: 1). the
SED of AT2018cow is a blackbody; 2). the photometric temperature evolves as a
power law $T\propto(t-t_{0})^{-0.5}$, as we derived from the early temperature
evolution. And then the bolometric luminosities before MJD 58288.44 are
estimated using the single band photometry data. We adopt a hybrid model which
includes 56Ni powering and the interaction of the SN ejecta with a dense CSM
with density profile as a power-law, i. e. $\rho_{\mathrm{CSM}}\propto
r^{-s}$, where the typical value of s is 2 and 0 (e.g., Chatzopoulos et al.,
2012, 2013; Wang et al., 2019). In our model, the density distribution of the
ejecta is uniform in the inner region ($\delta$=0), and follows a power-law
($\rho\propto r^{-12}$) in the outer region. The early fast rising light curve
of AT2018cow is mainly powered by CSI, while the slower declining tail is
dominated by radioactive decay (RD) of 56Ni . We first consider the case of
s=2, which corresponds to a steady-wind CSM. The best-fit light curve is shown
in Fig. 10, and the fitted parameters are presented in Tab. 3. As shown in
Fig. 10, our CSI+RD(s=2) model can fit the observations quite well. The mass
loss rate of the progenitor star can be estimated as
$0.1(v_{\mathrm{CSM}}/100$ km s-1) $\mathrm{M_{\odot}}$ yr-1, $\sim$ 1
$\mathrm{M_{\odot}}$ yr-1 with $v_{\mathrm{CSM}}\sim$ 1000 km s-1. Margutti et
al. (2019) also reach similar conclusion by analysing the optical and X-ray
data. Such a mass loss rate is much higher than that found from the radio
observations of AT2018cow ($\dot{M}\sim 10^{-4}-10^{-3}$ $\mathrm{M_{\odot}}$
yr-1) (Ho et al., 2019 had a similar conclusion). If we set limit on the mass
loss rate, the model can hardly fit the observations. Thus, we claim that the
early bright and fast evolving light curve of AT2018cow can not be produced by
CSI with a steady stellar wind.
We then try the other case where s=0, i. e. the density of the CSM is a
constant. The fitting result is shown in Fig. 10, and the fitted parameters
are presented in Tab. 3. As shown in Fig. 10, with $M_{\mathrm{ej}}\approx
3.16$ $\mathrm{M_{\odot}}$, $M_{\mathrm{CSM}}\approx 0.04$
$\mathrm{M_{\odot}}$, $M_{\mathrm{Ni}}\approx 0.23$ $\mathrm{M_{\odot}}$, the
CSM+56Ni (s=0) model can also provide a plausible fit for the observed
bolometric light curve. The inferred inner radius of the CSM gives a
constraint on the radius of the progenitor star $R<3~{}\mathrm{R_{\odot}}$,
which is consistent with the typical size of WR stars. The CSM shell extends
outwards to a radius of $8.70\times 10^{13}$ cm ($\approx 1200~{}R_{\odot}$),
implying that the CSM was formed shortly prior to the explosion. Such a CSM
shell can be produced by an episodic mass ejection from the progenitor star,
like a luminous blue variable (LBV) or from a common-envelope episode of a
binary system. Combining the mass and velocity of the ejecta, the kinetic
energy of the ejecta can be estimated as 6.6$\times 10^{51}$ erg, several
times higher than that of the ordinary SNe Ibc and rather similar to the broad
lined SNe Ic (SN Ic-BL) (Lyman et al., 2016), which are found to be possibly
associated with long gamma ray bursts (e.g. SN 1998bw (Iwamoto et al., 1998;
Nakamura et al., 2001)). The high velocity of the ejecta might be connected to
a relativistic jet.
Table 3: Parameters of the best fit CSM+56Ni models for AT2018cow. … | s=0 | s=2
---|---|---
$t_{0}$ | MJD 58284.5 | MJD 58284.7
$M_{\mathrm{ej}}(M_{\odot})$ | 3.16 | 1.69
$x_{0}$aaThe dimensionless radius of the division between inner and outer region of ejecta. | 0.63 | 0.76
$v_{\mathrm{ej}}$(km s-1) | 26000 | 13600
$M_{\mathrm{Ni}}(M_{\odot})$ | 0.23 | 0.14
$M_{\mathrm{CSM}}(M_{\odot})$ | 0.04 | 0.12
$r_{\mathrm{CSM,in}}$(cm)bbThe inner radius of CSM. | 2.11$\times 10^{11}$ | 1.03$\times 10^{13}$
$r_{\mathrm{CSM,out}}$(cm)ccThe outer radius of CSM. | 8.70$\times 10^{13}$ | 3.16$\times 10^{14}$
$\rho_{\mathrm{CSM,in}}$(g cm-3)ddThe CSM density at $r_{\mathrm{CSM,in}}$. | $2.9\times 10^{-11}$ | $5.6\times 10^{-10}$
$\epsilon$eeThe radiation efficiency. | 0.22 | 0.65
$\kappa$ | 0.14 | 0.15
$\kappa_{\gamma}$ | 0.015 | 0.014
Alternatively, the bolometric luminosity and effective temperature evolutions
can be explained by a magnetar-powered model (Nicholl et al., 2017). Assuming
that the opacities of ejecta are $\kappa=0.2~{}\mathrm{cm^{2}~{}g^{-1}}$ for
the optical photon and
$\kappa_{\mathrm{mag}}=0.013~{}\mathrm{cm^{2}~{}g^{-1}}$ for the magnetar
wind, respectively, the best-fit parameters for this model are
$t_{0}=58283.4$, $M_{\mathrm{ej}}=0.1$ $\mathrm{M_{\odot}}$,
$v_{\mathrm{ej}}=2.72\times 10^{4}$ km s-1, $P=4.5$ ms and $B=1.11\times
10^{14}$ G, where $P$ and $B$ are the initial spin period and magnetic field
strength of the nascent magnetar, respectively. We caution that the best-fit
$M_{\mathrm{ej}}$ = 0.1 $\mathrm{M_{\odot}}$ is the lower limit of the
magnetar model in our fitting program. If no limit is given, the fitting tends
to find a significantly lower value to fit the narrow light curve better. This
may imply that the magnetar-powered model requires a rather low ejecta mass
for AT2018cow.
## 6 Discussion: Progenitor Properties
In the previous section, we made analysis of the bolometric light curve of
AT2018cow based on an assumption that it is a supernova origin. While we do
not rule out other possibilities, especially the TDE origin. A main problem of
the supernova origin for AT2018cow is that the process of an expanding
photosphere is missing. In early phases, the photospheric velocity may be very
high ($\sim$0.1c) for AT2018cow in early phases. The photospheric radius keeps
decreasing since very early time. This can be a clue for the interpretation as
a TDE for AT2018cow. Although both Lyman et al. (2020) and Margutti et al.
(2019) find no evidence of the connection between the site of AT2018cow and an
IMBH. Nevertheless, one can notice that the measurements of photospheric
radius start after the peak, probably suggesting that the expanding phase is
not observed.
The magnetar-powered model can make a good fit to the bolometric light curve.
The best-fit $B$ and $P$ of the central engine lies in the range of SLSNe (Lin
et al., 2020 and references therein). Distinction between At2018cow and SLSNe
is the evolution timescale, which is related to the ejecta mass. Nicholl et
al. (2017) found $M_{\mathrm{ej}}\geqslant 2.2$$\mathrm{M_{\odot}}$ with an
average of 4.8$\mathrm{M_{\odot}}$. Besides, the low ejected mass
($M_{\mathrm{ej}}\sim 0.1$$\mathrm{M_{\odot}}$) required by the magnetar model
for AT2018cow is not likely favorable for a massive star, except for some
really extreme cases. Some studies find that massive stars can be ultra-
stripped by binary interaction with a compact neutron star (Tauris et al.,
2015). But in these cases, little H or He remains in the progenitor system,
which is not consistent with the observed spectral features of AT2018cow.
Thus, we disfavor the magnetar model for AT2018cow.
Our CSI+RD(s=0) model makes a plausible fit to the bolometric light curve of
AT2018cow. With $R<3R_{\odot}$, the progenitor star is most likely to be a
compact WR star. The ejected mass
($M_{\mathrm{ej}}\approx$3$\mathrm{M_{\odot}}$) is lower than that predicted
by single stellar evolution models (e.g., Georgy et al., 2012) but around the
mean value of SNe Ibc (Lyman et al., 2016). This might be a result of binary
interaction or episodic eruptive mass loss during the lifetime of massive
stars. It is hard to derive the mass of the progenitor star simply from the
ejecta mass, since the mass loss mechanisms of massive stars can be
complicated and ambiguous.
In the case of s=0, the CSM can be dense shells formed by strong stellar winds
of WR stars or an eruptive of LBV stars (Chevalier & Liang, 1989; Dwarkadas,
2011). According to our fitting result, with wind velocity of 100 km s-1, the
eruption started several months before core-collapse, and was possibly still
on when exploding. The average mass loss rate is
$\sim$0.15$\mathrm{M_{\odot}}$ yr-1, or even higher if the wind velocity is
higher, lying well in the range of LBV eruptions (Smith, 2014). Such mass loss
behaviour can be found in some SNe IIn and SNe Ibn(Gal-Yam et al., 2007;
Taddia et al., 2015; Pastorello et al., 2015b; Kiewe et al., 2012; Moriya et
al., 2014a). Under this scenario, the progenitor of AT2018cow might be a
massive star which is during eruptive state. However, with $R<3R_{\odot}$, the
progenitor star is most likely to be H-poor or even He-poor, so is the CSM.
While there is possibility that H and He are mixed into inner shells so that
the progenitor can keep some H/He at core-collapse.
Meanwhile, binary interaction might dominate the evolution of massive stars,
which are thought to be the progenitors of stripped envelope supernovae
(SESNe). Mass loss can be quite efficient in binaries (Eldridge et al., 2017).
SN 2006jc is a representative of interacting SNe originated from binary
massive stars (Maund et al., 2016; Sun et al., 2020). In the binary scenario,
the progenitors can be less massive stars, and the companion stars evolve
slower so that they can keep their H/He envelopes. A common-envelope episode
of a binary system can also form this dense CSM shell. The detection of H and
He lines in the spectra of AT2018cow indicates that the CSM is not H-free. So
it is quite possible that the CSM is from the companion star rather than the
progenitor itself. The slightly redshifted peaks of the emission lines in the
spectra of AT2018cow suggest asymmetry of the CSM, in favor of the common-
envelope picture.
The progenitor star could be a very massive star which have experienced
violent mass loss due to pulsational ejection. Recently Leung et al. (2020)
has proposed a scenario based on a pulsational pair-instability supernova
(PPISN) model, concluding that the rapidly evolving light curve of AT2018cow
can be explained by a 42 $\mathrm{M_{\odot}}$ He star exploding in a dense He-
rich CSM ($M_{\mathrm{CSM}}\sim 0.5$ $\mathrm{M_{\odot}}$). The proposed model
can fit the bolometric light curve well (at t<30 days). However, the presence
of H lines in the spectra of AT2018cow is inconsistent with the assumption
that both the ejecta and CSM formed around AT2018cow should be H-poor. Leung
et al. (2020) tested different compositions of the CSM and found that the
amount of H in the CSM only has slight effect on the bolometric light curve.
Our fitting result is in agreement with Leung et al. (2020) in terms of the
density and size of the CSM, but we find much lower CSM mass. We do not assume
any Ni-mixing, while Leung et al. (2020) assumes that Ni is fully mixed into
the outer layers of the ejecta. Nevertheless, both models may be plausible.
Our model can correspond to a massive progenitor in a binary system, while
Leung et al. (2020) requires a very massive star, whose zero-age-main-sequence
(ZAMS) mass is 80 $\mathrm{M_{\odot}}$. It is worth noting that Leung et al.
(2019) claims that a massive He core can only be formed under low metallicity
($Z\leq 0.5Z_{\odot}$), which is inconsistent with our measurement of a solar-
like metallicity environment for the progenitor of AT2018cow (see Sec. 4).
This may imply that the progenitor of AT2018cow did not undergo PPI.
The fast evolving light curves of AT2018cow may be related to a very low
ejecta mass, which is consistent with electron-capture supernovae (ECSNe)
(Nomoto, 1984, 1987; Nomoto & Kondo, 1991; Moriya et al., 2014b). Stars with
ZAMS mass of $\sim$8-12 $\mathrm{M_{\odot}}$ form degenerate cores of O, Ne
and Mg, which are susceptible to electron capture, leading to core collapse.
For KSN 2015K, an example of FELTs, Rest et al. (2018) prefers a CSI model,
while Tolstov et al. (2019) has found that the collapse of an ONeMg star
surrounded by an optically thick CSM can also explain the fast rise of the
light curve. However, the progenitors of ECSNe are thought to be super-AGB
stars, which have stellar winds with relatively low velocities ($\sim$10 km
s-1). According to theoretical predictions, ECSNe are usually faint and have
low explosion energies (e.g., Kawabata et al., 2010; Botticella et al., 2009
and a most recent study Hiramatsu et al., 2020). Thus, the ECSN scenario is
unlikely for AT2018cow.
## 7 Summary
In this paper, we present our photometric and spectroscopic observations on
the peculiar transient AT2018cow. The multi-band photometry covers from peak
to $\sim$70 days and spectroscopy ranges from 5 to $\sim$50 days after
discovery. The rapid rise ($t_{\mathrm{r}}\lesssim$2.9 days), luminous light
curves ($M_{V,\mathrm{peak}}\sim-$20.8 mag) and fast post-peak decline make
AT2018cow stand out of any other optical transients. After a thorough
analysis, we find that the light curves and color evolution show high
resemblances to some SNe Ibn. With detailed analysis of the spectral evolution
and line identifications, we find that AT2018cow shows similar properties to
the interacting SNe, like SNe IIn and SNe Ibn. Some broad emission lines due
to HI, HeI, HeII, CIII, OI, and OIII emerge at $t\sim 10$ days, with
$v_{\mathrm{FWHM}}$ decreasing from $\sim 11,000$ km s-1 to $\sim$3000 km s-1
at the end of our observations. At $t\sim 20$ days, narrow and weak He I lines
($v_{\mathrm{FWHM}}\sim$ 800-1000 km s-1) overlain on the broad lines. These
emission lines are evidence of interaction between the ejecta and a H-rich
CSM. Furthermore, we spotted the site of AT2018cow after it faded away and
find that it has a solar-like metallicity. The host galaxy of AT2018cow has
properties similar to those of GRBs and CCSNe, but is distinct from SLSNe and
SNe Ia. High star formation rate at the site of AT2018cow implies that
AT2018cow might originate from a massive star.
Based on the interpretation of a CSI supernova, we fit the bolometric light
curves with CSI+RD models. We find that in order to produce the fast and
bright early light curve of AT2018cow, the CSI model with a steady wind
requires much larger mass loss rate than that derived from radio observations.
While with a dense uniform CSM shell, the CSI+RD model can make plausible fit
with best-fit parameters $M_{\mathrm{ej}}\sim 3.16$ $\mathrm{M_{\odot}}$,
$M_{\mathrm{CSM}}\sim 0.04$ $\mathrm{M_{\odot}}$, $M_{\mathrm{Ni}}\sim 0.23$
$\mathrm{M_{\odot}}$. Such a CSM shell can be formed by eruptive mass ejection
of LBVs immediately before core-collapse or common envelope ejection in
binaries. With $Z\approx Z_{\odot}$, the progenitor is less likely to have
undergone PPI. We conclude that the progenitor of AT2018cow is likely to be a
less massive star in a binary system.
We acknowledge the support of the staff of the Xinglong 2.16 m and Lijiang 2.4
m telescope. This work is supported by the National Natural Science Foundation
of China (NSFC grants 12033003, 11761141001, and 11633002), and the National
Program on Key Research and Development Project (grant no. 2016YFA0400803).
This work was also partially supported by the Open Project Program of the Key
Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese
Academy of Sciences. This work is partially supported by the Scholar Program
of Beijing Academy of Science and Technology (DZ:BS202002). J.-J. Zhang is
supported by the National Science Foundation of China (NSFC, grants 11403096,
11773067), the Key Research Program of the CAS (Grant NO. KJZD-EW- M06), the
Youth Innovation Promotion Association of the CAS (grants 2018081), and the
CAS “Light of West China” Program. X.L. was supported by the China
Postdoctoral Science Foundation funded project (No: 2017m610866). This work
makes use of the LCO network. J.B., D.A.H., and D.H. were supported by NSF
grant AST-1911225, and NASA grant 80NSSC19kf1639. Research by S.V. is
supported by NSF grants AST C1813176 and AST-2008108. We thank the staff of
AZT for their observations and allowance of the use of the data. JV and his
group at Konkoly Observatory is supported by the project “Transient
Astrophysical Objects” GINOP 2.3.2-15-2016-00033 of the National Research,
Development and Innovation Office (NKFIH), Hungary, funded by the European
Union. LK received support from the Hungarian National Research, Development
and Innovation Office grant OTKA K-131508, and from the János Bolyai Research
Scholarship of the Hungarian Academy of Sciences. AB has been supported by the
Lendület Program of the Hungarian Academy of Sciences, project No.
LP2018-7/2019. ZsB acknowledges the support provided by the Hungarian National
Research, Development and Innovation Office (NKFIH), project No. PD-123910,
and the support by the János Bolyai Research Scholarship of the Hungarian
Academy of Sciences.
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|
11institutetext: LaBRI, University of Bordeaux, France22institutetext: Dept.
of Computer Science & Engineering IIT Bombay, India 33institutetext: Dept. of
Mathematics, Computer Science, and Physics, Udine University, Italy
# One-way Resynchronizability
of Word Transducers
Sougata Bose 11 S.N. Krishna 22 Anca Muscholl 11 Gabriele Puppis 33
###### Abstract
The origin semantics for transducers was proposed in 2014, and it led to
various characterizations and decidability results that are in contrast with
the classical semantics. In this paper we add a further decidability result
for characterizing transducers that are close to one-way transducers in the
origin semantics. We show that it is decidable whether a non-deterministic
two-way word transducer can be resynchronized by a bounded, regular
resynchronizer into an origin-equivalent one-way transducer. The result is in
contrast with the usual semantics, where it is undecidable to know if a non-
deterministic two-way transducer is equivalent to some one-way transducer.
###### Keywords:
String transducers Resynchronizers One-way transducers
## 1 Introduction
Regular word-to-word functions form a robust and expressive class of
transformations, as they correspond to deterministic two-way transducers, to
deterministic streaming string transducers [1], and to monadic second-order
logical transductions [11]. However, the transition from word languages to
functions over words is often quite tricky. One of the challenges is to come
up with effective characterizations of restricted transformations. A first
example is the characterization of functions computed by one-way transducers
(known as _rational functions_). It turns out that it is decidable whether a
regular function is rational [14], but the algorithm is quite involved [3]. In
addition, non-determinism makes the problem intractable: it is undecidable
whether the relation computed by a non-deterministic two-way transducer can be
also computed by a one-way transducer, [2]. A second example is the problem of
knowing whether a regular word function can be described by a first-order
logical transduction. This question is still open in general [16], and it is
only known how to decide if a _rational_ function is definable in first-order
logic [13].
Word transducers with origin semantics were introduced by Bojańczyk [4] and
shown to provide a machine-independent characterization of regular word-to-
word functions. The origin semantics, as the name suggests, means tagging the
output by the positions of the input that generated that output.
A nice phenomenon is that origins can restore decidability for some
interesting problems. For example, the equivalence of word relations computed
by one-way transducers, which is undecidable in the classical semantics [18,
19], is PSPACE-complete for two-way non-deterministic transducers in the
origin semantics [7]. Another, deeper, observation is that the origin
semantics provides an algebraic approach that can be used to decide fragments.
For example, [4] provides an effective characterization of first-order
definable word functions under the origin semantics. As for the problem of
knowing whether a regular word function is rational, it becomes almost trivial
in the origin semantics.
Ag$b$Ag$a$Ag$c$Ag$a$Ag$a$Ag$b$Ag$a$Ag$c$input:output:origins:Ag$b$Ag$a$Ag$c$Ag$a$Ag$a$Ag$b$Ag$a$Ag$c$input:output:resynchronizedorigins:
Figure 1: On the left, an input-output pair for a transducer $T$ that reads
$wd$ and outputs $dw$, $d\in\Sigma$, $w\in\Sigma^{*}$, the arrows denoting
origins. On the right, the same input-output pair, but with origins modified
by a resynchronizer $\mathcal{R}$. The resynchronized relation
$\mathcal{R}(T)$ is order-preserving, and $T$ is one-way resynchronizable.
A possible objection against the origin semantics is that the comparison of
two transducers in the origin semantics is too strict. Resynchronizations were
proposed in order to overcome this issue. A resynchronization is a binary
relation between input-output pairs with origins, that preserves the input and
the output, changing only the origins. Resynchronizations were introduced for
one-way transducers [15], and later for two-way transducers [7]. For one-way
transducers _rational_ resynchronizations are transducers acting on the
synchronization languages, whereas for two-way transducers, _regular_
resynchronizations are described by regular properties over the input that
restrict the change of origins. The class of bounded111“Bounded” refers here
to the number of source positions that are mapped to the same target position.
It rules out resynchronizations such as the universal one. regular
resynchronizations was shown to behave very nicely, preserving the class of
transductions defined by non-deterministic, two-way transducers: for any
bounded regular resynchronization $\mathcal{R}$ and any two-way transducer
$T$, the resynchronized relation $\mathcal{R}(T)$ can be computed by another
two-way transducer [7]. In particular, non-deterministic, two-way transducers
can be effectively compared modulo bounded regular resynchronizations.
As mentioned above, it is easy to know if a two-way transducer is equivalent
under the origin semantics to some one-way transducer [4], since this is
equivalent to being order-preserving. But what happens if this is not the
case? Still, the given transducer $T$ can be “close” to some order-preserving
transducer. What we mean here by “close” is that there exists some bounded
regular resynchronizer $\mathcal{R}$ such that $\mathcal{R}(T)$ is order-
preserving and all input-output pairs with origins produced by $T$ are in the
domain of $\mathcal{R}$. We call such transducers _one-way resynchronizable_.
Figure 1 gives an example.
In this paper we show that it is decidable if a two-way transducer is one-way
resynchronizable. We first solve the problem for bounded-visit two-way
transducers. A bounded-visit transducer is one for which there is a uniform
bound for the number of visits of any input position. Then, we use the
previous result to show that one-way resynchronizability is decidable for
arbitrary two-way transducers, so without the bounded-visit restriction. This
is done by constructing, if possible, a bounded, regular resynchronization
from the given transducer to a bounded-visit transducer with regular language
outputs. Finally, we show that bounded regular resynchronizations are closed
under composition, and this allows to combine the previous construction with
our decidability result for bounded-visit transducers.
_Related work and paper overview._ The synthesis problem for resynchronizers
asks to compute a resynchronizer from one transducer to another one, when the
two transducers are given as input. The problem was studied in [6] and shown
to be decidable for unambiguous two-way transducers (it is open for
unrestricted transducers). The paper [21] shows that the containment version
of the above problem is undecidable for unrestricted one-way transducers.
The origin semantics for streaming string transducers (SST) [1] has been
studied in [5], providing a machine-independent characterization of the sets
of origin graphs generated by SSTs. An open problem here is to characterize
origin graphs generated by aperiodic streaming string transducers [10, 16].
Going beyond words, [17] investigates decision problems of tree transducers
with origin, and regains the decidability of the equivalence problem for non-
deterministic top-down and MSO transducers by considering the origin
semantics. An open problem for tree transducers with origin is that of
synthesizing resynchronizers as in the word case.
We will recall regular resynchronizations in Section 3. Section 4 provides the
proof ingredients for the bounded-visit case, and the proof of decidability of
one-way resynchronizability in the bounded-visit case can be found in Section
5. Finally, in Section 6 we sketch the proof in the general case. Missing
proofs can be found in the appendix.
## 2 Preliminaries
Let $\Sigma$ be a finite input alphabet. Given a word $w\in\Sigma^{*}$ of
length $|w|=n$, a _position_ is an element of its domain
$\mathsf{dom}(w)=\\{1,\dots,n\\}$. For every position $i$, $w(i)$ denotes the
letter at that position. A _cut_ of $w$ is any number from $1$ to $|w|+1$, so
a cut identifies a position _between_ two consecutive letters of the input.
The cut $i=1$ represents the position just before the first input letter, and
$i=|w|+1$ the position just after the last letter of $w$.
#### Two-way transducers.
We use two-way transducers as defined in [3, 6], with a slightly different
presentation than in classical papers such as [22]. As usual for two-way
machines, for any input $w\in\Sigma^{*}$, $w(0)=\mathop{\vdash}$ and
$w(|w|+1)=\mathop{\dashv}$, where
$\mathop{\vdash},\mathop{\dashv}\notin\Sigma$ are special markers used as
delimiters.
A _two-way transducer_ (or just _transducer_ from now on) is a tuple
$T=(Q,\Sigma,\Gamma,\Delta,I,F)$, where $\Sigma,\Gamma$ are respectively the
input and output alphabets, $Q=Q_{\prec}\uplus Q_{\succ}$ is the set of
states, partitioned into left-reading states from $Q_{\prec}$ and right-
reading states from $Q_{\succ}$, $I\subseteq Q_{\succ}$ is the set of initial
states, $F\subseteq Q$ is the set of final states, and $\Delta\subseteq
Q\times(\Sigma\uplus\\{\mathop{\vdash},\mathop{\dashv}\\})\times\Gamma^{*}\times
Q$ is the finite transition relation. Left-reading states read the letter to
the left, whereas right-reading states read the letter to the right. This
partitioning will also determine the head movement during a transition, as
explained below.
As usual, to define runs of transducers we first define configurations. Given
a transducer $T$ and a word $w\in\Sigma^{*}$, a _configuration_ of $T$ on $w$
is a state-cut pair $(q,i)$, with $q\in Q$ and $1\leq i\leq|w|+1$. A
configuration $(q,i)$, $1\leq i\leq|w|+1$ means that the automaton is in state
$q$ and its head is between the $(i-1)$-th and the $i$-th letter of $w$. The
transitions that depart from a configuration $(q,i)$ and read $a$ are denoted
$(q,i)\stackrel{{\scriptstyle a}}{{\longrightarrow}}(q^{\prime},i^{\prime})$,
and must satisfy one of the following:
(1) $q\in Q_{\succ}$, $q^{\prime}\in Q_{\succ}$, $a=w(i)$,
$(q,a,v,q^{\prime})\in\Delta$, and $i^{\prime}=i+1$,
(2) $q\in Q_{\succ}$, $q^{\prime}\in Q_{\prec}$, $a=w(i)$,
$(q,a,v,q^{\prime})\in\Delta$, and $i^{\prime}=i$,
(3) $q\in Q_{\prec}$, $q^{\prime}\in Q_{\succ}$, $a=w(i-1)$,
$(q,a,v,q^{\prime})\in\Delta$, and $i^{\prime}=i$,
(4) $q\in Q_{\prec}$, $q^{\prime}\in Q_{\prec}$, $a=w(i-1)$,
$(q,a,v,q^{\prime})\in\Delta$, and $i^{\prime}=i-1$. When $T$ has only right-
reading states (i.e. $Q_{\prec}=\emptyset$), its head can only move rightward.
In this case we call $T$ a _one-way transducer_.
A _run_ of $T$ on $w$ is a sequence $\rho=(q_{1},i_{1})\stackrel{{\scriptstyle
a_{j_{1}}\mid v_{1}}}{{\longrightarrow}}(q_{2},i_{2})\stackrel{{\scriptstyle
a_{j_{2}}\mid v_{2}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
a_{j_{m}}\mid v_{m}}}{{\longrightarrow}}(q_{m+1},i_{m+1})$ of configurations
connected by transitions. Note that the positions $j_{1},j_{2},\dots,j_{m}$ of
letters do not need to be ordered from smaller to bigger, and can differ
slightly (by $+1$ or $-1$) from the cuts $i_{1},i_{2},\dots,i_{m+1}$, since
cuts take values in between consecutive letters.
A configuration $(q,i)$ on $w$ is _initial_ (resp. _final_) if $q\in I$ and
$i=1$ (resp. $q\in F$ and $i=|w|+1$). A run is _successful_ if it starts with
an initial configuration and ends with a final configuration. The _output_
associated with a successful run $\rho$ as above is the word $v_{1}v_{2}\cdots
v_{m}\in\Gamma^{*}$. A transducer $T$ defines a relation
${[\\![T]\\!]}\subseteq\Sigma^{*}\times\Gamma^{*}$ consisting of all the pairs
$(u,v)$ such that $v$ is the output of some successful run $\rho$ of $T$ on
$u$.
#### Origin semantics.
In the origin semantics for transducers [4] the output is tagged with
information about the position of the input where it was produced. If reading
the $i$-th letter of the input we output $v$, then all letters of $v$ are
tagged with $i$, and we say they have _origin_ $i$. We use the notation
$(v,i)$ for $v\in\Gamma^{*}$ to denote that all positions in the output word
$v$ have origin $i$, and we view $(v,i)$ as word over the alphabet
$\Gamma\times\mathbb{N}$. The outputs associated with a successful run
$\rho=(q_{1},i_{1})\stackrel{{\scriptstyle b_{1}\mid
v_{1}}}{{\longrightarrow}}(q_{2},i_{2})\stackrel{{\scriptstyle b_{2}\mid
v_{2}}}{{\longrightarrow}}(q_{3},i_{3})\cdots\stackrel{{\scriptstyle b_{m}\mid
v_{m}}}{{\longrightarrow}}(q_{m+1},i_{m+1})$ in the origin semantics are the
words of the form $\nu=(v_{1},j_{1})(v_{2},j_{2})\cdots(v_{m},j_{m})$ over
$\Gamma\times\mathbb{N}$ where, for all $1\leq k\leq m$, $j_{k}=i_{k}$ if
$q_{k}\in Q_{\succ}$, and $j_{k}=i_{k}-1$ if $q_{k}\in Q_{\prec}$. Under the
origin semantics, the relation defined by $T$, denoted ${[\\![T]\\!]}_{o}$, is
the set of pairs $\sigma=(u,\nu)$ —called _synchronized pairs_ — such that
$u\in\Sigma^{*}$ and $\nu\in(\Gamma\times\mathbb{N})^{*}$ is the output of
some successful run on $u$.
Equivalently, a synchronized pair $(u,\nu)$ can be described as a triple
$(u,v,\mathit{orig})$, where $v$ is the projection of $\nu$ on $\Gamma$, and
$\mathit{orig}:\mathsf{dom}(v)\to\mathsf{dom}(u)$ associates with each
position of $v$ its origin in $u$. So for
$\nu=(v_{1},j_{1})(v_{2},j_{2})\cdots(v_{m},j_{m})$ as above, $v=v_{1}\dots
v_{m}$, and, for all positions $i$ s.t. $|v_{1}\dots v_{k-1}|<i\leq|v_{1}\dots
v_{k}|$, we have $\mathit{orig}(i)=j_{k}$. Given two transducers
$T_{1},T_{2}$, we say they are _origin-equivalent_ if
${[\\![T_{1}]\\!]}_{o}={[\\![T_{2}]\\!]}_{o}$. Note that two transducers
$T_{1},T_{2}$ can be equivalent in the classical semantics,
${[\\![T_{1}]\\!]}={[\\![T_{2}]\\!]}$, while they can have different origin
semantics, so ${[\\![T_{1}]\\!]}_{o}\neq{[\\![T_{2}]\\!]}_{o}$.
#### Bounded-visit transducers.
Let $k>0$ be some integer, and $\rho$ some run of a two-way transducer $T$. We
say that $\rho$ is _$k$ -visit_ if for every $i\geq 0$, it has at most $k$
occurrences of configurations from $Q\times\\{i\\}$. We call a transducer $T$
_$k$ -visit_ if for every $\sigma\in{[\\![T]\\!]}_{o}$ there is some
successful, $k$-visit run $\rho$ of $T$ with output $\sigma$ (actually we
should call the transducer $k$-visit _in the origin semantics_ , but for
simplicity we omit this). For example, the relation $\\{(w,\overline{w})\mid
w\in\Sigma^{*}\\}$, where $\overline{w}$ denotes the reverse of $w$, can be
computed by a $3$-visit transducer. A transducer is called _bounded-visit_ if
it is $k$-visit for some $k$.
#### Common guess.
It is often useful to work with a variant of two-way transducers that can
guess beforehand some annotation on the input and inspect it consistently when
visiting portions of the input multiple times. This feature is called _common
guess_ [5], and strictly increases the expressive power of two-way
transducers, including bounded-visit ones.
## 3 One-way resynchronizability
### 3.1 Regular resynchronizers
Resynchronizations are used to compare transductions in the origin semantics.
A _resynchronization_ is a binary relation
$\mathcal{R}\subseteq(\Sigma^{*}\times(\Gamma\times\mathbb{N})^{*})^{2}$ over
synchronized pairs such that $(\sigma,\sigma^{\prime})\in\mathcal{R}$ implies
that $\sigma=(u,v,\mathit{orig})$ and
$\sigma^{\prime}=(u,v,\mathit{orig}^{\prime})$ for some origin mappings
$\mathit{orig},\mathit{orig}^{\prime}:\mathsf{dom}(v)\to\mathsf{dom}(u)$. In
other words, a resynchronization will only change the origin mapping, but
neither the input, nor the output. Given a relation
$S\subseteq\Sigma^{*}\times(\Gamma\times\mathbb{N})^{*}$ with origins, the
_resynchronized relation_ $\mathcal{R}(S)$ is defined as
$\mathcal{R}(S)=\\{\sigma^{\prime}\mid(\sigma,\sigma^{\prime})\in\mathcal{R},\;\sigma\in
S\\}$. For a transducer $T$ we abbreviate $\mathcal{R}({[\\![T]\\!]}_{o})$ by
$\mathcal{R}(T)$. The typical use of a resynchronization $\mathcal{R}$ is to
ask, given two transducers $T,T^{\prime}$, whether $\mathcal{R}(T)$ and
$T^{\prime}$ are origin-equivalent.
Regular resynchronizers (originally called MSO resynchronizers) were
introduced in [7] as a resynchronization mechanism that preserves definability
by two-way transducers. They were inspired by MSO (monadic second-order)
transductions [9, 12] and they are formally defined as follows. A _regular
resynchronizer_ is a tuple
$\mathcal{R}=(\overline{I},\overline{O},\mathsf{ipar},\mathsf{opar},(\mathsf{move}_{\tau})_{\tau},(\mathsf{next}_{\tau,\tau^{\prime}})_{\tau,\tau^{\prime}})$
consisting of
* •
some monadic parameters (colors) $\overline{I}=(I_{1},\dots,I_{m})$ and
$\overline{O}=(O_{1},\dots,O_{n})$,
* •
MSO sentences $\mathsf{ipar},\mathsf{opar}$, defining languages over expanded
input and output alphabets, i.e. over $\Sigma^{\prime}=\Sigma\times
2^{\\{1,\dots,m\\}}$ and $\Gamma^{\prime}=\Gamma\times 2^{\\{1,\dots,n\\}}$,
respectively,
* •
MSO formulas $\mathsf{move}_{\tau}(y,z)$,
$\mathsf{next}_{\tau,\tau^{\prime}}(z,z^{\prime})$ with two free first-order
variables and parametrized by expanded output letters $\tau,\tau^{\prime}$
(called types, see below).
To apply a regular resynchronizer as above, one first guesses the valuation of
all the predicates $I_{j},O_{k}$, and uses it to interpret the parameters
$\overline{I}$ and $\overline{O}$. Based on the chosen valuation of the
parameters $\overline{O}$, each position $x$ of the output $v$ gets an
associated _type_
$\tau_{x}=(v(x),b_{1},\dots,b_{n})\in\Gamma\times\\{0,1\\}^{n}$, where $b_{j}$
is $1$ or $0$ depending on whether $x\in O_{j}$ or not. We refer to the output
word together with the valuation of the output parameters as _annotated
output_ , so a word over $\Gamma\times\\{0,1\\}^{n}$. Similarly, the
_annotated input_ is a word over $\Sigma\times\\{0,1\\}^{m}$. The annotated
input and output word must satisfy the formulas $\mathsf{ipar}$ and
$\mathsf{opar}$, respectively.
The origins of output positions are constrained using the formulas
$\mathsf{move}_{\tau}$ and $\mathsf{next}_{\tau,\tau^{\prime}}$, which are
_parametrized by output types and evaluated over the annotated input_.
Intuitively, the formula $\mathsf{move}_{\tau}(y,z)$ states how the origin of
every output position of type $\tau$ changes from $y$ to $z$. We refer to $y$
and $z$ as _source_ and _target_ origin, respectively. The formula
$\mathsf{next}_{\tau,\tau^{\prime}}(z,z^{\prime})$ instead constrains the
target origins $z,z^{\prime}$ of any two consecutive output positions with
types $\tau$ and $\tau^{\prime}$, respectively.
Formally,
$\mathcal{R}=(\overline{I},\overline{O},\mathsf{ipar},\mathsf{opar},(\mathsf{move}_{\tau}),(\mathsf{next}_{\tau,\tau^{\prime}}))$
defines the resynchronization consisting of all pairs
$(\sigma,\sigma^{\prime})$, with $\sigma=(u,v,\mathit{orig})$,
$\sigma^{\prime}=(u,v,\mathit{orig}^{\prime})$, $u\in\Sigma^{*}$, and
$v\in\Gamma^{*}$, for which there exist $u^{\prime}\in{\Sigma^{\prime}}^{*}$
and $v^{\prime}\in{\Gamma^{\prime}}^{*}$ such that
* •
$\pi_{\Sigma}(u^{\prime})=u$ and $\pi_{\Gamma}(v^{\prime})=v$
* •
$u^{\prime}$ satisfies $\mathsf{ipar}$ and $v^{\prime}$ satisfies
$\mathsf{opar}$,
* •
$(u^{\prime},\mathit{orig}(x),\mathit{orig}^{\prime}(x))$ satisfies
$\mathsf{move}_{\tau}$ for all $\tau$-labeled output positions
$x\in\mathsf{dom}(v^{\prime})$, and
* •
$(u^{\prime},\mathit{orig}^{\prime}(x),\mathit{orig}^{\prime}(x+1))$ satisfies
$\mathsf{next}_{\tau,\tau^{\prime}}$ for all
$x,x+1\in\mathsf{dom}(v^{\prime})$ such that $x$ and $x+1$ have label $\tau$
and $\tau^{\prime}$, respectively.
###### Example 1
Consider the following resynchronization $\mathcal{R}$. A pair
$(\sigma,\sigma^{\prime})$ belongs to $\mathcal{R}$ if
$\sigma=(uv,uwv,\mathit{orig})$,
$\sigma^{\prime}=(uv,uwv,\mathit{orig}^{\prime})$, with $u,v,w\in\Sigma^{+}$.
The origins $\mathit{orig}$ and $\mathit{orig}^{\prime}$ are both the identity
over $u$ and $v$. The origin of every position of $w$ in $\sigma$ (hence a
source origin) is either the first or the last position of $v$. The origin of
every position of $w$ in $\sigma^{\prime}$ (a target origin) is the first
position of $v$.
This resynchronization is described by a regular resynchronizer that uses two
input parameters $I_{1},I_{2}$ to mark the last and the first positions of $v$
in the input, and one output parameter $O$ to mark the factor $w$ in the
output. The formula $\mathsf{move}_{\tau}(y,z)$ is either $(I_{1}(y)\vee
I_{2}(y))\wedge I_{2}(z)$ or $(y=z)$, depending on whether the type $\tau$
describes a position inside $w$ or a position outside $w$.
We now turn to describing some important restrictions on (regular)
resynchronizers. Let
$\mathcal{R}=(\overline{I},\overline{O},\mathsf{ipar},\mathsf{opar},(\mathsf{move}_{\tau}),(\mathsf{next}_{\tau,\tau^{\prime}}))$
be a resynchronizer.
* •
$\mathcal{R}$ is _$k$ -bounded_ (or just _bounded_) if for every annotated
input $u^{\prime}\in{\Sigma^{\prime}}^{*}$, every output type
$\tau\in\Gamma^{\prime}$, and every position $z$, there are at most $k$
positions $y$ such that $(u^{\prime},y,z)$ satisfies $\mathsf{move}_{\tau}$.
Recall that $y,z$ are input positions.
* •
$\mathcal{R}$ is _$T$ -preserving_ for a given transducer $T$, if every
$\sigma\in{[\\![T]\\!]}_{o}$ belongs to the domain of $\mathcal{R}$.
* •
$\mathcal{R}$ is _partially bijective_ if each $\mathsf{move}_{\tau}$ formula
defines a partial, bijective function from source origins to target origins.
Observe that this property implies that $\mathcal{R}$ is 1-bounded.
The boundedness restriction rules out resynchronizations such as the universal
one, that imposes no restriction on the change of origins. It is a decidable
restriction [7], and it guarantees that definability by two-way transducers is
effectively preserved under regular resynchronizations, modulo common guess.
More precisely, Theorem 16 in [7] shows that, given a bounded regular
resynchronizer $\mathcal{R}$ and a transducer $T$, one can construct a
transducer $T^{\prime}$ with common guess that is origin-equivalent to
$\mathcal{R}(T)$.
###### Example 1 (continued)
Consider again the regular resynchronizer $\mathcal{R}$ described in the
previous example. Note that $\mathcal{R}$ is $2$-bounded, since at most two
source origins are redirected to the same target origin. If we used an
additional output parameter to distinguish, among the positions of $w$, those
that have source origin in the first position of $v$ and those that have
source origin in the last position of $v$, we would get a $1$-bounded, regular
resynchronizer.
We state below two crucial properties of regular resynchronizers (the second
lemma is reminiscent of Lemma 11 from [21], which proves closure of bounded
resynchronizers with vacuous $\mathsf{next}_{\tau,\tau^{\prime}}$ relations).
###### Lemma 1 ()
Every bounded, regular resynchronizer is effectively equivalent to some
$1$-bounded, regular resynchronizer.
###### Lemma 2 ()
The class of bounded, regular resynchronizers is effectively closed under
composition.
### 3.2 Main result
Given a two-way transducer $T$ one can ask if it is origin-equivalent to some
one-way transducer. It was observed in [4] that this property holds if and
only if all synchronized pairs defined by $T$ are _order-preserving_ , namely,
for all $\sigma=(u,v,\mathit{orig})\in{[\\![T]\\!]}_{o}$ and all
$y,y^{\prime}\in\mathsf{dom}(v)$, with $y<y^{\prime}$, we have
$\mathit{orig}(y)\leq\mathit{orig}(y^{\prime})$. The decidability of the above
question should be contrasted to the analogous question in the classical
semantics: “is a given two-way transducer classically equivalent to some one-
way transducer?” The latter problem turns out to be decidable for functional
transducers [14, 3], but is undecidable for arbitrary two-way transducers [2].
Here we are interested in a different, more relaxed notion:
###### Definition 1
A transducer $T$ is called _one-way resynchronizable_ if there exists a
bounded, regular resynchronizer $\mathcal{R}$ that is $T$-preserving and such
that $\mathcal{R}(T)$ is order-preserving.
Note that if $T^{\prime}$ is an order-preserving transducer, then one can
construct rather easily a one-way transducer $T^{\prime\prime}$ such that
$T^{\prime}=_{o}T^{\prime\prime}$, by eliminating non-productive U-turns from
accepting runs.
Moreover, note that without the condition of being $T$-preserving every
transducer $T$ would be one-way resynchronizable, using the empty
resynchronization.
###### Example 2
Consider the transducer $T_{1}$ that moves the last letter of the input $wa$
to the front by a first left-to-right pass that outputs the last letter $a$,
followed by a right-to-left pass without output, and finally by a left-to-
right pass that produces the remaining $w$. Let $\mathcal{R}$ be the bounded
regular resynchronizer that redirects the origin of the last $a$ to the first
position. Assuming an output parameter $O$ with an interpretation constrained
by $\mathsf{opar}$ that marks the last position of the output, the formula
$\mathsf{move}_{(a,1)}(y,z)$ says that target origin $z$ (source origin $y$,
resp.) of the last $a$ is the first (last, resp.) position of the input. It is
easy to see that $\mathcal{R}(T_{1})$ is origin-equivalent to the one-way
transducer that on input $wa$, guesses $a$ and outputs $aw$. Thus, $T_{1}$ is
one-way resynchronizable. See also Figure 1.
###### Example 3
Consider the transducer $T_{2}$ that reads inputs of the form $u\\#v$ and
outputs $vu$ in the obvious way, by a first left-to-right pass that outputs
$v$, followed by a right-to-left pass, and a finally a left-to-right pass that
outputs $u$. Using the characterization with the notion of cross-width that we
introduce below, it can be shown that $T_{2}$ is not one-way resynchronizable.
In order to give a flavor of our results, we anticipate here the two main
theorems, before introducing the key technical concepts of cross-width and
inversion (these will be defined further below).
###### Theorem 3.1
For every bounded-visit transducer $T$, the following are equivalent:
* (1)
$T$ is one-way resynchronizable,
* (2)
the cross-width of $T$ is finite,
* (3)
no successful run of $T$ has inversions,
* (4)
there is a partially bijective, regular resynchronizer $\mathcal{R}$ that is
$T$-preserving and such that $\mathcal{R}(T)$ is order-preserving.
Moreover, condition (3) is decidable.
We will use Theorem 3.1 to show that one-way resynchronizability is decidable
for arbitrary two-way transducers (not just bounded-visit ones).
###### Theorem 3.2
It is decidable whether a given two-way transducer $T$ is one-way
resynchronizable.
Let us now introduce the first key concept, that of cross-width:
###### Definition 2 (cross-width)
Let $\sigma=(u,v,\mathit{orig})$ be a synchronized pair and let
$X_{1},X_{2}\subseteq\mathsf{dom}(v)$ be sets of output positions such that,
for all $x_{1}\in X_{1}$ and $x_{2}\in X_{2}$, $x_{1}<x_{2}$ and
$\mathit{orig}(x_{1})>\mathit{orig}(x_{2})$. We call such a pair
$(X_{1},X_{2})$ a _cross_ and define its _width_ as
$\min(|\mathit{orig}(X_{1})|,|\mathit{orig}(X_{2})|)$, where
$\mathit{orig}(X)=\\{\mathit{orig}(x)\mid x\in X\\}$ is the set of origins
corresponding to a set $X$ of output positions.
The _cross-width_ of a synchronized pair $\sigma$ is the maximal width of the
crosses in $\sigma$. A transducer has _bounded cross-width_ if for some
integer $k$, all synchronized pairs associated with successful runs of $T$
have cross-width at most $k$.
For instance, the transducer $T_{2}$ in Example 3 has unbounded cross-width.
In contrast, the transducer $T_{1}$ in Example 2 has cross-width one.
The other key notion of _inversion_ will be introduced formally in the next
section (page 3), as it requires a few technical definitions. The notion
however is very similar in spirit to that of cross, with the difference that a
single inversion is sufficient for witnessing a family of crosses with
arbitrarily large cross-width.
## 4 Proof overview for Theorem 3.1
This section provides an overview of the proof of Theorem 3.1, and introduces
the main ingredients.
We will use flows (a concept inspired from crossing sequences [22, 3] and
revised in Section 4.1) in order to derive the key notion of inversion.
Roughly speaking, an inversion in a run involves two loops that produce
outputs in an order that is reversed compared to the order on origins.
Inversions were also used in the characterization of one-way definability of
two-way transducers under the classical semantics [3]. There, they were used
for deriving some combinatorial properties of outputs. Here we are only
interested in detecting inversions, and this is a simple task.
Flows will also be used to associate factorization trees with runs (the
existence of factorization trees of bounded height was established by the
celebrated Simon’s factorization theorem [23]). We will use a structural
induction on these factorization trees and the assumption that there is no
inversion in every run to construct a regular resynchronization witnessing
one-way resynchronizability of the transducer at hand.
Another important ingredient underlying the main characterization is given by
the notion of dominant output interval (Section 4.2), which is used to
formalize the invariant of our inductive construction.
### 4.1 Flows and inversions
#### Intervals.
An _interval_ of a word is a set of consecutive positions in it. An interval
is often denoted by $I=[i,i^{\prime})$, with $i=\min(I)$ and
$i^{\prime}=\max(I)+1$. Given two intervals $I=[i,i^{\prime})$ and
$J=[j,j^{\prime})$, we write $I<J$ if $i^{\prime}\leq j$, and we say that
$I,J$ are adjacent if $i^{\prime}=j$. The union of two adjacent intervals
$I=[i,i^{\prime})$, $J=[j,j^{\prime})$, denoted $I\cdot J$, is the interval
$[i,j^{\prime})$ (if $I,J$ are not adjacent, then $I\cdot J$ is undefined).
#### Subruns.
Given a run $\rho$ of a transducer, a _subrun_ is a factor of $\rho$. Note
that a subrun of a two-way transducer may visit a position of the input
several times. For an input interval $I=[i,j)$ and a run $\rho$, we say that a
subrun $\rho^{\prime}$ of $\rho$ _spans over $I$_ if $i$ (resp. $j$) is the
smallest (resp. greatest) input position labeling some transition of
$\rho^{\prime}$. The left hand-side of the figure at page 4.1 gives an example
of an interval $I$ of an input word together with the subruns
$\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2},\beta_{3}$, $\gamma_{1}$
that span over it. Subruns spanning over an interval can be left-to-right,
left-to-left, right-to-left, or right-to-right depending on where the starting
and ending positions are w.r.t. the endpoints of the interval.
#### Flows.
Flows are used to summarize subruns of a two-way transducer that span over a
given interval. The definition below is essentially taken from [3], except for
replacing “functional” by “$K$-visit”. Formally, a _flow_ of a transducer $T$
is a graph with vertices divided into two groups, $\mathsf{L}$-vertices and
$\mathsf{R}$-vertices, labeled by states of $T$, and with directed edges also
divided into two groups, productive and non-productive edges. The graph
satisfies the following requirements. Edge sources are either an
$\mathsf{L}$-vertex labeled by a right-reading state, or an
$\mathsf{R}$-vertex labeled by a left-reading state, and symmetrically for
edge destinations; moreover, edges are of one of the following types:
$\mathsf{L}\mathsf{L}$, $\mathsf{L}\mathsf{R}$, $\mathsf{R}\mathsf{L}$,
$\mathsf{R}\mathsf{R}$. Second, each node is the endpoint of exactly one edge.
Finally, $\mathsf{L}$ ($\mathsf{R}$, resp.) vertices are totally ordered, in
such a way that for every $\mathsf{L}\mathsf{L}$ ($\mathsf{R}\mathsf{R}$,
resp.) edge $(v,v^{\prime})$, we have $v<v^{\prime}$. We will only consider
flows of $K$-visiting transducers, so flows with at most $2K$ vertices. For
example, the flow in the left-hand side of the figure at page 4.1 has six
$\mathsf{L}$-vertices on the left, and six $\mathsf{R}$-vertices on the right.
The edges $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ are $\mathsf{L}\mathsf{L}$,
$\mathsf{L}\mathsf{R}$, and $\mathsf{R}\mathsf{R}$, respectively.
Given a run $\rho$ of $T$ and an interval $I=[i,i^{\prime})$ on the input,
_the flow of $\rho$ on $I$_, denoted $\mathit{flow}_{\rho}(I)$, is obtained by
identifying every configuration at position $i$ (resp. $i^{\prime}$) with an
$\mathsf{L}$ (resp. $\mathsf{R}$) vertex, labeled by the state of the
configuration, and every subrun spanning over $I$ with an edge connecting the
appropriate vertices (this subrun is called the _witnessing subrun_ of the
edge of the flow). An edge is said to be _productive_ if its witnessing subrun
produces non-empty output.
#### Flow monoid.
The composition of two flows $F$ and $G$ is defined when the
$\mathsf{R}$-vertices of $F$ induce the same sequence of labels as the
$\mathsf{L}$-vertices of $G$. In this case, the composition results in the
flow $F\cdot G$ that has as vertices the $\mathsf{L}$-vertices of $F$ and the
$\mathsf{R}$-vertices of $G$, and for edges the directed paths in the graph
obtained by glueing the $\mathsf{R}$-vertices of $F$ with the
$\mathsf{L}$-vertices of $G$ so that states are matched. Productiveness of
edges is inherited by paths, implying that an edge of $F\cdot G$ is productive
if and only if the corresponding path contains at least one edge (from $F$ or
$G$) that is productive. When the composition is undefined, we simply write
$F\cdot G=\bot$. The above definitions naturally give rise to a _flow monoid_
associated with the transducer $T$, where elements are the flows of $T$,
extended with a dummy element $\bot$, and the product operation is given by
the composition of flows, with the convention that $\bot$ is absorbing. It is
easy to verify that for any two adjacent intervals $I<J$ of a run $\rho$,
$\mathit{flow}_{\rho}(I)\cdot\mathit{flow}_{\rho}(J)=\mathit{flow}_{\rho}(I\cdot
J)$. We denote by $M_{T}$ the _flow monoid_ of a $K$-visiting transducer $T$.
Let us estimate the size of $M_{T}$. If $Q$ is the set of states of $T$, there
are at most $|Q|^{2K}$ possible sequences of $\mathsf{L}$ and
$\mathsf{R}$-vertices; and the number of edges (marked as productive or not)
is bounded by ${2K\choose K}\cdot(2K)^{K}\cdot 2^{K}\leq(2K+1)^{2K}$.
Including the dummy element $\bot$ in the flow monoid, we get
$|M_{T}|\leq(|Q|\cdot(2K+1))^{2K}+1=:\mathbf{M}$.
#### Loops.
A loop of a run $\rho$ over input $w$ is an interval $I=[i,j)$ with a flow
$F=$ $\mathit{flow}_{\rho}(I)$ such that $F\cdot F=F$ (call $F$ _idempotent_).
The run $\rho$ can be pumped on a loop $I=[i,j)$ as expected: given $n>0$, we
let $\mathit{pump}^{n}_{I}(\rho)$ be the run obtained from $\rho$ by glueing
the subruns that span over the intervals $[1,i)$ and $[j,|w|+1)$ with $n$
copies of the subruns spanning over $I$ (see figure to the right). X
$I$$\boldsymbol{\alpha_{1}}$$~{}\boldsymbol{\alpha_{2}}$$\boldsymbol{\alpha_{3}}$$\boldsymbol{\beta_{1}}$$\boldsymbol{\beta_{2}}$$\boldsymbol{\beta_{3}}$$~{}\boldsymbol{\gamma_{1}}$01234560123456$I$$2$
more copies of
$I$$\boldsymbol{\alpha_{1}}$$~{}\boldsymbol{\alpha_{2}}$$\boldsymbol{\alpha_{1}}$$\boldsymbol{\alpha_{3}}$$~{}\boldsymbol{\alpha_{2}}$$\boldsymbol{\alpha_{1}}$$\boldsymbol{\alpha_{3}}$$~{}\boldsymbol{\alpha_{2}}$$\boldsymbol{\alpha_{3}}$$\boldsymbol{\beta_{1}}$$\boldsymbol{\beta_{2}}$$\boldsymbol{\beta_{1}}$$\boldsymbol{\beta_{3}}$$\boldsymbol{\beta_{2}}$$\boldsymbol{\beta_{1}}$$\boldsymbol{\beta_{3}}$$\boldsymbol{\beta_{2}}$$\boldsymbol{\beta_{3}}$$~{}\boldsymbol{\gamma_{1}}$$~{}\boldsymbol{\gamma_{1}}$$~{}\boldsymbol{\gamma_{1}}$
The lemma below shows that the occurrence order relative to subruns witnessing
$\mathsf{L}\mathsf{R}$ or $\mathsf{R}\mathsf{L}$ edges of a loop (called
_straight edges_ , for short) is preserved when pumping the loop. This
seemingly straightforward lemma is needed for detecting inversions and its
proof is surprisingly non-trivial. For example, the external edge connecting
the two $\mathsf{L}$-vertices $1,2$ in the figure above appears before edge
$\alpha_{2}$, and also before every copy of $\alpha_{2}$ in the run where loop
$I$ is pumped.
###### Lemma 3 ()
Let $\rho$ be a run of $T$ on $u$, let $J<I<K$ be a partition of the domain of
$u$ into intervals, with $I$ loop of $\rho$, and let
$F=\mathit{flow}_{\rho}(J)$, $E=\mathit{flow}_{\rho}(I)$, and
$G=\mathit{flow}_{\rho}(K)$ be the corresponding flows. Consider an arbitrary
edge $f$ of either $F$ or $G$, and a straight edge $e$ of the idempotent flow
$E$. Let $\rho_{f}$ and $\rho_{e}$ be the witnessing subruns of $f$ and $e$,
respectively. Then the occurrence order of $\rho_{f}$ and $\rho_{e}$ in $\rho$
is the same as the occurrence order of $\rho_{f}$ and any copy of $\rho_{e}$
in $\mathit{pump}^{n}_{I}(\rho)$.
We can now formalize the key notion of inversion:
###### Definition 3 (inversion)
An _inversion_ of $\rho$ is a tuple $(I,e,I^{\prime},e^{\prime})$ such that •
$I,I^{\prime}$ are loops of $\rho$ and $I<I^{\prime}$, • $e,e^{\prime}$ are
productive straight edges in $\mathit{flow}_{\rho}(I)$ and
$\mathit{flow}_{\rho}(I^{\prime})$ respectively, • the subrun witnessing
$e^{\prime}$ precedes the subrun witnessing $e$ in the run order (see the
figure to the right). X
$I$$\boldsymbol{\alpha_{1}}$$~{}\boldsymbol{\alpha_{2}}$$\boldsymbol{\alpha_{3}}$$\boldsymbol{\beta_{1}}$$\boldsymbol{\beta_{2}}$$\boldsymbol{\beta_{3}}$$~{}\boldsymbol{\gamma_{1}}$$I$$I^{\prime}$$\boldsymbol{e^{\prime}}\
\ \ $$\ \ \ \boldsymbol{e}$
### 4.2 Dominant output intervals
In this section we identify some particular intervals of the output that play
an important role in the inductive construction of the resynchronizer for a
one-way resynchronizable transducer.
Given $n\in\mathbb{N}$, we say that a set $B$ of output positions is _$n$
-large_ if $|\mathit{orig}(B)|>n$; otherwise, we say that $B$ is _$n$ -small_.
Recall that here we work with a $K$-visiting transducer $T$, for some constant
$K$, and that $\mathbf{M}=(|Q|\cdot(2K+1))^{2K}+1$ is an upper bound to the
size of the flow monoid $M_{T}$. We will extensively use the derived constant
$\mathbf{C}=\mathbf{M}^{2K}$ to distinguish between large and small sets of
output positions. The intuition behind this constant is that any set of output
positions that is $\mathbf{C}$-large must traverse a loop of $\rho$. This is
formalized in the lemma below. The proof uses algebraic properties of the flow
monoid $M_{T}$ [20] (see also Theorem 7.2 in [3], which proves a similar
result, but with a larger constant derived from Simon’s factorization
theorem):
###### Lemma 4 ()
Let $I$ be an input interval and $B$ a set of output positions with origins
inside $I$. If $B$ is $\mathbf{C}$-large, then there is a loop $J\subseteq I$
of $\rho$ such that $\mathit{flow}_{\rho}(J)$ contains a productive straight
edge witnessed by a subrun that intersects $B$ (in particular,
$\mathit{out}(J)\cap B\neq\emptyset$).
We need some more notations for outputs. Given an input interval $I$ we denote
by $\mathit{out}_{\rho}(I)$ the set of output positions whose origins belong
to $I$ (note that this might not be an output interval). An _output block_ of
$I$ is a maximal interval contained in $\mathit{out}_{\rho}(I)$.
The _dominant output interval_ of $I$, denoted $\mathit{bigout}_{\rho}(I)$, is
the smallest output interval that contains all $\mathbf{C}$-large output
blocks of $I$. In particular, $\mathit{bigout}_{\rho}(I)$ either is empty or
begins with the first $\mathbf{C}$-large output block of $I$ and ends with the
last $\mathbf{C}$-large outblock block of $I$. We will often omit the
subscript $\rho$ from the notations $\mathit{flow}_{\rho}(I)$,
$\mathit{out}_{\rho}(I),\mathit{bigout}_{\rho}(I)$, etc., when no confusion
arises.
We now fix a successful run $\rho$ of the $K$-visiting transducer $T$. The
rest of the section presents some technical lemmas that will be used in the
inductive constructions for the proof of the main theorem. _In the lemmas
below, we assume that all successful runs of $T$ (in particular, $\rho$) avoid
inversions._
###### Lemma 5 ()
Let $I_{1}<I_{2}$ be two input intervals and $B_{1},B_{2}$ output blocks of
$I_{1}$, $I_{2}$, respectively. If both $B_{1},B_{2}$ are $\mathbf{C}$-large,
then $B_{1}<B_{2}$.
###### Proof (sketch)
If the claim would not hold, then Lemma 4 would provide some loops
$J_{1}\subseteq I_{1}$ and $J_{2}\subseteq I_{2}$, together with some
productive edges in them, witnessing an inversion. ∎
###### Lemma 6 ()
Let $I=I_{1}\cdot I_{2}$, $B=\mathit{bigout}(I)$, and
$B_{i}=\mathit{bigout}(I_{i})$ for $i=1,2$. Then $B\setminus(B_{1}\cup B_{2})$
is $4K\mathbf{C}$-small.
###### Proof (sketch)
By Lemma 5, $B_{1}<B_{2}$. Moreover, all $\mathbf{C}$-large output blocks of
$I_{1}$ or $I_{2}$ are also $\mathbf{C}$-large output blocks of $I$, so $B$
contains both $B_{1}$ and $B_{2}$. Suppose, by way of contradiction, that
$B\setminus(B_{1}\cup B_{2})$ is $4K\mathbf{C}$-large. This means that there
is a $2K\mathbf{C}$-large set $S\subseteq B\setminus(B_{1}\cup B_{2})$ with
origins entirely to the left of $I_{2}$, or entirely to the right of $I_{1}$.
Suppose, w.l.o.g., that the former case holds, and decompose $S$ as a union of
maximal output blocks $B^{\prime}_{1},B^{\prime}_{2},\dots,B^{\prime}_{n}$
with origins either entirely inside $I_{1}$, or entirely outside. Since $S\cap
B_{1}=\emptyset$, every block $B^{\prime}_{i}$ with origins inside $I_{1}$ is
$\mathbf{C}$-small. Similarly, by Lemma 0.C.1 in Appendix 0.C, every block
$B^{\prime}_{i}$ with origins outside $I_{1}$ is $\mathbf{C}$-small too.
Moreover, since $\rho$ is $K$-visiting, we get $n\leq 2K$. Altogether, this
contradicts the assumption that $S$ is $2K\mathbf{C}$-large. ∎
###### Lemma 7 ()
Let $I=I_{1}\cdot I_{2}\cdots I_{n}$, such that $I$ is a loop and
$\mathit{flow}(I)=\mathit{flow}(I_{k})$ for all $k$. Then $\mathit{bigout}(I)$
can be decomposed as $B_{1}\cdot J_{1}\cdot B_{2}\cdot J_{2}\cdot\dots\cdot
J_{n-1}\cdot B_{n}$, where
1. 1.
for all $1\leq k\leq n$, $B_{k}=\mathit{bigout}(I_{k})$ (with $B_{k}$ possibly
empty);
2. 2.
for all $1\leq k<n$, the positions in $J_{k}$ have origins inside $I_{k}\cup
I_{k+1}$ and $J_{k}$ is $2K\mathbf{C}$-small.
###### Proof (sketch)
The proof idea is similar to the previous lemma. First, using properties of
idempotent flows, one shows that all output positions strictly between $B_{k}$
and $B_{k+1}$, for any $k=1,\dots,n-1$, have origin in $I_{k}\cup I_{k+1}$.
Then, one observes that every output block of $I_{k}$ disjoint from $B_{k}$ is
$\mathbf{C}$-small, and since $T$ is $K$-visiting there are at most $K$ such
blocks. This shows that every output interval $J_{k}$ between $B_{k}$ and
$B_{k+1}$ is $2K\mathbf{C}$-small. For an illustration see the figure to the
right. The $\mathbf{C}$-large blocks in $I_{1}$ are shown in red; in blue
those for $I_{2}$, in purple those for $I_{3}$. So $\mathit{bigout}(I_{1})$ is
the entire output between the two red dots, $\mathit{bigout}(I_{2})$ between
the two blue dots, and $\mathit{bigout}(I_{3})$ between the purple dots. All
three blocks are non-empty, and $\mathit{bigout}(I_{1}\cdot I_{2}\cdot I_{3})$
goes from the first red to the second purple dot. Black non-dashed arrows
stand for $\mathbf{C}$-small blocks. ∎ X
$I$$\boldsymbol{\alpha_{1}}$$~{}\boldsymbol{\alpha_{2}}$$\boldsymbol{\alpha_{3}}$$\boldsymbol{\beta_{1}}$$\boldsymbol{\beta_{2}}$$\boldsymbol{\beta_{3}}$$~{}\boldsymbol{\gamma_{1}}$$I_{1}$$I_{2}$$I_{3}$
## 5 Proof of Theorem 3.1
This section is devoted to proving the characterization of one-way
resynchronizability in the bounded-visit case. We will use the notion of
_bounded-traversal_ from [21], that was shown to characterize the class of
bounded regular resynchronizers, in as much as bounded-delay characterizes
rational resynchronizers [15].
###### Definition 4 (traversal [21])
Let $\sigma=(u,v,\mathit{orig})$ and
$\sigma^{\prime}=(u,v,\mathit{orig}^{\prime})$ be two synchronized pairs with
the same input and output words.
Given two input positions $y,y^{\prime}\in\mathsf{dom}(u)$, we say that _$y$
traverses $y^{\prime}$_ if there is a pair $(y,z)$ of source and target
origins associated with the same output position such that $y^{\prime}$ is
between $y$ and $z$, with $y^{\prime}\neq z$ and possibly $y^{\prime}=y$. More
precisely:
* •
$(y,y^{\prime})$ is a _left-to-right traversal_ if $y\leq y^{\prime}$ and for
some output position $x$, $\mathit{orig}(x)=y$ and
$z=\mathit{orig}^{\prime}(x)>y^{\prime}$;
* •
$(y,y^{\prime})$ is a _right-to-left traversal_ if $y\geq y^{\prime}$ and for
some output position $x$, $\mathit{orig}(x)=y$ and
$z=\mathit{orig}^{\prime}(x)<y^{\prime}$.
A pair $(\sigma,\sigma^{\prime})$ of synchronized pairs with input $u$ and
output $v$ is said to have _$k$ -bounded traversal_, with $k\in\mathbb{N}$, if
every $y^{\prime}\in\mathsf{dom}(u)$ is traversed by at most $k$ distinct
positions of $\mathsf{dom}(u)$.
A resynchronizer $\mathcal{R}$ has _bounded traversal_ if there is some
$k\in\mathbb{N}$ such that every $(\sigma,\sigma^{\prime})\in\mathcal{R}$ has
$k$-bounded traversal.
###### Lemma 8 ([21])
A regular resynchronizer is bounded if and only if it has bounded traversal.
###### Proof (of Theorem 3.1)
First of all, observe that the implication $4\rightarrow 1$ is
straightforward. To prove the implication $1\rightarrow 2$, assume that there
is a $k$-bounded, regular resynchronizer $\mathcal{R}$ that is $T$-preserving
and such that $\mathcal{R}(T)$ is order-preserving. Lemma 8 implies that
$\mathcal{R}$ has $t$-bounded traversal, for some constant $t$. We head
towards proving that $T$ has cross-width bounded by $t+k$. Consider two
synchronized pairs $\sigma=(u,v,\mathit{orig})$ and
$\sigma^{\prime}=(u,v,\mathit{orig}^{\prime})$ such that
$\sigma\in{[\\![T]\\!]}_{o}$ and $(\sigma,\sigma^{\prime})\in\mathcal{R}$, and
consider a cross $(X_{1},X_{2})$ of $\sigma$. We claim that
$|\mathit{orig}(X_{1})|$ or $|\mathit{orig}(X_{2})|$ is at most $t+k$. Let
$x_{1}=\min(\mathit{orig}(X_{1}))$,
$x^{\prime}_{1}=\max(\mathit{orig}^{\prime}(X_{1}))$,
$x_{2}=\max(\mathit{orig}(X_{1}))$, and
$x^{\prime}_{2}=\min(\mathit{orig}^{\prime}(X_{2}))$. Since $(X_{1},X_{2})$ is
a cross, we have $x_{1}>x_{2}$, and since $\sigma^{\prime}$ is order-
preserving, we have $x^{\prime}_{1}\leq x^{\prime}_{2}$. Now, if
$x^{\prime}_{1}>x_{2}$, then at least $|\mathit{orig}(X_{2})|-k$ input
positions from $X_{2}$ traverse $x^{\prime}_{1}$ to the right (the $-k$ term
is due to the fact that at most $k$ input positions can be resynchronized to
$x^{\prime}_{1}$). Symmetrically, if $x^{\prime}_{1}\leq x_{2}$, then at least
$|\mathit{orig}(X_{1})|-k$ input positions from $X_{1}$ traverse $x_{2}$ to
the left (the $-k$ term accounts for the case where some positions are
resynchronized to $x^{\prime}_{1}$ and $x^{\prime}_{1}=x_{2}$). This implies
$\min(|\mathit{orig}(X_{1})|,|\mathit{orig}(X_{2})|)\leq t+k$, as claimed.
The remaining implications rely on the assumption that $T$ is bounded-visit.
The implication $2\rightarrow 3$ is shown by contraposition: one considers a
successful run $\rho$ with an inversion, and shows that crosses of arbitrary
width emerge after pumping the loops of the inversion (here Lemma 3 is
crucial).
The proof of $3\rightarrow 4$ is more involved, we only sketch it here.
Assuming that no successful run of $T$ has inversions we build a partially
bijective, regular resynchronizer $\mathcal{R}$ that is $T$-preserving and
$\mathcal{R}(T)$ is order-preserving. The resynchronizer $\mathcal{R}$ uses
some parameters to guess a successful run $\rho$ of $T$ on $u$ and a
factorization tree of bounded height for $\rho$. Formally, a _factorization
tree_ for a sequence $\alpha$ of monoid elements (e.g. the flows
$\mathit{flow}_{\rho}([y,y])$ for all input positions $y$) is an ordered,
unranked tree whose yield is the sequence $\alpha$. The leaves of the
factorization tree are labeled with the elements of $\alpha$. All other nodes
have at least two children and are labeled by the monoid product of the child
labels (in our case by the flows of $\rho$ induced by the covered factors in
the input). In addition, if a node has more than two children, then all its
children must have the same label, representing an idempotent element of the
monoid. By Simon’s factorization theorem [23], every sequence of monoid
elements has some factorization tree of height at most linear in the size of
the monoid (in our case, at most $3|M_{T}|$, see e.g. [8]).
_Parameters._ We use input parameters to encode the successful run $\rho$ and
a factorization tree for $\rho$ of height at most $H=3|M_{T}|$. These
parameters specify, for each input interval corresponding to a subtree, the
start and end positions of the interval and the label of the root of the
subtree. Correctness of these annotations can be enforced by an MSO sentence
$\mathsf{ipar}$. The run and the factorization tree also need to be encoded
over the output, using output parameters. More precisely, given a level in the
tree and an output position, we need to be able to determine the flow and the
productive edge that generated that position. The technical details for
checking correctness of the output annotation using the formulas
$\mathsf{opar}$, $\mathsf{move}_{\tau}$ and
$\mathsf{next}_{\tau,\tau^{\prime}}$ can be found in Appendix 0.D.
_Moving origins._ For each level $\ell$ of the factorization tree, a partial
resynchronization relation $\mathcal{R}_{\ell}$ is defined. The relation is
partial in the sense that some output positions may not have a source-target
origin pair defined at a given level. But once a source-target pair is defined
for some output position at a given level, it remains defined for all higher
levels.
In the following we write $\mathit{bigout}(p)$ for the dominant output
interval associated with the input interval $I(p)$ corresponding to a node $p$
in the tree. For every level $\ell$ of the factorization tree, the
resynchronizer $\mathcal{R}_{\ell}$ will be a partial function from source
origins to target origins, and will satisfy the following:
* •
the set of output positions for which $\mathcal{R}_{\ell}$ defines target
origins is the union of the intervals $\mathit{bigout}(p)$ for all nodes $p$
at level $\ell$;
* •
$\mathcal{R}_{\ell}$ only moves origins within the same interval at level
$\ell$, that is, $\mathcal{R}_{\ell}$ defines only pairs $(y,z)$ of source-
target origins such that $y,z\in I(p)$ for some node $p$ at level $\ell$;
* •
the target origins defined by $\mathcal{R}_{\ell}$ are order-preserving within
every interval at level $\ell$, that is, for all output positions
$x<x^{\prime}$, if $\mathcal{R}_{\ell}$ defines the target origins of
$x,x^{\prime}$ to be $z,z^{\prime}$, respectively, and if $z,z^{\prime}\in
I(p)$ for some node $p$ at level $\ell$, then $z\leq z^{\prime}$;
* •
$\mathcal{R}_{\ell}$ is $\ell\cdot 4K\mathbf{C}$-bounded, namely, there are at
most $\ell\cdot 4K\mathbf{C}$ distinct source origins that are moved by
$\mathcal{R}_{\ell}$ to the same target origin.
The construction of $\mathcal{R}_{\ell}$ is by induction on $\ell$. For a
binary node $p$ at level $\ell$ with children $p_{1},p_{2}$, the
resynchronizer $\mathcal{R}_{\ell}$ inherits the source-origin pairs from
level $\ell-1$ for output positions that belong to
$\mathit{bigout}(p_{1})\cup\mathit{bigout}(p_{2})$. Note that
$\mathit{bigout}(p_{1})<\mathit{bigout}(p_{2})$ by Lemma 5, so
$\mathcal{R}_{\ell}$ is order-preserving inside
$\mathit{bigout}(p_{1})\cup\mathit{bigout}(p_{2})$. Output positions inside
$\mathit{bigout}(p)\setminus(\mathit{bigout}(p_{1})\cup\mathit{bigout}(p_{2}))$
are moved in an order-preserving manner to one of the extremities of $I(p)$,
or to the last position of $I(p_{1})$. Boundedness of $\mathcal{R}_{\ell}$ is
guaranteed by Lemma 6.
The case where $p$ is an idempotent node at level $\ell$ with children
$p_{1},p_{2},\dots,p_{n}$ follows a similar approach. For brevity, let
$I_{i}=I(p_{i})$ and $B_{i}=\mathit{bigout}(p_{i})$, and observe that, by
Lemma 5, $B_{1}<B_{2}<\dots<B_{n}$. Lemma 7 provides a decomposition of
$\mathit{bigout}(p)$ as $B_{1}\cdot J_{1}\cdot B_{2}\cdot J_{2}\cdot\dots\cdot
J_{n-1}\cdot B_{n}$, for some $2K\mathbf{C}$-small output intervals $J_{k}$
with origins inside $I_{k}\cup I_{k+1}$, for $k=1,\dots,n-1$. As before, the
resynchronizer $\mathcal{R}_{\ell}$ behaves exactly as $\mathcal{R}_{\ell-1}$
for the output positions inside the $B_{k}$’s. For any other output position,
say $x\in J_{k}$, the resynchronizer $\mathcal{R}_{\ell}$ will move the origin
either to the last position of $I_{k}$ or to the first position of $I_{k+1}$,
depending on whether the source origin of $x$ belongs to $I_{k}$ or $I_{k+1}$.
∎
## 6 Proof overview of Theorem 3.2
The main obstacle towards dropping the bounded-visit restriction from Theorem
3.1, while maintaining the effectiveness of the characterization, is the lack
of a bound on the number of flows. Indeed, for a transducer $T$ that is not
necessarily bounded-visit, there is no bound on the number of flows that
encode successful runs of $T$, and thus the proofs of the implications
$2\rightarrow 3\rightarrow 4$ are not applicable anymore. However, the proofs
of the implications $1\rightarrow 2$ and $4\rightarrow 1$ remain valid, even
for a transducer $T$ that is not bounded-visit.
The idea for proving Theorem 3.2 is to transform $T$ into an equivalent
bounded-visit transducer $\mathit{low}(T)$, so that the property of one-way
resynchronizability is preserved. More precisely, given a two-way transducer
$T$, we construct:
1. 1.
a bounded-visit transducer $\mathit{low}(T)$ that is classically equivalent to
$T$,
2. 2.
a $1$-bounded, regular resynchronizer $\mathcal{R}$ that is $T$-preserving and
such that $\mathcal{R}(T)=_{o}\mathit{low}(T)$.
We can apply our characterization of one-way resynchronizability in the
bounded-visit case to the transducer $\mathit{low}(T)$. If $\mathit{low}(T)$
is one-way resynchronizable, then by Theorem 3.1 we obtain another partially
bijective, regular resynchronizer $\mathcal{R}^{\prime}$ that is
$\mathit{low}(T)$-preserving and such that
$\mathcal{R}^{\prime}(\mathit{low}(T)))$ is order-preserving. Thanks to Lemma
2, the resynchronizers $\mathcal{R}$ and $\mathcal{R}^{\prime}$ can be
composed, so we conclude that the original transducer $T$ is one-way
resynchronizable. Otherwise, if $\mathit{low}(T)$ is not one-way
resynchronizable, we show that neither is $T$. This is precisely shown in the
lemma below.
###### Lemma 9 ()
For all transducers $T,T^{\prime}$, with $T^{\prime}$ bounded-visit, and for
every partially bijective, regular resynchronizer $\mathcal{R}$ that is
$T$-preserving and such that $\mathcal{R}(T)=_{o}T^{\prime}$, $T$ is one-way
resynchronizable if and only if $T^{\prime}$ is one-way resynchronizable.
There are however some challenges in the approach described above. First, as
$T$ may output arbitrarily many symbols with origin in the same input
position, and $\mathit{low}(T)$ is bounded-visit, we need $\mathit{low}(T)$ to
be able to produce arbitrarily long outputs within a single transition. For
this reason, we allow $\mathit{low}(T)$ to be a transducer with _regular
outputs_. The transition relation of such a transducer consists of finitely
many tuples of the form $(q,a,L,q^{\prime})$, with $q,q^{\prime}\in Q$,
$a\in\Sigma$, and $L\subseteq\Gamma^{*}$ a regular language over the output
alphabet. The semantics of a transition rule $(q,a,L,q^{\prime})$ is that,
upon reading $a$, the transducer can switch from state $q$ to state
$q^{\prime}$, and move its head accordingly, while outputting any word from
$L$. We also need to use transducers with common guess. Both extensions,
regular outputs and common guess, already appeared in prior works (cf. [5,
7]), and the proof of Theorem 3.1 in the bounded-visit case can be easily
adapted to these features.
There is still another problem: we cannot always expect that there exists a
bounded-visit transducer $\mathit{low}(T)$ classically equivalent to $T$.
Consider, for instance, the transducer that performs several passes on the
input, and on each left-to-right pass, at an arbitrary input position, it
copies as output the letter under its head. It is easy to see that the Parikh
image of the output is an exact multiple of the Parikh image of the input, and
standard pumping arguments show that no bounded-visit transducer can realize
such a relation.
A solution to this second problem is as follows. Before trying to construct
$\mathit{low}(T)$, we test whether $T$ satisfies the following condition on
vertical loops (these are runs starting and ending at the same position and at
the same state). There should exist some $K$ such that $T$ is _$K$ -sparse_,
meaning that the number of different origins of outputs generated inside some
vertical loop is at most $K$. If this condition is not met, then we show that
$T$ has unbounded cross-width, and hence, by the implication $1\rightarrow 2$
of Theorem 3.1, $T$ is not one-way resynchronizable. Otherwise, if the
condition holds, then we show that a bounded-visit transducer
$\mathit{low}(T)$ equivalent to $T$ can indeed be constructed.
## 7 Complexity
We discuss the effectiveness and complexity of our characterization. For a
$k$-visit transducer $T$, the effectiveness of the characterization relies on
detecting inversions in successful runs of $T$. It is not difficult to see
that this can be decided in space that is polynomial in the size of $T$ and
the bound $k$. We can also show that one-way resynchronizability is Pspace-
hard. For this we recall that the emptiness problem for two-way finite
automata is Pspace-complete. Let $A$ be a two-way automaton accepting some
language $L$, and let $\Sigma$ be a binary alphabet disjoint from that of $L$.
The function $\\{(w\cdot a_{1}\dots a_{n},a_{n}\dots a_{1})\mid w\in
L,a_{1}\dots a_{n}\in\Sigma^{*},n\geq 0\\}$ can be realized by a two-way
transducer $T$ of size polynomial in $|A|$, and $T$ is one-way
resynchronizable if and only if $L$ is empty.
In the unrestricted case, we showed that one-way resynchronizability is
decidable (Theorem 3.2). We briefly outline the complexity of the decision
procedure:
1. 1.
First one checks that $T$ is $K$-sparse for some $K$. To do this, we construct
from $T$ the regular language $L$ of all inputs with some positions marked
that correspond to origins produced within the same vertical loop. Bounded
sparsity is equivalent to having a uniform bound on the number of marked
positions in every input from $L$. Standard techniques for two-way automata
allow to decide this in space that is polynomial in the size of $T$. Moreover,
this also gives us a computable exponential bound to the largest constant $K$
for which $T$ can be $K$-sparse.
2. 2.
Next, we construct from the $K$-sparse transducer $T$ a bounded-visit
transducer $T^{\prime}$ that is classically equivalent to $T$ and has
exponential size.
3. 3.
Finally, we decide one-way resynchronizability of $T^{\prime}$ by detecting
inversions in successful runs of $T^{\prime}$ (Theorem 3.1).
Summing up, one can decide one-way resynchronizability of unrestricted two-way
transducers in exponential space. It is open if this bound is optimal. We also
do not have any interesting bound on the size of the resynchronizer that
witnesses one-way resynchronizability, both in the bounded-visit case and in
the unrestricted case. Similarly, we lack upper and lower bounds on the size
of the resynchronized one-way transducers, when these exist.
## 8 Conclusions
As the main contribution of this paper, we provided a characterization for the
subclass of two-way transducers that are one-way resynchronizable, namely,
that can be transformed by some bounded, regular resynchronizer, into an
origin-equivalent one-way transducer.
There are similar definability problems that emerge in the origin semantics.
For instance, one could ask whether a given two-way transducer can be
resynchronized, through some bounded, regular resynchronization, to a relation
that is origin-equivalent to a first-order transduction. This can be seen as a
relaxation of the first-order definability problem in the origin semantics,
namely, the problem of telling whether a two-way transducer is origin-
equivalent to some first-order transduction, shown decidable in [4]. It is
worth contrasting the latter problem with the challenging open problem whether
a given transduction is equivalent to a first-order transduction in the
classical setting.
#### Acknowledgments.
We thank the FoSSaCS reviewers for their constructive and useful comments.
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Appendix
## Appendix 0.A Proofs from Section 3.1
See 1
###### Proof
Let
$R=(\overline{I},\overline{O},\mathsf{ipar},\mathsf{opar},(\mathsf{move}_{\tau})_{\tau},(\mathsf{next}_{\tau,\tau^{\prime}})_{\tau,\tau^{\prime}})$
be a $k$-bounded, regular resynchronizer. Let $\hat{u}$ and $\hat{v}$ be a
pair of annotated input and output satisfying $\mathsf{ipar}$ and
$\mathsf{opar}$ respectively. To construct an equivalent $1$-bounded regular
resynchronizer $R^{\prime}$ we introduce additional output parameters.
Specifically, each output position will be annotated with an output type
$\tau$ from $R$ and an additional index in $\\{1,\dots,k\\}$. The intended
meaning of the index is as follows: if $(y,z)$ is the source/target origin
pair associated with an output position labeled by $(\tau,i)$,
$i\in\\{1,\dots,k\\}$, then then there are exactly $(i-1)$ positions
$y^{\prime}<y$ such that $(\hat{u},y^{\prime},z)\models\mathsf{move}_{\tau}$.
Note that this indexing depends on the choice of the target origin $z$.
Therefore, different indexing are possible for different choice of the target
origin $z$.
Based on the resynchronizer $R$, we define the new resynchronizer as
$R^{\prime}=(\overline{I},\overline{O}^{\prime},\mathsf{ipar},\mathsf{opar}^{\prime},(\mathsf{move}^{\prime}_{(\tau,i)})_{\tau,i},(\mathsf{next}_{(\tau,i),(\tau^{\prime},i^{\prime})})_{\tau,i,\tau^{\prime},i^{\prime}})$,
where
* •
$\overline{O}^{\prime}=\overline{O}\uplus\\{O^{\prime}_{1},\dots,O^{\prime}_{k}\\}$
consists of the old output parameters $\overline{O}$ of $R$ plus some new
parameters $O^{\prime}_{1},\dots,O^{\prime}_{k}$ for representing indices in
$\\{1,\dots,k\\}$;
* •
$\mathsf{opar}^{\prime}$ defines language of all output annotations whose
projections over $\Gamma^{\prime}$ (the output alphabet extended with the
parameters of $R$) satisfy $\mathsf{opar}$ and each position is marked by
exactly one index;
* •
given a type $\tau^{\prime}$ that encodes a type $\tau$ of $R$ and an index
$i\in\\{1,\dots,k\\}$, $\mathsf{move}^{\prime}_{\tau^{\prime}}(y,z)$ states
that $y$ is the $i$-th position $y^{\prime}$ satisfying
$\mathsf{move}_{\tau}(y^{\prime},z)$; This property can be expressed by the
MSO-formula
$\displaystyle\exists~{}y_{1}<\dots<y_{i}=y~{}\bigwedge\nolimits_{j}\mathsf{move}_{\tau}(y_{j},z)$
$\displaystyle~{}\wedge~{}\forall y^{\prime}\leq
y~{}\big{(}\mathsf{move}_{\tau}(y^{\prime},z)\rightarrow\bigvee\nolimits_{j}y^{\prime}=y_{j}\big{)};$
* •
$\mathsf{next}^{\prime}_{(\tau,i),(\tau^{\prime},i^{\prime})}(z,z^{\prime})$
enforces the same property as
$\mathsf{next}_{\tau,\tau^{\prime}}(z,z^{\prime})$.
The resynchronizer $R^{\prime}$ is $1$-bounded by definition of
$\mathsf{move}^{\prime}_{(\tau,i)}$. If for positions $y<y^{\prime}$,
$(\hat{u},y,z)\models\mathsf{move}^{\prime}_{(\tau,i)}$ and
$(\hat{u},y^{\prime},z)\models\mathsf{move}^{\prime}_{(\tau,i)}$, then $y$ and
$y^{\prime}$ are both the $i$-th source position in $\hat{u}$ satisfying
$\mathsf{move}_{\tau}$ with target $z$, which is a contradiction.
We now prove that $R$ and $R^{\prime}$ define the same relation between
synchronized pairs. First we show $R^{\prime}\subseteq R$. Consider
$((u,v),(u,v^{\prime}))\in R^{\prime}$. Therefore, there exists
$\hat{u}\models\mathsf{ipar}$ and $\hat{v}\models\mathsf{opar}^{\prime}$ such
that $\mathsf{move}^{\prime}$ applied to positions of $\hat{v}$ give the
$v^{\prime}$ witnessing $((u,v),(u,v^{\prime}))\in R^{\prime}$. By definition
of $\mathsf{opar}^{\prime}$, $\hat{v}_{\Gamma^{\prime}}\models\mathsf{opar}$.
Suppose, a position $x$ of output type $(\tau,i)$ is moved from origin $y$ in
$v$ to $z$ in $v^{\prime}$. This means
$(\hat{u},y,z)\models\mathsf{move}^{\prime}_{(\tau,i)}$. Then, by definition
of $\mathsf{move}^{\prime}_{(\tau,i)}$,
$(\hat{u},y,z)\models\mathsf{move}_{\tau}$. This shows $R^{\prime}\subseteq
R$.
For the containment $R\subseteq R^{\prime}$, consider
$((u,v),(u,v^{\prime}))\in R$. Therefore, there exists
$\hat{u}\models\mathsf{ipar}$ and $\hat{v}\models\mathsf{opar}$ such that
$\mathsf{move}$ applied to each position in $\hat{v}$ witnesses
$((u,v),(u,v^{\prime}))\in R$. This means for every position
$x\in\mathsf{dom}(\hat{v})$ with output-type $\tau$, there exist $y,z$, such
that $(\hat{u},y,z)\models\mathsf{move}_{\tau}$, $y=\mathit{orig}(v(x))$ and
$z=\mathit{orig}(v^{\prime}(x))$. For such a position
$x\in\mathsf{dom}(\hat{v})$ of output type $\tau$, let $i\in\\{1,\dots,k\\}$
be such that there are exactly $i-1$ positions $y_{1}<y_{2}<\dots y_{i-1}<y$
such that $(\hat{u},y_{j},z)\models\mathsf{move}_{\tau}$. Let
$\hat{v}^{\prime}$ be the annotation of $\hat{v}$ where every position $x$ is
annotated with the index $i$ as above. Clearly
$\hat{v}^{\prime}\models\mathsf{opar}^{\prime}$ and therefore,
$((u,v),(u,v^{\prime}))\in R^{\prime}$. We conclude $R=R^{\prime}$. ∎
See 2
###### Proof
Let
$R=(\overline{I},\overline{O},\mathsf{ipar},\mathsf{opar},(\mathsf{move}_{\tau})_{\tau},(\mathsf{next}_{\tau,\tau^{\prime}})_{\tau,\tau^{\prime}})$
and
$R^{\prime}=(\overline{I}^{\prime},\overline{O}^{\prime},\mathsf{ipar}^{\prime},\mathsf{opar}^{\prime}$,
$(\mathsf{move}^{\prime}_{\lambda})_{\lambda},(\mathsf{next}^{\prime}_{\lambda,\lambda^{\prime}})_{\lambda,\lambda^{\prime}})$
be two bounded, regular resynchronizers. In view of Lemma 1, we can assume
that both resynchronizers are $1$-bounded. The composition $R\circ R^{\prime}$
can be defined by combining the effects of $R$ and $R^{\prime}$ almost
component-wise. Some care should be taken, however, in combining the formulas
$\mathsf{next}$ and $\mathsf{next}^{\prime}$. Formally, we define the composed
resynchronizer
$R^{\prime\prime}=(\overline{I}^{\prime\prime},\overline{O}^{\prime\prime},\mathsf{ipar}^{\prime\prime},\mathsf{opar}^{\prime\prime},(\mathsf{move}^{\prime\prime}_{(\tau,\lambda)})_{\tau,\lambda},(\mathsf{next}^{\prime\prime}_{(\tau,\lambda),(\tau^{\prime},\lambda^{\prime})})_{\tau,\lambda,\tau^{\prime},\lambda^{\prime}})$,
where
* •
$\overline{I}^{\prime\prime}$ is the union of the parameters $\overline{I}$
and $\overline{I}^{\prime}$,
* •
$\overline{O}^{\prime\prime}$ is the union of the parameters $\overline{O}$
and $\overline{O}^{\prime}$,
* •
$\mathsf{ipar}^{\prime\prime}$ is the conjunction of the formulas
$\mathsf{ipar}$ and $\mathsf{ipar}^{\prime}$;
* •
$\mathsf{opar}^{\prime\prime}$ is the conjunction of the formulas
$\mathsf{opar}$ and $\mathsf{opar}^{\prime}$;
* •
$\mathsf{move}^{\prime\prime}_{(\tau,\lambda)}(y,z)$ states the existence of
some position $t$ satisfying both formulas $\mathsf{move}_{\tau}(t,z)$ and
$\mathsf{move}^{\prime}_{\lambda}(y,t)$;
* •
$\mathsf{next}_{(\tau,\lambda),(\tau^{\prime},\lambda^{\prime})}(z,z^{\prime})$
requires that $\mathsf{next}_{\tau,\tau^{\prime}}(z,z^{\prime})$ holds and,
moreover, that there exist some positions $t,t^{\prime}$ satisfying
$\mathsf{move}_{\tau}(t,z)$,
$\mathsf{move}_{\tau^{\prime}}(t^{\prime},z^{\prime})$, and
$\mathsf{next}_{\lambda,\lambda^{\prime}}(t,t^{\prime})$; note that these
positions $t,t^{\prime}$ are uniquely determined from $z,z^{\prime}$ since $R$
is $1$-bounded, and they act, at the same time, as source origins for $R$ and
as target origins for $R^{\prime}$.
By definition, $\mathsf{move}^{\prime\prime}_{(\tau,\lambda)}$ is $1$-bounded,
thus $z$ and $\tau$ determine a unique $t$, which together with $\lambda$
determines a unique $y$. It is also easy to see that $R^{\prime\prime}$ is
equivalent to $R\circ R^{\prime}$ as the positions corresponding to $t$ in
formulas $\mathsf{move}^{\prime\prime}_{(\tau,\lambda)}$ and
$\mathsf{next}^{\prime\prime}_{(\tau,\lambda),(\tau^{\prime},\lambda^{\prime})}$
correspond to the source origin of $R$ and target origin of $R^{\prime}$. ∎
## Appendix 0.B Proofs from Section 4.1
See 3
###### Proof
It is convenient to rephrase the claim of the lemma in terms of a
juxtaposition operation on flows and in terms of an induced accessibility
relation on edges. Formally, given two flows $F,G$, we define the
juxtaposition $FG$ in a way similar to concatenation, with the only exception
that in the result we maintain as an additional group of vertices the
$\mathsf{R}$ vertices of $F$, glued with the state-matching $\mathsf{L}$
vertices of $G$ (strictly speaking, the result of a juxtaposition of two flows
is not a flow, since it has three distinguished groups of vertices). We denote
by $E\dots E$ the $n$-fold juxtaposition of the flow $E$ with itself (this
must not be confused with the $n$-fold concatenation $E\cdot\ldots\cdot E$).
Let $F,E,G$, $f,e$, $\rho_{f},\rho_{e}$ be as stated in the lemma, and let
$\preceq$ denote the accessibility order between edges in a juxtaposition of
flows, e.g. $FEG$ (note that, due to the type of flows considered here,
$\preceq$ turns out to always be a total order on $FEG$). Observe that the
relative occurrence order of $\rho_{f}$ and $\rho_{e}$ inside $\rho$ is the
same as the accessibility order $\preceq$ of the edges $f$ and $e$ on the
graph $FEG$. A similar claim holds for the occurrence order of $\rho_{f}$ and
any copy of $\rho_{e}$ inside the pumped run $\mathit{pump}^{n}_{I}(\rho)$,
which corresponds to the accessibility order of $f$ and any copy of $e$ in the
graph $FE\dots EG$. Thanks to these correspondences, to prove the lemma it
suffices to consider any copy $e^{\prime}$ of $e$ in $FE\dots EG$, and show
that
$f\preceq e\text{ in }FEG\qquad\text{iff}\qquad f\preceq e^{\prime}\text{ in
}FE\dots EG.$
We thus prove the above claim. Consider the maximal path $\pi$ inside $E\dots
E$ that contains the edge $e^{\prime}$. Note that this path starts and ends at
some extremal vertices of $E\dots E$ (otherwise the path could be extended
while remaining inside $E\dots E$). Also recall that concatenation can be
defined from juxtaposition by removing the intermediate groups of vertices,
leaving only the extremal ones, and by shortcutting paths into edges. We call
this operation _flattening_ , for short. In particular, since $E$ is
idempotent, we have that $E=E\cdot\ldots\cdot E$ can be obtained from the
flattening of $E\dots E$, and this operation transforms the path $\pi$ into an
edge $e^{\prime\prime}$. By construction, we have that $f\preceq e^{\prime}$
in $FE\dots EG$ if and only if $f\preceq e^{\prime\prime}$ in $FEG$. So it
remains to prove that
$f\preceq e\text{ in }FEG\qquad\text{iff}\qquad f\preceq
e^{\prime\prime}\text{ in }FEG.$
Clearly, this latter claim holds if the edges $e$ and $e^{\prime\prime}$
coincide. This is indeed the case when $e$ is a straight edge and $E$ is
idempotent. The formal proof that this holds is rather tedious, but follows
quite easily from a series of results we have already proven in [3]. Roughly
speaking, one proves that:
* •
the edges of an idempotent flow $E$ can be grouped into _components_ (cf.
Definition 6.4 from [3]), so that each component contains exactly one straight
edge (cf. Lemma 6.6 from [3], see also the figure at page 4.1, where
components are represented by colors);
* •
every path inside the juxtaposition $EE$, with $E$ idempotent, consists of
edges from the same component, say $C$; moreover, after the flattening from
$EE$ to $E$, this path becomes an edge of $E$ that belongs again to the
component $C$ (cf. Claims 7.3 and 7.4 in the proof of Theorem 7.2 from [3]);
* •
every maximal path in $E\dots E$ that contains a straight edge starts and end
at opposite sides of $E\dots E$ (simple observation based on the definition of
concatenation and Lemma 6.6 from [3]).
To conclude, recall that $\pi$ is a path inside $E\dots E$ that contains a
copy $e^{\prime}$ of the straight edge $e$, and that becomes the edge
$e^{\prime\prime}$ after the flattening into $E$. The previous properties
immediately imply that $e=e^{\prime\prime}$.
## Appendix 0.C Proofs from Section 4.2
As explained in the body, the technical lemmas involving output blocks are
only applicable to transducers that avoid inversions. _Hereafter we assume
that $T$ is a transducer that avoids inversions, and we denote by $\rho$ an
arbitrary successful run of $T$._
See 5
###### Proof
$B_{1}$ and $B_{2}$ are clearly disjoint. By way of contradiction, assume that
$B_{1}$ and $B_{2}$ are $\mathbf{C}$-large, but $B_{1}>B_{2}$. By Lemma 4, we
can find for both $i=1$ and $i=2$ a loop $J_{i}\subseteq I_{i}$ and a
productive straight edge $e_{i}\in\mathit{flow}(J_{i})$ that is witnessed by a
subrun intersecting $B_{i}$. Clearly, we have $J_{1}<J_{2}$, and since
$B_{1}>B_{2}$, the subrun witnessing $e_{1}$ follows the subrun witnessing
$e_{2}$. Thus, $(J_{1},e_{1},J_{2},e_{2})$ is an inversion of $\rho$, which
contradicts the assumption that $T$ avoids inversions.
We now turn to the proofs of Lemmas 6 and 7, which both require auxiliary
lemmas relying again on the assumption that $T$ avoids inversion.
###### Lemma 0.C.1 ()
Let $I$ be an input interval, $B_{1}<B_{2}$ two output blocks of $I$, and $S$
the set of output positions strictly between $B_{1}$ and $B_{2}$ and with
origins outside $I$. If $B_{1},B_{2}$ are $\mathbf{C}$-large, then $S$ is
$2\mathbf{C}$-small.
###### Proof
By way of contradiction, suppose that $S$ is $2\mathbf{C}$-large. This means
that $|\mathit{orig}(S)\cap I^{\prime}|>\mathbf{C}$ for some interval
$I^{\prime}$ disjoint from $I$, say $I^{\prime}<I$ (the case of $I^{\prime}>I$
is treated similarly). By Lemma 4, we can find two loops $J\subseteq I$ and
$J^{\prime}\subseteq I^{\prime}$ and some productive straight edges
$e\in\mathit{flow}(J)$ and $e^{\prime}\in\mathit{flow}(J^{\prime})$ that are
witnessed by subruns intersecting $B_{1}$ and $S$, respectively. Since
$S>B_{1}$, we know that the subrun witnessing $e$ follows the subrun
witnessing $e^{\prime}$. As in the previous proof, this shows the inversion
$(J,e,J^{\prime},e^{\prime})$, which contradicts the assumption that $T$
avoids inversions.
See 6
###### Proof
By Lemma 5, we have $B_{1}<B_{2}$. Moreover, all $\mathbf{C}$-large output
blocks of $I_{1}$ or $I_{2}$ are also $\mathbf{C}$-large output blocks of $I$,
so $B$ contains both $B_{1}$ and $B_{2}$. Let $I_{0}$ be the maximal interval
to the left of $I_{1}$, and thus adjacent to it, and, similarly, let $I_{3}$
be the maximal interval to the right of $I_{2}$, and thus adjacent to it.
Suppose, by way of contradiction, that $B\setminus(B_{1}\cup B_{2})$ is
$4K\mathbf{C}$-large. This means that there is a $2K\mathbf{C}$-large set
$S\subseteq B\setminus(B_{1}\cup B_{2})$ with origins entirely inside
$I_{0}\cdot I_{1}$ or entirely inside $I_{2}\cdot I_{3}$. Suppose, w.l.o.g.,
that the former case holds, and decompose $S$ as a union of maximal output
blocks $B^{\prime}_{1},B^{\prime}_{2},\dots,B^{\prime}_{n}$ of either $I_{0}$
or $I_{1}$. Since $S\cap B_{1}=\emptyset$, we have that every block
$B^{\prime}_{i}$ with origins inside $I_{1}$ is $\mathbf{C}$-small. Similarly,
by Lemma 0.C.1, every block $B^{\prime}_{i}$ with origins inside $I_{0}$ is
$\mathbf{C}$-small too. Moreover, since $\rho$ is $K$-visiting, we have that
the number $n$ of maximal output blocks of either $I_{0}$ or $I_{1}$ that are
contained in $S$ is at most $2K$. All together, this contradicts the
assumption that $S$ is $2K\mathbf{C}$-large.
###### Lemma 0.C.2 ()
Let $I$ be a loop of $\rho$. Then $\mathit{flow}(I)$ has at most one
productive straight edge, and this edge must be $\mathsf{L}\mathsf{R}$.
###### Proof
Suppose, by way of contradiction, that there are two productive straight edges
in $\mathit{flow}(I)$, say $e$ and $f$, with $e$ before $f$ in $\rho$ (the
reader may refer again to the figure at page 4.1, and think of $e$ and $f$,
for instance, as the edges labeled by $\alpha_{2}$ and $\gamma_{1}$,
respectively). Suppose that we pump $I$ twice, and let $I_{1}<I_{2}$ be the
copies of $I$ in the pumped run $\rho^{\prime}$. Let also $e_{1},e_{2}$ (resp.
$f_{1},f_{2}$) be the corresponding copies of $e$ (resp. $f$), so that
$e_{j},f_{j}$ belong to the flow $\mathit{flow}_{\rho^{\prime}}(I_{j})$. It is
easy to check the following properties:
* •
if $e$ is an $\mathsf{L}\mathsf{R}$ edge, then the subrun witnessed by $e_{1}$
occurs in $\rho^{\prime}$ before the subrun witnessed by $e_{2}$ (and the
other way around if $e$ is $\mathsf{R}\mathsf{L}$);
* •
the subruns witnessed by $e_{1}$ and $e_{2}$ occur in $\rho^{\prime}$ before
the subruns witnessed by $f_{1},f_{2}$ (this property follows easily from the
observation that when building the product
$\mathit{flow}(I)\cdot\mathit{flow}(I)$, the edges $e_{1},e_{2}$ will be
“part” of the edge $e$ in the product, whereas $f_{1},f_{2}$ will be “part” of
the edge $f$).
Let us assume first that $e$ is an $\mathsf{R}\mathsf{L}$ edge. Then observe
that $(I_{1},e_{1},I_{2},e_{2})$ is an inversion in $\rho^{\prime}$. But this
contradicts $T$ being inversion-free. Therefore, both $e,f$ are
$\mathsf{L}\mathsf{R}$ edges. But then, $(I_{1},f_{1},I_{2},e_{2})$ is an
inversion in $\rho^{\prime}$, and we have again a contradiction.
###### Remark 0.C.1
The statement of Lemma 0.C.2 can be strengthened by observing the following
property of productive edges in an idempotent flow. Assume that $I$ is a loop
and $e$ is the unique productive straight edge in $\mathit{flow}(I)$. Let $f$
be some productive (non-straight) edge of $\mathit{flow}(I)$ with $f\not=e$.
When $I$ is pumped then the subruns witnessing the copies of $f$ are part of
the subrun witnessing $e$ in the product flow. This means for example, that in
the figure on page 4.1 the productive edges are either all among the blue
edges, or all among the gray edges (none of the red edges can be productive,
because the straight edge is $\mathsf{R}\mathsf{L}$, and would result in a
productive $\mathsf{R}\mathsf{L}$ edge on pumping).
See 7
###### Proof
By Lemma 0.C.2, we can assume that $\mathit{flow}(I)=\mathit{flow}(I_{k})$ has
a unique productive straight edge $e$, which is an $\mathsf{L}\mathsf{R}$
edge. Let $B^{\prime}_{k}$ be the output block corresponding to $e$ in
$\mathit{flow}(I_{k})$. Since $\mathit{flow}(I)$ is idempotent, any output
block of $I$ has one of the following shapes (see also Remark 0.C.1):
* (a)
A block $B=B^{\prime}_{1}\cdot J^{\prime}_{1}\cdot\dots J^{\prime}_{n-1}\cdot
B^{\prime}_{n}$, for some intervals $J^{\prime}_{1},\dots,J^{\prime}_{n-1}$
such that $\mathit{out}(I_{k})$ is included in $J^{\prime}_{k-1}\cdot
B^{\prime}_{k}\cdot J^{\prime}_{k}$ for all $1<k<n$,
* (b)
At most $2K$ output blocks $L_{1},\dots,L_{p},R_{1},\dots,R_{s}$, where each
$L_{i}$ and $R_{j}$ corresponds to an edge of $\mathit{flow}(I_{1})$ and
$\mathit{flow}(I_{n})$, respectively: the blocks $L_{i},R_{j}$ appear before,
respectively after the straight edge.
Moreover, the order of the output blocks of $I$ is
$L_{1},\dots,L_{p},B,R_{1},\dots,R_{s}$. To illustrate the statement (a)
above, the reader can take as example $p=s=2$, $L_{1}=\alpha$, $L_{2}=\beta$,
$R_{1}=\kappa$, $R_{2}=\zeta$,
$J^{\prime}_{1}=\cdots=J^{\prime}_{n-1}=\alpha\kappa\beta\zeta$ in Figure 2.
For statement (b), notice that in $I_{1}\cdot I_{2}\cdot I_{3}$, we have the
output blocks $L_{1}=\alpha,L_{2}=\beta$ of $I_{1}$, the straight edge
$(\gamma\alpha\kappa\beta\zeta)^{2}\gamma$ (the purple zigzag) followed by
$R_{1}=\kappa,R_{2}=\zeta$ of $I_{3}$.
Note that $B_{k}=\mathit{bigout}(I_{k})$ is contained in
$J^{\prime}_{k-1}\cdot B^{\prime}_{k}\cdot J^{\prime}_{k}$ for all $1<k<n$.
Moreover, $B_{1}=\mathit{bigout}(I_{1})$ is contained in $L_{1}\cdots
L_{p}\cdot B^{\prime}_{1}\cdot J^{\prime}_{1}$, and
$B_{n}=\mathit{bigout}(I_{n})$ is contained in $J^{\prime}_{n-1}\cdot
B^{\prime}_{n}\cdot R_{1}\cdots R_{s}$. Also by Lemma 5, $B_{j}$ precedes
$B_{j+1}$ for all $j$.
$I$$\boldsymbol{\alpha_{1}}$$~{}\boldsymbol{\alpha_{2}}$$\boldsymbol{\alpha_{3}}$$\boldsymbol{\beta_{1}}$$\boldsymbol{\beta_{2}}$$\boldsymbol{\beta_{3}}$$~{}\boldsymbol{\gamma_{1}}$$I_{1}$$I_{2}$$I_{3}$$\boldsymbol{\alpha}$$\boldsymbol{\beta}$$\boldsymbol{\gamma}$$\boldsymbol{\alpha}$$\boldsymbol{\kappa}$$\boldsymbol{\beta}$$\boldsymbol{\zeta}$$\boldsymbol{\gamma}$$\boldsymbol{\alpha}$$\boldsymbol{\kappa}$$\boldsymbol{\beta}$$\boldsymbol{\zeta}$$\boldsymbol{\zeta}$$\boldsymbol{\gamma}$$\boldsymbol{\kappa}$
Figure 2: Illustration for Lemma 7.
If one of the $L_{k}$ is $\mathbf{C}$-large, then $B_{1}$ is non-empty, hence
$\mathit{bigout}(I)$ is non-empty and starts at the first position of $B_{1}$.
Similarly, if one of the $R_{k}$ is $\mathbf{C}$-large then $B_{n}$ is non-
empty, hence $\mathit{bigout}(I)$ is non-empty and ends with the last position
of $B_{n}$. Otherwise, if all $L_{j},R_{j}$ are $\mathbf{C}$-small then
$\mathit{bigout}(I)$ is either empty or equal to $B$. In all cases we can
write $\mathit{bigout}(I)=B_{1}\cdot J_{1}\cdot B_{2}\cdot
J_{2}\cdot\dots\cdot J_{n-1}\cdot B_{n}$, with each $J_{k}$ consisting of at
most $K$ $\mathbf{C}$-small blocks of $I_{k}$ and $K$ $\mathbf{C}$-small
blocks of $I_{k+1}$, namely those left over after gathering the
$\mathbf{C}$-large blocks into $\mathit{bigout}(I_{k})$ and
$\mathit{bigout}(I_{k+1})$, respectively. Therefore, each $J_{k}$ is
$2K\mathbf{C}$-small.
## Appendix 0.D Proof of Theorem 3.1.
Recall that the implication $4\rightarrow 1$ is straightforward, and the
implication $1\rightarrow 2$ was already proven in full detail in the main
body. Below, we provide detailed proofs of the implications $2\rightarrow
3\rightarrow 4$.
The implication $2\rightarrow 3$ is shown by contradiction. Consider a
successful run $\rho$ of $T$ on some input $u$ and suppose there is an
inversion: $\rho$ has disjoint loops $I<I^{\prime}$, whose flows contain
productive straight edges, say $e$ in $\mathit{flow}_{\rho}(I)$ and
$e^{\prime}$ in $\mathit{flow}_{\rho}(I^{\prime})$, such that $e^{\prime}$
precedes $e$ in the run order. Let $u=u_{1}\,w\,u_{2}\,w^{\prime}\,u_{3}$ so
that $w$ and $w^{\prime}$ are the factors of the input delimited by the loops
$I$ and $I^{\prime}$, respectively. Further let $v$ and $v^{\prime}$ be the
outputs produced along the edges $e$ and $e^{\prime}$, respectively. Consider
now the run $\rho_{k}$ obtained from $\rho$ by pumping the input an arbitrary
number $k$ of times on the loops $I$ and $I^{\prime}$. This run is over the
input $u_{1}\,(w)^{k}\,u_{2}\,(w^{\prime})^{k}\,u_{3}$, and in the output
produced by $\rho_{k}$ there are $k$ (possibly non-consecutive) occurrences of
$v$ and $v^{\prime}$. By Lemma 3 all occurrences of $v^{\prime}$ precede all
occurrences of $v$. In particular, if $X_{1}$ (resp. $X_{2}$) is the set of
positions corresponding to all the occurrences of $v$ (resp. $v^{\prime}$) in
the output produced by $\rho_{k}$, then $(X_{1},X_{2})$ is a cross of width at
least $k$.
Now we prove the implication $3\rightarrow 4$. We assume that no run of $T$
has any inversion. We want to build a partially bijective, regular
resynchronizer $\mathcal{R}$ that is $T$-preserving and such that
$\mathcal{R}(T)$ is order-preserving. The resynchronizer $\mathcal{R}$ will
use input and output parameters to guess a successful run $\rho$ of $T$ on the
input $u$ and a corresponding _factorization tree_ for $\rho$ of height at
most $H=3|M_{T}|$ (see page 5 for the formal definition and the existence of a
factorization tree).
The resynchronizer $\mathcal{R}$ that we will define is _functional_ , which
means here that every source origin is mapped by each $\mathsf{move}_{\tau}$
formula to at most one target position.
#### Notations.
For a node $p$ of a factorization tree we write $I(p)$ for the input interval
which is the yield of the subtree of $p$. Recall that the leaves of the
factorization tree correspond to singleton intervals on the input. The set of
output positions with origins in $I(p)$ is denoted by $\mathit{out}(p)$ (note
that this might not be an interval).
Recall that an output block $B$ of $\mathit{out}(p)$ is a maximal interval of
output positions with origins in $I(p)$, and hence the position just before
and the position just after $B$ have origins outside $I(p)$. We also write
$\mathit{bigout}(p)$, instead of $\mathit{bigout}(I(p))$, for the dominant
output interval of $I(p)$ (see page 4 for the definition). Finally, given a
position $x$ in the output and a level $\ell$ of the factorization tree of
$\rho$, we denote by $p_{x,\ell}$ the unique node at level $\ell$ such that
$I(p_{x,\ell})$ contains the source origin of $x$.
#### Input Parameters.
The successful run $\rho$ together with its factorization tree of height at
most $H=3|M_{T}|$ can be easily encoded over the input using input parameters
$\mathsf{ipar}$. The parameters describe each input interval $I(p)$ and the
label $\mathit{flow}(I(p))$ of each node $p$ in the factorization tree.
Formally, an input interval $I(p)$ is described by marking the begin and end
with two distinguished parameters for the specific level. The label
$\mathit{flow}(I(p))$ annotates every position inside $I(p)$. This accounts
for $H(2+|M_{T}|)$ input parameters. Correctness of the annotations with the
above input parameters can be expressed by a formula $\mathsf{ipar}$. In
particular, on the leaves, $\mathsf{ipar}$ checks that every interval is a
singleton of the form $\\{y\\}$ and its flow is the one induced by the letter
$u(y)$. On the internal nodes, $\mathsf{ipar}$ checks that the label of a node
coincides with the monoid product of the labels of its children, which is a
composition of flows. It also checks that for every node with more than two
children, the node and the children are labelled by the same idempotent flow.
#### Output Parameters.
We also need to encode the run $\rho$ on the output, because the
resynchronizer will determine the target origin of an output position, not
only on the basis of the flow at the source origin, but also on the basis of
the productive transition that generated that particular position. The
annotation that encodes the run $\rho$ on the output is done using output
parameters (one for each transition in $\Delta$), and its correctness will be
enforced by a suitable combination of the formulas $\mathsf{opar}$,
$\mathsf{move}_{\tau}$, and $\mathsf{next}_{\tau,\tau^{\prime}}$. This will
take a significant amount of technical details and will rely on specific
properties of formulas $\mathsf{move}_{\tau}$, so we prefer to temporarily
postpone those details.
Below, we explain how the origins are transformed by a series of partial
resynchronizers $\mathcal{R}_{\ell}$ that “converge” in finitely many steps to
a desired resynchronization, under the assumption that the output annotation
correctly encodes the same run $\rho$ that is represented in the input
annotation.
#### Moving origins.
Here we will work with a fixed successful run $\rho$ and a factorization tree
for it, that we assume are correctly encoded by the input and output
annotations. For every level $\ell$ of the factorization tree, we will define
a functional, bounded, regular resynchronizer $\mathcal{R}_{\ell}$. Each
resynchronizer $\mathcal{R}_{\ell}$ will be _partial_ , in the sense that for
some output positions it will not define source-target origin pairs. However,
the set of output positions with associated source-target origin pairs
increases with the level $\ell$, and the top level resynchronizer
$\mathcal{R}_{*}$ will specify source-target origin pairs for all output
positions. The latter resynchronizer will almost define the resynchronization
that is needed to prove item (4) of the theorem; we will only need to modify
it slightly in order to make it $1$-bounded and to check that the output
annotation is correct.
To enable the inductive construction, we need the resynchronizer
$\mathcal{R}_{\ell}$ to satisfy the following properties, for every level
$\ell$ of the factorization tree:
* •
the set of output positions for which $\mathcal{R}_{\ell}$ defines target
origins is the union of the dominant output intervals $\mathit{bigout}(p)$ of
all nodes $p$ at level $\ell$;
* •
$\mathcal{R}_{\ell}$ only moves origins within the same interval at level
$\ell$, that is, $\mathcal{R}_{\ell}$ defines only pairs $(y,z)$ of source-
target origins such that $y,z\in I(p)$ for some node $p$ at level $\ell$;
* •
the target origins defined by $\mathcal{R}_{\ell}$ are order-preserving within
the same interval at level $\ell$, that is, for all output positions
$x<x^{\prime}$, if $\mathcal{R}_{\ell}$ defines the target origins of
$x,x^{\prime}$ to be $z,z^{\prime}$, respectively, and if $z,z^{\prime}\in
I(p)$ for some node $p$ at level $\ell$, then $z\leq z^{\prime}$.
* •
$\mathcal{R}_{\ell}$ is $\ell\cdot 4K\mathbf{C}$-bounded, namely, there are at
most $\ell\cdot 4K\mathbf{C}$ distinct source origins that are moved by
$\mathcal{R}_{\ell}$ to the same target origin.
The inductive construction of $\mathcal{R}_{\ell}$ will basically amount to
defining appropriate formulas $\mathsf{move}_{\tau}(y,z)$.
Base Case. The base case is $\ell=0$, namely, when the resynchronization is
acting at the leaves of the factorization tree. In this case, the regular
resynchronizer $\mathcal{R}_{\ell}$ is vacuous, as the input intervals $I(p)$
associated with the leaves $p$ are singletons, and hence all dominant output
intervals $\mathit{bigout}(p)$ are empty. Formally, for this resynchronizer
$\mathcal{R}_{\ell}$, we simply let $\mathsf{move}_{\tau}(y,z)$ be false,
independently of the underlying output type $\tau$ and of the source and
target origins. This resynchronization is clearly functional, $0$-bounded, and
order-preserving.
Inductive Step. For the inductive step, we explain how the origins of an
output position $x\in\mathit{bigout}(p)$ are moved within the interval $I(p)$,
where $p=p_{x,\ell}$ is the node at level $\ell$ that “generates” $x$. Even
though we explain this by mentioning the node $p_{x,\ell}$, the definition of
the resynchronization will not depend on it, but only on the level $\ell$ and
the underlying input and output parameters. In particular, to describe how the
origin of a $\tau$-labeled output position $x$ is moved, the formula
$\mathsf{move}_{\tau}(y,z)$ has to determine the productive edge that
generated $x$ in the flow that labels the node $p_{x,\ell}$. This can be done
by first determining from the output type $\tau$ the productive transition
$t_{x}$ that generated $x$, and then inspecting the annotation at the source
origin $y$ to “track” $t_{x}$ inside the productive edges of the flow
$\mathit{flow}(I_{p^{\prime}})$, for each node $p^{\prime}$ along the unique
path from the leaf $p_{x,0}$ to node $p_{x,\ell}$. In the case distinction
below, we implicitly rely on this type of computation, which can be easily
implemented in MSO.
1. 1.
$\boldsymbol{p_{x,\ell}}$ is a binary node. We first consider the case where
$p=p_{x,\ell}$ is a binary node (the annotation on the source origin $y$ will
tell us whether this is the case). Let $p_{1},p_{2}$ be the left and right
children of $p$. If $x$ belongs to one of the dominant output blocks
$\mathit{bigout}(p_{1})$ and $\mathit{bigout}(p_{2})$ (again, this information
is available at the input annotation), then the resynchronizer
$\mathcal{R}_{\ell}$ will inherit the source-target origin pairs associated
with $x$ from the lower level resynchronization $\mathcal{R}_{\ell-1}$. Note
that $\mathit{bigout}(p_{1})<\mathit{bigout}(p_{2})$ by Lemma 5, so
$\mathcal{R}_{\ell}$ is order-preserving at least for the output positions
inside $\mathit{bigout}(p_{1})\cup\mathit{bigout}(p_{2})$.
We now describe the source-target origin pairs when
$x\in\mathit{bigout}(p)\setminus(\mathit{bigout}(p_{1})\cup\mathit{bigout}(p_{2}))$.
The idea is to move the origin of $x$ to one of the following three input
positions, depending on the relative order between $x$ and the positions in
$\mathit{bigout}(p_{1})$ and in $\mathit{bigout}(p_{2})$:
* •
the first position of $I(p_{1})$, if $x<\mathit{bigout}(p_{1})$;
* •
the last position of $I(p_{1})$, if
$\mathit{bigout}(p_{1})<x<\mathit{bigout}(p_{2})$;
* •
the last position of $I(p_{2})$, if $x>\mathit{bigout}(p_{2})$.
Which of the above cases holds can be determined, again, by inspecting the
output type $\tau$ and the annotation of the source origin $y$, in a way
similar to the computation of the productive edge that generated $x$ at level
$\ell$. So the described resynchronization can be implemented by an MSO
formula $\mathsf{move}_{\tau}(y,z)$.
The resulting resynchronization $\mathcal{R}_{\ell}$ is functional and order-
preserving inside every interval at level $\ell$. It remains to argue that
$\mathcal{R}_{\ell}$ is $\ell\cdot 4K\mathbf{C}$-bounded. To see why this
holds, assume, by the inductive hypothesis, that $\mathcal{R}_{\ell-1}$ is
$(\ell-1)\cdot 4K\mathbf{C}$-bounded. Recall that the new source-target origin
pairs that are added to $\mathcal{R}_{\ell}$ are those associated with output
positions in
$\mathit{bigout}(p)\setminus(\mathit{bigout}(p_{1})\cup\mathit{bigout}(p_{2}))$.
Lemma 6 tells us that there are at most $4K\mathbf{C}$ distinct positions that
are source origins of such positions. So, in the worst case, at most
$(\ell-1)\cdot 4K\mathbf{C}$ source origins from $\mathcal{R}_{\ell-1}$ and at
most $4K\mathbf{C}$ new source origins from $\mathcal{R}_{\ell}$ are moved to
the same target origin. This shows that $\mathcal{R}_{\ell}$ is $\ell\cdot
4K\mathbf{C}$-bounded.
2. 2.
$\boldsymbol{p_{x,\ell}}$ is an idempotent node. The case where $p=p_{x,\ell}$
is an idempotent node with children $p_{1},p_{2},\dots,p_{n}$ follows a
similar approach. For brevity, let $I_{i}=I(p_{i})$ and
$B_{i}=\mathit{bigout}(p_{i})$. By Lemma 5, we have $B_{1}<B_{2}<\dots<B_{n}$.
Lemma 7 then provides a decomposition of $\mathit{bigout}(p)$ as $B_{1}\cdot
J_{1}\cdot B_{2}\cdot J_{2}\cdot\dots\cdot J_{n-1}\cdot B_{n}$, for some
$2K\mathbf{C}$-small output intervals $J_{k}$, for $k=1,\dots,n-1$, that have
origins inside $I_{k}\cup I_{k+1}$.
As before, the resynchronizer $\mathcal{R}_{\ell}$ behaves exactly as
$\mathcal{R}_{\ell-1}$ for the output positions inside the $B_{k}$’s. For any
other output position, say $x\in J_{k}$ for some $k=1,2,\dots,n-1$, we first
recall that the source origin $y$ of $x$ is either inside $I_{k}$ or inside
$I_{k+1}$. Depending on which of the two intervals contains $y$, the
resynchronizer $\mathcal{R}_{\ell}$ will define the target origin $z$ to be
either the last position of $I_{k}$ or the first position of $I_{k+1}$.
However, since we cannot determine using MSO the index $k$ of the interval
$J_{k}$ that contains $x$, we proceed as follows.
First observe that any block $B_{i}$ can be identified by some flow edge at
level $\ell-1$, and the latter edge can represented in MSO by suitable monadic
predicates over the input. Let $B,B^{\prime}$ be the two consecutive blocks
among $B_{1},\dots,B_{n}$ such that $B<x<B^{\prime}$. Note that these blocks
can be determined in MSO once the productive edge that generated $x$ is
identified. Further let $I$ be the interval among $I_{1},\dots,I_{n}$ that
contains the origin $y$ of $x$. By the previous arguments, we have that the
interval $I$ contains either all the origins of $B$ or all the origins
$B^{\prime}$. Moreover, which of the two sub-cases holds can again be
determined in MSO by inspecting the annotations. The formula
$\mathsf{move}_{\tau}(y,z)$ can thus define the target origin $z$ to be
* •
the last position of $I$, if $I$ contains the origins of $B$;
* •
the first position of $I$, if $I$ contains the origins of $B^{\prime}$.
The above construction yields a functional regular resynchronization
$\mathcal{R}_{\ell}$ that associates with any two output positions
$x<x^{\prime}$ with source origins in the same interval $I(p)$, some target
origins $z\leq z^{\prime}$. In other words, the resynchronization
$\mathcal{R}_{\ell}$ is order-preserving in each interval at level $\ell$.
It remains to show that $\mathcal{R}_{\ell}$ is $\ell\cdot
4K\mathbf{C}$-bounded, under the inductive hypothesis that
$\mathcal{R}_{\ell-1}$ is $(\ell-1)\cdot 4K\mathbf{C}$-bounded. This is done
using a similar argument as before, that is, by observing that the output
positions in $\mathit{bigout}(p)\setminus\big{(}\bigcup_{1\leq k\leq
n}\mathit{bigout}(p_{i})\big{)}$ belong to some $J_{k}$, and in the worst case
all source origins $y$ of positions from $J_{k}$ are moved to the last
position of $I_{k}$. By Lemma 7, there are at most $2K\mathbf{C}$ such
positions $y$.
#### Top level resynchronizer.
Let $\mathcal{R}_{*}$ be the the resynchronizer $\mathcal{R}_{\ell}$ obtained
at the top level $\ell$ of the factorization tree. Based on the above
constructions, $\mathcal{R}_{*}$ defines target origins for all output
positions, unless the dominant output interval $\mathit{bigout}(p)$ associated
with the root $p$ is empty (this can indeed happen when the number of
different origins in the output is at most $\mathbf{C}$, so not sufficient for
having at least one $\mathbf{C}$-large output factor). In particular, if
$\mathit{bigout}(p)\neq\emptyset$, then $\mathit{bigout}(p)$ is the whole
output, and $\mathcal{R}_{\ell}$ is basically the desired resynchronization,
assuming that the output annotations are correct.
Let us now discuss briefly the degenerate case where
$\mathit{bigout}(p)=\emptyset$, which of course can be detected in MSO. In
this case, the appropriate resynchronizer $\mathcal{R}_{*}$ should be
redefined so that it moves all source origins to the same target origin, say
the first input position. Clearly, this gives a functional, regular
resynchronizer that is order-preserving and $\mathbf{C}$-bounded.
#### Correctness of output annotation.
Recall that the properties of the top level resynchronizer $\mathcal{R}_{*}$,
in particular, the claim that $\mathcal{R}_{*}$ is bounded, were crucially
relying on the assumption that every output position $x$ is correctly
annotated with the productive transition that generated it. This assumption
cannot be guaranteed by the MSO sentence $\mathsf{opar}$ alone (the property
intrinsically talks about a relation between input and output annotations).
Below, we explain how to check correctness of the output annotation with the
additional help of the formulas $\mathsf{move}_{\tau}(y,z)$ (that will be
modified for this purpose) and
$\mathsf{next}_{\tau,\tau^{\prime}}(z,z^{\prime})$.
Let $\rho$ be the successful run as encoded by the input annotation. The idea
is to check that the sequence of productive transitions $t_{x}$ that annotate
the positions $x$ in the output is the maximal sub-sequence of $\rho$
consisting only of productive transitions. Besides the straightforward
conditions (concerning, for instance, the first and last productive
transitions of $\rho$, or the possible multiple symbols that could be produced
within a single transition), the important condition to be verified is the
following:
($\dagger$)
The above property is easily expressible by an MSO formula
$\varphi^{\dagger}_{\tau,\tau^{\prime}}(y,y^{\prime})$, assuming that
$\tau,\tau^{\prime}$ are the output types of $x,x+1$ and the free variables
$y$ and $y^{\prime}$ are interpreted by the _source origins_ of $x$ and $x+1$,
with $x$ ranging over all output positions. This is very close to the type of
constraints that can be enforced by the formula
$\mathsf{next}_{\tau,\tau^{\prime}}$ of a regular resynchronizer, with the
only difference that the latter formula can only access the _target origins_
$z,z^{\prime}$ of $x,x+1$.
We thus need a way to uniquely determine from the target origins
$z,z^{\prime}$ of $x$ the source origins $y,y^{\prime}$ of $x$. For this, we
could rely on the formulas $\mathsf{move}_{\tau}(y,z)$, if only they were
defining partial bijections between $y$ and $z$. Those formulas are in fact
close to define partial bijections, as they are functional and $k$-bounded,
for $k=H\cdot 4K\mathbf{C}$. The latter boundedness property, however, depends
again on the assumption that the output annotation is correct. We overcome
this problem by gradually modifying the resynchronizer $\mathcal{R}_{*}$ so as
to make it functional and $1$-bounded (i.e., partially bijective),
independently of the output annotations.
We start by modifying the formulas $\mathsf{move}_{\tau}(y,z)$ to make them
“syntactically” $k$-bounded. Formally, we construct from
$\mathsf{move}_{\tau}(y,z)$ the formula
$\displaystyle\mathsf{move}^{\prime}_{\tau}(y,z)$
$\displaystyle~{}=~{}\mathsf{move}_{\tau}(y,z)$
$\displaystyle\>~{}\wedge~{}\>\forall
y_{1},\dots,y_{k},y_{k+1}~{}\Big{(}\bigwedge\nolimits_{i}\mathsf{move}_{\tau}(y_{i},z)\Big{)}\rightarrow\Big{(}\bigvee\nolimits_{i\neq
j}y_{i}=y_{j}\Big{)}.$
Intuitively, the above formula is semantically equivalent to
$\mathsf{move}_{\tau}(y,z)$ when there are at most $k$ input positions
$y^{\prime}$ that can be paired with $z$ via the same formula
$\mathsf{move}_{\tau}$, and it is false otherwise.
Let $\mathcal{R}^{\prime}_{*}$ be the regular resynchronizer obtained from
$\mathcal{R}_{*}$ by replacing the formulas $\mathsf{move}_{\tau}$ by
$\mathsf{move}^{\prime}_{\tau}$, for every output type $\tau$. By
construction, $\mathcal{R}^{\prime}_{*}$ is functional and $k$-bounded,
independently of any assumption on the output annotations. We can then apply
Lemma 1 and obtain from $\mathcal{R}^{\prime}_{*}$ an equivalent regular
resynchronizer
$\mathcal{R}^{\prime\prime}_{*}=(\overline{I}^{\prime\prime},\overline{O}^{\prime\prime},\mathsf{ipar}^{\prime\prime},\mathsf{opar}^{\prime\prime},(\mathsf{move}^{\prime\prime}_{\tau})_{\tau},(\mathsf{next}^{\prime\prime}_{\tau,\tau^{\prime}})_{\tau,\tau^{\prime}})$
that is $1$-bounded. So each $\mathsf{move}^{\prime\prime}_{\tau}$ is a
partial bijection.
We are now ready to verify the correctness of the output annotation. Recall
that the idea is to enforce the property $(\dagger)$ by exploiting the
previously defined formula
$\varphi^{\dagger}_{\tau,\tau^{\prime}}(y,y^{\prime})$ and the partial
bijection between the source origings $y,y^{\prime}$ and the target origins
$z,z^{\prime}$, as defined by $\mathsf{move}^{\prime\prime}_{\tau}(y,z)$ and
$\mathsf{move}^{\prime\prime}_{\tau^{\prime}}(y^{\prime},z^{\prime})$.
Formally, we define
$\mathsf{next}^{\prime\prime\prime}_{\tau,\tau^{\prime}}(z,z^{\prime})~{}=~{}\mathsf{next}^{\prime\prime}_{\tau,\tau^{\prime}}(z,z^{\prime})~{}\wedge~{}\exists
y,y^{\prime}~{}\mathsf{move}^{\prime\prime}_{\tau}(y,z)~{}\wedge~{}\mathsf{move}^{\prime\prime}_{\tau^{\prime}}(y^{\prime},z^{\prime})~{}\wedge~{}\varphi^{\dagger}_{\tau,\tau^{\prime}}(y,y^{\prime}).$
To conclude, by replacing in $\mathcal{R}^{\prime\prime}$ the formulas
$\mathsf{next}^{\prime\prime}_{\tau,\tau^{\prime}}$ with
$\mathsf{next}^{\prime\prime\prime}_{\tau,\tau^{\prime}}$, we obtain a regular
resynchronizer $\mathcal{R}$ that is partially bijective, $T$-preserving and
such that $\mathcal{R}(T)$ is order-preserving. This completes the proof of
the implication $3\rightarrow 4$ of our main theorem. ∎
## Appendix 0.E Proof of Theorem 3.2
We provide here the missing details of the proof of Theorem 3.2, as sketched
in Section 6. We recall that the goal is to construct, from a given arbitrary
two-way transducer $T$:
1. 1.
a bounded-visit transducer $\mathit{low}(T)$ that is classically equivalent to
$T$,
2. 2.
partially bijective, regular resynchronizer $\mathcal{R}$ that is
$T$-preserving and such that $\mathcal{R}(T)=_{o}\mathit{low}(T)$.
We will reason with a fixed input $u$ at hand and with an induced
accessibility relation on productive transitions of $T$, tagged with origins.
Formally, a _tagged transition_ is any pair $(t,y)$ consisting of a transition
$t\in\Delta$ and a position $y$ on the input $u$, such that $t$ occurs at
position $y$ in some successful run on $u$. The accessibility preorder on
tagged transitions is such that $(t,y)\preceq_{u}(t^{\prime},y^{\prime})$
whenever $T$ has a run on $u$ starting with transition $t$ at position $y$ and
ending with transition $t^{\prime}$ at position $y^{\prime}$. This preorder
induces an equivalence relation, denoted $\sim_{u}$. Intuitively,
$(t,y)\sim_{u}(t^{\prime},y^{\prime})$ means that $T$ can cycle an arbitrary
number of times between these two tagged transitions (possibly
$(t,y)=(t^{\prime},y^{\prime})$). A $\sim_{u}$-equivalence class $C$ is called
_realizable on $u$_ if there is a successful run on $u$ that uses at least
once a tagged transition from the class $C$.
We say that $T$ is _$K$ -sparse_ if for every input $u$ and every realizable
$\sim_{u}$-equivalence class $C$, there are at most $K$ _productive_ tagged
transitions in $C$ (recall that a productive transition is one that produces
non-empty output). Intuitively, bounded sparsity means that the number of
origins of outputs produced by vertical loops in successful runs of $T$ is
uniformly bounded. If $T$ is not $K$-sparse for any $K$, then we say that $T$
has _unbounded sparsity_.
When $T$ is $K$-sparse, the productive tagged transitions from the same
realizable $\sim_{u}$-equivalence class can be lexicographically ordered and
distinguished by means of numbers from a fixed finite range, say
$\\{1,\dots,K\\}$. An important observation is that the equivalence $\sim_{u}$
is a regular property, in the sense that one can construct, for instance, an
MSO formula $\varphi^{\sim_{u}}_{t,t^{\prime}}(y,y^{\prime})$ that holds on
input $u$ if and only if $(t,y)\sim_{u}(t^{\prime},y^{\prime})$. In
particular, this implies that unbounded sparsity can be effectively tested: it
suffices to construct the regular language consisting of every possible input
$u$ with a distinguished realizable $\sim_{u}$-equivalence class marked on it,
and check whether this language contains words with arbitrarily many marked
positions that correspond to productive tagged transitions (this boils down to
detecting special loops in a classical finite-state automaton).
###### Lemma 0.E.1
If $T$ has unbounded sparsity, then $T$ is not one-way definable.
###### Proof
The assumption that $T$ has unbounded sparsity and the definition of
$\sim_{u}$ imply that, for every $n\in\mathbb{N}$, there exist an input $u$, a
successful run $\rho$ on $u$, and $2n$ tagged transitions
$(t_{1},y_{1}),\dots,(t_{n},y_{n})$,
$(t^{\prime}_{1},y^{\prime}_{1}),\dots,(t^{\prime}_{n},y^{\prime}_{n})$ such
that the $t_{i}$’s occur before the $t^{\prime}_{j}$ in $\rho$ and the
$y^{\prime}_{i}$ are to the right of the $y^{\prime}_{j}$. Since $n$ can grow
arbitrarily, this witnesses precisely the fact that $T$ has unbounded cross-
width. Thus, by the implication $1\rightarrow 2$ of Theorem 3.1, which is
independent of $T$ being bounded-visit, we know that $T$ is not one-way
resynchronizable. ∎
Let us now show how to construct a bounded-visit transducer $\mathit{low}(T)$
with regular outputs and common guess that is equivalent to $T$, under the
assumption that $T$ is $K$-sparse for some constant $K$. Intuitively,
$\mathit{low}(T)$ simulates successful runs of $T$ on input $u$ by
shortcutting maximal vertical loops. Formally, for an input $u$ and a tagged
transition $(t,y)$, a _vertical loop_ at $(t,y)$ is any run on $u$ that starts
and ends with transition $t$ at position $y$. We will tacitly focus on
vertical loops that are realizable on $u$, exactly as we did for
$\sim_{u}$-equivalence classes. The output of a vertical loop is the word
spelled out by the productive transitions in it.
Of course, all tagged transitions in a vertical loop at $(t,y)$ are
$\sim_{u}$-equivalent to $(t,y)$. In particular, as $T$ is $K$-sparse, there
are at most $K$ productive tagged transitions in a (realizable) vertical loop,
and hence the language $L_{t,y}$ of outputs of vertical loops at $(t,y)$ is
regular. In addition, there are only finitely many languages $L_{t,y}$ for
varying $(t,y)$. This can be seen as follows: we can assume an order on the
elements of the $\sim_{u}$-class $C$ of $(t,y)$, and a strongly connected
graph with nodes corresponding to $C$ and edges reflecting the accessibility
preorder. The correctness of the graph can be checked with regular annotations
on the input, and the graph itself can be turned into an automaton accepting
$L_{t,y}$. Therefore, using common guess in $\mathit{low}(T)$, we can assume
that every position $y$ carries as annotation the language $L_{t,y}$ for each
transition $t$. By definition, $L_{t,y}$ is non-empty if and only if there is
some productive vertical loop at $(t,y)$.
Consider an arbitrary successful run $\rho$ of $T$ on $u$. Let
$\mathit{low}(\rho)$ be the run obtained by replacing, from left to right,
every maximal vertical loop at $(t,y)$ by the single transition $t$. Here,
maximality refers to the subrun relation. We call $\mathit{low}(\rho)$ the
_normalization of $\rho$_ and we observe that this is a successful,
$|\Delta|$-visit run. This means that (i) $\mathit{low}(\rho)$ can be finitely
encoded on the input as a sequence of flows of height at most $|\Delta|$, and
(ii) the language consisting of inputs annotated with such encodings is
regular.
The transducer $\mathit{low}(T)$ guesses the encoding of a normalization
$\mathit{low}(\rho)$ and uses it to simulate a possible run $\rho$ of $T$. In
particular, every time $\mathit{low}(T)$ traverses a transition $t$ from the
flow of $\mathit{low}(\rho)$ at position $y$, it outputs a word from the
language $L_{t,y}$. However, in order to simplify later the construction of a
resynchronizer $\mathcal{R}$ such that $\mathcal{R}(T)=_{o}\mathit{low}(T)$,
it is convenient that $\mathit{low}(T)$ outputs the word from $L_{t,y}$ in a
possibly different origin, which is uniquely determined by the
$\sim_{u}$-equivalence class of $(t,y)$. Formally, we define the _anchor_ of a
$\sim_{u}$-equivalence class $C$, denoted $\mathit{an}(C)$, to be the leftmost
input position $z$ such that $(t^{\prime},z)\in C$ for some transition
$t^{\prime}$. After traversing a transition $t$ from the flow at position $y$,
and before outputting a word from $L_{t,y}$, the transducer $\mathit{low}(T)$
moves to the anchor $\mathit{an}([(t,y)]_{\sim_{u}})$. Then it outputs the
appropriate word and moves back to position $y$, where it can resume the
simulation of the normalized run $\mathit{low}(\rho)$. Note that the position
$y$ can be recovered from the anchor $\mathit{an}([(t,y)]_{\sim_{u}})$ since
the elements inside the equivalence class $[(t,y)]_{\sim_{u}}$ can be
identified by numbers from $\\{1,\dots,K\\}$ (recall that $T$ is $K$-sparse),
and since the relationship between any two such elements is a regular
property. It is routine to verify that the described transducer
$\mathit{low}(T)$ is equivalent to $T$ and bounded-visit.
Let us now explain how to construct a partially bijective, regular
resynchronizer $\mathcal{R}$ that is $T$-preserving and such that
$\mathcal{R}(T)=_{o}\mathit{low}(T)$. We proceed as in the construction of
$\mathit{low}(T)$ by annotating the input word $u$ with flows that encode the
normalization $\mathit{low}(\rho)$ of a successful run $\rho$ of $T$ on $u$.
As for the output word $v$, we annotate every position $x$ of $v$ with the
productive transition $t=(q,a,v,q^{\prime})$ of $\rho$ that generated $x$. For
short, we call $t$ _the transition of $x$_. In addition, we fix an MSO-
definable total ordering on tagged transitions (e.g. the lexicographic
ordering). Then, we determine from each output position $x$ the
$\sim_{u}$-equivalence class $C=[(t,y)]_{\sim_{u}}$, where $u$ is the
underlying input, $t$ is the productive transition that generated $x$, and $y$
is its origin, and we extend the annotation of $x$ with the index of the
element $(t,y)$ inside the equivalence class $C$, according to the fixed total
ordering on tagged transitions. This number $i$ is called _the index of $x$_.
The resynchronizer $\mathcal{R}$ needs to redirect the source origin $y$ of
any output position generated by a transition $t$ to a target origin $z$ that
is the anchor of the $\sim_{u}$-equivalence class of $(t,y)$. To simplify the
explanation, we temporarily assume that the input and output are correctly
annotated as described above. By inspecting the type $\tau$ of an output
position $x$, the formula $\mathsf{move}_{\tau}(y,z)$ of $\mathcal{R}$ can
determine the transition $t$ of $x$, and enforce that
$(t,y)\sim_{u}(t^{\prime},z)$, for some transition $t^{\prime}$, and that
$(t,y)\not\sim_{u}(t^{\prime\prime},z^{\prime})$, for all $z^{\prime}<z$ and
all transitions $t^{\prime\prime}$. _Under the assumption that the input and
output annotations are correct_ , this would result in a bounded
resynchronizer $\mathcal{R}$. Indeed, for every position $z$, there exist at
most $K\cdot|\Delta|$ positions $y$ that, paired with some productive
transition, turn out to be $\sim_{u}$-equivalent to $(t^{\prime},z)$ for some
transition $t^{\prime}$. Once again, we need to further constrain the relation
$\mathsf{move}_{\tau}(y,z)$ so that it describes a partial bijection between
source and target origins (this will be useful later). For this, it suffices
to additionally enforce that $(t,y)$ is the $i$-th element in its
$\sim_{u}$-equivalence class, accordingly to the fixed total ordering on
tagged transitions, where $i$ is the index specified in the output type $\tau$
of $x$. This latter modification also guarantees that $i$ is the correct index
of $x$.
Unless we further refine our constructions, we cannot claim that they always
result in a $1$-bounded resynchronizer $\mathcal{R}$, since the above
arguments crucially rely on the assumption that the input and output
annotations are correct. However, we can apply the same trick that we used in
the proof of Theorem 3.1, to make the resynchronizer $\mathcal{R}$
“syntactically” $1$-bounded, even in the presence of badly-formed annotations.
Formally, let $\mathsf{move}_{\tau}(y,z)$ be the formula that transforms the
origins in the way described above, and define
$\mathsf{move}^{\prime}_{\tau}(y,z)~{}=~{}\mathsf{move}_{\tau}(y,z)~{}\wedge~{}\forall
y^{\prime}~{}\big{(}\mathsf{move}_{\tau}(y^{\prime},z)\rightarrow
y^{\prime}=y\big{)}.$
By construction, the above formula defines a partial bijection entailing the
old relation $\mathsf{move}_{\tau}$ (in the worst case, when the annotations
are not correct, the above formula may not hold for some pairs of source and
target origins). In addition, if the annotations are correct, then
$\mathsf{move}^{\prime}_{\tau}(y,z)$ is semantically equivalent to
$\mathsf{move}_{\tau}(y,z)$, as desired. In this way, we obtain a regular
resynchronizer
$\mathcal{R}=(\overline{I},\overline{O},\mathsf{ipar},\mathsf{opar},\mathsf{move}^{\prime}_{\tau},\mathsf{next})$
that is always $1$-bounded, no matter how we define $\mathsf{ipar}$,
$\mathsf{opar}$, and $\mathsf{next}$.
We now explain how to check that the annotations are correct. The input
annotation does not pose any particular problem, since the language of inputs
annotated with normalized runs is regular, and can be checked using the first
formula $\mathsf{ipar}$ of the resynchronizer. As for the output annotation,
correctness of the indices was already enforced by the
$\mathsf{move}^{\prime}_{\tau}$ relation. It remains to enforce correctness of
the transitions. Once again, this boils down to verifying the following
property ($\dagger$):
($\dagger$)
From here we proceed exactly as in the proof of Theorem 3.1. We observe that
Property ($\dagger$) is expressible by an MSO formula
$\varphi_{\tau,\tau^{\prime}}^{\dagger}(y,y^{\prime})$, assuming that
$\tau,\tau^{\prime}$ are the output types of $x,x+1$, that $y,y^{\prime}$ are
interpreted by the source origins of $x,x+1$, and that $x$ ranges over all
output positions. We then recall that $\mathsf{move}_{\tau}(y,z)$ and
$\mathsf{move}_{\tau}(y^{\prime},z^{\prime})$ describe partial bijections
between source and target origins, and exploit this enforce ($\dagger$) by
means of the last formula of $\mathcal{R}$:
$\mathsf{next}_{\tau,\tau^{\prime}}(z,z^{\prime})~{}=~{}\exists
y,y^{\prime}~{}\mathsf{move}_{\tau}(y,z)~{}\wedge~{}\mathsf{move}_{\tau^{\prime}}(y^{\prime},z^{\prime})~{}\wedge~{}\varphi_{\tau,\tau^{\prime}}^{\dagger}(y,y^{\prime}).$
This guarantees that all annotations are correct, and proves that
$\mathcal{R}$ is a partially bijective, regular resynchronizer satisfying
$\mathcal{R}(T)=_{o}\mathit{low}(T)$. It is also immediate to see that
$\mathcal{R}$ is $T$-preserving.
We finally prove that one-way resynchronizability of $T$ reduces to one-way
resynchronizability of $\mathit{low}(T)$, which can be effectively tested
using Theorem 3.1 since $\mathit{low}(T)$ is bounded-visit:
See 9
###### Proof
For the right-to-left implication, suppose that
$T^{\prime}=_{o}\mathcal{R}(T)$ is bounded-visit and one-way resynchronizable.
Since $T^{\prime}$ is bounded-visit, we can use the implications $1\rightarrow
2\rightarrow 3\rightarrow 4$ in Theorem 3.1 to get the existence of a
partially bijective, regular resynchronizer $\mathcal{R}^{\prime}$ that is
$T^{\prime}$-preserving and such that $\mathcal{R}^{\prime}(T^{\prime})$ is
order-preserving. By Lemma 2, there is a bounded, regular resynchronizer
$\mathcal{R}^{\prime\prime}$ that is equivalent to
$\mathcal{R}^{\prime}\circ\mathcal{R}$. In particular,
$\mathcal{R}^{\prime\prime}(T)$ is order-preserving. It remains to verify that
$\mathcal{R}^{\prime\prime}$ is also $T$-preserving. Consider any synchronized
pair $\sigma\in{[\\![T]\\!]}_{o}$. Since $\mathcal{R}$ is $T$-preserving,
$\sigma$ belongs to the domain of $\mathcal{R}^{\prime}$, and hence
$(\sigma,\sigma^{\prime})\in\mathcal{R}$ for some synchronized pair
$\sigma^{\prime}\in{[\\![T^{\prime}]\\!]}_{o}$. Since $\mathcal{R}$ is
$T^{\prime}$-preserving, $\sigma^{\prime}$ belongs to the domain of
$\mathcal{R}$, and hence there is
$(\sigma,\sigma^{\prime\prime})\in(\mathcal{R}^{\prime}\circ\mathcal{R})=\mathcal{R}^{\prime\prime}$.
This shows that $\mathcal{R}^{\prime\prime}$ is $T$-preserving, and hence $T$
is one-way resynchronizable.
For the converse direction, suppose that $T^{\prime}$ is bounded-visit, but
not one-way resynchronizable. We apply again Theorem 3.1, but now we use the
contrapositives of the implications $2\rightarrow 3\rightarrow 4\rightarrow
1$, and obtain that $T^{\prime}$ has unbounded cross-width (see Definition 2).
We also recall that
$\mathcal{R}=(\overline{I},\overline{O},\mathsf{ipar},(\mathsf{move}_{\tau})_{\tau},(\mathsf{next}_{\tau,\tau^{\prime}})_{\tau,\tau})$
is partially bijective. This means that every formula
$\mathsf{move}_{\tau}(y,z)$ defines a partial bijection from source to target
positions. A useful property of every MSO-definable partial bijection is that,
for every position $t$, it can only define boundedly many pairs $(y,z)$ with
either $y\leq t<z$ or $z\leq t<y$ — for short, we say call such a pair $(y,z)$
_$t$ -separated_. This follows from compositional properties of regular
languages. Indeed, let $\mathcal{A}$ be a deterministic automaton equivalent
to the formula that defines the partial bijection. For every pair $(y,z)$ in
the partial bijection, let $q_{y,z}$ be the state visited at position $t$ by
the successful run of $\mathcal{A}$ on the input annotated with the pair
$(y,z)$. If $\mathcal{A}$ accepted more than $|Q|$ pairs that are
$t$-separated, where $Q$ is the state space of $\mathcal{A}$, then at least
two of them, say $(y,z)$ and $(y^{\prime},z^{\prime})$, would satisfy
$q_{y,z}=q_{y^{\prime},z^{\prime}}$. But this would imply that the pair
$(y,z^{\prime})$ is also accepted by $\mathcal{A}$, which contradicts the
assumption that $\mathcal{A}$ defines a partial bijection.
We now exploit the above result to prove that the property of having unbounded
cross-width transfers from $T^{\prime}$ to $T$. Consider a cross
$(X_{1},X_{2})$ of arbitrarily large width $h$ in some synchronized pair
$\sigma=(u,v,\mathit{orig})$ of $T^{\prime}$. Without loss of generality,
assume that all positions in $X_{1}\cup X_{2}$ have the same type $\tau$. Let
$Z_{i}=\mathit{orig}(X_{i})$, for $i=1,2$, and $t=\max(Z_{2})$. By definition
of cross, we have $X_{1}<X_{2}$ and $Z_{2}\leq t<Z_{1}$. Recall that
$\mathsf{move}_{\tau}$ defines a partial bijection, and that this implies that
there are only boundedly many pairs of source-target origins that are
$t$-separated, say $(y_{1},z_{1}),\dots,(y_{k},z_{k})$ for a constant $k$ that
only depends on $\mathcal{R}$. Moreover, since
$\mathcal{R}(T)=_{o}T^{\prime}$, the positions in $Z_{i}$ can be seen as
target origins of the formula $\mathsf{move}_{\tau}$ of $\mathcal{R}$. Now,
let $X^{\prime}_{i}=X_{i}\setminus\mathit{orig}^{-1}(\\{z_{1},\dots,z_{k}\\}$
and $Y^{\prime}_{i}=\mathit{orig}^{\prime}(X^{\prime}_{i})$, for any
synchronized pair $\sigma^{\prime}=(u,v,\mathit{orig}^{\prime})$ such that
$(\sigma,\sigma^{\prime})\in\mathcal{R}$. By construction, we have
$X^{\prime}_{1}<X^{\prime}_{2}$ and $Y^{\prime}_{2}\leq t<Y^{\prime}_{1}$ (the
latter condition follows from the fact that the source origins from
$Y^{\prime}_{i}$ can only be moved to target origins on the same side w.r.t.
$t$). This means that $(X^{\prime}_{1},X^{\prime}_{2})$ is a cross of width
$h-k$. As $h$ can be taken arbitrarily large and $k$ is constant, this proves
that $T$ has unbounded cross-width as well.
Finally, by the contrapositive of the implication $1\rightarrow 2$ of Theorem
3.1 (which does not need the assumption that $T$ is bounded-visit), we
conclude that $T$ is not one-way resynchronizable. ∎
Summing up, the algorithm that decides whether a given two-way transducer $T$
is one-way resynchronizable first verifies that $T$ is $K$-sparse for some $K$
(if not, it claims that $T$ is not one-way resynchronziable), then it
constructs a bounded-visit transducer $\mathit{low}(T)$ equivalent to $T$, and
finally decides whether $\mathit{low}(T)$ is one-way resynchronizable (which
happens if and only if $T$ is one-way resynchronizable). This concludes the
proof of Theorem 3.2. ∎
|
NANOGrav signal from first-order confinement/deconfinement phase transition in
different QCD matters
Shou-Long Li$\,{}^{1}$, Lijing Shao$\,{}^{2,3}$, Puxun Wu$\,{}^{1}$ and
Hongwei Yu$\,{}^{1}$
$\,{}^{1}$Department of Physics and Synergetic Innovation Center for Quantum
Effect and Applications, Hunan Normal University, Changsha 410081, China
$\,{}^{2}$Kavli Institute for Astronomy and Astrophysics, Peking University,
Beijing 100871, China
$\,{}^{3}$National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100012, China
ABSTRACT
Recently, an indicative evidence of a stochastic process, reported by the
NANOGrav Collaboration based on the analysis of 12.5-year pulsar timing array
data which might be interpreted as a potential stochastic gravitational wave
signal, has aroused keen interest of theorists. The first-order color charge
confinement phase transition at the QCD scale could be one of the cosmological
sources for the NANOGrav signal. If the phase transition is flavor dependent
and happens sequentially, it is important to find that what kind of QCD matter
in which the first-order confinement/deconfinement phase transition happens is
more likely to be the potential source of the NANOGrav signal during the
evolution of the universe. In this paper, we would like to illustrate that the
NANOGrav signal could be generated from confinement/deconfinement transition
in either heavy static quarks with a zero baryon chemical potential, or quarks
with a finite baryon chemical potential. In contrast, the gluon confinement
could not possibly be the source for the NANOGrav signal according to the
current observation. Future observation will help to distinguish between
different scenarios.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
## 1 Introduction
Recently, the North American Nanohertz Observatory for Gravitational Wave
(NANOGrav) Collaboration [1] has reported an analysis of 12.5-year pulsar
timing array (PTA) data. According to the analysis [1], a possible evidence is
found for a stochastic common-spectrum process which may be interpreted as a
gravitational wave (GW) signal with its frequency in $1$–$10$ nHz, and its
average GW energy density
$\langle\Omega_{\textup{GW}}h^{2}\rangle_{\textup{NANOGrav}}\sim 10^{-10}$
with an almost flat GW spectrum $\Omega_{\rm GW}h^{2}\sim f^{-1.5\pm 0.5}$ at
1-$\sigma$ level. Although the observational results need further analyses,
such as a joint analysis with data from the other PTA collaborations (such as
EPTA and PPTA) [2, 3, 4], the potentiality of its being a stochastic GW
background (SGWB) signal has aroused keen interest of theorists [5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32]. On the one hand, one may explain it as a potential SGWB
signal by considering different astrophysical and cosmological sources. On the
other hand, the possible SGWB signal could serve as a potential new probe to
studying new physics.
During the evolution of the universe, as the temperature decreases, the
universe may undergo several phase transitions from the metastable vacuums to
stable vacuums. At the quantum chromodynamics (QCD) energy scale, there are
two important transitions, i.e. the spontaneous chiral symmetry breaking and
the color charge confinement. For the confinement transition, the universe
will go from a quark-gluon plasma (QGP) phase to a hadron phase as the
temperature decreases. The numerical lattice simulation shows [33, 34, 35]
that the QCD transition is likely a crossover for three dynamical quark
flavors when the baryon and charge chemical potential is negligible, i.e., in
the absence of the baryon and lepton asymmetries. If the transition is first-
order, GWs could be produced due to the violent process of vacuum bubble
nucleation and subsequent bubble collisions [36, 42, 37, 38, 39, 40, 41],
sound waves [43, 44, 45, 46] and magnetohydrodynamic (MHD) turbulence [47, 48,
49, 50, 51, 52, 53]. The GWs produced are within the frequency range of the
PTA observation [54, 55, 56]. The three processes in the first-order phase
transition could be potential sources of the NANOGrav signal [19, 17, 22].
Therefore, the next issue is what are the QCD matters in which the first-order
phase transition can occur. In this regard, let us note that for the case of a
zero baryon chemical potential, the transition is first-order in a non-
dynamical (static) heavy quark system [57, 58]. For the case of a finite
baryon chemical potential, a large lepton asymmetry might affect the dynamics
of the QCD phase transition in a way to render it first-order in the early
universe [59]. Besides, for a pure gluon system, the first-order phase
transition might occur as well.
Different QCD matter has a different temperature of phase transition which is
a crucial parameter determining the features of the GWs produced. Then a
question arises naturally as to what kind of QCD matter in which the first-
order confinement/deconfinement phase transition happens is more likely to be
the potential source of the NANOGrav signal during the evolution of the
universe. In this paper, we will try to answer this question. We consider
three types of QCD matter systems: (i) heavy static quarks with a zero baryon
chemical potential, (ii) quarks with a finite baryon chemical potential, and
(iii) a pure gluon system. We match the NANOGrav signal with the GW spectra
from the first-order confinement/deconfinement phase transitions in three QCD
matter systems with holographic models [56, 60, 61]. We show that the GW
spectra from the phase transitions in pure quark systems, regardless of
whether the chemical potential is finite or zero, could explain the NANOGrav
signal according to the current observation. In contrast, the signal could not
possibly come from the gluon confinement.
## 2 GWs from first-order phase transitions
When first-order phase transition occurs, the universe transfers from a
metastable vacuum to a stable vacuum. This process can be described as bubble
nucleation. Generally, the GW signal from cosmological first-order phase
transitions mainly comes from three processes: collisions of vacuum bubble
walls, sound waves, and the MHD turbulence in the plasma [36, 42, 37, 38, 39,
40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53]. So, the total energy
density of the GW, which is a sum of the three, is given by
$h^{2}\Omega(f)\simeq
h^{2}\Omega_{\textup{en}}(f)+h^{2}\Omega_{\textup{sw}}(f)+h^{2}\Omega_{\textup{tu}}(f)\,,$
(2.1)
where $h=H_{0}/100\ \textup{km}^{-1}\ \textup{s}\ \textup{Mpc}$ in which
$H_{0}$ is the Hubble constant today. $h^{2}\Omega_{\textup{en}}$,
$h^{2}\Omega_{\textup{sw}}$ and $h^{2}\Omega_{\textup{tu}}$ are the
contributions from bubble collision, sound waves, and the turbulence,
respectively, which are given by [41, 42, 46, 52]
$\displaystyle h^{2}\Omega_{\textup{en}}$ $\displaystyle=$ $\displaystyle
3.6\times
10^{-5}\left(\frac{H_{\ast}}{\beta}\right)^{2}\left(\frac{\kappa_{1}\alpha}{1+\alpha}\right)^{2}\left(\frac{10}{g_{\ast}}\right)^{\frac{1}{3}}\left(\frac{0.11v_{w}^{3}}{0.42+v_{w}^{2}}\right)S_{\textup{en}}\,,$
$\displaystyle h^{2}\Omega_{\textup{sw}}$ $\displaystyle=$ $\displaystyle
5.7\times
10^{-6}\left(\frac{H_{\ast}}{\beta}\right)\left(\frac{\kappa_{2}\alpha}{1+\alpha}\right)^{2}\left(\frac{10}{g_{\ast}}\right)^{\frac{1}{3}}v_{w}S_{\textup{sw}}\,,$
$\displaystyle h^{2}\Omega_{\textup{tu}}$ $\displaystyle=$ $\displaystyle
7.2\times
10^{-4}\left(\frac{H_{\ast}}{\beta}\right)\left(\frac{\kappa_{3}\alpha}{1+\alpha}\right)^{2}\left(\frac{10}{g_{\ast}}\right)^{\frac{1}{3}}v_{w}S_{\textup{tu}}\,.$
(2.2)
In above equations, $\beta$ is the inverse time duration of the phase
transition, $v_{w}$ is the velocity of bubble wall, $g_{\ast}$ and $H_{\ast}$
represent the number of active degrees of freedom and Hubble parameter at the
time of production of GWs respectively,
$\alpha=30\epsilon_{\ast}/(\pi^{2}g_{\ast}T_{\ast}^{4})$ is the ratio of the
vacuum energy density $\epsilon_{\ast}=\big{[}\Delta F(T)-Td\Delta
F(T)/dT\big{]}\Big{|}_{T=T_{\ast}}$ over radiation energy density where
$T_{\ast}$ and $\Delta F(T)$ are the temperature of the thermal bath at time
of production of GWs and the difference of the free energy between two phases
respectively, and $S_{\textup{en}},S_{\textup{sw}}$, and $S_{\textup{tu}}$ are
spectral shapes of GWs which are characterized from numerical fits as [41, 42,
46, 52]
$\displaystyle S_{\textup{en}}$ $\displaystyle=$
$\displaystyle\frac{3.8\left(\frac{f}{f_{\textup{en}}}\right)^{2.8}}{1+2.8\left(\frac{f}{f_{\textup{en}}}\right)^{3.8}}\,,$
$\displaystyle S_{\textup{sw}}$ $\displaystyle=$
$\displaystyle\left(\frac{f}{f_{\textup{sw}}}\right)^{3}\left(\frac{7}{4+3\left(\frac{f}{f_{\textup{sw}}}\right)^{2}}\right)^{\frac{7}{2}}\,,$
$\displaystyle S_{\textup{tu}}$ $\displaystyle=$
$\displaystyle\frac{\left(\frac{f}{f_{\textup{tu}}}\right)^{3}}{\left(1+\frac{f}{f_{\textup{tu}}}\right)^{\frac{11}{3}}\left(1+\frac{8\pi
f}{h_{\ast}}\right)}\,,$ (2.3)
where $h_{\ast}$ is Hubble rate at $T_{\ast}$,
$f_{\textup{en}},f_{\textup{sw}}$ and $f_{\textup{tu}}$ are peak frequencies
in three cases, which are given by [39, 42, 52]
$\displaystyle h_{\ast}$ $\displaystyle=$ $\displaystyle 11.2\ \textup{nHz}\
\left(\frac{T_{\ast}}{100\textup{MeV}}\right)\left(\frac{g_{\ast}}{10}\right)^{\frac{1}{6}}\,,$
$\displaystyle f_{\textup{en}}$ $\displaystyle=$ $\displaystyle 11.2\
\textup{nHz}\
\left(\frac{0.62}{1.8-0.1v_{w}+v_{w}^{2}}\right)\left(\frac{\beta}{H_{\ast}}\right)\left(\frac{T_{\ast}}{100\textup{MeV}}\right)\left(\frac{g_{\ast}}{10}\right)^{\frac{1}{6}}\,,$
$\displaystyle f_{\textup{sw}}$ $\displaystyle=$ $\displaystyle 12.9\
\textup{nHz}\
\left(\frac{1}{v_{w}}\right)\left(\frac{\beta}{H_{\ast}}\right)\left(\frac{T_{\ast}}{100\textup{MeV}}\right)\left(\frac{g_{\ast}}{10}\right)^{\frac{1}{6}}\,,$
$\displaystyle f_{\textup{tu}}$ $\displaystyle=$ $\displaystyle 18.4\
\textup{nHz}\
\left(\frac{1}{v_{w}}\right)\left(\frac{\beta}{H_{\ast}}\right)\left(\frac{T_{\ast}}{100\textup{MeV}}\right)\left(\frac{g_{\ast}}{10}\right)^{\frac{1}{6}}\,.$
(2.4)
In Eq. (2.2), $\kappa_{1},\kappa_{2}$, and $\kappa_{3}$ are the fractions of
the vacuum energy converted to the kinetic energy of the bubbles, bulk fluid
motion, and the MHD turbulence, respectively. These factors are model-
dependent. In this work, we consider two cases of bubble: Jouguet detonations
and non-runaway bubbles. For the case of Jouguet detonations [62, 42, 39, 63,
64, 46], we have
$\displaystyle\kappa_{1}$ $\displaystyle=$
$\displaystyle\frac{0.715\alpha+0.181\sqrt{\alpha}}{1+0.715\alpha}\,,$
$\displaystyle\kappa_{2}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\alpha}}{0.135+\sqrt{\alpha+0.98}}\,,$
$\displaystyle\kappa_{3}$ $\displaystyle=$ $\displaystyle 0.05\kappa_{2}\,,$
$\displaystyle v_{w}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{1/3}+\sqrt{\alpha^{2}+2\alpha/3}}{1+\alpha}\,,$
(2.5)
and for the case of non-runaway bubbles [64, 42, 46, 39],
$\displaystyle\kappa_{1}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle\kappa_{2}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{0.73+0.083\sqrt{\alpha}+\alpha}\,,$
$\displaystyle\kappa_{3}$ $\displaystyle=$ $\displaystyle 0.05\kappa_{2}\,,$
$\displaystyle v_{w}$ $\displaystyle=$ $\displaystyle 0.95\,.$ (2.6)
With all these expressions, there are still four unknown parameters in Eq.
(2.1), i.e., $g_{\ast},\beta/H_{\ast}$, $T_{\ast}$ and $\alpha$, which
characterize the first-order cosmological QCD phase transition. Generally, for
QCD phase transitions, the temperature $T_{\ast}$ is around several hundreds
MeV, of which the concrete value depends on the types of the QCD matter and
the phase transition. We will obtain, in the following section, $T_{\ast}$ for
the phase transition in different QCD matters by the method of the holographic
QCD. One can also calculate $\alpha$ from different holographic models. For
simplicity, we assume that the GW can be generated soon after the
confinement/deconfinement phase transition occurs, so $T_{\ast}$ is
approximated by the critical temperature of the phase transition. Besides, we
fix $g_{\ast}$ and $\beta/H_{\ast}$ at their typical values which could be
chosen as $g_{\ast}=10$ and $\beta/H_{\ast}=10$ [56, 60, 61] at the QCD scale.
## 3 GWs from holographic QCD models and the NANOGrav signal
In this section, we will obtain the GW spectra from the first-order
confinement/deconfinement transition in heavy static quarks with a zero baryon
chemical potential, quarks with a finite baryon chemical potential, and a pure
gluon system, and match them with the NANOGrav signal. First, we start with
finding the corresponding critical temperatures by use of holographic QCD
models.
Following the Anti-de Sitter/conformal field theory (AdS/CFT) correspondence
principle [65, 66, 67], AdS/QCD offers some new insights to the non-
perturbative hadron dynamics from the dual gravitational field [68].
Especially, the first-order confinement/deconfinement phase transitions could
be interpreted by Hawking-Page (HP) phase transitions [69] in five-dimensional
spacetime in the AdS/QCD models [70], where the high-temperature QGP
corresponds to the AdS black hole, while the low-temperature hadron
corresponds to the thermal AdS space. In the absence of the baryon chemical
potential, the transition temperature calculated by the soft-wall model is
consistent with numerical results [70]. In the case of the finite baryon
chemical potential, AdS/QCD models can also give a good explanation of the
phase transition [71, 72, 73, 74], while the standard lattice QCD simulation
suffers from the famous sign problem [75, 76, 77] and could not provide many
useful results. The GW produced by the first-order QCD phase transition was
estimated in the case of the heavy static quark system with a zero baryon
chemical potential via hard-wall and soft-wall models of AdS/QCD in Ref. [56]
for the first time, and then studied in the finite chemical potential system
[60, 78] and pure gluon system [61] via different models. It is also worth
mentioning that the initial idea of explaining GWs generated from cosmological
phase transitions by holographic method could be traced back to Randall and
Servant’s seminal work [79] in 2006 to the best of our knowledge.
We consider three types of QCD matter systems in five different holographic
models: (i) heavy static quarks with a zero baryon chemical potential in hard-
wall model $S_{1}$ and (ii) soft-wall model $S_{2}$, (iii) quarks with a
finite baryon chemical potential in hard-wall model $S_{3}$ and (iv) soft-wall
model $S_{4}$, and (v) pure gluons in the quenched dynamical holographic model
$S_{5}$. The corresponding five-dimensional gravitational actions are given by
[70, 60, 56, 61]
$\displaystyle S_{1}$ $\displaystyle=$ $\displaystyle\frac{1}{2\kappa^{2}}\int
d^{5}x\sqrt{-g}\left(R+\frac{12}{\ell^{2}}\right)\,,$ (3.1) $\displaystyle
S_{2}$ $\displaystyle=$ $\displaystyle\frac{1}{2\kappa^{2}}\int
d^{5}x\sqrt{-g}e^{-\phi}\left(R+\frac{12}{\ell^{2}}\right)\,,$ (3.2)
$\displaystyle S_{3}$ $\displaystyle=$ $\displaystyle\int
d^{5}x\sqrt{-g}\left[\frac{1}{2\kappa^{2}}\left(R+\frac{12}{\ell^{2}}\right)-\frac{1}{4g_{5}^{2}}F^{2}\right]-\frac{1}{\kappa^{2}}\int
d^{4}x\sqrt{h}\nabla_{\mu}n^{\mu}\,,$ (3.3) $\displaystyle S_{4}$
$\displaystyle=$ $\displaystyle\frac{1}{2\kappa^{2}}\int
d^{5}x\sqrt{-g}e^{-\phi}\left[\frac{1}{2\kappa^{2}}\left(R+\frac{12}{\ell^{2}}\right)-\frac{1}{4g_{5}^{2}}F^{2}\right]-\frac{1}{\kappa^{2}}\int
d^{4}x\sqrt{h}e^{-\phi}\nabla_{\mu}n^{\mu}\,,$ (3.4) $\displaystyle S_{5}$
$\displaystyle=$ $\displaystyle\frac{1}{2\kappa^{2}}\int
d^{5}x\sqrt{-g}e^{-2\varphi}\left(R-\frac{4}{3}\partial_{\mu}\varphi\partial^{\mu}\varphi-V(\varphi)\right)\,,$
(3.5)
where $\kappa^{2}=8\pi G$, $g_{5}$ is coupling constant, $\ell$ is the radius
of five-dimensional AdS space, $g$ and $h$ are determinants of bulk and
boundary metrics respectively, $n^{\mu}$ is the unit vector normal to the
hypersurface, $\phi$ is a non-dynamical dilaton, and $\varphi$ is a dynamical
dilaton. The confinement/deconfinement transition in the cases of heavy static
quarks with a zero baryon chemical potential and pure gluons are analogous to
the HP transition between the static AdS black hole and the thermal AdS
vacuum. For the case of quarks with a finite baryon chemical potential, the
confinement/deconfinement transition is analogous to the HP transition between
the charged AdS black hole and the thermal charged AdS vacuum, and the
chemical potential is related to the electric charge of the black hole. For a
specific model, one can calculate the free energies of the black hole and AdS
vacuum respectively, and obtain the difference of the free energy $\Delta F$
between two phases. Then one can obtain the value of the temperature
$T_{\ast}$ and $\alpha$ via some holographic techniques. Here, we refer
readers to Refs. [70, 60, 56, 61] for detailed calculations and list the
corresponding critical temperatures in these models in Table 1.
Table 1: Critical temperatures from five holographical QCD models. QCD matters | Holographic QCD models | Temperature
---|---|---
heavy static quarks with | hard wall | 122 MeV [70, 56]
a zero chemical potential | soft wall | 191 MeV [70, 56]
quarks with a finite | hard wall | 112 MeV [60]
chemical potential | soft wall | 192 MeV [60]
pure gluons | quenched dynamical holographic QCD | 255 MeV [61]
We assume that GW is generated quickly after the phase transition occurs and
temperature $T_{\ast}$ is approximated as the critical phase transition
temperature. Now we match the NANOGrav results with the GW produced from the
confinement/deconfinement phase transition in different holographic models. We
plot the GW spectra in two bubble models: Jouguet detonations and non-runaway
bubbles. The results are illustrated respectively in Fig. 1 and Fig. 2.
Figure 1: GW spectra from QCD matter confinement/deconfinement phase
transition in the Jouguet detonation bubble case from five holographical
models. The results are compared with the current sensitivity of NANOGrav
12.5-yr [1, 8], EPTA [2], and PPTA [3].
Figure 2: Same as Fig. 1, for the non-runaway bubble case.
From Fig. 1 and Fig. 2, we show that the power spectra of GWs from the quark
confinement phase transitions enter the 95% confidence interval from the
NANOGrav 12.5-yr observation [1, 8], which indicates that the
confinement/deconfinement phase transition in pure quark systems (cases of
heavy static quarks with a zero baryon chemical potential and quarks with a
finite baryon chemical potential) could possibly be the potential cosmological
sources of the NANOGrav signal for both the Jouguet detonation and non-runaway
bubble cases. The power spectra of GWs calculated from different holographic
models (hard wall and soft wall) in a specific QCD matter system are
different, but the difference does not change the conclusion. In contrast, the
confinement/deconfinement transition in the pure gluon system could not
possibly be the source of the NANOGrav signal. Since the critical phase
transition temperatures for the cases of a finite chemical potential and a
zero chemical potential are very close, the chemical potential has little
effect on the power spectrum of GWs, and the quark confinement dominates the
cosmological QCD transition according to the current NANOGrav observation. But
these conclusions need to be further supported by more accurate observations
of GWs in the future. We also calculate the spectrum indices from different
holographic models, and find that the values are similar in the same bubble
model. The values are about 2.78 and 2.91 for the Jouguet and non-runaway
cases, respectively. These values could be used to check our conclusions with
more accurate future observations.
## 4 Conclusions and Discussions
In this paper, we have showed that a possible stochastic common-spectrum
process reported by the NANOGrav Collaboration based on the 12.5-yr PTA data
can be explained potentially as a GW signal from the first-order cosmological
confinement/deconfinement phase transition in the cases of (i) heavy static
quarks with a zero baryon chemical potential and (ii) quarks with a finite
baryon chemical potential. We also find that the gluon confinement could not
possibly be the potential source of the NANOGrav signal based on the current
observation. We match the GW spectra from the first-order phase transition in
five different holographic QCD models with the NANOGrav signal. By considering
both the Jouguet detonation and non-runaway bubble growth models, we find that
the GW spectra from the confinement/deconfinement phase transition in pure
quark systems, irrespective of whether the baryon chemical potential is finite
or zero, enter the 95% confidence interval from the NANOGrav 12.5-yr
observation [1, 8]. The baryon chemical potential has little influence on the
power spectra of GWs produced by confinement/deconfinement phase transitions
since the phase transition temperatures in both the finite baryon chemical
potential case and zero chemical potential case are very close.
We must point out that we have assumed in our analysis that GWs are generated
soon after the phase transition happens, so the temperature at which the GWs
are produced is approximately the transition temperature. Although this is an
acceptable assumption, it remains interesting to find a more credible
holographic method to calculate the temperature $T_{\ast}$, analogous to what
was done in Ref. [79]. Moreover, it is also worth considering the
confinement/deconfinement phase transitions in possible QCD matter systems
other than those we have looked at in the present paper and chiral symmetry
breaking phase transitions. We would like to leave these to future studies.
## Acknowledgement
We are grateful to Jie-Wen Chen, Muyang Chen, Chengjie Fu, Yong Gao, Long-
Cheng Gui, Hongbo Li, Chang Liu, Jing Liu, H. Lü, Xueli Miao, Shi Pi, Shao-
Jiang Wang, Hao Wei, Rui Xu, and Junjie Zhao for useful discussions. SL thanks
Lijing for his warm hospitality during the visit to KIAA. SL, PW and HY were
supported in part by the NSFC under Grants No. 11947216, No. 11690034, No.
11805063, No. 11775077 and No. 12075084, and China Postdoctoral Science
Foundation 2019M662785. LS was supported by the National SKA Program of China
(2020SKA0120300), the National Natural Science Foundation of China (11975027,
11991053, 11721303), the Young Elite Scientists Sponsorship Program by the
China Association for Science and Technology (2018QNRC001), and the Max Planck
Partner Group Program funded by the Max Planck Society.
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|
# Improved Signed Distance Function for 2D Real-time SLAM and Accurate
Localization
Xingyin Fu1, Zheng Fang2, Xizhen Xiao1, Yijia He1, Xiao Liu1 1Xingyin Fu,
Xizhen Xiao, Yijia He, and Xiao Liu are with the Megvii (Face++) Technology
Inc., Beijing, China, fuxingyin, xiaoxizhen, heyijia<EMAIL_ADDRESS>Fang is with the Faculty of Robot Science and Engineering, Northeastern
University, Shenyang, China, 110169<EMAIL_ADDRESS>
###### Abstract
Accurate mapping and localization are very important for many industrial
robotics applications. In this paper, we propose an improved Signed Distance
Function (SDF) for both 2D SLAM and pure localization to improve the accuracy
of mapping and localization. To achieve this goal, firstly we improved the
back-end mapping to build a more accurate SDF map by extending the update
range and building free space, etc. Secondly, to get more accurate pose
estimation for the front-end, we proposed a new iterative registration method
to align the current scan to the SDF submap by removing random outliers of
laser scanners. Thirdly, we merged all the SDF submaps to produce an
integrated SDF map for highly accurate pure localization. Experimental results
show that based on the merged SDF map, a localization accuracy of a few
millimeters (5mm) can be achieved globally within the map. We believe that
this method is important for mobile robots working in scenarios where high
localization accuracy matters.
## I INTRODUCTION
Nowadays, autonomous robots are widely used in industrial and logistic areas,
etc. To achieve autonomous navigation in those natural environments, the
robots usually need to firstly map the environments using onboard sensors and
then localize themselves in the map. For mapping the environments,
Simultaneously Localization And Mapping (SLAM) is commonly used for mobile
robots. Currently, many 2D mapping methods have been proposed , for instance,
GMapping [1], Cartographer [2] and Hector SLAM [3]. However, those methods
usually are grid-based methods that build maps in discrete grids. The accuracy
of the map highly depends on the resolution of the grid. For memory saving
issue, most time the resolution is set to 2-5 centimeters. Unfortunately, this
resolution is not good enough for highly accurate mapping and it is also not
good for highly accurate localization based on the map.
Localization based on a known map without mapping is usually called pure
localization. In the past two decays, many excellent localization methods had
been proposed, for example, AMCL[4], NDT-MCL[5] and Kalman Filter[6]. Among
those methods, AMCL that uses KLD-sampling idea is the most widely used method
for 2D pure localization. This method is very robust in various environments.
However, the localization accuracy of this method is usually around 2-5cm.
This is good enough for service robots but is not for industrial applications.
For industrial application, mobile robots usually need millimeter accuracy to
accomplish docking or manipulation.
Figure 1: The same SDF map is used for localization, mapping and pure
localization.
To achieve highly accurate mapping and localization, in this paper, we propose
a new 2D laser SLAM system. In our system, an SDF map instead of an occupancy
grid map is built when the robot works around in an unknown environment. When
the robot is working in the environment, the same SDF mapping generated is
also used for pure localization. Experimental results show that the global
localization accuracy of 5mm can be achieved, which is significantly higher
compared to AMCL. Our system is also considerably more efficient.
The rest of this paper is structured as follows. Section II describes the
related work. Section III presents the improved real-time 2D SLAM system. The
highly accurate SDF-based pure localization is presented in Section IV.
Experiments are introduced in Section III-B and we conclude in Section VI.
## II RELATED WORK
The SLAM system is often used to build a map of unknown environments. If a
robot works in a map-based environment, a pure localization module is
required. Typically, the SLAM system calculates the pose by aligning the laser
scan with the building map [3, 7, 2] or the points of previous frame [8].
Hector SLAM [3] and Cartographer [2] build probability maps, and laser scanner
pose is calculated by minimizing the probability costs of hit points. To
prevent the optimization get stuck in a local minimum, Hector SLAM builds a
multi-resolution map and uses a pyramid approach. And Cartographer uses
correlative scan matching [9] and integrates other sensors (odometry, IMU,
etc.) to pre-calculate the initial pose. The 2D-SDF-SLAM method [7] proposed
by Fossel et al. uses Sign Distance Function (SDF) for real-time localization
and mapping. SDF is popularly used in 3D mapping systems, for instance,
Kinectfusion [10, 11], to fuse depth received from consumer depth cameras such
as Microsoft Kinect and Google Tango. The 2D-SDF-SLAM method improves the
mapping method to handle the situation where the laser beam is perpendicular
to the surface, which is often seen in 2D laser mapping. We also notice that
Cartographer [2] implements an SDF mapping module following KinectFusion-style
map updates. PL-ICP [8] is a variant of ICP algorithm [12, 13] for 2D
registration. The algorithm first searches for the correspondences between two
sets of points. The pose between two frames is calculated by minimizing the
point-to-line distance error of all correspondences.
Particle filters [14] and pose graph optimization [15] are two commonly used
methods for the backend in 2D laser SLAM systems. Gmapping [14] and
Cartographer [2] are two typical systems based on particle filters and pose
graph optimization. Particle filter generates possible particle statuses
according to the pose predicted with other sensors, such as wheel odometry.
And the predicted poses are evaluated by aligning laser scan with the 2D map
to reduce error in system localization and mapping. Cartographer reduces
accumulated pose error by building submaps. And loop closures are detected by
matching frame points with previously built submap. Accumulated system
tracking error is distributed with sparse pose graph optimization [15].
Occupancy grid map is the most commonly used map in 2D robot localization and
path planning. An environment is represented by fixed-sized cells, and the
value stored in each cell represents the probability of the cell being an
obstacle. The occupancy map can be built with particle filter [14] or graph-
based optimization systems [3, 2]. Currently, AMCL based pure localization
using an occupancy grid is the most commonly used 2D pure localization
solution. However, the occupancy grid map can only represent the environment
with the accuracy of map resolution, This may reduce the system localization
and mapping accuracy.
Röwekämper et al. [16] achieve a few millimeters localization accuracy (5mm)
at the taught-in locations by combining AMCL and PL-ICP. Overall the system is
localized based on AMCL. The PL-ICP is only used when a robot is close to a
taught-in location. PL-ICP calculates pose by registering the current frame
with a neighbor reference keyframe. Switching between AMCL and PL-ICP may
cause trajectory discontinuities. And the system achieves millimeters
localization accuracy only when the system is close to a taught-in location.
Overall the system achieves 5cm localization accuracy with AMCL.
## III 2D REAL-TIME SLAM
### III-A Improved SDF-Base Mapping
We found that an accurate SDF map is especially important for achieving a
highly accurate localization. The 2D-SDF mapping method proposed by Fossel et
al. [7] achieves better results in building 2D laser maps than KinectFusion-
style map updates. Our SDF mapping approach is built upon the 2D-SDF mapping
method. We mainly make the following three improvements.
#### III-A1 Update SDF Map with Extended Neighbor Points
Since the laser scanner acquires laser points at equal angular intervals, the
points near the laser center is dense, while the far points are sparse. The
points away from the center are more conducive to improving the accuracy of
the pose angle. We improve the 2D-SDF-SLAM method for far away point mapping.
Figure 2: The search area is gradually extended to collect more points for
calculating the regression line.
To update the map with a new frame, following the 2D-SDF-SLAM, we first gather
the points falling in each cell and calculate the Deming-regression line [17]
using the points collected in each cell. At least three points are used when
calculating the regression line. If the point number collected in one cell is
fewer than three, the points hitting in the neighbor cells are also collected.
We gradually expand the scope to collect more points. As displayed in Figure
2, only one point fallen in the cell $c$. Therefore, we expand the search area
to collect more points. The points hit in the nearest eight cells around the
cell $c$ are also collected. However, only one additional point was found in
the nearest eight cells. Therefore, we expanded the search area again to the
green box. At the second extension, a total of four points were collected. The
four points collected were used to compute the regression line. And the
regression line was used to update the SDF values of the cell $c$ and its
neighbors.
To update the SDF values of neighbor cells, the cells whose distance is less
than $K$ are collected as indicated with the black circle in Figure 2. $K$ is
set to the truncation distance of the SDF map. We project the centers of the
cells enclosed in the black circle to the regression line and calculate the
projection point. We update the cells whose projection point on the regression
line falling within the range
$(c_{x}\pm(1+0.5*\rm{e})*\rm{r},c_{y}\pm(1+0.5*\rm{e})*\rm{r})$, where
$(c_{x},c_{y})$ indicates the 2D coordinate of the cell that causes the
update, and $e$ is the number of times that we expands the search area. $r$ is
the map resolution. More extensions will lead to a broader update range.
Figure 3: (a) Map update results for long distance cells. (b) is the map built
with expansion and (c) is without expansion.
We found that expanding the update scope is particularly important for
building high-resolution maps (for instance, 3cm or 5cm resolution). And it is
also crucial for updating the values of long-distance cells because of the
sparse hit points as displayed in Figure 3. As can be seen from the figure,
the points displayed in the red box are far away from the laser origin, The
updated result of our method is displayed in (b). The map is densely updated
even in places where there are only sparse hit points. In contrast, (c) is the
results updated without expansion.
In general, if less than three points are collected, the search area is only
allowed to expand once for map resolution 10cm to 20cm. For map resolution 5cm
or less, at most three expansion times are used. If less than two points are
collected when the maximum expansion is reached, we will give up the updates.
The collected points are used to calculate the regression line. The SDF values
of the cells are updated with the regression line as follows:
$\displaystyle\mathbf{F}_{k}(\mathbf{c})=\frac{\mathbf{W}_{k-1}(\mathbf{c})\mathbf{F}_{k-1}(\mathbf{c})+\mathbf{W}_{t}(\mathbf{c})\mathbf{F}_{t}(\mathbf{c})}{\mathbf{W}_{k-1}(\mathbf{c})+\mathbf{W}_{t}(\mathbf{c})},$
(1)
$\displaystyle\mathbf{W}_{k}(\mathbf{c})=\rm{min}(\mathbf{W}_{k-1}(\mathbf{c})+\mathbf{W}_{t}(\mathbf{c}),\
\mathbf{W}_{max})$
where $\mathbf{F}_{k-1}(c)$ is the previous accumulated SDF value of cell $c$,
and $\mathbf{W}_{k-1}(c)$ is the accumulated weight. $\mathbf{F}_{k}(c)$ is
the current update SDF value. $\mathbf{F}_{k}(c)$ is set as the distance of
the cell to the regression line. If the distance from laser origin to the cell
along the line normal is greater than the distance from the origin to the
line, $\mathbf{F}_{k}(c)$ is set to negative, otherwise it is positive.
$\mathbf{W}_{k}(c)$ is the current update weight. We found that there is
almost no difference of using different strategies for setting the weight, and
simply setting the current update weight to 1 works fine. We set the maximum
weight $\mathbf{W}_{max}$ to 10 in our system.
#### III-A2 Update Free Space
Figure 4: Update free space of an SDF map.
Typically, an occupancy map contains free space, unknown space and obstacle
space. We found that building free space is particularly important for
reducing the impact of dynamic objects. Dynamic objects maybe someone or a
robot walking around. Updated free space is resistant with dynamic scenes to
arise in the SDF map, which is important to reduce the outliers caused by the
dynamic scene when minimizing the cost functions.
Usually, free space is updated by starting from the laser origin and marching
alone a laser beam for a distance of $d_{l}-t_{d}$, where $d_{l}$ denotes the
hit distance of a laser point and $t_{d}$ is the truncation distance of the
SDF map. Since the laser beam may not be perpendicular to the geometry
surface, updating cells along the line from the point $d_{l}$ to $d_{l}-t_{d}$
may result in incorrect free space update. We propose a new method to
determine the free space.
As displayed in Figure 4, the gray cells are updated as free space. The free
space points are updated from the laser origin alone the ray to the point
$p_{i}$, where $p_{i}$ is the point at which laser beam $l_{b}$ intersects
line $l_{t}$. The point $p_{i}$ is calculated as follows. $\alpha$ is the
angle between the line $l_{b}$ and the $x$ axis. $\beta$ is the angle between
the normal of the regression line and the x-axis. We represent $\gamma$ as
$\beta-\alpha$, and the distance between point $c$ and $p_{i}$ is
$t_{d}/cos(\gamma)$. The free space is updated starting from the origin $o$,
marching along the laser beam and ended when reach to the distance
$d_{b}-t_{d}/cos(\gamma)$. The update SDF of free space is directly set to the
truncation distance. The SDF value and weight are also fused according to
Equation 1.
#### III-A3 Improve Priority Strategy
As presented in Section III-A1, every regression line is used to update the
SDF values of neighbor cells. Therefore, a cell may be updated multiple times
with different regression lines. To determine which regression line is used to
update for each cell, we calculate the priority at the same time we calculate
the SDF value. The priority is set to the distance of the cell center that
needs to be updated and the cell that brought the update. The cells closer to
the cell that gives rise to the update are put on a higher priority.
For the update of one laser frame, a set $S_{u}$ is used to store the current
update SDF values and priorities of all cells. The cells that need to be
updated are put on an update set $S_{u}$ one by one. If the cell has been
placed in the update set before, we compare the current update priority and
the previous one. And if the current priority is higher, we replace the
previously stored SDF value and priority with the current ones. If the current
priority and the previous one are equal, we fuse the SDF values using Equation
1, and use the fused SDF value and the current priority to replace the
previous values. If the current update has a lower priority, we give up the
current update. Finally, we iterate the update set and finish the one frame
update with Equation 1.
### III-B SDF-Based Lolicazation
#### III-B1 SDF Cost Function
In our system, both the SLAM localization module and the pure localization
module calculate pose by aligning the laser points with an SDF map. The pose
is calculated by minimizing the following equation:
$\displaystyle\mathbf{P}^{*}=\arg\min_{\mathbf{P}}\sum_{i=0,\ldots,m}\psi\left(\rm{W}(\mathbf{c}_{i})*\rm{F}(\mathbf{c}_{i})\right)^{2},$
(2) $\displaystyle where\ \ \mathbf{c}_{i}=\mathbf{P}\otimes\mathbf{d}_{i},$
where $\mathbf{d}_{i}$ is the 2D coordinate of laser point $i$. $\mathbf{P}$
is the laser pose to be estimated. $\rm{F}(\mathbf{c}_{i})$ is the SDF value
of cell $\mathbf{c}_{i}$, and $\rm{W}(\mathbf{c}_{i})$ is the weight. We use
the SDF weight in the cost function, as $\rm{W}(\mathbf{c}_{i})$ indicates the
confidence of the SDF value. Cells that updated with more frames would be more
reliable and impose a larger influence on the cost function. We use Huber
kernel $\psi$ to stabilize the optimization. The cost function is minimized
with the Gauss-Newton method.
#### III-B2 Iterative Optimization Strategy
Figure 5: Outliers displayed with red points come out often in-depth
discontinuity places.
Like other sensors, laser scanner takes systematic error and statistical
error. Take the SICK TiM5xx 2D LiDAR sensor as an example, the systematic
error is within $\pm$6cm according to the technical specifications and
statistical error is within $\pm$2cm. And we found that outliers come out
often in-depth discontinuity places as displayed in Figure 5. Outliers in the
free space of an SDF map do not affect the pose estimation. Because the cost
of hit point in the free space stays constant during the pose optimization.
The outliers falling inside the truncation range of the SDF map as displayed
in Figure 5 would impose a large influence on the optimization. Huber kernel
can alleviate the impact of outliers to a certain extent. We propose an
iterative optimization strategy to further reduce the impact of outliers.
First, an initial pose is calculated with all the laser points by minimizing
Equation 2. The optimization can be initialized with the pose that calculated
using a constant velocity model or predicted with other sensor data (odometry,
imu, etc. ). Second, we minimize the cost function again with the initial pose
calculated in the first step. In the second step, we trim the laser points to
a narrow range and discard the hit points where SDF value is greater than the
threshold. For SICK TIM561 laser range finder, we set the truncation range to
6cm, which is identical to its systematic error. Typically, in the second
step, 15% points which hit outside the truncation range are discarded in the
optimization. We use up to 10 iterations for the first step and up to 20
iterations for the second step. And the optimization is also ceased if the
error variation between two consecutive iterations falls below a threshold.
## IV SDF-BASED ACCURATE PURE LOCALIZATION
We build submaps, detect loop closures and optimize the pose graph to
distribute system tracking error following Cartographer. In the Cartographer
system, pure localization is accomplished following SLAM pipeline. Laser pose
is calculated by maintaining a sliding window of the newly created submaps,
and the system continuously detects loop closures and optimize the pose graph.
We propose a new method to do pure localization. When the system finishes a
mapping, we merge the submaps to produce an integrated SDF map. The merged map
contains the regions of all submaps, and pure localization is accomplished
based on the merged map. We present the submap merging method as follows.
### IV-A Merge SDF Submaps
Figure 6: All submaps are merged to produce an integral map.
As displayed in Figure 6, the inside black boxes are generated SDF submaps.
Due to the backend pose graph optimization, the submap boxes are not
horizontal or vertical to each other as they were when initialized. To merge
all submaps, we first calculate the position and size of the large bounding
box $b_{s}$ as shown by the outermost one. The large bounding box can cover
the area of all submaps. For each submap, the four corner coordinates are
calculated using the optimized pose. And the position and size of the enclosed
bounding box $b_{s}$ is initialized according to the maximum and minimum
corner coordinates of the submaps. The merged SDF map $S_{m}$ is created based
on the center coordinates and area of the bounding box $b_{s}$. And all
submaps are integrated into the map $S_{m}$.
For each submap, for instance, the blue one $S_{b}$ shown in Figure 6, we
first calculate its corresponding region in the large map $S_{m}$ as indicated
with the red box. Then, we iterate the cells of the map $S_{m}$ in the red box
and fuse the SDF values of the map $S_{m}$ and submap $S_{b}$. For each cell
$c_{m}$ in the red box of the large map $S_{m}$, we calculate its
corresponding point $p_{c}$ in the submap $S_{b}$. The SDF value at $p_{c}$ of
$S_{b}$ is bicubic interpolated using the values of neighbor cells. The SDF
value of the cell $c_{m}$ in the integrated map $S_{m}$ and the value at the
corresponding coordinate $p_{c}$ in the blue submap are fused according to
Equation 3,
$\displaystyle\mathbf{F}(\mathbf{c})=\frac{\mathbf{W}_{m}(\mathbf{c}_{m})\mathbf{F}_{m}(\mathbf{c}_{m})+\mathbf{W}_{b}(\mathbf{p}_{c})\mathbf{F}_{b}(\mathbf{p}_{c})}{\mathbf{W}_{m}(\mathbf{c}_{m})+\mathbf{W}_{b}(\mathbf{p}_{c})},$
(3)
$\displaystyle\mathbf{W}(\mathbf{c})=\rm{max}(\mathbf{W}_{b}(\mathbf{p}_{c}),\
\mathbf{W}_{m}(\mathbf{c}_{m})),$
where $\mathbf{W}_{m}(\mathbf{c}_{m})$ and $\mathbf{F}_{m}(\mathbf{c}_{m})$ is
previously accumulated weight and value of cell $\mathbf{c}$,
$\mathbf{W}_{b}(\mathbf{p}_{c})$ and $\mathbf{F}_{b}(\mathbf{p}_{c})$ is the
weight and value of point $\mathbf{p}_{c}$ in submap $S_{b}$. As submaps may
overlap, the cell in map $S_{m}$ may have multiple correspondences in submaps.
The submaps are gradually integrated into the large map $S_{m}$, and SDF
values are continually fused. And for the fused weight, we set the weight to
the maximum value of the corresponding points in all submaps.
### IV-B Pure Localization Based on Merged Submap
The merged submap is used to do pure localization. The initial pose can be
relocalized with other sensors, such as a visual camera. In the pure
localization module, we register laser points with the merged map without
building new submap. Laser pose is calculated following the pipeline of
localization as introduced in Section III-B. Since there is no map building
when pure localization is done, and usually only a few iterations (typically
5) are utilized when minimizing the cost function, we consume much less
computing resources compared to the pure localization method of Cartographer.
Pure localization accuracy and computing consumption are discussed in Section
V.
TABLE I: Trajectory evaluation results based on MIT Stata Center dataset [18] | Hector SLAM [3] | Cartographer Probability Grid [2] | Cartographer SDF [2] | Our System
---|---|---|---|---
2012-01-27-07-37-01_part_1 | 0.0114 | 0.0098 | 0.0102 | 0.0096
2012-01-27-07-37-01_part_3 | 0.0121 | 0.0101 | 0.0101 | 0.0097
2012-01-28-11-12-01 | 0.0616 | 0.0109 | 0.0106 | 0.0102
2012-04-02-10-54-41 | 0.4538 | 0.0265 | 0.0279 | 0.0262
2012-02-02-10-44-08 | - | 0.0141 | 0.1466 | 0.0140
2012-05-01-12-12-25_part_2 | 0.0122 | 0.0100 | 0.0100 | 0.0098
2012-05-02-06-23-02 | 0.0442 | 0.0221 | 0.0126 | 0.0123
2012-01-28-12-38-24_part_1 | 0.0956 | 0.0118 | 0.0105 | 0.0101
2012-01-28-12-38-24_part_4 | - | 0.0113 | 0.0112 | 0.0107
2012-01-28-11-12-01 | 0.0616 | 0.0109 | 0.0105 | 0.0102
## V EXPERIMENTS
We evaluated the trajectories generated by our systems and other state-of-the-
art 2D-SLAM systems, including Hector SLAM [3] and Cartographer [2] based on
the MIT Stata Center dataset [18]. Since Cartographer implements two mapping
method of probability grid and SDF, We evaluate the performance of both. We
also evaluate the position error to determine the accuracy that a robot
localizes itself in a known map. The position error is the joint error of the
pure localization and robot position systems as a whole.
### V-A Evaluate Odometry Accuracy
We turn off the loop closure detection and pose graph optimization modules
when generating the trajectories, and use Root Mean Square Error (RMSE) to
evaluate the accuracy of the trajectories. Because Hector SLAM is an odometry
system and RMSE can better reflect the performance of the localization and
mapping modules that we focus on. The evaluation results are presented in
Table I. As can be seen from the table, our system performs better than the
other three systems. Our SDF mapping method represents the surface more
accurately compared to the occupancy grid. Because the surface that crosses
the SDF map from positive to negative can be accurately located. In the
occupancy grid map, the surface can only be located in the accuracy of map
resolution. More importantly, the localization based on the SDF map has a
wider convergence range than the occupancy map. Optimization can find the
direction of convergence within the truncation distance. This is a key factor
that SDF map can be used robustly for pure localization.
Cartographer implements SDF mapping following a KinectFusion-style map update.
We also compare the performance of Cartographer SDF mapping method with ours.
During the evaluation, we use the same parameters, for example, the same
number of points used in pose solving and the same truncation distance used in
the mapping. As displayed in Table I, our system achieves better results than
Cartographer. When the laser beam is closely parallel to the geometric
surface, the SDF map generated by the Cartographer is not accurate, as shown
in Figure 7. Only a very narrow area alone the geometry surface was updated by
Cartographer. In contrast, we build an more accurate SDF map. On the other
hand, our iterative optimization strategy can trim outliers to some extent.
Figure 7: A comparison of the SDF maps generated by Cartographer and our
system. The one on the left is generated by Cartographer [2] and the one on
the right is by our system.
### V-B Evaluation the Pure Localization Accuracy
We evaluate the robot position error, which is measured relative to a
reference position following [16]. We arbitrarily specify two positions in the
SDF map and let the robot controller control the robot to move from one
position to another cyclically for reciprocation motion. The robot position is
calculated relative to a fixed coordinate system using an April tag every time
the robot was standing at a designated position. We summarize 30 locations and
use the deviation of each location from the center as the position error. The
evaluation results are presented in Figure 8.
Figure 8: A robot moves from two designated positions in the map from one to
the other as displayed in (a) and (d). (b) and (e) is the SDF map when the
robot stands still at the two position. (c) and (f) is the robot position
error at the two positions.
As can be seen from the figure, our system achieves a position error of
$\pm$5mm. There are always people moving during the robot position accuracy
evaluation experiments. And we found that small dynamic objects, such as few
people or robots moving around, have little affects on robot position
accuracy. In contrast, the system proposed by Röwekämper et al. [16] achieves
$\pm$5mm accuracy only in the taught-in locations using PL-ICP method [8].
Overall the system can only achieve 2 $\sim$ 5cm position accuracy with AMCL.
There are two reasons why our system performs better than AMCL. On the one
hand, our system uses SDF map while AMCL using occupancy map. SDF map is more
accurate as geometry surface can be precisely localized. One the other hand,
we use optimization to find the optimal solution which is more accurate than
particle filter used by AMCL.
We also evaluate the accuracy of robot position in a highly dynamic
environment, as displayed in Figure 9. The environment was changed greatly,
resulting in the large misalignment of the laser points and the SDF map. As
can be seen from the figure, the robot position error is $\pm$2cm. During all
the experiments, our system works properly. And we found that our system
achieves high robustness owning to the large convergence range of SDF map.
Figure 9: Evaluate the robot position accuracy in highly dynamic environment.
### V-C Compare the Pure Localization Efficiency
We also compare the pure localization efficiency of our system with AMCL. Both
the AMCL and our system are executed in an NVIDIA Jetson TX2 platform and
evaluated in the same environment. AMCL uses the occupancy map generated by
our system. And both the pure localization modules run in CPU with a single
thread. The statistical execution time of one frame is displayed in Table II.
As can be seen from the table, our system is about three times faster than
AMCL. Pure localization efficiency is critical for multitasking systems
running on embedded CPUs.
TABLE II: Compare the efficiency of AMCL and our system. | median | mean | max | std
---|---|---|---|---
AMCL | 0.0314 | 0.0351 | 0.1050 | 0.0262
Our system | 0.0078 | 0.0090 | 0.0345 | 0.0048
## VI CONCLUSION
In this paper, we improve the 2D-SDF-SLAM system and propose a new pure
localization method. We improve the 2D-SDF-SLAM method to build a more
accurate SDF map. To resolve random outliers of laser scanners, we propose a
new iterative registration method. Using fewer outliers can greatly improve
localization accuracy. The generated SDF map is also used for pure
localization. Globally, a few millimeters (5 mm) of positioning accuracy can
be achieved within the map while using much less computation resources. This
accuracy allows the creation of highly demanding robotic applications, for
example, precise maneuvering tasks for docking maneuvers or movements.
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|
# Size and Shape Fluctuations of Ultrasoft Colloids
Huarui Wu Department of Engineering Physics and Key Laboratory of Particle
and Radiation Imaging (Tsinghua University) of Ministry of Education, Tsinghua
University, Beijing 100084, China. Jing Song Department of Engineering
Physics and Key Laboratory of Particle and Radiation Imaging (Tsinghua
University) of Ministry of Education, Tsinghua University, Beijing 100084,
China. Wei-Ren Chen Neutron Scattering Division, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, United States Kun Song Department of
Engineering Physics and Key Laboratory of Particle and Radiation Imaging
(Tsinghua University) of Ministry of Education, Tsinghua University, Beijing
100084, China. Lionel Porcar Institut Laue-Langevin, B.P. 156, F-38042
Grenoble CEDEX 9, France Zhe Wang<EMAIL_ADDRESS>Department
of Engineering Physics and Key Laboratory of Particle and Radiation Imaging
(Tsinghua University) of Ministry of Education, Tsinghua University, Beijing
100084, China.
###### Abstract
Ultrasoft colloidal particle fluctuates due to its flexibility. Such
fluctuation is essential for colloidal structure and dynamics, but is
challenging to quantify experimentally. We use dendrimers as a model system to
study the fluctuation of ultrasoft colloids. By considering the dynamic
polydispersity in the small-angle neutron scattering (SANS) model, and
introducing the fluctuation of invasive water into the contrast in SANS, we
reveal the fluctuating amplitudes of the size and shape of the dendrimer of
generation 6 at finite concentrations. The size fluctuation is suppressed
while the shape fluctuation increases as the weight fraction of dendrimers
passes 11%. With neutron spin echo data, we suggest that such crossover
originates from the competition between the inter- and intra-particle
dynamics. Further investigation on lower-generation samples shows a contrary
result, which suggests a structural basis for these dynamic phenomena.
Ultrasoft colloids, such as dendrimers and star polymers, are featured by
extraordinary molecular flexibility [1, 2, 3, 4, 5, 6]. The elastic energy
stored by such a particle that undergoes a large strain can be just hundreds
of or even tens of the thermal energy [7, 8, 9, 10], which distinguishes them
from emulsions, most microgels, or other common deformable particles [11, 12,
13, 14, 10, 15, 16, 17]. Consequently, ultrasoft colloids exhibit significant
molecular fluctuations. These fluctuations are crucial in many physical
processes. Simulations suggest that the size and shape fluctuations
considerably affect the self-diffusion of particles [18]. At concentrations
close to the random close packing, it is proven that the shape fluctuation is
related to the stress-releasing and -building of particle, and plays an
essential role in the unusual dynamics [19]. On the practical level,
fluctuations can modulate the particle conformation [20] that profoundly
impacts a large variety of applications [21, 22, 23, 24]. For instance,
dendrimers have been established as drug carriers. The conformational
fluctuation directly affects the size, shape and internal cavity of the
molecule, which are important for the drug loading and release ability and the
permeability across the biomembrane [25, 26, 27, 28, 29]. Therefore, there is
a strong need to quantify the size and shape fluctuations to advance our
understanding on both the physics and applications of ultrasoft colloids.
In the past two decades, neutron spin echo (NSE) technique has been used to
measure the intra-particle motions of ultrasoft colloids at the dilute limit,
and the fluctuations of the structural unit and shape of the isolated particle
were revealed [30, 31]. While many ultrasoft colloidal suspensions of physical
and technological importance are with finite concentrations [10, 19, 32]. For
these systems, the NSE analysis on the molecular fluctuation is much
complicated by the difficulty in experimentally determining the collective
translational diffusion of particles and its coupling to the intra-particle
dynamics [33]. On the other hand, the fluctuations influence the distribution
of the particle conformation, which could be reflected in small-angle
scattering (SAS) patterns. Note that, most previous SAS studies of ultrasoft
colloids have not explicitly considered the effect of particle size and shape
fluctuations [1, 34]. Many SAS analyses deteriorate at volume fractions higher
than about 10%. For example, the calculated SAS curve may not well match the
position or height of the main peak [7, 9], or underestimate the intensity at
small $Q$ ($\bm{Q}$ is the scattering vector) [35]. Thus, it is possible that
the fluctuation effect is important in interpreting the SAS data of ultrasoft
colloids, and can be extracted by reasonable modelling.
In this letter, we investigate the size and shape fluctuations of ultrasoft
colloids by using neutral poly(amido amine) (PAMAM) dendrimers dissolved in
water as the model system. Six concentrations, $c=1$, 5, 10, 12.5, 15, and 20
wt% (weight fraction of dendrimers) were measured. We first introduce an
approach to examine these fluctuations in the dendrimer molecule by small-
angle neutron scattering (SANS). It will be seen that explicitly considering
the size fluctuation is of special importance in SANS analysis. Then, by
combining the NSE result, and investigating lower-generation samples, we
reveal the mechanism for the dependences of these fluctuations on the
molecular structure and dynamics.
Dendrimers with the same molecular weight possess different conformations due
to molecular fluctuations. Thus, dendrimer solutions can be regarded as
polydisperse. We introduce the dynamic polydispersity [36] to reflect this
fluctuation effect, which leads to the following expression of the SANS
intensity [37]:
$I(Q)=n_{\textrm{p}}AP(Q)S^{\prime}(Q),$ (1)
where $n_{\textrm{p}}$ is the number density of dendrimer molecules, $A$
denotes the contrast of the scattering length between solute particle and
solvent, $P(Q)$ is the average form factor normalized at $Q=0$, and
$S^{\prime}(Q)$ is the apparent structure factor given by
$S^{\prime}(Q)=1+\beta(Q)[S(Q)-1]$, where $\beta(Q)$ is the polydispersity
factor [37] that incorporates the size and shape fluctuations into the
analysis, and $S(Q)$ is the inter-particle structure factor. $S(Q)$ is
calculated by the Percus-Yevick closure of the Ornstein-Zernike equation [38]
with a Gaussian pair potential [9].
We first explore the size fluctuation by modeling the fluctuating dendrimers
as a collection of polydisperse spheres. So, $P(Q)$ is expressed as [36, 39]:
$\displaystyle
P_{\textrm{s}}(Q)=\sqrt{\frac{2}{\pi}}\frac{1}{\sigma_{R}}\left[1+\textrm{erf}\left(\frac{R}{\sqrt{2}\sigma_{R}}\right)\right]^{-1}$
$\displaystyle\times\int_{0}^{\infty}\left[\frac{3j_{1}(Qr)}{Qr}\right]^{2}\textrm{exp}\left[-\frac{(r-R)^{2}}{2\sigma_{R}^{2}}\right]dr+a_{\textrm{b}}P_{\textrm{b}}(Q),$
(2)
where $j_{1}(x)$ is the 1st order spherical Bessel function of the 1st kind,
$P_{\rm b}(Q)$ represents the contribution from the intra-particle density
variation with $a_{\rm b}$ denoting its amplitude [36, 40, 41], $R$ is the
average radius, and $\sigma_{R}$ is the standard deviation of the Gaussian
distribution of $R$. $\sigma_{R}$ reflects the fluctuating amplitude of the
dendrimer size. This form factor ignores the “fuzzy” profile of the radial
density distribution [36, 5]. It could be reasonable for high-generation
dendrimers, because their ratios of the fuzzy edge ($\sigma_{\rm f}$, see Eq.
(3)) to radius are relatively small [36]. The analyses for the samples of
generation 6 (G6) are displayed in Fig. 1 (a, blue lines), and are seen to be
satisfactory at the measured concentrations.
To highlight the effect of the size fluctuation, we analyze the same data with
the frequently-used monodisperse fuzzy-ball model [36, 42, 43]. Here, the size
fluctuation is not considered, so $\beta(Q)=1$ and $S^{\prime}(Q)=S(Q)$.
$P(Q)$ is given by [36]:
$P_{\textrm{f}}(Q)=\left[\frac{3j_{1}(QR)}{QR}\textrm{exp}\left(-\frac{Q^{2}\sigma_{\textrm{f}}^{2}}{4}\right)\right]^{2}+a_{\textrm{b}}P_{\textrm{b}}(Q),$
(3)
where $\sigma_{\textrm{f}}$ denotes the spatial range of the fuzzy edge. This
model has the same number of parameters as the preceding one. The fitting
results, shown in Fig. 1(b), exhibit clear discrepancies from the measured
spectra at $c\geq 10$ wt%. The low-$Q$ intensity is remarkably underestimated.
The peak position also deviates from the measured one. Comparing the two
models, we assert that the size fluctuation is significant, and should be
taken into account in analyzing the SAS data at $c\geq 10$ wt%. Numerically,
the incorporation of the size fluctuation lowers the low-$Q$ part of
$\beta(Q)$ from 1 since $\beta(Q=0)=\langle r^{3}\rangle^{2}/\langle
r^{6}\rangle$ [37], and thus lifts the low-$Q$ part of $S^{\prime}(Q)$ to
match the experimental data [44]. Notice that, at $c\leq 5$ wt%, both models
work well, leading to contradictory results on the existence of the size
fluctuation. This issue will be clarified in the scattering contrast analysis
shown later.
Figure 1: SANS analyses on the G6 PAMAM dendrimers dissolved in $\rm{D_{2}O}$
at $c=1$, 5, 10, 12.5, 15 and 20 wt%. Symbols in (a) and (b) denote the
measured spectra(they are vertically shifted for visibility). (a) Fits with
the size fluctuation model and the model with both the size and shape
fluctuations. (b) Fits with the monodisperse fuzzy-ball model and the model
only considering the shape fluctuation. These two models, without explicitly
considering the size fluctuation, underestimate the low-$Q$ intensity at
$c\geq 10$ wt%. (c) Size fluctuation ($\sigma_{R}$) of the G6 dendrimer as a
function of $c$. The results are obtained with the size fluctuation model and
the model with both the size and shape fluctuations. (d) Shape fluctuation
($\sigma_{\epsilon}$) of the G6 dendrimer as a function of $c$. The details of
the SANS fitting can be found in the Supplementary Material.
Considering that the dendrimer can deform to a spheroid-like form [45], we
incorporate the shape fluctuation by allowing the aspect ratio of a dendrimer
($\epsilon$) to fluctuate according to a Schultz distribution [37] with the
center at 1 and the standard deviation of $\sigma_{\epsilon}$. The fitting
results considering both the size and shape fluctuations are given in Fig.
1(a, magenta lines). It is seen that incorporating the shape fluctuation
improves the fit at high $Q$. Notice that, solely incorporating the shape
fluctuation cannot match the low-$Q$ spectra (Fig. 1(b)), since in this case
$\beta(Q\to 0)$ does not deviate from 1 [44].
Figure 1(c) and (d) show the fitting results for $\sigma_{R}$ and
$\sigma_{\epsilon}$ of the G6 dendrimer, respectively. At the dilute limit,
$\sigma_{R}$ is 4.7 $\mathrm{\mathring{A}}$, corresponding to a dynamic
polydispersity $\xi_{R}=\sigma_{R}/R$ of 15%. As $c$ crosses about 11 wt%, the
size fluctuation is suppressed while the shape fluctuation is strongly
enhanced. In later part we will show that this behavior corresponds to a
dynamic crossover.
As seen from Fig. 1, incorporating the size fluctuation is important in
decoding the structural information of ultrasoft colloids. Next, we will
verify its existence from another view. Since dendrimer has a water-accessible
architecture [46, 47], the contrast term $A$ in Eq. (1) is written as
$A=\langle(b_{\rm{pol}}+Nb_{\rm{w}}-n_{\rm{w}}V_{\rm{p}}b_{\rm{w}})^{2}\rangle$,
where $b_{\rm{pol}}$ is the total scattering length of a dry dendrimer,
$b_{\rm{w}}$ is the average scattering length of a water molecule, $N$ is the
number of water molecules inside a dendrimer, $V_{\rm{p}}$ is the volume of a
dendrimer in solution, $n_{\rm{w}}$ is the number density of bulk water, and
$\langle\dots\rangle$ denotes the average over all particles. It is
straightforward to find that:
$A(b_{\rm w})=[n_{\rm{w}}^{2}(\langle V_{\rm p}\rangle^{2}+\langle\Delta
V_{\rm p}^{2}\rangle)+\langle N\rangle^{2}+\langle\Delta
N^{2}\rangle-2n_{\rm{w}}(\langle V_{\rm p}\rangle\langle
N\rangle+\langle\Delta V_{\rm p}\Delta N\rangle)]b_{\rm
w}^{2}-2b_{\rm{pol}}(n_{\rm w}\langle V_{\rm p}\rangle-\langle N\rangle)b_{\rm
w}+b_{\rm pol}^{2},$ (4)
where $\Delta V_{\rm p}=V_{\rm p}-\langle V_{\rm p}\rangle$ and $\Delta
N=N-\langle N\rangle$. $\Delta N$ and $\Delta V_{\rm p}$ should be highly
correlated. To the first-order approximation, it can be assumed that $\Delta
N$ is proportional to $\Delta V_{\rm p}$ by $\Delta N=n_{\rm{in}}\Delta V_{\rm
p}$, where $n_{\rm{in}}$ should be smaller than $n_{\rm{w}}$ due to the
excluded volume of the constituent atoms of dendrimer. So that Eq. (4) is
rewritten as:
$A(b_{\rm w})=(\alpha^{2}+\theta^{2})b_{\rm w}^{2}-2b_{\rm pol}\theta b_{\rm
w}+b_{\rm pol}^{2},$ (5)
where $\theta=n_{\rm w}\langle V_{\rm p}\rangle-\langle N\rangle$, and
$\alpha=(n_{\rm w}/n_{\rm in}-1)\sigma_{N}$ with
$\sigma_{N}=\sqrt{\langle\Delta N^{2}\rangle}$ denoting the fluctuation of the
number of invasive water molecules. If no such fluctuation exists, then
$\alpha=0$, and Eq. (5) reduces to:
$A(b_{\rm w})=(\theta b_{\rm w}-b_{\rm pol})^{2}.$ (6)
Equations (5) and (6) provide an approach to verify the existence of the size
fluctuation ($\propto\sigma_{N}$) from a microscopic view. One can vary
$b_{\rm w}$ by changing the molar fraction of $\rm{D_{2}O}$ in solvent
($\gamma$), and fit the experimental $A(b_{\rm w})$ with Eqs.(5) and (6). If
the size fluctuation is considerable, the fitting quality with Eq. (5) will be
better than that with Eq. (6). We vary $\gamma$ from 100% to 60% for all
concentrations, and fit the experimental contrast term with Eqs.(5) and (6).
The results are shown in Fig. 2. It is seen that at $c\leq 10$ wt%, Eq. (5)
performs much better than Eq. (6), suggesting that the fluctuations of size
and invasive water exist and strongly affect the scattering contrast. At
$c>10$ wt%, this effect is less significant, implying a smaller fluctuation.
The values of $\alpha/N$, representing the fluctuation of invasive water, are
given in Fig. 3(a). In the same panel, we also plot the particle volume
fluctuation $\sigma_{V}/V_{\rm p}$ obtained from the model fitting of SANS
spectra as illustrated in Fig. 1(a). It is remarkable that these two
quantities display very similar behaviors, especially considering that they
are found with different approaches [48]. Such consistency confirms the
existence of the size fluctuation in the G6 dendrimer.
Figure 2: Scattering contrast $A$ as a function of the average scattering
length of solvent molecule $b_{\rm w}$ at $c\leq 10$ wt% (a) and $c>10$ wt%
(b). Symbols denote the experimental results. Solid and dashed lines denote
the fits using Eq. (5) (with size fluctuation) and Eq. (6) (without size
fluctuation), respectively [48]. They are shifted vertically for visibility.
Figure 3: (a) Fluctuations of volume ($\sigma_{V}/V_{\rm p}$) and invasive
water ($\alpha/N\propto\sigma_{N}/N$) of a G6 dendrimer as a function of $c$.
(b) Inter-particle collisional time $\uptau_{\rm{inter}}$ and intra-particle
relaxation time $\uptau_{\rm{intra}}$ of G6 dendrimers as a function of $c$
[49]. The dashed line marks the crossover concentration $c^{*}$.
As mentioned above, SAS analyses on neutral ultrasoft colloids usually
deteriorate when $c$ is larger than about 10 wt%. It has been tentatively
attributed to the failure of factorizing $I(Q)$ into the product of $P(Q)$ and
$S(Q)$ due to the inter-particle interpenetration or overlap [50, 7, 5].
Nevertheless, many studies show that strong interpenetration or overlap does
not occur at such low concentrations [51, 52, 53, 54, 55, 56, 57]. Our
analysis suggests that the failure of $P(Q)\cdot S(Q)$ factorization is due to
the molecular fluctuation, and can be corrected with Eq. (1). In fact,
Pedersen also found that the SANS intensity of block copolymer micelles is
expressed by Eq. (1) rather than a product of $P(Q)$ and $S(Q)$ by considering
the configuration distribution of the chains in corona [42]. We argue that the
particle conformational fluctuation is the basis for both Ref. [42] and our
work. This agreement suggests that Eq. (1) is the better expression to account
for the softness, which originates from the flexible architecture, of
ultrasoft particles such as dendrimers and starlike polymers. Note that, Eq.
(1) is based on the assumption that the particle conformation decouples with
the position [37]. It becomes invalid at concentrations close to random close
packing because of the considerable spatial heterogeneity of particle
deformation [19].
As seen from Fig. 1 and Fig. 3(a), all discussed fluctuations exhibit
crossovers at $c^{*}\approx 11$ wt%. Here, we seek for the dynamic origin of
this phenomenon. Figure 3(b) shows the results of an NSE study on the same
system [49]. By decomposing the motion of a dendrimer into the translational
diffusion and the internal relaxation, this NSE analysis gives two
characteristic times: the inter-particle collisional time $\uptau_{\rm inter}$
and the intra-particle relaxation time $\uptau_{\rm intra}$. Interestingly,
$\uptau_{\rm inter}$ and $\uptau_{\rm intra}$ also intersect at $c^{*}$. So,
we associate the fluctuation crossover with the dynamic process. At $c<c^{*}$,
conformational fluctuations emerge due to the enormous internal degrees of
freedom. The inter-particle collisions do not strongly perturb the internal
relaxation since $\uptau_{\rm intra}<\uptau_{\rm inter}$. While at $c>c^{*}$,
$\uptau_{\rm intra}>\uptau_{\rm inter}$, the frequent collisions hinder the
dendrimer from fully exploring its conformational space at the dilute limit,
and thus restrain the size fluctuation. Meanwhile, these collisions enhance
the particle deformation and suppress the internal relaxation, which results
in an increasing shape fluctuation.
Figure 4: (a) SANS analysis of G4 PAMAM dendrimers dissolved in $\rm D_{2}O$
at $c=1$, 5, 10, 12.5, 15 and 20 wt%. Symbols denote the measured spectra.
Solid and dashed lines respectively denote the fits using the monodisperse
fuzzy-ball model and the size fluctuation model. (b) Scattering contrast of G4
dendrimers as a function of $b_{\rm{w}}$. Symbols denote the experimental
results. Solid and dashed lines respectively denote the fits using Eq. (5) and
Eq. (6). Data are vertically shifted.
We further measure the samples of generation 4 (G4) to investigate the
structural origin of the observed fluctuations. The polydisperse sphere model
and monodisperse fuzzy-ball model are applied to fit the SANS spectra and the
results are shown in Fig. 4(a). In contrast to the G6 samples, for G4 samples
the fuzzy-ball model works well, while the polydisperse sphere model
overestimates the low-$Q$ intensity, and gives an inappropriate oscillation at
high $Q$. This result indicates that the size fluctuation is imperceptible for
G4 samples. To confirm this observation, we perform the contrast analysis
given by Eqs.(4)-(6). As seen from Fig. 4(b), Eqs.(5) and (6) give the same
fits, showing that the size fluctuation term $\alpha$ equals to zero. The
sharp difference between the G4 and G6 samples reveals the role of molecular
structure in determining the fluctuations. For our G4 sample, the ratio of the
fuzzy edge to radius ($\sigma_{\textrm{f}}/R$) is 74%, which demonstrates the
pronouncing open feature of its periphery. Intuitively, the strong fuzzyness
implies an unsharp boundary, which makes the particle volume and its
fluctuation not so well-defined. On the contrary, G6 dendrimers are found to
be alike spheres with a clearer boundary. This result agrees with the previous
finding that the dendrimer structure evolves from an open one to a compact one
as the generation increases [45, 46, 36, 58, 59]. The evident compactness in
G6 dendrimers leads to small intra-particle free volume, which enhances the
interaction between dendrons [60, 55, 5]. Therefore, it is reasonable that the
dendrons can fluctuate collectively with long-range correlation, and form the
size fluctuation of the whole particle. For G4 dendrimers, a simulation shows
that the internal fluctuations are uncorrelated at large distances [20] due to
the open structure. In this case, intra-particle motions involving long-range
correlated motion of monomers, such as the size fluctuation, should be weak.
In addition, an NSE study suggests that the molecular fluctuation of low-
generation starlike dendrimers is dominated by the breathing mode that does
not induce significant size and shape fluctuations [31]. This feature is also
attributed to the fuzzy nature of the radial density profile of the molecule
[31].
In summary, we investigate the fluctuation of ultrasoft colloids by using
PAMAM dendrimers as the model system. By considering the dynamic
polydispersity and the fluctuation of invasive water in SANS analysis, we
reveal the fluctuating amplitudes of the size and shape of the G6 dendrimer.
The size fluctuation is found to be of particular importance in interpreting
the SAS data. These fluctuations exhibit strong dependences on the dynamics
and structure. The competition between the intra- and inter-particle dynamics
introduces a crossover concentration above which the size fluctuation is
suppressed, and the shape fluctuation increases. The absence of size
fluctuation in G4 dendrimers highlights the importance of the structural
compactness in determining the intra-particle motion. Our approach provides a
basis for further exploration on the fluctuation effect on various structural
and dynamic properties of ultrasoft colloids.
###### Acknowledgements.
This research was supported by the National Natural Science Foundation of
China (No. 11975136, U1830205). We thank the EQ-SANS at Spallation Neutron
Source, Oak Ridge National Lab and the D22 beamline at Institut Laue-Langevin
for beamtime.
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|
# Adversarial Attacks for Tabular Data: Application to Fraud Detection and
Imbalanced Data
Francesco Cartella1 *, Orlando Anunciação 1 *, Yuki Funabiki 1 *, Daisuke
Yamaguchi 1,
Toru Akishita 1, Olivier Elshocht 1
###### Abstract
Guaranteeing the security of transactional systems is a crucial priority of
all institutions that process transactions, in order to protect their
businesses against cyberattacks and fraudulent attempts. Adversarial attacks
are novel techniques that, other than being proven to be effective to fool
image classification models, can also be applied to tabular data. Adversarial
attacks aim at producing adversarial examples, in other words, slightly
modified inputs that induce the Artificial Intelligence (AI) system to return
incorrect outputs that are advantageous for the attacker. In this paper we
illustrate a novel approach to modify and adapt state-of-the-art algorithms to
imbalanced tabular data, in the context of fraud detection. Experimental
results show that the proposed modifications lead to a perfect attack success
rate, obtaining adversarial examples that are also less perceptible when
analyzed by humans. Moreover, when applied to a real-world production system,
the proposed techniques shows the possibility of posing a serious threat to
the robustness of advanced AI-based fraud detection procedures.
## 1 Introduction
Fraud detection plays a crucial role in financial transactional systems such
as banks, insurances or online purchases. The ability to detect early whether
a transaction is fraudulent has a very high value and big investments have
been made to make these systems more effective. It is however important to
note that fraudsters are constantly developing new ways of fooling these
systems, a phenomenon known as concept drift (Widmer and Kubat 1996). A fraud
detection system therefore typically has high maintenance requirements.
Machine Learning (ML) is a classical approach for fraud detection systems
(Abdallah, Maarof, and Zainal 2016; Ngai et al. 2011). The ability to retrain
the models with new data helps in this need for adaptation to new fraud
patterns. However, given the possibility of errors in the models decisions,
which could lead to overlooking frauds or blocking licit transactions and
sales opportunities, fraud detection systems often do not rely solely on the
models but also contain one or more layers involving some form of human
intervention (Carcillo et al. 2018; Dal Pozzolo et al. 2017).
Risky transactions can be manually inspected and a decision is made whether
those transactions should go through or should be blocked.
Fraudsters may use a wide range of techniques to bypass fraud detection
systems. Among these techniques, adversarial attacks are novel and innovative
approaches that might be used as a next level of smart financial frauds. The
goal of adversarial attacks is to generate adversarial examples, i.e., inputs
that are almost indistinguishable from natural data and yet classified
incorrectly by the machine learning model (Madry et al. 2018).
Algorithms to build adversarial examples have recently been shown to be very
effective in fooling Machine Learning models, in particular Deep Neural
Networks (DNNs) in Image Recognition (Papernot et al. 2016b). This is a cause
of concern for many applications that rely on these technologies, such as
self-driving cars or facial recognition. The reason adversarial examples exist
is a consequence of the difference between the way humans and machines
represent knowledge and relations of visual elements in object recognition
tasks. This difference leads to the possibility for an attacker to perturb the
pixels of an image in a way that the change is imperceptible to a human, but
still induces an image classifier to produce a wrong interpretation. For
instance, an attacker can induce an image classifier to recognize with very
high confidence a gibbon in a picture that represents a panda, after the color
of a few pixels has been slightly modified (Goodfellow, Shlens, and Szegedy
2014).
Recent studies (Ballet et al. 2019) have shown that adversarial algorithms can
also be applied to other types of machine learning models using tabular data.
The positive results obtained by these studies highlighted the need and the
importance of investigating adversarial algorithms for a wider range of
domains and applications, so that effective defensive strategies can be
designed.
Motivated by the crucial role that security plays in the financial sector, in
this paper we deal with the problem of creating adversarial examples for
tabular data to effectively bypass fraud detection checks. In the particular
case of this research, bypassing fraud checks means either inducing the system
to classify a fraudulent transaction as non-fraud, or make the violation
unnoticed by a potential human inspection. These kinds of misclassifications
are particularly risky for fraud detection systems, as they would lead
attackers to succeed in their criminal intent and to obtain illegal economic
advantages.
It is a well-known fact that the security of a system is related to the
protection of multiple layers of an application (Zhu, Rieger, and Başar 2011).
Therefore, the security of a particular part of a system should be treated
independently, without relying on the integrity of other layers. In light of
this concept, in this paper we assume that the training set is available to
the attacker, as our main focus is the analysis of security risks affecting
the Machine Learning layer of a fraud detection system. Using this data, a
proper surrogate model can be created and used to evaluate the effectiveness
of the obtained adversarial examples, before submitting them to the real
system. Notice that we do not make any assumptions on the architecture of the
real model, which can be considered as unknown by the attacker.
To build successful attacks we tackled several problems, like adapting
adversarial algorithms to imbalanced fraud detection data and properly
treating non-editable variables. Moreover, as fraud detection systems often
involve human intervention, we also considered the problem of building
imperceptible adversarial examples, that are more difficult to be detected by
operators. Experimental results show that, with the modifications introduced
in this paper, it is possible to build realistic and imperceptible adversarial
examples for all the fraudulent transactions of the considered use case. In
comparison with state-of-the-art techniques, we achieved a drop of up to 64%
in the number of perturbed variables that are most commonly checked by human
investigators and, for the most successful cases, adversarial examples were
obtained by modifying just a minimum number of fields, reducing the
probability for an attack to be discovered. Finally, by obtaining a 13.6%
success rate in attacking a deployed production system, we also demonstrated
that the resulting adversarial examples were transferable to a target real-
world model, representing a real threat to businesses dealing with fraud
detection operations.
## 2 Related Works
Since the concept of adversarial examples was proposed (Szegedy et al. 2013),
it has been a main topic in the field of adversarial machine learning. This
topic is especially discussed in image recognition tasks using DNNs, but has
recently been discussed in other tasks such as audio recognition (Carlini and
Wagner 2018), text summarization (Cheng et al. 2020), neural network policy
for reinforcement learning (Huang et al. 2017) and so on. Furthermore, the
provision of tools to assist in the generation of adversarial examples is
being promoted. This is intended to make the training of models more robust
against adversarial machine learning. Adversarial examples can be easily
generated using tools like Adversarial Robustness Toolbox (Nicolae et al.
2018), CleverHans (Papernot et al. 2016a), Foolbox (Rauber, Brendel, and
Bethge 2017) and advertorch (Ding, Wang, and Jin 2019). However, despite such
trends, there are only a few studies of adversarial examples on tabular data.
To the best of our knowledge, the paper (a) (Ballet et al. 2019) was the first
systematic introduction to adversarial examples in tabular domain using recent
terminology. However, similar concepts, such as performing small changes in
features to get desired outputs or executing model inversion, were studied
before (Grosse et al. 2017; Papernot, McDaniel, and Goodfellow 2016; Bella et
al. 2010). Although it is very difficult to discuss the imperceptibility of
perturbations in the tabular domain as opposed to the image domain, the
authors of (Ballet et al. 2019) proposed to use the feature importance (the
contribution of each feature to the model) as an indicator and applied more
perturbations on the less important features. Another recent paper (b)
(Hashemi and Fathi 2020) treated the adversarial examples on tabular data as
counterfactual examples to help the explainability of the model. Another paper
(c) (Levy et al. 2020) proposed a method of conversion to a surrogate model
that maintains the properties of the target model in order to apply existing
generation methods.
The main differentiators of our research with respect to the contributions
mentioned above consist of: (a) they adopted a gradient-based attack method,
which is applicable to models like DNNs but not to architectures with discrete
gradients (such as decision trees), while we propose a model agnostic approach
applicable to any architecture; (b) they provide a counterfactual explanation
for model interpretation, while we assume a more realistic scenario of attack
attempts; (c) the generation method they used is a variant of the black-box
attack via surrogate model, while we assume that less information about the
model is available to the attacker.
## 3 Main Contributions
As discussed in Section 1, in this paper we present a novel approach to adapt
adversarial attack algorithms, that are commonly used in the image recognition
domain, to tabular data. In particular, we target fraud detection use cases.
Achieving this goal requires facing and solving several challenges that arise
from the different nature of data and model types used, compared to image
classification applications.
One of the main differences between image and fraud detection data is the
balance of samples that represent each class. While image classification is a
multiclass problem where each of the classes is represented by a relatively
similar amount of instances, fraud detection data sets are usually binary
(i.e., they contain only two classes) and are typically characterized by a
large imbalance between genuine and fraudulent transactions, the first being
in big majority. Fraud detection models return a risk score that represents
the estimated probability that the classified transaction is fraudulent. The
large imbalance in the data leads to highly biased models that tend to
attribute a risk score that takes into account the higher probability of
observing instances belonging to the most numerous class. Differently from
image classification, where the image is generally attributed to highest
predicted probability class (with some exception, like in the case of
diagnosis or fault detection applications), a decision threshold is normally
tuned for fraud detection, according to some business requirements. The input
transaction is deemed to be fraudulent if the predicted risk score is bigger
than the threshold. In this research, we introduced the concept of the
decision threshold within the attack algorithms, as it represents essential
information to verify if the perturbations correctly resulted in the creation
of adversarial samples.
Moreover, ensemble models and, in particular, Extreme Gradient Boosting
techniques are commonly used in applications handling imbalanced tabular data,
having proven to be particularly effective for these kind of settings.
Applying adversarial attack algorithms in a model agnostic fashion, rather
than closely targeted to Deep Neural Networks, was also one of the main
challenges of this research.
Another aspect that differentiates image and tabular data is the value range
that each feature can assume. Representing pixel values, image data can
normally vary within limited ranges and data types (i.e., integer numbers
between 0 and 255). On the other hand, tabular data can represent disparate
pieces of information, like email addresses, surnames or amounts. As such,
features representing a transaction can be extremely different from each
other. Even if ultimately encoded as numerical values, a proper handling and
representation of data types and range was essential to enable algorithms to
generate realistic adversarial transactions.
Field editability also represents a crucial aspect to take into account when
dealing with transactional data. Differently from the image domain, where an
attacker can potentially modify any of the pixels independently, for tabular
data there might be fields that are not directly controllable by the user but
that are rather automatically determined by the system. Examples of these
fields could be the historical amount borrowed in a loan application or the
discount rate applied for an online purchase. To simulate the fact that direct
changes to these values are not allowed by the system, specific constraints
were added to the algorithm to prevent the modification of non-editable
information.
Finally, we addressed imperceptibility as one of the main challenges of our
research. Differently from image data, where imperceptibility is an intuitive
concept related to human perception, for fraud detection we assume that
imperceptibility is related to the number and entity of changes made to
important features, such as fields that are most commonly checked by human
operators within the specific application context. From a purely practical
point of view, we define as imperceptible an adversarial attack that
ultimately passes the fraud check, remaining unnoticed. We approached
imperceptibility by introducing a custom norm as a measure of distance between
the original transaction and the adversarial sample. The distance is obtained
through weights assigned to each feature that are proportional to a novel
definition of importance that takes into account the propensity of a feature
to be inspected. We will show that the custom norm properly drives the
algorithm procedures to prioritize changes made on features that are rarely
checked by human operators, obtaining less perceptible attacks.
Details of the contributions described above will be given in Section 5,
together with other aspects introduced in this research, including some
algorithms specific solutions, like a novel loss function definition for the
Zeroth Order Optimization (ZOO) algorithm (Chen et al. 2017) and an improved
initialization strategy for Boundary attack (Brendel, Rauber, and Bethge 2018)
and HopSkipJumpAttack (Chen, Jordan, and Wainwright 2020).
## 4 Problem Statement
In this paper we address the problem of building adversarial examples for
fraud detection systems on financial data.
A financial transaction is a vector of $m$ variables, $v_{1},...,v_{m}$ with
each $v_{i}\in\mathbb{R}$111Some variables can be textual, booleans, naturals
or integers, but for the sake of simplicity we assume that there is a feature
processing step that transforms all values into real numbers..
Fraud detection systems in financial data analyze a set of $n$ transactions
$t_{1},...,t_{n}$ in a certain period of time. A model $M$ is used to label
each transaction $t_{i}$ with a corresponding class $c_{i}\in\\{0,1\\}$ in
which 0 corresponds to Non-Fraud and 1 corresponds to Fraud: $M(t_{i})=c_{i}$.
The first goal of an attacker is to find a perturbation vector
$p_{i}=[p_{i}^{1},...,p_{i}^{m}]$ such that:
$M(t_{i}+p_{i})=0$ (1)
for values of $i=1,...,n$ such that $M(t_{i})=1$ and $t_{i}$ is a real fraud.
If such a perturbation vector $p_{i}$ can be found then
$\tilde{t_{i}}=t_{i}+p_{i}$ is a successful adversarial sample.
For the purpose of our experiments, our goal is to create an adversarial
sample $\tilde{t_{i}}$ such that $M(\tilde{t_{i}})=0$ for each fraudulent
transaction $t_{i}$ that is correctly identified by model $M$ ($M(t_{i})=1$).
## 5 Algorithms modifications
In this section we describe the main problems that were faced to create
successful adversarial examples. We used the Adversarial Robustness Toolbox
(ART) (Nicolae et al. 2018) as the reference tool. ART is a Python library for
Machine Learning Security that provides tools to enable developers and
researchers to defend and evaluate Machine Learning models and applications
against adversarial attacks.
Even though some ART algorithms can be applied to tabular data, the majority
of the tool’s algorithms is designed to deal with image data. So it was no
surprise that it was necessary to make changes in order to build successful
adversarial examples for tabular data, and more specifically fraud detection.
### 5.1 A Generic Adversarial Algorithm
Adversarial algorithms can be used by attackers to retrieve the optimal
changes that, when applied to the fraudulent transactions they want to submit,
induce the fraud check to fail, by erroneously accepting the submitted
transactions as legitimate. To simulate this scenario in our experimental
setup, we applied adversarial algorithms on fraudulent samples that are
correctly detected as fraud by the model under attack. An adversarial
algorithm is considered successful if it outputs adversarial examples that are
classified as non-frauds by the same model. A generic adversarial algorithm
starts with an initial sample and makes perturbations to that sample until the
model misclassifies it. A second goal of adversarial algorithms is to make the
adversarial sample as similar as possible to the original sample. Algorithm 1
shows the pseudo-code of a simple generic algorithm that serves the purpose of
illustrating the main concepts. This algorithm receives a fraudulent
transaction $t$, a model $M$ such that $M(t)=1$ and a similarity threshold
$\rho$. The perturbations can be selected through many different ways. Two of
the most common approaches involve the use of distance metrics to get the
adversarial sample $\tilde{t}$ _closer_ to the original sample $t$ (e.g.:
Boundary or HopSkipJump) or calculations based on the gradient of the model
(e.g.: ZOO algorithm). Algorithm 1 uses a similarity function and a similarity
threshold $\rho$. The similarity function can be based on the distance between
$t$ and $\tilde{t}$, calculated using a norm such as $L_{2}$ or $L_{\infty}$.
Threshold $\rho$ can be provided explicitly, as is the case in our generic
algorithm. However, some algorithms calculate it in an indirect way. As an
example, Boundary attack algorithm converges when it is close enough to the
decision boundary. Algorithms usually also have a maximum number of allowed
steps in the while loop. This was not included in the pseudo-code for
simplicity reasons.
Algorithm 1 Generic Adversarial Algorithm
1:function generate_adv($t,M,\rho$)
2: $\tilde{t}\leftarrow initialize\\_sample()$
3: while $M(\tilde{t})=1\lor similarity(\tilde{t},t)<\rho$ do
4: $\tilde{t}=make\\_perturbation(\tilde{t},t)$
5: end whilereturn $\tilde{t}$
6:end function
### 5.2 Using Custom Threshold
In ART, adversarial algorithms are fed with the model that is being attacked.
Adversarial examples are iteratively refined and, at every iteration, the
model is used to evaluate the current samples’ success. Because fraud
detection is binary, the adversarial algorithms stop as soon as the current
adversarial example is deemed to be successful, i.e., when it is classified as
non-fraud with a score higher than $0.5$. While this may work well for an
image recognition model, it is problematic for a fraud detection model.
In fraud detection use cases, a decision threshold $\tau\in[0,1]$ is commonly
tuned and an input transaction is classified as fraud if
$[M(t)]_{1}>\tau$,where $[M(t)]_{1}$ is the probability that the transaction
$t$ belongs to class 1, i.e., fraud. The threshold $\tau$ is typically very
small and much lower than 0.5, to compensate for the tendency of the model to
attribute very low risk scores to new transactions, given the big majority of
non-fraud samples observed at training time.
Our initial results when applying the default ART algorithms were poor because
a threshold of 0.5 was used, misleading the algorithms by assuming that a
successful adversarial sample had been found. To correct the problem we
modified the Boundary (Brendel, Rauber, and Bethge 2018), HopSkipJumpAttack
(Chen, Jordan, and Wainwright 2020) and ZOO (Chen et al. 2017) attacks. These
algorithms are now fed with a custom threshold and whenever the model is
evaluated internally, the custom threshold is taken into account for the
models’ decision. With this correction, the adversarial algorithms have access
to true information about whether a sample is classified as fraud or not by
the model.
### 5.3 Specifying a Custom Loss Function for ZOO
To drive the creation of adversarial examples, the ZOO algorithm uses a
specific loss function that, as detailed below, implicitly considers a
balanced threshold of 0.5 in its standard formulation. For this reason, the
introduction of a novel loss function was essential to adapt the algorithm to
biased cases.
To adapt the ZOO algorithm formulation to the specific case of binary
classification and fraud detection, following the notation introduced in
Section 4, let us define the model under attack as a function $M(t)$ that
takes a transaction $t$ and returns a two dimensional vector
$M(t)\in[0,1]^{2}$. The two dimensions of this vector represent the
probability score of class 0 (not fraud) and of class 1 (fraud), respectively.
As a consequence, $[M(t)]_{0}+[M(t)]_{1}=1$
Given a fraudulent transaction $t_{f}$ correctly classified by the model, the
ZOO attack finds the corresponding adversarial sample $\tilde{t}$ by solving
the following optimization problem:
$\mbox{minimize}_{\tilde{t}}\left[||\tilde{t}-t_{f}||_{2}^{2}+r\cdot
f(\tilde{t})\right]$ (2)
where $||v||_{2}=\sqrt{\sum_{i=1}^{m}v_{i}^{2}}$ denotes the Euclidean norm
(or the $L_{2}$ norm) of the vector $v=[v_{1},...,v_{m}]^{T}$ and $r>0$ is a
regularization parameter. Equation 2 is expressed as a sum of two terms to be
minimized: the first term $||\tilde{t}-t_{f}||_{2}^{2}$ represents a measure
of distance between the adversarial example $\tilde{t}$ and the original
transaction $t_{f}$; the $f(\tilde{t})$ of the second term represents a loss
function that measures how unsuccessful an adversarial attack is. The
minimization of Equation 2 is performed using stochastic coordinate descent
methods (see (Chen et al. 2017) for details). The loss function proposed in
the standard formulation of the ZOO algorithm is the following:
$f(t)=\mbox{max}\left[(log[M(t)]_{1}-log[M(t)]_{0}),-\nu\right]$ (3)
where $\nu>=0$ is a tuning parameter for attack transferability, commonly set
to 0 for attacking a targeted model or to a larger value when performing a
transfer attack. If, for simplicity, we consider $\nu=0$, the loss function
above will return its minimum value of 0 for all the adversarial samples
$\tilde{t}$ having $[M(\tilde{t})]_{0}>=[M(\tilde{t})]_{1}$, i.e., probability
of not fraud bigger or equal than fraud. As explained previously, in the
context of biased models, assigning to a transaction a not fraud probability
higher than the probability of fraud, does not necessarily imply that the
transaction is classified as licit, but it is necessary that
$[M(t)]_{1}\leq\tau$, where $\tau\in[0,1]$ is the decision threshold. As a
consequence, the loss function of Equation 3 is minimized also by a set of
adversarial examples that, being still classified as fraud, are unsuccessful.
This is the set of adversarial examples $\tilde{t}$ for which
$[M(\tilde{t})]_{0}\geq[M(\tilde{t})]_{1}$ and $[M(t)]_{1}>\tau$.
As Equation 3 results inadequate for imbalanced use cases, we propose to use
the following loss function in the optimization of Equation 2:
$f(t)=\mbox{max}\left[\left([M(t)]_{1}-\tau\right),-\nu\right]$ (4)
The loss function above assures that minimum values are obtained only for
successful adversarial examples $\tilde{t}$, for which
$[M(\tilde{t})]_{1}<=\tau$ (i.e., classified as not frauds).
### 5.4 Creating Realistic Attacks with Editability Constraints
We analyzed the nature of the perturbations that were obtained by the
adversarial algorithms. In particular, tabular data has features of different
types: boolean, integer, hot-encoded variables, integers that only take
positive values, etc. Without imposing any constraint, the adversarial
algorithms created perturbations that led to illegal values, with respect to
the type of features that are taken into consideration (e.g.: a boolean
feature having value different from $0$ or $1$, or a positive integer feature
that becomes negative). It was then necessary to make sure that perturbations
assume only what we designated by _realistic_ values. Each variable $v_{i}$
can assume values from a specific domain $D_{i}$ (e.g.: for a real variable
$v_{i}$, $D_{i}=\mathbb{R}$). An adversarial sample $\tilde{t}$ has a
realistic value $x$ for variable $v_{i}\in D_{i}$ if $x\in D_{i}$. In the case
that $x\notin D_{i}$ a transformation needs to be made in order to ensure that
$x\in D_{i}$.
The inspection of adversarial samples raised awareness about the presence of
non-editable fields in the data (i.e., fields that cannot be directly modified
by the user), but are rather calculated automatically by the system. An
example of this could be the total amount of money borrowed by a customer in
the past, in the context of a loan management application. This value cannot
be changed when a new loan is requested. Adversarial algorithms should take
this into account and only make changes to variables that the user can have
access to. In order to address this we defined an editability vector that
contains the variables that can be changed by adversarial algorithms.
In order to address realistic and editability problems, we modified the
adversarial algorithms ZOO, Boundary and HopSkipJump. In the execution of each
algorithm, whenever a potential adversarial sample is modified, editability
and realistic properties are enforced by correcting the illegal values.
In order to make adversarial samples realistic we considered the data types
and the corresponding corrections for a specific value $x$ that are listed on
Table 1:
Type | Correction
---|---
Boolean | 0 if $x\leq 0.5$, 1 otherwise
Integer | $\mathrm{round}(x)$
Positive Integer | $\mathrm{round}(x)$ if $x\geq 0$, 0 otherwise
Positive Float | 0 if $x<0$, $x$ otherwise
Hot-encoded fields | 1 for field with maximum value.
| 0 for other fields of same group
Table 1: Data types and corresponding corrections for adversarial samples
In order to implement corrections listed on Table 1, adversarial algorithms
now receive a data specification dictionary containing a list of features for
each data type.
The editability constraints are enforced by defining a vector of editable
features and passing it to the adversarial algorithms. The editability vector
$e$ for variables $v_{i},...,v_{m}$ is defined as $e_{i}=1$ if $v_{i}$ is
editable, $0$ otherwise, for $i=1,...,m$. Algorithms will only allow
perturbations on features $v_{i}$, with $i=1,...,m$ for which $e_{i}=1$.
Features $v_{j}$, with $j=1,...,m$ for which $e_{j}=0$ are not perturbed and
forced to maintain their original values. Which feature $v_{i}$ are editable
is a property of the system under consideration.
### 5.5 Specifying a Custom Norm
After creating realistic adversarial samples and taking editability into
consideration, it was important to go one step further in terms of
imperceptibility of the attack. Besides editability considerations,
adversarial algorithms pick up any available feature as a candidate for a
perturbation. Within a specific application context, an attacker can guess
that, in the case of a hypothetical manual inspection, some features may
capture the attention of human operators more than others. For instance, in a
loan request application, the applicant salary information is usually more
informative than other fields, like the number of owned pets (Ballet et al.
2019). Nevertheless, less important features are also considered by the model
to estimate the request’s risk score. As a consequence, the attacker’s goal is
to minimize the perturbations made on features that have a bigger chance to be
checked.
Adversarial algorithms such as Boundary or HopSkipJump attacks use norms as
measures of distance between adversarial and original examples. These
algorithms try to minimize this distance as much as possible in order to make
the adversarial example imperceptible. $L_{2}$ norm considers the global
distance between the original and the adversarial sample, disregarding that
some features may have very large perturbations. Minimizing $L_{\infty}$ on
the other hand means that the algorithm will try to avoid having a big
perturbation on a single feature, giving preference to small perturbations on
many features. None of these norms completely satisfy the needs of an
imperceptible attack in the context of tabular data. In order to do that more
successfully it is necessary to consider features differently, depending on
whether they are checked by a human operator. This motivated the introduction
of a novel custom norm that is expressed in Equation 5:
$n=||p(\alpha h+\beta[(1-h)(1-v)+hv])||_{\gamma}$ (5)
where $p$ is the perturbation vector, $h$ is a Boolean vector indicating
whether a variable is checked, $v$ is a vector of feature importance,
$\alpha,\beta\in[0,1]$ are weights on the check and importance of a feature
respectively, and $||.||_{\gamma}$ is a $\gamma$-norm such as $L_{2}$ norm
that is being used. It is known that algorithms using gradient descent can
empirically derive values of coefficients such as $\alpha$ and $\beta$ in a
binary search (Carlini and Wagner 2017), but it is future work to verify
whether these techniques are applicable to our approach.
For the definition of the custom norm, two properties were considered: 1)
whether a feature is checked or ignored by the operators and 2) the importance
of the feature for the model. The idea behind the custom norm is that changes
to features that are checked and important lead to high values of the
distance, so that the optimization algorithm prefers other solutions.
Moreover, we also want to penalize solutions in which the feature is not
checked and not important, because it will not have a significant effect in
the attack. On the other hand, we would like the algorithm to prefer solutions
based on perturbing features that are not checked and have high importance for
the model. For these types of perturbations the custom norm returns low
values. Finally, if checked variables need to be perturbed it is preferable
that they are not important for the model, so we assign low distances for
these situations. In conclusion, the goal of the custom norm is to drive the
optimization procedure of the attack algorithms to obtain adversarial examples
that are imperceptible and unnoticed by human operators.
## 6 Experiments and Results
In this section we describe the experiments that were performed and the
obtained results. After having modified the ART algorithms as discussed in
Section 5, we applied them to the German Credit Dataset (Dua and Graff 2017)
use case. The strategy described in the following sections was also applied to
2 additional datasets with similar results. The results obtained using the
first dataset (the IEEE-CIS Fraud Detection dataset) are not detailed due to
space limitations. The second dataset is an internal dataset that cannot be
disclosed for confidentiality reasons.
### 6.1 Use Case and Data Preparation
German Credit Dataset (Dua and Graff 2017) is a publicly available dataset
used for building models that evaluate the risk of a loan application, given
account and customer information. Out of 1000 applications in total, 700 were
accepted while 300 were rejected and deemed to be risky in terms of low
propensity of the applicant of being able to pay back the loan. In the context
of adversarial attacks, we considered the rejected applications as fraudulent,
as the goal of a potential attacker would be to slightly modify their loan
request such that it eventually gets accepted. The data contains 20 features
with 7 integer and 13 categorical ones, such as age, sex, purpose of the loan
or if the customer is a foreign worker. We applied a one-hot encoding to
categorical features, obtaining a total of 61 numerical features for modeling.
#### Capabilities of Attackers
As discussed, the goal of attackers is to modify true-positive requests (i.e.,
applications that are deemed risky and should not be accepted) so that they
can be accepted. Our assumption is that the attacker can make reasonable
judgments about the importance of the features and estimate what are the
fields that a human investigator most probably checks to measure the
applicant’s ability to pay back the loan. In this experiment, we assumed that
human investigators would mainly check 10 of the total 20 features such as the
“Purpose (of the loan)” and “Credit amount”. Moreover, we assumed that the
features “Credit history”, “Personal status and sex”, “Other
debtors/guarantors” and “Age in years” are not directly modifiable by the
attacker and set them as non-editable. Although we conducted experiments under
these hypothesis, different settings can be considered as well, depending on
different assumptions on the system and the application context.
#### Model Construction
We used XGBoost (Chen and Guestrin 2016) as a learning algorithm. At first,
the dataset was split into train and test sets consisting of 70% and 30% of
the data, respectively. Furthermore, train set was split into training and
validation sets consisting 80% and 20% of the data, respectively. The training
data was used to generate a binary classification model and the validation
data was used to adjust the threshold. Before the threshold adjustment, the
accuracy on validation set was 75.7%, the recall was 38.1% and the precision
was 66.6%. An optimal threshold of $0.192$ was obtained using the F2 score
maximization as a target metric. With this threshold we obtained an accuracy
of 60.0%, a recall of 95.2% and a precision of 42.6%. Using the resulting
model on the test set, we were able to discriminate 82 true-positive data,
representing a recall of 91% and a precision of 42.5%. These results show
that, even without performing particularly sophisticated feature engineering,
we obtained a fair model with satisfactory performance that can be effectively
used to evaluate our study.
### 6.2 Results
In this subsection we summarize the results obtained. Our goal is to show that
the approaches we followed to build adversarial samples were successful.
We considered 4 parameters that can be switched on and off in our experiment
designs: threshold (Section 5.2), realistic (Section 5.4), editability
(Section 5.4) and custom norm (Section 5.5). As shown in Table 2, we obtained
5 different configurations.
ID | Threshold | Realistic | Custom | Editability
---|---|---|---|---
| | | Norm |
1 | OFF | OFF | OFF | OFF
2 | ON | OFF | OFF | OFF
3 | ON | ON | OFF | OFF
4 | ON | ON | ON | OFF
5 | ON | ON | ON | ON
Table 2: Configurations of the performed experiments
Experiment 1 is performed with ART as it is, without any changes or
adaptations. In Experiment 2 we use custom thresholds as described in Section
5.2 and in Experiment 3 we make the attacks realistic (Section 5.4). In
Experiment 4 we use the custom norm as described in Section 5.5 and in
Experiment 5 we add editability constraints (Section 5.4).
The results obtained are shown in Tables 3, 4 and 5.
Boundary | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
Success Rate (%) | 0 | 100 | 100 | 100 | 100
% of Unrealistic Values | - | 47.6 | 0 | 0 | 0
# Checked Fields | - | - | 592 | 214 | 228
# Non-Editable Fields | - | - | - | 63 | 0
Table 3: Results obtained with each experiment configuration for Boundary attack HopSkipJump | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
Success Rate (%) | 0 | 100 | 100 | 100 | 100
% of Unrealistic Values | - | 87.0 | 0 | 0 | 0
# Checked Fields | - | - | 554 | 465 | 418
# Non-Editable Fields | - | - | - | 159 | 0
Table 4: Results obtained with each experiment configuration for HopSkipJump attack ZOO | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
Success Rate (%) | 0 | 100 | 100 | 100 | 100
% of Unrealistic Values | - | 1.6 | 0 | 0 | 0
# Checked Fields | - | - | 159 | 133 | 153
# Non-Editable Fields | - | - | - | 37 | 0
Table 5: Results obtained with each experiment configuration for ZOO attack
Experiment 1 was very unsuccessful, with no adversarial samples found. This
means that the original algorithms cannot be applied directly to an unbalanced
problem. When changes are made to the algorithm to use a proper threshold in
Experiment 2, the success rate increases to 100% for the three algorithms.
This experiment however still generates unrealistic values for some features.
As an example, Table 4 shows that 87% of the values generated by HopSkipJump
are unrealistic. This makes it easy for a human operator to detect the attack.
In Experiment 3 this problem is solved and only realistic values are
generated. Experiment 3 does not use the custom norm, which is done in
Experiment 4. By observing the results we can check that, by using the custom
norm, the number of perturbed checked fields decreased in the application of
each of the 3 adversarial algorithms, thus increasing imperceptibility. For
instance, Table 3 shows that, in Experiment 3, a total of 592 fields that are
checked by human operators were perturbed by the boundary attack. When the
proposed custom norm is used in Experiment 4, only 214 of these fields are
modified, representing a drop of 64%, with respect to the state-of-the-art
norm. Experiment 5 considers editability constraints and we observe that the
number of non-editable fields that are changed is reduced to 0 in each
algorithm. For Boundary and ZOO attacks on Experiment 5 there is a slight
increase on the number of checked fields that are changed. This can be
explained by the fact that the algorithms are not allowed to change non-
editable fields and the pressure to change checked fields is higher.
Finally, it is important to mention that some successful adversarial examples
were obtained by changing just a few fields. Table 6 shows an example where
changing only the value of one attribute (Status checking account) caused the
model to return a lower risk score and flip its decision from rejection to the
acceptance of the loan application. It is evident that these types of
adversarial examples are highly imperceptible and that it is very probable
that they might remain unnoticed.
ZOO algorithm | Original | Adversarial
---|---|---
Status checking account | A12 | A14
Model’s Risk Score | 0.275 | 0.127
Table 6: One adversarial example obtained with ZOO algorithm.
## 7 Conclusions and Future Work
In this paper we illustrated the process we followed to adapt state-of-the-art
adversarial algorithms, that are commonly used in the image classification
domain, to imbalanced tabular data. In particular we targeted fraud detection
use cases.
After verifying the inadequacy of existing techniques to handle tabular data,
we introduced modifications to address the shortcomings. In particular (i) we
allowed adversarial algorithms to deal with biased model scores through the
usage of a custom threshold within the algorithms and the introduction of a
novel loss function for ZOO algorithm; (ii) we introduced constraints in the
allowed perturbation to obtain realistic adversarial examples and avoid out-
of-bound values; (iii) we improved imperceptibility through a proper
management of not editable fields and through the introduction of a custom
norm that drives the creation of adversarial examples that have a higher
chance to be unnoticed by human investigators.
In terms of results, the changes we made contributed to increase the attack
success rate from 0% to 100%. Moreover we showed examples of successful
imperceptible attacks that were obtained by changing the value of just a few
features.
Ultimately, we conducted a final experiment on the transferability of the
adversarial examples to a real-world production system. To this extent, we
could not perform attack transferability for the use case we considered in
this paper, given the lack of a real deployed AI system. For this reason, we
executed the full adversarial attack process on a real-world use case that is
currently in production. For confidentiality reasons and due to the
substantial economical dangers that sharing information on internal system
vulnerabilities might cause, only final results can be reported, without
disclosing any detail about the analyzed use case. We submitted 44 modified
fraudulent transactions, created using a surrogate side model, to the real
production system. For 35 transactions, representing 80% of the submitted
adversarial examples, the production model returned a lower risk score than
for the original transaction. More importantly, 6 cases, representing 13.6% of
the submitted transactions, were flagged as safe by the system and
automatically accepted, bypassing the human check. These results demonstrate
that the techniques introduced in this paper represent a real threat for many
AI-based fraud detection models, used in day-to-day business.
Future work will be conducted in the direction of performing more extensive
experiments on attack transferability, by setting a lower target threshold for
the adversarial algorithms, in order to increase the success probability of
attacks for the considered real-world use case.
On the other hand, these preliminary results highlighted the need of assuring
a better robustness of production fraud detection models. To this extent, we
started exploring the topic of defense techniques, with the goal of improving
their ability to detect and block also the most sophisticated adversarial
attacks. After conducting a survey of existing defensive methodologies, our
plan is to identify their shortcomings and, eventually, come up with new
approaches, following a similar process we used for the attack techniques.
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# Riemannian Manifold Optimization for Discriminant Subspace Learning
Wanguang Yin<EMAIL_ADDRESS>Zhengming Ma<EMAIL_ADDRESS>Quanying Liu<EMAIL_ADDRESS>Shenzhen Key Laboratory of Smart Healthcare
Engineering, Department of Biomedical and Information Engineering, Southern
University of Science and Technology, Shenzhen, 518055 China School of
Electronics and Information Technology, Sun Yat-sen University, Guangzhou,
Guangdong, 510006 China
###### Abstract
Discriminant analysis, as a widely used approach in machine learning to
extract low-dimensional features from the high-dimensional data, applies the
Fisher discriminant criterion to find the orthogonal discriminant projection
subspace. But most of the Euclidean-based algorithms for discriminant analysis
are easily convergent to a spurious local minima and hardly obtain an unique
solution. To address such problem, in this study we propose a novel method
named Riemannian-based Discriminant Analysis (RDA), which transforms the
traditional Euclidean-based methods to the Riemannian manifold space. In RDA,
the second-order geometry of trust-region methods is utilized to learn the
discriminant bases. To validate the efficiency and effectiveness of RDA, we
conduct a variety of experiments on image classification tasks. The numerical
results suggest that RDA can effectively extract low-dimensional features and
robustly outperform state-of-the-art algorithms in classification tasks.
###### keywords:
Dimensionality reduction , Discriminant analysis , Riemannian manifold
optimization , Stiefel manifold , Grassmannian manifold
††journal: arXiv
## 1 Introduction
Linear discriminant analysis (LDA) is an essential method for extracting
statistically significant features as a prerequisite for pattern recognition
and machine learning. LDA has broad applications ranging from text mining [1]
and image classification [2] to brain-computer interface (BCI) [3]. Generally,
LDA learns to discriminate different classes by computing the distance (or
similarity) metrics among the extracted features from training data, and then
assign the test data to a specific class based on the measured distance and
the learned threshold. Therefore, the performance of LDA largely relies on the
distance metrics defined on the features and the optimization strategy for
solving the loss function. However, most current methods for solving LDA are
based on the Euclidean space. However, these Euclidean-based methods easily
convergent to a spurious local minimum and hardly obtain a globally optimal
solution [4]. It motivates us to pursue alternative methods for solving the
LDA and ensuring an effective approximation of the high-dimensional input data
with a lower-dimensional representation.
To this end, by employing the specific nature of orthogonal constraints of the
discriminant bases, the LDA can be transformed from the Euclidean space to a
Riemannian manifold space and be solved by Riemannian manifold optimization.
Specifically, the Riemannian manifold optimization utilizes underlying
structures of the matrix manifold and optimizes the loss function by using the
Riemannian-based conjugate gradient and trust-region method, which benefits
from the Riemannian concepts, such as the tangent space, Riemannian metrics,
Retraction, connection, and transport parallel [5]. It’s worth noting that the
trust-region method can linearly approximate a local solution on the tangent
space in each iteration and eventually converge upon an extreme point as the
globally nonlinear solution, usually resulting in superior performance
compared to the traditional Euclidean-based methods [5].
In this way, we propose a family of discriminant analysis algorithms defined
on the Riemannian space, namely Riemannian-based discriminant analysis (RDA)
(Sec 3). The performance of RDA is compared with Euclidean-based methods and
several other existing Riemannian-based methods in terms of dimensionality
reduction and classification. Our results show that RDA algorithms are
superior in solving the multiclass, large-scale clustering tasks, as well as
the classification tasks, compared to the Euclidean-based discriminant
analysis. The main contributions of this paper can be concluded as the
following:
* 1.
First, RDA transforms the linear discriminant analysis from the Euclidean
space to the Riemannian manifold space and then employs the trust-region
method to learn the discriminant basis of the projection subspace (Sec 3). In
this way, the loss function can be converted from a division form to a
subtraction one (Eq.(10)). RDA can therefore effectively avoid calculating the
inverse of the Hessian matrix.
* 2.
Second, two types of Riemannian manifolds (i.e. Stiefel manifold and Grassmann
manifold) are investigated, and effects of the second-order approximation and
the sparsity regularization on the discriminant bases are constructed. The
numerical experiments suggest that the second-order geometry of the trust-
region method on the Riemannian manifold outperforms the first-order geometry
of the conjugate gradient method.
* 3.
Lastly, RDA achieves state-of-the-art (SOTA) performance in both clustering
experiments (Sec 4.2) and classification experiments (Sec 4.3). The numerical
experiments on multiple image datasets (e.g. COIL20, ETH80, MNIST, USPS, CMU
PIE) demonstrate that RDA can robustly obtain higher performance than
traditional Euclidean-based algorithms, as well as other existing Riemannian-
based algorithms.
## 2 Related Work
### 2.1 Subspace Learning
Subspace learning is essential for computer vision, pattern recognition [6],
biomedical engineering [7, 8], and bioinformatics [9]. It aims to map the
high-dimensional data to a lower-dimensional space with maximally maintaining
the information in the original data. The input data is usually represented as
vectors, matrices, or tensors, and subspace learning is to find an optimal
mapping, either linear or nonlinear, to project the input data to a low-
dimensional space. Linear subspace learning is a powerful tool for
dimensionality reduction and it provides a solid foundation for machine
learning algorithms [10]. A variety of methods have been proposed for linear
subspace learning, including the matrix or tensor decomposition [11, 12] and
the linear discriminant analysis [13].
The representative algorithms for matrix factorization include singular value
decomposition (SVD) [14], principle component ananlysis (PCA) [15], canonical
correlation analysis (CCA) [16], independent component analysis (ICA) [17],
and nonnegative matrix factorization (NMF) [18]. NMF assumes that the original
input data is nonnegative, and the components as a part-based representation
of the original data are also nonnegative. Moreover, tensor decomposition, as
an extension of matrix factorization to the higher-order arrays, is
ubiquitously used for linear subspace learning for high-order data. Tucker
decomposition and the canonical decomposition are two main types of tensor
decomposition methods. The former is usually used in machine learning, and the
latter is usually used in signal processing, also known as the parallel
factors (PARAFAC) decomposition. Higher-order orthogonal iteration (HOOI) is a
variant of Tucker decomposition with orthogonality constraints in the
projection matrices [19]. The higher-order singular value decomposition
(HOSVD) extends the matrix SVD to higher-order tensors, and its projection
matrices are column-wise orthogonality and the core tensor is orthogonal as
well. Its computation leads to the calculation of $N$ different matrix SVDs of
differently unfolded matrices [20, 21]. Multilinear CCA, as a multilinear
extension of the CCA algorithm, aims to find maximal correlations between the
weighted linear combinations of variables [22]. Multilinear PCA aims to find a
tensor to tensor projection that maximally captures the variations of the
original tensorial data [23] . Multilinear ICA model of tensor data learns the
statistically independent component of multiple factors [22]. When the
components of the raw input data is nonnegative, especially when it meets to
the nonnegative conditions have the physical meaning, such as spectrum,
energy, and probability, it hence that nonnegative tensor factorization (NTF)
is enforced nonnegative conditions on the PARAFAC model to find the
nonnegative factors or components [24]. Non-negative Tucker Decomposition
(NTD) is based on Tucker tensor decomposition and simultaneously enforces non-
negative constraints on the projection matrix and the core tensor[25]. Low-
rank regularized heterogeneous tensor decomposition (LRRHTD) adds the
orthogonal constraint for the first N-1 modes and the low-rank constraint for
the last mode of the projection matrix [26].
Alternatively, LDA and its variants are another popular way for subspace
learning, especially when the labelled data is available [27]. The target of
LDA is to find a discriminant subspace that maximizes the trace of the
between-class scatter while minimizing the trace of the within-class scatter.
Some variants of LDA have been proposed [28, 29], including the discriminant
analysis with tensor representation (DATER) [30] and the general tensor
discriminant analysis (GTDA) [31]. The discriminant analysis with tensor
representation (DATER) algorithm aims to find a tensor-to-tensor projection
while maximizing the tensor-based scatter ratio [30]. However, a limitation of
this algorithm is that it does not always converge over its iterations. The
general tensor discriminant analysis (GTDA) learns a discriminant subspace
with a tensor-to-tensor projection while maximizing the discriminant
information in a low-dimensional space [32]. Consider that independence
between extracted features is a desirable property in many real-world
applications, such that, uncorrelated multilinear discriminant analysis
(UMLDA) has been proposed to extract uncorrelated discriminative features
directly from tensorial data, with an assumption that each class is
represented by a single cluster and none of them can be solved by nonlinear
separation [33]. Moreover, the tensor rank-one discriminant analysis (TR1DA)
is to learn the projection subspace by repeatedly calculating the residues of
the original data with the scatter difference criterion, and eventually
obtains a set of rank-one projections [34]. The high order discriminant
analysis (HODA) is to find discriminative bases that based on the multilinear
structure of Tucker model [11]. The constrained multilinear discriminant
analysis (CMDA) seeks an optimal tensor-to-tensor projection for
discrimination in a lower-dimensional tensor subspace [32]. Theoretically, the
value of the scatter ratio criterion in CMDA approaches its extreme value, if
it is exists, with a bounded error.
Although the methods for linear subspace learning are well-studied, there are
still a number of open challenges, regarding the effectiveness and the
robustness in characterizing the nonlinear structures of the high-dimensional
data. In fact, several studies have reported that DATER could not guarantee
convergence to a stationary point during iterations [11, 32]. Another critical
issue of LDA-type algorithms are the singularity and instability of the
within-class scatter. Since LDA and its variants rely on the calculation of
the discriminant score, while the discriminant score requires computing the
inverse of the covariance matrix [28], thus it might meets the singularity
problem. To address such problems, Riemannian manifold optimization is
considered an candidate approach to learn the discriminant projection
subspace.
### 2.2 Riemannian Manifold Optimization
Riemannian manifold is actually a smooth subset of a vector space included in
the Euclidean space [35]. It abandons the flat Euclidean space and formulates
the optimization problem directly on the curved manifold. To describe a
general framework of Riemannian manifold optimization, it needs to define some
basic ingredients, such as the Riemannian matrix manifold $\mathcal{M}$,
smooth function $f:\mathcal{M}\rightarrow\mathbb{R}$ (i.e. along with its
Riemannian gradient i.e. $\mathrm{grad}f$, or Riemannian Hessian i.e.
$\mathrm{hess}f$ to perform the procedures of Riemannian manifold
optimization), projection operator i.e. $\mathrm{P}^{t}\left(\cdot\right)$,
Riemannian metric i.e. $\mathrm{g}\left(\cdot,\cdot\right)$, Riemannian
connection i.e. $\bigtriangledown_{\xi}\eta$, and retraction i.e.
$R_{x}\left(\xi\right)$. Concretely, we can define a projection operator to
project the embedded space (i.e. ambient space) to its tangent space, that is
obtained by subtracting the component in the orthogonal complement of the
tangent space (i.e. normal space $\mathcal{N}_{x}$). If the Riemannian
manifold is a quotient manifold, we can further define a projection operator
from the tangent space to the horizontal space, that is obtained by removing
the component in the orthogonal complement of the horizontal space (i.e.
vertical space $\mathcal{V}_{x}$). Note that connection is an important notion
that intimately relevant to the Riemannian Hessian and the vector transport,
and Levi-Civita connection is a unique affine connection used to define the
Riemannian Hessian of a loss function [36]. Vector transport allows movements
from a tangent space to another tangent space. Retraction is a mapping from
the tangent space back onto the manifold, ensuring that each update of
Riemannian manifold optimization remains on the manifold, and the exponential
retraction is the most expensive retraction, which describes the movement
along a geodesic. A geodesic is defined as a curve with the minimal length
connecting two points on the manifold. Figure 1 is the semantic illustration
of the Riemannian-based discriminant analysis. $T_{x}\mathcal{M}$ is the
tangent space of the embedded matrix manifold $\mathcal{M}$ endowed with a
bilinear, symmetric-positive form of Riemannian metric i.e.
$\mathrm{g}\left(\cdot,\cdot\right)$, that is termed as a Riemannian manifold.
In other words, a Riemannian manifold is a smooth manifold with a Riemannian
metirc. The Riemannian metirc defines a family of inner products on the
tangent spaces that smoothly vary with point $x$ on the manifold. Once that
Riemannian metric is defined, the distance, angle, and the curvature on the
manifold can be calculated.
Figure 1: A semantic illustration of the Riemannian-based Discriminant
Analysis. Here, $\mathrm{x}$ and $\mathrm{y}$ respectively represent the raw
input data and the reduced output data in the Euclidean space. For the
Riemannnian manifold optimization, $\mathcal{M}$ denotes the matrix manifold,
and its tangent space $T_{x}\mathcal{M}$ is a tangent plane on a point $x$ of
the manifold $\mathcal{M}$, which can be divided into the horizontal space
$\mathcal{H}_{x}$ and the vertical space $\mathcal{V}_{x}$. A retraction
$R_{x}\left(\xi_{x}\right)$ is a mapping from the tangent space back onto the
manifold $\mathcal{M}$. The tangent vector $\xi_{x}$ on the tangent space
$T_{x}\mathcal{M}$ denotes a possible movement direction at point $x$.
The Stiefel manifold and the Grassmann manifold are two popular manifolds to
conduct Riemannian manifold optimization. Specifically, the Stiefel manifold
$\mathrm{St}\left(D,d\right)$ is a set of $D\times d$ orthonormal matrices
$\left\\{U\in\mathbb{R}^{D\times d}:U^{T}U=I_{d}\right\\}$ [36]. Notably,
Stiefel manifold has no unique representation of $U$, for multiplying by any
orthogonal identity group does not change its original representation. Thus,
if $O_{d}$ is a set of $d\times d$ orthogonal matrices, then
$U_{1}=U_{2}O_{d}$. Otherwise, the Grassmann manifold
$\mathrm{Gr}\left(D,d\right)$ is a set of $d$-dimensional linear subspace of
$\mathbb{R}^{D}$ [36, 37]. If $d\leq D$, then the elements on the Grassmann
manifold $U\in\mathrm{Gr}\left(D,d\right)$ can also be represented as the
column space of Stiefel manifold $U\in\mathrm{St}\left(D,d\right)$, that is
identified with a set of equivalent classes
$\left[U\right]\in\mathrm{Gr}\left(D,d\right)$. Additionally, some notions
closely relevant to the Riemannian manifold (e.g. the Riemannian metric,
tangent space, and tangent vector) are worthy to clarify. When the columns of
equivalence class $\left[U\right]$ equals to the columns of $U$, such as for
given $U\in\mathrm{St}\left(D,d\right)$, the inner product of
$\mathrm{St}\left(D,d\right)$ also holds for $\mathrm{Gr}\left(D,d\right)$,
and the tangent space $T_{U}\mathrm{St}\left(D,d\right)$ of Stiefel manifold
is a vector space of all tangent vectors at point $U$, and the tangent vector
$\xi$ on the tangent space $T_{U}\mathrm{St}\left(D,d\right)$ is a possible
movement direction at point $U$, characterized as a matrix of $D\times d$.
Many existing methods for subspace learning in Sec 2.2 can be extended to
Riemannian manifold space. For example, manifold-based high order discriminant
analysis (MHODA) is an extension of HODA from Euclidean space to the
Riemannian manifold space [38]. Taking into account of the heterogeneity in
multimodal data, HTD Multinomial add the orthogonal constraints on the first
$N-1$ modes of the corresponding projection matrices, while the last mode of
the corresponding sample information is treated as the Multinomial manifold.
It results in an optimization problem on the Multinomial manifold which can be
solved by using the second-order geometry of trust-region method [39].
Naturally, using Riemannian manifold optimization can uncover the nonlinear
geometric structures of the high-dimensional data. It has valuable merits to
guarantee convergence to a globally optimal solution, whereas the traditional
methods (e.g. alternating least square (ALS) [40], multiplicative updating
rules (MURs) [25], and alternating direction method of multipliers (ADMM)
[41]) might be stuck into the local minima. In this work, we propose a novel
RDA method, which performs Riemannian manifold optimization for discriminant
subspace learning. Specifically, we define the loss function of RDA in the
Riemannian space, derive the Riemannian Hessian and present Riemannian
optimization algorithms for RDA.
## 3 Riemannian-based Discriminant Analysis (RDA)
### 3.1 The Loss Function of RDA
The target of linear discriminant analysis is to minimize the reconstruction
error in a mapping from high-dimensional data to a low-dimensional feature
space, while maximizing the discrimination between classes. In other words, it
aims to find an optimal discriminant bases $U\in\mathbb{R}^{D\times d}$ by
minimizing the within-class scatter $S_{W}$ and maximizing the between-class
scatter $S_{B}$, whereas the manipulation of projection operation
$\mathrm{y}=U^{T}\mathrm{x}$, and matrix $U$ is subject to the orthogonal
constraint i.e. $U^{T}U=I_{d}$. Here, we denote $\mathrm{x}\in\mathbb{R}^{D}$
as the input data with a high dimension $D$, $\mathrm{y}\in\mathbb{R}^{d}$ as
the low-dimensional representation of the input data. More concretely, the
loss function $f\left(U\right)$ can be formulated as the following:
$\displaystyle\mathop{\mathrm{min}}_{U}f\left(U\right)$ (1)
$\displaystyle=\sum_{c=1}^{C}\sum_{n\in
C_{c}}\left\|\mathrm{y}_{n}-\overline{\mathrm{y}}_{c}\right\|_{F}^{2}-\sum_{c=1}^{C}N_{c}\left\|\overline{\mathrm{y}}_{c}-\overline{\mathrm{y}}\right\|_{F}^{2}$
$\displaystyle=\sum_{c=1}^{C}\sum_{n\in
C_{c}}\left\|U^{T}\left(\mathrm{x}_{n}-\overline{\mathrm{x}}_{c}\right)\right\|_{F}^{2}-\sum_{c=1}^{C}N_{c}\left\|U^{T}\left(\overline{\mathrm{x}}_{c}-\overline{\mathrm{x}}\right)\right\|_{F}^{2}$
$\displaystyle=\left\|U^{T}\left(X-\overline{X}_{C}\right)\right\|_{F}^{2}-\left\|U^{T}\left(\overline{X}_{C}-\overline{X}\right)\right\|_{F}^{2}$
$\displaystyle=tr\left(U^{T}S_{W}U\right)-tr\left(U^{T}S_{B}U\right)$
$\displaystyle\mathrm{s.t.}\ U^{T}U=I_{d}$
where $N$ is the number of samples and $N_{c}$ is the number of samples from
class $c$. Obviously, the number of samples $N=\sum_{c=1}^{C}N_{c}$, and the
sample mean $\overline{\mathrm{y}}=\frac{1}{N}\sum_{n}\mathrm{y}_{n}$. The
mean of samples from class $c$, denoted as $\overline{\mathrm{y}}_{c}$, with
$\overline{\mathrm{y}}_{c}=\frac{1}{n_{c}}\sum_{n}\left[\mathrm{y}_{n}|n=c\right]$.
$S_{W}=\left(X-\overline{X}_{C}\right)\left(X-\overline{X}_{C}\right)^{T}$ is
a covariance matrix relative to the within-class scatter, and
$S_{B}=\left(\overline{X}_{C}-\overline{X}\right)\left(\overline{X}_{C}-\overline{X}\right)^{T}$
is a covariance matrix relative to the between-class scatter. Note that the
procedures of categorical alignment can promote transferable learning and
strengthen the generalization ability of the model.
An advantage of the loss function in Eq.(1) is that we convert the divisive
form to the subtractive one, thereby allows to effectively calculate the
Riemann gradient and Riemann Hessian. We can transform the constrained loss
function of Eq.(1) in Euclidean space to an unconstrained one in Stiefel
manifold $U\in{\mathrm{St}}\left(D,d\right)$, and then employ Riemannian
manifold optimization to solve the loss function. Therefore, the loss function
in the Stiefel manifold can be rewritten as:
$\displaystyle\mathop{\mathrm{min}}_{U\in{\mathrm{St}}(D,d)}f\left(U\right)=tr\left(U^{T}S_{W}U\right)-tr\left(U^{T}S_{B}U\right)$
(2)
According to the equivalence relation defined by the orthogonal group
$\mathcal{O}\left(d\right)$, the Grassmann manifold
$\mathrm{Gr}\left(D,d\right)$ can be formulated as the quotient space of
Stiefel manifold. In this case, the loss function of Eq.(1) can be formulated
on the Grassmann manifold:
$\mathop{\mathrm{min}}_{[U]\in{\mathrm{Gr}}(D,d)}f\left(U\right)=tr\left(U^{T}S_{W}U\right)-tr\left(U^{T}S_{B}U\right)$
(3)
where $\left[U\right]\in{\mathrm{Gr}}\left(D,d\right)$ is the equivalence
class for a given $U\in\mathrm{St}\left(D,d\right)$, and $\left[U\right]$
denotes a Grassmann point.
Since the covariance matrix is a symmetric-positive definite matrix, then the
optimization problem of Eq.(2) can also be formulated on the generalized
Stiefel manifold $\mathrm{GSt}\left(D,d;G\right)$ as
$\mathrm{GSt}\left(D,d;G\right)=\left\\{U\in\mathbb{R}^{D\times
d}:U^{T}GU=I_{d}\right\\}$ (4)
where $G$ denotes a covariance matrix.
Similarly, the optimization problem of Eq.(3) can be cast on the generalized
Grassmann manifold $\mathrm{GGr}\left(D,d;G\right)$ as
$\mathrm{GGr}\left(D,d;G\right)=\mathrm{GSt}\left(D,d;G\right)/\mathcal{O}\left(d\right)$
(5)
where $\mathcal{O}\left(d\right)$ represents the orthogonal group.
### 3.2 The Learning Algorithm for RDA
Here we present some of typical objects relative to the embedded submanifold
that utilized in the Riemannian manifold optimization. Firstly, the loss
function of Eq.(3) is reformulated as following:
$f\left(U\right)=tr\left(U^{T}\left(S_{W}-S_{B}\right)U\right)$ (6)
We use the inner product $\mathrm{g_{U}}:T_{U}\mathcal{M}\times
T_{U}\mathcal{M}\rightarrow\mathbb{R}$ as the Riemannian metric on the tangent
space of the manifold:
$\mathrm{g}_{U}\left(\xi,\eta\right)=tr\left(\xi^{T}\eta\right)$ (7)
In addition, the covariance matrix $G$ can be defined as the scaling matrix of
Riemannian preconditioning that regulates Riemannian metric on the tangent
space:
$\mathrm{g}_{U}\left(\xi,\eta\right)=tr\left(\xi^{T}\eta/G\right)$ (8)
We denote $\nabla f$ as the Euclidean gradient of the loss function Eq.(6),
and obtain the following expression:
$\nabla f\left(U\right)=2S_{W}U-2S_{B}U$ (9)
Once the computational space is split into two complementary spaces (i.e. the
tangent space and normal space), the Riemannian gradient of loss function,
denoted as $\mathrm{grad}f\left(U\right)$, can be obtained by the orthogonal
projection of the Euclidean gradient $\nabla f\left(U\right)$ to the tangent
space of the Riemannian manifold. For the Stiefel manifold,
$\mathrm{grad}f(U)$ can be calculated as follows
$\displaystyle\mathrm{grad}f\left(U\right)$
$\displaystyle=\mathrm{P}_{U}^{t}\left(\nabla f\left(U\right)\right)$ (10)
$\displaystyle=\nabla f\left(U\right)-U\mathrm{sym}\left(U^{T}\nabla
f\left(U\right)\right)$
where the function $\mathrm{sym}\left(X\right)$ is defined as
$\mathrm{sym}\left(X\right)=\left(X+X^{T}\right)/2$ to extract the symmetric
part of a square matrix $X$.
In the case that generalized Stiefel manifold, its orthogonal projection of
Euclidean gradient $\nabla f\left(U\right)$ from an ambient space to the
tangent space can be efficiently computed by the following
$\displaystyle\mathrm{grad}f\left(U\right)$ (11)
$\displaystyle=\mathrm{P}_{U}^{t}\left(\nabla f\left(U\right)\right)$
$\displaystyle=\nabla f\left(U\right)-U\mathrm{sym}\left(U^{T}G\nabla
f\left(U\right)\right)$
Likewise, the orthogonal projection from an ambient space to the tangent space
for the generalized Grassmann manifold can be formulated as following
$\mathrm{P}_{\left[U\right]}^{t}\left(U\right)=U-U\mathrm{sym}\left(U^{T}GU\right)$
(12)
Note that the second-order geometry of Riemannian Hessian is one of the most
important concepts relative to the connection i.e.
$\bigtriangledown_{\xi}\eta$, denoting the covariant derivative of the vector
field $\eta$ along the direction of another vector field $\xi$. Given a
concrete example, the covariant derivative of $D\nabla
f\left(U\right)\left[\xi\right]$ is the Euclidean directional derivative of
the Euclidean gradient $\nabla f\left(U\right)$ along the direction of the
tangent vector $\xi$ on the manifold.
Therefore, the Euclidean Hessian, i.e.
$\mathrm{Hess}f\left(U\right)\left[\xi\right]$, can be directly calculated
from Eq.(9) as following:
$\displaystyle\mathrm{Hess}f\left(U\right)\left[\xi\right]$
$\displaystyle=D\nabla f\left(U\right)\left[\xi\right]$ (13)
$\displaystyle=2S_{W}\xi-2S_{B}\xi$
And, the Riemannian Hessian, i.e.
$\mathrm{hess}f\left(U\right)\left[\xi\right]$, equals to the Euclidean
Hessian followed by the orthogonal projection onto the tangent space equipped
with Riemannian metric, thus
$\mathrm{hess}f\left(U\right)\left[\xi\right]=\mathrm{P}_{U}^{t}\left(\mathrm{Hess}f\left(U\right)\left[\xi\right]\right)$
(14)
For the Riemannian quotient manifold (e.g. Grassmann manifold), it requires to
further split the tangent space into other two orthogonal complementary
subspaces (i.e. the horizontal space and vertical space). Then, we can conduct
an orthogonal projection from the tangent space to the horizontal space along
the equivalence class of the vertical space to effectively isolate the extreme
point as the unique solution. More detailed discussions about the quotient
space refer to [36, 42]. For the implementation of Riemannian manifold
optimization, Riemannian version of conjugate gradient, and steepest descent,
and trust-region method have been constructed into some existing toolbox, such
as the Manopt toolbox [43]. Once the Riemannian gradient in Eq.(10) and
Riemannian Hessian in Eq.(14) are calculated, it is convenient to perform
Riemannian manifold optimization for solving the RDA.
### 3.3 Sparsity regularized discriminant analysis
In this subsection, we take into account of the model’s generalization
ability, and further incorporate an additional term about $U$ into the loss
function to prevent the model from overfitting the data. Specifically, the
loss function $f(U)$ for the sparsity regularized discriminant analysis is
formulated as follows:
$\mathop{\mathrm{min}}_{U}f(U)=tr\left(U^{T}\left(S_{W}-S_{B}\right)U\right)+\lambda\left\|U\right\|_{1}$
(15)
where $\lambda$ is a hyper-parameter to balance the discriminant performance
and the sparsity of $U$ in the model. Here, the loss function $f(U)$ can be
defined on either Stiefel manifold or Grassmann manifold.
To solve the sparsity regularized discriminant analysis, we have to derive the
first-order and second-order derivatives of regularization term w.r.t. $U$.
Naturally, we can obtain the Euclidean gradient of $\left\|U\right\|_{1}$
w.r.t. $U$ as $\nabla\left\|U\right\|_{1}=\mathrm{sgn}\left(U\right)$, where
$\mathrm{sgn}\left(U\right)=\left\\{\begin{matrix}1&if&\mathrm{U}\left(i,j\right)>0\\\
0&if&\mathrm{U}\left(i,j\right)=0\\\
-1&if&\mathrm{U}\left(i,j\right)<0\end{matrix}\right.$ (16)
And, the second-order derivatives of $\left\|U\right\|_{1}$ with respect to
$U$ in the Euclidean space is obtained as follows:
$\mathrm{Hess}\left\|U\right\|_{1}=2\sigma\left(U\right)$ (17)
where $\sigma\left(U\right)$ is defined as:
$\sigma\left(U\right)=\left\\{\begin{matrix}1&if\;\;U\left(i,j\right)=0\\\
0&otherwise\end{matrix}\right.$ (18)
Till now, all the deductions about loss function of the sparsity regularized
discriminant analysis have been completed. Algorithm 1 provides the pseudo
code of the optimization procedures. The code for RDA is available at
https://github.com/ncclabsustech/RDA-algorithm.
Algorithm 1 Riemannian-based Discriminant Analysis (RDA)
0: image dataset $X\in\mathbb{R}^{D\times N}$, sample labels
$L\in\mathbb{R}^{N\times 1}$
1: initial matrix $U$, gradient norm tolerance $\varepsilon^{1}=10^{-5}$, and
max iteration number $\mathrm{maxit}=200$. Let $0<c<1$, $\beta^{1}=0$,
$\xi^{0}=0$.
2: for $k\leq\mathrm{maxit}$ do
3: Compute Hessian in the Euclidean space,
$\mathrm{Hess}f\left(U\right)\left[\xi\right]$, by Eq.(13)
4: Compute the Riemannian Hessian,
$\mathrm{hess}f\left(U\right)\left[\xi\right]$, by Eq.(14)
5: Compute the weighted value
$\beta^{k}=tr\left(\eta^{kT}\eta^{k}\right)/tr\left(\eta^{(k-1)T}\eta^{k-1}\right)$.
6: Compute a transport direction $\mathcal{T}_{U^{k-1}\rightarrow
U^{k}\left(\xi^{k-1}\right)}=\mathcal{P}_{U^{k}}\left(\xi^{k-1}\right)$.
7: Compute a conjugate direction
$\xi^{k}=-\mathrm{grad}_{\mathrm{R}}f\left(U^{k}\right)+\beta^{k}\mathcal{T}_{U^{k-1}\rightarrow
U^{k}\left(\xi^{k-1}\right)}$.
8: Compute Armijo step size $\alpha^{k}$ using backtracking
$f\left(\mathrm{R}_{U^{k}}\left(\alpha^{k}\xi^{k}\right)\right)\geq
f\left(U^{k}\right)+c\alpha^{k}tr\left(\eta^{kT}\xi^{k}\right)$.
9: Terminate and output $U^{k+1}$ if one of the stopping conditions,
$\left\|\eta^{k+1}\right\|_{F}^{2}\leq\varepsilon^{1}$, or iteration number
$k\geq\mathrm{maxit}$ is met.
10: end for
11: OUTPUT $U$.
## 4 Numerical Experiments and Results
In this section, we test the effectiveness of RDA on feature extraction tasks
and classification tasks. RDA is compared with four variants of multilinear
discriminant analysis (i.e. HODA [11], DATER [30], CMDA [32], and MHODA [38])
and four variants of tensor decomposition (i.e. NTD [25], LRRHTD [26], HTD
Multinomial [39], and HOSVD [20]). All subsequent numerical experiments are
carried out on a desktop (Intel Core i5-5200U CPU with a frequency of 2.20 GHz
and a RAM of 8.00 GB). Each experiment is repeated 10 times, each time using
different random sampling data.
### 4.1 Datasets Description
Our experiment involves seven benchmark image datasets, namely the COIL20
Object, ETH80 Object, MNIST Digits, USPS Digits, ORL Faces, Olivetti Faces,
and CMU PIE Faces. Figure 2 shows some examples of sampling from these data
sets. We did not show the MNIST dataset here because it is a well-known
dataset.
Figure 2: Some examples from six datasets used in experiments. (a) Olivetti
dataset. (b) COIL20 dataset. (c) ETH80 dataset. (d) ORL dataset. (e) CMU PIE
dataset. (f) USPS dataset.
The COIL20 dataset contains 1420 grayscale images of 20 objects (72 images per
object). Objects in COIL20 have a variety of complex geometric and reflective
properties. In our experiments, the image from COIL20 is downsampled to a size
of $32\times 32$ with 0-255 grayscale.
The ETH80 dataset is a multi-view image dataset used for object
classification. It includes 8 categories: apple, car, cow, cup, dog, horse,
pear, and tomato. Each category contains 10 objects, and each object has 41
images from different views, resulting in a total of 3280 images. The
resolution of original images is $128\times 128$, and we adjust the size of
each image to $32\times 32$ pixel.
Both USPS and MNIST datasets are 0-9 handwritten digits. The USPS dataset has
11,000 images, with a size of $16\times 16$ pixels, while the MNIST dataset
has 60,000 images belonging to the training set, with a size of $28\times 28$
pixels. In our experiment, we randomly selected 2000 images (200 images per
category) from the USPS dataset, and 3000 images (300 images per category)
from the MNIST dataset.
The ORL dataset contains 400 images from 40 different people, each with 10
different images. These images were taken multiple times under different
lighting conditions and facial expressions (eyes open/closed; with/without
smile) and facial details (with/without glasses). All images were taken
against a dark uniform background, with the subject in an upright frontal
position (tolerance to certain lateral movements). We adjust the size of each
image to $32\times 32$ pixels.
The Olivetti dataset consists of 400 faces from 40 people (10 per person). The
viewing angle of those images changes very little, but people’s expressions
change a lot, and occasionally they wear glasses. The size of the image is
$64\times 64=4096$ pixels, and the data is labeled according to the identity.
The CMU PIE dataset is a gray-scale face dataset, including 68 people, and
each person has 141 face images. The images were taken under different
lighting conditions. We extracted a subset of 50 individuals and the
corresponding 50 facial images of each person, resulting in a total of 2500
images.
Table 1: Illustrations of the datasets dataset | #samples | sizeoriginal | sizefinal | #classes
---|---|---|---|---
ETH80 | 3280 | 32*32 | 8*8 | 8
MNIST | 3000 | 28*28 | 10*10 | 10
USPS | 2000 | 16*16 | 7*8 | 10
COIL20 | 1440 | 32*32 | 8*8 | 20
ORL | 400 | 32*32 | 6*6 | 40
Olivetti | 400 | 64*64 | 8*8 | 40
CMU PIE | 2500 | 32*32 | 8*8 | 50
Table 1 shows the general description of seven datasets, where the attributes
of each data set are the total number of samples, the dimensions of the
original data, the final dimension after dimensionality reduction, and the
number of classes we use experiments. Note that each sample has a real
category label (such as object, identity, or digit). We preprocess the dataset
in the following way: a) randomly shuffle all the data, b) normalize the gray
value of pixels to the unit.
In the following numerical experiments, the data is represented by a third-
order tensor, where the first two modes are associated with the spatial
information of image pixels, and the last mode represents the number of
samples. It is worth noting that RDA algorithm and its implementation are very
general, and there is no such restriction on the data format. In the tests, we
first perform subspace learning and reduce the dimensionality of the tensor
data (from sizeoriginal to sizefinal in Table 1), and then apply the $k$-means
clustering or $k$-nearest-neighbour classification on the extracted low-
dimensional features.
### 4.2 Clustering analysis
We first test whether the features in the low-dimensional subspace extracted
by RDA can cluster the data. Specifically, we use five supervised algorithms
(e.g. RDA, HODA, CMDA, MHODA, and DATER) to perform subspace learning for each
data set. Then we cluster the features on the subspace with $k\mathrm{-means}$
clustering. We randomly initialize 10 times and calculate the average result
of 10 times. The results are quantified by clustering accuracy (ACC) and
normalized mutual information (NMI) [39].
Table 2: $k\mathrm{-means}$ clustering results of RDA and four supervised
algorithms on 7 datasets.
Dataset | Metric | Riemannian-based optimization | Euclidean-based optimization
---|---|---|---
RDA | MHODA | HODA | CMDA | DATER
ETH80 | ACC | 0.5452±0.0048 | 0.5098±0.0000 | 0.4750±0.0039 | 0.4852±0.0108 | 0.4714±0.0219
NMI | 0.5094±0.0000 | 0.4691±0.0000 | 0.4523±0.0050 | 0.4598±0.0102 | 0.4155±0.0180
MNIST | ACC | 0.7552±0.0029 | 0.1888±0.1107 | 0.5563±0.0297 | * | *
NMI | 0.6314±0.0016 | 0.0830±0.1256 | 0.4902±0.0184 | * | *
USPS | ACC | 0.8482±0.0010 | 0.5074±0.0673 | 0.4580±0.0339 | 0.3377±0.0152 | 0.4912±0.0570
NMI | 0.7339±0.0000 | 0.4621±0.0718 | 0.4368±0.0289 | 0.2752±0.0142 | 0.4607±0.0447
COIL20 | ACC | 0.7948±0.0398 | 0.7244±0.0345 | 0.6144±0.0216 | 0.6563±0.0324 | 0.6337±0.0178
NMI | 0.8553±0.0199 | 0.8133±0.0139 | 0.7388±0.0118 | 0.7637±0.0093 | 0.7334±0.0144
ORL | ACC | 0.7380±0.0278 | 0.5817±0.0262 | 0.4437±0.0213 | 0.4390±0.0199 | 0.4690±0.0273
NMI | 0.8739±0.0112 | 0.7871±0.0114 | 0.6769±0.0089 | 0.6713±0.0149 | 0.6538±0.0194
Olivetti | ACC | 0.7508±0.0407 | 0.6627±0.0372 | 0.4900±0.0324 | 0.5045±0.0292 | 0.5727±0.0404
NMI | 0.8776±0.0146 | 0.8251±0.0154 | 0.7044±0.0152 | 0.7155±0.0151 | 0.7470±0.0255
CMU PIE | ACC | 0.7866±0.0220 | 0.5927±0.0193 | 0.1546±0.0034 | 0.1206±0.0042 | 0.3764±0.0299
NMI | 0.8776±0.0086 | 0.7472±0.0073 | 0.3686±0.0078 | 0.3014±0.0040 | 0.5690±0.0238
* The algorithm failed in the dataset, as the between-class matrix is singular.
Figure 3: The clustering accuracy varying with the number of classes in the
USPS digit dataset. RDA achieves the highest accuracy in digit clustering.
Figure 4: The clustering accuracy varying with the number of classes in the
CMU PIE face dataset. RDA achieves the highest accuracy in face clustering.
Table 2 shows the clustering results of RDA and four supervised methods on
seven datasets. We show the mean and standard deviation of ACC/NMI in 10
tests. The best result for each data set is highlighted in bold text.
Obviously, RDA achieves the best performance compared to HODA, CMDA, MHODA and
DATER. Especially, when the dataset is complex and multi-class, such as the
CMU PIE dataset, the Riemannian-based algorithms (both of RDA and MHODA)
provide better clustering results than Euclidean-based algorithms, implying
that Riemannian-based methods have a higher ability to extract complex
features.
We then further test the performance of RDA with four unsupervised tensor
decomposition methods, including a Riemannian-based method (e.g. HTD-
Multinomial), and three Euclidean-based clustering methods (e.g LRRHTD, NTD,
and HOSVD). Table 3 shows the experimental results, suggesting that RDA
outperforms all the other tested methods.
Table 3: $k\mathrm{-means}$ clustering results of RDA and four unsupervised tensor decomposition methods on 7 datasets. Dataset | Metric | Riemannian-based optimization | Euclidean-based optimization
---|---|---|---
RDA | HTD-Multinomial | LRRHTD | NTD | HOSVD
ETH80 | ACC | 0.5452±0.0048 | 0.4714±0.0219 | 0.4994±0.0062 | 0.4385±0.0042 | 0.4633±0.0025
NMI | 0.5094±0.0000 | 0.4155±0.0180 | 0.4764±0.0065 | 0.3968±0000 | 0.3773±0000
MNIST | ACC | 0.7552±0.0029 | 0.5040±0.0385 | 0.5365±0.0135 | 0.5090±0.0140 | 0.5101±0.0023
NMI | 0.6314±0.0016 | 0.4386±0.0247 | 0.4790±0.0054 | 0.4608±0.0053 | 0.4484±0.0024
USPS | ACC | 0.8482±0.0010 | 0.4912±0.0570 | 0.4625±0.0089 | 0.4186±0.0311 | 0.5200±0.0259
NMI | 0.7339±0.0000 | 0.4607±0.0447 | 0.4699±0.0064 | 0.4324±0.0199 | 0.4639±0.0142
COIL20 | ACC | 0.7948±0.0398 | 0.6337±0.0178 | 0.6633±0.0296 | 0.6317±0.0265 | 0.5928±0.0199
NMI | 0.8553±0.0199 | 0.7334±0.0144 | 0.7675±0.0116 | 0.7428±0.0122 | 0.7215±0.0153
ORL | ACC | 0.7380±0.0278 | 0.4690±0.0273 | 0.5215±0.0252 | 0.4397±0.0186 | 0.5915±0.0284
NMI | 0.8739±0.0112 | 0.6538±0.0194 | 0.7339±0.0127 | 0.6704±0.0112 | 0.7611±0.0239
Olivetti | ACC | 0.7508±0.0407 | 0.5727±0.0404 | 0.5300±0.0309 | 0.5627±0.0163 | 0.5693±0.0266
NMI | 0.8776±0.0146 | 0.7470±0.0255 | 0.7347±0.0166 | 0.7366±0.0092 | 0.7451±0.0156
CMU PIE | ACC | 0.7866±0.0220 | 0.3764±0.0299 | 0.1477±0.0041 | 0.1424±0.0025 | 0.3707±0.0277
NMI | 0.8776±0.0086 | 0.5690±0.0238 | 0.3521±0.0063 | 0.3420±0.0040 | 0.5994±0.0163
We further investigate the influence of the number of classes on the
performance of RDA clustering. We test the clustering ability of RDA on the
USPS digit dataset and CUM PIE face dataset, compared with other seven SOTA
algorithms. Figure 3-4 shows that the clustering accuracy varying with the
number of classes in the USPS dataset and CMU PIE dataset, respectively. These
results confirm that RDA robustly achieves the best performance on both data
sets regardless of the number of classes.
### 4.3 Classification
Here we test the classification performance using the learned features from
RDA with a standard classifier, namely $k$-nearest-neighbour ($k\mathrm{NN}$)
classifier. We calculate the projection matrix $U$ from the train samples
$X_{train}$, and then use the learned matrix $U$ to learn the low-dimensional
representation of the test data $X_{test}$. The class of the test samples is
predicted with the following equation:
$Y_{test}=U^{T}X_{test}$ (19)
We conduct classification experiments on five benchmark datasets (Sec 4.1),
including ETH80, MNIST, USPS, COIL 20 and CMU PIE. The data samples from the
datasets are assumed to have the uniform distribution in each experiment. A
3-fold cross validation is applied to the training data and a 5-fold cross
validation to the test data. We use the the ACC, NMI and $k\mathrm{NN}$
classification accuracy as the evaluation metrics.
Table 4 shows the classification results from RDA and other methods using a
$k\mathrm{NN}$ classifier. As shown in Table 4, RDA achieves better
performance than most existing algorithms. Interestingly, the MHODA algorithm,
optimizing via the product manifold, is consistently worse than RDA on the
Stiefel manifold, implying that the Stiefel manifold optimization might be
more robust than the product manifold.
Table 4: Comparisons of classification results on 5 datasets.
Dataset | Metric | Riemannian-based optimization | Euclidean-based optimization
---|---|---|---
RDA | MHODA | HODA | CMDA | DATER | HOSVD
ETH80 | ACC | 0.5405 | 0.5058 | 0.4784 | 0.5170 | 0.5104 | 0.4665
NMI | 0.5073 | 0.4692 | 0.4489 | 0.4565 | 0.4571 | 0.3816
$k\mathrm{NN}$ | 0.7355 | 0.6856 | 0.7621 | 0.7650 | 0.7686 | 0.7844
MNIST | ACC | 0.7631 | 0.2641 | 0.5494 | * | * | 0.5114
NMI | 0.6509 | 0.1700 | 0.4875 | * | * | 0.4565
$k\mathrm{NN}$ | 0.8445 | 0.4040 | 0.8505 | * | * | 0.8555
USPS | ACC | 0.8600 | 0.5266 | 0.4554 | 0.5688 | 0.5487 | 0.5026
NMI | 0.7565 | 0.4889 | 0.4363 | 0.5352 | 0.5285 | 0.4682
$k\mathrm{NN}$ | 0.8591 | 0.5952 | 0.7878 | 0.8808 | 0.8831 | 0.8458
COIL20 | ACC | 0.7777 | 0.6973 | 0.6155 | 0.7247 | 0.7442 | 0.6050
NMI | 0.8522 | 0.8385 | 0.7402 | 0.8264 | 0.8378 | 0.7163
$k\mathrm{NN}$ | 0.8771 | 0.8385 | 0.6729 | 0.8417 | 0.8302 | 0.7177
CMU PIE | ACC | 0.8024 | 0.5866 | 0.1708 | 0.6166 | 0.6018 | 0.3920
NMI | 0.8857 | 0.7435 | 0.3848 | 0.7638 | 0.7625 | 0.6061
$k\mathrm{NN}$ | 0.6713 | 0.5261 | 0.2147 | 0.6002 | 0.6205 | 0.1890
* The algorithm failed in the dataset, as the between-class matrix is singular.
To compare the first-order approximation and the second-order approximation,
we test the trust region methods (RDA and MHODA) and the conjugate gradient
methods (conj-RDA and conj-MHODA). As shown in Table 5. We find that RDA
reliably outperforms conj-RDA in all datasets, suggesting the trust region
method is better than the first-order approximation for RDA. The improvement
of the trust region method is not obvious for the manifold-based high-order
discriminant analysis (i.e MHODA vs conj-MHODA), which might be caused by the
sub-optimal solution in the product manifold optimization used in MHODA.
Table 5: Comparison of algorithmic performance on first-order approximation and the second-order approximation. Dataset | Metric | Stiefel manifold optimization | Product manifold optimization
---|---|---|---
RDA | conj-RDA | MHODA | conj-MHODA
ETH80 | ACC | 0.5000±0.0200 | 0.4362±0.0284 | 0.5099±0.0091 | 0.5012±0.0086
NMI | 0.5231±0.0050 | 0.4749±0.0197 | 0.4696±0.0059 | 0.4674±0.0052
$k\mathrm{NN}$ | 0.7156±0.0226 | 0.6935±0.0435 | 0.6699±0.0294 | 0.6747±0.0322
MNIST | ACC | 0.7696±0.0207 | 0.6559±0.0343 | 0.2148±0.1331 | 0.1679±0.0383
NMI | 0.6518±0.0145 | 0.5634±0.0160 | 0.1137±0.1431 | 0.0726±0.0617
$k\mathrm{NN}$ | 0.8420±0.0236 | 0.8110±0.0244 | 0.4410±0.1116 | 0.3620±0.0461
USPS | ACC | 0.8599±0.0062 | 0.6931±0.0609 | 0.4978±0.0816 | 0.5162±0.0249
NMI | 0.7555±0.0053 | 0.6097±0.0378 | 0.4682±0.0764 | 0.4888±0.0057
$k\mathrm{NN}$ | 0.8724±0.0414 | 0.8192±0.0380 | 0.6091±0.0516 | 0.6013±0.0411
COIL20 | ACC | 0.7666±0.0376 | 0.6473±0.0373 | 0.7111±0.0495 | 0.7085±0.0291
NMI | 0.8469±0.0177 | 0.7506±0.0204 | 0.8133±0.0191 | 0.8077±0.0127
$k\mathrm{NN}$ | 0.8625±0.0380 | 0.7438±0.0247 | 0.8385±0.0445 | 0.8229±0.0396
CMU PIE | ACC | 0.8543±0.0306 | 0.3526±0.0436 | 0.5784±0.0233 | 0.5934±0.0140
NMI | 0.9344±0.0100 | 0.5767±0.0343 | 0.7414±0.0128 | 0.7485±0.0095
$k\mathrm{NN}$ | 0.7973±0.0422 | 0.3923±0.0445 | 0.5512±0.0522 | 0.5659±0.0333
### 4.4 Sparse regularized RDA
The sparsity property has been reported in many real-world applications, and
using sparsity regularization have the advantages of being robust to noise and
thus might improve the classification performance especially for the high-
dimensional data [4]. In order to study the effect of sparsity regularization
on RDA-based classification, we apply the second-order geometry of the trust-
region method and the first-order geometry of the conjugate gradient to solve
the loss function in Eq. (15) on Stiefel manifold and Grassmann manifold,
respectively. Table 6 lists the classification performance of the sparsity
regularized RDA. StRDA and GrRDA represent to the RDA on Stiefel manifold and
Grassmann manifold, respectively. SStRDA and SGrRDA represent StRDA and GrRDA
with an additional sparsity regularization, while conj-SStRDA and conj-SGrRDA
denote SStRDA and SGrRDA solved by the first-order geometry of the conjugate
gradient method.
In theory, sparsity regularization on $U$ can reduce the learning parameters
and improve the generalization ability of algorithms [4]. Our experimental
evidence in (Table 5 & 6) also supports the sparsity regularization of Stiefel
manifold (StRDA vs SStRDA) and Grassmann manifold (GrRSA vs SGrRDA) in most
cases, demonstrating that the sparsity regularization can effectively enhance
model generalization, which is consistent with previous study [4].
Table 6: Comparison of classification results with/without a sparsity regularization term. Dataset | Metric | Stiefel manifold Optimization | Grassman manifold Optimization
---|---|---|---
StRDA | SStRDA | conj-SStRDA | GrRDA | SGrRDA | conj-SGrRDA
COIL20 | ACC | 0.7666 | 0.7851 | 0.7721 | 0.7818 | 0.7872 | 0.7713
NMI | 0.8469 | 0.8556 | 0.8488 | 0.8523 | 0.8573 | 0.8512
$k\mathrm{NN}$ | 0.8625 | 0.8562 | 0.8615 | 0.8562 | 0.8844 | 0.8406
ETH80 | ACC | 0.5000 | 0.4959 | 0.4970 | 0.5029 | 0.4935 | 0.4970
NMI | 0.5231 | 0.5260 | 0.5218 | 0.5260 | 0.5253 | 0.5240
$k\mathrm{NN}$ | 0.7156 | 0.7194 | 0.7156 | 0.7301 | 0.7226 | 0.7171
Dense ETH80 | ACC | 0.5486 | 0.5439 | 0.5333 | 0.5489 | 0.5451 | 0.5386
NMI | 0.5113 | 0.5095 | 0.5018 | 0.5105 | 0.5131 | 0.5059
$k\mathrm{NN}$ | 0.7338 | 0.7162 | 0.6920 | 0.7256 | 0.7350 | 0.6674
MNIST | ACC | 0.7696 | 0.7601 | 0.7695 | 0.7682 | 0.7570 | 0.7651
NMI | 0.6518 | 0.6510 | 0.6505 | 0.6532 | 0.6465 | 0.6491
$k\mathrm{NN}$ | 0.8420 | 0.8480 | 0.8395 | 0.8415 | 0.8465 | 0.8715
USPS | ACC | 0.8599 | 0.8375 | 0.8413 | 0.8498 | 0.8615 | 0.8655
NMI | 0.7555 | 0.7434 | 0.7492 | 0.7546 | 0.7578 | 0.7609
$k\mathrm{NN}$ | 0.8724 | 0.8872 | 0.8716 | 0.8680 | 0.8777 | 0.8715
CMU PIE | ACC | 0.8543 | 0.8619 | 0.8482 | 0.8602 | 0.8587 | 0.8407
NMI | 0.9344 | 0.9324 | 0.9310 | 0.9361 | 0.9381 | 0.9290
$k\mathrm{NN}$ | 0.7973 | 0.7954 | 0.7761 | 0.7813 | 0.8328 | 0.8095
## 5 Discussion and Conclusion
In this paper, we proposed a novel method using Riemannian manifold
optimization for discriminant subspace learning, namely Riemannian-based
discriminant analysis. The numerical results (Table 2-4) suggest that RDA
outperforms many other methods optimized in Euclidean space, as well as the
existing Riemannian-based methods.
Since the inter-class scatter matrix may be singularity (such as the CMDA and
DATER algorithms shown in Table 2&4), many traditional Euclidean-based methods
for subspace learning may not be able to guarantee monotonic convergence to
its optimal solution. The previous literature also reported similar results
[28]. In contrast, our proposed RDA can effectively avoid the singularity
problem. As RDA has the subtraction form of the loss function instead of a
division form in the traditional methods, RDA can effectively avoid
calculating the inverse of Hessian matrix, thereby reducing the amount of
calculation to the Riemannian Hessian.
Due to the discovery of non-linear structures, our proposed RDA is superior to
traditional methods optimized in Euclidean space (such as HODA, CMDA, DATER in
Table 2). In addition, the existing Riemannian-based algorithms (e.g. HTD-
Multinomial and MHODA), as well as the tensor decomposition methods (e.g.
LRRHTD and NTD), are not as good as RDA algorithms. It is worth noting that
RDA obtains higher performance for dealing with multi-class and complex
dataset (e.g. CMU PIE, COIL20). RDA can provide the higher clustering accuracy
regardless of the number of classes (Figure 3-4), suggesting that Riemannian-
based optimization reliably learns an optimal subspace. Generally, the
supervised learning methods are superior to unsupervised learning methods in
extracting and selecting features (Table 2 vs Table 3) due to the full
utilization of sample labels in the supervised learning, which is in line with
previous study [44].
The comparisons between the trust-region methods (RDA and MHODA) and the
conjugate gradient methods (conjRDA and conj-MHODA) shows that the second-
order geometry of the trust-region method improves the clustering and
classification performance of RDA, although it may be unreliable for MHODA
(Table 5). The advantage of RDA may stem from the use of equivalence classes
in vertical space, which can effectively isolate the optimal solution in the
quotient space [36].
Although we have shown that RDA benefits from the use of Riemannian geometry
in subspace learning, there are many aspects that have not been covered in
this study. For example, how to initialize the factor matrix to ensure the
convergence of the algorithm, how to design regularization terms other than
sparsity, and how to choose the best Riemann metric. It has been shown that
adjusting the Riemann metric according to the underlying structure and
constraints of the loss function can speed up the convergence speed and reduce
the running time [45]. Moreover, Riemannian-based discriminant analysis has a
limitation: it suffers from an expensive optimization process to find the
optimal subspace. In order to solve this problem, other methods, such as
Riemannian preconditioning [45], are worthy of further study.
Beyond our work, many other traditional methods solved in the Euclidean space
can be transformed into the Riemannian manifold space and employ Riemannian
manifold optimization. It is of particular interests to study how to design
the cost function and achieve super-linear convergence in the tangent space of
each iteration. When designing algorithms in Riemannian manifold, it is
important to balance the trade-off between effectiveness (e.g. accuracy) and
efficiency (e.g. computational complexity).
In summary, RDA provide a novel way to solve the LDA problem with Riemannian
manifold optimization. It is an effective method for dimensionality reduction,
feature extraction, and classification. We believe that RDA for subspace
learning method has great potential in many practical applications.
## Acknowledgements
The authors would like to thank anonymous reviewers for their detailed and
helpful comments. This work was funded in part by the National Natural Science
Foundation of China (62001205), Guangdong Natural Science Foundation Joint
Fund (2019A1515111038), Shenzhen Science and Technology Innovation Committee
(20200925155957004, KCXFZ2020122117340001, SGDX2020110309280100), Shenzhen Key
Laboratory of Smart Healthcare Engineering (ZDSYS20200811144003009).
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|
Lucky planets: how circum-binary planets survive the supernova in one of the
inner-binary components
Fedde Fagginger Auer,
Simon Portegies Zwart
Leiden Observatory, Leiden University, PO Box 9513, 2300 RA, Leiden, The
Netherlands
*<EMAIL_ADDRESS>
## Abstract
A planet hardly ever survives the supernova of the host star in a bound orbit,
because mass loss in the supernova and the natal kick imparted to the newly
formed compact object cause the planet to be ejected. A planet in orbit around
a binary has a considerably higher probability to survive the supernova
explosion of one of the inner binary stars. In those cases, the planet most
likely remains bound to the companion of the exploding star, whereas the
compact object is ejected. We estimate this to happen to $\sim 1/33$ the
circum-binary planetary systems. These planetary orbits tend to be highly
eccentric ($e\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 0.9$), and
$\sim 20$ % of these planets have retrograde orbits compared to their former
binary. The probability that the planet as well as the binary (now with a
compact object) remains bound is about ten times smaller ($\sim 3\cdot
10^{-3}$). We then expect the Milky way Galaxy to host $\
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ 10$ x-ray binaries that are
still orbited by a planet, and $\
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ 150$ planets that survived
in orbit around the compact object’s companion. These numbers should be
convolved with the fraction of massive binaries that is orbited by a planet.
## 1 Introduction
Since the discovery of exoplanets around pulsars [1] there has been a debate
on their origin. Popular scenarios include in situ formation [3, 2] or the
dynamical capture of a planet in a dense stellar system [4]. The possibility
of a planet surviving its host star’s supernova is often neglected, because a
planet in orbit around a single exploding star is not expected to survive the
supernova [5]. The combination of mass loss in the supernova explosion [6] and
the natal kick imparted to the new compact object [7, 8] mediates the
survivability of low-mass x-ray binaries [11, 9, 12, 10, 13, 14, 15], but
planetary orbits are too fragile to survive this process [16].
The survivability of a planet in orbit around a binary, of which one of the
components experiences a supernova is considerably larger than when the planet
directly orbits the exploding star. The lower relative mass loss in a binary
compared to a single star and the dilution of the velocity kick by dragging
along the companion star [17], may cause the planet to survive either in orbit
around the binary, or around the original secondary –non-exploding– star (in
which case the compact object is ejected from the system).
We calculate the probability that a circum-binary planet survives the first
supernova in a massive binary. In a first step, we perform binary population
synthesis calculations to determine the orbital phase-space distribution of
the pre-supernova binary system. We subsequently add a planet in orbit around
the binary and analytically calculate the supernova’s effect on the system to
determine the planet’s survivability and post-supernova orbital parameters.
## 2 Method
We approach the problem on the survivability of circum-binary planets using a
combination of techniques. First, we perform a series of population synthesis
calculations for binary stars. The binaries that survive until they experience
their first supernovae are, in the second step, provided with a circum-binary
planet in a stable orbit, after which we resolve the supernova explosion. This
second step is calculated analytically for each individual system, and a
population study is carried out through Monte Carlo sampling.
### 2.1 Population synthesis calculations of pre-supernova binaries
Binary evolution is a complicated non-linear problem that is not easily
performed analytically (see however [14]). Therefore we perform this part of
the calculation numerically, using the publicly available and well-tested
binary population-synthesis code SeBa [15, 18]. We adopt the version available
in the Astrophysics Multipurpose Software Environment (AMUSE, [19]). The code
takes the metallicities and masses of the two stars ($m_{1}$ and $m_{2}$, at
zero age) and together with the orbital period and the eccentricity ($e_{i}$),
the code gives the evolution of these parameters as a function of time. We
adopt Solar metallicity throughout this study.
There are quite a number of free parameters in a binary population synthesis
code [20]. The most important ones are the treatment of non-conservative mass
transfer, the amount of angular momentum per unit mass that is lost by the
mass leaving the binary system ($\beta$), and the common-envelope treatment
(often summarized in the parameters $\alpha\lambda$ and $\gamma$). We adopt
the model parameters as in [12] (their model C, see table 1,
$\alpha\lambda=0.5$, $\beta=6$ and $\gamma=1$), which matches with the
Galactic x-ray binary population. These parameters are classically identified
with $\alpha$, $\beta$, and $\gamma$, but in the next section we use the same
letters in a different context.
We determine the probability density function for pre-supernova parameters in
phase space through binary population synthesis, and subsequently, apply a
kernel-density estimator to smooth these distributions and bootstrap the
number of systems. In the next step, we use these smoothed distributions to
randomly select pre-supernova binaries to which we add a planet and
subsequently apply the effect of the supernova.
### 2.2 Analytic considerations of the post-supernova system
The effect of a supernova on a binary system was explored by [6]. Later [21,
15, 11] further studied the binary’s survivability through population
synthesis, expanding the original formulation for [6] to include elliptical
orbits and a wide range of velocity kicks imparted to the newly formed compact
object. [22] and [23] subsequently expanded the formalism to multiple systems
by considering a hierarchy of nested binaries. We expand on these studies by
adopting a planet around a binary system of which one component
instantaneously loses mass isotropically and receives a velocity kick.
Figure 1: The geometry of the hierarchical triple, with inner binary composed
o stars $A$ and $B$, and the outer planet $C$. Also indicated are the outer
binary semi-major axis $a_{o}$ and the true anomaly $\nu_{i}$ and $\nu_{o}$,
for the inner and outer orbit, respectively.
In figure 1 we sketch the adopted configuration, in which an inner binary
(star $A$ and star $B$) is orbited by a planet ($C$)111Following [15], we
write systems by assing the first three letters of the alphabet to the two
stars and the planet. Parenthesis indicate a bound system, and multiple braces
a hierachy. The initial triple, for example, then is written as $((A,B),C)$,
and a possible binary between the seconday star and the planet with an unbound
primary is written as $A,(B,C)$.. The properties of the inner (outer) binary
are labeled with subscript $i$ ($o$). The star $A$ undergoes the supernova. We
further adopt the notation from [21], and with that the remnant mass
$m^{\prime}_{A}$ and the average orbital velocity $v_{\text{orb}}$.
$\tilde{m}=\frac{m_{A}+m_{B}}{m^{\prime}_{A}+m_{B}},\hskip
14.22636pt\tilde{v}=\frac{|\mathbf{v}_{\text{kick}}|}{v_{\text{orb}}},\hskip
14.22636pt\mu_{B}=\frac{m_{B}}{m_{A}+m_{B}}.$
For a circular binary we have the expression for the orbital energy after the
supernova:
$\begin{split}E^{\prime}&=-\frac{Gm^{\prime}_{A}m_{B}}{2a}\Delta,\\\
\Delta&=\frac{2a}{|\mathbf{r}-\delta\mathbf{r}|}-\tilde{m}(1+\tilde{v}^{2}-2\tilde{v}\cos{(\theta)}\cos{(\phi)}).\end{split}$
Here $\theta,\phi$ give the kick direction (we define $\theta=\phi=0$ to be a
kick anti-parallel to orbital motion, and therefore have a minus sign
difference with [21]), $a$ is the semi-major axis of the binary before the
supernova, and $\mathbf{r}$ the relative position vector between the binary
components. The shift $\delta\mathbf{r}$ allows for an instantaneous
displacement of the center of mass of the inner binary. This displacement of
the inner binary results from supernova shell when it passes the companion
star (see [22]). A bound orbit requires $E^{\prime}<0$ and therefore
$\Delta>0$. This inequality gives the physical requirements on the mass loss
and velocity kick. Combining this with the method of [22] we can derive a
similar formula for the outer binary in the hierarchical triple. We assume
that both binaries are initially circular and co-planar (see figure 1). We
select the inner binary’s true anomaly $\nu_{i}=0$ by an appropriate rotation
of our coordinate system and assume $m_{C}\ll m_{A},m_{B}$. We define a
$\Delta_{i},\Delta_{o}$ for both binaries. The supernova in the inner binary
leads to an ‘effective’ supernova in the outer binary through an instantaneous
change in the mass, position, and velocity of the inner binary center of mass:
$\begin{split}\tilde{m}_{o}&=\frac{(m_{A}+m_{B})+m_{C}}{(m^{\prime}_{A}+m_{B})+m_{C}}\approx\frac{m_{A}+m_{B}}{m^{\prime}_{A}+m_{B}}=\tilde{m}_{i}\equiv\tilde{m},\\\
\delta\mathbf{r}_{\text{COM}}&=-\mu_{B}(\tilde{m}-1)a_{i}\hat{\mathbf{r}}_{i},\\\
\delta\dot{\mathbf{r}}_{\text{COM}}&=v_{\text{orb},i}[-\mu_{B}(\tilde{m}-1)\hat{\dot{\mathbf{r}}}_{i}+(1-\tilde{m}\mu_{B})\tilde{v}_{i}\hat{\mathbf{v}}_{\text{kick}}].\end{split}$
We define auxiliary variables:
$\begin{split}l&=a_{o}/a_{i},\\\
K_{1}&=\mu_{B}(\tilde{m}-1)=\mu_{B}(m_{A}-m^{\prime}_{A})/(m^{\prime}_{A}+m_{B}),\\\
K_{2}&=1-\tilde{m}\mu_{B}=m^{\prime}_{A}/(m^{\prime}_{A}+m_{B}),\\\
\alpha&=\left(1+(K_{1}l^{-1})^{2}+2K_{1}l^{-1}\cos{(\nu_{o})}\right)^{-1/2},\\\
\beta&=\left(1+K_{1}^{2}l+2K_{1}\sqrt{l}\cos{(\nu_{o})}\right)^{1/2},\\\
\gamma&=K_{2}\sqrt{l},\\\
\phi_{0}&=\arctan{\left(\frac{\sin{(\nu_{o})}}{\cos{(\nu_{o})}+K_{1}\sqrt{l}}\right)},\\\
&\hskip
28.45274pt+\frac{\pi}{2}\left(1-\text{sgn}\left(\cos{(\nu_{o})}+K_{1}\sqrt{l}\right)\right)\end{split}$
Letting $\tilde{v}=\tilde{v}_{i}$, this leads to expressions for
$\Delta_{i,o}$:
$\begin{split}\Delta_{i}&=2-\tilde{m}[1+\tilde{v}^{2}-2\tilde{v}\cos{(\theta)}\cos{(\phi)}],\\\
\Delta_{o}&=2\alpha-\tilde{m}[\beta^{2}+(\gamma\tilde{v})^{2}+2\beta\gamma\tilde{v}\cos{(\theta)}\cos{(\phi-\phi_{0})}].\end{split}$
Note that each of the terms in $\Delta_{o}$ are re-scaled versions of similar
terms in $\Delta_{i}$. Each of the factors $\alpha,\beta,\gamma$ gives the
effect of the supernova in the inner binary on the outer binary. The effect of
kick direction compared to the planet’s true anomaly on the outer binary
survivability is given by $\phi_{0}$. We can obtain limits on the magnitude of
the kick and its direction by following [21]. A new constraint for the triple
system is the existence of a maximum $\tilde{m}$. From the inequalities
$\Delta_{i}>0,\Delta_{o}>0$ we get requirements on the kick magnitude.
Specifically, $\tilde{v}$ must be in the intervals for the inner and outer
orbits:
$\left[1-\sqrt{\frac{2}{\tilde{m}}},1+\sqrt{\frac{2}{\tilde{m}}}\right]\hskip
1.42271pt\text{and}\hskip
1.42271pt\left[\frac{\beta}{\gamma}-\sqrt{\frac{2\alpha}{\tilde{m}\gamma^{2}}},\frac{\beta}{\gamma}+\sqrt{\frac{2\alpha}{\tilde{m}\gamma^{2}}}\right].$
In the limit $m^{\prime}_{A}\to 0$ the value of $\gamma^{-1}$ diverges, and
the lower bound of the second interval may exceed the upper bound of the first
interval (depending on its sign). In such a case, the system cannot remain
bound, leading to a maximum allowed mass loss $\tilde{m}_{\text{max}}$. This
value is limited by the constraint $m^{\prime}_{A}\geq 0$, which sets
$\tilde{m}\leq\tilde{m}_{\text{max}}\leq\mu_{B}^{-1}$. This is plotted in
figure 2.
Figure 2: Maximum relative mass loss
$\tilde{m}_{\text{max}}=(m_{A}+m_{B})/(m^{\prime}_{A,\text{min}}+m_{B})$ as a
function of semi-major axis ratio $l=a_{o}/a_{i}$ for various outer binary
true anomalies.
The survivability of the $(B,C)$ as a bound subsystem with $A$ ejected can be
investigated by letting $m^{\prime}_{A}\to 0$ in $\Delta_{o}$. Examining the
limits $\Delta_{(B,C)}=0$ gives the boundaries in $\mu_{B},l,\nu_{o}$ phase
space where $(B,C)$ remains bound. This is plotted in figure 3. For each pair
$\mu_{B},l$ we maximize $\Delta_{(B,C)}$ by picking $\nu_{o}=\pi$. Using that
$\cos{(x)}$ is an even function and that we maximize $\Delta_{(B,C)}$ by
setting $\nu_{o}=\pi$, we can define a $\nu_{\text{crit}}(\mu_{B},l)$ so that
$\Delta_{(B,C)}>0$ if
$\nu_{o}\in(\pi-\nu_{\text{crit}},\pi+\nu_{\text{crit}})$. For a uniformly
random value of $\nu_{o}$ at the time of the supernova the probability of the
planet remaining bound is $\nu_{\text{crit}}/\pi$.
Figure 3: Boundaries in $\mu_{B},l,\nu_{o}$ phase space for which the $(B,C)$
subsystem remains bound when the remnant star $A$ is ejected from the system.
The system $(B,C)$ remains bound if
$\nu_{o}\in(\pi-\nu_{\text{crit}},\pi+\nu_{\text{crit}})$.
## 3 Results
Having set out the framework for determining the pre-supernova binary
properties, and describing the effect of the supernova on the triple, we now
combine both to study the survivability of a circum-binary planet.
### 3.1 Specific choice of initial conditions
We initialize zero-age main-sequence binaries and evolve them using population
synthesis (see section 2.1). In table 1 we give the initial conditions.
Table 1: Initial parameter distributions and value ranges for the inner eccentricity $e_{i}$, mass $m_{1}$, mass ratio $m_{2}/m_{1}$, and orbital period $P$. the primary mass-function and eccentricity limits are adopted from [24] and [25], respectively. Parameter | Distribution shape | Value range
---|---|---
$m_{1}[\text{M}_{\odot}]$ | $(m_{1}[\text{M}_{\odot}])^{-2.3}$ | $[10,100]$
$m_{2}/m_{1}$ | $(m_{2}/m_{1})^{0.1}$ | $[6\cdot 10^{-3},1]$
$\log_{10}(P[\text{d}])$ | $\log_{10}(P[\text{d}])^{0.2}$ | $[0,4]$
$e_{i}$ | $e_{i}^{-0.6}$ | $[10^{-4},0.9]$
Binaries that merge or become unbound before the supernova occurs are
discarded. We subsequently bootstrap the number of systems by training an
sklearn kernel density estimator on $\ln{(m_{A})}$, $\ln{(m_{B})}$,
$\ln{(a_{i})},\ln{(e_{i}/(1-e_{i})},\ln{(a_{i,\text{max}})}$, and the
logarithm of the remnant mass. This ensures that the values drawn from this
distribution are positive and $e_{i}$ is in the range $(0,1)$. Here
$a_{i,\text{max}}$ is the largest semi-major axis encountered during the
evolution of the inner binary and $e_{i}$ the inner binary’s eccentricity.
The mass of a black hole is determined by SeBa, but for the neutron-star mass
we adopt a Gaussian distribution with mean $1.325$ $M_{\odot}$ and standard
deviation of $0.1125$ $M_{\odot}$ independent of the progenitor mass (figure 2
in [26]). The other parameters for the supernova kick are drawn randomly from
the distribution functions presented in table 2.
The inner binary is generally circularized due to mass transfer before the
supernova occurs. In some wide binaries (those with orbital periods $\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 20$ yr), however, this may
not be the case. These wide and eccentric binaries tend to be very fragile for
the supernova, in particular the planet in an even wider and stable orbit
around such binaries are susceptible to being ionized as a result of the
supernova explosion. In some cases, however, the combination of mass loss and
the natal supernova kick may cause the planet to remain bound. To accommodate
this possibility, we relax the assumption of a circular inner binary just
before the supernova. In our Monte Carlo sampling, we randomly select the true
anomaly of the inner binary to determine the effect of the supernova. A
uniformly distributed true anomaly introduces a bias towards smaller
separations and consequently higher orbital velocities in comparison with
adopting a uniform distribution in the mean anomaly. We correct for this
effect by determining, for each binary, the probability distribution of mean
anomalies by weighting the simulation results, and scale the supernova
survival probability accordingly.
Table 2: Distribution parameters for the hierarchical triple and velocity kick. Here $e$ refers to eccentricity, $i$ to inclination, and $\omega$ to argument of periapsis. The parameter $a_{i,\text{max}}$ is the maximum inner binary semi-major axis found during the SeBa evolution. The lower and upper limits to the semi-major axis are from [27] and [28], respectively. Parameter | Distribution shape | Value range /parameters
---|---|---
$a_{o}/a_{i,\text{max}}$ | $(a_{o}/a_{i,\text{max}})^{-1}$ | $[3.23,1000]$
$|\mathbf{v}_{\text{kick}}|[\text{km/s}]$ | Maxwellian | $\sigma=265$
Kick direction | Uniform | Sphere
$m_{C}[M_{\odot}]$ | Uniform | $[10^{-5},5\cdot 10^{-3}]$
$e_{o}$ | - | 0
$i_{o}$ | - | 0
$i_{i}$ | - | 0
$\nu_{o}$ | Uniform | $[0,2\pi]$
$\nu_{i}$ | Uniform | $[0,2\pi]$
$\omega_{i}$ | Uniform | $[0,2\pi]$
### 3.2 Evolution of the inner binary
After having determined the initial parameter space, we evolve a total of
$4.8\times 10^{4}$ zero age binaries up to the moment of the first supernova.
A fraction of $0.14$ experienced a supernova resulting in a black hole, $0.08$
produced a neutron star in a supernova explosion, $0.03$ left no remnant after
the supernova; the rest either resulted in a merger, an unbound system or did
not experience a supernova. We do not discuss the evolution of these systems
here, but adopt the eventual distributions of the orbital parameters of the
surviving binaries to further explore the possibility that a circum-binary
planet survives the supernova.
Figure 4 shows the parameter distributions for the simulations leaving a
remnant. Most of the supernova progenitors have a mass $m_{A}<10\,M_{\odot}$,
while there are secondary masses as high as $80\,M_{\odot}$. Most of the
supernovæ are stripped core-collapse of type Ibc that naturally result from
the mass transfer during the system evolution [29]. Most inner binaries have
semi-major axes $<1$ au.
(a) Pre-supernova masses for binaries with a neutron star.
(b) Pre-supernova masses for binaries with a black hole.
(c) Semi-major axis versus eccentricity for neutron-star binaries directly
after the supernova
(d) Semi-major axis versus eccentricity for black-hole binaries directly after
the supernov.
Figure 4: Distribution of the parameters for the inner binaries just before
and after the supernova for both neutron stars (left) and black holes (right).
### 3.3 Survivability of the planet after the supernova
In order to study the probability that a planet survives in orbit throughout
the supernova, we add a planet, with a mass chosen uniformly between $10^{-5}$
M⊙ and $5\times 10^{-3}$ M⊙ and orbital separation between
$a_{o}=3.23\,a_{i,\text{max}}$ to $1000\,a_{i,\text{max}}$ (flat in
$\log(a_{o})$). Here $a_{i,\text{max}}$ is the largest semi-major axis the
inner binary reaches during its pre-supernova evolution. The inner limit is
chosen to assure that the planet would remain stable throughout the evolution
of the inner binary ([30, 31], and see [28] for a more empirical
characterization). Mass lost from non-conservative mass transfer in the inner
binary or by the wind of one of the components will have driven the planet
further out [32], but we ignore that here. The planets are chosen to have
prograde circular orbits in the plane of the pre-supernova binary system.
The supernova is simulated applying an instantaneous mass loss and change in
the velocity vector of star $A$. This leads to a sudden change in the center
of mass and velocity of the inner binary, which has the effective of a
(diluted) supernova in the outer orbit, as discussed in [22].
Neutron stars and black holes acquire a velocity kick upon their formation in
a supernova. This kick’s magnitude and direction are crucial for the
survivability of the binary system and important for the orbital parameters
when the binary remanis bound. We adopt the distribution of pulsar kicks from
[27], which is described by a Maxwellian distribution with a dispersion of
$\sigma=265$ km/s. This distribution appears consistent with population
statistics of x-ray binaries in the Milky Way [12]. Black holes are expected
to receive a lower kick velocity, which we address by applying a momentum-
kick.
The supernova in a triple then results in one of the following:
* •
The entire triple becomes unbound: $A,B,C$,
* •
The triple becomes dynamically unstable,
* •
The inner binary remains bound, but the planet escapes: $(A,B),C$,
* •
$A$ and $C$ remain bound, but $B$ escapes: $(A,C),B$,
* •
$B$ and $C$ remain bound, but $A$ escapes: $A,(B,C)$,
* •
The entire triple remains bound: $((A,B),C)$,
We look at every possible combination of $A,B$ and $C$ and determine the
distribution of the corresponding orbital elements. We recognize two distinct
dynamically unstable configurations: 1) the entire triple remains bound but
violates the stability criterion [28], and 2) the is no clear hierarcy in the
surviving triple. In the latter case we often find that The planet is bound to
both stars, but the two stars are not bound in a binary.
Using the earlier mentioned binary population synthesis results, we generate
$4\times 10^{6}$ pre-supernova systems with a neutron star and $4\times
10^{6}$ with a black hole, each of which with a circum-binary planet.
In figure 5, we show, as a function of $a_{o}$, the fraction of simulations
that experience a supernova (see 2.1). The increasing noise for $a_{o}>10^{3}$
au is due to the smaller number of simulations in that region of parameter
space (note that $a_{0}$ was initially distributed randomly on a $\log$
scale). Another interesting feature is that for $a_{o}\
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ 10^{2}$ au the probability
of $(B,C)$ to remain bound is larger than for $(A,B)$ to stay bound. The
majority of triples that survive the supernova are dynamically stable, which
is a direct consequence of the adapted stable conditions before the supernova.
(a) Black hole remnants.
(b) Neutron star remnants.
Figure 5: Fraction of simulations per possible outcome of the supernova as a
function of the outer binary semi-major axis $a_{o}$ before the supernova. The
fraction of simulations in these figures is calculated as the ratio of two
histograms of $a_{o}$ with logarithmic bins, comparing the number of
simulation with a specific outcome to the total number of simulations in each
bin. The ‘triple unstable’ category refers to systems which remain fully bound
but for which the outer planet orbit is unstable [30].
In only 3 of the $8\times 10^{6}$ simulations, the planet remains bound to the
exploding star, leading to a binary composed of $(A,C)$. For this to happen,
the remnant must receive a kick in a narrow cone and with the velocity similar
to the planets orbital speed but away from the companion star so that it can
pick-up the planet on its escaping trajectory. Alternatively, the compact
object escapes, leaving it companion star $B$ bound to the planet. Both
processes are rather improbable, as expected, which is consistent with the
small probability in our simulations.
In figure 6 we present the branching ratios of all occurrences in our
simulations. Following the appropriate branches, the total probability for the
triple to survive is $2.7\cdot 10^{-3}$, and the probability that $(B,C)$
remains bound is $3\cdot 10^{-2}$. The currently detected number of x-ray
binaries is around 300 [33, 34], with an estimates for the Galactic population
of $\sim 1300$ [35] to $10^{4}$ [36]. With these last two estimates, the
expected number of x-ray binaries that after the supernova that still host a
circum-binary planet is $\ {\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\
10$: the Galaxy may host a few x-ray binaries with a circum-binary planet that
survived its host’s supernova. There is a comparable probability for the
entire triple to remain bound but with an unstable orbit for the planet. In
the latter case, the planet may either collide with one of the stars or be
ejected to become a rogue planet.
Figure 6: Branching ratios for each possible outcome for the triple system,
averaged over all input parameters. The upper branches give the results of the
binary population synthesis, and the lower branch gives the effect of the
supernova on the resulting systems.
In figure 7 we present the distribution of the semi-major axis and
eccentricity of the surviving planet’s orbit. The planet then either continues
to orbit the inner binary, or it orbits the original secondary star. Mass loss
in the inner binary generally induces a considerable eccentricity in the final
orbit. Surviving planets therefore typically have highly eccentric orbits,
with a wide range of semi-major axes.
There is an absence of stable triples with semi-major axes larger than
$10^{3}$ au and low eccentricities, because the number of planets with a pre-
supernova semi-major axis $>10^{3}$ au is low, and the post-supernova
periapsis of the planet can not exceed the pre-supernova apoapsis. No planets
are found with tight $\ll 1$ au orbits because these systems tend to be
dynamically unstable, and wider, up to $\sim 10$ au orbits tend to have high
eccentricities. Most of the orbits in the $(B,C)$ systems that remain bound
have $e_{(B,C)}\uparrow 1$ due to the extreme mass loss in the triple. We
predict systems that formed this way to have highly eccentric planetary
orbits, with no strong constraints on the semi-major axis.
(a) Black hole remnants.
(b) Neutron star remnants.
(c) Case for which the exploding star is ejected.
Figure 7: Distribution for semi-major axis and eccentricity for the orbiting
planet that survive the supernova of one of the binary components. The panels
a) and b) give the orbital elements for the circum-binary planet with a black
hole and a neutron star, respectively. Panel c) gives the orbital parameters
for the planet in orbit around star $B$.
In figure 8 we present the distribution of relative inclinations for the
planetary orbits with respect to the post-supernova binaries. In this case, we
used the results for the neutron stars, but the black hole distribution is
similar. The fraction of retrograde orbits for circum-binary planets with a
neutron star is 0.277 compared to 0.148 for binaries with a black hole. A
fraction of 0.206 of the planets around a companion of the exploding star in
$(B,C)$ has a retrograde orbit.
Figure 8: Relative inclination of the planet orbit with respect to the post-
supernova binary, presented in the case where the supernova resulted in a
neutron star. $\Delta i=0$ indicates that the planet still orbits in the plane
of the post-supernova binary system. The fraction of retrograde orbits is
between 0.148 (for binaries with a black hole) and 0.277 (for binaries with a
neutron star).
## 4 Observational implications
When both stars and the planet remain bound, the inner binary hosting the
compact object continues to evolve into an x-ray binary. In that case, the
circum-binary planet may be observed in the x-ray binary phase or around a
binary millisecond pulsar once the neutron star has been spun-up. In
principle, it is even possible that the planet survives a second supernova
explosion, in which case the planet remains bound to the inner compact-object
binary. This requires the triple to survive and remain stable through both
supernovæ. The probability of this to happen seems small ($\
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ (0.14\times 0.015)^{2}\sim
4.4\cdot 10^{-6}$, for a black hole binary and more than an order of magnitude
smaller for neutron star binaries, ($\
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ (0.08\times 0.00070)^{2}\sim
3.1\cdot 10^{-7}$, see figure 6). When the compact object is ejected, the
planet may still survive in a relatively wide and eccentric orbit around the
original secondary star. Such systems may be recognizable for their curious
planetary orbit and the mass-transfer affected stellar host.
The orbits of the surviving planets are wide, with semi-major axes ranging
from $\sim 1$ au to well over $10^{3}$ au, and over the entire range of
eccentricities, but skewed to $e\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 0.2$ orbits. The majority of
systems in which the planet remains bound to the companion of the exploding
star tends to have very highly eccentricities $e\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 0.9$ orbits (but skewed to
$e\sim 1)$. Such a system may lead to a merger between the planet and its
newly acquired companion star within a few years after the supernova. For the
surviving systems, the discovery of a massive star with a planet in a wide
($10$ to $10^{3}$ au) and eccentric ($\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 0.9$) orbit may be a
signature of a supernova survival. These planets will have experienced a
nearby supernova, possibly obliterating their atmosphere, or at the least
enriching it with a healthy radioactive mix of heavy decay-product of the
supernova blastwave.
A considerable fraction of the pre-supernova systems have eccentricities
$e>0.1$ ($0.24$ of systems with a neutron star and $0.32$ of systems with a
black hole). A uniformly random true anomaly, as we adopted in our analysis,
is biased to lower separations and higher orbital velocities compared to a
mean anomaly selected randomly from a uniform distribution for eccentric
systems. This impacts the survivability of the system, and introduces a bias
in our results. We correct for this by calculating how much the mean anomaly
of a specific system is overcounted (compared to a uniformly random mean
anomaly), and scaling the effect of that system on our results by the inverse
of this.
## 5 Conclusion
We simulated a population of massive zero-age binaries up to the moment of the
first supernova. The surviving binaries were equipped with a circum-binary
planet to determine the probability distributions of the planets’ orbital
parameters of the surviving binaries. We did this by analytically
investigating limiting factors on the planet’s survival and through Monte
Carlo sampling. The analytic expressions for the amount of mass lost to assure
that the planet remains bound and the probability of remaining bound are
presented in figures 2 and 3, respectively. The resulting numerically
calculated survivability for the planet is presented in figure 5.
From the total population of massive Galactic binaries that evolve into x-ray
binaries, we predict a fraction of $3\cdot 10^{-3}$ to keep its circum-binary
planet. This fraction should be perceived as an upper limit, because we
assumed 100% triplicity. Interestingly enough, the probability for the planet
to remain bound to the exploding star’s companion is $11$ ($0.03/0.0027$, see
the table in figure 6) times higher, or $3\cdot 10^{-2}$. These systems,
however, are probably harder to identify as post-supernova planetary systems
except maybe for the curious orbit of the single planet. More than 20% of
these planets have retrogade orbits compared the their pre-suprenova orbit.
This could potentially be observed in mass transfer in the pre-supernova epoch
has synchronized the secondary’s rotation.
We conclude that $\ {\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ 10$
x-ray binaries in the Galaxy may still harbor a circum-binary planet, and at
most $\sim 150$ massive stars may be orbited by a planet in a wide ($\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 10$ au) and highly eccentric
($\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 0.9$) orbit. Note,
however, that we assumed that every binary is orbited by a planet, which seems
unlikely (see [37]).
The survivability of the circum-binary planets is mediated by the large
proportion of stripped (post-mass transfer) supernovae in our simulations.
Such a type Ib/c supernova results in a relatively small mass loss, which
helps to keep the planet bound. Together with the probability of a low kick
velocity, mediates in the survivability of planets around post-supernova
systems. The adopted distribution functions for the kick velocity are
important for this study because of the planet’s survival. There are not many
constraints to the low-velocity tail of the supernova kick distribution [44].
In a future study, it may be worth exploring this part of parameter space more
exhaustively. Finally, our conclusions should be checked against observations.
## Energy consumption of this calculation
The population synthesis calculations for $4.8\times 10^{4}$ zero-age binaries
took about 3 seconds per binary on a single Xeon E-2176M core ($\sim 12$ Watt)
resulting in about 41 hours of CPU time or 0.5 kWh. Building the database and
repeating the calculations because of earlier errors or tuning the selected
initial conditions we should multiply the runtime by a factor of 2 or 3. With
$0.649$ kWh/kg (Dutch norm for gray electricity) results in $\sim 2$ kg CO2,
which is comparable to a daily commute.
## Software used for this study
In this work we used the following packages: python [38, 39], SeBa [15, 18],
AMUSE [19], numpy [40], scipy [41], sklearn [42], matplotlib [43], and
sqlite3.
## Acknowledgments
We thank the anonymous referee for spotting an error in our earlier version of
the manuscript, and for useful comments that helped us improving the
manuscript. This work was performed using resources provided by the Academic
Leiden Interdisciplinary Cluster Environment (ALICE).
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|
# Effects of internal dynamics on chemotactic aggregation of bacteria
Shugo YASUDA Graduate School of Information Science, University of Hyogo,
650-0047 Kobe, Japan<EMAIL_ADDRESS>
###### Abstract.
The effects of internal adaptation dynamics on the self-organized aggregation
of chemotactic bacteria are investigated by Monte Carlo (MC) simulations based
on a two-stream kinetic transport equation coupled with a reaction-diffusion
equation of the chemoattractant that bacteria produce.
A remarkable finding is a nonmonotonic behavior of the peak aggregation
density with respect to the adaptation time; more specifically, aggregation is
the most enhanced when the adaptation time is comparable to or moderately
larger than the mean run time of bacteria. Another curious observation is the
formation of a trapezoidal aggregation profile occurring at a very large
adaptation time, where the biased motion of individual cells is rather
hindered at the plateau regimes due to the boundedness of the tumbling
frequency modulation.
Asymptotic analysis of the kinetic transport system is also carried out, and a
novel asymptotic equation is obtained at the large adaptation-time regime
while the Keller-Segel type equations are obtained when the adaptation time is
moderate. Numerical comparison of the asymptotic equations with MC results
clarifies that trapezoidal aggregation is well described by the novel
asymptotic equation, and the nonmonotonic behavior of the peak aggregation
density is interpreted as the transient of the asymptotic solutions between
different adaptation time regimes.
This work was supported by the Japan-France Integrated Action Program
(SAKURA), Grant number JPJSBP120193219.
## 1\. Introduction
The collective motion of chemotactic bacteria, such as Escherichia coli, stems
from, at the individual level, continuous reorientations by runs and tumbles.
It has been established that the length of a run is determined by a stiff
response to the temporal variation of extracellular chemical cues via an
intracellular signal transduction pathway. The chemotactic response and the
intracellular signal transduction pathway for E. coli have been extensively
studied by various authors, and sophisticated mathematical models have been
proposed. [BL1997, KLBTS2005, TSB2008, JOT2010, H2012] However, the multiscale
mechanism between intracellular signal transduction, individual chemotactic
motion, and collective dynamics of cells is not yet well understood.
Currently, engineered bacteria (or genetically modified bacteria) are utilized
in a variety of industrial fields involving, for example, food, agriculture,
medicine, and the environment. Understanding the multiscale mechanism can
contribute to further advances in industrial technology to control the
collective motion of cells.
Kinetic transport models have been proposed to describe the multiscale
mechanism in the collective motion of cells; a kinetic transport model
describing the velocity jump process in the run-and-tumble motion of E. Coli
was first proposed in Ref. [ODA1998], and it was then further developed to
involve more detailed chemosensory systems [DS2005, EO2004, SWOT2012].
Although the chemosensory system involves complicated biochemical reaction
networks, in the simplified description, it can be constituted by two
essential steps, i.e., a rapid response to an external signal change called
“excitation” and a subsequent slow “adaptation”, in which the internal state
returns to the baseline, allowing the cell to respond to a further external
signal change. [SPO1997] A kinetic transport equation involving the simplified
description of the excitation and adaptation dynamics has been proposed in
Refs. [EO2004] and [EO2007], where two internal state variables are introduced
in the model. Since the excitation dynamics are much faster than the
adaptation dynamics, one can integrate the fast variable related to the
excitation and derive the kinetic transport equation involving only a single
internal state variable related to the slow adaptation dynamics [SWOT2012],
i.e.,
$\partial_{t}p+\mathbf{v}\cdot\nabla_{x}p+\partial_{m}[F(m,S)p]=\mathcal{Q}[m,S](p),$
(1)
where $p(t,\mathbf{x},\mathbf{v},m)$ is the density of cells with velocity
$\mathbf{v}\in\mathbb{V}$ and internal state $m>0$ at time $t>0$ and position
$\mathbf{x}\in\mathbb{R}^{d}$. Here, on the left-hand side, the $x$-divergence
term describes the change in density due to the “run” of the bacteria, and the
$m$-derivative term describes the evolution of the internal state $m$ at the
rate of change $F(m,S)$, where $S(t,x)$ is the concentration of the
extracellular chemical cue. On the right-hand side, $\mathcal{Q}[m,S](p)$ is
the tumbling operator described as
$\mathcal{Q}[m,S](p)=\frac{1}{||\mathbb{V}||}\int_{\mathbb{V}}[\lambda(m,S,{\bf
v},{\bf v}^{\prime})p(t,{\bf x},{\bf v}^{\prime},m)-\lambda(m,S,{\bf
v}^{\prime},{\bf v})p(t,{\bf x},{\bf v},m)]d{\bf v}^{\prime},$ (2)
where $\lambda(m,S,{\bf v},{\bf v}^{\prime})$ denotes the tumbling frequency
describing the reorientation from velocity ${\bf v}^{\prime}$ to the new
velocity ${\bf v}$. The velocity space $\mathbb{V}$ is the bounded domain of
$\mathbb{R}^{d}$ and $||\mathbb{V}||=\int_{V}\,dv$.
Since the bacteria communicate with each other via the extracellular chemical
cues they produce, to describe the observed self-organization phenomena
occurring in a population of chemotactic bacteria, for example, in Refs.
[BB1991, WTMMBB1995, MBBO2003], chemoattractant equations must be coupled with
the kinetic transport model (1). In this study, we consider a single species
of chemical cues whose concentration $S(t,\mathbf{x})$ is described as
$\partial_{t}S=D_{S}\Delta S-aS+b\rho,$ (3)
where $D_{S}$ is the diffusion coefficient of the chemical cue, $a$ is the
degradation rate of the chemical cue, $b$ is the production rate of the
chemical cue by bacteria, and $\rho(t,{\bf
x})=\int_{\mathbb{V}}\int_{0}^{\infty}p(t,{\bf x},{\bf v},m)dmd{\bf v}$ is the
population density of bacteria.
In the kinetic transport model, the microscopic characteristics at the
individual level are involved in the tumbling frequency $\lambda(m,S,{\bf
v},{\bf v}^{\prime})$ and the rate of change of the internal state $F(m,S)$.
Thus, by specifying the mathematical formulas for $\lambda(m,S,{\bf v},{\bf
v}^{\prime})$ and $F(m,S)$, one can address the multiscale mechanism between
the intracellular adaptation dynamics, individual chemotactic motion, and
collective dynamics of cells in the self-organization phenomena. One can also
derive macroscopic models for the population density of bacteria, e.g.,
Keller-Segel (KS)-type systems [KS1970, KS1971, HP2009], and kinetic transport
equations without internal state variables [DS2005] by using moment closure or
asymptotic analysis of Eqs. (1)–(3). [PTV2016, PVW2018, ST2017, X2015]
Investigations of the aggregation of chemotactic bacteria based on kinetic
transport models have been carried out in various studies. For example, in
Ref. [SM2011], the aggregation of chemotactic bacteria under a given
concentration gradient of a chemical cue was investigated based on the kinetic
transport model with internal states, and the volcano-like (bimodal)
aggregation of E. coli observed in an experiment [MBBO2003] was numerically
reproduced in one-dimensional space. In Ref. [XXT2018], the concentric stripe
patterns formed by engineered E. coli [Liuetal2011] were reproduced
numerically, and the role of intracellular signal transduction in stripe
pattern formation was clarified. In Ref. [PY2018], the instability of the
kinetic transport model describing colony pattern formation over a long period
of time due to proliferation was investigated, and stiff-response-induced
instability was uncovered at the kinetic level. Additionally, in Ref. [R2020],
the role of the hydrodynamic interaction in the self-organized aggregations
was numerically investigated by using a Monte Carlo method related to a
kinetic transport model without internal states. These studies have
established that kinetic transport models are useful for elucidating the
multiscale mechanism in the collective motion of chemotactic bacteria.
However, the multiscale mechanism between collective motions and internal
state dynamics in self-organized aggregation has yet to be clarified.
In this paper, we investigate the self-organized aggregation of chemotactic
bacteria in one-dimensional space based on a two-stream kinetic transport
model with an internal state. In contrast to the previous study [PY2018], this
paper concerns the internal dynamics of chemotactic bacteria and considers
self-organized aggregation, which may occur in a rather short period of time
without proliferation. In particular, we focus on the effect of the adaptation
time on the instability and aggregation behavior.
In the following text, the problem and the basic equations are given in Sec.
2. In Sec. 3, numerical analyses are carried out for a wide range of
adaptation times by using a Monte Carlo (MC) method, which is an extension of
the MC method previously developed in Refs. [Y2017, VY2020]. In Sec. 4, we
formally carry out asymptotic analysis of the kinetic transport model at
different scalings of the adaptation time and derive a KS-type model and a
novel asymptotic equation involving the internal state variable. The
asymptotic behavior is also numerically investigated over a wide range of
adaptation times, through which a suitable parameter regime for the KS-type
system and a remarkable numerical solution in the novel asymptotic regime are
uncovered. Finally, a summary and perspectives are given in Sec. 5.
## 2\. Problem and formulation
We consider the chemotactic bacteria moving in positive and negative
directions with a constant speed $V_{0}$, i.e., $v=\\{-V_{0},V_{0}\\}$, in
one-dimensional space $x\in[0,L]$ with periodic boundary conditions.
Initially, the bacteria are uniformly distributed, and the internal state $m$
is in the equilibrium state at $m=M(S)$, where $M(S)$ denotes the equilibrium
internal state determined by the extracellular chemical concentration. The
chemical concentration $S(t,x)$ is also uniformly distributed in the initial
state.
For the internal state dynamics, we consider the following linear adaptation
model:
$\frac{dm}{dt}=F(m,S)=\frac{M(S)-m}{\tau},$ (4)
where $\tau>0$ denotes the characteristic adaptation time. We also assume that
the bacteria tumble (i.e., change in moving direction) depends only on the
deviation of the internal state $m$ from the equilibrium state $M(S)$,
$M(S)-m$, [SWOT2012]:
$\lambda(m,S,v,v^{\prime})=\lambda_{0}\Lambda\left(\frac{M(S)-m}{\delta}\right),$
(5)
where $\lambda_{0}>0$ is the mean tumbling frequency,
$\Lambda(\frac{M(S)-m}{\delta})>0$ denotes the modulation of the tumbling
frequency, and $\delta>0$ denotes the stiffness of the chemotactic response.
In this study, we consider the following modulation function:
$\Lambda_{\delta}(y)=\Lambda\left(\frac{y}{\delta}\right),\quad\Lambda(y)=1-R(y),\quad
R(y)=\frac{\chi y}{\sqrt{1+y^{2}}},$ (6)
where $0<\chi<1$ denotes the modulation amplitude and $\delta$ denotes the
stiffness of the chemotactic response.
Then, the density of bacteria with positive and negative velocities,
$p^{\pm}(t,x,m)$, is described by the following two-stream kinetic transport
equation with the internal state:
$\partial_{t}p^{\pm}\pm
V_{0}\partial_{x}p^{\pm}+\partial_{m}\left(\frac{M(S)-m}{\tau}p^{\pm}\right)=\pm\frac{\lambda_{0}}{2}\Lambda_{\delta}\left(M(S)-m\right)(p^{-}-p^{+}).$
(7)
By introducing the nondimensional variables
$\widehat{x}=x/L_{0},\quad\widehat{t}=t/t_{0},\quad\widehat{v}=v/V_{0},$
where $L_{0}$, $V_{0}$, and $t_{0}$ are the characteristic length, speed, and
time, respectively, Eq. (7) is written in nondimensional form as
$\widehat{\sigma}\partial_{\widehat{t}}\widehat{p}^{\pm}\pm\partial_{\widehat{x}}\widehat{p}^{\pm}+\partial_{m}\left(\frac{M(S)-m}{\widehat{\tau}}p^{\pm}\right)=\frac{\widehat{\lambda}_{0}}{2}\Lambda_{\delta}\left(M(S)-m\right)(\widehat{p}^{\mp}-\widehat{p}^{\pm}).$
(8)
Here, the nondimensional parameters $\widehat{\sigma}$,
$\widehat{\lambda}_{0}$, and $\widehat{\tau}$ are defined as
$\widehat{\sigma}=L_{0}/(t_{0}V_{0}),\quad\widehat{\lambda}_{0}=\lambda_{0}/(V_{0}/L_{0}),\quad\widehat{\tau}=\tau/(L_{0}/V_{0}).$
(9)
We also define $\widehat{p}^{\pm}=p^{\pm}/\rho_{0}$, where $\rho_{0}$ is the
initial population density of bacteria. The characteristic time $t_{0}$ can be
chosen arbitrarily depending on the time scale with which we address the
problem. For example, a typical choice of $t_{0}$ is $t_{0}=L_{0}/V_{0}$,
which gives $\sigma$=1 and makes the kinetic transport equation (8) simpler,
reducing one free parameter. Another typical choice of $t_{0}$ is
$t_{0}=(\lambda_{0}L_{0}^{2})/V_{0}^{2}$, which denotes the characteristic
diffusion time and enables us to derive macroscopic continuum-limit equations
such as KS-type models by asymptotic analysis of the kinetic transport
equation. In this paper, we address both time scales; i.e., we set
$t_{0}=L_{0}/V_{0}$ for MC simulations of the kinetic transport model, while
we use the diffusion time scale $t_{0}=(\lambda_{0}L_{0}^{2})/V_{0}^{2}$ for
the asymptotic analysis to investigate the behaviors in the near-continuum
regime.
By the same token, the nondimensional form of (3) is written as
$\widehat{\sigma}_{S}\partial_{\widehat{t}}\widehat{S}=\widehat{D}_{S}\partial_{\widehat{x}\widehat{x}}\widehat{S}-\widehat{S}+\widehat{\rho},$
(10)
where the nondimensional quantities are defined as
$\widehat{\sigma}_{S}=1/(at_{0}),\quad\widehat{D}_{S}=D_{S}/(aL_{0}^{2}),\quad\widehat{S}=S/(b\rho_{0}/a),\quad\widehat{\rho}=\rho/\rho_{0},$
(11)
and the population density $\widehat{\rho}$ is calculated as
$\widehat{\rho}(x,t)=\int_{0}^{\infty}\frac{\widehat{p}^{+}(t,x,m)+\widehat{p}^{-}(t,x,m)}{2}dm.$
(12)
In this study, we fix the nondimensional diffusion constant
$\widehat{D}_{S}=1$ (although the notation $D_{S}$ retains in the following
equations for generality). This indicates that the characteristic length
$L_{0}$ denotes the diffusion length of chemoattractant $S$ within the
degradation time $a^{-1}$, i.e.,
$L_{0}=\sqrt{\frac{D_{S}}{a}}.$ (13)
In the rest of the paper, unless otherwise stated, all quantities are written
in nondimensional forms, and we drop the hat signs on the variables and
parameters for simplicity.
It is convenient to introduce the new internal state variable
$y=M(S)-m\in\mathbb{R}$ and change the variable to
$f^{\pm}(t,x,y=M(S)-m)=p^{\pm}(t,x,m)$. Then, we obtain
$\sigma\partial_{t}f^{\pm}\pm\partial_{x}f^{\pm}+\partial_{y}\left\\{\left(D^{\pm}_{t}M(S)-\frac{y}{\tau}\right)f^{\pm}\right\\}=\pm\frac{\lambda_{0}\Lambda_{\delta}(y)}{2}(f^{-}-f^{+}),$
(14)
where $D^{\pm}_{t}$ denotes the material derivative defined as
$D_{t}^{\pm}=\sigma\partial_{t}\pm\partial_{x}$. Here, the $y$-derivative term
of (14) describes the change in the internal state variable $y$, where
$D_{t}^{\pm}M(S)$ denotes the temporal variation of the extracellular chemical
cue sensed by bacteria moving in positive and negative directions. Since E.
coli cells respond to the spatial gradient of the logarithmic extracellular
chemical concentration [BB1974, KJTW2009], we model the logarithmic sensing by
$D_{t}^{\pm}M(S)=D_{t}^{\pm}\ln
S(x,t)=\frac{\sigma\partial_{t}S\pm\partial_{x}S}{S}.$ (15)
Thus, the change in the internal state $y$ of each bacterium is described as
$\dot{y}=\frac{D_{t}^{\pm}S}{S}-\frac{y}{\tau},$ (16)
where $D_{t}^{\pm}S$ denotes the temporal variation of extracellular chemical
cues sensed by bacteria along their moving trajectory.
It is clearly seen that the uniform state $f^{\pm}=\delta(y=0)$ and
$S=\rho=1$, where $\delta(y)$ is the Dirac delta function, solves the system
of Eqs. (10) and (14). We investigate the instability of the uniform state by
the Monte Carlo code explained in Sec. 3.1.
## 3\. Numerical analysis
### 3.1. Monte Carlo method
The one-dimensional space $0\leq x\leq L$ is divided into the uniform mesh
system $x_{i}=\Delta x\times i$ ($i=0,\cdots,I$) with the mesh width $\Delta
x=L/I$, where $I$ is the number of mesh intervals. Initially, Monte Carlo (MC)
particles are uniformly distributed in each mesh interval with the equilibrium
internal state at $y=0$. The velocities of each MC particle, $v=\pm 1$, are
randomly determined. The chemical concentrations in each mesh interval
$x\in[x_{i},x_{i+1}]$ ($i=0,\cdots,I-1$), $S_{i}$, are also uniformly given at
the initial state, i.e., $S_{i}^{0}=1$.
Then, the position $r^{k}_{l}$, velocity $v_{l}^{k}$, and internal state
$y^{k}_{l}$ of the $l$th MC particle at time $t=k\Delta t$ are determined as
follows:
1. (1)
Each MC particle moves as
$r^{k}_{l}=r^{k-1}_{l}+v_{l}^{k-1}\Delta t.$ (17)
2. (2)
Population density in the $i$th mesh interval $x\in[x_{i},x_{i+1}]$,
$\rho^{k}_{i}$, is calculated as
$\rho_{i}^{k}=\frac{1}{\overline{N}}\sum_{l=0}^{N}\int_{x_{i}}^{x_{i+1}}\delta(x-r_{l}^{k})dx,$
(18)
where $\overline{N}$ is the number of MC particles in each mesh interval in
the uniform state. Thus, the total number of MC particles is given by
$I\times\overline{N}$.
3. (3)
Concentration of chemical cues in the $i$th mesh interval, $S^{k}_{i}$, is
calculated explicitly as
$\sigma_{S}\frac{S_{i}^{k}-S_{i}^{k-1}}{\Delta t}=\frac{D_{S}}{\Delta
x^{2}}(S^{k-1}_{i+1}-2S^{k-1}_{i}+S^{k-1}_{i-1})-S_{i}^{k-1}+\rho_{i}^{k}.$
(19)
At the boundaries $x$=0 and $L$, we consider the periodic condition, i.e.,
$S_{-1}=S_{I-1}$ and $S_{I}=S_{0}$.
4. (4)
Internal state of the $l$th MC particle, $y^{k}_{l}$, is updated by following
Eq. (16) as
$\frac{y^{k}_{l}-y^{k-1}_{l}}{\Delta
t}=\frac{S^{k}_{(l)}-S^{k-1}_{(l)}}{\Delta
t{S^{k-1}_{(l)}}}-\frac{y^{k}_{l}}{\tau},$ (20)
where $S^{k}_{(l)}$ denotes the local concentration of chemical cues at the
position of the $l$th MC particle $x=r^{k}_{l}$, i.e., $S^{k}_{(l)}=S(k\Delta
t,r_{l}^{k})$, and is calculated by linear interpolation:
$S^{k}(l)=\left\\{\begin{array}[]{cc}S_{i}^{k}+\frac{S^{k}_{i}-S^{k}_{i-1}}{\Delta
x}(r_{l}^{k}-x_{i}-\frac{\Delta x}{2}),&\mathrm{if}\quad x_{i}\leq
r_{l}^{k}<x_{i}+\frac{\Delta x}{2},\\\
S_{i}^{k}+\frac{S^{k}_{i+1}-S^{k}_{i}}{\Delta x}(r_{l}^{k}-x_{i}-\frac{\Delta
x}{2}),&\mathrm{if}\quad x_{i}+\frac{\Delta x}{2}\leq
r_{l}^{k}<x_{i+1}.\end{array}\right.$ (21)
We note that in Eq. (20), the pathway derivative $D^{\pm}_{t}S$ in Eq. (16) is
given by the rate of change of $S$ sensed by each bacterium, i.e.,
$(S_{(l)}^{k}-S_{(l)}^{k-1})/\Delta t$.
5. (5)
Tumbling of the $l$th MC particle is decided by the probability $\frac{\Delta
t\lambda_{0}}{2}\Lambda_{\delta}(y_{l}^{k})$.
6. (6)
The particles that decide to make tumbles change their velocities as
$v_{l}^{k-1}\rightarrow v_{l}^{k}=-v_{l}^{k-1}$, and other particles retain
their velocities.
This MC method was applied for aggregation under a given constant spatial
gradient of chemical cues, i.e., $\partial_{x}M(S)=$const., in Ref. [VY2020],
and the accuracy of the MC method was confirmed throughout the comparison to
the asymptotic preserving schemes developed in the paper.
### 3.2. Numerical results
MC simulations are performed for various values of the mean tumbling frequency
$\lambda_{0}$, the adaptation time $\tau$, and the stiffness of the
chemotactic response $\delta$, while the diffusion constant $D_{S}=1$, the
length of the periodic interval $L=10$, the modulation amplitude $\chi=0.5$,
and the time scale parameters $\sigma$=$\sigma_{S}$=1 are fixed. As is
mentioned in Sec. 2, since the diffusion constant is fixed as $D_{S}=1$, the
length scale of the system corresponds to the diffusion length of
chemoattractant defined as Eq. (13).
The number of mesh intervals $I$ and the average number of MC particles in
each mesh interval $\overline{N}$ are set to $I$=50 and $\overline{N}$=28,800
except for the cases for $\lambda_{0}$=500 and 1000 at $\delta=0.01$, where
$I=100$ and $\overline{N}=7,400$ are used. The time step size is set to
$\Delta t=10^{-3}$ for $\lambda_{0}<100$, $\Delta t=2\times 10^{-4}$ for
$\lambda_{0}$=100 and 200, and $\Delta t=5\times 10^{-5}$ for
$\lambda_{0}=$500 and 1000.
In the following, we introduce a new parameter $\alpha$, which is defined by
the ratio of the adaptation time $\tau$ to the mean run time
$\lambda_{0}^{-1}$, i.e., $\alpha=\lambda_{0}\tau$, and call it the relative
adaptation time.
We also use the notation $t_{\lambda}$ for the scaled time
$t_{\lambda}=t/(\lambda_{0}L^{2})$. The macroscopic population density $\rho$
in the following numerical results is time-averaged over the interval $\delta
t_{\lambda}=0.05$ to remove the fluctuations caused by the Monte Carlo method.
#### 3.2.1. Instability and aggregation profile
Figure 1. The instability diagrams with respect to the scaled adaptation time
$\alpha$ and the stiffness $\chi/\delta$ at $\lambda_{0}=5$ (a),
$\lambda_{0}=10$ (b) and $\lambda_{0}=20$ (c). The upward triangles
$\bigtriangleup$ show the results where stationary patterns are clearly
observed, while the downward triangles $\bigtriangledown$ show the results
where no evident patterns are observed (where the time average of the maximum
deviation $\overline{\delta\rho}$ defined by (22) is less than 0.01). See also
Fig. 4. The squares $\Box$ show the intermediate results where nonstationary
sinusoidal waves with small amplitudes, i.e.,
$0.01<\overline{\delta\rho}<0.1$, are observed. The dotted line shows the
linear stability condition of the KS system, which is obtained in Sec. 4.1.1.
Under the critical line, the uniform solution to the KS system is linearly
stable.
Figure 1 shows the instability diagrams with respect to the relative
adaptation time $\alpha$ and the stiffness of the chemotactic response
$\chi/\delta$ at different values of the tumbling frequency, i.e.,
$\lambda_{0}=5$ in (a), $\lambda_{0}=10$ in (b), and $\lambda_{0}=20$ in (c).
To confirm the stable state, we carried out a long-term simulation over
$t_{\lambda}\in[0,T_{\lambda}]$ with $T_{\lambda}=10$ and measured the time
average of the maximum deviation of the population density from the uniform
state defined as
$\overline{\delta\rho}=\frac{2}{T_{\lambda}}\int_{\frac{T_{\lambda}}{2}}^{T_{\lambda}}{\displaystyle\max_{x}}|\rho-1|dt_{\lambda}.$
(22)
The stable uniform states shown by the downward triangles $\bigtriangledown$
are confirmed when $\overline{\delta\rho}<0.01$, and the intermediate states
shown by the squares $\Box$ are confirmed when
$0.01<\overline{\delta\rho}<0.1$.
It is clear that when the relative adaptation time $\alpha$ is fixed,
instability occurs when the stiffness $\chi/\delta$ is sufficiently large. At
small relative adaptation times, e.g., $\alpha\lesssim 1$, instability always
occurs when the stiffness $\chi/\delta$ is larger than the critical value of
the KS instability, which is obtained by Eq. (35), and the transition between
the stable and unstable regimes is very sharp.
It is also seen that the critical behavior for instability is not as affected
by the mean tumbling frequency $\lambda_{0}$ in the small $\alpha$ regime.
However, in the large $\alpha$ regime, e.g., $\alpha>10$, the instability
behavior is significantly affected by the mean tumbling frequency
$\lambda_{0}$, and the instability condition of the KS system (the dotted line
in Fig. 1) is no longer consistent with the MC results, especially at
$\lambda_{0}=5$ (Fig. 1(a)).
Figure 2. Time evolutions of the population density of bacteria with moderate
stiffness $\delta=0.1$ at different values of the relative adaptation time,
i.e., $\alpha=0.4$ (a), $\alpha=1$ (b), and $\alpha=100$ (c), at the mean
tumbling frequency $\lambda_{0}=10$. Here, $x_{0}$ represents the position
where the chemical cue $S$ takes the maximum value in the stationary state. We
note that only the result at $t=0$ is calculated from the snapshot of the
initial distribution of MC particles so that it involves relatively large
fluctuations. Figure 3. Time evolutions of the population density of
bacteria with large stiffness $\delta=0.01$ at different values of the
relative adaptation time, i.e., $\alpha=0.03$ (a), $\alpha=1$ (b), and
$\alpha=100$ (c), at the mean tumbling frequency $\lambda_{0}=10$. See also
the caption in Fig. 2.
The aggregation profiles and their time evolution are shown in Figs. 2 and 3.
Figure 2 shows the result at moderate stiffness $\delta=0.1$, while Figure 3
shows the result at large stiffness $\delta=0.01$. In both figures, the
tumbling frequency $\lambda_{0}=10$ is fixed. Initially, the population
density $\rho$ is uniformly distributed with small fluctuations, whose
amplitudes are at most 0.015.
The aggregation profiles are highly affected by the adaptation time and the
stiffness. At moderate stiffness $\delta=0.1$ (Fig. 2), sinusoidal-like curves
are generated in the stationary states at small and large adaptation times,
i.e., $\alpha=0.4$ and 100, while at $\alpha=1$, a sharp aggregation profile
is generated in the stationary state.
At large stiffness $\delta=0.01$ (i.e., Fig. 3), very sharp aggregation, i.e.,
the spike-like aggregation profile, occurs at $\alpha=1$ in the stationary
state. On the other hand, interestingly, at large adaptation time $\alpha=100$
(see Fig. 3(c)), aggregation is not enhanced but is rather hindered at the
central region of the aggregation profile such that the trapezoidal profile,
where the plateau regimes appear at the top and bottom of the aggregate, is
formed. Later, we will see that this remarkable profile is obtained at a large
stiffness when the adaptation time is as long as $\tau=O(\lambda_{0})$.
Note that in both Figs. 2 and 3, the maximum aggregation density is not
monotonically dependent on the adaptation time; aggregation is enhanced at
moderate relative adaptation time $\alpha=1$. The nonmonotonic dependency of
the maximum aggregation density on the adaptation time is discussed in Fig. 4.
The spatial profiles of chemical cue $S$ are shown in Fig. 7(c). In contrast
to the population density of bacteria $\rho$, the spatial distribution of $S$
is moderate and not significantly affected by either the relative adaptation
time $\alpha$ or the stiffness $\delta$.
Note that the parameter sets used in Fig. 2(a) and Fig. 3(a) are very close to
but slightly above the linear stability condition of the KS system (the dotted
line in Fig. 1), where, although the time evolution is much slower than other
cases, distinct sinusoidal aggregation occurs. The sharp transition between
the stable and unstable modes in the small relative adaptation-time regime
(i.e., $\alpha<1$) is also observed in Fig. 4.
The nonmonotonic behavior of the maximum aggregation density with respect to
the relative adaptation time is seen in Figure 4. It is clear that the
transition from the stable to unstable modes is very sharp in the small
$\alpha$ regime, i.e., $\alpha\lesssim 1$. On the other hand, in the large
$\alpha$ regime, the maximum aggregation density gradually decreases as
$\alpha$ increases, and the slope of the decrease increases as $\lambda_{0}$
decreases. This behavior is completely different from that in the KS system,
where the maximum aggregation density monotonically increases and saturates to
a certain value as the relative adaptation time $\alpha$ increases.
Note that at any fixed $\lambda_{0}$, there exists the optimal adaptation time
to enhance aggregation in the regime $1<\alpha<\lambda_{0}$. Importantly, this
nonmonotonic behavior is a distinguished result obtained by the kinetic system
but not by the KS system.
Figure 4. The dependency of the maximum aggregation density on the relative
adaptation time $\alpha$ at different values of the mean tumbling frequency
$\lambda_{0}$. Figure (a) shows the result at $\delta=0.1$, and Figure (b)
shows the result at $\delta=0.2$. Here, the vertical axis shows the difference
between the maximum aggregation density $\rho_{\mathrm{max}}$ and the uniform
state $\rho=1$, i.e., $\Delta\rho=\rho_{\mathrm{max}}-1$. The results of the
KS system are obtained by numerical computation with the finite difference
scheme on the staggered grid given in Ref. [CPY2018].
#### 3.2.2. Distribution of the internal state
Figure 5. The stationary distributions of the internal state $y$ at different
distances from the center of the aggregate $r=|x-x_{0}|$ at the moderate
stiffness $\delta=0.1$. Figures (a), (b), and (c) show the results at
different relative adaptation times, i.e., $\alpha=0.4$, $\alpha=1$, and
$\alpha=100$, respectively, at the mean tumbling frequency $\lambda_{0}=10$.
Here, $G$ denotes the maximum value of the spatial gradient of $M(S)$, i.e.,
$G=\max_{x}|\partial_{x}M(S)|$. Figure 6. The stationary distributions of
the internal state $y$ at different distances from the center of the aggregate
$r=|x-x_{0}|$ at the large stiffness $\delta=0.01$. Figures (a), (b), and (c)
show the results at different relative adaptation times, i.e., $\alpha=0.03$,
$\alpha=1$, and $\alpha=100$, respectively, at the mean tumbling frequency
$\lambda_{0}=10$. See also the caption in Fig. 5.
Figures 5 and 6 show the stationary distributions of the internal state $y$ at
different distances from the center of the aggregate $r=|x-x_{0}|$, which are
defined as
$f_{r}^{\pm}(y)=\frac{f^{+}(x_{0}\mp r,y)+f^{-}(x_{0}\pm r,y)}{2}.$
That is, $f_{r}^{\pm}(y)$ denotes the local distributions of internal state
$y$ for the bacterial moving toward and away from the maximum aggregation
density at the distance $r$ from the center of the aggregate. Figure 5 shows
the result at the moderate stiffness $\delta=0.1$, while Fig. 6 shows the
results at the large stiffness $\delta=0.01$. The parameter sets used in Figs.
5 and 6 are the same as those in Figs. 2 and 3, respectively.
The distribution of the internal state is highly affected by the relative
adaptation time. When the relative adaptation time is short (i.e., Figs. 5(a)
and 6(a)), $f_{r}^{+}$ and $f_{r}^{-}$ are symmetric to each other with
respect to $y=0$ and have steep peaks at different values of the internal
state according to the distance $r$. Table 1 shows the relation between the
peak position of $f^{\pm}_{r}(y)$ with respect to $y$, $y_{p}^{\pm}$, and the
local spatial gradient of $M(S)$ at the distance $r$. The following relation
almost holds at each distance $r$, i.e.,
$y_{p}^{\pm}=\pm\tau|\partial_{x}M(S)|.$
| Fig. 5(a) | | Fig. 6(a)
---|---|---|---
$r$ | $y^{+}_{p}/(\tau G)$ | $y^{-}_{p}/(\tau G)$ | $|\partial_{x}M/G|$ | | $y^{+}_{p}/(\tau G)$ | $y^{-}_{p}/(\tau G)$ | $|\partial_{x}M/G|$
0.1 | 0.05 | -0.03 | 0.07 | | 0.05 | 0.05 | 0.05
0.7 | 0.35 | -0.33 | 0.37 | | 0.33 | 0.33 | 0.35
1.3 | 0.63 | -0.61 | 0.63 | | 0.59 | 0.59 | 0.62
2.5 | 0.97 | -0.95 | 0.99 | | 0.93 | 0.93 | 0.97
Table 1. Relation between the peak positions of $f^{\pm}_{r}(y)$ with respect
to $y$, $y_{p}^{\pm}$ in Fig. 5(a) and Fig. 6(a), and the local spatial
gradient of $M(S)$. See also the caption in Fig. 5.
This behavior is intuitively explained as follows. From Eq. (4) and the
definition of $y$, the dynamics of the internal state $y$ of each bacterium is
described as
$\dot{y}=\dot{M}(S)-\frac{y}{\tau},$
where $\dot{M}(S)$ is the temporal derivative of $M(S)$ along the moving path
of each bacterium and is replaced with $|\partial_{x}M(S)|$ (or
$-|\partial_{x}M(S)|$) at each local position $r$ when the bacteria move
toward (or away from) the maximum aggregation density. Note that $\dot{M}(S)$
changes the sign due to the tumbling of bacteria. Thus, when we denote the
internal states of the bacteria moving toward (or away from) the maximum
aggregation density $y^{+}$ (or $y^{-}$), and the temporal evolution of
$y^{\pm}$ at each instant is written as
$\dot{y}^{\pm}=\pm|\partial_{x}M|-\frac{y}{\tau}.$ (23)
When the adaptation time $\tau$ is much smaller than the run time
$\lambda_{0}^{-1}$, i.e., $\lambda_{0}\tau\ll 1$, the internal state is
determined by the local equilibrium state to be
$y_{p}^{\pm}=\pm\tau|\partial_{x}M|$ in each run duration.
On the other hand, at large relative adaptation time $\alpha=100$ (i.e., Figs.
5(a) and 6(a)), both distributions $f_{r}^{\pm}$ concentrate around $y=0$ in
the scaled internal variable $y/(\tau G)$, while at the moderate relative
adaptation time $\alpha=1$ (i.e., Figs. 5(b) and 6(b)), the internal state
moderately concentrates around $y_{p}^{\pm}=\pm\tau|\partial_{x}M|$ at each
local position $r$.
These behaviors of the internal state variable $y$, according to the change in
the relative adaptation time $\alpha$, are consistent with the continuum-limit
solutions obtained by the asymptotic analysis, which is presented in the next
section and in Ref. [PSTY2020]; that is, the continuum limit solution at
$\lambda_{0}^{-1}\rightarrow 0$ is obtained as
$f_{0}^{\pm}(t,x,y)=\rho(t,x)\delta(\frac{y}{\tau}=\pm\partial_{x}M(S))$ when
$\alpha\ll 1$ and $f_{0}^{\pm}(t,x,y)=\rho(t,x)\delta(y=0)$ when $\alpha\gg
1$. When $\alpha=1$, the distribution of the internal state moderately
concentrates around $y=\pm\tau\partial_{x}M(S)$. This characteristic behavior
of the internal state variable with respect to the relative adaptation time
$\alpha$ is less affected by the change in the stiffness parameter $\delta$
compared to the dependency of the macroscopic population density $\rho$ on the
stiffness parameter $\delta$. In the next subsection, we consider how the
stiffness parameter $\delta$ affects the individual motions of bacteria to
create different aggregation profiles, as shown in Figs. 2 and 3.
#### 3.2.3. Local mean run length
Figure 7. Spatial distributions of the mean run length $\xi^{\pm}_{r}$,
defined by Eq. (24), and the chemical cue $S$ in the stationary state at
different values of the stiffness parameter $\delta$. Figures (a) and (b) show
the results of the mean run length at $\alpha=1$ and $\alpha=100$,
respectively, and Figure (c) shows the results of chemical cues at the same
parameter sets. The mean tumbling frequency $\lambda_{0}=10$ is fixed.
Figure 7 shows the spatial distributions of the local mean run length of
bacteria at moderate stiffness $\delta=0.1$ and at large stiffness
$\delta=0.01$. Here, the local mean run length $\xi_{r}^{\pm}$ is calculated
as
$\xi_{r}^{\pm}=\int\frac{f_{r}^{\pm}(y)}{\rho_{r}\lambda_{0}\Lambda(\frac{y}{\delta})}dy,$
(24)
where $\rho_{r}$ is the population density at distance $r$. Thus,
$\xi_{r}^{+}$ denotes the local mean run length of the bacteria moving toward
the maximum aggregation density at distance $r$, while $\xi_{r}^{-}$ denotes
that of the bacteria moving away from the maximum aggregation density at
distance $r$. Since the modulation function $\Lambda_{\delta}(y)$, which is
defined by Eq. (6), is bounded, the mean run length is also bounded. In Figs.
7 (a) and (b), the upper and lower bounds of the mean run length are shown by
leftward arrows on the vertical axis.
At moderate relative adaptation time $\alpha=1$ (Fig. 7(a)), there is a
significant difference between $\xi_{r}^{+}$ and $\xi_{r}^{-}$. This indicates
that the bacteria create highly biased motions according to the moving
directions. At moderate stiffness $\delta=0.1$, the biased motion is maximized
around the location of the maximum gradient of $M(S)$, i.e.,
$|\partial_{x}M(S)|=|\partial_{x}S/S|$ (see Fig. 7(c)), while at large
stiffness $\delta=0.01$, the biased motion is further enhanced over the whole
domain except the vicinities at $r=0$ and $r=5$ so that the spike-like
aggregation profile forms due to the highly biased motions of bacteria (see
also Fig. 3(b)).
On the other hand, at large relative adaptation time $\alpha=100$, both
$\xi_{r}^{+}$ and $\xi_{r}^{-}$ vary similarly according to the local
amplitude (not the spatial gradient) of the chemical cue $S$, so the biased
motion of bacteria is less prominent. This observation is consistent with the
$y$ distributions in Figs. 5(c) and 6(c), where the difference between
$f_{r}^{+}$ and $f_{r}^{-}$ is small. Thus, aggregation is weakened compared
to that at moderate relative adaptation time $\alpha=1$.
In contrast to the results at $\alpha$=1 [Fig. 7(a)], at $\alpha$=100 [Fig.
7(b)], the stiffness of the chemotactic response $\delta$ does not enhance the
biased motions in different moving directions but amplifies the spatial
modulation of the mean run lengths in both moving directions, although the
spatial profile of the chemical cue is less affected by the stiffness
parameter [Fig. 7(c)]. Remarkably, at large stiffness $\delta=0.01$, the mean
run lengths in both moving directions $\xi^{\pm}$ attain the upper and lower
bounds in the vicinities of $r=$0 and 5, respectively. Thus, the biased motion
is rather hindered due to the boundedness of the tumbling frequency modulation
$\Lambda_{\delta}(y)$ in the vicinities of $r=$0 and 5 such that the plateau
regimes of the trapezoidal aggregation profile in Fig. 3(c) are created.
The overall observations in the local mean run length indicate that the biased
motion of bacteria is mostly determined by the relative adaptation time
$\alpha$, while the stiffness parameter $\delta$ amplifies only the signal of
the internal state in the chemotactic response function. These orthogonal
effects of the relative adaptation time $\alpha$ and the stiffness of the
chemotactic response $\delta$ produce the variety of aggregation profiles
observed in Figs. 2 and 3.
## 4\. Asymptotic analysis
### 4.1. Keller-Segel limit
In this section, we formally derive the asymptotic equations at
$\varepsilon=\lambda_{0}^{-1}\rightarrow 0$ under the diffusive scalings of
(10) and (14), i.e.,
$\varepsilon\partial_{t}f^{\pm}_{\varepsilon}\pm\partial_{x}f^{\pm}_{\varepsilon}+\partial_{y}\left\\{\left(\varepsilon\partial_{t}M(S_{\varepsilon})\pm\partial_{x}M(S_{\varepsilon})-\frac{y}{\tau}\right)f^{\pm}_{\varepsilon}\right\\}=\pm\frac{\Lambda_{\delta}(y)}{2\varepsilon}(f^{-}_{\varepsilon}-f^{+}_{\varepsilon}).$
(25)
$\varepsilon\partial_{t}S_{\varepsilon}=D_{S}\partial_{xx}S_{\varepsilon}-S_{\varepsilon}+\rho_{\varepsilon},$
(26)
We consider three different adaptation time scalings, i.e., (I)
$\tau=O(\varepsilon)$ (i.e., $\alpha=O(1)$), (II) $\tau=O(\varepsilon^{2})$
(i.e., $\alpha=O(\varepsilon)$), and (III) $\tau=O(1)$. Here, we set the time
scaling parameters as $\sigma$=$\sigma_{S}$=$\varepsilon$.
In Ref. [PSTY2020], asymptotic analysis of the continuous velocity version of
Eq. (14) is carried out at the same scalings of the adaptation time by
assuming that the stiffness parameter $\delta$ is the same order as the
adaptation time, i.e., $\delta=O(\tau)$, and the spatial gradient of $M(S)$ is
uniform, i.e., $\nabla_{x}M(S)=G$, where $G$ is constant. The results of Ref.
[PSTY2020] are briefly summarized as follows:
* •
In Case I, supposing $\chi=O(\varepsilon)$, a hyperbolic model is found at
$\sigma=1$, while a novel type of flux-limited KS model is found at
$\sigma=\varepsilon$. Furthermore, in both time scalings ($\sigma$=1 and
$\varepsilon$), the leading order solution $f_{0}$ is obtained from a new type
of equilibrium equation (which is the continuous velocity version of Eq.
(40)).
* •
In Case II, supposing $\chi=O(\varepsilon)$, a FLKS model is obtained at
$\sigma=\varepsilon$, and the leading order solution is explicitly written as
$f_{0}=\rho(t,x)\delta(y=v\cdot G)$.
* •
In Case III, a classical KS-type model is obtained for $\sigma=\varepsilon$,
and the leading order solution is explicitly written as
$f_{0}=\rho(t,x)\delta(y=0)$.
In this paper, we consider the case where both stiffness and modulation are
moderate, i.e., $\delta=O(1)$ and $\chi=O(1)$. By taking the sum of Eq. (25),
we have
$\partial_{t}\left(\frac{f_{\varepsilon}^{+}+f_{\varepsilon}^{-}}{2}\right)+\partial_{x}\left(\frac{f_{\varepsilon}^{+}-f_{\varepsilon}^{-}}{2\varepsilon}\right)+\partial_{y}\left\\{\left(\varepsilon\partial_{t}M_{\varepsilon}-\frac{y}{\tau}\right)\left(\frac{f_{\varepsilon}^{+}+f_{\varepsilon}^{-}}{2}\right)+\partial_{x}M_{\varepsilon}\left(\frac{f_{\varepsilon}^{+}-f_{\varepsilon}^{-}}{2\varepsilon}\right)\right\\}=0.$
Integration of the above equation with respect to $y$ gives the following
macroscopic conservation law:
$\partial_{t}\rho_{\varepsilon}+\partial_{x}\left(\frac{J_{\varepsilon}}{\varepsilon}\right)=0,$
(27)
where the flux $J_{\varepsilon}$ is defined as
$J_{\varepsilon}=\int_{R}\frac{f_{\varepsilon}^{+}-f_{\varepsilon}^{-}}{2}dy.$
(28)
The continuum-limit equations for the population density $\rho_{0}$ can be
derived from the above formulas (27) and (28).
By taking the continuum limit at Eq. (26), we also obtain that $S_{0}$ is the
following equation for chemoattractant $S_{0}$:
$-D_{S}\partial_{xx}S_{0}+S_{0}=\rho_{0}.$ (29)
We carried out asymptotic analysis under different scalings of the adaptation
time, i.e., (i) $\tau=O(\varepsilon)$, (ii) $\tau=O(\varepsilon^{2})$, and
(iii) $\tau=O(1)$, and obtained the KS-type models as described below. The
detailed calculations are given in A.1–A.3. Here, we only summarize the main
results, i.e., for (i) $\tau=O(\varepsilon)$ (or $\alpha=O(1)$),
$\partial_{t}\rho_{0}-\partial_{xx}\rho_{0}+\partial_{x}\left(\frac{\alpha\chi\partial_{x}M(S_{0})}{\delta(1+\alpha)}\rho_{0}\right)=0,$
(30)
for (ii) $\tau=O(\varepsilon^{2})$ (or $\alpha=O(\varepsilon)$),
$\partial_{t}\rho_{0}-\partial_{xx}\rho_{0}=0,$ (31)
and for (iii) $\tau=O(1)$ (or $\alpha=O(1/\varepsilon)$),
$\partial_{t}\rho_{0}-\partial_{xx}\rho_{0}+\partial_{x}\left(\frac{\chi\partial_{x}M_{0}}{\delta}\rho_{0}\right)=0.$
(32)
Note that Eqs. (31) and (32) coincide with Eq. (30) at the limits
$\alpha\rightarrow 0$ and $\alpha\rightarrow\infty$, respectively. Thus, the
KS-type equation (30) can uniformly describe the continuum-limit behavior of
the kinetic transport equation when the adaptation time is at most moderate,
i.e., $\tau<O(1)$.
#### 4.1.1. Linear instability of the KS system
The linear instability of the KS system around the uniform solution $\rho=S=1$
is obtained as follows:
First, we consider a small perturbation of the uniform solution in the
following form:
$\rho(t,x)=1+\widetilde{\rho}(x)e^{\mu t},\quad
S(t,x)=1+\widetilde{S}(x)e^{\mu t},$
and linearize Eq. (30) as
$\displaystyle\mu\widetilde{\rho}(x)e^{\mu
t}-\widetilde{\rho}^{\prime\prime}(x)e^{\mu
t}+\partial_{x}\left(\frac{\alpha\Lambda^{\prime}_{\delta}(0)}{1+\alpha}\widetilde{S}^{\prime}(x)e^{\mu
t}(1+\widetilde{\rho}^{\prime}(x)e^{\mu t}\right)=0,$
$\displaystyle\mu\widetilde{\rho}(x)-\widetilde{\rho}^{\prime\prime}(x)+\left(\frac{\alpha\Lambda^{\prime}_{\delta}(0)}{1+\alpha}\right)\widetilde{S}^{\prime\prime}(x)=0.$
By taking the Fourier transform of the above equations, we obtain
$\mu\widetilde{\rho}_{k}+k^{2}\left(\widetilde{\rho}_{k}-\frac{\alpha\Lambda^{\prime}_{\delta}(0)}{1+\alpha}\widetilde{S}_{k}\right)=0,$
(33)
where $k$ is the Fourier variable and $\widetilde{\rho}_{k}$ and
$\widetilde{S}_{k}$ are the Fourier transforms of $\widetilde{\rho}(x)$ and
$\widetilde{S}(x)$, respectively, which are calculated as
$\widetilde{\rho}_{k}=\int_{R}\widetilde{\rho}(x)e^{-\mathrm{i}kx}dx$. By
inserting the Fourier transform of Eq. (29),
$\displaystyle(1+D_{S}k^{2})\widetilde{S}_{k}=\widetilde{\rho}_{k},$
$\displaystyle\widetilde{S}_{k}=\frac{\widetilde{\rho}_{k}}{1+D_{S}k^{2}},$
into Eq. (33), we obtain
$\left[\mu+k^{2}\left(1-\frac{\alpha\Lambda^{\prime}_{\delta}(0)}{(1+\alpha)(1+D_{S}k^{2})}\right)\right]\widetilde{\rho}_{k}=0.$
(34)
Hence, the condition in which the mode $k$ becomes linearly unstable ($\mu>0$)
is written as
$\Lambda^{\prime}_{\delta}(0)>\frac{1+\alpha}{\alpha}(1+D_{S}k^{2}).$ (35)
Thus, the instability of the mode $k$ occurs when the stiffness of the
chemotactic response $\Lambda^{\prime}_{\delta}(0)$ is larger than the right-
hand side of Eq. (35). This also indicates that when the stiffness of the
response $\Lambda^{\prime}_{\delta}(0)$ and the diffusion coefficient $D_{S}$
are fixed, instability more likely occurs as $\alpha$ increases.
### 4.2. A novel asymptotic equation at large adaptation time
In the previous section, we show that the KS-type model involving the relative
adaptation time $\alpha$, Eq. (30) is derived when the adaptation time is
scaled as $\tau=O(\varepsilon^{n})$ ($n=0,1,2$) at the continuum limit
$\varepsilon=\lambda_{0}^{-1}\rightarrow 0$.
In this section, we consider the case where the adaptation time is very large,
i.e., $\tau=O(\varepsilon^{-1})$. The formal asymptotic analysis at the large
adaptation-time regime, i.e., $\tau=\widetilde{\tau}/\varepsilon$ with
$\widetilde{\tau}=O(1)$, is carried out in B, and the following asymptotic
equation is obtained at the limit $\varepsilon\rightarrow 0$:
$\partial_{t}p_{0}-\partial_{x}\left(\frac{\partial_{x}p_{0}}{\Lambda_{\delta}(M(S)-m)}\right)+\partial_{m}\left(\frac{M(S)-m}{\widetilde{\tau}}p_{0}\right)=0,$
(36)
where $p_{0}=p_{0}(t,x,m)$ is the continuum-limit solution to Eq. (8) at the
large adaptation-time scaling $\tau=\widetilde{\tau}/\varepsilon$. This novel
asymptotic equation retains the internal state variable $m$ as an independent
variable. Hence, the population density $\rho_{0}$ is obtained by the
integration of $p_{0}$ with respect to the internal variable $m$:
$\rho_{0}(t,x)=\int_{-\infty}^{\infty}p_{0}(t,x,m)dm.$ (37)
Obviously, the novel asymptotic equation is completely different from the KS
system. However, we can confirm the consistency of this asymptotic equation
with the KS system (32) at the limit $\widetilde{\tau}\rightarrow 0$ (see also
B). We will also numerically confirm the robustness of the asymptotic solution
to Eq. (36) in the next section.
Since $\tau$ and $\lambda_{0}$ are nondimensionalized as Eq. (9), the large
adaptation-time scaling $\tau\sim\lambda_{0}$ is rewritten in dimensional form
as
$\tau\sim t_{d},\quad t_{d}=\frac{L_{0}^{2}}{D_{\rho}},$ (38)
where $t_{d}$ is the time for bacterial population to diffuse over the
characteristic length $L_{0}$, which is, in the present problem, determined by
the diffusion of chemoattractant in the medium (Eq. (13), and $D_{\rho}$ is
the diffusion constant of the macroscopic population density defined as
$D_{\rho}=V_{0}^{2}/\lambda_{0}$. Thus, the novel asymptotic equation (36) is
appropriate when the adaptation time is comparable to the diffusion time of
the population density in the characteristic length, $\tau=O(t_{d})$, while
the KS system is only valid when the adaptation time is much smaller than the
diffusion time, $\tau\ll t_{d}$.
### 4.3. Asymptotic behavior at a large adaptation time
In this section, we further discuss the asymptotic behavior of the population
density with respect to $\varepsilon=\lambda_{0}^{-1}$ at large adaptation
times, i.e., (i) $\tau=O(1)$ and (ii) $\tau=O(\varepsilon^{-1})$.
Figure 8. Asymptotic behaviors of the population density $\rho$ at moderate
stiffness $\delta=0.1$. Figures (a), (b), and (c) show the results at
different scalings of the adaptation time, i.e., $\tau=1$,
$\tau=1/\varepsilon$, and $\tau=2/\varepsilon$, respectively, with
$\varepsilon=\lambda_{0}^{-1}$. The inset in (c) shows the magnification
around the center of the aggregate. The solid line in (a) shows the result of
the KS model, while the solid lines in (b) and (c) show the results of the
novel asymptotic equation (53).
Figure 8 shows the asymptotic behaviors of the population density at moderate
stiffness $\delta=0.1$. At $\tau=1$, the MC results approach those of the KS
system as $\varepsilon$ decreases. This observation is consistent with the
asymptotic analysis in Sec. 4. However, the asymptotic convergence is very
slow; a significant deviation remains between MC results and the KS system
results, even at small $\varepsilon$, e.g., $\varepsilon\sim 0.01$.
This slow asymptotic convergence is also confirmed in Fig. 4, where the MC
results at $\tau=1$ line up in an upper-right direction and gradually approach
the KS limit as the relative adaptation time $\alpha$ increases. On the other
hand, it is also seen that when $\alpha$ is fixed at $\alpha=O(1)$, i.e.,
$\tau=O(\varepsilon)$, the MC results converge to the KS result more rapidly
as $\varepsilon$ decreases.
When the adaptation time is set to $\tau=\widetilde{\tau}/\varepsilon$ (e.g.,
$\widetilde{\tau}$=1 in Fig. 8(b) and $\widetilde{\tau}$=2 in Fig. 8(c)), the
asymptotic convergence of the MC results is much faster at
$\tau=O(\varepsilon^{-1})$ compared to that observed at $\tau=1$. Remarkably,
even at moderately small $\varepsilon$, e.g., $\varepsilon\sim 0.1$, the MC
results almost coincide with the numerical solutions of the novel asymptotic
equation (36). Convergence to the asymptotic solution is also observed in Fig.
9. These results numerically confirm that the asymptotic solution to Eq. (36)
is robust at the large adaptation-time scaling $\tau=O(\varepsilon^{-1})$.
Note that the maximum aggregation density decreases as $\widetilde{\tau}$
increases, as is already observed in Fig. 4. Thus, the nonmonotonic behavior
of the maximum aggregation density with respect to the adaptation time $\tau$
at each fixed $\varepsilon$ can be viewed as the transition from the KS-type
solution to the asymptotic solution to Eq. (36).
Figure 9. Asymptotic behaviors of the population density $\rho$ at large
stiffness $\delta=0.01$. Figures (a), (b), and (c) show the results at
different scalings of the adaptation time, i.e., $\tau=5$,
$\tau=1/\varepsilon$, and $\tau=2/\varepsilon$, respectively. The solid lines
in (b) and (c) show the results of the novel asymptotic equation (53).
Figure 9 shows the asymptotic behaviors of the population density at large
stiffness $\delta=0.01$. It is seen from Fig. 9(a) that when $\tau=5$ is
fixed, the width of the aggregate narrows as $\varepsilon$ decreases, so
asymptotic convergence is not confirmed from the present MC results.
Interestingly, a trapezoidal aggregate is robustly formed at the large
adaptation-time scaling $\tau=O(1/\varepsilon)$; in Figs. 9(b) and (c), the MC
results at moderately small $\varepsilon$, e.g., $\varepsilon\lesssim 0.1$,
are close to each other and asymptotically converge to the trapezoidal
profile. Furthermore, it is clear that the trapezoidal profile is related to
the novel asymptotic equation (36) at the large adaptation-time scaling
$\tau=\widetilde{\tau}/\varepsilon$.
## 5\. Summary and perspectives
We investigated the self-organized aggregation of chemotactic bacteria in one-
dimensional space with periodic boundary conditions based on a two-stream
kinetic transport model with an internal state coupled with the
chemoattractant equation. MC simulations were conducted for a wide range of
adaptation times $\tau$ at various values of the mean tumbling frequency
$\lambda_{0}$ and the stiffness of chemotactic response $\delta$ Asymptotic
analysis of the kinetic transport model was also carried out to complement the
MC results. Thus, the effect of the adaptation time on the aggregation
behavior was investigated both macroscopically and microscopically.
An important finding is the nonmonotonic dependence of the adaptation time on
the aggregation behavior. See, for example, Figs. 1 and 4. A sharp transition
between stable and unstable modes is observed when the relative adaptation
time $\alpha$, which is defined as $\alpha=\lambda_{0}\tau$, is small, e.g.,
$\alpha\lesssim 1$. In this small $\alpha$ regime, instability always occurs
when $\alpha$ is slightly larger than the critical value of the linear
instability condition of the KS model, which is derived from the kinetic
transport model by asymptotic analysis, and the maximum aggregation density
rapidly increases as $\alpha$ increases. However, when the relative adaptation
time is large, e.g., $\alpha\gtrsim\lambda_{0}$, the MC results deviate from
the linear stability condition of the KS model, and the maximum aggregation
density gradually decreases as $\alpha$ increases. Thus, there exists an
optimal adaptation time to enhance aggregation around $1<\alpha<\lambda_{0}$.
This nonmonotonic behavior is a significantly important feature that can be
described at the kinetic level but not at the KS level; in the KS model, the
maximum aggregation density monotonically increases and saturates to a certain
value as $\alpha$ increases.
We also investigated the microscopic behaviors in the variety of aggregation
profiles in terms of the local distribution of the internal state (Figs. 5 and
6) and the spatial distribution of the local mean run length (Fig. 7). We
found an orthogonal effect of the relative adaptation time $\alpha$ and the
stiffness of the chemotactic response $\delta$ on the microscopic dynamics.
That is, the relative adaptation time $\alpha$ significantly affects the
distribution of the internal state, while the stiffness $\delta$ does not
similarly affect the distribution of the internal state but only amplifies the
signal of the internal state in the response function $R_{\delta}(y)$.
From these microscopic perspectives, the optimal adaptation time to produce
sharp aggregation can be intuitively explained as follows. When the adaptation
time $\tau$ is smaller than the mean run duration $\lambda_{0}^{-1}$ (i.e.,
$\alpha<1$), the internal state $y$ is rapidly equilibrated at
$y_{p}^{\pm}=\pm\tau|\partial_{x}M|$ depending on the moving direction in each
run duration. Here, since the equilibrium state $M$ sensed by the bacteria is
temporally changed along the run of each bacterium, the internal state $y$ is
not equilibrated at $y=0$ but is equilibrated at $y=y_{p}$, which is linearly
proportional to the adaptation time $\tau$ unless the bacteria tumbles. Thus,
the amplitude of the chemotactic response $|R(y)|$ (see Eq. (6)) becomes
larger in each run duration at adaptation time $\tau$ when $\alpha<1$. On the
other hand, when the adaptation time $\tau$ is much larger than the run
duration $\lambda_{0}^{-1}$ (i.e., $\alpha>\lambda_{0}\gg 1$), the internal
state cannot be equilibrated in each run duration so that the biased motion
between the different moving directions is significantly reduced, as shown in
Fig. 7(b). Thus, aggregation is hindered when $\alpha$ is very large. These
competitive effects of the adaptation time on the chemotactic response lead to
the optimal behavior of chemotactic aggregation.
Figure 10. Schematic of the asymptotic regimes. The novel asymptotic equation
(36) is discovered along the red solid line while the KS regime is limited in
$\tau\lesssim O(1)$. The KS equation is also obtained from the novel
asymptotic equation by taking a consistent limit.
Another remarkable result of this paper is the discovery of the novel
asymptotic equation Eq. (36) at large adaptation-time scaling
$\tau=O(\varepsilon^{-1})$. This asymptotic regime physically indicates that
the adaptation time $\tau$ is comparable to the diffusion time of the
population density in the characteristic length [Eq. (38)]. A numerical
comparison clarified that the trapezoidal aggregates, which are robustly
formed at the large adaptation-time regime are well described by the novel
asymptotic equation. See Fig. 9.
Figure 10 is the cartoon of the asymptotic regimes of the novel asymptotic
equation (36) and the KS system (30). The KS system is not valid even near the
continuum limit $\varepsilon\ll 1$ unless the adaptation time $\tau$ is at
most moderate $\tau<O(1)$. Although the novel asymptotic equation is obtained
at the scaling $\tau=O(\varepsilon^{-1})$ (the red dashed line in the figure),
we also formally show that the novel asymptotic equation converges to the KS
system when $\widetilde{\tau}(=\varepsilon\tau)\rightarrow 0$ (the dashed left
arrow in the figure).
The asymptotic behavior of the MC results clearly illustrates the limitation
of the appropriate adaptation-time regime for the KS system and the transient
behavior from the KS solution to the novel asymptotic solution when the
adaptation time becomes large. Thus, the nonmonotonic behavior of the
aggregation density with respect to the adaptation time can be interpreted as
the transient behavior from the KS-type solution to the novel asymptotic
solution, although the detailed analysis of the transient behavior remains as
important future work.
One may think that such a large adaptation time is biologically unrealistic.
However, when the system size is as small as $L_{0}\sim 100\,\mu\mathrm{m}$
and the mean run length of bacteria is measured as $l_{0}\sim
20\,\mu\mathrm{m}$, the ratio of the mean run length of bacteria to the system
size, $\varepsilon$, is typically estimated to be $\varepsilon\sim 0.2$. In
this case, the relative adaptation time in the novel asymptotic regime is
estimated to be $\alpha\sim 25$. The adaptation time of bacteria is usually
much longer than the run time, so $\alpha\sim 25$ is not unrealistic but
rather commonly observed.
Indeed, when the system size is as small as $\varepsilon\sim 0.1$, unusual
aggregation behaviors of bacteria are observed in experiments. For example, in
Ref. [MBBO2003], the volcano-like aggregation profile of E. coli is observed
at $L_{0}\sim 100\,\mu\mathrm{m}$, and in Ref. [BM2003], swarm bands of marine
bacteria around a chemoattractant microbead with a distance of approximately
$L_{0}\sim 20\,\mu\mathrm{m}$ are observed. The common feature in these
curious aggregation behaviors is that the maximum aggregation density is not
located at the center of the aggregate but rather at a certain distance from
the center. It is interesting that the trapezoidal aggregation profile is
obtained in the present MC simulation in the same parameter regime as in the
experiments, although neither volcano-like aggregation nor swarm rings have
yet been confirmed.
In the present paper, to focus on the multiscale mechanism between collective
motion and individual motility involving internal adaptation dynamics, we
ignore several factors that should be important in reproducing experimental
results. Among them, the nutrients consumed by bacteria and proliferation due
to cell division should play a significant role in the collective motions and
pattern formations of chemotactic bacteria. For example, in Ref. [CMD2013],
the variety of patterns observed in experiments was successively reproduced by
an individual-based simulation coupled with the consumption of nutrients and
the secretion of chemoattractants. Inclusion of these factors in the present
kinetic transport model is rather straightforward and plays a significant role
in collective motions and pattern formations. Indeed, in our previous study
[PY2018], the instability that leads to Turing-like periodic pattern formation
was clarified based on the kinetic transport model involving the proliferation
of bacteria. The traveling pulse created by the bacteria pursuing the nutrient
is also successively illustrated based on the kinetic transport model coupled
with the reaction-diffusion equations of the nutrient and secreted
chemoattractant [SCBPBS2011, C2020]. In these previous kinetic studies, the
internal dynamics were completely ignored. Typically, the adaptation time is
comparable to the characteristic time of the collective traveling pulse of
bacteria pursuing the nutrient; for example, in the traveling pulse observed
in Ref. [SCBPBS2011], the characteristic time of the wave is estimated to be
$t_{0}\sim 20$ s. Thus, we can expect some coupling effects between the
internal adaptation dynamics and the collective traveling pulse pursuing the
nutrient. Investigation of the effect of the adaptation time on the traveling
waves and pattern formations coupled with the sensing of the nutrient and
secreted chemoattractant is important future work.
In the present kinetic transport model, we also utilize the simplified model
for intracellular signal transduction, where only the adaptation dynamics of
the internal state are considered while the excitation dynamics, which are
much faster than the adaptation dynamics, are ignored. The tumbling time
(i.e., the time required for the bacteria to change their moving direction) is
also ignored since it is much shorter than the run time. However, in small
systems, these fast-time-scale dynamics may bring about a delay in individual
motion and affect collective dynamics. Indeed, in the literature [BLL2007,
JJP2018, SM2011], the nonunimodal aggregates around a chemoattractant point
source are numerically reproduced by using a model involving either the
excitation dynamics of the internal state or the tumbling time.
It has also been reported that the noise arising in the intracellular signal
transduction process plays a significant role in the occurrence of chemotactic
aggregation around a chemoattractant point source [BCHYLS2019] and in the
fractional diffusion mode of chemotactic bacteria [KEVSC2004, PST2018].
Furthermore, in Refs. [MJD2009, MMCD2011], the origin of the noise in
intracellular signal transduction and its effect on the sensing behavior are
also argued based on an optimal biochemical network model proposed in Ref.
[KLBTS2005]. To uncover the multiscale mechanism between collective motion and
individual motility involving internal dynamics in a variety of aggregation
behaviors, further investigations based on the kinetic transport model
involving more sophisticated formulas of the intracellular signal transduction
pathway as well as the tumbling state will also be important future work.
## Appendix A Derivation of continuum-limit equations
### A.1. Fast adaptation
Here, we consider the case where the adaptation time is comparable to the run
duration, i.e., $\tau=O(\varepsilon)$. Thus, the relative adaptation time is
$\alpha=O(1)$. By changing the variable as
$g_{\varepsilon}(t,x,z)=f_{\varepsilon}(t,x,\tau z)$, Eq. (25) is written as
$\varepsilon^{2}\partial_{t}g_{\varepsilon}^{\pm}\pm\varepsilon\partial_{x}g_{\varepsilon}^{\pm}+\frac{1}{\alpha}\partial_{z}\left\\{\left(\varepsilon\partial_{t}M_{\varepsilon}\pm\partial_{x}M_{\varepsilon}-z\right)g_{\varepsilon}^{\pm}\right\\}=\pm\frac{\Lambda_{\delta}(\varepsilon\alpha
z)}{2}(g_{\varepsilon}^{-}-g_{\varepsilon}^{+}).$ (39)
Note that $\Lambda_{\delta}(\varepsilon\alpha
z)=1-R_{\delta}(\varepsilon\alpha z)$ and $|R_{\delta}(\varepsilon\alpha
z)|\leq\frac{\varepsilon\alpha|z|}{\delta}$. Thus, from the leading term, we
obtain
$\partial_{z}[(\pm
G_{0}-z)g_{0}^{\pm}]=\pm\frac{\alpha}{2}(g_{0}^{-}-g_{0}^{+}),$
where we write $G_{0}=\partial_{x}M_{0}$. Here, we assume that
$g_{\varepsilon}^{\pm}$ is compactly supported with respect to $z$. (This can
be proved when $|G_{0}|$ is bounded, as is done in Sec. 3 of Ref. [PSTY2020].)
By integrating the above equation with respect to $z$, we obtain $J_{0}=0$.
We seek the leading-order solution in the form of
$g_{0}^{\pm}=\rho_{0}(t,x)Q_{0}^{\pm}(z;G_{0})$, where $Q_{0}^{\pm}$ is
described as
$\partial_{z}[(\pm
G_{0}-z)Q_{0}^{\pm}]=\pm\frac{\alpha}{2}(Q_{0}^{-}-Q_{0}^{+}),$ (40)
with $\int_{R}Q_{0}^{\pm}dz=1$. Furthermore, $Q_{0}^{\pm}$ is compactly
supported on $z=[-|G_{0}|,|G_{0}|]$.
By using the leading order solution, we can write Eq. (39) as
$\displaystyle\pm\partial_{x}(\rho_{0}Q_{0}^{\pm})$
$\displaystyle+\frac{1}{\alpha}\partial_{z}\left\\{\partial_{t}M_{0}\rho_{0}Q_{0}^{\pm}+(\pm
G_{0}-z)g_{1}^{\pm}\right\\}$
$\displaystyle=\pm\frac{1}{2}(g_{1}^{-}-g_{1}^{+})\pm\frac{\alpha\chi\rho_{0}}{2\delta}z(Q_{0}^{+}-Q_{0}^{-})+O(\varepsilon),$
where we use $R_{\delta}(\varepsilon\alpha z)=\frac{\varepsilon\alpha
z}{\delta}+O(\varepsilon^{2})$ for $z\in[-|G_{0}|,|G_{0}|]$. By integrating
the above equation with respect to $z$ and taking the limit
$\varepsilon\rightarrow 0$, we obtain
$J_{1}=-\partial_{x}\rho_{0}+\frac{\alpha\chi\rho_{0}}{2\delta}\int_{-|G|}^{|G|}z(Q_{0}^{+}-Q_{0}^{-})dz.$
Here, we also assume $\int_{R}|z(g_{1}^{-}-g_{1}^{+})|dz<+\infty$. The last
term of the above equation is calculated as follows. By integrating Eq. (40)
multiplied by $z$, we obtain
$\int z\partial_{z}[(\pm G-z)Q_{0}^{\pm}]dz=\pm\frac{\alpha}{2}\int
z(Q_{0}^{-}-Q_{0}^{+}),$ $-\int(\pm G-z)Q_{0}^{\pm}dz=\pm\frac{\alpha}{2}\int
z(Q_{0}^{-}-Q_{0}^{+}),$ $\mp G+\int zQ_{0}^{\pm}dz=\pm\frac{\alpha}{2}\int
z(Q_{0}^{-}-Q_{0}^{+}),$ $\int z(Q_{0}^{+}-Q_{0}^{-})dz=\frac{2G}{1+\alpha}.$
Thus, we obtain
$J_{1}=-\partial_{x}\rho_{0}+\frac{\alpha\chi G\rho_{0}}{\delta(1+\alpha)}.$
Hence, from Eq. (27), we obtain the following KS equation in the continuum
limit $\varepsilon\rightarrow 0$,
$\partial_{t}\rho_{0}-\partial_{xx}\rho_{0}+\partial_{x}\left(\frac{\alpha\chi\partial_{x}M(S_{0})}{\delta(1+\alpha)}\rho_{0}\right)=0.$
(41)
### A.2. Very fast adaptation
Here, we consider the case where the adaptation time is much smaller than the
run duration, i.e., $\tau=O(\varepsilon^{2})$. Hence, $\alpha=O(\varepsilon)$.
By setting $\alpha=\alpha_{1}\varepsilon$ at Eq. (39), we obtain
$\varepsilon^{2}\partial_{t}g_{\varepsilon}^{\pm}\pm\varepsilon\partial_{x}g_{\varepsilon}^{\pm}+\frac{1}{\alpha_{1}\varepsilon}\partial_{z}\left\\{\left(\varepsilon\partial_{t}M_{\varepsilon}\pm\partial_{x}M_{\varepsilon}-z\right)g_{\varepsilon}^{\pm}\right\\}=\pm\frac{\Lambda_{\delta}(\varepsilon^{2}\alpha_{1}z)}{2}(g_{\varepsilon}^{-}-g_{\varepsilon}^{+}).$
(42)
From the leading term of the above equation, we obtain
$\partial_{z}[(\pm\partial_{x}M_{0}-z)g_{0}^{\pm}]=0.$
Thus, the leading order solution is written as
$g_{0}^{\pm}=\rho_{0}\delta(z=\pm\partial_{x}M_{0}),$ (43)
where $\delta(z)$ is the Dirac delta function. The leading order flux is
obtained as $J_{0}=0$.
By taking the difference of Eq. (42) and integrating it with respect to $z$,
we obtain
$\varepsilon\partial_{x}\rho_{\varepsilon}=-J_{\varepsilon}+\int_{R}\frac{R_{\delta}(\varepsilon^{2}\alpha_{1}z)}{2}(g_{\varepsilon}^{-}-g_{\varepsilon}^{+})dz+O(\varepsilon^{2}).$
By taking the limit $\varepsilon\rightarrow 0$ of the above equation under the
assumption $\int_{R}|z(g_{\varepsilon}^{-}-g_{\varepsilon}^{+})|dz<+\infty$,
we obtain
$J_{1}=-\partial_{x}\rho_{0}.$
Hence, we obtain the following diffusion equation at the continuum limit
$\varepsilon\rightarrow 0$:
$\partial_{t}\rho_{0}-\partial_{xx}\rho_{0}=0.$ (44)
### A.3. Moderate adaptation
Here, we consider the case where the adaptation time is order unity,
$\tau=O(1)$. Hence, $\alpha=O(1/\varepsilon)$. The leading term of Eq. (25)
gives us
$f_{0}^{+}=f_{0}^{-}=f_{0},$ (45)
and hence, $J_{0}=0$.
Thus, the next order term is written as
$\pm\partial_{x}f_{0}+\partial_{y}\left\\{\left(\pm\partial_{x}M_{0}-\frac{y}{\tau}\right)f_{0}\right\\}=\pm\frac{\Lambda_{\delta}(y)}{2}(f_{1}^{-}-f_{1}^{+}).$
(46)
By taking the sum of the above equation, we obtain
$\partial_{y}(yf_{0})=0.$
Hence, the leading order solution is written as
$f_{0}=\rho_{0}(t,x)\delta(y=0).$ (47)
Thus, from Eq. (46) with Eq. (47), the flux $J_{1}$ is calculated as
$\displaystyle J_{1}$
$\displaystyle=-\int\frac{\delta(y=0)}{\Lambda_{\delta}(y)}dy\partial_{x}\rho_{0}-\rho_{0}\partial_{x}M_{0}\int\frac{\delta^{\prime}(y=0)}{\Lambda_{\delta}(y)}dy,$
$\displaystyle=-\frac{1}{\Lambda_{\delta}(0)}\partial_{x}\rho_{0}-\frac{\Lambda_{\delta}^{\prime}(0)}{\Lambda_{\delta}^{2}(0)}\rho_{0}\partial_{x}M_{0}.$
Since $\Lambda_{\delta}(0)=1$ and
$\Lambda_{\delta}^{\prime}(0)=-\frac{\chi}{\delta}$, we obtain
$J_{1}=-\partial_{x}\rho_{0}+\frac{\chi\rho_{0}\partial_{x}M_{0}}{\delta},$
(48)
and hence,
$\partial_{t}\rho_{0}-\partial_{xx}\rho_{0}+\partial_{x}\left(\frac{\chi\partial_{x}M_{0}}{\delta}\rho_{0}\right)=0.$
(49)
## Appendix B Large adaptation time
We consider the kinetic transport equation with internal state $m$, i.e., Eq.
(8) under scaling at a very large adaptation time
$\tau=\widetilde{\tau}/\varepsilon$,
$\varepsilon\partial_{t}p^{\pm}_{\varepsilon}\pm\partial_{x}p^{\pm}_{\varepsilon}+\varepsilon\partial_{m}\left(\frac{M(S)-m}{\widetilde{\tau}}p^{\pm}_{\varepsilon}\right)=\pm\frac{\Lambda_{\delta}(M-m)}{2\varepsilon}(p_{\varepsilon}^{-}-p_{\varepsilon}^{+}).$
(50)
By taking the limit $\varepsilon\rightarrow 0$ in Eq. (50), we obtain, at the
leading order,
$p_{0}^{+}=p_{0}^{-}=p_{0},$ (51)
and, furthermore, by using Eq. (51), we also obtain
$\partial_{x}p_{0}=\Lambda_{\delta}(M-m)\frac{p_{1}^{-}-p_{1}^{+}}{2},$
$\frac{p_{1}^{+}-p^{-}_{1}}{2}=-\frac{\partial_{x}p_{0}}{\Lambda_{\delta}(M-m)}.$
(52)
By taking the sum of Eq. (50), we obtain
$\partial_{t}\left(\frac{p_{\varepsilon}^{+}+p_{\varepsilon}^{-}}{2}\right)+\partial_{x}\left(\frac{p_{\varepsilon}^{+}-p_{\varepsilon}^{-}}{2\varepsilon}\right)+\partial_{m}\left(\frac{M-m}{\widetilde{\tau}}\frac{p_{\varepsilon}^{+}+p_{\varepsilon}^{-}}{2}\right)=0.$
Thus, by taking the limit $\varepsilon\rightarrow 0$ in the above equation and
using Eq. (52), we obtain
$\partial_{t}p_{0}-\partial_{x}\left(\frac{\partial_{x}p_{0}}{\Lambda_{\delta}(M(S)-m)}\right)+\partial_{m}\left(\frac{M(S)-m}{\widetilde{\tau}}p_{0}\right)=0.$
(53)
The consistency with the Keller-Segel limit can be confirmed when taking the
limit $\widetilde{\tau}\rightarrow 0$ in Eq. (53); that is, at the limit
$\widetilde{\tau}\rightarrow 0$, we can obtain from Eq. (53)
$p_{0}=\rho_{0}\delta(m-M(S)),$
and
$\displaystyle\partial_{t}\rho_{0}\delta(m-M(S))-\partial_{x}\left(\frac{\partial_{x}\rho_{0}\delta(m-M(S))}{\Lambda_{\delta}(M(S)-m)}\right)=0,$
$\displaystyle\partial_{t}\rho_{0}\delta(m-M(S))-\partial_{x}\left(\frac{\delta(m-M(S))}{\Lambda_{\delta}(M(S)-m)}\partial_{x}\rho_{0}-\frac{\delta^{\prime}(m-M(S))}{\Lambda_{\delta}(M(S)-m)}\rho_{0}\partial_{x}M(S)\right)=0.$
Thus, by integrating the above equation with respect to $m$, we obtain Eq.
(32).
### B.1. Numerical Scheme
The one-dimensional space $x\in[-L/2,L/2]$ and the internal state $m\in[-Y,Y]$
are discretized as $x_{i}=-L/2+i\Delta x$ ($i=0,1,\cdots,I$) and
$m_{k}=-Y+k\Delta m$ ($k=0,1,\cdots,K$), where the mesh intervals are defined
as $\Delta x=L/I$ and $\Delta m=2Y/K$.
By integrating Eq. (53) over the unit cell
$[x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}]\times[m_{k-\frac{1}{2}},m_{k+\frac{1}{2}}]$
and time interval $[t_{n},t_{n+1}]$, where $t_{n}=n\Delta t$, we obtain
$\displaystyle p_{i,k}^{n+1}=p_{i,k}^{n}$ $\displaystyle+\frac{1}{\Delta
x\Delta
m}\left[\int_{t_{n}}^{t_{n+1}}dt\int_{m_{k-\frac{1}{2}}}^{m_{k+\frac{1}{2}}}dm\frac{\partial_{x}p_{0}}{\Lambda_{\delta}(M(S)-m)}\right]_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}$
$\displaystyle-\frac{1}{\Delta x\Delta
m}\left[\int_{t_{n}}^{t_{n+1}}dt\int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}dx\frac{M(S)-m}{\widetilde{\tau}}p_{0}\right]_{m_{k-\frac{1}{2}}}^{m_{k+\frac{1}{2}}},$
where $p_{i,k}^{n}$ is the value of $p_{0}$ at unit cell
$[x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}]\times[m_{k-\frac{1}{2}},m_{k+\frac{1}{2}}]$
at time $t_{n}$.
When we approximate $\partial_{x}p_{0}$ in the second term of the R.H. S by
the centered difference and apply the upwind scheme for the last term, we
obtain
$p_{i,k}^{n+1}=p_{i,k}^{n}+\frac{\Delta t}{\Delta
x^{2}}\left(\frac{p^{n}_{i+1,k}-p^{n}_{i,k}}{\Lambda_{i+\frac{1}{2},k}}-\frac{p^{n}_{i,k}-p^{n}_{i-1,k}}{\Lambda_{i-\frac{1}{2},k}}\right)-\frac{\Delta
t}{\tau\Delta m}\left(\psi_{i,k+\frac{1}{2}}-\psi_{i,k-\frac{1}{2}}\right),$
(54)
where $\Lambda_{i-\frac{1}{2},k}=\Lambda_{\delta}(M(S_{i-\frac{1}{2}})-m_{k})$
and the flux $\psi_{i,k-\frac{1}{2}}$ is defined as
$\psi_{i,k-\frac{1}{2}}=\left(M(S_{i})-m_{k-1}\right)^{+}p_{i,k-1}-\left(M(S_{i})-m_{k}\right)^{-}p_{i,k}.$
(55)
Here, we use the notation $u^{+}=\max\\{0,u\\}$ and $u^{-}=\max\\{0,-u\\}$.
## References
|
# The geodesic-transversal problem
Paul Manuela Boštjan Brešarb,c Sandi Klavžarb,c,d
###### Abstract
A maximal geodesic in a graph is a geodesic (alias shortest path) which is not
a subpath of a longer geodesic. The geodesic-transversal problem in a graph
$G$ is introduced as the task to find a smallest set $S$ of vertices of $G$
such that each maximal geodesic has at least one vertex in $S$. The minimum
cardinality of such a set is the geodesic-transversal number ${\rm gt}(G)$ of
$G$. It is proved that ${\rm gt}(G)=1$ if and only if $G$ is a subdivided star
and that the geodesic-transversal problem is NP-complete. Fast algorithms to
determine the geodesic-transversal number of trees and of spread cactus graphs
are designed, respectively.
a Department of Information Science, College of Computing Science and
Engineering, Kuwait University, Kuwait
<EMAIL_ADDRESS>
b Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia
<EMAIL_ADDRESS>
c Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
d Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
<EMAIL_ADDRESS>
Keywords: hitting set; geodesic-transversal problem; network centrality; tree;
cactus graph; algorithm
AMS Subj. Class.: 05C69; 05C85; 68R10
## 1 Introduction
Given a set $U$ and a family $S=\\{S_{1},\ldots,S_{k}\\}$, where
$S_{i}\subseteq U$, a subset $H$ of $U$ is a hitting set for the family $S$ if
$H\cap S_{i}$ $\neq$ $\emptyset$ for all $i\in\\{1,\ldots,k\\}$. The hitting
set problem is to find a smallest hitting set for $S$. The hitting set problem
is NP-complete [16] and has been studied in different terminologies. In
particular, in graph theory the term $S$-transversal problem presents the
quest for a minimum set of vertices that intersect every set of a given family
$S$ of subsets of the vertex set. When $S$ is a collection of maximal cliques
of a graph, the $S$-transversal problem is called the clique-transversal
problem [1, 8, 9, 11, 12, 13, 18], and when $S$ is a collection of fixed size
cliques, it is called the generalized clique transversal problem [11, 12]. The
clique-transversal problem is polynomially solvable for interval graphs and
NP-complete for chordal graphs [12]. Dahlhaus et al. [13] have studied the
$S$-transversal problem where $S$ is a collection of hyperedges in a
hypergraph. When $S$ is a collection of $k$-paths, the $S$-transversal problem
is called the $k$-path-transversal problem. This problem has been well-studied
under different terminologies [5, 6, 15, 19, 28, 30, 39, 47].
A geodesic in a graph $G$ is a shortest path between two vertices, and a
geodesic is maximal if it is not a subpath of a longer geodesic. When $S$ is a
collection of maximal geodesics, we call the $S$-transversal problem the
geodesic-transversal problem. A geodesic on $k$ vertices is a $k$-geodesic.
When $S$ is a collection of $k$-geodesics, the $S$-transversal problem is
called $k$-geodesic-transversal problem.
To our knowledge, there is no literature on the geodesic-transversal problem
and the $k$-geodesic-transversal problem. The geodesic-transversal number of
$G$, denoted by ${\rm gt}(G)$, is the minimum cardinality of a geodesic-
transversal set of $G$. A set $S$ of vertices is a gt-set of $G$ if $S$ is a
minimum cardinality geodesic-transversal set of $G$. Thus, the geodesic-
transversal problem of $G$ is to find a gt-set of $G$. It is easy to see that
the $2$-geodesic-transversal problem is the vertex cover problem.
In the next section, we provide further motivation for the new geodesic-
transversal problem. In Section 3, we determine the geodesic-transversal
number of some graphs and show that this number equals $1$ precisely for
subdivided stars. We also prove that the geodesic-transversal problem is NP-
complete for general graphs. In Section 4 we derive a polynomial algorithm for
arbitrary trees, while in Section 5 a fast algorithm is designed for spread
cactus graphs.
## 2 Motivation from (large-scale) network theory
The geodesic-transversal problem is not entirely new. The path version of this
problem is quite popular in graph theory and is well studied by graph theory
researchers [5, 6, 15, 19, 28, 30, 39, 47]. A set $S$ of vertices of a graph
$G$ is a $k$-path vertex cover if every path of order $k$ in $G$ contains at
least one vertex from $S$ [6]. It is not uncommon in graph theory that the
same concept is studied under different names. If indeed so, this indicates
that the concept is of wider interest. The $k$-path vertex cover has been
studied also as vertex $k$-path cover [5], $k$-path vertex cover [2, 6, 19,
28, 30], $VCP_{k}$-set [39], and $k$-path cover [15]. The $k$-path vertex
cover problem is to find the minimum cardinality of a $k$-path vertex cover.
The problem is NP-hard for cubic planar graphs of girth 3 [6, 39] and for
bipartite graphs [47]. The problem has applications in many areas, such as
traffic control [41] and wireless sensor networks [6]. Funke et al. [15] have
provided a list of applications of this problem on different domains. The
concepts of path transversal have also been generalized to the context of
hypergraphs [50]. The geodesic-transversal problem is a natural extension and
adaptation of the path-transversal problem. Note that the $k$-path vertex
cover problem and the $k$-geodesic transversal problem coincide in general
graphs when $k=2$, and coincide in triangle-free graphs when $k=3$.
Betweenness centrality and closeness centrality are key measures of large-
scale network analysis [32, 43]. The concepts of betweenness centrality and
closeness centrality play a vital role in the study of large-scale network
analysis including social networks [17, 26, 32], brain networks [14, 21, 25],
biological networks (gene regulatory networks, protein-protein interaction
network) [24, 25], chemical networks [49], communication networks [10],
transport networks [27, 35] and IoT networks [35, 43] etc. The betweeness
centrality $B(v)$ and closeness centrality $C(v)$ are defined as follows [32,
43]:
$B(v)=\sum_{s\neq v\neq t}\frac{\sigma_{st}(v)}{\sigma_{st}}$
$C(v)=\sum_{s\neq v\neq t}\sigma_{st}(v)$
where $\sigma_{st}$ is the total number of geodesics from node $s$ to node $t$
and $\sigma_{st}(v)$ is the number of those paths that pass-through $v$.
The scope of geodesic-transversal is wider than betweenness centrality and
closeness centrality. The geodesic load geo-load($v$) of a vertex $v$ of a
graph $G$ is defined as the number of maximal geodesics which traverse through
$v$. The concept of geo-load of a network is applied in the geodesics-based
routing algorithms [34, 37]. The concept is also used in load-balanced routing
of fixed interconnection networks [36, 46]. While the betweenness centrality
of a vertex focuses on all possible geodesics, the geodesic load of a vertex
concentrates on only maximal geodesics.
Some interesting combinatorial problems of large-scale network analysis are
propagation (malware propagation [48], immunization [33], disease propagation
[42] and data communication [20]), broadcasting, and gossiping problems [45].
An interesting research problem is to demonstrate how the geodesic-transversal
is a good model to represent these problems in large-scale network analysis.
## 3 Basic observations and NP-completeness
For a starting example consider the Petersen graph $P$. It is of diameter $2$,
therefore to hit all the five maximal geodesics on the outer $5$-cycle we need
at least two vertices. Similarly, we need at least two vertices to hit the
maximal geodesics which are subpaths of the inner $5$-cycle. Hence ${\rm
gt}(P)\geq 4$. On the other hand, in Fig. 1 a geodesic-transversal set with
four vertices is shown, hence we conclude that ${\rm gt}(P)=4$. Using a
similar reasoning we can deduce that if $r,s\geq 1$, then ${\rm
gt}(K_{r,s})=\min\\{r,s\\}$.
Figure 1: A gt-set of the Petersen graph
The following simple lemma will turn out to be quite useful.
###### Lemma 3.1
Let $Q$ be a geodesic of a graph $G$ and $x,y\in V(Q)$. If $u$ is a vertex
from $V(G)\setminus V(Q)$ such that $d(u,x)=d(u,y)$, then $Q$ does not extend
to a geodesic that contains $u$.
Proof. Suppose on the contrary that $Q^{\prime}$ is a geodesic such that $Q$
is contained in $Q^{\prime}$ and $u\in V(Q^{\prime})$. Clearly, on the
geodesic $Q^{\prime}$, the vertex $u$ cannot lie between $x$ and $y$.
Therefore, either $d(u,x)<d(u,y)$ or $d(u,y)<d(u,x)$, and each of the
possibilities in a contradiction with the lemma assumption. $\square$
Clearly, ${\rm gt}(P_{n})=1$ for all $n\in\mathbb{N}$. In particular, ${\rm
gt}(P_{n})=1$ because its only vertex forms a geodesic by itself and hence has
to lie in its unique gt-set. Considering an arbitrary edge $e$ of the complete
graph $K_{n}$, $n\geq 3$, and a vertex not on the edge, Lemma 3.1 implies that
at least one of the endpoint of $e$ must lie in a geodesic-transversal set of
$K_{n}$. Consequently, ${\rm gt}(K_{n})=n-1$ holds for $n\geq 2$. These two
examples generalize as follows, where by a subdivided star we mean the graph
obtained from $K_{1,k}$, $k\geq 1$, by subdividing each of the edges of
$K_{1,k}$ arbitrary number of times (possibly zero). If $k=1$, then the
subdivided stars coincide with the family of paths.
###### Proposition 3.2
If $G$ is a connected graph of order at least $2$, then $1\leq{\rm gt}(G)\leq
n(G)-1$. In addition, the lower bound is attained if and only if $G$ is a
subdivided star, and the upper bound is attained if and only if $G$ is a
complete graph of order at least $2$.
Proof. Since every graph $G$ has at least one maximal geodesic, we infer ${\rm
gt}(G)\geq 1$. Since every maximal geodesic of a non-trivial graph contains at
least two vertices, we infer ${\rm gt}(G)\leq n(G)-1$.
Suppose now that ${\rm gt}(G)=1$ and let $\\{u\\}$ be a gt-set of $G$. Let $T$
be a BFS-tree of $G$ with the root $u$.
We first claim that $G$ is bipartite. Suppose on the contrary that there
exists an edge $xy$ of $G$, where vertices $x$ and $y$ lie in the $k^{\rm th}$
distance level of $T$, for some $k\geq 1$. Then $d_{G}(u,x)=d_{G}(u,y)=k$.
Consider now an arbitrary maximal geodesic $Q$ of $G$ that contains the edge
$xy$. Then Lemma 3.1 implies, that $u$ does not belong to $Q$, a contradiction
with the assumption that $u$ forms a gt-set. Hence the claim.
We next claim that $G$ is a tree. Suppose on the contrary that $G$ contains at
least one cycle $C$. Since we already know that $G$ is bipartite, considering
the cycle $C$ we infer that there exist a vertex $x$ of $C$ which lies in some
$k^{\rm th}$ distance level of $T$ such that $x$ has two neighbors (in $G$),
say $y$ and $z$, in the $(k-1)^{\rm st}$ distance level of $T$. If $Q$ is an
arbitrary maximal geodesic of $G$ that contains as a subpath the path $y-x-z$,
then Lemma 3.1 again implies, that $u$ does not belong to $Q$, a
contradiction. Hence $G$ is a tree.
We finally claim that $G$ is a subdivided star. If this is not the case, then
in $T$ (which is just $G$, rooted in $u$), there exists a vertex $x$ which
lies in $k^{\rm th}$ distance level of $T$, $k\geq 1$, such that $x$ has two
neighbors, say $y$ and $z$, in the $(k+1)^{\rm st}$ distance level of $T$. As
in the previous paragraph we now see that a maximal geodesic of $G$ that
contains as a subpath the path $y-x-z$, yields a contradiction. It follows
that every vertex of $T$, except maybe $u$, is of degree either $2$ or $1$.
The latter is is equivalent to the fact that $T$ is a subdivided star. We
hence conclude that ${\rm gt}(G)=1$ holds if and only if $G$ is a subdivided
star.
Suppose now that $G$ is a an arbitrary graph that is not complete. Then there
exist vertices $x,y\in V(G)$ such that $xy\notin E(G)$. But then
$V(G)\setminus\\{x,y\\}$ form a geodesic-transversal set of $G$ and
consequently, ${\rm gt}(G)\leq n(G)-2$. We can hence conclude that ${\rm
gt}(G)=n(G)-1$ and and only if $G$ is a complete graph of order at least $2$.
$\square$
To conclude the section we are going to show that the geodesic-transversal
problem is NP-complete. In the study of vertex-deletion problems [47], the
concept of a dissociation set (see [4, 22, 40]) was considered, which was
shown in [6] to be the complement of a $3$-path vertex cover in any graph.
Since dissociation set problem is NP-complete even when restricted to
bipartite graphs [47], we infer the following.
###### Theorem 3.3
[6, 47] The $3$-path vertex cover problem is NP-complete for bipartite graphs.
For additional complexity results on the $3$-path vertex cover problem see [3,
23, 38, 44]. It is clear that in bipartite graphs the $3$-path vertex cover
and the $3$-geodesic transversal coincide. Thus, Theorem 3.3 can be restated
as follows:
###### Theorem 3.4
The $3$-geodesic-transversal problem is NP-complete for bipartite graphs.
Now we will prove that the geodesic-transversal problem is NP-complete for
general graphs. In order to prove this, we will provide a polynomial reduction
from the 3-geodesic-transversal problem to the geodesic-transversal problem.
Given a graph $G$, where $V(G)=[n]=\\{1,\ldots,n\\}$, the reduced graph is
denoted by $G^{\prime}$, where $V(G^{\prime})=V\cup\\{x,y,z\\}$ and
$E(G^{\prime})=E\cup\\{xz,zy\\}\cup\\{iz:\,i\in V\\}$. For an example see Fig.
2.
$1$$2$$3$$4$$5$$6$$7$$8$$1$$2$$3$$4$$5$$6$$7$$8$$z$$x$$y$ Figure 2: A graph
(left) and its reduced graph (right)
###### Property 3.5
A set $S$ of vertices is a $3$-geodesic-transversal of $G$ if and only if
$S\cup\\{z\\}$ is a geodesic-transversal of $G^{\prime}$.
Property 3.5 leads to the following conclusion:
###### Theorem 3.6
The geodesic-transversal problem is NP-complete for general graphs.
## 4 The geodesic-transversal problem of trees
In this section, we design an algorithm to locate a gt-set of a tree.
Let $T$ be a tree. A vertex of degree $1$ of a tree is a leaf. A neighbor of a
leaf is a support vertex. A support vertex $u$ is an end support vertex if $u$
is adjacent to at least $\deg(u)-1$ leafs.
###### Lemma 4.1
A tree of order at least $2$ has at least one end support vertex.
Proof. Let $T$ be a tree of order at least $2$ and let $u_{1},\ldots u_{k}$ be
the support vertices of $T$. Let $T^{\prime}$ be a tree obtained from $T$ by
removing all the leaves of $T$. Suppose that $\deg_{T^{\prime}}(u_{i})\geq 2$
for for each $i\in[k]$. Since the degree of every vertex of
$T^{\prime}\setminus\\{u_{1},\ldots,u_{k}\\}$ is the same in $T^{\prime}$ as
in $T$, we would have a tree $T^{\prime}$ whose every vertex is of degree at
least $2$. As this is clearly not possible, there exists a vertex $u_{i}$ such
that $\deg_{T^{\prime}}(u_{i})\leq 1$. This in turn means that $u_{i}$ is an
end support vertex of $T$. $\square$
Let $G$ be a graph, let $v\in V(G)$ be a vertex of degree $2$, and let $x$ and
$y$ be the neighbors of $u$. If $G^{\prime}$ is the graph obtained from $G$ be
removing the vertex $u$ and adding the edge $xy$, then we say that
$G^{\prime}$ is obtained from $G$ by smoothing the vertex $u$. Note that if
the vertices $u$, $x$, and $y$ induce a triangle in $G$, then there are two
parallel edges between $x$ and $y$ in $G^{\prime}$. Let further ${\rm SM}(G)$
denote a graph obtained from $G$ by smoothing all the vertices of $G$ of
degree $2$. Since the smoothing operation preserves the degree of vertices,
${\rm SM}(G)$ is well-defined, that is, unique up to isomorphism. In
particular, no matter in which order a smoothing of vertices of $C_{n}$,
$n\geq 3$, is performed, we end up with ${\rm SM}(C_{n})=C_{2}$. (The
$2$-cycle $C_{2}$ is the graph on the vertices with two parallel edges.) For
another example see Fig. 3.
Figure 3: A tree $T$ (above) and ${\rm SM}(T)$ (below)
###### Lemma 4.2
If $T$ is a tree, then ${\rm gt}(T)={\rm gt}({\rm SM}(T))$.
Proof. Let $S$ be a gt-set of $T$. Suppose that $S$ contains a vertex $u$ with
$\deg(u)=2$. Let $P$ be the maximal path of $T$ that contains $u$ and exactly
two vertices which are not of degree $2$. Such a path is indeed unique. To see
it, let $x$ and $y$ be the neighbors of $u$. If ${\rm deg}(x)=2$, then
continue the path until the first vertex which is not of degree $2$ is found.
Such a vertex exists since $T$ is a tree. Do the same procedure from the
vertex $y$. Now, every maximal geodesic in $T$ that contains $u$, also
contains $x$ and $y$. It follows that $(S\setminus\\{u\\})\cup\\{x\\}$ (or
$(S\setminus\\{u\\})\cup\\{y\\}$ for that matter) is also a gt-set of $T$.
Repeating this construction for every vertex of $S$ of degree $2$ we arrive at
a gt-set $S^{\prime}$ of $T$ which contains no vertex of degree $2$. Since
$S^{\prime}\subseteq V({\rm SM}(T))$ is also a gt-set of ${\rm SM}(T)$, it
follows that ${\rm gt}({\rm SM}(T))\leq{\rm gt}(T)$. On the other hand, if $S$
is a gt-set of ${\rm SM}(T)$, then we infer that $S$ is also a gt-set of $T$,
hence ${\rm gt}(T)\leq{\rm gt}({\rm SM}(T))$ also holds. $\square$
Lemma 4.2 does not hold for an arbitrary graph $G$, even when $SM(G)$ does not
contain parallel edges. See Fig. 4, where a graph $G$ is show for which we
have ${\rm gt}(G)=4$ and ${\rm gt}({\rm SM}(G))=3$.
Figure 4: A graph $G$ (left) with ${\rm gt}(G)=4$, and ${\rm SM}(G)$ (right)
with ${\rm gt}({\rm SM}(G))=3$
###### Lemma 4.3
Let $T$ be a tree with no vertices of degree $2$. Let $u$ be an end support
vertex of $T$ and $u_{1},\ldots,u_{s}$ the leaves adjacent to $u$. Then ${\rm
gt}(T)={\rm gt}(T\setminus\\{u,u_{1},\ldots,u_{s}\\})+1$. Moreover, there
exists a gt-set $S$ of $T$ such that $u\in S$.
Proof. Since $T$ has no vertices of degree $2$, the end support vertex $u$ is
adjacent to at least two leaves, that is, $s\geq 2$. If $T$ is a star, and
hence $u$ being the center of it, then the assertion of the lemma is clear. In
the rest of the proof we may thus assume that $u$ has at least one non-leaf
neighbor, and since $u$ is an end support vertex, it has only one non-leaf
neighbor. We denote the latter vertex by $w$, and let $T^{\prime}$ be the
component of $T-u$ that contains the vertex $w$.
Let $S$ be a gt-set of $T$. Since $s\geq 2$, we see that
$|S\cap\\{u,u_{1},\ldots,u_{s}\\}|\geq 1$, for otherwise the geodesic
$u_{1},u,u_{2}$ would not be hit. Moreover,
$|S\cap\\{u,u_{1},\ldots,u_{s}\\}|=1$. If $u_{i}\in S$ for some $i\in[s]$,
then $(S\setminus\\{u_{i}\\})\cup\\{u\\}$ is also a gt-set of $T$. This proves
the last assertion of the lemma and we may without loss of generality assume
in the rest that $u\in S$.
We claim now that $S\cap V(T^{\prime})$ is a gt-set of $T^{\prime}$. Indeed,
since $\deg_{T^{\prime}}(w)\geq 2$, no maximal geodesic of $T^{\prime}$ can be
hit by $u$. That is, only the vertices from $T^{\prime}$ can be used to hit
the maximal geodesics of $T^{\prime}$, hence the claim. It follows that ${\rm
gt}(T)=1+{\rm gt}(T^{\prime})=1+{\rm
gt}(T\setminus\\{u,u_{1},\ldots,u_{s}\\})$ and we are done. $\square$
Here, an algorithm is designed to construct a gt-set $S$ of an arbitrary tree
$T$.
Input: A tree $T$.
Output: A gt-set $S$ of $T$.
1 $S=\emptyset$;
2 $T=SM(T)$ (i.e., perform the smoothing operation on each vertex of degree
$2$ in $T$).
3 while _$|V(T)| >0$_ do
4 identify an arbitrary end support vertex $p$ of ${\rm SM}(T)$;
5 $S=S\cup\\{p\\}$;
6 $T=T\setminus\\{p,p_{1}\ldots,p_{t}\\}$, where $p_{1},\ldots,p_{t}$ are leaf
neighbors of $p$;
7 $T={\rm SM}(T)$.
8
Algorithm 1 A gt-set of a tree
###### Theorem 4.4
Given a tree $T$, Algorithm 1 determines a gt-set of $T$ in linear time.
The proof of correctness of Algorithm 1 follows from Lemmas 4.1, 4.2, and 4.3.
The time complexity of the algorithm is clearly linear.
To see that the smoothing operation performed in Line 2 and Line 7 of
Algorithm 1 is necessary, consider the tree $T$ in Fig. 5. Note first that
${\rm SM}(T)=4$. Assuming that Line 2 and Line 7 would be removed from the
algorithm, the modified algorithm would return a wrong value $5$. On the other
hand, Algorithm 1 first produces ${\rm SM}(T)$. Then, after two while loops
(after selecting two end support vertices), another smoothing operation at
Line 7 is needed. This in turn guarantees that the algorithm will end after
two additional selections of end support vertices, and hence will return the
correct value $4$.
Figure 5: Tree $T$
## 5 Fast algorithm on spread cactus graphs
A connected graph in which each edge belongs to at most one cycle is a a
cactus graph. We further restrict our attention to the subclass of cactus
graphs in which every vertex belongs to at most one cycle, and call them
spread cactus graphs. They are exactly the graphs that have neither a diamond
nor a butterfly as a topological minor [31]. Every block in these graphs is
either $K_{2}$ or a cycle, and cycle blocks do not intersect other cycle
blocks. The blocks in a spread cactus have a tree structure, and they contain
leaves or leaf-cycles, where the latter are defined as the cycle blocks, which
intersect only one $K_{2}$-block.
As usual, let $C_{n}$ denote an $n$-cycle. Let $C$ be an $n$-cycle with
vertices $\\{v_{1},\ldots,v_{n}\\}$, and let $I\subseteq[n]$ be a set of
indices of vertices in $V(C)$. By $C_{n}(I)$ we denote the graph obtained from
$C$ by attaching a leaf $v_{i}^{\prime}$ to the vertex $v_{i}\in V(C)$ for
every $i\in I$. If $I=\\{i_{1},\ldots,i_{k}\\}$, then we will simplify the
notation $C_{n}(\\{i_{1},\ldots,i_{k}\\})$ to $C_{n}(i_{1},\ldots,i_{k})$. For
instance, $C_{3}(1,2,3)$ denotes the net graph, $C_{3}(1,2)$ is known as the
bull graph, $C_{3}(1)$ is the paw graph, while $C_{4}(1)$ is the $P$-graph;
see Fig. 6 for the former three graphs.
$C_{3}(1,2,3)$$C_{3}(1,2)$$C_{3}(1)$ Figure 6: Net, bull, and paw
We start our discussion by constructing an algorithm that finds a minimum
geodesic transversal in the graphs $C_{n}(I)$ for all $n\geq 3$ and any index
set $I\subseteq[n]$. Note that $C_{n}(I)$ are spread cactus graphs with only
one cycle and no two $K_{2}$-blocks intersect.
Consider $C_{n}(I)$, where $I=\\{i_{1},\ldots,i_{k}\\}$ and
$i_{1}<i_{2}<\cdots<i_{k}$. In the following, these indices will be taken
modulo $k$. If $j\in[k]$, then we set $P^{j}$ to be a
$v_{i_{j}},v_{i_{j+1}}$-path along $C_{n}(I)$, that is, the path on vertices
$v_{i_{j}},v_{i_{j}+1},\ldots,v_{i_{j+1}}$. If $j=k$, this thus means that
$P^{k}$ is the path on vertices
$v_{i_{k}},v_{i_{k}+1},\ldots,v_{1},\ldots,v_{i_{1}}$.
We claim that there exists a gt-set $S$ of $C_{n}(I)$ such that each path
$P^{j}$, $j\in[k]$, contains a vertex in $S$. Indeed, if
$i_{j+1}-i_{j}\leq\Big{\lfloor}\frac{n}{2}\Big{\rfloor},$
then $P^{j}$ lies on the maximal geodesic between $v_{j}^{\prime}$ and
$v_{j+1}^{\prime}$. Now, if a gt-set $S$ contains $v_{j}^{\prime}$ (resp.,
$v_{j+1}^{\prime}$), then $S^{\prime}=(S-\\{v_{j}^{\prime}\\})\cup\\{v_{j}\\}$
(resp., $S^{\prime}=(S-\\{v_{j+1}^{\prime}\\})\cup\\{v_{j+1}\\}$) is clearly a
gt-set of $C_{n}(I)$. On the other hand, if
$i_{j+1}-i_{j}>\Big{\lfloor}\frac{n}{2}\Big{\rfloor},$
then either $P^{j}$ contains a maximal geodesic between two vertices in $C$,
or there is a maximal geodesic between $v_{j}^{\prime}$ and $v_{j+1}$. Hence
we may assume that $P^{j}$ contains a vertex in $S$.
To state the next lemma, we introduce the following concept. In the graph
$C_{n}(I)$, where $I=\\{i_{1},\ldots,i_{k}\\}$, we say that $j\in[k]$ is
lonely, if $i_{j+1}-i_{j-1}>\lfloor\frac{n}{2}\rfloor+1$.
###### Lemma 5.1
If $n\geq 3$ and $I=\\{i_{1},\ldots,i_{k}\\}$, where $0\leq k\leq n$, then
${\rm gt}(C_{n}(I))=\left\\{\begin{array}[]{ll}\vspace*{1mm}2;&k\leq 3,\\\
\vspace*{1mm}\frac{k+1}{2};&k\geq 5\textrm{ odd,}\\\
\vspace*{1mm}\frac{k}{2}+1;&k\geq 4\textrm{ even, and there exist lonely
}j_{1},j_{2}\in[k],j_{1}\textrm{ odd},j_{2}\textrm{ even},\\\
\frac{k}{2};&\textrm{otherwise.}\end{array}\right.$
Proof. Set $G=C_{n}(I)$ and use the notation for vertices of $G$ as
established before the lemma. Let $S$ be a gt-set of $G$. Then, as noted
above, we may assume that $S\cap V(G)\subseteq C$.
We start with the case $k=|I|=0$, that is, $G=C_{n}$. In this case,
$S=\\{v_{1},v_{i}\\}$, where $i=\lfloor\frac{n}{2}\rfloor$, is clearly a gt-
set of $G$, yielding ${\rm gt}(G)=2$. When $k\in\\{1,2\\}$, and assuming
without loss of generality that $1\in I$, again the set $S=\\{v_{1},v_{i}\\}$,
where $i=\lfloor\frac{n}{2}\rfloor$, is a gt-set of $G$. Next, let $k=3$, and
assume without loss of generality that $1\in I$. If the set
$S=\\{v_{1},v_{i}\\}$, where $i=\lfloor\frac{n}{2}\rfloor$, is not a gt-set of
$G$, then we may assume that $1<i_{2}<i_{3}<\lfloor\frac{n}{2}\rfloor$ (the
case when $\lfloor\frac{n}{2}\rfloor<i_{2}<i_{3}$ can be dealt with in a
similar way). However, then $S=\\{v_{2},v_{2}+\lfloor\frac{n}{2}\rfloor\\}$ is
a gt-set of $G$, yielding ${\rm gt}(G)=2$. The first line of the equality of
the lemma is thus established. We next consider $k\geq 4$ and distinguish two
cases.
Let $k$ be odd, $k\geq 5$. Assume that for every even $j\in[k]$, we have
$i_{j+1}-i_{j-1}\leq\lfloor\frac{n}{2}\rfloor+1$. Then the set
$S=\\{v_{i_{j}}:\,i_{j}\in I\textrm{ and }j\textrm{ odd}\\}$ is a gt-set of
$G$ with $|S|=\frac{k+1}{2}$. Indeed, since a maximal geodesic in $C_{n}$ is
of length $\lfloor\frac{n}{2}\rfloor$, every maximal geodesic in $C_{n}(I)$
has at least one leaf as an endvertex, from which we derive that it contains a
vertex $v_{i_{j}}$, where $j$ is odd. In the second case we may assume without
loss of generality that $i_{3}-i_{1}>\lfloor\frac{n}{2}\rfloor+1$. Then
$S=\\{v_{i}:\,i=i_{3}-\lfloor\frac{n}{2}\rfloor-1\textrm{ or }i>1\textrm{
odd}\\}$ is a gt-set of $G$ with $|S|=\frac{k+1}{2}$.
Finally, let $k$ be even, $k\geq 4$. Suppose first that for every even
$j\in[k]$ we have $i_{j+1}-i_{j-1}\leq\lfloor\frac{n}{2}\rfloor+1$. Then, we
derive in the same way as in the case of odd $k$ that the set
$S=\\{v_{i_{j}}:\,i_{j}\in I\textrm{ and }j\textrm{ odd}\\}$ is a gt-set of
$G$ with $|S|=\frac{k}{2}$. In a similar way we conclude that ${\rm
gt}(G)=\frac{k}{2}$ if for every odd $j\in[k]$ we have
$i_{j+1}-i_{j-1}\leq\lfloor\frac{n}{2}\rfloor+1$. In the second case there
exist a lonely odd $j_{1}\in[k]$ and a lonely even $j_{2}\in[k]$. Then the
path $P^{t}$ between $v_{i_{j_{t}-1}}$ and $v_{i_{j_{t}+1}}$ is of length at
least $\lfloor\frac{n}{2}\rfloor+2$, which implies that this path contains a
maximal geodesic of length $\lfloor\frac{n}{2}\rfloor$, which does not involve
$v_{i_{j_{t}+1}}$ nor $v_{i_{j_{t}-1}}$. Since a gt-set must hit both paths
$P^{t}$, we infer that ${\rm gt}(G)>\frac{k}{2}$. It is easy to see that ${\rm
gt}(G)\leq\frac{k}{2}+1$ by using a similar construction as in the case when
$k$ is odd. $\square$
From the proof it is also clear that a gt-set of a graph $C_{n}(I)$ can be
efficiently computed. If the set $I$ is a part of the input, the computation
can be done in time linear in the size of $I$.
Next, we determine a minimum geodesic transversal set $S$ in a graph
$C_{n}(I)$ in which some of the vertices are declared in advance to be in $S$.
This situation appears naturally in the construction of an algorithm for
determining a gt-set of a unicyclic graph presented later.
Let $A\subseteq[n]$ be the set of indices of the vertices of the cycle of
$C_{n}(I)$ such that every $v_{i}$, $i\in A$, is predetermined to be in a
geodesic transversal set $S$ of $C_{n}(I)$. Denote by $C_{n}(I,A)$ the graph
$C_{n}(I)$ together with the requirement that vertices indexed by elements
from $A$ must lie in a geodesic transversal set. The algorithm for
constructing a minimum geodesic transversal of $C_{n}(I,A)$ is based on the
constructions from the proof of Lemma 5.1. In Algorithm 2, the notation of
vertices $v_{i}\in V(C_{n})$ is simplified to $i$. The indices from
$A=\\{a_{1},\ldots,a_{t}\\}$ are ordered cyclically as follows:
$a_{1}<a_{2}<\cdots<a_{t}<a_{t+1}=a_{1},$
by which the main while loop is performed at least once (and is performed
exactly once when $A=\\{a_{1}\\}$). The correctness of Algorithm 2 can be
proved by using similar arguments as in the proof of Lemma 5.1.
Input: Cycle on $V(C_{n})=\\{1,\ldots,n\\}$, a leaf attached to $i$, where
$i\in I$, and $A\subseteq[n]$.
Output: Minimum geodesic transversal $S$ of $C_{n}(I)$ containing $A$.
1 $S=A$;
2 Order $A:a_{1}<a_{2}<\cdots<a_{t}<a_{t+1}=a_{1}$;
3 $i=1$;
4 while _$i\leq t$_ do
5 let $I_{i}=\\{j\in I:\,a_{i}<j<a_{i+1}\\}=\\{j_{1},\ldots,j_{k}\\}$ and
$j_{0}=a_{i},j_{k+1}=a_{i+1}$;
6 if _$k$ odd_ then
7 if
_$\forall\ell\in[\frac{k+1}{2}]:\,j_{2\ell}-j_{2(\ell-1)}\leq\lfloor\frac{n}{2}\rfloor+1$
_ then
8 $S=S\cup\\{j_{2\ell}:\,\ell\in[\frac{k-1}{2}]\\}$;
9 else
10 let $m\in[\frac{k+1}{2}]$, where
$j_{2m}-j_{2m-2}>\lfloor\frac{n}{2}\rfloor+1$;
11
$S=S\cup\\{j_{2\ell}:\,\ell\in[\frac{k-1}{2}]\\}\bigcup\\{j_{2m-2}+\lfloor\frac{n}{2}\rfloor+1\\}$;
12
13 else
14 let $\ell=0$;
15 while _$\ell\leq k$_ do
16 if _$j_{\ell+2}-j_{\ell}\leq\lfloor\frac{n}{2}\rfloor+1$_ then
17 $S=S\bigcup\\{j_{\ell+2}\\};\ell=\ell+2$;
18 else
19 $S=S\bigcup\\{j_{\ell}+\lfloor\frac{n}{2}\rfloor+1\\}$;
20 if _$j_{\ell+1}-j_{\ell}\leq\lfloor\frac{n}{2}\rfloor+1$_ then
21 $S=S\bigcup\\{j_{\ell+3}\\};\ell=\ell+3$;
22 else
23 $S=S\bigcup\\{j_{\ell+2}\\};\ell=\ell+2$;
24
25
26
27 $i=i+1$;
Algorithm 2 A minimum geodesic transversal of $C_{n}(I,A)$
We continue by presenting an algorithm for determining a gt-set of a unicyclic
graph. (This part is written mostly for intuition purposes. Algorithm 3 deals
also with the special case when $G$ is unicyclic.) Let $G$ be a unicyclic
graph, and $C$ the cycle in $G$ of length $n$. If $G$ is isomorphic to
$C_{n}$, then ${\rm gt}(G)=2$. Otherwise, let $G^{\prime}=G-E(C)$, let
$T_{1},\ldots,T_{r}$ be the nontrivial components of $G^{\prime}$, and let
$v_{1},\ldots,v_{r}$ be the vertices of $C$, where $v_{i}$ belongs to $T_{i}$
for all $i\in[r]$. Clearly, each $T_{i}$ is a tree on at least two vertices.
If $T_{i}$ is a path, then by the smoothing operation, and the fact that ${\rm
gt}({\rm SM}(T_{i}))={\rm gt}(T_{i})$, we may assume that $T_{i}$ is
isomorphic to $P_{2}$, that is, $v_{i}$ has a leaf attached. In this case we
set $S_{i}=\emptyset$. Otherwise, $T_{i}$ has vertices of degree at least $3$,
and we perform the algorithm for obtaining a gt-set $S_{i}$ of a tree $T_{i}$.
It is easy to see that the sets $S_{i}$, $i\in[r]$, are subsets of a gt-set of
$G$. There are three possibilities:
1. (i)
$v_{i}\in S_{i}$;
2. (ii)
$v_{i}\notin S_{i}$, but all neighbors of $v_{i}$ in $T_{i}$ are in $S_{i}$;
3. (iii)
$v_{i}\notin S_{i}$, and there is a neighbor of $v_{i}$ in $T_{i}$ that is not
in $S_{i}$.
Turning back our attention to $G$, after gt-sets of trees $T_{i}$ are
obtained, the above possibilities yield different cases by which we complete
the construction of a gt-set of $G$. Note that all maximal geodesics within
trees $T_{i}$ are hit by the sets $S_{i}$, hence it remains to consider the
maximal geodesics that pass some vertices of $C$. The problem can be
translated to determination of a minimum geodesic transversal of $C_{n}(I,A)$.
In particular, all vertices $v_{i}$ that are in $S_{i}$ (possibility (i)) are
considered to be in the set $A$, all vertices $v_{i}$ that are not in $S_{i}$
and have a neighbor in $T_{i}$ that is not in $S_{i}$ (possibility (iii)) are
considered to be in $I$. Finally, the vertices $v_{i}\notin S_{i}$ for which
possibility (ii) appears are in neither of the sets $A$ and $I$ (the same
holds for the vertices of $C$ that are isolated in $G^{\prime}$). Perform
Algorithm 2 on $C_{n}(I,A)$, and let $S$ be the output of the algorithm.
Finally, $S^{\prime}=S\cup\bigcup_{i=1}^{r}{S_{i}}$ is a gt-set of $G$.
We follow with two auxiliary results that will be a key for the algorithm for
determining a gt-set of a spread cactus graph. We need some more notation. A
vertex $v$ in a graph $G$ is heavy if $\deg_{G}(v)\geq 3$. Next, a heavy
vertex $v$ is a boundary heavy vertex if at most one component of $G-v$ is not
a path. If $v$ is a heavy vertex, then let $P^{v}$ denote the subset of $V(G)$
containing $v$ and every vertex of degree at most $2$ that can be reached from
$v$ on a path that does not contain heavy vertices.
###### Lemma 5.2
If $G$ is a graph and $v$ a boundary heavy vertex in $G$ such that $G-v$ has
more than two components, then ${\rm gt}(G)=1+{\rm gt}(G-P^{v})$.
Proof. Since $P_{t}$ contains two leaves, there is a maximal geodesic that
lies in $P_{v}$. Hence ${\rm gt}(G)\geq 1+{\rm gt}(G-P^{v})$. Since every
maximal geodesic in $G$ that contains a vertex in $P^{v}$ contains also $v$,
we infer ${\rm gt}(G)=1+{\rm gt}(G-P^{v})$. $\square$
Consider now a graph $G$ in which some of the vertices are declared to be in a
geodesic transversal, and denote by $A_{G}$ the set of such vertices in $G$.
(This situation appears naturally within an algorithm for determining a gt-set
of $G$, where in the process of building a gt-set some of the vertices are
already put in the set.) Let $C:v_{1},\ldots,v_{n},v_{1}$ be a cycle in $G$,
let $A=A_{G}\cap V(C)$, and let $I$ be the set of vertices $v_{i}$, $i\in[n]$,
which are adjacent to a leaf. We say that $C$ is a boundary cycle in $G$ if
there exists at most one vertex $v_{j}\in V(C)$, where $v_{j}\notin I\cup A$,
such that $v_{i}$ has a neighbor outside $C$.
###### Lemma 5.3
Let $G$ be a graph, $C$ a boundary cycle in $G$, $I$ support vertices of $C$,
$A$ the set of vertices in $C$ that belong to $A_{G}$, and $x\in V(C)$ be
adjacent to a non-leaf vertex outside $C$. Let $S_{C}$ be a minimum geodesic
transversal of $C_{n}(I,A)$ and $S_{C}^{\prime}$ a minimum geodesic
transversal of $C_{n}(I\cup\\{x\\},A)$. If $|S_{C}^{\prime}|=|S_{C}|$, then
$S_{C}^{\prime}$ belongs to a minimum geodesic transversal of $G$ that
contains $A_{G}$. Otherwise, $|S_{C}^{\prime}|=|S_{C}|+1$, and $S_{C}$ belongs
to a minimum geodesic transversal of $G$ that contains $A_{G}$.
Proof. Clearly, $|S_{C}|\leq|S_{C}^{\prime}|\leq|S_{C}|+1$. A (minimum)
geodesic transversal of $G$ must hit all maximal geodesics between two
vertices in $C_{n}(I)$. This implies that at least $|S_{C}|$ vertices from $C$
need to be in a minimum geodesic transversal of $G$ that contains $A_{G}$. If
$|S_{C}^{\prime}|=|S_{C}|$, then $S_{C}^{\prime}$ is a better choice than
$S_{C}$, since it hits not only all the maximal geodesics that lie between two
vertices in $C_{n}(I)$, but also all maximal geodesics that have one endvertex
in $C_{n}(I)$. Otherwise, when $|S_{C}^{\prime}|=|S_{C}|+1$, $S_{C}$ belongs
to a minimum geodesic transversal of $G$ that contains $A_{G}$. $\square$
A gt-set of a path clearly consist of a single vertex, hence we may
concentrate on spread cactus graphs that are not paths. Note that for such
graphs there exists a boundary heavy vertex or a boundary cycle. Hence, using
Lemmas 5.2 and 5.3, we propose Algorithm 3 for determining a gt-set of a
spread cactus graph.
Input: A spread cactus graph $G$, which is not a path.
Output: Minimum geodesic transversal $S$ of $G$.
1 $S=\emptyset$;
2 while _there is a heavy vertex in $G$_ do
3 if _there is a boundary heavy vertex $v$ that lies on no cycle_ then
4 $S=S\cup\\{v\\}$; $G=G-P^{v}$;
5 else if _there is a boundary cycle $C=C_{n}(I,A)$, where $A=V(C)\cap S$,_
then
6 if _$x$ a vertex in $C$ with a non-leaf neighbor_ then
7 let $S_{C}$ a minimum geodesic transversal of $C_{n}(I,A)$ and
$S_{C}^{\prime}$ a minimum geodesic transversal of $C_{n}(I\cup\\{x\\},A)$;
8 if _$|S_{C}|=|S^{\prime}_{C}|$_ then
9 $S=S\cup S^{\prime}_{C}$; remove from $G$ all vertices of $C_{n}(I)$ and all
vertices of degree $2$ reachable by a path from $x$;
10 else
11 $S=S\cup S_{C}$; $G=G-\Bigl{(}V(C_{n}(I))\setminus\\{x\\}\Bigr{)}$;
12
13 else
14 $G=C_{n}(I,A)$, where $A=V(G)\cap S$, and let $S^{\prime}$ be a minimum
geodesic transversal of $G$ containing $A$; $S=S\cup S^{\prime}$;
15
16 else
17 let $v$ be a boundary heavy vertex lying on a cycle;
18 if _$\deg(v)=3$_ then
19 smooth out the path $P_{v}$ so that $v$ is adjacent to a leaf
20 else
21 $S=S\cup\\{v\\}$;
22 $G=G-P^{v}$.
23
24
Algorithm 3 A minimum geodesic transversal of a spread cactus graph $G$.
###### Theorem 5.4
Given a spread cactus graph $G$, which is not a path, Algorithm 3 determines a
gt-set of $G$ in linear time.
Proof. By the above observations, if $G$ is a non-path spread cactus graph,
then $G$ contains a heavy vertex $v$. Now, there are three possibilities: $v$
is a boundary heavy vertex that does not lie on a cycle (Line 3), $v$ lies on
a cycle and its degree is at least $4$ (Line 18), or $v$ lies on a cycle and
its degree is $3$. (By Line 17, $v$ can be made adjacent to a leaf.) If the
latter holds for all heavy vertices of a cycle with at most one exception,
then we have a boundary cycle (Line 5). The correctness of the first case and
the second case (Line 3 and 18, resp.) follows from Lemma 5.2, the correctness
of the second case (Lines 5-13) follows from Lemma 5.3. The case when $v$ is a
boundary heavy vertex with degree $3$ that lies on a cycle (Lines 16-17)
follows similar arguments as in the proof of Lemma 4.2.
An implementation of the algorithm uses a tree-like structure of a spread
cactus graph, which can be obtained by a BFS search. Finding a boundary heavy
vertex can be done by using a reversed order of the BFS, and all cases of the
if-then-else condition can be checked in linear time with respect to the
number of vertices that they involve. In particular, the case when there is a
boundary cycle (lines 5-13) can be realized in linear time by applying
Algorithm 2 twice. $\square$
## 6 Conclusion and future work
A new concept of geodesic-transversal is introduced in this paper. In addition
to NP-completeness, polynomial time algorithms are derived for arbitrary trees
and spread cactus graphs. The potential future research is to investigate the
complexity status of this problem for important interconnection networks such
as butterfly networks and hypercubes, as well as for other classes of graphs
such as bipartite graphs and chordal graphs. As mentioned in the initial part
of the paper, it would be interesting to study how the geodesic-transversal
can be used to model distance-based combinatorial problems in large-scale
network analysis.
## Acknowledgments
This work was supported and funded by Kuwait University, Research Project No.
(QI 01/20).
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|
# Long term X-Ray Observations of Seyfert 1 Galaxy Ark 120: On the origin of
soft-excess
Prantik Nandi1, Arka Chatterjee1, Sandip K. Chakrabarti2, Broja G. Dutta2,3
1Department of Astrophysics & Cosmology, S. N. Bose National Centre for Basic
Science, Salt lake, Sector III, Kolkata 700091, India
2Indian Centre for Space Science, Garia Station Road, Kolkata 700084, India
3Department of Physics, Rishi Bankim Chandra College, Naihati, West Bengal,
743165, India E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
We present the long-term X-ray spectral and temporal analysis of a ‘bare-type
AGN’ Ark 120. We consider the observations from XMM-Newton, Suzaku, Swift, and
NuSTAR from 2003 to 2018. The spectral properties of this source are studied
using various phenomenological and physical models present in the literature.
We report (a) the variations of several physical parameters, such as the
temperature and optical depth of the electron cloud, the size of the Compton
cloud, and accretion rate for the last fifteen years. The spectral variations
are explained from the change in the accretion dynamics; (b) the X-ray time
delay between 0.2-2 keV and 3-10 keV light-curves exhibited zero-delay in
2003, positive delay of $4.71\pm 2.1$ ks in 2013, and negative delay of
$4.15\pm 1.5$ ks in 2014. The delays are explained considering Comptonization,
reflection, and light-crossing time; (c) the long term intrinsic luminosities,
obtained using nthcomp, of the soft-excess and the primary continuum show a
correlation with a Pearson Correlation Co-efficient of $0.922$. This indicates
that the soft-excess and the primary continuum are originated from the same
physical process. From a physical model fitting, we infer that the soft excess
for Ark 120 could be due to a small number of scatterings in the Compton
cloud. Using Monte-Carlo simulations, we show that indeed the spectra
corresponding to fewer scatterings could provide a steeper soft-excess power-
law in the 0.2-3 keV range. Simulated luminosities are found to be in
agreement with the observed values.
###### keywords:
galaxies: active – galaxies: Seyfert – X-rays: galaxies – X-rays: individual:
Ark 120
††pubyear: 2020††pagerange: Long term X-Ray Observations of Seyfert 1 Galaxy
Ark 120: On the origin of soft-excess–References
## 1 Introduction
Active Galactic Nuclei (AGNs) are the most energetic phenomena in the
universe. The emitted radiation is observed over the entire range of the
electromagnetic spectrum. The high energy X-rays are believed to be emitted
from the innermost region of an accretion disc which surrounds the central
black hole (Shakura & Sunyaev, 1973; Pringle et al., 1973). The X-ray spectra
of Seyfert 1 galaxies, a subclass of AGNs, is mostly fitted by a power-law
component with photon index in the range $\Gamma=1.6-2.2$ (Bianchi et al.,
2009; Sobolewska & Papadakis, 2009) and a high energy cut-off. The spectral
contribution which deviates from the power-law at lower energy (below $\sim 2$
keV) is known as ‘soft excess’ (Halpern, 1984; Arnaud et al., 1985; Singh et
al., 1985). The X-ray spectra are often associated with a Fe K$\alpha$ line,
which is observed near 6.4 keV, and a Compton hump in the energy range of 20.0
to 40.0 keV. It has been observed that the primary power-law emission is
produced by the Comptonization of low energy seed photons (Sunyaev &
Titarchuk, 1980; Titarchuk, 1994) emitted from the standard Keplerian disc.
The seed photons are processed from the accretion mechanism, and the peak
emission arises at optical/ultraviolet (UV) wavelengths (Pringle et al., 1973)
for a supermassive black hole (SMBH). However, the location, as well as the
geometry of the Compton reprocessing region, are still a matter of debate.
This Compton cloud can be situated above the accretion disc (Haardt &
Maraschi, 1991, 1993; Poutanen & Svensson, 1996) or at the base of the
relativistic jet (Chakrabarti & Titarchuk, 1995; Fender et al., 1999, 2004;
Markoff et al., 2005). The region could be a hot, radiatively inefficient and
behave like a quasi-Bondi flow as discussed initially by Ichimaru (1977). This
region could originate the thermal Comptonization of soft photons produced in
the optical/UV range from an optically thick Keplerian disc (Magdziarz et al.,
1998; Dewangan et al., 2007; Done et al., 2012; Lohfink et al., 2012) or a
blurred reflection from ionized disc (Fabian et al., 2002; Ross & Fabian,
2005; Crummy et al., 2006; García et al., 2014). The iron line is thought to
be originated by the photoelectric absorption followed by the fluorescence
line emission from a dense and relatively cold accretion disc. Moreover, it is
believed that the Compton hump could be due to the Compton scattering
dominated above 10 keV in a relatively cold dense medium. Nevertheless, the
complex broad-band spectrum of AGNs requires a proper physical explanation of
the flow dynamics and radiative properties around the central engine across
the soft and hard energy regime of the X-ray.
In this scenario, the Two-Component Advective Flow (TCAF) (Chakrabarti &
Titarchuk, 1995) model, which combines the essence of all the salient features
of a viscous transonic flow (Chakrabarti, 1989, 1990, 1995) around black holes
is worth exploring. It is a physical solution encompassing hydrodynamics and
radiative processes. The transonic flow solution allows two types of accretion
flows depending on how efficiently angular momentum is being transported: a
viscous, geometrically thin, optically thick standard Keplerian component
(Shakura & Sunyaev, 1973) and a weakly viscous, geometrically thick, optically
thin sub-Keplerian halo component (Chakrabarti & Titarchuk, 1995). The latter
is basically an inefficiently radiating generalized Bondi flow with high
radial velocity till it forms the centrifugal barrier after which it becomes
efficient in radiating at higher energies. The Keplerian disc is formally
truncated at the centrifugal barrier, the outer boundary of which is the shock
location (Chakrabarti, 1989). The post-shock region (i.e., the region between
the shock and the innermost sonic point) is known as CENtrifugal barrier
supported BOundary Layer or CENBOL and it acts as the Compton cloud. The soft
photons from the Keplerian disc are upscattered by Comptonization process in
the post-shock region and produce the high energy X-ray photons. TCAF, a self-
consistent model, is quantified by four flow parameters: two types of
accretion rates, namely, the disc rate ($\dot{m}_{d}$) and halo rate
($\dot{m}_{h}$), size and density of the Compton cloud, through the shock
location ($X_{s}$) and the compression ratio ($R$), ratio of the post-shock
and the pre-shock flow densities ($\frac{\rho_{+}}{\rho_{-}}$). It also
requires an intrinsic parameter, namely, the mass of the central black hole
(in the units of $M_{\odot}$), and an extrinsic parameter, namely, the
normalization which is required to place the observed spectrum over the
theoretical spectrum of TCAF. The broadband spectra of M87 was explained with
this model by fitting the data from multi-wavelength observations(Mandal &
Chakrabarti, 2008). Later, TCAF has been implemented in the xspec as a local
table model and has been successful to fit the data of the Galactic black
holes (Debnath et al., 2014) and has also been able to estimate the mass of
nearby Seyfert 1 galaxy NGC 4151 using NuSTAR data (Nandi et al., 2019).
Arakelian 120 (Ark 120) is a nearby ($z=0.03271$111The redshift is taken from
the NASA/Infrared Process and Analysis center (IPAC) Extragalactic Database.
https://ned.ipac.caltech.edu) radio-quiet Seyfert 1 AGN with radio-loudness
$R\approx 0.1$ (Condon et al., 1998; Ho, 2002). This source was intensely
monitored nearly in all wavelengths: optical/UV (Kollatschny et al., 1981,
1981; Schulz & Rafanelli, 1981; Alloin et al., 1988; Marziani et al., 1992;
Peterson et al., 1998; Stanic et al., 2000; Popović et al., 2001; Doroshenko
et al., 2008; Kuehn et al., 2008) and X-ray (Vaughan et al., 2004; Nardini et
al., 2016; Reeves et al., 2016; Gliozzi et al., 2017; Lobban et al., 2018) and
was found to be consistently bright in optical, UV, and X-rays displaying
substantial wavelength-dependent variability (Gliozzi et al., 2017; Lobban et
al., 2018). From the simultaneous UV/X-ray measurements, it was reported that
the observations are neither ‘contaminated’ by absorption signatures along the
line of sight (Vaughan et al., 2004; Reeves et al., 2016; Crenshaw et al.,
1999) nor by neutral intrinsic absorbers (Reeves et al., 2016) around the
central engine. Furthermore, Ark 120 is nearly free from intrinsic reddening
in the IR-optical-UV continuum (Ward et al., 1987; Vasudevan et al., 2009).
Therefore, it provides one of the cleanest views ($N_{H}\sim 3\times 10^{19}$
cm-2; (Vaughan et al., 2004)) of the central region. This type of AGNs are
called “bare nucleus” Seyferts or bare AGNs. The estimated mass of the central
black hole of Ark 120 is M${}_{BH}=1.50\pm 0.19\times 10^{8}$ M⊙ (Peterson et
al., 2004) which was measured using the reverberation-mapping technique. From
the spectroscopic monitoring data of Ark 120 during 1976 to 2013 using a 70 cm
telescope, Denissyuk et al. (2015) estimated the mass of the central SMBH to
be M${}_{BH}=1.675\pm 0.028\times 10^{8}$ M⊙. This source has a low Eddington
ratio of $L_{bol}/L_{E}\sim 0.05$ (Vasudevan & Fabian, 2007) with a strong
soft-excess (Matt et al., 2014; Porquet et al., 2004, 2019) and a significant
broad Fe Kα line (Vaughan et al., 2004; Nardini et al., 2011). Nardini et al.
(2011) analyzed Ark 120 spectra, where, in the absence of absorber of complex
morphology, soft-excess was explained by reflection from the centrally located
hot and cold medium located at a distance. Marinucci et al. (2019) used the
Monte-Carlo technique to investigate the favourable shape of the Compton cloud
considering the future polarimetric missions such as IXPE (Weisskopf et al.,
2016).
Although Ark 120 is a widely studied source, the evolution of the X-ray
spectra over the last two decades is yet to be understood. However, a
steepening of the X-ray spectrum was observed during six-month monitoring in
2014 with Swift. The observed spectral variability was attributed to the
possible existence of a large disc reprocessing region (Gliozzi et al., 2017).
Again during 2017-18, a longer time delay was observed (Lobban et al., 2018)
between longer wavelength difference (i.e., optical and X-ray). They predicted
that the accretion disc could exist in a longer scale as predicted by standard
accretion disc theory. The soft-excess part of Ark 120 could be originated due
to the Comptonization within the hot electron cloud of various shape
(Marinucci et al., 2019), reflection from a cold medium (Nardini et al., 2011)
or the shock heating near the inner edge of the disc (Fukumura et al., 2016).
We analyzed the long term X-ray archival data of Ark 120 which provides an
ideal testbed to understand the soft-excess as well as its interaction with
the harder (>2 keV) photons. Along with the observations, we perform Monte-
Carlo simulations to find the effect of Comptonizaton within the energy range
of soft-excess. We also study the X-ray variability of the source over a
longer period and to calculate the approximate time-delays in X-ray bands. For
the first time, we also find the flow and system parameters by fitting of the
X-ray data. The paper is structured in the following way: in Sec 2, we provide
the details of the observational data and their reduction procedure. The
results of the spectral and temporal analysis are presented in Sec 3 and 4. We
discuss our findings in Sec 5 and finally, draw our conclusions in Sec 6.
## 2 Observation and Data Reduction
We use the publicly available archival data of XMM-Newton, NuSTAR, Chandra,
and Suzaku using HEASARC222http://heasarc.gsfc.nasa.gov/. We reprocessed all
data using HEAsoft v6.26.1 (Arnaud, 1996), which includes XSPEC v12.10.1f.
Table 1: Observation Log ID | Date | Obs. ID | Instrument | Exposures
---|---|---|---|---
| (yyyy-mm-dd) | | | (ks)
XMM1 | 2003-08-24 | 0147190101 | XMM-Newton/EPIC-pn | 112.15
S1 | 2007-04-01 | 702014010 | Suzaku/XIS-HXD | 100.86
XRT1 | 2008-07-24 | 00037593001 | Swift/XRT | 10.86
| -2008-08-03 | -00037593003 | |
XMM2 | 2013-02-18 | 0693781501 | XMM-Newton/EPIC-pn | 130.46
N1 | 2013-02-18 | 60001044004 | NuSTAR/FPMA | 65.46
XMM3 | 2014-03-22 | 0721600401 | XMM-Newton/EPIC-pn | 124.0
N2 | 2014-03-22 | 60001044002 | NuSTAR/FPMA | 55.33
XRT2 | 2014-09-04 | 00091909002 | Swift/XRT | 22.81
| -2014-10-19 | -00091909022 | |
XRT3 | 2014-10-22 | 00091909023 | Swift/XRT | 20.18
| -2014-12-05 | -00091909044 | |
XRT4 | 2014-12-09 | 00091909045 | Swift/XRT | 23.48
| -2015-01-26 | -00091909068 | |
XRT5 | 2015-01-26 | 00091909069 | Swift/XRT | 21.66
| -2015-03-15 | -00091909090 | |
XRT6 | 2017-12-07 | 00010379001 | Swift/XRT | 44.14
| -2018-01-24 | -00010379048 | |
### 2.1 XMM-Newton
Ark 120 has been observed by XMM-Newton (Jansen et al., 2001) during three
epochs from 2003 to 2014. In 2003 and 2013, it has made $\sim$ 112 ks (XMM1)
and $\sim$ 130 ks (XMM2) observations respectively. The XMM1 data is used by
(Vaughan et al., 2004) and reported that the source Ark 120 is one of the
cleanest Sy1 type AGN. In 2014, XMM-Newton observed Ark 120 four times between
March 18 and March 24. Out of these, one (XMM3) was simultaneous with NuSTAR
observation. The details of the observation log are presented in Table 1. It
was observed that the X-ray flux of this source was about a factor of two
higher in 2014 than the XMM2 observation (Matt et al., 2014; Marinucci et al.,
2019) made in 2013. A similar trend of flux variation was also reported in
optical/UV (Lobban et al., 2018) band.
Due to the high brightness of the source, the European Photon Imaging Camera
(EPIC-pn (Strüder et al., 2001)) operated in Small Window (SW) mode to prevent
any pile-up. The details of the XMM-Newton/EPIC-pn observations of this source
are listed in Table-1. We reprocessed the raw data to level 1 data for EPIC-pn
by Scientific Analysis System (SAS
v16.1.0333https://www.cosmos.esa.int/web/xmm-newton/sas-threads) with
calibration files dated February 2, 2018. We have used only the unflagged
(FLAG == 0) events for excluding the edge of CCD and the edge of the bad
pixel. Besides this, we also use PATTERN $\leq 4$ for single and double pixel.
We exclude the photon flares by proper GTI files to acquire the maximum signal
to noise ratio. After that, we use an annular area of 30″ outer radii and 5″
inner radii centered at the source to extract the source event. For the
background, we use a circle of 60″ in the lower part of the window that
contains no (or negligible) source photons. The response files (arf and rmf
files) for each EPIC-pn spectral data set were produced with SAS tasks ARFGEN
and RMFGEN, respectively. The GRPPHA task is used with 100 counts per bin for
0.3 - 10.0 keV EPIC-pn spectra.
### 2.2 Suzaku
Suzaku observed Ark 120 on 2007 April 1 (Obs ID: 702014010) in HXD normal
position with exposure of $\sim$ 101 ks using X-ray imaging spectrometer
(Koyama et al., 2007) and $\sim$ 89 ks for Hard X-ray Detector (Takahashi et
al., 2007). The photons were collected in both $3\times 3$ and $5\times 5$
editing modes. From this observation, a presence of soft-excess emission in
soft X-ray was reported by (Nardini et al., 2011). Also, Fe K$\alpha$ emission
line with full-width at half maximum of $4700^{+2700}_{-1500}$ km s-1 was
previously reported by (Nardini et al., 2016) by using Suzaku observation
along with XMM-Newton, Chandrai, and NuSTAR.
We use the standard data reduction technique for Suzaku data analysis
illustrated in Suzaku Data Reduction
Guide444http://heasarc.gsfc.nasa.gov/docs/suzaku/analysis/abc/ and followed
the recommended screening criteria while extracting Suzaku/XIS spectrum and
light-curves. The latest calibration
files555http://www.astro.isas.jaxa.jp/suzaku/caldb/ available (2014-02-03)
using FTOOLS 6.25 is used to reprocess the event files. The source spectra and
lightcurves are extracted from a circular region of radius 200″centered on the
Ark 120 and the background region is selected on the same slit with a circular
region 250″. Finally, we merge the two front illuminated detectors (XIS0 and
XIS3) to produce the final spectra and lightcurves for Ark 120. We generated
the response files through XISRESP script.
As Suzaku has a high energy X-ray detector (HXD), we use the HXD/PIN data for
our analysis. We reprocessed the unfiltered event files using the standard
tools. The output spectrum and lightcurves are extracted by using the
hxdpinxbpi and hxdpinxblc, respectively. Further, we correct the spectrum to
take into account both the non-X-ray and the cosmic X-ray backgrounds and the
dead time correction.
### 2.3 NuSTAR
NuSTAR (Harrison et al., 2013) observed Ark 120 simultaneously with XMM-Newton
with FPMA and FPMB on 2013 February 18 (N1) and 2014 March 22 (N2) for the
exposure of $\sim$166 ks and $\sim$ 131 ks respectively. The details of the
observation log are given in Table 1. We consider both N1 and N2 observations
for our analysis. (Porquet et al., 2018, 2019) used this data along with XMM-
Newton and determined the spin $0.83^{+0.05}_{-0.03}$ and comment on the
dimension of the corona and temperature by analyzing these X-ray data.
The level 1 data is produced from the raw data by using the NuSTAR data
analysis software (NuSTARDAS v1.8.0). The cleaned event files are produced
with standard NUPIPELINE task and calibrated with the latest calibration files
available in the NuSTAR calibration database
(CALDB)666http://heasarc.gsfc.nasa.gov/FTP/caldb/data/nustar/fpm/. We chose
90″ radii for source and 180″ radii for the background region on the same
detector to avoid contamination and detector edges. For the final background-
subtracted lightcurves, we use 100s bin for both FPMA and FPMB. As both
detectors are identical, here we present the results of FPMA only. The
response files (arf and rmf files) are generated by using the numkrmf and
numkarf modules, respectively.
### 2.4 Swift data
Swift X-ray telescope (XRT; Burrows et al. (2005)), working in the energy
range of 0.2 to 10.0 keV, is an X-ray focusing telescope. XRT observed this
source in both WT (windowed timing) and pc (photon count) modes depending on
the brightness of the source. Ark 120 was observed over $\sim 130$ times from
2008-07-24 to 2018-01-24. In 2008, Swift observed three times, July 24, July
31 and August 3. We stack the spectra to produce a combined spectrum (XRT1).
Then, it again observed on 2014-03-22, which has a simultaneous observation
with XMM and NuSTAR. We consider the XMM3 observation over this particular XRT
observation. Swift observed Ark 120 from 2014-09-06 to 2015-03-15 on a nearly
daily basis. Further, we stack these observations into four observations
(XRT2, XRT3, XRT4, XRT5) with each observations spanning around 50 days. In
the last epoch, Swift observed Ark 120 from 2017-12-05 to 24-01-2018 over
$\sim$ 50 days. We stack the observations to produce the spectra of XRT6. The
details of the observation log are stated in Table 1. We use the online tool
“XRT product builder”777http://swift.ac.uk/user_objects/ Evans et al. (2009)
to extract the spectrum and light curves. This product builder performs all
necessary processing and calibration and produces the final spectra and
lightcurves of Ark 120 in WT and PC mode.
## 3 Spectral Analysis
We use XMM-Newton, Suzaku, NuSTAR, and Swift data for the spectral analysis
and explore the spectral variation over $\sim$15 years (2003-2018) period
using XSPEC v12.10.1f (Arnaud, 1996). We explore the broad spectral properties
with nthcomp model (Zdziarski, Johnson & Magdziarz, 1996). Later, we apply Two
Component Advective Flow (TCAF) model (Chakrabarti & Titarchuk, 1995) to
extract the physical flow parameters such as the accretion rates and size of
the Compton cloud.
Along with these models, we use a Gaussian component for the Fe fluorescent
emission line. While fitting the data, we use two absorption components,
namely TBabs and zTBabs (Wilms et al., 2000). The component, TBabs is used for
the Galactic absorption, where hydrogen column density ($N_{H,gal}$) is fixed
at $9.78\times 20^{20}$ cm-2 (Kalberla et al., 2005). To calculate the error
for each parameter in spectral fitting with 90% confidence level, we use
‘error’ command in XSPEC.
We use following cosmological parameters in this work: $H_{0}$ = 70 km s-1 Mpc
-1, $\Lambda_{0}$ = 0.73, $\Omega_{M}$ = 0.27 (Bennett et al., 2003). With the
assumed cosmological parameters, the luminosity distance of Ark 120 is 142
Mpc.
Table 2: nthcomp fitting result for the spectrum above 3.0 keV. The optical depth $\tau$ is calculated from equation-1. ID | MJD | $\Gamma^{nth}$ | $kT_{e}$ | Fe $K_{\alpha}$ | EW | $\chi/dof$ | $\tau^{*}$
---|---|---|---|---|---|---|---
| | | (keV) | (keV) | (eV) | |
XMM1 | $52875$ | $1.90^{+0.01}_{-0.01}$ | $159.45^{+81.68}_{-81.69}$ | $6.40^{+0.016}_{-0.017}$ | $116^{+3}_{-4}$ | 312.33/300 | $0.733\pm 0.003$
S1 | $54191$ | $2.08^{+0.03}_{-0.03}$ | $124.65^{+35.54}_{-35.21}$ | $6.38^{+0.052}_{-0.052}$ | $710^{+10}_{-10}$ | 1117.31/1093 | $0.726\pm 0.008$
XRT1 | $54676$ | $1.76^{+0.02}_{-0.08}$ | $217.72^{+105.6}_{-112.5}$ | - | - | 75.68/74 | $0.671\pm 0.030$
XMM2+N1 | $56341$ | $1.75^{+0.01}_{-0.02}$ | $221.56^{+105.3}_{-107.5}$ | $6.42^{+0.061}_{-0.062}$ | $136^{+8}_{-9}$ | 644.55/641 | $0.670\pm 0.074$
XMM3+N2 | $56738$ | $1.87^{+0.01}_{-0.01}$ | $205.95^{+100.6}_{-99.87}$ | $6.37^{+0.052}_{-0.052}$ | $227^{+12}_{-11}$ | 508.07/469 | $0.612\pm 0.003$
XRT2 | $56926$ | $1.60^{+0.01}_{-0.02}$ | $274.40^{+136.5}_{-130.8}$ | $-$ | - | 306.65/290 | $0.700\pm 0.008$
XRT3 | $56974$ | $1.84^{+0.02}_{-0.02}$ | $215.72^{+105.5}_{-105.8}$ | $-$ | - | 319.98/320 | $0.610\pm 0.006$
XRT4 | $57024$ | $1.72^{+0.02}_{-0.03}$ | $225.57^{+109.7}_{-109.9}$ | $-$ | - | 269.17/280 | $0.688\pm 0.011$
XRT5 | $57073$ | $1.88^{+0.02}_{-0.02}$ | $201.58^{+99.78}_{-99.20}$ | $-$ | - | 246.53/261 | $0.616\pm 0.006$
XRT6 | $58118$ | $1.65^{+0.02}_{-0.02}$ | $246.87^{+120.9}_{-122.8}$ | $-$ | - | 327.78/318 | $0.708\pm 0.008$
Table 3: Soft-excess spectral indices are generated while keeping the spectral slope of nthcomp ($\Gamma^{nth}$) frozen. Intrinsic luminosities are calculated for both of the components using clum in the energy energy 0.5 to 10.0 kev. ID | $\Gamma^{PC}$ | $Norm^{PC}$ | $L^{PC}$ | $\Gamma^{SE}$ | $Norm^{SE}$ | $L^{SE}$
---|---|---|---|---|---|---
| $=\Gamma^{nth}$ | $(10^{-2})$ | | | $(10^{-2})$ |
XMM1 | $1.90$ | $1.16^{+0.04}_{-0.05}$ | $44.18^{+0.06}_{-0.07}$ | $3.15^{+0.07}_{-0.06}$ | $0.58^{+0.02}_{-0.02}$ | $43.66^{+0.05}_{-0.04}$
S1 | $2.08$ | $18^{+20.3}_{-25.6}$ | $45.35^{+0.05}_{-0.05}$ | $2.52^{+0.02}_{-0.02}$ | $2105^{+10.6}_{-16.5}$ | $45.58^{+0.04}_{-0.04}$
XRT1 | $1.76$ | $0.66^{+0.03}_{-0.03}$ | $43.99^{+0.04}_{-0.04}$ | $4.11^{+0.22}_{-0.20}$ | $0.84^{+0.10}_{-0.10}$ | $43.87^{+0.02}_{-0.03}$
XMM2+N1 | $1.75$ | $0.57^{+0.03}_{-0.03}$ | $43.93^{+0.05}_{-0.05}$ | $3.03^{+0.03}_{-0.02}$ | $0.19^{+0.03}_{-0.05}$ | $43.16^{+0.03}_{-0.03}$
XMM3+N2 | $1.86$ | $1.21^{+0.01}_{-0.01}$ | $44.90^{+0.04}_{-0.04}$ | $4.23^{+0.02}_{-0.02}$ | $0.88^{+0.10}_{-0.10}$ | $43.19^{+0.04}_{-0.04}$
XRT2 | $1.60$ | $0.48^{+0.03}_{-0.03}$ | $44.92^{+0.04}_{-0.04}$ | $2.92^{+0.19}_{-0.20}$ | $0.57^{+0.03}_{-0.04}$ | $43.66^{+0.04}_{-0.04}$
XRT3 | $1.84$ | $0.79^{+0.03}_{-0.04}$ | $44.04^{+0.05}_{-0.05}$ | $3.27^{+0.27}_{-0.27}$ | $0.47^{+0.06}_{-0.06}$ | $43.57^{+0.05}_{-0.05}$
XRT4 | $1.72$ | $0.57^{+0.04}_{-0.05}$ | $43.94^{+0.04}_{-0.04}$ | $2.53^{+0.10}_{-0.12}$ | $0.39^{+0.05}_{-0.06}$ | $43.54^{+0.04}_{-0.04}$
XRT5 | $1.88$ | $0.76^{+0.03}_{-0.03}$ | $44.00^{+0.04}_{-0.04}$ | $3.17^{+0.34}_{-0.34}$ | $0.31^{+0.05}_{-0.05}$ | $43.37^{+0.05}_{-0.05}$
XRT6 | $1.65$ | $0.42^{+0.02}_{-0.03}$ | $43.84^{+0.03}_{-0.03}$ | $2.96^{+0.28}_{-0.29}$ | $0.23^{+0.02}_{-0.03}$ | $43.27^{+0.06}_{-0.05}$
Table 4: The TCAF parameter space is defined in the file lmod.dat.
Model parameters | Parameter units | Default value | Min. | Min. | Max. | Max. | Increment
---|---|---|---|---|---|---|---
$\rm M_{BH}$ | $\rm M_{Sun}$ | $\rm 1.0\times 10^{8}$ | $2\times 10^{6}$ | $2\times 10^{6}$ | $5.5\times 10^{9}$ | $5.5\times 10^{9}$ | $10.0$
$\dot{m}_{d}$ | $\rm Edd$ | $\rm 0.001$ | $0.0001$ | $0.0001$ | $1.0$ | $2.0$ | $0.0001$
$\dot{m}_{h}$ | $\rm Edd$ | $\rm 0.01$ | $0.0001$ | $0.0001$ | $2.0$ | $3.0$ | $0.0001$
$\rm X_{s}$ | $\rm r_{g}$ | $\rm 100.0$ | $10.0$ | $10.0$ | $1000.0$ | $1000.0$ | $2.0$
$\rm R$ | | $\rm 1.5$ | $1.1$ | $1.1$ | $6.8$ | $6.8$ | $0.1$
### 3.1 Nthcomp
We have started the spectral fitting with nthcomp model, and the model in
XSPEC reads as:
TBabs*zTBabs*(nthcomp+zGaussian)
nthcomp is a thermally Comptonized continuum model proposed by Zdziarski,
Johnson & Magdziarz (1996) and later extended by Zycki, Done & Smith (1999).
We fit all X-ray spectrum above 3.0 keV by this baseline model. The model
depends on the seed photon energy ($kT_{bb}$), which we consider at 3 eV for
all spectrum. Although, Marinucci et al. (2019) considered $kT_{bb}$ at 15 eV.
It is to be noted that, we vary $kT_{bb}$ from 1 eV to 50 eV, and failed to
notice any deviation in the residuals of the fitted spectra. We consider these
seed photons to be disc-blackbody type. For that, we have opted for the inp-
type is 1 for all fit. For the spectral fitting, first, we consider the energy
range 3.0 to 10.0 keV. The fitted asymptotic power-law photon index
$\Gamma=1.90$, electron temperature $kT_{e}=159.45$ keV and an iron K$\alpha$
line at 6.40 keV with equivalent width (EW) of $116^{+3}_{-4}$ eV with reduced
chi-square ($\chi^{2}/dof$)=1.04 for degrees of freedom (dof) = 300 is
obtained. Next, we analyse the data from the 2007 Suzaku observation. We have
combined the Suzaku/XIS observation with Suzaku/HXD and make a spectrum from
0.5 to 40.0 keV. But, we fit 3.0 to 40.0 keV spectrum using the baseline
model. The fitted parameters are $\Gamma=2.08$, $kT_{e}=124.65$ keV and iron
K$\alpha$ line at 6.38 keV with equivalent width (EW) of $710^{+10}_{-10}$ eV.
We are also in need of an additional powerlaw and Gaussian to take care of
high energy (above 10.0 keV) spectrum and emission lines. We have obtained the
reduced chi-square ($\chi^{2}/dof$)=1.02 for degrees of freedom (dof) = 1093
for this fitting. We have fitted the combined spectrum of XMM2+N1 (MJD-56341)
and XMM3+N2 (MJD-56738) spectrum using this model for the energy range 3.0 to
79.0 keV with the model parameters such as $\Gamma=1.75$ & $1.87$ and
corresponding $kT_{e}=221.56$ & $205.95$ respectively. We have applied a
zGaussian for a Fe K$\alpha$ line at $6.42^{+0.061}_{-0.062}$ &
$6.37^{+0.052}_{-0.052}$ keV with equivalent widths (EW) of $136^{+8}_{-9}$ &
$227^{+12}_{-11}$ eV for these combined spectra and the
($\chi^{2}/dof$)=644.55/641 & ($\chi^{2}/dof$)= 508.07/469 respectively. Next,
we analyse the data obtained from Swift/XRT observation for the energy range
of 3.0 to 10.0 keV. Fe K$\alpha$ line is not detected for all the six XRT
spectra. We have fitted the Swift/XRT spectra by removing Gaussian component
from the baseline model. The power-law index $\Gamma$ vary from 1.60 to 1.88
and the corresponding electron temperature $kT_{e}$ vary from 274.40 to 201.58
keV respectively. The nthcomp model fitted spectral analysis result is
presented in Table 2. Furthermore, we calculate the optical depth for each
observation using the formula:
$\tau=\sqrt{\frac{9}{4}+\frac{3}{\theta_{e}(\Gamma+2)(\Gamma-1)}}-\frac{3}{2},$
(1)
by inverting the relation A1 is presented in Zdziarski, Johnson & Magdziarz
(1996). Here, $\theta_{e}=\frac{kT_{e}}{m_{e}c^{2}}$ is the electron energy
with respect to the rest mass energy. The value of optical depth $\tau$ for
each observation is provided in Table 2. The maximum error in optical depth is
obtained from
$\Delta\tau\sim(\frac{1}{2}\frac{\Delta\theta_{e}}{\theta_{e}}+\frac{\Delta\Gamma}{\Gamma})\times\tau$,
where $\Delta\theta_{e}$ and $\Delta\Gamma$ are considered from the fitted
errors presented in Table 2.
We address the soft-excess (< 3 keV) part by adding another powerlaw
component. We freeze the $\Gamma$ obtained earlier while fitting the primary
continuum alone. The second power-law fits the soft-excess, and the results
are presented in Table 3. It should be noted that the spectral index of soft-
excess ($\Gamma^{SE}$) is higher than the spectral index of the primary
continuum ($\Gamma^{PC}$) for every observation.
Figure 1: Variation of $\chi_{red}$ is shown for each model components on
broadband spectra of Ark 120 during 2014 epoch. Primarily, we have started
with TCAF, and then added zGaussian and Pexrav upon necessity.
Figure 2: TCAF model fitted spectra of Ark 120 from the XMM-Newton, Suzaku,
NuSTAR and Swift observations along with the residuals obtained from the
spectral fitting. Table 5: TBabs*zTBabs*(TCAF+zGaussian) model fitted
Parameters in 0.2-79.0 keV energy band for Ark 120. The TBabs is fixed at
$N_{H_{gal}}$= $9.78\times 10^{20}$ cm-2. The second column shows the
variation of zTBabs for z = 0.033.
ID | MJD | $N_{H}$ | $M_{BH}$ | $\dot{m}_{d}$ | $\dot{m}_{h}$ | $X_{s}$ | $R$ | $N_{TCAF}$ | $\Gamma_{pexrav}$ | $R_{ref}$ | $N_{pexrav}$ | $\chi^{2}/dof$
---|---|---|---|---|---|---|---|---|---|---|---|---
| | $(10^{20}cm^{-2})$ | $\rm(10^{8}M_{\odot})$ | $(\dot{m}_{Edd})$ | $(\dot{m}_{Edd})$ | $(r_{g})$ | | $(10^{-3})$ | | | ($10^{-3}$) |
XMM1 | $\rm 52875$ | $1.0^{+0.2}_{-0.2}$ | $1.50^{+0.03}_{-0.03}$ | $\rm 0.063^{+0.002}_{-0.002}$ | $\rm 0.112^{+0.001}_{-0.001}$ | $20.36^{+4.46}_{-4.55}$ | $1.95^{+0.34}_{-0.33}$ | $0.16^{+0.05}_{-0.05}$ | $0.14^{+0.10}_{-0.08}$ | $1.96^{+0.05}_{-0.09}$ | $0.14^{+0.04}_{-0.05}$ | $1026.20/842$
| | | | | | | | | | | |
S1 | $\rm 54191$ | $1.4^{+0.1}_{-0.2}$ | $1.49^{+0.04}_{-0.04}$ | $\rm 0.126^{+0.002}_{-0.001}$ | $\rm 0.191^{+0.001}_{-0.001}$ | $21.44^{+4.96}_{-4.85}$ | $1.66^{+0.54}_{-0.57}$ | $233.6^{+54.26}_{-58.65}$ | $1.46^{+0.15}_{-0.41}$ | $0.642^{+0.71}_{-0.51}$ | $5.09^{+1.05}_{-1.03}$ | $1869.89/1673$
| | | | | | | | | | | |
XRT1 | $\rm 54675$ | $1.6^{+0.6}_{-0.4}$ | $1.49^{+0.19}_{-0.15}$ | $\rm 0.064^{+0.005}_{-0.005}$ | $\rm 0.110^{+0.004}_{-0.003}$ | $30.08^{+5.36}_{-6.24}$ | $2.80^{+0.54}_{-0.56}$ | $0.54^{+0.01}_{-0.02}$ | $-$ | $-$ | $-$ | $307.78/283$
| | | | | | | | | | | |
XMM2+N1 | $\rm 54341$ | $1.9^{+0.3}_{-0.2}$ | $1.50^{+0.08}_{-0.07}$ | $\rm 0.068^{+0.006}_{-0.005}$ | $\rm 0.111^{+0.004}_{-0.004}$ | $52.83^{+8.65}_{-8.56}$ | $2.83^{+0.55}_{-0.52}$ | $0.57^{+0.02}_{-0.02}$ | $0.90^{+0.09}_{-0.10}$ | $0.254^{+0.47}_{-0.06}$ | $0.25^{+0.06}_{-0.05}$ | $1230.15/1112$
| | | | | | | | | | | |
XMM3+N2 | $\rm 54738$ | $2.3^{+0.3}_{-0.3}$ | $1.51^{+0.09}_{-0.10}$ | $\rm 0.103^{+0.006}_{-0.005}$ | $\rm 0.126^{+0.004}_{-0.004}$ | $28.24^{+5.04}_{-5.25}$ | $2.43^{+0.55}_{-0.58}$ | $0.31^{+0.01}_{-0.01}$ | $1.66^{+0.19}_{-0.19}$ | $0.96^{+0.05}_{-0.56}$ | $0.12^{+0.06}_{-0.06}$ | $1578.92/1359$
| | | | | | | | | | | |
XRT2 | $\rm 56926$ | $2.5^{+0.1}_{-0.1}$ | $1.49^{+0.18}_{-0.20}$ | $\rm 0.068^{+0.005}_{-0.006}$ | $\rm 0.110^{+0.003}_{-0.003}$ | $53.56^{+8.27}_{-8.87}$ | $2.73^{+0.51}_{-0.47}$ | $0.36^{+0.01}_{-0.02}$ | $-$ | $-$ | $-$ | $579.89/555$
| | | | | | | | | | | |
XRT3 | $\rm 56974$ | $1.9^{+0.5}_{-0.5}$ | $1.51^{+0.19}_{-0.20}$ | $\rm 0.068^{+0.006}_{-0.006}$ | $\rm 0.110^{+0.004}_{-0.005}$ | $55.16^{+8.57}_{-8.80}$ | $2.74^{+0.45}_{-0.41}$ | $0.25^{+0.01}_{-0.01}$ | $-$ | $-$ | $-$ | $630.08/594$
| | | | | | | | | | | |
XRT4 | $\rm 57024$ | $1.1^{+0.2}_{-0.3}$ | $1.50^{+0.15}_{-0.18}$ | $\rm 0.061^{+0.006}_{-0.006}$ | $\rm 0.110^{+0.005}_{-0.003}$ | $56.86^{+10.97}_{-10.89}$ | $2.69^{+0.48}_{-0.47}$ | $0.11^{+0.01}_{-0.01}$ | $-$ | $-$ | $-$ | $702.39/548$
| | | | | | | | | | | |
XRT5 | $\rm 57073$ | $1.4^{+0.4}_{-0.3}$ | $1.49^{+0.16}_{-0.15}$ | $\rm 0.069^{+0.006}_{-0.007}$ | $\rm 0.110^{+0.005}_{-0.005}$ | $57.87^{+12.99}_{-12.08}$ | $2.77^{+0.52}_{-0.54}$ | $0.28^{+0.02}_{-0.01}$ | $-$ | $-$ | $-$ | $551.49/531$
| | | | | | | | | | | |
XRT6 | $\rm 58118$ | $2.0^{+0.4}_{-0.4}$ | $1.51^{+0.16}_{-0.15}$ | $\rm 0.081^{+0.006}_{-0.005}$ | $\rm 0.140^{+0.006}_{-0.007}$ | $42.95^{+8.98}_{-8.20}$ | $2.69^{+0.59}_{-0.61}$ | $0.22^{+0.01}_{-0.01}$ | $-$ | $-$ | $-$ | $612.9/589$
| | | | | | | | | | | |
Figure 3: Variation of different model parameters with time are presented.
### 3.2 TCAF
From the nthcomp model fitting, we have extracted several valuable information
on the spectral hardness and electron temperature of the emitting system in a
time duration of $\sim$ 15 years. We have also calculated the optical depths
from these parameters, which are shown in Table 2. However, the fundamental
properties, such as the central black hole mass, accretion rates, the size of
the Compton cloud radius could provide a deeper physical understanding of the
system. To estimate these quantities, we use the Two-Component Advective Flow
(TCAF) model (Chakrabarti & Titarchuk, 1995) for our spectral analysis. For
the spectral fitting, the model in XSPEC reads as:
TBabs*zTBabs*(TCAF+zGaussian)
TCAF is based on one black hole parameter and four flow parameters: (i) black
hole mass in units of the solar mass ($M_{\odot}$); (ii) Keplerian disc
accretion rate ($\dot{m}_{d}$) in the unit of the Eddington rate
($\dot{M}_{EDD}$); (iii) Sub-Keplerian halo accretion rate ($\dot{m}_{h}$) in
units of Eddington rate ($\dot{M}_{EDD}$); (iv) shock compression ratio (R)
and (v) shock location ($X_{s}$) in units of the Schwarzschild radius
($r_{g}=2GM/c^{2}$). The upper and lower limits of all the parameters are put
in a data file called lmodel.dat provided in Table-4 as an input to run the
source code using initpackage and lmod task in XSPEC. For the final spectral
fitting of a specified observation, we run the source code for a vast number
of times and select the best spectrum from many spectra using minimization of
$\chi$ method. First, we have started fitting by the baseline model described
as above. Some spectra, like XMM1, S1, XMM2+N1, XMM3+N2 have high reduced
$\chi^{2}$ ($\chi^{2}_{red}>2$) value. We noticed that the model has deviated
from the actual data at the high energy end. To compensate for that, we have
added a powerlaw/pexrav with the baseline model. Thus the model became:
TBabs*zTBabs*(TCAF+powerlaw/pexrav+zGaussian).
We have fitted the spectra with this model and found $\chi^{2}_{red}\approx
1$. Further, to investigate the source of this power-law (whether it is from
reflection or not), we have replaced the powerlaw component by pexrav
(Magdziarz & Zdziarski, 1995). The pexrav model has a power-law continuum with
a reflected component from an infinite neutral slab. We have estimated the
relative reflection coefficient ($R_{ref}$) with photon index
($\Gamma_{pexrav}$) and cosine of inclination angle $\cos\theta$ from the
model fitting. We find $\theta$ to vary from $40\degree$ to $72\degree$. We
fix abundances for heavy elements, such as iron at the Solar value (i.e., 1).
For the photon index ($\Gamma_{pexrav}$), first, we freeze its value to the
value of $\Gamma$ obtained from nthcomp. For this, we have found
$\chi^{2}_{red}>2$. Thereafter, we thaw this parameter and fit it again which
have resulted $\chi^{2}_{red}\approx 1$ with new value of $\Gamma_{pexrav}$.
We first fit the XMM-Newton observation (XMM1) during 2003 (MJD-52875) in the
energy range of 0.2 to 10.0 keV with TBabs*zTBabs*(TCAF+zGaussian) model.
However, we found a high $\chi^{2}_{red}$. The model has deviated after 9.2
keV from the actual data. As mentioned above, we then add a powerlaw with the
baseline model, and then the powerlaw is replaced by pexrav. The fitted
parameters are, $M_{BH}=1.5\times 10^{8}M_{\odot}$, $\dot{m}_{d}=0.063$,
$\dot{m}_{h}=0.112$, $X_{s}=20.36$, $R=1.95$ with $\Gamma_{pexrav}=0.14$,
$R_{ref}=1.96$, and $E_{fold}=16.08$ keV and the corresponding
$\chi^{2}=1026.20$ with degrees of freedom (dof)= 842. The Fe line is found at
$6.4$ keV with an equivalent width of $116$ eV.
Next, we consider the Suzaku observation (S1) of 2007 (MJD-54191). We combine
the Suzaku/XIS and Suzaku/HXD spectra and make a broadband spectrum in the
energy range of 0.5 to 40 keV. We follow the similar steps as described in
XMM1 fitting and the fitted parameters are $M_{BH}=1.49\times
10^{8}M_{\odot}$, $\dot{m}_{d}=0.126$, $\dot{m}_{h}=0.191$, $X_{s}=21.44$,
$R=1.66$ with $\Gamma_{pexrav}=1.46$, $R_{ref}=0.642$, and the corresponding
$\chi^{2}_{red}/dof=1869.89/1673$. The position of Fe line is $6.38$ keV with
an equivalent width of $710$ eV. It is to be noted that, within $6-7$ keV
range, Nardini et al. (2011) reported the possibility of three lines for XMM1
and two lines for S1 observation respectively.
Following a similar procedure, we fit the broadband spectra of Ark 120 for the
observations during 2013 XMM2+N1 (MJD-56341) and 2014 XMM3+N2 (MJD-56738). For
these, we have obtained $M_{BH}=1.50$ & $1.51\times 10^{8}M_{\odot}$,
$\dot{m}_{d}=0.068$ & $0.103$, $\dot{m}_{h}=0.111$ & $0.126$, $X_{s}=52.83$ &
$28.24$, $R=2.83$ & $2.43$ with $\Gamma_{pexrav}=0.96$ & $1.66$ respectively.
The details of data fitting are given in Table-5.
We fit all the six Swift/XRT spectra using the baseline model. Here, we do not
find any Fe line in all these spectra. From the fitting, it is noticed that
the mass of the central black hole $M_{BH}$ =$1.5\times 10^{8}M_{\odot}$, the
disc $\dot{m}_{d}\sim 0.065$ and halo accretion rates $\dot{m}_{h}$$\sim
0.110$ are more or less constant except XRT6 observation. Here, we find
$\dot{m}_{d}=0.081$ & $\dot{m}_{h}=0.14$ and the corresponding shock location
has moved inward from 57.87 to 42.95 $r_{g}$. Therefore, the shock location
($X_{s}$) has varied in between $30.0$ to $57.87~{}r_{g}$, and the
corresponding variation of the compression ratio (R) is in between $2.6$ to
$2.8$ within September 2014 to January 2018. Here, we do not require any
additional powerlaw to fit the high energy spectra. The details of the
parameter variations are presented in Table-5. In Figure 2, we plot the model
fitted spectrum with the variation of $\chi$. Detailed discussions on spectral
properties are demonstrated in Sec 5.1.
Table 6: Variability statistics in various energy ranges are shown in this Table. We have opted for 100s time bins for variability analysis. In some cases, the average error of observational data exceeds the limit of $1\sigma$, resulting negative excess variance. In such cases, we have imaginary $F_{var}$, which are not shown in the table. ID | Energy band | $N$ | $x_{max}$ | $x_{min}$ | $\frac{x_{max}}{x_{min}}$ | $\sigma^{2}_{NXS}$ | $F_{var}$
---|---|---|---|---|---|---|---
| keV | | Count/s | Count/s | | $(10^{-2})$ | $(10^{-2})$
XMM1 | 0.2-2.0 | 1117 | 21.95 | 19.58 | 1.12 | $0.57\pm 0.003$ | $1.6\pm 0.14$
XMM2 | 0.2-2.0 | 1294 | 10.24 | 8.40 | 1.21 | $2.9\pm 0.015$ | $5.6\pm 0.40$
XMM3 | 0.2-2.0 | 1309 | 21.37 | 17.11 | 1.25 | $8.22\pm 0.011$ | $6.4\pm 0.41$
XMM1 | 3-10.0 | 1117 | 1.95 | 1.53 | 1.79 | $0.18\pm 0.023$ | $3.2\pm 0.04$
XMM2 | 3-10.0 | 1294 | 1.22 | 0.94 | 1.30 | $0.137\pm 0.035$ | $3.5\pm 0.55$
XMM3 | 3-10.0 | 1309 | 3.64 | 2.92 | 1.24 | $0.56\pm 0.0011$ | $4.1\pm 0.34$
N1 | 10.0-78.0 | 722 | 0.711 | 0.105 | 6.748 | $0.093\pm 0.007$ | $7.5\pm 1.6$
N2 | 10.0-78.0 | 667 | 1.712 | 0.239 | 7.143 | $0.147\pm 0.024$ | $3.5\pm 3.6$
XMM1 | 0.5-10.0 | 1117 | 20.46 | 15.26 | 1.341 | $10.11\pm 0.06$ | $2.36\pm 0.14$
S1 | 0.5-10.0 | 586 | 9.80 | 5.58 | 1.76 | $5.64\pm 0.31$ | $8.62\pm 0.31$
XRT1 | 0.5-10.0 | 8 | 3.22 | 1.67 | 3.01 | -$22.3\pm 5.6$ | $-$
XMM2 | 0.5-10.0 | 1294 | 10.36 | 6.73 | 1.54 | $2.93\pm 0.02$ | $5.85\pm 0.19$
XMM3 | 0.5-10.0 | 1309 | 20.13 | 13.67 | 1.47 | $5.80\pm 0.01$ | $5.79\pm 0.15$
XRT2 | 0.5-10.0 | 50 | 2.13 | 0.70 | 3.02 | $5.50\pm 0.44$ | $20.70\pm 2.3$
XRT3 | 0.5-10.0 | 43 | 2.79 | 1.52 | 2.79 | $3.61\pm 0.42$ | $15.31\pm 2.1$
XRT4 | 0.5-10.0 | 43 | 1.90 | 0.88 | 2.16 | $4.50\pm 0.46$ | $18.01\pm 2.3$
XRT5 | 0.5-10.0 | 42 | 1.77 | 0.81 | 2.18 | $2.72\pm 0.25$ | $14.03\pm 1.8$
XRT6 | 0.5-10.0 | 72 | 1.63 | 0.52 | 3.09 | $6.02\pm 0.49$ | $23.40\pm 2.2$
Figure 4: Top panel: The light-curves of the energy ranges of $0.2$ to $2.0$
keV and $3.0$ to $10.0$ keV observed by XMM-Newton are plotted for three
epochs. The high energy count always remained a fraction of low energy
counterpart. In 2013, the low-energy count dropped to nearly 50% as compared
to 2003. Again in 2014, the $0.2-2$ keV count doubled from its value observed
in 2013. Middle panel: Corresponding discrete cross-correlations between
light-curves of $0.2-2$ keV and $3-10$ keV are plotted. All three epochs
exhibited different patterns where zero, positive, and negative delays are
observed in 2003, 2013, and 2014 respectively. We have also presented the ICF
(solid-blue line) for XMM3 observation. Lower panel: $\zeta$-discrete cross-
correlations (light-green) are plotted for light-curves of $0.2-2$ keV and
$3-10$ keV. While 2003 and 2013 patterns remain similar to what have been
observed from DCF, the pattern obtained from 2014 data develops twin peak. The
likelihoods (dark-green), simulated using 12000 points, are plotted along with
the ZDCF.
## 4 Timing Analysis
### 4.1 Variability
X-ray variability of an AGN provides a powerful probe of the nearby regions of
the central black hole. Since Ark 120 has a ‘bare-type nucleus’, the X-ray
comes from the Compton cloud and is not intercepted by any clouds such as BLR,
NLR or molecular torus. Thus, the X-ray variability is originated from the
varying Compton cloud and the central accretion disc. To analyze the temporal
variability in X-ray of Ark 120 in different energy bands, we have estimated
different parameters for the duration of 2003 (MJD-52875) to 2018 (MJD-58118).
The fractional variability $F_{var}$ ((Edelson et al., 1996); (Nandra et al.,
1997); (Edelson et al., 2001); (Edelson et al., 2012); (Vaughan et al., 2003);
(Rodríguez-Pascual et al., 1997)) of lightcurves of $x_{i}$ count/s with
finite measurement error $\sigma_{i}$ of length $N$ with a mean $\mu$ and
standard deviation $\sigma$ is given by:
$F_{var}=\sqrt{\frac{\sigma^{2}_{XS}}{\mu^{2}}}$ (2)
where, $\sigma^{2}_{XS}$ is excess variance (Nandra et al. (1997); Edelson et
al. (2002)), an estimator of the intrinsic source variance and is given by:
$\sigma^{2}_{XS}=\sigma^{2}-\frac{1}{N}\sum_{i=1}^{N}\sigma^{2}_{i}.$ (3)
The normalized excess variance is given by
$\sigma^{2}_{NXS}=\sigma^{2}_{XS}/\mu^{2}$. The uncertainties in
$\sigma^{2}_{NXS}$ and $F_{var}$ are taken from Vaughan et al. (2003) and
Edelson et al. (2012).
The X-ray variability of Ark 120 in different energy bands ($0.5-10.0$ keV;
$0.2-2.0$ keV; $3.0-10.0$ keV) have demonstrated different degrees of
variabilities (Table 6) while the time binsize is kept constant at $100$s.
From XMM1, the lower energy ($0.2-2.0$ keV) count rate was initially high
($X_{max}=21.95$) in 2003 observation. Then, in 2013 (XMM2), it became half
($X_{max}=10.24$) from its initial value. In 2014 (XMM3), the count increased
($X_{max}=21.37$). The fractional variability in this energy range increased
from $0.016$ to $0.064$ from 2003 to 2014 observations. A similar trend is
shown by $\sigma^{2}_{NXS}$ ($0.006$ to $0.082$) in this energy band for each
observation of XMM (Table 6). Like low energy part, the high energy
($3.0-10.0$ keV) follow the similar type of trend for the count rate and
fractional variability. The average value of $\sigma^{2}_{NXS}$ is $0.003$,
with a range from $0.0014$ to $0.0056$.
We calculate the variability in $0.5-10.0$ keV range from the Suzaku data. We
find higher variability $F_{var}=8.62\pm 0.31$ in the 2007 Suzaku data as
compared to the previous XMM observations. The variability for XRT
observations in $0.5-10.0$ keV range is shown in Table 6. Due to the lack of
data points, XRT1 observation yields an imaginary value of $F_{var}$, and is
not shown in Table 6. From the other observations of Swift/XRT, we observe
high fractional variability ($F_{var}$) from $0.14$ to $0.23$ with
$<F_{var}>=18.22$. The average value of $x_{max}/x_{min}$ and
$\sigma^{2}_{NXS}$ for these observations are $2.65$ and $0.045$ with a range
from $2.16$ to $3.09$ and $0.027$ to $0.060$ respectively.
### 4.2 Delay Estimation
For temporal analysis of the long term archival data of Ark 120, we stress
three epochs of XMM-Newton, 2003, 2013, and 2014 out of which the latter two
have high energy (3-80 keV) counterparts observed by NuSTAR. We have performed
cross-correlation analysis using DCF (Edelson & Krolik, 1988) and
$\zeta$-discrete cross-correlation function (ZDCF888ZDCF:
http://www.weizmann.ac.il/particle/tal/research-activities/software,
Alexzander (1997)) for comparison. The likelihood is calculated using 12000
simulation points in the ZDCF code for the lightcurves obtained by XMM-Newton.
The peak error is calculated using the formula provided by Gaskell & Peterson
(1987). We have followed a similar procedure as in Chatterjee et al. (2020).
The time resolution of each light curve is $1000$s. The $0.2-2$ keV lightcurve
obtained from 2003 data yields an acceptable $\chi^{2}_{red}<1.5$ when fit
with a straight line. However, data procured in 2013 and 2014 in a similar
energy band have a high residual and are not suitable for linear fitting. All
three high energy lightcurves (3-10 keV) have $\chi^{2}_{red}<1.5$ when fitted
with straight lines. We have carried out the delay estimation using the XMM-
Newton/Epic-pn data to ensure the simultaneity in their procurements.
Table 7: Parameters used in delay estimations are presented. The error in measurement of delay is considered as the larger between binsize and $\epsilon_{\tau}$. $\epsilon^{d}_{\tau}$ and $\epsilon^{z}_{\tau}$ represents errors for DCF and ZDCF patterns. Id | Epochs | Bin size | $\epsilon^{d}_{\tau}$ | $\Delta\tau^{dcf}_{d}$ | $\epsilon^{z}_{\tau}$ | $\Delta\tau^{zdcf}_{d}$
---|---|---|---|---|---|---
| Year | (ks) | (ks) | (ks) | (ks) | (ks)
blackXMM1 | 2003 | 1 | 0.388 | $0.16\pm 1$ | 0.936 | $-0.057\pm 1$
blackXMM2 | 2013 | 1 | 0.862 | $4.71\pm 1$ | 2.11 | $6.76\pm 2.11$
blackXMM3 | 2014 | 1 | 0.622 | $-4.15\pm 1$ | 1.54 | $-4.56\pm 1.54$
blackXMM3 | 2014 | 1 | $-do-$ | $-do-$ | 1.58 | $-49.2\pm 1.58$
The DCF (Edelson & Krolik, 1988), performed using the lightcurves, have
generated three distinct patterns. The 2003 data has produced $2.78\pm 16.67$
minutes or $\sim 0.16$ ks delay. We have fitted the peak using a Gaussian
model (dotted line in Fig. 4). Considering the error, no delay can be seen
between two bands of X-ray. Similar delay pattern is also observed from ZDCF,
and the likelihood density also maximizes around zero. Likewise, we have
performed Gaussian fitting for 2013 data where a positive delay of $78.51\pm
35.17$ minutes or $\sim 4.7$ ks has been seen between soft and hard X-ray
photons using DCF. But, the ZDCF peak maximizes around $112.68\pm 35.22$
minutes or 6.7 ks and likelihood peak coincides with that (see Fig. 4). In
2014, the delay sign have switched, and we find a negative delay of $-69.19\pm
25.67$ minutes or $\sim-4.1$ ks between the soft and hard band from DCF
analysis. However, ZDCF peaks maximize around two positions, $-76.19\pm 25.67$
($-4.56\pm 1.54$ ks) and $-820.19\pm 26.46$ ($-49.2\pm 1.58$ ks) minutes
having peak values of 0.664 and 0.722 respectively. Between these two, the
former coincides with the DCF pattern (see, Table 7 for details). For all
three cases, we find the peak values of ZDCF patterns are lesser than the
corresponding peak values obtained from DCF patterns.
## 5 Discussions
We have studied the central region of Ark 120 through X-ray (above 0.2 keV)
using the data of XMM, Suzaku, NuSTAR and Swift/XRT in the period 2003
(MJD-52875) to 2018 (MJD-58118). As it is a bare type AGN, the X-ray spectra
mainly generated from the nearby region of the central engine.
### 5.1 Evolution of the Source: Primary Continuum
The ‘bare-type AGN’ Ark 120 was observed for a period of fifteen years, 2003
to 2018 using various X-ray satellites. During these observations, the source
has exhibited variabilities in both spectral and temporal domain. The
luminosity of the source in the energy range of 2.0 to 10.0 keV varied within
$\sim 10^{43.5}-10^{45.5}$ erg/s throughout these observations. From the
nthcomp model, we report the variation of the spectral index
(1.6<$\Gamma$<2.08) where the harder spectra were observed after 2014.
Following Vaughan et al. (2004), we have fitted the 2003 spectrum of Ark 120
with (nthcomp + Gaussian) model. The fitted $\Gamma=1.90^{+0.01}_{-0.01}$
agrees with the spectral index previously observed (Table 4 of Vaughan et al.
(2004)). Corresponding temperature of the Compton cloud is
$kT_{e}=159.45^{+81.68}_{-81.69}$ keV. The (TCAF + Gaussian) model provided a
few previously unknown parameters like accretion rates, disc rate
$\dot{m}_{d}=0.063\pm 0.002$ and halo rate $\dot{m}_{h}=0.112\pm 0.001$. This
suggests that the the source was initially halo dominated. This is normal for
an AGN. The shock location or the size of the CENBOL ($X_{s}$), estimated from
the fits, is $20.36\pm 4.4~{}r_{g}$. The shock is found to be moderately
strong with a compression ratio of $R=1.95\pm 0.05$.
The softest spectrum, having $\Gamma=2.08^{+0.03}_{-0.03}$ is seen during the
Suzaku observation in 2007. It is to be noted that, Nardini et al. (2011)
found the spectral index to be $\Gamma=2.03^{+0.01}_{-0.04}$ for the Suzaku
data using blurred reflection model. We have estimated the temperature of the
Compton cloud to be $kT_{e}=124.65^{+35.54}_{-35.21}$ keV. This is the least
of all temperatures obtained from all the observations. Using a single
Gaussian, we find the presence of a broad iron line ($6.38^{+0.052}_{-0.052}$)
keV having an equivalent width of $EW=710^{+10}_{-10}$ eV. The derived optical
depth is $\tau=0.726^{+0.008}_{-0.008}$. This suggests an optically thin
Compton cloud. From the TCAF fits, we find that the size of the Compton cloud
has slightly increased to $X_{s}=21.44\pm 4.9~{}r_{g}$ from the earlier
observation. Corresponding disc rate, which enhances the soft seed photons,
has increased to $\dot{m}_{d}=0.126$. Also, the halo rate has increased to
$\dot{m}_{h}=0.191$. However, shock strength has decreased (see Table 5). The
drop in the $kT_{e}$ could be understood easily from TCAF, where the increase
in disc rate leads to an enhanced cooling fraction. Thus, within the epochs of
2003 and 2007, the temperature of the Compton cloud was varied from 159.45 to
124.65 and as a result the spectrum softened.
Figure 5: Correlation of fitted parameters are plotted. Fig. (a) represents
the correlation between $\Gamma$ vs $kT_{e}$ and the corresponding PCC is
-0.95. It is also noted that the $kT_{e}$ is losely bound with $\Gamma$. Fig.
(b) is the correlation between $\tau$ and R. The PCC for these parameters is
-0.72. In Fig. (c) represents the correlation between $\tau$ vs $X_{s}$ and
the corresponding PCC is -0.45. Fig. (d) provides the correlation of $\Gamma$
vs $X_{s}$ with PCC -0.53.
Later, in 2008, Swift observed the source where the spectrum hardened from the
previous observation having $\Gamma=1.76^{+0.02}_{-0.08}$,
$kT_{e}=217.72^{+105.6}_{-112.5}$ keV, and optical depth
$\tau=0.671^{+0.030}_{-0.030}$. The iron line could not be detected from the
XRT spectrum. Corresponding TCAF fitted parameters, such as the shock location
$30.08~{}r_{g}$ and $R=2.80$ while $\dot{m}_{d}$ and $\dot{m}_{h}$ have
changed to $0.064$ and $0.11$ respectively.
Significant variation of spectral properties is also noted during 2013 and
2014. The broad-band spectra (3-78) keV are fitted with (nthcomp + Gaussian)
having the spectral indices $1.75^{+0.01}_{-0.02}$ and $1.87^{+0.01}_{-0.01}$
and are in good agreement with parameters obtained by Porquet et al. (2018);
Marinucci et al. (2019). The optical depth is reduced from $\tau=0.670\pm
0.074$ to $\tau=0.612\pm 0.003$. The flux in 2-10 keV band has doubled within
a year. The spectral softening could be explained by the drop of temperature
of the Compton cloud. However, the decrease in the optical depth for March
2014 data with respect to 2013 has also been seen from Monte-Carlo simulations
(Marinucci et al., 2019). From TCAF fitting, we find a distinct variation of
the flow parameters. The $\dot{m}_{d}$ changed from $0.068$ to $0.103$,
$\dot{m}_{h}$ changed from $0.111$ to $0.126$, and $X_{s}$ changed from
$52.83$ to $28.24$ within 2013 and 2014 observations respectively. As the disc
accretion rate increases, Compton cooling increases, and this lead to the
decrease in the $X_{s}$ which finally softens the spectrum. Considering TCAF,
the lower optical depth for softer spectrum could be explained by the
weakening of the shock ($R=2.43$ as compared to $R=2.83$ in February 2013) for
this observation. The stronger shock creates a distinct boundary between the
halo and CENBOL region where the majority of the hard photons are produced.
However, for the weaker shock, the CENBOL boundary is less sharp and a
fraction of inverse Comptonization could occur within the halo component.
Thus, the effective optical depth of the medium could become lower even though
the spectrum has softened.
Ark 120 has shown significant variabilities after February 2014 and is
monitored by Swift. We have tabulated the spectral and temporal variabilities
in Table 2 and 5. During September-October of 2014, we find that the spectral
slope was $\Gamma=1.60^{+0.01}_{-0.02}$ and the corresponding temperature was
$274.40\pm 130.0$ keV, which was maximum within the duration of our
observation. From the TCAF fitting, we find $\dot{m}_{d}$ and $\dot{m}_{h}$
has changed to $0.068$ and $0.11$ respectively and the corresponding shock
location has changed to $53.56\pm 8.2~{}r_{g}$ and the shock strength has
increased from $2.43$ to $2.73\pm 0.5$ as observed during February 2014\.
Later, in December 2014, the spectrum has softened with $\Gamma=1.84\pm 0.02$
with the temperature of Compton cloud $215.72\pm 105.5$ keV. The corresponding
shock has moved outward and observed at $55.16~{}r_{g}$ and $R=2.74$. Like
previous observations, we see the halo rate and disc rates are fixed at $0.11$
and $0.068$, respectively.
XRT4 and XRT5 observations were made starting from the end of December 2014 to
March of 2015. During this time, the spectral indices are $1.74$ and $1.88$
respectively. The temperature and optical depths have also varied during this
time. From TCAF fitting, we find the halo rate has decreased to $0.061$ in the
XRT4 observation. However, the disc rate was constant. Again in XRT5
observation, halo rate has increased to $0.069$ while the disc rate remained
the same. The shock location and the compression ratio remained constant
(considering the errors) within this period. Thus, we can see that Ark 120
exhibited spectral variability (see Fig. 3) within $\sim$200 days (since
September 2014-March 2015).
In XRT6, which was observed from December 2017 to January 2018, the spectrum
of Ark 120 has hardened with respect to the earlier observations during
January 2015. The spectral index and temperature of Compton cloud are $1.65\pm
0.02$ and $246.87\pm 121$ keV respectively. From TCAF fitting, we find the
disc and halo rates have increased to $\dot{m}_{d}=0.081$ & $\dot{m}_{d}=0.14$
respectively and the corresponding shock location settled at $42.95\pm
8.0~{}r_{g}$.
In Figure 5, we have plotted the correlations of a few spectral parameters. We
find the spectral index and the temperature of the Compton cloud is anti-
correlated (Fig. 5a with Pearson Correlation Co-efficient (PCC) = -0.9542) for
the long term observation. However, the values of $kT_{e}$ are poorly
constrained with respect to spectral indices. This is a well-established
relation and is generally found in case of AGNs and Galactic black holes. In
Fig. 5b, we have presented the correlation between shock compression ratio and
optical depth. We find $R-\tau$ produces anti-correlation having PCC=-0.721.
In general, stronger shocks are associated with the harder spectra where the
optical depth is expected to be less (Chatterjee et al., 2016) and the
corresponding shock location is also expected to be bigger. Keeping that
argument, we also show the $X_{s}-\tau$ correlation where an anti-correlation
(PCC=-0.457) has been observed from the long term data and presented in Fig.
5c. As a consequence, the spectral softens due to the reduction of the shock
location $X_{s}$ i.e., the size of the Compton cloud, we find a global trend
of anti-correlation (PCC=-0.562) between $X_{s}-\Gamma$ (see Fig. 5) for Ark
120.
From the nthcomp fitting, it can be found that the Compton cloud of the source
was optically thin for the entire period of observation. Overall, we also
noticed that the disc and halo rate is nearly constant and they are $\sim
0.07$ and $\sim 0.11$ respectively for the majority of observations. But, we
find a higher disc and halo rate in 2007 and 2014 observation. The shock
location and the compression ratio have varied with time. The variation of
these parameters is shown in Figure 3. First, the shock location increases
with time from 20 to 52 $r_{g}$ in the first $\sim 10$ years. Then the shock
location falls to 26.7$r_{g}$ within the next $\sim$ 13 months. Later, we find
that the shock location again moves outward from 26.7 to 57.8 $r_{g}$ before
moving inward again, and finally settling at 42.95$r_{g}$ in January 2018. The
Compression ratio (R) also varies as the shock location ($X_{s}$). First, the
compression ratio increased from 1.95 to 2.83 in $\sim 10$ years. Then, the
value of $R$ decreased to $1.67$ within next 1 year. After that, it increased
to 2.73 within less than six months and finally reached 2.69 at the end of
January 2018.
### 5.2 Evolution of the Source: Delay patterns
The Compton delay (Payne, 1980; Sunyaev & Titarchuk, 1980) for an electron
cloud of size $\mathcal{R}$ having an optical depth $\tau$ and temperature
$\theta_{e}=kT_{e}/m_{e}c^{2}$ can be described by,
$t_{c}=\frac{\mathcal{R}}{c(1+\tau)}\frac{ln(E_{h}/E_{ss})}{ln[1+4\theta_{e}(1+4\theta_{e})]},$
where, $c$ is the velocity of light, $E_{h}$ and $E_{ss}$ are the energy of
hard photons and soft seed photons respectively. For AGNs having a central
black hole mass of $1.5\times 10^{8}$ (Peterson et al., 2004), the seed
temperature of the photons remains in the 1-10 eV range. The maximum of the
hard and soft energy band is considered to be $10$ keV and $1$ keV and the
seed photon temperature is $E_{ss}=3$ eV. The light-crossing time for 1$r_{g}$
is $r_{g}/c=$1.5 ks for Ark 120. We calculated the delays for the combined
parameters obtained from nthcomp and TCAF model.
We have calculated the Compton delay for XMM1 observation where the size of
the Compton cloud is $\sim 20~{}r_{g}$, optical depth $0.733$, and
$\theta_{e}=0.311$. Substituting the values, we find $t^{h}_{c}=105.3$ ks and
$t^{s}_{c}=75.3$ ks which produces a positive theoretical delay of
$\Delta\tau=t^{h}_{c}-t^{s}_{c}=30$ ks. However, from the observed DCF
pattern, we fail to notice any such delay for this case. Here, we find light
crossing delay ($\tau_{lc}$) of 30 ks for a $\sim 20~{}r_{g}$ Compton cloud.
The observed zero-delay could be a combined result of $\tau_{c}$ and
$\tau_{lc}$. In that case, it is to be noted that $\tau_{lc}$ becomes crucial
in presence of a significant contribution of reflection component
($R_{ref}=1.96$, see Table 5).
Figure 6: Correlation of intrinsic luminosities of 0.5-10.0 keV obtained using
nthcomp. Left: shows a correlation (PCC=0.92) between the observed intrinsic
luminosities of primary continuum and soft-excess (blue-circle). Monte-Carlo
simulated luminosities for both energy ranges are presented with red-diamond
points. Right: No correlation between intrinsic luminosities and $N_{H}$ from
long term observations.
For the broadband observation (XMM2+N1), the size of the Compton cloud is
$\mathcal{R}\sim 50$ $r_{g}$, having an optical depth of $0.67$ and
temperature $\theta_{e}=0.434$. Combining all these, the maximum hard and soft
energy delay which can be generated via Compton scatterings are
$t^{h}_{c}=208$ ks and $t^{s}_{c}=148$ ks respectively. Thus, the maximum
delay between hard and soft bands of X-ray can be
$\Delta\tau=t^{h}_{c}-t^{s}_{c}=60$ ks. The light crossing delay is around
$\tau_{lc}=75$ ks. The combined effects of $\Delta\tau$ and $\tau_{lc}$ should
yield a negative delay of 15 ks. However, as discussed previously, $\tau_{lc}$
could dominate if reflection becomes dominating (here $R_{ref}=0.25$). Also,
the size of the Compton cloud is much bigger than the what should be the
‘transition radius’ (see, Dutta & Chakrabarti (2016); Dutta, Pal & Chakrabarti
(2018) for details) of an AGN having mass $1.5\times 10^{8}~{}M_{\odot}$.
Being an intermediate inclination angle source (Nardini et al., 2011;
Marinucci et al., 2019), Comptonization dominates the time delay when the size
of the Compton cloud is bigger. The theoretical structure of Compton cloud is
somewhat deviated from the sphere (see, Chakrabarti & Titarchuk (1995)) and
the thermodynamical fluctuations within the inhomogeneous Compton cloud (see,
Chatterjee et al. (2017b)) contributes to the delay patterns. Considering
this, the effect of light crossing delay would be much less and Comptonization
could be considered as the core process, which generates $0.2-2$ keV photons
during 2013 observations.
In a similar way, we calculate the Compton delay for broadband observation in
2014 (XMM3+N2). For that, the size of the Compton cloud
$\mathcal{R}=28~{}r_{g}$, the optical depth is $\tau=0.612$, and
$\theta_{e}=0.403$. We have obtained $t^{h}_{c}=123.4$ ks and $t^{s}_{c}=88.2$
ks which produces $\Delta\tau=t^{h}_{c}-t^{s}_{c}=35$ ks. Contrary to that,
the observed delay is $-69.19\pm 16.67$. Clearly, the Comptonization may not
be the dominating radiative process for this observation. From Table 5, we see
that the reflection co-efficient $R_{ref}=0.96$, which refers to a stronger
reflection. It is also to be noted that Lobban et al. (2018) found the X-ray
to be leading the U-band by $2.4\pm 1.8$ days which they have explained with
the light crossing delay. Considering the Compton cloud only, $\tau_{lc}$
becomes 42 ks, which is comparable to compensate for the positive lag obtained
from Comptonization. In this particular case, the maximum possible negative
delay would be $\Delta\tau-\tau_{lc}\sim-7$ ks or -116 minutes. However, as
the size of the Compton cloud has become bigger and $R_{ref}$ is much less
than the XMM1 observation. Thus, the contribution from $\tau_{lc}$ could be
less effective and we observe a negative delay much less than the maximum
allowed delay.
Thus, along with the spectral variations, we find the delay patterns have
varied over the three epochs (2003, 2013, and 2014) in which XMM-Newton
observed Ark 120. A significant change in the delay pattern is observed within
a year (2013-2014) where the positive delay changed sign and becomes negative
with a similar magnitude.
### 5.3 Soft Excess
The origin of ubiquitous soft-excess (Arnaud et al., 1985; Singh et al., 1985;
Brandt et al., 1993; Fabian et al., 2002; Gierliński & Done, 2004) remains
debated. A plausible cause of soft-excess was given using reflection
Sobolewska & Done (2007). The multi-wavelength campaign of Mrk 509 (Mehdipour
et al., 2011) revealed the correlation of soft-excess with the optical-UV part
both in the spectral and temporal domains where they concluded that the soft-
excess was generated due to Comptonization by a warm optically thick region
surrounding the accretion disc. Done et al. (2012) proposed that the high mass
accretion rate of the disc could generate the soft-excess. For lower
$L/L_{EDD}$, the energy dependent variability in the soft-excess part was
found to be less in case of Narrow line Seyfert 1 galaxies. Lohfink et al.
(2012) studied Seyfert 1 galaxy Fairfall 9 where the origin of the soft-excess
component was found to be connected with source which generates the broad iron
line. However, they implied that another source of Comptonization might be
responsible for the formation of the soft-excess.
Figure 7: Monte-Carlo simulated spectra for Ark 120 are presented. We have
considered $M_{BH}=1.5\times 10^{8}~{}M_{\odot}$ for Ark 120. Simulation
boundary extends up to $100~{}r_{g}$. For left panel $\dot{m}_{d}$ = 0.06;
$\dot{m}_{h}$ = 0.1, $X_{s}=60~{}r_{g}$, and maximum $kT_{e}=270$ keV. For
right panel $\dot{m}_{d}$ = 0.1; $\dot{m}_{h}$ = 0.1, $X_{s}=40~{}r_{g}$, and
maximum $kT_{e}=100$ keV. Notice the spectral contributions due to increasing
number of scatterings. $L_{\nu}$ has been normalized with respect to the
observed spectrum.
A strong soft-excess present in the X-ray spectrum of Ark 120 was reported by
Brandt et al. (1993); Matt et al. (2014); Porquet et al. (2004). This soft-
excess is also free from the absorbers and was reported by Nardini et al.
(2011). As a first step, we investigate the spectral slopes and the relative
contribution of the soft-excess from 2003 to 2018 using the
nthcomp+zGaussian+powerlaw model and the results are presented in Table 3.
Subsequently, we freeze the $\Gamma^{nth}$ obtained from nthcomp while fitting
the soft excess below 3 keV. The $\Gamma^{pl}$ fits the soft-excess < 3 keV.
For every observation, we find a soft-excess steeper than the primary
continuum (see, Table 3) which is a characteristic associated with the Narrow
line Seyfert 1 galaxies. Apart from the steeper power law, the variation of
soft-excess luminosity and spectral index can be observed from long term
observations presented in Table 3. We have calculated the intrinsic
luminosities of nthcomp and powerlaw within the energy range 0.5 to 10.0 keV.
In Fig. 6a, we see a strong correlation (PCC=0.9227) between the intrinsic
luminosities of soft-excess ($L^{SE}_{int}$) and primary continuum
($L^{PC}_{int}$). However, as a “bare” type AGN, Ark 120 has not shown any
correlation (Fig. 6b) among the intrinsic luminosities and the line of sight
hydrogen column density ($N_{H}$).
While nthcomp provides a good fit in the high energy range, we have used
TCAF+zGaussian+pexrav model (presented in Table 5) in the entire range. We
find that the TCAF fits well in the range of $0.2-10$ and requires no other
additional model for the soft-excess part with the range of $0.2-3$ keV. The
fitted results and residuals are presented in Fig. 2. From the spectral
fitting using TCAF, one recognizes that the soft-excess could be originated
from the photons which are rarely scattered in the Compton cloud. The
surrounding halo will contribute to this energy band (0.2 - 2 keV). Also, some
high energy photons from the Compton cloud which could be reflected from the
disc will appear in this energy range after losing their energy through
reflection from the cold disc. We have performed Monte-Carlo simulations to
show the spectral variations with $N_{s}$. This is briefly discussed in Sec.
5.3.1.
#### 5.3.1 Simulated spectra
Radiative and hydrodynamic origin of soft-excess has been investigated in
Fukumura et al. (2016) where they proposed that the shock heating near the
ISCO could produce the soft-excess. The model reproduced the spectra of “bare”
Seyfert 1 galaxy, Ark 120. We have inspected the possibility of scattering
dependent spectral contribution from the pre-shock and the post-shock regions
(Chakrabarti & Titarchuk, 1995). We extend the work of Ghosh et al. (2011);
Chatterjee et al. (2018) in case of AGNs considering Ark 120. Using the Total
Variation Diminishing (TVD) scheme (Ryu et al., 1997), we inject matter having
a halo rate of $0.1$ from the outer boundary at $200~{}r_{g}$. TCAF fitted
parameters are used for the simulation setup and are mentioned in the Fig. 7.
Considering the Keplerian disc in the equatorial plane ($z=0$), we construct
the profile of the accretion disc following Shakura & Sunyaev (1973). The
Monte-Carlo simulation ($0<r<100~{}r_{g}$) has followed the process provided
by Pozdnyakov et al. (1983) and later extended by Ghosh et al. (2009);
Chatterjee et al. (2017a). The simulations are performed using $10^{7}$
injected photons for each case. The emergent Comptonized spectra are plotted
in Fig. 7. We show the variation of spectral components with respect to the
number of scatterings (see also Ghosh et al. (2011)) within the region. From
Fig. 7, we find that the spectra harden as the number of scatterings increase.
The spectra of the primary component within the energy range 2.0 to 10.0 keV
is dominated by the photons where the number of scatterings are $\geq 10$.
However, the soft-excess, the red long-dashed line within 0.2-2 keV, is
dominated by the contribution from photons which have suffered $\leq 10$
scatterings. A steeper spectral slope ($\Gamma^{SE}$) for soft-excess is
achieved with respect to the primary component ($\Gamma^{PC}$) for both of the
spectrum. This is similar to what has been observed for Ark 120 (Table 3). It
is to be noted that, Boissay et al. (2016) studied the AGN 102 Sy1 and found
that there is no link between the reflection and the soft excess. Instead,
they indicated that the soft-excess could be related to the thermodynamical
properties of Compton cloud and associated medium.
## 6 Conclusions
We have studied $\sim 15$ years of X-ray data of Ark 120. We find the source
varied considerably within that time span. This source was previously reported
to be a ‘bare-type AGN’ and we also find a similar nature of this source from
the long term analysis. The X-ray count rate has increased by a factor of two
in a few years, and it is not found to be related to the Hydrogen column
density ($N_{H}$) since it is a ‘bare-type AGN’. Following are the major
findings from our work.
1. 1.
The spectral slopes of the primary continuum ($\Gamma^{PC}$) and the soft-
excess ($\Gamma^{SE}$) are not constant throughout our observational time
span. $\Gamma^{PC}$ has varied between 1.60 and 2.08 whereas $\Gamma^{SE}$
between 2.52 and 4.23 from 2003 to 2018.
2. 2.
The variation is reflected in fitted parameters of TCAF, namely, the accretion
rates and properties of the Compton cloud. From the spectral fitting using
TCAF, we find that the disc rate ($\dot{m}_{d}$) and the halo rate
($\dot{m}_{h}$) have varied between $0.061$ and $0.126$ and between $0.108$
and $0.191$ respectively. The shock location ($X_{s}$) or the size of the
Compton cloud and compression ratio ($R$) vary correspondingly. $X_{s}$ varies
between $20.36$ and $57.87$, whereas $R$ varies between $1.66$ and $2.73$.
3. 3.
We focussed on the simultaneous observations in low ($0.2-2.0$ keV) and high
($3.0-10.0$ keV) energy X-ray band from XMM-Newton to calculate the time delay
between them. We find that in XMM1 observation, there is no delay between the
low and high energy band, while a positive delay of $4.71\pm 1$ ks is detected
in XMM2 observation and a negative delay of $4.15\pm 1$ ks is seen in XMM3
observation. A correlated variability among the optical, UV, and X-ray bands
have already been reported Lobban et al. (2020). Also, (Dutta & Chakrabarti,
2016; Chatterjee et al., 2017b) reported in a different context that the X-ray
lag has a strong dependency on the geometric structure of the Comptonization
region and orientation of the Keplerian disc. The net delay is a resultant
effect of different physical mechanisms, e.g., Comptonization, reflection,
focusing, and jet/outflow emission (Chatterjee et al., 2019; Patra et al.,
2019). For the lower inclination and radio-quiet nature of Ark 120, the
positive delay could be attributed to the Compton delay while reflection and
light-crossing delay could contribute to the negative delay.
4. 4.
From the analysis of the long term data, we report that the luminosity is
independent of Hydrogen column density ($N_{H}$). This is expected as the
source has a negligible line-of-sight hydrogen column density ($N_{H}<5\times
10^{20}$). The luminosity of the primary continuum is highly correlated
(PCC$\sim 0.92$) with the soft excess emission. From TCAF fitting and Monte-
Carlo simulations using TCAF flow configurations, we show that the soft-excess
spectral slope ($\Gamma^{SE}$) is the result of a fewer Compton scatterings in
the Compton cloud and the primary continuum ($\Gamma^{PC}$) is the result of
the higher number of Compton scatterings. Corresponding intrinsic luminosities
obtained from simulations corroborate with the observed pattern.
## Acknowledgements
PN acknowledges CSIR fellowship for this work. AC acknowledges Post-doctoral
fellowship of S. N. Bose National Centre for Basic Sciences, Kolkata India,
funded by Department of Science and Technology (DST), India. BGD acknowledges
Inter-University Centre for Astronomy and Astrophysics (IUCAA) for the
Visiting Associateship Programme. This research has made use of data and/or
software provided by the High Energy Astrophysics Science Archive Research
Center (HEASARC), which is a service of the Astrophysics Science Division at
NASA/GSFC and the High Energy Astrophysics Division of the Smithsonian
Astrophysical Observatory. This work has made use of data obtained from the
Suzaku, a collaborative mission between the space agencies of Japan (JAXA) and
the USA (NASA). This work made use of data supplied by the UK Swift Science
Data Centre at the University of Leicester. This work has made use of data
obtained from the NuSTAR mission, a project led by Caltech, funded by NASA and
managed by NASA/JPL, and has utilized the NuSTARDAS software package, jointly
developed by the ASDC, Italy and Caltech, USA. This research has made use of
the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet
Propulsion Laboratory, California Institute of Technology, under contract with
the National Aeronautics and Space Administration. This research has made use
of the SIMBAD database, operated at CDS, Strasbourg, France.
## Data Availability
We have used archival data for our analysis in this manuscript. All the
softwares used in this manuscript are publicly available. Appropriate links
are given in the manuscript.
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|
# Bayesian Optimization Assisted Meal Bolus Decision Based on Gaussian
Processes Learning and Risk-Sensitive Control[1]
Deheng Cai Wei Liu Linong Ji Dawei Shi11footnotemark: 1<EMAIL_ADDRESS>State Key Laboratory of Intelligent Control and Decision of Complex Systems,
School of Automation, Beijing Institute of Technology, Beijing, China
Department of Endocrine and Metabolism, Peking University People’s Hospital,
Beijing, China
###### Abstract
Effective postprandial glucose control is important to glucose management for
subjects with diabetes mellitus. In this work, a data-driven meal bolus
decision method is proposed without the need of subject-specific glucose
management parameters. The postprandial glucose dynamics is learnt using
Gaussian process regression. Considering the asymmetric risks of hyper- and
hypoglycemia and the uncertainties in the predicted glucose trajectories, an
asymmetric risk-sensitive cost function is designed. Bayesian optimization is
utilized to solve the optimization problem, since the gradient of the cost
function is unavailable. The proposed approach is evaluated using the 10-adult
cohort of the FDA-accepted UVA/Padova T1DM simulator and compared with the
standard insulin bolus calculator. For the case of announced meals, the
proposed method achieves satisfactory and similar performance in terms of mean
glucose and percentage time in [70, 180] mg/dL without increasing the risk of
hypoglycemia. Similar results are observed for the case without the meal
information (assuming that the patient follows a consistent diet) and the case
of basal rate mismatches. In addition, advisory-mode analysis is performed
based on clinical data, which indicates that the method can determine safe and
reasonable meal boluses in real clinical settings. The results verify the
effectiveness and robustness of the proposed method and indicate the
feasibility of achieving improved postprandial glucose regulation through a
data-driven optimal control method.
###### keywords:
Meal bolus decision , Gaussian processes , risk-sensitive control , Bayesian
optimization.
††journal: Control Engineering Practice11footnotetext: This work was supported
in part by the National Natural Science Foundation of China under Grant
61973030, and in part by the Peking University People’s Hospital Scientific
Research Development Funds RDY2019-05.
## 1 Introduction
Diabetes mellitus (DM) is a chronic metabolic disease that is characterized by
absolute insulin deficiency (type 1 diabetes (T1D)) or relative lack of
insulin secretion and sensitivity (type 2 diabetes (T2D)). Patients with DM
tend to suffer from the long term complications, e.g., retinopathy and
nephropathy, due to poor blood glucose (BG) managements [1]. Nowadays,
external insulin administration through a basal-bolus strategy with multiple
daily injections (MDI) or continuous subcutaneous insulin infusion (CSII) pump
is a popular way among the patients to control BG [2]. With improved modern
diabetes technologies, a closed-loop control system for BG regulations, named
artificial pancreas (AP), is further developed through integrating the pumps
and continuous glucose monitoring (CGM) sensors. An AP automatically delivers
insulin to achieve desired BG levels, based on CGM-driven feedback control
algorithms [3, 4, 5, 6].
However, for the therapies above, the effective control of postprandial
glucose still remains a challenge [7]. At present, the feedforward control
action in terms of preprandial insulin bolus is widely adopted to counteract
the hyperglycemia associated with meal intakes. Specifically, a meal bolus is
determined by a bolus calculator according to carbohydrate content of the
meal, the BG level and subject-specific settings (carbohydrate ratio (CR) and
correction factor (CF) profiles). Multiple efforts have been devoted to
improve the performance of the bolus advisors. Considering the repetitive
nature of the daily activities of the patient, Owens _et al._ [8] proposed a
Run-to-Run (R2R) algorithm to update the insulin bolus amount and timing
daily. In Schiavon _et al._ [9], a novel optimization method for CR was
developed based on a validated index of insulin sensitivity estimated form CGM
and CSII data. Combining case-based reasoning (CBR) with R2R, an advanced
insulin bolus advisor through adapting CR and CF was presented in Herrero _et
al._ [10]. Similarly, Torrent-Fontbona [11] investigated a bolus insulin
recommend system based on CBR, but provided a new reuse, revise and retain
mechanism to adapt CR and CF. Besides CBR, other artificial intelligence
techniques including fuzzy logic [12] and reinforcement learning (RL) [13]
have been investigated for the bolus advisors. Moreover, several algorithms
have also been explored to calculate the required bolus within the framework
of an AP. In Turksoy _et al._ [14], a meal bolus calculation method was
developed for unannounced meals based on the Bergman’s minimal model and
unscented Kalman filter. In Toffanin _et al._ [15], R2R was implemented in the
AP to adapt CR. Shi _et al._ [16] explored a Bayesian optimization assisted
learning framework to adapt CR profiles for the AP, using historical CGM and
CSII data.
Most of the methods above improve postprandial glucose control through
updating CR and CF according to the designed mechanisms. These methods do not
capture the dynamics of glucose metabolism, and ignore the upcoming
postprandial glucose situations in the optimization of bolus dosage, thus
leading to potentially sub-optimal glycemic control. With the improved glucose
sensor accuracy and accessibility, data-driven optimal control provides a
promising way to achieve optimal glycemic control and reduce the burden of
optimizing CR and CF. The data-driven optimal control can explicitly exploit
the modeled dynamics for the bolus decision by taking account of the
preprandial glucose levels, and optimizing the bolus dosage using the
predicted postprandial glucose situations. This forms the motivation of our
work. To implement the data-driven optimal control for the meal bolus
decision, glucose prediction is a critical role. Along this direction, many
methods have been explored in the literature [17]. For example, an
autoregressive with exogenous input model was presented in Romero-Ugaldein _et
al._ [18] to predict interstitial glucose. In Yu _et al._ [19], four different
adaptive filters and a fusion mechanism were proposed for the online glucose
concentration predictions. Combining feature ranking with support vector
regression or Gaussian processes, Georga _et al._ [20] investigated the short-
term glucose prediction. To improve long-term glucose prediction, Montaser _et
al._ [21] presented an integrated predicting method based on seasonal local
models and fuzzy _c_ -means. Different from the predictions above, using the
Gaussian process (GP) regression, we provide multi-step predictions (e.g., 2
hours) for the postprandial glucose trajectories corresponding to various
preprandial glucose situations, meal boluses and meal information
(carbohydrate content), so that the optimization for the meal bolus can be
done via the predicted glucose trajectories.
Since GP is a data-efficient and robust modeling method, different approaches
have concerned the research of GP-based control. For example, in [22], a
framework of probabilistic inference for learning control was proposed based
on the GP and applied in real robotics and control tasks. In [23], a GP-based
model predictive controller (MPC) was investigated for building energy control
and demand response. To enhance effective online learning and control, a risk-
sensitive cost was introduced in the MPC with GP models [24], as well as in
the RL [25]. Inspired by the work in [24, 25], we utilize the predicted
information in a risk-sensitive fashion, but construct an asymmetric risk-
sensitive cost with the consideration of asymmetric risks in hyper- and
hypoglycemia. Finally, based on the designed cost, a constrained stochastic
optimization problem is proposed for the meal bolus decision. Since the
gradient of the cost cannot be computed analytically, we utilize Bayesian
optimization [26] and Monte-Carlo method to solve the optimization problem.
For safety reasons, the final solution for the meal bolus enforces an insulin
on board (IOB) constraint.
The effectiveness and robustness of the proposed method are evaluated using
different _in silico_ protocols on the 10-adult cohort of the US Food and Drug
Administration (FDA) accepted Universities of Virginia (UVA)/Padova T1DM
simulator, and compared with the standard insulin bolus calculator. For the
case of announced meals, the proposed method achieves satisfactory and similar
performance for scenarios of nominal basal rates, in terms of mean glucose
level and percent time in the safe range, without increasing the risk of
hypoglycemia. Similar results are observed for the case without the meal
information (assuming that the patient follows a consistent diet) and the
scenarios of over/under-estimated basal rates. In addition, advisory-mode
analysis [27] based on clinical data from a T1DM subject show that the
proposed method can determine reasonable meal boluses by explicitly taking
account of the preprandial glucose levels in the data-driven optimal control.
Figure 1: Schematic of the proposed method.
## 2 Materials and Methods
The overall structure of the proposed data-driven meal bolus decision method
is illustrated in Fig. 1. The method builds on three key components: model
learning, asymmetric risk-sensitive control, and Bayesian optimization.
Model learning is responsible for constructing the postprandial glucose
dynamics. Here, the aim is to provide a robust description for postprandial
glucose dynamics using GPs. Specifically, fed with the serialized data samples
(including preprandial glucose measurements, the corresponding meal
information, bolus dosage and postprandial glucose measurements), the GPs are
trained offline and then applied online to provide the prediction and the
uncertainty estimation of postprandial glucose trajectories. Considering the
asymmetric risks of hyper- and hypoglycemia and the uncertainties in the
predicted glucose trajectories, we develop an asymmetric risk-sensitive cost
function to favor safe control actions. Finally, a constrained stochastic
optimization problem is formulated for the meal bolus decision based on the
designed cost function. Since the gradient of the cost function is
unavailable, we solve the optimization problem using Bayesian optimization and
Monte-Carlo simulations. To ensure the safety of the method, IOB constraints
are also incorporated.
### 2.1 Gaussian Processes for Model Learning
#### 2.1.1 Gaussian Processes
A GP assumes a distribution over random functions
$f(x):\mathbb{R}^{n}\rightarrow\mathbb{R}$, such that values of $f$ at any
input $x$ have a joint Gaussian distribution [28, 29], which is denoted as
$\displaystyle f(x)\sim\mathcal{GP}(m(x),k(x,x^{\prime})),$ (1)
where $m(x)$ and $k(x,x^{\prime})$ are the mean function and positive
semidefinite covariance function, respectively, which have the form:
$\displaystyle m(x)$ $\displaystyle=\mathbb{E}_{f}[f(x)],$ (2) $\displaystyle
k(x,x^{\prime})$
$\displaystyle=\text{cov}_{f}[f(x),f(x^{\prime})],~{}x,x^{\prime}\in\mathbb{R}^{n}.$
(3)
In GPs, we can encode our prior knowledge about the process by designing
corresponding prior mean and covariance functions, which are parameterized by
the parameter vector $\theta_{f}$. Considering the noisy observations $y$ of
$f$ with the form of $y=f(x)+\omega$, where
$\omega\sim\mathcal{N}(0,\sigma_{\omega}^{2})$ is a white Gaussian noise, the
covariance function of $y$ has the form:
$\displaystyle
k_{y}(x,x^{\prime})=k(x,x^{\prime})+\sigma_{\omega}^{2}\delta(x,x^{\prime}),$
(4)
where $\delta(x,x^{\prime})$ is the Kronecker delta function which is one if
and only if $x=x^{\prime}$ and zero otherwise. Given $N$ training inputs
$X=[x_{1},x_{2},...,x_{N}]^{\top}$ and the corresponding observations
$Y=[y_{1},y_{2},...,y_{N}]^{\top}$, the posterior GP hyper-parameters
$\theta=[\theta_{f},\sigma_{\omega}]$ are learned by maximizing the log-
marginal likelihood: $\mathop{\arg\max}_{\theta}\log(p(Y|X,\theta))$ [29].
With the determined hyper-parameters, the GP can infer the posterior
distribution of $y_{*}$ corresponding to a new input $x_{*}$:
$y_{*}\sim\mathcal{N}(m_{*},\sigma_{*}^{2})$ with
$\displaystyle m_{*}$
$\displaystyle=m(x_{*})+K_{*}^{\top}(K+\sigma_{\omega}I)^{-1}(Y-m(X)),$ (5)
$\displaystyle\sigma_{*}^{2}$
$\displaystyle=k(x_{*},x_{*})-K_{*}^{\top}(K+\sigma_{\omega}I)^{-1}K_{*},$ (6)
where $K_{*}=[k(x_{*},x_{1}),...,k(x_{*},x_{N})]^{\top}$ and $K$ is the
covariance matrix with elements $K_{ij}=k(x_{i},x_{j})$. As shown in (5) and
(6), for a new input, a GP can provide a determined prediction via predictive
mean, c.f., (5), along with an estimate of uncertainty or confidence in the
prediction via the predictive variance, c.f., (6).
#### 2.1.2 Postprandial Glucose Prediction
Physiologically-based compartmental models are commonly utilized to describe
glucose dynamics using first order differential equations [30, 31, 32];
however, the parameter identification of these models is time-consuming and
sometimes is even impossible with incomplete information, e.g., when only the
CGM data is available. Here, we utilize GPs to perform robust modeling for
postprandial glucose dynamics with incomplete and noisy information.
Specifically, by feeding autoregressive, or time-delayed, input and output
signals back to the model as regressors, the GPs can be used for modeling
nonlinear dynamical control systems [22, 23].
To do this, the postprandial glucose (PG) dynamics is described in a multistep
form using autoregressive models, and each step is separately represented by a
nonlinear function with additive noise, which has the form:
$\displaystyle P_{t+1}=$ $\displaystyle f_{t}(z_{t})+w_{t},$ $\displaystyle
z_{t}=$ $\displaystyle[P_{t-l},...,P_{t},u,d]^{\top};$ $\displaystyle
P_{t+2}=$ $\displaystyle f_{t+1}(z_{t+1})+w_{t+1},$ $\displaystyle z_{t+1}=$
$\displaystyle[P_{t-l+1},...,P_{t+1},u,d]^{\top};$ $\displaystyle...$
$\displaystyle P_{t+n}=$ $\displaystyle f_{t+n-1}(z_{t+n-1})+w_{t+n-1},$
$\displaystyle z_{t+n-1}=$
$\displaystyle[P_{t+n-1-l},...,P_{t+n-1},u,d]^{\top},$ (7)
where $t$ denotes the time of the meal intake, $w$ is a white Gaussian noise,
$P$ is the glucose measurement, and $l$ is the lag for autoregressive outputs;
$u$ is the meal bolus, and $d$ is the carbohydrate intake. To convey the most
information of glucose situations, the lag of $l=7$ and the sampling period of
$T=15\min$ are considered. This corresponds to the lag of 2 hours.
Correspondingly, we take $n=8$; since the sampling period is 15 $\min$, this
corresponds to the duration of 2 hours.
The utilization of GPs is divided into two stages: offline modeling and online
prediction. In the offline modeling stage, based on (7), the GPs are
separately used to model the PG dynamics in each step following a similar way.
For example, for the time step $t+1$, we use
$z_{t}=[P_{t-7},...,P_{t},u,d]^{\top}$ as training inputs, and the differences
$\Delta P_{t}=P_{t+1}-P_{t}$ as training targets to reduce the prediction
uncertainty [22]. A linear mean function and a commonly-used covariance
function known as squared exponential (SE) covariance kernel are considered:
$\displaystyle m(z_{t})=$ $\displaystyle a^{\top}z_{t}+b,$ (8) $\displaystyle
k_{z}(z_{t},z_{t}^{\prime})=$
$\displaystyle\sigma_{f}^{2}\exp\left(-\frac{1}{2}(z_{t}-z_{t}^{\prime})^{\top}\Omega^{-1}(z_{t}-z_{t}^{\prime})\right)+\delta(z_{t},z_{t}^{\prime})\sigma_{\omega}^{2},$
(9)
where $\Omega=\text{diag}\\{[l_{1}^{2},l_{2}^{2},...,l_{10}^{2}]\\}$,
$\sigma_{f}^{2}$ denotes the signal variance, and the characteristic length
scales for input space $l_{1},l_{2},...,l_{10}$ describe the smoothness of the
function. Note that $u$ and $d$ stay the same for all steps. Moreover, to
highlight the effects of $u$ and $d$ on the glucose regulation, the glucose
measurements in the training inputs for each step are separately normalized
into $[0,1]$ using min-max normalization.
In the online prediction stage, given a bolus dosage and known carbohydrate
amount intakes, by iteratively feeding back the predictive mean of previous
step into the input for the prediction, we can obtain the prediction for each
step using corresponding trained GPs. This prediction corresponds to the
difference between the current step and previous step. We then add this
prediction with the predicted mean of the previous step to determine the final
prediction in current step. Uniting the predictions for the all steps, the GPs
are able to provide 8-step predictions for the PG trajectories that correspond
to the corresponding preprandial glcucose situations, carbohydrate intakes and
meal boluses. Note that when the eating habit of a subject is approximately
consistent in terms of timing and sizes of meal intake, we can approximate the
effect of similar food intake as an invariant disturbance and the meal size
information can be optional for postprandial glucose prediction, utilizing the
robust prediction ability of the GPs; this allows the design of a bolus
decision algorithm without meal announcements.
### 2.2 Asymmetric Risk-Sensitive Control
#### 2.2.1 Asymmetric Risk-Sensitive Cost
The glucose control problem is highly asymmetric in the sense that
consequences of hypoglycemia are immediate and more detrimental in comparison
with those of (temporary) hyperglycemia; therefore, we try to correct the
postprandial hyperglycemia while taking extra care of hypoglycemia. One
feasible method to address this issue is to construct asymmetric cost
functions [33, 34]. Inspired by the work in [34], we penalize deviations above
and below the target asymmetrically; in addition, considering the
uncertainties in postprandial glucose prediction, the cost function is built
in the risk-sensitive (RS) framework.
To do this, we denote the 8-step predictions provided by the GPs as
$G_{i}\sim\mathcal{N}(m_{i},\sigma_{i}^{2})$, $i\in\\{1,2,...,8\\}$,
respectively, and collect them as a vector state
$G=[G_{1},G_{2},...,G_{8}]^{\top}$, which describes the probability
distribution of postprandial glucose trajectories. According to the principle
of risk-sensitive analysis [35], an asymmetric RS cost is designed as follows:
$\displaystyle\mathcal{L}_{ARS}=-\frac{2}{\gamma}\log$
$\displaystyle\mathbb{E}\left[\exp\left(-\frac{\gamma}{2}(G-G_{r})_{+}^{\top}Q^{+}(G-G_{r})_{+}\right.\right.$
$\displaystyle\left.\left.-\frac{\gamma}{2}(G-G_{r})_{-}^{\top}Q^{-}(G-G_{r})_{-}\right)\right],$
(10)
where
$\displaystyle(G-G_{r})_{+}=[(G_{1}-G_{r1})\mathbf{1}(G_{1}-G_{r1}\geq
0),...,$ $\displaystyle(G_{8}-G_{r8})\mathbf{1}(G_{8}-G_{r8}\geq 0)]^{\top},$
(11)
$\displaystyle(G-G_{r})_{-}=[(G_{1}-G_{r1})\mathbf{1}(G_{1}-G_{r1}<0),...,$
$\displaystyle(G_{8}-G_{r8})\mathbf{1}(G_{8}-G_{r8}<0)]^{\top},$ (12)
and $\mathbf{1}(\cdot)$ denotes the indicator function; $G_{r}$ is the target
for the postprandial glucose management; $Q^{+}$ is a positive penalty matrix
for the glucose excursions above the target; $Q^{-}$ is a negative penalty
matrix for the glucose excursions below the target; $\gamma<0$ is a risk
sensitivity parameter that determines the system’s attitude towards
uncertainty [24, 25]. With the risk-sensitive cost function, the optimizer is
able to fully exploit the experience learned from the historical data while
keeping its own decision-making ability.
As for the design of asymmetric penalty, $Q^{+}$ is designed as a constant
diagonal matrix. Based on the designed $Q^{+}$, the diagonal elements of
$Q^{-}$ are devised to increase exponentially with the increase of the
absolute deviation from target while being restricted by upper and lower
bounds. Specifically, the $i$th diagonal element of $Q^{-}$ has the form of
$\displaystyle
Q^{-}_{i}:=Q^{+}_{i}\left(\frac{c_{1}}{1+\exp\\{\alpha(\beta-|G_{i}-G_{ri}|)\\}}+c_{2}\right),$
(13)
where $Q^{+}_{i}$ is the $i$th diagonal element of $Q^{+}$;
$\Gamma:=[\alpha,\beta,c_{1},c_{2}]$ is a quadruple determines the penalty
intensity, which is designed same for the all diagonal elements. The lower and
upper bounds are determined by $c_{2}$ and $c_{1}+c_{2}$, respectively, and
the rate of increase is controlled by $\alpha$. The parameter design will be
discussed in Section 3.
Based on the above design, the quadratic penalty using constant $Q^{+}$ is
scaled on the excursions above the target to maintain a reasonable but active
response to hyperglycemia. Comparatively, a quadratic penalty with
exponentially weighted coefficients is applied to the glucose excursions below
the target to have a reasonably conservative response to the glucose
excursions near the target, while maintaining the ability to respond quickly
to larger glucose excursions and safely compensate for hypoglycemia.
#### 2.2.2 GP-Based Asymmetric Risk-Sensitive Control
Given the designed asymmetric risk-sensitive cost $\mathcal{L}_{ARS}$ in (10),
the GP-based asymmetric risk-sensitive control for the meal bolus decision is
formulated as the following constrained stochastic optimization problem,
$\displaystyle\min_{u}$
$\displaystyle~{}\mathcal{L}(u):=\mathcal{L}_{ARS}+Ru^{2},$ (14) s.t.
$\displaystyle~{}G_{i}\sim\mathcal{N}(m_{i},\sigma_{i}),~{}i\in\\{1,...,8\\}$
(15) $\displaystyle~{}0\leq u\leq u_{\max},$ (16)
where $u$ is the meal bolus to be optimized, $R$ is the input weighting to
compromise the asymmetric RS cost and the actual needed bolus dosage, $m_{i}$
and $\sigma_{i}$ are parameters provided by the GPs.
### 2.3 Bayesian Optimization for Meal Bolus Decision
Since the gradient of the cost function cannot be obtained analytically,
Bayesian Optimization (BO) [26] is employed to solve the above constrained
stochastic optimization problem.
#### 2.3.1 Model Learning for the Cost Function
Here, we utilize a new GP to construct an approximation of a complex map from
the decision variable $u$ to the cost function value $\mathcal{L}(u)$ in (14).
Specifically, we consider a prior zero mean function and the SE covariance
function in (9) with a scalar input. As discussed in Section 2.1, given the
set of $N$ past observations
$\mathcal{D}_{1:N}=\\{u_{1:N},\mathcal{L}(u_{1:N})\\}$, the GP is trained and
then applied to predict the cost function value for a candidate meal bolus
$u_{*}$ according to (5)-(6). The prediction is denoted as
$\hat{\mathcal{L}}(u_{*})$ and will be utilized to construct the acquisition
function (see Section 2.3.2).
Note that the value of the cost function corresponding to the decision
variable is estimated by Monte-Carlo simulations. Specifically, given a bolus
dosage, we generate 1000 samples for the postprandial glucose trajectory based
on the joint Gaussian distribution provided by the GP in Section 2.1. The
average cost for these samples is further calculated, which is then regarded
as the observed value of the cost corresponding to the bolus dosage.
#### 2.3.2 Acquisition Function
As a critical ingredient of the BO, the acquisition function guides the
optimization by determining the optimum candidate point for the next
evaluation. Specifically, utilizing the prediction information offered by the
model learning phase, the acquisition function is constructed to determine the
candidate point by maintain a trade-off exploration of the search space and
exploitation of current promising areas. Up to now, there are rich literature
concerning the acquisition function design [26], where several structures have
been developed, e.g., probability of improvement, expected improvement, upper
confidence bound, and entropy search. In this work, the expected improvement
is considered.
Compared with probability of improvement, the expected improvement (EI)
acquisition function [36] also incorporates the amount of improvement in
selecting the candidate points. Specifically, to minimize the cost in (14),
the improvement function of the EI acquisition function is defined as
$\displaystyle
I(u_{*}):=(\mathcal{L}^{p}_{\min}-\hat{\mathcal{L}}(u_{*}))\mathbf{1}(\mathcal{L}^{p}_{\min}>\hat{\mathcal{L}}(u_{*})),$
(17)
where $\mathcal{L}^{p}_{\min}$ denotes the minimum observed value of the cost
so far, $\hat{\mathcal{L}}(u_{*})\sim\mathcal{N}(m(u_{*}),\sigma^{2}(u_{*}))$
is the prediction at candidate point $u_{*}$. Note that the term
$\mathcal{L}^{p}_{\min}-\hat{\mathcal{L}}(u_{*})$ represents the amount of
improvement, and the other term denotes the probability of that improvement.
Based on the improvement function, the EI acquisition function is further
defined as
$\displaystyle\alpha_{EI}(u_{*}):=\mathbb{E}(I(u_{*})).$ (18)
By calculating the expectation in (18), we have
$\displaystyle\alpha_{EI}(u_{*})=$
$\displaystyle\left\\{\begin{array}[]{ll}(\mathcal{L}^{p}_{\min}-m(u_{*}))\Phi(U)+\sigma(u_{*})\phi(U),&\textrm{if~{}}m(u_{*})>0,\\\
0,&\textrm{otherwise,}\end{array}\right.$ (21)
where
$\displaystyle U=\frac{\mathcal{L}^{p}_{\min}-m(u_{*})}{\sigma(u_{*})},$ (22)
Algorithm 1 Bayesian optimization for meal bolus decision
1: $\mathcal{D}\leftarrow$ Initialize: $\\{u_{1:8},\mathcal{L}(u_{1:8})\\}$,
where $u_{1:8}$ are selected equidistantly among the bound in (16)
2: while the number of iterations $\leq M$ do
3: Train a GP model from $\mathcal{D}$
4: Compute the prediction at candidate $u_{*}$ by the GP (Eq. (5)-(6))
5: Determine candidate $u_{*}$ by maximizing the acquisition function (Eq.
(2.3.2))
6: Observe $\mathcal{L}(u_{*})$ at determined $u_{*}$ using the Monte-Carlo
method (see Sec. 2.3.1)
7: Append $\\{u_{*},\mathcal{L}(u_{*})\\}$ to $\mathcal{D}$
8: end while
9: Select the input value that corresponds to the minimum observed value of
the cost as the final solution
and $\Phi(\cdot)$ is the standard normal cumulative distribution function, and
$\phi(\cdot)$ denotes the standard normal probability density function. The
candidate point for the next evaluation is determined as the one that
maximizes $\alpha_{EI}$.
The BO algorithm that solves the optimization problem in (14)-(16) is
summarized in Algorithm 1. After $M$ sequential operations of the BO, the
final solution $\widetilde{u}_{b}$ is determined. For safety concern, an IOB
constraint [37] is enforced to prevent overbolus based on insulin delivery
history. Denoting the IOB constraint as $u_{\text{IOB}}$, the final meal bolus
is determined as
$\displaystyle u_{b}=\widetilde{u}_{b}-u_{\text{IOB}}.$ (23)
## 3 Data collection and Parameter design
The parameters of the proposed method are designed and evaluated using the
UVA/Padova T1DM metabolic simulator [38]. The “average patient” of the
simulator is selected to perform the parameter design, and the obtained
parameters are further evaluated using the 10 virtual adult patients.
### 3.1 Data Collection
Before the parameter design and evaluation, the data samples needed for the PG
model learning are collected for each patient. The samples are collected under
the situation where the _in silico_ patients combine basal insulin and bolus
insulin to control BG. To do this, a protocol with nominal basal rate and
announced meals are designed for the patients. The protocol starts from 5:00
on day one and lasts one week (7 days). Considering the daily activities of
diabetic patients tend to form similar patterns, e.g., with respect to meal
timing and meal amount, but to mimic lifestyle disturbances, we assume the
patients take breakfast, lunch, and dinner with normally distributed meal
sizes (with means and standard deviations equal to [50, 75, 75] g and [3, 4,
4] g CHO) and meal times uniformly distributed in [07:00, 09:00], [11:00,
13:00], and [18:00, 20:00], respectively. The meals are all announced but the
meal boluses are calculated with potentially inappropriate CR, such that the
variation of the bolus is uniformly distributed in [-30%, +30%].
Table 1: Parameters for the proposed method Variables | Value
---|---
$\gamma$ | -2
$R$ | 4
$u_{\max}$ | 15
$M$ | 25
$Q^{+}$ | $\text{diag}\\{[0.01,\cdots,0.01,0.02,0.02]\\}$
$\Gamma$ | $[1,10,5,1]$
$G_{r}$ | $[100,120,140,160,160,150,140,140]^{\top}$
With the generated data by the protocol, we totally collect 7 samples for the
breakfast, lunch and dinner, respectively. Each sample includes preprandial
glucose measurements, corresponding bolus dosage, carbohydrate amount, and
postprandial glucose measurements. As the consumed CHO for lunch and dinner
are similar but different from the breakfast, we construct the same PG model
for both lunch and dinner, and build a separate model for the breakfast using
corresponding samples, although it is also possible to build separate models
for breakfast, lunch and dinner.
### 3.2 Parameter Design
Using the collected samples, the GPs are trained offline to provide PG
predictions for the online control of the method. In the online control, the
parameter design and its evaluation are performed. In the parameter design
phase, a 12-hour _in silico_ protocol starting from 6:00 is employed, where
breakfast (50 g CHO), lunch (75 g CHO) are consumed at 8:00 and 12:00,
respectively.
The parameters are designed using a trial-and-error approach based on the
glucose data obtained from the “average patient”, such that satisfactory
postprandial regulation performance in terms of average glucose, percent time
in [70, 180] mg/dL, and percent time below 70 mg/dL can be achieved. The
obtained parameters are summarized in Table 1. In the evaluation phase,
different scenarios are designed to evaluate the performance of the proposed
method with the obtained parameters. The results are reported in Section 4.
## 4 Performance Analysis
As mentioned above, the proposed method is evaluated on the 10-adult cohort of
the UVA/Padova T1DM simulator. Besides, the advisory-mode analysis [27] is
performed for the method based on the clinical data from a particular T1DM
subject who undertook the MDI therapy.
For the _in silico_ evaluations, two protocols are designed to perform the
comparison of the proposed method and the standard insulin bolus calculator
(denoted as “Control”), which has the form:
$\displaystyle
u_{\text{bolus}}=\frac{\text{CHO}}{\text{CR}}+\frac{G_{c}-G_{sp}}{\text{CF}}-u_{\text{IOB}},$
(24)
where $G_{c}$ is the current glucose level (mg/dL); $G_{sp}$ is the glucose
set-point (mg/dL), selected as $140$ in this comparison. One of the designed
protocol (denoted as “Protocol A”) begins at 5:00 on day 1 and lasts two days
(48 hours) where breakfast (55 g and 45 g CHO), lunch (65 g and 85 g CHO) and
dinner (85 g and 65 g CHO) are consumed at 8:00, 12:00 and 18:00 for two days,
respectively. The other protocol (denoted as “Protocol B”) begins at 5:00 and
lasts one day (24 hours), including breakfast, lunch and dinner with 45 g, 85
g, and 65 g of CHO content at 8:00, 12:00 and 18:00, respectively. Based on
protocol A, _in silico_ experiments are performed on the whole 10-adult cohort
with nominal basal rates using additive CGM noises for scenario (not)
utilizing the meal information for the prediction, thus a total of 10
simulations are performed for each scenario and each method. The statistical
results are summarized in Table 2, and the comparison performance in terms of
glucose regulation and meal bolus is illustrated in Fig. 2, where the 5%, 25%,
75% and 95% quartile curves together with the median curves are presented.
Besides, to illustrate the robust decision-making ability of the proposed
method, _in silico_ evaluations without meal information for scenarios of
basal-rate mismatches (110% and 80% of the nominal basal rate) are performed
on the same cohort using protocol B, respectively. The results are presented
in Table 3 and Fig. 3, and the corresponding discussions for the both protocol
are provided in Section 4.1.
Table 2: Comparison results with nominal basal rate Scenario (#Simulations=10) | With meal information | Without meal information
---|---|---
Metric | Control | Proposed | $p$ value | Control | Proposed | $p$ value
% time | | | | | |
$<$54 mg/dL | 0.0 (0.0) | 0.0 (0.0) | 1.000 | 0.0 (0.0) | 0.0 (0.0) | 1.000
$<$70 mg/dL | 0.0 (0.0) | 0.0 (0.0) | 0.500 | 0.0 (0.0) | 0.0 (0.0) | 0.500
70-180 mg/dL | 95.8 (5.7) | 96.6 (7.1) | 0.627 | 95.8 (5.7) | 95.4 (7.8) | 0.264
$>$180 mg/dL | 4.2 (5.7) | 3.4 (5.6) | 0.605 | 4.2 (5.7) | 3.7 (7.8) | 0.223
$>$250 mg/dL | 0.0 (0.0) | 0.0 (0.0) | 1.000 | 0.0 (0.0) | 0.0 (0.0) | 1.000
Mean glucose (mg/dL) | 132.8 (8.3) | 126.7 (13.3) | 0.037 | 132.8 (8.3) | 131.0 (10.9) | 0.084
SD glucose (mg/dL) | 26.2 (7.2) | 25.9 (8.9) | 1.000 | 26.2 (7.2) | 27.0 (5.8) | 0.020
Mean glucose at 07:00 | 118.8 (11.0) | 116.5 (9.5) | 0.008 | 118.8 (11.0) | 117.3 (7.0) | 0.172
Data in this table are shown as median (inter quartile range), and $p$ values
are calculated based on the number
of virtual patients. Statistically significant ($p<0.05$) changes are
highlighted in bold.
(a) With meal information (b) Without meal information
Figure 2: Performance comparison with nominal basal rate in terms of glucose regulation and meal bolus. Yellow, blue, green and purple triangles denote meals of 45 g, 55 g, 65 g and 85 g CHO, respectively. Table 3: Comparison results with under/over-estimated basal rate Scenario (#Simulations=10) | 80% of the nominal basal rate | 110% of the nominal basal rate
---|---|---
Metric | Control | Proposed | $p$ value | Control | Proposed | $p$ value
% time | | | | | |
$<$54 mg/dL | 0.0 (0.0) | 0.0 (0.0) | 1.000 | 0.0 (0.0) | 0.0 (0.0) | 1.000
$<$70 mg/dL | 0.0 (0.0) | 0.0 (0.0) | 1.000 | 0.0 (0.0) | 0.0 (3.1) | 0.625
70-180 mg/dL | 87.8 (3.5) | 87.7 (8.0) | 0.389 | 96.5 (3.8) | 95.0 (9.7) | 0.586
$>$180 mg/dL | 12.2 (3.5) | 12.3 (8.0) | 0.389 | 3.5 (4.9) | 4.7 (7.3) | 0.945
$>$250 mg/dL | 0.0 (0.0) | 0.0 (0.0) | 1.000 | 0.0 (0.0) | 0.0 (0.0) | 1.000
Mean glucose (mg/dL) | 151.2 (4.9) | 146.6 (10.4) | 0.131 | 120.9 (6.9) | 119.0 (7.6) | 0.064
SD glucose (mg/dL) | 24.7 (5.9) | 26.8 (5.9) | 0.131 | 28.8 (4.8) | 32.0 (8.9) | 0.275
Mean glucose at 07:00 | 151.0 (21.0) | 145.0 (27.0) | 0.047 | 93.0 (11.0) | 93.0 (12.0) | 0.559
Data in this table are shown as median (inter quartile range), and $p$ values
are calculated based on the number
of virtual patients. Statistically significant ($p<0.05$) changes are
highlighted in bold.
(a) Under-estimated basal rates (b) Over-estimated basal rates
Figure 3: Performance comparison for the scenario of under-/over-estimated
basal rates.
The advisory-mode analysis [27] allows the comparisons with insulin
recommendations made by clinicians, through feeding the identical glucose data
obtained in the clinical trial to the system. Here, by feeding the historical
preprandial glucose data to the proposed method, the corresponding meal
boluses are determined and compared with the ones following the clinician’s
advice. The obtained meal boluses have no causal impact on historical data and
are only for the comparison. The collection of the historical clinical data
are presented in Fig. 4, and the comparison performance are illustrated in
Fig. 5. The corresponding discussions are provided in Section 4.2.
### 4.1 In Silico Performance Evaluation
In this subsection, using Protocol A, the performance of the proposed method
are evaluated for the cases with the meal information (IM) and without the
meal information (NM), respectively. From Table 2, for the both cases, the
proposed method achieves comparable glucose regulation performance in
comparison with the standard insulin bolus calculator, which is equipped with
the well-designed CR and CF. This is reflected in percent time in the
euglycemic range of 70-180 mg/dL (96.6% vs. 95.8%, $p=0.627$ for IM; 95.4% vs.
95.8%, $p=0.264$ for NM), mean glucose (126.7 mg/dl vs. 132.8 mg/dL, $p=0.037$
for IM; 131.0 mg/dL vs. 132.8 mg/dL, $p=0.084$ for NM), percent time $>$ 250
mg/dL (0.0% vs. 0.0%, $p=1.000$ for IM; 0.0% vs. 0.0%, $p=1.000$ for NM) and
glucose standard deviation (SD) (25.9 vs. 26.2, $p=1.000$ for IM; 27.0 vs.
26.2, $p=0.020$ for NM). No increase in the risk of hypoglycemia is observed
(percent time $<$ 70 mg/dL, 0.0% vs. 0.0%, $p=0.500$ for IM; 0.0% vs. 0.0%,
$p=0.500$ for NM). These results show the method is robust to the meal
information if the CHO amount intakes are similar to the standard amount used
for the model learning, and also illustrate the learning ability of proposed
method, as the samples for the model learning are collected with the
inappropriate CR. The discussions of control performance are consistent with
the quartile curves in Fig. 2. Besides, from the quartile curves of meal bolus
in Fig. 2, we observe that compared with the case of NM, the method in the
case of IM tends to increase (decrease) the meal bolus for the known large
(small) CHO. This implies that when the accurate meal information are
available, the method can react reasonably to the CHO amount, but this will
increase the burden of data collection for model learning in return.
Using Protocol B, we also perform additional tests considering realistic
scenarios of under/over-estimated basal rates with/without meal information to
evaluate the robustness of the proposed method. The results are also compared
with those obtained for the standard insulin bolus calculator. Since similar
results are observed for both cases, here we only present the results of the
case without meal information. From Table 3 and Fig. 3, it is observed that
for the scenario of under-estimated basal rate, the proposed method tends to
increase the meal bolus for the elevated preprandial glucose levels, and
achieves similar performance in terms of percent time in [70, 180] mg/dL
(87.7% vs. 87.8%, $p$ = 0.389) and mean glucose (146.6 mg/dL vs. 151.2 mg/dL,
$p=0.131$) without causing risk of hypoglycemia (percent time below 70 mg/dL,
0.0% vs. 0.0%, $p=1.000$). Similar results are observed for the scenario of
over-estimated basal rate, which are reflected in percent time in [70, 180]
mg/dL (95.0% vs. 96.5%, $p$ = 0.586) and mean glucose (119.0 mg/dL vs. 120.9
mg/dL, $p=0.064$) without causing risk of hypoglycemia (0.0% vs. 0.0%,
$p=0.625$). These results further illustrate the robust decision-making
ability of the method for the extreme preprandial glucose situations.
Finally, we would like to note that the extremely satisfactory glucose control
performance obtained by the proposed approach and the standard bolus
calculator is partially attributed to the simulator dynamics and should not be
over-emphasized, as we observe that the time-in-range achieved using the
standard bolus calculator (using the CR and CF values provided by the
simulator) goes beyond $95\%$ in Table 2. The aim of presenting the _in
silico_ evaluation results, however, is to compare the proposed data-driven
method with the standard approach that is built on CR and CF information.
Figure 4: The collection of the clinical data from a T1DM subject. Meals are
denoted by green triangles with sizes below them, and the corresponding meal
boluses determined by the clinicians are displayed in the second panel. Figure
5: Performance evaluation of the proposed method based on the clinical data.
Meals are denoted by green triangles with sizes below them.
### 4.2 Advisory-Mode Comparisons Using Clinical Data
In this subsection, the historical clinical data from a T1DM subject who
undertook the MDI therapy are utilized to evaluate the performance of the
proposed method. Flash glucose monitoring (FGM) was worn to collect the
glucose measurements. The data for the seven days (see Fig. 4) were collected
at hospital, where the patient was managed to have a consistent diet for every
day, in terms of similar meal timing and meal intakes, and the corresponding
meal boluses were determined by the clinicians. The study was approved by
institutional review board at Peking University People’s Hospital and written
informed consent of the participant was obtained. Since the meal intakes are
almost identical for every day, we use the data of the first five days to
model the PG dynamics for the breakfast and lunch-dinner, respectively,
without the meal information. At last, the performance of the proposed method
equipped with the trained GPs is evaluated by feeding the data for the next
two days.
From Fig. 5, compared with the fixed meal boluses (denoted as blue bars)
following the clinician’s advice, the proposed method can determine reasonable
meal boluses (denoted as red bars) according to the preprandial glucose
levels. We observe that the method would suggest additional 1-2 units of the
insulin bolus for the lunch or dinner of the two days due to the elevated
preprandial glucose level. This reasonable increase bolus would help reduce
the later happened hyperglycemia. Besides, the breakfast meal boluses are the
same as those determined by the clinicians; observing that the historical
breakfast boluses are almost all identical, this indicates that the risk-
sensitive control mechanism tends to maintain the decisions in the database to
ensure safety when the risk of taking a different value is not clear.
## 5 Conclusion
In this work, a GP-based asymmetric risk-sensitive (ARS) control method is
proposed for the personalized meal bolus decision. With the formulation of the
ARS cost function, the method is capable to apply the experience learned form
the samples, while keeping own decision-making ability, e.g., taking extra
care of the hyperglycemia and increasing (decreasing) the meal bolus for the
elevated (lowered) preprandial glucose levels. Besides, the method is robust
to the meal variability within a tolerable range, which reduces the burdens of
estimating the CHO amount for each meal. The effectiveness and robustness of
the controller are evaluated using the 10-adult cohort of the UVA/Padova
simulator through comparisons with the standard insulin bolus calculator.
Also, advisory-mode analysis is performed based on the clinical data from a
T1DM subject. For future work, since the model learning is relied on the
offline samples, an adaptive PG model will be developed for the method to take
the long-term variations of the physiological dynamics into account.
## Acknowledgment
Access to the distributed version of the University of Virginia (UVA)/Padova
metabolic simulator for research purposes is acknowledged.
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|
# Extended Reality (XR) Remote Research: a Survey of Drawbacks and
Opportunities
Jack Ratcliffe 0000-0001-5729-3252 Queen Mary, University of LondonMile End
RoadLondonUKE1 4NS , Francesco Soave 0000-0002-7136-2135 Queen Mary,
University of LondonMile End RoadLondonUKE1 4NS , Nick Bryan-Kinns Queen
Mary University of LondonMile End RoadLondonUKE1 4NS , Laurissa Tokarchuk
Queen Mary University of LondonMile End RoadLondonUKE1 4NS and Ildar
Farkhatdinov Queen Mary University of LondonMile End RoadLondonUKE1 4NS
(2021)
###### Abstract.
Extended Reality (XR) technology - such as virtual and augmented reality - is
now widely used in Human Computer Interaction (HCI), social science and
psychology experimentation. However, these experiments are predominantly
deployed in-lab with a co-present researcher. Remote experiments, without co-
present researchers, have not flourished, despite the success of remote
approaches for non-XR investigations. This paper summarises findings from a
30-item survey of 46 XR researchers to understand perceived limitations and
benefits of remote XR experimentation. Our thematic analysis identifies
concerns common with non-XR remote research, such as participant recruitment,
as well as XR-specific issues, including safety and hardware variability. We
identify potential positive affordances of XR technology, including leveraging
data collection functionalities builtin to HMDs (e.g. hand, gaze tracking) and
the portability and reproducibility of an experimental setting. We suggest
that XR technology could be conceptualised as an interactive technology and a
capable data-collection device suited for remote experimentation.
Extended Reality, Virtual Reality, Augmented Reality, literature review,
expert interviews
††journalyear: 2021††copyright: acmlicensed††conference: CHI Conference on
Human Factors in Computing Systems; May 8–13, 2021; Yokohama,
Japan††booktitle: CHI Conference on Human Factors in Computing Systems (CHI
’21), May 8–13, 2021, Yokohama, Japan††price: 15.00††doi:
10.1145/3411764.3445170††isbn: 978-1-4503-8096-6/21/05††ccs: Human-centered
computing Mixed / augmented reality††ccs: Human-centered computing Virtual
reality
## 1\. Introduction
Extended reality (XR) technology - such as virtual, augmented, and mixed
reality - is increasingly being examined and utilised by researchers in the
HCI and other research communities due to its potential for creative, social
and psychological experiments (Blascovich et al., 2002). Many of these studies
take place in laboratories with the co-presence of the researcher and the
participant (Kourtesis et al., 2020). The XR research community has been slow
to embrace recruiting remote participants to take part in studies running
outside of laboratories - a technique which has proven useful for non-XR HCI,
social and psychological research (Preece, 2016)(Paolacci et al., 2010).
However, the current Covid-19 pandemic has highlighted the importance and
perhaps necessity of understanding and deploying remote recruitment methods
within XR research.
There is also limited literature about remote XR research, although what
reports exist suggest that the approach shows promise: data-collection is
viable (Steed et al., 2016), results are similar to those found in-lab
(Mottelson and Hornbæk, 2017) even when the participants are unsupervised
(Huber and Gajos, 2020), and recruiting is possible (Ma et al., 2018).
Researchers have also suggested using existing communities for these
technologies, such as customisable social VR experiences, as combined
platforms for recruitment and experimentation (Saffo et al., 2020). With the
increasingly availability of consumer XR devices (estimates show five million
high-end XR HMDs sold in 2020, raising to 43.5 million by 2025(Tankovska,
2020)), and health and safety concerns around in-lab experimentation,
particularly for research involving head-mounted displays (HMDs), it seems an
important time to understand the conceptions around remote research from
researchers who use XR technologies.
This paper outlines the methodology and results from the first (that we are
aware of) survey of XR researchers regarding remote XR research. The results
have been derived from 46 respondents answering 30 questions regarding
existing research practice. It offers three core contributions: (1) we
summarise existing research on conducting remote XR experiments. (2) We
provide an overview of the status quo, showing that many of the concerns
regarding remote XR are those also applicable to other remote studies; and
that the unique aspects of remote XR research could offer more benefits than
drawbacks. (3) We set out recommendations for advancing remote XR research,
and outline important questions that should be answered to create an evidence-
backed experimentation process.
## 2\. Literature
We present a literature review of relevant publications on XR research, remote
research and remote XR research. We use ”XR” as the umbrella term for virtual
reality (VR), augmented reality (AR) and mixed reality (MR) (Ludlow, 2015).
This space is also sometimes referred to as spatial or immersive computing.
The chapter is organised in three parts. First, we explore conventional XR
experiments under ‘normal’ conditions (e.g. in laboratory andor directly
supervised by the researcher). We then summarise existing literature on remote
experiments in XR research. Finally, we report the main findings in previous
publications on remote data collection and experimentation.
### 2.1. Conventional XR experiments
#### 2.1.1. Experiment types and fields of interest
According to Suh and Prohpet’s 2018 systematic literature review (Suh and
Prophet, 2018), XR experiments involving human participants can broadly be
categorised into two groups: (1) studies about XR, and (2) studies about
_using_ XR. The first group focuses on the effects of XR system features on
the user experience (e.g. if enhancing embodiment could affect presence
outcomes (Ratcliffe and Tokarchuk, 2020b)), whereas the second category
examines how the use of an XR technology modifies a measurable user attribute
(e.g. if leveraging XR embodiment could affect learning outcomes (Ratcliffe
and Tokarchuk, 2020a)). Across these categories there have been a variety of
explorations on different subjects and from different academic fields. These
include social psychological (Blascovich et al., 2002), including social
facilitation–inhibition (Hoyt et al., 2003), conformity and social comparison
(Blascovich, 2002), social identity (Kilteni et al., 2013); neuroscience and
neuropsychology (Kourtesis et al., 2020), visual perception (Wilson and
Soranzo, 2015), multisensory integration (Choi et al., 2016), proxemics (Sanz
et al., 2015), spatial cognition (Waller et al., 2007), education and training
(Radianti et al., 2020), therapeutic applications (Freeman et al., 2018), pain
remediation (Gromala et al., 2015), motor control (Connelly et al., 2010),
terror management (Josman et al., 2006) and media effects such as presence
(Bailey et al., 2012).
The theoretical approaches behind these studies are also disparate, including
theories such as conceptual blending, cognitive load, constructive learning,
experiential learning, flow, media richness, motivation, presence, situated
cognition, the stimuli-organism-response framework and the technology
acceptance model (Suh and Prophet, 2018).
#### 2.1.2. Data collection, approaches and techniques
According to Suh and Prophet’s meta-analysis, the majority of XR research
explorations have been experiments (69%) (Suh and Prophet, 2018). Other types
of explorations include surveys (24%), interviews (15%) and case studies (9%).
These approaches have been used both alone and in combination with each other.
Data collection methods are predominantly quantitative (78%), although
qualitative and mixed approaches are also used. Another systematic review of
XR research (focused on higher education) (Radianti et al., 2020) adds focus
group discussion and observation as research methods, and presents two
potential subcategories for experiments: mobile sensing and ”interaction log
in VR app”, in which the XR application logs the user’s activities and the
researcher uses the resulting log for analysis.
The types of data logging found in XR experiments are much the same as those
listed in Weibel’s exploration of physiological measures in non-immersive
virtual reality (Weibel et al., 2018), with studies using skin conductance
(Yuan and Steed, 2010), heart rate (Egan et al., 2016), blood pressure
(Hoffman et al., 2003), as well as electroencephalogram (EEG) (Amores et al.,
2018). Built-in inertial sensors that are integral to providing an XR
experience, such as head and hand position for VR HMDs, have also been widely
used for investigations, including posture assessment (Brookes et al., 2019),
head interaction tracking (Zhang and Healey, 2018), gaze and loci of attention
(Piumsomboon et al., 2017) and gesture recognition (Kehl and Van Gool, 2004),
while velocity change (Warriar et al., 2019) has also been used in both VR and
AR interventions.
#### 2.1.3. Benefits of XR experiments
There are many suggested benefits to using XR technology as a research tool:
it allows researchers to control the mundane-realism trade-off (Aronson, 1969)
and thus increase the extent to which an experiment is similar to situations
encountered in everyday life without sacrificing experimental control
(Blascovich et al., 2002); to create powerful sensory illusions within a
controlled environment (particularly in VR), such as illusions of self-motion
and influence the proprioceptive sense (Soave et al., 2020); improve
replication (Blascovich et al., 2002) by making it easier to recreate entire
experimental environments; and allow representative samples(Blascovich et al.,
2002) to experience otherwise inaccessible environments, when paired with
useful distribution and recruitment networks.
#### 2.1.4. Challenges of XR experiments
Pan (Pan and Hamilton, 2018) explored some of the challenges facing
experiments in virtual worlds, which continue to be relevant in immersive XR
explorations. These include the challenge of ensuring the experimental design
is relevant for each technology and subject area; ensuring a consistent
feeling of self‐embodiment to ensure engaged performance (Kilteni et al.,
2012); avoid uncanny valley, in which characters which look nearly-but-not-
quite human are judged as uncanny and are aversive for participants (Mori et
al., 2012); simulation sickness and nausea during VR experiences(Moss and
Muth, 2011); cognitive load (Sweller, 2010) which may harm participation
results through over-stimulation, particularly in VR (Steed et al., 2016)
(Makransky et al., 2019); novelty effects of new technology interfering with
results (Clark and Craig, 1992) (Ely and Minor, 1994); and ethics, especially
where experiences in VR could lead to changes in participants’ behaviour and
attitude in their real life (Banakou et al., 2016) and create false memories
(Segovia and Bailenson, 2009).
### 2.2. Remote XR experiments
There has been little research into remote XR experimentation, particularly
for VR and AR HMDs. By remote, we mean any experiment that takes place outside
of a researcher-controlled setting. This is distinct from field or in-the-wild
research, which is research ”that seeks to understand new technology
interventions in everyday living” (Rogers and Marshall, 2017), and so is
dependent on user context. These definitions are somewhat challenged in the
context of remote VR research, as for VR, remote and field/in-the-wild are
often the same setting, as the location where VR is most used outside the lab
is also where it is typically experienced (e.g. home users, playing at home
(Ma et al., 2018)). For AR, there is a greater distinction between remote,
which refer to any AR outside of the controlled setting of the lab; and
field/in-the-wild, which require a contextual deployment.
In terms of remote XR research outcomes, Mottelson and Hornbæk (Mottelson and
Hornbæk, 2017) directly compared in-lab and remote VR experiment results. They
found that while the differences in performance between the in-lab and remote
study were substantial, there were no significant differences between effects
of experimental conditions. Similarly, Huber and Gajos explored uncompensated
and unsupervised remote VR samples and were able to replicate key results from
the original in-lab studies, although with smaller effect sizes (Huber and
Gajos, 2020). Finally, Steed et al. showed that collecting data in the wild is
feasible for virtual reality systems (Steed et al., 2016).
Ma et al. (Ma et al., 2018) is perhaps the first published research on
recruiting remote participants for VR research. The study, published in 2018,
used the Amazon Mechanical Turk (AMT) crowdsourcing platform, and received 439
submissions over a 13-day period, of which 242 were eligible. The participant
demographics did not differ significantly from previously reported
demographics of AMT populations in terms of age, gender, and household income.
The notable difference was that the VR research had a higher percentage of
U.S.-based workers compared to others. The study also provides insight into
how remote XR studies take place: 98% of participants took part at home, in
living rooms (24%), bedrooms (18%), and home offices (18%). Participants were
typically alone (84%) or in the presence of one (14%) or two other people
(2%). Participants reported having “enough space to walk around” (81%) or “run
around (10%)”. Only 6% reported that their physical space would limit their
movement.
While Ma et al’s work is promising in terms of reaching a representative
sample and the environment in which participants take part in experiments, it
suggests a difficulty in recruiting participants with high-end VR systems,
which allow six-degrees of freedom (the ability to track user movement in real
space) and leverage embodied controllers (e.g. Oculus Rift, HTC Vice). Only 18
(7%) of eligible responses had a high-end VR system. A similar paucity of
high-end VR equipment was found by Mottelson and Hornbæk (Mottelson and
Hornbæk, 2017), in which 1.4% of crowdworkers had access to these devices
(compared to 4.5% for low-end devices, and 83.4% for Android smartphones).
This problem is compounded if we consider Steed et al’s finding that only 15%
of participants provide completed sets of data (Steed et al., 2016).
An alternative approach to recruiting participants is to create experiments
inside existing communities of XR users, such as inside the widely-used VR
Chat software (Saffo et al., 2020). This allows researchers to enter into
existing communities of active users, rather than attempt to establish their
own. However, there are significant limitations for building experiments on
platforms not designed for experimentation, such as programming limitations,
the ability to communicate with outside services for data storage, and the
absence of bespoke hardware interfaces.
### 2.3. Remote data collection and experimentation
#### 2.3.1. Validity, benefits, drawbacks and differences
Using networks for remote data collection from human participants has been
proven valid in some case studies (Gosling et al., 2004; Krantz and Dalal,
2000). In Gosling et al’s comprehensive and well-cited study (Gosling et al.,
2004), internet-submitted samples were found to be diverse, generalise across
presentation formats, were not adversely affected by non-serious or repeat
respondents, and present results consistent with findings from in-lab methods.
There is similar evidence for usability experiments, in which both the lab and
remote tests captured similar information about the usability of websites
(Tullis et al., 2002).
That said, differences in results for lab and remote experiments are common
(Stern and Faber, 1997; Buchanan, 2000; Senior et al., 1999). The above
website usability study also found that in-lab and remote experiments offered
their own advantages and disadvantages in terms of the usability issues
uncovered (Tullis et al., 2002). The factors that influence differences
between in-lab and remote research are still being understood, but even beyond
experiment design, there is evidence that even aspects such as the
participant-perceived geographical distance between the participant and the
data collection system influences outcomes (Moon, 1998).
Reips’ (Reips, 2000) well-cited study outlined 18 advantages of remote
experiments, including (l) easy access to a demographically and culturally
diverse participant population, including participants from unique and
previously inaccessible target populations; (2) bringing the experiment to the
participant instead of the opposite; (3) high statistical power by enabling
access to large samples; (4) the direct assessment of motivational
confounding; and (5) cost savings of lab space, person-hours, equipment, and
administration. He found seven disadvantages: (l) potential for multiple
submissions, (2) lack of experimental control, (3) participant self-selection,
(4) dropout, (5) technical variances, (6) limited interaction with
participants and (7) technical limitations.
#### 2.3.2. Supervised vs unsupervised
With the increasing availability of teleconferencing, it has become possible
for researchers to be co-”tele”present and supervise remote experiments
through scheduling webcam experiment sessions. This presents a distinction
from the unsupervised internet studies discussed above, and brings its own
opportunities and limitations.
Literature broadly suggests that unsupervised experiments provide suitable
quality data collection (Ryan et al., 2013; Hertzum et al., 2015; Kettunen and
Oksanen, 2018). A direct comparison between a supervised in-lab experiment and
a large, unsupervised web-based experiment found that the benefits outweighed
its potential costs (Ryan et al., 2013); while another found that a higher
percentage of high-relevance responses came from unsupervised participants
than supervised ones in a qualitative feedback setting (Hertzum et al., 2015).
There is also evidence that unsupervised participants react faster to tasks
over the internet than those observed in the laboratory (Kettunen and Oksanen,
2018).
For longitudinal studies, research in healthcare has found no significant
difference between task adherence rates between unsupervised and supervised
groups (Creasy et al., 2017). However, one study noted that supervised studies
had more effective outcomes (Lacroix et al., 2015).
#### 2.3.3. Crowdworkers: Viable?
Remote data collection was theorised to bring easy access to participants,
including diverse participants and large samples (Reips, 2000). Researchers
have found that recruiting crowdworkers, people who work on tasks distributed
to them over the internet, allowed them access to a large participant
pool(Paolacci et al., 2010), with enough diversity to facilitate cross-
cultural and international research (Buhrmester et al., 2016). Research has
found that crowdworkers were significantly more diverse than typical American
college samples and more diverse than other internet recruitment methods
(Buhrmester et al., 2016), at an affordable rate (Paolacci et al.,
2010)(Buhrmester et al., 2016). This has allowed researchers a faster theory-
to-experiment cycle (Mason and Suri, 2012).
Results from crowdworker-informed studies have been shown to reproduce
existing results from historical in-lab studies (Paolacci et al., 2010)
(Buhrmester et al., 2016) (Sprouse, 2011), while a direct comparison between
experiment groups of crowdworkers, social media-recruited participants and on-
campus recruitment, found almost indistinguishable results (Casler et al.,
2013).
Some distinctions between crowdworkers and in-lab have been discovered,
however. Comparative experiments between crowdworkers and in-person studies
have suggested slightly higher participant rejection rates (Sprouse, 2011),
while participants have been shown to report shorter narratives than other
groups of college students (both online and in-person) and use proportionally
more negative emotion terms than college students reporting verbally to an
experimenter (Grysman, 2015).
Distinctions also exist within crowdworker recruitment sources. A study of
AMT, CrowdFlower (CF) and Prolific Academic (ProA) found differences in
response rate, attention-check question results, data quality, honesty,
diversity and how successfully effects were reproduced (Peer et al., 2017).
Data quality is a common concern regarding crowdworkers (Goodman et al.,
2013). However, attention-check questions used to screen out inattentive
respondents or to increase the attention of respondents have been shown to be
effective in increasing the quality of data collected (Aust et al., 2013), as
have participant reputation scores (Peer et al., 2014).
A growing concern regarding crowdworkers is non-naivete, in which participants
having some previous knowledge of the study or similar studies that might bias
them in the experiment. Many workers report having taken part in common
research paradigms (Paolacci and Chandler, 2014), and there are concerns that
if researchers continue to depend on this resource, the problem may expand. As
such, further efforts are needed by researchers to identify and prevent non-
naive participants from participating in their studies (Buhrmester et al.,
2018).
### 2.4. Summary
It is clear that remote methods have been usefully deployed for non-XR
research, and seemingly bring benefits such as easier participant recruitment,
reduced recruitment cost and broadened diversity, without introducing major
biases. However, there is still a paucity of research regarding the extent to
which remote XR research can and has been used to leverage the unique benefits
of both XR (environmental control, sensory illusions, data collection,
replication) and remote (participation, practicality, cost-savings) methods,
as well as the potential impact of their combined limitations. Therefore a
survey of XR researcher experiences and beliefs regarding remote XR research
could help us understand how these apply practically at the current time, and
understand the key areas for future developments in this field.
## 3\. Methodology
### 3.1. Survey
We surveyed current practice to outline the researcher-perceived benefits and
drawbacks of lab-based and remote XR research. We used a 30-item qualitative
questionnaire that enquired about participants’ existing lab-based and remote
research practices; thoughts on future lab-based and remote research; and
potential benefits and drawbacks for each area. The survey was circulated
through relevant mailing lists<EMAIL_ADDRESS>BCS-HCI@
jiscmail.ac.uk, chi-announcements@listserv.acm.org), to members of groups
thinking of or currently running remote studies, and to members of
universities’ virtual and augmented reality groups found via search engines.
Responses were thematically analysed using an inductive approach based upon
Braun and Clarke’s six phases of analysis (Braun and Clarke, 2006). The coding
and theme generation process was conducted twice by independent researchers;
themes were then reviewed collaboratively to create the final categorisations.
### 3.2. Participants
We received 46 responses to our survey from 36 different (predominantly
academic) institutions. Most responses came from researchers based in Europe
and North America, but responses also came from Asia. The majority of
participants were either PhD students (18) or lecturers, readers or professors
(11) at universities. Other roles were academic/scientific researcher (5),
masters student (5), corporate researcher (4) and undergraduate student (2). A
diverse set of ages responded to the survey: 18-24 (5), 25-34 (22), 35-44
(11), 45+ (6), and gender skewed male (29) over female (16) or other (1).
## 4\. Participant XR setup results
Figure 1. Type of XR medium explored by survey respondents. Figure 2. Features
used by respondents in their user studies. (A) Embodied Interactivity: using
embodied controller/camera-based movement. (B) Embodied Movement:using your
body to move/”roomscale”. (C) Abstract Movement: using a gamepad or keyboard
and mouse to move. (D) Sound 3D: binaural acoustics. E) Spoken Input. (F)
Abstract Interactivity: using a gamepad or keyboard and mouse to interact. (G)
Sound non-3D: mono/stereo audio. (H) Unique features: e.g. haptics, hand
tracking, scent.
Participants were more likely to have previously ran in-lab studies (37) than
remote studies (14). Twenty-seven participants noted that, because of the
Covid-19 pandemic, they have considered conducting remote XR experiments. In
the next six months, more researchers were planning to run remote studies (24)
than lab-based (22).
Participants predominantly categorised their research as VR-only (28) over AR-
only (5). Ten participants considered their research as both VR and AR (and
three did not provide an answer). This result is illustrated in Fig. 1. In
terms of research hardware, the majority of VR research leveraged virtual
reality HMD-based systems with six degrees of freedom (32), that tracks
participants’ movements inside the room, over three degrees of freedom (15) or
CAVE systems (1). Nineteen researchers made use of embodied or gesture
controllers, where the position of handheld controllers are tracked in the
real world and their position virtualised. For AR, HMDs were the predominant
medium (13) over smartphones (9), with some researchers (5) using both.
An array of supplementary technologies and sensors were also reported by 13
respondents, including gaming joysticks, haptic actuators, a custom haptic
glove, motion capture systems, e-textiles, eye-trackers, microphones, computer
screens, Vive body trackers, brain-computer interfaces, EEG and
electrocardiogram (ECG) devices, galvanic skin response sensors and hand-
tracking cameras, as well as other spatial audio and hardware rigs.
The use of a variety of different off-the-shelf systems was also reported:
Vive, Vive Pro, Vive Eye, Vive Index, Vive Pucks, Quest, Go, Rift, Rift S,
DK2, Cardboard, Magic Leap One, Valve Knuckles, Hololens. Predominantly used
devices are part of HTC Vive (25) and Oculus (23) family.
Respondents outlined numerous features of immersive hardware that they used in
their research, visible in Fig. 2. The most prominent were embodiment aspects,
including embodiment interactivity, in which a user’s hand or body movements
are reflected by a digital avatar (37) and embodiment movement (35), where
participants can move in real space and that is recognised by the environment.
Abstract movement (13), where a user controls an avatar via an abstracted
interface (like a joystick) and abstract interactivity (8) were less popular.
Spoken input was also used (10), as well as 3D sound (13) and non-3D sound
(6). Scent was also noted (1) along with other unique features.
## 5\. Thematic analysis results
In this section, we present and discuss the themes found in our survey study.
The key points of each theme are summarised in a table at the start each
subsection. Some of these points were found across multiple themes as they
touch various aspects of user-based XR research.
### 5.1. Theme: Study Sub-types
Table 1. Summary of XR Study Sub-types Method | Summary
---|---
In-lab (vital) | Experiment requires features only feasible in-lab, e.g. bespoke hardware, unique data collection
In-lab (preferred) | Concerns about integrity of data collected remotely, high value on controlled setting
Remote (vital) | User’s natural (in-the-wild) environment is important (e.g. Social VR, naturally experienced at home and online)
Remote (preferred) | Priority to get cross-cultural feedback or reach large number of participant; lab provides limited benefits
Our analysis suggests that in-lab and remote studies can be additionally
distinguished by whether the setting type is vital or preferred (summarised in
Table 1). Broadly, in-lab (vital) studies require experimental aspects only
feasible in-lab, such as bespoke hardware or unique data collection processes;
in-lab (preferred) studies could take place outside of labs, but prefer the
lab-setting based upon heightened concerns regarding the integrity of data
collected and place a high value on a controlled setting. Remote (vital)
studies are required when a user’s natural environment is prioritised, such as
explorations into behaviour in Social VR software; and remote (preferred)
studies are used when cross-cultural feedback or a large number of
participants are needed, or if the benefits offered by an in-lab setting are
not required.
Beyond these, another sub-type emerged as an important consideration for user
studies: supervised or unsupervised. While less of an important distinction
for in-lab studies (which are almost entirely supervised), participant
responses considered both unsupervised ”encapsulated” studies, in which
explanations, data collection and the study itself exist within the software
or download process, and supervised studies, in which researchers schedule
time with the remote participant to organise, run and/or monitor the study.
These distinctions will be discussed in more detail throughout the analysis
below, as the sub-types have a distinct impact on many of the feasibility
issues relating to remote studies.
### 5.2. Theme: Study Participants
Table 2. Study Participants Key Points Key Point | Issue | Lab | Remote
---|---|---|---
Recruitment Scope | Sample size | Usually smaller numbers | Potential for larger number
Recruitment Scope | Sample balance | Might be easier to ensure balance | How to ensure balance? (e.g. who mostly owns XR equipment?)
Efficiency | Time | Requires setup time and organise participants | Potential less time especially if encapsulated and unsupervised
Precursor Requirements | Requisites | Pre-test and linguistic/culture comprehension conditions are ensured | Not clear how to verify conditions in remote studies
#### 5.2.1. Recruitment scope
Twenty-nine respondents stated the well-known challenge of recruiting a
satisfactory number of participants for lab-based studies. Issues were
reported both with the scale of available participants, and the problem of
convenience sampling and WEIRD - Western, educated, industrialized, rich and
democratic societies - participants(Henrich et al., 2010).
Participant recruitment was mentioned by 27 respondents as the area in which
remote user studies could prove advantageous over labs. Remote studies could
potentially provide easier recruitment (in terms of user friction: accessing
to lab, arriving at the correct time), as well as removing geographic
restrictions to the participant pool.
Removing the geographic restrictions also simplifies researchers’ access to
cross-cultural investigations (R23, R43). While cross-cul-
tural lab-based research would require well-developed local recruitment
networks, or partnerships with labs in target locations, remote user studies,
and more specifically, systems built deliberately for remote studies,
introduce cross-cultural scope at no additional overhead.
There are, however, common concerns over the limitations to these benefits due
to the relatively small market size of XR technologies. For AR, this is not a
strong limitation for smartphones-based explorations, but the penetration of
HMD AR and VR technology is currently limited, and it is possible that those
who currently have access to these technologies will not be representative of
the wider populations. Questions remain over who the AR/VR HMD owners are, if
they exhibit notable differences from the general population, and if those
differences are more impactful than those presented by existing convenience
sampling.
Despite the belief that designing for remote participants will increase
participant numbers, and therefore the power of studies, it seems unclear how
researchers will reach HMD-owning audiences. Thirty respondents who have, or
plan to, run remote XR studies have concerns about the infrastructure for
recruiting participants remotely. Unlike other remote studies, the requirement
for participants to own or have access to XR hardware greatly reduces the pool
(around 5 million XR HMDs were sold in 2020 (Tankovska, 2020)). A major
outstanding question is how researchers can access these potential
participants, although some platforms for recruiting XR participants have
emerged in the past few months such as XRDRN.org.
Nine respondents noted that remote XR experiments may encourage participation
from previously under-represented groups, including introverts and those who
cannot or do not wish to travel into labs to take part (e.g. people who
struggle to leave their homes due to physical or mental health issues).
However, respondents with research-specific requirements also raised concerns
that recruitment of specific subsets of participants could be more difficult
remotely. For example, when recruiting for a medical study of those with age-
related mobility issues, it is unlikely that there will be a large cohort with
their own XR hardware.
#### 5.2.2. Theme: Efficiency
Twenty-five respondents noted the potential for remote studies to take up less
time, particularly if remote studies are encapsulated and unsupervised. They
stated that this removes scheduling concerns for both the researcher and the
participant, and allows experiments to occur concurrently, reducing the total
researcher time needed or increasing the scale of experiment. However, there
are concerns this benefit could be offset by increased dropouts for
longitudinal studies, due to a less ”close” relationship between research and
participant (R17, R25).
#### 5.2.3. Participant precursor requirements
One respondent noted they needed to run physiological precursor tests (i.e.
visual acuity and stereo vision) that have no remote equivalent. Transitioning
to remote research has meant this criteria must now be self-reported.
Similarly, experiments have general expectations of linguistic and cultural
comprehension, and opening research to a global scale might introduce
distinctions from typically explored population. One respondent cautioned that
further steps should be taken to ensure participants are able to engage at the
intended level, as in-lab these could be filtered out by researcher intuition.
### 5.3. Data Collection
Table 3. Data Collection Key Points Key Point | Lab | Remote
---|---|---
Hardware | Access custom and/or reliable hardware | Limited access to devices (e.g. EEG, ECG, computational power, etc.)
Data | Collection can be supervised, more detailed, real-time, more space for qualitative | Mostly unsupervised (less control), human expressions (e.g. facial) are generally lost, qualitative feedback is harder to collect
Behaviour | Likely more serious, richer (qualitative) data | Lack of detailed feedback, potentially less honest
The overwhelming drawback of remote XR research, as reported by the majority
respondents, was that of data collection. Excluding changes to participant
recruitment, as mentioned above, the issues can broadly be categorised as: (1)
bespoke hardware challenges, (2) monitoring/sensing challenges, and (3) data
transmission and storage.
The use of bespoke hardware in any type of remote user study is a well-known
issue, predominantly regarding the difficulty of managing and shipping bespoke
technology to participants and ensuring it works in their test environments.
In the context of XR technologies, 13 respondents voiced concerns about the
complicated and temperamental system issues that could arise, particularly
surrounding the already strenuous demands of PC-based VR on consumer-level XR
hardware, without additional overheads (e.g. recording multiple cameras).
Four respondents felt it was unreasonable to ask remote participants to
prepare multiple data-collection methods that may be typical in lab-studies,
such as video recording and motion tracking. There were also concerns
regarding the loss of informal, ad-hoc data collection (e.g. facial
expressions, body language, casual conversations).
Finally, concerns were also raised regarding the efforts required to
encapsulate all data capture into the XR experience, the effects this might
have on data collection (for example, a recent study highlighted a difference
on the variability of presence when participants recorded it from inside the
VR experience versus outside (Schwind et al., 2019)), the reliability of
transferring large amounts of data from participants, and how sensitive
information (especially in the context of medical XR interventions) can
securely be transferred and stored. This areas perhaps presents the biggest
area for innovation for remote XR research, as it is reasonable to assume the
academic community could create efficient, easy-to-use toolkits for remote
data collection in XR environments which integrate to ethics-compliant data
archives.
Many data collection methods were deemed infeasible for remote
experimentation: EEG, ECG, eye/hand tracking, GSR, as well as body language
and facial expressions. Five researchers noted adaptions they had been working
on to overcome these, including using HMD orientation to replace eye tracking,
and using built-in HMD microphones to record breaths instead of ECG monitoring
to determine exertion, or using the HMD controllers to perform hand tracking.
Respondents also noted some behavioural concerns and changes for remote,
unsupervised participants. These included a lack of participation in
qualitative feedback (6 respondents); for one researcher (R20), participants
were ”encouraged to provide feedback but few took the initiative.” Another
researcher (R31) stated ”Debriefing is such a good space to collect
unstructured interview data. Users relax after the questionnaire/debriefing …
produc[ing] a … meta-narrative where participants consider your questions and
their experiences together”. The lack of supervision raised concerns regarding
whether participants were being ”truthful” in their responses, with one
researcher (R41) stating that participants attempted to ”game” their study in
order to claim the participation compensation. However, others stated that
unsupervised studies could reduce research bias arising from their perception
of the participants’ appearance and mannerisms.
### 5.4. Theme: Experiment Processes
Table 4. Experiment Process Key Points Key Point | Issue | Lab | Remote
---|---|---|---
Process & Guidance | Control | Full control over setup and participants | No control and guidance over participants
Process & Guidance | Participants | Rapport with researcher, welcoming, more serious, attentive | Different attitude, potential cheating
Environment | Setting | Can be distracting (e.g. outside noise) but generally more controlled | Might be distracting or overwhelming but likely more realistic/natural for participants
Hardware & software | Hardware | Access to custom devices, normal calibration process | No calibration (by researcher), potential for unknown errors, no custom tools
Hardware & software | Software | Allows for Wizard of Oz, adjust setting in real time | Issues harder to spot and influence results, longer development time
Research questions | Topics | Unchanged, if we go back to normal research conditions | Remote setup might influence research questions and topics
Cost | Expenditures | More time consuming, more expensive to run | Potentially cheaper but potentially more work for implementation
#### 5.4.1. Process & Guidance
Many respondents were concerned that unsupervised participants may conduct the
experiments incorrectly, or have incorrect assumptions, or misunderstand
processes or target actions. Twenty-four respondents felt that guidance would
be better provided (introduction, explanations, etc) in a lab setting that
also allows ad-hoc guidance and real-time corrections.
There were also concerns over the mental state of participants: remote
participants ”may not take it seriously” or not focus (lack of motivation and
engagement) or approach the study with a specific mood unknown to the
researcher (R19, R30). Contrasting opinions suggested that participants may
feel that the in-lab experience is ”overly formal and uncomfortable” (R32).
Some respondents stated that remote experiments risk losing the ”rapport”
between researcher and participant, which might negatively influence the way a
participant performs a remote study. However, one respondent stated that the
transition to remote experimentation allowed them different, deeper, on-going
connection with their participants. Their research was for a VR machine
learning tool, and they found that moving away from in-person experimentation
and to a remote workshop process encouraged the up-take of longitudinal
community-building tools. The chosen communication method between researcher
and user - Discord servers - became a place for unsupervised interaction
between participants, and led to an on-going engagement with the research
(R33). However it should be considered that any ”rapport” between participant
and researcher might introduce bias.
#### 5.4.2. Environment
Concern was raised around participants’ environments, and their potential
varying unsuitability for remote experimentation, compared with controlled
laboratory settings. For example, one respondent (R20) stated: ”one user
reported walking into their whiteboard multiple times, causing low presence
scores.” The concern is particularly strong for unsupervised remote
experiments, as distractions could enter into the experiment and affect data
without the researcher being aware.
This concern was not universal, however. Four respondents noted that their
laboratories space was far from distraction free, and even suggested that a
remote space could prove freer of interruptions than the space available to
them in their research setting; while others stated that researchers should be
mindful that the laboratory itself is an artificial space, far more so than
where people will typically use their VR setups - in their homes. Five
respondents highlighted how XR research could benefit from being deployed in
”the participants’ own environment”.
The immediate environment of the user was also raised as a concern for VR
experiment design: the choice of being able to move freely in an open space in
a laboratory against a more adaptive solution for the unknown variables of
participants’ home environments.
Respondents noted that supporting the different VR and AR setups to access a
larger remote audience would also prove more labour-intensive, and would
introduce more variables compared with the continuity of the tech stack
available in-lab. With remote experiments, and more so for encapsulated
unsupervised ones, 10 respondents believe there will be more time spent in
developing the system.
#### 5.4.3. Hardware and software
A concern regarding remote experiments, particularly unsupervised, is that
calibration processes are harder to verify (R30). This could either cause
participants to unknowingly have faulty experiences, and therefore report
faulty data; or it will increase time taken to verify user experiences are
correct. Unknown errors can effect data integrity or participant behaviour.
Respondents noted that this type of remote error are often much more difficult
and labour-intensive to fix compared with in-lab. This issue is compounded by
individual computer systems introducing other confounding factors (for both
bug-fixing and data collection) such as frame-rates, graphic fidelity,
tracking quality and even resolution can vary dramatically.
Five respondents reflected that overcoming these issues could lead to more
robust research plans, as well as better development and end-product software
to overcome problems listed. This encapsulation could also lead to easier
opportunities for reproducability, as well as the ability for researchers to
share working versions of the experiment with other researchers, instead of
just the results. It could also help with the versioning of experiments,
allowing researchers to build new research on-top of previous experiment
software.
Four respondents were aware these advantages are coupled with longer
development times. The increased remote development requirements could also be
limiting for researchers who face constrained development resources,
particularly those outside of computer science departments. This is compounded
by the fact that the infrastructure for recruiting remote XR participants,
data capture, data storage and bug fixing is not particularly developed. Once
these are established, however, respondents felt these might make for a higher
overall data quality compared with the current laboratory-based status quo,
due to more time spent creating automated recording processes, and not relying
on researcher judgement. There are also arguments that the additional
development time is offset by the potential increase in participants and, if
unsupervised, the reduction in experiment supervision requirements.
Six respondents that use specific hardware in their research, noted that it
was currently difficult to measure physiological information in a reliable
way, and included hand tracking in this. However, we are aware that some
consumer VR hardware (Oculus Quest) allows hand-tracking, and so there is an
additional question of whether researchers are being fully supported in
knowing what technologies are available to them.
To alleviate issues with reaching participants, two respondents wrote about
potentially sending equipment to participants. The limitations of this were
noted as hardware having gone missing (which had happened, R35), and
participants being unable to use equipment on their own (which had not
happened yet).
#### 5.4.4. Research questions
Five respondents noted that their research questions changed or could change
depending on whether they were aiming for a laboratory or remote settings. For
example, one respondent (R31) suggested that ”instead of the relationship of
the physical body to virtual space, I’d just assess the actions in virtual
space”. Others explored the potentiality of having access to many different
system setups, for example, now being able to easily ask questions like ”are
there any systematic differences in cybersickness incidence across different
HMDs?”. (R39)
Nine respondents speculated that remote research has potential for increasing
longitudinal engagement, due to lower barriers to entry for researcher (room
booking, time) and participant (no commute), and that rare or geographically
based phenomena could be cheaply studied using remote research; as providing
those communities access to VR may be cheaper than relocating a researcher to
them.
#### 5.4.5. Costs
Eight respondents noted the potential of remote experimentation for reducing
some of the cost overheads for running experiments. Laboratories have
important costs that are higher than remote studies: lab maintenance, hardware
maintenance, staff maintenance. Without these, costs per participant are lower
(and for unsupervised studies, almost nil). As experiment space availability
was also noted as a concern for laboratory-based experiments, this seems a
potentially under-explored area of benefit, provided remote participant
recruitment is adequate.
### 5.5. Theme: Health & Safety
Table 5. Health and Safety Key Points Key Point | Summary
---|---
Protocols | Missing standard protocols (to work safely with participants in-lab)
Equipment | Sanitizing of in-lab equipment and spaces
Remote | Concerns for remote participants (e.g. accidents during a user study)
Real-Time Aid | Not available for remote participants (e.g. motion sickness)
The leading benefit given for remote user studies was that of health and
safety, citing shared HMDs and controllers as a potential vector for Covid-19
transmission, as well as more general issues such as air quality in enclosed
lab spaces. Concerns were raised for both viral transmission between
participants, and between participant and the researcher. This concern has
also increased administration overheads, with 6 respondents stating it could
be more time consuming to prepare the lab and organise the studies or using
new contract-tracing methods for lab users.
However, respondents also raised concerns about additional safety implications
for remote participants. The controlled lab environment is setup to run the
study, whereas remote participants are using a general-purpose space. One AR
researcher who conducts research that requires participants to move quickly
outside in fields noted his study could be considered ”incredibly unsafe” if
unsupervised or run in an inappropriate location. Additionally, for health and
mental health studies, in-lab allows for researcher to provide support,
especially with distressing materials. Finally, VR environment design has a
direct impact on the level of simulator sickness invoked in participants.
There were questions about the responsibility of researchers to be present to
aid participants who could be made to feel unwell from a system they build.
### 5.6. Theme: Ethics
Three ethics concerns were reported by respondents: encouraging risky
behaviours, responsibility for actions in XR and data privacy. An example of
this might be the ethical implications of paying participants, and therefore
incentivising them, to take part in what could be considered a high-risk
behaviour: entering an enclosed space with a stranger and wearing a VR HMD.
Respondent (R30) raised the question of liability for participants who are
injured in their homes while taking part in an XR research project. The
embodied nature of XR interventions - and most respondents used this
embodiment in their studies - could put participants at a greater risk of
harming themselves than with other mediums.
Finally, while cross-cultural recruitment was seen as a potential boom for
remote research, questions were raised about ethics and data storage and
protection rules when participants are distributed across different countries,
each with different data storage laws and guidelines. Although not limited to
XR, due to the limited number of VR users, and the disproportionate
distribution of their sales, it seems the majority of remote VR participants
will originate from North America, and ethics clarification from non-US-based
universities are needed.
## 6\. Covid-19 Implications
Table 6. Covid-19 Implications Key Point | Summary
---|---
Suspensions | No user studies at the moment
Facilities | Sanitizing of equipment and spaces
Recruitment | Harder/impossible to recruit in-lab participants
Exclusion | Bias and high risk participants
While Covid-19 has impacted most studies around the world, the dependence on
shared hardware for XR research, especially HMDs, has led to many implications
reported by our respondents. These concerns are particularly related to
Covid-19, and therefore be reduced as the pandemic is resolved. However, as it
is currently unclear when the pandemic will end, we felt it was useful to
discuss them in a dedicated section.
Most respondents noted that Covid-19 had caused a suspension of studies and
that they were unclear how long the suspension would last for, resulting in an
overall drop in the number of studies being conducted, with 30 respondents
stating it will change the research they conduct (e.g. moving to online
surveys). The continuation of lab studies was eventually expected, but with
added sanitizing steps. However for many, it was unclear what steps they
should take in order to make XR equipment sharing safe. These concerns
extended beyond the XR hardware to general facility suitability, including
room airflow and official protocols which may vary for each country and/or
institution.
Five respondents also had concerns about participants. There were worries that
lab-based recruitment would be slow to recover, as participants may be put off
taking part in experiments because of the potential virtual transfer vectors.
Similarly, respondents were concerned about being responsible for
participants, and putting them in a position in which their is a chance they
could be exposed to the virus.
There was also concerns around Covid-19 and exclusion, as researchers who are
at high risk of Covid-19 or those who are in close contact with high risk
populations, would now have to self-exclude from lab-based studies. This might
introduce a participant selection bias towards those willing to attend a small
room and sharing equipment,
It should be noted that not all labs are facing the same problems - some of
our respondents had continued lab-based experimentation during this period,
with Covid-19 measures ensuring that participants wore face masks during
studies. This was considered a drawback as combined with an HMD, it covered
the participant’s entire face and was cumbersome. These measures are also
known not to be 100% protective.
## 7\. Discussion
In the previous section we presented the results as themes we found in our
analysis. Some of these presented common characteristics and some issues were
reported in multiple themes. We now summarise the results, highlight the key
points and suggest important questions for future research.
### 7.1. Recruitment and participants
As with non-XR experiments, researchers are interested in the potential
benefits of remote research for increasing the amount, diversity and
segmentation of participants compared with in-lab studies. However, with many
respondents reporting that it has been difficult to recruit XR participants,
it seems there is a gap between potential and practice. The unanswered
question is how to build a pool of participants that is large and diverse
enough to accommodate various XR research questions and topics, given that
there are few high-end HMDs circulating in the crowdworker community (Ma et
al., 2018)(Mottelson and Hornbæk, 2017). So far, we have found three potential
solutions for participant recruitment, although each requires further study:
(1) Establish a dedicated XR crowdworker community. However, concerns of non-
naivety(Paolacci and Chandler, 2014), which are already levied at the much
larger non-XR crowdworker participant pools, would surely be increased. We
would also have to understand if the early version of this community would be
WEIRD(Henrich et al., 2010) and non-representative, especially given the cost
barrier to entry for HMDs.
(2) Leverage existing consumer XR communities on the internet, such as the
large discussion forums on Reddit. These should increase in size further as
they shift from early-adopter to general consumer communities. However, these
communities may also have issues with representation.
(3) Establish hardware-lending schemes to enable access to a broader base of
participants (Steed et al., 2020). However, the cost of entry and risk of
these schemes may make them untenable for smaller XR research communities.
It is also not clear, beyond HMD penetration, what the additional obstacles
are that XR poses for online recruitment. Technical challenges (e.g. XR
applications needing to run on various devices, on different computers,
requiring additional setup beyond simple software installation) and
unintuitive experiment procedures (e.g. download X, do an online survey at Y,
run X, record Z) for participants are notable distinct issues for remote XR
research. It is also unclear if the use of XR technology has an impact on what
motivates participants to take part in remote studies, an area of study that
has many theoretical approaches even in the non-XR area(Keusch, 2015).
### 7.2. Data collection
Respondents feel that many types of physiological data collection are not
feasible with either XR or non-XR remote research. For remote XR research,
there are unique concerns over video and qualitative data collection as using
XR technologies can make it (technically) difficult to reliably video or
record the activity, as well as moving participants’ loci of attention away
from the camera or obscuring it behind an HMD. However, the hardware involved
in creating XR experiences provides a variety of methods to gather data, such
as body position, head nodding, breath-monitoring, hand tracking, HMD angle
instead of eye tracking. These can be used to explore research topics that are
often monitored via other types of physiological, video or qualitative data,
such as attention, motivation, engagement, enjoyment, exertion or focus of
attention. It would be useful for XR researchers to build an understanding of
what the technologies that are built into XR hardware can tell us about
participant experiences, so as to allow us to know the data collection
affordances and opportunities of XR hardware.
That said, the infrastructure for collecting and storing this (mass) of XR
data remotely is currently not fully implemented, and we are not aware of any
end-to-end standardised framework. However, work is being done to simplify the
data collection step for XR experiments build in Unity (Brookes et al., 2019).
There are also opportunities to further develop web-based XR technologies that
could send and store data on remote servers easily. There are also ethical
concerns, as respondents were unclear on guidance regarding data collection
from participants located in other nations, particularly when they should be
paid. This includes how the data is collected, where it should be stored, and
how can be manipulated.
### 7.3. Health, safety and Covid-19
At the time of writing, many laboratories are considered unsafe for running
user studies. Although some respondents reported being able to work in-lab,
the limitations mean it is not currently feasible to run user studies under
normal conditions. The main concern for the near future is the lack of
standardised protocols to ensure safety of researchers and participants while
running user studies and the issue with the ethics protocols of the research
institutions. For XR research, it is unclear how to adequately sanitize
equipment and tools, as well as how to maintain physical distancing. There are
also concerns about the comfort of participants if they are required to wear
masks alongside HMDs. Finally, respondents reported concerns about a potential
long-term fall in user motivation to take part in such experiments, when HMDs
are a notable infection vector. There are distinctly different safety and
ethics concerns around remote XR experiments, including the research
responsibility for not harming participants (e.g. ensuring environments are
safe for the movements, and not inducing simulator sickness), which, while
also true of in-lab experiments, are considered a greater challenge when a
participant is not co-located with the researcher.
### 7.4. Mediated impact
Respondents reported framing their research questions and experiments
differently depending on the target experiment setting. The strongest
transition was that of an in-lab study of participants using an AR HMD
(Hololens), which changed to a remote study that had participants watch a pre-
recorded video of someone using the AR HMD. It seems these kinds of
transitions will continue to be necessary depending on how esoteric the
hardware is, with fewer concerns for AR smartphone investigations.
A concern for respondents was that remote settings introduce additional
uncontrolled variables that need to be considered by researchers, such as
potential unknown distractions, trust in participants and their motivation,
and issues with remote environmental spaces. However, previous research shows
that most HMD-wearing remote participants engage in space well-known to them
(the home) and predominantly when they are alone (Ma et al., 2018), which
could alleviate some of the environmental space and distraction concerns.
Further research into how a home environment could impact XR studies is
needed, and the creation of well-defined protocols to alleviate uncontrolled
influences remote XR results. Beyond this, we also need to understand any
impact that remote experiments may have on results compared with in-lab
experiences, especially if we are to be able to reliably contrast lab and
remote research. Previous research for non-XR experiments suggest that
distinctions between lab and remote settings exist (Buchanan, 2000)(Senior et
al., 1999) (Stern and Faber, 1997), but it has been theorised that the impact
might be less for XR experiments, as you ”take the experimental environment
with you” (Blascovich et al., 2002).
### 7.5. ”Encapsulated” experiments: the ideal?
Respondents stated that creating remote XR experiments might encourage better
software development and experimental processes. If experiments are able to be
deployed as all-in-one experience and data collection bundles that can run
unsupervised, the time-saving implications for researchers (and participants)
are huge, especially when paired with the potential increase in participants.
This type of ”encapsulated experiment” can also improve replication and
transparency, as theorised by Blascovich (Blascovich et al., 2002), and allow
for versioning of experiments, in which researchers can build on perfect
replicas of other’s experimental environments and processes. Finally, due to
the similar nature of XR hardware, data logging techniques could easily be
shared between system designers or standardised; something we have seen with
the creation of the Unity Experiment Framework (Brookes et al., 2019).
However, there are some limitations to this approach. It is likely it will
require additional development time from the researchers, especially as a
comprehensive experiment framework is established. In addition, there are data
collection limitations for remote XR studies, as discussed in previous
sections. It is also interesting to consider how encapsulation might work for
AR investigations, as the environment will only partially be controlled by the
designer.
We believe that the potential for remote XR experiments lies in understanding
the data collection affordances of the hardware; collectively building
frameworks to ease the collection of this data; and to design research
questions that maximise their use; all inside encapsulated experiences. This
might be a mindset shift for researchers, who according to our survey, are
predominantly lab-orientated.
## 8\. Limitations
Our goal with this research was to provide an overall insight into the XR
researcher community. However, this approach means that insights from sub-
communities may not have been found. For example, we had no responses from
researchers involved in topics such as vulnerable populations. Further
investigation into sub-communities is needed to uncover potential insights for
those areas.
## 9\. Conclusion and Recommendations
It is clear from our survey that respondents believe that remote XR research
has the potential to be a useful research approach. However, it currently
suffers from numerous limitations regarding data collection, system
development and a lack of clarity around participant recruitment. Analysis of
our survey results and literature around remote and remote XR research suggest
that, to better understand the boundaries of remote XR experimentation,
researchers need answers to the following questions:
* (1)
Who are the potential remote XR participants, and are they representative?
* (2)
How can we access a large pool of remote XR participants?
* (3)
To what extent do remote XR studies affect results compared with in-lab?
* (4)
What are the built-in XR data collection affordances of XR hardware, and what
can they help us study?
* (5)
How can we lower the barriers to creating encapsulated experiment software, to
maximise the potential of remote XR research?
We believe there is an opportunity to reconceptualise approaches to XR and
remote research. XR experiments, as it stands, are predominantly used to study
a participant’s experience with an XR system, in an artificial but controlled
setting (laboratory) using external data collection methods (surveys, cameras,
etc.). However, if we consider XR devices primarily as data-collection
hardware with set properties, we can work backwards to understand what
research questions are suitable with the existing data collection afforded by
XR hardware. Additionally, we also believe that there is potential to
reconceptualise, for suitable applications, the home as a natural research
location and move away from the laboratory as the default location for user
studies. This is a potentially unique opportunity for XR compared with non-XR
studies as, for many investigations, the XR experiment takes the environment
with it.
###### Acknowledgements.
This work is supported by the EPSRC and AHRC Centre for Doctoral Training in
Media and Arts Technology (EP/L01632X/1).
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|
∎
11institutetext: ∗Corresponding author
Mostafa Ghadampour 22institutetext: Department of Mathematics, Lorestan
University, Lorestan, Khoramabad, Iran.
22email<EMAIL_ADDRESS>33institutetext: Donal O′Regan
44institutetext: School of Mathematics, Statistics, National University of
Ireland, Galway, Ireland.
44email<EMAIL_ADDRESS>55institutetext: Ebrahim Soori
66institutetext: Department of Mathematics, Lorestan University, Lorestan,
Khoramabad, Iran.
66email<EMAIL_ADDRESS>77institutetext: Ravi P. Agarwal88institutetext:
Department of Mathematics Texas A$\&$M University-Kingsville 700 University
Blvd., MSC 172 Kingsville, Texas, USA.
88email<EMAIL_ADDRESS>
# Two generalized strong convergence algorithms for the variational
inequality problems in Banach spaces
Mostafa Ghadampour1 Ebrahim Soori∗,2
(Received: date / Accepted: date)
###### Abstract
In this paper, two generalized algorithms for solving the variational
inequality problem in Banach spaces are proposed. Then the strong convergence
of the sequences generated by these algorithms will be proved under the
suitable conditions. Finally, using MATLAB software, we provide some numerical
examples to illustrate our results.
###### Keywords:
Variational inequality Relatively nonexpansive mapping Monotone mapping
Asymptotical fixed point
## 1 Introduction
Let $C$ be a nonempty closed convex subset of a Banach space $E$ with norm
$\|.\|$ and let $E^{*}$ denotes the dual of $E$. The variational inequality
problem (VIP) is to find a point $x\in C$ such that
$\langle Ax,y-x\rangle\geq 0\;\;\;\forall\;\;y\in C,$ (1)
where $A$ is a mapping of $C$ into $E^{*}$ and $\langle.,.\rangle$ denotes the
pairing between $E$ and $E^{*}$. The solutions set of (1) is denoted by
$VI(A,C)$.
It is well known that variational inequalities cover a variety of fields in
optimal control, optimization, mathematical programming, operational research,
partial differential equations, engineering, and equilibrium models and hence,
it have been studied by many authors in the recent yearsjlotaa ; tdvh ; vpts ;
cgga .
The operator $A$ of $C$ to $E^{*}$ is said to be
(i) monotone if
$\langle x-y,Ax-Ay\rangle\geq 0,\;\;\forall x,y\in C;$
(ii) $\alpha-$inverse strongly monotone if there exists a constant $\alpha>0$
such that
$\langle x-y,Ax-Ay\rangle\geq\alpha\|Ax-Ay\|^{2}\;\;\forall x,y\in C;$
(iii) $L$-Lipchitz continuous if there exists $L>0$ such that
$\displaystyle\|Ax-Ay\|\leq L\|x-y\|,\;\;\forall x,y\in C.$
Let $f:C\times C\rightarrow\mathbb{R}$ be a bifunction. The equilibrium
problem (GEP) is as follows: Find $x\in C$ such that
$f(x,y)+\langle Ax,y-x\rangle\geq 0,\;\;\forall\;y\in C.$ (2)
The set of solutions of (2) is denoted by $GEP(f,A)$. Clearly, the problem (2)
is equivalent to (VIP) if $f\equiv 0$.
Korpelevichko proposed the following algorithm for solving the problem (VIP)
that is known as extragradient method as (3). Let $x_{1}$ be an arbitrarily
element in $H$:
$\begin{cases}y_{n}=P_{C}(x_{n}-\lambda Ax_{n}),\\\
x_{n+1}=P_{C}(x_{n}-\lambda Ay_{n}),\end{cases}$ (3)
Tseng tp proposed the following algorithm which was introduced using the
modified front-to-back (F-B) method.
$\begin{cases}y_{n}=P_{C}(x_{n}-\lambda Ax_{n}),\\\
x_{n+1}=P_{X}(y_{n}-\lambda(Ay_{n}-Ax_{n})),\end{cases}$ (4)
where $X=C$ and $X=H$ if $A$ is Lipschitz continuous. Thong et al th proposed
the following convergent algorithm based on the Tseng algorithm.
$\begin{cases}y_{n}=P_{C}(x_{n}-\lambda_{n}Ax_{n}),\\\
z_{n}=y_{n}-\lambda_{n}(Ay_{n}-Ax_{n}),\\\
x_{n+1}=\alpha_{n}f(x_{n})+(1-\alpha_{n})z_{n},\end{cases}$ (5)
where the operator $A$ is monotone and Lipschitz continuous,
$\gamma>0,\;l\in(0,1),\;\mu\in(0,1)$ and $\lambda_{n}$ is chosen to be the
largest $\lambda\in\\{\gamma,\gamma l,\gamma l^{2},...\\}$ satisfying
$\displaystyle\lambda\|Ax_{n}-Ay_{n}\|\leq\mu\|x_{n}-y_{n}\|.$ (6)
In this paper, we present our algorithms in Banach spaces motivated by the
Thong algorithm and prove the strong convergence of the sequences generated by
these algorithms. Finally, using MATLAB software, we provide some numerical
examples to illustrate our claims.
## 2 Preliminaries
Let $E$ be a real Banach space with norm $\|.\|$ and let $E^{*}$ be the dual
space of $E$. The strong convergence and the weak convergence of the sequence
$\\{x_{n}\\}$ to $x$ in $E$ are denoted by $x_{n}\rightarrow x$ and
$x_{n}\rightharpoonup x$ through in the paper, respectively. The modulus
$\delta$ of convexity of $E$ is defined by
$\delta(\epsilon)=\inf\\{1-\frac{\|x+y\|}{2}:\|x\|\leq 1,\|y\|\leq
1,\|x-y\|\geq\epsilon\\}$
for every $\epsilon\in[0,2]$. $A$ Banach space $E$ is said to be uniformly
convex if $\delta(0)=0$ and $\delta(\epsilon)>0$ for every $\epsilon>0$. It is
known that a Banach space $E$ is uniformly convex if and only if for any two
sequences $\\{x_{n}\\}$ and $\\{y_{n}\\}$ in $E$ such that
$\lim_{n\rightarrow\infty}\|x_{n}\|=\lim_{n\rightarrow\infty}\|y_{n}\|=1\;and\;\lim_{n\rightarrow\infty}\|x_{n}+y_{n}\|=2,$
$\lim_{n\rightarrow\infty}\|x_{n}-y_{n}\|=0$ holds. Suppose that $p$ is a
fixed real number with $p\geq 2$. A Banach space $E$ is said to be
$p$-uniformly convextyhk , if there exists a constant $c>0$ such that
$\delta\geq c\epsilon^{p}$ for all $\epsilon\in[0,2]$. It is also known that a
uniformly convex Banach space has the Kadec-Klee property, that is,
$x_{n}\rightharpoonup u$ and $\|x_{n}\|\rightarrow\|u\|$ imply that
$x_{n}\rightarrow u$(see cii ; res ).
The normalized duality mapping $J:E\rightarrow E^{*}$ is defined by
$\displaystyle J(x)=\\{f\in E^{*}:\langle x,f\rangle=\|x\|^{2}=\|f\|^{2}\\},$
for each $x\in E$. Suppose that $S(E)=\\{x\in E:\|x\|=1\\}$. A Banach space
$E$ is called smooth if for all $x\in S(E)$, there exists a unique functional
$j_{x}\in E^{*}$ such that $\langle x,j_{x}\rangle=\|x\|$ and $\|j_{x}\|=1$(
see Ag ).
The norm of $E$ is said to be $G\hat{a}teaux$ differentiable if for each
$x,y\in S(E)$, the limit
$\lim_{t\rightarrow 0}\frac{\|x+ty\|-\|x\|}{t}$ (7)
exists. In this case, $E$ is called smooth and $E$ is said to be uniformly
smooth if the limit (7) is attained uniformly for all $x,y\in S(E)$tn . If a
Banach space $E$ is uniformly convex, then $E$ is reflexive and strictly
convex, and $E^{*}$ is uniformly smoothAg . It is well known that if $E$ is a
reflexive, strictly convex and smooth Banach space and $J^{*}:E^{*}\rightarrow
E$ be the duality mapping on $E^{*}$, then $J^{-1}=J^{*}$, also, if $E$ is a
uniformly smooth Banach space, then $J$ is uniformly norm to norm continuous
on bounded sets of $E$ and $J^{-1}=J^{*}$ is also uniformly norm to norm
continuous on bounded sets of $E^{*}$. Let $E$ be a smooth Banach space and
let $J$ be the duality mapping on $E$. The function $\phi:E\times
E\rightarrow\mathbb{R}$ is define by
$\phi(x,y)=\|x\|^{2}-2\langle x,Jy\rangle+\|y\|^{2},\;\;\;\forall x,y\in E.$
(8)
Clearly, from (8), it is concluded that
$(\|x\|-\|y\|)^{2}\leq\phi(x,y)\leq(\|x\|+\|y\|)^{2}.$ (9)
If $E$ is a reflexive, strictly convex and smooth Banach space, then for all
$x,y\in E$
$\phi(x,y)=0\Leftrightarrow x=y.$ (10)
Also, It is obvious from the definition of the function $\phi$ that the
following conditions hold for all $x,y,z,w\in E$,
$\phi(x,y)=\phi(x,z)+\phi(z,y)+2\langle x-z,Jz-Jy\rangle,$ (11) $2\langle
x-y,Jz-Jw\rangle=\phi(x,w)+\phi(y,z)-\phi(x,z)-\phi(y,w).$ (12)
$\phi(x,y)=\langle x,Jx-Jy\rangle+\langle y-x,Jy\rangle\leq\|x\|\|Jx-
Jy\|+\|y-x\|\|y\|.$ (13)
Now, the function $V:E\times E^{*}\rightarrow\mathbb{R}$ is defined as follows
$V(x,x^{*})=\|x\|^{2}-2\langle x,x^{*}\rangle+\|x^{*}\|^{2},$
for all $x\in E$ and $x^{*}\in E$. Moreover, $V(x,x^{*})=\phi(x,J^{-1}x^{*})$
for all $x\in E$ and $\in E$. If $E$ is a reflexive strictly convex and smooth
Banach space with $E^{*}$ as its dual, it is concluded that
$V(x,x^{*})+2\langle J^{-1}x^{*}-x,y^{*}\rangle\leq V(x,x^{*}+y^{*}),$ (14)
for all $x\in E$ and all $x^{*},y^{*}\in E^{*}$kt .
An operator $A:C\rightarrow E^{*}$ is hemicontinuous at $x_{0}\in C$, if for
any sequence $\\{x_{n}\\}$ converging to $x_{0}$ along a line implies that
$Tx_{n}\rightharpoonup Tx_{0}$, i.e., $Tx_{n}=T(x_{0}+t_{n}x)\rightharpoonup
Tx_{0}$ as $t_{n}\rightarrow 0$ for all $x\in C$.
The generalized projection $\Pi_{C}:E\rightarrow C$ is a mapping that assigns
to an arbitrary point $x\in E$, the minimum point of the functional
$\phi(y,x)$; that is, $\Pi_{C}x=x_{0}$, where $x_{0}$ is the solution of the
minimization problem
$\phi(x_{0},x)=\min_{y\in C}\phi(y,x).$ (15)
The existence and uniqueness of the operator $\Pi_{C}$ follows from the
properties of the functional $\phi(x,y)$ and strict monotonicity of the
mapping $J$ayl . Suppose that $C$ is a nonempty closed convex subset of $E$,
and $T$ is a mapping from $C$ into itself. A point $p\in C$ is called an
asymptotically fixed point of $T$ if $C$ contains a sequence $\\{x_{n}\\}$
which converges weakly to $p$ such that $Tx_{n}-x_{n}\rightarrow 0$Ag . The
set of asymptotical fixed points of $T$ will be denoted by $\hat{F}(T)$. A
mapping $T$ from $C$ into itself is said to be relatively nonexpansive if
$\hat{F}(T)=F(T)$ and $\phi(p,Tx)\leq\phi(p,x)$ for all $x\in C$ and $p\in
F(T)$. The asymptotic behavior of a relatively nonexpansive mapping was
studied in bdrs ; bdrs1 ; cyrs .
We need the following lemmas for the proof of our main results.
###### Lemma 1
(ktw ) Let $E$ be a smooth and uniformly convex Banach space and let
$\\{x_{n}\\}$ and $\\{y_{n}\\}$ be two sequences of $E$. If
$\phi(x_{n},y_{n})\rightarrow 0$ and either $\\{x_{n}\\}$ or $\\{y_{n}\\}$ is
bounded, then $x_{n}-y_{n}\rightarrow 0$.
###### Lemma 2
(ayl ) Let $C$ be a nonempty closed convex subset of a smooth, strictly convex
and reflexive Banach space $E$, let $x\in E$ and let $z\in C$. Then
$z=\Pi_{C}x\Leftrightarrow\langle y-z,Jx-Jz\rangle\leq 0$, for all $y\in C$.
###### Lemma 3
(ayl ) Let $C$ be a nonempty closed convex subset of a smooth, strictly convex
and reflexive Banach space $E$ and let $y\in E$. Then
$\phi(x,\Pi_{C}y)+\phi(\Pi_{C}y,y)\leq\phi(x,y),\;\;\forall x\in C$.
###### Lemma 4
(bb ; xuh ) Let $E$ be a 2-uniformly convex and smooth Banach space. Then, for
all $x$, $y\in E$, it is concluded that
$\|x-y\|\leq\frac{2}{c^{2}}\|Jx-Jy\|$,
where $\frac{1}{c}(0\leq c\leq 1)$is the 2-uniformly convex constant of $E$.
###### Lemma 5
(Xu xuh ). Let $E$ be a uniformly convex Banach space and $r>0$. Then there
exists a continuous strictly increasing convex function
$g:[0,2r]\rightarrow[0,\infty)$ such that $g(0)=0$ and
$\displaystyle\|tx+(1-t)y\|^{2}\leq
t\|x\|^{2}+(1-t)\|y\|^{2}-t(1-t)g(\|x-y\|),$
for all $x,y\in B_{r}(0)=\\{z\in E:\|z\|\leq r\\}$ and $t\in[0,1]$.
###### Lemma 6
(ktw ). Let $E$ be a uniformly convex Banach space and $r>0$. Then there
exists a continuous strictly increasing convex function
$g:[0,2r]\rightarrow[0,\infty)$ such that $g(0)=0$ and
$\displaystyle g(\|x-y\|)\leq\phi(x,y),$
for all $x,y\in B_{r}(0)=\\{z\in E:\|z\|\leq r\\}$.
Throughout this paper, we assume that $f:C\times C\rightarrow\mathbb{R}$ be a
bifunction satisfying the following conditions
1. (A1)
$f(x,x)=0$ for all $x\in C$,
2. (A2)
f is monotone, i.e. $f(x,y)+f(y,x)\leq 0$, for all $x,y\in C$,
3. (A3)
$\displaystyle\lim_{t\downarrow 0}f(tz+(1-t)x,y)\leq f(x,y)$, for all
$x,y,z\in C$,
4. (A4)
for each $x\in C,y\mapsto f(x,y)$ is convex and lower semicontinuous.
###### Lemma 7
(lyc ) Let $C$ be a nonempty closed convex subset of a smooth, strictly convex
and reflexive Banach space $E$. Let $A:C\longrightarrow E^{*}$ be an
$\alpha-$inverse-strongly monotone operator and $f$ be a bifunction from
$C\times C$ to $\mathbb{R}$ satisfying $(A_{1})-(A_{4})$. Then for all $r>0$
hold the following
1. (i)
for $x\in E$, there exists $u\in C$ such that
$f(u,x)+\langle Au,y-u\rangle+\frac{1}{r}\langle y-u,Ju-Jx\rangle\geq
0,\;\;\;\forall y\in C,$
2. (ii)
if $E$ is additionally uniformly smooth and $K_{r}:E\longrightarrow C$ is
defined as
$K_{r}(x)=\\{u\in C\;\;:\;\;f(u,y)+\langle Au,y-u\rangle+\frac{1}{r}\langle
y-u,Ju-Jx\rangle\geq 0,\;\;\;\forall y\in C\\}$
Then, the following conditions hold:
1. (1)
$K_{r}$ is single-valued,
2. (2)
$K_{r}$ is firmly nonexpansive, i.e., for all $x,y\in E$,
$\langle K_{r}x-K_{r}y,JK_{r}x-JK_{r}y\rangle\leq\langle K_{r}x-K_{r}y,Jx-
Jy\rangle,$
3. (3)
$F(K_{r})=\hat{F(K_{r})}=GEP(f,A)$,
4. (4)
$GEP$ is a closed convex subset of $C$,
5. (5)
$\phi(p,K_{r}x)+\phi(K_{r}x,x)\leq\phi(p,x),\;\;\forall\;\;p\in F(K_{r})$.
The normal cone for $C$ at a point $\upsilon\in C$ is denoted by
$N_{C}(\upsilon)$, that is $N_{C}(\upsilon):=\\{x^{*}\in
E^{*}:\langle\upsilon-y,x^{*}\rangle\geq 0,\forall y\in C\\}$.
###### Lemma 8
(rrt ) Let $C$ be a nonempty closed convex subset of a Banach space $E$ and
let $T$ be monotone and hemicontinuous operator of $C$ into $E^{*}$ with
$C=D(T)$. Let $B\subset E\times E^{*}$ be an operator define as follows:
$Bv=\left\\{\begin{array}[]{lr}Tv+N_{C}v,\qquad v\in C,\\\
\emptyset,\qquad\qquad\qquad v\notin C.\end{array}\right.$
Then $B$ is maximal monotone and $B^{-1}(0)=$ SOL$(T,C)$.
## 3 Main results
In this section, we introduce a new iterative algorithms for solving monotone
variational inequality problems which are based on Tseng’s intergradient
method.We prove strong convergence theorems for generated sequences by
presented intergradient algorithms, under suitable conditions.
Throughout this section, we assume that $C$ is a nonempty closed convex subset
of a real 2-uniformly convex and uniformly smooth Banach space $E$ and $E^{*}$
is the dual space of $E$, $A:C\rightarrow E^{*}$ is an $\alpha$-inverse
strongly monotone operator. Assume that $\\{\lambda_{n}\\}$ is a sequence of
real numbers such that $0<\lambda_{n}<\frac{c^{2}\alpha}{2}$ for all
$n\in\mathbb{N}$, where $\frac{1}{c}$ is the 2-uniformly convexity constant of
$E$.
###### Theorem 3.1
Let $x_{0}\in C$, $\Gamma:=VI(C,A)\cap F(f)\neq\emptyset$ and
$\begin{cases}y_{n}=\Pi_{C}J^{-1}(Jx_{n}-\lambda_{n}Ax_{n}),\\\
z_{n}=J^{-1}(Jy_{n}-\lambda_{n}Ay_{n}),\\\
x_{n+1}=\Pi_{C}J^{-1}(\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}),\end{cases}$
(16)
where $\\{\lambda_{n}\\}\subseteq[0,1]$ such that
$\displaystyle\lim_{n\rightarrow\infty}\lambda_{n}=0$. Let
$\\{\alpha_{n,i}\\}\subset(0,1)$ for $i=1,2,3$,
$\alpha_{n,1}+\alpha_{n,2}+\alpha_{n,3}=1$ and
$\displaystyle\liminf_{n\rightarrow\infty}\alpha_{n,2}\alpha_{n,3}>0$. Let f
be a relatively nonexpansive self-mapping on $C$ and $\|Ax\|\leq\|Ax-Au\|$ for
all $x\in C$ and $u\in\Gamma$. Consider the sequence $\\{x_{n}\\}$ generated
by the algorithm (16). Then the sequence $\\{x_{n}\\}$ converges strongly to
$q=\Pi_{VI(C,A)}\circ f(q)$, where $P_{VI(C,A)}\circ f:H\rightarrow VI(C,A)$
is the mapping defined by $P_{VI(C,A)}\circ f(x)=P_{VI(C,A)}(f(x))$ for each
$x\in H$.
###### Proof
Let $\hat{u}\in\Gamma$. From the definition of function $V$ and the inequality
(14), it is concluded that
$\displaystyle\phi(\hat{u},z_{n})=$
$\displaystyle\phi(\hat{u},J^{-1}(Jy_{n}-\lambda_{n}Ay_{n}))$ $\displaystyle=$
$\displaystyle V(\hat{u},Jy_{n}-\lambda_{n}Ay_{n})$ $\displaystyle\leq$
$\displaystyle V(\hat{u},Jy_{n})-2\langle
J^{-1}(Jy_{n}-\lambda_{n}Ay_{n})-\hat{u},\lambda_{n}Ay_{n}\rangle$
$\displaystyle=$ $\displaystyle\phi(\hat{u},y_{n})+2\langle
J^{-1}(Jy_{n}-\lambda_{n}Ay_{n})-J^{-1}(Jy_{n}),-\lambda_{n}Ay_{n}\rangle$
$\displaystyle-2\langle y_{n}-\hat{u},\lambda_{n}Ay_{n}\rangle,$ (17)
then from Lemma 4 and the condition $\|Ax\|\leq\|Ax-A\hat{u}\|$ for all $x\in
C$, it is followed that
$\displaystyle 2\langle J^{-1}(Jy_{n}-$
$\displaystyle\lambda_{n}Ay_{n})-J^{-1}(Jy_{n}),-\lambda_{n}Ay_{n}\rangle$
$\displaystyle\leq$ $\displaystyle
2\|J^{-1}(Jy_{n}-\lambda_{n}Ay_{n})-J^{-1}(Jy_{n})\|\|-\lambda_{n}Ay_{n}\|$
$\displaystyle\leq$
$\displaystyle\frac{4\lambda_{n}^{2}}{c^{2}}\|Ay_{n}\|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{4\lambda_{n}^{2}}{c^{2}}\|Ay_{n}-A\hat{u}\|^{2}.$ (18)
Since $A$ is $\alpha$-inverse strongly monotone and the fact that $\hat{u}\in
VI(C,A)$, we have
$\displaystyle-2\langle y_{n}-\hat{u},$
$\displaystyle\lambda_{n}Ay_{n}\rangle$ $\displaystyle=$
$\displaystyle-2\lambda_{n}\langle
y_{n}-\hat{u},Ay_{n}-A\hat{u}\rangle-2\lambda_{n}\langle
y_{n}-\hat{u},A\hat{u}\rangle$ $\displaystyle\leq$
$\displaystyle-2\lambda_{n}\langle y_{n}-\hat{u},Ay_{n}-A\hat{u}\rangle$
$\displaystyle\leq$ $\displaystyle-2\lambda_{n}\alpha\|Ay_{n}-A\hat{u}\|^{2},$
(19)
substituting (3) and (3) in (3) and using our assumptions, we obtain
$\displaystyle\phi(\hat{u},z_{n})\leq$
$\displaystyle\phi(\hat{u},y_{n})+(\frac{4\lambda_{n}^{2}}{c^{2}}-2\lambda_{n}\alpha)\|Ay_{n}-Au\|^{2}$
$\displaystyle=$
$\displaystyle\phi(\hat{u},y_{n})+2\lambda_{n}(\frac{2\lambda_{n}}{c^{2}}-\alpha)\|Ay_{n}-A\hat{u}\|^{2}$
$\displaystyle\leq$ $\displaystyle\phi(\hat{u},y_{n}),$
hence,
$\phi(\hat{u},z_{n})\leq\phi(\hat{u},y_{n}).$ (20)
From Lemma 3 and the inequality (14), we have
$\displaystyle\phi(\hat{u},y_{n})=$
$\displaystyle\phi(\hat{u},\Pi_{C}J^{-1}(Jx_{n}-\lambda_{n}Ax_{n}))$
$\displaystyle\leq$
$\displaystyle\phi(\hat{u},J^{-1}(Jx_{n}-\lambda_{n}Ax_{n}))=V(\hat{u},Jx_{n}-\lambda_{n}Ax_{n})$
$\displaystyle\leq$ $\displaystyle V(\hat{u},Jx_{n})-2\langle
J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})-\hat{u},\lambda_{n}Ax_{n}\rangle$
$\displaystyle=$ $\displaystyle\phi(\hat{u},x_{n})-2\lambda_{n}\langle
x_{n}-\hat{u},Ax_{n}\rangle$ $\displaystyle+2\langle
J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})-J^{-1}(Jx_{n}),-\lambda_{n}Ax_{n}\rangle,$
(21)
since $A$ is $\alpha-$inverse strongly monotone and $\hat{u}\in VI(C,A)$, it
follows that
$\displaystyle-2\lambda_{n}\langle x_{n}-\hat{u},$ $\displaystyle
Ax_{n}\rangle$ $\displaystyle=$ $\displaystyle-2\lambda_{n}\langle
x_{n}-\hat{u},Ax_{n}-A\hat{u}\rangle-2\lambda_{n}\langle
x_{n}-\hat{u},A\hat{u}\rangle$ $\displaystyle\leq$
$\displaystyle-2\lambda_{n}\langle x_{n}-\hat{u},Ax_{n}-A\hat{u}\rangle$
$\displaystyle\leq$ $\displaystyle-2\lambda_{n}\alpha\|Ax_{n}-A\hat{u}\|^{2}.$
(22)
From Lemma 4 and our assumptions, we can conclude
$\displaystyle 2\langle J^{-1}(Jx_{n}-\lambda_{n}$ $\displaystyle
Ax_{n})-J^{-1}(Jx_{n}),-\lambda_{n}Ax_{n}\rangle$ $\displaystyle\leq$
$\displaystyle
2\|J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})-J^{-1}(Jx_{n})\|\|-\lambda_{n}Ax_{n}\|$
$\displaystyle\leq$
$\displaystyle\frac{4\lambda_{n}^{2}}{c^{2}}\|Ax_{n}\|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{4\lambda_{n}^{2}}{c^{2}}\|Ax_{n}-A\hat{u}\|^{2}.$ (23)
By applying (3) and (3) in (3) and our assumptions, it is implied that
$\phi(\hat{u},y_{n})\leq\phi(\hat{u},x_{n})+2\lambda_{n}(\frac{2\lambda_{n}}{c^{2}}-\alpha)\|Ay_{n}-Ax_{n}\|^{2}\leq\phi(\hat{u},x_{n}).$
(24)
Hence, from (20) and (24), we have
$\phi(\hat{u},z_{n})\leq\phi(\hat{u},x_{n}).$ (25)
Next, it will be shown that the sequence $\\{\phi(\hat{u},x_{n})\\}$ is
decreasing. From the relatively nonexpansiveness condition of $f$, convexity
of $\|.\|^{2}$, Lemma 3 and the inequality (25), it is implied that
$\displaystyle\phi(\hat{u},x_{n+1})\leq$
$\displaystyle\phi(\hat{u},J^{-1}(\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n})$
$\displaystyle=$
$\displaystyle\|\hat{u}\|^{2}-2\langle\hat{u},\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}\rangle$
$\displaystyle+\|\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}\|^{2}$
$\displaystyle\leq$
$\displaystyle\|\hat{u}\|^{2}-2\alpha_{n,1}\langle\hat{u},Jx_{n}\rangle-2\alpha_{n,2}\langle\hat{u},Jf(x_{n})\rangle-2\alpha_{n,3}\langle\hat{u},Jz_{n}\rangle$
$\displaystyle+\alpha_{n,1}\|x_{n}\|^{2}+\alpha_{n,2}\|f(x_{n})\|^{2}+\alpha_{n,3}\|z_{n}\|^{2}$
$\displaystyle=$
$\displaystyle\alpha_{n,1}\phi(\hat{u},x_{n})+\alpha_{n,2}\phi(\hat{u},f(x_{n}))+\alpha_{n,3}\phi(\hat{u},z_{n})$
$\displaystyle\leq$
$\displaystyle\alpha_{n,1}\phi(\hat{u},x_{n})+\alpha_{n,2}\phi(\hat{u},x_{n})+\alpha_{n,3}\phi(\hat{u},x_{n})$
$\displaystyle=$ $\displaystyle\phi(\hat{u},x_{n}),$ (26)
so $\\{\phi(\hat{u},x_{n})\\}$ is decreasing. Then it is implied that
$\\{\phi(\hat{u},x_{n})\\}$ is bounded, hence
$\displaystyle\lim_{n\rightarrow\infty}\phi(\hat{u},x_{n})$ exists. Then from
(9), $\\{x_{n}\\}$ is bounded. It follows from the relatively nonexpansiveness
condition of $f$, (24) and (25) that $\\{f(x_{n})\\}$, $\\{y_{n}\\}$ and
$\\{z_{n}\\}$ are bounded. From Lemmas 3, 4, the inequality (14) and the
condition $\displaystyle\lim_{n\rightarrow\infty}\lambda_{n}=0$, we have
$\displaystyle\phi(x_{n},y_{n})$
$\displaystyle\leq\phi(x_{n},J^{-1}(Jx_{n}-\lambda_{n}Ax_{n}))$
$\displaystyle=$ $\displaystyle V(x_{n},Jx_{n}-\lambda_{n}Ax_{n})$
$\displaystyle\leq$ $\displaystyle V(x_{n},Jx_{n})-2\langle
J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})-x_{n},\lambda_{n}Ax_{n})$ $\displaystyle=$
$\displaystyle\phi(x_{n},x_{n})-2\langle
J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})-J^{-1}(Jx_{n}),\lambda_{n}Ax_{n})\rangle$
$\displaystyle\leq$ $\displaystyle
2\|J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})-J^{-1}(Jx_{n})\|\|\lambda_{n}Ax_{n}\|$
$\displaystyle\leq$
$\displaystyle\frac{4\lambda_{n}^{2}}{c^{2}}\|Ax_{n}\|^{2}\rightarrow
0\;\;\;as\;n\rightarrow\infty.$ (27)
By Lemma 1, it is implied that
$\displaystyle\lim_{n\rightarrow\infty}\|x_{n}-y_{n}\|=0.$ (28)
Next, from (13), (28), the boundedness of the sequences $\\{x_{n}\\}$ and
$\\{y_{n}\\}$, and using uniformly norm-to-norm continuity of $J$ on bounded
sets, it is obvious that
$\phi(y_{n},x_{n})\leq\|y_{n}\|\|Jy_{n}-Jx_{n}\|+\|x_{n}-y_{n}\|\|x_{n}\|\rightarrow
0\;\;\;as\;n\rightarrow\infty.$ (29)
By Lemmas 3, 4, the inequality (14) and the condition
$\displaystyle\lim_{n\rightarrow\infty}\lambda_{n}=0$, we have
$\displaystyle\phi(y_{n},z_{n})$
$\displaystyle=\phi(y_{n},J^{-1}(Jy_{n}-\lambda_{n}Ay_{n}))$ $\displaystyle=$
$\displaystyle V(y_{n},Jy_{n}-\lambda_{n}Ay_{n})$ $\displaystyle\leq$
$\displaystyle V(y_{n},Jy_{n})-2\langle
J^{-1}(Jy_{n}-\lambda_{n}Ay_{n})-y_{n},\lambda_{n}Ay_{n})$ $\displaystyle=$
$\displaystyle\phi(y_{n},y_{n})-2\langle
J^{-1}(Jy_{n}-\lambda_{n}Ay_{n})-J^{-1}(Jy_{n}),\lambda_{n}Ay_{n})\rangle$
$\displaystyle\leq$ $\displaystyle
2\|J^{-1}(Jy_{n}-\lambda_{n}Ay_{n})-J^{-1}(Jy_{n})\|\|\lambda_{n}Ay_{n}\|$
$\displaystyle\leq$
$\displaystyle\frac{4\lambda_{n}^{2}}{c^{2}}\|Ay_{n}\|^{2}\rightarrow
0\;\;\;as\;n\rightarrow\infty.$ (30)
By Lemma 1, it is implied that
$\displaystyle\lim_{n\rightarrow\infty}\|y_{n}-z_{n}\|=0.$ (31)
Since $\\{f(x_{n})\\}$ and $\\{z_{n}\\}$ are bounded. Now, setting
$r_{1}=sup\\{\|f(x_{n})\|,\|z_{n}\|\\}$, by Lemma 5 there exists a continuous
strictly increasing and convex function
$g_{1}:[0,2r_{1}]\longrightarrow[0,\infty]$ with $g_{1}(0)=0$. From (25),
Lemmas 3, 5 and the condition relatively nonexpansiveness of $f$, it is
concluded for each $\hat{u}\in\Gamma$ that
$\displaystyle\phi(\hat{u},x_{n+1})\leq$
$\displaystyle\phi(\hat{u},J^{-1}(\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n})$
$\displaystyle=$
$\displaystyle\|\hat{u}\|^{2}-2\langle\hat{u},\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}\rangle$
$\displaystyle+\|\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}\|^{2}$
$\displaystyle\leq$
$\displaystyle\|\hat{u}\|^{2}-2\alpha_{n,1}\langle\hat{u},Jx_{n}\rangle-2\alpha_{n,2}\langle\hat{u},Jf(x_{n})\rangle-2\alpha_{n,3}\langle\hat{u},Jz_{n}\rangle$
$\displaystyle+\alpha_{n,1}\|x_{n}\|^{2}+\alpha_{n,2}\|f(x_{n})\|^{2}+\alpha_{n,3}\|z_{n}\|^{2}$
$\displaystyle-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|)$
$\displaystyle=$
$\displaystyle\alpha_{n,1}\phi(\hat{u},x_{n})+\alpha_{n,2}\phi(\hat{u},f(x_{n}))+\alpha_{n,3}\phi(\hat{u},z_{n})$
$\displaystyle-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|)$
$\displaystyle\leq$
$\displaystyle\alpha_{n,1}\phi(\hat{u},x_{n})+\alpha_{n,2}\phi(\hat{u},x_{n})+\alpha_{n,3}\phi(\hat{u},x_{n})$
$\displaystyle-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|)$
$\displaystyle=$
$\displaystyle\phi(\hat{u},x_{n})-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|),$
therefore
$\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|)\leq\phi(\hat{u},x_{n})-\phi(\hat{u},x_{n+1}).$
Since $\liminf_{n\rightarrow\infty}\alpha_{n,2}\alpha_{n,3}>0$, we have
$\lim_{n\rightarrow\infty}g_{1}(\|Jf(x_{n})-Jz_{n}\|)=0,$ (32)
because $\\{\phi(\hat{u},x_{n})\\}$ is Cauchy and
$\displaystyle\lim_{n\rightarrow\infty}\alpha_{n,2}\alpha_{n,3}>0$. Since
$g_{1}$ is a continuous function, so
$g_{1}(\lim_{n\rightarrow\infty}\|Jf(x_{n})-Jz_{n}\|)=\lim_{n\rightarrow\infty}g_{1}(\|Jf(x_{n})-Jz_{n}\|)=0=g_{1}(0),$
(33)
and also $g_{1}$ is strictly increasing, hence
$\lim_{n\rightarrow\infty}\|Jf(x_{n})-Jz_{n}\|)=0.$ (34)
On the other hand, since $J^{-1}$ is uniformly norm-to-norm continuous on
bounded sets, we obtain that
$\lim_{n\rightarrow\infty}\|f(x_{n})-z_{n}\|=\lim_{n\rightarrow\infty}\|J^{-1}(Jf(x_{n}))-J^{-1}(Jz_{n})\|=0.$
(35)
Next, from (13) and (35), we have
$\displaystyle\lim_{n\rightarrow\infty}\phi(z_{n},f(x_{n}))=0.$ (36)
Similarly, from (13), (28) and (31), we obtain
$\displaystyle\lim_{n\rightarrow\infty}\phi(z_{n},x_{n})=0.$ (37)
Moreover, from Lemma 3, the inequalities (36), (37) and the convexity of
$\|.\|^{2}$, it is concluded that
$\displaystyle\phi(z_{n},x_{n+1})\leq$
$\displaystyle\phi(z_{n},J^{-1}(\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}))$
$\displaystyle=$ $\displaystyle\|z_{n}\|^{2}-2\langle
z_{n},\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}\rangle$
$\displaystyle+\|\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}\|^{2}$
$\displaystyle\leq$ $\displaystyle\|z_{n}\|^{2}-2\alpha_{n,1}\langle
z_{n},Jx_{n}\rangle-2\alpha_{n,2}\langle
z_{n},Jf(x_{n})\rangle-2\alpha_{n,3}\langle z_{n},Jz_{n}\rangle$
$\displaystyle+\alpha_{n,1}\|x_{n}\|^{2}+\alpha_{n,2}\|f(x_{n})\|^{2}+\alpha_{n,3}\|z_{n}\|^{2}$
$\displaystyle=$
$\displaystyle\alpha_{n,1}\phi(z_{n},x_{n})+\alpha_{n,2}\phi(z_{n},f(x_{n}))+\alpha_{n,3}\phi(z_{n},z_{n})$
$\displaystyle=$
$\displaystyle\alpha_{n,1}\phi(z_{n},x_{n})+\alpha_{n,2}\phi(z_{n},f(x_{n}))\rightarrow
0\;\;\;as\;n\rightarrow\infty,$
then using Lemma 1, we get
$\displaystyle\lim_{n\rightarrow\infty}\|z_{n}-x_{n+1}\|=0.$ (38)
It follows from (28), (31) and (38) that
$\|x_{n+1}-x_{n}\|\leq\|x_{n+1}-z_{n}\|+\|z_{n}-y_{n}\|+\|y_{n}-x_{n}\|\rightarrow
0\;\;\;as\;n\rightarrow\infty.$ (39)
So $\\{x_{n}\\}$ is a Cauchy sequence, thus $\\{x_{n}\\}$ converges strongly
to a point $q\in C$. It follows from (28) and (31) that the sequences
$\\{y_{n}\\}$ and $\\{z_{n}\\}$ are convergent to $q$. Next, it will be shown
that $q\in VI(C,A)$. Let $B\subset E\times E^{*}$ be an operator defined as
follows:
$Bv=\left\\{\begin{array}[]{lr}\lambda_{n}Av+N_{C}v,\qquad v\in C,\\\
\emptyset,\qquad\qquad\qquad v\notin C.\end{array}\right.$ (40)
Since $\lambda_{n}A$ is $\lambda_{n}\alpha$-inverse strongly monotone, it is
followed that $\lambda_{n}A$ is $\frac{1}{\lambda_{n}\alpha}$-Lipschitz
continuous, hence $\lambda_{n}A$ is hemicontinuous. Therefore, by Lemma 8 $B$
is maximal monotone and $B^{-1}(0)=VI(C,\lambda_{n}A)=VI(C,A)$. Let
$(\upsilon,w)\in G(B)$ with $w\in
B\upsilon=\lambda_{n}A\upsilon+N_{C}(\upsilon)$. Then
$w-\lambda_{n}A\upsilon\in N_{C}(\upsilon)$, hence
$\langle\upsilon-y_{n},w-\lambda_{n}A\upsilon\rangle\geq 0,$ (41)
because $y_{n}\in C$. On the other hand by Lemma 2, it is concluded that
$\langle\upsilon-y_{n},J(J^{-1}(Jx_{n}-\lambda_{n}Ax_{n}))-Jy_{n}\rangle\leq
0,$
so
$\langle\upsilon-y_{n},\lambda_{n}Ax_{n}+Jy_{n}-Jx_{n}\rangle\geq 0.$ (42)
From (41), (42) and using the definition $A$, we get
$\displaystyle\langle\upsilon-y_{n},w\rangle$ $\displaystyle\geq$
$\displaystyle\lambda_{n}\langle\upsilon-
y_{n},A\upsilon\rangle-\langle\upsilon-
y_{n},\lambda_{n}Ax_{n}+Jy_{n}-Jx_{n}\rangle$ $\displaystyle=$
$\displaystyle\lambda_{n}\langle\upsilon-y_{n},A\upsilon-
Ay_{n}\rangle+\lambda_{n}\langle\upsilon-y_{n},Ay_{n}\rangle$
$\displaystyle-\langle\upsilon-y_{n},\lambda_{n}Ax_{n}+Jy_{n}-Jx_{n}\rangle$
$\displaystyle\geq$ $\displaystyle\lambda_{n}\langle\upsilon-
y_{n},Ay_{n}-Ax_{n}\rangle-\langle\upsilon-y_{n},Jy_{n}-Jx_{n}\rangle$
$\displaystyle\geq$ $\displaystyle-\lambda_{n}\|\upsilon-
y_{n}\|\|Ax_{n}-Ay_{n}\|-\|\upsilon-y_{n}\|\|Jx_{n}-Jy_{n}\|.$ (43)
Hence, using uniformly norm-to-norm continuity of $J$ on bounded sets and
(28), $\langle\upsilon-y_{n},w\rangle\geq 0$ as $n\rightarrow\infty$, i.e.
$\langle\upsilon-q,w\rangle\geq 0$. Therefore $\langle
q-\upsilon,0-w\rangle\geq 0$, it is concluded from Lemma 8 that $q\in
B^{-1}(0)=VI(C,A)$, because $B$ is a maximal monotone operator.
Next, we show that $q\in F(f)$. From (28), (31) and (35), we have
$\|f(x_{n})-x_{n}\|\leq\|f(x_{n})-z_{n}\|+\|z_{n}-y_{n}\|+\|y_{n}-x_{n}\|\rightarrow
0\;\;\;as\;n\rightarrow\infty,$ (44)
and since $x_{n}\rightharpoonup q$, then $q$ is an asymptotic fixed point of
$f$. Moreover, $\hat{F}(f)=F(f)$, because $f$ is a relatively nonexpansive
mapping, hence $q\in F(f)$. Therefore,
$\Pi_{VI(C,A)}of(q)=\Pi_{VI(C,A)}(q)=q$.
###### Theorem 3.2
Suppose that $\tilde{F}$ is a bifunction from $C\times C$ to $\mathbb{R}$
which satisfies the conditions $(A_{1})-(A_{4})$. Let f be a relatively
nonexpansive self-mapping on $C$ and $\|Ax\|\leq\|Ax-Au\|$ for all $x\in C$
and $u\in\Omega:=VI(C,A)\cap GEP(\tilde{F},A)\cap F(f)$. Let $x_{0}$ be an
arbitrary point in $C$ and $\\{x_{n}\\}$ be a sequence generated by
$\left\\{\begin{array}[]{lr}u_{n}\in C\;\;s.t\;\;\tilde{F}(u_{n},y)+\langle
Au_{n},y-u_{n}\rangle+\frac{1}{r_{n}}\langle y-u_{n},Ju_{n}-Jx_{n}\rangle\geq
0,\\\ w_{n}=\Pi_{C}J^{-1}(Ju_{n}-\lambda_{n}Au_{n}),\\\
y_{n}=\Pi_{C}J^{-1}(Jx_{n}-\lambda_{n}Ax_{n}),\\\ C_{n}=\\{v\in
C:\phi(v,w_{n})\leq\phi(v,x_{n})\\},\\\
z_{n}=\Pi_{C_{n}}J^{-1}(Jy_{n}-\lambda_{n}Ay_{n}),\\\
x_{n+1}=\Pi_{C}J^{-1}(\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}).\end{array}\right.$
(45)
where $\\{\lambda_{n}\\}\subseteq[0,1]$ such that
$\displaystyle\lim_{n\rightarrow\infty}\lambda_{n}=0$, and
$\\{r_{n}\\}\subset[a,\infty)$ for some $a>0$. If
$\\{\alpha_{n,i}\\}\subset[0,1]$ for $i=1,2,3,4$ such that
$\sum_{i=1}^{4}\alpha_{n,i}=1$ and
$\displaystyle\liminf_{n\to\infty}\alpha_{n,2}\alpha_{n,3}>0$ and
$\displaystyle\liminf_{n\to\infty}\alpha_{n,2}\alpha_{n,4}>0$. Then the
sequence $\\{x_{n}\\}$ generated by (45) converges strongly to
$q=\Pi_{VI(C,A)\cap GEP(\tilde{F},A)}\circ f(q)$.
###### Proof
Clearly, by part $(i)$ of lemma 7, the sequence $\\{u_{n}\\}$ exists. Now, it
will be checked that $C_{n}$ is closed and convex for each $n\geq 1$.
Obviously, by the definition of $C_{n}$, it is clear that $C_{n}$ is closed.
Applying the definition of $\phi$, the inequality
$\phi(v,w_{n})\leq\phi(v,x_{n})$ is equivalent to
$2\langle v,Jx_{n}-Jw_{n}\rangle\leq\|x_{n}\|^{2}-\|w_{n}\|^{2}.$ (46)
It is obvious from (46) that $C_{n}$ is convex for all $n\geq 1$.
Now, it will be verified that $\\{x_{n}\\}$ is well defined. Suppose that
$p\in\Omega$. By Lemma 7, it may be put $u_{n}=K_{r_{n}}x_{n}$. So, by the
condition (5) of Lemma 7, it is concluded that
$\phi(p,u_{n})=\phi(p,K_{r_{n}}x_{n})\leq\phi(p,x_{n}).$ (47)
Moreover, from Lemma 3 and the inequality (14), it follows that
$\displaystyle\phi(p,w_{n})=$
$\displaystyle\phi(p,\Pi_{C}J^{-1}(Ju_{n}-\lambda_{n}Au_{n}))$
$\displaystyle\leq$ $\displaystyle\phi(p,J^{-1}(Ju_{n}-\lambda_{n}Au_{n}))$
$\displaystyle\leq$ $\displaystyle V(p,Ju_{n}-\lambda_{n}Au_{n})$
$\displaystyle\leq$ $\displaystyle V(p,Ju_{n})-2\langle
J^{-1}(Ju_{n}-\lambda_{n}Au_{n})-p,\lambda_{n}Au_{n}\rangle$ $\displaystyle=$
$\displaystyle\phi(p,u_{n})-2\lambda_{n}\langle u_{n}-p,Au_{n}\rangle$
$\displaystyle+2\langle
J^{-1}(Ju_{n}-\lambda_{n}Au_{n})-J^{-1}(Ju_{n}),-\lambda_{n}Au_{n}\rangle,$
(48)
since $A$ is an $\alpha-$inverse strongly monotone operator, it is proved that
$\displaystyle-2\lambda_{n}\langle u_{n}-p,$ $\displaystyle Au_{n}\rangle$
$\displaystyle=$ $\displaystyle-2\lambda_{n}\langle
u_{n}-p,Au_{n}-Ap\rangle-2\lambda_{n}\langle u_{n}-p,Ap\rangle$
$\displaystyle\leq$ $\displaystyle-2\lambda_{n}\alpha\|Au_{n}-Ap\|^{2}.$ (49)
From Lemma 4 and the condition $\|Ax\|\leq\|Ax-Ap\|$ for all $x\in C$, it is
demonstrated that
$\displaystyle 2\langle J^{-1}(Ju_{n}-\lambda_{n}$ $\displaystyle
Au_{n})-J^{-1}(Ju_{n}),-\lambda_{n}Au_{n}\rangle$ $\displaystyle\leq$
$\displaystyle
2\|J^{-1}(Ju_{n}-\lambda_{n}Au_{n})-J^{-1}(Ju_{n})\|\|\lambda_{n}Au_{n}\|$
$\displaystyle=$ $\displaystyle\frac{4\lambda_{n}^{2}}{c^{2}}\|Au_{n}\|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{4\lambda_{n}^{2}}{c^{2}}\|Au_{n}-Ap\|^{2}.$ (50)
By substituting (3) and (3) in (3) and the assumption
$0<\lambda_{n}<\frac{c^{2}\alpha}{2}$, it is implied that
$\phi(p,w_{n})\leq\phi(p,u_{n})+2\lambda_{n}(\frac{2}{c^{2}}\lambda_{n}-\alpha)\|Au_{n}-Ap\|^{2}\leq\phi(p,u_{n}).$
(51)
From (47) and (51), it is evident that
$\phi(p,w_{n})\leq\phi(p,x_{n}).$ (52)
Then $p\in C_{n}$ and hence $\\{x_{n}\\}$ is well defined.
Let $\Omega\neq\emptyset$ and $\hat{u}\in\Omega$. From Lemma 3, the convexity
of $\|.\|^{2}$ and the relatively nonexpansiveness of $f$, it follows
$\displaystyle\phi(\hat{u},x_{n+1}$ $\displaystyle)$ $\displaystyle\leq$
$\displaystyle\phi(\hat{u},J^{-1}(\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}))$
$\displaystyle=$
$\displaystyle\|\hat{u}\|^{2}-2\langle\hat{u},\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}\rangle$
$\displaystyle+\|\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}\|^{2}$
$\displaystyle\leq$
$\displaystyle\|\hat{u}\|^{2}-2\alpha_{n,1}\langle\hat{u},Jx_{n}\rangle-2\alpha_{n,2}\langle\hat{u},Jf(x_{n})\rangle-2\alpha_{n,3}\langle\hat{u},Jz_{n}\rangle-2\alpha_{n,4}\langle\hat{u},Jw_{n}\rangle$
$\displaystyle+\alpha_{n,1}\|x_{n}\|^{2}+\alpha_{n,2}\|f(x_{n})\|^{2}+\alpha_{n,3}\|z_{n}\|^{2}+\alpha_{n,4}\|w_{n}\|^{2}$
$\displaystyle=$
$\displaystyle\alpha_{n,1}\phi(\hat{u},x_{n})+\alpha_{n,2}\phi(\hat{u},f(x_{n}))+\alpha_{n,3}\phi(\hat{u},z_{n})+\alpha_{n,4}\phi(\hat{u},w_{n})$
$\displaystyle\leq$
$\displaystyle\alpha_{n,1}\phi(\hat{u},x_{n})+\alpha_{n,2}\phi(\hat{u},x_{n})+\alpha_{n,3}\phi(\hat{u},z_{n})+\alpha_{n,4}\phi(\hat{u},w_{n})$
$\displaystyle=$
$\displaystyle(\alpha_{n,1}+\alpha_{n,2})\phi(\hat{u},x_{n})+\alpha_{n,3}\phi(\hat{u},z_{n})+\alpha_{n,4}\phi(\hat{u},w_{n}).$
Similarly, using Lemma 3, the inequality (25) holds for the algorithm (45),
too. Hence, from (25) and (52), it is implied that
$\phi(\hat{u},x_{n+1})\leq\phi(\hat{u},x_{n}).$ (53)
It is concluded that $\\{\phi(\hat{u},x_{n})\\}$ is decreasing, so from the
boundedness of the sequence $\\{\phi(\hat{u},x_{n})\\}$,
$\displaystyle\lim_{n\rightarrow\infty}\phi(\hat{u},x_{n})$ exists. Also from
(9), $\\{x_{n}\\}$ is bounded and hence from (47) and the relatively
nonexpansiveness of $f$, $\\{u_{n}\\}$ and $\\{f(x_{n})\\}$ are bounded.
Similarly, using Lemma 3, the inequalities (28) and (31) hold for the
algorithm (45). Hence, it is concluded from (28) and (31) that the sequences
$\\{y_{n}\\}$ and $\\{z_{n}\\}$ are bounded. Now, let
$r_{1}=\sup\\{\|z_{n}\|,\|f(x_{n})\|\\}$, by Lemma 5, there exists a
continuous strictly increasing and convex function
$g_{1}:[0,2r_{1}]\longrightarrow[0,\infty)$ with $g_{1}(0)=0$. We get
$\displaystyle\phi(\hat{u},x_{n+1}$ $\displaystyle)$ $\displaystyle\leq$
$\displaystyle\phi(\hat{u},J^{-1}(\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}))$
$\displaystyle=$
$\displaystyle\|\hat{u}\|^{2}-2\langle\hat{u},\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}\rangle$
$\displaystyle+\|\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}\|^{2}$
$\displaystyle\leq$
$\displaystyle\|\hat{u}\|^{2}-2\alpha_{n,1}\langle\hat{u},Jx_{n}\rangle-2\alpha_{n,2}\langle\hat{u},Jf(x_{n})\rangle-2\alpha_{n,3}\langle\hat{u},Jz_{n}\rangle$
$\displaystyle-2\alpha_{n,4}\langle\hat{u},Jw_{n}\rangle+\alpha_{n,1}\|x_{n}\|^{2}+\alpha_{n,2}\|f(x_{n})\|^{2}+\alpha_{n,3}\|z_{n}\|^{2}$
$\displaystyle+\alpha_{n,4}\|w_{n}\|^{2}-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|)$
$\displaystyle=$
$\displaystyle\alpha_{n,1}\phi(\hat{u},x_{n})+\alpha_{n,2}\phi(\hat{u},f(x_{n}))+\alpha_{n,3}\phi(\hat{u},z_{n})+\alpha_{n,4}\phi(\hat{u},w_{n})$
$\displaystyle-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|)$
$\displaystyle\leq$
$\displaystyle\alpha_{n,1}\phi(\hat{u},x_{n})+\alpha_{n,2}\phi(\hat{u},x_{n})+\alpha_{n,3}\phi(\hat{u},z_{n})+\alpha_{n,4}\phi(\hat{u},w_{n})$
$\displaystyle-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|)$
$\displaystyle=$
$\displaystyle(\alpha_{n,1}+\alpha_{n,2})\phi(\hat{u},x_{n})+\alpha_{n,3}\phi(\hat{u},z_{n})+\alpha_{n,4}\phi(\hat{u},w_{n})$
$\displaystyle-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|).$
Now from (25) and (52), we have
$\phi(\hat{u},x_{n+1})\leq\phi(\hat{u},x_{n})-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|).$
(54)
So
$\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|)\leq\phi(\hat{u},x_{n})-\phi(\hat{u},x_{n+1}).$
Since $\liminf_{n\rightarrow\infty}\alpha_{n,2}\alpha_{n,3}>0$, using the
method as in the proof of Theorem 3.1, we conclude that the inequality (36)
and (37) hold.
By Lemma 3 and convexity of $\|.\|^{2}$, it is obtained that
$\displaystyle\phi(z_{n},x_{n+1}$ $\displaystyle)$ $\displaystyle\leq$
$\displaystyle\phi(z_{n},J^{-1}(\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}))$
$\displaystyle=$ $\displaystyle\|z_{n}\|^{2}-2\langle
z_{n},\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}\rangle$
$\displaystyle+\|\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}\|^{2}$
$\displaystyle\leq$ $\displaystyle\|z_{n}\|^{2}-2\alpha_{n,1}\langle
z_{n},Jx_{n}\rangle-2\alpha_{n,2}\langle
z_{n},Jf(x_{n})\rangle-2\alpha_{n,3}\langle z_{n},Jz_{n}\rangle$
$\displaystyle-2\alpha_{n,4}\langle
z_{n},Jw_{n}\rangle+\alpha_{n,1}\|x_{n}\|^{2}+\alpha_{n,2}\|f(x_{n})\|^{2}+\alpha_{n,3}\|z_{n}\|^{2}$
$\displaystyle+\alpha_{n,4}\|w_{n}\|^{2}$ $\displaystyle=$
$\displaystyle\alpha_{n,1}\phi(z_{n},x_{n})+\alpha_{n,2}\phi(z_{n},f(x_{n}))+\alpha_{n,3}\phi(z_{n},z_{n})+\alpha_{n,4}\phi(z_{n},w_{n})$
$\displaystyle\leq$
$\displaystyle(\alpha_{n,1}+\alpha_{n,4})\phi(z_{n},x_{n})+\alpha_{n,2}\phi(z_{n},f(x_{n})),$
because $z_{n}\in C_{n}$. Using (36), (37) and taking the limit in the above
as $n\rightarrow\infty$, it is deduced that
$\phi(z_{n},x_{n+1})\rightarrow 0.$
Then, from Lemma 1, we have
$\displaystyle\lim_{n\rightarrow\infty}\|x_{n+1}-z_{n}\|=0,$
therefore, it follows from (28), (31) that
$\displaystyle\|x_{n+1}-x_{n}\|\leq\|x_{n+1}-z_{n}\|+\|z_{n}-y_{n}\|+\|y_{n}-x_{n}\|\rightarrow
0\;\;as\;\;n\rightarrow\infty,$
hence, $\\{x_{n}\\}$ is a cauchy sequence. Thus, $\\{x_{n}\\}$ converges
strongly to a point $q\in C$. Obviously, the relations (40), (41), (42) and
(3) are valid for the algorithm (45). Hence, as in the proof of Theorem 3.1,
it is understood that $q\in VI(C,A)$.
Now, it will be proved that $q\in GEP(\tilde{F},A)$. From (37) and the fact
that $z_{n}\in C_{n}$, it is induced that $\phi(z_{n},w_{n})\rightarrow 0$ as
$n\rightarrow\infty$. Therefore, by Lemma 1, we have
$\displaystyle\lim_{n\rightarrow\infty}\|z_{n}-w_{n}\|=0.$ (55)
From (28), (31) and (55), it is evident that
$\displaystyle\lim_{n\rightarrow\infty}\|x_{n}-w_{n}\|=0.$ (56)
Assume that $r_{2}=\sup\\{\|u_{n}\|,\|x_{n}\|\\}$. From Lemma 6, there exists
a continuous, convex and strictly increasing function
$g_{2}:[0,2r_{2}]\longrightarrow[0,\infty)$ such that $g_{2}(0)=0$ and
$\displaystyle g_{2}(\|u_{n}-x_{n}\|)\leq\phi(u_{n},x_{n}).$ (57)
Since $u_{n}=K_{r_{n}}(x_{n})$ and by using (51), (57) and condition $(5)$ of
Lemma 7, it is implied that
$\displaystyle g_{2}(\|u_{n}-x_{n}\|)\leq$ $\displaystyle\phi(u_{n},x_{n})$
$\displaystyle\leq$ $\displaystyle\phi(u,x_{n})-\phi(u,u_{n})$
$\displaystyle\leq$ $\displaystyle\phi(u,x_{n})-\phi(u,w_{n})$
$\displaystyle=$ $\displaystyle\|u\|^{2}-2\langle
u,Jx_{n}\rangle+\|x_{n}\|^{2}-\|u\|^{2}+2\langle
u,Jw_{n}\rangle-\|w_{n}\|^{2}$ $\displaystyle=$
$\displaystyle\|x_{n}\|^{2}-\|w_{n}\|^{2}+2\langle u,Jw_{n}-Jx_{n}\rangle$
$\displaystyle\leq$
$\displaystyle\|x_{n}\|^{2}-\|w_{n}\|^{2}+2\|u\|\|Jw_{n}-Jx_{n}\|$
$\displaystyle\leq$
$\displaystyle(\|x_{n}-w_{n}\|+\|w_{n}\|)^{2}-\|w_{n}\|^{2}+2\|u\|\|Jw_{n}-Jx_{n}\|$
$\displaystyle\leq$
$\displaystyle\|x_{n}-w_{n}\|^{2}+2\|w_{n}\|\|x_{n}-w_{n}\|+2\|u\|\|Jw_{n}-Jx_{n}\|,$
from (56) and the condition uniformly norm-to-norm continuity of $J$ on
bounded sets, we have
$\displaystyle\lim_{n\rightarrow\infty}g_{2}(\|u_{n}-x_{n}\|)=0$. Then it is
followed from the conditions that $g_{2}$ is a strictly increasing and
continuous function that $\|u_{n}-x_{n}\|\rightarrow 0$ as
$n\rightarrow\infty$. Then
$\lim_{n\rightarrow\infty}\|Ju_{n}-Jx_{n}\|\rightarrow 0.$ (58)
Since $u_{n}=K_{r_{n}}x_{n}$, we concluded that
$\tilde{F}(u_{n},y)+\langle Au_{n},y-u_{n}\rangle+\frac{1}{r_{n}}\langle
y-u_{n},Ju_{n}-Jx_{n}\rangle\geq 0,$ (59)
for all $y\in C$. From the condition $(A_{2})$, we have
$\tilde{F}(y,u_{n})\leq-\tilde{F}(u_{n},y)\;\;\text{for}\;\text{all}\;y\in C.$
(60)
From (59) and (60), it is implied that
$\displaystyle\tilde{F}(y,u_{n})\leq-\tilde{F}(u_{n},y)\leq\langle
Au_{n},y-u_{n}\rangle+\frac{1}{r_{n}}\langle y-u_{n},Ju_{n}-Jx_{n}\rangle,$
for all $y\in C$. Letting $n\rightarrow\infty$, using condition $(A_{4})$ and
by (58), it can be concluded that
$\tilde{F}(y,q)\leq\langle Aq,y-q\rangle\;\;\;\text{for}\;\text{all}\;y\in C.$
(61)
Put $y_{\lambda}=\lambda y+(1-\lambda)q$ for all $y\in C$ and
$\lambda\in(0,1)$. Now from the conditions $(A_{1})$, $(A_{4})$, the
inequality (61), the monotonicity of $A$ and the convexity of $\tilde{F}$, we
have
$\displaystyle 0=$ $\displaystyle\tilde{F}(y_{\lambda},y_{\lambda})+\langle
Ay_{\lambda},y_{\lambda}-y_{\lambda}\rangle$ $\displaystyle\leq$
$\displaystyle\lambda\tilde{F}(y_{\lambda},y)+(1-\lambda)\tilde{F}(y_{\lambda},q)+\langle
Ay_{\lambda},\lambda y+(1-\lambda)q-y_{\lambda}\rangle$ $\displaystyle=$
$\displaystyle\lambda\tilde{F}(y_{\lambda},y)+(1-\lambda)\tilde{F}(y_{\lambda},q)+\lambda\langle
Ay_{\lambda},y-y_{\lambda}\rangle+(1-\lambda)\langle
Ay_{\lambda},q-y_{\lambda}\rangle$ $\displaystyle=$
$\displaystyle\lambda\tilde{F}(y_{\lambda},y)+(1-\lambda)\tilde{F}(y_{\lambda},q)+\lambda\langle
Ay_{\lambda},y-y_{\lambda}\rangle+(1-\lambda)\langle
Ay_{\lambda}-Aq,q-y_{\lambda}\rangle$ $\displaystyle+(1-\lambda)\langle
Aq,q-y_{\lambda}\rangle$ $\displaystyle\leq$
$\displaystyle\lambda\tilde{F}(y_{\lambda},y)+\lambda\langle
Ay_{\lambda},y-y_{\lambda}\rangle,$
for all $y\in C$. So $0\leq\tilde{F}(y_{\lambda},y)+\langle
Ay_{\lambda},y-y_{\lambda}\rangle$. Now by taking limit as $\lambda\rightarrow
0$ and by using the condition $(A_{3})$, it is followed that
$0\leq\tilde{F}(q,y)+\langle Aq,y-q\rangle$ for all $y\in C$. Therefore $q\in
GEP(\tilde{F},A)$.
Now, we show that $q\in F(f)$. Let $r_{3}=\sup\\{\|w_{n}\|,\|f(x_{n})\|\\}$,
hence, in a similar way with (54), there exists a continuous, convex and
strictly increasing function $g_{3}:[0,2r_{3}]\longrightarrow[0,\infty)$ whit
$g_{3}(0)=0$, such that
$\phi(\hat{u},x_{n+1})\leq\phi(\hat{u},x_{n})-\alpha_{n,2}\alpha_{n,4}g_{3}(\|Jf(x_{n})-Jw_{n}\|),$
hence
$\alpha_{n,2}\alpha_{n,4}g_{3}(\|Jf(x_{n})-Jw_{n}\|)\leq\phi(\hat{u},x_{n+1})-\phi(\hat{u},x_{n}).$
Taking the limit as $n\longrightarrow\infty$ and using our assumptions, we
obtain
$\displaystyle\lim_{n\rightarrow\infty}g_{3}(\|Jf(x_{n})-Jw_{n}\|)=0,$
since $g_{1}$ is a continuous function, it is easy to see that
$\displaystyle\lim_{n\rightarrow\infty}\|Jf(x_{n})-Jw_{n}\|=0.$ (62)
Therefore
$\lim_{n\rightarrow\infty}\|f(x_{n})-w_{n}\|=\lim_{n\rightarrow\infty}\|J^{-1}(Jf(x_{n}))-J^{-1}(Jw_{n})\|=0,$
(63)
because $J^{-1}$ is uniformly norm-to-norm continuous on bounded sets. From
(56) and (63), it is concluded that
$\|f(x_{n})-x_{n}\|\leq\|f(x_{n})-w_{n}\|+\|w_{n}-x_{n}\|\rightarrow
0\;\;\;as\;n\rightarrow\infty,$
and since $x_{n}\rightharpoonup q$, then $q\in\hat{F(f)}=F(f)$. Hence
$\\{x_{n}\\}$ is strongly convergent to a point $q\in\Omega$, and also we have
$q=\Pi_{VI(C,A)\cap GEP(\tilde{F},A)}\circ f(q)$.
## 4 Numerical example
Now, some examples are given to illustrate Theorem 3.2. Then the behaviors of
the sequences $\\{x_{n}\\},\\{y_{n}\\},\\{z_{n}\\}$ and $\\{w_{n}\\}$ are
investigated which were generated by the algorithm (45).
###### Example 1
Let $E=\mathbb{R}$, $C=[-5,5]$, $A=I$, $\lambda_{n}=\frac{1}{n}$, $c=1$,
$\alpha=1$ and $f$ be a self-mapping on $C$ defined by $f(x)=\frac{x}{3}$ for
all $x\in C$. Consider the function $\tilde{F}:C\times C\rightarrow\mathbb{R}$
defined by
$\tilde{F}(u,y):=16y^{2}+9uy-25u^{2},$
for all $u$, $y\in C$. We see that $f$ satisfies in the conditions (A1) - (A4)
as follows:
(A1) $\tilde{F}(u,u)=16u^{2}+9u^{2}-25u^{2}=0$ for all $u\in[-5,5]$,
(A2) $\tilde{F}$ is monotone, because
$\tilde{F}(u,y)+\tilde{F}(y,u)=-9(u-y)^{2}\leq 0$ for all $y,u\in[-5,5]$,
(A3) for each $u,y,z\in[-5,5],$
$\displaystyle\lim_{\lambda\to 0}\tilde{F}(\lambda z+(1$
$\displaystyle-\lambda)u,y)$ $\displaystyle=$ $\displaystyle\lim_{\lambda\to
0}(16y^{2}+9(\lambda z+(1-\lambda)u)y-25(\lambda z+(1-\lambda)u)^{2})$
$\displaystyle=$ $\displaystyle 16y^{2}+9uy-25u^{2}$ $\displaystyle=$
$\displaystyle\tilde{F}(u,y).$
(A4) Obviously, for each $u\in[-5,5]$, $y\rightarrow(16y^{2}+9uy-25u^{2})$ is
convex and lower semicontinuous.
Let $u\in K_{r}x$, hence, it is concluded from Lemma 7 that
$\displaystyle\tilde{F}(u,y)+\langle Au,y-u\rangle+\frac{1}{r}\langle y-u,Ju-
Jx\rangle\geq 0,$
for all $y\in[-5,5]$ and $r>0$, i.e.,
$\displaystyle 0\leq 16ry^{2}+9ruy-25ru^{2}+$ $\displaystyle ruy-
ru^{2}+uy-u^{2}+ux-xy$ $\displaystyle=$ $\displaystyle
16ry^{2}+(10ru+u-x)y-26ru^{2}-u^{2}+ux.$
Let $a=16r$, $b=10ru+u-x$ and $c=-26ru^{2}-u^{2}+ux$. Then, it is implied that
$\triangle=b^{2}-4ac\leq 0$, i.e.,
$\displaystyle 0\geq(10ru+u-x)^{2}-64r(-26$ $\displaystyle ru^{2}-u^{2}+ux)$
$\displaystyle=$ $\displaystyle 1764r^{2}u^{2}+84ru^{2}+u^{2}-84rux-2ux+x^{2}$
$\displaystyle=$ $\displaystyle((42r+1)u-x)^{2}.$
It follows that $u=\frac{x}{42r+1}$. It is concluded from Lemma 7 that $K_{r}$
is single valued. Hence, $K_{r}x=\frac{x}{42r+1}$. Now by applying in theorem
3.2, it is implied that $u_{n}=\frac{x_{n}}{42r_{n}+1}$ where $\\{x_{n}\\}$ is
a sequence generated by the algorithm (45). Since $F(K_{r_{n}})=\\{0\\}$, from
condition $(3)$ of Lemma 7, we have $GEP(\tilde{F},I)=\\{0\\}$.
Obviously, $F(f)=\\{0\\}$ and $\phi(0,f(x))\leq\phi(0,x)$, for all $x\in C$.
Now, let $x_{n}\rightharpoonup q$ and also
$\lim_{n\to\infty}(f(x_{n})-x_{n})=0$, hence $q=0$ and
$\hat{F(f)}=\\{0\\}=F(f)$. Therefore, $f$ is a relatively nonexpansive
mapping. Moreover, it is obvious that $0\in VI(C,I)$. Therefore,
$0=\Pi_{\\{0\\}}of(0)=\Pi_{VI(C,I)\cap GEP(\tilde{F},I)}of(0)$.
Next, assume that
$\alpha_{n,1}=\frac{1}{4}+\frac{1}{4n},\alpha_{n,2}=\frac{1}{4}-\frac{1}{6n},\alpha_{n,3}=\frac{1}{4}+\frac{1}{12n},\alpha_{n,4}=\frac{1}{4}-\frac{1}{6n}$,
for all $n\in\mathbb{N}$ and $u_{0}=0$, so clearly $\alpha_{n}$, $\beta_{n}$
and $\gamma_{n}$ satisfy in the conditions of Theorem 3.2. Since $x_{n}\in C$,
we have
$\left\\{\begin{array}[]{lr}w_{n}=\Pi_{C}J^{-1}(u_{n}-\frac{1}{n}u_{n})=\frac{n-1}{n}u_{n}=\frac{n-1}{2n}x_{n},\\\
y_{n}=\Pi_{C}J^{-1}(x_{n}-\frac{1}{n}x_{n})=\Pi_{C}\frac{n-1}{n}x_{n}=\frac{n-1}{n}x_{n},\\\
C_{n}=\\{v\in C:|v-w_{n}|\leq|v-x_{n}|\\},\\\
z_{n}=\Pi_{C_{n}}J^{-1}(y_{n}-\frac{1}{n}y_{n})=\frac{n-1}{n}y_{n}=(\frac{n-1}{n})^{2}x_{n},\\\
x_{n+1}=\Pi_{C}J^{-1}((\frac{1}{4}+\frac{1}{4n})x_{n}+(\frac{1}{4}-\frac{1}{6n})\frac{1}{3}x_{n}+(\frac{1}{4}+\frac{1}{12n})(\frac{n-1}{n})^{2}x_{n}\\\
\hskip
31.2982pt+(\frac{1}{4}-\frac{1}{6n})\frac{n-1}{2n}x_{n}).\end{array}\right.$
See the table LABEL:tableexample1 and Figure LABEL:pp1 with the initial point
$x_{1}=5$ of the sequence $\\{x_{n}\\}$.
## Declarations
Not applicable.
## Funding
No funding is applicable to this article.
## Conflict of interest
The authors declare that they have no conflict of interest.
## Authors’ contributions
The two authors equally contributed, read, and approved the final manuscript.
## Competing interests
The authors declare that they have no competing interests.
## Acknowledgements
The authors would like to thank the referees for their esteemed comments and
suggestions.
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|
# Exploring Design and Governance Challenges in the Development of Privacy-
Preserving Computation
Nitin Agrawal<EMAIL_ADDRESS>University of OxfordOxford, UK ,
Reuben Binns<EMAIL_ADDRESS>University of OxfordOxford, UK , Max
Van Kleek<EMAIL_ADDRESS>University of OxfordOxford, UK , Kim
Laine<EMAIL_ADDRESS>Microsoft ResearchRedmond, CA, USA and Nigel
Shadbolt<EMAIL_ADDRESS>University of OxfordOxford, UK
(2021)
###### Abstract.
Homomorphic encryption, secure multi-party computation, and differential
privacy are part of an emerging class of Privacy Enhancing Technologies which
share a common promise: to preserve privacy whilst also obtaining the benefits
of computational analysis. Due to their relative novelty, complexity, and
opacity, these technologies provoke a variety of novel questions for design
and governance. We interviewed researchers, developers, industry leaders,
policymakers, and designers involved in their deployment to explore
motivations, expectations, perceived opportunities and barriers to adoption.
This provided insight into several pertinent challenges facing the adoption of
these technologies, including: how they might make a nebulous concept like
privacy computationally tractable; how to make them more usable by developers;
and how they could be explained and made accountable to stakeholders and wider
society. We conclude with implications for the development, deployment, and
responsible governance of these privacy-preserving computation techniques.
privacy-enhancing technologies, expert interview, cryptography, policy
††journalyear: 2021††copyright: acmlicensed††conference: CHI Conference on
Human Factors in Computing Systems; May 8–13, 2021; Yokohama,
Japan††booktitle: CHI Conference on Human Factors in Computing Systems (CHI
’21), May 8–13, 2021, Yokohama, Japan††price: 15.00††doi:
10.1145/3411764.3445677††isbn: 978-1-4503-8096-6/21/05††ccs: Security and
privacy Usability in security and privacy††ccs: Security and privacy Social
aspects of security and privacy††ccs: Security and privacy Privacy protections
## 1\. Introduction
HCI research on privacy has traditionally focused on end-users: understanding
their privacy attitudes and mental models, studying their privacy-related
behaviours, and designing tools to help them manage data disclosure according
to their preferences. While important, this paradigm of end-user privacy also
has limitations. First, individuals may have their data processed in remote
and opaque ways by dint of being taxpayers, credit risks, or suspected
terrorists — not ‘end users’ as traditionally conceived. In such cases we
still need to understand how privacy as a human right and public good can be
reflected and governed in such systems. Second, the end-user privacy paradigm
neglects the many other entities who play an important role in articulating,
navigating, and embedding privacy in a range of contexts. If HCI is about
reflecting human values in computer systems more broadly (Sellen et al.,
2009), it is equally important to study those people, whether they are
developers (Balebako et al., 2014), designers, risk managers, or policy
makers. Finally, addressing privacy as a problem of end-user interaction often
yields depressing results due to the sheer complexity of personal data
processing making it difficult for end-users to comprehend the choices and
tools available. This is perhaps especially true when such complexity is the
result of modern cryptographic techniques designed to _protect_ privacy
(Whitten and Tygar, 1999).
These three limitations are particularly salient in the context of this paper,
which addresses technologies for _privacy-preserving computation_. These are a
subset of Privacy Enhancing Technologies (PETs) which have emerged in recent
years. These include Homomorphic encryption (HE), secure multi-party
computation (SMPC), and differential privacy (DP). These foundational
technologies share a common promise: to preserve privacy while also obtaining
the benefits of computational analysis. HE enables computation on encrypted
data, making it possible to outsource computation to another entity without
them ever having access to the input data in the clear. SMPC allows multiple
parties to jointly perform a computation based on multiple respective inputs
without revealing those inputs to each other. DP refers to a way of measuring
the extent to which the output of a computation reveals information about an
individual, and a range of associated techniques for reducing it. While these
technologies have been available in some form for years, and to some extent
already are in deployment, recent progress in their foundational techniques
and computational tractability has led some to anticipate their imminent
adoption.111Recent industry analyst reports have suggested that PETs are
‘experiencing a renaissance’(CPOmagazine.Com, 2020), and that 2020 was the
‘the year of PETs’(Deloitte.com, 2020)
Compared to systems and contexts typically studied in privacy-related HCI
research, these privacy-preserving computation techniques may be far removed —
conceptually, operationally and experientially — from the entities whose
privacy they purport to protect. While HE, SMPC, and DP are sometimes touted
in the marketing campaigns of some device makers,222See e.g. Apple
https://www.apple.com/privacy/docs/Differential_Privacy_Overview.pd and
Google’s (Erlingsson et al., 2014) DP initiatives for the most part these
technologies are deployed as invisible infrastructure rather than being
positioned as features which end-users are expected to value, let alone
understand or control themselves. In many other (actual or envisioned)
deployment contexts, the data being kept private may relate to individual data
subjects who are not informed or engaged with its processing; and even if they
were aware, they may have no ability to distinguish between whether such
processing was genuinely ‘privacy-enhancing’ or not. Furthermore, the
mathematical and computational complexity underpinning these techniques raises
particular challenges to explaining them to various stakeholders; not only end
users and/or data subjects, but also developers, investors, product managers,
and policymakers.
These differences make privacy-preserving computation technologies a prime
case study for an expanded understanding of privacy within HCI beyond
traditional paradigms of user attitudes and behaviours(Fallman, 2011), to
consider developers(Acar et al., 2016), managers, policymakers and others
(Oudshoorn and Pinch, 2003), and the roles they play in defining and
operationalising goals like security and privacy. For better or worse, the
development and adoption of these technologies, and the political values and
consequences they reflect may ultimately have relatively little to do with
‘end users’ as traditionally conceived. With these considerations in mind,
this paper aims to explore the following:
1. (1)
What challenges are associated with the adoption of privacy-preserving
computation techniques for different stakeholders?
2. (2)
What are the motivations for adopting them?
3. (3)
Why and how should privacy-preserving computation technologies be explained,
governed, and made accountable to data subjects and wider society?
To gain insight into these questions, we undertook a series of interviews with
a variety of stakeholders involved in various ways in the development and
adoption of privacy-preserving computation technologies (PPCTs). These
included cryptographers and theoretical computer scientists working on
foundational PPC techniques, developers of practical tools and libraries for
non-expert developers, senior managers and policymakers assessing and
identifying real-world use cases, practitioners building PPC products, and
designers working with PPCs as a design material. Our aim was to draw out
implications for HCI and design raised by this new class of technologies.
## 2\. Background
We begin by briefly introducing emerging privacy-preserving computation
techniques. We then situate our approach to studying them in relation to prior
related work in HCI.
### 2.1. Overview of Privacy-Preserving Computation Technologies
Privacy-preserving computation is a subset of Privacy-Enhancing Technologies
(PETs). PETs are a broad category which could include everything from a
sticker placed over a webcam (Machuletz et al., 2018) to advanced
cryptographic techniques (Menezes et al., 1996). Existing and well-established
examples include encryption schemes used to secure data at rest, end-to-end
encryption protecting data over the network, and anonymous routing protocols
to prevent interactions between identities from being revealed. Such
technologies are already widespread, embedded in products and as part of the
global internet infrastructure. While they each have different underlying
approaches and motivations, these technologies are primarily concerned with
the protection of data, at rest and in transit. They generally assume that
once data is safely transferred to a secure endpoint, it can be decrypted and
computed on in the clear; that a single entity performs the computation; and
that whether or not the result of the computation is ‘private’ has a binary
answer.
A more recent wave of PETs — including homomorphic encryption, secure multi-
party computation, and differential privacy —allow these assumptions to be
relaxed or even abandoned altogether. We briefly introduce them here.
#### 2.1.1. Homomorphic Encryption
Informally, homomorphic encryption (HE) enables computation over encrypted
data without ever ‘seeing’ the input or the output. This is realized through a
specific encryption and decryption scheme. In effect, a user could send their
encrypted data to a service provider who could then perform the desired
computation and send back the output to the user, while remaining oblivious to
both the input and the output. More formally, homomorphic encryption is an
encryption primitive that enables secure evaluation of an arbitrary circuit
$f$ on an encryption $C(x)$ of a plaintext $x$, without decrypting $C(x)$ in
the process, and without requiring any information about the private key. Such
an encrypted evaluation results in an encryption $C(f(x))$ (Rivest et al.,
1978; Gentry, 2009), which can at a later point be decrypted by the owner of
the private key, to reveal the result $f(x)$, as if $f$ had been evaluated on
the plaintext data. In principle, homomorphic encryption can be used to
evaluate any circuit on encrypted data, but often a weaker functionality
called leveled homomorphic encryption is used instead, which allows only
circuits of a predetermined (but arbitrarily high) depth to be evaluated on
encrypted data. In practice, the encryption scheme must be parameterised
according to a desired depth bound of some interesting class of circuits.
Homomorphic encryption, and often leveled homomorphic encryption, has found
its application in problems such as secure data retrieval (Yi et al., 2012;
Angel et al., 2018; Chen et al., 2018; Akavia et al., 2019), outsourced
computation (Barbosa and Farshim, 2012; Kerschbaum, 2012) and secure machine
learning as a service for sensitive data (Hall et al., 2011; Gilad-Bachrach et
al., 2016; Sanyal et al., 2018), amongst others.
#### 2.1.2. Secure Multi-Party Computation
Secure Multi-Party Computation (SMPC) is a class of cryptographic primitives
which enables secure evaluation of a function over data shared across multiple
parties. It was formally introduced in 1982 as a 2 party protocol for the
Millionaire’s problem (Yao, 1982). Informally, SMPC primitives allow multiple
parties to come together and jointly compute a function on their combined
inputs while remaining oblivious to each other’s inputs; the Millionaire’s
problem involves two parties learning which has greater wealth without
revealing their respective fortunes. Formally, in an $n-$party setting, party
$P_{i}$ possess an input $x_{i}$ and gets an output $y_{i}$ upon computation
of function $f$ over the combined set of $x_{i}s$ $(i\in\\{1...n\\})$. The
secure computation guarantees privacy of the individual inputs $x_{i}s$. Most
SMPC protocols could be defined by the choice of the circuit for computing a
particular function and the type of secret sharing scheme. Use cases for SMPC
include secure operations over distributed sensitive data such machine
learning (Gao et al., 2018; Agrawal et al., 2019; Mohassel and Zhang, 2017;
Riazi et al., 2019; Hussain et al., 2020), genomic comparison (Jha et al.,
2008; Evans et al., 2011) and private set operations (Hazay, 2018; Hazay and
Lindell, 2008).
#### 2.1.3. Differential Privacy
: Differential Privacy (DP) (Dwork, 2008) is a framework for sharing
information based on a dataset while statistically limiting information
exposure about the individuals in the dataset. More broadly, the idea of
differential privacy is to deploy a mechanism where the effect of a single
substitution in a dataset is very small. In effect, a query on a dataset with
such a mechanism in place does not reveal anything substantial about a single
individual. Differential privacy may not always be considered a privacy-
enhancing technology _per se_ , but rather a theory for _measuring_ privacy in
a particular way. However there are several techniques which are closely
associated with differential privacy, all of which involve adding noise to
results according to differentially private constraints; we therefore refer to
this family of techniques loosely as differential privacy technologies.
Formally, a randomized function $f$ gives $(\epsilon,\delta)$-differential
privacy for all databases $D$ and $D^{\prime}$, non-negative values
$\epsilon$, $\delta$ and $\forall S\subseteq$ range of $f$, where $D$ and
$D^{\prime}$ differs by at most one record iff,
$Pr[f(D)\in S]\leq\delta+e^{\epsilon}Pr[f(D^{\prime})\in S]$
Here $\delta$ and $\epsilon$ are the privacy parameters. Differential privacy
is one of the more widely deployed privacy-preserving computation
technologies. It can be applied to querying databases(Johnson et al., 2018),
building differentially private machine learning models (Abadi et al., 2016;
Wei et al., 2020) and performing statistical analysis (Dwork and Lei, 2009;
Dwork and Smith, 2010) with privacy guarantees. More recently, the US census
used DP in 2020333https://www.census.gov/newsroom/blogs/research-
matters/2018/08/protecting_the_confi.html, Apple has deployed local DP for a
number of
features444https://www.apple.com/privacy/docs/Differential_Privacy_Overview.pdf
and Google has been using DP for collecting data over its Chrome browser
(Erlingsson et al., 2014) in a privacy preserving manner.
#### 2.1.4. ‘Privacy-preserving computation’
Despite their differences, these technologies have all been classed as ‘tools
for privacy-preserving computation’(Team, 2020), which enable ‘the derivation
of useful results from data without giving other people access’ to such data
(Society, 2019). These privacy-preserving computation technologies are still
an emerging technology. While significant theoretical progress has been made,
this has yet to be translated into widespread adoption. However, numerous
libraries exist for HE, SMPC and DP555For HE: Microsoft SEAL (sea, 2020),
HElib (https://github.com/homenc/HElib), PALISADE (https://palisade-
crypto.org); for SMPC: Crypten (https://crypten.ai), emp-toolkit (Wang et al.,
2016), SPDZ (https://github.com/bristolcrypto/SPDZ-2); for DP: Google’s DP
framework (https://github.com/google/differential-privacy), Diffprivlib
(https://github.com/IBM/differential-privacy-library), Pysyft (Ryffel et al.,
2018)., the number of government-funded PPC projects is increasing (Team,
2020), and various industry and policymaking forums have publicly heralded
their potential. Recent reports and working papers have catalogued actual or
potential use-cases, as well as noting possible usability barriers including
programming complexity, computational overhead, and parameter selection
(Cammarota et al., 2020; Team, 2020).
### 2.2. Related work on PETs in HCI
Much research on privacy in HCI is concerned with how end-users value,
negotiate, and manage privacy in the context of their interactions with
computers. Work in this vein involves: understanding the attitudes (Ur et al.,
2012; King, 2013; Shklovski et al., 2014; Van Kleek et al., 2017; Kumaraguru
and Cranor, 2005), expectations (Lin et al., 2012; Balebako et al., 2013) and
mental models (Kang et al., 2015) of end-users regarding how their data is
collected and used; studying privacy-related behaviours such as willingness to
share data (Leon et al., 2013; Acquisti et al., 2015) and use of protective
measures (Baruh et al., 2017); and evaluating and designing tools for privacy
management such as permission settings (Leon et al., 2012; Lin et al., 2014),
privacy notices (Schaub et al., 2015; Balebako et al., 2015), and privacy
assistants (Liu et al., 2016). Related research in usable privacy and security
addresses the usability of various end-user PETs tools. These include privacy
and security aspects of ubiquitous tools e.g. web browsers (Herzberg, 2009;
Cranor et al., 2002), as well as more advanced specialist tools, such as end-
to-end encryption (Whitten and Tygar, 1999) and anonymous communication and
routing tools (Clark et al., 2007). Such work is highly relevant to contexts
in which end users directly interact with systems in ways that may affect
their privacy, and where there are opportunities to (re)design tools and
interfaces to give them more control. Such work is premised on the ideal of
individual users being able to understand at least some aspects of how their
data is processed, and having the potential to exert some meaningful choices
over it.
In some cases, privacy-preserving computation technologies might be usefully
studied from this end-user perspective. Bullek et al. (Bullek et al., 2017)
studied people’s comprehension of the randomized response method for local
differential privacy (Warner, 1965). Participants were asked a series of
questions, the answers to which were perturbed with noise to provide privacy.
In response to a final question about a particularly sensitive topic, they
were able to choose how much perturbation to add (i.e. the value of
$\epsilon$). While most participants selected the lowest (most privacy-
preserving) value for $\epsilon$, surprisingly, 20% chose the highest (least
privacy-preserving) value for $\epsilon$. Some participants explained this was
because adding more noise felt like lying. Xiong et al. (Xiong et al., 2020)
also studied participants’ willingness to share data with a hypothetical
differentially private system. They examined the effect of different
descriptions of differential privacy (including real descriptions provided by
technology companies and the U.S. Census bureau) on willingness to share, and
their findings suggest that certain descriptions (in particular, implication
descriptions) are more understandable and increase willingness to share data
as a result. Finally, Qin et al. explored usability and understanding in the
context of privacy-preserving data aggregation initiatives based on MPC,
finding that using various analogies to explain the process of additive secret
sharing increased participants’ confidence in the scheme (Qin et al., 2019).
However, in many contexts, the ‘data subjects’ are not co-extensive with the
‘users’. In the case of the PPCTs mentioned above, there may be several
primary users (which may include developers and others) and many wider
‘stakeholders’ (e.g. commercial and government partners, the wider public).
Rather than studying end users who are also data subjects, then, we might
instead follow previous HCI research on privacy and security which focuses
instead on other actors, such as developers (e.g. (Balebako et al., 2014; Acar
et al., 2016; Gorski et al., 2020; Assal and Chiasson, 2019)). Balebako et al.
note that while users may be concerned about privacy, they are generally not
‘empowered to protect themselves’; by contrast ‘the decisions made by app
developers have great impact’ (Balebako et al., 2014). Studying developers,
designers, and others can reveal both practical and organisational challenges
hindering the deployment of privacy and security technologies (Acar et al.,
2016; Furnell et al., 2009), highlight discrepancies between privacy research
and privacy engineering (Gurses and Van Hoboken, [n.d.]; Kostova et al.,
2020), as well as elucidate the moral dimensions of design. While some studies
of software developers suggest that they may ‘not have sufficient knowledge
and understanding of the concept of informational privacy’ (Hadar et al.,
2018), others do explicitly engage with the ethical and political
ramifications of their work; e.g. Rogaway (Rogaway, 2015) who acknowledges how
the field of cryptographic privacy technologies has ‘an intrinsically moral
dimension’.
This kind of reflexivity on the part of developers and designers is something
acknowledged and addressed in approaches like Value Sensitive Design (VSD),
which aim to ‘illuminate the ethical and moral responsibility on the part of
the designer rather than the user’ (Fallman, 2011). To understand how
particular technologies are imagined as solutions to problems (Jasanoff and
Kim, 2009), we may need to study a wide variety of actors involved in their
development, not only engineers but also those involved in the business of
marketing them (Oudshoorn and Pinch, 2003). By encompassing the full breadth
of different actors involved in creating and deploying these systems, we are
also able to grapple in different ways with the trade-offs and tensions
inherent in the field of privacy-preserving computation, and ask questions
like ‘“Who is making the design decision?”, “Who is paying for it?”, “What is
this saying about the user?”’ (Harrison et al., 2007).
Finally, there is also work which critically addresses Privacy-Enhancing
Technologies from a philosophical and conceptual perspective. For instance,
Tavani and Moor (Tavani and Moor, 2001) assess how earlier PETs such as PGP
and anonymity tools may address privacy as individual control, but do not
provide ‘external’ control beyond the user, which they argue is necessary to
protect privacy in the round. Gurses and Berendt point to the limitations of
PETs that stem from understanding privacy solely in terms of confidentiality
(Gürses and Berendt, 2010). Stalder points to the ways that PETs designed for
individual use may occlude broader social meanings of privacy (Stalder, 2002),
while Phillips notes how PETs designed to assist businesses with automating
compliance with privacy laws reinforce a restricted notion of privacy as
unwanted intrusion (Phillips, 2004).
## 3\. Research Approach
Given that the technologies being addressed here are still emerging, and the
broad and exploratory nature of our research questions, we chose to undertake
in-depth semi-structured interviews with a select range of experts from a
range of backgrounds and roles (Bogner et al., 2009). All had direct
experience of working on projects relating to privacy-preserving computation,
and occupied different strategic positions in the developing ecosystem. They
included: researchers working across HE, SMPC and DP research; industry
practitioners and designers with experience delivering practical applications
of these technologies; as well policy experts with experience in PETs. We
deliberately selected some experts whose careers and roles bridged between the
domains of research, industry, or policy, some having moved from one to the
other over the course of a career, while others maintaining feet in multiple
domains simultaneously. These participants can be seen as ‘boundary workers’,
working between the boundaries of science and policy to facilitate the co-
production of knowledge and innovation (Hoppe, 2009) and ‘knowledge brokers’
who facilitate connections between scientific and other audiences (Meyer,
2010). Including a variety of different roles also reflects the nature of
these technologies as ‘use-inspired basic research’(Oulasvirta and Hornbæk,
2016) operating between ‘basic’ and ‘applied’ research paradigms (Stokes,
2011). This enabled us to not only understand how the knowledge surrounding
these technologies is made in specific places (e.g. research labs, technology
companies, government) but also ‘how transactions occur between
places’(Shapin, 1998).
Because these technologies are still emerging, we inevitably could only draw
from a small class of professionals, whose roles in the production of these
technologies are to some extent ill-defined. As is typical with expert
interviews, there was no comprehensive list of relevant experts to sample
from; we therefore built a ‘sample frame’ based on publicly available
materials from a wide variety of sources including research papers, industry
and policymaking fora, and press (Goldstein, 2002), to identify potentially
relevant experts, and also used snowball sampling. As a result of the variety
of roles and experiences of our 9 experts (see Table 1), we used a semi-
structured interview format slightly tailored to four different roles
(research, industry, policy, design). We invited participants to discuss their
experiences, motivations, perceived opportunities and barriers, relating to
this space. Open ended questions allowed us to have relatively free reign to
explore these issues (Page et al., 2005). The interviews were conducted over
video chat during Spring and Summer 2020. The 9 interviews varied in length
between 35-75 minutes, with the average taking 55 minutes, producing 8.3 hours
of audio recordings in total, which were transcribed. All parts of the study
were approved by our institution’s ethics review committee. We used thematic
analysis (Braun and Clarke, 2006) to identify key themes and ideas discussed
by experts. Two of the authors independently developed a set of codes based on
close reading of disjoint subsets of the interview transcripts, using an open
coding process. The two authors then discussed and consolidated their codes to
derive a common set, which was then applied by both authors to all the
interview transcripts, and memo notes were taken to record observations about
the codes and their relation to one another (Lempert, 2007). A final round of
discussion based on this data resulted in a set of themes and sub-themes,
presented in the following section.
Participants | Description (& years experience in PETs)
---|---
P1[R] | Cryptography Researcher and an industrial PETs library developer (20-30y experience)
P2[R] | Cryptography Researcher at company specialising in privacy preserving computation (5-10y experience)
P3[R] | Cryptography Researcher and an industrial PETs library developer (5-10y experience)
P4[P] | Law and Policy professional working on industrial adoption of privacy-preserving ML (4-6y experience)
P5[P] | Senior government adviser on technology, with strong interest in PETs and their applications (5-10y experience)
P6[I] | Security and Privacy Researcher working in executive role at large tech company (2-5y experience)
P7[I] | Data Scientist working on privacy at a ‘big four’ accounting firm (5-10y experience)
P8[I] | Researcher at consultancy specialising in privacy preserving computation as a service (2-5y experience)
P9[D] | Designer at a tech and design agency specialising in ethical use of data & AI (2-5y experience)
Table 1. A summary table of the total participant sample from Research (R),
Policy (P), Industry (I) and Design (D).
## 4\. Findings
We divide the findings from our interviews into two main areas. The first
addresses the technical challenges and opportunities around the adoption of
privacy-preserving computation techniques - primarily concerning their
transition from theoretical research into practical application. The second
concerns the motivations and goals for deploying these techniques to address
commercial and societal goals, and how the institutions that deploy them might
explain and be held accountable for their use.
### 4.1. Technical challenges and opportunities
#### 4.1.1. From Theory to Practice
Many participants brought up how advances in the theoretical grounding on
which privacy-preserving computation techniques are based may not translate
straightforwardly into specific real-world applications. This was acknowledged
by both research scientists working on those foundations, and practitioners
attempting to deploy the technologies in particular contexts. Many
participants expressed confidence that those theoretical advances would
translate into practice in time. P3, a researcher, argued that while it is
_‘early stages, from the business development perspective of homomorphic
encryption’_ , he was nevertheless _‘confident that [the] technology is
useful, practical’_ (P3[R]).
It was generally accepted that much of the research had only reached the stage
of proofs-of-concept rather than deployments; while _‘doable in principle’_ ,
_‘there is a lot of work to do before that bigger picture potential can be
realized’_ (P1[R]). Similarly, P9[D] explained that as a designer she was
_‘trying to make them more widely understood within the design and tech
community … There’s lots of research in academia at the moment, but not very
many examples of them being used in practice’_.
The policy experts we interviewed were optimistic that the technology was
already nearly ready for practical deployment. For P4[P] _‘the technology has
scaled to the point that … it’s definitely commercially deployable’_. For
P5[P], while practical deployment would require a series of _‘reasonable
engineering and architectural compromises’_ , he was still optimistic that
_‘existing approaches to homomorphic encryption are tractable’_.
While both research scientists and policy experts were optimistic about the
big picture, those trying to bridge theory and practice on the ground
expressed frustration that much of the research was not directly relevant. In
some cases, this was because the scientific work made simplifying assumptions
that were rarely satisfied in real use cases. In the context of trying to
apply differential privacy techniques to a project involving time series data,
P7[I] admitted that he _‘really struggled, you know, seeing the value in all
those techniques that academia likes to talk about … how am I going to use
that with time series?’_.
Similarly, several participants pointed to the variety of messy underlying
data and software issues that exist on the ground that hamper deployment.
P7[I] explained that the initial challenges are around _‘how do we list data
assets, and manage access, at an enterprise scale?’_ ; while P8[I] spoke of
clients with various custom systems and data formats, so _‘while we’re solving
the security aspect of the communication between counterparties … we still
haven’t fixed this engineering problem - it’s not going away’_.
P2[R], a research scientist who had worked on both foundational theory and
engineering, explained how applying techniques in practice involved moving
carefully between theory and engineering:
> ‘Engineers do need to learn a lot to deploy these kind of technologies and
> what makes the whole thing complicated is that they need to acquire a kind
> of knowledge that is not something that the university professor knows … a
> lot of low level optimizations make a huge difference, and yeah, from theory
> to practice they need to somehow be invented.’
In P2’s opinion, such work would not come from a _‘linear transmission of
knowledge’_ , but rather through continuous iteration and _‘course
correction’_ between theoreticians and engineers.
#### 4.1.2. Interdisciplinarity and translation between roles
As well as theoreticians and engineers to bridge the gap between theory and
practice, several participants discussed the need for people with different
skills, backgrounds and motivations to work together. This included not only
the combinations of different expertise involved in foundational privacy-
preserving computation research such as math, statistics, and cryptography,
but also specialists in specific application domains. As P2[R] noted,
successful deployment depends on a _‘component of multidisciplinarity’_ :
> ‘In my experience this is very hard to get the right people and crucially,
> with the right incentives in the same room … You not only need data
> scientists, but also security engineers, mathematicians and then experts in
> the application domain.’
The differing motivations and cultures of these different communities was seen
by some as as a problem as it leads to certain important problems being
neglected. Commenting on the misalignment between incentives of academic
researchers and industry, P3(R), who had worked in both sectors, lamented how
research is _‘driven by the need to get published; it favors more …
performance breakthroughs, functional breakthroughs’_ , meanwhile, topics like
usability are _‘not so interesting for the basic core research community’_.
The need for an even broader range of disciplinary expertise and professional
skills was articulated by P4[P], who described her role as _‘to bridge the
lexical gap between technologists, lawyers, and policymakers to defragment the
current initiatives in PETs’_. Drawing from previous experience working on AI
in government, where _‘insulated development’_ led by technologists failed to
account for the _‘constitutional implications’_ of these technologies, she
warned that _‘the same could happen for PETs without this sort of …
interdisciplinary discourse’_.
#### 4.1.3. Usability for Developers
A common theme among both researchers and industry practitioners was the
complexity of applying privacy-preserving PETs from a software engineering
perspective. They discussed a set of inherent challenges facing developers
around flexibility, performance, and specifying appropriate parameters.
Several participants described how PPCTs, in particular homomorphic
encryption, can be very _‘brittle’_ (P1[R]): small changes in parameters can
result in drastic reductions in performance, security, or privacy guarantees.
Such sensitivity can be hard for developers to anticipate and manage,
especially as there are many different parameters to tune. This was contrasted
against other PETs, like public key cryptoschemes, where there is one main
parameter — key size— which has a fairly predictable relationship with
security guarantees and computational overheads:
> ‘I mean RSA, you have the bit length and that’s pretty much it. These things
> [homomorphic encryption applications] you have a ton of decisions to make
> when it comes to how to instantiate it, and they have implications for both
> speed and for the actual function that you will need to compute.’ (P1[R])
This results in problems for developers not just in the initial implementation
of a privacy-preserving technique, but also as they inevitably need to update
a system to _‘evolve when you need to change various details … performance and
tractability can be so highly dependent on small details’_ (P1[R]).
Participants articulated this as a trade off between approaches to building
PPCTs that either work out-of-the-box but have poor and unpredictable
performance, _or_ that have reasonable performance, but require fine-tuning by
engineers. For instance, while the invention of fully homomorphic encryption
enables both addition and multiplication and therefore arbitrary computation,
specific applications still need to be converted into those arithmetic
operations and may incur great computational costs depending on how that is
implemented. While it may be possible to ‘come up with a system with adequate
performance for your application’, this often requires having an application
which is _‘fully specified and well defined in mind, and you have a team of
experts working for you’_ (P1[R]).
Several participants spoke about the development of privacy-preserving
computation software libraries for developers (see 2.1.4), often contrasting
two approaches which reflect the trade-off articulated above: on the one hand,
libraries which create an abstraction layer which obscures the underlying
complexity; and on the other, libraries which expose all of that complexity so
that the developers still need to create bespoke solutions for their
application context. P1[R] noted that many developers expect a library to
provide _‘abstractions that are convenient’_ ; otherwise _‘It’s like telling
people: OK, I’ll give you transistors and you’ll build from them … people
don’t think this way and for good reason’_. P2[R] made a comparison to machine
learning frameworks (e.g. Tensorflow and Scipy):
> They have a very nice abstraction layer that allows them to say ”OK, here’s
> my function over the reals: optimize it, under the hood”… We will have
> worried about implementing matrix multiplication super quickly over floating
> point numbers so that the data scientists can assume [it’s] like doing math
> on their on their notebook, right? This level of abstraction doesn’t exist
> yet [for privacy-preserving computation]
However, P2[R] cautioned against such an approach for privacy-preserving
computation libraries, because it would preclude _‘a lot of optimizations that
come from understanding the underlying protocol’_. P1 echoed this, stating
that:
> ‘The only way that we know now of making the computation go reasonably fast
> is to use a lot of tricks and the developer needs to know about those
> tricks’
As a result, P2[R] felt that _‘general purpose tools’_ would inevitably fail
to meet developer’s performance expectations and thus give them a mistaken
impression about the true potential of PETs.
While many participants were in favor of some form of standardisation via
libraries, P8[I] explained that the prospects for a standard platform depends
on where the technologies are being deployed to. In the context of SMPC,
because smartphone operating system providers _‘control the platform, they can
decide … this is how it’s going to work’_ ; whereas P8[I]’s work involved
deploying SMPC into a wide range of different clients’ environments where _‘we
can’t really dictate to them how they store their data’_ ; as a result the
possibility for standardisation was small.
### 4.2. Motivating, explaining and governing privacy-preserving computation
Our participants also raised various insights relating to the motivations for
adopting PPCTs, and challenges relating to explanation and accountability.
#### 4.2.1. Motivations
Unsurprisingly, ‘privacy’ was often cited as the motivation for developing and
deploying privacy-preserving computation technologies. However, some subtly
different articulations and understandings of privacy emerged from our
interviews, as well as some other motivations which went beyond privacy
altogether.
Some very directly motivated the adoption of privacy-preserving computation by
reference to the interests of individuals in privacy and the protection of
their personal data: _‘it’s individual privacy - it’s human rights’_ (P6[I]).
In comparison to the push for similar technologies in other markets, privacy-
preserving computation was more a response to individual privacy:
> ‘People do understand when their privacy is violated. So … the push for
> these technologies is very different to the push the semiconductor
> industries have had … So I give a lot of time on the examples that are user-
> centric’ (P6[I]).
However, appeals to individual privacy were often mediated via other
pressures. First, organisations deploying PETs may not have a direct
relationship with those individuals, but are instead concerned with third-
parties affected through business-to-business relations:
> ‘You have customers: these may be business to business customers, but that
> also extends to customers of customers and therefore it boils down to
> individuals’ (P6[I])
Second, some cited the existence of privacy and data protection regulation as
an incentive to provide and deploy PETs: _‘because of GDPR_ [the E.U. General
Data Protection Regulation]_… all the regulatory environment is … very
favorable for providers’_ (P7[I]). This regulatory pressure meant that
investment in PPC could be accounted for in terms of corporate risk
management: _‘to have compliance at least formally speaking with GDPR … it’s
really protecting assets of a company’_ (P6[I]).666A sentiment echoed in
(Cammarota et al., 2020)
Third, P9[D] argued that rather than just enabling existing data processing to
be done in a more privacy-preserving way, these technologies could enable new
insights which _‘you might not have been able to gain before because of the
sensitivities around the data that you are using’_. P4[P] highlighted a range
of _‘missed opportunities’_ for privacy-preserving computation _‘for a good
purpose’_. These included cases such as the Boston Women’s Workforce Council
who _‘used secure multiparty computation to confidentially analyze gender wage
gaps without … disclosing who the salary belonged
to’_.777https://thebwwc.org/mpc P5[P] noted the opportunities for government
national security services to use HE techniques like private set intersection
to identify suspects without combining certain databases in the clear,
something that might not otherwise be undertaken due to the _‘intrusiveness’_
of sharing data of large numbers of innocent citizens between departments.888A
use case discussed in (De Cristofaro and Tsudik, 2010).
While individual privacy was cited by all participants as an important
motivator, it was often an indirect motivator, and in some cases perhaps
insufficient on its own (e.g. without being coupled with new opportunities to
extract value from data). Other participants articulated motivations for
pursuing privacy-preserving computation which had nothing to do with
individual privacy as such. For example, for some researchers (e.g. P1, P2),
it was basic intellectual curiosity (_‘somebody thinks of something that …
looks interesting to them’_ (P1)). Other cases included where competing
businesses would have a mutual interest in the output of some computation on
their respective data, but would not otherwise share that data out of _‘fear
of losing a competitive edge’_ (P6[I]). Intellectual property protection was
also frequently cited as a key motivation for many business applications.
Privacy-preserving computation techniques were also seen by some as offering
the possibility to navigate regulatory obligations and trade-offs in different
ways. First, they have the potential to fulfill obligations to protect data in
new, more ‘technological’ ways, offering _‘technological safeguards that can’t
be easilly overridden’_ , the kind of protection that _‘paper safeguards, like
contractual guarantees and policies, just can’t provide’_ (P4[P]). They were
seen as especially promising in cases where different regulatory obligations
might appear to be in conflict, as P4[P] explained:
> ‘Anti money laundering regulations are very data maximalist; they want you
> to collect more data [to prevent] financial crimes. But in the meantime the
> GDPR is quite the opposite; it wants you to minimize data, … and this really
> conflicts with the regime of AML. I think that PETs could actually cut
> through these legal conflicts and really provide a practical solution … it’s
> not actually transferring PII, but it still allows for banks to prepare for
> AML protocols’
Similarly, for P5[P], privacy-preserving methods had the capacity to change
what is possible without sharing data and thereby shift the scales in legal
balancing tests (Brown and Korff, 2009) that might otherwise make certain data
analysis unlawful:
> ‘ UK law … sets out a test for those of us in national security which is
> _necessity_ and _proportionality_. So if you can shift the proportionality,
> then you’re in a better position so you can avoid intruding, you can avoid
> privacy risk’.
In these ways, such techniques were envisioned by P4[P] and P5[P] as enabling
organisations in the public and private sector to break free of what P4[P]
called _‘legal gridlocks’_ that currently are (or are perceived to) exist
around data use and enable new kinds of analysis.999While a legal analysis of
potential conflicts between these two areas of law is beyond the scope of this
paper, we note that data protection and financial services regulators have, at
least in the UK, affirmed their compatibility in general terms: see
https://www.fca.org.uk/news/statements/fca-and-ico-publish-joint-update-gdpr.
#### 4.2.2. Explanation
Our participants discussed various facets relating to _explaining_ privacy-
preserving computation, including _how_ they go about explaining it to
different audiences (and in some cases, why they don’t even try).
The researchers described a variety of contexts in which they had had to
explain underlying techniques and their strategies for doing so. For a general
audience, P2’s strategy was to explain simplified versions of protocols, such
as simulating a secure multi-party computation for dating using playing cards
(see (Marcedone et al., 2015)). While these were _‘fun to explain’_ , P2[R]
was unsure about the effectiveness of such explanations:
> ‘Then in the future, [the audience] will be like: ”Oh yeah, multi-party
> computation, the thing with the cards.” That doesn’t mean that my
> explanation was effective… My feeling is that people tend to end up amused
> and satisfied.’ (P2[R])
Such explanations were offered as a starting point to encourage people _‘who
are attracted by that kind of magic’_ and would _‘go into Wikipedia
immediately after’_ (P2[R]). However, P3[I] felt that there was a lack of
accessible educational material: _‘there is certainly not enough material and
the classical crypto papers are essentially useless for someone who is not an
encryption expert’_ ; they suggested that explanations of core concepts might
be more effective if tackled as part of a standardisation process and included
within libraries.
Several participants also cautioned that the kind of explanations offered (if
any) need to be tailored to the audience. On the one hand, explanations could
be too technical: _‘If you start with equations … you lose 99 percent of the
audience right away’_ (P7[I]). On the other hand, short intuitive explanations
might be too simplistic for informing executive decisions:
> ‘So one thing is getting people interested, and the other one is informing,
> like, executive decisions. I don’t think they should be informed … by two
> minute stories… I don’t think decisions about encryption are made based on
> an intuitive understanding of crypto’
For P9[D], designers have a role to play in explaining privacy-preserving
techniques through prototyping their use in specific contexts. This included
explanations to end users, but also _‘a different language to explain it to
those designers as well’_. Previously, their design agency hadn’t _‘seen much
demand for them on the industry side’_ ; however, that changed after
publishing a blog post explaining visually how differential privacy could work
in the context of a project on identifying inequalities in urban mobility:
> ‘Each step of the randomized response process … we had an image to go with
> it, so that you could see … the noise that you are adding to data. Visually
> seeing it was really helpful for me as a designer and then tying it to sort
> of real life stories so that I could see how you wouldn’t be able to re-
> identify someone. Imagining what that makes possible forces you to think
> about the qualities of that technique, what it now enables you to do’
When it came to explaining these systems to end-users, however, some
participants questioned whether this was a worthwhile goal. P9[D] couldn’t
imagine _‘many scenarios where it’s necessary to explain what privacy
preserving techniques are being used to an end user who is trying to do
something with their phone’_. P7[I] asked himself whether end-users understood
these techniques, and answered: _‘Well in general, not. Is it a problem? I’m
not sure it’s a problem’_. In such cases, it was seen as sufficient that end-
users _‘trust the provider of the solution that they do a good job’_(P7[I]).
#### 4.2.3. Governance and Accountability
A final theme was around the challenges of _governance_ and _accountability_
of privacy-preserving computation. These topics often followed organically
from discussions of explanation; attempts to explain these systems were often
made in the course of trying to _justify_ their use to affected stakeholders,
and justification is a key element of accountability (Bovens et al., 2014)).
But even if explanations don’t lead to real understanding on an individual
level, it might still be possible to justify them to the public. P5[P] put it
this way:
> ‘These technologies are extremely difficult to understand… Do they
> meaningfully address genuine privacy issues? Yes they do. Do they address
> public concern? That’s not to do with the technologies _per se_ , [but] how
> the technologies are explicated and made available. If you told the public:
> ”As a result of using these technologies, we are able to limit the amount of
> your personal information that’s shared, and are still able to offer you
> valuable services”, they would be enthusiasts.’
Other participants expressed scepticism that the public would take such
guarantees at face value. In the context of proposals for privacy-preserving
facial recognition in border control, P6 asked:
> ‘if someone publicises this new system … just by saying: ”and by the way the
> privacy of the information is very well handled because we use the state of
> the art cryptography”, what does that mean to a citizen?’
Both P6 and P7 suggested that certifications and trust marks applied to
services which use these techniques could enable individuals to seek out more
trustworthy systems. However, expecting individuals to exercise meaningfully
informed choices in relation to different services involving privacy-
preserving computation was seen by some as adding to the burden of
responsibility unhelpfully placed on individuals. P9 reflected on how
_‘constantly making decisions about data in the technology that we use is just
not sustainable’_ ; instead, they suggested that _‘collective consent models
and other governance mechanisms … that can make decisions on behalf of
people’_ might be a better approach. Similarly, P2[R] felt decisions about the
technical details of the adoption of these technologies ought to be made by
_‘using experts or authorities’_ who can act as _‘proxies … [who] understand
their communities’_.
While most of our participants pointed to the positive potential of privacy-
preserving computation techniques, a few were also concerned about the power
imbalances they might reinforce. When the stakeholders are individuals, they
are _‘by definition, the weaker party’_ , and _‘lack the resources … to induce
changes; every time we talk about privacy there is some asymmetry that is
implied by it.’_ (P1[R]). For P9[D], it is important to recognise the
limitations of PPCs as they are just:
> ‘a technical solution to protecting people’s privacy … you have to think
> about the wider system that they sit within and what other kind of power
> dynamics are in that system.’ (P9[D])
## 5\. Discussion
The findings from our interviews raised several important implications for the
design and governance of privacy-preserving computation. They reveal how these
techniques are being not only technically but also socio-technically
constructed and constituted by a variety of actors, each pursuing overlapping
and sometimes diverging agendas. Clearly, privacy-preserving computation
techniques entail a variety of human-centric challenges which HCI research
could seek to address. These challenges are multifaceted and will require
diverse approaches; something that HCI as a methodologically diverse field is
well-positioned to reflect. Furthermore, these challenges are inter-related:
for instance, the way in with these technologies are translated from theory to
practice may well affect how they can be explained and held accountable; while
closer inspection of how ‘privacy’ and other motivations are unpacked might
reconfigure what kinds of interdisciplinary collaborations are required in a
particular context. Our aim in this section is to reflect on these, to
understand both the design problems facing these techniques, and the
challenges they raise in relation to the interests of a variety of users and
wider society. This discussion is not intended as direct ‘implications for
design’; rather, we hope to draw attention to issues which require further
research, as well as interdisciplinary discussion.
### 5.1. Whither the end user?
While our experts generally acknowledged the individuals whose personal data
is being privately computed on as an important stakeholder group, few seemed
to prioritise seeking their understanding and acceptance. This is in contrast
to the small number of existing HCI studies that investigate ‘user
acceptability’ of particular privacy-preserving computation techniques such as
differential privacy (Bullek et al., 2017; Xiong et al., 2020) and MPC (Qin et
al., 2019). User acceptability could and should be further examined in
particular contexts; for instance, Colnago et al. suggest further work is
needed to explore whether such techniques embedded in Internet-of-Things
privacy assistants might ‘help mitigate people’s reservations about data
collection practices and reduce the chance they opt out’ (Colnago et al.,
2020). There is clearly great scope for important research within this
paradigm of user acceptability.
However, our experts spoke about privacy-preserving computation technologies
more as tools enabling _organisations_ to achieve a variety of goals
(including managing privacy risks, but also protecting corporate assets and
secrets), rather than as a means of directly serving users’ interests. While
user acceptance was not entirely disregarded, it did not appear to be a
primary concern; even P9, a designer well-versed in user-centred design,
doubted that people could or should be expected to understand and make
decisions about privacy-preserving computation. Privacy may be important, and
these techniques may have the potential to meaningfully embed it, but whether
or not individuals understand and accept them seemed to be almost a secondary
issue. In many of the use cases they mentioned, individuals whose data is
being computed may not have any direct interaction with the system, nor any
choice about whether to use it. In expressing such doubts, our interviewees
might appear to be denying a sacrosanct tenet of HCI as a _human_ -centred
discipline. However, rather than denying the importance of user acceptance, we
believe that these doubts should in fact point us towards alternative human-
centric approaches to the development of privacy-preserving computation, in
addition to solely looking at end users as data subjects.
First, our findings point towards studying the needs of different kinds of end
users; specifically, those developers and designers who attempt to apply
foundational privacy-preserving computation techniques in real-world
applications. This echoes recent calls to acknowledge that ‘developers are
users too’, as Green and Smith argue in relation to crypto and security
libraries (Green and Smith, 2015). Similarly, P9[D] pointed to the relative
lack of awareness and understanding of these techniques among designers. As
with the application of other complex methods in computer science, such as
machine learning, it may be difficult for designers to use privacy-preserving
computation techniques in design practice due to unfamiliarity with how they
work and awareness of what they can achieve(Dove et al., 2017). P9[D] made the
case for technical specialists and designers to work together to translate
these technologies into ‘design material’ which design practitioners can use
to explore real use cases.
In addition to understanding developers, designers, and others as _users_ of
privacy-preserving computation techniques, studying them also allows us to
explore how a human value like privacy shapes the construction of complex
computational systems. This perspective accords with ‘third wave’ approaches
to HCI which orient attention towards the ethical obligations and values of
designers (Fallman, 2011), and incorporate different disciplinary perspectives
which examine how social and political dimensions are embedded and reflected
in systems (Bardzell and Bardzell, 2015). As such, rather than just
considering whether end users or laypeople understand, trust, and accept
privacy-preserving computation technologies, we might also benefit from
considering the perspectives of the various people involved in constructing
their technical, commercial and regulatory foundations. Assessing whether an
innovative technology will be acceptable to users through lab and field
studies may be valuable, but such approaches often neglect the ways in which
such technologies are interpreted, shaped, and mutually constructed over time
through their designers, users, and broader political, economic and regulatory
forces(MacKenzie and Wajcman, 1999; Jasanoff and Kim, 2009). As a result, it
is equally important to consider the plurality of different actors and broader
contexts through which values like privacy will be understood, traded-off, and
embedded in these systems (or not).
### 5.2. The Limits of Abstraction
If, as suggested above, we are to consider the needs of developers and
designers as users of underlying privacy-preserving computation techniques,
then how might those needs be met? Many of the interviewees identified the
need to create building blocks for privacy-preserving computation. In an ideal
world, these building blocks would allow developers to _abstract away_ the
technical details and apply them to applications in different contexts.
Creating such abstractions is fundamental to progress in computer science and
programming; in Edsger Dijkstra’s words, it is ‘our only mental aid … to
organize and master complexity’ (Dijkstra, 1982). However, many of the experts
expressed uncertainties about the form such abstractions should take and the
extent to which they could reasonably be made in the domains of privacy-
preserving computation. Especially with homomorphic encryption, abstracting
away the details of implementation could mean losing the ability to optimise
performance through engineering ‘tricks’ (P1[R]).
Attempts to create tools for developers to enable them to integrate privacy-
preserving computation techniques into their products may therefore need to
grapple with this need to balance abstraction and engagement with the
implementation details. Specific applications will always require some
‘intimacy with the details’ (Steimann, 2018) that might otherwise be
abstracted away. Some of our interviewees argued that the necessary education
required for developers could potentially be integrated into standardised
APIs. This suggests that broader adoption of privacy-preserving computation
may benefit from work in HCI which considers APIs and libraries as ‘first
class design objects’ (Myers and Stylos, 2016; Zibran, 2008), with the goal of
‘driving adoption of software components’ (Mechtley, 2020). This could involve
(re)designing them around the typical ways programmers learn, e.g. on-the-fly,
via information foraging, and trial and error (Lawrance et al., 2010; Kelleher
and Ichinco, 2019).
However, the nature and extent to which developers need to become intimate
with the details, and how they might do so, will clearly depend on the
particular technique in question. For instance, a DP library might implement a
variety of noise sampling and injection techniques, but this is relatively
simple compared to the much more complex mathematics and reasoning involved in
deciding on and managing an appropriate privacy budget, which requires case-
by-case human consideration. For SMPC, libraries might take care of some of
the networking details, but leave difficult decisions regarding the protocol
up to the developer. The nature and value of these standardised building
blocks will therefore vary greatly between approaches.
Ultimately, the design and adoption of these privacy-preserving computation
building blocks may need to reckon with the messy realities of underlying
enterprise IT infrastructure, agile and iterative approaches to software
development (Gurses and Van Hoboken, [n.d.]), and service-oriented
architectures (Kostova et al., 2020). Given these practical considerations,
the full complexity of these technologies might instead need to be mediated
via a two-step process: general-purpose libraries which expose all of the
complexity of a domain (e.g. homomorphic encryption) that enable specialist
privacy engineers to create particular privacy-preserving computation
components for common operations or use cases (e.g. private set intersection
for contact discovery); those components could then be adapted and deployed
with minimal configuration by non-specialist developers as microservices.
### 5.3. Privacy-Enhanced Technocracy
The way privacy-enhancing technologies are sometimes described can make them
seem esoteric, exotic, and mysterious. For instance, in industry press they
have been described as ‘black magic’ 101010
https://dualitytech.com/tag/homomorphic-encryption/ and a ‘holy
grail’111111https://www.fastcompany.com/90314942/duality-homomorphic-
encryption. Such language suggests that their development is entirely in the
hands of a small and specialised cabal of cryptographers and engineers, much
like the early programmers who regarded themselves as ‘high priests’ of
assembly code.121212In the words of Rear Admiral Grace Hopper (Williams, 2012)
It is possible to imagine how in these respects, they might end up sharing the
same ‘rampant hyperbole and political envisioning’ (Elsden et al., 2018) of a
higher-profile cryptographic technology — blockchain.
While our interviewees avoided such language, and even criticised the
perceived ‘hype’ around PETs, they did reflect the highly specialised
knowledge required to make use of the underlying mathematics, and drew
parallels with magic. P6[I] described feeling like ‘Gandalf the wizard’ upon
telling people that computation on encrypted data was possible, while P1[R]
described the need for engineering ‘tricks’ to optimise performance within
reasonable levels. From this perspective, the technical work of applying
privacy-preserving computation seems more like craft than science, which the
guild of cryptographers and engineers are uniquely capable of performing
(Prak, 2006).
However, the mystery of their inner workings could easily serve as an excuse
for not making these systems accountable to affected stakeholders. When
reflecting on the challenges laypeople face in trying to understand PPCTs on
any meaningful level, both P5[P] and P7[I] expressed some doubts about the
possibility that individuals could ever be expected to really understand _how_
they work. However, without some form of explanation, and absent any other
mechanism for meaningfully communicating their risks and opportunities, there
is a risk that privacy-preserving computation becomes not just a _technical_
but a _technocratic_ solution imposed on populations without popular consent
by grey eminences operating behind the scenes.
However, several of the experts did acknowledge the need for mechanisms of
accountability and governance to developed as these technologies are rolled
out. P6[I] and P7[I] suggested this could involve certification schemes.
Similarly, while P1[R] and P9[D] were doubtful about individuals being able to
meaningfully consent to these technologies, they proposed alternative forms of
collective governance, where the interests of affected individuals could be
represented by relevant representatives and experts who can make informed
choices and demands on their behalf. These and other democratic mechanisms
will need to be explored in order to counter a privacy-enhanced technocracy,
and methods from HCI — such as participatory design (Ehn, 1988), futures
workshops (Jungk and Müllert, 1987), and other governance approaches — may
have much to offer.
### 5.4. Secrets, Assets, Human Rights: Unpacking ‘Privacy’
Our findings attest to the many varied interpretations and uses of the term
privacy. As previous work has explored, and as discussed above, the notion of
privacy in Privacy-Enhancing Technologies is often a narrow interpretation of
what is a multi-faceted and contested concept (Tavani and Moor, 2001; Gürses
and Berendt, 2010; Stalder, 2002; Phillips, 2004). This is certainly the case
for the subset of privacy-preserving computation PETs studied here. They turn
privacy into something mathematically formalisable, e.g. in terms of entropy
in cryptographic approaches, or indistinguishability in statistical
approaches, which can all be understood as variations of ‘confidentiality’, a
pillar of the security triad (Dhillon and Backhouse, 2001). This means that
other ways of understanding privacy may be de-emphasised and de-prioritised.
There are continuities here with earlier PETs, such as de-identification
techniques based on hashing personal data. Phillips argues that that these
techniques embody privacy as protection ‘from unwanted intrusion’(Phillips,
2004). However, they leave in place the ability of powerful observers to
produce ‘panoptic’ knowledge which can be used to sort and discipline
populations (Gandy Jr, 1993; Cohen, 2012). Similarly, if we understand privacy
as confidentiality, this can be engineered through architectures of data
minimisation (Spiekermann and Cranor, 2008); but this can lead to design
choices which preclude alternative understandings of privacy (e.g. privacy-as-
control), and hinder the exercise of related rights afforded by data
protection law (Veale et al., 2018). In our expert’s discussions, these
alternative understandings of privacy were conspicuous by their absence.
Our findings also demonstrate that even while discourse around privacy-
preserving computation restricts certain interpretations of privacy, it also
stretches the meaning of privacy to incorporate unorthodox meanings, such as
competitive secrecy, corporate asset protection, and government security.
These are clearly significant and important use cases for the technology, but
they arguably bear only a family resemblance to privacy as it relates to
individuals and society. Indeed, intellectual traditions which value privacy
as an individual right and public good have often been associated with
opposition to corporate and government secrecy; according to them, privacy
should be reserved for the weak, while transparency should be an obligation
required of the strong (De Hert and Gutwirth, 2006). In referring to all of
these things as ‘privacy’, privacy-preserving computation technologies may
elide significant political tensions between them. This is not to deny that
they may have a powerful role to play in supporting privacy as an individual
right and as a public good (Kwecka et al., 2014; Fairfield and Engel, 2015);
but this confluence of quite different values under one banner complicates the
narrative around whose interests they serve.
As well as tending to address narrow and perhaps unorthodox conceptualisations
of privacy, it is important to recognise that these technologies do not
protect other important values and interests. If our aim is to build and shape
systems encompassing multiple social goals, where privacy is just one such
goal, then privacy-preserving computation techniques have to be considered in
relation to the whole system and the social context. The danger is that the
societal problems of data processing technologies — such as the ways they
create distinctions and hierarchies that reinforce power, shape politics, or
facilitate abuse — are sidelined, redefined, or collapsed under the banner of
‘privacy’, so that privacy-preserving computation techniques can be positioned
as _the_ solution (what Pinch and Bijker term ‘closure by problem
redefinition’ (Pinch and Bijker, 1984)). This danger was alluded to in P6[I]’s
example of privacy-preserving border control (where people are still
ultimately at the mercy of a powerful state), and in P9[D]’s concern about
considering the wider power dynamics in the context of deployment.
## 6\. Conclusions
Like other technologies which have been touted as potentially revolutionary in
recent years, the concrete impact of these privacy-preserving computation
techniques remains to be seen. New technologies often emerge in unexpected
ways, at unpredictable times from niches of computer science: hypertext,
Merkle trees, and neural networks were once confined to their respective
research sub-fields before they became known more widely as the world wide
web, blockchain, and ‘AI’ (in its latest guise of deep learning). Prior to
their take-up in wider society, these specialised areas of research were
conceived as laying the groundwork for purely technical pieces of invisible
infrastructure, whose implications for human-computer interaction were remote
and unclear.
However, we believe it is worth HCI researchers studying such technologies
prior to their widespread adoption. Whatever technical and institutional forms
they take, the journey of privacy-preserving computation techniques from the
annals of cryptography into production code will be shaped in substantial part
by the approach they take to a variety of human and societal challenges.
Indeed, these challenges directly implicate some fundamental concerns of HCI,
including: multifaceted (re)conceptualisations of the notion of ‘the user’;
helping people navigate and manage computational complexity and its
consequences; exploring how values like privacy can be reflected in the
systems we build; and examining how different political agendas, economic
rationale, and user groups shape and are shaped by those systems. These
concerns all cohere and overlap in the emerging space of privacy-preserving
computation.
This paper has aimed to provide a preliminary and partial outline of those
challenges, laying some of the groundwork for substantial further exploratory
and in-depth work to be done. In addition to several recent studies which
focused on people’s understanding of these techniques and their willingness to
disclose personal data in the presence of them, we have outlined a broader set
of research questions prompted for HCI by PPCs. These include understanding
specific application contexts; usability of PPC libraries and tools from a
non-specialist developer’s perspective; and understanding the explanation and
governance challenges associated with these techniques.
###### Acknowledgements.
This work was funded by EPSRC grant EP/S035362/1 and Callsign Inc.
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|
# Evaluating uncertainties in electrochemical impedance spectra of solid oxide
fuel cells
Luka Žnidarič<EMAIL_ADDRESS>Gjorgji Nusev Bertrand Morel Julie
Mougin Đani Juričić Pavle Boškoski Jožef Stefan Institute, Jamova cesta 39,
SI-1000 Ljubljana, Slovenia Jožef Stefan International Postgraduate School,
Jamova cesta 39, SI-1000 Ljubljana, Slovenia Laboratoire des Technologies
Hydrogèn Commissariat à l’Énergie Atomique et aux Énergies Alternatives - CEA,
CEA/LITEN/DTBH/STHB/LTH 17 rue des Martyrs – 38054 GRENOBLE CEDEX 9, France
###### Abstract
Electrochemical impedance spectroscopy (EIS) is a widely used tool for
characterization of fuel cells and other electrochemical conversion systems.
When applied to the on-line monitoring in the context of in-field
applications, the disturbances, drifts and sensor noise may cause severe
distortions in the evaluated spectra, especially in the low-frequency part.
Failure to ignore the random effects can result in misinterpreted spectra and,
consequently, in misleading diagnostic reasoning. This fact has not been often
addressed in the research so far. In this paper, we propose an approach to the
quantification of the spectral uncertainty, which relies on evaluating the
uncertainty of the equivalent circuit model (ECM). We apply the
computationally efficient variational Bayes (VB) method and compare the
quality of the results with those obtained with the Markov chain Monte Carlo
(MCMC) algorithm. Namely, MCMC algorithm returns accurate distributions of the
estimated model parameters, while VB approach provides the approximate
distributions. By using simulated and real data we show that approximate
results provided by VB approach, although slightly over-optimistic, are still
close to the more realistic MCMC estimates. A great advantage of the VB method
for online monitoring is low computational load, which is several orders of
magnitude lower compared to MCMC. The performance of VB algorithm is
demonstrated on a case of ECM parameters estimation in a 6 cell solid oxide
fuel cell (SOFC) stack. The complete numerical implementation for recreating
the results can be found at https://repo.ijs.si/lznidaric/variational-bayes-
supplementary-material.
###### keywords:
Variational Bayes, Monte Carlo, Solid oxide fuel cells, Fractional-order
systems
## 1 Introduction
Currently available tools for characterising electrochemical energy systems
predominantly rely on Nyquist curves obtained through electrochemical
impedance spectroscopy (EIS) [1]. Conceptually simple and well understood, EIS
analysis has become a standard tool for characterising the health condition of
cells and stacks. Correct evaluation of the EIS characteristic is subject to
several requirements.
First, the perturbation signal must have low amplitude in order not to excite
the nonlinear modes of the cells dynamics. If the amplitudes are too small,
there is a risk that the noise-to-signal ratio will decrease, which will
reflect in an increased variance of the EIS estimates. Second, the cells
should operate in stationary, stable, and repeatable test conditions. This is
mandatory for a correct evaluation of the EIS curves. The internal conditions
should remain constant during the perturbation session and no external
disturbance should corrupt the measurements. Under stable laboratory
conditions, the effect of disturbances on stack are minimised, which usually
results in smooth Nyquist curves. Correctly evaluated EIS spectra ensures
accurate detection of fault and stack degradation. Detection is done by
checking their similarity to the reference EIS spectra obtained in the healthy
state. An anomaly in the online spectra might be ascribed to the fault instead
to the disturbances, which results in false fault alarm. Third, to obtain
comparable results, measurements on a cell or a stack should be performed at
the same operating point. The above requirements are not easily met in
applications outside laboratories.
In the in-field operation, the presence of disturbances in the balance of
plant (BoP) and system environment is likely to come into play. In the
laboratory conditions there are opportunities to apply expensive
instrumentation (e.g. on-line gas analyser), high-performance actuators, power
electronics and measurement devices. In commercial applications, a trade-off
between performance and cost of implementation should normally be sought. Cost
optimization dictates reduction of the number and types of implemented sensors
to minimum and use of standard industrial modules for HW realisation.
Variations in gas channel, drift and noise in flows and temperatures can
affect the low-frequency part of the Nyquist curve and can lead to non-smooth
results. Increased noise in current and voltage measurements can also
contribute to the non-smooth EIS evaluation. In addition, improper operating
conditions, such as increased fuel utilization, can cause dispersed values of
impedance in the low-frequency part of the spectra. The reason is the local
fuel starvation, which then affects the voltage variance.
A strong motivation for the work stems on one hand from experience with some
cases of practical implementations and, on the other hand, by the fact that it
has not been explicitly addressed in the literature. For better illustration,
Figure 1 shows two sets of evaluated EIS curves obtained from successive
measurements. The first set was obtained on a short solid oxide fuel cell
(SOFC) stack consisting of six planar anode-supported cells installed in an
electrical furnace. The stack was operated in the laboratory conditions under
nominal current of 32 $\mathrm{A}$ and FU=77.5 %. More details can be found in
[2]. The second case is an example of a commercial stack with anode-supported
cells operated at around $750^{o}$C and at the power $1$kW. As fuel, natural
gas is used in a way that it is first partly transformed to syngas by steam
reforming. During the long-term run there were occasional issues related to
the periodic component in the fuel and steam flow, which was attributed to the
control system issues. Note that perturbations do not significantly affect the
high frequency range arcs in the EIS curve.
It seems the problem is relevant for the SOFC systems domain. With the
increasing need for automated condition monitoring, robust solutions are
mandatory to ensure reliable diagnosis. This is important since the mitigation
actions are taken based on the diagnostic results. Unreliable EIS evaluations
can result in misleading diagnosis and countermeasures that can even worsen
the condition of a cell or a stack.
An idea of how to deal with the problem in SOFCs was recently proposed in
articles [3, 4]. The authors perform inference on the equivalent circuit model
(ECM) based on a combination of EIS data smoothing and EIS averaging. Instead
of using a single EIS measurement, the data from several consecutive EIS
measurements are averaged to reduce the dispersion of the EIS estimates in the
low-frequency region. This solution, while simple, requires a certain number
of measurements spanned on a time window to reliably assess the change in EIS
characteristic.
In this paper, we propose a novel approach, not pursued before, which is able
to account for the influence of perturbations on the evaluated EIS
characteristic, and consequently on the equivalent circuit model, by a
statistical modelling approach. Properly quantified uncertainties lead to
diagnostic solutions that, rather than making a fault statement in clear
yes/no categories, suggest the probability that a particular fault is present
[5]. This is essential for cautious and more reliable diagnosis of SOFCs,
which can be additionally enhanced with the operator’s prior knowledge.
Figure 1: Evaluated EIS characteristics in laboratory conditions (above) and
during an in-field application (below).
In the area of electrochemical energy conversion systems, there have been only
limited attempts to analyze the stochastic nature of the parameters. Since
parameters of the ECM model are usually obtained by constrained nonlinear
optimization, the uncertainty of the estimated parameters is obtained from the
local properties of the objective function around the optimal parameters, c.f.
[6, 7]. The problem with the approach is that the approximation is rather
rough and might suggest a misleading uncertainty region [8]. To obtain a
realistic estimate of the parameters uncertainty, more elaborated approaches
should be applied. Of the available methods, the Markov chain Monte Carlo
(MCMC) [9] provides the most accurate probability distributions for estimated
model parameters. The key problem with the approach is the overwhelming
computational time, which increases rapidly with the number of unknown
parameters. That renders the approach inappropriate for in-field on-line
condition monitoring. Instead, a computationally less excessive but still
sufficiently accurate method is desired.
A remedy for the issues imposed by MCMC is the variational Bayes (VB) [10].
Unlike the MCMC algorithm that results in the “true” posterior distribution,
the VB approach provides the closest approximation of the posterior
distribution using the probability distribution functions from the family of
exponential functions. Consequently, instead of solving the typically
intractable evidence integral in pure Bayesian approach, the posterior
distribution in VB algorithm is found by means of optimization. The result
contains inherent bias, which is a low price to pay considering the impressive
computational efficiency of the approach even for multidimensional cases. VB
approach has been extensively applied in various areas, such as Gaussian
process modeling [11, 12], deep generative models [13], compressed sensing
[14, 15], Hidden Markov models [16, 17], reinforcement learning and control
[18]. VB approach is also applicable to energy management problems [19, 20,
21, 22, 23].
In this paper we analyse the nature of the uncertainty regions for the ECM
parameters and compare the results of the computationally feasible VB approach
with the results of MCMC approach. To the best of the authors’ knowledge, the
first and only attempt to study the uncertainty of ECM estimates has been done
in [24]. That work is the first report on the probability distribution
functions of the ECM parameters under nominal operating conditions.
Unfortunately, the method is infeasible for on-line monitoring since it takes
several hours even with a strong HPC infrastructure.111HPC Meister,
capability: $244$ TFLOPs For example, 25 hours are needed to obtain the
correct distributions of parameters for the experimental measurement presented
later on. What we suggest below is to approximate the true posterior
probability density function with a closed form distribution that best
captures the nature of the true posterior. The benefits of having the
uncertainties of the model parameters explained in the closed form are
twofold. First, we get a useful indicator of the quality of the selected ECM
model structure and the quality of the experimental data leading to the model.
Second, it becomes possible to rather easily employ statistical reasoning
tools for detecting changes in the ECM parameters and thus the deterioration
of the system behaviour. That is the first such approach in the domain of
SOFC.
The organisation of the paper is as follows. A brief introduction to the MCMC
and VB approach is given in section 2. The performance of the method in terms
of computational efficiency and accuracy is first demonstrated on a simulated
ECM in section 3. Finally, the VB algorithm is applied to the identification
of ECM parameters based on data obtained on a SOFC in section 4. The main body
of the paper is followed by 3 appendices. In A the complete numerical
implementation for recreating the results is described. Additional results on
VB on simulated measurements with varying degrees of noise can be found in B.
Finally, some results of the VB approach on experimental measurements under
different conditions are delivered in C.
## 2 Methodology
Assume we want to describe a process with a model $\mathcal{M}_{\bm{\theta}}$
parameterized by a vector $\bm{\theta}$. Based on measurements
$\bm{\mathbf{x}}$ obtained from the process, one would like to get the unknown
$\bm{\theta}$ from $\bm{\mathbf{x}}$. Assume also there is some prior
knowledge (or guess) about $\bm{\theta}$ expressed in terms of a probability
distribution (also called prior distribution222A prior probability
distribution of an uncertain parameter $\bm{\theta}$ is the probability
distribution that reflect one’s belief about the parameter before evidence in
terms of data is taken into account. ) $p(\bm{\theta})$. Prior knowledge can
be updated (improved) with information contained in the data by means of Bayes
rule. The result is the posterior distribution333Posterior probability is the
updated probability for the event, after taking into account the prior
knowledge and information contained in the data.
$p(\bm{\theta}|\bm{\mathbf{x}})$ as follows
$\underbrace{p(\bm{\theta}|\mathbf{x})}_{\text{Posterior}}=\frac{\overbrace{p(\mathbf{x}|\bm{\theta})}^{\text{Likelihood}}\overbrace{p(\bm{\theta})}^{\text{Prior}}}{\underbrace{p(\mathbf{x})}_{\text{Evidence}}}$
(1)
The likelihood can be calculated from the model and the prior is specified as
a design input. The normalisation factor (evidence) is the following integral:
$p(\mathbf{x})=\int_{\bm{\theta}}p(\mathbf{x}|\bm{\theta})p(\bm{\theta})d\bm{\theta}.$
(2)
For general multidimensional distributions the integral (2) becomes
intractable. Consequently, getting the posterior in (1) becomes infeasible,
hence the need for the apximate approaches.
### 2.1 Markov Chain Monte Carlo
MCMC algorithm is a ubiquitous method for solving integration problems (2) in
various areas, e.g. statistics, physics and econometrics. Central in the
approach is the way of taking samples, say a set of $N$ samples
$\bm{\theta}^{(i)}\in\mathbb{R}^{m},\ i=1,...,N$, from a target density
$p(\bm{\theta})$. Using these $N$ samples, we can approximate
$p(\bm{\theta})$, by calculating the empirical point-mass function
$p_{N}(\bm{\theta})=\frac{1}{N}\sum_{i=1}^{N}\delta_{\bm{\theta}^{(i)}}(\bm{\theta}),$
where $\bm{\theta}^{(i)}$ is the $i^{th}$ sample in our set $\mathbf{\theta}$
and $\delta_{\bm{\theta}^{(i)}}$ is its delta-Dirac mass.
In our case we used MCMC with No-U-Turn sampling (NUTS) [25], implemented in
Python with the PyMC3 [26] library. NUTS sampling is an extension of the well
known Hamiltonian Monte Carlo algorithm (HMC) [27]. HMC tends to be sensitive
to the required user inputs. This is mostly avoided since the NUTS algorithm
stops automatically when it starts retracing its steps. Additionally, authors
of the NUTS developed a method for automatic adaptation of the step size, so
that the sampler needs minimal user-defined parameters on entry. For the
experimental data described later on, we have let the algorithm run for
$300.000$ iterations, which proved sufficient to guarantee convergence.
### 2.2 Variational Bayes
Using MCMC methods for solving (2) can produce results very close to the true
posterior distribution. However, for the multidimensional cases, the
computational load and sheer number of samples required for obtaining proper
estimate of the posterior blows up. One solution to this problem is to find a
sufficiently close approximation of the posterior with the significantly lower
computational load.
The main idea of the VB approach is finding a candidate distribution
$q_{\bm{\lambda}}(\bm{\theta})$ (parameterized with the hyperparameters 444A
hyperparameter is a parameter of a prior distribution; the term is used to
distinguish them from parameters of the model.
$\bm{\lambda}\in\mathbb{R}^{\nu}$), that is a sufficiently close approximation
of the true posterior $p(\bm{\theta}|\mathbf{x})$. The distribution
$q_{\bm{\lambda}}(\bm{\theta})$ is usually referred to as the _variational
distribution_. The variational distribution is typically selected from the
mean-field variational family [28]. This means that one can assume
independence among the latent variables of the variational distribution.
A variational distribution that is the best fit for the true posterior can be
obtained by minimizing the Kullback-Leibler (KL) divergence [28], which can be
re-arranged as follows:
$\begin{split}\lx@glossaries@gls@link{acronym}{kl}{\leavevmode
KL}(q_{\bm{\lambda}}(\bm{\theta})||p(\bm{\theta}|\mathbf{x}))&=\mathbb{E}_{q}\left[\log\frac{q_{\bm{\lambda}}(\bm{\theta})}{p(\bm{\theta}|\mathbf{x})}\right]\\\
&=\mathbb{E}_{q}[\log q_{\bm{\lambda}}(\bm{\theta})]-\mathbb{E}_{q}[\log
p(\bm{\theta}|\mathbf{x})]\\\ &=\mathbb{E}_{q}[\log
q_{\bm{\lambda}}(\bm{\theta})]-\mathbb{E}_{q}[\log
p(\mathbf{x},\bm{\theta})-\log p(\mathbf{x})]\\\ &=\mathbb{E}_{q}[\log
q_{\bm{\lambda}}(\bm{\theta})]-\mathbb{E}_{q}[\log
p(\bm{\theta},\mathbf{x})]+\log p(\mathbf{x})\\\ &=-({\mathbb{E}_{q}[\log
p(\bm{\theta},\mathbf{x})]-\mathbb{E}_{q}[\log
q_{\bm{\lambda}}(\bm{\theta})]}+\log p(\mathbf{x})\\\
&=-\underbrace{(\mathbb{E}_{q}[\log
p(\mathbf{x}|\bm{\theta})p(\bm{\theta})]-\mathbb{E}_{q}[\log
q_{\bm{\lambda}}(\bm{\theta})])}_{L(\bm{\lambda},\bm{x})}+\log
p(\mathbf{x})\end{split}$ (3)
Second term $\log p(\mathbf{x})$ is constant. The first term
$L(\bm{\lambda})$is known as evidence lower bound (ELBO) and by maximising it
one can minimize the KL divergence between the variational distribution and
the true posterior. So the goal is to solve the following optimization
problem:
$\bm{\lambda}^{*}=\underset{\bm{\lambda}\in\Omega}{\arg\min}\lx@glossaries@gls@link{acronym}{kl}{\leavevmode
KL}(q_{\bm{\lambda}}(\bm{\theta})||p(\bm{\theta}|\mathbf{x}))=\underset{\bm{\lambda}\in\Omega}{\arg\max}{L(\bm{\lambda},\bm{x})},$
(4)
where ${\Omega}$ is the set of all possible values of the hyperparameters
$\bm{\lambda}$. The overall idea is schematically presented in Figure 2. It is
assumed that the model parameters are random variables. However, their true
distribution is almost always unknown. In such a case, the best option is to
select an approximate candidate distribution that will be used instead. When
selecting the candidate distribution, we try to incorporate prior knowledge as
much as possible, in particular the support interval, previous empirical
observations, etc. Some mismatch between the selected distributions and true
posterior is usually always present, which causes a bias in the VB solution.
However, this is a small price to pay compared to the substantial increase in
the computational efficiency of the VB approach. The VB algorithm was
implemented in Python using the PyTorch [29] library. For a link to the
implementation and an example data set sufficient to recreate a numerical
example, see A.
Figure 2: Optimization process of finding the closest variational distribution
$q_{\bm{\lambda}}(\bm{\theta})$ over the set of latent variables
$\bm{\lambda}$.
### 2.3 Finding the optimal hyper-parameters
Using VB we convert the problem of finding the posterior distributions from
statistical inference to a much simpler task of optimizing a cost function for
a variational family parameterised by the vector of hyperparameters
$\bm{\lambda}$. The cost function in this case is ELBO. For the optimization
part of our algorithm we use the adaptive moment estimation (ADAM) optimizer
[30].
ADAM allows an adaptive correction of the learning rates for each element
$\lambda^{(s)},s=1,...,\nu$ of the vector $\bm{\lambda}$ during the run of the
algorithm by calculating the first and second moment of the gradient, denoted
as ${m}_{t}^{(s)}$ and $v_{t}^{(s)}$ respectively. This is done using
exponentially moving averages computed on the gradient $g_{t}^{(s)},\
s=1,...,\nu$
$\begin{split}m_{t}^{(s)}&=\beta_{1}m_{t-1}^{(s)}+(1-\beta_{1})g_{t}^{(s)}\\\
v_{t}^{(s)}&=\beta_{2}v_{t-1}^{(s)}+(1-\beta_{2})\left(g_{t}^{(s)}\right)^{2},\end{split}$
(5)
where $t$ denotes the iteration number and $\beta_{1,2}$ are the parameters of
the moving average. The default values for $\beta$ parameters are 0.9 and
0.999 and the initial values of first and second moment of the gradient are
both set to $0$, since it turns out this does not impact the convergence rate
significantly. For a set of measured data (impedance in our case) $\bm{x}$,
the gradients can be presented with the following vector
$\bm{g}_{t}=\left(g_{t}^{(1)},\ldots,g_{t}^{(\nu)}\right)^{T}=\nabla_{\bm{\lambda}_{t}}L(\bm{\lambda}_{t},\bm{x}).$
(6)
The first and the second moment are only estimated with $m^{(s)}$ and
$v^{(s)}$, therefore we want them to satisfy the following condition
$\begin{split}\mathbb{E}\\{m_{t}^{(s)}\\}&=\mathbb{E}\left(g_{t}^{(s)}\right)\\\
\mathbb{E}\\{v_{t}^{(s)}\\}&=\mathbb{E}\left((g^{(s)}_{t})^{2}\right)\end{split}$
(7)
Above conditions ensure that we are dealing with unbiased estimates. First
moment at step $t$ in the recursive equation (5) is
$m_{t}^{(s)}=\left(1-\beta_{1}\right)\sum_{i=1}^{t}\beta_{1}^{t-i}{g}_{i}^{(s)}.$
(8)
We can see some bias still occurs in this estimate. Applying expected value
operator to equation (8) gives us
$\begin{split}E_{\lambda}\left[m_{t}^{(s)}\right]&=\left(1-\beta_{1}\right)\sum_{i=1}^{t}\beta_{1}^{t-1}E_{\lambda}\left[{g}_{i}^{(s)}\right]\\\
&\approx\left(1-\beta_{1}\right)\left(\sum_{i=1}^{t}\beta_{1}^{t-1}\right)E_{\lambda}\left[{g}_{t}^{(s)}\right]\\\
&=\left(1-\beta_{1}^{t}\right)E_{\bm{\lambda}}\left[{g}_{t}^{(s)}\right].\end{split}$
(9)
Bias correction is done automatically by ADAM during the evaluation of $m$ and
$v$ as follows
$\begin{split}\hat{m}_{t}^{(s)}&=\frac{m_{t}^{(s)}}{1-\beta_{2}^{t}}\\\
\hat{v}_{t}^{(s)}&=\frac{v_{t}^{(s)}}{1-\beta_{2}^{t}}.\end{split}$ (10)
Optimization steps are also adjusted. For each individual parameter the
algorithm updates
$\lambda_{t}^{(s)}=\lambda_{t-1}^{(s)}-\eta\frac{\hat{m}_{t}^{(s)}}{\sqrt{\hat{v}_{t}^{(s)}}+\eta},$
(11)
where $\eta=0.001$.
The use of ADAM is widely used across many areas and applications since it
delivers efficient and fast results. It should be noted, however, that as a
heuristic optimization algorithm there is no general guarantee for
convergence. Fortunately, Kingma and Ba [30] proved that the algorithm
converges globally in the convex settings. This was further refined and proved
by Reddi et al. [31]. We used the default settings of $\beta_{1}=0.9$ and
$\beta_{2}=0.999$, while setting the learning rate depending on the data set
at hand. The number of iterations is at least $8000$ but no more than $35000$.
An additional stopping criterion was defined by converged ELBO values, i.e.
when the relative change of less than $1\%$ is achieved over $1000$
iterations. In all of the optimization runs the maximum number of iterations
were almost never reached. The average number of iterations was around
$13000$.
## 3 Numerical example
Since we focus on solid oxide fuel cells in this paper, we will first
demonstrate the performance of VB algorithm using simulated data. The
rationale for the model structure arises from the electrochemical arguments
explained in more detail in section 4. The model can be represented as a
serial connection of resistance, inductance and RQ elements
$Z(\omega)=R_{s}+\sum_{i=1}^{N}\frac{R_{i}}{(j\omega)^{\alpha_{i}}Q_{i}R_{i}+1}+j\omega
L$ (12)
where $R_{s}$ stands for serial resistance, $R_{i}$ is parallel resistance and
$Q_{i}$ constant-phase parameter, fractional order of $i^{th}$ pole is denoted
with $\alpha_{i}\in(0,1]$, $\omega=2\pi f$ and $f$ is frequency. The schematic
of the ECM structure is given in Figure 3. For the simulation study below we
take $N=3$.
$L$$R_{S}$$R_{1}$$Q_{1}$$R_{2}$$Q_{2}$$\cdots$$\cdots$$R_{n}$$Q_{n}$ Figure 3:
The ECM as a series of RQ-elements.
The “true” impedance (12) was simulated on the frequency interval
$f\in[10^{-4},10^{4}]$ Hz by using parameters listed in Table 1. The additive
measurement noise was applied separately to the voltage $u(t)$ and current
$i(t)$ as $n_{1}(t)$ and $n_{2}(t)$ respectively. It should be noted that
$n_{1}(t)$ and $n_{2}(t)$ are zero-mean and uncorrelated. Having the simulated
input and output signals, the transfer function was estimated from Morlet
wavelet transform of the input and output as described in [32]. The entire
process is presented in Figure 4.
$i(t)$$H(s)$$u(t)$++$n_{1}\sim\mathcal{N}(0,\sigma_{1})$$n_{2}\sim\mathcal{N}(0,\sigma_{2})$ CWT [32] CWT [32] Impedance evaluation EIS curve as shown in Figure 5 measured voltage measuredcurrent Figure 4: Simulation of EIS data with additive noise in input and output. Table 1: Model parameters and variational distributions calculated using equations (13) and (14). True parameter | Variational distribution
---|---
$R_{s}$ = $3$ $\Omega$ | $\sim\text{Lognormal}(0.84,0.39)$
$R_{1}=1$ $\Omega$, $R_{2}=2$ $\Omega$, $R_{3}=3$ $\Omega$ | $\sim\text{Lognormal}(1.59,0.20)$
$Q_{1}=0.1$ $Fs^{\alpha_{1}}$ | $\sim\text{Lognormal}(-0.35,0.83)$
$Q_{2}=5$ $Fs^{\alpha_{2}}$ | $\sim\text{Lognormal}(1.96,0.83)$
$Q_{3}=150$ $Fs^{\alpha_{3}}$ | $\sim\text{Lognormal}(4.99,0.55)$
$\alpha_{1}=0.88$, $\alpha_{2}=0.82$, $\alpha_{3}=0.99$ | $\sim\text{Beta}(13.91,5.68)$
$L=100~{}nH$ |
Figure 5: Results of the VB algorithm (above). Current and voltage signals
used in simulation (below).
The first step in applying the VB approach is the definition of variational
distributions that approximate the true posterior $p(\bm{\theta}|\mathbf{x})$.
They are listed in Table 1. For our case $\bm{\theta}\in\mathbb{R}_{+}^{10}$,
with additional limitation as $\alpha_{i}\in[0,1]$ . Beta distribution was
therefore chosen as the best fit for the $\alpha_{i},i\in\\{1,2,3\\}$
parameters and Log-normal distribution for the rest. We selected the means and
variances for our variational distributions and then determined their
parameters using (13) and (14).
Rough estimates of variational distributions can be assessed from the EIS
curve in Figure 5, previous experience and integrating experts knowledge. In
the upper plot in Figure 5 one can notice that the real part of the curve
starts around $3$ $\Omega$. This provides a rough estimate for the parameter
$R_{s}$. Hence selecting variational distribution with mean $2.5$ $\Omega$ and
variance to $1$ seems a reasonable choice. The next step is setting up the
variational distributions for the parameters $R_{1,2,3}$. Just by looking at
the axis values, it is apparent that they should be located within the
interval (1,10) $\Omega$. Mean values of their distributions were therefore
set in the middle of the interval. Variational distributions of the parameters
$Q_{1,2,3}$ were roughly estimated with the help of experiences gained from
previous experiments and by incorporating experts knowledge. Lastly, the
fractional order powers $\alpha_{1,2,3}$ are expected to have values in the
interval [0.5,1] so we set their variational distributions with means at
$0.75$.
Once we have determined the mean and variance of our variational
distributions, we have to transform them into the correct parameters for the
chosen distributions. To obtain the parameters of the log-normal distribution
$(\sigma_{ln},\mu_{ln})$ from the mean and variance $(\sigma,\mu)$, the
following equations can be used:
$\sigma_{ln}=\ln\left(\frac{\sigma^{2}}{\sqrt{\sigma^{2}+\mu^{2}}}\right)\hskip
56.9055pt\mu_{ln}=\ln\left(1+\frac{\mu^{2}}{\sigma^{2}}\right)$ (13)
and for beta distribution parameters $\alpha$ and $\beta$ e.g.
$\text{Beta}(\alpha,\beta)$ we use:
$\alpha=\frac{\sigma^{2}-\sigma^{3}}{\mu}-\sigma\hskip
56.9055pt\beta=\frac{\alpha}{\sigma}-\alpha$ (14)
The optimization was performed using a global learning rate of $0.05$ and was
completed after $25000$ steps. ELBO loss curve can be observed in Figure 6. By
sampling the resulting posterior distributions and simulating the model (12)
it is possible to visualize the obtained results. For this purpose, $1000$
samples were drawn for each parameter from their respective posterior
distribution. Simulating the $1000$ EIS curves with the sampled sets of
parameters gave us an estimate of the confidence interval of our results,
which can be found in Figure 5. The EIS curve obtained from the means of the
posterior distributions for the parameters can also be found on the same
Figure. The variance of the posterior distributions is effectively presented
with the sampled curves and it is bound by the noise variance parameter. The
mean EIS curve seems to be a sufficiently good fit for the input data.
Additional results demonstrating the performance of the VB algorithm on
simulated data with heavy additive noise can be found in B.
Figure 6: The evolution of ELBO loss during optimization process.
The evolution of the optimization process for each parameter is presented in
Figure 7, where we can also find the true parameters used for the simulation
of our input data. We can see that each parameter reaches its true value,
except $Q_{3}$. We can assume this is the effect of noise, which seems to be
greater in the third arc, as seen in Figure 5. Progress in confidence of the
algorithm can be easily observed with the help of the evolution of estimated
spread in parameter distributions. As seen in the Figure 7, the spread is wide
at the beginning and as the optimization progresses and explores larger search
space it steadily narrows.
Figure 7: Progress of parameter estimation during the optimization process,
true values of parameters used for simulation are represented with dashed
lines. The variance of the estimated distribution(pink) is quite low. The
selected optimal shape of the posterior distributions are shown in Figure 8.
The resulting posterior distributions of the model parameters as well as the
posteriors obtained with MCMC are shown in Figure 8. Three key observations
can be made. First, comparing posterior and variational distributions, the
optimization process was capable of reaching the “true” mean values even for
cases when the variational means were “far” from the simulated values. Second,
the scales (variance) of the posterior distributions are low, leading to the
conclusion that the obtained model has low uncertainty. Table 2 lists all the
resulting posterior parameter distributions. Third, the empirical posteriors
obtained through MCMC method are similar to the ones obtained by VB approach.
It should be noted that in some cases VB posteriors can be under-dispersed
(lower variance) compared to the ones obtained by MCMC. Being overconfident is
a well known behavioural trait of VB algorithm and a mathematical proof of
this behaviour under certain conditions is provided by Blei and Wang [8, 33].
Our results seem to fit the MCMC results reasonably well, with minor
overconfidence in the posterior distributions of $R_{1,2}$ and $Q_{3}$.
Table 2: Posterior distributions of parameters.
$R_{s}\sim\text{Lognormal}(1.0,0.0006)$
---
$R_{1}\sim\text{Lognormal}(-0.011,0.003)$ | $R_{2}\sim\text{Lognormal}(0.72,0.002)$ | $R_{3}\sim\text{Lognormal}(1.12,0.003)$
$Q_{1}\sim\text{Lognormal}(-2.28,0.0007)$ | $Q_{2}\sim\text{Lognormal}(1.6,0.007)$ | $Q_{3}\sim\text{Lognormal}(4.94,0.004)$
$\alpha_{1}\sim\text{Beta}(969.35,144.11)$ | $\alpha_{2}\sim\text{Beta}(1016.18,225.57)$ | $\alpha_{3}\sim\text{Beta}(4724.72,144.39)$
Figure 8: Posterior distributions of parameters derived from simulated data.
## 4 Experimental validation
The VB was validated using EIS curves measured on stack of $6$ anode supported
solid oxide fuel cells which were installed in an insulated ceramic housing.
Stack operated for a total of 4500 hours at the operating temperature of $750$
°C. The active area of a single cell was 80 cm2. The SOFC short stack was fed
with a mixture of hydrogen and nitrogen with a flow rate H2/N2=0.216/0.144 Nl
h-1cm-2 whereas the air flow rate was 4 Nl h-1cm-2. Stack was operated at
nominal current of 32 $\mathrm{A}$ (0.4
$\mathrm{A}\text{\,}{\mathrm{cm}}^{-2}$) with fuel utilization of 77.5 %.
The EIS curves were obtained by current excitation (galvanostatic mode) with
discrete random binary signal. The excitation current had DC value of
I${{}_{\text{DC}}}$=32 A with peak-to-peak amplitude of 2 A. The amplitude was
chosen low to ensure linearised stack response, but still large enough to
guarantee sufficient signal-to-noise ratio.
The current and voltage sensor have sufficiently wide frequency bandwidth with
cut-off frequency of 240 kHz. The cell voltages were measured independently
using a differential 16-bit NI USB-6215 data acquisition system. The analogue
signals were firstly low pass filtered at 10.8 kHz and sampled with sampling
frequency $f_{s}=50$ kHz. Additional information about the SOFC short stack in
question can be found in [2].
### 4.1 Experimental results
The measured EIS curve is shown in Figure 10. Note here that the noise in
measurements is quite low thanks to the high-quality data acquisition
equipment and the well controlled laboratory conditions for the experiment.
The first step in model identification is the selection of the model
structure. In our case that breaks down to the selection of $N$, the number of
RQ elements and to do so physical arguments are used. In case of hydrogen as a
fuel one could expect five dominant processes divided into three main groups
[34]: gas conversion, cathode processes and anode processes. Their time
constants span the frequency band from 0.1 Hz to 1 Mhz. The gas conversion
processes are related to the low frequency part of the EIS curve, the cathode
processes are mainly visible in the mid-frequency range and the high-frequency
part can be attributed to the anode related processes. The contribution of
each group can be seen in Figure 9.
Figure 9: Main groups of processes within the hydrogen-powered SOFC.
Variational distributions were selected by examining the measured EIS curve.
If we look at the Figure 10, we can see that the high frequency part of the
real component has values around $3\text{m}\Omega$, which will serve as the
mean value for the $R_{s}$ parameter. The scale of its distribution was set
quite large to allow the optimization process to explore broad intervals in
the search for the optimum. Variational distributions of $R_{1,2,3}$
parameters can be inferred from the complex and real values our EIS curve
takes that their values should be roughly between the interval [0.1,3]
m$\Omega$. From previous testing and by incorporating experts knowledge we can
assume that the $Q_{1,2,3}$ parameters will be at least by an order of $10$
apart, so we assumed they are located at $1,50$ and $500$ $Fs^{\alpha_{1}}$
respectively. The scales for their variational distributions were set
respectively large. Finally, for $\alpha_{1,2,3}$ we can assume from
experience that they probably take values between $0.5$ and $1.0$, therefore
the means of their variational distributions were set at $0.75$. All the
variational distributions can be seen on Table 3.
Table 3: Variational distributions of parameters used for estimation of parameters for experimental data. $R_{s}\sim\text{Lognormal}(-5.86,0.32)$ | $R_{1,2,3}\sim\text{Lognormal}(-8.41,1.27)$ | $\alpha_{1,2,3}\sim\text{Beta}(12,3)$
---|---|---
$Q_{1}\sim\text{Lognormal}(-2.31,2.14)$ | $Q_{2}\sim\text{Lognormal}(3.57,0.83)$ | $Q_{3}\sim\text{Lognormal}(6.20,0.20)$
Figure 10: Measured EIS and the estimated mean EIS values from MCMC and VB.
The learning rate was set to $0.005$. This value is lower than in the
numerical example, since the goal was to allow steady and efficient
estimation. The optimization finished after $35000$ steps.
The performance of the algorithm can best be observed by looking at the EIS
curves obtained from the posterior distributions of the estimated parameters.
In the same manner as with numerical example, we also took $1000$ samples from
each of parameters posterior distribution, to simulate $1000$ EIS curves. The
confidence interval is shown in Figure 10, together with the EIS curve
obtained from the posterior means. The results confirm that despite the low
number of iterations, the resulting parameters represent an accurate fit for
the measured EIS data.
The convergence of ELBO can be seen in Figure 11. Comparing with the Figure 6
we can notice that the rate of convergence is slower, which can be associated
with the much smaller learning rate and smaller number of iterations used for
the run on experimental data. As optimization approaches the end, the ELBO
value changes minimally, which means the optimal estimates have been found.
Figure 11: The evolution of ELBO for experimental data.
The evolution of parameter estimation is shown in Figure 13. The algorithm
steadily converges towards the optimum, while the estimated spread for each
parameter continues to decrease. The parameter evolution is very smooth for
each parameter. Posterior distributions are shown in Figure 14, where we
compare them with results of MCMC algorithm. We can see the expected slight
overconfidence of VB, while the results are still inline with MCMC results. It
is apparent that scales (variance) of the posterior distributions are quite
low. Posterior parameter distributions are listed in Table 4.
Additional VB approach results on different settings for SOFC can be found in
C, including a result on a measurement recorded after a leakage fault has
occured.
Table 4: Posterior distributions of the parameters.
$R_{s}\sim\text{Lognormal}(-5.75,8.3e^{-4})$
---
$R_{1}\sim\text{Lognormal}(-7.995,0.011)$ | $R_{2}\sim\text{Lognormal}(-6.33,0.0039)$ | $R_{3}\sim\text{Lognormal}(-5.61,0.0075)$
$Q_{1}\sim\text{Lognormal}(2.25,0.0086)$ | $Q_{2}\sim\text{Lognormal}(6.22,0.0025)$ | $Q_{3}\sim\text{Lognormal}(3.96,0.0028)$
$\alpha_{1}\sim\text{Beta}(1081.56,41.42)$ | $\alpha_{2}\sim\text{Beta}(2216.36,187.46)$ | $\alpha_{3}\sim\text{Beta}(399.28,0.495)$
### 4.2 Comparison of Variational Bayes with averaging spectra
Averaging the EIS obtained from successive measurements is a pragmatic way to
reduce the effects of measurement noise. The more spectra are averaged, the
better the filtration of noise is. However, in practice that means many
repeated perturbations are needed.
Using VB approach, we can compute a comparable result by using only one
measurement. For the comparison, we averaged 10 consecutive measured spectra
and compared with the results of the VB approach on the last among them.
The results are presented in Figure 12, where measured EIS data are in the
upper part, while the mean estimate from VB approach and the evaluated average
are presented in the lower part. The results of both methods are very similar,
even though the VB approach used information from only one measurement. That
is practical benefit in the context of online health monitoring.
Figure 12: Comparison of VB approach and averaging EIS spectra. Ten
consecutive EIS evaluations are used (upper part). The compared VB posterior
and averaged spectra are on the lower graph.
### 4.3 Discussion
Results in Figure 10 that the ECM identified through the VB approach provide
an accurate fit for the measured data.
From ELBO plot in Figure 11 it is evident the algorithm converges. The same
observation holds true for the estimated parameters, as shown in Figure 13.
Comparing the resulting posterior distributions from both VB and MCMC in
Figure 14 we can conclude the following. First, results show close match for
every parameter, with only a slight mismatch noticeable in $R_{s},\alpha_{1}$
and $Q_{1}$. Second, the posterior distributions obtained with VB have lower
dispersion (variance) than the ones obtained through MCMC. As mentioned above,
this is common for VB approach.
One of the reasons for having overconfident results might be in the selection
of the variational distributions. A possible improvement might be achieved by
using more elaborated approximations in terms of a mixture of bounded
distributions [35]. That is a topic of further research.
Figure 13: The evolution of the parameter estimates for experimental data.
Figure 14: Posterior distributions of parameters inferred from experimental
data by VB and MCMC algorithm.
## 5 Conclusion
In this paper a novel approach to the statistical estimation of the ECM
parameters of SOFC impedance spectra is presented. From a single EIS
evaluation, the approach is able to evaluate the uncertainty in the model
parameters caused by noise and disturbances in the system. The VB approach was
validated on simulated data as well as on experimental measurements.
The simulation reveals that even in the case of excessive degree of noise the
VB algorithm still produces accurate estimates. Main benefit of using the
method is low computational load when compared to the commonly used MCMC
approach. In particular, to obtain comparable results the time needed was
approximately 5 minutes for VB approach and around 25 hours for MCMC on the
same computational power. Validation tests indicate that the difference in
performance between two methods is minimal, which renders VB as an option for
on-line SOFC monitoring.
Full information on the estimated value and accompanying uncertainty helps
avoid misinterpretation of the EIS data corrupted by noise and disturbances.
It is shown that the results of VB might be comparable with the results
obtained by averaging the spectra. The key advantage of VB is that only one
evaluated EIS is needed to get the information for which several spectra in
the averaging approach are needed (more spectra means more stack
perturbation).
The future work will concentrate on the design of the detection approach that
fully exploits the information provided by the VB. It is expected that
evaluated uncertainties in the estimated parameters will result in a more
cautious diagnosis, which will be insensitive to the illogical shapes of the
impedance characteristics.
## Acknowledgements
The authors acknowledge the support from the Slovenian Research Agency through
the programme P2-0001 and the project NC-0003. Part of the support is received
through the project RUBY (grant agreement No. 875047) within the framework of
the Fuel Cells and Hydrogen 2 Joint Undertaking under the European Union’s
Horizon 2020 research and innovation programme, Hydrogen Europe and Hydrogen
Europe research. We are grateful to the anonymous referees for their valuable
comments and suggestions.
## References
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## Appendix A Supplementary material: numerical implementation
Supplementary material containing numerical implementation of the VB
algorithm can be found at https://repo.ijs.si/lznidaric/variational-bayes-
supplementary-material. Python Jupyter notebook file
SupplementaryMaterialSVI.ipynb contains the main algorithm. Data set used to
recreate the results presented in this article can be found in NumPy file
dataset.npz. Repository contains all the needed scripts for implementation of
VB algorithm. Simulation of data was however done separately, so the code is
not included.
## Appendix B Additional results on simulated measurements
Additional simulation runs with the model from section 3 are performed under
different noise levels in current and voltage. Noise $n_{1}(t)$ and $n_{2}(t)$
was added both to the voltage $u(t)$ as well as current $i(t)$ respectively.
It should be noted that $n_{1}(t)$ and $n_{2}(t)$ are zero-mean and
uncorrelated. Most interesting results were selected and are presented here.
Table 5 contains the information about selected measurements. The results
suggest that even in the case of high noise level the VB produces plausible
estimate of the true EIS characteristic.
Table 5: Noise levels used in simulation. Measurement number | $1$ | $2$ | $3$
---|---|---|---
Input noise | $\mathcal{N}(0.0,0.0)$ | $\mathcal{N}(0.0,0.00005)$ | $\mathcal{N}(0.0,0.05)$
Output noise | $\mathcal{N}(0,0.0)$ | $\mathcal{N}(0,0.001)$ | $\mathcal{N}(0,0.05)$
Figure 15: Simulated current, voltage and the resulting EIS curves with
different noise settings along with VB estimated mean and the confidence
region. Figure 16: Posterior parameter distributions obtained by VB approach
on simulated noisy data. Uncertainty in the parameters increases with
increasing level of noise used for simulation.
## Appendix C Additional results on experimental measurements
The method was tested on a set of measured EIS data at different currents in
healthy state and on data under leakage fault. Measurement presented in the
main article is obtained during the nominal running conditions of the SOFC
which are 77% fuel utilization and current of 32 A. Main operating conditions
of additional measurements are presented in Table 6. Measurement 1 and
Measurement 2 are obtained during different fuel utilization settings. For
Measurement 1, fuel utilization was increased from 77% to 72% by decreasing
the gas flow and keeping the current steady at 32 A. On the flip side, fuel
utilization in Measurement 2 was increased from 72% to 87% by increasing the
electrical current from 32 A to 36.4 A. Measurement 3 was recorded after a
leakage fault occured in the plant. In Figure 17 the results of VB algorithm
are presented. The VB algorithm returns good estimates for each of the
measurement, irrespective of the fact that the same set of variational
distributions are applied. The resulting posterior distributions for each
parameter are presented in Figure 18.
Table 6: Operational conditions for the measurements. Measurement number | $1$ | $2$ | $3$
---|---|---|---
Current [A] | $32.21$ | $36.39$ | $32.04$
Fuel utilization [$\%$] | $82$ | $87$ | $77$
Figure 17: VB results for 3 EIS measurements at different operational
conditions. Figure 18: Posterior distributions of ECM parameters obtained on 3
measurements under different conditions explained in Table 6.
|
# Weak Convergence for Variational Inequalities with Inertial-Type Method
Yekini Shehu111Department of Mathematics, Zhejiang Normal University, Jinhua,
321004, People’s Republic of China; Institute of Science and Technology (IST),
Am Campus 1, 3400, Klosterneuburg, Austria; e-mail<EMAIL_ADDRESS>Olaniyi. S. Iyiola222Department of Mathematics, Computer Science and
Information Systems, California University of Pennsylvania, PA, USA;
<EMAIL_ADDRESS>
###### Abstract
Weak convergence of inertial iterative method for solving variational
inequalities is the focus of this paper. The cost function is assumed to be
non-Lipschitz and monotone. We propose a projection-type method with inertial
terms and give weak convergence analysis under appropriate conditions. Some
test results are performed and compared with relevant methods in the
literature to show the efficiency and advantages given by our proposed
methods.
## 1 Introduction
Suppose $C$ is a nonempty, closed and convex subset of a real Hilbert space
$H$ and $F:C\rightarrow H$ a continuous mapping. The variational inequality
problem (for short, VI($F,C$)) is defined as: find $x\in C$ such that
$\displaystyle\langle F(x),y-x\rangle\geq 0,\quad\forall y\in C.$ (1)
We shall denote by SOL the solution set of VI($F,C$) in (1). Various
applications of variational inequality can be found in [7, 8, 23, 24, 25, 33,
34, 35, 43].
Projection-type method for solving VI($F,C$) (1) have been considered
severally in the literature (see, for example, [16, 17, 18, 22, 29, 40, 41,
42, 45, 48, 56]). Several other related methods to extragradient method and
(5) for solving VI($F,C$) (1) in real Hilbert spaces when $F$ is monotone and
$L$-Lipschitz-continuous mapping have been studied in the literature (see, for
example, [16, 17, 18, 22, 29, 36, 40, 41, 42, 45, 56]). Some of these methods
involve computing projection onto the feasible set $C$ twice per iteration and
this can affect the efficiency of the methods.
In [19], Censor et al. introduced the subgradient extragradient method:
$x_{1}\in H$,
$\displaystyle\left\\{\begin{array}[]{llll}&y_{n}=P_{C}(x_{n}-\lambda
F(x_{n})),\\\ &T_{n}:=\\{w\in H:\langle x_{n}-\lambda
F(x_{n})-y_{n},w-y_{n}\rangle\leq 0\\},\\\ &x_{n+1}=P_{T_{n}}(x_{n}-\lambda
F(y_{n}))\end{array}\right.$ (5)
and gave weak convergence result when $F$ is monotone and $L$-Lipschitz-
continuous mapping where $\lambda\in(0,\frac{1}{L})$.
In order to accelerate the convergence of subgradient extragradient method (5)
and using the idea of in [2, 3, 4, 5, 6, 10, 12, 13, 20, 37, 38, 46, 47],
Thong and Hieu [55] introduced the following inertial subgradient
extragradient method: $x_{0},x_{1}\in H$,
$\displaystyle\left\\{\begin{array}[]{llll}&w_{n}=x_{n}+\alpha_{n}(x_{n}-x_{n-1}),\\\
&y_{n}=P_{C}(w_{n}-\lambda F(w_{n})),\\\ &T_{n}:=\\{w\in H:\langle
w_{n}-\lambda F(w_{n})-y_{n},w-y_{n}\rangle\leq 0\\},\\\
&x_{n+1}=P_{T_{n}}(w_{n}-\lambda F(y_{n}))\end{array}\right.$ (10)
and proved that $\\{x_{n}\\}$ generated by (10) converges weakly to a solution
of VI($F,C$) (1) when $F$ is monotone and $L$-Lipschitz-continuous mapping $F$
where $0<\lambda
L\leq\frac{\frac{1}{2}-2\alpha-\frac{1}{2}\alpha^{2}-\delta}{\frac{1}{2}-\alpha+\frac{1}{2}\alpha^{2}}$
for some $0<\delta<\frac{1}{2}-2\alpha-\frac{1}{2}\alpha^{2}$ and
$\\{\alpha_{n}\\}$ is a non-decreasing sequence with
$0\leq\alpha_{n}\leq\alpha<\sqrt{5}-2$.
The step-sizes in above methods (5) and (10) are bounded by the inverse of the
Lipschitz constant and this is quite inefficient, since in most applications a
global Lipschitz constant (if it indeed exists at all) of $F$ cannot be
accurately estimated, and is usually overestimated. This leads to too small
step-sizes, which, of course, is not practical. Therefore, algorithms (5) and
(10) are not applicable in most cases of interest. This can be overcome by
using an Armijo type line search procedure (see [33, 43, 53]).
We provide a simple example of a variational inequality problem where the
method (5) proposed in [19] and method (10) proposed in [55] cannot be
applied.
###### Example 1.1.
Suppose $F:[0,\infty)\rightarrow\mathbb{R}$ is defined by
$F(x):=e^{x},\leavevmode\nobreak\ x\in[0,\infty)$. It is easy to see that $F$
is not Lipschitz continuous on $[0,\infty)$. By the mean value theorem, one
has for an arbitrary $r>0$,
$|F(x)-F(y)|\leq e^{r}|x-y|$
with $|x|,|y|\leq r$. Hence, $F$ is uniformly continuous on bounded subsets of
$C:=[0,\infty)$. Consequently, one can easily see that $F$ is monotone on
$[0,\infty)$ since
$\langle F(x)-F(y),x-y\rangle=(F(x)-F(y))(x-y)\geq 0,\leavevmode\nobreak\
\leavevmode\nobreak\ \forall x,y\in[0,\infty).$
Finally, SOL of VI$(F,C)$ is nonempty since $0\in\text{SOL}$.
Motivated by Example 1.1, it would be of interest to propose an iterative
method for solving VI($F,C$) (1) for which the underline cost function $F$ is
uniformly continuous on bounded subsets of $C$ but not Lipschitz continuous on
$C$.
Our interest in this paper is to obtain weak convergence results using
inertial projection-type algorithm for VI($F,C$) (1) when the underline
operator $F$ is monotone and uniformly continuous. We do not assume the cost
function to be Lipschitz continuous as assumed in [18, 19, 22, 40, 41, 55].
Our proposed method is much more practical and outperforms the methods (5) and
(10) numerically.
We organize the paper as follows: Basic definitions and results are given in
Section 2 and the proposed method is introduced in Section 3. We give weak
convergence analysis of the proposed method in Section 4 and give some
numerical comparisons of our method with methods (5) and (10) in Section 5.
Finally, we some concluding remarks in Section 6.
## 2 Preliminaries
Suppose we take $H$ as a real Hilbert space and $X\subseteq H$ be a nonempty
subset.
###### Definition 2.1.
A mapping $F:X\to H$ is called
* (a)
monotone on $X$ if $\langle F(x)-F(y),x-y\rangle\geq 0$ for all $x,y\in X$;
* (b)
Lipschitz continuous on $X$ if there exists a constant $L>0$ such that
$\|F(x)-F(y)\|\leq L\|x-y\|,\ \forall x,y\in X.$
* (c)
sequentially weakly continuous if for each sequence $\\{x_{n}\\}$ we have:
$\\{x_{n}\\}$ converges weakly to $x$ implies $\\{F(x_{n})\\}$ converges
weakly to $F(x)$.
Given any point $u\in H$, there exists a unique point $P_{C}u\in C$ (see,
e.g., [9]) such that
$\|u-P_{C}u\|\leq\|u-y\|,\leavevmode\nobreak\ \forall y\in C.$
This $P_{C}$ is called the metric projection of $H$ onto $C$. It is known
that $P_{C}$ is a nonexpansive mapping of $H$ onto $C$ and satisfies
$\langle
x-y,P_{C}x-P_{C}y\rangle\geq\|P_{C}x-P_{C}y\|^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \forall x,y\in H.$ (11)
In particular, we get from (11) that
$\langle x-y,x-P_{C}y\rangle\geq\|x-P_{C}y\|^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \forall x\in C,y\in H.$ (12)
Another property of $P_{C}x$ is :
$P_{C}x\in C\quad\text{and}\quad\langle x-P_{C}x,P_{C}x-y\rangle\geq
0,\leavevmode\nobreak\ \forall y\in C.$ (13)
More details on $P_{C}$ can be found, for example, in Section 3 of [26].
The following results are needed in the next section.
###### Lemma 2.2.
The following statements hold in $H$:
* (a)
$\|x+y\|^{2}=\|x\|^{2}+2\langle x,y\rangle+\|y\|^{2}$ for all $x,y\in H$;
* (b)
$2\langle x-y,x-z\rangle=\|x-y\|^{2}+\|x-z\|^{2}-\|y-z\|^{2}$ for all
$x,y,z\in H$;
* (c)
$\|\alpha
x+(1-\alpha)y\|^{2}=\alpha\|x\|^{2}+(1-\alpha)\|y\|^{2}-\alpha(1-\alpha)\|x-y\|^{2}$
for all $x,y\in H$ and $\alpha\in\mathbb{R}$.
###### Lemma 2.3.
(see [1, Lem. 3]) Let $\\{\psi_{n}\\}$, $\\{\delta_{n}\\}$ and
$\\{\alpha_{n}\\}$ be the sequences in $[0,+\infty)$ such that
$\psi_{n+1}\leq\psi_{n}+\alpha_{n}(\psi_{n}-\psi_{n-1})+\delta_{n}$ for all
$n\geq 1$, $\sum_{n=1}^{\infty}\delta_{n}<+\infty$ and there exists a real
number $\alpha$ with $0\leq\alpha_{n}\leq\alpha<1$ for all $n\geq 1$. Then the
following hold:
$(i)\leavevmode\nobreak\ \leavevmode\nobreak\ \sum_{n\geq
1}[\psi_{n}-\psi_{n-1}]_{+}<+\infty$, where $[t]_{+}=\max\\{t,0\\}$;
(ii) there exists $\psi^{*}\in[0,+\infty)$ such that
$\lim_{n\rightarrow+\infty}\psi_{n}=\psi^{*}$.
###### Lemma 2.4.
(see [9, Lem. 2.39]) Let $C$ be a nonempty set of $H$ and $\\{x_{n}\\}$ be a
sequence in $H$ such that the following two conditions hold:
(i) for any $x\in C$, $\lim_{n\rightarrow\infty}\|x_{n}-x\|$ exists;
(ii) every sequential weak cluster point of $\\{x_{n}\\}$ is in $C$.
Then $\\{x_{n}\\}$ converges weakly to a point in $C$.
###### Lemma 2.5.
([28]) Let $C$ be a nonempty closed and convex subset of $H$. Let $h$ be a
real-valued function on $H$ and define $K:=\\{x:h(x)\leq 0\\}$. If $K$ is
nonempty and $h$ is Lipschitz continuous on $C$ with modulus $\theta>0$, then
${\rm dist}(x,K)\geq\theta^{-1}\max\\{h(x),0\\},\leavevmode\nobreak\ \forall
x\in C,$
where ${\rm dist}(x,K)$ denotes the distance function from $x$ to $K$.
###### Lemma 2.6.
Let $C$ be a nonempty closed and convex subset of $H$, $y:=P_{C}(x)$ and
$x^{*}\in C$. Then
$\|y-x^{*}\|^{2}\leq\|x-x^{*}\|^{2}-\|x-y\|^{2}.$ (14)
###### Lemma 2.7.
([31, Prop. 2.11], [30, Prop. 4]) Let $H_{1}$ and $H_{2}$ be two real Hilbert
spaces. Suppose $F:H_{1}\rightarrow H_{2}$ is uniformly continuous on bounded
subsets of $H_{1}$ and $M$ is a bounded subset of $H_{1}$. Then $F(M)$ is
bounded.
###### Lemma 2.8.
([54, Lem. 7.1.7]) Let $C$ be a nonempty, closed, and convex subset of $H$.
Let $F:C\rightarrow H$ be a continuous, monotone mapping and $z\in C$. Then
$z\in{\rm SOL}\Longleftrightarrow\langle F(x),x-z\rangle\geq 0\quad\text{for
all }x\in C.$
## 3 Proposed Method
We give some assumptions on the feasible set $C$, the cost function $F$ and
the iterative parameter $\\{\alpha_{n}\\}$ below.
###### Assumption 3.1.
Suppose that the following hold:
* (a)
The feasible set $C$ is a nonempty closed affine subset of the real Hilbert
space $H$.
* (b)
$F:C\to H$ is monotone and uniformly continuous on bounded subsets of $H$.
* (c)
The solution set SOL of VI$(F,C)$ is nonempty.
###### Assumption 3.2.
Suppose the real sequence $\\{\alpha_{n}\\}$ satisfy the following condition:
* •
$\\{\alpha_{n}\\}\subset(0,1)$ with
$0\leq\alpha_{n}\leq\alpha_{n+1}\leq\alpha<\frac{1}{3}$ for all $n$.
Suppose we define
$r(x):=x-P_{C}(x-F(x))$
as the residual equation. Then if $y=x-F(x)$ in (12), we obtain
$\langle F(x),r(x)\rangle\geq\|r(x)\|^{2},\leavevmode\nobreak\ \forall x\in
C.$ (15)
We next give our proposed inertial projection-type method.
Algorithm 1 Inertial Projection Method
1:Choose sequence $\\{\alpha_{n}\\}$ and $\sigma\in(0,1)$ such that the
conditions from Assumption 3.2 hold, and take $\gamma\in(0,1)$. Let
$x_{0}=x_{1}\in H$ be a given starting point. Set $n:=1$.
2:Set $w_{n}:=x_{n}+\alpha_{n}(x_{n}-x_{n-1}).$ Compute
$z_{n}:=P_{C}(w_{n}-F(w_{n}))$. If $r(w_{n})=w_{n}-z_{n}=0$: STOP.
3:Compute $y_{n}=w_{n}-\gamma^{m_{n}}r(w_{n})$, where $m_{n}$ is the smallest
nonnegative integer satisfying $\langle
F(y_{n}),r(w_{n})\rangle\geq\frac{\sigma}{2}\|r(w_{n})\|^{2}.$ (16) Set
$\eta_{n}:=\gamma^{m_{n}}$.
4:Compute $x_{n+1}=P_{C_{n}}(w_{n}),$ (17) where $C_{n}=\\{x:h_{n}(x)\leq
0\\}$ and $h_{n}(x):=\langle F(y_{n}),x-y_{n}\rangle.$ (18)
5:Set $n\leftarrow n+1$ and goto 2.
If $r(w_{n})=0$, then $w_{n}$ is a solution of VI($F,C$) (1). In the analysis
we assume that $r(w_{n})\neq 0$ for infinitely many iterations, so that
Algorithm 1 generates an infinite sequence satisfying $r(w_{n})\neq 0$ for all
$n\in\mathbb{N}$.
###### Remark 3.3.
(a) Our proposed Algorithm 1 requires, at each iteration, only one projection
onto the feasible set $C$ and another projection onto the half-space $C_{n}$
(which has a closed form solution, [15]) and this is numerically less
expensive than the twice computation of projection onto $C$ per iteration in
extragradient method [36].
(b) As we have mentioned before, Algorithm 1 is much more applicable than (5)
and (10) because the Lipschtz constant of the cost function $F$ is not needed
during implementations. $\Diamond$
###### Lemma 3.4.
Let the function $h_{n}$ be defined by (18). Then
$h_{n}(w_{n})\geq\frac{\sigma\eta_{n}}{2}\|w_{n}-z_{n}\|^{2}.$
In particular, if $w_{n}\neq z_{n}$, then $h_{n}(w_{n})>0$. If
$x^{*}\in\text{SOL}$, then $h_{n}(x^{*})\leq 0$.
###### Proof.
Since $y_{n}=w_{n}-\eta_{n}(w_{n}-z_{n})$, using (16) we have
$\displaystyle h_{n}(w_{n})$ $\displaystyle=\langle
F(y_{n}),w_{n}-y_{n}\rangle$ $\displaystyle=\eta_{n}\langle
F(y_{n}),w_{n}-z_{n}\rangle\geq\eta_{n}\frac{\sigma}{2}\|w_{n}-z_{n}\|^{2}\geq
0.$
If $w_{n}\neq z_{n}$, then $h_{n}(w_{n})>0$. Furthermore, suppose
$x^{*}\in\text{SOL}$. Then by Lemma 2.8 we have $\langle
F(x),x-x^{*}\rangle\geq 0\quad\text{for all }x\in C.$ In particular, $\langle
F(y_{n}),y_{n}-x^{*}\rangle\geq 0$ and hence $h_{n}(x^{*})\leq 0.$ ∎
## 4 Convergence Analysis
Let us give weak convergence analysis of our proposed Algorithm 1 in this
section.
###### Lemma 4.1.
Let $\\{x_{n}\\}$ be generated by Algorithm 1. Then under Assumptions 3.1 and
3.2, we have that
(i) $\\{x_{n}\\}$ is bounded, and
(ii) $\lim_{n\rightarrow\infty}\|x_{n+1}-w_{n}\|=0$.
###### Proof.
Let $x^{*}\in\text{SOL}$. By Lemma 2.6 we get (since $x^{*}\in C_{n}$) that
$\displaystyle\|x_{n+1}-x^{*}\|^{2}$ $\displaystyle=$
$\displaystyle\|P_{C_{n}}(w_{n})-x^{*}\|^{2}\leq\|w_{n}-x^{*}\|^{2}-\|x_{n+1}-w_{n}\|^{2}$
$\displaystyle=$ $\displaystyle\|w_{n}-x^{*}\|^{2}-{\rm
dist}^{2}(w_{n},C_{n}).$
Now, using Lemma 2.2 (c), we have
$\displaystyle\|w_{n}-x^{*}\|^{2}$ $\displaystyle=$
$\displaystyle\|(1+\alpha_{n})(x_{n}-x^{*})-\alpha_{n}(x_{n-1}-x^{*})\|^{2}$
(20) $\displaystyle=$
$\displaystyle(1+\alpha_{n})\|x_{n}-x^{*}\|^{2}-\alpha_{n}\|x_{n-1}-x^{*}\|$
$\displaystyle+\alpha_{n}(1+\alpha_{n})\|x_{n}-x_{n-1}\|^{2}.$
Also,
$\displaystyle\|x_{n+1}-w_{n}\|^{2}$ $\displaystyle=$
$\displaystyle\|x_{n+1}-(x_{n}+\alpha_{n}(x_{n}-x_{n-1}))\|^{2}$ (21)
$\displaystyle=$
$\displaystyle\|x_{n+1}-x_{n}\|^{2}+\alpha^{2}_{k}\|x_{n}-x_{n-1}\|^{2}-2\alpha_{n}\langle
x_{n+1}-x_{n},x_{n}-x_{n-1}\rangle$ $\displaystyle\geq$
$\displaystyle\|x_{n+1}-x_{n}\|^{2}+\alpha^{2}_{k}\|x_{n}-x_{n-1}\|^{2}-2\alpha_{n}\|x_{n+1}-x_{n}\|\|x_{n}-x_{n-1}\|$
$\displaystyle\geq$
$\displaystyle\|x_{n+1}-x_{n}\|^{2}+\alpha^{2}_{k}\|x_{n}-x_{n-1}\|^{2}-\alpha_{n}\|x_{n+1}-x_{n}\|^{2}$
$\displaystyle-\alpha_{n}\|x_{n}-x_{n-1}\|^{2}$ $\displaystyle=$
$\displaystyle(1-\alpha_{n})\|x_{n+1}-x_{n}\|^{2}+(\alpha_{n}^{2}-\alpha_{n})\|x_{n}-x_{n-1}\|^{2}.$
Combining (4), (20) and (21), we get
$\displaystyle\|x_{n+1}-x^{*}\|^{2}$ $\displaystyle\leq$
$\displaystyle(1+\alpha_{n})\|x_{n}-x^{*}\|^{2}-\alpha_{n}\|x_{n-1}-x^{*}\|^{2}$
(22)
$\displaystyle+\alpha_{n}(1+\alpha_{n})\|x_{n}-x_{n-1}\|^{2}-(1-\alpha_{n})\|x_{n+1}-x_{n}\|^{2}$
$\displaystyle-(\alpha_{n}^{2}-\alpha_{n})\|x_{n}-x_{n-1}\|^{2}$
$\displaystyle=$
$\displaystyle(1+\alpha_{n})\|x_{n}-x^{*}\|^{2}-\alpha_{n}\|x_{n-1}-x^{*}\|^{2}$
$\displaystyle-(1-\alpha_{n})\|x_{n+1}-x_{n}\|^{2}+(\alpha_{n}(1+\alpha_{n})-(\alpha_{n}^{2}-\alpha_{n}))\|x_{n}-x_{n-1}\|^{2}$
$\displaystyle=$
$\displaystyle(1+\alpha_{n})\|x_{n}-x^{*}\|^{2}-\alpha_{n}\|x_{n-1}-x^{*}\|^{2}$
$\displaystyle-(1-\alpha_{n})\|x_{n+1}-x_{n}\|^{2}+2\alpha_{n}\|x_{n}-x_{n-1}\|^{2}.$
Using the fact that $\alpha_{n}\leq\alpha_{n+1}$, we obtain from (22) that
$\displaystyle\|x_{n+1}-x^{*}\|^{2}$ $\displaystyle\leq$
$\displaystyle(1+\alpha_{n+1})\|x_{n}-x^{*}\|^{2}-\alpha_{n}\|x_{n-1}-x^{*}\|^{2}$
(23)
$\displaystyle-(1-\alpha_{n})\|x_{n+1}-x_{n}\|^{2}+2\alpha_{n}\|x_{n}-x_{n-1}\|^{2}.$
By (23), we get
$\displaystyle\|x_{n+1}-x^{*}\|^{2}-\alpha_{n+1}\|x_{n}-x^{*}\|^{2}+2\alpha_{n+1}\|x_{n+1}-x_{n}\|^{2}\leq\|x_{n}-x^{*}\|^{2}-\alpha_{n}\|x_{n-1}-x^{*}\|^{2}$
$\displaystyle+2\alpha_{n}\|x_{n}-x_{n-1}\|^{2}+2\alpha_{n+1}\|x_{n+1}-x_{n}\|^{2}-(1-\alpha_{n})\|x_{n+1}-x_{n}\|^{2}$
$\displaystyle=$
$\displaystyle\|x_{n}-x^{*}\|^{2}-\alpha_{n}\|x_{n-1}-x^{*}\|^{2}+2\alpha_{n}\|x_{n}-x_{n-1}\|^{2}+(2\alpha_{n+1}-1+\alpha_{n})\|x_{n+1}-x_{n}\|^{2}.$
Therefore,
$\displaystyle\|x_{n+1}-x^{*}\|^{2}$ $\displaystyle\leq$
$\displaystyle\|x_{n}-x^{*}\|^{2}-\alpha_{n}\|x_{n-1}-x^{*}\|^{2}+2\alpha_{n}\|x_{n}-x_{n-1}\|^{2}$
(24) $\displaystyle+(2\alpha_{n+1}-1+\alpha_{n})\|x_{n+1}-x_{n}\|^{2}.$
Let us define
$\Gamma_{n}:=\|x_{n}-x^{*}\|^{2}-\alpha_{n}\|x_{n-1}-x^{*}\|^{2}+2\alpha_{n}\|x_{n}-x_{n-1}\|^{2}.$
Then we have from (24) that
$\Gamma_{n+1}-\Gamma_{n}\leq(2\alpha_{n+1}-1+\alpha_{n})\|x_{n+1}-x_{n}\|^{2}.$
(25)
Since $0\leq\alpha_{n}\leq\alpha_{n+1}\leq\alpha<\frac{1}{3}$, we get
$-2\alpha_{n+1}\geq-2\alpha$ and $-\alpha_{n}\geq-\alpha$. This implies that
$-(2\alpha_{n+1}-1+\alpha_{n})=-2\alpha_{n+1}+1-\alpha_{n}\geq-2\alpha+1-\alpha\geq
1-3\alpha>0$ since $\alpha<\frac{1}{3}$. Now, let us define
$\sigma:=1-3\alpha$. Then
$2\alpha_{n+1}-1+\alpha_{n}\leq-\sigma.$ (26)
Putting (26) into (25), we have
$\Gamma_{n+1}-\Gamma_{n}\leq-\sigma\|x_{n+1}-x_{n}\|^{2}.$ (27)
From (27), we see that $\\{\Gamma_{n}\\}$ is monotone nonincreasing.
Furthermore,
$\displaystyle\Gamma_{n}$ $\displaystyle=$
$\displaystyle\|x_{n}-x^{*}\|^{2}-\alpha_{n}\|x_{n-1}-x^{*}\|^{2}+2\alpha_{n}\|x_{n}-x_{n-1}\|^{2}$
(28) $\displaystyle\geq$
$\displaystyle\|x_{n}-x^{*}\|^{2}-\alpha_{n}\|x_{n-1}-x^{*}\|^{2}.$
So,
$\displaystyle\|x_{n}-x^{*}\|^{2}$ $\displaystyle\leq$
$\displaystyle\alpha_{n}\|x_{n-1}-x^{*}\|^{2}+\Gamma_{n}$ (29)
$\displaystyle\leq$ $\displaystyle\alpha\|x_{n-1}-x^{*}\|^{2}+\Gamma_{1}$
$\displaystyle\vdots$ $\displaystyle\leq$
$\displaystyle\alpha^{k}\|x_{0}-x^{*}\|^{2}+(1+\alpha+\alpha^{2}+\ldots+\alpha^{k-1})\Gamma_{1}$
$\displaystyle=$
$\displaystyle\alpha^{k}\|x_{0}-x^{*}\|^{2}+\frac{\Gamma_{1}}{1-\alpha}.$
From (29), we can infer that $\\{x_{n}\\}$ is bounded. Using the definition of
$\Gamma_{n}$, we have
$\displaystyle\Gamma_{n+1}$ $\displaystyle=$
$\displaystyle\|x_{n+1}-x^{*}\|^{2}-\alpha_{n+1}\|x_{n}-x^{*}\|^{2}+2\alpha_{n+1}\|x_{n+1}-x_{n}\|^{2}$
(30) $\displaystyle\geq$ $\displaystyle-\alpha_{n+1}\|x_{n}-x^{*}\|^{2}.$
Using (29) in (30), we get
$\displaystyle-\Gamma_{n+1}$ $\displaystyle\leq$
$\displaystyle-\alpha_{n+1}\|x_{n}-x^{*}\|^{2}\leq\alpha\|x_{n}-x^{*}\|^{2}$
(31) $\displaystyle\leq$
$\displaystyle\alpha^{k+1}\|x_{0}-x^{*}\|^{2}+\frac{\alpha\Gamma_{1}}{1-\alpha}.$
From (27), we get
$\sigma\|x_{n+1}-x_{n}\|^{2}\leq\Gamma_{n}-\Gamma_{n+1}$
and so
$\displaystyle\sigma\sum_{j=1}^{n}\|x_{j+1}-x_{j}\|^{2}$ $\displaystyle\leq$
$\displaystyle\Gamma_{1}-\Gamma_{n+1}$ $\displaystyle\leq$
$\displaystyle\Gamma_{1}+\alpha^{n+1}\|x_{0}-x^{*}\|^{2}+\frac{\alpha\Gamma_{1}}{1-\alpha}$
$\displaystyle\leq$
$\displaystyle\alpha^{n+1}\|x_{0}-x^{*}\|^{2}+\frac{\Gamma_{1}}{1-\alpha}$
$\displaystyle\leq$
$\displaystyle\|x_{0}-x^{*}\|^{2}+\frac{\Gamma_{1}}{1-\alpha}.$
Therefore, since $x_{0}=x_{1}$, we get
$\displaystyle\sum_{k=1}^{\infty}\|x_{n+1}-x_{n}\|^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{\sigma}\Big{(}\|x_{0}-x^{*}\|^{2}+\frac{\Gamma_{1}}{1-\alpha}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{\sigma}\|x_{0}-x^{*}\|^{2}+\frac{1-\alpha_{1}}{1-\alpha}\|x_{0}-x^{*}\|^{2}$
$\displaystyle=$
$\displaystyle\Big{(}\frac{1}{1-3\alpha}+\frac{1-\alpha_{1}}{1-\alpha}\Big{)}\|x_{0}-x^{*}\|^{2}<\infty.$
Observe that
$\displaystyle\|x_{n+1}-w_{n}\|$ $\displaystyle=$
$\displaystyle\|x_{n+1}-x_{n}-\alpha_{n}(x_{n}-x_{n-1})\|$ (32)
$\displaystyle\leq$
$\displaystyle\|x_{n+1}-x_{n}\|+\alpha_{n}\|x_{n}-x_{n-1}\|$
$\displaystyle\leq$ $\displaystyle\|x_{n+1}-x_{n}\|+\alpha\|x_{n}-x_{n-1}\|.$
Using (32), we obtain
$\lim_{n\rightarrow\infty}\|x_{n+1}-w_{n}\|=0.$ (33)
∎
###### Lemma 4.2.
Let $\\{x_{n}\\}$ generated by Algorithm 1 above and Assumptions 3.1 and 3.2
hold. Then
* (a)
$\displaystyle\lim_{n\rightarrow\infty}\eta_{n}\|w_{n}-z_{n}\|^{2}=0$;
* (b)
$\displaystyle\lim_{n\rightarrow\infty}\|w_{n}-z_{n}\|=0.$
###### Proof.
Let $x^{*}\in\text{SOL}$. Since $F$ is uniformly continuous on bounded subsets
of $X$, then $\\{F(x_{n})\\},\\{z_{n}\\},\\{w_{n}\\}$ and $\\{F(y_{n})\\}$ are
bounded. In particular, there exists $M>0$ such that $\|F(y_{n})\|\leq M$ for
all $n\in\mathbb{N}$. Combining Lemma 2.5 and Lemma 3.4, we get
$\displaystyle\|x_{n+1}-x^{*}\|^{2}$ $\displaystyle=$
$\displaystyle\|P_{C_{n}}(w_{n})-x^{*}\|^{2}\leq\|w_{n}-x^{*}\|^{2}-\|x_{n+1}-w_{n}\|^{2}$
(34) $\displaystyle=$ $\displaystyle\|w_{n}-x^{*}\|^{2}-{\rm
dist}^{2}(w_{n},C_{n})$ $\displaystyle\leq$
$\displaystyle\|w_{n}-x^{*}\|^{2}-\Big{(}\frac{1}{M}h_{n}(w_{n})\Big{)}^{2}$
$\displaystyle\leq$
$\displaystyle\|w_{n}-x^{*}\|^{2}-\Big{(}\frac{1}{2M}\sigma\eta_{n}\|r(w_{n})\|^{2}\Big{)}^{2}$
$\displaystyle=$
$\displaystyle\|w_{n}-x^{*}\|^{2}-\Big{(}\frac{1}{2M}\sigma\eta_{n}\|w_{n}-z_{n}\|^{2}\Big{)}^{2}.$
Since $\\{x_{n}\\}$ is bounded, we obtain from (34) that
$\displaystyle\Big{(}\frac{1}{2M}\sigma\eta_{n}\|w_{n}-z_{n}\|^{2}\Big{)}^{2}$
$\displaystyle\leq$ $\displaystyle\|w_{n}-x^{*}\|^{2}-\|x_{n+1}-x^{*}\|^{2}$
(35) $\displaystyle=$
$\displaystyle\Big{(}\|w_{n}-x^{*}\|-\|x_{n+1}-x^{*}\|\Big{)}\Big{(}\|w_{n}-x^{*}\|+\|x_{n+1}-x^{*}\|\Big{)}$
$\displaystyle\leq$ $\displaystyle\|w_{n}-x^{*}\|-\|x_{n+1}-x^{*}\|M_{1}$
$\displaystyle\leq$ $\displaystyle\|w_{n}-x_{n+1}\|M_{1},$
where $M_{1}:=\sup_{n\geq 1}\\{\|w_{n}-x^{*}\|+\|x_{n+1}-x^{*}\|\\}$. This
establishes (a).
To establish (b), We distinguish two cases depending on the behaviour of (the
bounded) sequence of step-sizes $\\{\eta_{n}\\}$.
Case 1: Suppose that $\liminf_{n\to\infty}\eta_{n}>0$. Then
$0\leq\|r(w_{n})\|^{2}=\frac{\eta_{n}\|r(w_{n})\|^{2}}{\eta_{n}}$
and this implies that
$\displaystyle\limsup_{n\to\infty}\|r(w_{n})\|^{2}$
$\displaystyle\leq\limsup_{n\to\infty}\bigg{(}\eta_{n}\|r(w_{n})\|^{2}\bigg{)}\bigg{(}\limsup_{n\to\infty}\frac{1}{\eta_{n}}\bigg{)}$
$\displaystyle=\bigg{(}\limsup_{n\to\infty}\eta_{n}\|r(w_{n})\|^{2}\bigg{)}\frac{1}{\liminf_{n\to\infty}\eta_{n}}$
$\displaystyle=0.$
Hence, $\limsup_{n\to\infty}\|r(w_{n})\|=0$. Therefore,
$\lim_{n\rightarrow\infty}\|w_{n}-z_{n}\|=\lim_{n\rightarrow\infty}\|r(w_{n})\|=0.$
Case 2: Suppose that $\liminf_{n\to\infty}\eta_{n}=0$. Subsequencing if
necessary, we may assume without loss of generality that
$\lim_{n\to\infty}\eta_{n}=0$ and
$\lim_{n\rightarrow\infty}\|w_{n}-z_{n}\|=a\geq 0$.
Define
$\bar{y}_{n}:=\frac{1}{\gamma}\eta_{n}z_{n}+\Big{(}1-\frac{1}{\gamma}\eta_{n}\Big{)}w_{n}$
or, equivalently, $\bar{y}_{n}-w_{n}=\frac{1}{\gamma}\eta_{n}(z_{n}-w_{n})$.
Since $\\{z_{n}-w_{n}\\}$ is bounded and since $\lim_{n\to\infty}\eta_{n}=0$
holds, it follows that
$\lim_{n\to\infty}\|\bar{y}_{n}-w_{n}\|=0.$ (36)
From the step-size rule and the definition of $\bar{y}_{k}$, we have
$\langle
F(\bar{y}_{n}),w_{n}-z_{n}\rangle<\frac{\sigma}{2}\|w_{n}-z_{n}\|^{2},\
\forall n\in\mathbb{N},$
or equivalently
$2\langle F(w_{n}),w_{n}-z_{n}\rangle+2\langle
F(\bar{y}_{n})-F(w_{n}),w_{n}-z_{n}\rangle<\sigma\|w_{n}-z_{n}\|^{2},\ \forall
n\in\mathbb{N}.$
Setting $t_{n}:=w_{n}-F(w_{n})$, we obtain form the last inequality that
$2\langle w_{n}-t_{n},w_{n}-z_{n}\rangle+2\langle
F(\bar{y}_{n})-F(w_{n}),w_{n}-z_{n}\rangle<\sigma\|w_{n}-z_{n}\|^{2},\ \forall
n\in\mathbb{N}.$
Using Lemma 2.2 (b) we get
$2\langle
w_{n}-t_{n},w_{n}-z_{n}\rangle=\|w_{n}-z_{n}\|^{2}+\|w_{n}-t_{n}\|^{2}-\|z_{n}-t_{n}\|^{2}.$
Therefore,
$\|w_{n}-t_{n}\|^{2}-\|z_{n}-t_{n}\|^{2}<(\sigma-1)\|w_{n}-z_{n}\|^{2}-2\langle
F(\bar{y}_{n})-F(w_{n}),w_{n}-z_{n}\rangle\ \forall n\in\mathbb{N}.$
Since $F$ is uniformly continuous on bounded subsets of $H$ and (36), if $a>0$
then the right hand side of the last inequality converges to $(\sigma-1)a<0$
as $n\to\infty$. From the last inequality we have
$\limsup_{n\to\infty}\left(\|w_{n}-t_{n}\|^{2}-\|z_{n}-t_{n}\|^{2}\right)\leq(\sigma-1)a<0.$
For $\epsilon=-(\sigma-1)a/2>0$, there exists $N\in\mathbb{N}$ such that
$\|w_{n}-t_{n}\|^{2}-\|z_{n}-t_{n}\|^{2}\leq(\sigma-1)a+\epsilon=(\sigma-1)a/2<0\quad\forall
n\in\mathbb{N},n\geq N,$
leading to
$\|w_{n}-t_{n}\|<\|z_{n}-t_{n}\|\quad\forall n\in\mathbb{N},n\geq N,$
which is a contradiction to the definition of $z_{n}=P_{C}(w_{n}-F(w_{n}))$.
Hence $a=0$, which completes the proof. ∎
###### Lemma 4.3.
Let Assumptions 3.1 and 3.2 hold. Furthermore let $\\{x_{n_{k}}\\}$ be a
subsequence of $\\{x_{n}\\}$ converging weakly to a limit point $p$. Then
$p\in\text{SOL}$.
###### Proof.
By the definition of $z_{n_{k}}$ together with (13), we have
$\langle w_{n_{k}}-F(w_{n_{k}})-z_{n_{k}},x-z_{n_{k}}\rangle\leq 0,\ \forall
x\in C,$
which implies that
$\langle w_{n_{k}}-z_{n_{k}},x-z_{n_{k}}\rangle\leq\langle
F(w_{n_{k}}),x-z_{n_{k}}\rangle,\ \forall x\in C.$
Hence,
$\langle w_{n_{k}}-z_{n_{k}},x-z_{n_{k}}\rangle+\langle
F(w_{n_{k}}),z_{n_{k}}-w_{n_{k}}\rangle\leq\langle
F(w_{n_{k}}),x-w_{n_{k}}\rangle,\ \forall x\in C.$ (37)
Fix $x\in C$ and let $k\rightarrow\infty$ in (37). Since
$\lim_{k\to\infty}\|w_{n_{k}}-z_{n_{k}}\|=0$, we have
$0\leq\liminf_{k\to\infty}\langle F(w_{n_{k}}),x-w_{n_{k}}\rangle$ (38)
for all $x\in C$. It follows from (37) and the monotonicity of $F$ that
$\displaystyle\langle w_{n_{k}}-z_{n_{k}},x-z_{n_{k}}\rangle+\langle
F(w_{n_{k}}),z_{n_{k}}-w_{n_{k}}\rangle$ $\displaystyle\leq$
$\displaystyle\langle F(w_{n_{k}}),x-w_{n_{k}}\rangle$ $\displaystyle\leq$
$\displaystyle\langle F(x),x-w_{n_{k}}\rangle\quad\forall x\in C.$
Letting $k\to+\infty$ in the last inequality, remembering that
$\lim_{k\to\infty}\|w_{n_{k}}-z_{n_{k}}\|=0$ for all $k$, we have
$\langle F(x),x-p\rangle\geq 0\quad\forall x\in C.$
In view of Lemma 2.8, this implies $p\in\text{SOL}$. ∎
###### Theorem 4.4.
Let Assumptions 3.1 and 3.2 hold. Then the sequence $\\{x_{n}\\}$ generated by
Algorithm 1 weakly converges to a point in SOL.
###### Proof.
We have shown that
(i) $\lim_{n\to\infty}\|x_{n}-x^{*}\|$ exists;
(ii) $\omega_{w}(x_{n})\subset\text{SOL}$, where
$\omega_{w}(x_{n}):=\\{x:\exists x_{n_{j}}\rightharpoonup x\\}$ denotes the
weak $\omega$-limit set of $\\{x_{n}\\}$.
Then, by Lemma 2.4, we have that $\\{x_{n}\\}$ converges weakly to a point in
SOL. ∎
###### Remark 4.5.
(a) One can still obtain weak convergence for Algorithm 1 when $C$ is a
nonempty, closed and convex subset of $H$.
(b) In finite-dimensional spaces, Theorem 4.4 holds when $F$ is monotone and
continuous.
(c) Lemmas 3.5, 4.1, 4.2 and Theorem 4.4 can be obtained when $F$ pseudo-
monotone and weakly sequentially continuous (i.e., for all $x,y\in H$,
$\langle F(x),y-x\rangle\geq 0\Longrightarrow\langle F(y),y-x\rangle\geq 0;$).
The reader can see, for example, [49]. $\Diamond$
###### Remark 4.6.
Our proposed method in this paper gives weak convergence results in infinite
dimensional Hilbert space. There exists strong convergence methods in the
literature for solving variational inequality problem in infinite dimensional
Hilbert space (see, for example, [16, 18, 32, 39, 42, 44, 45, 52]). These
methods use ideas of viscosity terms, Halpern iterations and hybrid methods.
It has been shown numerically in [32] that viscosity and Halpern-type strongly
convergent methods outperform those of hybrid methods. Nonetheless, proposed
viscosity and Halpern-type strongly convergent methods involve the iterative
parameter that is both diminishing and non-summable. These conditions on the
iterative parameters make the viscosity and Halpern-type strongly convergent
methods to be slower than our proposed method in this paper in terms number of
iterations and CPU time.
## 5 Numerical Experiments
In this section, we discuss the numerical behaviour of Algorithm 1 using
different test examples taken from the literature which are describe below and
compare our method with (5), (10) and Shehu and Iyiola algorithm 3.2 in [51].
###### Example 5.1.
Equilibrium-optimization Model
In this example, we consider an equilibrium-optimization model (see, for
example, [50]) which can be regarded as an extension of a Nash-Cournot
oligopolistic equilibrium model in electricity markets.
In this equilibrium model, we assume that there are $m$ companies, each
company $i$ may possess $I_{i}$ generating units. Suppose we denote by $x$,
the vector whose entry $x_{j}$ stands for the power generating by unit $j$.
Suppose the price $p_{i}(s)$ is a decreasing affine function of $s$ where
$s:=\sum_{j=1}^{N}x_{j}$ where $N$ is the number of all generating units.
Thus, $p_{i}(s):=\alpha-\beta_{i}s$. Then the profit made by company $i$ is
given by $f_{i}(x):=p_{i}(s)\sum_{j\in I_{i}}x_{j}-\sum_{j\in
I_{i}}c_{j}(x_{j})$, where $c_{j}(x_{j})$ is the cost for generating $x_{j}$
by generating unit $j$ . Let us assume that $K_{i}$ is the strategy set of
company $i$, which implies that $\sum_{j\in I_{i}}x_{j}\in K_{i}$ for each
$i$. Then the strategy set of the model is $C:=K_{1}\times
K_{2}\times\ldots\times K_{m}$.
A commonly used approach when each company wants to maximize its profit by
choosing the corresponding production level under the presumption that the
production of the other companies are parametric input is the Nash equilibrium
concept.
We recall that a point $x^{*}\in C=K_{1}\times K_{2}\times\ldots\times K_{m}$
is an equilibrium point if
$f_{i}(x^{*})\geq f_{i}(x^{*}[x_{i}])\forall x_{i}\in
K_{i},\leavevmode\nobreak\ \leavevmode\nobreak\ i=1,2,\ldots,m,$
where the vector $x^{*}[x_{i}]$ stands for the vector obtained from $x^{*}$ by
replacing $x^{*}_{i}$ with $x_{i}$. Define
$f(x,y):=\psi(x,y)-\psi(x,x)$
with
$\psi(x,y):=-\sum_{i=1}^{n}f_{i}(x^{*}[y_{i}]).$
Then the problem of finding a Nash equilibrium point of our model can be
formulated as
$\displaystyle X^{*}\in C:f(x^{*},x)\geq 0\leavevmode\nobreak\
\leavevmode\nobreak\ \forall x\in C.$ (39)
Suppose for every $j$, the cost $c_{j}$ for production and the environmental
fee $g$ are increasingly convex functions. The convexity assumption here means
that both the cost and fee for producing a unit production increases as the
quantity of the production gets larger. Under this convexity assumption, it is
not hard to see that (39) is equivalent to (see, [58])
$\displaystyle x\in C:\langle Bx-a+\nabla\varphi(x),y-x\rangle\geq
0\leavevmode\nobreak\ \leavevmode\nobreak\ \forall y\in C,$ (40)
where
$\displaystyle a:=(\alpha,\alpha,\ldots,\alpha)^{T}$ $\displaystyle
B_{1}=\left(\begin{array}[]{ccccc}\beta_{1}&0&0&\ldots&0\\\
0&\beta_{2}&0&\ldots&0\\\ \ldots&\ldots&\ldots&\ldots&\ldots\\\
0&0&0&0&\beta_{m}\\\
\end{array}\right)B=\left(\begin{array}[]{ccccc}0&\beta_{1}&\beta_{1}&\ldots&\beta_{1}\\\
\beta_{2}&0&\beta_{2}&\ldots&\beta_{2}\\\
\ldots&\ldots&\ldots&\ldots&\ldots\\\
\beta_{m}&\beta_{m}&\beta_{m}&\ldots&\beta_{m}\\\ \end{array}\right)$
$\displaystyle\varphi(x):=x^{T}B_{1}x+\sum_{j=1}^{N}c_{j}(x_{j}).$
Note that when $c_{j}$ is differentiable convex for every $j$.
We tested the proposed algorithm with the cost function given by
$c_{j}(x_{j})=\frac{1}{2}x_{j}^{T}Dx_{j}+d^{T}x_{j}.$
The parameters $\beta_{j}$ for all $j=1,\ldots,m$, matrix $D$ and vector $d$
were generated randomly in the interval $(0,1]$, $[1,40]$ and $[1,40]$
respectively.
We perform numerical implementations using different choices of $10$, and
$20$, different initial choices $x_{1}$ generated randomly in the interval
$[1,40]$ and $m=10$ with the stopping criterion as $\|x_{n+1}-x_{n}\|\leq
10^{-2}$. Let us assume that each company have the same lower production bound
1 and upper production bound 40, that is,
$K_{i}:=\\{x_{i}:1\leq x_{i}\leq 40\\},\leavevmode\nobreak\
\leavevmode\nobreak\ i=1,\ldots,10.$
We compare our proposed Algorithm 1 with algorithm 3.2 proposed by Shehu and
Iyiola in [51].
Table 1: Example 5.1 Comparison: Proposed Alg. 1 and Shehu & Iyiola Alg. 3.2 (SI Alg.) for $\sigma=0.5$ | N=10 | | N=20
---|---|---|---
| No. of Iter. | CPU time ($10^{-2}$) | | No. of Iter. | CPU time ($10^{-2}$)
$\gamma$ | Alg. 1 | SI Alg. | Alg. 1 | SI Alg. | | Alg. 1 | SI Alg. | Alg. 1 | SI Alg.
$0.01$ | 223 | 435 | $2.6822$ | $8.8916$ | | 228 | 520 | $7.9536$ | 22.352
$0.1$ | 38 | 518 | $1.4433$ | 13.108 | | 36 | 473 | $1.0797$ | 14.871
$0.5$ | 10 | 434 | $0.8232$ | $7.9301$ | | 9 | 285 | $0.4196$ | $9.6452$
$0.8$ | 9 | 514 | 12.3590 | 13.1390 | | 8 | 320 | $0.4596$ | $8.3524$
Figure 1: Example 5.1: $\gamma=0.01$, $N=10$
Figure 2: Example 5.1: $\gamma=0.1$, $N=10$
Figure 3: Example 5.1: $\gamma=0.5$, $N=10$
Figure 4: Example 5.1: $\gamma=0.7$, $N=10$
Figure 5: Example 5.1: $\gamma=0.01$, $N=20$
Figure 6: Example 5.1: $\gamma=0.1$, $N=20$
Figure 7: Example 5.1: $\gamma=0.5$, $N=20$
Figure 8: Example 5.1: $\gamma=0.7$, $N=20$
###### Example 5.2.
This example is taken from [27] and has been considered by many authors for
numerical experiments (see, for example, [29, 42, 53]). The operator $A$ is
defined by $A(x):=Mx+q$, where $M=BB^{T}+S+D$, with
$B,S,D\in\mathbb{R}^{m\times m}$ randomly generated matrices such that $S$ is
skew-symmetric (hence the operator does not arise from an optimization
problem), $D$ is a positive definite diagonal matrix (hence the variational
inequality has a unique solution) and $q=0$. The feasible set $C$ is described
by linear inequality constraints $Bx\leq b$ for some random matrix
$B\in\mathbb{R}^{k\times m}$ and a random vector $b\in\mathbb{R}^{k}$ with
nonnegative entries. Hence the zero vector is feasible and therefore the
unique solution of the corresponding variational inequality. These projections
are computed using the MATLAB solver fmincon. Hence, for this class of
problems, the evaluation of $A$ is relatively inexpensive, whereas projections
are costly. We present the corresponding numerical results (number of
iterations and CPU times in seconds) using six different dimensions $m$ and
two different numbers of inequality constraints $k$.
We choose the stopping criterion as $\|x^{k}\|\leq\epsilon=0.001.$ The size
$k=30,50$ and $m=10,20,30,40,50,60$. The matrices $B,S,D$ and the vector $b$
are generated randomly. We choose $\gamma=0.8$, $\sigma=0.5$, $\alpha_{n}=0.2$
in Algorithm (1). In (5), we choose $\sigma=0.8$, $\rho=0.1$, $\mu=0.2$. In
(10), we choose $L=\|M\|$. Here, we compare our proposed Algorithm 1 with the
subgradient extragradient method (SEM) (5), and the inertial subgradient
extragradient method (Thong & Hieu) (10).
Table 2: Example 5.2 Comparison: Proposed Alg. 1 vs SEM (2) vs Thong & Hieu (3) (T & H (3)) | | No. of Iterations | | CPU time | | Norm sol. ($10^{-3}$)
---|---|---|---|---|---|---
| $m$ | Alg. 1 | SEM (2) | T & H (3) | | Alg. 1 | SEM (2) | T & H (3) | | Alg. 1 | SEM (2) | T & H (3)
$k=30$ | 10 | 344 | 3867 | 4123 | | 2.8707 | 31.5956 | 30.3300 | | 0.99605 | 0.99927 | 0.99938
| 20 | 747 | 14683 | 10493 | | 5.6957 | 117.7458 | 84.7406 | | 0.99608 | 0.99996 | 0.99984
| 30 | 1777 | 31668 | 24968 | | 13.891 | 269.2987 | 211.1937 | | 0.99955 | 0.99999 | 0.99980
| 40 | 2612 | 40224 | 36119 | | 21.5972 | 358.3453 | 320.2933 | | 0.99790 | 1.00000 | 0.99994
| 50 | 3710 | 70321 | 51143 | | 32.074 | 655.8354 | 469.0297 | | 0.99981 | 0.99997 | 0.99995
| 60 | 5619 | 56670 | 50619 | | 50.4537 | 554.3951 | 491.5552 | | 0.99929 | 0.99992 | 0.99998
$k=50$ | 10 | 200 | 6213 | 5518 | | 1.90869 | 60.1212 | 47.6937 | | 0.98471 | 0.99969 | 0.99945
| 20 | 835 | 14354 | 10372 | | 6.4942 | 126.429 | 96.9197 | | 0.99909 | 0.99980 | 0.99991
| 30 | 1978 | 25519 | 19357 | | 16.4674 | 240.61 | 208.910 | | 0.99794 | 0.99991 | 0.99990
| 40 | 2832 | 47661 | 26790 | | 30.3799 | 539.729 | 314.588 | | 0.99734 | 0.99991 | 0.99938
| 50 | 3933 | 43773 | 53055 | | 45.8745 | 562.959 | 800.371 | | 0.99985 | 0.99925 | 0.99999
| 60 | 6025 | 97772 | 65820 | | 100.304 | 1515.76 | 589.180 | | 0.99955 | 0.99995 | 0.99994
Figure 9: Example 5.2: $k=30$, $m=10$
Figure 10: Example 5.2: $k=30$, $m=20$
Figure 11: Example 5.2: $k=30$, $m=30$
Figure 12: Example 5.2: $k=30$, $m=40$
Figure 13: Example 5.2: $k=30$, $m=50$
Figure 14: Example 5.2: $k=30$, $m=60$
Figure 15: Example 5.2: $k=50$, $m=10$
Figure 16: Example 5.2: $k=50$, $m=20$
Figure 17: Example 5.2: $k=50$, $m=30$
Figure 18: Example 5.2: $k=50$, $m=50$
Figure 19: Example 5.2: $k=50$, $m=60$
Clearly, from both Examples, our proposed algorithm 1 outperforms and highly
improves Shehu and Iyiola Algorithm (3.2) in [51], subgradient extragradient
method (SEM) (5), and the inertial subgradient extragradient method (Thong &
Hieu) (10) with respect to number of iterations required and CPU time and
achieved norm of the solution. See Tables 1 \- 2 and Figures 2 \- 19.
We give an example in infinite dimensional Hilbert spaces. We give comparison
of our proposed Algorithm 1 with Algorithm (5), Algorithm (10) and the non-
inertial case of Algorithm 1 (when $\theta_{n}=0$).
###### Example 5.3.
Let $H:=L^{2}([0,1])$ with norm
$\|x\|:=\Big{(}\int_{0}^{1}x(t)^{2}dt\Big{)}^{\frac{1}{2}}$ and inner product
$\langle x,y\rangle:=\int_{0}^{1}x(t)y(t)dt,\leavevmode\nobreak\
\leavevmode\nobreak\ x,y\in H$. Let $C:=\\{x\in
L^{2}([0,1]):\int_{0}^{1}tx(t)dt=2\\}$. Let us define the Volterra integral
operator $F:L^{2}([0,1])\rightarrow L^{2}([0,1])$ by
$Fx(t):=\int_{0}^{t}x(s)ds,\leavevmode\nobreak\ \leavevmode\nobreak\ x\in
L^{2}([0,1]),t\in[0,1]$. Then, $F$ is monotone, bounded and linear with
$L=\frac{2}{\pi}$ (see Exercises 20.12 of [9]) . Observe that
$\text{SOL}\neq\emptyset$ since $0\in\text{SOL}$. Observe that (see [15])
$P_{C}(x)(t):=x(t)-\frac{\int_{0}^{1}tx(t)dt-2}{\int_{0}^{1}t^{2}dt}t,\leavevmode\nobreak\
\leavevmode\nobreak\ t\in[0,1].$
## 6 Final Remarks
We propose an inertial projection method for solving variational inequality
problem and give weak convergence result. The cost function is assumed to be
monotone and non-Lipschitz continuous. Our numerical implementations show that
our method is more efficient and outperforms some other related methods in the
literature. Our result is more applicable than the results on variational
inequality where the Lipschitz constant of the cost function is needed. Our
future project is focused on how to extend the range of inertial factor
$\alpha_{n}$ beyond $1/3$ and extend our results to infinite dimensional
Banach spaces.
## Acknowledgments
The authors are grateful to the anonymous referee and editor whose insightful
comments and suggestions improve the earlier version of this paper.
## Disclosure statement
No potential conflict of interest was reported by the author(s).
## Funding
The project of the first author has received funding from the European
Research Council (ERC) under the European Union’s Seventh Framework Program
(FP7 - 2007-2013) (Grant agreement No. 616160).
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|
Electric dipole moment of the tau lepton revisited
Werner Bernreuther<EMAIL_ADDRESS>Long Chen<EMAIL_ADDRESS>and Otto Nachtmann<EMAIL_ADDRESS>
$^a$Institut für Theoretische Teilchenphysik und Kosmologie,
RWTH Aachen University, 52056 Aachen, Germany
$^b$ Institut für Theoretische Physik, Universität Heidelberg, 69120 Heidelberg, Germany
We reconsider the issue of the search for a nonzero electric dipole form factor (EDM) $d_\tau(s)$ using optimal observables
in $\tau^+\tau^-$ production
by $e^+ e^-$ collisions in the center-of-mass energy range from the $\tau$-pair threshold to about $\sqrt{s} \sim 15$ GeV.
We discuss the general formalism of optimal observables and apply it
to two $CP$-odd observables that are sensitive to the real and imaginary part of $d_\tau(s)$, respectively.
We compute the expectation values and covariances of these optimal $CP$ observables for
$\tau$-pair production at $\sqrt{s}=10.58$ GeV
with subsequent decays of $\tau^\pm$ into major leptonic
or semihadronic modes. For the $\tau$ decays to two pions and three charged pions we take the full kinematic information of the hadronic system into account.
Assuming that the Belle II experiment at the KEKB accelerator will eventually
analyze data corresponding to an integrated luminosity of 50 ab$^{-1}$
and applying acceptance cuts on the final-state pions we find that
1 s.d. sensitivities $\delta \Re d_\tau = 6.8 \times 10^{-20} \ecm$ and $\delta \Im d_\tau = 4.0 \times 10^{-20} \ecm$ can be obtained
with events where both $\tau$'s decay semihadronically.
We consider also the ideal case that no cuts on the final-state particles are applied. With 50 ab$^{-1}$ at $\sqrt{s}=10.58$ GeV corresponding to
$4.5 \times 10^{10}$ $\tau^+ \tau^-$ events
we find the 1 s.d. sensitivities $\delta \Re d_\tau = 5.8 \times 10^{-20} \ecm$ and $\delta \Im d_\tau = 3.2 \times 10^{-20} \ecm$, again for events
where both $\tau$ leptons decay semihadronically.
Furthermore, we analyze the potential magnitude of the $\tau$ EDM form factor
in the type-II two-Higgs doublet extension and in two scalar leptoquark extensions of the Standard Model,
taking into account phenomenological constraints.
§ INTRODUCTION
The search for electric dipole moments (EDMs) of fundamental fermions is an important
aspect of experimental investigations hunting for physics beyond the Standard Model (SM)
of particle physics, in particular for $CP$ violation beyond the Kobayashi-Maskawa mechanism.
So far only upper bounds for these EDMs exist [1]. For the electron an impressive
upper limit was obtained rather recently by the ACME Collaboration [2]. The best muon EDM limit to date
was set by the Muon $(g-2)$ Collaboration [3]. These limits are
\begin{align}\label{Eq.01.01}
|d_{e}| &< 1.1 \times 10^{-29} \, \ecm \, \text{
at }\, 90\% \, {\rm C.L.} \, , \\ \label{Eq.01.02}
|d_{\mu}|& < 1.8\phantom{1}\times 10^{-19}\, \ecm \, \text{
at }\, 95\% \, {\rm C.L.} \, .
\end{align}
The lifetime of the $\tau$ lepton is too short to allow for the measurement of its
static moments. Instead information on the nonstatic $\tau$ EDM form factor[In this paper we use the acronym EDM for both the
static moment and the form factor at $q^2\neq 0$.] can be retrieved, for
instance, from the measurement
of $CP$-violating correlations in $\tau$-pair production by $e^+ e^-$ collisions.
The $\tau$ EDM form factor can be a complex quantity
for timelike momentum transfer. The best limits to date on its real and imaginary parts were obtained
by the Belle I Collaboration [4] at $q^{2}=(10.58~{\rm GeV})^{2}$:
\begin{align} \label{Eq.01.03}
-2.2 \times 10^{-17} \, {\ecm} &<\Re~d_{\tau}(q^{2})<4.5\times10^{-17} {\ecm} \, \text{
at }\, 95\% \,{\rm C.L.} \, , \notag \\
-2.5 \times 10^{-17} \, {\ecm} &<\Im~d_{\tau}(q^{2})<0.8\times10^{-17} {\ecm} \, \text{
at }\, 95\% \, {\rm C.L.} \,.
\end{align}
In a series of articles where two of the authors of this paper were involved,
ways of searching for $CP$-violating effects in $e^{+}e^{-}$ collisions, in particular for a nonzero $\tau$ EDM,
were proposed [5, 6, 7, 8, 9].
The observables and results of [6, 9] were used
in the experimental searches for an EDM form factor of the $\tau$ lepton by [4]
and earlier by the ARGUS Collaboration [10] that obtained the results
\begin{align}\label{Eq.01.05}
|\Re~d_{\tau}(q^{2})|&<4.6\times10^{-16} {\ecm} \, \text{
at }\, 95\% \,{\rm C.L.}\, , \notag \\
|\Im~d_{\tau}(q^{2})|&<1.8\times10^{-16} {\ecm} \, \text{
at }\, 95\% \,{\rm C.L.}
\end{align}
at a c.m. energy $\sqrt{s}=\sqrt{q^{2}}=10~{\rm GeV}$ of the reaction $e^+e^- \to \tau^+\tau^-$.
For reviews of the search results for the $\tau$ EDM and its weak dipole
form factor (the analogue of the EDM for the coupling of the Z boson to fermions);
see, for instance, [11, 12].
Further discussions of possible measurements of the anomalous magnetic
moment and the EDM of the $\tau$ lepton can be found in [13, 14, 15, 16, 17]
and references therein.
The experimentation at Belle II [18] which started recently at the KEKB accelerator
offers new possibilities for
measuring the $\tau$ EDM form factor, in particular, because a huge number of
recorded $\tau$-pair events are expected at the end of data taking [19].
Also the BES III experiment, where $e^+e^-$ collisions at a center-of-mass (c.m.) energy $\sqrt{s} \sim 4$ GeV are studied,
expects to collect and analyze a large number of $\tau^+\tau^-$ pairs [20].
Therefore, we reconsider the issue with particular emphasis on using optimal
observables [21, 22, 23] for tracing the $\tau$ EDM form factor
in $\tau$-pair production at c.m. energies from threshold up to about 15 GeV where the contribution from $Z$-boson exchange
is negligible.
In our numerical analysis we consider $\tau$-pair production at $\sqrt{s}=10.58$ GeV.
Moreover, we analyze this form factor in a few SM extensions
that can induce a potentially sizable $\tau$ EDM [24].
Our paper is organized as follows. In Section <ref> we recall the form factor
decomposition of the $\gamma\tau\tau$ vertex and in particular the definition of the $\tau$ EDM
form factor.
In section <ref> we discuss the production and decay matrices for the
process $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ with the $\tau$'s decaying into
one, two, or three particles that are measured in an experiment. Section <ref> deals with simple and
optimal observables [21, 22, 23]
for tracing the EDM of the $\tau$ lepton.
Section <ref> contains our numerical results, in particular
our estimates of the sensitivities with which the real and the imaginary parts of
the $\tau$ EDM form factor can be measured in various $\tau$ decay channels.
In Section <ref> we consider the $\tau$ EDM form factor in a type-II two-Higgs doublet extension and in two
leptoquark extensions of the SM and analyze the potential magnitude of the $\tau$ EDM taking into account
experimental constraints.
Moreover, we show that within these models $CP$-violating box contributions to the $S$-matrix element of $e^+e^-\to \tau^+ \tau^-$
are negligible as compared to that of the $\tau$ EDM form factor.
We conclude in Section <ref>. In Appendix <ref>
we list the density matrices for several major decays of polarized $\tau^\pm$ leptons. In particular, we present the explicit
form of the differential decay density matrices for $\tau \to 2 \pi \nu_\tau$ and $\tau \to 3 \pi \nu_\tau$.
Appendix <ref> contains a detailed analysis of the expectation values and covariances of the $CP$-odd optimal observables
used in Sec. <ref> in various $\tau^+\tau^-$ decay channels.
§ FORM FACTORS
We consider $\tau^+\tau^-$ production in $e^+ e^-$ collisions at c.m. energies $\sqrt{s}$ from threshold up to about 15 GeV,
with $\tau^-$ and $\tau^+$ decaying into a final state $A$ and ${\overline B}$, respectively,
\begin{equation}\label{Eq.01.04}
e^{+}(p_{+}) + e^{-}(p_{-}) \rightarrow \tau^{+}(k_{+},\alpha) +\tau^{-}(k_{-},\beta) \rightarrow {\overline B} \; + \; A \, .
\end{equation}
The four-momenta and the corresponding three-momenta are denoted in the $e^+e^-$ c.m.
frame by $p_\pm =(p^0_\pm, {\pp}_\pm)^T$, $k_\pm =(k^0_\pm, {\kk}_\pm)^T$.
We consider unpolarized electrons and positrons and neglect their masses; the labels $\alpha,\beta\in \{\pm 1/2\}$
denote the spin indices of the tau leptons. In the c.m. frame we have ${\pp}_+ + {\pp}_- = {\kk}_+ + {\kk}_- = 0$.
For unpolarized $e^{+}$ and $e^{-}$ the initial state
is described by a $CP$-invariant density matrix.
Thus, any non zero $CP$-odd correlation observed in the final state
indicates a genuine $CP$-violating effect that
can be located in the production and/or in
the decays of the $\tau$'s. We consider tau-pair production by
one-photon-exchange only. At the energies considered here $Z$-boson exchange is negligible.
This will be justified at the end of this section.
The diagram shown in Fig. <ref> exhibits this approximation with the full photon
\begin{equation} \label{eq:fullphpr}
i \Delta_{\mu\nu}^{(\gamma)}(q) = \frac{- i g_{\mu\nu}}{q^2[ 1 + e^2 \Pi_c(q^2)]} \, ,
\end{equation}
where $\Pi_c(q^2)$ is the vacuum-polarization function; see e.g. Eq. (19.45) of [25].
For instance, at the mass of the $\Upsilon(4{\rm S})$ resonance, at $\sqrt{q^2}= 10.58$ GeV, this vacuum polarization effect
produces an enhancement of the cross section. For a detailed discussion of the $\tau$-pair cross section at this
energy, including radiative corrections, we refer to [26]. Below we consider only normalized
expectation values of $CP$ observables where such resonance enhancements enter only through the number of events which we
take as input from experiment.
The reaction (<ref>) in the one-photon-exchange approximation.
In the following we assume that the only source of $CP$ violation in the diagram of Fig. <ref> is due to
a nonzero EDM form factor in the $\gamma\tau\tau$ vertex.
This vertex is given by the following one-particle irreducible (1PI) matrix element
of the electromagnetic current $J_{\lambda}^{em}$ between the vacuum and the $\tau^+\tau^-$ final state:
\begin{eqnarray}\label{Eq.02.03}
\big{\langle}\tau^{-}(k_{-},\beta), \, \tau^{+}(k_{+},\alpha) \, {\rm out}|J_{\lambda}^{em}(0) |0 \big{\rangle} = & \nonumber \\
- \overline{u}_{\beta}(k_{-}) \Bigl[ eF_{1}(q^{2})\gamma_{\lambda}+\dfrac{i}{2m_{\tau}}\sigma_{\lambda\mu}q^{\mu} e F_{2}(q^{2})
+d_{\tau}(q^{2})\sigma_{\lambda\mu}q^{\mu}\gamma_{5} \Bigr. & \nonumber \\
\Bigl. +\dfrac{1}{8\pi}A(q^{2})(q^{2}\gamma_{\lambda}-2m_{\tau}q_{\lambda})\gamma_{5} \Bigr] {\v}_{\alpha}(k_{+})\;, &
\end{eqnarray}
where $q = k_+ + k_-$. The right-hand side of (<ref>) represents the most general decomposition of this matrix element taking
into account the conservation of the
electromagnetic current. Moreover, $e=\sqrt{4\pi\alpha_{em}}>0$ denotes the $\tau^{+}$ charge and
we use the $\gamma$-matrix conventions of [25].
Note that the order of $\tau^{-}$ and $\tau^{+}$ in the matrix element (<ref>) matters because we are dealing with fermions.
The form factors $F_{1,2}(q^2),$ $d_\tau(q^2)$, and $A(q^{2})$ are
analytic functions of $q^{2}$ in the complex $q^{2}$ plane with a cut on the positive real axis satisfying
\begin{align}\label{Eq.02.04}
F_{i}(q^{2*})^{*}&=F_{i}(q^{2})\, , \quad i=1,2 \,, \notag \\
d_{\tau}(q^{2*})^{*}&=d_{\tau}(q^{2})\,, \notag \\
\end{align}
That is, on the real $q^{2}$ axis, the form factors are real functions for $q^{2}<0$ and can have
imaginary parts for $q^{2}>0$. At higher order in $\alpha_{em}$ these cuts start at $q^{2}=0$ due to cut diagrams of the type
shown in Fig. <ref> with three photons in the intermediate state.
In the decomposition (<ref>) we have $q^{2}\geq 4m_{\tau}^{2}$
and we have to set $q^{2}+i\varepsilon$, that is, to take $q^{2}$ above the cut.
A cut diagram leading to an imaginary part of the form factors in (<ref>) for $q^{2}>0$.
Next we recall the transformation properties of the $\gamma\tau\tau$ coupling terms associated with the four form factors
in (<ref>) under charge conjugation ($C$), parity ($P$), and $CP$. Assuming that the interaction is invariant under these
transformations and using the transformation of $J_{\lambda}^{em}(x)$ under $C$, $P$, and $CP$, one
gets the transformation properties listed in Table <ref>.
The $e^+e^-\to \tau^+\tau^-$ amplitude can receive also $CP$-odd 1PI box contributions, for instance contributions with
Lorentz structure $({\bar e} e) ({\bar\tau} i\gamma_5\tau)$. We do not take such contributions into account in the following.
We discuss a few SM extensions in Section <ref> that can induce sizable $\tau$ EDM form factors.
For these models we show in Section <ref> that the $CP$-violating box contributions can be neglected
as compared to that of the induced $\tau$ EDM form factor.
Transformation properties of the $\gamma\tau\tau$ coupling terms corresponding to the four form factors
in the decomposition of the matrix element (<ref>) of the
electromagnetic current.
$C$ $P$ $CP$
$F_1(q^2)$ $+$ $+$ $+ $
$F_2(q^2)$ $+$ $+$ $+ $
$d_\tau(q^2)$ $+$ $-$ $-$
$A(q^2)$ $-$ $-$ $+$
For the matrix elements of the current between $\tau^{-}$ and $\tau^{+}$ states, respectively,
we get, using the standard crossing relations:
\begin{equation}\label{Eq.02.06}
\big{\langle}\tau^{-}(k',\beta') |J_{\lambda}^{em}(0) |\tau^{-}(k,\beta) \big{\rangle}
\end{equation}
\begin{equation}\label{Eq.02.07}
\big{\langle}\tau^{+}(k',\alpha') |J_{\lambda}^{em}(0) |\tau^{+}(k,\alpha) \big{\rangle}
\end{equation}
where the vertex function $\Gamma_{\lambda}(q)$ is given by the expression in the square brackets of Eq. (<ref>)
with $q=k'-k$ and $q^2\leq 0.$
The form factor $F_{1}(q^2)$ is the electric or Dirac form factor with the normalization
\begin{equation}\label{Eq. 02.10}
\end{equation}
The magnetic or Pauli form factor $F_{2}(q^2)$ at $q^2=0$ yields the $\tau$ anomalous magnetic moment:
\begin{equation}\label{Eq.02.11}
\end{equation}
The $\tau^{-}$ and $\tau^{+}$ electric dipole moments, respectively, are obtained from the EDM form factor
$d_{\tau}(q^2)$ at $q^2=0$:
\begin{equation}\label{Eq.02.12}
\end{equation}
The form factor $A(q^{2})$ at $q^2=0$ defines the anapole moment [27, 28, 29, 30]
of the $\tau^{-}$:
\begin{equation}\label{Eq.02.13}
A_{\tau^{-}}= A(0)\;.
\end{equation}
For a $\tau^-$ at rest, $k=k_{R}=(m_\tau,\boldsymbol{0})^T$, one has
\begin{equation}\label{Eq.02.14}
\big{\langle}\tau^{-}(k_R,\beta') |(-\pi)\int d^{3}x|\boldsymbol{x}|^2 \boldsymbol{J}^{em}(\boldsymbol{x} , 0)|\tau ^{-}(k_{R},\beta) \big{\rangle}
=\frac{1}{2}{\ssig}_{\beta '\beta} A_{\tau^-}\;.
\end{equation}
A comment on the gauge invariance of the form-factor decomposition of the vertex function
(<ref>) is in order. Electromagnetic gauge invariance is obvious, because conservation of the electromagnetic current
was used in the decomposition of (<ref>). As to the invariance with respect to the electroweak gauge group
${\rm SU(2)}\times{\rm U(1)}$: The static moments at $q^2=0$, in particular the anomalous magnetic and electric dipole moment
and the anapole moment are gauge invariant, as they correspond to terms in the $\tau\to \tau$
$S$-matrix element in the soft-photon limit. Yet, for obtaining a gauge-invariant amplitude for $e^+e^- \to \tau^+\tau^-$
one cannot, of course, use (<ref>) in isolation, but must take into account all contributions (including box
contributions at one-loop order and beyond) to the $S$-matrix element order by order in the electroweak couplings.
However, in the following we use only the tree-level $\gamma\tau\tau$ vertex supplemented by the $\tau$ EDM form factor.
The $\tau$ EDM is extremely small in the SM, as will be briefly reviewed at the beginning of Section <ref>.
Thus, a sizable value for $d_\tau$ must come from “beyond the Standard Model” (BSM) physics.
In Section <ref> we discuss a few BSM extensions that can induce a sizable $\tau$ EDM form factor and compute
it at one-loop order. The form factors $d_\tau(q^2)$ given in that section are invariant with respect to the electroweak
gauge group.
As is well-known one may introduce a $\tau$ EDM, together with an analogous $CP$-violating weak dipole moment (WDM)
$d_\tau^Z$ in the $Z\tau\tau$ vertex, by using a ${\rm SU(3)}\times{\rm SU(2)}\times{\rm U(1)}$ invariant effective
Lagrangian approach for BSM couplings. Imposing baryon and lepton number conservation the leading gauge-invariant operators
have mass dimension 6 [31] and the relevant effective Lagrangian takes the
form (see, for instance, [32]):
\begin{eqnarray}\label{Eq.Leffinv}
\mathcal{L}_{\textup{eff}}(x) = &-i \frac{c_1}{\Lambda^2}~{\bar\tau_R}(x)\sigma^{\mu\nu}
\phi^\dagger(x)\left[g'\frac{\tau^a}{2}W_{\mu\nu}^a(x) - \frac{g}{2}B_{\mu\nu}(x)\right] L_L(x) \nonumber \\
& -i \frac{c_2}{\Lambda^2}~{\bar\tau_R}(x)\sigma^{\mu\nu}
\phi^\dagger(x)\left[g\frac{\tau^a}{2}W_{\mu\nu}^a(x) + \frac{g'}{2}B_{\mu\nu}(x)\right] L_L(x) + {\rm H.c.} \, .
\end{eqnarray}
Here $c_1$ and $c_2$ are dimensionless real coupling constants, $\Lambda\gg {\rm v}_0$ denotes the energy scale of
new physics that is assumed to
be considerably larger than the electroweak symmetry breaking scale ${\rm v}_0=246$ GeV, $g$ and $g'$ are
the ${\rm SU(2)}$ and ${\rm U(1)}$
gauge couplings, respectively, $W_{\mu\nu}$ and $B_{\mu\nu}$ are the gauge field strength tensors corresponding to these groups,
$\phi$ is the Higgs doublet field, and $\tau_R$ and $L_L^T=(\nu_\tau, \tau)_L^T$ are the right-handed singlet
and left-handed lepton doublet fields
of the third generation. (Our notation follows [33].)
After spontaneous symmetry breaking the effective Lagrangian (<ref>) contains the EDM interactions
\begin{equation}\label{Eq.EWeff}
\mathcal{L}_{\textup{eff}}(x) \supset -\frac{i}{2}d_\tau~\overline{\tau}(x)\sigma^{\mu\nu}\gamma_{5}\tau(x)F_{\mu\nu}(x)
-\frac{i}{2}d^Z_\tau~\overline{\tau}(x)\sigma^{\mu\nu}\gamma_{5}\tau(x)Z_{\mu\nu}(x) \, ,
\end{equation}
where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and $Z_{\mu\nu}=\partial_{\mu}Z_{\nu}-\partial_{\nu}Z_{\mu}$
are, respectively, the Abelian field strength tensors of the photon and $Z$ boson and $d_\tau$ and $d^Z_\tau$ the electric and weak dipole moments
of the $\tau$ lepton:
\begin{equation}\label{Eq.dEWeff}
d_\tau = \frac{{\rm v}_0}{\Lambda^2}\frac{\sqrt{g^2+g'^2}}{\sqrt{2}} c_1 \, , \qquad
d^Z_\tau = \frac{{\rm v}_0}{\Lambda^2}\frac{\sqrt{g^2+g'^2}}{\sqrt{2}} c_2 \, .
\end{equation}
This approach constitutes a possibility to introduce the $\tau$ EDM and WDM in a way that respects electroweak
gauge invariance.
Using the Hermitian Lagrangian (<ref>)
to leading order corresponds to setting
\begin{equation}\label{Eq.02.16}
\Re d_{\tau}(q^{2})=d_{\tau} \,, \quad \Im d_{\tau}(q^{2})=0 \,
\end{equation}
and likewise for $d_\tau^Z$. In this framework imaginary parts of $d_{\tau}$ and $d_{\tau}^Z$ will be generated by diagrams
involving both ${\cal L}_{\textup{eff}}$ of Eq. (<ref>) and SM couplings at higher order.
We shall take into account in the following that $\Im d_\tau$ can be nonzero for $q^2>0$ but we neglect, as already mentioned above,
the contribution
from $Z$-boson exchange, in particular the contribution from $d_\tau^Z$.
This can be justified as follows. Eq. (<ref>) shows that $d_\tau$ and $d^Z_\tau$ will be of the same order of magnitude if the coupling constants
$c_1$ and $c_2$ are of comparable size. This is the case, for instance, in the BSM models considered in Section <ref>,
as was shown in [24]. At energies $\sqrt{s}\ll m_Z$ that we consider in this paper, the effects of $d_\tau^Z$ resulting from $Z$-boson exchange
are then negligible compared to those of $d_\tau$, as is the contribution resulting from the interference of the SM $Z$-boson exchange amplitude with the amplitude
involving $d_\tau$. One can see this explicitly from the formulas given in
[9] where both $\gamma$ and $Z$-boson exchange were taken into account. From Eq. (3.10) of [9] we find that
for c.m. energies $\sqrt{s} \approx 10$ GeV that we are considering $Z$-boson exchange contributions are suppressed by a factor of order
\begin{equation} \label{eq:suppr}
s/{m_Z^2} \approx 10^{-2} \, .
\end{equation}
This holds for $\sqrt{s}$ in the continuum and at the $\Upsilon(4{\rm S})$ resonance where the suppression factor (<ref>)
is a few percent smaller
because there the photon contribution is enhanced as compared to the continuum value; see [26].
§ MATRIX ELEMENTS, PRODUCTION AND DECAY MATRICES
We are interested in analyzing $CP$-violating effects in $\tau$-pair production
generated by a nonzero $\tau$ EDM form factor. Therefore we shall analyze the reactions (<ref>)
by considering on-shell $\tau$-pair production by one-photon exchange,
including the $\tau$ EDM form factor in the $\gamma\tau\tau$ vertex, followed
by the decays of $\tau^-$ and $\tau^+$ into the final states $A$ and $\overline{B}$, respectively. The $\tau$ spin correlations and
polarizations will be taken into account. (The reactions (<ref>) were investigated in [9]
for arbitrary c.m. energies for photon and
$Z$-boson exchange including besides the EDM also the weak dipole form factor of the $\tau$ lepton.)
As to the decay channels $A$ and $\overline{B}$, we consider two cases:
i) Only one charged particle of $A$ and one of $\overline{B}$ are measured:
\begin{equation}
\label{eq:1prong}
\tau^- \rightarrow a(q_-) + X \, , \qquad \tau^+ \rightarrow {\bar b}(q_+) + X' \, ,
\end{equation}
Examples from the main decay modes of $\tau^-$ are
\begin{align}\label{Eq.03.02}
\tau^{-} \rightarrow &e^{-}(q_{-})\overline{\nu}_{e}\nu_{\tau}\, ,\quad \,\mu^{-}(q_{-})\overline{\nu}_{\mu}\nu_{\tau} \,,\notag \\
&\pi^{-}(q_{-})\nu_{\tau}\, ,\quad \pi^{-}(q_{-})\pi^{0}\nu_{\tau} \, ,
\quad \pi^{-}(q_{-})\pi^{0}\pi^{0}\nu_{\tau} \, , \quad \pi^{-}(q_{-})\pi^{+}\pi^{-}\nu_{\tau} \, ,
\end{align}
and the respective charge-conjugate $\tau^+$ decays. The decay modes (<ref>) include, in particular,
$\tau$ decays to one charged prong.
ii) We shall also treat the case where more than one particle from $\tau$ decay is observed,
specifically the decay to two pions via a $\rho$ and $\rho'$ meson and
the decay to the $a_{1}$ meson, respectively to three charged pions:
\begin{align}
\tau^{-} \rightarrow & \pi^{-}(q_1)\pi^{0}(q_2) \nu_{\tau} \, , &
\tau^{+} \rightarrow &\pi^{+}(\bar{q}_1)\pi^{0}(\bar{q}_2) \bar{\nu}_{\tau} \, , \label{eq:tdec2p} \\
\tau^{-} \rightarrow & \pi^{-}(q_1)\pi^{-}(q_2)\pi^{+}(q_3) \nu_{\tau} \, , &
\tau^{+} \rightarrow &\pi^{+}(\bar{q}_1)\pi^{+}(\bar{q}_2)\pi^{-}(\bar {q}_3) \bar{\nu}_{\tau} \,.\label{Eq.03.03}
\end{align}
For on-shell $\tau$-pair production and decay the cross section of (<ref>) can be written as a product of the
production density matrix $R$ for $e^+ e^-\rightarrow \tau^+ \tau^-$
times the density matrices $\mathcal{D}^{\bar B}_{\alpha' \alpha}$ and
$\mathcal{D}^A_{\beta'\beta}$ that describe the decays of polarized $\tau^+ \to \overline{B}$ and $\tau^-\to A$,
The production density matrix $R$ is defined as follows:
\begin{eqnarray}
\label{Eq.03.04}
R_{\alpha\alpha'\beta\beta'} = & \dfrac{1}{4}\sum_{\gamma,\delta} \big{\langle}\tau^{+}({k_{+}}, \alpha),
\tau^{-}({k_{-}}, \beta)|{\cal T}|e^{+}({p_{+}}, \gamma),e^{-}({p_{-}}, \delta)\big{\rangle} \nonumber \\
& \times~\big{\langle}\tau^{+}({k_{+}}, \alpha'), \tau^{-}({k_{-}}, \beta')|{\cal T}|e^{+}({p_{+}}, \gamma),e^{-}({p_{-}},
\delta)\big{\rangle}^*\, \, ,
\end{eqnarray}
where $\gamma,\delta$ are the spin indices of $e^+$ and $e^-$, respectively.
For a decay of $\tau^-$ according to case i) above the corresponding decay density matrix is given by
\begin{eqnarray}\label{Eq.03.05}
\mathcal{D}^a_{\beta'\beta}\bigl(\tau^{-}(k_{-})\rightarrow a(q_{-})+X\bigr)& =\Gamma^{-1}(\tau^{-}\rightarrow A) \dfrac{1}{2m_{\tau}}
\int d\Gamma_ {X}(2\pi)^{4}\delta^{(4)}(k_{-}-q_{-}-q_X) \nonumber \\
& \times~ \big{\langle}a(q_{-}), X|{\cal T}|\tau^{-}(k_{-}, \beta)\big{\rangle}
\big{\langle}a(q_{-}), X|{\cal T}|\tau^{-}(k_{-}, \beta')\big{\rangle}^{*} \, .
\end{eqnarray}
Here the normalization is chosen such that
\begin{equation}\label{Eq.03.07}
\int \frac{d^{3}q_{-}}{(2\pi)^{3}2q_{-}^{0}} \,\mathcal{D}^a_{\beta'\beta}\bigl(\tau^{-}(k_{-})\rightarrow a(q_{-})+X)
=\delta_{\beta'\beta}\langle n_{a}\rangle_{A}\,, \\
\end{equation}
where $\langle{n_a}\rangle_{A}$ is the mean multiplicity of particle $a$ in channel $A$. Formulas analogous to (<ref>)
and (<ref>) apply if decays $\tau^+ \to \bar{b} + X'$ according to case i) are considered.
Thus the cross section for the two-particle inclusive reactions
\begin{equation} \label{eq:2pincl}
e^+e^- \rightarrow \tau^+\tau^-\rightarrow \bar{B} + A \, ,
\end{equation}
\begin{equation} \label{eq:1prdec}
A = a(q_-) \, + \, X \, , \qquad {\bar B} = {\bar b}(q_+) \, + \, X' \, ,
\end{equation}
is given in the narrow-width
approximation of the intermediate $\tau$ leptons by
\begin{eqnarray}\label{Eq.03.10}
d\sigma_{a\bar{b}}=\dfrac{\sqrt{1-4m_{\tau}^{2}/s}}{16\pi s} \;
\dfrac{d\Omega_{k_{+}}}{4\pi} \, {\rm Br}(\tau^{-}\rightarrow A) \, {\rm Br}(\tau^{+}\rightarrow \overline{B}) & \nonumber \\
\times ~ R_{\alpha\alpha'\beta\beta'}
\frac{d^{3}q_{-}}{(2\pi)^{3}2q_{-}^{0}} \mathcal{D}^a_{\beta'\beta}\bigl[\tau^{-}\rightarrow a (q_{-})+X\bigr]
\frac{d^{3}q_{+}}{(2\pi)^{3}2q_{+}^{0}} \mathcal{D}^{\bar b}_{\alpha'\alpha}\bigl[\tau^{+}\rightarrow \bar{b} (q_{+})+X'\bigr] \, , &
\end{eqnarray}
where $s=(p_+ + p_-)^2$,
the solid angle element $d\Omega_{k_{+}}$ corresponds to the momentum vector
${\kk}_{+}$ in the $e^+e^-$ c.m. frame,
and ${\rm Br}(\tau^{-}\rightarrow A)$ and ${\rm Br}(\tau^{+}\rightarrow\overline{B})$ denote the branching fractions
for the decays $\tau^{-}\rightarrow A$ and $\tau^{+}\rightarrow\overline{B}$, respectively.[Formula (4.3)
of Ref. [9] contains a typo. These branching fraction factors
are missing. However, they were taken into account in the numerical results given in that paper. Moreover, the variable $q_0^*$
on the l.h.s. of Eq. (4.4) of that reference should be replaced by $|{\bq}^*|/\langle n_A \rangle$.]
For $\tau$ decay to three charged pions whose four-momenta are all measured in an experiment we define the corresponding
decay density matrix by
\begin{eqnarray}\label{Eq.03.08}
\mathcal{D}^A_{\beta'\beta}\bigl(\tau^{-}(k_{-})\rightarrow \pi^{-}(q_{1})\pi^{-}(q_{2})\pi^{+}(q_{3}) \nu_{\tau}\bigr) = & \nonumber \\
\Gamma^{-1}(\tau^{-} \rightarrow \pi^{-}\pi^{-}\pi^{+}{\nu_{\tau}})\frac{1}{2m_{\tau}}
\int \frac{d^{3}q_{4}}{(2\pi)^{3}2q_{4}^{0}} (2\pi)^{4} \delta^{(4)} (k_{-} -q_{1}-q_{2}-q_{3}-q_{4}) & \nonumber \\
\times~\big{\langle}\pi^{-}(q_1) \pi^{-}(q_2) \pi^{+}(q_3) \nu_{\tau}|{\cal T}|\tau^{-}(k_{-}, \beta)\big{\rangle}
\big{\langle}\pi^{-} (q_1) \pi^{-}(q_2) \pi^{+} (q_3) \nu_{\tau}|{\cal T}|\tau^{-}(k_{-}, \beta')\big{\rangle}^* & \, ,
\end{eqnarray}
where $q_4$ is the four-momentum of $\nu_\tau$,
and analogously for the decay $\tau^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}\overline{\nu}_{\tau}$.
The normalization is
\begin{equation}\label{Eq.03.09}
\int \prod_{i=1}^{3} \frac{d^{3}q_{i}}{(2\pi)^{3}2q_{i}^{0}}
\mathcal{D}^A_{\beta'\beta}\bigl(\tau^{-}(k_{-})\rightarrow \pi^{-}(q_{1})\pi^{-}(q_{2})\pi^{+}(q_{3}) \nu_{\tau}\bigr)
=2\delta_{\beta'\beta} \, ,
\end{equation}
corresponding to the $\pi^{-}$ multiplicity $2$ in this channel.
If the analysis is restricted to three pions in a suitably defined invariant mass region around
the nominal $a_{1}$ mass one has to take into account the corresponding phase-space cuts
in $\Gamma(\tau^{-}\rightarrow\pi^{-}\pi^{-}\pi^{+}{\nu}_{\tau})$ and in (<ref>).
For the $\tau$ decay (<ref>) to two pions, where both the charged and the neutral pion are measured, the
respective decay density matrix is defined accordingly by integrating the corresponding
squared matrix element over the four-momentum of the neutrino.
In order to get the inclusive cross section for case ii), considering, for instance,
the decay of the $\tau^{-}$ into three observed pions, we have to make in (<ref>) the replacement
\begin{equation}\label{Eq. 03.11}
\frac{d^{3}q_{-}}{(2\pi)^{3}2q_{-}^{0}} \mathcal{D}^a_{\beta'\beta}\bigl[\tau^{-}\rightarrow a+X\bigr]\rightarrow
\prod_{i=1}^{3} \frac{d^{3}q_{i}}{(2\pi)^{3}2q_{i}^{0}} \mathcal{D}^A_{\beta'\beta}\bigl[\tau^{-}\rightarrow \pi^{-} \pi^{-} \pi^{+} \nu_\tau\bigr]\;.
\end{equation}
Analogous replacements apply if the decay of $\tau^{+}$ to three observed pions or the decay of $\tau^\mp$ to two observed pions
are analyzed.
The production density matrix $R$ in (<ref>) is computed in the $e^+ e^-$ c.m. system, see below.
Instead of calculating the decay density matrices also in this frame we can determine them in
the $\tau^{-}$ and $\tau^{+}$ rest systems, respectively, if we use the following:
* We consider rotation-free Lorentz transformations (boosts) from the c.m. frame to the $\tau^{-}$ and $\tau^{+}$ rest systems, respectively.
* We use standard spinors $u_{\beta}(k)$, ${\v}_{\alpha}(k)$ for the $\tau$'s with $\beta, \alpha$
denoting the spin components in a given $z$ direction (see, e.g., [33]).
As is well known, these spin components are not changed by boost transformations.
Let $\Lambda_{\kk}$ be the boost transforming the $\tau^{-}$ momentum $k_{-}$ from the $e^+ e^-$ c.m. system
to rest, $\Lambda_{\kk} k_- = k_{-}^{*}$, where $k_{-}^{*} = (m_\tau,\boldsymbol{0})^T.$
We have then with $\Lambda_{\kk} q_{-}=q_{-}^{*}$; see (<ref>) and (<ref>),
\begin{equation}\label{Eq.03.18}
\langle a(q_{-}), X|{\cal T}|\tau^{-}(k_{-},\beta)\rangle=\langle a(q_{-}^{*}), X|{\cal T}|\tau^{-}(k_{-}^{*},\beta)\rangle \;.
\end{equation}
Insertion into the decay matrix (<ref>) proves our statements above. The analogous argumentation applies to the $\tau^+$
decay density matrices.
In Appendix <ref> we give the explicit forms of the $\tau^\mp$ decay
density matrices in the respective rest frames for
the decay modes listed in (<ref>) – (<ref>).
Finally, using the one-photon-exchange approximation and setting
\begin{equation}\label{eq:setFF}
F_1(q^2) = 1\, , \quad F_2(q^2) = 0 \, , \quad A(q^2) = 0 \, ,
\end{equation}
the production density matrix $R$ is given in the $e^+e^-$ c.m. frame by
\begin{equation} \label{eq:ProRchi}
R = \frac{\chi}{| 1 + e^2 \Pi_c(s)|^2} \, ,
\end{equation}
where [9]
\begin{equation}\label{Eq.03.12}
\chi=\chi_{SM}+ \Re \hat{d}_\tau ~\chi_{CP}^R +\Im\hat{d}_\tau ~\chi_{CP}^I +\chi_{{\hat d}^2} \, ,
\end{equation}
\begin{eqnarray}\label{Eq.03.13}
\chi_{SM} &= & \frac{e^4}{s} \left\{ [k_0^2 + m_\tau^2 +|\kk|^2 (\hk \cdot\hp)^2]\one -({\sip}\cdot{\ssim})|\kk|^2[1-(\hk \cdot \hp)^2] \right. \nonumber \\
& & \left. + 2 (\hk \cdot \sip)(\hk \cdot \ssim)[|\kk|^2 +(k_0-m_\tau)^2 (\hk \cdot \hp)^2] + 2 k_0^2 (\hp \cdot \sip)(\hp \cdot \ssim) \right. \nonumber\\
& & \left. - 2 k_0(k_0-m_\tau) (\hk \cdot \hp)[ (\hk \cdot \sip)(\hp \cdot \ssim)
+ (\hk \cdot \ssim)(\hp \cdot \sip)] \right \} \, ,
\end{eqnarray}
\begin{eqnarray}\label{Eq.03.14}
\chi_{CP}^R & = & - 2 e^4 \frac{|\kk|}{s} \left\{ -[m_\tau + (k_0-m_\tau) (\hk \cdot\hp)^2](\sip \times \ssim)\cdot\hk \right. \nonumber \\
& & \left. + k_0(\hk \cdot\hp) (\sip \times \ssim)\cdot\hp \right\} \, ,
\end{eqnarray}
\begin{eqnarray}\label{Eq.03.14a}
\chi_{CP}^I & = & 2 e^4 \frac{|\kk|}{s} \left\{ -[m_\tau + (k_0-m_\tau) (\hk \cdot\hp)^2](\sip - \ssim)\cdot\hk \right. \nonumber \\
& & \left. + k_0(\hk \cdot\hp) (\sip - \ssim)\cdot\hp \right\} \, ,
\end{eqnarray}
\begin{equation}\label{Eq.03.15}
\chi_{{\hat d}^{2}}= e^4 [(\Re {\hat d}_\tau)^2 + (\Im {\hat d}_\tau)^2] \frac{ |\kk|^2}{s}[1- (\hk \cdot \hp)^2](\one - \sip \cdot \ssim) \, .
\end{equation}
Compared to Eqs. (3.8) – (3.10) of [9] we neglect here the contributions from $Z$-boson exchange because we restrict
ourselves to the kinematic range $s\ll m_Z^2$, but we have included the photon vacuum polarization effects.
In (<ref>) – (<ref>) we put $\pp = {\pp}_+,~ \kk = {\kk}_+$, and $\hp$ and $\hk$ denote the respective unit vectors. We have introduced in (<ref>)
and (<ref>) dimensionless EDM form factors defined by
\begin{equation} \label{eq:dimlFF}
\Re {\hat d}_\tau(s) = \frac{\sqrt{s}}{e} \Re d_\tau(s) \, , \qquad \Im {\hat d}_\tau(s) = \frac{\sqrt{s}}{e} \Im d_\tau(s) \, .
\end{equation}
Moreover, we use in the equations above the notation [9]
\begin{eqnarray}\label{Eq.03.16}
\one\equiv(\one\otimes\one)_{\alpha\alpha'\beta\beta'}=\delta_{\alpha\alpha'}\delta_{\beta\beta'}\;, \notag \\
\sip \equiv(\ssig\otimes\one)_{\alpha\alpha'\beta\beta'}={\ssig}_{\alpha\alpha'}\delta_{\beta\beta'} \;, \notag \\
\ssim\equiv(\one\otimes\ssig)_{\alpha\alpha'\beta\beta'}=\delta_{\alpha\alpha'}{\ssig}_{\beta\beta'} \;,
\end{eqnarray}
where the first and second factors in these tensor products refer to the spin spaces of
$\tau^+$ and $\tau^-$, respectively.
The density matrices $\chi_{\rm SM}$ and $\chi_{{\hat d}^{2}}$ are $CP$-even, whereas $\chi_{CP}^R$
is $CP$- and $T_N$-odd while $\chi_{CP}^I$ is
$CP$-odd and $T_N$-even. Here and below $T_N$-even/odd refers to the behavior with respect to the naive
“time reversal” transformation, that is, reflections
of three-momenta and spins.
Equation (<ref>) shows that a nonzero $\Re {d}_\tau$ generates $CP$-odd $\tau^+ \tau^-$ spin correlations in the $\pp, \kk$
scattering plane while a nonzero imaginary part of $d_\tau$ leads to a $CP$-odd asymmetry of the $\tau^+$ and $\tau^-$ polarizations
with projections along $\pp$ and $\kk$, cf. (<ref>). The $\tau$ leptons autoanalyze their spin directions via their
parity-violating weak decays. In this way these $\tau$ spin correlations and polarization asymmetries induce $CP$-odd angular correlations
among the $\tau^\pm$ decay products, to which we now turn.
§ SIMPLE AND OPTIMAL $CP$ OBSERVABLES
In this chapter we discuss simple and optimal observables
for studying $CP$ violation in the reactions (<ref>). Let us first consider the case i) above where only one charged
particle is measured from $\tau^-$ and $\tau^+$ decay, respectively, i.e.,
$\tau^- \to a(q_-)+X$ and $\tau^+ \to \bar{b}(q_+)+X'$.
Simple $CP$ observables for this case were given in Ref. [9].
Observables sensitive to $\Re d_{\tau}(s)$ are, for instance, the tensors
\begin{equation}\label{Eq.04.01}
\widehat{T}^{ij}=(\hqp-\hqm)^{i}\,\frac{(\hqp\times\hqm)^{j}}{|\hqp\times\hqm|} \, + \, (i\leftrightarrow j) \;,
\end{equation}
\begin{equation}\label{Eq.04.02}
T^{ij}=(\qp-\qm)^{i}\,(\qp\times\qm)^{j} \, + \, (i\leftrightarrow j) \;.
\end{equation}
Observables sensitive to $\Im d_{\tau}(s)$ are, for instance,
\begin{equation}\label{Eq.04.03}
\widehat{Q}^{ij}=(\hqp+\hqm)^{i}\, (\hqp -\hqm)^{j}+(i\leftrightarrow j) \;,
\end{equation}
\begin{equation}\label{Eq.04.04}
Q^{ij}=(\qp + \qm)^{i}\,(\qp -\qm)^{j}-\dfrac{1}{3}\delta^{ij}(\qp^2 - \qm^2)+(i\leftrightarrow j) \;.
\end{equation}
The momenta ${\bq}_\mp$ in (<ref>) – (<ref>) are defined in the $e^+ e^-$ c.m. frame, and ${\hbq}_{\pm}={\bq}_{\pm}/|{\bq}_{\pm}|$
and $i,j\in\lbrace1,2,3\rbrace$ are the Cartesian vector indices.
These observables, denoted generically by $\cO(\qp,\qm)$, have the property to be odd under $CP$:
\begin{equation}\label{Eq.04.05}
\cO(\qp,\qm)=-\cO(-\qm,-\qp)\;.
\end{equation}
Moreover, Eqs. (<ref>) and (<ref>) are $T_N$-odd while (<ref>) and (<ref>) are $T_N$-even.
A nonzero expectation value of any such observable of the form
\begin{eqnarray}\label{Eq.04.06}
\langle\cO\rangle_{ab}& \equiv & \dfrac{1}{2}\bigl{\lbrace}\langle\cO\rangle_{a\bar{b}}+\langle\cO\rangle_{b\overline{a}}\bigr{\rbrace} \nn \\
& = & \dfrac{1}{2}\Bigl{\lbrace}\dfrac{\int d\sigma_{a\bar{b}}\cO}{\int d\sigma_{a\bar{b}}}+\dfrac{\int d\sigma_{b\overline{a}}
\cO}{\int d\sigma_{b\overline{a}}}\Bigr{\rbrace}
\end{eqnarray}
is a genuine signature of $CP$ violation. Here $d\sigma_{a\bar{b}}$ is the cross section (<ref>)
of the reaction (<ref>) and $d\sigma_{b\overline{a}}$ the corresponding one for the charge-conjugate channel.
We assume that any phase-space cuts that may be applied are made in a $CP$-symmetric way.
Observables of the type (<ref>)-(<ref>) were studied extensively in [9].
In Sec. <ref> we give an update of the sensitivities achievable with these observables at the KEKB accelerator with
Belle II. A discussion of the sensitivities achievable with the BES III experiment at the Beijing Electron-Positron Collider II
is deferred to a future publication.
We shall now turn to optimal observables [21, 22, 23] and we follow
here Ref. [23].
We denote the measured phase-space variables generically by $\phi$ and the $CP$-transformed ones by $\overline{\phi}$:
\begin{equation}\label{Eq.04.07}
\end{equation}
Phase-space cuts are assumed to be $CP$-symmetric. In the following we denote the dimensionless $CP$-violating
EDM form factors (cf. Eq. (<ref>)) that are to be measured by
\begin{equation}\label{Eq.04.08}
g_{1}= \Re \,{\hat d}_{\tau} \,,\quad g_{2}= \Im \, {\hat d}_{\tau} \;.
\end{equation}
From experiment we know that these couplings are small, $|g_{1,2}|\ll 1$. From (<ref>)
we get $|g_{1,2}| \le 2.4 \times 10^{-2}$
for $\sqrt{s}=10.58$ GeV.
Therefore, we shall work to leading order in these couplings.
The cross section (<ref>) can be expanded in the $g_{i}$ as follows,
neglecting terms of second order in these couplings:
\begin{equation}\label{Eq.04.09}
S^{a\bar{b}}(\phi)=\dfrac{d\sigma_{a\bar{b}}(\phi)}{d \phi}=S_{0}^{a\bar{b}}(\phi)+g_{i}\,S_{1,i}^{a\bar{b}}(\phi)\;.
\end{equation}
Here and in the following we use the summation convention. Moreover, in order not to overload the notation,
the labels $a$ and $\bar{b}$ denote in (<ref>) and in what follows
decays of $\tau^-$ and $\tau^+$ to one, two, or three measured particles, respectively.
The $CP$ properties of $S_{0}$ and $S_{1}$ in (<ref>) are:
\begin{equation}\label{Eq.04.09a}
S_{0}^{a\bar{b}}(\phi)= S_{0}^{b\overline{a}}(\overline{\phi}) \;, \qquad
S_{1,i}^{a\bar{b}}(\phi)=- S_{1,i}^{b\overline{a}}(\overline{\phi})\;.
\end{equation}
We define now the observables
\begin{equation}\label{Eq.04.10}
\cO_{i}^{a\bar{b}}(\phi)=S_{1,i}^{a\bar{b}}(\phi)\big{/}S_{0}^{a\bar{b}}(\phi)\;.
\end{equation}
Their expectation value $E_{0}$ for $g_i=0$ is
\begin{equation}\label{Eq.04.11}
E_{0}(\cO_{i}^{a\bar{b}})=\int d\phi S_{0}^{a\bar{b}}(\phi)\cO_{i}^{a\bar{b}}(\phi)\bigg{/}\int d\phi' S_{0}^{a\bar{b}}(\phi') \;.
\end{equation}
We set
\begin{equation}\label{Eq.04.12}
\cO_{i}^{\prime a\bar{b}}(\phi)=\cO_{i}^{a\bar{b}}(\phi)-E_{0}(\cO_{i}^{a\bar{b}})
\end{equation}
and get for the expectation value of ${\cO_{i}'}^{a\bar{b}}$:
\begin{equation}\label{Eq.04.13}
E(\cO_{i}^{\prime a\bar{b}})=\int d\phi S^{a\bar{b}}(\phi)\cO_{i}^{\prime a\bar{b}}(\phi)\Bigg{/}\int d\phi'S^{a\bar{b}}(\phi')
=V_{ij}(\cO^{\prime a\bar{b}})g_{j}\;.
\end{equation}
The expression on the right-hand side is obtained by expanding the ratio to first order in the $g_{j}$.
Here $V({\cO'}^{a\bar{b}})=\bigl(V_{ij}({\cO'}^{a\bar{b}})\bigr)$ is the covariance matrix
of the quantities $\cO'$ for $g_j=0$.
\begin{equation}\label{Eq.04.14}
V_{ij}(\cO^{\prime a\bar{b}})=E_{0}(\cO_{i}^{\prime a\bar{b}}\cO_{j}^{\prime a\bar{b}})
\frac{S_{1,j}^{a\bar{b}}}{S_{0}^{a\bar{b}}}\biggr)-E_{0}
\biggl(\frac{S_{1,i}^{a\bar{b}}}{S_{0}^{a\bar{b}}}\biggr)E_{0}\biggl(\frac{S_{1,j}^{a\bar{b}}}{S_{0}^{a\bar{b}}}\biggr)\;.
\end{equation}
The covariance matrix $V({\cO'}^{a\bar{b}})$ is positive definite.
From (<ref>) we obtain
\begin{equation}\label{Eq.04.15}
g_{i}=V^{-1}_{ij}(\cO^{\prime a\bar{b}})E(\cO_{j}^{\prime a\bar{b}})\;.
\end{equation}
In the remainder of this section we recall from [23] some general relations for optimal observables in order to
make our article self-contained. Also, we shall discuss that in the nondiagonal case $a\neq b$ the theoretically optimal estimators
may not always be “optimal” from a practical point of view (see the discussion after Eq. (<ref>) below).
We consider first the diagonal case, $a=b$, and assume that $n$ events of this type are analyzed.
The density function is then
\begin{align}\label{Eq.04.16}
F(\phi_{1},\dots , \phi_{n})=\prod_{k=1}^{n} f(\phi_{k})\;, \nn \\
f(\phi)=S^{a\overline{a}}(\phi)\Bigg{/}\int d\phi'S^{a\overline{a}}(\phi')\;.
\end{align}
The information matrix $I=(I_{ij})$ is defined by
\begin{equation}\label{Eq.04.17}
I_{ij}=E\Biggl{[}\Bigl{(}\frac{\partial}{\partial g_{i}}\ln F\Bigr{)}\Bigl{(}\frac{\partial}{\partial g_{j}}\ln F\Bigr{)}\Biggr{]} \, .
\end{equation}
The optimal estimators for the couplings $g_i$ are in this case:
\begin{equation}\label{Eq.04.18}
\gamma_{i}(\phi)=V^{-1}_{ij}(\cO^{\prime a \overline{a}})\overline{\cO'}_j^{ a \overline{a}}(\phi) \, ,
\end{equation}
where $\overline{\cO_j'}$ denotes the mean value of $\cO_j'$.
From Eqs. (<ref>) and (<ref>) we obtain the expectation values
\begin{equation}\label{Eq.04.19}
\end{equation}
and the covariance matrix of the $\gamma_{i}$, evaluated for $g_{i}=0$, is
\begin{equation}\label{Eq. 04.20}
V_{ij}(\gamma) = E_0(\gamma_{i}\gamma_{j})=\dfrac{1}{n} V^{-1}_{ij}(\cO^{\prime a \overline{a}}) \;.
\end{equation}
We get for the information matrix (<ref>):
\begin{equation}\label{Eq. 04.21}
I|_{g=0}=nV(\cO^{\prime a \overline{a}}) \;.
\end{equation}
Therefore, we have here
\begin{equation}\label{Eq.04.22}
\end{equation}
and the estimators (<ref>) are optimal for small $g_j$. That is, the error ellipse obtained with the estimators
$\gamma_{i}$ in (<ref>) is given by the one obtained from $I$ which is the smallest one possible.
We note that due to the $CP$ properties (<ref>) of $S_{0}$ and $S_{1,i}$
we have in the diagonal case $a=b$, assuming possible cuts in phase space to be $CP$-symmetric:
\begin{equation}\label{Eq.04.23}
E_{0}(\cO_{i}^{a \bar{a}})=0 \;, \qquad
\cO_{i}^{\prime a \bar{a}}(\phi)=\cO_{i}^{a \bar{a}}(\phi)\;, \qquad
V(\cO^{\prime a \bar{a}})=V(\cO^{ a \bar{a}})\;,
\end{equation}
and the optimal estimators are
\begin{equation}\label{Eq.04.24}
\gamma_{i}(\phi)=V^{-1}_{ij}(\cO^{a \overline{a}}) \overline{\cO}_{j}^{a\overline{a}}(\phi)\;;
\end{equation}
see (<ref>), (<ref>), and (<ref>).
Finally, we treat the nondiagonal case, $a\neq b$.
We assume that any phase-space cuts made for the channel $a\bar{b}$ are applied to $b\bar{a}$ in a $CP$-conjugate way.
We get then from the $CP$ relations (<ref>):
\begin{equation}\label{Eq.04.25}
\int d\phi S_{0}^{a\bar{b}}(\phi)=\int d\overline{\phi} S_{0}^{b\bar{a}}(\overline{\phi})\;,
\end{equation}
\begin{equation}\label{Eq.04.26}
\end{equation}
\begin{equation}\label{Eq.04.27}
V\bigl(\cO^{\prime a \bar{b}}\bigr)=V\bigl(\cO^{\prime b\bar{a}}\bigr)\;.
\end{equation}
We assume that $n_{1}$ events of the type $a\bar{b}$ and $n_{2}$ events $b\bar{a}$ are analyzed.
The density function is then
\begin{equation}\label{Eq.04.28}
F(\phi_{1}, \dots ,\phi_{n_{1}},\overline{\phi}_{1}, \dots , \overline{\phi}_{n_{2}})
=\prod_{k=1}^{n_{1}} f_{ a \bar{b}}(\phi_{k})\prod_{l=1}^{n_{2}} f_{b \overline{a}}(\overline{\phi}_{l})
\end{equation}
\begin{equation}\label{Eq.04.29}
f_{a\bar{b}}(\phi)=S^{a\bar{b}}(\phi)\Big{/}\int d\phi'S^{a\bar{b}}(\phi') \;, \qquad
f_{b\overline{a}}({\overline\phi})=S^{b\overline{a}}(\overline{\phi})\Big{/}\int d\overline{\phi}'S^{b\overline{a}}(\overline{\phi}') \, .
\end{equation}
Here the information matrix $I=(I_{ij})$ is given for $g_{i}=0$ by
\begin{equation}\label{Eq.04.30}
I_{ij}\big|_{g=0}=E\Bigl{[}\Bigl{(}\frac{\partial}{\partial g_{i}}\ln F\Bigr{)}
\Bigl{(}\frac{\partial}{\partial g_{j}}\ln F\Bigr{)}\Bigr{]}\Big|_{g=0}
= n~ V _{ij}(\cO^{\prime a\bar{b}}) \;,
\end{equation}
where $n=n_1 + n_2$. Here it is convenient to use as estimators for
the couplings $g_i$, with $\cO_{j}$ from (<ref>):
\begin{equation}\label{Eq.04.31}
\gamma_{i}(\phi,\overline{\phi})=\dfrac{1}{4}\Bigl[V^{-1}_{ij}(\cO^{\prime a \bar{b}}) + V^{-1}_{ij}(\cO^{\prime b \overline{a}})\Bigr]
\Bigl[\overline{\cO}_j^{ a \bar{b}}(\phi)+\overline{\cO}_j^{ b \overline{a}}(\overline{\phi})\Bigr]\;.
\end{equation}
We have
\begin{eqnarray} \label{eq:68a}
E\left(\frac{1}{2} \overline{\cO}_i^{ a \bar{b}}+\frac{1}{2}\overline{\cO}_i^{ b \overline{a}} \right) & = &
E\left(\frac{1}{2} {\cO}_i^{ a \bar{b}}+\frac{1}{2}{\cO}_i^{ b \overline{a}} \right)
= V_{ij}(\cO'^{ a \bar{b}}) g_j \, , \nonumber \\
E(\gamma_i) & = & V^{-1}_{ik}(\cO'^{ a \bar{b}}) V_{kj}(\cO'^{ a \bar{b}}) g_j = g_i \, .
\end{eqnarray}
The covariance matrix of these estimators is obtained as
\begin{equation}\label{Eq.04.32}
V(\gamma)=\dfrac{n_{1}+n_{2}}{4n_{1}n_{2}}V^{-1}(\cO^{\prime a \bar{b}}) \, ,
\end{equation}
which implies
\begin{equation}\label{Eq.04.33}
V^{-1}(\gamma)=n\biggl(1-\dfrac{(n_{1}-n_{2})^{2}}{n^{2}}\biggr) V(\cO^{\prime a \bar{b}})
=\biggl(1-\dfrac{(n_{1}-n_{2})^{2}}{n^{2}}\biggr)I\big|_{g=0} \;.
\end{equation}
The $\gamma_i$ in Eq. (<ref>) are the optimal estimators for $n_{1}=n_{2}=n/2$. For $n_{1}\neq n_{2}$
they are not quite optimal, but for the theoretically optimal estimators one would need in this case the precise knowledge
of $E_{0}(S_{1,i}^{a\bar{b}}/S_{0}^{a\bar{b}})$.
This would introduce an unnecessary source of uncertainty in the measurements.
To conclude this section we remark on the following. A more elaborate description of $\tau$-pair production and decay would take
higher-order radiative corrections into account. Let us denote the resulting differential cross section by $\tilde{S}^{a\bar{b}}$,
\begin{equation}\label{Eq.04.34}
\dfrac{d\sigma_{a\bar{b}}}{d\phi}(\phi)=\tilde{S}^{a\bar{b}}(\phi)\; .
\end{equation}
If it is $CP$-invariant, we have
\begin{equation}\label{Eq.04.34a}
\tilde{S}^{a\bar{b}}(\phi)=\tilde{S}^{b\overline{a}}(\overline{\phi}) \; .
\end{equation}
Then the corresponding expectation values $\tilde{E}$ of the estimators $\gamma_{i}$ defined in
(<ref>) and (<ref>) and constructed with the expressions $S_{0}$, $S_{1,i}$ from (<ref>)
will, of course, be zero due to (<ref>):
\begin{equation}\label{Eq.04.35}
\tilde{E}(\gamma_{i}) = 0 \, .
\end{equation}
That is, the observables $\gamma_i$ given in (<ref>) and (<ref>) are in all cases genuine $CP$ observables.
They cannot get nonzero expectation values, neither from $CP$-conserving radiative SM corrections nor from $CP$-conserving interactions
beyond the SM.
§ NUMERICAL RESULTS AT $\SQRT{S} = 10.58$ GEV
We consider now $\tau$-pair production and decay
at the $\Upsilon(4{\rm S})$ resonance at $\sqrt{s}=10.58~\GeV$ and compute the
expectation values of the simple and optimal $CP$ observables discussed in the previous section and estimate the resulting
1 s.d. (standard deviation) statistical sensitivities to the EDM form factors $\Re d_\tau$ and $\Im d_\tau$ at this c.m. energy.
The expectation values of the $CP$ observables are computed to leading order in the real and imaginary parts of the $\tau$ EDM form factor
using the expression (<ref>) for the differential cross section with (<ref>) – (<ref>) and
several of the decay density matrices given in Appendix <ref>.
First, no phase-space cuts are applied. At the end of this section we analyze also the effects of cuts.
The expectation values of the observables (<ref>) – (<ref>) at the $\Upsilon(4{\rm S})$ resonance
in the decay channels where only one charged particle from $\tau^-$ and one from $\tau^+$ decay is measured (case i) above)
are of the form:
\begin{eqnarray}
\langle T^{ij}\rangle_{ab} = c_{ab}(s) \, \Re {\hat d}_\tau(s)~ s^{ij} \, , &
\qquad \langle \widehat{T}^{ij}\rangle_{ab}= {\tilde c}_{ab}(s) \, \Re {\hat d}_\tau(s)~ s^{ij} \, , \label{eq:exsimT} \\
\langle Q^{ij}\rangle_{ab} = \kappa_{ab}(s) \, \,\Im {\hat d}_\tau(s)~ s^{ij} \, , &
\qquad \langle \widehat{Q}^{ij}\rangle_{ab}= {\tilde\kappa}_{ab}(s) \, \Im {\hat d}_\tau(s)~ s^{ij} \, .
\label{eq:exsimQ}
\end{eqnarray}
In the case of nondiagonal decay channels $a\neq b$ the expectation values are calculated as averages defined in (<ref>).
The expectation values of the symmetric traceless tensors (<ref>) – (<ref>) must be proportional to
a tensor $s^{ij}$ with the same property. Using the $e^+$ beam direction $\hp$ in the $e^+ e^-$ c.m. frame we have
\begin{equation} \label{eq:defsij}
(s^{ij})= \frac{1}{2} \left({\hat p}^i {\hat p}^j - \frac{1}{3} \delta^{ij} \right)
= {\rm diag}\left(-\frac{1}{6}, -\frac{1}{6}, \frac{1}{3} \right) \, .
\end{equation}
The right-hand side of (<ref>) follows from identifying $\pp$ with the $z$ axis which we do in the following.
Equation (<ref>) is identical to the tensor polarization of the intermediate photon state.
Because the diagonal elements
of the above tensor observables are not independent, we consider only their $3,3$ components that have the largest expectation values.
Naive “time reversal" invariance $T_N$ implies that the
expectation values (<ref>) and (<ref>) do not depend on $\Im {\hat d}_\tau$ and $\Re {\hat d}_\tau$, respectively.
That is, the covariance matrix of the $T$ and $Q$ tensors is diagonal; see Appendix <ref>.
In order to estimate the statistical error in the measurement of the expectation values of the observables $\mathcal{O}$ we compute
also the respective standard deviation $\Delta \mathcal{O} =\sqrt{\langle \mathcal{O}^2 \rangle - \langle \mathcal{O} \rangle^2 }$
of the distribution of $\mathcal{O}$ in the SM for the various decay channels. As discussed
in Appendix <ref> the SM expectation values of the tensors
$T^{ij}$, ${\widehat T}^{ij}$
vanish for the differential cross section as used by us. (Cf. Section <ref>.) For the tensors $Q^{ij}$, ${\widehat Q}^{ij}$
this is also true in the diagonal case $a = b$. In the nondiagonal case, $a \neq b$, their SM expectation values need not be zero, but
are found numerically to be negligibly small.
Tables <ref> and <ref>
we assume that the momenta of
$\rho^\mp$ mesons can be experimentally determined and we treat them as on-shell particles with the $\tau$-spin analyzing power given in (<ref>).
The symbols $\ell$ and $\ell'$ denote either the electron or muon, both are taken to be massless.
We sum over the diagonal and nondiagonal $\ell \ell'$ channels for estimating the respective sensitivity to the real and imaginary parts of the $\tau$ EDM.
In a diagonal decay channel the number of events
is $N_{aa} = N_{\tau\tau} ({\rm Br}(\tau\to a))^2$, while for a nondiagonal channel including its charge-conjugate mode we have
$N_{ab} = 2 N_{\tau\tau} {\rm Br}(\tau\to a){\rm Br}(\tau\to b)$. The $\tau$ branching ratios are taken from [1].
We assume that the Belle II experiment will eventually record $N_{\tau\tau} = 4.5 \times 10^{10}$ $\tau$ pairs [19]. Considering as an example
the measurements of $T_{33}$ and $Q_{33}$ in the decay channels $a\bar{b}$ and $b\bar{a}$ the resulting ideal 1 s.d.
statistical errors of the dimensionful EDM
couplings $\Re d_\tau$ and $\Im d_\tau$ are given by
\begin{equation} \label{eq:1sdsenTQ}
\delta \Re d_\tau(s) = \frac{e}{\sqrt{s}}
\frac{1}{\sqrt{N_{ab}}} \frac{3 \left[\langle T_{33}^2 \rangle_{ab}\right]^{1/2}}{|c_{ab}|} \, , \quad
\delta \Im d_\tau(s) = \frac{e}{\sqrt{s}}
\frac{1}{\sqrt{N_{ab}}} \frac{3 \left[\langle Q_{33}^2 \rangle_{ab}\right]^{1/2}}{|\kappa_{ab}|} \, .
\end{equation}
Equation (<ref>) yields the absolute value that $\Re d_\tau$ $(\Im d_\tau)$ must have in order that $\langle T_{33} \rangle_{ab}$
$(\langle Q_{33} \rangle_{ab})$ deviates from its SM prediction, namely zero, by 1 s.d.
obtained from the square root of its SM variance. Formulas analogous to (<ref>) hold for the
dimensionless observables $\widehat{T}_{33}$ and $\widehat{Q}_{33}$.
Observables $T^{ij}$ and $\widehat{T}^{ij}$ at $\sqrt{s}=10.58\, \GeV$ $(N_{\tau\tau} = 4.5 \times 10^{10})$.
$\tau^-\to$ $\tau^+\to$ $c_{ab}$ $\sqrt{\langle T_{33}^2 \rangle_{ab}}$ $\delta \Re d_\tau$ ${\tilde c}_{ab}$ $\sqrt{\langle \widehat{T}_{33}^2 \rangle_{ab}}$ $\delta \Re d_\tau$
$[\GeV^3]$ $ [\GeV^3]$ $(\times \, 10^{-19} \ecm)$ $(\times \, 10^{-19} \ecm)$
$\pi^-\nu$ $\pi^+ \bar\nu$ $4.46$ $11.34$ $6.21$ $0.332$ $1.02$ $7.50$
$\rho^-\nu$ $\rho^+ \bar\nu$ $0.71 $ $10.07$ $14.7$ $0.043$ $1.06$ $25.5$
$\pi^-\nu$ $\rho^+ \bar\nu$ $1.79$ $ 10.71 $ $6.74$ $0.110$ $1.03$ $10.5$
$\ell^-\nu\bar\nu$ ${\ell'}^+ {\bar\nu}\nu$ $ 0.36 $ $4.68$ $9.86$ $0.037$ $0.98$ $19.9$
$\ell^-\nu\bar\nu$ $\pi^+ \bar\nu$ $-1.27$ $ 6.66 $ $5.05$ $-0.111$ $0.96$ $8.3$
$\ell^-\nu\bar\nu$ $\rho^+ \bar\nu$ $-0.51$ $6.78$ $8.32$ $-0.037$ $1.00$ $16.9$
Observables $Q^{ij}$ and $\widehat{Q}^{ij}$ at $\sqrt{s}=10.58\, \GeV$ $(N_{\tau\tau} = 4.5 \times 10^{10})$.
$\tau^-\to$ $\tau^+\to$ $\kappa_{ab}$ $\sqrt{\langle Q_{33}^2 \rangle_{ab}}$ $\delta \Im d_\tau$ ${\tilde\kappa}_{ab}$ $\sqrt{\langle \widehat{Q}_{33}^2 \rangle_{ab}}$ $\delta \Im d_\tau$
$[\GeV^3]$ $ [\GeV^3]$ $(\times \, 10^{-19} \ecm)$ $(\times \, 10^{-19} \ecm)$
$\pi^-\nu$ $\pi^+ \bar\nu$ $-5.26$ $6.56$ $3.04$ $-0.601$ $0.59$ $2.38$
$\rho^-\nu$ $\rho^+ \bar\nu$ $-2.28 $ $7.01$ $3.18$ $-0.171$ $0.34$ $2.05$
$\pi^-\nu$ $\rho^+ \bar\nu$ $-3.77$ $7.07$ $2.11$ $-0.386$ $0.52$ $1.52$
$\ell^-\nu\bar\nu$ ${\ell'}^+ {\bar\nu}\nu$ $ 1.40$ $4.90$ $2.64$ $0.201$ $0.64$ $2.40$
$\ell^-\nu\bar\nu$ $\pi^+ \bar\nu$ $-1.93$ $7.24$ $3.61$ $-0.200$ $0.65$ $3.14$
$\ell^-\nu\bar\nu$ $\rho^+ \bar\nu$ $-0.44$ $7.32$ $10.4$ $0.015$ $0.54$ $22.5$
Tables <ref> and <ref> contain our results for the expectation values (as defined in Eqs. (<ref>)
and (<ref>)) and square roots of the variances of the observables (<ref>) – (<ref>) for several one-prong decays of $\tau^\mp$ where the
charged particle has a sizable $\tau$-spin analyzing power. Moreover, the resulting 1 s.d. sensitivities to the real and imaginary parts of the $\tau$ EDM form factor
are listed.[The last digit of the expectation values and variances listed in Tables <ref>, <ref>, and <ref> is rounded. The
sensitivities $\delta \Re d_\tau$ and $\delta \Im d_\tau$ listed in these tables are computed with these rounded numbers.]
Results for $T^{ij}$ and $\widehat{Q}^{ij}$ were previously given in [9]
and agree with those in Tables <ref> and <ref>. The accuracies $\delta\Re d_\tau$ and $\delta\Im d_\tau$ attainable in the various $\tau^\mp$ decay channels
listed in Tables <ref> and <ref> show that the dimensionful observable $T_{33}$ is more sensitive than $\widehat{T}_{33}$ while in the
case of $Q_{33}$ and $\widehat{Q}_{33}$ it is the other way around – except for the $\ell \rho$ decay channel which has,
in any case, a rather poor sensitivity compared to the
other decay modes.
Next we apply the optimal observables (<ref>) for measuring $\Re{\hat d}_{\tau}$ and $\Im{\hat d}_{\tau}$ to the reactions of Sec. <ref>.
As in Eq. (<ref>) and in the following equations, the labels $a,b$ refer here to the decays of $\tau^-$ and/or $\tau^+$
to one, two, or three measured particles. In particular, we take now the differential decay density matrices for $\tau\to 2 \pi \nu_\tau$
and $\tau\to 3 \pi \nu_\tau$ given in Appendix <ref> into account.
Using (<ref>), (<ref>), and (<ref>)
and the respective decay matrices $\mathcal{D}^a$ and $\mathcal{D}^{\bar b}$ we define
\begin{equation}
\cO_{R}^{a\bar{b}} = \frac{{\rm Tr}[\chi^R_{CP} \mathcal{D}^a \mathcal{D}^{\bar b}]}{{\rm Tr}[\chi_{SM} \mathcal{D}^a \mathcal{D}^{\bar b}]} \, , \qquad
\cO_{I}^{a\bar{b}} = \frac{{\rm Tr}[\chi^I_{CP} \mathcal{D}^a \mathcal{D}^{\bar b}]}{{\rm Tr}[\chi_{SM} \mathcal{D}^a \mathcal{D}^{\bar b}]} \, ,
\label{eq:optORI}
\end{equation}
where the trace is taken with respect to the spin indices of $\tau^-$ and $\tau^+$. Both observables are $CP$-odd
and $\cO_{R}^{a\bar{b}}$ is also $T_N$-odd while $\cO_{I}^{a\bar{b}}$ is $T_N$-even. As already
emphasized we compute the expectation values by integrating over the whole phase space.
According to the general theory discussed in Sec. <ref> and Appendix <ref>
the covariance matrix for a decay channel $a \bar{b}$ is given, for zero $\tau$ EDM, by (<ref>), (<ref>):
\begin{equation} \label{eqR:V}
V\bigl(\cO^{\prime a \bar{b}}\bigr)=V\bigl(\cO^{\prime b\bar{a}}\bigr) =
\left(\begin{array}{cc}
E_0({\cal O}_R^{' a\bar{b}} {\cal O}_R^{' a\bar{b}}) & E_0({\cal O}_R^{' a\bar{b}} {\cal O}_I^{' a\bar{b}}) \\
E_0({\cal O}_I^{' a\bar{b}} {\cal O}_R^{' a\bar{b}}) & E_0({\cal O}_I^{' a\bar{b}} {\cal O}_I^{' a\bar{b}})
\end{array}\right) \, ,
\end{equation}
\begin{equation} \label{eqR:defE}
E_0({\cal O}_R^{' a\bar{b}} {\cal O}_R^{' a\bar{b}}) \equiv \langle ({\cal O}_R^{' a\bar{b}})^2\rangle_0 \, , \quad
E_0({\cal O}_I^{' a\bar{b}} {\cal O}_I^{' a\bar{b}}) \equiv \langle ({\cal O}_I^{' a\bar{b}})^2\rangle_0 \, ,
\end{equation}
etc., denote the expectation values for $d_\tau = 0$. The expectation values for nonzero $\tau$ EDM are given by
\begin{equation} \label{eqR:Edtau}
\left(\begin{array}{c}
E\left(\frac{1}{2} {\cO}_R^{ a \bar{b}}+\frac{1}{2}{\cO}_R^{ b \overline{a}} \right) \\
E\left(\frac{1}{2} {\cO}_I^{ a \bar{b}}+\frac{1}{2}{\cO}_I^{ b \overline{a}} \right)\end{array}\right)
\, \equiv \, \left(\begin{array}{c}\langle {\cal O}_R^{a b} \rangle \\
\langle {\cal O}_I^{a b} \rangle \end{array}\right) \, = \, V\bigl(\cO^{\prime a \bar{b}}\bigr)
\left(\begin{array}{c} \Re{\hat d}_{\tau}(s) \\ \Im{\hat d}_{\tau}(s) \end{array} \right) \, .
\end{equation}
We get for the covariance matrix of the optimal estimators of $\Re{\hat d}_{\tau}(s)$ and $\Im{\hat d}_{\tau}(s)$;
see (<ref>), (<ref>) and (<ref>), (<ref>):
\begin{equation} \label{eqR:covG}
V(\gamma) \, = \, \frac{1}{N_{ab}} V^{-1}\bigl(\cO^{\prime a \bar{b}}\bigr) \, .
\end{equation}
Here $N_{ab}$ is the number of events
in the diagonal channels $a=b$ whereas for $a\neq b$ it is the sum of the events $a{\bar b}$
and ${\bar b}a$, assuming that their numbers are equal.
However, with the form of the differential cross section used in this paper considerable simplifications occur.
In the case where the $\tau$ leptons decay to one measured particle and/or to $2 \pi \nu_\tau$ where both pions
are measured we have, as shown in Appendix <ref>:
\begin{equation}\label{eqR:O1p}
\langle {\cal O}_i^{a\bar{b}}\rangle_0 = 0 \, , \quad {\cal O}_i^{' a\bar{b}} = {\cal O}_i^{a\bar{b}} \, , \quad (i=R,I) \, ,\qquad
\langle {\cal O}_R^{ a\bar{b}} {\cal O}_I^{a\bar{b}} \rangle_0 = 0 \, .
\end{equation}
That is, for these channels the respective covariance matrix (<ref>) is diagonal.
When $\tau^-$, $\tau^+$, or both $\tau$ leptons
decay to three measured pions, $\langle {\cal O}_i^{a\bar{b}}\rangle_0 = 0$ $(i=R,I)$ still holds in the one-photon approximation
(see Appendix <ref>), but the covariance matrix is no longer diagonal.
Yet we find for these decay modes that
$\langle {\cal O}_R^{ a\bar{b}} {\cal O}_I^{a\bar{b}} \rangle_0 < {\rm a\; few}\times 10^{-4}$ with numerical uncertainties
below $10^{-3}$. Therefore, within the precision of our numerical analysis the relations (<ref>) hold also for these
decay channels, and (<ref>) simplifies to[The left-hand sides of (<ref>) denote
averages according to (<ref>).]
\begin{equation} \label{eq:expoptO}
\langle \cO_{R}^{a b} \rangle = w_{a\bar b}(s)~\Re {\hat d}_\tau(s) \, , \qquad
\langle \cO_{I}^{a b} \rangle = \omega_{a\bar b}(s)~\Im {\hat d}_\tau(s) \, ,
\end{equation}
where we used the abbreviations
\begin{equation}\label{eqR:abbrev}
w_{a\bar b} \equiv \langle ({\cal O}_R^{a\bar{b}})^2 \rangle_0 \, , \qquad \omega_{a\bar b} \equiv \langle ({\cal O}_I^{a\bar{b}})^2 \rangle_0 \, .
\end{equation}
The resulting 1 s.d.
errors of the dimensionful EDM
couplings $\Re d_\tau$ and $\Im d_\tau$ are given by
\begin{equation} \label{eq:1sdsenopt}
\delta \Re d_\tau(s) = \frac{e}{\sqrt{s}}
\frac{1}{\sqrt{N_{ab}}} \frac{1}{\sqrt{\langle ({\cal O}_R^{a\bar{b}})^2 \rangle_0}} \, , \quad
\delta \Im d_\tau(s) = \frac{e}{\sqrt{s}}
\frac{1}{\sqrt{N_{ab}}} \frac{1}{\sqrt{\langle ({\cal O}_I^{a\bar{b}})^2 \rangle_0}} \, .
\end{equation}
Optimal observables $\cO_{R}^{a{\bar b}}$ and $\cO_{I}^{a{\bar b}}$ at $\sqrt{s}=10.58\, \GeV$ $(N_{\tau\tau} = 4.5 \times 10^{10})$.
$\tau^-\to$ $\tau^+\to$ $w_{a{\bar b}}$ $\sqrt{\langle (\cO_{R}^{a{\bar b}})^2 \rangle_0}$ $\delta \Re d_\tau$ ${\omega}_{a{\bar b}}$ $\sqrt{\langle (\cO_{I}^{a{\bar b}})^2 \rangle_0}$ $\delta \Im d_\tau$
$(\times \, 10^{-19} \ecm)$ $(\times\, 10^{-19} \ecm)$
$\pi^-\nu$ $\pi^+ \bar\nu$ $0.111$ $0.333$ $2.45$ $0.352$ $0.593$ $1.37$
$\pi^- \pi^0\nu$ $\pi^+ \pi^0 \bar\nu$ $0.111$ $0.333$ $1.04 $ $0.352$ $0.593$ $0.58 $
$\pi^-\pi^-\pi^+\nu$ $\pi^+ \pi^+\pi^-\bar\nu$ $0.111$ $0.333$ $2.84$ $0.352$ $0.593$ $1.59$
$\pi^-\nu$ $\pi^+ \pi^0 \bar\nu$ $0.111$ $0.333$ $1.13$ $0.352$ $0.593$ $0.63 $
$\pi^-\nu$ $ \pi^+ \pi^+\pi^-\bar\nu$ $0.111$ $0.333$ $1.86$ $0.352$ $0.593$ $1.05$
$\pi^-\pi^0\nu$ $ \pi^+ \pi^+\pi^- \bar\nu$ $0.111$ $0.333$ $1.21 $ $0.352$ $0.593$ $0.68 $
$\ell^-\nu\bar\nu$ ${\ell'}^+ {\bar\nu}\nu$ $ 0.004 $ $0.064 $ $4.04$ $0.055 $ $0.235 $ $1.08$
$\ell^-\nu\bar\nu$ $\pi^+ \bar\nu$ $0.020 $ $0.142 $ $2.26$ $0.162 $ $0.402 $ $0.80$
$\ell^-\nu\bar\nu$ $\pi^+\pi^0 \bar\nu$ $0.020 $ $0.142 $ $1.47$ $0.162 $ $0.402 $ $0.52 $
$\ell^-\nu\bar\nu$ $ \pi^+ \pi^+\pi^-\bar\nu$ $0.020 $ $0.142 $ $2.43$ $0.162 $ $0.402 $ $0.86$
Table <ref> contains our results for the expectation values defined in Eq. (<ref>)
and for the square roots of the variances of the observables (<ref>) for several $\tau^\mp$ decays to one, two and/or three measured particles.
The numbers in this table show that taking into account the full kinematic information on the hadronic system in the $\tau\to 2 \pi \nu_\tau$
and $\tau\to 3 \pi \nu_\tau$ decays results in maximal $\tau$-spin analyzing power [34, 35], as is the case in the decay $\tau\to \pi \nu_\tau$.
In addition, the resulting 1 s.d. sensitivities to the real and imaginary parts of the $\tau$ EDM form factor
are given in Table <ref>, assuming again $4.5 \times 10^{10}$ $\tau$-pair events.
The 1 s.d. statistical errors $\delta\Re d_\tau$ and $\delta\Im d_\tau$ exhibited in Table <ref> signify that taking into account the channels where one or both $\tau$ leptons
decay to two and/or three measured pions yields a significant improvement in the sensitivity to the $\tau$ EDM form factor.
Comparing for each channel the accuracies $\delta\Re d_\tau$ and $\delta\Im d_\tau$ exhibited in Table <ref> with those in
Tables <ref> and <ref> shows that, as expected, the optimal observables (<ref>) are significantly more sensitive to the $\tau$ EDM
than the observables $T_{33}$ and $\widehat{Q}_{33}$.
If the measurement errors of the various exclusive $\tau^+ \tau^-$ decay modes are uncorrelated, we may add in quadrature
the statistical errors of $\Re d_\tau$ and $\Im d_\tau$
attainable for each channel:
\begin{equation} \label{dredab}
\delta\Re d_\tau = \left(\sum\limits_{ab} \frac{1}{\left(\delta\Re d_\tau\right)^2_{ab}} \right)^{-1/2} \, ,
\end{equation}
and analogously for $\delta\Im d_\tau$.
Performing these quadratures with the uncertainties listed in Tables <ref>, <ref>,
and <ref> yields
the 1 s.d. errors $\delta\Re d_\tau$ and $\delta\Im d_\tau$ given in Table <ref>.
As to the optimal observables we assumed here for the purpose of comparison that they are measurable for
all channels listed in Table <ref>.
For the leptonic modes this may not be possible in an unambiguous way; see below.
The numbers in Table <ref> show that
the sensitivity to
$\Re d_\tau$ is improved by a factor of about 6 with the optimal observable $\cO_R$ as compared to using
the simple ones, whereas the
sensitivity to $\Im d_\tau$ is improved by a factor of about 4.
Ideal 1 s.d. statistical errors on $\Re d_\tau$ and $\Im d_\tau$ that result from adding the
respective uncertainties attainable in the various decay channels in quadrature.
$\delta\Re d_\tau \; [\ecm]$ $\delta\Im d_\tau \; [\ecm]$
$\langle T_{33}\rangle_{ab}$ $\langle\widehat{T}_{33}\rangle_{ab}$ $\langle \cO_{R}^{a b} \rangle$
$ \langle Q_{33}\rangle_{ab}$ $ \langle \widehat{Q}_{33}\rangle_{ab}$ $\langle \cO_{I}^{a b} \rangle$
$ 2.93 \times 10^{-19}$ $ 4.53 \times 10^{-19} $ $ 5.1 \times 10^{-20} $
$1.23 \times 10^{-19} $ $9.4 \times 10^{-20}$ $2.4 \times 10^{-20} $
We briefly discuss the measurability of the observables used in this section.
The KEKB accelerator is an asymmetric $e^+ e^-$ collider; particle momenta measured in
the laboratory frame can of course be transformed to the $e^+ e^-$ c.m. frame.
The simple $CP$ observables (<ref>) – (<ref>) applied to the $\tau^+ \tau^-$ decay channels
listed in Tables <ref>, <ref> require the momenta of charged mesons and of $ e, \mu$ in the
$e^+ e^-$ c.m. frame. They can be straightforwardly measured, except for the momentum of $\rho^\pm$ whose
determination requires
the reconstruction of the decay $\rho^\pm \to \pi^\pm \pi^0$.
The optimal observables involve the momenta of various particles from $\tau^\pm$ decay in the
respective $\tau^\pm$ rest frame.
This requires
the knowledge of the $\tau^\pm$ momenta in the $e^+ e^-$ c.m. frame. If both $\tau^+$ and $\tau^-$ decay
semihadronically their momenta can be
reconstructed in an unambiguous way [36]. If one of the $\tau$ leptons decays semihadronically and
the other one to either $e$ or $\mu$, one
may discard radiative events in this class such that the $\tau^+$ and $\tau^-$ in the remaining events are,
to good approximation, back to back
and carry half of the c.m. energy in the $e^+ e^-$ frame. If the $\tau$ momentum can
be reconstructed in the semihadronic decay, e.g. by reconstructing
the $\tau$ production and decay vertices, the momentum of the leptonically decaying $\tau$ can be inferred. If both $\tau$ leptons decay
leptonically the determination of their momenta is not possible in an unambiguous way.
Therefore, we discard the results for the $\ell \ell'$ channels in Table <ref>
and add in quadrature the statistical errors of $\Re d_\tau$ and $\Im d_\tau$
attainable with the events listed in Table <ref> where both $\tau$'s decay semihadronically
and for the case where the semihadronic-leptonic decays of $\tau^+ \tau^-$
are added to the purely semihadronic events. The resulting 1 s.d. errors are given in Table <ref>.
The numbers in this table and in Table <ref> show that restriction to purely semihadronic
$\tau^+ \tau^-$ decays does not lead to a significant decrease in sensitivity to $\Re d_\tau$ and $\Im d_\tau$.
Ideal 1 s.d. statistical errors on $\Re d_\tau$ and $\Im d_\tau$ that result from adding in quadrature the
respective uncertainties attainable with the optimal observables $\cO_{R}^{a b}$
and $\cO_{I}^{a b}$ in the semihadronic decays $( h h)$
and in the semihadronic and semihadronic-leptonic $( h h + h \ell)$ decays of $\tau^+ \tau^-$.
$\delta\Re d_\tau \; [\ecm]$ $\delta\Im d_\tau \; [\ecm]$
$ h h:$ $ 5.8 \times 10^{-20}$ $3.2 \times 10^{-20} $
$ h h + h \ell:$ $ 5.1 \times 10^{-20}$ $2.5 \times 10^{-20} $
Next we investigate the effects of cuts on the sensitivities to the $\tau$ EDM. A full-fledged Monte Carlo analysis
with detailed cuts is beyond the scope of this paper. We analyze in the following only the expectation values of the optimal observables
in the channels where both $\tau$ leptons decay semihadronically, as these observables and decay
modes appear to have the highest sensitivity to $d_\tau$ and allow for an unambiguous reconstruction of the $\tau^\pm$ momenta.
We apply the following $CP$-invariant phase-space cuts on the final-state pions in the $e^+e^-$ c.m. frame:
\begin{equation}\label{eq:picuts}
23^\circ < \theta^* < 157^\circ \, , \qquad p_T > 0.2 \, \GeV \, ,
\end{equation}
where $\theta^*$ is the polar angle of a pion with respect to the $e^+e^-$ beam
and $p_T$ its transverse momentum.[The cut on $\theta^*$ is inspired by the acceptance of the
Belle II detector in the KEKB laboratory frame [19].]
Table <ref> contains the resulting
coefficients $w_{a{\bar b}}$ and ${\omega}_{a{\bar b}}$ of
the expectation values of $\cO_{R}^{a{\bar b}}$ and $\cO_{I}^{a{\bar b}}$, respectively, defined in (<ref>).
The event numbers and sensitivities given in Table <ref> are
estimated by assuming an integrated luminosity of $50~{\rm ab}^{-1}$ that corresponds to
assuming $N_{\tau\tau}=4.5\times 10^{10}$ in the case of no cuts. The expectation values are somewhat increased by the cuts while
the event numbers are, of course, diminished.
The resulting overall sensitivities are given in Table <ref>. Comparing these numbers with those of Table <ref> shows
that the cuts (<ref>) lead only to a slight decrease in sensitivity to the $\tau$ EDM.
Optimal observables $\cO_{R}^{a{\bar b}}$ and $\cO_{I}^{a{\bar b}}$ at $\sqrt{s}=10.58\, \GeV$ for the semihadronic $\tau$ decay channels
with cuts specified in (<ref>). In the case of nondiagonal channels
the event numbers $N_{ab}$ include those of the charge-conjugate mode.
$\tau^-\to$ $\tau^+\to$ $N_{ab}$ $w_{a{\bar b}}$ $\delta \Re d_\tau$ ${\omega}_{a{\bar b}}$ $\delta \Im d_\tau$
$(\times \, 10^{-19} \ecm)$ $(\times\, 10^{-19} \ecm)$
$\pi^-\nu$ $\pi^+ \bar\nu$ $4.21\times 10^8$ $0.128$ $2.54 $ $0.359$ $1.52 $
$\pi^- \pi^0\nu$ $\pi^+ \pi^0 \bar\nu$ $16.88 \times 10^8$ $0.137 $ $1.23 $ $0.390$ $0.73 $
$\pi^-\pi^-\pi^+\nu$ $\pi^+ \pi^+\pi^-\bar\nu$ $1.73 \times 10^8$ $0.139 $ $3.81$ $0.408 $ $2.22 $
$\pi^-\nu$ $\pi^+ \pi^0 \bar\nu$ $16.53\times 10^8$ $0.135$ $1.25$ $0.386 $ $0.74$
$\pi^-\nu$ $ \pi^+ \pi^+\pi^-\bar\nu$ $5.18 \times 10^8$ $0.137 $ $2.21 $ $0.401 $ $1.29$
$\pi^-\pi^0\nu$ $ \pi^+ \pi^+\pi^- \bar\nu$ $10.74 \times 10^8$ $0.138 $ $1.53$ $0.401 $ $0.90 $
Ideal 1 s.d. statistical errors on $\Re d_\tau$ and $\Im d_\tau$ that result from adding in quadrature the
respective uncertainties attainable with the optimal observables $\cO_{R}^{a b}$
and $\cO_{I}^{a b}$ in the semihadronic decays $( h h)$ of $\tau^+ \tau^-$ given in Table <ref>.
$\delta\Re d_\tau \; [\ecm]$ $\delta\Im d_\tau \; [\ecm]$
$ h h:$ $ 6.8 \times 10^{-20}$ $4.0 \times 10^{-20} $
Moreover, the following remark is in order. As already indicated below Eq. (<ref>) our results for the normalized
expectation values listed in Tables <ref>, <ref>, <ref>, and <ref> do not depend
on the fact that there is a resonance enhancement at $\sqrt{s} = 10.58$ GeV; these numbers hold also for the
direct continuum production of $\tau$ pairs.
In addition, we emphasize again that the event numbers, respectively the integrated luminosity that we use for our
sensitivity estimates to the $\tau$ EDM
are expectations taken from [19].
The sensitivity to the $\tau$ EDM that the Belle II experiment may eventually achieve with
purely semihadronic $\tau^+ \tau^-$ decays
was investigated also in [16]. The authors of this paper use the term proportional to $d_\tau$ of the
matrix element for $e^+e^-\to \tau^+ \tau^- \to h \nu_\tau h' {\bar\nu}_\tau$ as optimal observable. It is evaluated with
the momenta of the mesons and the reconstructed one of the neutrinos. The real and imaginary parts of the $\tau$ EDM are not separately
determined. Assuming the same $\tau^+\tau^-$ event number as we did above,
the authors of Ref. [16] find that a 1 s.d. statistical sensitivity $\delta|d_\tau| = 2 \times 10^{-19} \ecm$
can be achieved with their approach.
§ THE $\TAU$ EDM FORM FACTOR IN SOME SM EXTENSIONS
In the SM the EDM $d_\ell$ of a charged lepton is extremely tiny and generated only at high loop order.
The dominant short-distance contribution to $d_\ell$ is thought to arise via Kobayashi-Maskawa phase induced four-loop
contributions that contain, for instance, the induced EDM form factor of the $W$ boson. It can be estimated to be of the order
$d_\tau \sim {\cal O}(10^{-42})~\ecm$.
(One may take, for instance, the estimate of [37] for $d_e$ and apply it to the
$\tau$ lepton.) Recently it was pointed out that
long-distance hadronic contributions are considerably larger [38]. For the $\tau$ EDM is was found that these
contributions amount to $d_\tau \simeq -7.3\times 10^{-38}~\ecm$ [38]. Nevertheless, this is undetectable for the time being.
Thus, the detection of a nonzero particle EDM, in particular of the $\tau$ lepton, in a present-day experiment or
one in the foreseeable future would be evidence for a new type of $CP$ violation.
In this section we consider three SM extensions with $CP$-violating interactions that generate EDM
form factors of fundamental fermions already at one loop. The models we are interested in have
$CP$-violating Yukawa couplings. These interactions can induce a $\tau$ EDM that can be much larger
than the electron EDM generated in these models.[We recall that in models with Higgs-Yukawa-like $CP$-violating couplings
the dominant contribution to the electron EDM occurs at two loops [39].]
We compute the $\tau$ EDM at one loop in a type-II two-Higgs-doublet model and in two scalar leptoquark models and investigate its
potential magnitude in the timelike region $q^2 \sim (10~\GeV)^2$, taking into account phenomenological constraints, in particular the
tight upper bound (<ref>) on the electron EDM.
One-loop diagrams that contribute to the $\tau$ EDM form factor in the models considered in Section <ref>.
In the type-II 2HDM only diagram a) contributes and the dashed and solid internal lines correspond to $h_j$ $(j=1,2,3)$ and $\tau$, respectively.
In the leptoquark models both diagrams contribute and the dashed and solid internal lines correspond to a spin-zero leptoquark and the top quark, respectively.
§.§ Type-II two-Higgs doublet extension
In two-Higgs doublet models (2HDM) the field content of the SM is extended by an additional Higgs doublet $H_2$. We consider
here as an example
the so-called type-II model. It is defined by its Yukawa coupling structure: the
doublet $H_1$ is coupled to right-chiral down-type quarks and
charged leptons, while $H_2$ is coupled to right-chiral up-type
quarks only. By construction, flavor-changing neutral currents are absent at tree
level in this model. Assuming a $CP$-violating
Higgs potential $V(H_1, H_2)$ the particle spectrum of the 2HDM contains three neutral
Higgs bosons $h_j$ $(j=1,2,3)$ that are $CP$ mixtures. In flavor-conserving 2HDM their Yukawa couplings to quarks
and leptons are of the form
\begin{equation}
{\cal L}_{Y,f} \; = \; - (\sqrt{2} G_F)^{1/2} m_f \left[ a_{f,j} {\bar f} f \, - \, b_{f,j} {\bar f}i \gamma_5 f \right] h_j \, ,
\label{yukint}
\end{equation}
where $f=q,\ell$, $G_F$ is the Fermi constant, and the reduced Yukawa couplings $ a_{f,j}$ and $b_{f,j}$ depend on the specific type of 2HDM.
In the type-II model the reduced couplings of the mass eigenstates $h_j$ to the $\tau$ lepton are
(we use here the conventions of [40]):
\begin{equation} \label{eq:abHiko}
a_{\tau,j} = R_{j1}/\cos\beta \, , \qquad b_{\tau,j} = R_{j3}\tan\beta \, .
\end{equation}
Here $\tan\beta = {\v}_2/{\v}_1$ is the ratio of the vacuum expectation values of the two Higgs doublet fields, and $(R_{ij})$
is a
real orthogonal matrix that relates the $CP$ eigenstates and the mass eigenstates of the three physical neutral Higgs bosons.
The relations (<ref>) hold also for the other charged leptons and the down-type quarks.
(For up-type quarks, see for instance [40].) If $ a_{f,j} b_{f,j} \neq 0$ then (<ref>) violates $CP$.
Here we identify $h_1$ with the $125\, \GeV$ Higgs boson and assume that $h_2$ and $h_3$ are heavier than $400\, \GeV$.
The exchange of the $h_j$ induces a $\tau$ EDM at one loop shown by the diagram Fig. <ref>a.
With the convention of Eq. (<ref>) we get[The real and imaginary parts of the EDM form factor of a fermion were
computed for a class of 2HDM including the type-II model in [41] and evaluated for the top quark.]
\begin{equation} \label{eqM:dtcomp}
d_\tau(s) = \sum\limits_{j=1}^3 a_{\tau,j} b_{\tau,j }d^{(j)}_\tau(s) \, ,
\end{equation}
\begin{equation} \label{eqM:d2hdm}
d^{(j)}_\tau(s) = - \frac{e \sqrt{2}G_F m_\tau^3}{4 \pi^2 s \beta_\tau^2}
\left[B_0(s,m_\tau^2,m_\tau^2) -B_0(m_\tau^2,m_j^2, m_\tau^2)
+ m_j^2 C_0(s,m_\tau^2,m_j^2,m_\tau^2)\right] \, ,
\end{equation}
where $\beta_\tau=(1-4 m_\tau^2/s)^{1/2}$ and $m_j$ is the mass of $h_j$.
The functions $B_0$ and $C_0$ denote the standard scalar one-loop two-point and three-point functions [42].
For $s \ge 4 m_\tau^2$ the
EDM form factor (<ref>) has both a real and an imaginary part.
However, apart from the upper bound (<ref>) on the electron EDM
existing constraints from experiments at the LHC preclude a $\tau$ EDM of order $10^{-20}\ecm$ or larger in this model.
A recent analysis of the decay of the 125 GeV Higgs boson to $\tau^+\tau^-$ by the CMS experiment
restricts the size of a potentially existing pseudoscalar coupling of $h_1$ to the $\tau$ lepton:
$|b_{\tau,1}/a_{\tau,1}| \le 0.38$ at $68\%$ C.L. [43]. Searches for additional neutral Higgs bosons
with decays to $\tau^+\tau^-$ exclude Higgs-boson masses of about 400 GeV and below for a large range of Higgs coupling to $\tau$ leptons;
see, for example, [44, 45] and references therein.
We exemplify the order of magnitude of $d_\tau$ that is compatible with these constraints by assuming the
masses of the Higgs bosons $h_2$ and $h_3$ to be $m_2= 500$ GeV and $m_3=800$ GeV, respectively. Moreover, we choose
$\tan\beta = 1$ and the angles of the mixing matrix $R$, in the parametrization of [40],
to be $\alpha_1=\alpha_3= 0.785$, $\alpha_2=0.209$. The resulting real and imaginary parts of the $\tau$ EDM (<ref>)
are given in Table <ref>
for several c.m. energies in the energy range considered in this paper.
Values of the real and imaginary parts of the $\tau$ EDM form factor (<ref>) in the type-II
2HDM, evaluated with the parameter choice given in the text.
$\sqrt{s}$ [GeV] 3.6 4 10.58 12
$\Re d_\tau(s)$ $[10^{-24} \ecm]$ 2.24 2.13 1.38 1.30
$\Im d_\tau(s)$ $[10^{-24} \ecm]$ 0.13 0.38 0.77 0.78
By and large the order of magnitude of the $\tau$ EDM form factor listed in Table <ref> is characteristic for a large class of Higgs models.
Significantly larger values of $\Re d_\tau(s)$ and $\Im d_\tau(s)$ would be possible if, for instance, Higgs bosons exist
with exclusive $CP$-violating couplings to the third generation of quarks and leptons, such that the stringent constraint (<ref>)
on the electron EDM can be evaded.
§.§ Spin-zero leptoquarks
Leptoquarks, whose interactions connect a lepton and a quark, occur naturally in unified models of strong and electroweak interactions.
In recent years they have come again into the focus of numerous investigations
in the context of possible explanations of semileptonic $B$ and $D$
meson decay and muon $(g-2)$ anomalies; see, for instance, [46, 47, 48, 49, 50] and references therein.
Here we are interested in spin-zero leptoquarks with $CP$-violating Yukawa couplings.
They can generate EDMs of the muon and tau lepton that are significantly larger than that of the electron,[A recent analysis of the effects
of spin-zero leptoquarks on the EDMs of leptons, quarks, and nucleons
was made in [51].] as pointed out some time ago
in [52, 24] (cf. also [53]).
We consider in the following two different spin-zero leptoquark models, namely the SM extended by
a weak ${\rm SU(2)}$ leptoquark doublet $\Phi$ with ${\rm SU(3)_c\times SU(2)_L \times U_Y(1)}$ quantum numbers
$\Phi(3,2,7/6)$ (model I) and a SM extension by a weak singlet $S$ with quantum numbers $S(3,1,-1/3) $ (model II).
The gauge-invariant interaction Lagrangians are [54]
\begin{equation}
\label{eq:LIdoub}
{\cal L}_I = [\overline{L_L} \Lambda_L \epsilon u_R + \overline{e_R} \Lambda_R Q_L]~\Phi^\dagger \; + \; {\rm H.c.} \,,
\end{equation}
\begin{equation}
\label{eq:LIsing}
{\cal L}_{II} = [\overline{L_L^c} Y_L \epsilon Q_L + \overline{e^c_R} Y_R u_R]~S^\dagger \; + \; {\rm H.c.} \, .
\end{equation}
Here $L_L =(\nu_{iL}, e_{i,L})^T$, $Q_L = (u_{i,L}, d_{i,L})^T$, $e_R =(e_{i,R})$, $u_R=(u_{i,R})$, where $i=1,2,3$ is a generation index.
The label $c$ denotes charge conjugation. The $2\times 2$ matrix $\epsilon= i \tau_2$ acts on the SU(2) indices.
The electric charge (in units of $e>0$) of $S$ is $Q_S=-1/3$. For the components of the doublet $\Phi = (\varphi, \varphi')^T$
we have $Q_\varphi = 5/3$ and $Q_{\varphi'} = 2/3.$
The $\Lambda_L$, $\Lambda_R$ and $Y_L$, $Y_R$ denote complex
$3\times 3$ matrices in flavor space. Usually the interactions (<ref>) and (<ref>) are defined in the weak basis
and are rotated, after electroweak symmetry breaking, to the mass basis. We can choose a basis in which the Yukawa matrices of the up-type
quark and of the charged-lepton couplings to the SM Higgs boson are already diagonal. Then only the down-type quark and neutrino fields must be
rotated with their respective
mixing matrices when one transforms to the mass basis. The interactions in (<ref>) involving charged leptons and up-type quarks, with
which we are concerned here, remain unaffected.
We assume that the off-diagonal elements of the matrices $\Lambda_L$, $\Lambda_R$ and $Y_L$, $Y_R$ in generation space are very small and can
be neglected. Let us denote
\begin{equation} \label{eqM:defcop}
\lambda_J = (\Lambda_J)_{33} \quad \text{and} \quad y_J = (Y_J)_{33} \, , \quad J=L, R \, ,
\end{equation}
\begin{equation}
f_{\rm I} = {\rm Im}(\lambda_L^* \lambda_R) \quad \text{and} \quad f_{\rm II} = {\rm Im}(y_R^* y_L) \, .
\end{equation}
If $ f_{\rm I}\neq 0$ $(f_{\rm II} \neq 0)$ then the interaction Eq. (<ref>) (Eq. (<ref>)) generates a nonzero $\tau$ EDM at one loop.
It is represented by Figs. <ref> a) and <ref> b) where the internal fermion and boson lines correspond to the top
quark $t$ and the $\varphi$ leptoquark in model I and to $t$ and $S$ in model II, respectively.
The $\tau$ EDM form factor is given by [24]
\begin{equation}\label{eq:tauEDMdouS}
d_\tau(s) = e m_t N_c \frac{f_\kappa}{8 \pi^2}
\frac{1}{s \beta_\tau^2} \left[ Q_t K_t(s) - Q_\chi K_\chi(s) \right] \, , \quad \kappa = {\rm I, II} \,
\end{equation}
where $N_c=3$, $m_t$ is the mass of the top quark which provides the chirality flip, $Q_t=2/3$ and $Q_\chi = 5/3~(-1/3)$
in case of model I (II), where $\chi$ denotes either $\varphi$ or $S$. Moreover
\begin{eqnarray}
K_t(s) & = & B_0(s,m_t^2,m_t^2) - B_0(m_\tau^2,m_t^2,m_\chi^2)
+(m_\chi^2+m_\tau^2 - m_t^2) C_0(s, m_t^2,m_\chi^2,m_t^2) \, , \label{eqM:Kt}\\
K_\chi(s) & = & B_0(s,m_\chi^2,m_\chi^2) - B_0(m_\tau^2,m_t^2,m_\chi^2)
+(s/2 + m_t^2 - m_\chi^2- m_\tau^2) C_0(s, m_\chi^2,m_t^2,m_\chi^2) \, . \label{eqM:Kchi}
\end{eqnarray}
Here $m_\chi$ is the mass of $\varphi~ (S)$ in the case of model I (II). Because
$m_t, m_\varphi, m_S \gg \sqrt{s}$ in the kinematic range that we consider here, the $\tau$ EDM form factor (<ref>)
is real.
In order to estimate the potential size of $d_\tau$ we choose the leptoquark masses $m_\chi=1.5$ TeV ($\chi = \varphi, S$) which are compatible with the
experimental bounds from LHC [55, 56] and the constraints from the anomalous magnetic moments of the electron
and muon [49]. For comparison we evaluate (<ref>) also for $m_\chi =1$ TeV and $2$ TeV.
With $m_t=172.4$ GeV [1] we get from (<ref>) the values listed in
Table <ref>.
Values of the $\tau$ EDM form factor (<ref>)
in the doublet (I) and singlet (II) leptoquark model. The numbers in the first, second, and third row of each model
are obtained with
$m_\chi =1$, $1.5$, and $2$ TeV ($\chi = \varphi, S$). Moreover, we use $m_t=172.4$ GeV.
$\sqrt{s}$ [GeV] 3.6 4 10.58 12
Model I: $\Re d_\tau(s)$ $[10^{-20} f_{\rm I} ~ \ecm]$ 14.44 14.44 14.45 14.45
7.89 7.89 7.89 7.89
5.04 5.04 5.04 5.04
Model II: $\Re d_\tau(s)$ $[10^{-20} f_{\rm II} ~ \ecm]$ 8.85 8.85 8.86 8.86
5.24 5.24 5.25 5.25
3.51 3.51 3.51 3.51
The numbers in Table <ref> show that for a given leptoquark mass the form factor $\Re d_\tau(s)$ is essentially flat in the
kinematic range considered here. So far, the experimental bounds on the $CP$ parameters $f_{\rm I}$, $f_{\rm II}$ are not stringent.
Using the experimental bound (<ref>) and the numbers given in Table <ref> for $\sqrt{s}=10.58$ GeV and $m_\chi =1.5$ TeV,
we get
\begin{equation} \label{eqM:bound12}
|f_{\rm I}| < 570 \, , \qquad |f_{\rm II}| < 857 \quad \text{for} \; \; m_\chi = 1.5~{\rm TeV} \, .
\end{equation}
If leptoquark couplings to the $\tau$ lepton and the $c$ quark are taken into account in (<ref>) and (<ref>) then $d_\tau(s)$ develops also
an imaginary part for $\sqrt{s} > 2 m_c$. However, away from the charm threshold, the $c$-quark contribution to $d_\tau$ is suppressed in magnitude by the factor $m_c/m_t\sim 10^{-2}$ as compared
to the leading contribution (<ref>),
regardless of additional suppression due to small off-diagonal Yukawa couplings.
One may expect that the Yukawa couplings of the spin-zero leptoquarks are of the Higgs-boson type. Then the (diagonal) couplings of $\varphi$ and $S$
in (<ref>) and (<ref>) will be proportional to the right-handed fermion involved. That is,
\begin{equation} \label{eqM:YukHL}
\lambda_L \sim m_t/M_{\rm I}\, , \quad \lambda_R \sim m_\tau/M_{\rm I} \, , \qquad y_L \sim m_\tau/M_{\rm II} \, , \quad y_R \sim m_t/M_{\rm II} \, ,
\end{equation}
where $ M_{\rm I}$ and $ M_{\rm II}$ are mass scales that are expected to be larger than the electroweak symmetry breaking scale $\v=246$ GeV.
In this case the magnitude of $\Re d_\tau(s)$ will be smaller by a factor of at least $10^{-2}$ than the numbers
listed in Table <ref>.
§.§ Box contributions
The one-loop $S$-matrix element of $e^+e^-\to \tau^+ \tau^-$ can receive in SM extensions also one-particle
irreducible $CP$-violating
box contributions that involve Lorentz structures such as $({\bar e} e) ({\bar\tau}i\gamma_5 \tau)$.
Here we argue that in the models considered in
Sections <ref> and <ref> these contributions that are depicted in
Fig. <ref> can be neglected compared to those of the $\tau$ EDM form factors.
One-loop box diagrams that contribute to the $S$-matrix element of $e^+e^-\to \tau^+ \tau^-$ in the models considered in
Sections <ref> and <ref>.
In the type-II 2HDM only diagram <ref> a) contributes and the dashed and solid internal lines correspond to $h_j$ $(j=1,2,3)$
and $f=e$, $f'=\tau$, respectively.
In the leptoquark models both diagrams can contribute and the dashed and solid internal lines correspond
to a spin-zero leptoquark and an up-type quark, respectively. Crossed diagrams are not shown.
In the type-II 2HDM only diagram a) appears. From the Yukawa interaction (<ref>) one obtains that this
contribution is proportional to $G_F m^2_e$.
Thus this contribution to the $S$-matrix element of $e^+e^-\to \tau^+ \tau^-$ is negligible compared to that
of the $\tau$ EDM form factor (<ref>).
As to the spin-zero leptoquark models: If one considers interactions (<ref>), (<ref>) that are diagonal
in generation space, then only
diagram a) contributes with $f=u$, $f'=t$ and this contribution entails a suppression factor $m_u/\sqrt{s}$ where $m_u$
is the mass of the $u$ quark.
In the case of interactions that are nondiagonal in generation space, diagrams <ref> a) and <ref> b) contribute,
but those contributions that involve leptoquark couplings
between the electron and the $c$ and $t$ quark contain off-diagonal matrix elements
$(\Lambda_J)_{1 j}$ or $(Y_J)_{1j}$ $(J=L,R, j\neq 1)$ that are
small due to experimental constraints (see, e.g., [49, 50]).
Thus we conclude that within the above SM extensions the $CP$-violating part of the one-loop $S$-matrix element
of $e^+e^-\to \tau^+ \tau^-$ is
given to very good approximation by the contribution from the $\tau$ EDM form factor.
In addition, we remark that the one-loop EDM form factors computed
in Sections <ref> and <ref> are gauge invariant.
Needless to say, the contribution of the electron EDM form
factor to this matrix element is completely irrelevant.
§ CONCLUSIONS
The huge data samples of $\tau^+ \tau^-$ production and decay that will eventually be recorded at existing low-energy $e^+ e^-$ colliders will
allow, among other investigations, the search for a $\tau$ electric dipole form factor $d_\tau(s)$ with a precision that is significantly
higher than existing bounds.
We reconsidered the issue of using simple and optimal $CP$ observables for such measurements. We discussed the general
formalism of optimal observables and applied it
to two $CP$-odd observables based on $CP$-odd
$\tau$-spin correlations and polarization asymmetries that are sensitive to the real and imaginary parts of $d_\tau(s)$, respectively.
Special emphasis was put on the covariance of these observables.
In our numerical analysis we computed the expectation values and covariances of the optimal $CP$ observables for
$\tau$-pair production in $e^+ e^-$ collisions at the $\Upsilon(4{\rm S})$ resonance with subsequent decays of $\tau^\pm$ to major
leptonic or semihadronic modes.
These results hold also for the continuum production of $\tau$ pairs at $\sqrt{s}=10.58$ GeV.
For the $\tau$ decays to two pions and three charged pions we took the full kinematic information
of the hadronic system into account by
incorporating the respective differential $\tau^\pm$ decay density matrices into the optimal observables. In this
way the maximal $\tau$-spin analyzing power
is obtained also with these decay modes.
Assuming that the Belle II experiment will eventually record and analyze $4.5 \times 10^{10}$ $\tau^+ \tau^-$ events
at $\sqrt{s}=10.58$ GeV we found that
with purely semihadronic $\tau^+\tau^-$ decays
1 s.d. sensitivities $\delta \Re d_\tau = 5.8 \times 10^{-20} \ecm$ and $\delta \Im d_\tau = 3.2 \times 10^{-20} \ecm$ can be
obtained with these optimal observables.
For $\Re d_\tau$ this is better than a factor of 5 and for $\Im d_\tau$ better than a factor of 3 as the
sensitivities attainable with the simple $CP$-odd observables that
we analyzed, too. Including events where one (or both) $\tau$ leptons decay leptonically
does not lead to a significant increase in sensitivity to $\Re d_\tau$ and $\Im d_\tau$.
These results were obtained without cuts. We analyzed also the sensitivity of the optimal observables to $\Re d_\tau$ and $\Im d_\tau$
in the purely semihadronic $\tau^+\tau^-$ decay channels by applying cuts on the final-state pions. Assuming an integrated luminosity
of 50 ab$^{-1}$, which corresponds to the above number of $\tau^+ \tau^-$ events in the case of no cuts,
we obtained $\delta \Re d_\tau = 6.8 \times 10^{-20} \ecm$ and $\delta \Im d_\tau = 4.0 \times 10^{-20} \ecm$.
That is, the 1 s.d. sensitivities decrease only slightly.
Furthermore, we discussed a few SM extensions with nonstandard $CP$ violation that predict an nonzero $\tau$ EDM already
at one-loop order. The tight experimental upper bound on the
electron EDM, experimental results from the LHC
on the $CP$ nature of the 125 GeV Higgs boson, and bounds on the mass and couplings of new particles severely
constrain the potential magnitude of $d_\tau$.
Within the type-II 2HDM, which we consider in this context to be exemplary for a large class of two-Higgs
doublet extensions of the SM, the $\tau$ EDM form factor turns out to
be too small to be detected in the foreseeable future. However, in scalar leptoquark extensions of
the SM $\Re d_\tau(s) \sim 10^{-20} \ecm$ is still possible in the energy range considered
in this paper. In any case, future $\tau$ EDM measurements with the Belle II and also the BES III
experiment using optimal observables will provide significant
information about new sources of $CP$ violation.
§ ACKNOWLEDGMENTS
The authors thank M. Diehl, R. Karl, F. M. Krinner, F. Nerling, A. Rostomyan, and A. Szczurek for discussions
and correspondence, and C. Ewerz for help with one figure. The work of L.C. was supported by the Deutsche Forschungsgemeinschaft under Grant No.
396021762-TRR 257.
§ $\TAU$ DECAY DENSITY MATRICES
Here we list the density matrices that describe several major decays of polarized $\tau^\mp$.
Most of them given below are
used in section <ref>. The kinematic variables in this
appendix are defined in the respective $\tau^\pm$ rest frame unless stated otherwise.
The decay density matrices are computed in the Standard Model; potential $CP$-violating effects
in $\tau$ decays are not taken into account.
First we consider $\tau^\mp$ decays into one charged prong, $\tau^\mp\to a^\mp +X$
with particle multiplicity $\langle n_a\rangle = 1$. The charged particle $a^\mp$ acts as the $\tau^\mp$
spin analyzer.
Assuming $CP$ invariance in the decays of $\tau^\mp$ we have in the $\tau$ rest frame:
\begin{equation} \label{eq:cptaupm}
\big{\langle}a^-({\bq}), X|{\cal T}|\tau^{-}_\beta\big{\rangle} =
\eta_a \big{\langle}a^+(-{\bq}), X^{CP}|{\cal T}|\tau^{+}_\beta\big{\rangle} \, ,
\end{equation}
where $X^{CP}$ is the $CP$ transform of $X$ and $\eta_a =\pm 1$. If $a$ denotes a lepton, we have $\eta_a =1$; for $a=$ meson,
$\eta_a$ is the product of the intrinsic parity and charge-parity quantum numbers of $a$. Thus, for a pion ($\rho$ meson)
we get $\eta_\pi= -1$ $(\eta_\rho =+1)$.
Equation (<ref>) implies for the $\tau^\mp$ decay density matrices
\begin{equation} \label{eq:cpDD}
\mathcal{D}^{a^-}_{\beta'\beta}(\tau^- \to a^-({\bq})+X)
= \mathcal{D}^{a^+}_{\beta'\beta}(\tau^+ \to a^+(-{\bq})+X^{CP}) \, .
\end{equation}
The respective decay density
matrix $\mathcal{D}^a= (\mathcal{D}^a_{\beta'\beta})$ defined in (<ref>) and (<ref>) is of the form
\begin{equation} \label{eqA:1prong}
\frac{d^{3}q_{\mp}}{(2\pi)^{3}2E_{a^\mp}} \,\mathcal{D}^{a^\mp}(\tau^\mp \to a^\mp(q_{\mp})+X)
= dE_{a^\mp} \frac{d\Omega_{a^\mp}}{4\pi}~n(E_{a^\mp})\bigl[\one \pm h(E_{a^\mp}){\hbq}_\mp \cdot \ssig \bigr] \, ,
\end{equation}
where $E_{a^\mp}$ is the energy of $a^\mp$ and $d\Omega_{a^\mp}= d\cos\theta_{a^\mp}d\varphi_{a^\mp}$.
In (<ref>) the symbol $\one$ denotes the two-dimensional unit matrix and
$\ssig = (\sigma_1, \sigma_2, \sigma_3)$ is the
vector of Pauli matrices.
The function $n(E_a)$ determines the energy spectrum of $\tau\to a$ while $h(E_a)$ encodes the $\tau$-spin analyzing power of the
charged prong. Equation (<ref>) is used in the calculations of Sec. <ref>.
If the right-hand side of (<ref>) is integrated over $E_{a^\mp}$, it takes,
due to the normalization convention (<ref>),
the form
\begin{equation} \label{eqA:1pronga}
\frac{d\Omega_{a^\mp}}{4\pi} \int dE_{a^\mp} ~n(E_{a^\mp})\bigl[\one \pm h(E_{a^\mp}){\hbq}_\mp \cdot \ssig \bigr] \, = \,
\frac{d\Omega_{a^\mp}}{4\pi}~\bigl[\one \pm \alpha_a{\hbq}_\mp \cdot \ssig \bigr] \, ,
\end{equation}
where $\alpha_a$ $(|\alpha_a|\leq 1)$ is a measure of the $\tau$ spin-analyzing power of $a$.
Next we list the spectral functions $n(E_a)$ and $h(E_a)$ of several
decay density matrices (<ref>).
The functions $n(E_a)$ have dimension 1/energy while
the functions $h(E_a)$ are dimensionless.
§.§ The decay $\tau^{\mp} \to \ell^{\mp}(q_\mp) + \nu_{\ell} \nu_{\tau}$
In the leptonic decays $\tau^{\mp} \to \ell^{\mp} \nu_{\ell} \nu_{\tau}$
the mass of $\ell=e, \mu$ can be
neglected. (Here and below the symbol $\nu$ denotes a neutrino or antineutrino, depending on the case.)
Using $x = 2E_{\ell} / m_{\tau}$, where $E_{\ell}$ is
defined in the $\tau$ rest frame, one has [57]
\begin{eqnarray}
& = &
\frac{4}{m_{\tau}} x^{2}\,\left(3-2x\right)
\, ,
\qquad
\frac{1-2\, x}{3-2\, x}
\label{eq:lep_nx_bx}
\end{eqnarray}
with $0 \le x \le 1$.
Integrating over the charged lepton energy
in (<ref>) yields (<ref>) with the $\tau$-spin analyzing power
\begin{equation} \label{eq:sppoel}
\alpha_\ell = -\frac{1}{3} \, .
\end{equation}
The value of $\alpha_\ell$ can be increased by a suitable cut on $E_\ell$.
§.§ The decay $\tau^\mp \to\pi^\mp(q_\mp) +\nu_{\tau}$
In the two-body decay $\tau\to\pi+\nu_{\tau}$ the energy $E_{\pi}$
in the $\tau$ rest frame is fixed and the functions
$n_{\pi}(E_{\pi})$ and $h_{\pi}(E_{\pi})$ are given by [57]:
\begin{eqnarray} \label{eq:1pincpi}
& = &
\delta\left(E_{\pi} - \frac{m_{\tau}^2 + m_{\pi}^2}{2m_{\tau}}\right)
\, ,
\qquad
\, .
\end{eqnarray}
Here the $\tau$-spin analyzing power is maximal,
\begin{equation} \label{eq:sppopi}
\alpha_\pi = 1 \, .
\end{equation}
§.§ The decay $\tau^{\mp} \to \rho^{\mp}(q_\mp) + \nu_{\tau}$
If the four-momentum of the intermediate $\rho$ meson can be determined in
the decay $\tau^\mp \to \pi^\mp \pi^0\nu_{\tau}$ by measuring the energies and momenta of
both $\pi^\mp$ and $\pi^0$, the $\rho$ meson can be used as $\tau$-spin analyzer.
It is well known that in the two-body decay of a polarized $\tau$ to a transversely or longitudinally
polarized spin-1 meson and $\nu_{\tau}$ the $\tau$-spin analyzing power of the meson is maximal [58]. However, the polarization
of the vector meson cannot be determined event by event. Summing over the polarizations of the $\rho$ meson
and treating it as an on-shell particle, one obtains $\tau^\mp$ decay density matrices of the form (<ref>)
with the spectral functions [57, 58]
\begin{eqnarray} \label{eq:1pincro}
& = &
\delta\left(E_{\rho} - \frac{m_{\tau}^2 + m_{\rho}^2}{2m_{\tau}}\right)
\, ,
\qquad
h_{\rho}(E_{\rho})\,\,=\,\frac{m_\tau^2 - 2 m_\rho^2}{m_\tau^2 + 2 m_\rho^2}
\, .
\end{eqnarray}
Using $m_\rho = 0.775~\GeV$ we obtain
\begin{equation} \label{eq:spporo}
\alpha_\rho = 0.45 \, .
\end{equation}
We use this two-body decay mode with (<ref>) and (<ref>) in our
analysis of the simple $CP$ observables in Sec. <ref>.
§.§ The decay $\tau^{\mp} \to \pi^{\mp}(q_1) + \pi^{0}(q_2) + \nu_{\tau}$
The differential rate of the decay of polarized $\tau$ leptons to a
charged and neutral pion via a $\rho$ meson was calculated in [57]
in the on-shell approximation for the intermediate $\rho$ meson.
A more elaborate description of this decay mode takes the $\rho$ and $\rho'$ resonances
and their finite widths as intermediate states into account [59, 60, 61].
We use the matrix element of [59, 60] for the decay chain
$\tau \to \rho~(\rho') \to 2 \pi \nu_\tau$. Exact isospin invariance is assumed. In the $\tau^-$ rest frame
we obtain for the $\tau^-\to \pi^- \pi^0 \nu_\tau$ decay density matrix
$\mathcal{D}^{2\pi}$ that is differential in the pion momenta:
\begin{eqnarray}\label{Eq.tau2pidec}
\prod_{i=1}^{2} \frac{d^{3}q_{i}}{(2\pi)^{3}2q_{i}^{0}}
\mathcal{D}^{2 \pi}\bigl(\tau^{-}(k)\rightarrow \pi^{-}(q_{1})\pi^{0}(q_{2}) \nu_{\tau}\bigr) =
\frac{1}{2 m_\tau \Gamma_{2\pi}} d\Phi_2 |\mathcal{M}_2|^2 \, ,
\end{eqnarray}
where $\Gamma_{2\pi}=\Gamma(\tau^{-}\to\pi^{-}\pi^{0}{\nu}_{\tau})$ and
\begin{equation} \label{eq:M2pi0}
|\mathcal{M}_2|^2 = G_F^2 |V_{ud}|^2 |F_\pi(Q^2)|^2\left( A_2 \one + \boldsymbol{H}_2 \cdot \ssig \right) \, .
\end{equation}
Here $G_F$ and $V_{ud}$ denote the Fermi constant and the $ud$ Cabibbo-Kobayashi-Maskawa matrix element, respectively.
The terms in the squared matrix element are
\begin{eqnarray}
A_2 & = & 4 \left[ 2 (k\cdot q)^2 + q^2(Q^2-k\cdot Q) \right] \, ,\label{eq:A2sq} \\
H_2^j & = & 4 m_\tau \left[2 (k \cdot q) q^j + q^2 Q^j \right] \, , \label{eq:B2sq}
\end{eqnarray}
where $j=1,2,3$, $k=(m_\tau, \boldsymbol{0})^T$ in the $\tau$ rest frame, $Q=q_1 + q_2$, and $q= q_1 -q_2$.
The phase-space measure $d\Phi_2$ can be parametrized as follows:
\begin{eqnarray}\label{eqrecPhas3}
d\Phi_2 = & \frac{1}{ 64~(2\pi)^5}
dQ^2 ~\theta(m_\tau^2-Q^2) \theta(Q^2-4 m_\pi^2)
\displaystyle{ d\Omega_Q \frac{\lambda^{1/2}(m_\tau^2,Q^2,0)}{m_\tau^2} d\Omega_1^* \frac{\lambda^{1/2}(Q^2,m_\pi^2\,m_\pi^2)}{Q^2} }\, ,
\end{eqnarray}
where $d\Omega_Q=d\cos\theta_Qd\varphi_Q$ is the solid angle element of $Q$, i.e. of $\rho~(\rho')$, in the $\tau$ rest frame and
$d\Omega_1^*$ is the solid angle element of the charged pion $\pi^-(q_1)$ in the rest frame of $\rho~(\rho')$.
\begin{equation}
\lambda(x,y,z) = x^2 + y^2 + z^2 - 2xy -2xz -2yz \, .
\end{equation}
The form factor $F_\pi$ in (<ref>) can be parametrized by [59]
\begin{equation}
\label{eq:deffpion2}
F_\pi(Q^2) = \frac{B_\rho(Q^2) + \beta_2 B_{\rho'}(Q^2)}{1+\beta_2} \, ,
\end{equation}
\begin{equation}
\label{eq:defBrho2}
B_\rho(x) = \frac{m^2_{\rho} }{m^2_{\rho} - x -i m_{\rho} \Gamma_{\rho}(x) } \; \qquad \text{and} \; \rho \to \rho' \, .
\end{equation}
The label $\rho$ ($\rho'$) refers to the $\rho$ ($\rho'$) resonance and $\beta_2$ is a tuning parameter (see below).
We use for the energy-dependent off-shell widths of the $\rho$ and $\rho'$ that are needed in (<ref>):
\begin{equation}
\label{eq:widrhopr}
\Gamma_{\rho}(x) = \Gamma_{\rho}(m_{\rho}^2) \frac{m_{\rho}}{\sqrt{x}}\left(\frac{p(x)}{p(m_{\rho}^2)}\right)^3
\theta(x - 4 m_\pi^2) \, ,
\end{equation}
\[ p(x) = \frac{1}{2} \sqrt{x - 4 m_\pi^2} \; , \]
and $\Gamma_{\rho'}(x)$ is given by the same formula with label $\rho \to \rho'$.
A value for the on-shell width $\Gamma_{\rho}(m_{\rho}^2)$ and $\Gamma_{\rho'}(m_{\rho'}^2)$,
respectively, is given in (<ref>).
We use the following input values for the computations of the optimal $CP$ observables in Sec. <ref>:
\begin{eqnarray}
m_\tau=1.777~{\rm GeV},\quad m_\pi =0.140~{\rm GeV}, & \nonumber \\
m_\rho =0.775~{\rm GeV}, \quad m_{\rho'} =1.465~{\rm GeV}, & \nonumber \\
G_F= 1.1664 \times 10^{-5}~({\rm GeV})^{-2}, \quad V_{ud}=0.974, & \nonumber \\
\Gamma_\rho = 0.149~{\rm GeV}, \quad
\Gamma_{\rho'} = 0.400~{\rm GeV} \, . &
\label{eq:inpar2}
\end{eqnarray}
With this input, agreement
with the experimental width $\Gamma(\tau^-\to \pi^-\pi^0\nu_\tau)_{\rm exp.} = 5.78 \times 10^{-13}~{\rm GeV}$
is obtained when the tuning parameter $\beta_2$ in
Eq. (<ref>) is chosen to be
\begin{equation} \label{eq:beta2}
\beta_2 = -0.175 \, .
\end{equation}
The differential decay density matrix for the charge-conjugate decay
\[ \tau^{+}(k) \rightarrow\pi^{+}(q_1) \, \pi^{0} (q_2) \; \bar{\nu}_{\tau} \]
is of the same form as (<ref>) with the squared matrix element
\begin{equation} \label{eq:M2pip}
|\mathcal{M}'_2|^2 = G_F^2 |V_{ud}|^2 |F_\pi(Q^2)|^2 \left( A_2 \one - \boldsymbol{H}_2 \cdot \ssig \right) \, ,
\end{equation}
and $A_2$ and $\boldsymbol{H}_2$ are given in Eqs. (<ref>) and (<ref>), respectively.
One may also determine the $\tau$-spin analyzing power of the “resonance” ${Q^\mp}$
in the $\tau^\mp\to \pi^\mp\pi^0\nu_\tau$
decay mode by computing the following decay density matrix:
\begin{equation} \label{eqA:2pipQ}
dx~ \frac{d\Omega_Q}{4 \pi} \,\mathcal{D}^{Q^\mp}
= dx~ \frac{d\Omega_Q}{4\pi}~\bigl[a_{2,Q}(x)\one \pm b_{2,Q}(x)
{\boldsymbol{\hat Q}} \cdot \ssig \bigr] \, ,
\end{equation}
where $4(m_\pi/m_\tau)^2 \leq x \equiv Q^2/m_\tau^2 \leq 1$
and $\boldsymbol{\hat Q}=(\boldsymbol{q_1}+ \boldsymbol{q_2})|/|(\boldsymbol{q_1}+ \boldsymbol{q_2})|$.
The spectral functions $a_{2,Q}$ and $b_{2,Q}$ are shown in Fig. <ref>.
Integrating the right-hand side of (<ref>) over $x$ the decay density matrix takes the form
(<ref>) with $\alpha_a \to \alpha_{2,Q}$ and
${\hbq}_\mp \to \boldsymbol{\hat Q}.$
We get for $\alpha_{2,Q}$:
\begin{equation} \label{eq:al2Q}
\alpha_{2,Q} = 0.42 \, .
\end{equation}
Comparison with (<ref>) shows that taking into account the finite widths of the intermediate resonances
and the whole kinematic range of $Q^2$ leads to a slightly smaller $\tau$-spin analyzing power. Nevertheless, we will
use the value (<ref>) in the computation of the expectation values of the simple $CP$
observables in Sec. <ref>.
For completeness we determine also the $\tau$-spin analyzing power of the charged pion
in $\tau^\mp \to \pi^\mp(q_1) + \pi^0 \nu_\tau$.
The respective 1-prong decay density matrix is given by
\begin{equation} \label{eqA:pip2p}
\frac{d^{3}q_1}{(2\pi)^3 2E_1} \,\mathcal{D}^{\pi^\mp}(\tau^\mp\to \pi^\mp(q_1) + \pi^0 \nu_\tau)
= dx_1 \frac{d\Omega_1}{4\pi}~\bigl[a_1(x_1)\one \pm b_1(x_1){\hbq}_1 \cdot \ssig \bigr] \, ,
% = \int \frac{d\Omega_1}{4\pi}~\bigl[\one \pm \alpha_1{\hbq}_1 \cdot \ssig \bigr] \, ,
\end{equation}
where $x_1 = 2 E_1/m_\tau$ and $2 m_\pi/m_\tau \leq x_1 \leq 1$.
The spectral functions $a_1$ and $b_1$ are shown in Fig. <ref>.
Integrating the right-hand side of (<ref>) over $x_1$ the decay density matrix takes the form
(<ref>) with $\alpha_a \to \alpha_1$ and ${\hbq}_\mp \to {\hbq}_1$.
We get for the $\tau$-spin analyzing power $\alpha_1$ of the charged pion[This
decay mode was analyzed in [9] using only the intermediate $\rho$ in the narrow-width
\begin{equation}
\label{eq:al1cp}
\alpha_1 = -0.036\, .
\end{equation}
Figure <ref> shows that negative and positive contributions cancel to a large extent when $b_1$ is integrated
over the whole kinematic range, leading to the small value (<ref>). The value of $\alpha_1$ can be enhanced by
a suitable cut on $x_1$. We do not use (<ref>) in our analysis of Sec. <ref>.
The spectral functions $a_{2,Q}$ (solid curve) and $b_{2,Q}$ (dotted curve) defined in Eq. (<ref>).
The spectral functions $a_1$ (solid curve) and $b_1$ (dotted curve) defined in Eq. (<ref>).
§.§ The decay $\tau^{\mp} \to a_1^{\mp} \to
\pi^{\mp}(q_1) + \pi^{\mp}(q_2) + \pi^{\pm} (q_3) + \nu_{\tau}$
The decay mode to three charged prongs proceeds mainly via an intermediate $a_1$ resonance. If one
approximates the $\tau \to 3 \pi$ decay mode by $\tau$ decay to an on-shell $a_1$,
the $\tau$-spin analyzing power of this resonance would be maximal, as stated above,
if the $a_1$ polarization states can be separated efficiently [58, 34].
If one sums over the $a_1$ polarizations the $\tau\to a_1 \nu_\tau$ decay density matrix is of the form
(<ref>) and (<ref>) with the label $\rho \to a_1$. The $a_1$ mass is not precisely determined
but, in any case, the $\tau$-spin analyzing power of this resonance is poor in the on-shell approximation. Using
(<ref>) (with $m_\rho \to m_{a_1}$) with the value $m_{a_1} = 1.230~\GeV$
given by the Particle Data Group [1] one obtains $\alpha_{a_1} = 0.02$.
However, maximal sensitivity to the $\tau$ polarization can be obtained with the $3 \pi$ decay mode if the full
decay dynamics is exploited and the energies and momenta of the three pions are measured. We use the $\tau \to 3 \pi \nu_\tau$
matrix element given in [59] (cf. also [60, 61]) where this decay is described by
the decay chain $\tau\to a_1 \to \rho \, (\rho')\, \pi \to 3 \pi$ with off-shell intermediate resonances.
Exact isospin invariance is assumed.[The $\tau\to 3\pi \nu_\tau$ decay was analyzed in
[62] within the resonance chiral theory using an elaborate description of the $a_1$
off-shell width.] We obtain for the differential $\tau^-\to 2\pi^- \pi^+ \nu_\tau$ decay density matrix
$\mathcal{D}^{3\pi}$ in the $\tau^-$ rest frame with the normalization conventions (<ref>) and (<ref>):
\begin{eqnarray}\label{Eq.tau3pidec}
\prod_{i=1}^{3} \frac{d^{3}q_{i}}{(2\pi)^{3}2q_{i}^{0}}
\mathcal{D}^{3 \pi}\bigl(\tau^{-}(k)\rightarrow \pi^{-}(q_{1})\pi^{-}(q_{2})\pi^{+}(q_{3}) \nu_{\tau}\bigr) =
\frac{1}{2 m_\tau \Gamma_{3\pi}} d\Phi_3 |\mathcal{M}_3|^2 \, ,
\end{eqnarray}
where $\Gamma_{3\pi}=\Gamma(\tau^{-}\to\pi^{-}\pi^{-}\pi^{+}{\nu}_{\tau})$ and
the phase-space measure is given in the recursive phase-space parametrization by
\begin{eqnarray}\label{eqrecPhase}
d\Phi_3 = \frac{1}{ 2^{9}~(2\pi)^8 }
dQ^2 du~\theta(m_\tau^2 -Q^2)\theta(Q^2- 9 m_\pi^2)
\theta((\sqrt{Q^2}-m_\pi)^2 -u) \theta(u- 4 m_\pi^2) & \nonumber \\
\times ~ d\Omega_Q \frac{\lambda^{1/2}(m_\tau^2,Q^2,0)}{m_\tau^2} d\Omega_3^* \frac{\lambda^{1/2}(Q^2,u,m_\pi^2)}{Q^2}
d\Omega_2^{**} \frac{\lambda^{1/2}(u,m_\pi^2,m_\pi^2)}{u} \, . &
\end{eqnarray}
Here $Q=q_1 + q_2 + q_3$,
$u=(q_1+q_2)^2$ and $d\Omega_Q=d\cos\theta_Qd\varphi_Q$ is the solid angle element of $Q$, i.e. $a_1$, in the $\tau$ rest frame,
$d\Omega_3^*$ is the solid angle element of $\pi^+(q_3)$ in the rest frame of $a_1$, and
$d\Omega_2^{**}$ is the solid angle element of $\pi^0(q_2)$ in the rest frame of $\rho$, i.e., the zero-momentum frame of
$q_1 + q_2$.
Note that the statistics factor $1/2$ for two identical particles in the final state is compensated
here by the normalization convention (<ref>).
The squared matrix element is given by
\begin{equation} \label{eq:M3pim}
|\mathcal{M}_3|^2 = G_F^2 |V_{ud}|^2 \left( A_3 \one + \boldsymbol{H}_3 \cdot \ssig \right) \, ,
\end{equation}
\begin{eqnarray}
A_3 = & ~|F_1|^2\left[ 4 (k\cdot V_1)^2 - 2 (k\cdot Q - Q^2) V_1^2\right] \nonumber \\
& + |F_2|^2\left[ 4 (k\cdot V_2)^2 - 2 (k\cdot Q - Q^2) V_2^2\right ] \nonumber \\
& + {\rm Re}(F_1 F_2^*)\left[8 k\cdot V_1 k\cdot V_2 - 4(k\cdot Q - Q^2) V_1 \cdot V_2\right]\nonumber \\
& - 2 i(F_2 F_1^* - F_1 F_2^*) \epsilon(k, Q, V_1, V_2) \, ,
\label{eq:formA}
\end{eqnarray}
\begin{eqnarray}
H_3^j = & 2 m_\tau \left\{ |F_1|^2\left[ 2k\cdot V_1 V_1^j + V_1^2 Q^j \right] \right. \nonumber \\
& + |F_2|^2\left[ 2k\cdot V_2 V_2^j + V_2^2 Q^j \right ] \nonumber \\
& \left. + 2 {\rm Re}(F_1 F_2^*)\left[ k\cdot V_2 V_1^j + k\cdot V_1 V_2^j + V_1 \cdot V_2 Q^j \right] \right\} \nonumber \\
& + 2 i m_\tau (F_2 F_1^* - F_1 F_2^*) \epsilon(q', j, V_1, V_2) \, ,
\label{eq:formHj}
\end{eqnarray}
and $j=1,2,3$, $k=(m_\tau, \boldsymbol{0})^T$, $q'=k-Q$,
\begin{equation}
\label{eq:defv12}
V_1^\mu= \left( g^{\mu\nu} - \frac{Q^\mu Q^\nu}{Q^2} \right) (q_1-q_3)_\nu \, , \quad
V_2^\mu=\left( g^{\mu\nu} - \frac{Q^\mu Q^\nu}{Q^2} \right) (q_2-q_3)_\nu \, ,
\end{equation}
and $\epsilon(k, Q, V_1, V_2)=\epsilon_{\mu\nu\alpha\beta} k^\mu Q^\nu V_1^\alpha V_2^\beta$,
$\epsilon(q', j, V_1, V_2) = \epsilon_{\mu j\alpha\beta}q'^{\mu} V_1^\alpha V_2^\beta$
and we use the convention $\epsilon_{0123} = +1$.
\begin{equation}
\label{eq:deff12}
F_1 = F(Q^2,s) \, , \qquad F_2 = F(Q^2,t) \, ,
\end{equation}
where $s =(q_1+q_3)^2$ and $t=(q_2+q_3)^2$. The function $F$ is given by [59]:
\begin{equation}
\label{eq:deffunF}
F(Q^2,x) = \frac{2\sqrt{2}}{3 f_\pi} B_{a_1}(Q^2) F_\pi(x) \, ,
\end{equation}
where $f_\pi$ is the pion decay constant (in the convention $f_\pi = 0.093$ GeV) and
$B_{a_1}$ denotes the Breit-Wigner enhancement factor of the $a_1$ meson:
\begin{equation}
\label{eq:defBa1}
B_{a_1}(Q^2) = \frac{m^2_{a_1} }{m^2_{a_1} - Q^2 -i m_{a_1} \Gamma_{a_1}(Q^2) } \, .
\end{equation}
We use as a model for the energy-dependent off-shell width of the $a_1$ meson:
\begin{equation}
\label{eq:gama1wid}
\Gamma_{a_1}(Q^2) = \Gamma_{a_1}(m^2_{a_1}) \frac{g(Q^2)}{g(m^2_{a_1})} \, ,
\end{equation}
where $\Gamma_{a_1}(m^2_{a_1})$ is the on-shell width (see below) and the function $g$ is given in Eq. (3.16) of
Ref. [59].
Moreover, the pion “form factor” $F_\pi(x)$ is given by the formulas (<ref>) – (<ref>) above
where now the tuning parameter $\beta_2$ is to be replaced by $\beta_3$ that will be determined below.
The differential decay density matrix for the charge-conjugate decay
\[ \tau^{+}(k) \rightarrow\pi^{+}(q_1) \, \pi^{+}(q_2) \, \pi^{-}(q_3) \; \bar{\nu}_{\tau} \]
is of the same form as Eqs. (<ref>) with the squared matrix element
\begin{equation} \label{eq:M3pip}
|\mathcal{M}'_3|^2 = G_F^2 |V_{ud}|^2 \left( A_3 \one - \boldsymbol{H}_3 \cdot \ssig \right) \, ,
\end{equation}
and $A_3$ and $\boldsymbol{H}_3$ are given in Eqs. (<ref>) and (<ref>), respectively.
To the best of our knowledge the differential $\tau\to 3\pi \nu$ density matrix (<ref>) – (<ref>)
was so far not given in this explicit form in the literature.
For our computation of the expectation values of the optimal observables
in Sec. <ref> we use the above formulas with the input values (<ref>) and
\begin{equation}
f_\pi =0.093~{\rm GeV}, \quad m_{a_1} =1.230~{\rm GeV}, \quad \Gamma_{a_1} = 0.483~{\rm GeV} \, .
\label{eq:inpar3}
\end{equation}
It remains to fix the tuning parameter $\beta_3$. Using the above squared matrix element and
input parameters we find agreement with the experimental width
$\Gamma(\tau^-\to 2\pi^- \pi^+ \nu_\tau)_{\rm exp.} = 2.11 \times 10^{-13}~{\rm GeV} $
when the tuning parameter $\beta_3$ is chosen to be
\begin{equation}
\label{eq:beta3}
\beta_3 = -0.204 \, .
\end{equation}
§.§ The decay $\tau^{\mp} \to a_1^{\mp} \to
\pi^{0}(q_1) + \pi^{0}(q_2) + \pi^{\mp} (q_3) + {\nu}_{\tau}$
For completeness we discuss here also this decay mode, although we do not use it in the
analysis of Sec. <ref>. Assuming exact isospin invariance the
differential decay density matrices for $\tau^\mp\to 2 \pi^\mp \pi^\pm \nu_\tau$
derived in the previous subsection can be used also for these decay modes.
Using the above input parameters with the exception
$m_\pi = m_{\pi^+} = 0.140~\GeV \rightarrow m_\pi = m_{\pi^0} = 0.135~\GeV$,
agreement with the experimental width $\Gamma(\tau^-\to 2\pi^0 \pi^-\nu_\tau)_{\rm exp.} = 2.10 \times 10^{-13}~{\rm GeV}$
is obtained with the following value of the tuning parameter, here denoted by $\beta'_3$:
\begin{equation}
\label{eq:betpr}
\beta'_3 = -0.190 \, .
\end{equation}
Moreover, the $\tau$-spin analyzing power of the charged pion in this decay mode is also of interest.
The 1-prong decay density matrix for $\tau^\mp \to \pi^\mp(q_3) + 2 \pi^0 \nu_\tau$, normalized to the charged
particle multiplicity $n_{\pi^\pm}=1$, is given by
\begin{equation} \label{eqA:pip3p}
\frac{d^{3}q_3}{(2\pi)^3 2E_3} \,\mathcal{D}^{\pi^\mp}(\tau^\mp \to \pi^\mp(q_3) + 2 \pi^0 \nu_\tau)
= dx_3 \frac{d\Omega_3}{4\pi}~\bigl[a_3(x_3)\one \pm b_3(x_3){\hbq}_3 \cdot \ssig \bigr] \, ,
\end{equation}
where $x_3 = 2 E_3/m_\tau$ and $2 m_\pi/m_\tau \leq x_3 \leq 1 - 3 (m_\pi/m_\tau)^2$.
The spectral functions $a_3$ and $b_3$ are shown in Fig. <ref>.
Integrating the right-hand side of (<ref>) over $x_3$ the decay density matrix takes the form
(<ref>) with $\alpha_a \to \alpha_3$ and
${\hbq}_\mp \to {\hbq}_3.$
We get for the $\tau$-spin analyzing power $\alpha_3$ of the charged pion:[A simpler description of this
decay mode was used in [9] and the value $\alpha_\pi = -0.18$ was obtained.]
\begin{equation}
\label{eq:al3cp}
\alpha_3 = -0.144\, .
\end{equation}
This number is rather small because $b_3$ has both negative and positive contributions that cancel
to a large extent when integrated over $x_3$.
The analyzing power can be enhanced by a suitable cut on $x_3$.
The spectral functions $a_3$ (solid curve) and $b_3$ (dotted curve) defined in Eq. (<ref>).
§ EXPECTATION VALUES AND COVARIANCES OF $CP$-ODD OBSERVABLES
In this appendix we discuss general properties of expectation values and covariances
of the $CP$-odd observables introduced in Sec. <ref> and computed in Sec. <ref>.
We treat first case i) of Sec. <ref> where only one charged particle is measured from
$\tau^-$ and $\tau^+$ decays, respectively.
The differential cross section of the two-particle inclusive reaction (<ref>) and (<ref>)
as used in this paper is given by (<ref>):
\begin{eqnarray} \label{Eq.apBdsig}
d\sigma_{a\bar{b}} & = & \dfrac{\sqrt{1-4m_{\tau}^{2}/s}}{16\pi s}
{\rm Br}(\tau^{-}\rightarrow A) \, {\rm Br}(\tau^{+}\rightarrow \overline{B})
\nonumber \\
& & \times ~{\rm Tr}\left[R {\cal D}^a {\cal D}^{\bar b}\right]
\frac{|\qm^*|}{(2\pi)^2}\frac{|\qp^*|}{(2\pi)^2} dE_-^* dE_+^*
\dfrac{d\Omega_{k_+}}{4\pi} \frac{d\Omega_-^*}{4\pi} \frac{d\Omega_+^*}{4\pi} \, ,
\end{eqnarray}
where we have used in (<ref>) the momenta of the charged particles
$a$ and ${\bar b}$ and the corresponding phase-space measures in the respective $\tau^-$ and $\tau^+$ rest frame.
The one-particle inclusive decay density matrices in the $\tau^\mp$ rest frames are given in (<ref>).
We recall the relation between the respective rest-frame momenta $q_\mp^*$ and $k_\mp^*=(m_\tau,\boldsymbol{0})^T$
and the momenta $q_\mp$ and $k_\mp$ in the $e^+ e^-$ c.m. frame. With the Lorentz boost
\begin{equation}\label{eq:Lorboost}
\Lambda_{\kk} =
\left(\begin{array}{cc}
\frac{k^0}{m_\tau} & \frac{ k^j}{m_\tau} \\
\frac{k^i}{m_\tau} & \delta^{ij} + {\hat k}^i {\hat k}^j \left(\frac{k^0 - m_\tau}{m_\tau}\right)
\end{array}\right) \, ,
\end{equation}
where $\kk$ is the three-momentum of $\tau^+$ in the $e^+ e^-$ c.m. frame, we have
\begin{equation} \label{eq:rebocm}
\Lambda_{\pm \kk}~k_\mp = k_\mp^* \, , \qquad \Lambda_{\pm \kk}~q_\mp = q^*_\mp \, .
\end{equation}
Next we decompose (<ref>) according to (<ref>) and (<ref>), neglecting terms
quadratic in ${\hat d}_\tau$. Here our phase-space variables are
\begin{equation} \label{eq:phaphi}
\phi = (E^*_-, E^*_+, \hk, \hqm^*, \hqp^*) \, ,
\end{equation}
and the measure is
\begin{equation}\label{eq:phamea}
d\phi = dE_-^* dE_+^*
\dfrac{d\Omega_{k}}{4\pi} \frac{d\Omega_-^*}{4\pi} \frac{d\Omega_+^*}{4\pi} \, .
\end{equation}
We get
\begin{equation} \label{eq:decsab}
d\sigma_{a\bar{b}} =\left\{S_{\rm SM}^{a\bar{b}}(\phi)+
S_{CP,R}^{a\bar{b}}(\phi)\Re \,{\hat d}_{\tau} + S_{CP,I}^{a\bar{b}}(\phi) \Im \, {\hat d}_{\tau} \right\} d\phi \, ,
\end{equation}
where, using (<ref>),
\begin{eqnarray} \label{eq:chrSMa}
S_{\rm SM}^{a\bar{b}}(\phi) & = & \dfrac{\sqrt{1-4m_{\tau}^{2}/s}}{16\pi s}
{\rm Br}(\tau^{-}\rightarrow A) \, {\rm Br}(\tau^{+}\rightarrow \overline{B})
\frac{\chi_{\rm SM,\alpha\alpha'\beta\beta'}}{| 1 + e^2 \Pi_c(s)|^2} \nonumber \\
& & \times~ n_a(E_-^*)\bigl[\delta_{\beta'\beta} + h_a(E_-^*){\hbq}_-^* \cdot \ssig_{\beta'\beta} \bigr] ~
n_b(E_+^*)\bigl[\delta_{\alpha'\alpha} - h_b(E_+^*){\hbq}_+^* \cdot \ssig_{\alpha'\alpha} \bigr] \, .
\end{eqnarray}
The quantities $S_{CP,R}^{a\bar{b}}$ and $S_{CP,I}^{a\bar{b}}$ are obtained from (<ref>) by the replacements
\begin{equation} \label{eq:Brepl}
\chi_{\rm SM} \rightarrow \chi_{CP}^R \qquad \text{and} \qquad \chi_{\rm SM} \rightarrow \chi_{CP}^I \, ,
\end{equation}
respectively; see (<ref>) – (<ref>).
We can now perform the traces in (<ref>). With (<ref>) we see that this amounts to make the following
replacements in (<ref>) – (<ref>):
\begin{eqnarray} \label{eq:BTrrepl}
\one & \rightarrow & 4 ~n_b ~n_a \, , \nonumber \\
\sip & \rightarrow & -4 ~n_b h_b ~n_a ~{\hbq}_+^* \, , \nonumber \\
\ssim & \rightarrow & 4 ~n_b ~n_a h_a ~ {\hbq}_-^*\, , \nonumber \\
\sigma_+^r \sigma_-^s & \rightarrow & - 4 ~ n_b h_b ~n_a h_a ~{\hat q}_+^{* r} {\hat q}_-^{* s} \, .
\end{eqnarray}
Next, we consider the transformation
\begin{equation} \label{eq:tracms}
\kk \rightarrow -\kk \, , \qquad \qp \rightarrow -\qp \, , \qquad \qm \rightarrow -\qm \, ,
\end{equation}
which, using (<ref>), implies
\begin{equation} \label{eq:traresf}
\hk \rightarrow -\hk \, , \qquad \hqp^* \rightarrow -\hqp^* \, , \qquad \hqm^* \rightarrow -\hqm^* \, ,
\end{equation}
and vice versa. These transformations correspond to the naive “time reversal” transformation[One may also transform
$\pp \to - \pp$, but this is irrelevant here.]
$T_N$ referred to in
Sec. <ref>.
Inspection of $\chi_{\rm SM}$, $\chi_{CP}^R$, and $\chi_{CP}^I$, i.e., of Eqs. (<ref>) – (<ref>) with
the replacements (<ref>), shows that applying (<ref>) we have
\begin{equation} \label{eq:traSSS}
S_{\rm SM}^{a\bar{b}}(\phi) \rightarrow S_{\rm SM}^{a\bar{b}}(\phi) \, , \qquad
S_{CP,R}^{a\bar{b}}(\phi) \rightarrow - S_{CP,R}^{a\bar{b}}(\phi) \, ,
\qquad S_{CP,I}^{a\bar{b}}(\phi) \rightarrow S_{CP,I}^{a\bar{b}}(\phi) \, .
\end{equation}
We turn to the simple and optimal observables of Sec. <ref>. We assume integration over the whole phase space or,
if cuts are applied, we assume the cuts to be $CP$-symmetric. In addition we assume the cuts to be invariant under
(<ref>). The tensors ${\widehat T}^{ij}$ and ${T}^{ij}$ of (<ref>) and (<ref>)
are odd whereas
${\widehat Q}^{ij}$ and ${Q}^{ij}$ of (<ref>) and (<ref>) are even under the transformation (<ref>).
Let us first consider the case $a=b$. The $CP$ properties of the observables $T$ and $Q$ imply
\begin{eqnarray} \label{eqB:CPTQ}
E_0(T^{ij}) = 0 \, , & \qquad & E_0( {\widehat T}^{ij}) = 0 \, , \nonumber \\
E_0(Q^{ij}) = 0 \, , & \qquad & E_0( {\widehat Q}^{ij}) = 0 \, .
\end{eqnarray}
Moreover, turning to the covariance matrix of one of the $T$ and one of the $Q$ variables,
the transformation (<ref>) implies
\begin{equation} \label{eqB:covS}
E_0(T^{ij}Q^{k l}) = 0
\end{equation}
and likewise for the other $T Q$ correlations. That is, the covariance matrix of the $T, Q$ variables
is diagonal.
The optimal $CP$ observables are in the case $a = b$:
\begin{equation} \label{eqB:optO}
{\cal O}_R^{a\bar{a}}(\phi) = \frac{S_{CP,R}^{a\bar{a}}(\phi)}{S_{\rm SM}^{a\bar{a}}(\phi) } \, ,\qquad
{\cal O}_I^{a\bar{a}}(\phi) = \frac{S_{CP,I}^{a\bar{a}}(\phi)}{S_{\rm SM}^{a\bar{a}}(\phi) } \, .
\end{equation}
The $CP$ transformation properties of these observables imply
\begin{equation} \label{eqB:E0optO}
E_0({\cal O}_R^{a\bar{a}}) = 0 \, , \qquad E_0( {\cal O}_I^{a\bar{a}}) = 0 \, ,
\end{equation}
and applying the $T_N$ transformation (<ref>) it follows that
\begin{equation} \label{eqB:covoO}
E_0({\cal O}_R^{a\bar{a}} {\cal O}_I^{a\bar{a}}) = 0 \, .
\end{equation}
Thus, the covariance matrix is diagonal in this case:
\begin{equation}\label{eqB:covaa}
V({\cal O}^{a\bar{a}}) =
\left(\begin{array}{cc}
E_0({\cal O}_R^{a\bar{a}} {\cal O}_R^{a\bar{a}}) & 0 \\
0 & E_0({\cal O}_I^{a\bar{a}} {\cal O}_I^{a\bar{a}})
\end{array}\right) \, .
\end{equation}
Next we turn to the case $a\neq b$. As this final state is no longer $CP$-symmetric, the $CP$ transformation properties
of the observables are no longer of immediate use. Let us first consider the simple observables, for instance,
$ T^{ij}$ and ${\widehat Q}^{ij}$. Applying the $T_N$ transformation (<ref>) we get
\begin{equation} \label{eqB:Tmat0}
E_0^{a\bar{b}}(T^{ij}) \equiv \langle T^{ij} \rangle_{0, a\bar{b}} = 0.
\end{equation}
The transformation (<ref>) implies also that expectation values of the form (<ref>)
vanish in the nondiagonal case.
As to the SM expectation value
\begin{equation} \label{eqB:Qhmat0}
E_0^{a\bar{b}}({\widehat Q}^{ij}) \equiv \langle {\widehat Q}^{ij} \rangle_{0, a\bar{b}}
\end{equation}
there is, however, in the case $a\neq b$ no symmetry argument implying that it vanishes, too.
Therefore, one should use in this case
in general the observables
\begin{equation} \label{eqB:QprO}
{\widehat Q}^{~' ij} = {\widehat Q}^{ij} - \langle {\widehat Q}^{ij} \rangle_{0, a\bar{b}} \, .
\end{equation}
From the $CP$ property of ${\widehat Q}^{ij}$ one gets, of course,
\begin{equation} \label{eqB:Qhimag}
\langle {\widehat Q}^{ij} \rangle_{0, a\bar{b}} + \langle {\widehat Q}^{ij} \rangle_{0, b \bar{a}} = 0 \, .
\end{equation}
Thus, the respective quantity to probe for $CP$ violation is (<ref>):
\begin{equation}
\frac{1}{2}\left\{\langle {\widehat Q}^{ij} \rangle_{a\bar{b}} + \langle {\widehat Q}^{ij} \rangle_{b \bar{a}} \right\} \, .
\end{equation}
But the corresponding variance in the SM, for instance of the $i=j=3$ components, has to be calculated in general as
\begin{equation} \label{eqB:QQcov}
\langle {\widehat Q}^{~' 33}{\widehat Q}^{~' 33} \rangle_{0, a\bar{b}} \, = \,
\langle {\widehat Q}^{~' 33}{\widehat Q}^{~' 33} \rangle_{0, b\bar{a}} \, .
\end{equation}
The above statements apply, of course, also to $Q^{ij}$.
Yet in our analysis where we use the SM matrix element of the form (<ref>) and integrate over the whole phase space
we find that (<ref>) vanishes within our numerical uncertainties of order $10^{-4}$. This holds also for
the respective expectation values of $Q^{ij}$.
The optimal observables are in the case $a\neq b$ (cf. (<ref>)):
\begin{equation} \label{eqB:OORI}
\cO_{R}^{a\bar{b}}(\phi) = \frac{S_{CP,R}^{a\bar{b}}(\phi)}{ S_{\rm SM}^{a\bar{b}}(\phi) } \, ,
\qquad \cO_{I}^{a\bar{b}} (\phi) = \frac{S_{CP,I}^{a\bar{b}}(\phi)}{ S_{\rm SM}^{a\bar{b}}(\phi)} \, ,
\end{equation}
where $\cO_{R}^{a\bar{b}}(\phi)$ and $\cO_{I}^{a\bar{b}} (\phi)$ are odd and even under the transformation
(<ref>), respectively; see (<ref>). Therefore, we have
\begin{equation} \label{eqB:E0OR}
E_0(\cO_{R}^{a\bar{b}}) \, = \, 0 \, .
\end{equation}
For analyzing $E_0(\cO_{I}^{a\bar{b}})$ we perform in $d\sigma_{a\bar{b}}$, Eq. (<ref>),
the variable transformation
\begin{equation} \label{eq:ApBvark}
\kk \rightarrow -\kk \, .
\end{equation}
The term $S_{\rm SM}^{a\bar{b}}(\phi)$ remains invariant, while $\cO_{i}^{a\bar{b}} (\phi)$ $(i=R,I)$ change sign;
see (<ref>) – (<ref>) and (<ref>), (<ref>).
\begin{equation} \label{eqB:E0OI1p}
E_0(\cO_{I}^{a\bar{b}}) \, = \, 0
\end{equation}
if one integrates over the whole angular range of $\kk$ or over a range that is symmetric with respect to (<ref>).
Beyond the one-photon approximation
$E_0(\cO_{i}^{a\bar{b}})$ $(i=R,I)$ will in general be nonzero. Therefore, one should use in general (cf. (<ref>)):
\begin{eqnarray}\label{eqB:ExORI}
\cO_{R}^{' a\bar{b}}(\phi) & = & \cO_{R}^{ a\bar{b}}(\phi) - E_0(\cO_{R}^{a\bar{b}}) \, , \nonumber \\
\cO_{I}^{' a\bar{b}}(\phi) & = & \cO_{I}^{ a\bar{b}}(\phi) - E_0(\cO_{I}^{a\bar{b}}) \, .
\end{eqnarray}
In our case here the transformations (<ref>) and (<ref>) imply that the
covariance matrix $V(\cO^{' a\bar{b}})$
is still diagonal and is given by
\begin{equation}\label{eqB:covab}
V({\cal O}^{' a\bar{b}}) = V({\cal O}^{ a\bar{b}}) =
\left(\begin{array}{cc}
E_0({\cal O}_R^{ a\bar{b}} {\cal O}_R^{ a\bar{b}}) & 0 \\
0 & E_0({\cal O}_I^{ a\bar{b}} {\cal O}_I^{ a\bar{b}})
\end{array}\right) \, .
\end{equation}
This covariance matrix is then used in (<ref>)
and (<ref>) for the estimators $\gamma_i$ and their covariance matrix $V(\gamma)$.
We come now to the final states of case ii) of Sec. <ref> to which we apply the optimal $CP$ observables.
The channels where $\tau\to 3 \pi \nu_\tau$
is involved require a more detailed discussion.
In our models for the hadronic $\tau$ decays outlined in Appendix <ref> the squared matrix element
of this decay mode given in (<ref>) – (<ref>) differs from the
respective squared matrix element of $\tau\to 2 \pi \nu_\tau$ and those of
the one-particle inclusive decays in that it contains contributions from absorptive parts caused by the finite widths of
the intermediate resonances. This implies that a $T_N$ transformation can no longer be used to discriminate between
the optimal observables $\cO_R$ and $\cO_I$. In order to see this explicitly let us for definiteness consider
the case where $\tau^-$ decays to three observed pions (labeled by the symbol $A$), while in the decay of $\tau^+$
only one charged particle is measured (label ${\bar b}$).
The differential cross section is obtained by inserting the respective
decay density matrices into (<ref>), taking into account (<ref>) – (<ref>).
Using (<ref>) the matrix element $S_{\rm SM}^{A\bar{b}}$ is obtained from (<ref>), up to an overall factor,
by replacing
\begin{equation} \label{eqB:repl}
n_a(E_-^*)\bigl[\delta_{\beta'\beta} + h_a(E_-^*){\hbq}_-^* \cdot \ssig_{\beta'\beta} \bigr]
\rightarrow \bigl[A_3 \delta_{\beta'\beta} + \boldsymbol{H}_3 \cdot \ssig_{\beta'\beta} \bigr] \, ,
\end{equation}
The matrix elements $S_{CP,R}^{A\bar{b}}$ and $S_{CP,I}^{A\bar{b}}$ are obtained in the same fashion.
Inspection of the functions $A_3$ and $H_3^j$ shows that neither has a definite behavior under the
following transformation that is analogous to (<ref>):
\begin{equation}\label{eqB:tra123}
\kk \rightarrow -\kk \, , \qquad \bq^*_i \rightarrow -\bq^*_i \quad (i=1,2,3) \, .
\end{equation}
The dispersive terms in $A_3$ and $H_3^j$ are even under (<ref>) whereas the absorptive terms are odd.
Therefore, the matrix elements $S_{\rm SM}^{A\bar{b}}$, $S_{CP,R}^{A\bar{b}}$, and $S_{CP,I}^{A\bar{b}}$
do not have a definite transformation behavior under (<ref>), too. Hence we expect that
\begin{equation} \label{eqB:corRI3}
E_0(\cO_{CP,R}^{A{\bar b}}\cO_{CP,I}^{A{\bar b}}) \neq 0 \, .
\end{equation}
Thus, the covariance matrix $V(\cO^{' A {\bar b}})$ can have nondiagonal elements that are nonvanishing.
On the other hand, applying the transformation (<ref>), $\kk \to - \kk$, to $S_{\rm SM}^{A\bar{b}}$ and
to $S_{CP,i}^{A\bar{b}}$ $(i=R,I)$
shows that the first term remains invariant while the two others change sign.
Therefore, with our matrix elements we have
\begin{equation} \label{eqB:E0nze}
E_0(\cO_{CP,R}^{A{\bar b}}) = 0 \, , \qquad E_0(\cO_{CP,I}^{A{\bar b}}) = 0 \, .
\end{equation}
Beyond the one-photon approximation (<ref>) will no longer hold.
Thus when $\tau$-pair
decays to final states $A{\bar b} + b{\bar A}$, $A{\bar A}$, and $A {\bar B}+ B {\bar A}$ are considered,
one should in general apply
– especially in experimental analyses – the full formalism of the optimal observable method as
explained in Sec. <ref>.
The nondiagonal elements of the respective covariance matrix of the optimal $CP$ observables computed
with the matrix elements for $\tau$-pair production and decay used in this paper
at $\sqrt{s}=10.58$ GeV are very small and can be neglected in view of our numerical uncertainties; see
Sec. <ref>.
At last a remark that applies if the full formalism of Sec. <ref> has to be used. Suppose the parameters $\Re d_\tau$
and $\Im d_\tau$ have been measured in $k$ decay channels. Let us denote the results for their mean values in the channel $\kappa$ by
\begin{equation}\label{eqB:meanvd}
\overline{\boldsymbol{X}}^{(\kappa)} =
\left(\begin{array}{c}
\Re {\bar d}_\tau^{(\kappa)} \\
\Im {\bar d}_\tau^{(\kappa)}
\end{array}\right) \, , \qquad (\kappa =1, \cdots,k) \, ,
\end{equation}
and for the respective covariance matrix by $V^{(\kappa)}$. Furthermore, we assume these $k$ measurements to be independent, i.e.,
uncorrelated. We define the matrix
\begin{equation} \label{eqB:MatVm1}
V^{-1} = \sum\limits_{\kappa=1}^k \left(V^{(\kappa)}\right)^{-1}
\end{equation}
and the overall mean
\begin{equation} \label{eqB:ovmea}
\overline{\boldsymbol{X}} = V \sum\limits_{\kappa=1}^k \left(V^{(\kappa)}\right)^{-1} \overline{\boldsymbol{X}}^{(\kappa)} \, .
\end{equation}
The covariance matrix is then given by $V$.
That is, the 1 s.d. error ellipse for the mean values (<ref>)
in the $\Re d_\tau - \Im d_\tau$ plane is given by
\begin{equation} \label{eqB:erell}
\left({\boldsymbol{X}} - \overline{\boldsymbol{X}}\right)^T V^{-1} \left({\boldsymbol{X}} - \overline{\boldsymbol{X}}\right) = 1 \, .
\end{equation}
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|
# Flocking and Collision Avoidance for a Dynamic Squad of Fixed-Wing UAVs
Using Deep Reinforcement Learning
Chao Yan, Xiaojia Xiang, Chang Wang*, and Zhen Lan This work was supported in
part by the Science and Technology Innovation 2030-Key Project of New
Generation Artificial Intelligence under Grant 2020AAA0108200, in part by the
National Natural Science Foundation of China under Grant 61825305, Grant
61803377, and Grant 61906203.College of Intelligence Science and Technology,
National University of Defense Technology, Changsha 410073, China, {yanchao17,
xiangxiaojia, wangchang07<EMAIL_ADDRESS>*Corresponding author
###### Abstract
Developing the flocking behavior for a dynamic squad of fixed-wing UAVs is
still a challenge due to kinematic complexity and environmental uncertainty.
In this paper, we deal with the decentralized flocking and collision avoidance
problem through deep reinforcement learning (DRL). Specifically, we formulate
a decentralized DRL-based decision making framework from the perspective of
every follower, where a collision avoidance mechanism is integrated into the
flocking controller. Then, we propose a novel reinforcement learning algorithm
PS-CACER for training a shared control policy for all the followers. Besides,
we design a plug-n-play embedding module based on convolutional neural
networks and the attention mechanism. As a result, the variable-length system
state can be encoded into a fixed-length embedding vector, which makes the
learned DRL policy independent with the number and the order of followers.
Finally, numerical simulation results demonstrate the effectiveness of the
proposed method, and the learned policies can be directly transferred to semi-
physical simulation without any parameter finetuning.
## I Introduction
Due to the capability limitation of a single unmanned aerial vehicle (UAV),
multi-UAV collaboration has attracted increasing attention [1, 2, 3]. One of
the fundamental and challenging problems is the flocking control of UAVs
without collision [4]. Traditional methods such as model predictive control
[5] and consensus theory [6] usually depend on precise physical models, which
are complex and difficult to obtain in practice.
As an alternative, reinforcement learning (RL) [7], [8] can be used for the
flocking control problem. For example, Hung et al. used the
Dyna-Q($\lambda{}$) algorithm [9] and the Q($\lambda{}$) algorithm [10] to
learn flocking control policies for fixed-wing UAVs. Speck et al. [11]
combined the SARSA algorithm with object-focused learning to implement the
formation control of UAV swarms. However, the above methods discretized the
state and action spaces, which was inappropriate for controlling the UAVs in
more realistic environments. Besides, the UAVs were assumed to fly at
different fixed-altitudes to simplify the collision avoidance problem.
Deep reinforcement learning (DRL) has been proved effective for high-
dimensional and continuous control problems in robotics [12, 13, 14]. In our
previous work, we proposed a DRL algorithm, i.e., continuous actor-critic with
experience replay (CACER), for flocking with fixed-wing UAVs in continuous
state and action spaces [15, 16]. Following the previous work [9, 10], we also
simplified the collision avoidance problem by assuming that the UAVs
maintained the same constant speed but different altitudes. However, some
real-world applications such as surveying and mapping require the UAVs to
flock at the same altitude. This requirement makes flocking control more
challenging, because the collision avoidance problem must be considered. In
contrast to the previous work [9], [10], [15], this paper solves a more
challenging flocking control problem. Specifically, we not only allow the UAVs
to change speed as needed, but also design a collision avoidance mechanism and
integrate it into the flocking controller.
Besides, it is difficult to construct a leader-follower flocking control model
for a dynamic squad of UAVs. The main reason is that the length of the system
state is relevant with the number of followers, while the DRL-based control
policies typically require a fixed-length input for deep neural networks. Sui
et al. [17] followed the method in [18] where long short-term memory network
(LSTM) [19] was applied to process the states of other followers sequentially
in the reverse order of their distances to the decision-making follower, under
the assumption that the nearest neighbor had the biggest effect on the
follower. However, this assumption is not always true due to the influence of
other factors such as speed and heading. Instead, we design a customized
network module to deal with the collision avoidance problem for a dynamic
squad of fixed-wing UAVs.
The main contributions of this paper are as follows:
* •
A decentralized DRL-based framework is developed to address the flocking
control and collision avoidance problem for a dynamic squad of fixed-wing
UAVs.
* •
A novel DRL algorithm is proposed for training the flocking controller, where
a plug-n-play embedding module based on convolutional neural networks and the
attention mechanism is designed to handle variable-length system state.
* •
The proposed method can be directly transferred from numerical simulation to
semi-physical simulation without any parameter finetuning.
The rest of this paper is organized as follows. Section II formulates the
flocking and collision avoidance problem, followed by the proposed PS-CACER
algorithm and the designed SEMP embedding module in Section III. Section IV
discusses the simulation results. Finally, Section V concludes this paper.
## II Problem Formulation
In this section, we describe the flocking problem and the kinematics of fixed-
wing UAV. Then, we formulate this problem as a Markov decision process (MDP)
in the RL framework.
### II-A Problem Description
In our flocking scenario, a unique leader is followed by a variable number of
followers. These followers are homogeneous and fly at the same altitude [10,
20]. The leader is remotely controlled by a human operator via a ground
control station. We assume that each UAV is able to obtain the state of other
UAVs through the inner communication channel [21]. Each follower has to select
its steering commands independently to maintain a certain distance $\rho$ from
the leader ( $d_{1}<\rho<d_{2}$ ) while avoiding collisions with other
followers simultaneously, as shown in Figure 1.
Figure 1: The top view of the relationship between the leader and followers.
Note that the number of followers, i.e., $n$, is a variable.
### II-B Kinematics of Fixed-Wing UAVs
We use the same numerical kinematic model of fixed-wing UAVs with stochastic
disturbances as in [16] to generate simulated samples for training the DRL-
based control policy. We rewrite the established kinematics as follows:
${\rm{\dot{\xi}}}=\frac{d}{{dt}}\left\\{{\begin{array}[]{*{20}{l}}x\\\ y\\\
\psi\\\ \phi\\\
v\end{array}}\right\\}=\left\\{{\begin{array}[]{*{20}{c}}{v\cos\psi+{\eta_{x}}}\\\
{v\sin\psi+{\eta_{y}}}\\\
{-\left({{\alpha_{g}}/v}\right)\tan\phi+{\eta_{\psi}}}\\\
{f_{\phi}\left({\phi,{\phi_{d}}}\right)}\\\
{f_{v}\left({v,{v_{d}}}\right)}\end{array}}\right\\},$ (1)
where $\left({x,\;y}\right)$ is the planar position; $\psi$ is the heading
angle; $\phi$ is the roll-angle; $v$ is the airspeed; ${\alpha_{g}}$ is the
gravitational acceleration;
$\left({{\eta_{x}},\;{\eta_{y}},\;{\eta_{\psi}}}\right)$ are disturbance terms
that follow normal distributions; the roll dynamics
$f_{\phi}\left({\phi,\;{\phi_{d}}}\right)$ and the airspeed dynamics
$f_{v}\left({v,\;{v_{d}}}\right)$ describe the response relationship between
(i) the desired airspeed ${v_{d}}$ and the actual airspeed $v$, and (ii) the
roll-angle setpoint ${\phi_{d}}$ and the actual roll-angle $\phi$,
respectively.
### II-C MDP of Flocking with Collision Avoidance
#### II-C1 State Representation
As mentioned above, there are one leader and $n$ followers. From the
perspective of an arbitrary follower, e.g., follower-$i$ ($i=1,\ 2,\ \ldots,\
n$), the flock of UAVs can be divided into three groups: the follower-i itself
(termed as the ego-follower), the leader, and the other followers
{follower-$j$ $|$ $j=1,\ 2,\ldots,\ n,\ j\not=i\\}$. Accordingly, the system
state is composed of three parts: the ego-follower’s state
${{\rm{\xi}}_{e}}={\rm{\xi}}_{f}^{i}$, the leader’s state ${{\rm{\xi}}_{l}}$,
and the other followers’ states ${\rm{\xi}}_{o}=\\{{\rm{\xi}}_{f}^{j}$ $|$
$j=1,\ 2,\ \ldots,\ n,\ j\not=i\\}$, where
${\rm{\xi}}:=(x,\;y,\;\psi,\;\phi,\;v)$ is a tuple that describes the state of
a single fixed-wing UAV according to its kinematics.
To reduce the redundancy of the representation of the system state, the ego-
follower and leader joint state
${s^{e}}:=\left({s_{1}^{e},\;s_{2}^{e},\;s_{3}^{e},\;s_{4}^{e},\;s_{5}^{e},\;s_{6}^{e},\;s_{7}^{e},\;s_{8}^{e},\;s_{9}^{e}}\right)$
is defined as:
$\begin{aligned} \left[{\begin{array}[]{*{20}{l}}{s_{1}^{e}}\\\
{s_{2}^{e}}\end{array}}\right]&=\left[{\begin{array}[]{*{20}{c}}{\cos{\psi_{l}}}&{\sin{\psi_{l}}}\\\
{-\sin{\psi_{l}}}&{\cos{\psi_{l}}}\end{array}}\right]\left[{\begin{array}[]{*{20}{l}}{{x_{e}}-{x_{l}}}\\\
{{y_{e}}-{y_{l}}}\end{array}}\right]\\\ s_{3}^{e}&={\psi_{e}}-{\psi_{l}}\\\
\vspace{0.5ex}s_{4}^{e}&={\phi_{e}}\\\ \vspace{0.5ex}s_{5}^{e}&={\phi_{l}}\\\
\vspace{0.5ex}s_{6}^{e}&=\phi_{d}^{l}\\\ \vspace{0.5ex}s_{7}^{e}&={v_{e}}\\\
\vspace{0.5ex}s_{8}^{e}&={v_{l}}\\\
\vspace{0.5ex}s_{9}^{e}&=v_{d}^{l}\end{aligned},$ (2)
where $(s_{1}^{e},s_{2}^{e})$ is the planar position of the ego-follower
relative to the leader; $s_{3}^{e}\in[-\pi,\pi)$ is the difference in the
heading between the leader and the follower. Besides, $\phi_{d}^{l}$ and
$v_{d}^{l}$ are the roll-angle and velocity setpoints of the leader,
respectively.
The ego-follower and other followers joint state
$s^{o}=\\{s_{j}^{o}:=\left({s_{1,j}^{o},\;s_{2,j}^{o},\;s_{3,j}^{o},\;s_{4,j}^{o},\;s_{5,j}^{o}}\right)$$|{}$
$j=1,\ 2,\ \ldots,\ n,\ j\not=i\\}$ is defined as:
$\begin{aligned} \left[{\begin{array}[]{*{20}{l}}{s_{1,j}^{o}}\\\
\vspace{0.5ex}{s_{2,j}^{o}}\end{array}}\right]&=\left[{\begin{array}[]{*{20}{c}}{\cos{\psi_{e}}}&{\sin{\psi_{e}}}\\\
\vspace{0.5ex}{-\sin{\psi_{e}}}&{\cos{\psi_{e}}}\end{array}}\right]\left[{\begin{array}[]{*{20}{l}}{{x_{f}^{j}}-{x_{e}}}\\\
\vspace{0.5ex}{{y_{f}^{j}}-{y_{e}}}\end{array}}\right]\\\
s_{3,j}^{o}&={\psi_{f}^{j}}-{\psi_{e}}\\\
\vspace{0.5ex}s_{4,j}^{o}&={\phi_{f}^{j}}\\\
\vspace{0.5ex}s_{5,j}^{o}&=v_{f}^{j}\\\ \end{aligned},$ (3)
where $\left({s_{1,j}^{o},s_{2,j}^{o}}\right)$ represents the planar position
of another follower-$j$, relative to the ego-follower;
$s_{3,j}^{o}\in[-\pi,\pi)$ denotes the difference in the heading between the
ego-follower and follower-$j$.
Consequently, concatenating the two joint states above, we construct the state
representation of the system $s:=\left({s^{e},\;s^{o}}\right)$. We note that
its dimensionality depends on the number of followers.
#### II-C2 Action Space
The followers are maneuvered by executing the roll and velocity actions. In
this paper, we define the action $a:=\left({a_{r},\;a_{v}}\right)$ in
continuous spaces, where the roll action
${a_{r}}\in[-\frac{\pi}{18},\frac{\pi}{18}]$ and the velocity action
${a_{v}}\in\left[{-1,\;+1}\right]$. The current roll-angle and airspeed of a
follower are denoted by $\phi$ and $v$, respectively. The next roll-angle
setpoint ${\phi_{d}}$ can be calculated by:
$\phi_{d}=\left\\{\begin{array}[]{ll}\vspace{0.5ex}r_{\mathrm{bd}}&\mathrm{if}\
\phi+a_{r}>r_{\mathrm{bd}}\\\ \vspace{0.5ex}-r_{\mathrm{bd}}&\mathrm{if}\
\phi+a_{r}<-r_{\mathrm{bd}}\\\
\vspace{0.5ex}\phi+a_{r}&\mathrm{otherwise}\end{array}\right.,$ (4)
where $[-{r_{{\rm{bd}}}},\;{r_{{\rm{bd}}}}]$ is the allowed range of the roll-
angle setpoint.
Similarly, the next velocity setpoint ${v_{d}}$ is defined by:
$v_{d}=\left\\{\begin{array}[]{ll}\vspace{0.5ex}{v_{\max}}&\mathrm{if}\
v+a_{v}>{v_{\max}}\\\ \vspace{0.5ex}{v_{\min}}&\mathrm{if}\
v+a_{v}<{v_{\min}}\\\
\vspace{0.5ex}v+a_{v}&\mathrm{otherwise}\end{array}\right.,$ (5)
where the ${v_{\max}}$ and ${v_{\min}}$ denote the maximum velocity and the
minimum velocity of the follower, respectively.
#### II-C3 Reward Function
In this paper, the followers aim to avoid collision between each other while
flocking with the leader. Therefore, the reward function consists of two
pieces: the flocking reward and the collision penalty. Specifically, in order
to facilitate the agent (i.e., the ego-follower) to maintain a suitable
distance from the leader, the flocking reward ${r_{l}}$ is designed as follows
[10]:
$r_{l}=-\max\left\\{{{d_{e}},\;\frac{{{d_{1}}\left|{s_{3}^{e}}\right|}}{{\pi\left({1+\omega{d_{e}}}\right)}}}\right\\},$
(6)
where
${d_{e}}=\max\left\\{{m\left({{d_{1}}-\rho}\right),0,\rho-{d_{2}}}\right\\}$.
$d_{1}$ and $d_{2}$ are the inner and outer radius of the desired annulus
shown in Figure 1, respectively.
$\rho=\sqrt{{{\left|{s_{1}^{e}}\right|}^{2}}+{{\left|{s_{2}^{e}}\right|}^{2}}}$
denotes the distance between the leader and the agent. Both $\omega$ and $m$
are tuning parameters.
Besides, the collision penalty $r_{c}^{j}$ is defined to prevent the agent
from colliding with the follower-$j$, as follows:
$r_{c}^{j}=-\max\left\\{{m\left({{d_{1}}-\rho_{j}^{o}}\right),0}\right\\},$
(7)
where
$\rho_{j}^{o}=\sqrt{{{\left|{s_{1,j}^{o}}\right|}^{2}}+{{\left|{s_{2,j}^{o}}\right|}^{2}}}$
is the distance from the agent to follower-$j$.
Lastly, the final reward function $r$ is specified as:
$r={r_{l}}+\sum\nolimits_{j}{r_{c}^{j}}.$ (8)
## III Approach
### III-A PS-CACER
In our previous work [15], [16], we proposed an algorithm called continuous
actor-critic with experience replay (CACER). In contrast to other actor-critic
algorithms, e.g., DDPG [22], one distinctive feature of CACER is in its
positive-temporal difference (TD) update scheme [23] for training the policy
(actor). In other words, CACER updates its policy only when the TD-error is
positive [24]. In this paper, we propose a novel algorithm PS-CACER by
extending CACER to a multi-agent scenario (see Algorithm 1). Specifically, PS-
CACER has a plug-n-play embedding module (see Section III-B) based on
convolutional neural networks and the attention mechanism to handle variable-
length system state, while CACER only deals with fixed-length system state.
Algorithm 1 PS-CACER
0: $N_{s}\ -$ maximum time steps; $N_{b}\ -$ training batch size; $M\ -$
desired number of training episodes
1: Empty replay memory $D$ with capacity $N$
2: Initialize actor $Act_{\theta^{A}}(s)$ and critic $V_{\theta^{V}}(s)$
randomly
3: for $episode=1$ to $M$ do
4: Initialize the number of followers $n$ randomly
5: Represent initial state $s_{i}$ for each follower $i$
6: for $t=1$ to $N_{s}$ do
7: for follower $i=1$ to $n$ do
8: Select action ${a_{i}}\leftarrow
Act_{\theta^{A}}\left(s_{i}\right)+\mathcal{N}$ with the current policy and
Gaussian exploration noise
9: Execute the selected action ${a_{i}}$
10: end for
11: for follower $i=1$ to $n$ do
12: Represent new system state ${s^{\prime}_{i}}$
13: Calculate immediate reward ${r_{i}}$
14: Store tuple $({s_{i}},{a_{i}},{r_{i}},s^{\prime}_{i})$ in $D$
15: $s_{i}\leftarrow s_{i}^{\prime}$
16: end for
17: Sample a minibatch of $N_{b}$ tuples
$({s_{k}},{a_{k}},{r_{k}},s^{\prime}_{k})$ from $D$
18: Empty temporal buffer $D^{\prime}=\varnothing$
19: for tuple $k=1$ to $N_{b}$ do
20: Calculate TD-error: ${\delta_{k}}={r_{k}}+\gamma\cdot
V_{\theta^{V}}\left({s^{\prime}}_{k}\right)-V_{\theta^{V}}\left(s_{k}\right)$
21: Store tuple $k$ to $D^{\prime}$ if ${\delta_{k}}>0$
22: end for
23: Optimize actor by minimizing the loss:
$\frac{1}{{\left\|{D^{\prime}}\right\|}}{\sum\nolimits_{k^{\prime}}{\left\|{{a_{k^{\prime}}}-Act_{\theta^{A}}\left({s_{k^{\prime}}}\right)}\right\|}^{2}}$
24: Optimize critic by minimizing the loss of
$\frac{1}{{{N_{b}}}}{\sum\nolimits_{k}{\left\|{{\delta_{k}}}\right\|}^{2}}$
25: end for
26: end for
As mentioned in Section II-A, the followers are homogeneous. Thus, we adopt
the parameter sharing (PS) approach [25] and allow all the followers to share
the parameters of a single policy. This shared policy can be optimized more
efficiently with the data of experiences collected by all the followers. We
note that the PS-CACER algorithm follows the centralized-learning and
decentralized-execution fashion [26]. In other words, during the learning
phase, experiences obtained by all the followers simultaneously are stored
into a shared experience replay memory, and the shared policy in terms of deep
neural networks are trained with the stored experiences centrally
(centralized-learning). Nevertheless, during the execution phase, each
follower independently selects and executes its action by following the shared
policy based on its perceived state (decentralized-execution).
### III-B Network Architecture
Inspired by [27] and [28], we design a customized network module based on
convolutional neural networks and attention mechanisms, which enables our
algorithm to adapt to the changes in the number of followers.
As illustrated in Figure 2, the Embedding module is composed of two
convolutional layers (Conv), two Squeeze-and-Excitation (SE) blocks [28], a
transpose layer (Transpose), a max-pooling layer (MaxPooling), and a flatten
layer (Flatten). First, the variable-length input (i.e., the ego-follower and
other followers joint state ${s^{o}}$) is passed by two convolutional layers
(i.e., Conv1 and Conv2) successively. The filter size of Conv1 is equal to the
length of $s_{j}^{o}$ , while the filter size of Conv2 is equal to the number
of filters of Conv1. With this specialized structure, each feature map
extracted by convolutional layers only depends on one follower (one of the
other followers). Besides, benefitting from the properties of CNNs, i.e.,
weight sharing and shift invariance [28], the sequencing operations according
to the distances between the leader and the followers as in [17, 18] are no
longer required. In other words, the output are invariant to the indexing of
the followers.
Figure 2: Embedding module. Each convolutional layer (Conv) is featured by the
number of filters, filter size, stride, and activation mode. Other layers are
represented by their type/name and output size. Figure 3: Network
architecture. Each fully-connected layer (Dense) is featured by its type,
number of neurons, and activation mode. Other layers are represented by their
type/name and output size.
After each convolutional layer, a SE block is added to improve the capacity
for feature extraction of networks. The SE block is a channel attention
mechanism. Each SE block uses a global average pooling layer
(GlobalAvgPooling) in the squeeze phase and two fully-connected layers (Dense)
with different activation functions in excitation phase, followed by a
channel-wise scaling operation (Multiply). By explicitly modelling the
relationship between channels, this architectural unit can enhance its
selectivety to the more informative features while suppressing the less useful
ones, boosting the representational power of the network [28]. After the
second Multiply layer, a max-pooling layer is added to create a fixed-size
output, independent of the input order and length. Lastly, the output of the
max-pooling layer is flattened by a flatten layer to create a fixed-length
embedding vector. In this way, the variable-length input can be embedded into
a fixed-length embedding vector.
We term our scheme of the Embedding module as SE-MaxPooling (SEMP). The
designed SEMP embedding module can handle inputs with arbitrary length and its
output is independent of the input order. We emphasize that this module is
plug-n-play, which means the SEMP embedding module can be easily integrated
with other network architectures, as well as any reinforcement learning
algorithm.
The entire architecture of the actor network and the critic network is
depicted in Figure 3. We process the ego-follower and leader joint state
${s^{e}}$ and the ego-follower and other followers joint state ${s^{o}}$
separately [17]. Specifically, a dense layer with ReLU activation function is
used to extract the features of ${s^{e}}$, and the Embedding module is used to
encode the ${s^{o}}$ into a fixed-length vector. After that, the two outputs
above are merged and then subsequently fed into 3 dense layers with different
activation functions. Note that both the actor and the critic use the same
network architecture up to the last output layer. The output layer of the
critic uses a linear activation function, while the output payer of the actor
uses a hyperbolic tangent (tanh) activation function.
## IV Simulation Results
In this section, we evaluate the proposed PS-CACER algorithm with the SEMP
scheme in both numerical simulation and semi-physical simulation.
### IV-A Simulation Setup
In this paper, the PS-CACER algorithm was trained with a total of 30000
episodes ($M=$ 30000), in which each episode had a maximal number of 60 time
steps ($N_{s}=$ 60). All network parameters were updated with 64 batch size
($N_{b}=$ 64), Adam optimizer with 0.001 (for actor) and 0.0001 (for critic)
learning rates. The exploration parameter was annealed exponential from 0.5 to
0.05 over a period of 2000 episodes, and then fixed at 0.05 thereafter. The
empirical values of essential parameters are listed in Table I [15, 16].
TABLE I: Parameter Settings Name | Value | Name | Value
---|---|---|---
d1 | 40 | d2 | 65
$\omega$ | 0.05 | m | 2
${\alpha_{g}}$ | 9.8 | ${r_{{\rm{bd}}}}$ | $\frac{\pi}{6}$
${v_{\max}}$ | 18 | ${v_{\min}}$ | 12
$N$ | 100000 | $\gamma$ | 0.95
### IV-B Numerical Simulation
In order to verify the effectiveness of the proposed SEMP scheme, three
existing state-of-the-art approaches, LSTM [17], SA (social attentive pooling)
[29], and CNNMP (convolutional neural networks with max pooling)[27], were
selected as the benchmarks. To make a fair comparison, we kept their network
architectures the same except for the embedding module. Besides, the above
solutions also used the same learning algorithm, i.e., the PS-CACER algorithm,
to update their network parameters. We note that, i) the number of LSTM cells
was set to 64, in accordance with the length of the fixed-length vector
encoded by SEMP. ii) the network and its parameters used by SA were the same
as [29]. iii) The network structure of CNNMP was the same as SEMP, except that
SEMP merged a channel attention mechanism by adding two SE blocks.
In the training phase, we evaluated the performance of the proposed algorithm
using the average reward $G_{{\rm{Avg}}}$ obtained by one agent within a
certain number of $n$e episodes:
${G_{{\rm{Avg}}}}=\frac{1}{{n{\kern
1.0pt}{N_{e}}{N_{s}}}}\sum\limits_{i=1}^{n}{\sum\limits_{p=1}^{{N_{e}}}{\sum\limits_{t=1}^{{N_{s}}}{{}^{i}r_{t}^{p}}}},$
(9)
where ${}^{i}r_{t}^{p}$ is the immediate reward obtained by agent $i$ at the
time step $t$ of the episode $p$ according to (8). Without loss of generality,
we set $N_{e}=$ 100\. At each episode, the number of followers $n$ was
randomly selected from 3 to 10. At each time step, the initial state of the
leader and followers were initialized randomly, and the steer commands of the
leader were generated randomly from the action spaces defined in Section
II-C2.
Figure 4: Learning curves of LSTM, SA, CNNMP, and SEMP. TABLE II: Comparison Results of LSTM, SA, CNNMP, and SEMP Method | Average Reward [Avg / Var] | Collision Rate (%) [Avg / Var]
---|---|---
$n$ = 4 | $n$ = 6 | $n$ = 8 | $n$ = 10 | $n$ = 4 | $n$ = 6 | $n$ = 8 | $n$ = 10
LSTM [17] | -70.99/330.43 | -76.07/292.45 | -81.89/97.87 | -92.70/152.85 | 0.179/0.019 | 0.300/0.058 | 0.325/0.026 | 0.429/0.029
SA [29] | -49.25/97.86 | -59.01/104.62 | -67.20/101.59 | -79.72/155.76 | 0.282/0.120 | 0.272/0.016 | 0.375/0.025 | 0.554/0.038
CNNMP [27] | -48.88/127.72 | -57.51/133.02 | -63.39/112.21 | -72.98/122.63 | 0.215/0.048 | 0.282/0.026 | 0.294/0.025 | 0.404/0.036
SEMP (Ours) | -36.95/58.32 | -45.65/79.99 | -51.91/60.34 | -60.54/49.67 | 0.140/0.011 | 0.192/0.016 | 0.285/0.021 | 0.317/0.021
The learning curves of the four above approaches are shown in Figure 4. As can
be seen, the LSTM scheme has the worst performance in terms of both learning
efficiency and the obtained average reward. One reasonable explanation is that
LSTM has too many parameters to optimize. Besides, the learning curves of SA
and CNNMP are almost overlapped, indicating that the two schemes have similar
performance. Additionally, the average reward obtained by SEMP grows as fast
as CNNMP in the early stage. However, with the increase of training episodes,
the average reward of SEMP still gradually increases, and finally obtains a
higher reward. This comparison results validate that the attention mechanism
with SE blocks used by SEMP is effective. Overall, SEMP finally obtains the
highest reward with a similar learning efficiency. This result demonstrates
that our SEMP scheme has advantage over the three benchmarks.
After training, we tested the control policy learned by the PS-CACER algorithm
with the SEMP scheme (hereafter referred to as the learned SEMP policy for
short) in a flocking task with eight followers lasting for 200 seconds.
Initially, the leader was followed by four followers (Follower #1 $\sim$ #4 in
Figure 5). Since the time step 100, the other four followers (Follower #5
$\sim$ #8 in Figure 5) joined the flock simultaneously. The testing results
are visualized in Figure 5. The generated trajectories illustrate that the
eight followers can successfully keep up with the leader while avoiding
collisions with each other. This result indicates that the learned SEMP policy
can handle a variable number of followers.
Figure 5: The visualized results of the eight followers following the leader
using the learned SEMP policy, i.e., trajectories (left), the distances
($\rho$) to the leader (top-right), and the minimum distance ($mindis$) among
the followers (bottom-right).
To further quantitatively evaluate the performance of the learned SEMP policy,
we defined collision rate as a metric aside from the average reward. We
considered that a collision would happen if the distance between two followers
was less than a certain value, e.g., 2 meters. Thus, the collision rate meant
the percentage of being too close (i.e., less than 2 meters) among the
followers during the testing episode. We compared the learned SEMP policy with
three baselines by averaging the results of 200 episode. The parameter
settings are the same as the training phase, but the maximum time step
($N_{s}$) for each episode was set to 180. Table II and Figure 6 compares the
results of the four methods.
As can be seen from Table II, the learned policies with one set of network
parameters can adapt to a dynamic squad of UAVs, without the need of
retraining in a new environment with different number of followers. When the
number of followers increases, the average reward decreases and the collision
rate increases. The reason is that increasing the number of followers leads to
a higher probability of collision. However, compared with the other three
baselines, our SEMP method enables the follower to obtain the largest average
reward, the lowest collision rate, and the lowest variance, regardless of the
number of followers. This result shows that the SEMP method outperforms the
three existing state-of-the-art methods.
Figure 6: Box-and-whisker plot for comparison results of LSTM, SA, CNNMP, and
SEMP. Left: Average reward. Right: Collision rate.
### IV-C Semi-Physical Simulation
In addition to the numerical simulation, we also tested the generalization
performance of the learned control policy by conducting a hardware-in-loop
(HIL) experiment in a high fidelity semi-physical simulation system [15],
[16]. Instead of the kinematic model of fixed-wing UAVs with stochastic
disturbances used in the training phase, we selected the professional tool
X-Plane 10111https://www.x-plane.com/manuals/desktop/ as the flight simulator
for the testing. X-Plane 10 can simulate complex environmental conditions,
such as weather changes and wind disturbances.
In this experiment, the five followers used the learned SEMP policy to flock
with the leader lasting for 200 time steps (seconds). At each time step, the
leader selected its roll-angle setpoint according to the pre-planned paths and
changed its velocity randomly, then broadcasted its state information to all
the followers. Each follower used the actor network with the trained
parameters to determine its steering commands according to its own state, the
leader’s state, and the other followers’ states.
Figure 7: The trajectory results in the semi-physical simulation. Note that
the three representative snapshots displayed on the top are captured from the
ground control station. The top-left and top-right snapshots show a third
person perspective of the UAV flock, while the top-middle snapshot is top
view. Figure 8: The distances between the leader and five followers ($\rho$)
and the minimum distance among five followers ($mindis$) in the semi-physical
simulation.
The trajectory results are depicted in Figure 7; the distances between the
leader and the five followers ($\rho$) as well as the minimum distance among
the five followers ($mindis$) are shown in Figure 8. As can be seen, the
distances between the leader and the followers were maintained around
75$\,\rm{m}$ most of the time (the average distance is 74.44$\,\rm{m}$). This
means that the five followers were able to keep up with the leader steadily,
even if the leader changed its heading sharply. More importantly, the minimum
value of $mindis$ is larger than 2$\,\rm{m}$, which means that there were no
collisions among the followers during this experiment. The above results
demonstrate that the proposed SEMP method enables the followers to avoid
collisions between each other while flocking with the leader. We note that the
control policy employed by the five followers in this semi-physical simulation
is the SEMP policy learned from the previous numerical simulation, without any
parameter finetuning. This result demonstrates that the learned SEMP policy
can be directly generalized to new situations.
## V Conclusion
In this paper, we have proposed a deep reinforcement learning algorithm to
solve the leader-follower flocking control and collision avoidance problem for
a dynamic squad of fixed-wing UAVs. Specifically, we have designed a
decentralized DRL-based framework where the collision avoidance policy is
integrated into the flocking controller for a variable number of UAVs. Then,
we have proposed the PS-CACER algorithm for training multiple agents, in which
a customized embedding module, SEMP, is integrated to handle the variable-
length input for deep neural networks. With this module, the learned policies
are adaptive to the changes of the number of UAVs without the need of
retraining. Numerical simulation results have demonstrated that the proposed
method outperforms three existing state-of-the-art methods, i.e., LSTM, SA,
and CNNMP. Finally, the learned policy can be directly transferred to a semi-
physical simulation without any parameter finetuning. In the future, we will
further evaluate our method with fixed-wing UAVs in real-world environments.
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|
# Upstream mobility Finite Volumes for the Richards equation in heterogenous
domains
Sabrina Bassetto IFP Energies nouvelles, 1 et 4 avenue de Bois Préau, 92852
Rueil-Malmaison Cedex, France<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Clément Cancès Inria,
Univ. Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, 59000 Lille, France.
<EMAIL_ADDRESS>Guillaume Enchéry∗ Quang-Huy Tran∗
###### Abstract
This paper is concerned with the Richards equation in a heterogeneous domain,
each subdomain of which is homogeneous and represents a rocktype. Our first
contribution is to rigorously prove convergence toward a weak solution of
cell-centered finite-volume schemes with upstream mobility and without
Kirchhoff’s transform. Our second contribution is to numerically demonstrate
the relevance of locally refining the grid at the interface between
subregions, where discontinuities occur, in order to preserve an acceptable
accuracy for the results computed with the schemes under consideration.
Keywords
Richards’ equation, heterogeneous domains, finite-volume schemes, mobility
upwinding
Mathematics subject classification
65M08, 65M12, 76S05
## 1 Presentation of the continuous model
The Richards equation [41] is one of the most well-known simplified models for
water filtration in unsaturated soils. While it has been extensively studied
in the case of a homogeneous domain, the heterogeneous case seems to have
received less attention in the literature, at least from the numerical
perspective. The purpose of this paper is to investigate a class of
discretization scheme for a special instance of heterogeneous domains, namely,
those with piecewise-uniform physical properties.
Before stating our objectives in a precise manner, a few prerequisites must be
introduced regarding the model in §1.1–§1.2 and the scheme in §2.1–§2.2. The
goal of the paper is fully described in §1.3, in relation with other works.
Practical aspects related to the numerical resolution are detailed in §5 and
results on illustrative test cases are shown in §6. A summary of our main
results is provided in §2.3, together with the outline of the paper.
### 1.1 Richards’ equation in heterogeneous porous media
Let $\Omega\subset\mathbb{R}^{d}$, where $d\in\\{2,3\\}$, be a connected open
polyhedral domain with Lipschitz boundary $\partial\Omega$. A porous medium
defined over the region $\Omega$ is characterized by
* –
the porosity $\phi:\Omega\rightarrow(0,1]$;
* –
the permeability $\lambda:\Omega\rightarrow\mathbb{R}_{+}^{*}$;
* –
the mobility function $\eta:[0,1]\times\Omega\rightarrow\mathbb{R}_{+}$;
* –
the capillary pressure law
$\mathcal{S}:\mathbb{R}\times\Omega\rightarrow[0,1]$.
The conditions to be satisfied by $\phi$, $\lambda$, $\eta$ and $\mathcal{S}$
will be elaborated on later. In a homogeneous medium, these physical
properties are uniform over $\Omega$, i.e.,
$\phi(x)=\phi_{0},\qquad\lambda(x)=\lambda_{0},\qquad\eta(s,x)=\eta_{0}(s),\qquad\mathcal{S}(p,x)=\mathcal{S}_{0}(p)$
for all $x\in\Omega$. In a heterogeneous medium, the dependence of $\phi$,
$\lambda$, $\eta$ and $\mathcal{S}$ on $x$ must naturally be taken into
account. The quantity $s$, called saturation, measures the relative volumic
presence of water in the medium. The quantity $p$ is the pressure.
Let $T>0$ be a finite time horizon. We designate by $Q_{T}=(0,T)\times\Omega$
the space-time domain of interest. Our task is to find the saturation field
$s:Q_{T}\rightarrow[0,1]$ and the pressure field
$p:Q_{T}\rightarrow\mathbb{R}$ so as to satisfy
* •
the interior equations
$\displaystyle\phi(x)\,\partial_{t}s+{\text{div}}\,F$ $\displaystyle=0$
$\displaystyle\text{in}\;Q_{T},$ (1.1a) $\displaystyle
F+\lambda(x)\,\eta(s,x)\,\nabla(p-\varrho g\cdot x)$ $\displaystyle=0$
$\displaystyle\text{in}\;Q_{T},$ (1.1b) $\displaystyle s-\mathcal{S}(p,x)$
$\displaystyle=0$ $\displaystyle\text{in}\;Q_{T};$ (1.1c)
* •
the boundary conditions
$\displaystyle F\cdot n(x)$ $\displaystyle=0$
$\displaystyle\text{on}\;(0,T)\times\Gamma^{\rm N},$ (1.1d) $\displaystyle
p(t,x)$ $\displaystyle=p^{\rm D}(x)$
$\displaystyle\text{on}\;(0,T)\times\Gamma^{\rm D};$ (1.1e)
* •
the initial data
$\displaystyle s(0,x)$ $\displaystyle=s^{0}(x)$
$\displaystyle\text{in}\;\Omega.$ (1.1f)
The partial differential equation (1.1a) expresses the water volume balance.
The flux $F$ involved in this balance is given by the Darcy-Muskat law (1.1b),
in which $g$ is the gravity vector and $\varrho$ is the known constant density
of water, assumed to be incompressible. It is convenient to introduce
$\psi=-\varrho g\cdot x,\qquad\vartheta=p+\psi,$ (1.2)
referred to respectively as gravity potential and hydraulic head. In this way,
the Darcy-Muskat law (1.1b) can be rewritten as
$F+\lambda(x)\,\eta(s,x)\,\nabla(p+\psi)=F+\lambda(x)\,\eta(s,x)\,\nabla\vartheta=0.$
Equation (1.1c) connecting the saturation $s$ and the pressure $p$ is the
capillary pressure relation. The boundary $\partial\Omega$ is split into two
non-overlapping parts, viz.,
$\partial\Omega=\Gamma^{\rm N}\cup\Gamma^{\rm D},\qquad\Gamma^{\rm
N}\cap\Gamma^{\rm D}=\emptyset,$ (1.3)
where $\Gamma^{\rm N}$ is open and $\Gamma^{\rm D}$ is closed, the latter
having a positive $(d-1)$-dimensional Hausdorff measure $\nu^{d-1}(\Gamma^{\rm
D})>0$. The no-flux Neumann condition (1.1d) is prescribed on
$(0,T)\times\Gamma^{\rm N}$, where $n(x)$ is the outward normal unit vector at
$x\in\Gamma^{\rm N}$. The Dirichlet condition (1.1e) with a known Lipschitz
function $p^{\rm D}\in W^{1,\infty}(\Omega)$ is imposed on
$(0,T)\times\Gamma^{\rm D}$. Note that, in our theoretical development, the
function $p^{\rm D}$ is assumed to be defined over the whole domain $\Omega$,
which is stronger than a data $p^{\rm D}\in L^{\infty}(\Gamma^{\rm D})$ given
only on the boundary. The assumption that $p^{\rm D}$ does not depend on time
can be removed by following the lines of [14], but we prefer here not to deal
with time-dependent boundary data in order to keep the presentation as simple
as possible. Finally, the initial data $s^{0}\in L^{\infty}(\Omega;[0,1])$ in
(1.1f) is also a given data.
In this work, we restrict ourselves to a specific type of heterogeneous media,
defined as follows. We assume that the domain $\Omega$ can be partitioned into
several connected polyhedral subdomains $\Omega_{i}$, $1\leq i\leq I$.
Technically, this means that if $\Gamma_{i,j}$ denotes the interface between
$\Omega_{i}$ and $\Omega_{j}$ (which can be empty for some particular choices
of $\\{i,j\\}$), then
$\Omega_{i}\cap\Omega_{j}=\emptyset,\quad\overline{\Omega}_{i}\cap\overline{\Omega}_{j}=\Gamma_{i,j},\;\text{if}\;i\neq
j,\quad\Omega=\Big{(}\bigcup_{1\leq i\leq I}\Omega_{i}\Big{)}\cup\Gamma,$
(1.4)
with $\Gamma=\bigcup_{i\neq j}\Gamma_{i,j}$. Each of these subdomains
corresponds to a distinctive rocktype. Inside each $\Omega_{i}$, the physical
properties are homogeneous. In other words,
$\phi(x)=\phi_{i},\qquad\lambda(x)=\lambda_{i},\qquad\eta(s,x)=\eta_{i}(s),\qquad\mathcal{S}(p,x)=\mathcal{S}_{i}(p)$
for all $x\in\Omega_{i}$. Therefore, system (1.1) is associated with
$\displaystyle\phi(x)$ $\displaystyle=\sum_{1\leq i\leq
I}\phi_{i}\,\mathbf{1}_{\Omega_{i}}(x),$ $\displaystyle\qquad\eta(s,x)$
$\displaystyle=\sum_{1\leq i\leq I}\eta_{i}(s)\,\mathbf{1}_{\Omega_{i}}(x),$
(1.5a) $\displaystyle\lambda(x)$ $\displaystyle=\sum_{1\leq i\leq
I}\lambda_{i}\,\mathbf{1}_{\Omega_{i}}(x),$
$\displaystyle\qquad\mathcal{S}(p,x)$ $\displaystyle=\sum_{1\leq i\leq
I}\mathcal{S}_{i}(p)\,\mathbf{1}_{\Omega_{i}}(x),$ (1.5b)
where $\mathbf{1}_{\Omega_{i}}$ stands for the characteristic function of
$\Omega_{i}$. For all $i\in\\{1,\ldots,I\\}$, we assume that
$\phi_{i}\in(0,1]$ and $\lambda_{i}>0$. Furthermore, we require that
$\eta_{i}\text{ is increasing on
}[0,1],\qquad\eta_{i}(0)=0,\qquad\eta_{i}(1)=\frac{1}{\mu},$ (1.6a) where
$\mu>0$ is the (known) viscosity of water. In addition to the assumption that
$\mathcal{S}(\cdot,x)$, defined in (1.5b), is absolutley continuous and
nondecreasing, the functions $\mathcal{S}_{i}$ are also subject to some
generic requirements commonly verified the models available in the literature:
for each $i\in\\{1,\ldots,I\\}$, there exists $\overline{p}_{i}\leq 0$ such
that $\mathcal{S}_{i}\text{ is increasing on
}(-\infty,\overline{p}_{i}],\qquad\lim_{p\to-\infty}\mathcal{S}_{i}(p)=0,\qquad\mathcal{S}_{i}\equiv
1\text{ on }[\overline{p}_{i},+\infty).$ (1.6b)
This allows us to define an inverse
$\mathcal{S}_{i}^{-1}:(0,1]\to(-\infty,\overline{p}_{i}]$ such that
$\mathcal{S}_{i}\circ\mathcal{S}_{i}^{-1}(s)=s$ for all $s\in(0,1]$. We
further assume that for all $i\in\\{1,\dots,I\\}$ the function
$\mathcal{S}_{i}$ is bounded in $L^{1}(\mathbb{R}_{-})$, or equivalently, that
$\mathcal{S}_{i}^{-1}\in L^{1}(0,1)$. It thus makes sense to consider the
capillary energy density functions
$\text{{€\,}}_{i}:\mathbb{R}\times\Omega_{i}\to\mathbb{R}_{+}$ defined by
$\text{{€\,}}_{i}(s,x)=\int_{\mathcal{S}_{i}(p^{\rm
D}(x))}^{s}\phi_{i}(\mathcal{S}_{i}^{-1}(\varsigma)-p^{\rm D}(x))\,{\rm
d}\varsigma.$ (1.7)
For all $x\in\Omega_{i}$, the function $\text{{€\,}}_{i}(\cdot,x)$ is
nonnegative, convex since $\mathcal{S}_{i}^{-1}$ is monotone, and bounded on
$[0,1]$ as a consequence of the integrability of $\mathcal{S}_{i}$. For
technical reasons that will appear clearly later on, we further assume that
$\sqrt{\eta_{i}\circ\mathcal{S}_{i}}\in L^{1}(\mathbb{R}_{-}),\qquad\forall
i\in\\{1,\dots,I\\}.$ (1.8)
Let $Q_{i,T}=(0,T)\times\Omega_{i}$ be the space-time subdomains for $1\leq
i\leq I$. The interior equations (1.1a)–(1.1c) then boil down to
$\displaystyle\phi_{i}\,\partial_{t}s+{\text{div}}\,F$ $\displaystyle=0$
$\displaystyle\text{in}\;Q_{i,T},$ (1.9a) $\displaystyle
F+\lambda_{i}\,\eta_{i}\,\nabla(p+\psi)$ $\displaystyle=0$
$\displaystyle\text{in}\;Q_{i,T},$ (1.9b) $\displaystyle s-\mathcal{S}_{i}(p)$
$\displaystyle=0$ $\displaystyle\text{in}\;Q_{i,T}.$ (1.9c)
At the interface $\Gamma_{i,j}$ between $\Omega_{i}$ and $\Omega_{j}$, $i\neq
j$, any solution of (1.1a)–(1.1c) satisfies the matching conditions
$\displaystyle F_{i}\cdot n_{i}+F_{j}\cdot n_{j}$ $\displaystyle=0$
$\displaystyle\text{on}\;(0,T)\times\Gamma_{i,j},$ (1.10a) $\displaystyle
p_{i}-p_{j}$ $\displaystyle=0$
$\displaystyle\text{on}\;(0,T)\times\Gamma_{i,j}.$ (1.10b)
In the continuity of the normal fluxes (1.10a), which is enforced by the
conservation of water volume, $n_{i}$ denotes the outward normal to
$\partial\Omega_{i}$ and $F_{i}\cdot n_{i}$ stands for the trace of the normal
component of $F_{|_{Q_{i,T}}}$ on $(0,T)\times\partial\Omega_{i}$. In the
continuity of pressure (1.10b), which also results from (1.1a)–(1.1c), $p_{i}$
denotes the trace on $(0,T)\times\partial\Omega_{i}$ of the pressure
$p_{|_{Q_{i,T}}}$ in the $i$-th domain.
### 1.2 Stability features and notion of weak solutions
We wish to give a proper sense to the notion of weak solution for problem
(1.1). To achieve this purpose, we need a few mathematical transformations the
definition of which crucially relies on a fundamental energy estimate at the
continuous level. The calculations below are aimed at highlighting this energy
estimate and will be carried out in a formal way, in constrast to those in the
fully discrete setting.
Multiplying (1.9a) by $p-p^{\rm D}$, invoking (1.7), integrating over
$\Omega_{i}$ and summing over $i$, we end up with
$\frac{{\rm d}}{{\rm
d}t}\,\sum_{i=1}^{I}\int_{\Omega_{i}}\text{{€\,}}_{i}(s,x)\,{\rm
d}x+\sum_{i=1}^{I}\int_{\Omega_{i}}{\text{div}}\,F(p-p^{\rm D})\,{\rm d}x=0.$
(1.11)
We now integrate by parts the second term. Thanks to the matching conditions
(1.10) and the regularity of $p^{\rm D}$, we obtain
$\mathtt{A}:=\sum_{i=1}^{I}\int_{\Omega_{i}}{\text{div}}\,F(p-p^{\rm D})\,{\rm
d}x=-\sum_{i=1}^{I}\int_{\Omega_{i}}F\cdot\nabla(p-p^{\rm D})\,{\rm d}x.$
It follows from the flux value (1.9b) that
$\displaystyle\mathtt{A}$
$\displaystyle=\sum_{i=1}^{I}\int_{\Omega_{i}}\lambda_{i}\eta_{i}(s)\nabla(p+\psi)\cdot\nabla(p-p^{\rm
D})\,{\rm d}x$
$\displaystyle\,=\sum_{i=1}^{I}\int_{\Omega_{i}}\lambda_{i}\eta_{i}(s)|\nabla
p|^{2}\,{\rm
d}x-\sum_{i=1}^{I}\int_{\Omega_{i}}\lambda_{i}\eta_{i}(s)\nabla\psi\cdot\nabla
p^{\rm D}\,{\rm d}x$
$\displaystyle\,+\;\sum_{i=1}^{I}\int_{\Omega_{i}}\lambda_{i}\eta_{i}(s)\nabla
p\cdot\nabla(\psi-p^{\rm D})\,{\rm d}x.$
Young’s inequality, combined with the boundedness of $\nabla p^{\rm D}$,
$\nabla\psi$, $\lambda$ and $\eta$, yields
$\mathtt{A}\,\geq\,\frac{1}{2}\sum_{i=1}^{I}\int_{\Omega_{i}}\lambda_{i}\eta_{i}(s)|\nabla
p|^{2}\,{\rm d}x-C$
for some $C\geq 0$ depending only on $\lambda$, $\eta$, $\psi$, $\mu$,
$\Omega$ and $p^{\rm D}$.
Let us define the energy $\mathfrak{E}:[0,T]\to\mathbb{R}_{+}$ by
$\mathfrak{E}(t)=\sum_{i=1}^{I}\int_{\Omega_{i}}\text{{€\,}}_{i}(s(t,x),x)\,{\rm
d}x,\qquad 0\leq t\leq T.$
Integrating (1.11) w.r.t. time results in
$\mathfrak{E}(T)+\frac{1}{2}\sum_{i=1}^{I}\int\\!\\!\\!\int_{Q_{i,T}}\lambda_{i}\eta_{i}(s)|\nabla
p|^{2}\,{\rm d}x\,{\rm d}t\leq\mathfrak{E}(0)+CT.$ (1.12)
Estimate (1.12) is the core of our analysis. However, it is difficult to use
in its present form since $\eta_{i}(s)=\eta_{i}(\mathcal{S}_{i}(p))$ vanishes
as $p$ tends to $-\infty$, so that the control of $\nabla p$ degenerates. To
circumvent this difficulty, we resort to the nonlinear functions (customarily
referred to as the Kirchhoff transforms) $\Theta_{i}:\mathbb{R}\to\mathbb{R}$,
$\Phi_{i}:\mathbb{R}\to\mathbb{R}$, and
$\Upsilon:\mathbb{R}\times\Omega\to\mathbb{R}$ respectively defined by
$\displaystyle\Theta_{i}(p)$
$\displaystyle=\int_{0}^{p}\sqrt{\lambda_{i}\eta_{i}\circ\mathcal{S}_{i}(\pi)}\,{\rm
d}\pi,$ $\displaystyle\qquad p$ $\displaystyle\in\mathbb{R},$ (1.13a)
$\displaystyle\Phi_{i}(p)$
$\displaystyle=\int_{0}^{p}{\lambda_{i}\eta_{i}\circ\mathcal{S}_{i}(\pi)}\,{\rm
d}\pi,$ $\displaystyle\qquad p$ $\displaystyle\in\mathbb{R},$ (1.13b)
$\displaystyle\Upsilon(p)$ $\displaystyle=\int_{0}^{p}\min_{1\leq i\leq
I}\sqrt{\lambda_{i}\eta_{i}\circ\mathcal{S}_{i}(\pi)}\,{\rm d}\pi,$
$\displaystyle\qquad p$ $\displaystyle\in\mathbb{R},$ (1.13c)
the notion of $\Upsilon$ being due to [23]. Bearing in mind that
$\mathfrak{E}(T)\geq 0$, estimate (1.12) implies that
$\sum_{i=1}^{I}\int\\!\\!\\!\int_{Q_{i,T}}|\nabla\Theta_{i}(p)|^{2}\,{\rm
d}x\,{\rm d}t\leq 2(\mathfrak{E}(0)+CT)<+\infty.$ (1.14)
As $\Phi_{i}\circ\Theta_{i}^{-1}$ is Lipschitz continuous, this also gives
rise to a $L^{2}(Q_{i,T})$-estimate on $\nabla\Phi_{i}(p)$. The functions
$\sum_{i}\Theta_{i}(p)\mathbf{1}_{\Omega_{i}}$ and
$\sum_{i}\Phi_{i}(p)\mathbf{1}_{\Omega_{i}}$ are in general discontinuous
across the interfaces $\Gamma_{i,j}$, unlike $\Upsilon(p)$. Since the
functions $\Upsilon\circ\Theta_{i}^{-1}$ are Lipschitz continuous, we can
readily infer from (1.14) that
$\int\\!\\!\\!\int_{Q_{T}}|\nabla\Upsilon(p)|^{2}\,{\rm d}x\leq C$ (1.15)
for some $C$ depending on $T$, $\Omega$, $\lVert\nabla p^{\rm
D}\rVert_{\infty}$, the
$\lVert\mathcal{S}_{i}\rVert_{L^{1}(\mathbb{R}_{-})}$’s and
$\overline{\lambda}=\lVert\lambda\rVert_{L^{\infty}(\Omega)}=\max_{1\leq i\leq
I}\lambda_{i},\qquad\overline{\eta}=\lVert\eta\rVert_{L^{\infty}(\Omega)}=\max_{1\leq
i\leq I}\lVert\eta_{i}\rVert_{L^{\infty}(\Omega)}=\frac{1}{\mu},$
the last equality being due to (1.6a).
Moreover, $\Upsilon(p)-\Upsilon(p^{\rm D})$ vanishes on
$(0,T)\times\Gamma^{\rm D}$. Poincaré’s inequality provides a
$L^{2}(Q_{T})$-estimate on $\Upsilon(p)$ since $\Gamma^{\rm D}$ has positive
measure and since $\Upsilon(p^{\rm D})$ is bounded in $\Omega$. In view of
assumption (1.8), the functions $\Theta_{i}$ and $\Upsilon$ are bounded on
$\mathbb{R}_{-}$. Besides, for $p\geq 0$,
$\eta_{i}\circ\mathcal{S}_{i}(p)=1/\mu$, so that
$\Theta_{i}(p)=p\sqrt{\lambda_{i}/\mu}$ and $\Upsilon(p)=\min_{1\leq i\leq
I}p\sqrt{\lambda_{i}/\mu}$. It finally comes that
$\Theta_{i}(p)\leq C(1+\Upsilon(p)),\qquad\forall p\in\mathbb{R},\;1\leq i\leq
I,$ (1.16)
from which we infer a $L^{2}(Q_{i,T})$-estimate on $\Theta_{i}(p)$. Putting
$V=\big{\\{}u\in H^{1}(\Omega)\;|\;u_{|_{\Gamma^{\rm D}}}=0\big{\\}},$
the above estimates suggest the following notion of weak solution for our
problem.
###### Definition 1.1.
A measurable function $p:Q_{T}\to\mathbb{R}$ is said to be a weak solution to
the problem (1.9a)–(1.9c) if
$\displaystyle\Theta_{i}(p)$ $\displaystyle\in
L^{2}((0,T);H^{1}(\Omega_{i})),\qquad\text{for }\;1\leq i\leq I,$ (1.17a)
$\displaystyle\qquad\Upsilon(p)-\Upsilon(p^{\rm D})$ $\displaystyle\in
L^{2}((0,T);V)$ (1.17b) and if for all $\varphi\in
C^{\infty}_{c}([0,T)\times(\Omega\cup\Gamma^{\rm N}))$, there holds
$\int\\!\\!\\!\int_{Q_{T}}\phi\,\mathcal{S}(p,x)\partial_{t}\varphi\,{\rm
d}x\,{\rm d}t+\int_{\Omega}\phi\,s^{0}\varphi(\cdot,0)\,{\rm
d}x+\int\\!\\!\\!\int_{Q_{T}}F\cdot\nabla\varphi\,{\rm d}x\,{\rm d}t=0,$
(1.17c) with
$F=-\nabla\Phi_{i}(p)+\lambda_{i}\eta_{i}(\mathcal{S}_{i}(p))\,\varrho
g\qquad\text{in }\;Q_{i,T},\;1\leq i\leq I.$ (1.17d)
The expression (1.17d) is a reformulation of the original one (1.9b) in a
quasilinear form which is suitable for analysis, even though the physical
meaning of the Kirchhoff transform $\Phi_{i}(p)$ is unclear. While the
formulation (1.17c) should be thought of as a weak form of (1.9a), (1.10a),
(1.1f), and (1.1d), the condition $\Upsilon(p)-\Upsilon(p^{\rm D})\in
L^{2}((0,T);V)$ contains (1.10b) and (1.1e).
### 1.3 Goal and positioning of the paper
We are now in a position to clearly state the two objectives of this paper.
The first objective is to put forward a rigorous proof that, for problem (1.1)
with heterogeneous data (1.5), cell-centered finite-volume schemes with
upstream mobility such as described in §2.2, do converge towards a weak
solution (in the sense of Definition 1.1) as the discretization parameters
tend to $0$. Such mathematically assessed convergence results are often
dedicated to homogeneous cases: see for instance [3, 30, 40] for schemes
involving the Kirchhoff transforms for Richards’ equation, [1] for a upstream
mobility CVFE approximation of Richards’ equation in anisotropic domains, [17,
18, 19] for schemes for two-phase flows involving the Kirchhoff transform, and
[31, 34] for upstream mobility schemes for two-phase porous media flows. For
flows in highly heterogeneous porous media, rigorous mathematical results have
been obtained for schemes involving the introduction of additional interface
unknowns and Kirchhoff’s transforms (see for instance [23, 11, 12, 6]), or
under the non-physical assumption that the mobilities are strictly positive
[28, 29]. It was established very recently in [7] that cell-centered finite-
volumes with (hybrid) upwinding also converge for two-phase flows in
heterogeneous domains, but with a specific treatment of the interfaces located
at the heterogeneities. Here, the novelty lies in the fact that we do not
consider any specific treatment of the interface in the design of the scheme.
The second objective is of more practical nature. Even though our analysis
still holds without any specific treatment of the interface, it is well-known
that cell-centered upstream mobility finite-volumes can be inaccurate in the
presence of heterogeneities. This observation motivated several contributions
(see for instance [24, 23, 35, 29]) where skeletal (i.e., edge or vertex)
unknowns where introduced in order to enforce the continuity of the pressures
at the interfaces $\Gamma_{i,j}$. By means of extensive numerical simulations
in §6, we will show that without local refinement of the grid at the
interface, the method still converges, but with a degraded order. Our ultimate
motivation is to propose an approach which consists in adding very thin cells
on both sides of the interface before using the cell centered scheme under
study. Then the scheme appears to behave better, with first-order accuracy.
Moreover, one can still make use of the parametrized cut-Newton method
proposed in [4] to compute the solution to the nonlinear system corresponding
to the scheme. This method appears to be very efficient, while it avoids the
possibly difficult construction of compatible parametrizations at the
interfaces as in [8, 9, 7].
## 2 Finite-volume discretization
The scheme we consider in this paper is based on two-point flux approximation
(TPFA) finite-volumes. Hence, it is subject to some restrictions on the mesh
[26, 33]. We first review the requirements on the mesh in §2.1. Next, we
construct the upstream mobility finite-volume scheme for Richards’ equation in
§2.2. The main mathematical results of the paper, which are the well-posedness
of the nonlinear system corresponding to the scheme and the convergence of the
scheme, are then summarized in §2.3.
### 2.1 Admissible discretization of $Q_{T}$
Let us start by discretizing w.r.t. space.
###### Definition 2.1.
An _admissible mesh of $\Omega$_ is a triplet
$(\mathscr{T},\mathscr{E},{(x_{K})}_{K\in\mathscr{T}})$ such that the
following conditions are fulfilled:
1. (i)
Each control volume (or cell) $K\in\mathscr{T}$ is non-empty, open, polyhedral
and convex, with positive $d$-dimensional Lebesgue measure $m_{K}>0$. We
assume that
$K\cap L=\emptyset\quad\text{if}\;K,L\in\mathscr{T}\;\text{with}\;K\neq
L,\qquad\text{while}\quad\bigcup_{K\in\mathscr{T}}\overline{K}=\overline{\Omega}.$
Moreover, we assume that the mesh is adapted to the heterogeneities of
$\Omega$, in the sense that for all $K\in\mathscr{T}$, there exists
$i\in\\{1,\dots,I\\}$ such that $K\subset\Omega_{i}$.
2. (ii)
Each face $\sigma\in\mathscr{E}$ is closed and is contained in a hyperplane of
$\mathbb{R}^{d}$, with positive $(d-1)$-dimensional Hausdorff measure
$\nu^{d-1}(\sigma)=m_{\sigma}>0$. We assume that
$\nu^{d-1}(\sigma\cap\sigma^{\prime})=0$ for
$\sigma,\sigma^{\prime}\in\mathscr{E}$ unless $\sigma^{\prime}=\sigma$. For
all $K\in\mathscr{T}$, we assume that there exists a subset $\mathscr{E}_{K}$
of $\mathscr{E}$ such that $\partial
K=\bigcup_{\sigma\in\mathscr{E}_{K}}\sigma$. Moreover, we suppose that
$\bigcup_{K\in\mathscr{T}}\mathscr{E}_{K}=\mathscr{E}$. Given two distinct
control volumes $K,L\in\mathscr{T}$, the intersection
$\overline{K}\cap\overline{L}$ either reduces to a single face
$\sigma\in\mathscr{E}$ denoted by $K|L$, or its $(d-1)$-dimensional Hausdorff
measure is $0$.
3. (iii)
The cell-centers $(x_{K})_{K\in\mathscr{T}}$ are pairwise distinct with
$x_{K}\in K$, and are such that, if $K,L\in\mathscr{T}$ share a face $K|L$,
then the vector $x_{L}-x_{K}$ is orthogonal to $K|L$.
4. (iv)
For the boundary faces $\sigma\subset\partial\Omega$, we assume that either
$\sigma\subset\Gamma^{\rm D}$ or $\sigma\subset\overline{\Gamma}{}^{\rm N}$.
For $\sigma\subset\partial\Omega$ with $\sigma\in\mathscr{E}_{K}$ for some
$K\in\mathscr{T}$, we assume additionally that there exists
$x_{\sigma}\in\sigma$ such that $x_{\sigma}-x_{K}$ is orthogonal to $\sigma$.
In our problem, the standard Definition 2.1 must be supplemented by a
compatibility property between the mesh and the subdomains. By “compatbility”
we mean that each cell must lie entirely inside a single subregion. Put
another way,
$\forall
K\in\mathscr{T},\quad\exists!\;i(K)\in\\{1,\ldots,I\\}\;\;|\;\;K\subset\Omega_{i(K)}.$
(2.1)
This has two consequences. The first one is that, if we define
$\mathscr{T}_{i}=\\{K\in\mathscr{T}\;|\;K\subset\Omega_{i}\\},\qquad 1\leq
i\leq I,$ (2.2)
then $\mathscr{T}=\bigcup_{i=1}^{I}\mathscr{T}_{i}$. The second one is that
the subdomain interfaces $\Gamma_{i,j}$ for $i\neq j$ coincide necessarily
with some edges $\sigma\in\mathscr{E}$. To express this more accurately, let
$\mathscr{E}_{\Gamma}=\\{\sigma\in\mathscr{E}\;|\;\sigma\subset\Gamma\\}$ be
the set of the interface edges, $\mathscr{E}_{\rm ext}^{\rm
D}=\\{\sigma\in\mathscr{E}\;|\;\sigma\subset\Gamma^{\rm D}\\}$ be the set of
Dirichlet boundary edges, and $\mathscr{E}_{\rm ext}^{\rm
N}=\\{\sigma\in\mathscr{E}\;|\;\sigma\subset\overline{\Gamma}{}^{\rm N}\\}$ be
the set of Neumann boundary edges. Then,
$\Gamma=\bigcup_{\sigma\in\mathscr{E}_{\Gamma}}\sigma$, while $\Gamma^{\rm
D}=\bigcup_{\sigma\in\mathscr{E}_{\rm ext}^{\rm D}}\sigma$ and
$\overline{\Gamma}{}^{\rm N}=\bigcup_{\sigma\in\mathscr{E}_{\rm ext}^{\rm
N}}\sigma$. For later use, it is also convenient to introduce the subset
$\mathscr{E}_{i}\subset\mathscr{E}$ consisting of those edges that correspond
to cells in $\mathscr{T}_{i}$ only, i.e.,
$\mathscr{E}_{i}=\bigg{(}\bigcup_{K\in\mathscr{T}_{i}}\mathscr{E}_{K}\bigg{)}\setminus\mathscr{E}_{\Gamma},\qquad
1\leq i\leq I,$ (2.3a) and the subset $\mathscr{E}_{\rm int}$ of the internal
edges, i.e., $\mathscr{E}_{\rm int}=\mathscr{E}\setminus(\mathscr{E}_{\rm
ext}^{\rm D}\cup\mathscr{E}_{\rm ext}^{\rm
N})=\bigcup_{K,L\in\mathscr{T}}\\{\sigma=K|L\\}.$ (2.3b)
Note that $\mathscr{E}_{\Gamma}\subset\mathscr{E}_{\rm int}$.
To each edge $\sigma\in\mathscr{E}$, we associate a distance $d_{\sigma}$ by
setting
$d_{\sigma}=\begin{cases}\,|x_{K}-x_{L}|&\text{if}\;\sigma=K|L\in\mathscr{E}_{\rm
int},\\\
\,|x_{K}-x_{\sigma}|&\text{if}\;\sigma\in\mathscr{E}_{K}\cap(\mathscr{E}_{\rm
ext}^{\rm D}\cup\mathscr{E}_{\rm ext}^{\rm N}).\end{cases}$ (2.4)
We also define $d_{K\sigma}={\rm dist}(x_{K},\sigma)$ for all
$K\in\mathscr{T}$ and $\sigma\in\mathscr{E}_{K}$. The transmissivity of the
edge $\sigma\in\mathscr{E}$ is defined by
$a_{\sigma}=\frac{m_{\sigma}}{d_{\sigma}}.$ (2.5)
Throughout the paper, many discrete quantities ${\boldsymbol{u}}$ will be
defined either in cells $K\in\mathscr{T}$ or on Dirichlet boundary edges
$\sigma\in\mathscr{E}_{\rm ext}^{\rm D}$, i.e.
${\boldsymbol{u}}=(\left(u_{K}\right)_{K\in\mathscr{T}},\left(u_{\sigma}\right)_{\sigma\in\mathscr{E}_{\rm
ext}^{\rm D}})\in\mathbb{X}^{\mathscr{T}\cup\mathscr{E}_{\rm ext}^{\rm D}}$,
where $\mathbb{X}$ can be either $\mathbb{R}^{\ell}$, $\ell\geq 1$, or a space
of functions. Then for all $K\in\mathscr{T}$ and $\sigma\in\mathscr{E}_{K}$,
we define the mirror value $u_{K\sigma}$ by
$u_{K\sigma}=\begin{cases}\,u_{L}&\text{if}\;\sigma=K|L\in\mathscr{E}_{\rm
int},\\\ \,u_{K}&\text{if}\;\sigma\in\mathscr{E}_{K}\cap\mathscr{E}_{\rm
ext}^{\rm N},\\\
\,u_{\sigma}&\text{if}\;\sigma\in\mathscr{E}_{K}\cap\mathscr{E}_{\rm ext}^{\rm
D}.\end{cases}$ (2.6)
The diamond cell $\Delta_{\sigma}$ corresponding to the edge $\sigma$ is
defined as the convex hull of $\\{x_{K},x_{K\sigma},\sigma\\}$ for $K$ such
that $\sigma\in\mathscr{E}_{K}$, while the half-diamond cell
$\Delta_{K\sigma}$ is defined as the convex hull of $\\{x_{K},\sigma\\}$.
Denoting by $m_{\Delta_{\sigma}}$ the Lebesgue measure of $\Delta_{\sigma}$,
the elementary geometrical relation
$m_{\Delta_{\sigma}}=d\,m_{\sigma}d_{\sigma}$ where $d$ stands for the
dimension will be used many times in what follows.
Another notational shorthand is worth introducing now, since it will come in
handy in the sequel. Let
$f(\cdot,x)=\sum_{1\leq i\leq I}f_{i}(\cdot)\mathbf{1}_{\Omega_{i}}(x)$ (2.7a)
be a scalar quantity or a function whose dependence of $x\in\Omega$ is of the
type (1.5). Then, for $K\in\mathscr{T}$, we slightly abuse the notations in
writing $f_{K}(\cdot):=f(\cdot,x_{K})=f_{i(K)}(\cdot),$ (2.7b)
where the index $i(K)$ is defined in (2.1). The last equality in the above
equation holds by virtue of the compatibility property. For example, we will
have not only $\phi_{K}=\phi(x_{K})$, $\lambda_{K}=\lambda(x_{K})$,
$\eta_{K}(s)=\eta(s,x_{K})$, $\mathcal{S}_{K}(p)=\mathcal{S}(p,x_{K})$ but
also $\text{{€\,}}_{K}(s)=\text{{€\,}}(s,x_{K})$. Likewise, we shall be
writing $f_{K\sigma}(\cdot)=f(\cdot,x_{K\sigma})$ for the mirror cell without
any ambiguity: if $\sigma\in\mathscr{E}_{\rm int}\cup\mathscr{E}_{\rm
ext}^{\rm N}$, then $x_{K\sigma}$ is a cell-center; if
$\sigma\in\mathscr{E}_{\rm ext}^{\rm D}$, then $x_{K\sigma}$ lies on the
boundary but does not belong to an interface between subdomains.
The size $h_{\mathscr{T}}$ and the regularity $\zeta_{\mathscr{T}}$ of the
mesh are respectively defined by
$h_{\mathscr{T}}=\max_{K\in\mathscr{T}}\,{\rm
diam}(K),\qquad\zeta_{\mathscr{T}}=\min_{K\in\mathscr{T}}\;{\bigg{(}\frac{1}{\operatorname{Card}\,\mathscr{E}_{K}}}\,\min_{\sigma\in\mathscr{E}_{K}}\frac{d_{K\sigma}}{{\rm
diam}(K)}{\bigg{)}}.$ (2.8)
The time discretization is given by $\left(t^{n}\right)_{0\leq 1\leq N}$ with
$0=t^{0}<t^{1}<\dots<t^{N}=T$. We denote by ${\Delta t^{n}}=t^{n}-t^{n-1}$ for
all $n\in\\{1,\dots,N\\}$ and by ${\boldsymbol{{\Delta t}}}=\left({\Delta
t^{n}}\right)_{1\leq n\leq N}$.
### 2.2 Upstream mobility TPFA Finite Volume scheme
Given a discrete saturation profile
$(s_{K}^{n-1})_{K\in\mathscr{T}}\in[0,1]^{\mathscr{T}}$ at time $t^{n-1}$,
$n\in\\{1,\dots,N\\}$, we seek for a discrete pressure profile
$(p_{K}^{n})_{K\in\mathscr{T}}\in\mathbb{R}^{\mathscr{T}}$ at time $t^{n}$
solution to the following nonlinear system of equations. Taking advantage of
the notational shorthand (2.7b), we define
$s_{K}^{n}=\mathcal{S}_{K}(p_{K}^{n}),\qquad K\in\mathscr{T},\;n\geq 1.$ (2.9)
The volume balance (1.9a) is then discretized into
$m_{K}\phi_{K}\frac{s_{K}^{n}-s_{K}^{n-1}}{{\Delta
t}^{n}}+\sum_{\sigma\in\mathscr{E}_{K}}m_{\sigma}F_{K\sigma}^{n}=0,\qquad
K\in\mathscr{T},\;n\geq 1,$ (2.10)
using the approximation
$F_{K\sigma}^{n}=\frac{1}{d_{\sigma}}\lambda_{\sigma}\eta_{\sigma}^{n}(\vartheta_{K}^{n}-\vartheta_{K\sigma}^{n}),\quad\sigma\in\mathscr{E}_{K},\;K\in\mathscr{T},\;n\geq
1,$ (2.11a) for the flux (1.1b), with
$\vartheta_{K}^{n}=p_{K}^{n}+\psi_{K},\qquad\vartheta_{K\sigma}^{n}=p_{K\sigma}^{n}+\psi_{K\sigma},$
(2.11b) where the mirror values $p_{K\sigma}^{n}$ and $\psi_{K\sigma}$ are
given by (2.6). In the numerical flux (2.11a), the edge permeabilities
$(\lambda_{\sigma})_{\sigma\in\mathscr{E}}$ are set to
$\lambda_{\sigma}=\begin{cases}\,\displaystyle\frac{\lambda_{K}\lambda_{L}d_{\sigma}}{\lambda_{K}d_{L,\sigma}+\lambda_{L}d_{K,\sigma}}&\text{if}\;\sigma=K|L\in\mathscr{E}_{\rm
int},\\\ \,\lambda_{K}&\text{if}\;\sigma\in\mathscr{E}_{K}\cap\mathscr{E}_{\rm
ext},\end{cases}$ while the edge mobilities are upwinded according to
$\eta_{\sigma}^{n}=\begin{cases}\,\eta_{K}(s_{K}^{n})&\text{if}\;\vartheta_{K}^{n}>\vartheta_{K\sigma}^{n},\\\
\frac{1}{2}(\eta_{K}(s_{K}^{n})+\eta_{K\sigma}(s_{K\sigma}^{n}))&\text{if}\;\vartheta_{K}^{n}=\vartheta_{K\sigma}^{n},\\\
\,\eta_{K\sigma}(s_{K\sigma}^{n})&\text{if}\;\vartheta_{K}^{n}<\vartheta_{K\sigma}^{n}.\end{cases}$
(2.11c)
In practice, the definition of $\eta_{\sigma}^{n}$ when
$\vartheta_{K}^{n}=\vartheta_{K\sigma}^{n}$ has no influence on the scheme. We
choose here to give a symmetric definition that does not depend on the
orientation of the edge $\sigma$ in order to avoid ambiguities.
The boundary condition $p^{\rm D}$ is discretized into
$\begin{cases}\,p_{K}^{\rm D}=\frac{1}{m_{K}}\int_{K}p^{\rm D}(x)\,{\rm
d}x&\text{for}\;K\in\mathscr{T},\\\ \,p_{\sigma}^{\rm
D}=\;\,\frac{1}{m_{\sigma}}\int_{\sigma}p^{\rm D}(x)\,{\rm
d}\nu^{d-1}(x)&\text{for}\;\sigma\in\mathscr{E}_{\rm ext}^{\rm D},\end{cases}$
(2.12)
whereas the initial condition is discretized into
$s_{K}^{0}=\frac{1}{m_{K}}\int_{K}s^{0}(x)\,{\rm
d}x,\qquad\text{for}\;K\in\mathscr{T}.$ (2.13)
The Dirichlet boundary condition is encoded in the fluxes (2.11a) by setting
$p_{\sigma}^{n}=p^{\rm D}_{\sigma},\qquad\forall\sigma\in\mathscr{E}_{\rm
ext}^{\rm D},\;n\geq 1.$ (2.14)
Bearing in mind the definition (2.6) of the mirror values for
$\sigma\in\mathscr{E}_{\rm ext}^{\rm N}$, the no-flux boundary condition
across $\sigma\in\mathscr{E}_{\rm ext}^{\rm N}$ is automatically encoded,
i.e., $F_{K\sigma}^{n}=0$ for all
$\sigma\in\mathscr{E}_{K}\cap\mathscr{E}_{\rm ext}^{\rm N}$, $K\in\mathscr{T}$
and $n\geq 1$.
In what follows, we denote by
${\boldsymbol{p}}^{n}=\left(p_{K}^{n}\right)_{K\in\mathscr{T}}$ for $1\leq
n\leq N$, and by
${\boldsymbol{s}}^{n}=\left(s_{K}^{n}\right)_{K\in\mathscr{T}}$ for $0\leq
n\leq N$. Besides, we set ${\boldsymbol{p}}^{\rm D}=((p_{K}^{\rm
D})_{K\in\mathscr{T}},(p_{\sigma}^{\rm D})_{\sigma\in\mathscr{E}^{\rm D}})$.
### 2.3 Main results and organization of the paper
The theoretical part of this paper includes two main results. The first one,
which emerges from the analysis at fixed grid, states that the schemes admits
a unique solution $({\boldsymbol{p}}^{n})_{1\leq n\leq N}$.
###### Theorem 2.2.
For all $n\in\\{1,\dots,N\\}$, there exists a unique solution
${\boldsymbol{p}}^{n}$ to the scheme (2.9)–(2.11c).
With Theorem 2.2 at hand, we define the approximate pressure
$p_{\mathscr{T},{\Delta t}}$ by
$p_{\mathscr{T},{\boldsymbol{{\Delta
t}}}}(t,x)=p_{K}^{n}\qquad\text{for}\;(t,x)\in(t^{n-1},t^{n}]\times K.$
(2.15a) We also define the approximate saturation as
$s_{\mathscr{T},{\boldsymbol{{\Delta
t}}}}=\mathcal{S}(p_{\mathscr{T},{\boldsymbol{{\Delta t}}}},x).$ (2.15b)
The second main result guarantees the convergence towards a weak solution of
the sequence of approximate solutions as the mesh size and the time steps tend
to $0$. Let
$(\mathscr{T}_{m},\mathscr{E}_{m},\left(x_{K}\right)_{K\in\mathscr{T}_{m}})_{m\geq
1}$ be a sequence of admissible discretizations of the domain $\Omega$ in the
sense of Definition 2.1 such that
$h_{\mathscr{T}_{m}}\underset{m\to\infty}{\longrightarrow}0,\qquad\sup_{m\geq
1}\;\zeta_{\mathscr{T}_{m}}=:\zeta<+\infty,$ (2.16)
where the size $h_{\mathscr{T}_{m}}$ and the regularity
$\zeta_{\mathscr{T}_{m}}$ are defined in (2.8). Let $({\boldsymbol{{\Delta
t}}}_{m})_{m\geq 1}$ be time discretizations of $(0,T)$ such that
$\lim_{m\to\infty}\;\max_{1\leq n\leq N_{m}}{\Delta t}^{n}_{m}=0.$ (2.17)
###### Theorem 2.3.
There exists a weak solution $p:Q_{T}\to\mathbb{R}$ in the sense of Definition
1.1 such that, up to a subsequence,
$\displaystyle s_{\mathscr{T}_{m},{\boldsymbol{{\Delta t}}}_{m}}$
$\displaystyle\underset{m\to\infty}{\longrightarrow}\mathcal{S}(p,x)$
$\displaystyle\text{a.e. in}\;Q_{T},$ (2.18a)
$\displaystyle\Upsilon(p_{\mathscr{T}_{m},{\boldsymbol{{\Delta t}}}_{m}})$
$\displaystyle\underset{m\to\infty}{\longrightarrow}\Upsilon(p)$
$\displaystyle\text{weakly in}\;L^{2}(Q_{T}).$ (2.18b)
The rest of this paper is outlined as follows. Section §3 is devoted to the
numerical analysis at fixed grid. This encompasses the existence and
uniqueness result stated in Theorem 2.2 as well as a priori estimates that
will help proving Theorem 2.3. The convergence of the scheme, which is taken
up in §4, relies on compactness arguments, which require a priori estimates
that are uniform w.r.t. the grid. These estimates are mainly adaptations to
the discrete setting of their continuous counterparts that arised in the
stability analysis sketched out in §1.2. These estimates are shown in §4.1 to
provide some compactness on the sequence of approximate solutions. In §4.2, we
show that these compactness properties together with the a priori estimates
are sufficient to identify any limit of an approximate solution as a weak
solution to the problem.
In §5, we provide some details about the practical numerical resolution by
laying emphasis on the switch of variable for selecting the primary unknown
and on the mesh refinement at an interface in order to better enforce pressure
continuity. Finally, in §6, numerical experiments on two configurations
(drying and filling cases) for two capillary pressure models (Brooks-Corey and
van Genuchten-Mualem) testify to the relevance of the local refinement
strategy as a simple technique to preserve accuracy.
###### Remark 2.4.
Theorem 2.3 only states the convergence of the scheme up to a subsequence. In
the case where the weak solution is unique, then the whole sequence of
approximate solutions would converge towards this solution. As far as we know,
uniqueness of the weak solutions to Richards’ equation is in general an open
problem for heterogeneous media where $x\mapsto\mathcal{S}(p,x)$ is
discontinuous. Uniqueness results are however available in the one-dimensional
setting for a slightly more restrictive notion of solutions, cf. [12], or
under additional assumptions on the nonlinearities $\eta_{i},\mathcal{S}_{i}$,
cf. [11].
## 3 Analysis at fixed grid
### 3.1 Some uniform a priori estimates
In this section, our aim is to derive a priori estimates on the solutions to
the scheme (2.9)–(2.13). These estimates will be at the core of the existence
proof of a solution to the scheme. They will also play a key role in proving
the convergence of the scheme.
The main estimate on which our analysis relies is a discrete counterpart of
(1.12). We recall that $a_{\sigma}$ is the transmissivity introduced in (2.5).
###### Proposition 3.1.
There exist two constants $C_{1}$, $C_{2}$ depending only on $\lambda$, $\mu$,
$p^{\rm D}$, $\psi$, $\zeta$, $\Omega$, $T$, $\phi$, and
$\lVert\mathcal{S}_{i}\rVert_{L^{1}(\mathbb{R}_{-})}$ such that
$\displaystyle\sum_{n=1}^{N}{\Delta
t^{n}}\sum_{\sigma\in\mathscr{E}}a_{\sigma}\lambda_{\sigma}\eta_{\sigma}^{n}(p_{K}^{n}-p_{K\sigma}^{n})^{2}$
$\displaystyle\leq C_{\ref{c:main}},$ (3.1a)
$\displaystyle\sum_{n=1}^{N}{\Delta
t^{n}}\sum_{\sigma\in\mathscr{E}}a_{\sigma}\lambda_{\sigma}\eta_{\sigma}^{n}(\vartheta_{K}^{n}-\vartheta_{K\sigma}^{n})^{2}$
$\displaystyle\leq C_{\ref{c:dissip.2}}.$ (3.1b)
In (3.1), the relationship between $\sigma$ and $K$ is to be understood as
follows. For an inner edge $\sigma\in\mathscr{E}_{\rm int}$, although it can
be written as $\sigma=K|L$ or $L|K$, only one of these contributes to the sum.
For a boundary edge $\sigma\in\mathscr{E}_{\rm ext}$, there is only one cell
$K$ such that $\sigma\in\mathscr{E}_{K}$, so there is no ambiguity in the sum.
###### Proof.
Multiplying (2.10) by $\Delta t^{n}(p_{K}^{n}-p_{K}^{\rm D})$, summing over
$K\in\mathscr{T}$ and $n\in\\{1,\dots,N\\}$, and carrying out discrete
integration by parts yield
$\mathtt{A}+\mathtt{B}=0,$ (3.2)
where we have set
$\displaystyle\mathtt{A}$
$\displaystyle=\sum_{n=1}^{N}\sum_{K\in\mathscr{T}}m_{K}\phi_{K}(s_{K}^{n}-s_{K}^{n-1})(p_{K}^{n}-p_{K}^{\rm
D}),$ (3.3a) $\displaystyle\mathtt{B}$ $\displaystyle=\sum_{n=1}^{N}\Delta
t^{n}\sum_{\sigma\in\mathscr{E}}a_{\sigma}\lambda_{\sigma}\eta_{\sigma}^{n}(\vartheta_{K}^{n}-\vartheta_{K\sigma}^{n})(p_{K}^{n}-p_{K}^{\rm
D}-p_{K\sigma}^{n}+p_{K\sigma}^{\rm D}).$ (3.3b)
The discrete energy density function
$\text{{€\,}}_{K}:[0,1]\to\mathbb{R}_{+}$, defined by means of the notation
(2.7) from the functions $f_{i}=\text{{€\,}}_{i}$ introduced in (1.7), is
convex by construction. Consequently,
$\text{{€\,}}_{K}(s_{K}^{n-1})-\text{{€\,}}_{K}(s_{K}^{n})\geq\text{{€\,}}^{\prime}_{K}(s_{K}^{n})(s_{K}^{n-1}-s_{K}^{n})=\phi_{K}(p_{K}^{n}-p_{K}^{\rm
D})(s_{K}^{n-1}-s_{K}^{n}).$
Therefore, the quantity $\mathtt{A}$ of (3.3a) can be bounded below by
$\qquad\qquad\mathtt{A}\,\geq\,\sum_{n=1}^{N}\sum_{K\in\mathscr{T}}m_{K}(\text{{€\,}}_{K}(s_{K}^{n})-\text{{€\,}}_{K}(s_{K}^{n-1}))\\\
=\sum_{K\in\mathscr{T}}m_{K}(\text{{€\,}}_{K}(s_{K}^{N})-\text{{€\,}}_{K}(s_{K}^{0}))\,\geq\,-C_{\mathtt{A}},\qquad\qquad$
(3.4)
the last inequality being a consequence of the boundedness of
$\text{{€\,}}_{K}$ on $[0,1]$.
Writing $\vartheta=p+\psi$ and expanding each summand of (3.3b), we can split
$\mathtt{B}$ into
$\mathtt{B}=\mathtt{B}_{1}+\mathtt{B}_{2}+\mathtt{B}_{3},$
with
$\displaystyle\mathtt{B}_{1}$ $\displaystyle=\phantom{-}\sum_{n=1}^{N}\Delta
t^{n}\sum_{\sigma\in\mathscr{E}}a_{\sigma}\lambda_{\sigma}\eta_{\sigma}^{n}(p_{K}^{n}-p_{K\sigma}^{n})^{2},$
$\displaystyle\mathtt{B}_{2}$ $\displaystyle=\phantom{-}\sum_{n=1}^{N}\Delta
t^{n}\sum_{\sigma\in\mathscr{E}}a_{\sigma}\lambda_{\sigma}\eta_{\sigma}^{n}(p_{K}^{n}-p_{K\sigma}^{n})(\psi_{K}-\psi_{K\sigma}-p_{K}^{\rm
D}+p_{K\sigma}^{\rm D}),$ $\displaystyle\mathtt{B}_{3}$
$\displaystyle=-\sum_{n=1}^{N}\Delta
t^{n}\sum_{\sigma\in\mathscr{E}}a_{\sigma}\lambda_{\sigma}\eta_{\sigma}^{n}(\psi_{K}-\psi_{K\sigma})(p_{K}^{\rm
D}-p_{K\sigma}^{\rm D}).$
It follows from [27, Lemma 9.4] and from the boundedness of $\eta$ that there
exists a constant $C$ depending only on $\lambda$, $\mu$,
$\zeta_{\mathscr{T}}$ and $\Omega$ such that
$\displaystyle\sum_{\sigma\in\mathscr{E}}a_{\sigma}\lambda_{\sigma}\eta_{\sigma}^{n}(p_{K}^{\rm
D}-p_{K\sigma}^{\rm D})^{2}$ $\displaystyle\leq C\,\lVert\nabla p^{\rm
D}\rVert^{2}_{L^{2}(\Omega)^{d}},$ (3.5a)
$\displaystyle\sum_{\sigma\in\mathscr{E}}a_{\sigma}\lambda_{\sigma}\eta_{\sigma}^{n}(\psi_{K}-\psi_{K\sigma})^{2}$
$\displaystyle\leq C\,\lVert\nabla\psi\rVert^{2}_{L^{\infty}(\Omega)^{d}}.$
(3.5b)
Thanks to these estimates and to the Cauchy-Schwarz inequality, we have
$\mathtt{B}_{3}\geq-CT\,\lVert\nabla p^{\rm
D}\rVert_{L^{2}(\Omega)^{d}}\,\lVert\nabla\psi\rVert_{L^{\infty}(\Omega)^{d}}.$
On the other hand, Young’s inequality provides
$\mathtt{B}_{2}\geq-\frac{1}{2}\mathtt{B}_{1}-CT\,\big{(}\lVert\nabla p^{\rm
D}\rVert^{2}_{L^{2}(\Omega)^{d}}+\lVert\nabla\psi\rVert^{2}_{L^{\infty}(\Omega)^{d}}\big{)}.$
Hence,
$\mathtt{B}\,\geq\,\frac{1}{2}\sum_{n=1}^{N}\Delta
t^{n}\sum_{\sigma\in\mathscr{E}}a_{\sigma}\lambda_{\sigma}\eta_{\sigma}^{n}(p_{K}^{n}-p_{K\sigma}^{n})^{2}-C_{\mathtt{B}},$
(3.6)
by setting $C_{\mathtt{B}}=CT\,(\lVert\nabla p^{\rm
D}\rVert^{2}_{L^{2}(\Omega)^{d}}+\lVert\nabla\psi\rVert^{2}_{L^{\infty}(\Omega)^{d}}+\lVert\nabla
p^{\rm
D}\rVert_{L^{2}(\Omega)^{d}}\,\lVert\nabla\psi\rVert_{L^{\infty}(\Omega)^{d}})$.
Inserting (3.4) and (3.6) into (3.2), we recover (3.1a) with
$C_{\ref{c:main}}=2(C_{\mathtt{A}}+C_{\mathtt{B}})$. From (3.1a), we can
deduce (3.1b) by elementary manipulations. ∎
So far, we have not used the upwind choice (2.11c) for the mobilities
$\eta_{\sigma}^{n}$. This will be done in the next lemma, where we derive a
more useful variant of estimate (3.1a), in which $\eta_{\sigma}^{n}$ is
replaced by $\overline{\eta}_{\sigma}^{n}$ defined below. In a homogeneous
medium, $\overline{\eta}_{\sigma}^{n}\geq\eta_{\sigma}^{n}$ so that the new
estimate (3.8) seems to be stronger than (3.1a).
We begin by introducing the functions
$\widecheck{\eta}_{\sigma}:\mathbb{R}\to(0,1/\mu]$ defined for
$\sigma\in\mathscr{E}$ by
$\widecheck{\eta}_{\sigma}(p)=\min\big{\\{}\eta_{K}\circ\mathcal{S}_{K}(p),\eta_{K\sigma}\circ\mathcal{S}_{K\sigma}(p)\big{\\}},\qquad\forall
p\in\mathbb{R}.$ (3.7a) By virtue of assumptions (1.6), each argument of the
minimum function is nondecreasing and positive function of $p\in\mathbb{R}$.
As a result, $\widecheck{\eta}_{\sigma}$ is also a nondecreasing and positive
function of $p\in\mathbb{R}$. Note that
$\widecheck{\eta}_{\sigma}=\eta_{i}\circ\mathcal{S}_{i}$ for all
$\sigma\in\mathscr{E}_{i}$, while for interface edges
$\sigma\subset\Gamma_{i,j}$, the mere inequality
$\widecheck{\eta}_{\sigma}\leq\eta_{i}\circ\mathcal{S}_{i}$ holds. Next, we
consider the intervals $\mathfrak{J}_{\sigma}^{n}=[p_{K}^{n}\bot
p_{K\sigma}^{n},\,p_{K}^{n}\top p_{K\sigma}^{n}],\qquad\text{for
}\,\sigma\in\mathscr{E}_{K},\;K\in\mathscr{T},\;1\leq n\leq N,$ (3.7b) with
the notations $a\bot b=\min(a,b)$ and $a\top b=\max(a,b)$. At last, we set
$\overline{\eta}_{\sigma}^{n}=\max_{p\in\mathfrak{J}_{\sigma}^{n}}\widecheck{\eta}_{\sigma}(p),\qquad\text{for
}\,\sigma\in\mathscr{E},\;1\leq n\leq N.$ (3.7c)
###### Lemma 3.2.
There exists a constant $C_{3}$ depending on the same data as
$C_{\ref{c:main}}$ such that
$\sum_{n=1}^{N}{\Delta
t}^{n}\sum_{\sigma\in\mathscr{E}}a_{\sigma}\overline{\eta}_{\sigma}^{n}\left(p_{K}^{n}-p_{K\sigma}^{n}\right)^{2}\leq
C_{\ref{c:dissip.max}}.$ (3.8)
###### Proof.
We partition the set $\mathscr{E}$ of edges into three subsets, namely,
$\mathscr{E}^{n}_{+}=\big{\\{}\sigma\,|\;\vartheta_{K}^{n}>\vartheta_{K\sigma}^{n}\big{\\}},\quad\mathscr{E}^{n}_{-}=\big{\\{}\sigma\,|\;\vartheta_{K}^{n}<\vartheta_{K\sigma}^{n}\big{\\}},\quad\mathscr{E}^{n}_{0}=\big{\\{}\sigma\,|\;\vartheta_{K}^{n}=\vartheta_{K\sigma}^{n}\big{\\}}.$
Invoking
$\widecheck{\eta}_{\sigma}=\min(\eta_{K}\circ\mathcal{S}_{K},\,\eta_{K\sigma}\circ\mathcal{S}_{K\sigma})$,
we can minorize the left-hand side of (3.1a) to obtain
$\sum_{n=1}^{N}{\Delta
t}^{n}\Big{[}\sum_{\sigma\in\mathscr{E}^{n}_{+}}a_{\sigma}\lambda_{\sigma}\widecheck{\eta}_{\sigma}(p_{K}^{n})(p_{K}^{n}-p_{K\sigma}^{n})^{2}+\sum_{\sigma\in\mathscr{E}^{n}_{-}}a_{\sigma}\lambda_{\sigma}\widecheck{\eta}_{\sigma}(p_{K\sigma}^{n})(p_{K}^{n}-p_{K\sigma}^{n})^{2}\\\
+\sum_{\sigma\in\mathscr{E}^{n}_{0}}a_{\sigma}\lambda_{\sigma}\textstyle\frac{1}{2}(\widecheck{\eta}_{\sigma}(p_{K}^{n})+\widecheck{\eta}_{\sigma}(p_{K\sigma}^{n}))(p_{K}^{n}-p_{K\sigma}^{n})^{2}\Big{]}\leq
C_{\ref{c:main}}.$
Starting from this inequality and using the boundedness of $\eta_{i}$ and
$\psi$, we can readily show that there exists a constant $C$ depending on the
same data as $C_{\ref{c:main}}$ such that
$\mathtt{D}_{1}:=\sum_{n=1}^{N}{\Delta
t}^{n}\Big{[}\sum_{\sigma\in\mathscr{E}^{n}_{+}}a_{\sigma}\lambda_{\sigma}\widecheck{\eta}_{\sigma}(p_{K}^{n})(p_{K}^{n}-p_{K\sigma}^{n})(\vartheta_{K}^{n}-\vartheta_{K\sigma}^{n})\\\
+\sum_{\sigma\in\mathscr{E}^{n}_{-}}a_{\sigma}\lambda_{\sigma}\widecheck{\eta}_{\sigma}(p_{K\sigma}^{n})(p_{K}^{n}-p_{K\sigma}^{n})(\vartheta_{K}^{n}-\vartheta_{K\sigma}^{n})\Big{]}\leq
C,$
in which the sum over $\mathscr{E}_{0}^{n}$ was omitted because all of its
summands vanish. Simlarly to what was pointed out in equation 2.9 in [1], we
notice that since $\eta_{\sigma}$ is nondecreasing w.r.t. $p$, it is
straightforward to check that the definition
$\widecheck{\eta}_{\sigma}^{n}:=\begin{cases}\;\widecheck{\eta}_{\sigma}(p_{K}^{n})&\text{
if }\vartheta^{n}_{K}>\vartheta^{n}_{K\sigma},\\\
\;\textstyle\frac{1}{2}(\widecheck{\eta}_{\sigma}(p_{K}^{n})+\widecheck{\eta}_{\sigma}(p_{K\sigma}^{n}))&\text{
if }\vartheta^{n}_{K}=\vartheta^{n}_{K\sigma},\\\
\;\widecheck{\eta}_{\sigma}(p_{K\sigma}^{n})&\text{ if
}\vartheta^{n}_{K}<\vartheta^{n}_{K\sigma}\end{cases}$ (3.9)
exactly amounts to
$\widecheck{\eta}_{\sigma}^{n}=\begin{cases}\;\max_{p\in\mathfrak{J}_{\sigma}^{n}}\widecheck{\eta}_{\sigma}(p)&\text{
if
}(p^{n}_{K}-p^{n}_{K\sigma})(\vartheta^{n}_{K}-\vartheta^{n}_{K\sigma})>0,\\\
\;\textstyle\frac{1}{2}(\widecheck{\eta}_{\sigma}(p_{K}^{n})+\widecheck{\eta}_{\sigma}(p_{K\sigma}^{n}))&\text{
if
}(p^{n}_{K}-p^{n}_{K\sigma})(\vartheta^{n}_{K}-\vartheta^{n}_{K\sigma})=0,\\\
\;\min_{p\in\mathfrak{J}_{\sigma}^{n}}\widecheck{\eta}_{\sigma}(p)&\text{ if
}(p^{n}_{K}-p^{n}_{K\sigma})(\vartheta^{n}_{K}-\vartheta^{n}_{K\sigma})<0.\end{cases}$
(3.10)
Taking advantage of this equivalence, we can transform $\mathtt{D}_{1}$ into
$\mathtt{D}_{1}=\sum_{n=1}^{N}{\Delta
t}^{n}\Big{[}\sum_{\sigma\in\mathscr{E}^{n}_{>}}a_{\sigma}\lambda_{\sigma}\max_{\mathfrak{J}_{\sigma}^{n}}\widecheck{\eta}_{\sigma}(p_{K}^{n}-p_{K\sigma}^{n})(\vartheta_{K}^{n}-\vartheta_{K\sigma}^{n})\\\
+\sum_{\sigma\in\mathscr{E}^{n}_{<}}a_{\sigma}\lambda_{\sigma}\min_{\mathfrak{J}_{\sigma}^{n}}\widecheck{\eta}_{\sigma}(p_{K}^{n}-p_{K\sigma}^{n})(\vartheta_{K}^{n}-\vartheta_{K\sigma}^{n})\Big{]}\leq
C,$ (3.11)
where
$\mathscr{E}^{n}_{>}=\\{\sigma\,|\,(p^{n}_{K}-p^{n}_{K\sigma})(\vartheta^{n}_{K}-\vartheta^{n}_{K\sigma})>0\\}$
and
$\mathscr{E}^{n}_{<}=\\{\sigma\,|\,(p^{n}_{K}-p^{n}_{K\sigma})(\vartheta^{n}_{K}-\vartheta^{n}_{K\sigma})<0\\}$.
The second sum over $\mathscr{E}^{n}_{<}$ contains only negative summands and
can be further minorized if
$\min_{\mathfrak{J}^{n}_{\sigma}}\widecheck{\eta}_{\sigma}$ is replaced by
$\max_{\mathfrak{J}^{n}_{\sigma}}\widecheck{\eta}_{\sigma}$. In other words,
$\mathtt{D}_{2}:=\sum_{n=1}^{N}{\Delta
t}^{n}\sum_{\sigma\in\mathscr{E}}a_{\sigma}\lambda_{\sigma}\overline{\eta}_{\sigma}^{n}(p_{K}^{n}-p_{K\sigma}^{n})(\vartheta_{K}^{n}-\vartheta_{K\sigma}^{n})\leq\mathtt{D}_{1}\leq
C.$
Writing $\vartheta=p+\psi$, expanding each summand of $\mathtt{D}_{2}$ and
applying Young’s inequality, we end up with
$\frac{1}{2}\sum_{n=1}^{N}{\Delta
t}^{n}\sum_{\sigma\in\mathscr{E}}a_{\sigma}\lambda_{\sigma}\overline{\eta}_{\sigma}^{n}\,[(p_{K}^{n}-p_{K\sigma}^{n})^{2}-(\psi_{K}^{n}-\psi_{K\sigma}^{n})^{2}]\leq\mathtt{D}_{2}\leq
C.$
Estimate (3.8) finally follows from the boundedness of $\eta$, $1/\lambda$ and
$\psi$. ∎
The above lemma has several important consequences for the analysis. Let us
start with discrete counterparts to estimations (1.14) and (1.15).
###### Corollary 3.3.
Let $C_{\ref{c:dissip.max}}$ be the constant in Lemma 3.2. Then,
$\displaystyle\sum_{n=1}^{N}{\Delta
t^{n}}\sum_{i=1}^{I}\sum_{\sigma\in\mathscr{E}_{i}}a_{\sigma}(\Theta_{i}(p_{K}^{n})-\Theta_{i}(p_{K\sigma}^{n}))^{2}$
$\displaystyle\leq C_{\ref{c:dissip.max}},$ (3.12a)
$\displaystyle\sum_{n=1}^{N}{\Delta
t^{n}}\sum_{\sigma\in\mathscr{E}}a_{\sigma}(\Upsilon(p_{K}^{n})-\Upsilon(p_{K\sigma}^{n}))^{2}$
$\displaystyle\leq C_{\ref{c:dissip.max}}.$ (3.12b)
Moreover, there exists two constants $C_{4}$, $C_{5}$ depending on the same
data as $C_{\ref{c:main}}$ and additionnally on
$\lVert\sqrt{\eta_{i}\circ\mathcal{S}_{i}}\rVert_{L^{1}(\mathbb{R}_{-})}$,
$1\leq i\leq I$, such that
$\displaystyle\sum_{n=1}^{N}{\Delta
t}^{n}\sum_{K\in\mathscr{T}}m_{K}|\Upsilon(p_{K}^{n})|^{2}\leq
C_{\ref{c:L2.Upsilon}},$ (3.13a) $\displaystyle\sum_{n=1}^{N}{\Delta
t}^{n}\sum_{i=1}^{I}\sum_{K\in\mathscr{T}_{i}}m_{K}|\Theta_{i}(p_{K}^{n})|^{2}\leq
C_{\ref{c:L2.xi}}.$ (3.13b)
###### Proof.
Consider those edges $\sigma\in\mathscr{E}_{i}$ —defined in (2.3a)—
corresponding to some fixed $i\in\\{1,\ldots,I\\}$, for which
$\widecheck{\eta}_{\sigma}=\eta_{i}\circ\mathcal{S}_{i}=|\Theta^{\prime}_{i}|^{2}$
and
$\overline{\eta}^{n}_{\sigma}=\max_{\mathfrak{J}^{n}_{\sigma}}|\Theta^{\prime}_{i}|^{2}$
due to (1.13a). By summing the elementary inequality
$(\Theta_{i}(p_{K}^{n})-\Theta_{i}(p_{K\sigma}^{n}))^{2}\leq\overline{\eta}^{n}_{\sigma}\;(p_{K}^{n}-p_{K\sigma}^{n})^{2},$
over $\sigma\in\mathscr{E}_{i}$, $i\in\\{1,\ldots,I\\}$ and
$n\in\\{1,\ldots,N\\}$ using appropriate weights, we get
$\sum_{n=1}^{N}{\Delta
t^{n}}\sum_{i=1}^{I}\sum_{\sigma\in\mathscr{E}_{i}}a_{\sigma}(\Theta_{i}(p_{K}^{n})-\Theta_{i}(p_{K\sigma}^{n}))^{2}\leq\sum_{n=1}^{N}{\Delta
t^{n}}\sum_{i=1}^{I}\sum_{\sigma\in\mathscr{E}_{i}}a_{\sigma}\overline{\eta}^{n}_{\sigma}\;(p_{K}^{n}-p_{K\sigma}^{n})^{2},$
whose right-hand side is obviously less than $C_{\ref{c:dissip.max}}$, thanks
to (3.8). This proves (3.12a).
Similarly, the respective definitions of $\overline{\eta}_{\sigma}^{n}$ and
$\Upsilon$ have been tailored so that
$\max_{\mathfrak{J}^{n}_{\sigma}}|\Upsilon^{\prime}|^{2}\leq\overline{\eta}^{n}_{\sigma}$
for all $\sigma\in\mathscr{E}$. As a consequence,
$(\Upsilon(p_{K}^{n})-\Upsilon(p_{K\sigma}^{n}))^{2}\leq\overline{\eta}_{\sigma}^{n}(p_{K}^{n}-p_{K\sigma}^{n})^{2}.$
Summing these inequalities over $\sigma\in\mathscr{E}$ and
$n\in\\{1,\ldots,N\\}$ with appropriate weights and invoking (3.8), we prove
(3.12b).
The argument for (3.13a) is subtler. Starting from the basic inequality
$\qquad(\Upsilon(p_{K}^{n})-\Upsilon(p_{K}^{\rm
D})-\Upsilon(p_{K\sigma}^{n})+\Upsilon(p_{K\sigma}^{\rm D}))^{2}\\\ \leq
2(\Upsilon(p_{K}^{n})-\Upsilon(p_{K\sigma}^{n}))^{2}+2(\Upsilon(p_{K}^{\rm
D})-\Upsilon(p_{K\sigma}^{\rm D}))^{2},\qquad$
we apply the discrete Poincaré inequality of [27, Lemma 9.1] —which is
legitimate since $\Gamma^{\rm D}$ has positive measure— followed by [27, Lemma
9.4] to obtain
$\sum_{n=1}^{N}{\Delta
t}^{n}\sum_{K\in\mathscr{T}}m_{K}(\Upsilon(p_{K}^{n})-\Upsilon(p_{K}^{\rm
D}))^{2}\\\ \leq 2C_{{\rm
P},\mathscr{T}}\big{(}C_{\ref{c:dissip.max}}+C_{\zeta}T\lVert\Upsilon^{\prime}\rVert_{\infty}\lVert\nabla
p^{\rm D}\rVert^{2}\big{)},$
where $C_{{\rm P},\mathscr{T}}$ denotes the discrete Poincaré constant, and
$C_{\zeta}$ is the quantity appearing in [27, Lemma 9.4] and only depends on
$\zeta_{\mathscr{T}}$. $C_{\ref{c:L2.Upsilon}}=4C_{{\rm
P},\mathscr{T}}\big{(}C_{\ref{c:dissip.max}}+C_{\zeta}T\lVert\Upsilon^{\prime}\rVert_{\infty}\lVert\nabla
p^{\rm D}\rVert^{2}\big{)}+2m_{\Omega}T\lVert\Upsilon(p^{\rm
D})\rVert_{\infty}^{2}.$
The last estimate (3.13b) results from the comparison (1.16) of the
nonlinearities $\Theta_{i}$ and $\Upsilon$. ∎
The purpose of the next lemma is to work out a weak estimate on the discrete
counterpart of $\partial_{t}s$, which will lead to compactness properties in
§4.1. For $\varphi\in C^{\infty}_{c}(Q_{T})$, let
$\varphi_{K}^{n}=\frac{1}{m_{K}}\int_{K}\varphi(t^{n},x)\,{\rm
d}x,\qquad\forall K\in\mathscr{T},\;1\leq n\leq N.$
###### Lemma 3.4.
There exists a constant $C_{6}$ depending on the same data as
$C_{\ref{c:main}}$ such that
$\sum_{n=1}^{N}\sum_{K\in\mathscr{T}}m_{K}\phi_{K}(s_{K}^{n}-s_{K}^{n-1})\varphi_{K}^{n}\leq
C_{\ref{c:s2}}\lVert\nabla\varphi\rVert_{L^{\infty}(Q_{T})^{d}},\quad\forall\varphi\in
C^{\infty}_{c}(Q_{T}).$ (3.14)
###### Proof.
Multiplying (2.10) by ${\Delta t^{n}}\leavevmode\nobreak\ \varphi_{K}^{n}$,
summing over $K\in\mathscr{T}$ and $n\in\\{1,\cdots,N\\}$ and carrying out
discrete integration by parts, we end up with
$\mathtt{A}:=\\!\sum_{n=1}^{N}\\!\sum_{K\in\mathscr{T}}m_{K}\phi_{K}(s_{K}^{n}-s_{K}^{n-1})\varphi_{K}^{n}=-\\!\sum_{n=1}^{N}{\Delta
t^{n}}\\!\sum_{\sigma\in\mathscr{E}}a_{\sigma}\lambda_{\sigma}\eta_{\sigma}^{n}(\vartheta_{K}^{n}-\vartheta_{K\sigma}^{n})(\varphi_{K}^{n}-\varphi_{K\sigma}^{n}).$
Applying the Cauchy-Schwarz inequality and using (3.1b), we get
$\mathtt{A}^{2}\leq
C_{\ref{c:dissip.2}}\frac{\max_{i}\lambda_{i}}{\mu}\sum_{n}{\Delta
t^{n}}\sum_{\sigma\in\mathscr{E}}a_{\sigma}(\varphi_{K}^{n}-\varphi_{K\sigma}^{n})^{2}.$
(3.15)
The conclusion (3.14) is then reached by means of the property (see [2,
Section 4.4])
$\sum_{n=1}^{N}{\Delta
t^{n}}\sum_{\sigma\in\mathscr{E}}a_{\sigma}(\varphi_{K}^{n}-\varphi_{K\sigma}^{n})^{2}\leq
C\lVert\nabla\varphi\rVert_{L^{\infty}(Q_{T})^{d}}^{2}$
for some $C$ depending only on $\Omega$, $T$ and the mesh regularity
$\zeta_{\mathscr{T}}$. ∎
### 3.2 Existence of a solution to the scheme
The statements of the previous section are all uniform w.r.t. the mesh and are
meant to help us passing to the limit in the next section. In contrast, the
next lemma provides a bound on the pressure that depends on the mesh size and
on the time-step. This property is needed in the process of ensuring the
existence of a solution to the numerical scheme.
###### Lemma 3.5.
There exist two constants $C_{7}$, $C_{8}$ depending on $\mathscr{T}$,
${\Delta t}^{n}$ as well as on the data of the continuous model $\lambda$,
$\mu$, $p^{\rm D}$, $\psi$, $\zeta$, $\Omega$, $T$, $\phi$,
$\lVert\mathcal{S}_{i}\rVert_{L^{1}(\mathbb{R}_{-})}$ and
$\lVert\sqrt{\eta_{i}\circ\mathcal{S}_{i}}\rVert_{L^{1}(\mathbb{R}_{-})}$,
$1\leq i\leq I$, such that
$-C_{\ref{c:pb}}\leq p_{K}^{n}\leq C_{\ref{c:pb2}},\quad\forall
K\in\mathscr{T},\;n\in\\{1,\dots,N\\}.$ (3.16)
###### Proof.
From (3.13a) and from $\Upsilon(p)=p\sqrt{\min_{i}{\lambda_{i}}/\mu}$ for
$p\geq 0$, we deduce that
$p_{K}^{n}\leq\sqrt{\frac{\mu C_{\ref{c:L2.Upsilon}}}{{\Delta
t}^{n}m_{K}\min_{i}\lambda_{i}}},\qquad\forall K\in\mathscr{T},\;1\leq n\leq
N.$
Hence, the upper-bound $C_{\ref{c:pb2}}$ is found by maximizing the right-hand
side over $K\in\mathscr{T}$ and $n\in\\{1,\dots,N\\}$.
To show that $p_{K}^{n}$ is bounded from below, we employ a strategy that was
developed in [13] and extended to the case of Richards’ equation in [1, Lemma
3.10]. From (2.12), (2.14) and the boundedness of $p^{\rm D}$, it is easy to
see that
$p_{\sigma}^{n}\geq\inf_{x\in\partial\Omega}p^{\rm
D}(x),\quad\forall\sigma\in\mathscr{E}_{\rm ext}^{\rm D}.$
Estimate (3.8) then shows that for all $K\in\mathscr{T}$ such that
$\mathscr{E}_{K}\cap\mathscr{E}_{\rm ext}^{\rm D}\neq\emptyset$, we have
$p_{K}^{n}\geq p_{\sigma}^{n}-\sqrt{\frac{C_{\ref{c:dissip.max}}}{{\Delta
t}^{n}a_{\sigma}\widecheck{\eta}_{\sigma}(p_{\sigma}^{n})}}=:\pi_{K}^{n},\quad\forall\sigma\in\mathscr{E}_{K}\cap\mathscr{E}_{\rm
ext}^{\rm D}.$
The quantity $\pi_{K}^{n}$ is well-defined, since
$\widecheck{\eta}_{\sigma}(p^{n}_{\sigma})>0$ for $p^{n}_{\sigma}>-\infty$,
and does not depend on time, as $p^{\rm D}$ does not either. Furthermore, if
$p_{K}^{n}$ is bounded from below by some $\pi_{K}$, then the pressure in all
its neighboring cells $L\in\mathscr{T}$ such that
$\sigma=K|L\in\mathscr{E}_{K}$ is bounded from below by
$p_{L}^{n}\geq\pi_{K}^{n}-\sqrt{\frac{C_{\ref{c:dissip.max}}}{{\Delta
t}^{n}a_{\sigma}\widecheck{\eta}_{\sigma}(\pi_{K}^{n})}}=:\pi_{L}^{n}.$
Again, $\pi_{L}^{n}$ is well-defined owing to
$\widecheck{\eta}_{\sigma}(\pi_{K}^{n})>0$. Since the mesh is finite and since
the domain is connected, only a finite number of edge-crossings is required to
create a path from a Dirichlet boundary edge $\sigma\in\mathscr{E}_{\rm
ext}^{\rm D}$ to any prescribed cell $K\in\mathscr{T}$. Hence, the lower bound
$C_{\ref{c:pb}}$ is found by minimizing $\pi^{n}_{K}$ over $K\in\mathscr{T}$
and $n\in\\{1,\ldots,N\\}$. ∎
Lemma 3.5 is a crucial step in the proof of the existence of a solution
${\boldsymbol{p}}^{n}=(p_{K}^{n})_{K\in\mathscr{T}}$ to the scheme
(2.9)–(2.14).
###### Proposition 3.6.
Given
${\boldsymbol{s}}^{n-1}=(s_{K}^{n-1})_{K\in\mathscr{T}}\in[0,1]^{\mathscr{T}}$,
there exists a solution ${\boldsymbol{p}}^{n}\in\mathbb{R}^{\mathscr{T}}$ to
the scheme (2.9)–(2.14).
The proof relies on a standard topological degree argument and is omitted
here. However, we make the homotopy explicit for readers’ convenience. Let
$\gamma\in[0,1]$ be the homotopy parameter. We define the nondecreasing
functions $\eta_{i}^{(\gamma)}:[0,1]\to\mathbb{R}_{+}$ by setting
$\eta_{i}^{(\gamma)}(s)=(1-\gamma)/\mu+\gamma\eta_{i}(s)$ for $s\in[0,1]$, and
we seek a solution
${\boldsymbol{p}}^{(\gamma)}={(p_{K}^{(\gamma)})}_{K\in\mathscr{T}}$ to the
problem
$\gamma
m_{K}\phi_{K}\frac{\mathcal{S}_{K}(p_{K}^{(\gamma)})-s_{K}^{n-1}}{{\Delta
t}_{n}}+\sum_{\sigma\in\mathscr{E}_{K}}m_{\sigma}F_{K\sigma}^{(\gamma)}=0,\qquad
K\in\mathscr{T},\;\gamma\in[0,1],$ (3.17a) where the fluxes
$F_{K\sigma}^{(\gamma)}$ are defined by
$F_{K\sigma}^{(\gamma)}=\frac{1}{d_{\sigma}}\lambda_{\sigma}\eta_{\sigma}^{(\gamma)}\big{(}\vartheta_{K}^{(\gamma)}-\vartheta_{K\sigma}^{(\gamma)}\big{)},\qquad\sigma\in\mathscr{E}_{K},\;K\in\mathscr{T},\;\gamma\in[0,1]$
(3.17b) with $\vartheta^{(\gamma)}=p^{(\gamma)}+\psi$ and using the upwind
mobilities
$\eta_{\sigma}^{(\gamma)}=\begin{cases}\,\eta_{K}^{(\gamma)}(\mathcal{S}_{K}(p_{K}^{(\gamma)}))&\text{if
}\;\vartheta_{K}^{(\gamma)}>\vartheta_{K\sigma}^{(\gamma)},\\\
\,\frac{1}{2}(\eta_{K}^{(\gamma)}(\mathcal{S}_{K}(p_{K}^{(\gamma)}))+\eta_{K\sigma}^{(\gamma)}(\mathcal{S}_{K\sigma}(p_{K}^{(\gamma)})))&\text{if
}\;\vartheta_{K}^{(\gamma)}=\vartheta_{K\sigma}^{(\gamma)},\\\
\,\eta_{K\sigma}^{(\gamma)}(\mathcal{S}_{K\sigma}(p_{K}^{(\gamma)}))&\text{if
}\;\vartheta_{K}^{(\gamma)}<\vartheta_{K\sigma}^{(\gamma)}.\end{cases}$
(3.17c)
At the Dirichlet boundary edges, we still set
$p_{\sigma}^{(\gamma)}=p_{\sigma}^{\rm D}$. For $\gamma=0$, the system is
linear and invertible, while for $\gamma=1$, system (3.17) coincides with the
original system (2.9)–(2.14). A priori estimates on
${\boldsymbol{p}}^{(\gamma)}$ that are uniform w.r.t. $\gamma\in[0,1]$ (but
not uniform w.r.t. $\mathscr{T}$ nor ${\Delta t}^{n}$) can be derived on the
basis of what was exposed previously, so that one can unfold Leray-Schauder’s
machinery [37, 20] to prove the existence of (at least) one solution to the
scheme.
### 3.3 Uniqueness of the discrete solution
To complete the proof of Theorem 2.2, it remains to show that the solution to
the scheme is unique. This is the purpose of the following proposition.
###### Proposition 3.7.
Given
${\boldsymbol{s}}^{n-1}=(s_{K}^{n-1})_{K\in\mathscr{T}}\in[0,1]^{\mathscr{T}}$,
the solution ${\boldsymbol{p}}^{n}\in\mathbb{R}^{\mathscr{T}}$ to the scheme
(2.9)–(2.14) is unique.
###### Proof.
The proof heavily rests upon the monotonicity properties inherited from the
upwind choice (2.11c) for the mobilities. Indeed, due to the upwind choice of
the mobility, the flux $F_{K\sigma}^{n}$ is a function of $p_{K}^{n}$ and
$p_{K\sigma}^{n}$ that is nondecreasing w.r.t. $p_{K}^{n}$ and nonincreasing
w.r.t. $p_{K\sigma}^{n}$. Moreover, by virtue of the monotonicity of
$\mathcal{S}_{K}$, the discrete volume balance (2.10) can be cast under the
abstract form
$\mathcal{H}_{K}^{n}(p_{K}^{n},(p_{K\sigma}^{n})_{\sigma\in\mathscr{E}_{K}})=0,\qquad\forall
K\in\mathscr{T},$ (3.18)
where $\mathcal{H}_{K}^{n}$ is nondecreasing w.r.t its first argument
$p_{K}^{n}$ and nonincreasing w.r.t each of the remaining variables
$(p_{K\sigma}^{n})_{\sigma\in\mathscr{E}_{K}}$.
Let
$\widetilde{{\boldsymbol{p}}}^{n}=\left(\widetilde{p}_{K}^{n}\right)_{K\in\mathscr{T}}$
be another solution to the system (2.9)–(2.14), i.e.,
$\mathcal{H}_{K}^{n}(\widetilde{p}_{K}^{n},(\widetilde{p}_{K\sigma}^{n})_{\sigma\in\mathscr{E}_{K}})=0,\qquad\forall
K\in\mathscr{T}.$ (3.19)
The nonincreasing behavior of $\mathcal{H}_{K}^{n}$ w.r.t. all its variables
except the first one implies that
$\mathcal{H}_{K}^{n}(p_{K}^{n},(p_{K\sigma}^{n}\top\widetilde{p}_{K\sigma}^{n})_{\sigma\in\mathscr{E}_{K}})\leq
0,\qquad\mathcal{H}_{K}^{n}(\widetilde{p}_{K}^{n},(p_{K\sigma}^{n}\top\widetilde{p}_{K\sigma}^{n})_{\sigma\in\mathscr{E}_{K}})\leq
0,$
for all $K\in\mathscr{T}$, where $a\top b=\max(a,b)$. Since
$p_{K}^{n}\top\widetilde{p}_{K}^{n}$ is either equal to $p_{K}^{n}$ or to
$\widetilde{p}_{K}^{n}$, we infer from the above inequalities that
$\mathcal{H}_{K}^{n}(p_{K}^{n}\top\widetilde{p}_{K}^{n},(p_{K\sigma}^{n}\top\widetilde{p}_{K\sigma}^{n})_{\sigma\in\mathscr{E}_{K}})\leq
0,\qquad\forall K\in\mathscr{T}.$ (3.20)
By a similar argument, we can show that
$\mathcal{H}_{K}^{n}(p_{K}^{n}\bot\widetilde{p}_{K}^{n},(p_{K\sigma}^{n}\bot\widetilde{p}_{K\sigma}^{n})_{\sigma\in\mathscr{E}_{K}})\geq
0,\qquad\forall K\in\mathscr{T},$ (3.21)
where $a\bot b=\min(a,b)$. Subtracting (3.21) from (3.20) and summing over
$K\in\mathscr{T}$, we find
$\sum_{K\in\mathscr{T}}m_{K}\phi_{K}\frac{|s_{K}^{n}-\widetilde{s}_{K}^{n}|}{\Delta
t^{n}}+\sum_{\sigma\in\mathscr{E}_{\rm ext}^{\rm
D}}a_{\sigma}\lambda_{\sigma}\mathtt{R}_{\sigma}^{n}\leq 0,$ (3.22)
where $s_{K}^{n}=\mathcal{S}_{K}(p_{K}^{n})$,
$\widetilde{s}_{K}^{n}=\mathcal{S}_{K}(\widetilde{p}_{K}^{n})$ and
$\displaystyle\mathtt{R}_{\sigma}^{n}$
$\displaystyle=\eta_{K}(s_{K}^{n}\top\widetilde{s}_{K}^{n})(\vartheta_{K}^{n}\top\widetilde{\vartheta}_{K}^{n}-\vartheta_{\sigma}^{n})^{+}-\eta_{K}(s_{\sigma}^{n})(\vartheta_{\sigma}^{n}-\vartheta_{K}^{n}\top\widetilde{\vartheta}_{K}^{n})^{+}$
$\displaystyle-\eta_{K}(s_{K}^{n}\bot\widetilde{s}_{K}^{n})(\vartheta_{K}^{n}\bot\widetilde{\vartheta}_{K}^{n}-\vartheta_{\sigma}^{n})^{+}+\eta_{K}(s_{\sigma}^{n})(\vartheta_{\sigma}^{n}-\vartheta_{K}^{n}\bot\widetilde{\vartheta}_{K}^{n})^{+},$
(3.23)
with $s_{\sigma}^{n}=\mathcal{S}_{K}(p^{n}_{\sigma})$. The top line of (3.23)
expresses the upwinded flux of (3.20), while the bottom line of (3.23) is the
opposite of the upwinded flux of (3.21). Note that, since
$p^{n}_{\sigma}=p^{\rm D}_{\sigma}$ is prescribed at
$\sigma\in\mathscr{E}^{\rm D}_{\rm ext}$, we have
$\vartheta^{n}_{\sigma}=\vartheta^{n}_{\sigma}\top\widetilde{\vartheta}^{n}_{\sigma}=\vartheta^{n}_{\sigma}\bot\widetilde{\vartheta}^{n}_{\sigma}$.
Upon inspection of the rearrangement
$\displaystyle\mathtt{R}_{\sigma}^{n}$
$\displaystyle=[\eta_{K}(s_{K}^{n}\top\widetilde{s}_{K}^{n})-\eta_{K}(s_{K}^{n}\bot\widetilde{s}_{K}^{n})](\vartheta_{K}^{n}\top\widetilde{\vartheta}_{K}^{n}-\vartheta_{\sigma}^{n})^{+}$
$\displaystyle+\eta_{K}(s_{K}^{n}\bot\widetilde{s}_{K}^{n})[(\vartheta_{K}^{n}\top\widetilde{\vartheta}_{K}^{n}-\vartheta_{\sigma}^{n})^{+}-(\vartheta_{K}^{n}\bot\widetilde{\vartheta}_{K}^{n}-\vartheta_{\sigma}^{n})^{+}]$
$\displaystyle+\eta_{K}(s_{\sigma}^{n})\big{[}(\vartheta_{\sigma}^{n}-\vartheta_{K}^{n}\bot\widetilde{\vartheta}_{K}^{n})^{+}-(\vartheta_{\sigma}^{n}-\vartheta_{K}^{n}\top\widetilde{\vartheta}_{K}^{n})^{+}\big{]},$
(3.24)
it is trivial that $\mathtt{R}_{\sigma}^{n}\geq 0$. As a consequence, (3.22)
implies that $\mathtt{R}_{\sigma}^{n}=0$ for all $\sigma\in\mathscr{E}_{\rm
ext}^{\rm D}$ and that $s_{K}^{n}=\widetilde{s}_{K}^{n}$ for all
$K\in\mathscr{T}$. At this stage, however, we cannot yet claim that
$p_{K}^{n}=\widetilde{p}_{K}^{n}$, as the function $\mathcal{S}_{K}$ is not
invertible.
Taking into account $s_{K}^{n}=\widetilde{s}_{K}^{n}$, the residue (3.24)
becomes
$\displaystyle\mathtt{R}_{\sigma}^{n}$
$\displaystyle=\eta_{K}(s_{K}^{n})[(\vartheta_{K}^{n}\top\widetilde{\vartheta}_{K}^{n}-\vartheta_{\sigma}^{n})^{+}-(\vartheta_{K}^{n}\bot\widetilde{\vartheta}_{K}^{n}-\vartheta_{\sigma}^{n})^{+}]$
$\displaystyle+\eta_{K}(s_{\sigma}^{n})\big{[}(\vartheta_{\sigma}^{n}-\vartheta_{K}^{n}\bot\widetilde{\vartheta}_{K}^{n})^{+}-(\vartheta_{\sigma}^{n}-\vartheta_{K}^{n}\top\widetilde{\vartheta}_{K}^{n})^{+}\big{]},$
(3.25)
which can be lower-bounded by
$\mathtt{R}_{\sigma}^{n}\geq\min(\eta_{K}(s_{K}^{n}),\,\eta_{K}(s_{\sigma}^{n}))|\vartheta_{K}^{n}-\widetilde{\vartheta}_{K}^{n}|$
(3.26)
thanks to the algebraic identities $a^{+}-(-a)^{+}=a$ and $a\top b-a\bot
b=|a-b|$. In view of the lower-bound on the discrete pressures of Lemma 3.5,
we deduce from (1.6b) that $s_{K}^{n}>0$ and $\widetilde{s}_{K}^{n}>0$. The
increasing behavior of $\eta_{K}$ implies, in turn, that
$\eta_{K}(s_{K}^{n})>0$ and $\eta_{K}(\widetilde{s}_{K}^{n})>0$. Therefore,
the conjunction of $\mathtt{R}_{\sigma}^{n}=0$ and (3.26) yields
$\vartheta_{K}^{n}=\widetilde{\vartheta}_{K}^{n}$ and hence
$p_{K}^{n}=\widetilde{p}_{K}^{n}$ for all cells $K$ having a Dirichlet
boundary edge, i.e., $\mathscr{E}_{K}\cap\mathscr{E}_{\rm ext}^{\rm
D}\neq\emptyset$.
It remains to check that $p_{K}^{n}=\widetilde{p}_{K}^{n}$, or equivalently
$\vartheta_{K}^{n}=\widetilde{\vartheta}_{K}^{n}$ for those cells
$K\in\mathscr{T}$ that are far away from the Dirichlet part of the boundary.
Subtracting (3.19) from (3.18) and recalling that
$s_{K}^{n}=\widetilde{s}_{K}^{n}$, we arrive at
$\displaystyle\sum_{\sigma\in\mathscr{E}_{K}}a_{\sigma}\lambda_{\sigma}\Big{\\{}\eta_{K}(s_{K}^{n})$
$\displaystyle\big{[}\left(\vartheta_{K}^{n}-\vartheta_{K\sigma}^{n}\right)^{+}-(\widetilde{\vartheta}_{K}^{n}-\widetilde{\vartheta}_{K\sigma}^{n})^{+}\big{]}$
$\displaystyle+\eta_{K\sigma}(s_{K\sigma}^{n})$
$\displaystyle\big{[}(\widetilde{\vartheta}_{K\sigma}^{n}-\widetilde{\vartheta}_{K}^{n})^{+}-(\vartheta_{K\sigma}^{n}-\vartheta_{K}^{n})^{+}\big{]}\Big{\\}}=0.$
(3.27)
Consider a cell $K\in\mathscr{T}$ where
$\vartheta_{K}^{n}-\widetilde{\vartheta}_{K}^{n}$ achieves its maximal value,
i.e.,
$\vartheta_{K}^{n}-\widetilde{\vartheta}_{K}^{n}\geq\vartheta_{L}^{n}-\widetilde{\vartheta}_{L}^{n},\qquad\forall
L\in\mathscr{T}.$ (3.28)
This entails that
$\vartheta_{K}^{n}-\vartheta_{K\sigma}^{n}\geq\widetilde{\vartheta}_{K}^{n}-\widetilde{\vartheta}_{K\sigma}^{n},\qquad\forall\sigma\in\mathscr{E}_{K},$
so that the two brackets in the right-hand side of (3.27) are nonnegative. In
fact, they both vanish by the positivity of $\eta_{K}(s_{K}^{n})$ and
$\eta_{K\sigma}(s_{K\sigma}^{n})$. As a result,
$\vartheta_{K}^{n}-\vartheta_{K\sigma}^{n}=\widetilde{\vartheta}_{K}^{n}-\widetilde{\vartheta}_{K\sigma}^{n}$
for all $\sigma\in\mathscr{E}_{K}$. This implies that
$\vartheta_{K}^{n}-\widetilde{\vartheta}_{K}^{n}=\vartheta_{L}^{n}-\widetilde{\vartheta}_{L}^{n}$
for all the cells $L\in\mathscr{T}$ sharing an edge $\sigma=K|L$ with $K$, and
thus that the cell $L$ also achieves the maximality condition (3.28). The
process can then be repeated over and over again. Since $\Omega$ is connected,
we deduce that $\vartheta_{K}^{n}-\widetilde{\vartheta}_{K}^{n}$ is constant
over $K\in\mathscr{T}$. The constant is finally equal to zero since
$\vartheta_{K}^{n}=\widetilde{\vartheta}_{K}^{n}$ on the cells having a
Dirichlet edge. ∎
## 4 Convergence analysis
Once existence and uniqueness of the discrete solution have been settled, the
next question to be addressed is the convergence of the discrete solution
towards a weak solution of the continuous problem, as the mesh-size and the
time-step are progressively refined. In accordance with the general philosophy
expounded in [27], the proof is built on compactness arguments. We start by
highlighting compactness properties in §4.1, before identifying the limit
values as weak solutions in §4.2.
### 4.1 Compactness properties
Let us define $G_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}:Q_{T}\to\mathbb{R}^{d}$ and $J_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}:Q_{T}\to\mathbb{R}^{d}$ by
$G_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}(t,x)=\begin{cases}\;d\displaystyle\frac{\Theta_{i}(p_{K\sigma}^{n})-\Theta_{i}(p_{K}^{n})}{d_{\sigma}}n_{K\sigma},&\text{if
}\;(t,x)\in(t_{m}^{n-1},t_{m}^{n}]\times\Delta_{\sigma},\\\
\;0&\text{otherwise},\end{cases}$ (4.1)
for $\sigma\in\mathscr{E}_{i,m}$, $1\leq n\leq N_{m}$ and, respectively,
$J_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}(t,x)=d\displaystyle\frac{\Upsilon(p_{K\sigma}^{n})-\Upsilon(p_{K}^{n})}{d_{\sigma}}n_{K\sigma},\quad\text{if
}\;(t,x)\in(t_{m}^{n-1},t_{m}^{n}]\times\Delta_{\sigma},$ (4.2)
for $\sigma\in\mathscr{E}_{m}$, $1\leq n\leq N_{m}$. We remind that
$s_{\mathscr{T}_{m},{\boldsymbol{{\Delta
t}}}_{m}}=\mathcal{S}(p_{\mathscr{T}_{m},{\boldsymbol{{\Delta t}}}_{m}},x)$ is
the sequence of approximate saturation fields computed from that of
approximate pressure fields $p_{\mathscr{T}_{m},{\boldsymbol{{\Delta
t}}}_{m}}$ by (2.15b).
###### Proposition 4.1.
There exists a measurable function $p:Q_{T}\to\mathbb{R}$ such that
$\Upsilon(p)-\Upsilon(p^{\rm D})\in L^{2}((0,T);V)$ and $\Theta_{i}(p)\in
L^{2}((0,T);H^{1}(\Omega_{i}))$, $1\leq i\leq I$, such that, up to a
subsequence,
$\displaystyle s_{\mathscr{T}_{m},{\boldsymbol{{\Delta t}}}_{m}}$
$\displaystyle\underset{m\to+\infty}{\longrightarrow}\mathcal{S}(p,x)$
$\displaystyle\text{a.e. in }\;Q_{T},$ (4.3a) $\displaystyle
G_{\mathscr{E}_{m},{\boldsymbol{{\Delta t}}}_{m}}$
$\displaystyle\underset{m\to+\infty}{\longrightarrow}\nabla\Theta_{i}(p)$
$\displaystyle\text{weakly in }\;L^{2}(Q_{i,T})^{d},$ (4.3b) $\displaystyle
J_{\mathscr{E}_{m},{\boldsymbol{{\Delta t}}}_{m}}$
$\displaystyle\underset{m\to+\infty}{\longrightarrow}\nabla\Upsilon(p)$
$\displaystyle\text{weakly in }\;L^{2}(Q_{T})^{d}.$ (4.3c)
###### Proof.
We know from Corollary 3.3 that
$\Theta_{i}(p_{\mathscr{T}_{m},{\boldsymbol{{\Delta t}}}_{m}})$ and
$\Upsilon(p_{\mathscr{T}_{m},{\boldsymbol{{\Delta t}}}_{m}})$ are bounded
w.r.t. $m$ in $L^{2}(Q_{i,T})$ and $L^{2}(Q_{T})$ respectively, while
$G_{\mathscr{E}_{m},{\boldsymbol{{\Delta t}}}_{m}}$ and
$J_{\mathscr{E}_{m},{\boldsymbol{{\Delta t}}}_{m}}$ are respectively bounded
in $L^{2}(Q_{i,T})^{d}$ and $L^{2}(Q_{T})^{d}$. In particular, there exist
$\widehat{\Theta}_{i}\in L^{2}(Q_{i,T})$, $\widehat{\Upsilon}\in
L^{2}(Q_{T})$, $J\in L^{2}(Q_{i,T})^{d}$, and $J\in L^{2}(Q_{T})^{d}$ such
that
$\displaystyle\Theta_{i}(p_{\mathscr{T}_{m},{\boldsymbol{{\Delta t}}}_{m}})$
$\displaystyle\underset{m\to+\infty}{\longrightarrow}\widehat{\Theta}_{i}$
$\displaystyle\text{weakly in}\;L^{2}(Q_{i,T}),$ (4.4a)
$\displaystyle\Upsilon(p_{\mathscr{T}_{m},{\boldsymbol{{\Delta t}}}_{m}})$
$\displaystyle\underset{m\to+\infty}{\longrightarrow}\widehat{\Upsilon}$
$\displaystyle\text{weakly in}\;L^{2}(Q_{T}),$ (4.4b) $\displaystyle
G_{\mathscr{E}_{m},{\boldsymbol{{\Delta t}}}_{m}}$
$\displaystyle\underset{m\to+\infty}{\longrightarrow}G$
$\displaystyle\text{weakly in }\;L^{2}(Q_{i,T})^{d},$ (4.4c) $\displaystyle
J_{\mathscr{E}_{m},{\boldsymbol{{\Delta t}}}_{m}}$
$\displaystyle\underset{m\to+\infty}{\longrightarrow}J$
$\displaystyle\text{weakly in }\;L^{2}(Q_{T})^{d}.$ (4.4d)
Establishing that $\widehat{\Theta}_{i}\in L^{2}((0,T);H^{1}(\Omega_{i}))$ and
$\widehat{\Upsilon}\in L^{2}((0,T);H^{1}(\Omega))$ with
$G=\nabla\widehat{\Theta}_{i}$ and $J=\nabla\widehat{\Upsilon}$ is now
classical, see for instance [25, Lemma 2] or [16, Lemma 4.4].
The key points of this proof are the identification
$\widehat{\Theta}_{i}=\Theta_{i}(p)$ and $\widehat{\Upsilon}=\Upsilon(p)$ for
some measurable $p$, as well as the proofs of the almost everywhere
convergence property (4.3a). The identification of the limit and the almost
everywhere convergence can be handled simultaneously by using twice [2,
Theorem 3.9], once for $\Theta_{i}(p)$ and once for $\Upsilon(p)$. More
precisely, Lemma 3.4 provides a control on the time variations of the
approximate saturation $s_{\mathscr{T}_{m},{\boldsymbol{{\Delta t}}}_{m}}$,
whereas Corollary 3.3 provides some compactness w.r.t. space on
$\Theta_{i}(p_{\mathscr{T}_{m},{\boldsymbol{{\Delta t}}}_{m}})$ and
$\Upsilon(p_{\mathscr{T}_{m},{\boldsymbol{{\Delta t}}}_{m}})$. Using further
that $s_{\mathscr{T}_{m},{\boldsymbol{{\Delta
t}}}_{m}}=\mathcal{S}_{i}\circ\Theta_{i}^{-1}\left(\Theta_{i}(p_{\mathscr{T}_{m},{\boldsymbol{{\Delta
t}}}_{m}})\right)$ with $\mathcal{S}_{i}\circ\Theta_{i}^{-1}$ nondecreasing
and continuous, then one infers from [2, Theorem 3.9] that
$s_{\mathscr{T}_{m},{\boldsymbol{{\Delta
t}}}_{m}}\underset{m\to+\infty}{\longrightarrow}\mathcal{S}_{i}\circ\Theta_{i}^{-1}(\widehat{\Theta}_{i})\quad\text{a.e.
in}\;Q_{i,T}.$
Let $p=\Theta_{i}^{-1}(\widehat{\Theta}_{i})$. Then, (4.3a) and (4.3b) hold.
Proving (4.3a) and (4.3c) is similar, and the properties (4.3) can be assumed
to hold for the same function $p$ up to the extraction of yet another
subsequence.
Finally, by applying the arguments developed in [6, §4.2], we show that
$\Upsilon(p)$ and $\Upsilon(p^{\rm D})$ share the same trace on
$(0,T)\times\Gamma^{\rm D}$, hence $\Upsilon(p)-\Upsilon(p^{\rm D})\in
L^{2}((0,T);V)$. ∎
Let us now define
$\eta_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}(t,x)=\eta_{\sigma}^{n}\quad\text{if
}\;(t,x)\in(t_{m}^{n-1},t_{m}^{n}]\times\Delta_{\sigma}$ (4.5)
for $\sigma\in\mathscr{E}_{m}$, $1\leq n\leq N_{m}$.
###### Lemma 4.2.
Up to a subsequence, the function $p$ whose existence is guaranteed by
Proposition 4.1 satisfies
$\eta_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}\underset{m\to\infty}{\longrightarrow}\eta(\mathcal{S}(p,x))\qquad\text{in
}\;L^{q}(Q_{T}),\;1\leq q<+\infty.$ (4.6)
###### Proof.
Because of (4.3a), $\eta_{\mathscr{T}_{m},{\boldsymbol{{\Delta
t}}}_{m}}=\eta(s_{\mathscr{T}_{m},{\boldsymbol{{\Delta t}}}_{m}},x)$ converges
almost everywhere to $\eta(\mathcal{S}(p,x),x)$. Since $\eta$ is bounded,
Lebesgue’s dominated convergence theorem ensures that the convergence holds in
$L^{q}(Q_{T})$ for all $q\in[1,+\infty)$. The reconstruction
$\eta_{\mathscr{E}_{m},{\boldsymbol{{\Delta t}}}_{m}}$ of the mobility is also
uniformly bounded, so we have just to show that
$\lVert\eta_{\mathscr{T}_{m},{\boldsymbol{{\Delta
t}}}_{m}}-\eta_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}\rVert_{L^{1}(Q_{T})}\rightarrow 0$ as $m\rightarrow+\infty$. Letting
$\Delta_{K\sigma}=K\cap\Delta_{\sigma}$ denote the half-diamond cell, we have
$\lVert\eta_{\mathscr{T}_{m},{\boldsymbol{{\Delta
t}}}_{m}}-\eta_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}\rVert_{L^{1}(Q_{T})}\leq\;\sum_{n=1}^{N_{m}}{\Delta
t}^{n}_{m}\sum_{K\in\mathscr{T}_{m}}\sum_{\sigma\in\mathscr{E}_{K}}m_{\Delta_{K\sigma}}|\eta_{K}(s_{K}^{n})-\eta_{\sigma}^{n}|\\\
\leq\;\sum_{n=1}^{N_{m}}{\Delta
t}^{n}_{m}\sum_{\sigma\in\mathscr{E}_{m}}m_{\Delta_{\sigma}}|\eta_{K}(s_{K}^{n})-\eta_{K\sigma}(s_{K\sigma}^{n})|\,\leq\;\sum_{i=1}^{I}\mathtt{R}_{i,m}+\mathtt{R}_{\Gamma,m},$
where
$\displaystyle\mathtt{R}_{i,m}=\;$ $\displaystyle\sum_{n=1}^{N_{m}}{\Delta
t}^{n}_{m}\sum_{\sigma\in\mathscr{E}_{i,m}}m_{\Delta_{\sigma}}|\eta_{K}(s_{K}^{n})-\eta_{K\sigma}(s_{K\sigma}^{n})|,$
$\displaystyle\mathtt{R}_{\Gamma,m}=\;$
$\displaystyle\sum_{n=1}^{N_{m}}{\Delta
t}^{n}_{m}\sum_{\sigma\in\mathscr{E}_{\Gamma,m}}m_{\Delta_{\sigma}}|\eta_{K}(s_{K}^{n})-\eta_{K\sigma}(s_{K\sigma}^{n})|.$
Let us define
$r_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}(t,x)=|\eta_{K}^{n}-\eta_{K\sigma}^{n}|=r_{\sigma}^{n}\quad\text{if}\;(t,x)\in(t_{m}^{n-1},t_{m}^{n}]\times\Delta_{\sigma},$
then $r_{\mathscr{E}_{m},{\boldsymbol{{\Delta t}}}_{m}}$ is uniformly bounded
by $\lVert\eta\rVert_{\infty}=1/\mu$. Therefore,
$\mathtt{R}_{\Gamma,m}\leq\frac{T}{\mu}\sum_{\sigma\in\mathscr{E}_{\Gamma,m}}m_{\Delta_{\sigma}}\leq\frac{2T\,\nu^{d-1}(\Gamma)}{\mu
d}\,h_{\mathscr{T}_{m}}$
where $h_{\mathscr{T}_{m}}$ is the size of $\mathscr{T}_{m}$ as defined in
(2.8). Besides, for $i\in\\{1,\dots,I\\}$,
$\eta_{i}\circ\mathcal{S}_{i}\circ\Theta_{i}^{-1}$ is continuous, monotone and
bounded, hence uniformly continuous. This provides the existence of a modulus
of continuity $\varpi_{i}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ with
$\varpi_{i}(0)=0$ such that
$r_{\sigma}^{n}:=|\eta\circ\mathcal{S}\circ\Theta_{i}^{-1}(\Theta_{K}^{n})-\eta\circ\mathcal{S}\circ\Theta_{i}^{-1}(\Theta_{K\sigma}^{n})|\leq\varpi_{i}(|\Theta_{K}^{n}-\Theta_{K\sigma}^{n}|)$
(4.7)
for $\sigma\in\mathscr{E}_{i,m}$. Therefore, if the function
$q_{\mathscr{E}_{i,m},{\boldsymbol{{\Delta
t}}}_{m}}(t,x)=\begin{cases}\,|\Theta_{i}(p_{K}^{n})-\Theta_{i}(p_{K\sigma}^{n})|&\text{if}\;(t,x)\in(t_{m}^{n-1},t_{m}^{n}]\times\Delta_{\sigma},\\\
\,0&\text{otherwise},\end{cases}$ (4.8a) for $\sigma\in\mathscr{E}_{i,m}$,
$1\,\leq\,n\leq N_{m}$,
could be proven to converge to $0$ almost everywhere in $Q_{i,T}$, then it
would also be the case for $r_{\mathscr{E}_{m},{\boldsymbol{{\Delta t}}}_{m}}$
and $\mathtt{R}_{i,m}$ as $m\rightarrow+\infty$, thanks to Lebesgue’s
dominated convergence theorem. Now, it follows from (3.12a) and from the
elementary geometric relation
$m_{\Delta_{\sigma}}=\frac{a_{\sigma}}{d}d_{\sigma}^{2}\,\leq\,4\frac{a_{\sigma}}{d}h_{\mathscr{T}_{m}}^{2},$
that
$\lVert q_{\mathscr{E}_{i,m},{\boldsymbol{{\Delta
t}}}_{m}}\rVert^{2}_{L^{2}(Q_{i,T})}=\sum_{n=1}^{N_{m}}{\Delta
t}^{n}_{m}\sum_{\sigma\in\mathscr{E}_{i,m}}m_{\Delta_{\sigma}}|\Theta_{i}(p_{K}^{n})-\Theta_{i}(p_{K\sigma}^{n})|^{2}\leq\frac{4C_{\ref{c:dissip.max}}}{d}h_{\mathscr{T}_{m}}^{2}.$
Therefore, $q_{\mathscr{E}_{i,m},{\boldsymbol{{\Delta t}}}_{m}}\rightarrow 0$
in $L^{2}(Q_{i,T})$, thus also almost everywhere up to extraction of a
subsequence. This provides the desired result. ∎
### 4.2 Identification of the limit
So far, we have exhibited some “limit” value $p$ for the approximate solution
$p_{\mathscr{T}_{m},{\boldsymbol{{\Delta t}}}_{m}}$ in Proposition 4.1. Next,
we show that the scheme is consistent with the continuous problem by showing
that any limit value is a weak solution.
###### Proposition 4.3.
The function $p$ whose existence is guaranteed by Proposition 4.1 is a weak
solution of the problem (1.9a)–(1.9c) in the sense of Definition 1.1.
###### Proof.
Let $\varphi\in C_{c}^{\infty}(\\{\Omega\cup\Gamma^{\rm N}\\}\times[0,T))$ and
denote by $\varphi_{K}^{n}=\varphi(t^{n}_{m},x_{K})$, for all
$K\in\mathscr{T}_{m}$ and all $n\in\\{0,\dots,N_{m}\\}$. We multiply (2.10) by
${\Delta t}^{n}_{m}\varphi_{K}^{n-1}$ and sum over $n\in\\{1,\dots,N_{m}\\}$
and $K\in\mathscr{T}_{m}$ to obtain
$\mathtt{A}_{m}+\mathtt{B}_{m}=0,\qquad m\geq 1,$ (4.9)
where we have set
$\displaystyle\mathtt{A}_{m}=$
$\displaystyle\sum_{n=1}^{N_{m}}\sum_{K\in\mathscr{T}_{m}}m_{K}\phi_{K}(s_{K}^{n}-s_{K}^{n-1})\varphi_{K}^{n-1},$
(4.10a) $\displaystyle\mathtt{B}_{m}=$ $\displaystyle\sum_{n=1}^{N_{m}}{\Delta
t}^{n}_{m}\sum_{\sigma\in\mathscr{E}_{m}}a_{\sigma}\lambda_{\sigma}\eta_{\sigma}^{n}(\vartheta_{K}^{n}-\vartheta_{K\sigma}^{n})(\varphi_{K}^{n-1}-\varphi_{K\sigma}^{n-1}).$
(4.10b)
The quantity $\mathtt{A}_{m}$ in (4.10a) can be rewritten as
$\displaystyle\mathtt{A}_{m}$ $\displaystyle=-\sum_{n=1}^{N_{m}}{\Delta
t}^{n}_{m}\sum_{K\in\mathscr{T}_{m}}m_{K}\phi_{K}s_{K}^{n}\frac{\varphi_{K}^{n}-\varphi_{K}^{n-1}}{{\Delta
t}_{m}^{n}}-\sum_{K\in\mathscr{T}_{m}}m_{K}\phi_{K}s_{K}^{0}\varphi_{K}^{0}$
$\displaystyle=-\int\\!\\!\\!\int_{Q_{T}}\phi\,s_{\mathscr{T}_{m},{\boldsymbol{{\Delta
t}}}_{m}}\delta\varphi_{\mathscr{T}_{m},{\boldsymbol{{\Delta t}}}_{m}}\,{\rm
d}x\,{\rm
d}t-\int_{\Omega}\phi\,s_{\mathscr{T}_{m}}^{0}\varphi_{\mathscr{T}_{m}}^{0}\,{\rm
d}x$
where
$\displaystyle\delta\varphi_{\mathscr{T}_{m},{\boldsymbol{{\Delta
t}}}_{m}}(t,x)$
$\displaystyle=\frac{\varphi_{K}^{n}-\varphi_{K}^{n-1}}{{\Delta t}_{m}^{n}},$
if $\displaystyle(t,x)\in(t_{m}^{n-1},t_{m}^{n})\times K,$
$\displaystyle\varphi_{\mathscr{T}_{m}}^{0}$ $\displaystyle=\varphi(0,x_{K})$
if $\displaystyle x\in K.$
Thanks to the regularity of $\varphi$, the function
$\delta\varphi_{\mathscr{T}_{m},{\Delta t}_{m}}$ converges uniformly to
$\partial_{t}\varphi$ on $\Omega\times[0,T]$. Moreover, by virtue of (4.3a)
and the boundedness of $s_{\mathscr{T}_{m},{\boldsymbol{{\Delta t}}}_{n}}$ we
can state that
$\int\\!\\!\\!\int_{Q_{T}}\phi\,s_{\mathscr{T}_{m},{\Delta
t}_{m}}\delta\varphi_{\mathscr{T}_{m},{\Delta t}_{m}}\,{\rm d}x\,{\rm
d}t\underset{m\to+\infty}{\longrightarrow}\int\\!\\!\\!\int_{Q_{T}}\phi\,\mathcal{S}(p,x)\partial_{t}\varphi\,{\rm
d}x\,{\rm d}t,$
and, in view of the definition (2.13) of $s^{0}_{\mathscr{T}_{m}}$ and of the
uniform convergence of $\varphi_{\mathscr{T}_{m}}^{0}$ towards
$\varphi(0,\cdot)$,
$\int_{\Omega}\phi_{\mathscr{T}_{m}}s_{\mathscr{T}_{m}}^{0}\varphi^{0}_{\mathscr{T}_{m}}\,{\rm
d}x\underset{m\to+\infty}{\longrightarrow}\int_{\Omega}\phi\,s^{0}\varphi(0,\cdot)\,{\rm
d}x.$
From the above, we draw that
$\lim_{m\to+\infty}\mathtt{A}_{m}=-\int\\!\\!\\!\int_{Q_{T}}\phi\,\mathcal{S}(p,x)\partial_{t}\varphi\,{\rm
d}x\,{\rm d}t-\int_{\Omega}\phi\,s^{0}\varphi(0,\cdot)\,{\rm d}x.$ (4.11)
Let us now turn our attention to the quantity $\mathtt{B}_{m}$ of (4.10b),
which can be split into $\mathtt{B}_{m}=\mathtt{B}_{m}^{1}+\mathtt{B}_{m}^{2}$
using
$\displaystyle\mathtt{B}_{m}^{1}=$ $\displaystyle\sum_{n=1}^{N_{m}}{\Delta
t}_{m}^{n}\sum_{\sigma\in\mathscr{E}_{m}}a_{\sigma}\lambda_{\sigma}\eta_{\sigma}^{n}(p_{K}^{n}-p_{K\sigma}^{n})(\varphi_{K}^{n-1}-\varphi_{K\sigma}^{n-1}),$
$\displaystyle\mathtt{B}_{m}^{2}=$ $\displaystyle\sum_{n=1}^{N_{m}}{\Delta
t}_{m}^{n}\sum_{\sigma\in\mathscr{E}_{m}}a_{\sigma}\lambda_{\sigma}\eta_{\sigma}^{n}(\psi_{K}-\psi_{K\sigma})(\varphi_{K}^{n-1}-\varphi_{K\sigma}^{n-1}).$
Consider first the convective term $\mathtt{B}_{m}^{2}$. It follows from the
definition of the discrete gravitational potential
$\psi_{K}=-\varrho g\cdot x_{K},\quad\psi_{\sigma}=-\varrho g\cdot
x_{\sigma},\qquad K\in\mathscr{T}_{m},\;\sigma\in\mathscr{E}_{{\rm
ext},m}^{\rm D}$
and from the orthogonality of the mesh that
$\psi_{K}-\psi_{K\sigma}=d_{\sigma}\varrho g\cdot
n_{K\sigma},\qquad\forall\sigma\in\mathscr{E}_{K}\setminus\mathscr{E}_{\rm
ext}^{\rm N},\;K\in\mathscr{T}_{m}.$
Therefore, $\mathtt{B}^{2}_{m}$ can be transformed into
$\displaystyle\mathtt{B}_{m}^{2}=$ $\displaystyle\sum_{n=1}^{N_{m}}{\Delta
t}_{m}^{n}\sum_{\sigma\in\mathscr{E}_{m}}m_{\Delta_{\sigma}}\lambda_{\sigma}\eta_{\sigma}^{n}d\frac{\varphi_{K}^{n-1}-\varphi_{K\sigma}^{n-1}}{d_{\sigma}}n_{K\sigma}\cdot\varrho
g$ $\displaystyle=$
$\displaystyle-\int\\!\\!\\!\int_{Q_{T}}\lambda_{\mathscr{E}_{m}}\eta_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}H_{\mathscr{E}_{m},{\boldsymbol{{\Delta t}}}_{m}}\cdot\varrho
g\,\,{\rm d}x\,{\rm d}t,$ (4.12)
where
$\displaystyle\lambda_{\mathscr{E}_{m}}(x)$ $\displaystyle=\lambda_{\sigma}$
if $\displaystyle x\in\Delta_{\sigma},\;\sigma\in\mathscr{E}_{m},$
$\displaystyle H_{\mathscr{E}_{m},{\boldsymbol{{\Delta t}}}_{m}}(t,x)$
$\displaystyle=(d/d_{\sigma})(\varphi_{K\sigma}^{n-1}-\varphi_{K}^{n-1})n_{K\sigma}$
if $\displaystyle(t,x)\in[t^{n-1}_{m},t^{n}_{m})\times\Delta_{\sigma}.$
After [16, Lemma 4.4], $H_{\mathscr{E}_{m},{\boldsymbol{{\Delta t}}}_{m}}$
converges weakly in $L^{2}(Q_{T})^{d}$ towards $\nabla\varphi$, while
$\lambda_{\mathscr{E}_{m}}$ and $\eta_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}$ converge strongly in $L^{4}(\Omega)$ and $L^{4}(Q_{T})$ towards
$\lambda$ and $\eta(\mathcal{S}(p,x))$ respectively (cf. Lemma 4.2). Thus, we
can pass to the limit in (4.12) and
$\lim_{m\to+\infty}\mathtt{B}_{m}^{2}=-\int\\!\\!\\!\int_{Q_{T}}\lambda\eta(\mathcal{S}(p,x))\varrho
g\cdot\nabla\varphi\,{\rm d}x\,{\rm d}t.$ (4.13)
The capillary diffusion term $\mathtt{B}_{m}^{1}$ appears to be the most
difficult one to deal with. Taking inspiration from [13], we introduce the
auxiliary quantity
$\displaystyle\widetilde{\mathtt{B}}{}_{m}^{1}$
$\displaystyle=\sum_{i=1}^{I}\widetilde{\mathtt{B}}_{i,m}^{1}$
$\displaystyle=\sum_{i=1}^{I}\sum_{n=1}^{N_{m}}{\Delta
t}_{m}^{n}\\!\\!\sum_{\sigma\in\mathscr{E}_{i,m}}\\!\\!a_{\sigma}\sqrt{\lambda_{i}\eta_{\sigma}^{n}}(\Theta_{i}(p_{K}^{n})-\Theta_{i}(p_{K\sigma}^{n}))(\varphi_{K}^{n-1}-\varphi_{K\sigma}^{n-1}).$
Analogously to [22], we can define a piecewise-constant vector field
$\overline{H}_{\mathscr{E}_{m},{\boldsymbol{{\Delta t}}}_{m}}$ such that
$\overline{H}_{\mathscr{E}_{m},{\boldsymbol{{\Delta t}}}_{m}}(t,x)\cdot
n_{K\sigma}=\varphi_{K\sigma}^{n-1}-\varphi_{K}^{n-1},\qquad\text{if
}\;(t,x)\in[t^{n-1}_{m},t^{n}_{m})\times\Delta_{\sigma},\;\sigma\in\mathscr{E}_{m},$
and such that $\overline{H}_{\mathscr{E}_{m},{\boldsymbol{{\Delta t}}}_{m}}$
converges uniformly towards $\nabla\varphi$ on $\overline{Q}_{T}$. Under these
circumstances, $\widetilde{\mathtt{B}}_{i,m}^{1}$ reads
$\widetilde{\mathtt{B}}_{i,m}^{1}=\int_{0}^{T}\\!\\!\int_{\Omega_{i,m}}\sqrt{\lambda_{i}\eta_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}}\,G_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}\cdot\overline{H}_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}\,{\rm d}x\,{\rm d}t$
where
$\Omega_{i,m}=\bigcup_{\sigma\in\mathscr{E}_{i,m}}\Delta_{\sigma}\subset\Omega_{i}$.
The strong convergence of $\sqrt{\eta_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}}$ in $L^{2}(Q_{i,T})$ towards $\sqrt{\eta_{i}(\mathcal{S}_{i}(p))}$
directly follows from the boundedness of $\eta_{i}$ combined with (4.3a).
Combining this with (4.3b) results in
$\widetilde{\mathtt{B}}_{i,m}^{1}\underset{m\to+\infty}{\longrightarrow}\int\\!\\!\\!\int_{Q_{i,T}}\\!\\!\\!\\!\\!\\!\sqrt{\lambda_{i}\eta_{i}(\mathcal{S}_{i}(p))}\nabla\Theta_{i}(p)\cdot\nabla\varphi\,{\rm
d}x\,{\rm
d}t=\int\\!\\!\\!\int_{Q_{i,T}}\\!\\!\\!\\!\\!\\!\nabla\Phi_{i}(p)\cdot\nabla\varphi\,{\rm
d}x\,{\rm d}t.$ (4.14)
Therefore, to finish the proof of Proposition 4.3, it only remains to check
that $B^{1}_{m}$ and $\widetilde{B}^{1}_{m}$ share the same limit. To this
end, we observe that, by the triangle inequality, we have
$|\mathtt{B}_{m}^{1}-\widetilde{\mathtt{B}}_{m}^{1}|\leq\mathtt{R}_{\Gamma,m}+\sum_{i=1}^{I}\mathtt{R}_{i,m},$
(4.15)
where
$\displaystyle\mathtt{R}_{\Gamma,m}$ $\displaystyle=\sum_{n=1}^{N_{m}}{\Delta
t}_{m}^{n}\\!\\!\\!\sum_{\sigma\in\mathscr{E}_{\Gamma,m}}\\!\\!\\!a_{\sigma}\lambda_{\sigma}\eta_{\sigma}^{n}|p_{K}^{n}-p_{K\sigma}^{n}||\varphi_{K}^{n-1}-\varphi_{K\sigma}^{n-1}|,$
$\displaystyle\mathtt{R}_{i,m}$ $\displaystyle=\sum_{n=1}^{N_{m}}{\Delta
t}_{m}^{n}\\!\\!\\!\sum_{\sigma\in\mathscr{E}_{i,m}}\\!\\!\\!a_{\sigma}\sqrt{\lambda_{i}\eta_{\sigma}^{n}}|\Theta_{i}(p_{K}^{n})-\Theta_{i}(p_{K\sigma}^{n})-\sqrt{\lambda_{i}\eta_{\sigma}^{n}}(p_{K}^{n}-p_{K\sigma}^{n})||\varphi_{K}^{n-1}-\varphi_{K\sigma}^{n-1}|.$
Applying the Cauchy-Schwarz inequality and using Proposition 3.1, we find
$|\mathtt{R}_{\Gamma,m}|^{2}\leq C_{\ref{c:main}}\sum_{n=1}^{N_{m}}{\Delta
t}_{m}^{n}\\!\\!\sum_{\sigma\in\mathscr{E}_{\Gamma,m}}\\!\\!a_{\sigma}\lambda_{\sigma}\eta_{\sigma}^{n}|\varphi_{K}^{n-1}-\varphi_{K\sigma}^{n-1}|^{2}\leq
2C_{\ref{c:main}}T\lVert\nabla\varphi\rVert_{\infty}^{2}\frac{\max_{i}\lambda_{i}}{\mu}\nu^{d-1}(\Gamma)h_{\mathscr{T}_{m}},$
so $\mathtt{R}_{\Gamma,m}\rightarrow 0$ as $m\rightarrow+\infty$. Besides, we
also apply the Cauchy-Schwarz inequality to $\mathtt{R}_{i,m}$ in order to
obtain
$\displaystyle|\mathtt{R}_{i,m}|^{2}$ $\displaystyle\leq
C_{\ref{c:main}}\sum_{n=1}^{N_{m}}{\Delta
t}_{m}^{n}\\!\\!\sum_{\sigma\in\mathscr{E}_{i,m}}\\!\\!a_{\sigma}\,\lambda_{i}\big{|}\sqrt{\eta_{\sigma}^{n}}-\sqrt{\widetilde{\eta}_{\sigma}^{n}}\,\big{|}^{2}\,\big{|}\varphi_{K}^{n-1}-\varphi_{K\sigma}^{n-1}\big{|}^{2}$
$\displaystyle\leq
d\lambda_{i}C_{\ref{c:main}}\lVert\nabla\varphi\rVert_{\infty}^{2}\sum_{n=1}^{N_{m}}{\Delta
t}_{m}^{n}\\!\\!\sum_{\sigma\in\mathscr{E}_{i,m}}\\!\\!m_{\Delta_{\sigma}}\left|{\eta_{\sigma}^{n}}-{\widetilde{\eta}_{\sigma}^{n}}\right|,$
where we have set
$\widetilde{\eta}_{\sigma}^{n}=\begin{cases}\,\eta_{i}(s_{K}^{n})&\text{if}\;p_{K}^{n}=p_{K\sigma}^{n},\\\
\,\displaystyle\bigg{[}\frac{\Theta_{i}(p_{K}^{n})-\Theta_{i}(p_{K\sigma}^{n})}{\sqrt{\lambda_{i}}(p_{K}^{n}-p_{K\sigma}^{n})}\bigg{]}^{2}&\text{otherwise}.\end{cases}$
Define
$\widetilde{\eta}_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}(t,x)=\begin{cases}\,\widetilde{\eta}_{\sigma}^{n}&\text{if}\;(t,x)\in(t_{m}^{n-1},t_{m}^{n}]\times\Delta_{\sigma},\;\sigma\in\bigcup_{i=1}^{I}\mathscr{E}_{i,m},\\\
\,0&\text{otherwise}.\end{cases}$
Reproducing the proof of Lemma 4.2, we can show that
$\widetilde{\eta}_{\mathscr{E}_{m},{\boldsymbol{{\Delta
t}}}_{m}}\underset{m\to\infty}{\longrightarrow}\eta(\mathcal{S}(p,x))\qquad\text{in
}\;L^{q}(Q_{T}),\;1\leq q<+\infty.$
Therefore, $\mathtt{R}_{i,m}\rightarrow 0$ as $m\rightarrow+\infty$. Putting
things together in (4.15), we conclude that $\mathtt{B}_{m}^{1}$ and
$\widetilde{\mathtt{B}}_{m}^{1}$ share the same limit, which completes the
proof of Proposition 4.3. ∎
## 5 Practical aspects of numerical resolution
We provide some details on the resolution strategy for the discrete problem
(2.9)–(2.11c). It is based on a parametrization technique to automatically
choose the most convenient variable during the Newton iterations (§5.1) and on
the addition of cells on the interfaces between different rock types (§5.2) to
improve the pressure continuity.
### 5.1 Switch of variable and parametrization technique
A natural choice to solve the nonlinear system (2.9)–(2.11c) is to select the
pressure $(p_{K})_{K\in\mathscr{T}}$ as primary unknown and to solve it via an
iterative method such as Newton’s one. Nevertheless, the pressure variable is
known to be an inefficient choice for $s\ll 1$ because of the degeneracy of
Richards’ equation. For dry soils, this strategy is outperformed by schemes in
which saturation is the primary variable. On the other hand, the knowledge of
the saturation is not sufficient to describe the pressure curve in saturated
regions where the pressure-saturation relation cannot be inverted. This
motivated the design of schemes involving a switch of variable [21]–[32]. In
this work, we adopt the technique proposed by Brenner and Cancès [5], in which
a third generic variable $\tau$ is introduced to become the primary unknown of
the system. Then the idea is to choose a parametrization of the graph
$\\{p,\mathcal{S}(p)\\}$, i.e., to construct two functions
$\mathfrak{s}:I\rightarrow[s_{\mathrm{rw}},1-s_{\mathrm{rn}}]$ and
$\mathfrak{p}:I\rightarrow\mathbb{R}$ such that
$\mathfrak{s}(\tau)=\mathcal{S}(\mathfrak{p}(\tau))$ and
$\mathfrak{s}^{\prime}(\tau)+\mathfrak{p}^{\prime}(\tau)>0$ for all $\tau\in
I\subset\mathbb{R}$. Such a parametrization is not unique, for instance one
can take $I=\mathbb{R}$, $\mathfrak{p}=Id$ which amounts to solving the system
always in pressure, but this is not recommended as explained before. Here, we
set $I=(s_{\mathrm{rw}},+\infty)$ and
$\displaystyle\mathfrak{s}(\tau)=\begin{cases}\tau&\text{if}\;\tau\leq s_{\rm
s},\\\ \mathcal{S}\left(p_{\rm s}+\displaystyle\frac{\tau-s_{\rm
s}}{\mathcal{S}^{\prime}(p_{\rm s}^{-})}\right)&\text{if}\;\tau\geq s_{\rm
s},\\\
\end{cases}\qquad\mathfrak{p}(\tau)=\begin{cases}\mathcal{S}^{-1}(\tau)&\text{if}\;\tau\leq
s_{\rm s},\\\ p_{\rm s}+\displaystyle\frac{\tau-s_{\rm
s}}{\mathcal{S}^{\prime}(p_{\rm s}^{-})}&\text{if}\;\tau\geq s_{\rm
s},\end{cases}$
where $\mathcal{S}^{\prime}(p_{\rm s}^{-})$ denotes the limit as $p$ tends to
$p_{\rm s}=\mathcal{S}(s_{\rm s})$ from below of $\mathcal{S}^{\prime}(p)$.
Since the switch point $s_{\rm s}$ is taken as the inflexion point of
$\mathcal{S}$, both $\mathfrak{s}$ and $\mathfrak{p}$ are $C^{1}$ and concave,
and even $C^{2}$ if $\mathcal{S}$ is given by the Van Genuchten model.
Moreover, for all $p\in\mathbb{R}$, there exists a unique $\tau\in(s_{\rm
rw},+\infty)$ such that
$(p,\mathcal{S}(p))=(\mathfrak{p}(\tau),\mathfrak{s}(\tau))$. The resulting
system $\mathcal{F}_{n}(\boldsymbol{\tau}^{n})=\mathbf{0}$ made up of
$N_{\mathscr{T}}={\rm Card}(\mathscr{T})$ nonlinear equations admits a unique
solution $\boldsymbol{\tau}^{n}$, since it is fully equivalent to
(2.9)–(2.11c). More details about the practical resolution of this nonlinear
system via the Newton method can be found in [4].
### 5.2 Pressure continuity at rock type interfaces
Physically, the pressure should remain continuous on both sides of an
interface between two different rock types. But this continuity is here not
imposed at the discrete level. The two-point flux approximation based on the
cell unknowns is strongly dependent on the mesh resolution and can induce a
large error close to the rock type interface. We here propose a very simple
method to improve this continuity condition in pressure. It consists in adding
two thin cells of resolution $\delta$ around the rock-type interface with
$\delta\ll\Delta x$ as shown in Figure 1.
Figure 1: Mesh refinement on both sides of an interface face for a 2D case.
The idea is here to add two unknowns in the neighborhood of the interface to
have a more precise approximation of the pressure gradient on each side of the
faces where changes of rock types occur. In this way, we avoid the
introduction of face unknowns in our solver which remains unchanged.
Alternatives that strictly impose the pressure continuity will be studied and
compared with this approach in a forthcoming work.
## 6 Numerical results
In this section, we present the results obtained for different test cases. For
all these cases, we consider a two-dimensional layered domain
$\Omega=[0\mathrm{m},5\mathrm{m}]\times[-3\mathrm{m},0\mathrm{m}]$ made up of
two rock types denoted by RT0 and RT1 respectively, RT0 being less permeable
than RT1. Using these two lithologies, the domain $\Omega$ is partitioned into
three connected subdomains:
$\Omega_{1}=[1\mathrm{m},4\mathrm{m}]\times[-1\mathrm{m},0\mathrm{m}]$,
$\Omega_{2}=[0\mathrm{m},5\mathrm{m}]\times[-3\mathrm{m},-2\mathrm{m}]$ and
$\Omega_{3}=\Omega\leavevmode\nobreak\ \setminus\leavevmode\nobreak\
(\Omega_{1}\cup\Omega_{2})$, as depicted in Figure 2.
Figure 2: Simulation domain
$\Omega=[0\mathrm{m},5\mathrm{m}]\times[-3\mathrm{m},0\mathrm{m}]$.
The Brooks-Corey [10] and van Genuchten-Mualem [42] petro-physical models are
used to model the flow characteristics of both rock types. In these models,
the water saturation and the capillary pressure are linked pointwise by the
relation $s=\mathcal{S}(p)$ where $\mathcal{S}:\mathbb{R}\rightarrow[0,1]$ is
nondecreasing and satisfies $\mathcal{S}(p)=1-s_{\mathrm{rn}}$ if $p\geq
p_{b}$ and $\mathcal{S}(p)\rightarrow s_{\mathrm{rw}}$ as
$p\rightarrow-\infty$, $s_{\mathrm{rw}}$ being the residual wetting
saturation, $s_{\mathrm{rn}}$ the residual non-wetting saturation and $p_{b}$
the entry pressure. More precisely, we have,
* •
for the Brooks-Corey model,
$\displaystyle
s=\mathcal{S}(p)=\begin{cases}s_{\mathrm{rw}}+(1-s_{\mathrm{rn}}-s_{\mathrm{rw}})\left(\frac{p}{p_{b}}\right)^{-n}&\text{if}\leavevmode\nobreak\
p\leq p_{b},\\\ 1-s_{\mathrm{rn}}&\text{if}\leavevmode\nobreak\
p>p_{b},\end{cases}$ $\displaystyle
k_{r}(s)=s_{\mathrm{eff}}^{3+\frac{2}{n}},\qquad\qquad
s_{\mathrm{eff}}=\frac{s-s_{\mathrm{rw}}}{1-s_{\mathrm{rn}}-s_{\mathrm{rw}}};$
* •
for the van Genuchten-Mualem model,
$\displaystyle
s=\mathcal{S}(p)=\begin{cases}s_{\mathrm{rw}}+(1-s_{\mathrm{rn}}-s_{\mathrm{rw}})\left[1+\left|\frac{\alpha}{\rho
g}p\right|^{n}\right]^{-m}&\text{if}\leavevmode\nobreak\ p\leq 0,\\\
1-s_{\mathrm{rn}}&\text{if}\leavevmode\nobreak\ p>0,\end{cases}$
$\displaystyle
k_{r}(s)=s_{\mathrm{eff}}^{\frac{1}{2}}\\{1-[1-s_{\mathrm{eff}}^{\frac{1}{m}}]^{m}\\}^{2},\qquad
s_{\mathrm{eff}}=\frac{s-s_{\mathrm{rw}}}{1-s_{\mathrm{rn}}-s_{\mathrm{rw}}},\qquad\qquad
m=1-\frac{1}{n};$
where $\eta(\cdot)=k_{r}(\cdot)/\mu$, $\mu=10^{-3}\leavevmode\nobreak\
\textrm{Pa}\cdot\textrm{s}$ being water viscosity, is the relative
permeability. The parameters used for both rock types are given in Table 1 for
the Brooks-Corey model and in Table 2 for the van Genuchten-Mualem model. With
these choices of parameters, water is more likely to be in RT1 than in RT0, in
the sense tha, at a given pressure, the water saturation is higher in RT1 than
in RT0, as it can be seen on the plots of the capillary-pressure functions
depicted in Figures 3–4 for these two petro-physical models. Figures 3–4 also
show the relative permeability functions. Note the non-Lipschitz character of
the relative permeability in the van Genuchten-Mualem framework. For the
numerical tests, in order to avoid infinite values for the derivative of
$k_{r}(s)$ when $s\rightarrow 1-s_{\mathrm{rn}}$, we approximate it for
$s\in[s_{lim},1-s_{\mathrm{rn}}]$ using a second degree polynomial
$\widetilde{k_{r}}(s)$. Such a polynomial satisfies the following constraints:
$k_{r}(s_{lim})=\widetilde{k_{r}}(s_{lim})$ and
$\widetilde{k_{r}}(1-s_{\mathrm{rn}})=1$. The value $s_{lim}$ corresponds to
$s_{\text{eff}}=0.998$.
| $1-s_{rn}$ | $s_{rw}$ | $p_{b}[\textrm{Pa}]$ | $n$ | $\lambda[\mathrm{m}^{2}]$ | $\phi$
---|---|---|---|---|---|---
RT0 | $1.0$ | $0.1$ | $-1.4708\cdot 10^{3}$ | $3.0$ | $10^{-11}$ | $0.35$
RT1 | $1.0$ | $0.2$ | $-3.4301\cdot 10^{3}$ | $1.5$ | $10^{-13}$ | $0.35$
Table 1: Parameters used for the Brooks-Corey model
Figure 3: Capillary pressure and relative permeability curves for the Brooks-Corey model | $1-s_{\rm rn}$ | $s_{\rm rw}$ | $n$ | $\lambda\leavevmode\nobreak\ [\mathrm{m}^{2}]$ | $\alpha\leavevmode\nobreak\ [\mathrm{m}^{-1}]$ | $\phi$
---|---|---|---|---|---|---
RT0 (Sand) | $1.0$ | $0.0782$ | $2.239$ | $6.3812\cdot 10^{-12}$ | $2.8$ | $0.3658$
RT1 (Clay) | $1.0$ | $0.2262$ | $1.3954$ | $1.5461\cdot 10^{-13}$ | $1.04$ | $0.4686$
Table 2: Parameters used for the van Genuchten-Mualem model
Figure 4: Capillary pressure and relative permeability curves for the van
Genuchten-Mualem model
### 6.1 Configurations of the test cases
For both petro-physical models, we consider two configurations further
referred as filling and drainage cases, which are described in the following.
#### 6.1.1 Filling case
The filling test case has already been considered in [36, 32, 39, 15].
Starting from an initially dry domain $\Omega$, whose layers’ composition is
reported in Figure 5, water flows from a part of the top boundary during one
day. A no-flow boundary condition is applied elsewhere. More precisely, the
initial capillary pressure is set to $-47.088\cdot 10^{5}\textrm{Pa}$ and the
water flux rate to $0.5\textrm{m/day}$ through
$\Gamma_{N}=\\{(x,y)\,|\,x\in[1\mathrm{m},4\mathrm{m}],y=0\mathrm{m}\\}$. For
this simulation an homogeneous time-step $\Delta t=1000\mathrm{s}$ is
prescribed for the test using the Brooks-Corey model and $\Delta
t=500\mathrm{s}$ for the one using the van Genuchten-Mualem model.
Figure 5: Boundary condition for the filling case
The test case follows the following dynamics. Water starts invading the void
porous space in $\Omega_{1}$. When it reaches the interface with $\Omega_{3}$,
capillarity involves a suction force on water from $\Omega_{1}$ to
$\Omega_{3}$. Since clay (RT1) has low permeability, water encounters
difficulties to progress within $\Omega_{3}$. This yields a front moving
downward in $\Omega_{1}$ which is stiffer for the Brooks-Corey model than for
the van Genuchten-Mualem one. In both cases, the simulation is stopped before
water reaches the bottom part corresponding to $\Omega_{2}$. In Figure 6 we
can observe the evolution of the saturation profile during the simulation
performed on a $50\times 30$ cells mesh with Brooks and Corey model, whereas
the evolution corresponding to van Genuchten-Mualem nonlinearities is depicted
in Figure 7.
Figure 6: Evolution of the saturation profile for $t\in\\{0s,20\cdot
10^{3}s,40\cdot 10^{3}s,60\cdot 10^{3}s,86\cdot 10^{3}s\\}$ for filling case,
using Brooks and Corey model, Method A and the $50\times 30$ cells mesh.
Figure 7: Evolution of the saturation profile for $t\in\\{0s,20\cdot
10^{3}s,40\cdot 10^{3}s,60\cdot 10^{3}s,86\cdot 10^{3}s\\}$ for filling case
using Van Genuchten model, Method A and the $50\times 30$ cells mesh.
#### 6.1.2 Drainage case
This test case is designed as a two-dimensional extension of a one-dimensional
test case proposed by [38] and addressed in [39, 15]. We simulate a vertical
drainage starting from initially and boundary saturated conditions during
$105\cdot 10^{4}\leavevmode\nobreak\ \mathrm{s}$. At the initial time, the
pressure varies with depth with $p^{0}(z)=-\rho gz$. A Dirichlet boundary
condition $p_{D}=0\leavevmode\nobreak\ \textrm{Pa}$ is imposed on the bottom
of the domain, more precisely on
$\Gamma_{D}=\\{(x,y)\,|\,x\in[0\mathrm{m},5\mathrm{m}],y=-3\mathrm{m}\\}$. The
layers’ composition of $\Omega$ is reported in Figure 8. For this simulation
an homogeneous time-step $\Delta t=2000\mathrm{s}$ is used for the test with
the Brooks-Corey model and $\Delta t=800\mathrm{s}$ for the one with the van
Genuchten-Mualem model.
Figure 8: Boundary condition for the drainage test
At the top interface between $\Omega_{1}$ and $\Omega_{3}$, capillarity acts
in opposition to gravity and to the evolution of the system into a dryer
configuration. The interface between $\Omega_{2}$ and $\Omega_{3}$ acts in the
reverse way: suction accelerates the gravity driven drainage of RT0.
In Figure 9 we can observe the evolution of the saturation profile during the
simulation performed on a $50\times 30$ cells mesh with Brooks and Corey
model, whereas the evolution corresponding to van Genuchten-Mualem
nonlinearities is depicted in Figure 10.
Figure 9: Evolution of the saturation profile for $t\in\\{0s,26.2\cdot
10^{4}s,52.4\cdot 10^{4}s,78.6\cdot 10^{4}s,105\cdot 10^{4}s\\}$ for drying
case, using Brooks and Corey model, Method A and the $50\times 30$ cells mesh.
Figure 10: Evolution of the saturation profile for $t\in\\{0s,26.16\cdot
10^{4}s,52.4\cdot 10^{4}s,78.56\cdot 10^{4}s,105\cdot 10^{4}s\\}$ for drying
case using Van Genuchten model, Method A and the $50\times 30$ cells mesh.
### 6.2 Comparisons of the numerical treatments of the interfaces
For each petro-physical model and configuration, a numerical convergence
analysis is carried out for the schemes with (method A) or without (method B)
thin cells, whose thickness is fixed to $\delta=10^{-6}\mathrm{m}$, at rock
type interfaces. Five structured meshes with the following resolutions are
considered for this analysis: $50\times 30$, $100\times 60$, $200\times 120$,
$400\times 240$, $800\times 480$. The evolution of the error is measured using
the $L^{2}([0,T],\Omega)$-norm of the relative difference between the
saturations obtained on a given mesh and a reference solution obtained with
Method A and the mesh $800\times 480$. The number of Newton iterations
obtained with both methods is also compared.
#### 6.2.1 Brooks-Corey model: drainage case
For the drainage case with the Brooks-Corey model, the convergence error is
given in Figure 11. First we notice that, for all meshes, the error is smaller
with method A than with method B and that we have a linear rate of convergence
with the first one whereas this rate is smaller with the latter one. The
total, average and maximal number of Newton iterations are also given in Table
3. Method A appears to be slightly more expensive.
50x30100x60200x120400x240$0.00316$$0.01$$0.0316$MeshL2 relative errorMethod B Method A order $0.7$order $1.11$ Figure 11: $L^{2}(Q_{T})$ relative error in saturation for the drainage case using Brooks and Corey model. | $\sharp$ total | $\sharp$ avg | $\sharp$ max
---|---|---|---
Method A | $2038$ | $3$ | $29$
Method B | $1927$ | $3$ | $29$
Table 3: Newton’s iterations for the mesh $200\times 120$ for the drainage
case using Brooks and Corey model.
Let us now evaluate the saturation absolute error between results obtained
with Method A and Method B. In Figure 12 we plot the absolute-error
distribution over the domain at three different times: when the cells line in
$\Omega_{1}$ above the interface between $\Omega_{1}$ and $\Omega_{3}$ starts
drying, when the cells line in $\Omega_{2}$ below the interface between
$\Omega_{3}$ and $\Omega_{2}$ starts drying and at final time.
Figure 12: Saturation absolute error between Method A and Method B for the
drainage case with Brooks and Corey model at $t\in\\{23\cdot 10^{4}s,53.2\cdot
10^{4}s,105\cdot 10^{4}s\\}$.
#### 6.2.2 Brooks-Corey model: filling case
For the filling case with the Brooks-Corey model, the convergence error is
given in Figure 13. As for the previous case, Method A enables to recover a
linear convergence rate. Except for the first two meshes where the error
obtained with Method A is slightly larger, for all other meshes, this error is
smaller than the one obtained with method B. The total, average and maximal
number of Newton iterations are given in Table 4. The algorithm behaves here
in the same way as before.
50x30100x60200x120400x240$0.0251$$0.0398$$0.0631$$0.1$$0.158$MeshL2 relative errorMethod B Method A order $0.3$order $0.8$ Figure 13: $L^{2}(Q_{T})$ relative error in saturation for the filling case using Brooks and Corey model. | $\sharp$ total | $\sharp$ avg | $\sharp$ max
---|---|---|---
Method A | $788$ | $9$ | $32$
Method B | $659$ | $7$ | $31$
Table 4: Newton’s iterations for the mesh $200\times 120$ for the filling case
using Brooks and Corey model.
Let us now evaluate the saturation absolute error between results obtained
with Method A and Method B. In Figure 14 we plot the absolute-error
distribution over the domain at three different times: when water crosses the
interface between $\Omega_{1}$ and $\Omega_{3}$, when cells around this
interface are almost saturated and at final time.
Figure 14: Saturation absolute error between Method A and Method B for the
filling case with Brooks and Corey model at $t\in\\{20\cdot 10^{3}s,30\cdot
10^{3}s,86\cdot 10^{3}s\\}$.
#### 6.2.3 Van Genuchten-Mualem model: filling case
For the filling case with the Van Genuchten model, the convergence error is
given in Figure 15. Both methods exhibit a linear rate of convergence. On the
other hand, the error is slightly larger with method A than with method B. The
total, average and maximal number of Newton iterations are given in Table 5.
50x30100x60200x120400x240$0.0251$$0.0398$$0.0631$$0.1$$0.158$MeshL2 relative errorMethod B Method A order $0.78$order $0.77$ Figure 15: $L^{2}(Q_{T})$ relative error in saturation for the filling case using the Van Genuchten model. | $\sharp$ total | $\sharp$ avg | $\sharp$ max
---|---|---|---
Method A | $959$ | $5$ | $15$
Method B | $782$ | $4$ | $15$
Table 5: Newton’s iterations for the mesh $200\times 120$ for the filling case
using the Van Genuchten model.
Figure 16 shows the localization of the difference for the numerical solutions
provided by the two methods A and B. Unsurprisingly, the difference is located
in the neighborhood of the interfaces. Moreover, as suggested by Figures 13
and 15, the influence of the introduction of additional interface unknowns
(method A) has a lower impact for van Genuchten-Mualem nonlinearities than for
Brook-Corey nonlinearities.
Figure 16: Saturation absolute error between Method A and Method B for the
filling case with the Van Genuchten model at $t\in\\{27.5\cdot 10^{3}s,45\cdot
10^{3}s,86\cdot 10^{3}s\\}$.
#### 6.2.4 Van Genuchten-Mualem model: drainage case
For the drainage case with the Van Genuchten model, the convergence error is
given in Figure 17. Both methods exhibit a linear rate of convergence.
Moreover, the error is slightly larger with method B than with method A. The
total, average and maximal number of Newton iterations are given in Table 6.
50x30100x60200x120400x240$0.001$$0.01$MeshL2 relative errorMethod B Method A order $1.27$order $1.05$ Figure 17: $L^{2}(Q_{T})$ relative error in saturation for the drainage case using the Van Genuchten model. | $\sharp$ total | $\sharp$ avg | $\sharp$ max
---|---|---|---
Method A | $3523$ | $2$ | $20$
Method B | $2845$ | $2$ | $29$
Table 6: Newton’s iterations for the mesh $200\times 120$ for the filling case
using the Van Genuchten model.
Let us now evaluate the saturation absolute error between results obtained
with Method A and Method B. In Figure 18 we plot the absolute-error
distribution over the domain at three different times: when the cells line in
$\Omega_{1}$ above the interface between $\Omega_{1}$ and $\Omega_{3}$ starts
drying, when the cells line in $\Omega_{2}$ below the interface between
$\Omega_{3}$ and $\Omega_{2}$ starts drying and at final time.
Figure 18: Saturation absolute error between Method A and Method B for the
drainage case with the van Genuchten-Mualem model at $t\in\\{1.12\cdot
10^{4}s,10.56\cdot 10^{4}s,105\cdot 10^{4}s\\}$.
#### 6.2.5 Influence of the parameter $\delta$
Let us now analyze how the thickness of the thin cells employed in Method A
affects the accuracy of the solution obtained with this method. We consider
the filling and drainage cases along with the Brooks and Corey model and
evaluate the relative $L^{2}(Q_{T})$ error between the solution obtained on
the $200\times 120$ cells mesh using
$\delta\in\\{10^{-2}m,10^{-4}m,10^{-6}m\\}$ with respect to the reference
solution obtained on the $800\times 480$ cells mesh using
$\delta_{ref}=10^{-6}m$. As shown in Figure 19, the value of $\delta$ does not
have a significant influence on the overall error as soon as $\delta$ is small
enough. We also observe a moderate influence on the robustness of the non-
linear solver for the values considered here.
1e-61e-41e-2$0.00316$$0.01$$0.0316$Thickness of thin cells, Method AL2
relative errorFilling casedrainage case Figure 19: $L^{2}(Q_{T})$ relative
error in saturation as a function of the thickness $\delta$ of the thin cells
with Method A using the $200\times 120$ cells mesh.
## 7 Conclusions and perspectives
This article aimed at proving that standard upstream mobility finite volume
schemes for variable saturated porous media flows still converge in highly
heterogeneous contexts without any specific treatment of the rock type
discontinuities. The scheme is indeed shown to satisfy some energy stability
which provides enough a priori estimates to carry out its numerical analysis.
First, the existence of a unique solution to the nonlinear system stemming
from the scheme is established thanks to a topological degree argument and
from the monotonicity of the scheme. Besides, a rigorous mathematical
convergence proof is conducted, based on compactness arguments. No error
estimate can then be deduced from our analysis.
Because of the choice of a backward Euler in time discretization and from the
upwind choice of the mobilities, a first order in time and space accuracy is
expected in the case of homogeneous computational domains. We show in
numerical experiments that without any particular treatment of the interfaces
at rock discontinuities, this first order accuracy can be lost, especially in
the case of Brooks-Corey nonlinearities. This motivates the introduction of a
specific treatment of the interfaces. The approach we propose here is based on
the introduction of additional unknowns located in fictitious small additional
cells on both sides of each interface. Even though the rigorous convergence
proof of this approach is not provided here in the multidimensional setting –
such a proof can for instance be done by writing the scheme with the specific
treatment of the interface (method A) as a perturbation of the scheme without
any particular treatment of the interface (method B)–, the numerical
experiments show that it allows to recover the first order accuracy without
having major impacts on the implementation and on the behavior of the
numerical solver.
For future researches, we suggest to test the so-called method A on a two-
phase flow test and to compare it to the approaches presented in [7].
Moreover, in a forthcoming work, we propose two other methods to really impose
the pressure continuity condition at interfaces. A comparison between all
methods will be shown.
## References
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|
# NB-IoT Random Access for Non-Terrestrial Networks: Preamble Detection and
Uplink Synchronization
Houcine Chougrani, Steven Kisseleff, , Wallace A. Martins, ,
and Symeon Chatzinotas H. Chougrani, S. Kisseleff, W.A. Martins, and S.
Chatzinotas are with the Interdisciplinary Centre for Security, Reliability
and Trust (SnT), University of Luxembourg, Luxembourg. W.A. Martins is also
with the Federal University of Rio de Janeiro (UFRJ), Brazil. E-mails:
{houcine.chougrani, steven.kisseleff, wallace.alvesmartins,
<EMAIL_ADDRESS>research was funded in whole by the Luxembourg
National Research Fund (FNR) in the frameworks of the FNR-IPBG INSTRUCT
project ”INSTRUCT: Integrated Satellite-Terrestrial Systems for Ubiquitous
Beyond 5G Communications” (Grant no. FNR/IPBG19/14016225) and of the FNR-CORE
project ”MegaLEO: Self-Organised Lower Earth Orbit Mega-Constellations” (Grant
no. C20/IS/14767486). For the purpose of open access, the authors have applied
a Creative Commons Attribution 4.0 International (CC BY 4.0) license to any
Author Accepted Manuscript version arising from this submission.
###### Abstract
The satellite component is recognized as a promising solution to complement
and extend the coverage of future Internet of things (IoT) terrestrial
networks (TNs). In this context, a study item to integrate satellites into
narrowband-IoT (NB-IoT) systems has been approved within the 3rd Generation
Partnership Project (3GPP) standardization body. However, as NB-IoT systems
were initially conceived for TNs, their basic design principles and operation
might require some key modifications when incorporating the satellite
component. These changes in NB-IoT systems, therefore, need to be carefully
implemented in order to guarantee a seamless integration of both TN and non-
terrestrial network (NTN) for a global coverage. This paper addresses this
adaptation for the random access (RA) step in NB-IoT systems, which is in fact
the most challenging aspect in the NTN context, for it deals with multi-user
time-frequency synchronization and timing advance for data scheduling. In
particular, we propose an RA technique which is robust to typical satellite
channel impairments, including long delays, significant Doppler effects, and
wide beams, without requiring any modification to the current NB-IoT RA
waveform. Performance evaluations demonstrate the proposal’s capability of
addressing different NTN configurations recently defined by 3GPP for the 5G
new radio system.
###### Index Terms:
NB-IoT, 3GPP, Random Access, 5G, NTN, Satellite.
## I Introduction
In recent years, a high connectivity demand has started to be experienced in
wireless communications. Virtually everyone and everything need to be
connected. This universal connectivity poses serious challenges to terrestrial
radio coverage. In fact, the geography of the current uncovered areas is far
more diverse than the covered ones, with fjords, mountains, islands, ice,
deserts, and vast distances, leading to stringent constraints in terms of
backhauling, power availability, site construction and maintenance [1]. In
this context, non-terrestrial networks (NTNs) are a promising solution to
complement terrestrial networks (TNs) for global coverage extension [2]. An
NTN refers to a network, or segment of networks, using radio frequency (RF)
resources on board of a satellite or unmanned aerial system (UAS) platform.
The primary role of an NTN in this context is to complement the TN services in
under-served areas, to improve the TN service reliability, especially for
mission-critical services, and to enable the network scalability by means of
efficient multicast/broadcast resources for data delivery [2, 3].
In 2017, the 3rd Generation Partnership Project (3GPP) started to study the
integration of satellites into the 5G ecosystem [4]. The study was completed
in 2019 during the corresponding 3GPP November meeting with the technical
reports TR 38.821 [5] and TR 38.811 [6]. Then, a new normative working item
(WI) for 5G new radio (NR) in NTN was approved in 3GPP Release 17 [7].
Recently, a new study WI for narrowband Internet of things (NB-IoT) support in
NTN has also been approved in 3GPP Release 17 [8].
NB-IoT is a recent cellular technology standardized by 3GPP that aims to
provide improved coverage for a massive number of low-throughput low-cost
devices with low device power consumption in delay-tolerant applications.
Prospective applications include utility metering, environment monitoring,
asset tracking, municipal light, and waste management, to name but a few [9].
However, in many cases these devices are deployed in remote areas which are
out of reach of the TN. By integrating NB-IoT into an NTN, 3GPP aims to
provide a standardized solution enabling global IoT operation anywhere on
Earth. In this direction, numerous companies (e.g. Eutelsat and MediaTek) and
start-ups (e.g. OQTechnology, Sateliot, and Kepler Communication Inc.) have
already started the integration of NB-IoT into NTNs via satellite. However,
the current NB-IoT protocol [10, 11] was initially designed for TNs and does
not account for satellite constraints, such as larger propagation delays,
stronger Doppler effects (in case of non-geostationary satellites), limited
on-board power availability, etc.
In order to adopt NB-IoT in satellite systems, some modifications to the NB-
IoT protocol are therefore required. In this context, and from the physical
layer (PHY) perspective, one of the major challenges is the _random access_
(RA) which manages the uplink synchronization and requests for scheduling data
transmissions. In fact, the current NB-IoT RA preamble namely _Narrowband
Physical Random Access Channel_ (NPRACH) cannot support the satellite channel
impairments (e.g. strong Doppler, long delays) which are more severe compared
to those faced by TNs.
The above problem is a new challenge to be faced at PHY level. The most
relevant related work in this area is by MediaTek Inc. [12]. The authors have
considered a user equipment (UE) without built-in global navigation satellite
system (GNSS) capabilities and proposed a new solution to tackle the
relatively large residual frequency offset during the RA reception with
relatively mild changes to the existing NB-IoT RA preamble. The proposed
solution relies on the introduction of a fractional frequency-hopping sequence
per preamble to remove the ambiguity among UEs in presence of large frequency
offsets. The authors, however, have not explicitly pointed out the solution
for managing the relatively large differential delay that may impact the
orthogonality of UEs in the network. Another work in [13] has analyzed the RA
procedure and proposed some system level recommendations, such as reducing the
beam width (i.e. cell) in order to handle the large differential delay and
residual frequency offset of UEs within this beam using the current preamble,
and/or the proposal of a new preamble with longer cyclic prefix (CP) and
larger bandwidth to cope with differential delay and frequency offsets. The
first recommendation (i.e. small cells) is not straightforward, for it
requires a complex antenna technology that may lead to increased size and
power consumption of the low Earth orbit (LEO) satellite. The second solution
requires considerable changes in the standard, which is something that both
3GPP and stakeholders try to avoid for flexibility, time-to-market, and
backward compatibility reasons.
In this work, we address the aforementioned issues without any modification to
the current NB-IoT RA preamble. The proposed method consists of the
combination of a system level solution that overcomes the problem of a large
residual frequency offset with a signal processing solution that tackles the
large differential delay. In order to reduce the residual frequency offset, we
take advantage of the initial downlink synchronization. More specifically, the
UE uses the estimated frequency offset during the downlink to pre-compensate
the uplink RA transmission. For the differential delay problem, the inherent
$2\pi$-ambiguity problem is solved using a discrimination criterion based on
the estimation of Doppler rate.111The time-varying Doppler shift. In summary,
the main contributions of this paper are:
1. 1.
A detailed mathematical analysis including all satellite impairments, i.e.
delay, frequency offset, and Doppler rate, is presented with an efficient RA
receiver design for integrating NB-IoT into NTN via LEO satellites. The
designed receiver supports a coverage up to five times the original maximum
coverage in TN, and does not require any modifications to the current preamble
waveform, which makes it a seamless solution when integrating the satellite
into the NB-IoT system.
2. 2.
The Doppler rate corresponding to each UE is estimated within the satellite
coverage. This allows for the NTN system to take this information into account
for UE data scheduling, which is very useful for the system to preserve uplink
multiple access orthogonality.
3. 3.
PHY-based simulations and performance evaluations in terms of preamble
detection probability as well as Doppler and time-of-arrival (ToA) estimation
accuracy are provided under realistic scenarios recently defined by 3GPP in
the context of NTN.
The remainder of the paper is organized as follows. In Section II, a
background on the 3GPP-based solutions and recommendations to integrate
satellites into a 5G system is provided. Section III contains a system
overview, including the system parameters, link budget, LEO channel
impairments, and the NB-IoT RA preamble. In Section IV, the preamble detection
and ToA estimation method for TN is briefly described. Then, a novel method
for preamble detection and ToA estimation in NTN is proposed in Section V. A
comprehensive set of simulation results is provided and discussed in Section
VI. The concluding remarks are drawn in Section VII.
## II 3GPP-based satellite integration into 5G
NB-IoT is a recent cellular technology standardized by 3GPP that inherits from
the existing long term evolution (LTE) technology. The radio access is based
on orthogonal frequency-division multiple access (OFDMA) for downlink and
single-carrier frequency-division multiple access (SC-FDMA) for uplink with
180-kHz bandwidth.
As we pointed out in the previous section, the adoption of NB-IoT in satellite
systems requires some modifications either to the system and/or to the
protocols. However, since NB-IoT is a 3GPP-based protocol, the recent 3GPP
recommendations in [5, 6] to integrate 5G-NR into NTN could be adopted for NB-
IoT. Those recommendations are based on the type of UE in the NTN system, as
follows [5, 6]:
* •
_UE with GNSS capability_ : the UE performs the pre-compensation of Doppler
shift222Shift of the signals’ frequency content due to the motion of either
the receiver, the transmitter, or both of them. and delay propagation. This is
possible thanks to the knowledge of satellite _ephemeris_ and available UE
location. Depending on the prediction accuracy, additional delay and frequency
offsets are assumed to be handled by the protocol [6].
* •
_UE without GNSS capability_ : the satellite performs the pre-compensation of
Doppler shift at the center of the beam on the ground and broadcasts the
_common delay_ 333This common delay corresponds to the propagation time
between the next generation node B (gNB) and the closest point on Earth in the
satellite beam coverage. to all UEs inside the concerned beam to be taken into
account for the uplink transmission, as illustrated in Fig. 1. In this case,
the additional delay, called _differential delay (DD)_ in 3GPP terminology,
and frequency offset in the beam (e.g. UE1-DD and $f_{\rm off/UE_{1}}$ in Fig.
1) are assumed to be handled by the protocol [7].
Figure 1: Doppler shift pre-compensation mechanism and common and differential
delay illustration for NTN.
In this context, one of the major challenges encountered in the integration of
5G-NR into NTN from a PHY perspective is RA. In 5G-NR as well as in NB-IoT, RA
manages the uplink synchronization and requests for scheduling data
transmissions. The uplink synchronization means that the base station (BS) has
both to detect (and identify) all active UEs in its coverage area and to
estimate their round-trip delays (RTDs). Through this, the propagation delay
between each UE and the BS is acquired/estimated, which enables attaining a
common timing reference. The acquired delay allows the BS to perform timing
advance needed to keep the orthogonality among multiple UEs, which is
typically required in SC-FDMA systems. The estimation of RTD refers to a ToA
estimation, whereas the user detection refers to NPRACH preamble detection. It
is worth noting that this operation is essential for a successful system
operation. In fact, RA is the first phase of system operation and covers the
first messages from each UE to the BS. Hence, a wrong detection and/or an
erroneous ToA estimation would lead to increased system latency, overall
performance degradation, and even service outage in extreme cases.
Regarding 5G-NR RA, it was concluded in [5] that, in the case of UEs with GNSS
capability (assumption for pre-compensation of timing and frequency offset at
the UE side based on GNSS and _ephemeris_ information), existing RA preambles
can be reused if the knowledge of UE’s geo-location is available and meets the
prescribed accuracy level. It was also stated that additional enhancements,
such as repetitions and/or larger subcarrier spacing (SCS), are necessary to
ensure uplink coverage. In contrast, for UEs without GNSS capabilities, [5]
concluded that enhanced/new RA preambles should be proposed. This paper
addresses the same problem but for NB-IoT integration into NTNs considering
LEO satellites. More specifically, the paper provides a solution to integrate
the NB-IoT NPRACH into the NTN context. We focus here on UEs without GNSS
capabilities, since the applicability of GNSS solution for NB-IoT in NTN is
questionable. In fact, the GNSS solution poses additional challenges, such as:
* •
unclear update procedure for the satellite _ephemeris_ at the UE level.
* •
reduced battery lifetime for the UEs due to the frequent estimation of Doppler
and delay along with their compensation, thus requiring substantial additional
signal processing.
Furthermore, 3GPP targets both cases, i.e. UE with and without GNSS
capability, which eventually means that the GNSS solution should not be viewed
as the key strategy.
## III System model
In this section, we describe the system model adopted in this work. The
description includes the system architecture, the link budget, the satellite
channel impairments, and the NPRACH preamble. For clarity’s sake, Table I
summarizes the mathematical symbols used in this section.
TABLE I: Mathematical symbols used in the system model Symbol | Definition
---|---
$G/T$ | Antenna-gain-to-noise-temperature
EIRP | Effective isotopic radiated power
$k_{\rm B}$ | Boltzmann constant
$PL_{\rm FS}$ | Free space propagation loss
${\rm PL}_{\rm A}$ | Atmospheric gas losses
${\rm PL}_{\rm S}$ | Shadowing margin
${\rm PL}_{\rm AD}$ | Additional losses due to the scintillation phenomena
${\rm BW}_{\text{NB-IoT}}$ | NB-IoT channel bandwidth
${\rm BW}_{\mathrm{NPRACH}}$ | NPRACH bandwidth
$g$ | Ratio between NB-IoT channel and NPRACH bandwidths
CNR | Carrier-to-noise ratio
SNR | Signal-to-noise ratio
TCP | CP length in ms
L | Number of identical symbols within one symbol group
TSEQ | Length of identical symbols within one symbol group in ms
P | Number of symbol groups in one preamble basic unit
$\Delta f$ | Subcarrier spacing between NB-IoT tones/SGs
$N_{\rm rep}$ | Number of preamble basic unit repetition
### III-A System architecture
In the 3GPP framework, six reference scenarios have been specified depending
on the orbit nature—whether geosynchronous Earth orbit (GEO) or LEO—, payload
type—whether transparent or regenerative—, and beam type—whether steerable or
fixed [5, 6]. We consider in this work the scenario referred to as _D2_ in [5]
among the NTN reference scenarios from 3GPP. TABLE II provides the satellite
parameters for this scenario. Note that 3GPP has defined two sets of beam
layout and RF parameters of the payload, namely: Set-1 and Set-2. Both
parameter sets are taken into account in this work. The satellite is placed at
600-km altitude (LEO) with a regenerative payload, which means that the
satellite implements fully or partially the functionalities of the classical
base station (i.e. gNB). The system is operating in S-band at a carrier
frequency of 2 GHz. The satellite generates several spot-beams over its
coverage area to increase its capacity. These spot-beams are moving along with
the satellite. Each spot-beam size, however, depends on the elevation angle
and the satellite configuration, whether it is Set-1 or Set-2 (depending on
the 3-dB beam width angular value). Moreover, antenna-gain-to-noise-
temperature ($G/T$) in Set-1 is greater than in Set-2, which comes as a
byproduct of the reduction of the equivalent satellite antenna aperture. For
more details related to the payload configuration and beam layout aspect, we
refer the reader to [14, 15, 5], where a thorough discussion can be found.
TABLE II: Satellite parameters for UL transmissions Parameters | Set-1 | Set-2
---|---|---
Altitude | 600 km | 600 km
Payload | Regenerative | Regenerative
Operating band | S-band (2 GHz) | S-band (2 GHz)
Moving beams | Yes | Yes
Satellite antenna aperture | 2 m | 1 m
$G/T$ | 1.1 dB$\cdot$K-1 | -4.9 dB$\cdot$K-1
3 dB beam width | $\approx$ 4.4127∘ | $\approx$ 8.8320∘
### III-B Link budget
In this subsection, we calculate the link budget of the aforementioned system
architecture for the RA in UL transmissions. We consider a minimum elevation
angle of 30 degrees, which represents a typical elevation angle in satellite-
based systems [6]. The general formula for the link budget is derived from
[16]. It accounts for all the gains and losses in the propagation medium from
transmitter to receiver. Accordingly, the carrier-to-noise ratio (CNR) can be
computed as follows:
$\displaystyle\mathrm{CNR[dB]}=$
$\displaystyle\mathrm{EIRP[dBW]}+G/T\mathrm{[dB]}-k_{\rm B}{\rm[dBW/K/Hz]}$
$\displaystyle-{\rm PL}_{\rm FS}{\rm[dB]}-{\rm PL}_{\rm A}{\rm[dB]}-{\rm
PL}_{\rm S}{\rm[dB]}$ $\displaystyle-{\rm PL}_{\rm
AD}{\rm[dB]}-10\log_{10}({\rm BW}_{\text{NB-IoT}}{\rm[Hz]}),$ (1)
where $\mathrm{EIRP}$ is the effective isotopic radiated power of the
transmitting antenna, $k_{\rm B}=-228.6$ dBW/K/Hz is the Boltzmann constant,
${\rm PL}_{\rm FS}$ represents the free space propagation loss, ${\rm PL}_{\rm
A}$ corresponds to atmospheric gas losses, ${\rm PL}_{\rm S}$ is a shadowing
margin, ${\rm PL}_{\rm AD}$ denotes some underlying additional losses due to
the scintillation phenomena, and ${\rm BW}_{\text{NB-IoT}}$ is the NB-IoT
channel bandwidth. TABLE III illustrates the link budget for both payload
configurations (i.e. Set-1 and Set-2) in the UL. The values in the table are
similar to the ones suggested in [5, 16]. To obtain the signal-to-noise ratio
(SNR), the NPRACH bandwidth (i.e. ${\rm BW}_{\mathrm{NPRACH}}$) of 3.75 kHz is
considered. This leads to a gain factor of $g=48$ (i.e. $\frac{{\rm
BW}_{\text{NB-IoT}}}{{\rm BW}_{\mathrm{NPRACH}}}$). Following the
recommentation in [5], 6-dB degradation is introduced in the SNR calculation
as an additional margin. Hence, the SNR is given by:
$\displaystyle\mathrm{SNR[dB]}={\rm CNR[dB]}+10\log_{10}(g)-6.$ (2)
TABLE III: Link budget for NPRACH Transmission mode | UL Set-1 | UL Set-2
---|---|---
UE elevation angle | 30∘ | 90∘ | 30∘ | 90∘
Frequency [GHz] | 2 | 2 | 2 | 2
TX EIRP [dBm] | 23.01 | 23.01 | 23.01 | 23.01
RX $G/T$ [dB/K] | 1.10 | 1.10 | -4.9 | -4.9
BW${}_{\text{NB-IoT}}$ [kHz] | 180 | 180 | 180 | 180
Free space path loss [dB] | 159.10 | 154.03 | 159.10 | 154.03
Atmospheric loss [dB] | 0.07 | 0.07 | 0.07 | 0.07
Shadowing margin [dB] | 3.00 | 3.00 | 3.00 | 3.00
Scintillation loss [dB] | 2.20 | 2.20 | 2.20 | 2.20
Polarization loss [dB] | 0.00 | 0.00 | 0.00 | 0.00
Additional losses [dB] | 0.00 | 0.00 | 0.00 | 0.00
CNR [dB] | 5.79 | 10.85 | -0.21 | 4.86
BWNPRACH [kHz] | 3.75 | 3.75 | 3.75 | 3.75
Additional margin [dB] | -6 | -6 | -6 | -6
SNR [dB] | 16.60 | 21.66 | 10.60 | 15.67
### III-C Satellite channel impairments
Figure 2: Beam layout for LEO satellite at 600 km.
#### III-C1 Long propagation delay
In satellite systems, the propagation delays are long when compared to TN’s
delays due to much longer distances between the UE and the satellite. In fact,
the propagation delay depends on the satellite orbit, type of payload, and
elevation angle. In our scenario (i.e. LEO at 600 km with regenerative
payload), the minimum and maximum RTDs at 30∘ and 90∘ are 7.17 ms and 4 ms,
respectively. Note that the RTD is approximated as twice the propagation delay
between the satellite and the UE. Adopting the recommendations of [5]
regarding the RTD—i.e., broadcasting the common RTD (C-RTD) to all UEs within
each beam—, the maximum differential RTD (MD-RTD) will depend on the beam
size, which itself depends on the elevation angle, as shown in Fig. 2. TABLE
IV provides the MD-RTD values at different elevation angles (after taking into
account the C-RTD) for both types of payload configurations, viz. Set-1 and
Set-2. At higher elevation angles, the beam is small (e.g. 46-km and 92-km
beam diameters at 90∘ for Set-1 and Set-2, respectively), whereas at lower
elevation the beam becomes larger (e.g. 144-km and 257-km beam diameters at
30∘ for Set-1 and Set-2, respectively). For Set-1 configuration, the MD-RTD
varies from 3.25 $\mu$s to 807.13 $\mu$s, whereas for Set-2 configuration, it
starts from 13.04 $\mu$s and reaches up to 1.40 ms. As it will be discussed
later, this is a problematic situation for the RA in NB-IoT.
TABLE IV: MD-RTD after common delay compensation Elevation | Set-1 | Set-2
---|---|---
Beam diameter [km] | MD-RTD [$\mu$s] | Beam diameter [km] | MD-RTD [$\mu$s]
30∘ | 144.02 | 807.13 | 256.91 | 1397.23
45∘ | 82.10 | 369.53 | 153.63 | 658.85
90∘ | 46.23 | 3.25 | 92.70 | 13.04
#### III-C2 Significant Doppler effects
Figure 3: Doppler variation depending on the elevation angle.
The high speed motion of LEO satellites introduces significant Doppler shifts
on the NB-IoT signal as compared to those expected in TN systems. In our
scenario, up to $\pm 41$-kHz Doppler shift (see Fig. 3) is observed (with 30∘
minimum elevation angle). Furthermore, the received signals are impacted by a
significant Doppler rate (see Fig. 3) that varies between $-101$ Hz/s (for 30∘
elevation) and $-594$ Hz/s (for 90∘ elevation), thus leading to additional
degradation of the detection performance in case of longer preambles/data
packets. Using the Doppler shift pre-compensation technique as defined in [5],
the maximum residual frequency offsets (normalized by the carrier frequency)
at the satellite receiver for RA are shown in TABLE V. This maximum residual
frequency offset is twice the offset of one-way propagation from satellite to
UE. Note that in NB-IoT, the UE motion is negligible, and the residual
frequency offsets account only for satellite motion and the precision of local
oscillators (LOs).
TABLE V: Residual frequency offset after Doppler pre-compensation Elevation | Set-1 | Set-2
---|---|---
Beam diameter [km] | Max. residual freq. offsets [ppm] | Beam diameter [km] | Max. residual freq. offsets [ppm]
30∘ | 144.02 | $\pm$ 1.28 | 256.91 | $\pm$ 2.68
45∘ | 58.15 | $\pm$ 1.54 | 153.63 | $\pm$ 3.24
90∘ | 46.23 | $\pm$ 1.94 | 92.70 | $\pm$ 3.88
### III-D NB-IoT random access
TABLE VI: NPRACH preamble formats for FDD mode format | $\Delta f$ | P | L | TCP | TSEQ
---|---|---|---|---|---
0 | 3.75 kHz | 4 | 5 | 66.67 $\mu$s | 1.33 ms
1 | 3.75 kHz | 4 | 5 | 266.67 $\mu$s | 1.33 ms
2 | 1.25 kHz | 6 | 3 | 800 $\mu$s | 2.4 ms
Figure 4: NPRACH preamble format 1 structure.
The RA preamble in NB-IoT, known as NPRACH preamble, was originally proposed
in [17, 18, 19, 20] and then adopted by 3GPP and integrated into the NB-IoT
Release 13 [21], followed by an enhancement in Release 15 [22, 11]. In the
latter, a new preamble format is introduced for range enhancement and time-
division multiplexing (TDD). Note that, in Releases 13 and 14, only frequency-
division multiplexing (FDD) mode was supported. It is worth noting here that
our work focuses only on FDD since it is the preferable mode in NTN, as
described in [5]. This is because TDD mode requires a guard time that directly
depends on the propagation delay between UE and satellite to prevent UE from
transmitting and receiving simultaneously. However, such a guard time might be
excessive in NTN and would lead to a very inefficient radio interface.
In FDD mode, there are three NPRACH formats, namely: format 0, format 1, and
format 2, as shown in TABLE VI. The NPRACH formats with different CP, symbol
group sizes, and symbol group repetitions have been designed according to the
targeted cell sizes. The preamble consists of a set of _symbol groups_ (SGs).
An SG consists of a CP of length $T_{\rm CP}$ and a sequence of $L$ identical
symbols with total length $T_{\rm SEQ}$. In the 3GPP standard, $P$ SGs are
treated as the basic unit of the preamble. The basic unit can be repeated up
to $N_{\rm rep}=2^{j},j\in\\{0,1,\ldots,7\\}$ times for coverage extension.
Accordingly, the length of a preamble equals $P\cdot N_{\rm rep}$ SGs. The
NPRACH transmission supports either a 3.75-kHz or a 1.25-kHz subcarrier
spacing (SCS, i.e. $\Delta f$).
The hopping pattern is fixed within the basic unit of $P$ SGs. Symbol groups
in preamble format 0 and 1 (with 3.75-kHz SCS) hop by one or six subcarriers
in frequency, whereas symbol groups in format 2 (with 1.25-kHz SCS) hop by
one, three, or eighteen subcarriers in frequency [22]. Fig. 4 shows an example
of preamble format 1 with two preamble basic unit repetitions (i.e. $N_{\rm
rep}=2$). Note that when repetitions are configured, the hopping between the
basic units is no longer fixed, but it follows a pseudo-random selection
procedure defined in [22].
Currently, there are three possible CP lengths. CP lengths of 66.7 $\mu$s (for
format 0), 266.67 $\mu$s (for format 1), and 800 $\mu$s (for format 2) are
designed to support cell radius of up to 10 km, 40 km, and 120 km,
respectively. In terms of propagation delay, the three formats account for
maximum RTDs of 66.67 $\mu$s, 266.67 $\mu$s, and 800$\mu$s, respectively.
## IV Preamble detection and ToA estimation in TN
In this section, we provide a brief review of our recently proposed method
[23] for NPRACH detection and ToA estimation in TN. This is an essential part
to understand the proposed solution in this paper. Note that in NB-IoT TN
system, the Doppler is negligible due to very low mobility conditions. For
clarity’s sake, we employ the same notation and the same variable definitions
as provided in the 3GPP standard [22] and in our previous work [23] in the
rest of this paper. These mathematical notations are summarized in Appendix
Appendix A Mathematical symbols used in Preamble Detection and ToA Estimation
in both TN and NTN.
### IV-A Signal model
Based on [23], the transmitted NPRACH baseband signal can be written as
$s_{m,i}[n]=\sum_{k}{S_{m,i}[k]}{\rm e}^{{\rm j}2\pi\frac{k}{N}n},$ (3)
where $s_{m,i}[n]$ is the $n$-th sample of the time-domain waveform of $i$-th
symbol in the $m$-th SG, whereas $S_{m,i}[k]$ denotes the $i$-th symbol on the
$k$-th subcarrier during the $m$-th SG; in this case, $S_{m,i}[k]=1$ for all
$m,~{}i$. Furthermore, the sample index $n$ belongs to the set
$\\{N_{m,i}-N_{\rm CP},...,N_{m,i}+N-1\\}$, in which $i\in\\{0,...,L-1\\}$,
with $L$ being the number of symbols in one SG, and $N_{m,i}=mN_{g}+iN$, with
$N_{g}=N_{\rm CP}+LN$ being the size of one SG, $N_{\rm CP}$ denoting the CP
size, and $N$ denoting the size of a symbol.
It was shown in [23], that the received signal after OFDM demodulation (i.e.
CP removal and discrete Fourier transform, DFT, application) assuming a
negligible inter-carrier interference (ICI) can be expressed as
$\displaystyle{Y_{m,i}}=\,$ $\displaystyle h_{m}{\rm e}^{{\rm j}2\pi f_{\rm
off}(mN_{g}+iN-D)}$ $\displaystyle\times{\rm e}^{-{\rm j}2\pi n_{\rm SC}^{\rm
RA}(m)\frac{D}{N}}\left(\frac{1-{\rm e}^{{\rm j}2\pi f_{\rm off}N}}{1-{\rm
e}^{{\rm j}2\pi f_{\rm off}}}\right)+W_{m,i}\,,$ (4)
where $f_{\rm off}$ is the carrier frequency offset (CFO) normalized by the
sampling frequency, $D$ is the RTD normalized by the symbol duration, $h_{m}$
is the channel coefficient for the $m$-th SG, $n_{\rm SC}^{\rm RA}(m)$ is the
subcarrier occupied by the $m$-th SG, and $W_{m,i}$ is the noise term.
Combining the symbols within the same $m$-th SG, we get the so-called SG-sum
(SG-S) as follows:
$\displaystyle Y_{m}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{L-1}{{{Y_{m,i}}}}=h_{m}{\rm e}^{{\rm j}2\pi f_{\rm
off}(mN_{g}-D)}$ (5) $\displaystyle\times$ $\displaystyle{\rm e}^{-{\rm j}2\pi
n_{\rm SC}^{\rm RA}(m)\frac{D}{N}}\left(\frac{1-{\rm e}^{{\rm j}2\pi f_{\rm
off}LN}}{1-{\rm e}^{{\rm j}2\pi f_{\rm off}}}\right)~{}{+~{}W_{m}}{.}$
### IV-B Preamble detection and ToA estimation
For the clarity of exposition, the noise term and some constant factors are
omitted here. For more details, we refer the reader to [23]. The process to
detect the preamble and to estimate the ToA is as follows:
1. 1.
Perform differential processing by multiplying the $m$-th SG-S with the
complex conjugated $(m+1)$-th SG-S. Defining
$\Delta(m)=n_{SC}^{RA}(m+1)-n_{\rm SC}^{\rm RA}(m)$ as the hopping step
between the $m$-th and $(m+1)$-th SGs, the differential processing gives
$\displaystyle Z_{m,1}$ $\displaystyle=$
$\displaystyle{Y_{m}Y_{m+1}^{*}}\propto{\rm e}^{-{\rm j}2\pi f_{\rm
off}N_{g}}{\rm e}^{{\rm j}2\pi\Delta(m)\frac{D}{N}}$ (6)
2. 2.
Construct an array $v$ such that the position of $Z_{m,1}$ in $v$ corresponds
to the hopping step $\Delta(m)$. If there are multiple $Z_{m,1}$ with equal
value of $\Delta(m)$, i.e. $\Delta(m_{1})=\Delta(m_{2})$ with $m_{1}\neq
m_{2}$, then their values are first summed up before being inserted in $v$.
3. 3.
Perform one dimensional DFT (1D-DFT) with $N_{\rm DFT}$ points on $v[n]$ to
get
$\begin{split}V[k]&=\sum_{n=0}^{N_{\rm DFT}-1}{v[n]}{\rm e}^{-{\rm j}2\pi
k\frac{n}{N_{\rm DFT}}},\end{split}$ (7)
4. 4.
Combine the results non-coherently over $N_{\rm RX}$ receive antennas to get
$\begin{split}X[k]&=\sum_{N_{\rm RX}}\lvert{V[k]}\rvert^{2}.\end{split}$ (8)
5. 5.
Determine $k_{\max}={\rm argmax}_{k}\\{X[k]\\}$ and define
$X_{\max}=X[k_{\max}]$.
6. 6.
Compare $X_{\max}$ to a predefined threshold (cf. [23]). If $X_{\max}$ is
greater than or equal to the threshold the preamble presence is declared,
otherwise it is considered absent.
7. 7.
If the preamble is declared as present, the ToA is estimated as
$\hat{D}=\frac{k_{\max}}{N_{\rm DFT}\Delta f}.$ (9)
Note that the most interesting property of the above method is its
insensitivity to CFO when estimating ToA. This is due to the perfect
elimination of the CFO effect in step 4). Indeed, the term ${\rm e}^{-{\rm
j}2\pi f_{\rm off}N_{g}}$ is common for all $Z_{m,1}$ symbols (i.e. common for
all $v[n]$) and affects only the phase of $v[n]$, not its magnitude.
Similarly, since the DFT operation is linear, this factor affects only the
phase of $V[k]$, not its magnitude, such that by taking the absolute of
$V[k]$, the impact of CFO is completely eliminated [23].
## V Preamble detection and ToA estimation in NTN
In this section, we first provide the solution to handle high frequency
offsets observed in NTN-LEO-based systems. Then, a detailed mathematical
analysis is provided before describing the solution for preamble detection and
ToA estimation. Before describing the proposed method, we notice here that in
NTN systems, UEs may experience multiple visible satellites, leading to inter-
beam interference. Although this issue concerns the whole NTN operation system
including the actual data transmissions (not only the preamble detection), we
believe that this issue can be tackled using the classical color schemes,
usually employed in practical satellite communication where different carriers
are configured for the adjacent satellites/beams. Through this the inter-beam
interference will be reduced to a minimum, which would have a negligible
impact on the performance of the proposed method.
### V-A Solution for residual frequency offset
We described in Section II the 3GPP solution to reduce large Doppler shifts
due to LEO satellites’ motions. However, the residual frequency offset seen by
the satellite (in uplink) is still high and it can even exceed the largest SCS
in NPRACH, i.e. 3.75 kHz (see TABLE V). This frequency misalignment causes ICI
over the subcarriers and also a mismatch between the sampling frequencies used
at transmission and reception. To tackle this issue, we propose a system level
solution to reduce the overall frequency offsets seen by the satellite to a
value that could be handled by the RA preamble. Accordingly, in addition to
the Doppler pre-compensation at the satellite, we propose to pre-compensate
the residual frequency offset at the UE level using the initial downlink
frequency synchronization (known as _initial cell-search procedure_). Fig. 5
illustrates the different steps of the proposed system level solution. The
general idea is that the UE estimates the frequency offset in the downlink
synchronization process. Hence, it can shift the central frequency to the
opposite side of the _estimated frequency offset_ for uplink transmission.
Figure 5: Residual CFO reduction scheme.
The process is done as follows:
* •
UE acquires a downlink frequency synchronization during initial cell-search
procedure, which is not perfectly accurate. In practice, the frequency offset
seen by the UE at this level consists of 3 components. The first one is the
residual Doppler shift that depends on the UE location within the beam. The
second and third components are frequency offsets introduced by imperfect
accuracy of both satellite and UE LOs.
* •
UE tracks frequency after the initial downlink synchronization due to the high
Doppler rate (e.g. during the demodulation of narrowband physical broadcast
channel).
* •
UE updates the initial estimated frequency offset considering the tracking
information just before the UL transmission.
* •
UE pre-compensates the calculated frequency offset in the previous step and
sends its RA signal.
At the end, the residual frequency offset seen by the satellite receiver will
depend on the performance of downlink frequency synchronization, frequency
tracking, and LO stability (LO frequency drift in time). To the best of our
knowledge, there is no work providing the downlink synchronization performance
for NB-IoT under a LEO satellite scenario. However, if we can consider the
downlink synchronization results for 5G-NR reported recently in [24], the
residual frequency offset after downlink frequency synchronization was about
600 Hz (at 99% of cases). In such an NTN scenario, note that preamble format 2
is not suitable because of its small SCS (i.e. only 1.25 kHz) which makes it
too sensitive to a frequency offset. A CFO of 600 Hz represents about half of
its SCS. For this reason, we adopt in this work the preamble format 1 since it
has the largest SCS (i.e. 3.75 kHz) associated with the longest CP (i.e. 267
$\mu$s). However, if future NB-IoT UEs guarantees lower frequency offsets,
preamble format 2 can be adopted with our proposed method. It is worth
pointing out that the UE hardware does not require any modification, if the
Doppler pre-compensation can be done at the satellite as assumed in our
method, the maximum CFO seen at the UE level will be comparable to the
terrestrial scenario and, therefore, resolvable by typical NB-IoT UE
implementation.
### V-B Solution for Differential RTD
Regarding the differential RTD (D-RTD), one can see that the current NPRACH
preambles (see TABLE VI) cannot address beams at lower elevation, since the
timing uncertainty (up to 1.4 ms) is much larger than the current CP length
(266.67 $\mu$s for format 1). In order to better understand the proposed
solution to support long D-RTDs, the reasons behind the design of NPRACH
preamble are listed below:
* •
A CP length needs to be at least equal to the expected maximum RTD.
* •
The frequency hopping is used to facilitate the ToA (i.e. RTD) estimation at
the BS.
* •
The maximum ToA that can be estimated corresponds to the reciprocal minimum
hopping step (i.e. ToA $\in\\{0,\dots,\frac{1}{\Delta(m)_{\min}}\\}$).
Figure 6: OFDM demodulation examples with the proposed CP extension scheme.
Regarding the CP, we propose to extend its length by considering the current
one (with 266.67 $\mu$s) plus additional symbols from the SG. The number of
additional symbols depends on the MD-RTD (i.e. beam width). Note that this is
possible since the current CP consists of multiple repetitions of the same
symbol. Fig. 6 shows, from the receiver perspective, three examples
representing three scenarios where the D-RTD belongs to three different
coverage levels with MD-RTD of 266.67 $\mu$s, 533.34 $\mu$s and 800 $\mu$s,
respectively. For example, if the MD-RTD is 800 $\mu$s (Example 3 in Fig. 6),
the receiver considers the new CP, namely CPnew, to be the current one plus 2
symbols from the sequence within the SG. Accordingly, the receiver will
discard (i.e. remove) in this case the CPnew when the OFDM demodulation is
done. Interestingly, the aforementioned processing does not imply any change
neither on the transmitter nor on the standard. Hence, it does not limit the
practicality of the proposed solution.
Figure 7: Flowchart of the overall solution for preamble detection and ToA
estimation in NTN.
The other issue that we need to solve, is the estimation of ToA larger than
266.67 $\mu$s (in case of using the preamble format 1), which represents the
maximum ToA that can be estimated according to the minimum hopping distance of
3.75 kHz. A flowchart illustrating the overall solution for ToA estimation as
well as the preamble detection is depicted in Fig. 7. The flowchart aims to
provide a roadmap for the reader to refer and follow the different steps
detailed in the forthcoming description. The step numbers that will refer to
in the description correspond to the numbers in the flowchart in Fig. 7,
unless otherwise specified.
First, let us take a look at the received signal in the presence of Doppler
rate. Omitting the noise term (for the clarity of exposition), the received
signal can be reformulated as
$\displaystyle y_{m,i}[n]$ $\displaystyle=$ $\displaystyle h_{m}{\rm e}^{{\rm
j}2\pi[f_{\rm off}(n-D)+\frac{1}{2}\alpha(n-D)^{2}]}s_{m,i}[n-D]$ (10)
$\displaystyle=$ $\displaystyle h_{m}{\rm e}^{{\rm j}2\pi[f_{\rm
off}(n-D)+\frac{1}{2}\alpha(n-D)^{2}]}$ $\displaystyle\times$
$\displaystyle\sum_{k}{\rm e}^{{\rm j}2\pi\frac{k}{N}(n-D)},$
where $\alpha$ is the Doppler rate normalized by the squared sampling
frequency. By removing the CP (i.e. $N_{\rm CP}$ samples) and performing DFT
(Step 1), we obtain
$\displaystyle Y_{m,i}[l]$ $\displaystyle=$
$\displaystyle\sum_{n=N_{m,i}}^{N_{m,i}+N-1}{y_{m,i}[n]{\rm e}^{-{\rm j}2\pi
ln/N}}$ (11) $\displaystyle=$
$\displaystyle\sum_{n=N_{m,i}}^{N_{m,i}+N-1}h_{m}{\rm e}^{{\rm j}2\pi[f_{\rm
off}(n-D)+\frac{1}{2}\alpha(n-D)^{2}]}$ $\displaystyle\times$
$\displaystyle\sum_{k}{{\rm e}^{{\rm j}2\pi\frac{k-l}{N}n}{\rm e}^{-{\rm
j}2\pi\frac{k}{N}D}}.$
Assuming (for the analytical derivation) that the energy leakage from other
subcarriers is small, and when $l=n_{\rm SC}^{\rm RA}(m)$, (11) can be
expressed as
$\displaystyle Y_{m,i}$ $\displaystyle=$ $\displaystyle h_{m}{\rm e}^{{\rm
j}2\pi f_{\rm off}(N_{m,i}-D)}{\rm e}^{{\rm
j}\pi\alpha(N_{m,i}^{2}+D^{2}-2N_{m,i}D)}$ (12) $\displaystyle\times$
$\displaystyle{\rm e}^{-{\rm j}2\pi n_{\rm SC}^{\rm
RA}(m)\frac{D}{N}}\sum_{n^{\prime}=0}^{N-1}{\rm e}^{{\rm j}2\pi f_{\rm
off}n^{\prime}}$ $\displaystyle\times$ $\displaystyle{\rm e}^{{\rm
j}\pi\alpha(n^{\prime 2}+2n^{\prime}(N_{m,i}-D))},$
where $n^{\prime}=n-N_{m,i}$.
Similarly to [25, 26], since $\alpha\ll 1$, the approximation ${\rm e}^{{\rm
j}\pi\alpha(n^{\prime 2}-2n^{\prime}D)}\approx 1$ can be made in (12) for
$0\leq n^{\prime}\leq N-1$. Correspondingly, we obtain
$\displaystyle Y_{m,i}$ $\displaystyle=$ $\displaystyle h_{m}{\rm e}^{{\rm
j}2\pi f_{\rm off}(N_{m,i}-D)}{\rm e}^{{\rm
j}\pi\alpha(N_{m,i}^{2}+D^{2}-2N_{m,i}D)}$ (13) $\displaystyle\times$
$\displaystyle{\rm e}^{-{\rm j}2\pi n_{\rm SC}^{\rm
RA}(m)\frac{D}{N}}\left(\frac{1-{\rm e}^{{\rm j}2\pi(f_{\rm off}+\alpha
N_{m,i})N}}{1-{\rm e}^{{\rm j}2\pi(f_{\rm off}+\alpha N_{m,i})}}\right).$
By combining the signals within the same $m$-th SG, we get the SG-S (Step 2)
$\displaystyle Y_{m}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{L^{\prime}-1}{{{Y_{m,i}}}}=h_{m}{\rm e}^{{\rm j}2\pi
f_{\rm off}(mN_{g}-D)}$ (14) $\displaystyle\times$ $\displaystyle{\rm e}^{{\rm
j}\pi\alpha[(mN_{g})^{2}+D^{2}-2mN_{g}D]}{\rm e}^{-{\rm j}2\pi n_{\rm SC}^{\rm
RA}(m)\frac{D}{N}}$ $\displaystyle\times$
$\displaystyle\sum_{i=0}^{L^{\prime}-1}{\rm e}^{{\rm j}2\pi f_{\rm
off}iN}{{\rm e}^{{\rm j}\pi\alpha[(iN)^{2}+2mN_{g}iN-2iND]}}$
$\displaystyle\times$ $\displaystyle\frac{1-{\rm e}^{{\rm j}2\pi(f_{\rm
off}+\alpha(mN_{g}+iN))N}}{1-{\rm e}^{{\rm j}2\pi(f_{\rm
off}+\alpha(mN_{g}+iN))}},$
where $L^{\prime}=L-L_{\rm CP}$, with $L_{\rm CP}$ denoting the number of
symbols employed to extend the CP within the SG.
Reasoning as with (13) (i.e. $\alpha\ll 1$), approximations444 Approximations
in (13) and (14) are possible only for indexes representing a short time
duration. With indexes representing a long time duration (e.g. index $m$)
those approximations are no longer reasonable. ${\rm e}^{{\rm
j}\pi\alpha[(iN)^{2}-2iND]}\approx 1$, ${\rm e}^{{\rm j}2\pi\alpha
iN^{2}}\approx 1$ and ${\rm e}^{{\rm j}2\pi\alpha iN}\approx 1$ can be made in
(14) for $0\leq i\leq L^{\prime}-1$. This leads to
$\displaystyle Y_{m}$ $\displaystyle=$ $\displaystyle h_{m}{\rm e}^{{\rm
j}2\pi f_{\rm off}(mN_{g}-D)}{\rm e}^{-{\rm j}2\pi n_{\rm SC}^{\rm
RA}(m)\frac{D}{N}}$ (15) $\displaystyle\times$ $\displaystyle{\rm e}^{{\rm
j}\pi\alpha[(mN_{g})^{2}+D^{2}-2mN_{g}D]}$ $\displaystyle\times$
$\displaystyle\frac{1-{\rm e}^{{\rm j}2\pi(f_{\rm off}+\alpha
mN_{g})L^{\prime}N}}{1-{\rm e}^{{\rm j}2\pi(f_{\rm off}+\alpha mN_{g})}}\,.$
The preamble detection and ToA estimation method in NTN is performed in the
same way as described in Section IV for TN systems. However, one modification
may be included if the Doppler rate is too high. This is because the
assumption of having a constant CFO during the entire NPRACH period no longer
holds. Hence, when constructing the array $v$ with $Z_{m,1}$ symbols (Step 2
in the TN method), we cannot sum up multiple $Z_{m,1}$ with equal value of
$\Delta(m)$ (particularly for far apart symbols), since they do not have the
same phase due to the variant frequency offset. Furthermore, to eliminate the
CFO impact on the ToA estimation, the factor ${\rm e}^{-{\rm j}2\pi f_{\rm
off}N_{g}}$ needs to be a common factor for all $Z_{m,1}$, which is not the
case with time-varying CFO. Nevertheless, for the analytical derivation, we
assume that the CFO variation within one preamble basic unit555The preamble
basic unit contains $P$ SGs, where $P$ can be set according to TABLE VI. is
negligible. Accordingly, the proposed method for the NTN can be outlined as
follows:
* •
Perform differential processing by multiplying the $m$-th SG-S with the
complex conjugated $(m+1)$-th SG-S (Step 3). This gives:
$\displaystyle Z_{m,1}$ $\displaystyle=$
$\displaystyle{Y_{m}Y_{m+1}^{*}}\propto{\rm e}^{-{\rm j}2\pi f_{\rm
off}N_{g}}{\rm e}^{{\rm j}2\pi\Delta(m)\frac{D}{N}}$ (16)
$\displaystyle\times$ $\displaystyle{\rm e}^{-{\rm
j}\pi\alpha(N_{g}^{2}+2mN_{g}^{2}-2N_{g}D)}{\rm e}^{-{\rm j}\pi\alpha
N_{g}(L^{\prime}N-1)}.$
* •
For each preamble basic unit,666There are $N_{\rm rep}$ basic units in a
preamble. we construct an array $v_{u}$ with the $Z_{m,1}$ symbols belonging
to the basic unit $u$. Their position in $v_{u}$ corresponds to their hopping
step $\Delta(m)$ (Steps 4 and 5). Thus, we end up with $N_{\rm rep}$ arrays.
* •
Perform 1D-DFT with $N_{\rm DFT}$ points for each $v_{u}[n]$ to get (Step 6)
$\begin{split}V_{u}[k]&=\sum_{n=0}^{N_{\rm DFT}-1}{v_{u}[n]}{\rm e}^{-{\rm
j}2\pi k\frac{n}{N_{\rm DFT}}},\end{split}$ (17)
* •
Combine non-coherently over $N_{\rm rep}$ and $N_{\rm RX}$ receive antennas to
get (Step 7)
$\begin{split}X[k]&=\sum_{u=0}^{N_{\rm rep}-1}\sum_{N_{\rm
RX}}\lvert{V_{u}[k]}\rvert^{2}.\end{split}$ (18)
* •
Perform Steps 5 and 6 of the TN method (Steps 8 to 12 in Fig. 7).
### V-C Correct ToA selection
#### V-C1 ToA selection concept
As mentioned earlier, we need to solve the problem of estimating ToAs larger
than the maximum allowed by the NPRACH preamble (i.e.
$\mathrm{ToA}>\frac{1}{\Delta(m)_{\min}}$). In the proposed method, the
estimated ToA will always be in the range $[0,\frac{1}{\Delta(m)_{\min}}]$.
But, if we analyze the ToA term in (16), i.e. ${\rm e}^{{\rm
j}2\pi\Delta(m)\frac{D}{N}}$, we notice that there may be a 2$\pi$-phase-
ambiguity problem. In other words, the phase rotation caused by D-RTD is prone
to a 2$\pi$-ambiguity by multiples of $\frac{1}{\Delta(m)_{\min}}$ in the time
estimates. Given that $\Delta(m)_{\min}=$ 3.75 kHz (for format 1), the time
domain ambiguity is in multiples of 266.67 $\mu$s. This means that all
possible D-RTDs will be estimated as D-RTD modulo 266.67 $\mu$s. For example,
D-RTDs of 66.67 $\mu$s, 333.34 $\mu$s and 600 $\mu$s will all have the same
phase rotation (i.e. $2\pi\Delta(m)\frac{D}{N}$) and a ToA estimate of 66.67
$\mu$s. Note that depending on the CP length, a list of ToA candidates can be
obtained. In principle, if the new CP is equal to $c\times 266.67~{}\mu$s, we
will end up with $c$-ToA candidates. To select the right one, we propose a
discrimination criteria based on the estimation of the underlying Doppler
rate.
It is well-known that the Doppler rate depends on the position of the UE
within the beam, similarly to the D-RTD. Also, the satellite knows the Doppler
rate and D-RTD of each geographical location within its coverage area (thanks
to _ephemeris_ and beam layout knowledge). Hence, the estimation of Doppler
rate during the NPRACH reception allows us to select the correct ToA (Step 16)
by comparing the estimated Doppler rate with the available one predicted by
the satellite. More explicitly, the estimated Doppler rate will be compared to
all Doppler rates corresponding to all D-RTD candidates (Step 15). To
illustrate this strategy, consider the following example. In a given beam,
assume that the MD-RTD is 800 $\mu$s and the estimated ToA is 66.67 $\mu$s.
Due to the phase ambiguity there will be three D-RTD candidates, namely 66.67
$\mu$s, 333.34 $\mu$s, and 600 $\mu$s with corresponding Doppler rate (e.g.
Set-2 beam with minimum elevation angle of 42.1∘) of -305 Hz/s, -261 Hz/s, and
-225 Hz/s, respectively. Next, assume that the estimated Doppler rate is -275
Hz/s. Since -275 Hz/s is closer to -261 Hz/s than the other two Doppler rate
candidates, we can conclude that the most likely ToA is 333.34 $\mu$s.
Note that in this solution, the Doppler rate estimation does not require a
very high accuracy, especially at higher elevation angles. In practice, the
estimation error needs only to be less than half of the minimum difference
between any two ToA candidates. At lower elevation angles, where wider beams
are expected (i.e. higher D-RTD), Doppler rates are small and the number of
ToA candidates is high. In this situation, a very good Doppler rate estimation
accuracy is required, since the Doppler rate separating the ToA candidates is
small. Nonetheless, for higher elevation angles, where high Doppler rates and
low ToA candidates are expected, the accuracy is relaxed due to larger Doppler
rate separation between the ToA candidates. In addition, note that the Set-1
configuration requires better accuracy compared to Set-2 configuration, since
it has a narrower beam. In the former example, the error should be smaller
than 18 Hz/s.
#### V-C2 Doppler rate estimation
After we outlined the concept of selecting the correct ToA candidate based on
the Doppler rate estimation, we describe in the following how the Doppler rate
estimator is designed. Note that the Doppler rate is always negative and has a
maximum value at 90∘ elevation angle.
We start by compensating the estimated ToA in (16) as follows (Step 13):
$\displaystyle t_{m}$ $\displaystyle=$ $\displaystyle Z_{m,1}{\rm e}^{-{\rm
j}2\pi\Delta(m)\frac{\hat{D}}{N}}$ (19) $\displaystyle\propto$
$\displaystyle{\rm e}^{-{\rm j}2\pi f_{\rm off}N_{g}}{\rm e}^{-{\rm
j}\pi\alpha(N_{g}^{2}+2mN_{g}^{2}-2N_{g}D)}$ $\displaystyle\times$
$\displaystyle{\rm e}^{-{\rm j}\pi\alpha N_{g}(LN-1)}.$
Interestingly, one can notice that all the terms in (19) are constants except
the term ${\rm e}^{-{\rm j}2\pi\alpha N_{g}^{2}m}$ that depends on $m$.
Similarly to the ToA estimator, the Doppler rate $\alpha$ can be estimated
using an approximate maximum-likelihood (ML) estimator based on 1D-DFT with
$N_{\rm DFT}$ points. In this case, we obtain
$\displaystyle T[p]$ $\displaystyle=\sum_{n=0}^{N_{\rm DFT}-1}{t[n]}{\rm
e}^{-{\rm j}2\pi p\frac{n}{N_{\rm DFT}}}\,.$ (20)
In case of multiple receive antennas, the combination of the signals is done
non-coherently by summing up squared magnitudes of $T[p]$ over $N_{\rm RX}$
antennas, i.e. $J[p]=\sum_{N_{\rm RX}}\lvert{T[p]}\rvert^{2}$. Note that by
taking the absolute square of $T[p]$, we eliminate the impact of the CFO given
by ${\rm e}^{-{\rm j}2\pi f_{\rm off}N_{g}}$ as well as the other constant
terms, since they affect only the phase of $T[p]$ and not its magnitude.
Let $p_{\max}$ be the index of the maximum value of $J[p]$. The Doppler rate
can be estimated as
$\displaystyle\hat{\alpha}$ $\displaystyle=-\frac{p_{\max}}{N_{\rm
DFT}N_{g}^{2}}.$ (21)
However, one can observe from (19) that $w_{\alpha}=2\pi\alpha N_{g}^{2}$ in
$t_{m}$ symbols is very small, since $\alpha\ll 1$. Hence, to be able to
estimate $\alpha$ by the above method (i.e. 1D-DFT) with a good estimation
accuracy, a very large number of DFT points is needed. For example, if a
resolution of 6 Hz/s is desirable, $N_{\rm DFT}=\frac{1}{6\times
N_{g}^{2}}\approx 2^{16}$ is required. This leads to a very high computational
complexity, namely $\mathcal{O}\left(N_{\rm DFT}\log_{2}(N_{\rm
DFT})\right)\approx 2^{20}$ (considering radix-2 fast Fourier transform
implementations), which may become a burden for the system operation,
especially due to the limited computational resources and power available on-
board of the satellite.
To reduce this computational complexity and enable an efficient implementation
of the method, we consider the approximate ML estimator via a numerical
evaluation of the discrete-time Fourier transform (DTFT). The key idea herein
is to restrict the search space in frequency domain only to the possible
Doppler rate values (depending on the scenario) rather than the whole
spectrum. More specifically, note that the maximum Doppler rate that can be
estimated with the above method corresponds to
$\alpha_{\max}=-\frac{1}{N_{g}^{2}}=-390.625$ kHz/s, whereas the maximum value
in the adopted scenario is only $\alpha_{\max}^{\rm sc}=-594$ Hz/s, which
occupies a very small part of the considered spectrum. In addition, note that
the minimum Doppler rate is $\alpha_{\min}^{\rm sc}=-101$ Hz/s (see Section
III-C). Furthermore, each beam, depending on its size, has a minimum and
maximum Doppler frequency drift between [$\alpha_{\min}^{\rm
sc},\alpha_{\max}^{\rm sc}$] depending on the elevation angle. All these
aspects reduce further the space search in the spectrum, which can be
exploited in order to considerably reduce the computational complexity and
increase the estimation accuracy.
At first, the DTFT of $t_{m}$ with $N_{\rm DTFT}$ points is calculated:
$\displaystyle T({\rm e}^{{\rm j}w_{q}})$
$\displaystyle=\sum_{n=0}^{N_{t}-1}{t[n]}{\rm e}^{-{\rm j}w_{q}n},$ (22)
where $N_{t}=PN_{\rm rep}-1$ is the number of symbols in $t[n]$, and
$w_{q}=-2\pi\delta_{\alpha}q$ with $\delta_{\alpha}=\frac{(\alpha_{\max}^{\rm
sc}-\alpha_{\min}^{\rm sc})N_{g}^{2}}{N_{\rm DTFT}}$ is the desired resolution
controlled by the number of DTFT points $N_{\rm DTFT}$, and
$q\in\left\\{\frac{\alpha_{\min}^{\rm sc}}{\alpha_{\max}^{\rm
sc}-\alpha_{\min}^{\rm sc}},...,\frac{\alpha_{\max}^{\rm
sc}}{\alpha_{\max}^{\rm sc}-\alpha_{\min}^{\rm sc}}\right\\}\cdot N_{\rm
DTFT}$.
Then, similarly to the 1D-DFT based method, we combine non-coherently over
$N_{\rm RX}$ antennas:
$\displaystyle J[q]=\sum_{N_{\rm RX}}\lvert{T({\rm e}^{{\rm
j}w_{q}})}\rvert^{2}.$ (23)
Let $q_{\max}\in[1,N_{\rm DTFT}]$ be the index of the maximum value of $J[q]$.
Then the Doppler rate is given by (Step 14):
$\displaystyle\hat{\alpha}$
$\displaystyle=\frac{q_{\max}\delta_{\alpha}}{N_{g}^{2}}+\alpha_{\min}^{\rm
sc}.$ (24)
The complexity of the DTFT-based approximate ML estimator will be ${\cal
O}(N_{\rm DTFT}\times N_{t})$. To show the gain of the proposed ML estimator
based on DTFT compared to the standard DFT (implemented via FFT), we consider
the same example as before with a target resolution of 6 Hz/s. For the largest
spotbeam from Set-2 configuration at 30∘ (see TABLE IV), the maximum and
minimum Doppler rate will be -206 Hz/s and -101 Hz/s, respectively. In this
case, we need to have $N_{\rm DTFT}=\frac{\alpha_{\max}^{\rm
sc}-\alpha_{\min}^{\rm sc}}{-6}\approx 18$. Assuming a preamble with $N_{\rm
rep}=64$ (i.e. $N_{t}=255$), the foreseen complexity will be only ${\cal
O}(N_{\rm DTFT}\times N_{t})\approx 4600$. Thus, we reduce the complexity by a
factor of $\frac{2^{20}}{4600}\approx 228$, leading to a substantial
improvement in term of computational complexity.
Ultimately, based on the estimate of the Doppler rate, it is possible to
discriminate the candidates of ToA (Step 15) and resolve the corresponding
ambiguity (Step 16), as described in Section V-C1.
## VI Simulation Results and discussion
TABLE VII: Simulation parameters Parameter | Value
---|---
Antenna config. [27] | 1 Tx, 2 Rx
No. of active UEs | 1 & 3
Channel | AWGN
Preamble format | 1
Subcarrier spacing | 3.75 kHz
No. of repetitions | 32 and 64
Min. Elevation | 30∘ (for Set-1) and 31∘ (for Set-2)
MD-RTD | 533.3 $\mu$s, 800 $\mu$s, 1066.7 $\mu$s, and 1333.4 $\mu$s
Timing uncertainty | Uniformly drawn from [0, MD-RTD] $\mu$s
Residual Freq. offset | Uniformly drawn from [-600, 600] Hz
$N_{\rm DFT}$ (ToA estim.) | 256 points
$N_{\rm DTFT}$ ($\alpha$ estim.) | 62 points
In this section, we provide simulation results to assess the performance of
the proposed method. The PHY-based simulation parameters are summarized in
TABLE VII. The antenna configuration is set to 1 transmitter antenna (at the
UE) and 2 receiver antennas (at the satellite) similarly to the TN system. One
active UE is considered first to obtain an upper-bound performance for the
proposed method. Then, three active UEs with adjacent subcarriers are
simulated to represent a realistic scenario specifically accounting for ICI
effects [28]. Regarding the channel model, it is worth noting here that the
NTN propagation environment can exhibit NLOS/ multipath effects. 3GPP study
related to the channel model in TR 38.811 [6] has developed a channel model
for 5G-NR that supports a range of deployment including urban, suburban and
rural scenarios. However, in contrast to terrestrial propagation environment,
it was highlighted in TR 38.811 [6] that the large distance in NTN leads to
lower delay spread and thus to higher coherence bandwidth. In fact, TR 38.811
[6] defined 100 ns to be the maximum Delay spread for our considered scenario
(i.e. LEO at 600 km). This corresponds in the worst case (i.e. performance
lower bound) to 200 kHz of coherence bandwidth. Hence, the NB-IoT NTN channel
is considered to be flat due to its narrow band nature (i.e. 180 kHz),
especially for the random access (i.e. 45 kHz considering maximum hopping of
12 subcarriers). Consequently, and as stated in TR 38.811 [6], the link level
simulation under flat fading does not require any channel modeling, only
additive white Gaussian noise (AWGN) is considered. Both 32 and 64 repetitions
($N_{\rm rep}$) are configured. The considered minimum elevation angles for
Set-1 and Set-2 payload configurations are 30∘ and 31∘, respectively. This
latter represents the MD-RTD that the proposed method can address in case of
Set-2 configuration. The MD-RTDs for our scenario that exceed the typical
maximum RTD in the TN system (i.e. 266.67 $\mu$s) are considered to show the
coverage extension provided by the proposed method. Accordingly, the timing
uncertainty is uniformly chosen from [0, MD-RTD] $\mu$s. The maximum residual
frequency offset is chosen to be 600 Hz according to our discussion in Section
V-A. Finally, 256 points DFT (implemented via FFT) and 62 points DTFT are set
for ToA and Doppler rate estimation, respectively.
### VI-A 3GPP performance metrics
3GPP has not yet defined performance metrics for NB-IoT in NTN. However, we
adopt in this work the same 3GPP performance metrics as in TN. Accordingly,
the performance metrics are the probability of preamble detection and false
alarm. More specifically, the probability of preamble detection should be
equal to $99\%$ (i.e. missed detection rate below $1\%$), and false alarm
probability should be less than or equal to $0.1\%$. According to [27], the
probability of detection is defined as the conditional probability of correct
detection of the preamble when the signal is present. There are several error
cases:
* •
detection of a wrong preamble (different than the one sent);
* •
no detection of any preamble;
* •
correct preamble detection but with the wrong timing (ToA) estimation.
This latter occurs if the ToA estimation error is larger than $3.646~{}\mu$s.
The false alarm probability is defined as the conditional total probability of
erroneous detection of the preamble when the input contains only noise, i.e.
in absence of the useful signal.
(a) NTN beam with MD-RTD of 1333.4 $\mu$s
(b) NTN beam with MD-RTD of 1066.7 $\mu$s
(c) NTN beam with MD-RTD of 800 $\mu$s
(d) NTN beam with MD-RTD of 533.3 $\mu$s
Figure 8: NPRACH preamble detection performance with different MD-RTDs in our
NTN scenario.
### VI-B Obtained results
(a) CDF of absolute Doppler rate estimation error with 32 repetitions
(b) CDF of absolute Doppler rate estimation error with 64 repetitions
Figure 9: CDF of absolute Doppler rate estimation error at the missed
detection target of 10-2.
(a) CDF of absolute ToA estimation error with 32 repetitions
(b) CDF of absolute ToA estimation error with 64 repetitions
Figure 10: CDF of absolute ToA estimation error at the missed detection target
of 10-2.
The overall obtained results are first analyzed and discussed. The performance
of ToA and Doppler rate estimators are provided in terms of cumulative
distribution function (CDF) of the absolute ToA estimation error and Doppler
rate estimation error. All results are obtained with a false alarm probability
being less than 0.1%.
#### VI-B1 Overall performance
The obtained results are depicted in Fig. 8 and presented in terms of missed
detection rate versus SNR. We start first by presenting and analyzing the case
of single UE. For an MD-RTD of 1333.4 $\mu$s (e.g. scenario for beam at
minimum elevation angle of 31∘ for Set-2 configuration with 246-km diameters)
the missed detection target of 10-2 is achieved at SNR of 15.5 dB and 9.3 dB
for 32 and 64 repetitions, respectively (see Fig. 8a). In case of an MD -RTD
of 1067 $\mu$s (e.g. scenario for beam at minimum elevation angle of 30∘ for
Set-1 configuration with 144-km diameters) the target is reached at SNR of
14.8 dB and 8.1 dB for 32 and 64 repetitions, respectively (see Fig. 8b). For
lower MD-RTDs, lower SNR values are observed at the missed detection target
(see Fig. 8c and Fig. 8d). For an MD-RTD of 800 $\mu$s the target is reached
at 7.9 dB and 2.8 dB for 32 and 64 repetitions, respectively, whereas for MD-
RTD of 533.4 $\mu$s these values are 6.4 dB and 2.6 dB for 32 and 64
repetition, respectively. When analyzing the results, we notice that the
performance is enhanced when we reduce the MD-RTD. This is an expected
behavior, since the number of useful symbols ($\notin$ CP) is inversely
proportional to the CP length. This leads to SNR enhancement (i.e. performance
enhancement) when combining within the same SG in (14). Furthermore, this is
also related to the fact that at lower elevation angles, where higher D-RTDS
are expected, better Doppler rate accuracy is needed compared to higher
elevation angles, as explained in Section V-C.
Regarding the case of three active UEs, naturally, we observe some performance
degradation compared to the single-UE performance. This degradation is not the
same for all cases. For MD-RTDs above 1000 $\mu$s, we observe a relatively
substantial degradation before the missed detection target in case of 32
repetitions, and about 0.5-dB degradation at the missed detection target in
case of 64 repetitions. On the other hand, for MD-RTDs below 1000 $\mu$s, we
notice about 0.5-dB degradation in case of 32 repetitions and negligible
degradation in case of 64 repetitions. The low impact of ICI at lower SNRs is
understandable. In fact, at low SNR regime, the noise is the dominant
impairment. But when moving to moderate and high SNR regimes, the ICI starts
to be the dominant impairment leading to performance degradation. The
substantial degradation in case of 32 repetitions at higher MD-RTDs ($>$ 1000
$\mu$s) can be explained by the lack of signal length (less repetitions) to
estimate the Doppler rate with enough accuracy in presence of ICI. On the
other hand, it seems sufficient to have 64 repetitions to reach the required
Doppler rate accuracy, leading to better overall performance.
#### VI-B2 Estimators’ performance
Fig. 9 and Fig. 10 respectively show the performance of Doppler rate and ToA
estimators around the missed detection target of 10-2 for both 32 and 64
repetitions when three UEs are active.
First, when observing the Doppler estimation accuracy (see Fig. 9), one can
notice that the estimation accuracy required to achieve the missed detection
target is high in case of large D-RTDs (i.e. lower elevation angles), compared
to small D-RTDs (i.e. higher elevation angles) for all repetitions. For 32
repetitions, the Doppler rate errors (at 99% confidence level) are 9.8 Hz/s,
8.3 Hz/s, 17 Hz/s, 30.5 Hz/s for MD-RTDs of 1333.4 $\mu$s, 1066.7 $\mu$s, 800
$\mu$s and 533.3 $\mu$s, respectively. In case of 64 repetitions, those values
are 8.7 Hz/s, 7.8 Hz/s, 17 Hz/s, and 31 Hz/s for MD-RTDs of 1333.4 $\mu$s,
1066.7 $\mu$s, 800 $\mu$s and 533.3 $\mu$s, respectively. As explained in
Section V-C, this reflects the fact that for large D-RTD (i.e. lower elevation
angles) one expects wider beams, smaller Doppler rates, and a high number of
ToA candidates to be discriminated, thus requiring a very good accuracy
compared to higher elevation angles. In addition, one can notice that the
accuracy needed for MD-RTD of 1333.4 $\mu$s is less than the one for MD-RTD of
1066.7 $\mu$s. This is because the latter MD-RTD represents Set-1
configuration, which has a narrower beam compared to the case of MD-RTD of
1333.4 $\mu$s in Set-2 configuration. Note that for 32 repetitions and in case
of MD-RTDs of 1333.4 $\mu$s and 1066.7 $\mu$s, the required accuracy (expected
to be $\approx$ 9 Hz/s and $\approx$ 8 Hz/s, respectively) to attain the
target missed detection is not reached. This is why in those cases (i.e. 32
repetitions with MD-RTD of 1333.4 $\mu$s and 1066.7 $\mu$s) the missed
detection target is not achieved.
Regarding the ToA estimator, as shown in Fig. 10, it is indeed very accurate
at the target missed detection rate. For 32 and 64 repetitions, the accuracy
of the estimator is less than 1 $\mu$s except for cases where MD-RTD is equal
to 1333.4 $\mu$s and 1066.7 $\mu$s with 32 repetitions. Those exceptions stem
from the aforementioned lack of Doppler rate accuracy, which leads to a poor
discrimination among the ToA candidates.
#### VI-B3 Concluding remarks on the results
The overall performance demonstrates the ability of the proposed method to
address the NTN scenario described in Section III. Explicitly, the method can
address UEs at minimum elevation angle of 31∘ for Set-2 configuration and UEs
at 30∘ elevation angles and even lower for Set-1 configuration. For both
configurations, 64 preamble repetitions are needed for UEs at lower elevation
angles, whereas for higher ones, the UEs can be served with lower preamble
repetitions. It is worth noting here that that the obtained results may
constitute a new reference for future investigations of NPRACH detection
performance in the NTN context, where the available literature is rather
scarce.
### VI-C Discussion
The proposed method in this paper is designed to work with UEs without GNSS
but could also work with UEs integrating GNSS capability. The method can be
useful in terms of ToA estimation accuracy by eliminating the CFO as well as
the Doppler rate effects when estimating the ToA. Furthermore, independently
of the UE type (with or without GNSS), the estimation of Doppler rate can be
reported to upper layers for the scheduling algorithm, which could use it to
preserve the UEs’ orthogonality for data transmission [29].
On the other hand, the actual performance of the method depends on the
satellite altitude. Satellites at higher altitudes have larger spotbeams and
lower speed, leading to larger D-RTD and lower Doppler rates, assuming the
same payload configurations. First, the method (i.e. without GNSS) cannot work
if D-RTD exceeds the MD-RTD that the system can handle (i.e. $T_{\rm SEQ}$).
This issue can be tackled by considering higher elevation angles (e.g. for a
LEO at 2000 km, the minimum elevation angle becomes around 42∘ for Set-1
configuration) for the system operations. Now, when the Doppler rate is very
small, the method may not be able to discriminate between the possible D-RTDs,
since it is limited by the precision of the Doppler rate estimator. For
example, with a LEO at 2000 km and minimum elevation angle of 42 degree with
Set-1 configuration, the Doppler rate variation between the two edges of the
spotbeam is around 12 Hz/sec (assuming a maximum Doppler rate of 125 Hz/sec).
Accordingly, in order to discriminate 5 D-RTDs, we need a precision better
than 12/5/2 = 1.2 Hz/sec, which is difficult to achieve for 99% of cases. In
this configuration, adopting higher carrier frequency may solve this issue.
## VII Conclusion
In this work, a new receiver method was designed for the integration of NB-IoT
random access in non-terrestrial networks. A detailed scenario definition
along with link budget description as well as some design aspects related to
the integration of NB-IoT random access via LEO satellites were provided. The
proposed method is designed to minimize the impact of frequency offset and
Doppler rate while extending the coverage beyond the typical limit of the NB-
IoT system in terrestrial networks. Performance evaluation showed how the
proposed method can address the different configurations defined by 3GPP
standardization body for non-terrestrial networks. Since numerous companies
are interested into the integration of NB-IoT via satellite, the proposed
method can constitute a practical and seamless solution for the random access
channel.
## Appendix A
Mathematical symbols used in Preamble Detection and ToA Estimation in both TN
and NTN
Symbol | Definition
---|---
$s_{m,i}[n]$ | Transmit signal at the $n$-th sample of the $i$-th symbol in $m$-th symbol group
$S_{m,i}[k]$ | $i$-th symbol on the $k$-th subcarrier during the $m$-th SG
$N_{g}$ | Size of one symbol group in sample
$N_{\rm CP}$ | CP size in sample
$N$ | Symbol size in sample
$y_{m,i}[n]$ | Received time domain signal of the $n$-th sample of the $i$-th symbol in the $m$-th symbol group
$Y_{m,i}[l]$ | Received frequency domain signal of the $l$-th subcarrier of the $i$-th symbol in the $m$-th symbol group
$W_{m,i}$ | Additive noise signal in the $i$-th symbol in the $m$-th symbol group
$Y_{m}$ | Sum of frequency domain symbols of the $m$-th symbol group
$L_{\rm CP}$ | Number of symbols employed to extend the CP within one symbol group
$L^{\prime}$ | Number of remaining symbols after CP removal
$f_{\rm off}$ | Carrier frequency offset normalized by the sampling frequency
$D$ | RTD normalized by the symbol duration
$\alpha$ | Doppler rate normalized by squared sampling frequency
$\alpha_{\max}$ | Maximum Doppler rate that can be theoretically estimated
$\alpha_{\max}^{\rm sc}$ | Maximum Doppler rate for the considered scenario
$\alpha_{\min}^{\rm sc}$ | Minimum Doppler rate for the considered scenario
$h_{m,i}$ | Channel gain of the $i$-th symbol in the $m$-th symbol group.
$h_{m}$ | Channel gain of the the $m$-th symbol group
$n_{\rm SC}^{\rm RA}(m)$ | Subcarrier occupied by the $m$-th symbol group
$\Delta(m)$ | Hopping step between the $m$-th and $(m+1)$-th symbol groups
$\Delta(m)_{min}$ | Minimum hopping step in the preamble
$Z_{m,1}$ | Differential symbol resulting from $m$-th and $(m+1)$-th symbol group differential processing
$v$ | Array containing differential symbols with corrected frequency hopping
$v_{u}$ | Array containing differential symbols of $u$-th preamble basic unit with corrected frequency hopping
$N_{\rm DFT}$ | Number of DFT points
$N_{\rm RX}$ | Number of receive antennas
$V$ | Post-DFT vector of corrected differential symbols
$V_{u}$ | Post-DFT vector of corrected differential symbols of the $u$-th preamble basic unit
$X[k]$ | NPRACH detection metric vector
$X_{\max}$ | Maximum value of the NPRACH detection metric vector
$\hat{D}$ | Estimated ToA (i.e. RTD)
$t_{m}$ | ToA corrected differential symbol
$T$ | Post-DTFT vector of ToA corrected differential symbols
$N_{t}$ | Number of symbols in $t$ vector
$N_{\rm DTFT}$ | Number of DTFT points
$J$ | Post-DTFT vector of corrected differential symbols summed across receive antennas
$\hat{\alpha}$ | Estimate of Doppler rate
$\delta_{\alpha}$ | Resolution of Doppler rate estimation
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# Recursive Trellis Processing of Large Polarization Kernels
Peter Trifonov Peter Trifonov ITMO University, Russia
Email<EMAIL_ADDRESS>
###### Abstract
A reduced complexity algorithm is presented for computing the log-likelihood
ratios arising in the successive cancellation decoder for polar codes with
large kernels of arbitrary dimension. The proposed algorithm exploits
recursive trellis representation of the codes generated by submatrices of the
polarization kernel, and enables codes based on large kernels to provide
better performance compared to the codes based on Arikan kernel with the same
decoding complexity.
## I Introduction
Polar codes, introduced by Arikan in [1], have already made their way into the
5G specification. However, their role is limited there for encoding of small
data blocks, since long polar codes do not compete well with other modern
codes, such as LDPC. Polar codes with large kernels were shown to have
asymptotically optimal scaling exponent [2]. These codes have decoding
complexity $O(n\log n)$ operations of computing LLRs. However, each such
operation, called kernel processing or kernel marginalization [3], has
complexity $O(2^{l}l)$ if implemented straightforwardly. An approximate kernel
processing method based on box-and-match decoder was suggested in [4]. Some
kernels of dimension 16 and 32 were published together with efficient
processing algorithms [5]. It was shown that the corresponding polar codes
have lower decoding complexity compared to codes based on Arikan kernel with
the same performance. However, no efficient processing techniques for generic
polarization kernels have been published so far.
In this paper we present such algorithm, and report its complexity for some
kernels available in the literature. We show that the proposed approach has
much lower complexity compared to the Viterbi algorithm.
## II Background
### II-A Polar codes
Polar code [6] is a set of vectors $c_{0}^{n-1}=u_{0}^{n-1}K^{\otimes l}$,
where $K$ is a non-singular $l\times l$ matrix called polarization kernel,
$n=l^{m}$, $u_{i}=0$ for $i\in\mathcal{F}$, $\mathcal{F}\subset[n]$ is a
frozen set, and $[n]=\left\\{{0,\dots,n-1}\right\\}$. This definition can be
generalized to obtain mixed kernel polar codes with the codewords given by
$c_{0}^{n-1}=u_{0}^{n-1}(K_{l_{1}}\otimes K_{l_{2}}\otimes\cdots\otimes
K_{l_{m}})$ [7, 8], where $K_{l_{i}}$ is a kernel of dimension $l_{i}$. Unless
stated otherwise, we consider here the case of all kernels being the same.
Decoding of polar codes can be implemented by the successive cancellation
algorithm, which makes decisions
$\hat{u}_{i}=\begin{cases}0,&i\in\mathcal{F}\\\
\arg\max_{u_{i}\in\operatorname{\mathbb{F}}_{2}}\operatorname{\mathcal{W}}_{m}^{(i)}(\hat{u}_{0}^{i-1}\bullet
u_{i}|r_{0}^{n-1}),&i\notin\mathcal{F},\end{cases}$
where
$\operatorname{\mathcal{W}}_{m}^{(i)}(u_{0}^{i}|r_{0}^{n-1})=\sum_{u_{i+1}^{n-1}}\prod_{j=0}^{n-1}W_{0}^{(0)}((u_{0}^{n-1}K^{\otimes
l})_{i}|r_{i}),$
$\operatorname{\mathcal{W}}_{0}^{(0)}(c|r)=W_{0}^{(0)}(c|r)=\frac{W(r|c)}{2W(r)}$,
$W(r|c)$ is the channel transition probability, and $\bullet$ denotes the
concatenation operator. These probabilities can be recursively computed as
$\displaystyle\operatorname{\mathcal{W}}_{m}^{(li+s)}(u_{0}^{li+s}|r_{0}^{n-1})=\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle\sum_{u_{s+1}^{l-1}}\prod_{j=0}^{l-1}\operatorname{\mathcal{W}}_{m-1}^{(i)}\left((u_{lt}^{l(t+1)-1}K),t\in[i+1]|r_{0,j}^{n-1}\right),$
(1)
where $r_{0,j}^{n-1}=(r_{j},r_{j+l},\dots,r_{j+n-l})$. This operation is known
as kernel processing or kernel marginalization [9, 3].
It was suggested in [4] to approximate the above probabilities as
$\operatorname{\mathcal{W}}_{m}^{(i)}(u_{0}^{i}|r_{0}^{n-1})\approx
W_{m}^{(i)}(u_{0}^{i}|r_{0}^{n-1})=\max_{u_{i+1}^{n-1}}\prod_{j=0}^{n-1}W_{0}^{(0)}((u_{0}^{n-1}K^{\otimes
l})_{i}|r_{i}),$ so that
$\displaystyle
W_{m}^{(li+s)}(u_{0}^{li+s}|r_{0}^{n-1})=\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle\max_{u_{s+1}^{l-1}}\prod_{j=0}^{l-1}W_{m-1}^{(i)}\left((u_{lt}^{l(t+1)-1}K),t\in[i+1]|r_{0,j}^{n-1}\right).$
(2)
Decoding can be implemented in the LLR domain using LLRs
$S_{m}^{(i)}(u_{0}^{i-1},r_{0}^{n-1})=\ln\frac{W_{m}^{(i)}(u_{0}^{i-1}\bullet
0|r_{0}^{n-1})}{W_{m}^{(i)}(u_{0}^{i-1}\bullet 1|r_{0}^{n-1})}.$
Assume for the sake of simplicity $m=1$. It can be seen that
$\displaystyle S_{1}^{(i)}(u_{0}^{i-1},r_{0}^{n-1})=$
$\displaystyle\max_{u_{i+1}^{l-1}}\sum_{j=0}^{l-1}\ln
W_{0}^{(0)}\left(((u_{0}^{i-1}\bullet 0\bullet
u_{i+1}^{l-1})K)_{j}|r_{j}\right)$ $\displaystyle-$
$\displaystyle\max_{u_{i+1}^{l-1}}\sum_{j=0}^{l-1}\ln
W_{0}^{(0)}\left(((u_{0}^{i-1}\bullet 1\bullet
u_{i+1}^{l-1})K)_{j}|r_{j}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\max_{u_{i+1}^{l-1}}Q((u_{0}^{i-1}\bullet 0\bullet
u_{i+1}^{l-1})K,r_{0}^{l-1})$ $\displaystyle-$
$\displaystyle\frac{1}{2}\max_{u_{i+1}^{l-1}}Q((u_{0}^{i-1}\bullet 1\bullet
u_{i+1}^{l-1})K,r_{0}^{l-1}),$ (3)
where
$Q(c_{0}^{l-1},r_{0}^{l-1})=\sum_{j=0}^{l-1}(-1)^{c_{j}}S_{0}^{(0)}(r_{j})$ is
the correlation function. The objective of the present paper is to provide an
efficient method for computing LLRs $S_{1}^{(i)}(u_{0}^{i-1},r_{0}^{n-1})$.
### II-B Recursive maximum likelihood decoding algorithm
It can be seen that computing (3) reduces to ML decoding in the cosets of the
code generated by $l-i-1$ last rows of matrix $K$, provided that all symbols
$u_{s},s\in[l]$, are equiprobable binary random values. An efficient recursive
maximum likelihood decoding algorithm for linear block codes was suggested in
[10]. The idea is to recursively partition the received noisy vector into a
number of sections $[x,y)$, identify for each section a number of most likely
vectors $c_{x}^{y-1}\in\operatorname{\mathbb{F}}_{2}^{y-x}$ corresponding to
the received values $r_{x}^{y-1}$, and recursively combine them to obtain most
likely vectors for longer sections.
Given a linear code $C$, let $C_{h,h^{\prime}}$ be its subcode, such that all
its codewords have non-zero symbols only in positions $h\leq i<h^{\prime}$.
Let $p_{h,h^{\prime}}(C)$ be a linear code obtained by puncturing all symbols,
except those in positions $h\leq i<h^{\prime}$, from codewords of $C$. Let us
further define $s_{h,h^{\prime}}(C)=p_{h,h^{\prime}}(C_{h,h^{\prime}})$, i.e.
a code obtained from $C$ by shortening it on all symbols except those with
indices $h\leq i<h^{\prime}$. The codes $s_{h,h^{\prime}}(C)$ and
$p_{h,h^{\prime}}(C)$ will be referred to as section codes. Consider a minimal
trellis of code $C$, and its section corresponding to symbols from $x$ to $y$.
It is possible to show that the paths between two adjacent states in this
section correspond to a coset in $p_{x,y}(C)/s_{x,y}(C)$ [10]. This coset may
appear in the trellis several times. Hence, one can simplify maximum
likelihood (ML) decoding by pre-computing the metrics of these paths. That is,
for each coset $D\in p_{x,y}(C)/s_{x,y}(C)$ one needs to identify the most
probable element $l(D)$, and store its correlation $Q(D)=Q(l(D),r_{x}^{y-1})$.
Let the composite branch table (CBT) $T_{x,y}$ be an array containing values
$T_{x,y}[v].l=l(D)$ and $T_{x,y}[v].q=Q(D),$ where $v$ is an index of $D$. In
the case of conventional decoding of an $(n,k)$ code, $p_{0,n}(C)/s_{0,n}(C)$
contains a single element, so the corresponding CBT has one entry
$T_{0,n}[0]$, which gives a solution of the ML decoding problem.
The straightforward approach to construction of a composite branch table for
some code $C$ is to enumerate all codewords of $p_{x,y}(C)$, and find the most
probable one for each coset in $p_{x,y}(C)/s_{x,y}(C)$. We assume that this
method is used for $y-x\leq 2$. However, more efficient approach was suggested
in [10] for the case of $y-x\geq 2$. Consider some $z:x<z<y$. Let the
generator matrix of $p_{x,y}(C)$ be represented as
$G^{(p)}_{x,y}=\begin{pmatrix}G^{(s)}_{x,z}&0\\\ 0&G^{(s)}_{z,y}\\\
G^{(00)}_{x,y}&G^{(01)}_{x,y}\\\ \hline\cr G^{(10)}_{x,y}&G^{(11)}_{x,y}\\\
\end{pmatrix},$ (4)
where $G^{(s)}_{x,y}=\begin{pmatrix}G^{(s)}_{x,z}&0\\\ 0&G^{(s)}_{z,y}\\\
G^{(00)}_{x,y}&G^{(01)}_{x,y}\end{pmatrix}$ is a generator matrix of
$s_{x,y}(C)$, and $G_{x,y}^{(00)},G_{x,y}^{(01)}$ are some
$k_{x,y}^{\prime\prime}\times(z-x)$ and $k_{x,y}^{\prime\prime}\times(y-z)$
matrices, respectively, where $k_{x,y}^{\prime}=k_{x,y}^{\prime}(C)$ and
$k_{x,y}^{\prime\prime}=k_{x,y}^{\prime\prime}(C)$ are some code-dependent
integers. There is an one-to-one correspondence between vectors
$vG_{x,y}^{\prime}$, where
$G_{x,y}^{\prime}=\begin{pmatrix}G^{(10)}_{x,y}&G^{(11)}_{x,y}\end{pmatrix}$
is a $k^{\prime}_{x,y}\times(y-x)$ matrix, and cosets $D\in
p_{x,y}(C)/s_{x,y}(C)$. Here $k_{x,y}^{\prime},k_{x,y}^{\prime\prime}$ are
some integers, which depend on code structure, and can be obtained from the
minimum span form of its generator matrix.
Hence, we write $T_{x,y}[v].l:=l(D)$ and $T_{x,y}[v].q=Q(D)$, with $D$ being a
coset corresponding to $v$. It can be seen that
$\displaystyle T_{x,y}[v].q=$ $\displaystyle\max_{c_{x}^{y-1}\in
D}Q(c_{x}^{y-1},r_{x}^{y-1})$ $\displaystyle=$
$\displaystyle\max_{w\in\operatorname{\mathbb{F}}_{2}^{k_{x,y}^{\prime\prime}}}\left(T_{x,z}[a].q+T_{z,y}[b].q\right),v\in\operatorname{\mathbb{F}}_{2}^{k^{\prime}_{x,y}}$
(5)
where $a$ and $b$ are indices of the cosets $D^{\prime}\in
p_{x,z}(C)/s_{x,z}(C)$ and $D^{\prime\prime}\in p_{z,y}(C)/s_{z,y}(C)$,
respectively, such that
$\begin{pmatrix}w&v\end{pmatrix}\begin{pmatrix}G^{(00)}_{x,y}\\\
G^{(10)}_{x,y}\end{pmatrix}\in D^{\prime}$ and
$\begin{pmatrix}w&v\end{pmatrix}\begin{pmatrix}G^{(01)}_{x,y}\\\
G^{(11)}_{x,y}\end{pmatrix}\in D^{\prime\prime}$. Such values $a,b$ can be
identified from the system of equations
$\displaystyle\begin{pmatrix}a^{\prime}&a\end{pmatrix}\begin{pmatrix}G_{x,z}^{(s)}\\\
G_{x,z}^{\prime}\end{pmatrix}=$
$\displaystyle\begin{pmatrix}w&v\end{pmatrix}\begin{pmatrix}G^{(00)}_{x,y}\\\
G^{(10)}_{x,y}\end{pmatrix}$
$\displaystyle\begin{pmatrix}b^{\prime}&b\end{pmatrix}\begin{pmatrix}G_{z,y}^{(s)}\\\
G_{z,y}^{\prime}\end{pmatrix}=$
$\displaystyle\begin{pmatrix}w&v\end{pmatrix}\begin{pmatrix}G^{(01)}_{x,y}\\\
G^{(11)}_{x,y}\end{pmatrix},$
where $a^{\prime},b^{\prime}$ are some irrelevant values. Obviously, the
solutions are given by $a=\begin{pmatrix}w&v\end{pmatrix}\widehat{G}_{x,y}$
and $b=\begin{pmatrix}w&v\end{pmatrix}\widetilde{G}_{x,y}$ for some matrices
$\widehat{G}_{x,y}$ and $\widetilde{G}_{x,y}$. The corresponding most likely
coset representatives are given by $T_{x,y}[v].l=T_{x,z}[\hat{a}].l\bullet
T_{x,z}[\hat{b}].l$, where $\hat{a},\hat{b}$ are the values of $a$ and $b$,
which deliver maximum in (5).
The complexity of this calculation is
$O(2^{k_{x,y}^{\prime}+k_{x,y}^{\prime\prime}})$. It can be further reduced by
exploiting the tricks suggested in [10]. The overall decoding complexity
strongly depends on the sectionalization method being used, i.e. a rule for
selection of the partitioning point $z$ for some $x,y$. This approach, known
as recursive maximum likelihood decoding (RMLD), was shown to be more
efficient compared to the Viterbi algorithm [10].
## III Recursive trellis processing
### III-A Extended kernel codes
Polar codes can be considered as a result of recursive application of the
construction of generalized concatenated codes [11, 12]. These codes rely on
non-systematic inner codes, as well as the corresponding soft-decision
decoding algorithms. An optimal soft-input soft-output decoding algorithm for
non-systematically encoded linear block codes was presented in [13]. This
algorithm can be easily tailored to implement computation of (3). To do this,
let us consider an extended $(l+1,l-i)$ code $\overline{\mathcal{C}}^{(i)}$
generated by matrix $G^{(i)}$. First $l$ columns of this matrix are obtained
by taking $l-i$ last rows of kernel $K$. The last column has $1$ in the $0$-th
row, and zeroes in the remaining positions. Assuming that $u_{0}^{i-1}=0$, one
obtains that computing (3) is equivalent to finding the most probable
codewords of code $\overline{\mathcal{C}}^{(i)}$ having $0$ and $1$ in the
last symbol. This can be implemented by running the Viterbi algorithm over the
trellis of $\overline{\mathcal{C}}^{(i)}$, assuming that the last codeword
symbol is erased. The same trellises, although with different labeling, can be
used to implement decoding in the cosets of the extended codes, which arise in
the case of $u_{0}^{i-1}\neq 0$.
###### Example 1.
Consider Arikan kernel $F_{2}=\begin{pmatrix}1&0\\\ 1&1\end{pmatrix}$. One
obtains $G^{(0)}=\begin{pmatrix}1&0&1\\\ 1&1&0\end{pmatrix}$ and
$G^{(0)}=\begin{pmatrix}1&1&1\end{pmatrix}$.
### III-B Recursive processing of polarization kernels
We propose to compute (3) by applying the RMLD algorithm to the cosets of the
extended kernel codes. Let
$\overline{\operatorname{{\mathcal{C}}}}(u_{0}^{i-1})=w+\overline{\operatorname{{\mathcal{C}}}}^{(i)}$
be the coset of $\operatorname{{\mathcal{C}}}^{(i)}$ given by prior decisions
$u_{0}^{i-1}$, where $w=(u_{0}^{i-1}K_{0,\dots,i-1},0)$, and $K_{0,\dots,i-1}$
is the matrix consisting of $i$ top rows of $K$. For any section $[x,y)$, the
cosets associated with states in the recursive trellis for
$\overline{\operatorname{{\mathcal{C}}}}(u_{0}^{i-1})$ are obtained from those
for $\overline{\operatorname{{\mathcal{C}}}}^{(i)}$ as
$D(u_{0}^{i-1})=\left\\{{f+w_{x}^{y-1}|f\in D}\right\\},D\in
p_{x,y}(\overline{\operatorname{{\mathcal{C}}}}^{(i)})/s_{x,y}(\overline{\operatorname{{\mathcal{C}}}}^{(i)})$.
It can be seen that
$p_{0,l}(\overline{\operatorname{{\mathcal{C}}}}^{(i)})/s_{0,l}(\overline{\operatorname{{\mathcal{C}}}}^{(i)})$
contains two cosets, which correspond to $u_{i}=0$ and $u_{i}=1$. Hence,
$S_{1}^{(i)}(u_{0}^{i-1},r_{0}^{l-1})=\frac{T_{0,l}[0].q-T_{0,l}[1].q}{2},$
(6)
where $T_{0,l}$ is the composite branch table constructed for
$\overline{\operatorname{{\mathcal{C}}}}(u_{0}^{i-1})$ given noisy vector
$r_{0}^{l-1}$. It is assumed in what follows that $CBT$ entries contain only
$q$ values, and the corresponding coset representatives $l(D)$ are omitted.
Furthermore, we propose to reuse the composite branch tables, or their parts,
obtained at successive phases $i$. To do this, we need to identify how CBTs
evolve with $i$, find a way to handle prior decisions $u_{0}^{i-1}$, design
efficient algorithms for construction of CBTs for short sections, and obtain
an optimal sectionalization strategy.
#### III-B1 Reusing the CBTs
Assume that the same sectionalization is used for all phases $i$. Obviously,
$p_{x,y}(\overline{C}^{(i+1)})\subset p_{x,y}(\overline{C}^{(i)})$ and
$s_{x,y}(\overline{C}^{(i+1)})\subset s_{x,y}(\overline{C}^{(i)}),i\in[l-1]$
for any $x,y$, such that $0\leq x<y\leq l$.
###### Lemma 1.
If $p_{x,y}(\overline{C}^{(i+1)})=p_{x,y}(\overline{C}^{(i)})$ and
$s_{x,y}(\overline{C}^{(i+1)})=s_{x,y}(\overline{C}^{(i)})$, then for any
$u_{i}\in\operatorname{\mathbb{F}}_{2}$ the composite branch table $T_{x,y}$
constructed for $\overline{\operatorname{{\mathcal{C}}}}(u_{0}^{i-1})$ is
identical to the one constructed for
$\overline{\operatorname{{\mathcal{C}}}}(u_{0}^{i})$, denoted
$T_{x,y}^{\prime}$, for the same received vector $r_{0}^{l-1}$.
###### Proof.
Since both section codes are identical for phases $i$ and $i+1$, the CBTs have
the same size. $p_{x,y}(\overline{C}^{(i+1)})=p_{x,y}(\overline{C}^{(i)})$
implies that $\kappa=(K_{i,x},\dots,K_{i,y-1})\in
p_{x,y}(\overline{C}^{(i+1)})$. Hence, for any
$u_{i}\in\operatorname{\mathbb{F}}_{2}$ both
$p_{x,y}(\overline{\operatorname{{\mathcal{C}}}}^{(i)})/s_{x,y}(\overline{\operatorname{{\mathcal{C}}}}^{(i)})$
and
$p_{x,y}(\overline{\operatorname{{\mathcal{C}}}}^{(i+1)})/s_{x,y}(\overline{\operatorname{{\mathcal{C}}}}^{(i+1)})$
have cosets containing $u_{i}\kappa,u_{i}\in\operatorname{\mathbb{F}}_{2}$.
This implies that both $T_{x,y}$ and $T_{x,y}^{\prime}$ contain the same
values. ∎
The above lemma suggests that one does not need to recompute the CBTs for
those sections, where section codes do not change from phase $i$ to phase
$i+1$.
In what follows, we assume that
$s_{x,z}(\operatorname{{\mathcal{C}}}^{(i+1)})=s_{x,z}(\operatorname{{\mathcal{C}}}^{(i)})\text{\,and\,}s_{z,y}(\operatorname{{\mathcal{C}}}^{(i+1)})=s_{z,y}(\operatorname{{\mathcal{C}}}^{(i)}).$
(7)
Let us assume temporarily that $u_{0}^{i-1}=0$.
Even if section codes do change, it is still possible to reuse some results
obtained at prior phases. Let
$k_{i,x,y}^{\prime}=k_{x,y}^{\prime}(\operatorname{{\mathcal{C}}}^{(i)})$ and
$k_{i,x,y}^{\prime\prime}=k_{x,y}^{\prime\prime}(\operatorname{{\mathcal{C}}}^{(i)})$.
First, observe that if $k_{i+1,x,y}^{\prime\prime}=k_{i,x,y}^{\prime\prime}$,
but $k_{i+1,x,y}^{\prime}<k_{i,x,y}^{\prime}$, then
$p_{x,y}(\operatorname{{\mathcal{C}}}^{(i+1)})/s_{x,y}(\operatorname{{\mathcal{C}}}^{(i+1)})\subset
p_{x,y}(\operatorname{{\mathcal{C}}}^{(i)})/s_{x,y}(\operatorname{{\mathcal{C}}}^{(i)})$,
so that the corresponding CBT at phase $i+1$ can be obtained as a subvector of
the CBT at phase $i$.
Second, we propose to implement maximization recursively, and keep all
intermediate results. More specifically, we propose to rewrite (5) as
$\displaystyle T_{x,y}[v].q=$
$\displaystyle\max_{w\in\operatorname{\mathbb{F}}_{2}^{k_{i,x,y}^{\prime\prime}}}\left(T_{x,z}[a].q+T_{z,y}[b].q\right)$
$\displaystyle=$
$\displaystyle\max_{w_{k_{i,x,y}^{\prime\prime}-1}}\dots\max_{w_{1}}\max_{w_{0}}\left(T_{x,z}[a].q+T_{z,y}[b].q\right).$
(8)
Instead of storing in the CBT the final results of maximization in (8), we
propose to keep the intermediate results of maximization for all $w$. These
values can be arranged in a binary tree for each
$v\in\operatorname{\mathbb{F}}_{2}^{k^{\prime}_{x,y}}$, so that a path from a
root in this tree can be specified by values
$w_{k_{i,x,y}^{\prime\prime}-1},w_{k_{i,x,y}^{\prime\prime}-2},\dots,w_{0}$.
By maximization forest we denote the set of such trees obtained at some phase
for a given section. The subtrees within a forest can be indexed by variables
$w,v$.
We propose to use the above described maximization tree constructed at some
phase $i_{0}$ to obtain CBTs for all $i\geq i_{0}$, where (7) holds. Let
$i_{1}>i_{0}$ be the smallest integer, where this does not hold.
###### Lemma 2.
Let
$G_{j,x,y}^{\prime\prime}=\begin{pmatrix}G^{(00)}_{x,y}&G^{(01)}_{x,y}\end{pmatrix}$
and
$G_{j,x,y}^{\prime}=\begin{pmatrix}G^{(10)}_{x,y}&G^{(11)}_{x,y}\end{pmatrix}$
be the matrices obtained from (4) for code
$\operatorname{{\mathcal{C}}}^{(j)}$ for any $j$. If all matrices
$G_{i,x,y}^{\prime\prime}$ are nested, so that $G_{i+1,x,y}^{\prime\prime}$
occupies top rows of $G_{i,x,y}^{\prime\prime}$ for any $i:i_{0}\leq i<i_{1}$,
then the maximization forest for phase $i:i_{0}\leq i\leq i_{1}$ can be
obtained by taking the subtrees of the trees in the forest constructed at
phase $i_{0}$, given by values $w$ and $v$ satisfying the equation
$\begin{pmatrix}w&v\end{pmatrix}\begin{pmatrix}G_{i_{0},x,y}^{\prime\prime}\\\
G_{i_{0},x,y}^{\prime}\end{pmatrix}=\begin{pmatrix}\overline{w}&\overline{v}\end{pmatrix}\begin{pmatrix}G_{i,x,y}^{\prime\prime}\\\
G_{i,x,y}^{\prime}\end{pmatrix},$ (9)
where
$\overline{w}\in\operatorname{\mathbb{F}}_{2}^{k_{i}^{\prime\prime},x,y},\overline{v}\in\operatorname{\mathbb{F}}_{2}^{k_{i}^{\prime},x,y}$
denote the subtree indices in the forest at phase $i$.
###### Proof.
Assumption (7) ensures that the maximization trees for all $i$ are obtained
from the same values.
Observe that the matrices in (9) together with the generator matrices of
$s_{x,z}(\operatorname{{\mathcal{C}}}^{(i_{0})})$ and
$s_{z,y}(\operatorname{{\mathcal{C}}}^{(i_{0})})$ constitute generator
matrices of $p_{x,y}(\operatorname{{\mathcal{C}}}^{(i_{0})})$ and
$p_{x,y}(\operatorname{{\mathcal{C}}}^{(i)})$, given by (4). Since
$p_{x,y}(\operatorname{{\mathcal{C}}}^{(i)})\subset
p_{x,y}(\operatorname{{\mathcal{C}}}^{(i_{0})})$, for any
$\overline{w},\overline{v}$ there is a unique solution
$\begin{pmatrix}w&v\end{pmatrix}=\begin{pmatrix}\overline{w}&\overline{v}\end{pmatrix}M_{i,x,y}$
of (9), where $M_{i,x,y}$ is a matrix, which can be constructed at the design
time. Since $G_{i,x,y}$ occupies $k_{i,x,y}^{\prime\prime}$ top rows of
$G_{i_{0},x,y}^{\prime\prime}$, one obtains $w_{j}=\overline{w}_{j},0\leq
j<k_{i,x,y}^{\prime\prime}$. Hence, for each $\overline{v}$ the corresponding
maximization tree indeed appears as a subtree indexed by $w_{j}$ of the $v$-th
maximization tree. ∎
###### Remark 1.
Observe that it is always possible to obtain $G_{j,x,y}^{\prime\prime}$ in the
form required by Lemma 2 by applying elementary row operations.
#### III-B2 Handling prior decisions
The SC decoder needs to take into account at phase $i$ the values
$u_{0}^{i-1}$. This reduces to decoding in cosets of section codes. The
corresponding coset representative for section $[x,y)$ can be computed as a
linear combination of subvectors
$\begin{pmatrix}K_{j,x}&K_{j,x+1}&\dots&K_{j,y-1}\end{pmatrix},0\leq j<i$, of
the kernel.
Consider first the case of section $[x,y)$, where the maximization forest (8)
is constructed from scratch. If there exists a solution of
$\begin{pmatrix}f_{j,x,y}&h_{j,x,y}\end{pmatrix}\begin{pmatrix}G^{(s)}_{i,x,y}\\\
G_{i,x,y}^{\prime}\end{pmatrix}=\begin{pmatrix}K_{j,x}&K_{j,x+1}&\dots&K_{j,y-1}\end{pmatrix},$
(10)
then the CBT for section $[x,y)$ contains at position $h_{j,x,y}$ the required
coset representative. In this case we assume $h_{j,x^{\prime},y^{\prime}}=0$
for all $x<x^{\prime}<y^{\prime}<y$. Otherwise, we assume $h_{j,x,y}=0$. Given
a vector of prior decisions $u_{0}^{i-1}$, we obtain the corresponding
position offset at section $[x,y)$ as
$h_{x,y}=\sum_{j=0}^{i-1}u_{j}h_{j,x,y}$, so that (8) becomes
$\displaystyle
T_{x,y}[v].q=\max_{w}\left(T_{x,z}[a+h_{x,z}].q+T_{z,y}[b+h_{z,y}].q\right).$
(11)
The values $h_{x,y}$ are similar to partial sums, which arise in the SC
decoder of Arikan polar codes.
For those sections, where the CBT is obtained by taking subtrees of (8), we
need to check if there is a solution of
$\omega_{j,x,y}\begin{pmatrix}G^{(s)}_{i_{0},x,y}\\\
G_{i_{0},x,y}^{\prime}\end{pmatrix}=\begin{pmatrix}K_{j,x}&K_{j,x+1}&\dots&K_{j,y-1}\end{pmatrix}.$
(12)
If this equation does not have a solution for some $j<i$, the corresponding
coset representatives have already been accounted for at smaller sections
while constructing the CBT for section $[x,y)$ at phase $i_{0}$. and we assume
$\omega_{j,x,y}=0$. Otherwise, the corresponding CBT can be obtained from the
maximization forest constructed at phase $i_{0}<i$ by taking entries
$\begin{pmatrix}w&v\end{pmatrix}=\begin{pmatrix}\overline{w}&\overline{v}\end{pmatrix}M_{i,x,y}+\sum_{j=0}^{i-1}u_{i}\omega_{j,x,y}.$
#### III-B3 Special trellises
It can be seen that the result of (6) does not change if the same value is
subtracted from all CBT entries at any section. Doing this enables one to
compute some CBTs with smaller number of operations compared to (11). The
following special cases were identified:
1. 1.
$k_{i,x,y}^{\prime}=k_{i,x,y}^{\prime\prime}=1,\hat{G}_{i,x,y}=\begin{pmatrix}1\\\
0\end{pmatrix},\tilde{G}_{i,x,y}=\begin{pmatrix}1\\\ 1\end{pmatrix}$. Let
$a=\frac{T_{x,z}[0].q-T_{x,z}[1].q}{2}$,
$b=\frac{T_{z,y}[0].q-T_{z,y}[1].q}{2}$. We propose to set
$T_{x,y}[0].q=\operatorname{sgn}(a)\operatorname{sgn}(b)\min(|a|,|b|)$ and
$T_{x,y}[1].q=-T_{x,y}[0].q$. If one can guarantee that
$T_{x,z}[1].q=-T_{x,z}[0].q$, then it is possible to further simplify
computation by setting $a=T_{x,z}[0].q$. Similar simplification applies to
$b$. This trick allows one to construct section CBT using just 1 comparison
operation, instead of 2 comparison and 4 summations for a straightforward
implementation.
2. 2.
$k_{i,x,y}^{\prime}=1,k_{i,x,y}^{\prime\prime}=0,\hat{G}_{i,x,y}=\tilde{G}_{i,x,y}=(1)$.
Using the same definitions as above, one obtains
$T_{x,y}[0].q=(-1)^{h_{x,z}}a+(-1)^{h_{z,y}}b,T_{x,y}[1].q=-T_{x,y}[0].q$.
3. 3.
$k_{i,x,y}^{\prime}=0,k_{i,x,y}^{\prime\prime}=2,\hat{G}_{i,x,y}=\begin{pmatrix}1\\\
0\end{pmatrix},\tilde{G}_{i,x,y}=\begin{pmatrix}0\\\ 1\end{pmatrix}$. Using
the same definitions as above, we propose to set
$T_{x,y}[0].q=a+b,T_{x,y}[1].q=b-a,T_{x,y}[2].q=-T_{x,y}[1].q,T_{x,y}[3].q=-a$.
4. 4.
If there is all-1 row in $G_{i,x,y}^{(s)}$, then one step of maximization can
be avoided in (8) by taking the absolute values of the corresponding terms.
This reduces the complexity of construction of the maximization tree by a
factor of 2.
Simplification tricks 1 and 2 together with the results in [14] fully
establish the equivalence of the proposed approach and the min-sum SC
algorithm for the case of $K=B_{\mu}\begin{pmatrix}1&0\\\
1&1\end{pmatrix}^{\otimes\mu}$, where $B_{\mu}$ is the bit reversal
permutation matrix.
#### III-B4 Optimal sectionalization
The total complexity of kernel processing is equal to
$\mathbf{C}=\delta_{i}+\sum_{i=0}^{l-1}c_{i,0,l},$ where $\delta_{i}$ is the
complexity of computing the final LLR from the obtained CBT, and $c_{i,x,y}$
is the complexity of construction of the CBT for section $[x,y)$ at phase $i$.
In most cases the former operation reduces to computing (6), i.e. costs 1
subtraction. However, if above described special trellises 1 or 2 arise at
section $[0,l)$, then $\delta_{i}=0$. Furthermore,
$c_{i,x,y}=\begin{cases}m_{i,x,y},\text{\, if CBTs for subsections can be
reused}\\\ m_{i,x,y}+c_{i,x,z}+c_{i,z,y},\text{\,otherwise,}\end{cases}$
where
$m_{ixy}=\begin{cases}0,\text{\,\,\,\,\,\, if forest reuse is possible}\\\
M_{j},\text{\, if type-$j$ special trellis is encountered,}\\\
2^{k_{ixy}^{\prime}+k_{ixy}^{\prime\prime}-f_{ixy}}+2^{k_{ixy}^{\prime}}(2^{k_{ixy}^{\prime\prime}-f_{ixy}}-1),\text{otherwise,}\end{cases}$
$f_{ixy}$ is equal 1 if the above described 4-th simplification trick is
applicable, and 0 otherwise, and $M_{j}$ is the complexity of type-$j$ special
trellis, $1\leq j\leq 3$. The first and second terms in the latter expression
represent the number of summations and comparisons, respectively. These
expressions can be used in the optimization algorithm given in [10] to obtain
splitting position $z$ for each section $[x,y)$, so that the overall
complexity is minimized. Observe that sectionalization should be optimized
jointly for codes $\overline{\operatorname{{\mathcal{C}}}}^{(i)}$ at all
phases $i$.
## IV Numeric results
Table I presents processing complexity for some polarization kernels. Here
$K_{l}$ denotes a kernel of size $l$.
TABLE I: Kernel processing complexity Kernel $K_{l}$ | $E(K_{l})$ | $\mu(K_{l})$ | State of the art | Proposed
---|---|---|---|---
| | Method | Add | Comp. | Add | Comp.
$K_{16}B_{4}$ [15] | $0.51828$ | $3.45$ | window | 95 | 86 | 131 | 105
$K_{32}B_{5}$ [15] | $0.521936$ | $3.417$ | window | 297 | 274 | 406 | 262
$K_{32}^{r}$[16] | $0.52194$ | $3.42111$ | Viterbi | 4536 | 9072 | 355 | 191
$K_{32}^{bch}$ [17] | $0.53656$ | $3.1221$ | Viterbi | 99745 | 199490 | 31079 | 28337
$K_{32}^{enbch^{\prime}}$ [18] | $0.53656$ | $3.1221$ | window | 2864420 | 32183 | 29873
$K_{20}^{\ast}$ [16] | $0.506169$ | $3.43827$ | Viterbi | 7524 | 15054 | 2893 | 2001
$K_{20}$ [16] | $0.49943$ | $3.64931$ | Viterbi | 1866 | 3756 | 289 | 189
$K_{24}^{\ast}$ [16] | $0.51577$ | $3.3113$ | Viterbi | 9922 | 19860 | 1621 | 1207
$K_{24}$ [16] | $0.502911$ | $3.61903$ | Viterbi | 2102 | 4218 | 241 | 124
For comparison, we present the complexity of the window-based and Viterbi
processing algorithms. To the best of our knowledge, no other processing
algorithm was published for these kernels. It can be seen that for kernels
given in [15] the proposed approach has slightly higher complexity compared to
the window processing algorithm. However, extension of the latter algorithm to
kernels of size other than $2^{\mu}$ is non-obvious, while the recursive
trellis algorithm can be applied to any kernel. Observe that the the proposed
approach provides huge complexity reduction compared to the Viterbi algorithm.
Figure 1: Performance and SCL decoding complexity Figure 2: Performance of
polar codes under SCL decoding
Figure 1 presents the performance and complexity of the successive
cancellation list decoding algorithm with various list size for some polar
subcodes [19]. It can be seen that the proposed approach allows one to obtain
better performance compared to the codes based on Arikan kernel with the same
decoding complexity. More specifically, $(1024,512)$ code based on kernel
$K_{32}^{r}$ appears to be better than the code based on Arikan kernel
starting from $FER=3\cdot 10^{-2}$, which corresponds to list size 3 and 7 for
$K_{32}^{r}$ and Arikan kernels, respectively. Polar subcode $(576,288)$ based
on kernel $K_{24}$ outperform punctured Arikan polar subcode starting from
$FER=3\cdot 10^{-3}$, which corresponds to list size 8 and 20, respectively,
and approaches the performance of the chained polar subcode [20]. High
processing complexity of kernel $K_{24}^{\ast}$ allows the corresponding polar
subcode to outperform punctured polar subcode with Arikan kernel only at
$FER=4\cdot 10^{-4}$, which corresponds to list size 14 and 192, respectively.
Figure 3: Performance and decoding complexity of long codes
Figure 3 presents performance (solid lines) and average decoding complexity
(dashed lines) of rate $1/2$ polar subcodes and 5G LDPC codes. Sequential [21]
and shuffled belief propagation [22] algorithms were used for decoding of
polar subcodes and LDPC codes, respectively. Complexity is reported in terms
of the number of summation and comparison operations for polar subcodes, and
number of summations and calls to $\log\tanh(x/2)$ for LDPC codes. Polar
subcodes were constructed using a mixture of kernels, as shown in the plot. It
can be seen that polar subcodes provide almost the same performance as LDPC
codes. At sufficiently high SNR, the decoding complexity of polar subcodes
appears to be lower compared to the corresponding LDPC codes. Observe also,
that the code based on the Arikan kernel requires very large list size to
obtain the performance comparable to the LDPC code. It also has higher
decoding complexity compared to the code based on large kernels, and has
inferior performance at high SNR.
## V Conclusions
In this paper a novel processing algorithm for large polarization kernels was
proposed. This algorithm relies on extensive reuse of the intermediate results
arising in the recursive maximum likelihood decoding algorithm for the codes
generated by submatrices of the considered kernel. The proposed algorithm can
be applied to kernels of arbitrary dimension, and has much lower complexity
compared to the Viterbi algorithm. Derivation of a processor for a given
kernel according to the proposed method involves matrix manipulations, which
should be performed once at the design time. Actual kernel processing reduces
to summation and comparison of the elements of some arrays, where the indices
of the operands are obtained as XOR of some pre-computed values, and partial
sums given by the decisions of the SC algorithm.
The proposed algorithm enables polar subcodes with well-designed large kernels
to provide better performance/complexity tradeoff compared to the codes based
on Arikan kernel as well as LDPC codes. However, finding large kernels with
good polarization properties and low processing complexity remains, in
general, an open problem.
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* [14] P. Trifonov, “Trellis-based decoding techniques for polar codes with large kernels,” in _Proceedings of IEEE Information Theory Workshop_ , 2019, invited paper.
* [15] G. Trofimiuk and P. Trifonov, “Window processing of binary polarization kernels,” _IEEE Transactions on Communications_ , 2021, to appear.
* [16] G. Trofimiuk, “A search method for large polarization kernels,” ArXiV:2101.10269, 2021.
* [17] E. Moskovskaya and P. Trifonov, “Design of BCH polarization kernels with reduced processing complexity,” _IEEE Communications Letters_ , vol. 24, no. 7, pp. 1383–1386, July 2020.
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|
# Qudits for Witnessing Quantum Gravity Induced Entanglement of Masses Under
Decoherence
Jules Tilly<EMAIL_ADDRESS>Department of Physics and Astronomy,
University College London, WC1E 6BT London, United Kingdom Ryan J. Marshman
Department of Physics and Astronomy, University College London, WC1E 6BT
London, United Kingdom Anupam Mazumdar Van Swinderen Institute, University
of Groningen, 9747 AG Groningen, The Netherlands Sougato Bose Department of
Physics and Astronomy, University College London, WC1E 6BT London, United
Kingdom
###### Abstract
Recently a theoretical and an experimental protocol known as quantum gravity
induced entanglement of masses (QGEM) has been proposed to test the quantum
nature of gravity using two mesoscopic masses each placed in a superposition
of two locations. If, after eliminating all non-gravitational interactions
between them, the particles become entangled, one can conclude that the
gravitational potential is induced via a quantum mediator, i.e. a virtual
graviton. In this paper, we examine a range of different experimental set-ups,
considering different geometries and the number of spatially superposed states
taken, in order to determine which would generate entanglement faster. We
conclude that without decoherence, and given a maximum distance $\Delta x$
between any two spatial states of a superposition, a set of two qubits placed
in spatial superposition parallel to one another will outperform all other
models given realistic experimental parameters. Furthermore, when a
sufficiently high decoherence rate is introduced, multi-component
superpositions can outperform the two-qubit set-up. This is further verified
with an experimental simulation, showing that $O(10^{3})$ measurements are
required to reject the no entanglement hypothesis with a parallel qubits set-
up without decoherence at a 99.9$\%$ confidence level. The number of
measurements increases when decoherence is introduced. When the decoherence
rate reaches $0.125$ Hz, 6-dimensional qudits are required as the two-qubit
system entanglement cannot be witnessed anymore. However, in this case,
$O(10^{6})$ measurements will be required. One can group the witness operators
to measure in order to reduce the number of measurements (up to ten-fold).
However, this may be challenging to implement experimentally.
††preprint: APS/123-QED
## I Introduction
Testing the quantum aspect of gravity is a central question of modern physics.
While many theories of quantum gravity have been developed, there remains no
consensus on how to unify the theories of general relativity and quantum
physics. However, the lack of experimental evidence to gravity being quantum
still remains an impediment in ongoing research [1].
A number of recent experimental proposals have focused on trying to unveil
General-relativity (GR) and post GR evidence [2, 3, 4, 5, 6, 7, 8, 9, 10], and
there has also been an initial attempt to rule out the semi-classical
treatment of quantum gravity in [11]. Even attempts of detecting the B-mode
polarisation of stochastic gravitational waves have uncertainties in the
initial conditions for the Universe, which does not provide a concrete test
for quantum nature of gravity [12].
In this regard, there has been recent progress in providing a razor-sharp
witness to test the existence of the quantum nature of a graviton in a table-
top experiment based on the observation that [13]:
* •
The mediator of the universal gravitational interaction occurs via a spin-2
massless graviton, and if the graviton is quantum, it will entangle the two or
more quantum matter states, and provide the static gravitational potential at
the lowest order in the graviton/matter loop expansion.
* •
The above statement strictly relies on two main assumptions: special
relativity and perturbative quantum field theory, which allows an off-
shell/virtual exchange of a graviton to mediate the gravitational force.
Based on this fact, a bonafide test for quantum nature of the gravitational
interaction has been proposed in [14], where the two mesoscopic masses were
allowed to interact in a spatially superposed quantum state via gravity. A
similar proposal has also been made in [15]. This has attracted significant
interest from the research community [16, 17, 18, 21, 19, 20, 21, 22, 23, 24,
25, 26], and an experimental initiative in creating macroscopic superposition
with Stern-Gerlach set-up [27]. The above proposal has been coined as Quantum
Gravity induced Entanglement of Masses (QGEM), which exploits the loophole
that as local operations and classical communications are unable to entangle
the two quantum states if they were not entangled, to begin with, quantum
communication is required to generate the entanglement as highlighted in [13].
How a non-local gravitational interaction [28, 29] entangles the two quantum
states of matter is also shown in [13].
This paper aims at analysing different possible set-ups for the QGEM proposal
in order to determine which will be most efficient to implement in a real
experiment. In particular, we consider how quickly different set-ups can
generate entanglement according to a generalised model of the QGEM experiment
and how many measurements would be required to witness that entanglement. In
addition, proposals surrounding the QGEM experiment have been limited to
qubits, so far. In this paper, we assess the implications of using quantum
objects of higher dimensions (three: qutrits, $D$: qudits). We test our
findings in the presence of decoherence, furthering the analyses presented in
[19], [21], [20] and [22].
Our key findings are that: the parallel set-ups [19] of the experiment
entangle faster than any other set-up considered; in the presence of
decoherence, using qudits may be beneficial for the experiment, and could even
be necessary; We provide an order of magnitude for the number of measurements
required to reach a 99.9% level of confidence (about 3.4$\sigma$) for
different decoherence rates (up to $0.125Hz$).
This paper will proceed as follows: Section II presents a generalised version
of the QGEM experiment, with arbitrary geometries and allowing for the use of
spatial qudits. In Section III we analyse the different set-ups proposed using
the entanglement entropy under the assumption of no decoherence. More
practical entanglement witnesses are introduced in Section IV and used to re-
analyse the experiment allowing for decoherence in Section V. Finally, a
statistical simulation of the results is presented in Section VI to
demonstrate how the required number of runs varies with the dimension of the
qudits, and the decoherence rate assumed.
## II QGEM with Qudits
Linear set-up.
Parallel set-up.
Figure 1: Schematic representation of the experiment, shown here using qubits.
On the left, the liner set-up presented in [14, 15, 20]. On the right, a
schematic for the parallel set-up initially presented in [19]
The QGEM experimental protocol [14, 20] ensured that the gravitational
interaction would dominate over the electromagnetic interactions and the
Casimir induced vacuum fluctuations. This also provides one of the primary
constraints on the experimental set-up, which we will not seek to modify in
this paper. Specifically, there must be some minimum distance $d$ maintained
between the two particles. A schematic of two potential forms of the
experiment is presented in Figure 1.
The superposition width (or distance between the left-most and right-most
superposition instance of each qubit) is labelled $\Delta x$, and was
originally suggested to be $\sim 250\mu m$[14]. While the superposition is
held, the qubits are maintained at a distance such that their inner-most
superposition instances are a least $d$ apart (with $d\sim 200\mu m$ [14]). If
$d<200\mu m$ then forces such as Casimir-Polder forces and van der Waals
forces can affect the overall state of the system: gravity is no longer the
only possible quantum mediator for interactions between the two objects. There
has been further work considered to mediate this (for instance, see: [20]),
however, as it is beyond the primary considerations made here, we will not
look to include this here as it does not affect any final conclusions.
The two qubits are held in this superposition state for a time $\tau$ ([14]
suggests $\tau=2.5s$) after which the spatial states are brought together. The
qubits are then measured to determine whether they were entangled by their
gravitational interactions during the superposition period.
Nguyen and Bernards proposed a nearly identical scheme [19], in which the
superposition positions of each qubit are aligned parallel to each other as
opposed to linearly in [14, 20]. A schematic for this can be found in Figure
1. This scheme was motivated by the fact that maintaining the distance between
the two qubits would be easier in the parallel case than in the linear case.
In the remainder of this paper, we take the experimental parameters to match
those proposed in Bose et al., that is, $d$ is always set to $\sim 200\mu m$
and the masses of the two qudits are always $\sim 10^{-14}kg$. Furthermore,
unless otherwise stated $\Delta x\sim 250\mu m$.
While both [14] and [19] have discussed the implementation of the respective
set-ups, none of the intermediary models has been considered, and no direct
comparison has been drawn with respect to their impact on how fast the qubit
pair entangles or whether qudits would result in faster entanglement.
We present a generalised model for the QGEM experiment, considering rotations
of each interferometer, centred on the innermost spatial states of the two
qudits (see Figure 2). The allowed space of the rotation angles, $\theta_{1}$
and $\theta_{2}$ must be restricted so that at no point the two masses come
within a distance $d$ of one another. The linear set-up and the parallel set-
up are special cases of the above, using the angles $\theta_{1}=\theta_{2}=0$
and $\theta_{1}=\frac{3\pi}{2},\theta_{2}=\frac{\pi}{2}$ respectively.
Our model also allows for the use of qudits, rather than qubits. That is,
using $D$ spatial superposition states (and equivalently spin states, assuming
Stern-Gerlach interferometry is used as previously proposed) where $D\geq
2|D\in\mathbb{N}$. Under these conditions, the generalised state of the system
resulting from the QGEM experiment can be written as:
$\Ket{\psi(t=\tau)}=\frac{1}{D}\sum_{p=0}^{D-1}(\Ket{p}\otimes\sum_{q=0}^{D-1}e^{i\phi_{pq}}\Ket{q}).$
(1)
Where: $\phi_{pq}$ is defined by:
$\phi_{pq}\sim\frac{Gm_{1}m_{2}\tau}{\hbar C_{pq}}.$ (2)
Figure 2: Top view for generalised QGEM model showing the maximum spread of
the qudits (Distance $C_{pq}$ in red). The parametrization of the set-up is
controlled by $\theta_{1}$ and $\theta_{2}$.
The value $C_{pq}$ for each superposition pair can be derived using simple
trigonometric rules and is given by
$C_{pq}=\sqrt{A_{p}^{2}+B_{q}^{2}-2A_{p}B_{q}\cos(\theta_{3})},$ (3)
where
$A_{p}=((D-1)-p)\frac{\Delta x}{(D-1)},$ (4) $\displaystyle B_{q}$
$\displaystyle=[d^{2}+\bigg{(}q\frac{\Delta x}{(D-1)}\bigg{)}^{2}$
$\displaystyle-2d\bigg{(}q\frac{\Delta
x}{(D-1)}\bigg{)}\cos{(\pi-\theta_{2})}]^{\frac{1}{2}},$ (5)
$\theta_{3}=\pi-\theta_{1}+\arcsin{\bigg{(}\frac{q\frac{\Delta
x}{(D-1)}\sin(\theta_{2})}{B_{q}}\bigg{)}}.$ (6)
## III Entanglement entropy test
To assess different set-ups for the QGEM experiment, we begin by comparing the
von Neumann entropy (VNE), or entanglement entropy, of the output state. We
recall that noting $\rho_{1}$ the partial trace over the first qudit of the
two-qudit system $\rho$, the entanglement entropy is given by [30]:
$S(\rho)=-Tr[\rho_{1}log_{2}\rho_{1}],$ (7)
or, using the eigen-decomposition of $\rho_{1}$:
$\rho_{1}=\sum_{j}\lambda_{j}\Ket{j}\Bra{j}$ we can re-write $S(\rho)$ as:
$S(\rho)=-\sum_{j}\lambda_{j}log_{2}(\lambda_{j}).$ (8)
For a $D$ level system, the entanglement entropy is bound by:
$0\leq S(\rho)\leq log_{2}(D),$ (9)
From which it follows that more entanglement can be produced from higher
dimension systems. However, this does not provide any information on how
quickly such entanglement is created. Here we focus on the linear and parallel
set-ups, a discussion regarding all other alternatives is presented in
Appendix A.
Parallel set-up
Linear set-up
Figure 3: Entanglement entropy scalling with the state dimension for $D=2$ to
$D=6$. On the left, we present the results for the parallel set-up, on the
right we present the results for the linear set-up.
In order to assess whether qubits perform better or worse than higher
dimensional qudits, we compute the entanglement entropy for $\rho_{para}$ and
$\rho_{lin}$ using the generalised version of the model for 2 to 6 dimensional
qudits. The resulting entropy scaling with time is plotted in Figure 3. The
parallel set-up appears to perform significantly better than the linear set-up
in realistic experiment times.
At the proposed experimental time of $2.5$s, the qubit parallel case achieves
a entanglement entropy of $0.152$, much larger than that for the qutrit case
of $0.084$. Going to higher dimensions further reduces the entanglement
entropy to $0.068$, $0.060$ and $0.056$ for 4, 5 and 6 dimensions
respectively111Note that, at $\tau=2.5s$, as $D\to\infty$ the entanglement
entropy converges to $\sim 0.039$ suggesting that, under this model, multi-
component superpositions do not entangle faster through gravity.
Of course, given the superposition phases for the state resulting from the
QGEM experiment are periodic, the linear set-up will achieve a higher
entanglement entropy than the parallel set-up for sufficiently large
experiment times.
The entanglement entropy is not a valid metric for entanglement if classical
mixing or decoherence are affecting the system as these are indistinguishable
from entanglement as a source of entanglement entropy. Therefore, entanglement
entropy is only used when assuming the system is at all times in a pure state.
As such, for more realistic experiments, it is necessary to consider other
methods of witnessing the entanglement.
## IV Entanglement Witness tests
Multiple external factors can affect the system state throughout the
experiment, these include decoherence and classical uncertainties introduced
by the hardware used to implement the experiment. Entanglement Witnesses
provide a convenient testing system in the context of an experiment.
The Positive Partial Transpose (PPT) entanglement witness is an appropriate
witness for two for Negative Partial Transpose (NPT) entangled qudits:
$\mathcal{W}_{ppt}=\Ket{\lambda_{-}}\Bra{\lambda_{-}}^{T}$ (10)
In the parallel qubit case, this witness is simply [21]:
$\mathcal{W}_{ppt}=\frac{1}{4}[\mathbb{I}-X\otimes X-Z\otimes Y-Y\otimes Z]$
(11)
It is worth noting however that although this witness is not optimal in the
linear set-up, having fewer terms to measure results in less variance when
conducting the experiment (for a given number of measurements) (see Section
VI). Besides, all of the operators in the witness commute, as a result, there
exist a measurement basis in which one can derive the expectation value of all
these operators, reducing the number of terms to measure to one.
When considering the qudit case, there exist states which are entangled but
can not be detected by a PPT witness (see for example [32, 33]). These states
have also been referred to as bound entanglement, and multiple methods have
been developed to assess them [34, 35, 36, 37, 38, 39]. The witnesses
described above may therefore not be sufficient to test entanglement of
$D\otimes D$ systems (with $D>2$) as they would, by construction, fail to
detect Positive Partial Transpose entangled states (PPTES). These can
nonetheless be useful in case the theoretical density matrix of the state
created by the experiment is indeed NPT.
There is currently no general witness construction strategy for detecting
PPTES (also known as entangled bound states) and research is focused on
designing witnesses specific to certain families of states. Thankfully bound
entanglement represents only a small proportion of all entangled states, and
remains unlikely to occur in the proposed experiment.
We computed the PPT entanglement witness expectation value for $D$ dimensional
qudits with $D=2$ to $6$. The results are presented in Figure 4 for both the
parallel and linear cases.
Figure 4: Expectation value of PPT entanglement witness for the parallel set-
up with state dimension for $D=2$ to $D=6$
The expectation value of the PPT entanglement witness is nearing $-0.148$ at
$\tau=2.5s$ for qubits. The witness values are slightly higher for higher
dimensional qudits, echoing the findings of the entanglement entropy test.
These witnesses also appear to be finest for the set of states that are
produced by each set-up. A witness $\mathcal{W}_{A}$ is set to be finer than
another witness $\mathcal{W}_{B}$ if it detects as entangled all the states
detected by $\mathcal{W}_{B}$ and at least one more (see Ref. [40]). In this
case, the witnesses computed are clearly able to detect all entangled states
produced (for any $\tau$) albeit with values very close to zero for low values
of $\tau$.
Following the above, we can also note that the high dimension states generated
by the QGEM experiment set-ups considered must be NPT entangled states;
otherwise, $\mathcal{W}_{ppt}$ would fail to detect them as entangled.
A further consideration regarding the entanglement witness is how many
operators it needs to be broken into to be measured experimentally. For this,
we can consider the scaling of generalised Gell-Mann matrices, which is a
generalisation of the Pauli basis to quantum states of dimensions higher than
2. There are $D^{2}$ element for a set of Gell-Mann matrices for a $D$
dimensional quantum state. Because we build a system composed of two quantum
objects, we are therefore looking at a maximum total number of operators of
$D^{4}$ for the entanglement witness.
We found that the entanglement witness derived from the PPT principle has, in
general, a slightly lower than that number (due to the weights of certain
operators being negligible or equal to zero in the decomposition of the
entanglement witness).
We can further significantly improve on this number by grouping together the
operators that can be jointly measured. In general, operators can be jointly
measured if they can be diagonalised together in a specific Tensor Product
Basis (TPB) (for an overview of the method: [41]. Conveniently this is
equivalent to saying that operators can be jointly measured if they commute.
Commutation is not transitive, and as such, there may be many different
solutions to grouping operators together. To find a good solution, we use the
Largest Degree-first Colouring (LDFC) algorithm (for a good summary of the
LDFC algorithm, use the Supplementary materials in [42]), whereby group are
composed starting from one of the operators which have the highest number of
commuting operators. The results we obtained are summarised in Table 1.
D | PPT witness | PPT witness (grouped)
---|---|---
2 | 4 | 1
3 | 77 | 14
4 | 244 | 28
5 | 613 | 53
6 | 1272 | 94
Table 1: Number of operators in the generalised Pauli decomposition of the
witnesses, that must be measured to estimate the expectation value of the
entanglement witness in the parallel case for $D=2$ to $D=6$. The grouped
column present the number of operator groups that can be jointly measured in a
single Tensor Product Basis, obtained using the Largest Degree First Coloring
(LDFC) algorithm. Given this algorithm is a heuristic, one could find a
different set and number of groups.
One point to note is that measuring operators jointly requires finding and
implementing a joint measurement basis. While straightforward in some cases,
this could yield some significant complications in an actual experiment.
## V Testing models with decoherence
So far we have considered the case where both qubits are only coupled with
each other through their positional superposition. Real experimental setting
cannot however fully remove the potential for coupling of the studied quantum
system with the environment.
The particles’ coherence and hence joint entanglement erodes over time due to
interaction with the environment. This results in decoherence of the
positional qudits into a single, defined position or a classical mixture of
differing but well defined positions.
We schematically incorporate this in our model by adding a time dependent
exponential decay to all off diagonal term of each qudit’s density matrix,
parametrised by the decoherence rate $\gamma$. This is under the assumption
that in any experiment $\frac{\Delta x}{D}\gg\lambda_{\textrm{dB}}$ where
$\lambda_{\textrm{dB}}$ is the masses de Broglie wavelength. For a generic
qudit, with dimension $D$ and density matrix $\rho_{d}$, we can write:
$\leavevmode\resizebox{173.44534pt}{}{{\hbox{ \rho_d = \begin{bmatrix}c_{11}
&c_{12} &\vdots&c_{1(d-1)} &c_{1d} \\\ c_{21} &c_{22} &\vdots&c_{2(d-1)}
&c_{2d}\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\ c_{(d-1)1} &c_{(d-1)2}
&\vdots&c_{(d-1)(d-1)} &c_{(d-1)d} \\\ c_{d1} &c_{d2} &\vdots&c_{d(d-1)}
&c_{dd} \end{bmatrix} }}},$ (12)
as such, following the model described above, decoherence is incorporated as:
$\leavevmode\resizebox{195.12767pt}{}{{\hbox{ \rho_d
\rightarrow\begin{bmatrix}c_{11} &c_{12}e^{-\gamma\tau}
&\vdots&c_{1(d-1)}e^{-\gamma\tau} &c_{1d}e^{-\gamma\tau} \\\
c_{21}e^{-\gamma\tau} &c_{22} &\vdots&c_{2(d-1)}e^{-\gamma\tau}
&c_{2d}e^{-\gamma\tau}\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
c_{(d-1)1}e^{-\gamma\tau} &c_{(d-1)2}e^{-\gamma\tau} &\vdots&c_{(d-1)(d-1)}
&c_{(d-1)d}e^{-\gamma\tau} \\\ c_{d1}e^{-\gamma\tau} &c_{d2}e^{-\gamma\tau}
&\vdots&c_{d(d-1)}e^{-\gamma\tau} &c_{dd} \end{bmatrix} }}}.$ (13)
The overall two-qudit system density matrix is then computed using:
$\rho_{system}$ = $\rho_{d}^{(1)}\otimes\rho_{d}^{(2)}$.
We computed the expectation value of $\mathcal{W}_{ppt}$ for dimensions 2 to 6
for incremental values of $\gamma$. The results are plotted in Figure 5.
Figure 5: Expectation value of PPT entanglement witness as a function of the
decoherence rate in the parallel set-up and with $D$ ranging from 2 to 6.
Interestingly, higher dimension models appear more resilient to decoherence
than the qubit case. It is worth noting that increasing decoherence in the
model also reduces the ‘optimal’ time for the experiment, that is, the time at
which the entanglement witness is most negative, and the ability to detect
entanglement with longer experiment times. To be better understand the
interplay between time, decoherence and the number of dimensions, the expected
value of $\mathcal{W}_{ppt}$ was computed at two different values for the
decoherence rate: $0.1$Hz and $0.125$Hz.
$\gamma=0.1$Hz
$\gamma=0.125$Hz
Figure 6: Entanglement witness expectation value over time for the parallel
set up and with $D$ ranging from 2 to 6. We present two different possible
values of $\gamma$: 0.1Hz, and 0.125Hz
The advantage of higher-dimensional models, when decoherence is increased,
become significant when $\gamma\gtrsim 0.1$Hz. This suggest that in a real run
of the experiment, multi-component superposition may be preferable, or even
necessary, if decoherence is sufficiently high. We can also observe as
expected that longer time becomes detrimental for high decoherence rates. We
also considered the problem of optimising the experiment run-time to maximise
the decoherence rate for which the entangled witness could detect the state as
entangled. In Appendix C we plot the expectation value of $\mathcal{W}_{ppt}$
for different values of $\tau$, including decoherence. The result is that
$\tau=2.5$s is near optimal for qubits and qudits, though lower time performs
marginally better for the latter. Of course the experiment can be run for
shorter times without any negative impacts provided a detectable level of
entanglement has developed.
While higher dimension set-ups are more resilient to decoherence, one key
question in this analysis is to determine what number of measurements will be
required to reject that the experiment state is not entangled. In order to
further this analysis, we now need to fully simulate the experiment in order
to determine the required number of measurements and run-time of the QGEM
experiment. This is the object of the next section.
## VI Experiment simulation
In this section, we present experiment simulations used to estimate the number
of measurements that will be required to reject the hypothesis that the qubit
pair is not entangled. Failure to reject only means that it is impossible at
this stage to confidently test for gravitationally mediated entanglement.
To compare the entanglement witnesses, and the different model proposed, we
simulate an experiment, as described below:
1. 1.
The entanglement witness is decomposed into a weighted sum of generalised
Pauli tensors terms (Gell-Mann matrices for qutrits, and generalised $d$
dimension Pauli operators for qudits). (We used the method for generalisation
of Pauli operators described in Thew et al. [43]). In some cases, we group the
operators (following the groups described in Table 1).
2. 2.
The quantum state resulting from the experiment is measured a pre-determined
number of times against each of the Pauli terms or group, to compute (1) their
expectation value, (2) the standard error of the measurement series. To
minimise the variance of the observable for a given number of measurements, we
have weighted the number of measurement in proportion to the weight of each
Pauli tensor terms (or group) in the decomposition of the witness.
3. 3.
Details on how confidence levels are computed can be found in appendix D.
The plots presented in this section are an average over many single runs of
each numerical experiment simulation. As such, these should be representative
of a typical run of the experiment. The conclusions drawn from these are only
meant to indicate the order of magnitude of the number of measurements
required in order to define the most adequate experiment set-up.
We first describe the result obtained from the experiment simulation with
qubits. The simulation is first run comparing the linear and parallel set-ups
with no decoherence (Figure 7 followed by plots of the simulation for
increasing decoherence rate $\gamma$ in Figure 8. We have used
$\mathcal{W}_{ppt}$ with full basis decomposition in this first experiment
simulations, holding $\Delta x=250\mu m$ and $\tau=2.5s$.
Figure 7: QGEM experiment simulation - linear and parallel set-ups (no
decoherence) - Expected level of confidence (probability of the witness value
actually being negative) in the qubit case. We added the case in which the
parallel witness is used for the linear set-up showcasing the benefits of
having a lower number of terms to compute
As we can see from Figure 7, with no decoherence, it takes about 500
measurements to reject the hypothesis that the two states are not entangled at
a 99.9% confidence level in the parallel case and at least 3,000 measurements
for the linear set-up. Using the witness derived in the parallel set-up
marginally improves the results of the linear case, although not sufficient
for it to be comparable to the parallel version of the experiment. This
further confirms that the parallel set-up will be preferable in a real
experiment and we, therefore, discard the linear set-up in the remainder of
the simulations.
We can expect that incorporating decoherence will increase the number of
measurements required as it pushes the expectation value of
$\mathcal{W}_{ppt}$ upward. The result of the qubit experiment simulations
with decoherence are presented in Figure 8. It illustrates the rapid increase
in the number of measurements required as the decoherence rate is raised. For
$\gamma=0.05$Hz, the parallel set-up still only require about $\sim 2,000$
measurements. This figure goes up to $\sim 6,000$ measurements for
$\gamma=0.075$Hz. At $\gamma=0.1$Hz, the experiment would required at least
$25,000$ measurements. At this decoherence rate, qubits have fewer negative
expectation values for $\mathcal{W}_{ppt}$ than some qudits, and at
decoherence rate above $\gamma=0.12$Hz, the expectation value of the witness
is positive (that is, no entanglement is detected and therefore the results
are not plotted).
Figure 8: QGEM Experiment simulation with qubits - Confidence levels for three
decoherence level (0.05Hz, 0.075Hz and 0.1Hz) as a function of the number of
measurements available. Here the witness as expectation value of -0.074,
-0.043, and -0.016, respectively
We repeated the experiment simulations as described above with differing
decoherence rates in the case of qudits. We have used the $D=6$ qudit case for
illustration. For brevity, we will denote them as 6-qudits, and use an
analogous nomenclature for other dimensions. As for the qubits simulations,
all the experiments use the parallel set-up, $\Delta x=250\mu$m and
$\tau=2.5$s.
Figure 9: QGEM Experiment simulation with D=6 qudits - Confidence levels for
three decoherence level (0.05Hz, 0.075Hz and 0.1Hz) as a function of the
number of measurements available. Here the witness as expectation value of
-0.045, -0.032, and -0.021, respectively
As expected, for the 6-qudits case, the number of measurements required is
significantly higher. This is primarily due to the large number of additional
terms to compute. Our results are presented in Figure 9 and show that for
$\gamma=0.05$Hz, over 200,000 measurements would be needed, while nearly
400,000 would be required if $\gamma=0.075$Hz, and about 600,000 for
$\gamma=0.1$Hz.
One last test that is worth considering is the situation in which grouping of
terms is allowed (this is subject to being able to produce the relevant
measurement bases experimentally, as mentioned in Section IV).
Figure 10 presents the results for the qubits case. We can already see the
drastic reduction in the total number of measurements required, resulting from
the reduction of terms to measure from 4 (in reality 3 since the identity term
does not need to be measured), to 1. Less than 1,000 measurements and 2,000
measurements are necessary when $\gamma=0.05$Hz and $\gamma=0.075$Hz,
respectively. Similarly, only about 12,000 measurements is required if
$\gamma=0.1$Hz.
Figure 10: QGEM Experiment simulation with qubits (operators grouped) -
Confidence levels for three decoherence level (0.05Hz, 0.075Hz and 0.1Hz) as a
function of the number of measurements available. Here the witness as
expectation value of -0.074, -0.043, and -0.016, respectively
A very similar pattern can be seen in the qudit case. The results for 6-qudits
are presented in Figure 11. number of measurements are drastically reduced
when operators are grouped. About 25,000 measurements are needed for
$\gamma=0.05$Hz, while $\gamma=0.075$Hz requires less than 40,000
measurements, and $\gamma=0.1$Hz less than 80,000.
Figure 11: QGEM Experiment simulation with 6-qudits (operators grouped) -
Confidence levels for three decoherence level (0.05Hz, 0.075Hz and 0.1Hz) as a
function of the number of measurements available. Here the witness as
expectation value of -0.045, -0.032, and -0.021, respectively
There are clearly no reasons to use high dimensional qudits unless the
decoherence rate is such that the entanglement witness for qubits always has a
positive expectation value. In the experiment setting, we have assumed, this
happens at about $\gamma\approx 0.12$Hz. We therefore present in Figure 12 an
experiment simulation showing how 6-qudits perform in the window of
decoherence rate in which high dimensional qudits become relevant.
Figure 12: QGEM Experiment simulation with 6-qudits - Confidence levels for a
decoherence rate of 0.125Hz as a function of the number of measurements
available in a case where the operators are grouped, and not.
At $\gamma=1.25$Hz, it is clear that the 6-qudits states still require a very
large number of measurements in order to reject the null hypothesis. However,
grouping the operators to measure in joint measurement bases allow reducing
the number of measurements required from nearly 2,000,000 to slightly above
200,000. It is worth noting that at this rate, the qubits case would not work
as decoherence pushes the expectation value of the witness above zero, and the
4-qudits and 5-qudits case are too close to zero to reach confidence of
$99.9\%$ in a realistic number of measurements.
## VII Conclusion
This paper considered modifying the set-up proposed in the original QGEM
experiments to determine how best to generate, protect and detect entanglement
in a future experiment. We looked at two aspects, in particular, the geometric
set-up of the experiment, and the number of dimensions of the quantum objects
used and developed a generalised mathematical model of the experiment.
Based on this model, and using entanglement entropy, we concluded that the
parallel qubits set-up generates entanglement the fastest for realistic
experiment run-times ($\tau$ $O\left(1\mathrm{s}\right)$). As entanglement
entropy cannot account for entanglement once decoherence is introduced, we
presented an entanglement test based on entanglement witnesses. We concluded
that as the decoherence rate is increased higher dimension, qudits finally
out-perform qubits by providing lower expectation values for the witnesses.
To estimate which set-up would require the least number of measurements to
evidence entanglement, we simulated experiments to define a confidence level
for the negativity of the expectation value of the witness given a certain
number of measurements. Without noise, at a 99.9$\%$ certainty level, the
parallel qubits set-up requires less than 2,000 measurements (1,000 when
grouped) to reject the null hypothesis (that the state is not entangled). The
number of measurements required increases rapidly when decoherence is
introduced.
Qudits of higher dimensions only become more useful than qubits when the
expectation value of the witness for qubits becomes non-negative. That is
because the number of basis elements to be estimated to calculate the witness
expectation value increases quadratically in the number of dimensions
($d^{4}$). In the experiment settings proposed, at $\tau=2.5$s and $\Delta
x=250\mu$m, qudits of dimension 6 are more favourable than qubits when the
decoherence rate is $\gamma\sim 0.125$Hz, however, in this case, over
2,000,000 measurements will be required (200,000 when operators are grouped).
Thus to further improve the experiment design, the following points should be
considered. Clearly, reducing the qudit-pair exposure to the environment to
decoherence would render the experiment more economical. The total expected
decoherence rates in specific, realistic experimental designs must be
estimated in order to confirm whether higher dimension qudits will be
required. Finally, any increase in the superposition width $\Delta x$ is
significantly beneficial as it both improves the entanglement generation rate,
allowing a significant reduction in overall run-time, which therefore also
minimises the impact of decoherence.
## VIII Acknowledgements
JT is supported by the UK EPSRC grount No. EP/R513143/1. RJM is supported by a
University College London departmental studentship. AM’s research is funded by
the Netherlands Organisation for Science and Research (NWO) grant number
680-91-119. SB would like to acknowledge EPSRC grants No. EP/N031105/1 and
EP/S000267/1.
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* [48] Nguyen and Bernards provide a threshold value for the phase of the parallel set-up for which it would violate CHSH inequalities, including decoherence [19]. Further research would be necessary in order to conduct a similar test for qudits.
APPENDICES
## Appendix A Discussion on the geometrically generalised model
There are a number of parameters in the generalised formula that can be
modified and tested for more effective entanglement generation. We have
isolated two which offer particular insights, the superposition width ($\Delta
x$) in Section A.0.2 and the rotation angles ($\theta_{1}$ and $\theta_{2}$)
in section A.0.1. Prior to this, however, there are a few points that are
worth noting:
1. 1.
It is clear that more time can allow achieving maximally entangled states,
however increasing the experimental run time is unrealistic due to greater
risk of decoherence. This point is covered in more details in Section V.
2. 2.
Reducing the minimum distance $d$ also clearly generates much faster
entanglement. However, as mentioned before this is not necessarily useful in
practice. This has been considered by others [20], and any results here will
also hold for such modified set-ups.
3. 3.
Using more massive quantum objects result in higher relative phases and in
faster entanglement generation. However, more massive objects would also mean
more challenging implementation for the interferometry, larger particle radii
and hence higher Casimir-Polder forces. This could, in turn, increase the
minimum distance $d$ and overall negatively impact entanglement growth. This
is not something we will consider further.
### A.0.1 Comparing rotation angle set-ups
In the main body of the paper, we have only presented the linear and parallel
cases. Based on our results, the parallel set-up entangles the qudits faster.
For completeness, and to allow for further modification in the implementation
by experimentalists, we also considered a range of additional geometries for
the set-up that can be implemented using the generalised QGEM model derived
previously.
The heat-map presented in Figure 13 displays the entanglement entropy for all
possible combinations of $\theta_{1}$ and $\theta_{2}$ as defined in Figure 2,
having set $\Delta x=250\mu$m and $\tau=2.5$s. The dark blue dot represents
the two possible parallel set-ups
($\theta_{1}=\frac{3\pi}{2},\theta_{2}=\frac{\pi}{2}$ and
$\theta_{1}=\frac{\pi}{2},\theta_{2}=\frac{3\pi}{2}$), while the yellow dot
represents the linear set-up ($\theta_{1}=\theta_{2}=0$). On the heat-map,
blue represents low entanglement entropy while white represents entanglement
entropy nearing 1.0. It appears from this figure that higher entanglement
entropy could possibly be achieved with alternative set-ups; however, this is
without considering that some combinations result in some superposition
instances being under $200\mu m$ and therefore subject to non-negligible
Casimir-Polder forces (represented by the shaded areas on the figure).
Figure 13: Entanglement entropy for all possible combinations of $\theta_{1}$
and $\theta_{2}$. The areas of reduced contrast (the region around the center
$+$) are forbidden based on the requirement that the states do not come too
close.
### A.0.2 Impact of superposition width ($\Delta x$) on entanglement entropy
Figure 14 shows that a larger superposition width results in faster
entanglement entropy growth. To illustrate this point we can consider the
phases for the qudit parallel set-up: $\phi_{pq}$ is smaller or equal to
$\phi$, reaching equality at $\Delta x=0$m, as $\Delta x$ increases,
$\phi_{pq}$ decreases resulting in all the phase factors $\Delta_{\phi_{pq}}$
to become more negative than they already are and as such accelerating
entanglement generation.
Figure 14: Entanglement entropy for the parallel and linear set-ups in the
qubit case as a function of the superposition width ($\Delta x$)
The superposition width, however, is limited in practice by experimental
considerations such as the magnetic field gradient achievable. This suggests
that by further modifying the arrangement into a parallel set-up one can
further reduce the magnetic field gradient and/ or other experimental
parameters while maintaining a detectable level of entanglement.
Using higher dimension spin quantum objects would, in theory, result in larger
possible superposition width due to the spin-dependent nature of the magnetic
field gradient coupling. This suggests that higher-dimensional objects would
perform better than qubits if their maximum superposition width is higher.
However, once a larger spin object is created, one can simply initialise the
state including only the outer spin states and hence recreating a qubit state
which performs better at an equivalent value of $\Delta x$. For illustration
purposes, Figure 15 represents a plot of entanglement entropy against time for
qudits in which $\Delta x$ is scaled to the number of dimensions. This
somewhat unrealistic set-up leads to higher-dimensional objects perform
significantly better.
Figure 15: Entanglement entropy as a function of time for $D$ ranging from 2
to 6. In this case, we have scaled $\Delta x$ to the number of dimensions
As such, the analysis allows to conclude that:
1. 1.
Qubits entangle faster than qudits for values of $\tau$ and $\Delta x$
realistic for the experiment.
2. 2.
Parallel set-ups entangle faster for all dimensions.
3. 3.
The superposition width $\Delta x$ will be the variable of interest in terms
of obtaining faster entanglement or a more easily implementable experiment.
## Appendix B Alternative witness: Vicinity witness for Negative Partial
Transpose entangled states
We consider an alternative entanglement witness, which is built to detect
entangled states in the vicinity of a known entangled pure state. Constructing
this witness amounts to finding a value for $\alpha$ such that
$Tr(\mathcal{W}_{vic}\varrho)\geqslant 0$ for all separable states, with
$\mathcal{W}_{vic}$ given by:
$\mathcal{W}_{vic}=\alpha\mathbb{I}-\Ket{\psi}\Bra{\psi}$ (14)
The maximum value of $\alpha$ is then derived as the square of the maximum
Schmidt coefficient of the pure state [45]. Noting $\lambda_{m}$ the highest
of these coefficients, we can re-write the witness as:
$\mathcal{W}_{vic}=\lambda_{m}^{2}\mathbb{I}-\Ket{\psi}\Bra{\psi}$ (15)
We can now compare the performance of the PPT based witness
$\mathcal{W}_{ppt}$ and the ‘vicinity’ witness $\mathcal{W}_{vic}$. This
analysis is restricted to the qubit, parallel version of the QGEM experiment
as it appears to be the most optimal set-up.
Figure 16: Expectation value of PPT and Vicinity entanglement witnesses as a
function of time in the qubit case
Comparing the expectation value of the PPT-based witness and the vicinity
witness, the former exhibits clearly much lower values on short time frames.
As such, it is the recommended entanglement witness for the implemented
experiment. In Table 2, we show that the vicinity witness, in general,
requires to measure fewer terms than the PPT witness - it is not sufficient
however to make it more advantageous than the PPT witness in an experimental
model, and therefore we have mostly not included it in our analysis.
There also exist numerous other entanglement witnesses, often based on
existing separability criteria, which will not be treated here as not directly
relevant to the proposed research.222For an overview of several entanglement
witnesses, Ref. [44]
D | PPT (para.) | PPT (lin.) | Vicinity (para.)
---|---|---|---
2 | 4 | 9 | 6
3 | 77 | 81 | 60
4 | 244 | 256 | 211
5 | 613 | 625 | 547
6 | 1272 | 1296 | 1166
Table 2: Number of operators to be measured to estimate the expectation value
of the entanglement witness in the parallel case for $D=2$ to $D=6$ for the
parallel case and linear case using the PPT entanglement witness, and for the
parallel case using the Vicinity entanglement witness.
A possible way to reduce the number of operators to be measured would be to
find a sub-optimal witness with a lower number of terms to measure. One
example is an approach similar to Bell inequalities. These do not detect all
the entangled state as it is focused only on identifying states which cannot
be explained through Local Hidden Variable (LHV) models. In [47], Hyllus et
al. show that it is possible to convert an optimal entanglement witness into a
CHSH type inequality. The resulting inequality detects non-LHV states
optimally but not entangled LHV states.
Therefore, there is a trade-off between the optimality of the entanglement
witness (the finest entanglement witness being the optimal witness for a given
entanglement detection problem) and the overall number of measurements
required to test entanglement. The conversion method developed in Hyllus et
al. only relates to qubits and does not take into account the impact of
decoherence on the detectability of entanglement [47]. However, for the qubit
case, given the witness used only as three terms that need to be measured,
CHSH types inequality are unlikely to provide a significant benefit.333Nguyen
and Bernards provide a threshold value for the phase of the parallel set-up
for which it would violate CHSH inequalities, including decoherence [19].
Further research would be necessary in order to conduct a similar test for
qudits.
## Appendix C Time trade-off with decoherence
In the original paper, Bose et al. suggest an experiment run-time of
$\tau=2.5s$. In this appendix, We consider a simple way to estimate which run-
time produce the highest resilience to decoherence (which run-time results in
the expectation value of the entanglement witness reaching $0$ for the highest
decoherence rate). Figure 17 presents the case of qubits, while Figure 18
present the case of 6-dimensional qudits.
Figure 17: Expectation value of PPT entanglement witness for different runtime
of the experiment and as a function of the decoherence rate in the case $D=2$
On the qubit figure we can that additional time as little impact on how high
the decoherence rate can be allowed to be. It does, however, offer more
negative expectation values for the entanglement witness. If the experiment
decoherence rate is estimated, one can then verify whether additional time can
reduce the overall number of measurements required.
Figure 18: Expectation value of PPT entanglement witness for different runtime
of the experiment and as a function of the decoherence rate in the case $D=6$
The effect is more pronounced in the 6-dimensional qudits case as lower time
offer negative expectation values of the witness for higher decoherence rate.
## Appendix D Computing confidence interval for a quantum observable
In order to compute statistics related to the expectation value of a quantum
observable, we first need to deconstruct the witness we are trying to estimate
into a weighted sum of observable that can be directly measured, i.e. a set of
Pauli strings.
In particular, in the case of our two qudit system we have at most $D^{4}$
terms. These terms and the tensor product composite of the list of Gell-Mann
matrices of dimension $D$ which are numbered $\mathcal{D}=D^{2}$. Any witness
$\mathcal{W}$ can then be written as:
$\mathcal{W}=\sum_{i}^{\mathcal{D}}\sum_{j}^{\mathcal{D}}c_{ij}\lambda_{i}^{(1)}\otimes\lambda_{j}^{(2)}$
(16)
With $\lambda$ representing any Gell-Mann matrix (in the qubit case, these are
the Pauli matrices: $\lambda\in\\{\mathbb{I},X,Y,Z\\}$). Any tensor
$w_{ij}=\lambda_{i}^{(1)}\otimes\lambda_{j}^{(2)}$ is a quantum observable
that can be directly measured in an experiment.
For a given number of measurements $M$, we can partially improve the overall
variance of the observable by distributing these measurements in proportion to
the weight of each term in the decomposition of the witness, such that, noting
$c=\sum_{i}\sum_{j}\lvert c_{ij}\rvert$, we have:
$\displaystyle M=\sum_{i}\sum_{j}M_{ij}$ (17) $\displaystyle
M_{ij}=\frac{\lvert c_{ij}\rvert}{c}M$ (18)
Considering that we conduct $M_{ij}$ measurements we can then determine the
mean and variance of each of the term as follows:
$\overline{w_{ij}}=\sum_{m}^{M_{ij}}\frac{w_{ij}^{(m)}}{M_{ij}}$ (19)
$\sigma_{ij}^{2}=\sum_{m}^{M_{ij}}\frac{(\overline{w_{ij}}-w_{ij}^{(m)})^{2}}{(M_{ij}-1)}$
(20)
From there, we can compute the witness’ mean and variance:
$\overline{\mathcal{W}}=\sum_{i}^{\mathcal{D}}\sum_{j}^{\mathcal{D}}\sum_{m}^{M_{ij}}\frac{c_{ij}w_{ij}^{(m)}}{M_{ij}}$
(21)
$\sigma_{\mathcal{W}}^{2}=\sum_{i}^{\mathcal{D}}\sum_{j}^{\mathcal{D}}\sum_{m}^{M_{ij}}\lvert
c_{ij}\rvert^{2}\sigma_{ij}^{2}$ (22)
Finally, noting $\mathcal{M}$ the average number of measurements per term, we
can compute the standard error of normally distributed measurement population
as:
$s_{\mathcal{W}}=\frac{\sigma_{\mathcal{W}}}{\sqrt{\mathcal{M}}}$ (23)
We then compute the confidence interval as:
$CI_{\mathcal{W}}=[\overline{\mathcal{W}}-\alpha
s_{\mathcal{W}},\overline{\mathcal{W}}+\alpha s_{\mathcal{W}}]$ (24)
With $\alpha$ the t value corresponding to the desired level of confidence.
For computation of the confidence level, we test against the null hypothesis
$\mathcal{W}\geq\mu_{0}$, with $\mu_{0}=0$, and compute the t values following
the traditional methods for one-sided t-test:
$t=\frac{\lvert\mathcal{W}-\mu_{0}\rvert}{s_{\mathcal{W}}}$ (25)
Confidence is then computed as $1-p$, with $p$ the p-value corresponding to
the t value obtained.
## Appendix E Python model for the QGEM experiment
Our model for computation and modelling of the QGEM experiment is available on
github under the name Generalised QGEM
|
# An effective field theory of holographic dark energy
Chunshan Lin
###### Abstract
A general covariant local field theory of the holographic dark energy model is
presented. It turns out the low energy effective theory of the holographic
dark energy is the massive gravity theory whose graviton has 3 polarisations,
including one scalar mode and two tensor modes. The Compton wavelength is the
size of the future event horizon of the universe. The UV-IR correspondence in
the holographic dark energy model stems from the scalar graviton’s strong
coupling at the energy scale that marks the breaking down of the effective
field theory.
## 1 Introduction
The holographic dark energy model [1][2] is based on the simple idea that the
vacuum energy density arising from the quantum fluctuation of the UV-cut-off
quantum field theory should relate to the boundary surface of a system in the
way
$\displaystyle\rho_{\Lambda}\sim M_{p}^{2}L^{-2},$ (1.1)
where $M_{p}^{2}=8\pi G$ is the reduced Planck mass, $L$ is the size of a
system (thus $L^{2}$ is essentially the area of the boundary surface). This
relation between the energy density in a bulk and the area of its space-time
boundary surface is rooted in the holographic principle [6]. The argument that
leads to the above relation is the following. The zero-point energy diverges
quartically and thus the energy density scales as
$\rho_{\Lambda}\sim\Lambda^{4}$ if we simply cut-off the divergence at the UV
scale $\Lambda$ 111Noted that there are some works [3][4][5] that challenge
this perspective.. However, this simple scaling is violated when the total
energy in the system with size $L$, i.e. $L^{3}\Lambda^{4}$, approaches to the
mass of the black hole in the same size $LM_{p}^{2}$. In fact, the total
energy must be bounded by the mass of the black hole from above, as the
effective quantum field theory breaks down at the Schwarzschild radius scale.
This energy bound is stronger than the Bekenstein entropy bound [7] which
inspired the proposal of the holographic principle [6]. Nowadays it is widely
believed that the holographic principle is one of the most important
cornerstones of quantum gravity.
Assuming that the energy bound is saturated on the cosmological background,
one obtains the important UV-IR correspondence
$\displaystyle\Lambda\sim\sqrt{M_{p}/L}$ (1.2)
and the vacuum energy $\rho_{\Lambda}$ comparable to our current critical
energy density. The related idea was discussed in Ref. [8][9] with the Hubble
radius as the IR cut-off. However, as pointed out in Ref. [10], the resultant
equation of state is greater than $-1/3$ and thus the vacuum energy fails to
accelerate the cosmic expansion, if we simply take the Hubble radius $L=1/H$
as the IR cut-off. Later in the same year, by adopting the future event
horizon as the IR cut-off, an accelerated expanding solution in the Friedman
equation was eventually obtained in Ref.[2]. Interestingly, the cosmic
coincidence problem can also be resolved by inflation in this scenario,
provided the minimal number of inflationary e-foldings. Since then, the
holographic dark energy has drawn a lot of attention and has been widely
studied, see Ref. [11] for a comprehensive review on the topic. In passing, I
shall mention that it was pointed out very recently that the holographic dark
energy model may alleviate the Hubble tension problem [12].
Given the phenomenological success, however, one of key pieces for our
holographic jigsaw puzzle is still missing. Namely we do not know how to write
down the general covariant action for the holographic dark energy model. An
attempt was made in the unpublished work [13] in which a mini superspace
action was given. Nevertheless, due to the absence of the general covariant
action, one may easily spot several conceptual problems. For instance, the
evolution of our universe in the past is dependent on the one in the future.
This apparent causality violation may not imply the pathology of the model,
but rather that the low energy effective field theory is missing. This
causality violation may be partially addressed at the cosmological background
evolution level, given the mini superspace action [13]. However, it is still
not quite clear whether the local physics violates the causality, as the
perturbation theory is also missing222A preliminary analysis on the
perturbative stability was conducted in Ref. [14].
In the current work, I aim at finding this important missing piece for our
holographic jigsaw puzzle. As we will see in the remain of this paper, the low
energy effective field theory (EFT) for the holographic dark energy model is
actually a massive gravity theory. A graviton has three polarisations, namely
one scalar mode, and two tensor modes. The UV-IR correspondence eq. (1.2)
stems from the strong coupling of the scalar mode above the energy scale where
our effective field theory breaks down. One may get puzzled at this point as
the Poincare symmetry in 4 dimensional space-time implies a massive spin-2
particle has 5 polarisations (including helicity 0, $\pm 1$, $\pm 2$).
However, the massive gravity theory that we are going to engage is different
from the conventional one [15][16], as the global Lorentz invariance is broken
on the cosmological background and thus a graviton does not necessary to be
well equipped with 5 polarisations [17][18]. See Ref. [19][20] for two
comprehensive reviews on the topic of massive gravity.
This paper is organised in the following. I will start from the mini
superspace action in the section 2, and then write down the covariant action
by adopting the Stueckelberg trick. Upon writing down the general covariant
action, an Ostrogradsky ghost is spotted in the decoupling limit. The ghost is
eliminated in the section 3, in which the linear perturbation and Hamiltonian
structure are analyzed in detail. The conclusive remarks and outlooks are
given in the section 4.
## 2 From mini superspace to general covariance
We take the flat FLRW ansatz $ds^{2}=-N^{2}dt^{2}+a^{2}d\textbf{x}^{2}$ and
the following mini superspace action [13] as our starting point,
$\displaystyle S=\frac{M_{p}^{2}}{2}\int
dt\left[\sqrt{-g}\left(\mathcal{R}-\frac{2c}{a^{2}L^{2}}\right)-\lambda\left(\dot{L}+\frac{N}{a}\right)\right]+S_{m},$
(2.1)
where $M_{p}^{2}\equiv 8\pi G$ is the reduced Planck mass, $\mathcal{R}$ is
the 4 dimensional Ricci scalar, and $L$ is the variable with length dimension
and subject to the constraint equation enforced by the Lagrangian multiplier
$\lambda$, i.e. $\dot{L}=-N/a$. We may integrate this constraint equation from
the infinite past $-\infty$ to nowadays, namely
$\displaystyle
L=\int_{-\infty}^{t}\frac{-Ndt^{\prime}}{a(t^{\prime})}+L(-\infty),$ (2.2)
where $L(-\infty)$ is the initial condition in the infinite past, which
requires the input from new physics (such as quantum gravity) to fully
determine its value, in light of the Hawking-Penrose’s singularity theorem
[21]. Therefore $L(-\infty)$ remains unknown to me due to my ignorance and
obtuseness. On the other hand, one may try to take the integration from the
other side, namely
$\displaystyle
L=\int^{\infty}_{t}\frac{Ndt^{\prime}}{a(t^{\prime})}+L(+\infty),$ (2.3)
given the asymptotic solution in the infinite future $t=+\infty$. It turns out
$L(+\infty)=0$ for the asymptotic solution derived in Ref. [13] and later is
confirmed by the numerical computation in Ref. [22]. Therefore, $R_{h}\equiv
aL$ is exactly the size of the future event horizon333Mathematically a more
general solution to $\dot{L}=-N/a$ is
$L=\int_{t_{0}}^{t}\frac{-Ndt^{\prime}}{a(t^{\prime})}+L(t_{0}).$ We may input
the value of $L$ at the moment $t_{0}$ in the past by hand, and then treat it
as the initial condition of the equations of motion, which is equivalent to
introducing an assumption about the UV completion of the theory.. It follows
the energy density of the dark sector
$\displaystyle\rho_{\text{dark}}=M_{p}^{2}\left(\frac{c}{a^{2}L^{2}}+\frac{\lambda}{2a^{4}}\right),$
(2.4)
where the first term is the holographic term, and the second term is the dark
radiation as it scales as $a^{-4}$. In passing, I shall mention that in some
cases, for instance, a cyclic universe realised by means of some exotic matter
content, the future event horizon may not exist as the integral in the eq.
(2.3) may not converge to a finite and definite value. The effective field
theory of the holographic dark energy, which will be developed in the remains
of the this paper, does not apply to these cases. Therefore, we will only
focus on the universe models whose future event horizon has finite, definite
and non-vanishing size.
The general covariance can be recovered by Stueckelberging the mini superspace
action [23]. We may follow the following dictionary,
$\displaystyle\frac{1}{a^{2}}\to\frac{1}{3}g^{ij}\delta_{ij}\to
g^{\mu\nu}\partial_{\mu}\phi^{a}\partial_{\nu}\phi^{b}\delta_{ab},$
$\displaystyle\qquad\text{where}\qquad\langle\phi^{a}\rangle=\frac{x^{i}\delta_{ia}}{\sqrt{3}},$
$\displaystyle L\to\varphi(t,\textbf{x}),$
$\displaystyle-\frac{\dot{L}^{2}}{N^{2}}\to
g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi,$
$\displaystyle\qquad\text{where}\qquad\langle\varphi\rangle=\varphi(t),$ (2.5)
where $\delta_{ab}$ is the metric of the Stueckelberg scalar field space,
$\delta_{ia}$ is the pullback mapping between the physical space-time and the
scalar field space, and $i,j,k...$ are adopted as 3 dimensional spatial
coordinate indices, while $a,b,c...$ are adopted as the field space indices.
These 4 Stueckelberg fields are not yet canonically normalized and thus they
are of the length dimension. The field space is flat, and respects the $SO(3)$
rotational invariance. In passing, I shall mention that the global Lorentz
invariance is broken in this scalar configuration. Nevertheless, one should
not be worried about this Lorentz-violating vacuum expectation values (VEVs)
of scalars as the Lorentz invariance is broken anyway on the cosmological
background. For instance, CMB is not invariant under a Lorentz boost.
Before writing down the general covariant action, which is pretty easy and
straightforward at this point, I shall remind readers the hierarchical
structure of the theory. We have the Planck scale as the genetic fundamental
scale in the first place, and a secondary fundamental scale for our EFT of the
dark sector, which is expected to be generated via the non-trivial VEV of the
time-like Stueckelberg field $\varphi$, namely
$\Lambda\sim\sqrt{M_{p}/\varphi}$. The low energy effective field theory that
I begin with is the following,
$\displaystyle S=\int
d^{4}x\sqrt{-g}\left\\{\frac{1}{2}M_{p}^{2}\mathcal{R}-M_{p}^{2}\varphi^{-2}\left[\left(c+\lambda\right)\cdot\partial^{\mu}\phi^{a}\partial_{\mu}\phi^{b}\delta_{ab}+\lambda\partial^{\mu}\varphi\partial_{\mu}\varphi\right]\right\\}+S_{m},$
(2.6)
where $c$ is a constant, $\lambda$ is a Lagrangian multiplier, and $S_{m}$ is
the action of the matter sector which minimally couples to gravity. The theory
respects the $SO(3)$ rotational invariance, $Z_{2}$ symmetry, and the
rescaling invariance
$\displaystyle\phi^{a}\to\ell\cdot\phi^{a},\qquad\varphi\to\ell\cdot\varphi,$
(2.7)
where $\ell$ is a constant. The Lagrangian multiplier $\lambda$ generates a
constraint between the temporal Stueckelberg field and the spatial
Stueckelberg fields. A similar example of this sort can be found in the
generalized unimodular gravity [24]. The hierarchical structure of the theory
is manifest at the action level. One would expect that the hierarchy
disappears if we trace the cosmic evolution all the way backward in time to
the infinite past, and it was generated later as the universe expands. The
physical significance of the energy scale $\Lambda\sim\sqrt{M_{p}/\varphi}$
will be shown in the next section.
I adopt the FLRW ansatz, and the four Stueckelberg scalar fields take the
space-time VEVs in the eq. (2). The background equations of motion read,
$\displaystyle 3M_{p}^{2}H^{2}$ $\displaystyle=$
$\displaystyle\frac{c}{a^{2}\varphi^{2}}+\frac{\lambda}{2a^{4}}+\rho_{m},$
$\displaystyle-M_{p}^{2}\dot{H}$ $\displaystyle=$
$\displaystyle\frac{c}{3a^{2}\varphi^{2}}+\frac{\lambda}{3a^{4}}+\frac{1}{2}\left(\rho_{m}+p_{m}\right),$
(2.8)
and
$\displaystyle\dot{\varphi}=-\frac{1}{a},\qquad\qquad\dot{\lambda}=\frac{-4ca}{\varphi^{3}},$
(2.9)
where $\rho_{m}$ and $p_{m}$ are the energy density and pressure of the matter
sector, the lapse has been absorbed into the redefinition of time, namely
$Ndt\to dt$, and I have rescaled the Lagrangian multiplier
$\lambda\to\lambda\cdot\frac{\varphi^{2}}{4a^{2}M_{p}^{2}}$ for convenience.
All equations of motion presented in Ref. [13] have been reproduced444 The
constraint equation associated with the Lagrangian multiplier $\lambda$ has
two solutions $\dot{\varphi}=\pm 1/a$, which are equivalent due to the $Z_{2}$
symmetry of the theory. Without losing generality, we adopt the solution
$\dot{\varphi}=-1/a$.. Some detailed analyses about this set of equations,
including the analytical one and the numerical one, have already been
conducted in Ref. [13] and Ref. [22], and I shall not repeat it in my current
work.
Before moving on to study the local dynamics, I shall briefly comment on the
causality. The $apparent$ causality violation stems from our ignorance of the
UV completion of the theory, which compels us to integrate the differential
equation (2.9) backward from $t=\infty$ to the moment of interest, as the
asymptotic value of $\varphi$ at the infinite future is known to be
$\varphi(\infty)=0$ [13][22]. Assuming we had enough knowledge about the
quantum gravity and we were able to specify the initial condition at
$t=-\infty$, we might just integrate the differential equation from
$t=-\infty$ to the moment of interest, as we normally do, and there is no
apparent causality violation in this setup. The result obtained from this
setup should be equivalent to the one integrated backward from $t=\infty$. In
other words, the $apparent$ causality violation appears as an artifactual and
mathematical treatment for our equations of motion, rather than a real
phenomenon.
It is quite remarkable to see that the holographic dark energy, which is
$seemingly$ non local and causality-violating, actually originates from a
simple and well defined local field theory. The trick of the game is the
spontaneous symmetry breaking, namely that I start from the general covariant
theory, then 4 scalar fields take the non-trivial space-time VEVs and break
the spatial and temporal diffeomorphism invariance around this vacuum
solution. According the Goldstone theorem, there is a massless boson for each
generator of the symmetry that is broken. I shall start to introduce the
Goldstone excitations around this asymmetric state,
$\displaystyle\phi^{a}=\frac{1}{\sqrt{3}}\left(x^{i}\delta_{i}^{a}+\pi^{a}\right),\qquad\qquad\varphi=\varphi(t)+\pi^{0},$
(2.10)
where $\pi^{a}$ is the space-like Nambu-Goldstone boson, and $\pi^{0}$ is the
time-like one. Due to the $SO(3)$ rotational symmetry, we can decompose the
helicity $\pi^{a}=\delta^{i}_{a}\partial_{i}\pi+\hat{\pi}^{a}$, where $\pi$ is
the longitudinal mode, and $\hat{\pi}^{a}$ are two transverse modes satisfying
the transverse condition $\partial_{i}\hat{\pi}^{i}=0$.
In a gauge field theory, the Goldstone pions decouple from the gauge bosons at
the scale much shorter than the Compton wavelength. The physics becomes very
transparent in the decoupling limit. A similar decoupling limit can also be
adopted, where the Goldstone bosons decouple from the graviton, which is
regarded as a gauge boson in this case. The limit is defined by setting
$M_{p}^{2}\to\infty$ while keeping $\Lambda=\sqrt{M_{p}/\varphi}$ fixed (I
neglect the time dependence of $\varphi$ for the time being for the schematic
analysis). It turns out this decoupling limit is a very illuminating
perspective allowing us to take a quick peek at the microscopic dynamics of
the dark sector. Perturbatively expanding the action eq. (2.6) up to quadratic
order in the Goldstone excitations, I get the following Goldstone action in
the decoupling limit,
$\displaystyle
S_{\pi}\simeq\Lambda^{4}\int\frac{\lambda}{4}\left(\dot{\pi}^{0}\dot{\pi}^{0}-\partial_{i}\pi^{0}\partial_{i}\pi^{0}\right)+\frac{4c+\lambda}{12}\left(\partial_{i}\dot{\pi}\partial_{i}\dot{\pi}-\partial^{2}\pi\partial^{2}\pi\right)+\frac{1}{6}\delta\lambda\left(3\dot{\pi}^{0}-\partial^{2}\pi\right),$
(2.11)
where $\partial^{2}\equiv\partial_{i}\partial_{j}\delta^{ij}$. Seemingly so
far so good, but unfortunately there is a pitfall lies in the Goldstone
action. Taking variation w.r.t $\delta\lambda$ we get the constraint
$\partial^{2}\pi=3\dot{\pi}^{0}$. Inserting it back into the eq. (2.11), we
get a higher order temporal derivative term 555The operator $1/\partial^{2}$
should be understood in the momentum space.
$\displaystyle
S_{\pi}\supset\frac{12c+3\lambda}{4}\int\ddot{\pi}^{0}\frac{1}{\partial^{2}}\ddot{\pi}^{0}+...$
(2.12)
This term yields to a fourth order equation of motion, which requires 4
initial conditions to fully determine its evolution. It implies there are 2
scalar degrees of freedom in the system, instead of one. One of them is
actually the infamous Ostrogradsky ghost [25][26]. The ghost instability
spoils the validity of the low energy effective field theory, and renders the
theory inconsistent. I will devote the next section to kill the ghost.
## 3 ghost elimination and perturbation analysis
### 3.1 Stueckelberg fields and gauge transformation
I define the linear perturbation of our FLRW metric as follows,
$\displaystyle g_{00}$ $\displaystyle=$
$\displaystyle-N(t)^{2}\left(1+2\alpha\right),$ $\displaystyle g_{0i}$
$\displaystyle=$ $\displaystyle N(t)a(t)\left(\partial_{i}\beta+S_{i}\right),$
$\displaystyle g_{ij}$ $\displaystyle=$ $\displaystyle
a(t)^{2}\left[\delta_{ij}+2\psi\delta_{ij}+\partial_{i}\partial_{j}E+\frac{1}{2}\left(\partial_{i}F_{j}+\partial_{j}F_{i}\right)+\gamma_{ij}\right].$
(3.1)
where $\alpha,\beta,\psi$ and $E$ are the scalar perturbations, $S_{i}$ and
$F_{i}$ are the vector perturbations satisfying the transverse condition
$\partial_{i}S_{i}=\partial_{i}F_{i}=0$, and $\gamma_{ij}$ is the tensor
perturbation satisfying the transverse and traceless condition
$\gamma_{ii}=\partial_{i}\gamma_{ij}=0$.
Under the linear gauge transformation
$\displaystyle x^{\mu}\to x^{\mu}+\xi^{\mu}\left(t,\textbf{x}\right),$ (3.2)
these four Stueckelberg fields transform accordingly,
$\displaystyle\pi^{0}\to\pi^{0}+\dot{\varphi}\xi^{0},\qquad\pi^{a}\to\pi^{a}+\xi^{i}\delta_{i}^{a}.$
(3.3)
On the other hand, the vector $Z^{\mu}$ defined by [27]
$\displaystyle Z^{0}\equiv\frac{-a}{N}\beta+\frac{a^{2}}{2N^{2}}\dot{E},\qquad
Z^{i}\equiv\frac{1}{2}\delta^{ij}\left(\partial_{j}E+F_{j}\right),$ (3.4)
transforms in the same manner
$\displaystyle Z^{\mu}\to Z^{\mu}+\xi^{\mu}.$ (3.5)
Therefore, the combination $Z^{i}-\pi^{i}$ and $\dot{\varphi}Z^{0}-\pi^{0}$
are gauge invariant. It is very convenient to transform to the unitary gauge
by simply muting all Goldstone bosons, while keeping all perturbation
variables in the eq. (3.1). It can be achieved by adopting a proper
diffeomorphism $\xi^{\mu}(t,\textbf{x})$. In this gauge, the Goldstone bosons
are eaten by the graviton and the graviton develops a mass gap at the low
energy spectrum. The graviton turns into a massive spin-2 particle with at
most 5 polarizations (it can be less), if the Ostrogradsky ghost had been
eliminated (which will be done soon!).
On the other hand, we can also transform back to the Goldstone bosons’ gauge
by muting the vector $Z^{\mu}$, which can also be done by adopting a proper
diffeomorphism $\xi^{\mu}$. Noted that the way back to the Goldstone bosons’
gauge is not unique, we may choose to mute some other perturbation variables
in the eq. (3.1), based on their transformations under the diffeomprhism.
### 3.2 Hamiltonian analysis in the unitary gauge
The Hamiltonian analysis was introduced by P. Dirac in the 50s and 60s [28],
as a way of counting dynamical degrees of freedom, and quantizing mechanical
systems such as gauge theories (for busy/lazy readers see the appendix A in
the Ref. [29] for a digested version of the method).
It is more convenient to perform the Hamiltonian analysis in the unitary gauge
where $\pi^{0}=\pi^{i}=0$, as all terms introduced by Stueckelberg fields
appear only in the potential sector in the action and thus the Legendre
transformation can be easily done. In the unitary gauge, the action eq. (2.6)
reduces to (I set $M_{p}^{2}=1$ in this subsection)
$\displaystyle S=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}\mathcal{R}-\frac{c}{3}\Lambda_{1}(t)g^{ij}\delta_{ij}+\frac{\lambda}{4}\left(\frac{\Lambda_{2}(t)}{N^{2}}-\frac{1}{3}g^{ij}\delta_{ij}\right)\right].$
(3.6)
where $\Lambda_{1}=\varphi(t)^{-2}$ and $\Lambda_{2}(t)=\dot{\varphi}(t)^{2}$,
namely $\varphi$ and $\dot{\varphi}$ should not be treated as a canonical
variable and its velocity, and instead, they should be treated as two time-
dependent functions subject to some constraint equations which will be derived
later. The reason is that In the unitary gauge, all 4 Stueckelberg fields are
eaten by the graviton, and turn themselves into ADM variables, or some space-
time functions, and cease to be the independent dynamical degrees of freedom.
The Hamiltonian can be obtained by performing the Legendre transformation. To
this end, I need to adopt the ADM decomposition,
$\displaystyle
ds^{2}=-N^{2}dt^{2}+h_{ij}\left(dx^{i}+N^{i}dt\right)\left(dx^{j}+N^{j}dt\right),$
(3.7)
the inverse of the metric reads,
$\displaystyle g^{00}=-\frac{1}{N^{2}},\qquad
g^{0i}=\frac{N^{i}}{N^{2}},\qquad g^{ij}=h^{ij}-\frac{N^{i}N^{j}}{N^{2}}.$
(3.8)
The conjugate momenta are defined in the following,
$\displaystyle\Pi^{ij}=\frac{\partial\mathcal{L}}{\partial\dot{h}_{ij}}=\frac{1}{2}\sqrt{h}\left(K_{ij}-Kh_{ij}\right),\qquad\Pi_{N}=\frac{\partial\mathcal{L}}{\partial\dot{N}}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\Pi_{i}=\frac{\partial\mathcal{L}}{\partial\dot{N}^{i}}=0,\qquad\Pi_{\lambda}=\frac{\partial\mathcal{L}}{\partial\dot{\lambda}}$
$\displaystyle=$ $\displaystyle 0.$ (3.9)
The Hamiltonian is obtained by performing the Legendre transformation,
$\displaystyle H$ $\displaystyle=$ $\displaystyle\int
d^{3}x\left(\Pi^{ij}\dot{h}_{ij}-\mathcal{L}+\varrho_{N}\Pi_{N}+\varrho^{i}\Pi_{i}+\varrho_{\lambda}\Pi_{\lambda}\right)$
(3.10) $\displaystyle=$ $\displaystyle\int
d^{3}x\sqrt{h}\left(\mathcal{H}+\varrho_{N}\Pi_{N}+\varrho^{i}\Pi_{i}+\varrho_{\lambda}\Pi_{\lambda}\right)$
where $\varrho_{N},\varrho^{i},\varrho_{\lambda}$ are Lagrangian multipliers,
and
$\displaystyle\Pi_{N}\approx 0,\qquad\Pi_{i}\approx
0,\qquad\Pi_{\lambda}\approx 0$ (3.11)
are 5 primary constraints, and
$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\frac{2N}{h}\left(\Pi^{ij}\Pi_{ij}-\frac{1}{2}\Pi^{2}\right)-\frac{NR}{2}+N\left(\frac{c}{3}\Lambda_{1}(t)+\frac{\lambda}{4}\right)\left(h^{ab}-\frac{N^{a}N^{b}}{N^{2}}\right)\delta_{ab}-\frac{\lambda}{4}\frac{\Lambda_{2}(t)}{N}$
(3.12)
$\displaystyle-2\nabla_{j}\left(\frac{\Pi^{j}_{~{}i}}{\sqrt{h}}\right)N^{i},$
where $R$ is the 3-dimensional Ricci scalar, $\nabla_{i}$ is the covariant
derivative compatible with 3-dimensional induced metric $h_{ij}$, and
$\approx$ denotes the “weak equivalence”, namely equalities hold on the
constraint surface. Caution should be paid to the terms with different types
of indices. For instance, in the 3 dimensional hyperspace $h^{ab}\equiv
h^{ij}\delta_{i}^{a}\delta_{j}^{b}$ and $N^{a}\equiv N^{i}\delta_{i}^{a}$ are
scalars (as their space-time indices are dummy indices), $\delta^{a}_{i}$ is a
vector, while $h^{ij}$ and $N^{i}$ are tensor and vector respectively. The
consistency conditions of these 5 primary constraints give rise to the
following 5 secondary constraints,
$\displaystyle\frac{d\Pi_{N}}{dt}$ $\displaystyle=$
$\displaystyle\\{\Pi_{N},H\\}=-\mathcal{H}_{0}\approx 0,$
$\displaystyle\frac{d\Pi_{i}}{dt}$ $\displaystyle=$
$\displaystyle\\{\Pi_{i},H\\}=-\mathcal{H}_{i}\approx 0,$
$\displaystyle\frac{d\Pi_{\lambda}}{dt}$ $\displaystyle=$
$\displaystyle\\{\Pi_{\lambda},H\\}=-\mathcal{C}_{\lambda}\approx 0,$ (3.13)
where $\mathcal{H}_{0}$ is the Hamiltonian constraint, $\mathcal{H}_{i}$ is
the momentum constraint, and $\mathcal{C}_{\lambda}$ is the constraint
introduced by hand at the action level,
$\displaystyle\mathcal{H}_{0}$ $\displaystyle\equiv$
$\displaystyle\frac{2}{h}\left(\Pi^{ij}\Pi_{ij}-\frac{1}{2}\Pi^{2}\right)-\frac{R}{2}+\left(\frac{c}{3}\Lambda_{1}(t)+\frac{\lambda}{4}\right)\left(h^{ab}+\frac{N^{a}N^{b}}{N^{2}}\right)\delta_{ab}+\frac{\lambda}{4}\frac{\Lambda_{2}(t)}{N^{2}},$
$\displaystyle\mathcal{H}_{i}$ $\displaystyle\equiv$
$\displaystyle-2\nabla_{j}\left(\frac{\Pi^{j}_{~{}i}}{\sqrt{h}}\right)-\frac{2N^{b}\delta_{ab}\delta_{i}^{a}}{N}\left(\frac{c}{3}\Lambda_{1}(t)+\frac{\lambda}{4}\right),$
$\displaystyle\mathcal{C}_{\lambda}$ $\displaystyle\equiv$
$\displaystyle\frac{N}{4}\left(h^{ab}-\frac{N^{a}N^{b}}{N^{2}}\right)\delta_{ab}-\frac{\Lambda_{2}(t)}{4N}.$
(3.14)
These 5 secondary constraints must be conserved in time, which yields to the
following 5 consistency conditions,
$\displaystyle\frac{d\mathcal{H}_{0}}{dt}=\frac{\partial\mathcal{H}_{0}}{\partial
t}+\\{\mathcal{H}_{0},H\\}$ $\displaystyle\approx$ $\displaystyle 0,$
$\displaystyle\frac{d\mathcal{H}_{i}}{dt}=\partial_{t}\mathcal{H}_{i}+\\{\mathcal{H}_{i},H\\}$
$\displaystyle\approx$ $\displaystyle 0,$
$\displaystyle\frac{d\mathcal{C}_{\lambda}}{dt}=\partial_{t}\mathcal{C}_{\lambda}+\\{\mathcal{C}_{\lambda},H\\}$
$\displaystyle\approx$ $\displaystyle 0.$ (3.15)
Whether these 5 consistency conditions generate some tertiary constraints, or
they only fix the Lagrangian multipliers in the Hamiltonian, crucially depends
on the rank of the matrix
$\displaystyle\mathcal{M}_{AB}\equiv\\{\phi_{A},\phi_{B}\\},$ (3.16)
where $\phi_{A}$ is the whole set of constraints that we have so far, i.e.
$\phi_{A}=(\Pi_{N},\Pi_{i},\Pi_{\lambda},\mathcal{H}_{0},\mathcal{H}_{i},\mathcal{C}_{\lambda})$.
The commutation relations among these constraints are showed in the following,
$\displaystyle\\{\Pi_{N},\Pi_{N}\\}$ $\displaystyle\approx$ $\displaystyle
0,\qquad\\{\Pi_{N},\Pi_{i}\\}\approx
0,\qquad\\{\Pi_{N},\Pi_{\lambda}\\}\approx 0,$
$\displaystyle\\{\Pi_{N},\mathcal{H}_{0}\\}$ $\displaystyle\neq$
$\displaystyle 0,\qquad\\{\Pi_{N},\mathcal{H}_{i}\\}\neq
0,\qquad\\{\Pi_{N},\mathcal{C}\\}\neq 0,$ $\displaystyle\\{\Pi_{i},\Pi_{j}\\}$
$\displaystyle\approx$ $\displaystyle
0,\qquad\\{\Pi_{i},\Pi_{\lambda}\\}\approx
0,\qquad\\{\Pi_{i},\mathcal{H}_{0}\\}\neq 0,$
$\displaystyle\\{\Pi_{i},\mathcal{H}_{j}\\}$ $\displaystyle\neq$
$\displaystyle 0,\qquad\\{\Pi_{i},\mathcal{C}_{\lambda}\\}\neq 0,$
$\displaystyle\\{\Pi_{\lambda},\Pi_{\lambda}\\}$ $\displaystyle\approx$
$\displaystyle 0,\qquad\\{\Pi_{\lambda},\mathcal{H}_{0}\\}\neq
0,\qquad\\{\Pi_{\lambda},\mathcal{H}_{i}\\}\approx 0,$
$\displaystyle\\{\Pi_{\lambda},\mathcal{C}_{\lambda}\\}$
$\displaystyle\approx$ $\displaystyle 0,$
$\displaystyle\\{\mathcal{H}_{0},\mathcal{H}_{0}\\}$ $\displaystyle\neq$
$\displaystyle 0,\qquad\\{\mathcal{H}_{0},\mathcal{H}_{i}\\}\neq
0,\qquad\\{\mathcal{H}_{0},\mathcal{C}_{\lambda}\\}\neq 0,$
$\displaystyle\\{\mathcal{H}_{i},\mathcal{H}_{j}\\}$ $\displaystyle\neq$
$\displaystyle 0,\qquad\\{\mathcal{H}_{i},\mathcal{C}_{\lambda}\\}\neq 0,$
$\displaystyle\\{\mathcal{C}_{\lambda},\mathcal{C}_{\lambda}\\}$
$\displaystyle\approx$ $\displaystyle 0.$ (3.17)
The rank of the matrix $\mathcal{M}_{AB}$ is 10. Therefore the consistency
conditions eq. (3.2) only fix the Lagrangian multipliers $\varrho^{\prime}$s,
instead of generating new tertiary constraints. Let’s collect all of primary
and secondary constraints in the total Hamiltonian and treat them on the same
footing,
$\displaystyle H_{tot}=\int
d^{3}x\sqrt{h}\left(\mathcal{H}+\varrho_{N}\pi_{N}+\varrho^{i}\Pi_{i}+\varrho_{\lambda}\Pi_{\lambda}+\varrho^{0}\mathcal{H}_{0}+\tilde{\varrho}^{i}\mathcal{H}_{i}+\varrho_{\lambda}\mathcal{C}_{\lambda}\right)$
(3.18)
where
$\left(\varrho_{N},\varrho^{i},\varrho_{\lambda},\varrho^{0},\tilde{\varrho}^{i},\varrho_{\lambda}\right)$
are Lagrangian multipliers, and we have absorbed the shift vector into the
$\tilde{\varrho}^{i}$. The algebra closes here.
The rank of the matrix also implies all of these 10 constraints are second
class. This is the consequence of that the gauge symmetries in GR, namely the
space-time diffeomorphisms are all broken in the unitary gauge. Now let’s
count the number of degrees. In the phase space we have got 22 degrees in the
first place, where 20 degrees are from the 10 independent components of
$g^{\mu\nu}$ and their conjugate momenta, and 2 degrees are from Lagrangian
multiplier $\lambda$ and its conjugate momentum. On the other hand, those 10
second class constraints $\phi_{A}$ remove 10 degrees in the phase space, and
the leftover is
$\displaystyle 22-10=12~{}\text{ degrees in the phase space},$ (3.19)
which corresponds to 6 degrees of freedom in the physical space-time. The
sixth mode is the Ostrogradsky ghost.
### 3.3 Ghost elimination in the Goldstone action in the decoupling limit
The Goldstone action in the decoupling limit is a very convenient perspective
for us to hunt and eventually execute the ghost. Taking a closer look at the
Goldstone action eq. (2.11), I notice that the ghost arises from the kinetic
term of the Goldsteon boson $\pi$, and the constraint equating the temporal
derivative of $\pi^{0}$ and the gradient of the $\pi$, namely
$\displaystyle\int\partial_{i}\dot{\pi}\partial_{i}\dot{\pi}~{}\to~{}\int\ddot{\pi}^{0}\frac{1}{\partial^{2}}\ddot{\pi}^{0}\qquad\text{as}\qquad\partial^{2}\pi=3\dot{\pi}^{0}.$
(3.20)
Therefore, a direct way to eliminate the higher order temporal derivative term
is to eliminate the kinetic term $\partial_{i}\dot{\pi}\partial_{i}\dot{\pi}$,
which can be achieved by introducing the symmetry [17]
$\displaystyle\pi^{i}(t,\textbf{x})\to\pi^{i}(t,\textbf{x})+\xi^{i}\left(t\right),$
(3.21)
where $\xi^{i}$ is an arbitrary function of time. This symmetry prohibits all
temporal derivative terms of the Goldstone pion $\pi^{i}$, including the ones
of the longitudinal mode $\pi$ and the ones of the transverse mode
$\hat{\pi}^{i}$. At the lowest dimensional operator level, the building block
which respects this symmetry is the following,
$\displaystyle
Z^{ab}\equiv\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{b}-\frac{\left(\partial_{\mu}\varphi\partial^{\mu}\phi^{a}\right)\left(\partial_{\nu}\varphi\partial^{\nu}\phi^{b}\right)}{\partial_{\mu}\varphi\partial^{\mu}\varphi}.$
(3.22)
In the unitary gauge where all Goldstone bosons are muted, we have
$Z^{ab}=\frac{1}{3}h^{ij}\delta_{i}^{a}\delta_{j}^{b}$, namely the residual
symmetry eq. (3.21) strips the shift off the $g^{ij}$, and leaves us with only
3 dimensional induced metric $h^{ij}$. Now let’s look at the leftover action,
which reads
$\displaystyle
S\simeq\Lambda^{4}\int-c\dot{\pi}^{0}\dot{\pi}^{0}-\frac{1}{3}\left(\lambda+c\right)\partial_{i}\pi^{0}\partial_{i}\pi^{0},$
(3.23)
where $c$ and $\lambda$ are assumed to be positive, to ensure that the energy
density of dark energy and dark radiation are positive. Therefore, the kinetic
term of the leftover scalar mode $\pi^{0}$ has a wrong sign. Unfortunately
after removing the problematic higher order temporal derivative term, the
theory still contains a ghost. This ghost arises from the gradient term of the
Goldstone boson $\pi$, namely
$\displaystyle\int-\frac{4c+\lambda}{12}\partial^{2}\pi\partial^{2}\pi~{}\to~{}\int-\left(3c+\frac{3\lambda}{4}\right)\dot{\pi}^{0}\dot{\pi}^{0}\qquad\text{as}\qquad\partial^{2}\pi=3\dot{\pi}^{0}.$
(3.24)
I have to flip the sign of the term $\partial^{2}\pi\partial^{2}\pi$ to cure
this ghost pathology. The remedy is the operator introduced in Ref. [30],
$\displaystyle\bar{\delta}Z^{ab}\equiv
Z^{ab}-3\frac{Z^{ac}Z^{db}\delta_{cd}}{Z^{cd}\delta_{cd}},$ (3.25)
which is traceless up to the linear perturbation level, and thus it does not
contribute to the background evolution. Additionally I shall stick to the
$Z_{2}$ symmetry and the global scaling invariance eq. (2.7) to tighten up the
structure and reduce the arbitrariness of the theory. At the lowest
dimensional operator level, I write down the general covariant and ghost free
action which respects the residual symmetry eq. (3.21), the global symmetry
eq. (2.7) and the $SO(3)$ rotational symmetry in the field space as follows,
$\displaystyle S=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left\\{\frac{M_{p}^{2}}{2}\mathcal{R}-M_{p}^{2}\varphi^{-2}\left[\left(c+\lambda\right)\cdot
Z+\lambda\partial^{\mu}\varphi\partial_{\mu}\varphi+\frac{3d}{8Z}\cdot\bar{\delta}Z^{ab}\bar{\delta}Z^{cd}\delta_{ac}\delta_{bd}\right]\right\\}$
(3.26) $\displaystyle\qquad\qquad+S_{m}\,,$
where $Z\equiv Z^{ab}\delta_{ab}$, the coefficient $3/8$ is chosen for the
later convenience and $S_{m}$ is the action of the matter sector which
minimally couples to gravity. This is the main result of my current work.
Compared with the original action eq. (2.6), in the unitary gauge the
$g^{ab}$, which is produced by the spatial Stueckelberg fields
$\partial^{\mu}\phi^{a}\partial_{\mu}\phi^{b}\delta_{ab}$, is replaced by one
of our building blocks $h^{ab}$ defined in the eq. (3.22). Take the note of
that $h^{ab}$ differs from $g^{ab}$ only at the perturbation level. On the
other hand, the operator $\bar{\delta}Z^{ab}$ only appears at the perturbation
level too as it is traceless. Therefore, all modifications are operated at the
perturbation level, and do not alter the background evolution which is subject
to the set of equations in the eq. (2) (2.9). The Goldstone action for the
scalar graviton reads
$\displaystyle
S\supset\int-\left(c+d\right)\dot{\pi}^{0}\dot{\pi}^{0}-\frac{1}{3}\left(\lambda+c\right)\partial_{i}\pi^{0}\partial_{i}\pi^{0},$
(3.27)
with the ghost free condition,
$\displaystyle c+d<0.$ (3.28)
This ghost free condition will be reproduced later in the full perturbation
analysis in the unitary gauge, where the Goldstone bosons are muted.
### 3.4 Full perturbation analysis in the unitary gauge
We adopt the metric perturbation decomposition in the eq. (3.1). Due to the
background $SO(3)$ rotational invariance, the scalar perturbations, the vector
perturbations and the tensor perturbations completely decouple from each other
at the linear perturbation level. We also adopt the unitary gauge and mute all
Goldstone bosons.
The massive gravity is an analog of the Higgs mechanism in the particle
physics, where the Higgs field resides in an asymmetric state. In the unitary
gauge, the Goldstone pions in this symmetry broken phase are eaten by the
gauge bosons and consequently the gauge bosons become massive. We expect the
same phenomena should also occur in the massive gravity, namely once we mute
the Goldstone bosons, the scalar graviton and gravitational waves are massive
in the unitary gauge.
I will only focus on the perturbation analysis of the pure gravity, while the
cases with matter included will be discussed in the future work.
#### 3.4.1 Scalar perturbation
The quadratic action of the scalar perturbation is obtained after a
straightforward computation,
$\displaystyle S_{\text{scalar}}^{(2)}=\int
d^{4}x\left(\mathcal{L}_{\text{EH}}+\mathcal{L}_{\text{mass}}\right),$ (3.29)
where $\mathcal{L}_{\text{EH}}$ is the contribution from the Einstein-Hilbert
action,
$\displaystyle\frac{\mathcal{L}_{EH}}{M_{p}^{2}a^{3}}$ $\displaystyle=$
$\displaystyle-3\dot{\psi}^{2}+\frac{k^{2}}{a^{2}}\left[\psi^{2}-\frac{1}{2}a^{2}\dot{E}\left(H\psi-2\dot{\psi}\right)-\frac{1}{2}a^{2}HE\left(\dot{\psi}+3H\psi\right)\right]$
(3.30) $\displaystyle+\alpha\left[\frac{2k^{2}\beta
H}{a}+\frac{k^{2}}{a^{2}}\left(2\psi-a^{2}H\dot{E}\right)+6H\dot{\psi}-3H^{2}\alpha\right]-\frac{2k^{2}\beta\dot{\psi}}{a},$
and the $\mathcal{L}_{\text{mass}}$ is the contribution from the dark sector,
or in other word the graviton mass term,
$\displaystyle\mathcal{L}_{\text{mass}}$ $\displaystyle=$ $\displaystyle
M_{p}^{2}\varphi^{-2}\left[-\frac{3c+d}{36}k^{4}aE^{2}-\frac{1}{6}ck^{2}aE\left(2\alpha+\psi\right)+ca\psi\left(2\alpha+\psi\right)\right]$
(3.31)
$\displaystyle+\lambda\left[-\frac{k^{4}E^{2}}{48a}+\frac{k^{2}E\left(\psi-\alpha\right)}{12a}+\frac{\left(\alpha+\psi\right)^{2}}{4a}\right]+\delta\lambda\left(\frac{\psi-\alpha}{2a}-\frac{k^{2}E}{12a}\right).$
I have absorbed the lapse into the redefinition of time, and rescaled the
Lagrangian multiplier
$\lambda\to\lambda\cdot\frac{\varphi^{2}}{4a^{2}M_{p}^{2}}$ for convenience
once again. Taking the variation of the action with respect to the non-
dynamical variables $\alpha$, $\delta\lambda,$ and $~{}\beta$, I get the
following three constraint equations,
$\displaystyle
H\left(6\dot{\psi}-k^{2}\dot{E}\right)+\frac{2c\psi}{a^{2}\varphi^{2}}+\frac{k^{2}}{a^{2}}\left(2\psi-\frac{cE}{3a^{2}\varphi^{2}}\right)+\frac{\lambda}{12a^{4}M_{p}^{2}}\left(6\psi-k^{2}E\right)$
$\displaystyle-\frac{\delta\lambda}{2a^{4}M_{p}^{2}}+\alpha\left(\frac{\lambda}{2a^{4}M_{p}^{2}}-6H^{2}\right)+\frac{2k^{2}H\beta}{a}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
k^{2}E+6{\color[rgb]{1,0,0}\alpha}-6\psi$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle H\alpha-\dot{\psi}$ $\displaystyle=$ $\displaystyle 0.$ (3.32)
Substituting the solution of the above equations into the action, the
quadratic action of the scalar perturbation takes the following form,
$\displaystyle
S_{\text{scalar}}^{(2)}=\int-\frac{\left(c+d\right)aM_{p}^{2}}{H^{2}\varphi^{2}}\dot{\psi}^{2}+...$
(3.33)
where the eclipse denotes the potential term and gradient term. The ghost free
condition for the scalar perturbation reads
$\displaystyle c+d<0.$ (3.34)
We have reproduced the ghost free condition eq. (3.28), in the different
gauge. Let me redefine a new constant $b$ via
$\displaystyle b\equiv-2\left(c+d\right),\qquad\text{and we demand}\qquad
b>0.$ (3.35)
The scalar graviton is canonically normalised as
$\displaystyle\psi^{c}\equiv\frac{\sqrt{b}M_{p}}{aH\varphi}\psi.$ (3.36)
The canonically normalised scalar action reads
$\displaystyle S_{\text{scalar}}^{(2)}=\frac{1}{2}\int
dtd^{3}ka^{3}\left(\dot{\psi}^{c}\dot{\psi}^{c}-\frac{c_{s}^{2}k^{2}}{a^{2}}\psi^{c}\psi^{c}-M^{2}_{s}\psi^{c}\psi^{c}\right),$
(3.37)
where the sound speed reads
$\displaystyle
c_{s}^{2}=\frac{c}{b}\cdot\frac{\rho_{\text{dark}}}{\rho_{\text{hde}}}\cdot\left(1+w_{\text{dark}}\right),$
(3.38)
where $\rho_{\text{dark}}$ is energy density of the dark sector defined in the
eq. (2.4), $w_{\text{dark}}$ is its equation of state, and
$\rho_{hde}=cM_{p}^{2}/a^{2}\varphi^{2}$ is the energy density of the
holographic dark energy. It is clear that the $c_{s}^{2}$ becomes negative
when the equation of state of the dark sector drops below $-1$, this is
precisely the case for $c<6$ in the asymptotic future [13][22]. Therefore, if
we demand the freeness of the gradient instability throughout the whole cosmic
expansion history, the parameter region with $c<6$ needs to be excluded.
However, $c<6$ is still allowed, if we only demand the freeness of the
gradient instability at present and in the past, and allow the instability to
occur in the future. The mass of the scalar mode reads
$\displaystyle M_{s}^{2}$ $\displaystyle=$ $\displaystyle
2H^{2}\left[\frac{6c}{b}-\Omega_{\text{hde}}^{2}+\Omega_{\text{hde}}\left(-\frac{1}{2}-\frac{3}{c}+\frac{3c}{b}-4\Omega_{\text{rad}}\right)+\sqrt{\frac{3\Omega_{\text{hde}}}{c}}\left(1-\frac{2c}{b}+2\Omega_{\text{rad}}\right)\right.$
(3.39)
$\displaystyle\qquad\left.+\left(1+\frac{8c}{b}\right)\Omega_{\text{rad}}+\Omega_{\text{rad}}^{2}\left(\frac{4c}{b\Omega_{\text{hde}}}-4\right)\right],$
where $\Omega_{rad}$ is the fraction of the dark radiation defined by
$\Omega_{rad}\equiv\rho_{rad}/3M_{p}^{2}H^{2}$, and $\Omega_{\text{hde}}$ is
the fraction of the holographic dark energy defined by
$\Omega_{\text{hde}}\equiv\rho_{\text{hde}}/3M_{p}^{2}H^{2}$. The Compton
wavelength of the scalar mode is about Hubble radius size, up to a coefficient
whose value is dependent on the parameters $b$ and $c$.
One may wonder whether the mass of scalar graviton is always positive along
the cosmic evolution to ensure the stability in the scalar sector. This
problem may require a thorough numerical analysis, and thus it will not be
carried out in the current work. Instead, I would like to show the tachyon
freeness condition in the asymptotic de-Sitter phase where $c=6$ [13] and
$w_{\text{dark}}\to-1$. The scalar mass in the asymptotic phase reads
$\displaystyle M_{s}^{2}\to 8H^{2}\left[\frac{6}{b}-1\right].$ (3.40)
The tachyon freeness condition translates to
$\displaystyle 0<b<6.$ (3.41)
The stability analysis for general space-time solution will be performed in
the future work.
#### 3.4.2 Tensor perturbation
The quadratic action for the tensor perturbation reads
$\displaystyle S_{\text{tensor}}^{(2)}=\frac{M_{p}^{2}}{8}\int
dtd^{3}ka^{3}\left[\dot{\gamma}^{ij}\dot{\gamma}_{ij}-\left(\frac{k^{2}}{a^{2}}+M^{2}_{GW}\right)\gamma_{ij}\gamma^{ij}\right],$
(3.42)
where
$\displaystyle
M_{GW}^{2}=\frac{1}{6R_{h}^{2}}\left[4c-b+3c\frac{\rho_{\text{dark}}}{\rho_{\text{hde}}}\left(1+w_{dark}\right)\right]$
(3.43)
The Compton wavelength of the tensor graviton is about the size of the future
event horizon. The following tachyon freeness condition is required
$\displaystyle
4c-b+3c\frac{\rho_{\text{dark}}}{\rho_{\text{hde}}}\left(1+w_{dark}\right)>0$
(3.44)
to ensure the stability of the tensor sector. In the asymptotic de-Sitter
phase where $c=6,~{}w_{\text{dark}}=-1$, the stability condition translates to
$0<b<24$, which is weaker than the stability condition eq. (3.41) derived in
the preceding subsection of the scalar mode analysis.
The non-vanishing mass modifies the dispersion relation of the gravitational
waves, and the group velocity is constrained with high precision $-3\cdot
10^{-15}<c_{\text{gw}}-1<7\cdot 10^{-16}$ at low redshift regime by the multi-
messenger observation GW170817 [31]. It gives us an upper bound on the
graviton mass around $m_{g}<10^{-23}\text{eV}$. Our graviton mass is much
lower than this upper bound by around 10 orders of magnitude. It remains
challenging to directly probe the non-vanishing graviton mass effects at the
late time epoch. However, the size of the future event horizon is much shorter
during the early universe, and thus it leads to a much larger graviton mass,
which may give rise to some interesting observational effects on the
stochastic gravitational waves background.
#### 3.4.3 Vector perturbation
I do not expect to find the dynamical degrees of freedom in the vector sector
at the linear perturbation level, as the residual symmetry eq. (3.21) has
projected out all temporal derivative of the Goldstone vector bosons. I will
try to confirm it in this subsection. The quadratic action of the vector
perturbation reads
$\displaystyle S_{\text{V}}^{(2)}=\frac{M_{p}^{2}}{16}\int
dtd^{3}kk^{2}a^{3}\left[\dot{F}_{i}\dot{F}^{i}-\frac{4S_{i}\dot{F}^{i}}{a}+\frac{4S_{i}S^{i}}{a^{2}}+\left(\frac{b-6c}{6a^{2}\varphi^{2}}-\frac{\lambda}{3M_{p}^{2}a^{4}}\right)F_{i}F^{i}\right].$
(3.45)
Taking the variation with respect to the Lagrangian multiplier $S_{i}$, I get
$\displaystyle a\dot{F}_{i}-2S_{i}=0.$ (3.46)
Substituting the solution of this constraint equation back to the action, I
get
$\displaystyle S_{\text{V}}^{(2)}=\frac{1}{16}M_{p}^{2}\int
dtd^{3}kk^{2}a^{3}\left(\frac{b-6c}{6a^{2}\varphi}-\frac{\lambda}{3M_{p}^{2}a^{4}}\right)F_{i}F^{i}.$
(3.47)
The kinetic term of the vector mode has vanished, and the action is subject to
a new constraint $F_{i}=0$. After substituting this solution back to the
action, the whole action vanishes and I conclude that no dynamical degree of
freedom is found in the vector sector at the linear perturbation level.
Whether there exists dynamical degrees of freedom at the higher order level
requires a fully non-linear analysis, which will be provided in the next
subsection.
### 3.5 The Hamiltonian analysis confirms the total number of degrees
I have performed the Hamiltonian analysis in the subsection (3.2), and I found
that the original theory contains 6 modes. Compared to the Hamiltonian of the
original theory (2.6), the modified one (3.26) strips off the term that is
non-linear in the shift vector. Consequently the primary constraint
$\Pi_{i}\approx 0$ commutes with all primary and secondary constraints. The
consistency conditions for the momentum constraints conserve in time yield to
another 3 tertiary constraints,
$\displaystyle\mathcal{T}_{i}\equiv\\{\mathcal{H}_{i},H\\}\approx 0.$ (3.48)
Now let’s add these 3 new tertiary constraints into the constraint set
$\phi^{A}$, and remove the $\Pi_{i}\approx 0$ as it is first class, we have
the new set
$\displaystyle\phi_{A}=(\Pi_{N},\Pi_{\lambda},\mathcal{H}_{0},\mathcal{H}_{i},\mathcal{C}_{\lambda},\mathcal{T}_{i}).$
(3.49)
We can show that the rank of the following $10\times 10$ matrix
$\displaystyle\mathcal{M}_{AB}\equiv\\{\phi_{A},\phi_{B}\\}$ (3.50)
is 10, no new constraints are generated and the algebra closes here.
Therefore, all constraints in the eq. (3.49) are second class, and they remove
10 degrees in the phase space. On the other hand, the constraints
$\Pi_{i}\approx 0$ remove 6 degrees in the phase space as they are first
class. The total number of the leftover dynamical degrees of freedom in the
phase space is
$\displaystyle 22-3\times 2-10=6,$ (3.51)
which corresponds to 3 dynamical degrees of freedom in the physical space-
time. As we have shown in our perturbation analysis, 2 of them are tensor
modes, and the rest one is a scalar graviton. The additional degrees of
freedom are fewer than the broken symmetry generators, because the Lorentz-
invariance of the background space-time is broken and therefore the Goldstone
theorem needs to be understood in the extended sense.
### 3.6 Physical interpretation of the UV-IR correspondence
The UV-IR correspondence is based on the assumption that the vacuum energy
arising from the quartic divergence should not exceed the energy density of a
black hole in the same size, and the effective quantum field theory should
break down at the Schwarzschild radius scale. Take the note of that the Hubble
radius of our universe coincides with its Schwarzschild radius, and thus the
bound should be saturated on the cosmological background and we have the UV-IR
relation $\Lambda\sim\sqrt{M_{p}/L}$.
One may ask how this UV-IR correspondence appears naturally in our effective
field theory framework? The answer is that it should be interpreted as the
scale where the scalar graviton strongly couples to itself. Let’s look at the
4-votex interaction in the decoupling limit,
$\displaystyle\int
M_{p}^{2}\varphi^{-2}\partial^{2}\pi\partial^{2}\pi\partial^{2}\pi\partial^{2}\pi\sim\int
M_{p}^{2}\varphi^{-2}\dot{\pi}^{0}\dot{\pi}^{0}\dot{\pi}^{0}\dot{\pi}^{0}\sim\int\frac{\omega^{4}}{M_{p}^{2}\varphi^{-2}}\pi^{0}_{c}\pi^{0}_{c}\pi^{0}_{c}\pi^{0}_{c},$
(3.52)
where $\pi^{0}_{c}$ is the canonical normalised Goldstone boson, and $\omega$
is the frequency of $\pi^{0}_{c}$. The strength of the coupling exceeds unity
if $\omega>\sqrt{M_{p}/\varphi}$, and our effective field theory breaks down.
As shown in our perturbation analysis, we have learnt that $\varphi$ is the
graviton’s Compton’s wavelength (remember $\varphi$ has the length dimension,
and we set the scale factor $a=1$ in the decoupling limit). Therefore, a
sensible and physical interpretation of this UV-IR correspondence is that the
UV cut-off scale of our EFT is inversely proportional to the Compton
wavelength of our graviton.
## 4 Conclusion and Discussion
In the current work, a general covariant local field theory is proposed for
the holographic dark energy model. I started from the mini superspace action
proposed a few years ago [13], showed that the direct, and perhaps the
simplest way of covariantizing the action gives rise to additional 4 degrees
of freedom (6 in total including 2 degrees in the gravitational wave sector),
which leads to the Ostrogradsky ghost instability. To remedy the ghost
pathology, I introduced a new symmetry to prohibit the problematic terms in
the theory. It has turned out that the low energy effective field theory of
the holographic dark energy model is actually the Lorentz-violating massive
gravity theory, whose graviton has 3 polarisations including 1 helicity 0 mode
and 2 helicity 2 modes. The Compton wavelength of the graviton is about the
size of the future event horizon of our universe, as shown in the linear
perturbation analysis in the unitary gauge. To confirm the total number of
dynamical degrees of freedom of our theory at the fully non-linear level, I
have performed the Hamiltonian analysis and found that the momentum
constraints yield to another 3 tertiary constraints. These 6 constraints,
which are all second class, eliminate 3 out of 4 additional degrees and
therefore there are only 3 dynamical degrees of freedom in total in the
gravity sector, including a scalar mode and two tensor modes. Our effective
field theory breaks down at the scale $\Lambda\sim\sqrt{M_{p}/L}$, where $L$
is the graviton’s Compton wavelength, due to the strong coupling of the scalar
graviton above this scale, which offers a natural and physical interpretation
for the UV-IR correspondence in the holographic dark energy model.
Our effective field theory provides a general framework in which the
holographic dark energy model can be tested. For instance, the non-vanishing
mass of the gravitational waves, which is small at late time epoch but
sizeable during the early universe, leads to a modified stochastic
gravitational waves background. On the other hand, the existence of the scalar
graviton whose Compton wavelength is about the size of the Hubble radius (up
to an order $\mathcal{O}(1)\sim\mathcal{O}(10)$ coefficient) during inflation
may be tested by the cosmological collider physics [32]. It is also very
intriguing to ask what would happen if the Stueckelberg field, say the time
like one $\varphi$, couples to the standard model fields. Our EFT framework
also provides a general setup where the perturbation theory of the holographic
dark energy can be developed, which allows us to investigate its impacts on
the cosmological structure formation in detail. All these possibilities
warrant further scrutiny.
I shall end by commenting that the global Lorentz invariance is broken in our
scalar field configuration. As an analog to the Higgs mechanism in the
particle physics, one may expect a similar symmetry restoration to occur at
high energy scale, where the Lorentz invariance is recovered and the graviton
becomes massless again. However, this Higgs-like mechanism is till missing. In
fact, to my best knowledge it is still an open question for all massive
gravity theories.
## Acknowledgement
This work is supported by the grant No. UMO-2018/30/Q/ST9/00795 from the
National Science Centre, Poland. The author would like to thank Yi-fu Cai,
Alexander Ganz, A. Emir Gümrükçüoğlu, Qing-Guo Huang, Miao Li, Yin-zhe Ma, and
Yi Wang for the useful discussions and suggestions.
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|
ad short = AD, long = automatic differentiation cbpv short = CBPV, long =
call-by-push-value
# Automatic Differentiation via Effects and Handlers
An Implementation in Frank
Jesse Sigal 0000-0002-5117-8752 University of EdinburghEdinburghU.K.
<EMAIL_ADDRESS>
###### Abstract.
ad is an important family of algorithms which enables derivative based
optimization. We show that ad can be simply implemented with effects and
handlers by doing so in the Frank language. By considering how our
implementation behaves in Frank’s operational semantics, we show how our code
performs the dynamic creation of programs during evaluation.
automatic differentiation, algebraic effects, effect handlers, differentiable
programming
††copyright: none††conference: Partial Evaluation and Program Manipulation;
January 18–19; Online††ccs: Theory of computation Operational semantics††ccs:
Theory of computation Control primitives††ccs: Mathematics of computing
Automatic differentiation
## 1\. Introduction
Machine learning, artificial intelligence, scientific modelling, information
analysis, and other data heavy fields have driven the demand for tools which
enable derivative based optimization. The family of algorithms known as ad is
the foundation of the tools which achieve this. The family can be coarsely
divided into forward mode and reverse mode. Multiple modes exist because their
asymptotics depend on different features of the differentiated programs.
Forward mode ad was introduced in 1964 by Wengert (Wengert, 1964), and reverse
mode ad was created by Speelpenning in his 1980 thesis (Speelpenning, 1980).
It is not surprising that, given its history, ad has been implemented in many
different ways. Many popular tools such as ADIFOR (Bischof et al., 1996), ADIC
(Bischof et al., 1997), and Tapenade (Pascual and Hascoët, 2008; Hascoët and
Pascual, 2013) work via source transformation. These transformations take
place on languages such as C and FORTRAN, and thus all of the aforementioned
tools work externally from the program being written. We shall show here that
the recent Frank language (Lindley et al., 2017; Convent et al., 2020) and its
operational semantics, which leverages effects and handlers, can be informally
seen as dynamically performing partial evaluation and program manipulation.
## 2\. Background
### 2.1. Automatic Differentiation
We are most interested in showing the structure of ad algorithms, so we shall
only give a short intuition for ad. Let $f,g\colon\mathbb{R}\to\mathbb{R}$ be
smooth functions (i.e. infinitely differentiable at all points). The chain
rule states that $\left(f\circ
g\right)^{\prime}(x)=f^{\prime}\left(g(x)\right)\cdot g^{\prime}(x)$. ad
algorithms use this compositional property to incrementally calculate the
derivative of an entire program one basic operation at a time during
evaluation. We refer the reader to the textbook of Griewank and Walther
(Griewank and Walther, 2008) for general knowledge and to Hascoët and Araya-
Polo (Hascoët and Araya-Polo, 2006) for checkpointed reverse mode, our most
interesting example.
### 2.2. Effects and Handlers
Effects and handlers are a structured method of including side-effects into
programs. Algebraic effects were introduced in 2001 by Plotkin and Power
(Plotkin and Power, 2001) handlers for them in 2009 by Plotkin and Pretnar
(Plotkin and Pretnar, 2009). Effects and handlers can be viewed as an
extension of the common feature of catchable exceptions. Catching an exception
terminates the program delimited by the exception handling code, but effect
handlers can resume the handled code and pass a value to it. Effects and
handlers can implement many common side effects such as state, exceptions,
non-determinism, logging, and input-output.
### 2.3. Frank
We will be using the Frank language to implement ad. Frank’s typing and
operational semantics are inspired by call-by-push-value (Levy, 2003), meaning
there is a distinction between values and computations. We note Frank has a
fixed left-to-right evaluation order. Frank combines the concepts of functions
and handlers by unifying them into what Frank refers to as operators, which
act by application. However, we shall usually say handler for operators which
handle effects and functions otherwise. We shall also simplify certain aspects
for ease of exposition, see Convent et al. (Convent et al., 2020) for a
tutorial and details.
Let us consider a simple example of a handler for a program which uses state
of type S.
| state : {S -> <State S>X -> [Console]X}| state _ x = print "end"; x| state s
<get -> k> = print "get"; state s (k s)| state _ <put s -> k> = print "put";
state s (k unit)We first explain the type of state. The handler state takes
two arguments, one of type S and one of type X. In order for state to be used,
the context in which it is called must support the ability [Console] which is
a snoc-list containing exactly one instance Console of the interface Console
(the ability [Console, Console] contains two distinct instances of the same
interface). The ability [Console] means we can use the command print defined
by the interface Console. In the term state s x, the value produced by s can
only be computed using commands from the instances in the ability [Console].
On the other hand, the value produced by x can use commands from [Console,
State S]. The value x has access to State commands because the adjustment
<State S> extends the ambient ability [Console]. We note that the adjustment
<State S> guarantees that state handles all commands of the State interface
(get and set). The full type of state includes braces, showing that state is a
suspended computation. Frank automatically inserts these if they are absent.
We shall briefly explain some aspects of Frank’s operational semantics before
we go into more detail during ad examples. Consider the example top-level use
of state where semicolon is sequencing and postfix ! is nullary function
application.
| 2 + (state 1 (put (_get!_ \+ get!); get!))The ability [Console] is permitted
at the top-level as Frank’s implementation will handle it. As the program
executes, the underlined get is encountered and a continuation of the program
delimited by state is captured, namely the operator {r -> (put (r + get!);
get!)}, and bound to k in the body of the second line of state’s definition.
Once the execution of (put (get! + get!); get!) finishes, the first line of
state’s definition is matched and state exits.
## 3\. Algorithm Implementations
We will cover the implementation of four different handlers in Frank:
evaluate:
: the most basic handler which dispatches to builtin arithmetic operations;
diff:
: an implementation of forward mode ad;
reverse:
: an implementation of reverse mode ad which makes use of the builtin mutable
state interface; and
reversec:
: an implementation of checkpointed reverse mode ad which extends reverse.
Each of the handlers handle the interface Smooth, which conceptually
corresponds to smooth functions on the real numbers. We only include
constants, negation, addition, and multiplication for simplicity, but any
number of other smooth functions could be included. Additionally, Frank
currently does not support floats, so we use integers instead, however with
language support floats could be used.
| data Nullary = constE Int| data Unary = negateE| data Binary = plusE | timesE| | interface Smooth X =| ap0 : Nullary -> X| | ap1 : Unary -> X -> X| | ap2 : Binary -> X -> X -> XThe above definition says the Smooth interface is parameterized by X and has three effectful commands. The command apN is the N-ary application of a smooth function. Note that the nullary functions are constants. For ease of use, we define the following helper functions.
| c : Int -> [Smooth X] X| c i = ap0 (constE i)| | n : X -> [Smooth X] X| n x = ap1 negateE x| | p : X -> X -> [Smooth X] X| p x y = ap2 plusE x y| | t : X -> X -> [Smooth X] X| t x y = ap2 timesE x yThe operational semantics of Frank allows us to treat the above helper functions as if they were commands themselves, which we will do throughout. We will also define helper functions for the dispatching of $n$-ary functions and their derivatives to make the similarity between different ad modes more apparent.
| op0 : Nullary -> [Smooth X] X| op1 : Unary -> X -> [Smooth X] X| op2 : Binary -> X -> X -> [Smooth X] X| op0 (constE i) = c i| op1 negateE x = n x| op2 plusE x y = p x y| op2 timesE x y = t x y| | der1 : Unary -> X -> [Smooth X] X| der2L : Binary -> X -> X -> [Smooth X] X| der2R : Binary -> X -> X -> [Smooth X] X| der1 negateE x = n (c 1) $\nicefrac{{d}}{{dx}}(-x)\ =\ -1$| der2L plusE x y = c 1 $\nicefrac{{d}}{{dx}}(x\ +\ y)\ =\ 1$| der2L timesE x y = y $\nicefrac{{d}}{{dx}}(x\ \cdot\ y)\ =\ y$| der2R plusE x y = c 1 $\nicefrac{{d}}{{dy}}(x\ +\ y)\ =\ 1$| der2R timesE x y = x $\nicefrac{{d}}{{dy}}(x\ \cdot\ y)\ =\ x$
### 3.1. Evaluation
The most basic handler we will consider is the evaluate handler. It only
handles Smooth X where X is instantiated to Int.
| evaluate : <Smooth Int> X -> X| evaluate x = x| evaluate <ap0 (constE i) ->
k> = evaluate (k i)| evaluate <ap1 negateE x -> k> = evaluate (k (-x))|
evaluate <ap2 plusE x y -> k> = evaluate (k (x + y))| evaluate <ap2 timesE x y
-> k> = evaluate (k (x * y))In the case of constE i, its integer parameter i
is returned. Each other case of evaluate takes the integer arguments passed to
the command and performs the corresponding integer operation.
The evaluate handler will always be our top-level handler, and it is the only
way to remove all Smooth interfaces. We shall evaluate an example program
where evaluate is the top-level handler to illustrate how Frank executes. We
will be paying special attention to how delimited continuations are captured.
We will use underlining to show what term is currently at the focus of
evaluation.
Our initial program is below, and represents the term $1+x^{3}+-y^{2}$
evaluated at $x=2$ and $y=4$, which equals $-7$ .
| evaluate (p (c 1) (p (t (t 2 2) 2) (n (t 4 4))))The current focus of
evaluation is the command c 1.
| evaluate (p _(c 1)_ (p (t (t 2 2) 2) (n (t 4 4))))The argument 1 is in
normal form (fully evaluated). Therefore, we can handle the command c 1. The
handling process begins by capturing the proper delimited continuation by
incrementally freezing the stack of evaluation frames. We represent freezing
by highlighting and boldface.
| evaluate (p _(c 1)_ (p (t (t 2 2) 2) (n (t 4 4))))
| evaluate (p _(c 1)_ (p (t (t 2 2) 2) (n (t 4 4))))We have now reached a
handler, evaluate, for the command in focus. The frozen command (highlighted)
is the captured delimited continuation. The ap0 case of evaluate is then
matched to the command c 1, where k is bound to the continuation with c 1
removed and i is bound to 1. The bound variables k and i are then substituted
into the corresponding body of evaluate.
| evaluate _({x -> (p x (p (t (t 2 2) 2) (n (t 4 4))))} 1)_The next step
applies the continuation to 1.
| evaluate (p 1 (p (t (t 2 2) 2) (n (t 4 4))))The focus of evaluation now
moves to t 2 2, and a new delimited continuation is dynamically captured.
| evaluate (p 1 (p (t _(t 2 2)_ 2) (n (t 4 4))))
| evaluate (p 1 (p (t _(t 2 2)_ 2) (n (t 4 4))))
| evaluate (p 1 (p (t _(t 2 2)_ 2) (n (t 4 4))))
| evaluate (p 1 (p (t _(t 2 2)_ 2) (n (t 4 4))))
| evaluate (p 1 (p (t _(t 2 2)_ 2) (n (t 4 4))))We have now again reached the
evaluate handler, and this time match the ap2 case, resulting in the
following.
| evaluate ({x -> (p 1 (p (t x 2) (n (t 4 4))))} _(2 * 2)_)
| evaluate _({x -> (p 1 (p (t x 2) (n (t 4 4))))} 4)_
| evaluate (p 1 (p (t 4 2) (n (t 4 4))))Evaluation will continue as such until
the final answer of $-7$ is calculated.
We have now seen how the evaluate handler interprets Smooth commands with the
builtin arithmetic operations. Even though evaluate is simple, it allows us to
write our other handlers in a polymorphic fashion independent of Int.
### 3.2. Forward mode
Our next handler is the diff handler, which implements forward mode ad via a
method known as dual numbers. A dual number is a pair of real numbers where
the second number represents the derivative of the first. The diff handler
handles commands with dual number arguments. The mathematical justification of
ad is not our focus, and thus we shall just focus on the patterns of
computation present without proving their correctness. We define the Dual
datatype and diff below.
| data Dual X = dual X X| | v : Dual X -> X| v (dual x _) = x| | dv : Dual X -> X| dv (dual _ dx) = dx| | diff : <Smooth (Dual X)> Y -> [Smooth X] Y| diff x = x| diff <ap0 n -> k> =| let r = dual (op0 n) (c 0) in| diff (k r)| diff <ap1 u (dual x dx) -> k> =| let r = dual (op1 u x) (t (der1 u x) dx) in| diff (k r)| diff <ap2 b (dual x dx) (dual y dy) -> k> =| let r = dual (op2 b x y) (p (t (der2L b x y) dx)| (t (der2R b x y) dy)) in| diff (k r)Notice the similarities between each of the apN cases. The command being handled by diff is evaluated with opN in the first component of Dual, and a calculation involving derivatives creates the second component.
We will evaluate an example program similar to our previous one. The program
will represent the same mathematical term $1+x^{3}+-y^{2}$ evaluated at $x=2$
and $y=4$, but additionally we shall be calculating the derivative with
respect to $x$ at this point, which is $12$. This is achieved by setting $x$
to dual 2 1 and $y$ to dual 4 0, where $x$ has its second component set to $1$
to treat it as the differentiated variable and $y$ has its second component
set to $0$ to treat it as a constant.
| evaluate (diff (| p (c 1) (p (t (t (dual 2 1) (dual 2 1)) (dual 2 1))| (n (t
(dual 4 0) (dual 4 0))))| ))
Evaluation begins as before, with the c 1 command being in focus and a
delimited continuation being captured.
| evaluate (diff (| p _(c 1)_ (p (t (t (dual 2 1) (dual 2 1)) (dual 2 1))| (n
(t (dual 4 0) (dual 4 0))))| ))Note how the continuation captured is delimited
by diff and not evaluate. This behavior is due to the effect typing system of
Frank. There are two different instances of the Smooth interface available to
the portion of the program being handled. By default, the innermost handler
provides the instance being used by extending the ambient ability with an
adaptor. As we shall see later, Frank provides constructs allowing us to
select handlers other than the innermost. The top case of diff is matched by c
1 with the following result.
| evaluate (| let r = dual _(op0 (constE 1))_ (c 0) in| diff (| {x -> (p x (p
(t (t (dual 2 1) (dual 2 1)) (dual 2 1))| (n (t (dual 4 0) (dual 4 0)))))} r)|
)
| evaluate (| let r = dual (c 1) (c 0) in| diff (| {x -> (p x (p (t (t (dual 2
1) (dual 2 1)) (dual 2 1))| (n (t (dual 4 0) (dual 4 0)))))} r)| )We now have
two c commands which will be be handled by evaluate, producing dual 1 0 for
r’s value. After handling, r will be be substituted and the continuation
applied.
| evaluate (diff (| p (dual 1 0) (p (t _(t (dual 2 1) (dual 2 1))_ (dual 2
1))| (n (t (dual 4 0) (dual 4 0))))| ))Evaluation will then continue in a
similar manner for all remaining commands. Each command will first be handled
by diff, and the commands in the body of each diff case handled by evaluate,
eventually producing dual -7 12.
We will now focus on Frank’s ability to dynamically determine which handler
handles a command. First, we define two auxiliary functions.
| lift : X -> [Smooth X, Smooth (Dual X)] (Dual X)| lift x = dual x (<Smooth> (c 0))| | d : {(Dual X) -> [Smooth X, Smooth (Dual X)] (Dual X)}| -> X -> [Smooth X] X| d f x = dv (diff (f (dual x (<Smooth> (c 1)))))The adaptor <Smooth> in lift causes the command c 0 to be associated with Smooth X and not the rightmost instance Smooth (Dual X). The d function returns the derivative of a unary function and lift will enable us to nest d. Note that as in lift, <Smooth> in d causes the command c 1 to be associated with Smooth X. Consider the expression $\nicefrac{{d}}{{dx}}(x\cdot\nicefrac{{d}}{{dy}}(x+y)|_{y=1})|_{x=1}$ (which equals $1$). The corresponding program requires lift.
| evaluate (d {x -> t x (d {y -> p (lift x) y} (c 1))} _(c 1)_)We evaluate
until the delimited continuation is captured.
| evaluate _(d{x -> t x (d {y -> p (lift x) y} (c 1))} 1)_
| evaluate (dv (diff (| {x -> t x (d {y -> p (lift x) y} (c 1))}| (dual 1
(<Smooth> _(c 1)_))| )))
| evaluate (dv (diff (| {x -> t x (d {y -> p (lift x) y} (c 1))}| (dual 1
(<Smooth> _(c 1)_))| )))The continuation for c 1 is delimited by evaluate due
to <Smooth>. We conclude by noting that Frank will reject the nested program
if lift is not present.
### 3.3. Reverse mode
The evaluate and diff handlers manipulate programs by capturing delimited
continuations, but only in quite simple ways. They each eventually compute a
value based on the command being handled and then continue with the original
program with the computed value substituted in. The reverse handler will be
different, and will build up a secondary program during the evaluation of the
initial program.
Reverse mode ad works by creating a mutable cell for each value which
accumulates contributions to its derivative. The method of accumulation is a
generalized version of the backpropagation algorithm made prominent by machine
learning. We define the datatype Prop for backpropagation where Ref X is a
reference to a mutable cell containing a value of type X. The reverse handler
handles commands containing Prop’s.
| data Prop X = prop X (Ref X)| | fwd : Prop X -> X| fwd (prop x _) = x| | deriv : Prop X -> Ref X| deriv (prop _ r) = r| | reverse : <Smooth (Prop X)> Unit -> [RefState, Smooth X] Unit| reverse x = x| reverse <ap0 n -> k> =| let r = prop (op0 n) (new (c 0)) in| reverse (k r)| reverse <ap1 u (prop x dx) -> k> =| let r = prop (op1 u x) (new (c 0)) in| reverse (k r);| write dx (p (read dx) (t (der1 u x) (read (deriv r))))| reverse <ap2 b (prop x dx) (prop y dy) -> k> =| let r = prop (op2 b x y) (new (c 0)) in| reverse (k r);| write dx (p (read dx) (t (der2L b x y) (read (deriv r))));| write dy (p (read dy) (t (der2R b x y) (read (deriv r))))The reverse handler makes use of the same op and der functions as diff, but is different from evaluate and diff in two important ways. Firstly, the type of reverse shows that it requires access to the RefState interface of mutable state (a builtin effect of Frank that can be handled by the language implementation). Secondly, the body of the ap1 and ap2 cases contains code after the use of the captured delimited continuation k. We shall see these writes to memory will form the secondary program which actually accumulates derivatives.
To properly calculate derivatives with reverse, we require a small helper
function which starts the process of backpropagation, which we call grad for
gradient.
| grad : {(Prop X)| -> [RefState, Smooth X, Smooth (Prop X)] (Prop X)}| -> X
-> [RefState, Smooth X] X| grad f x =| let z = prop x (new (c 0)) in| reverse
(write (deriv (f z)) (<Smooth> (c 1)));| read (deriv z)We evaluate the same
term as before.
| evaluate (grad ({x ->| let y = c 4 in p (c 1) (p (t (t x x) x) (n (t y y)))|
} 2))
| evaluate (| let z = prop 2 (new (c 0)) in| reverse (write (deriv ({x ->| let
y = c 4 in p (c 1) (p (t (t x x) x) (n (t y y)))| } z)) (<Smooth> (c 1)));|
read (deriv z))The term new (c 0) is handled first by evaluate for c 0
(returning 0), and the command new 0 is handled by the Frank implementation
and returns a new reference <z> whose cell contains 0. The result is then
substituted for z.
| evaluate (| reverse (write (deriv _({x ->_| _let y = c 4 in p (c 1) (p (t (t x x) x) (n (t y y)))_ | _} (prop 2 <z>))_) (<Smooth> (c 1)));| read (deriv (prop 2 <z>)))Next, the anonymous function is applied to prop 2 <z>.
| evaluate (| reverse (write (deriv (| let y = _c 4_ in| p (c 1) (p (t (t
(prop 2 <z>) (prop 2 <z>)) (prop 2 <z>))| (n (t y y)))| )) (<Smooth> (c 1)));|
read (deriv (prop 2 <z>)))The command c 4 is handled by the ap0 case of
reverse, which as before creates a new reference <r1>, and thus y is
substituted by prop 4 <r1>. The command c 1 will create <r2>.
| evaluate (| reverse (write (deriv (| p (prop 1 <r2>)| (p (t _(t (prop 2
<z>) (prop 2 <z>))_ (prop 2 <z>))| (n (t (prop 4 <r1>) (prop 4 <r1>))))| ))
(<Smooth> (c 1)));| read (deriv (prop 2 <z>)))We have now reached the first
interesting command, which matches the ap2 case of reverse. The captured
delimited continuation is now explicitly highlighted.
| evaluate (| reverse (write (deriv (| p (prop 1 <r2>) | (p (t _(t (prop 2 <z>) (prop 2 <z>))_ (prop 2 <z>))| (n (t (prop 4 <r1>) (prop 4 <r1>)))) | )) (<Smooth> (c 1)));| read (deriv (prop 2 <z>)))The result of reverse handling the command produces a new reference <r3>.
| evaluate (| reverse (write (deriv (| p (prop 1 <r2>)| (p (t (prop 4 <r3>)
(prop 2 <z>))| (n (t (prop 4 <r1>) (prop 4 <r1>))))| )) (<Smooth> (c 1)));|
write <z> (p (read <z>)| (t (der2L timesE 2 2) (read (deriv (prop 4
<r3>)))));| write <z> (p (read <z>)| (t (der2R timesE 2 2) (read (deriv (prop
4 <r3>)))));| read (deriv (prop 2 <z>)))We see that the evaluation of the
initial program has produced new expressions to be evaluated after the initial
program finishes. The handling by reverse will eventually handle all commands
meant for it, producing the following.
| evaluate (| reverse (write <r8> (<Smooth> (c 1)));| write <r2> (p (read
<r2>)| (t (der2L plusE 1 -8) (read (deriv (prop -7 <r8>)))));| write <r7> (p
(read <r7>)| (t (der2R plusE 1 -8) (read (deriv (prop -7 <r8>)))));| write
<r4> (p (read <r4>)| (t (der2L plusE 8 -16) (read (deriv (prop -8 <r7>)))));|
write <r6> (p (read <r6>)| (t (der2R plusE 8 -16) (read (deriv (prop -8
<r7>)))));| write <r5> (p (read <r5>)| (t (der1 negateE 16) (read (deriv (prop
-16 <r6>)))));| write <r1> (p (read <r1>)| (t (der2L timesE 4 4) (read (deriv
(prop 16 <r5>)))));| write <r1> (p (read <r1>)| (t (der2R timesE 4 4) (read
(deriv (prop 16 <r5>)))));| write <r3> (p (read <r3>)| (t (der2L timesE 4 2)
(read (deriv (prop 8 <r4>)))));| write <z> (p (read <z>)| (t (der2R timesE 4
2) (read (deriv (prop 8 <r4>)))));| write <z> (p (read <z>)| (t (der2L timesE
2 2) (read (deriv (prop 4 <r3>)))));| write <z> (p (read <z>)| (t (der2R
timesE 2 2) (read (deriv (prop 4 <r3>)))));| read (deriv (prop 2 <z>)))The
above code is the secondary program created by reverse, which performs
backpropagation. Furthermore, if a user wished to capture this secondary
program, the definition of reverse could be changed to return a suspended
computation. Thus, we could also partially evaluate the whole program (initial
and backpropagation) by running only the initial program and capturing the
backpropagation computation.
It could also be possible to use multi-stage programming by reifying the
initial and secondary programs as a computation graph in the style of Wang et
al. (Wang et al., 2019). Their approach uses delimited continuations and
combines normal execution with building an intermediate representation. As
effects and handlers are essentially a structured use of delimited
continuations, a similar story for Frank may be possible.
### 3.4. Checkpointed reverse mode
The final algorithm we shall cover is checkpointed reverse mode. Reverse mode
has maximum memory residency proportional to the number of operations (as seen
in the definition of reverse). Checkpointed reverse mode allows a trade-off
between space and time by recomputing checkpointed subprograms, once without
allocating memory and an additional time with memory. However, any memory
allocated in between these two runs can be safely deallocated, as it
corresponds to code after the checkpointed subprogram in the original program,
thus reducing maximum memory residency.
To define our new handler, we introduce a Checkpoint effect which takes a
suspended computation that will be run multiple times. We also define a simple
evaluate like handler, evaluatet (see appendix for definition).
| interface Checkpoint X =| checkpoint :| {[Checkpoint X , Smooth (Prop X)]
Prop X} -> Prop XFrank also contains a catch-all pattern match <m> which
matches values and commands not handled above it. We use this feature to
extend reverse by delegating any Smooth commands received to reverse and only
adding a case for checkpoint.
| reversec : <Checkpoint X, Smooth (Prop X)> Unit| -> [RefState, Smooth X]
Unit| reversec x = x| reversec <checkpoint p -> k> =| let s = new (c 0) in|
let res = <RefState> (evaluatet s (| <Smooth(s a b -> s b)> p!| )) in| let r =
prop (fwd res) (new (c 0)) in| reversec (k r);| reversec (write| (deriv
(<Smooth(s a b -> s b), RefState> p!))| (read (deriv r)))| reversec <m> =
reversec (<Smooth(s a -> s)> (reverse m!))Note how the checkpointed subprogram
(the suspended computation p which is the argument of checkpoint) is called
twice, once with evaluatet as the handler and once with reversec as the
handler. Additionally, the last case will match every Smooth command, and then
reinvoke the captured computation with a new reverse handler to handle the
command.
Consider the following program where gradc is grad with reversec in the place
of reverse.
| evaluate (gradc ({x ->| let y = c 2 in| let z = _checkpoint{p x y}_ in| let
a = checkpoint {let w = checkpoint {t x z} in p w y} in| p a x| } (c 2)))The
first interesting evaluation step is after the underlined checkpoint has been
handled.
| evaluate (| reversec (<Smooth(s a -> s)> (reverse (write (deriv (| let z =
prop 4 <r2> in| let a = checkpoint {| let w = checkpoint t (prop 2 <z>) z in|
p w (prop 2 <r1>)} in| p a (prop 2 <z>)| )) (<Smooth> (c 1)))));| reversec
(write (deriv (<Smooth(s a b -> s b), RefState>| _{ p (prop 2 <z>) (prop 2
<r1>)}_!))| (read (deriv (prop 4 <r2>))));| read (deriv (prop 2 <z>)))Note how
on the second line the reverse handler has been made the innermost delimiter
of the remainder of the initial program, via the catch-all case of reversec.
Additionally, note how the checkpointed code (underlined) is stored as a thunk
to be run after the initial program in the second use of reversec. After the
initial program has been evaluated away, we obtain the following.
| evaluate (| reversec (<Smooth(s a -> s)> (| reverse (write <r4> 1);| write
<r3> (p (read <r3>)| (t (der2L plusE 10 2) (read (deriv (prop 12 <r4>)))));|
write <z> (p (read <z>)| (t (der2R plusE 10 2) (read (deriv (prop 12
<r4>))))));| reversec (write (deriv (<Smooth(s a b -> s b), RefState>| {let w
= _checkpoint t (prop 2 <z>) (prop 4 <r2>)_ in| p w (prop 2 <r1>)}!))| (read
(deriv (prop 10 <r3>))));| reversec (write (deriv (<Smooth(s a b -> s b),
RefState>| {p (prop 2 <z>) (prop 2 <r1>)}!))| (read (deriv (prop 4 <r2>))));|
read (deriv (prop 2 <z>)))The remaining checkpoint command illustrates the
recursive nature of reversec. It shows how even nested checkpointing in
checkpointed code can be properly evaluated by delaying the program
transformation happening via evaluation.
## 4\. Conclusion
We have seen the implementation and evaluation of ad in Frank via Frank’s
operation semantics and four handlers: evaluate, diff, reverse, and reversec.
While evaluate and diff do effectively no program transformations, reverse and
reversec build up ancillary programs via delimited continuations. The effects
and handler style of Frank allowed us to compose and nest our defined
handlers, which is especially apparent in the modular definition of reversec
which delegates all Smooth commands to reverse. It may also be possible to
integrate multi-stage programming by using the system of Wang et al.. In
conclusion, we have illustrated in Frank that effects and handlers are a good
match for ad, and that effects and handlers can be seen as a form of program
manipulation.
###### Acknowledgements.
I would like to thank Sam Lindley for answering my many questions about Frank,
my supervisor Chris Heunen for his support, and Ohad Kammar for conversations
about this work and encouragement to improve it. I would also like to thank
the reviewers for their valuable feedback.
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## Appendix
The following is the definition of evaluatet use in reversec. Note the
similarities with evaluate.
| evaluatet : Ref X| -> <Checkpoint X, Smooth (Prop X)> Y| -> [Smooth X] Y|
evaluatet _ x = x| evaluatet s <checkpoint p -> k> =| let res = evaluatet s
(<Smooth(s a b -> s b)> p!) in| evaluatet s (k (prop (fwd res) s))| evaluatet
s <ap0 n -> k> =| evaluatet s (k (prop (<Smooth> (op0 n)) s))| evaluatet s
<ap1 u (prop x dx) -> k> =| evaluatet s (k (prop (<Smooth> (op1 u x)) s))|
evaluatet s <ap2 b (prop x dx) (prop y dy) -> k> =| evaluatet s (k (prop
(<Smooth> (op2 b x y)) s))
|
# The role of reflections in the generation of a time delay in strong field
ionization
Daniel Bakucz Canário<EMAIL_ADDRESS>Michael Klaiber
Karen Z. Hatsagortsyan<EMAIL_ADDRESS>Christoph H. Keitel
Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg,
Germany
(August 27, 2024)
###### Abstract
The problem of time delay in tunneling ionization is revisited. The origin of
time delay at the tunnel exit is analysed, underlining the two faces of the
concept of the tunnelling time delay: the time delay around the tunnel exit
and the asymptotic time delay at a detector. We show that the former time
delay, in the sense of a delay in the peak of the wavefunction, exists as a
matter of principle and arises due to the sub-barrier interference of the
reflected and transmitted components of the tunneling electronic wavepacket.
We exemplify this by describing the tunnelling ionization of an electron bound
by a short-range potential within the strong field approximation in a “deep
tunnelling” regime. If sub-barrier reflections are extracted from this
wavefunction, then the time delay of the peak is shown to vanish. Thus, we
assert that the disturbance of the tunnelling wavepacket by the reflection
from the surface of the barrier causes a time delay in the neighbourhood of
the tunnel exit.
## I Introduction
In both classical and quantum mechanics, the passage of time is always defined
with respect to some dynamical variable (for instance, the hands of a clock).
In the absence of a canonical quantum mechanical time operator, the definition
and measurement of time in quantum mechanics is challenging since dynamical
observables are inherently non-deterministic. Despite the conceptual
difficulties involved, operational techniques with which to measure time have
been developed (see e.g. [1, 2, 3, 4, 5, 6, 7, 8, 9] for overviews).
One of the main applications of these time measurement protocols is the study
of the time taken for a particle to tunnel through a barrier, known as
tunnelling time. Several definitions of the tunneling time exist, such as the
Wigner time [10, 11, 12, 13, 14], the Büttiker-Landauer time[15], the Pollak-
Miller time [16], the Larmor time [17, 18, 19, 20, 21], the dwell time [22],
etc. [23, 24, 25, 26, 27]. Each definition corresponds to a specific aspect of
the measurement process and, in general, these do not coincide.
The tunneling phenomenon plays an essential role in the strong field
ionization process of atoms and in the related field of attoscience. State-of-
the-art techniques in attosecond science are now able to provide exceptional
time and space resolution, reaching tens of attoseconds time- and Angström
space-resolution. In particular, the attoclock technique provides time
resolution of tunneling ionization [28], and has inspired researchers to
experimentally address the challenging problem of tunneling time [29, 30, 31,
32, 33], i.e., how much time, if any, elapses during the quantum tunnelling
process.
This question strikes at the fundamentals of quantum mechanics but experiments
have often been followed by controversial discussion [29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 26, 46, 47, 48, 49, 50, 51, 52,
53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66] in the strong field
community arguing either for, or against, the existence of a tunnelling
induced time delay between ionised electron wavepacket and ionising laser
field.
In this respect we emphasise the distinction between two concepts of the
tunneling time in strong field ionization, namely, the time delay near the
tunnel exit (the classically expected coordinate for the tunneled electron to
emerge), and the asymptotic time delay [37]. While the latter is relevant to
attoclock experiments, the former, known also as the Wigner time delay, can be
calculated theoretically and measured in a Gedanken experiment with a so-
called virtual detector [67, 68]. In both cases, time delay arises due to sub-
barrier dynamics.
The asymptotic time delay is derived from the asymptotic photoelectron
momentum distribution (PMD). It is defined by the classical backpropagation of
the peak of the photoelectron asymptotic wavefunction up to the tunnel exit
[42, 45, 43, 44]. A theoretical description of the experimental asymptotic
time delay is challenging because of the entanglement of Coulomb field effects
with those of the tunneling delay. In a regime far from over-the-barrier
ionization (OTBI), the asymptotic time delay is vanishing [36, 37] However, in
regimes approaching the OTBI the asymptotic delay is not negligible; it was in
fact found to be negative [42, 43, 44] and shown to arise due to interference
of direct and the under-the-barrier recolliding trajectories [51].
In the tunnelling region, the tunneling time delay is deduced by following the
peak of the electron wavefunction, the so-called Wigner trajectory. A few de-
Broglie wavelengths away from the tunnel exit, the electron dynamics are
quasiclassical and the classical trajectory is accurately described by the
eikonal of the wave function in the Wentzel-Kramers-Brillouin (WKB)
approximation. However, near the tunnel exit, the WKB approximation fails and
a full quantum mechanical description is given by the Wigner trajectory. Near
the tunnel exit this description diverges strongly from the classical one but,
nevertheless approaches it asymptotically, as one moves toward a detector. The
Wigner time delay has been calculated numerically with the virtual detector
method in Refs. [36, 34, 35, 33], showing positive time delay and nonvanishing
longitudinal velocity at the tunnel exit.
In this paper we investigate the Wigner time delay in tunnelling ionization,
which a virtual detector in a Gedanken experiment would observe. We concern
ourselves chiefly with understanding the principles of the tunnelling delay,
and to this end we consider a simple, time-dependent model of a one-
dimensional (1D) atom, with an electron bound by a short-range potential,
which is ionized by a half-cycle laser pulse.
At first sight, such abstraction may seem overly simplistic; nonetheless, it
takes into account all the necessary features of tunnelling ionization.
Firstly, ionization occurs mainly in the direction of the electric field so a
1D treatment is appropriate. Likewise, no real laser pulse is a half-cycle
sinusoid but by considering one such laser we ensure no continuum electron-to-
atomic core recollisions take place, which convolute the physical picture.
Such an interpretation is rendered clearer by having analytic expressions,
unattainable with a pure Coulomb atomic potential but feasible with a short
range potential.
In this manner, the time-dependent wavefunction is described within the strong
field approximation (SFA). The components of the wavefunction which are
reflected by the tunnelling barrier are identified by analyzing the saddle
points of the time-integral of the wavefunction. We show that the interference
of reflected and transmitted components of the wavefunction generates the
Wigner time delay, while the same delay vanishes when the reflected components
are neglected. Additionally, the scalings with respect to the laser field
strength of the Wigner time delay and group velocity at the tunnel exit are
derived and interpreted.
We work well below the OTBI regime, in what we term the deep tunneling regime.
While in this regime the asymptotic tunneling time delay is vanishing we show
that the Wigner time at the tunnel exit (calculated in the first-order SFA) is
not only nonzero, but sizeable, as already noted in [37]. This is in contrast
to the asymptotic time delay, where the first-order SFA result is vanishing
and the second-order SFA is required to obtain a nonvanishing asymptotic time
delay (attributable to sub-barrier recollisions and interference of paths).
The structure of the paper is as follows: in Sec. II we introduce the basic
ionization model, calculate the SFA time-dependent wavefunction, and construct
the Wigner trajectory from the latter. To elucidate our method of analysing
the contribution of reflections to the wavefunction, in Sec. III we consider a
simpler, analytic, model of ionization: an electron in a constant electric
field. The analyticity of this model allows us to relate the reflected
components of the wavefunction to the contribution of the specific saddle
points of the integral representation of the wavefunction. We apply this
concept to the SFA wavefunction in Sec. IV, whereby we show that neglecting
reflections results in a zero Wigner time delay. Lastly, in Sec. V, we
summarize and interpret our results and discuss the implications of these on
the interpretation of attosecond streaking experiments. For the reader
unacquainted with strong field physics, the Wigner time delay for scattering
by a square potential barrier was previously analyzed in [69, 7], where the
significance of reflections to the time delay was alluded to. In Appendix A
this problem is revisited and the role of reflections is explicitly presented.
Atomic units (a.u.) are used exclusively throughout this work.
## II Time Delay in Strong Field Ionization
The Wigner time, developed for scattering problems [10, 11, 13], considers the
time taken for the peak of the wavepacket to travel a given distance. For
tunneling through a time-independent barrier, the Wigner time can be derived
via the energy derivative of the phase of the tunnelling wavefunction $\psi$:
$\tau_{W}(x)=-i\,\frac{\partial}{\partial\,\mathcal{E}}\ln\frac{\psi(x)}{|\psi(x)|}.$
(1)
The peak of the ionized wave packet leads to the asymptotic formation of a
well-defined peak in the attoclock PMD, which is subsequently measured and
interpreted.
As the peak of the wavepacket follows the trajectory $\tau_{W}(x)$, the Wigner
group delay velocity of the wave packet can be defined
$v_{W}(x)=\left(\frac{\partial\tau_{W}(x)}{\partial x}\right)^{-1}.$ (2)
When a monochromatic wave is incident on a finite potential barrier, the
propagation inside the barrier is a superposition of an exponentially
suppressed wave (transmission) with a growing exponential wave, namely the
reflection by the surface of the barrier (see e.g. Appendix A). The goal of
this paper is to establish the causal effect of wavefunction refections inside
the barrier on the time delay in strong field ionization.
In order to do this, we calculate the time-dependent ionized electron
wavefunction at the position where the electron appears in the continuum,
thereby allowing us to deduce the peak of the wavepacket with respect to time.
The Wigner time delay can be calculated as the energy derivative of the phase
of the wave function via Eq. 1 in a static field but in a time-dependent laser
field this prescription fails as energy is not conserved. Correspondingly, we
derive the Wigner time delay at the tunnel exit explicitly as the delay of the
peak of the wavepacket, rather than from the energy derivative of the phase of
the wavefunction.
### II.1 The Model
Consider an electron initially bound in a 1D short-range potential
$V(x)=-\kappa\>\delta(x)$ (3)
with a binding energy $I_{p}=\kappa^{2}/2$, and with a corresponding ground
state wavefunction
$\psi_{0}(x,t)=\sqrt{\kappa}\exp\left(-\kappa|x|+iI_{p}t\right)$.
To describe the generation of the peak of the ionized wavepacket at the tunnel
exit, not the usual PMD at a detector, a half-cycle laser field is modelled,
avoiding recollisions in the continuum. The laser electric field,
$\displaystyle E(t)$ $\displaystyle=$ $\displaystyle-E_{0}\cos^{2}(\omega t),$
(4)
is switched on at $\omega t_{0}=-\pi/2$. In a half-cycle laser field the
ionization happens mostly near the peak of the field. Here, nonadiabatic
effects are governed by the second-derivative of the field at the peak
$E^{\prime\prime}(0)/E(0)=2\omega^{2}\equiv\omega_{eff}^{2}$, where we define
the effective frequency $\omega_{eff}=\sqrt{2}\omega$.
We consider the tunnel ionization regime and keep constant the Keldysh
parameter [70]
$\gamma=\sqrt{\frac{I_{p}}{2U_{p}}}\ll 1,$ (5)
where $U_{p}=E_{0}^{2}/(4\,\omega_{eff}^{2})$ is the ponderomotive potential.
Thus, when varying the field strength $E_{0}$, we vary the frequency
$\omega=\gamma E_{0}/(\sqrt{2}\kappa)$ accordingly.
Moreover, we consider the so-called deep tunneling regime, wherein $E_{0}\ll
E_{th}$, and $E_{th}$ is the threshold field of OTBI. This threshold can be
estimated as the field strength where the coordinate-saddle point of the SFA-
matrix element, i.e. the starting point $x_{s}\approx\sqrt{\kappa/E_{0}}$ of
the quantum orbit, becomes comparable the tunnel exit $x_{e}$, which for the
short range potential corresponds to the condition $E_{th}\approx\kappa^{3}/4$
[71].
The superposition of the laser field with the atomic potential creates a
potential barrier, through which an electron can tunnel, with a characteristic
classical tunnel exit $x_{e}=I_{p}/E_{0}=10$ a.u. 111 A more precise
definition for the tunnel exit can be given by the SFA, namely
$x^{SFA}_{e}=\int_{t_{s}}^{\rm
Re\\{t_{s}\\}}(A(t^{\prime})-A(0))\;dt^{\prime}\approx(1-\gamma^{2}/4)\,I_{p}/E_{0}=0.995\,I_{p}/E_{0}$,
where $t_{s}$ is the temporal saddle point of the SFA integral. The minuscule
discrepancy between the classical tunnel exit, $x_{e}$, and its SFA correction
for the parameters in this work ($\gamma=\sqrt{2}/10$) leads us to use the
simpler of the two definitions.. While this is a rudimentary estimate, it is
in agreement with a tunnel exit derived quantum mechanically in our parameter
regime as shown in Section IV.2.
(a) (b)
Figure 1: (a) Ionization amplitude $|\psi_{i}(x,t)|^{2}$ as defined in Eq.
(17). At every cross-section in $x$, the temporal peak of the wavefunction was
determined (plotted in blue) which can be interpreted as the trajectory of the
peak. (b) Comparison of trajectories in the region near the tunnel exit,
$x_{e}$. The displayed curves are: the peak trajectory (blue, given by the
maximum of $|\psi_{i}(x,t)|^{2}$), the classical trajectory (orange, given by
the Newton equation starting at the tunnel exit), and the Wigner trajectory
(green, given by Eq. (25)). Under the barrier ($x<10$, shaded blue), there is
an increasing delay of the peak w.r.t the laser peak. Outside the barrier, the
peak trajectory rapidly converges with the classical and Wigner electron
trajectory. Near the atomic core ($x<2$), the bound state,
$|\psi_{0}(x)|\gg|\psi_{i}(x)|$, dominates the full wavefunction
$\psi(x)=\psi_{0}(x)+\psi_{i}(x)$ so the meaning of a trajectory for
ionization is lost. Figure 2: Probability distribution
$|\psi_{i}(x_{e},t)|^{2}$ vs time at the classical tunnel exit
$x_{e}=I_{p}/E_{0}$. This distribution is peaked at a greater time,
$t_{m}\approx 7.6$ a.u. $\approx 183$ as, marked by the dot, than the peak of
the laser pulse.
### II.2 Time Evolution of the Wavefunction
Our investigation is based on the SFA wavefunction describing the electron
dynamics in tunneling ionization. We define the Wigner trajectory and
tunneling time delay employing the virtual detector approach [34, 35, 67, 68]
based on the SFA wavefunction. The ionization dynamics are described by the
Schrödinger equation
$i\frac{\partial}{\partial t}\Psi(x,t)=(H_{0}+H_{i})\Psi(x,t),$ (6)
with the atomic Hamiltonian
$H_{0}=-\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}+V(x),$ (7)
and the interaction Hamiltonian with the laser field
$H_{i}=xE(t)\;$ (8)
The standard SFA is employed for the solution of the Schrödinger equation. In
the interaction picture with Hamiltonian $H=H_{0}+H_{i}$, the time evolution
operator $U(t,t_{0})$ satisfies the Dyson integral equations [73]
$U(t,t_{0})=U_{0}(t,t_{0})-i\int^{t}_{t_{0}}\>\mathop{}\\!\mathrm{d}t^{\prime}\>U(t,t^{\prime})H_{i}(t^{\prime})U_{0}(t^{\prime},t_{0}),$
(9)
where $U_{0}$ is the evolution operator corresponding to the Hamiltonian
$H_{0}$. A perturbation series can be constructed by replacing the full
evolution operator $U(t,t_{0})$ in the Dyson integral with the evolution
operator corresponding to the Hamiltonian $H_{i}$, namely
$U_{f}(t,t^{\prime})=\int\mathop{}\\!\mathrm{d}p\ket{\Psi_{p}(t)}\bra{\Psi_{p}(t^{\prime})},$
(10)
where $\ket{\Psi_{p}}$ are the Volkov states [74]. Each order in the
perturbation theory corresponds to a sub-barrier recollision in our half-cycle
laser field. As each sub-barrier recollision implies a longer tunnelling path,
each term in this series is suppressed by the Keldysh exponent
$\exp(\nicefrac{{-2\kappa^{3}}}{{3E_{0}}})$ and so, in contrast to the near
OTBI regime, in the deep tunnelling regime a first order expansion in the SFA
suffices (i.e. neglecting sub-barrier recollisions) [51].
Consequently, the electron state during the interaction takes the form
$\ket{\psi(t)}=\ket{\psi_{0}(t)}+\ket{\psi_{i}(t)}$ where $\ket{\psi_{0}(t)}$
is the eigenstate of the atomic Hamiltonian $H_{0}$ and
$\displaystyle\ket{\psi_{i}(t)}$ $\displaystyle=$
$\displaystyle-i\int^{\infty}_{-\infty}dp\int^{t}_{t_{0}}dt^{\prime}\ket{\Psi_{p}(t)}\braket{\Psi_{p}(t^{\prime})}{H_{i}(t^{\prime})}{\psi_{0}(t^{\prime})},\qquad$
(11)
describes the electron dynamics induced by the laser field, which generates
the Wigner trajectory. It is the dominant part of the wavefunction at
distances far from the core, and therefore, we concentrate on the analysis of
its behavior.
(a) (b)
Figure 3: Log-log plots of the scaling w.r.t. field strength, $E_{0}$, of the
(a) Wigner time delay and (b) group velocity at the classical tunnel exit,
$x_{e}=I_{p}/E_{0}$, for the time evolved SFA wavefunction (red dots) and the
adiabatic constant field wavefunction (blue lines). For the time evolved
wavefunction, the maxima in time, $t_{m}$, of the probability distribution
$|\psi(x_{e},t)|^{2}$ and its derivative were calculated; for the adiabatic
case, results follow from Eqs. (26) and (27). The agreement in the trends is
good, deteriorating as one approaches the OTBI threashold $E_{th}\approx 0.25$
a.u.
Denoting the electronic kinetic momentum as
${\cal P}(t)\equiv p+A(t),$ (12)
with the laser vector potential
$A(t)=-\int_{-t_{0}}^{t}E(t^{\prime})\,dt^{\prime}$, the 1D Volkov state in
the length gauge is
$\Ket{\Psi_{p}(t)}=\Ket{{\cal P}(t)}e^{-i\,S(t)}$ (13)
with a plane wave component
$\Braket{x}{p}=(2\pi)^{-\frac{1}{2}}\exp({i\,p\,x})$ and the Volkov-phase
$S(t)=\frac{1}{2}\int^{t}\mathop{}\\!\mathrm{d}\tau\,{\cal P}(\tau)^{2}\;.$
(14)
We can expand the overlap in (11) using a resolution of identity in a dummy
variable $x^{\prime}$
$\braket{\Psi_{p}(t^{\prime})}{H_{i}(t^{\prime})}{\psi_{0}(t^{\prime})}=\frac{e^{i\left(I_{p}t^{\prime}-S(t^{\prime})\right)}}{\sqrt{2\pi}}E(t^{\prime})\int^{\infty}_{-\infty}dx^{\prime}\,e^{i\,\mathcal{P}(t^{\prime})x^{\prime}-\kappa|x^{\prime}|}x^{\prime}$
(15)
While this integral is soluble, we do not evaluate it but rather first
integrate (11) w.r.t. to $p$ and then $x^{\prime}$. With the notation $\Delta
f_{n}(t,t^{\prime})=f_{n}(t)-f_{n}(t^{\prime})$, and defining the integrals
$f_{n}(t)=\int^{t}dt^{\prime}A(t^{\prime})^{n}$, we may perform the initial
integral over $p$ which is a simple Gaussian integral. The resulting
expression
$\psi_{i}(x,t)=\int^{t}_{t_{0}}\mathop{}\\!\mathrm{d}t^{\prime}\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}x^{\prime}\frac{x^{\prime}\sqrt{\kappa}\,E(t^{\prime})}{\sqrt{2\pi\,i(t-t^{\prime})}}\exp\left[i\left(I_{p}t^{\prime}+i\kappa|x^{\prime}|+x(A(t)-A(t^{\prime}))+\frac{i\left((x-x^{\prime})-\Delta
f_{1}(t,t^{\prime})\right)^{2}}{2(t-t^{\prime})}-\frac{\Delta
f_{2}(t,t^{\prime})}{2})\right)\right]$ (16)
may then be finally integrated in $x^{\prime}$ to yield
$\displaystyle\psi_{i}(x,t)$
$\displaystyle=\int^{t}_{t_{0}}dt^{\prime}\,\sqrt{\kappa}\,E(t^{\prime})\left(\frac{z_{-}^{2}(1+\mathop{}\\!\mathrm{Erf}(z_{-}))e^{-z_{+}^{2}}}{z-\kappa}+\frac{z_{+}^{2}(1-\mathop{}\\!\mathrm{Erf}(z_{+}))e^{-z_{-}^{2}}}{z+\kappa}\right)\exp\left(i\zeta\right)\equiv\int^{t}_{t_{0}}dt^{\prime}\exp({i\,\Phi(x,t,t^{\prime})})$
(17)
where we have defined $z_{\pm}=\sqrt{\frac{i}{2}(t-t^{\prime})}(z\pm\kappa)$
for
$z=-i\,\left(A(t^{\prime})+\frac{x-\Delta
f_{1}(t,t^{\prime})}{t-t^{\prime}}\right)$ (18)
and where
$\displaystyle\zeta$ $\displaystyle=I_{p}t^{\prime}+x\,A(t)+$ (19)
$\displaystyle(t-t^{\prime})\left(\frac{(x-\Delta
f_{1}(t,t^{\prime}))^{2}}{2(t-t^{\prime})^{2}}+\kappa^{2}+z^{2}\right)-\frac{\Delta
f_{2}(t,t^{\prime})}{2}$ (20)
Our aim is to calculate the wavefunction and Wigner time delay in the
interaction region around the tunnel exit by studying the phase $\Phi$. Thus,
in deviation to standard SFA studies, the time integral is calculated up to
the finite observation time $t$, meaning the integration must be performed
numerically. The usual saddle-point integration method is valid only for
sufficiently large values of $t$, placing one well outside the tunnelling
region.
### II.3 Numerical Integration
For the numerical integration of the SFA wavefunction Eq. (17) an electric
field strength $E_{0}=0.05$ a.u. and Keldysh parameter
$\gamma=\sqrt{2}/10\approx 0.14$ were chosen to ensure deep tunnelling is
considered. With these parameters, our half-cycle laser pulse spans
$[-314,314]$ a.u. We present results for the case of hydrogen, for which
$I_{p}=\kappa^{2}/2=1/2$ a. u., implying an OTBI threshold field
$E_{th}\approx 0.25$ a.u. The time integration was carried out with the
standard numerical integration routine of Mathematica 12 to a precision of 30
digits.
The space-time probability distribution, $|\psi_{i}(x,t)|^{2}$, is presented
in Fig. 1. From this distribution the Wigner trajectory is derived as follows:
the probability time-distribution at each space point $x$ is invoked (see Fig.
2 for the case of the tunnel exit coordinate $x=x_{e}$) and the maximum
$t_{m}(x)$ of this distribution is derived. The Wigner trajectory is
represented by the function $t_{m}(x)$, which runs along all maximum points of
the space-time distribution. Near the core, $|x|\lesssim 3$ a.u., the bound
state dominates the wavefunction $|\psi_{0}|\gg|\psi_{i}|$, so the concept of
the Wigner trajectory loses its meaning.
We underline that the wavefunction $\psi_{i}$ vanishes after the laser field
is switched off. Thus, $\psi_{i}$ does not include the possible net
excitations of the atomic states due to interaction with the laser field. This
is a feature of the SFA, where the exact time-evolution operator in the final
state is replaced by the Volkov propagator, representing the continuum
electron in the laser field.
We note the maximum of the wavefunction displays a time delay with respect to
the peak of the field. The tunneling time delay under the barrier ($x<10$)
increases when moving towards the tunnel exit. Outside the barrier, this
trajectory rapidly approaches the classical electron trajectory (beginning at
the classical tunnel exit with zero momentum). The slight deviation is due to
quantum mechanical corrections to the quasiclassical wave function not far
from the tunnel exit.
The main idea advocated in this paper is that the Wigner time delay during
tunneling is closely related to reflections arising during tunneling dynamics.
It is straightforward in the simple case of tunneling through a box potential
to show that reflections are responsible for the tunneling time delay, as is
done in Appendix A. However, unlike in the separable box potential case, a
given wavefunction is not easily deconstructible as a simple superposition of
reflected and transmitted components corresponding to simple decaying /growing
exponentials. Instead, the transmitted and reflected components are completely
encapsulated in the wavefunction making it rather more problematic to
disentangle. This is ultimately achieved in Sec. V for the SFA wavefunction
presented in this paper but in order to highlight the main aspects of the
method we discuss beforehand a simpler example, that of tunneling in a
constant electric field.
## III Time delay in a constant field
The adiabatic model of ionization, namely atomic ionization in a constant
field, will be helpful for our purposes as it is analytically tractable.
Moreover, it will provide us with a reference with which to compare our
earlier time dependent model.
### III.1 Ionisation in an adiabatic field
Consider a bound electron of energy $-I_{p}$ in 1D $\delta$-potential ionized
by a constant electric field $E_{0}$. The continuum eigenstate of the electron
in this field is given by the solution to the time-independent Schrödinger
equation:
$-\frac{1}{2}\frac{\mathop{}\\!\mathrm{d}^{2}\psi}{\mathop{}\\!\mathrm{d}x^{2}}+\left(I_{p}-E_{0}\>x\right)\psi(x)=0,$
(21)
which has as a general analytical solution as a superposition of the Airy
functions of the first and second kind
$\psi(x)=c_{1}\mathop{}\\!\mathrm{Ai}(\tilde{x})+c_{2}\mathop{}\\!\mathrm{Bi}(\tilde{x}),$
(22)
where
$\tilde{x}=\left(\frac{2}{E_{0}^{2}}\right)^{\frac{1}{3}}\left(I_{p}-E_{0}\>x\right).$
(23)
The requirement that the wavefunction be a travelling wave as
$x\rightarrow\infty$ imposes the condition $c_{1}=-ic_{2}$ so that the
wavefunction takes the form
$\psi(x)=T\left[\mathop{}\\!\mathrm{Bi}(\tilde{x})+i\,\mathop{}\\!\mathrm{Ai}(\tilde{x})\right],$
(24)
where $T$ can be determined by matching this wavefunction to the bound-state
solution of the atom; this prefactor plays a role in the ionization amplitudes
but is irrelevant to the phase of the wavefunction. We calculate the Wigner
time delay for this wavefunction via the energy derivative of the phase
$\displaystyle\tau_{W}(x)=i\frac{\partial}{\partial{I_{p}}}\ln\frac{\psi}{|\psi|}=\frac{2^{\frac{1}{3}}}{\pi
E_{0}^{\frac{2}{3}}}\;\frac{1}{\mathop{}\\!\mathrm{Ai}(\tilde{x})^{2}+\text{Bi}(\tilde{x})^{2}}.$
(25)
The constant field model allows one to estimate the scaling of the Wigner time
delay, given by Eq. (1), at the tunnel exit $x_{e}=I_{p}/E_{0}$:
$\tau_{W}(x_{e})=\frac{3^{4/3}\,\Gamma\left(\frac{2}{3}\right)^{2}}{2^{5/3}\pi}\frac{1}{E_{0}^{2/3}}.$
(26)
Using Eq. (2), we derive the scaling of the Wigner group velocity of the
electron at the tunnel exit,
$v_{W}(x_{e})=\frac{2\left(\frac{2}{3}\right)^{2/3}\sqrt{\pi}\,\Gamma\left(\frac{7}{6}\right)}{\Gamma\left(\frac{2}{3}\right)^{2}}\,E_{0}^{1/3},$
(27)
and find it consistent with the estimate of the scaling of the electron
momentum $p_{e}\sim E_{0}\,\tau_{W}$ in Ref. [51].
We compare the Wigner time delay and the group delay velocity in the constant
field case with the exact time dependent SFA calculations in Fig. 3. The time
delay in the SFA scales with the electric field in a similar manner to the
adiabatic case, $\tau_{W}(x_{e})\propto 1/E_{0}^{2/3}$, but shifted in time by
a constant. This shift stems from the fact that the Wigner trajectory is
derived via the energy derivative of the phase of the wave function (which
corresponds to the following the peak of the wave packet, neglecting its
spreading), while the time-integration responsible for the formation of the
ionization wave packet in SFA is calculated exactly numerically.
The scaling of the Wigner group velocity using the SFA is also in agreement
with the adiabatic estimate, $v_{W}(x_{e})\propto E_{0}^{1/3}$. However, the
relative error between these two grows with the field strength, becoming
appreciable at higher field strengths when approaching the OTBI threshold.
This is consistent with the simple estimate for the threshold for OTBI,
$E_{th}\approx 0.25$ a.u., where a tunnelling description no longer applies.
Note that the Wigner trajectory in the constant field case is determined from
the solution of the time-independent Schrödinger equation (energy eigenstate),
while in the SFA case from the time-dependent wavefunction. In the first case,
the Wigner trajectory is defined as the derivative of the wavefunction phase
with respect to the energy, which corresponds to determining the coordinate of
the peak of the electron wavepacket at a fixed time moment, i.e. determining
the motion of the wavepacket peak. In contrast, in the SFA case we explicitly
determine the maximum of the wavefunction in time for a fixed coordinate.
Figure 4: The solution to the electron in a constant field problem is given
by a superposition of Airy functions of the first and second kind,
$\mathop{}\\!\mathrm{Ai}(\tilde{x})$ and $\mathop{}\\!\mathrm{Bi}(\tilde{x})$,
plotted in blue and orange respectively. These functions have very accurate
asympotic expansions, shown dashed, which can be derived by considering the
saddle points of the Airy integrals (28)-(29). Under the barrier, $x<10$
(shaded blue), these expansions show that the wavefunction components
$\mathop{}\\!\mathrm{Ai}(\tilde{x})$ and $\mathop{}\\!\mathrm{Bi}(\tilde{x})$
respectively correspond to the reflected and transmitted components of the
wavefunction.
(a) (b) (c)
Figure 5: The Airy integral $\int_{\gamma}\,ds\exp(\tilde{x}\,s-s^{3}/3)$ is
defined on the complex $s$ plane; it converges when the endpoints of its
infinite contour, $\gamma$, lie in the shaded areas. The canonical contours
defining the Airy function of the first ($\gamma_{1}$) and second
($\gamma_{2}-\gamma_{3}$) kind are shown in (a). Contributions to the Airy
integral arise principally from portions of the contour near the saddle-points
of the integrand function $\exp(\tilde{x}\,s-s^{3}/3)$. The configuration of
the saddles $s_{\pm}(x)=\pm\sqrt{\tilde{x}}$ (red dots) in the complex plane
is shown for positions (b) inside the barrier ($\tilde{x}>0$), and (c) outside
the barrier ($\tilde{x}<0$). Through each saddle point pass two perpendicular
level curves, the path of steepest descents (solid blue) and the path of
steepest ascents (dashed gray). Deforming the defining contours in (a) through
the steepest descents contours in (b) and (c) provides asymptotic expansions
for the Airy function, as in Eqs. (31) and (32).
### III.2 Reflections in constant field
To investigate the effect of under-the-barrier reflected components, we need
to isolate their contribution to the full wavefunction. The analytical wave
function of Eq. (24) consists of a superposition of Airy functions
$\mathop{}\\!\mathrm{Ai}(\tilde{x})$ and $\mathop{}\\!\mathrm{Bi}(\tilde{x})$
which can be given interpretation as the under-the-barrier reflected and
transmitted components of the tunneling wavefunction, respectively.
This interpretation can be generally understood by considering the Airy
functions, as shown in Fig. 4. The $\mathop{}\\!\mathrm{Bi}$-component of the
wavefunction decays exponentially as it approaches the tunnel exit, $x=x_{e}$,
from the atomic core at $x=0$, and hence can be seen to correspond to the
transmission component; likewise, the exponentially growing
$\mathop{}\\!\mathrm{Ai}$-component corresponds to wavefunction reflection.
This interpretation can be formally established by considering the integral
representation of the Airy functions
$\displaystyle\mathop{}\\!\mathrm{Ai}(\tilde{x})$ $\displaystyle=\frac{1}{2\pi
i}\int_{\gamma_{1}}ds\;\exp(\tilde{x}\,s-\frac{s^{3}}{3}),$ (28)
$\displaystyle\mathop{}\\!\mathrm{Bi}(\tilde{x})$
$\displaystyle=\frac{1}{2\pi}\int_{\gamma_{2}-\gamma_{3}}ds\;\exp(\tilde{x}\,s-\frac{s^{3}}{3}),$
(29)
where the complex plane integration paths $\gamma_{i}$ are indicated in Fig. 5
(a). These integrals converge when their endpoints lie in the slices of the
complex plane defined by $-\frac{\pi}{6}<\theta<\frac{\pi}{6}$,
$\frac{\pi}{2}<\theta<\frac{5\pi}{6}$, and
$\frac{7\pi}{6}<\theta<\frac{3\pi}{2}$, where in polar $\\{r,\theta\\}$
coordinates $s=r\exp(i\,\theta)$. In these regions, shaded blue in Fig. 5, the
integrand vanishes rapidly as $r\rightarrow\infty$.
The majority of the contributions to the Airy integrals thus come from around
the saddle points,
$\displaystyle s_{\pm}=\pm\sqrt{\tilde{x}},$ (30)
of the argument of integrand. The Airy contours, $\gamma_{i}$, can be deformed
into paths that to go through these saddle-points and, using the standard
technique of saddle-point integration method [75], asymptotic expressions for
the Airy integrals can be determined.
The saddle-points, and the respective paths of steepest descents and ascents,
are illustrated in Figs. 5 (b) and (c); since these are dependent on
$\tilde{x}$ the application of the saddle point method is different for the
two cases of inside ($\tilde{x}>0$) and outside ($\tilde{x}<0$) the potential
barrier. As shown in Fig. 5 (b), for $\tilde{x}>0$ we may deform the contour
$\gamma_{1}$ smoothly into the path of steepest descents for the saddle point
$s_{-}$ and in doing so obtain the asymptotic approximation
$\displaystyle\mathop{}\\!\mathrm{Ai}(\tilde{x})$
$\displaystyle=\frac{\exp\left(-\frac{2}{3}\tilde{x}^{\frac{3}{2}}\right)}{2\,\sqrt{\pi}\,\tilde{x}^{\frac{1}{4}}}+\mathop{}\\!\mathcal{O}(\tilde{x}^{-\frac{3}{2}}).$
(31)
Likewise, we may deform the contours $\gamma_{2}$ and $-\gamma_{3}$ to both go
through the steepest descent path of the saddle point $s_{+}$ and hence deduce
the asymptotic form of the $\mathop{}\\!\mathrm{Bi}(x)$-function under the
potential barrier
$\displaystyle\mathop{}\\!\mathrm{Bi}(\tilde{x})$
$\displaystyle=\frac{\exp\left(+\frac{2}{3}\tilde{x}^{\frac{3}{2}}\right)}{\sqrt{\pi}\,\tilde{x}^{\frac{1}{4}}}+\mathop{}\\!\mathcal{O}(\tilde{x}^{-\frac{3}{2}}).$
(32)
From a tunnelling particle’s perspective, one may identify the
$\mathop{}\\!\mathrm{Ai}(x)$ function and its saddle-point $s_{-}$ with the
reflected part of the wavefunction, and the saddle-point $s_{+}$, or the
$\mathop{}\\!\mathrm{Bi}(x)$ function with the transmitted part of the
wavefunction: the transmitted wave decays as it moves to the tunnel exit,
while the reflected wave propagates from the exit decaying exponentially
toward the atomic core, as shown in Fig. 4.
In the following section, this relation between sub-barrier reflections and
the contribution of one saddle point to the time-integrand of the wavefunction
amplitude is applied to the problem of ionization in a time-dependent field.
This then allows us to investigate the role of sub-barrier reflection in the
formation of a tunneling time delay.
## IV Wigner Time and Reflections in the SFA
Figure 6: Complex $t^{\prime}$ plane of the argument $\Phi(x,t,t^{\prime})$
of the wavefunction integral
$\psi_{i}=\int^{t}_{t_{0}}dt^{\prime}\exp(i\,\Phi)$, for parameters $x=9$ and
$\omega t=\pi/40$. For these configurations there are two saddle-points,
denoted by a circle and triangle, the former which, $t_{+}$, corresponds to
reflections. The saddle-point method can be used to solve the wavefunction
integral, when the path is taken over both saddle-points. We use a partial
path given by (33) to remove the contribution of reflections by integrating
over only one of the saddle-points. Other possible configurations of the
saddle points are shown in Fig 11.
As shown in Sec. III, we may reasonably ascribe the saddle-point of the
integrand of the wavefunction to under the barrier reflection or transmission.
To reveal the contribution of the reflection to the tunneling time delay, we
investigate the complex continuation of the integrand function of the SFA
wavefunction, Eq. 17. The insight developed previously, namely the relation of
a specific saddle-point of the integrand function to reflection-like behaviour
in the wavefunction, is used to extract the contribution of reflections to the
wavefunction by modifying the path of integration the complex plane.
Figure 7: Log-log plot of the relative contributions to the wavefunction
amplitude from the portion of the integration path up to the saddle point
(blue) and from the saddle point to the given real time $t$ (orange) for
various positions $x$ under the classical barrier (i.e. far from the atomic
core, but smaller than the tunnel exit). The behaviours are approximately
decaying and growing exponentials, respectively, analogous to transmitted and
reflected parts of the wavefunction.
(a) (b)
Figure 8: (a) Temporal probability distributions at the position
$x=8\textnormal{ a. u.}$ (under the barrier, near tunnel exit), for the
ionised wavefunction, $\psi_{i}$, and the pseudo-wavefunction neglecting
reflections, $\psi_{nr}$. The physical wavefunction has accrued a Wigner time
delay with respect to the laser field, whereas the maximum of the pseudo-
wavefunction is synchronous with the laser field peak. (b) Trajectories in the
$(x,t)$ plane via: the maximum of the wavefunction $\psi_{i}$ (given by
$t_{m}(x)$), the maximum of the wavefunction neglecting reflections
$\psi_{nr}$, and the energy derivative of the constant field wavefunction
$\tau_{W}(x)$. The classical electron trajectory, $x_{cl}(t)$, starting at the
tunnel exit at the peak of the laser field is also shown. Under the barrier,
the probability distribution neglecting reflections $|\psi_{nr}|^{2}$ shows
zero time delay. In regions where Eq (34) does not apply, mostly near the
classical tunnel exit, we may not identify reflections nor plot a subsequent
trajectory.
In general, the complex $t^{\prime}$-plane picture of the phase $\Phi$ in Eq.
(17) has many similarities to that of adiabatic ionisaion. However, it depends
continuously not only on the coordinate $x$ but now also on observation time
$t$, making its analysis somewhat more involved. In adiabatic ionisation,
there was only one degree of freedom, $\tilde{x}$, and the saddle points were
either purely real or imaginary.
As shown in Fig. 6, in the time dependent case there are also two saddle-
points of relevance in the SFA integral denoted $t_{\pm}$. Their arrangement
in the complex $t^{\prime}$-plane is dependent on $x$ and $t$ but, as in the
adiabatic case, the real coordinate $x$ determines whether they are vertically
or horizontally aligned (in or outside the barrier); the observation time $t$
merely shifts the axis of symmetry which is always observed around the line
$t^{\prime}=t$. A more detailed discussion of the configuration of saddle
points in $(x,t)$ parameter space may be found in Appendix B.
### IV.1 Extracting Reflections
In the adiabatic case, a partial integration over only contour containing the
saddle point $s_{-}$ would exclude the exponentially growing contributions to
the wavefunction, i.e. the reflections, associated with the saddle point
$s_{+}$.
We use the same principle in the time dependent case; in Fig. 6 we identify
the saddle point corresponding to reflections, denoted $t_{+}$ (represented by
a circle), and the secondary saddle point $t_{-}$ (represented by a triangle).
When we consider the wavefunction under the potential barrier, we can split
the integration contour in Eq. (17) into two parts: from the start of the
pulse, $t_{0}=-\pi/(2\omega)$, to the upper saddle point, $t_{+}$, and from
$t_{+}$ to the real time $t$.
The contributions of these two contours to the wavefunction are shown in Fig.
7. The former integral has the form of a growing exponential and so can
ostensibly be identified as the reflected part of the wavefunction while the
latter is a decaying exponential identifiable as the transmitted part. To
neglect contributions from the reflections to the full wavefunction we
identify the following wavefunction:
$\displaystyle\psi_{nr}(x,t)=\begin{dcases}\quad\int_{t_{0}}^{t_{+}}dt^{\prime}\exp(i\,\Phi(x,t,t^{\prime}))&x>x_{e}\\\\[14.0pt]
\quad\int^{t}_{t_{+}}dt^{\prime}\exp(i\,\Phi(x,t,t^{\prime}))&x<x_{e}\\\
\end{dcases}$ (33)
Our method of distinguishing the reflection contribution is based on the
distinguishability of the saddle-points contributions to the wavefunction
integral. This is not possible when the two saddle points are so close that
the cubic term in the expansion of the phase $\Phi(x,t,t^{\prime})$ becomes
non-negligible [76], i.e. when
$\left\lvert\frac{\frac{\partial^{3}}{\partial t^{\prime
3}}\Phi(x,t,t^{\prime})}{\left(\frac{\partial^{2}}{\partial t^{\prime
2}}\Phi(x,t,t^{\prime})\right)^{\frac{3}{2}}}\right\rvert_{t^{\prime}=t_{\pm}}\gtrsim
1.$ (34)
for either saddle point $t_{\pm}$. This occurs in the region very near the
barrier boundary and at the peak of the laser pulse, as discussed further in
Sec. IV.2. In this case, the reflection cannot be separated in the
wavefunction in a meaningful manner.
We find that when reflections are extracted from the wavefunction using Eq.
(33) the Wigner time delay under the barrier vanishes, as evident in both
panels of Fig. 8. That is, when reflections are neglected, there is no time
delay between the peak of the electron probability distribution and the peak
of the laser field. Moreover, outside the barrier, the trajectory of the peak
rapidly becomes classical.
Plotted also in Fig. 8(b) is the trajectory for an electron in a constant
field, given by Eq. (25). For parameters in the deep tunnelling regime, the
SFA trajectory (given by the peak of the wavefunction), and the classical
trajectory, the constant field trajectory display very similar behaviour as
the pulse elapses (the trajectories do not completely coincide far away due to
the time dependence of the field and the need to match a tunnel exit,
respectively).
At a detector infinitely far away such as one in an attoclock experiment, a
signature of time delay would be absent for the deep tunneling conditions
detailed in this work. Thus, it is of importance to distinguish between a time
delay of the peak of the wavefunction near the tunnel exit and an asymptotic
time delay at a detector. Since in attoclock experiments the asymptotic
momentum distribution is measured, i.e., the asymptotic time delay, one
expects to find a zero time delay in the deep tunnelling regime. However, near
the tunnel exit, quantum mechanical considerations must be taken into account
and a quantum mechanical treatment such as the one exposed here exhibits an
explicit non-zero Wigner delay.
### IV.2 Tunnel Exit from the SFA
We may also use the saddle point analysis of Eq. (17) to extract relevant
physical information about the electron dynamics. In particular, and in
analogy to the constant field scenario, the topology of the saddle points
through the complex $t^{\prime}$ plane reveal the functional behaviour of the
electronic wavepacket.
The paths of these two saddle point, for two representative choices of time
$t$ while varying $x$, are shown in Fig. 9, closely resembling those
corresponding to Figs. 5 (b) and (c). Much as in the constant field case,
where the saddles are purely real or imaginary, as given by Eq. (30) and Fig.
5, the pair of saddle points in the SFA is aligned around the line given by
$\textnormal{Re}[t^{\prime}]=t$; this alignment is either vertical or
horizontal depending on the coordinate $x$. For a fixed observation time $t$,
there corresponds a coordinate $x$ of closest approach between the saddle
points. For the peak of the pulse, $t=0$ there exists a coordinate $x_{t}$
where the saddle points merge completely, as they do in the case $\tilde{x}=0$
for the constant field electron.
Thus, two lines, $t=0$ and $x=x_{t}$, define separatrices for the behaviour of
the saddle points and their contributions to the integral: the behaviour of
the wave function can be seen to change depending on whether $x>x_{t}$. This
is a phenomenon exactly analogous to the merging of the saddle points for
constant field case of Sec. III: there, the merging of the saddle points
occurs at the classical exit, $\tilde{x}=0$, and separates the regions of
evanescence and oscillation in the wave function.
We may use this parallel to identify the merging point as a measure of the
tunnel exit. In our study, for a field strength of $E_{0}=0.05$ and
$\gamma=0.14$, we find that this exit takes the value $x_{t}\approx 10.2$
a.u.; this is in good agreement with the classically expected tunnel exit
$x_{e}=I_{p}/E_{0}=10$ a.u. More precise agreement is achieved with the
probability averaged tunnel exit
$x_{K}=\dfrac{\int
dt^{\prime}\frac{I_{p}}{|E(t^{\prime})|}\exp\left(-\frac{2\kappa^{3}}{3|E(t^{\prime})|}\right)}{\int
dt^{\prime}\exp\left(-\frac{2\kappa^{3}}{3|E(t^{\prime})|}\right)}\approx
10.35\;\textnormal{a.u.},$ (35)
where the integration runs over the whole laser pulse.
Figure 9: Positions of the upper and lower saddle-points of the integral (17)
in the complex $t^{\prime}$ plane with varying $x$ and for fixed
$\omega\,t=\pm 0.1\pi/2$ (blue and red, respectively). The saddles draw smooth
curves with varying $x$, where the arrows indicate growing values of $x$. For
values of $x\lesssim 7$, the lower saddle-point disappears below the imaginary
axis. Each pair of curves is centred around the line $\textnormal{Re}(\omega
t^{\prime})=\omega t$, shown in dashed. For the case $\omega t=0$, not shown,
the two lines meet one point corresponding to $x=x_{t}$.
## V Conclusions
In this work, we have analyzed the tunneling time delay in strong field
ionization for a simple model of an atom with a short-range potential. The
wave function was calculated to first-order in the SFA for any intermediate
time, enabling a quantum treatment of dynamics in the region where
quasiclassical descriptions break down, viz. around the classical tunnel exit.
For a given coordinate, the peak of the wave packet shows a time delay with
respect to the peak of the laser field and this time delay is positive at the
classical tunnel exit.
We argue that reflections of the electron wavepacket under the tunneling
barrier are fundamentally responsible for this non-zero time delay around the
tunnel exit. This is a general phenomenon, present in any regime of strong
field ionization, as well as in any tunneling process. In particular, a simple
showcase is presented in Appendix A for the case of tunneling through a box
potential.
To identify and separate contributions of reflected components to the
wavefunction, we first considered the analytically soluble case of an electron
in a constant electric field. In addition to providing a benchmark to our SFA
wavefunction, the constant field wavefunction can be deconstructed by
analysing its definition as an integral in the complex plane; this integral is
dominated by two saddle points, allowing us to unambiguously identify the
transmitted and reflected wavefunction components as contributions to the
integral from the regions around each saddle point.
We apply this observation to the SFA wavefunction by defining an integration
contour that purposefully neglects the contribution of the reflection saddle
point, and obtain a pseudo-wavefunction distribution that has zero time delay
with respect to the peak of the laser field everywhere under the barrier.
Moreover, the tunnel exit is unequivocally determined from the SFA theory as
the spatial co-ordinate at which the saddle points merge at the peak of the
laser field.
Finally, we emphasize the distinction between the time delay present at the
tunnel exit (theoretically measurable using a virtual detector) and an
(experimentally observed) asymptotic delay. We have shown within the first-
order SFA that the signature of the time delay in the electron wavefunction
vanishes since the ionized electron propagates to the detector as the
classical (i.e. with zero time delay) and the quantum mechanical electron
share equivalent asymptotic trajectories. Thus, the asymptotic time delay
associated with the direct ionization path is always vanishing. Nevertheless,
a more accurate estimate of the asymptotic time delay via the second-order SFA
in Ref. [51] has shown that interference of the direct and the sub-barrier
ionization paths generates non-vanishing negative asymptotic time delay in the
near-threshold of the OTBI regime, which fades out in the deep tunneling
regime.
## VI Acknowledgements
The authors are grateful to the referees for their useful remarks, which
generated fruitful discussions. This article comprises parts of the PhD thesis
work of Daniel Bakucz Canário, submitted to Heidelberg University, Germany.
## Appendix A Tunneling through a box potential barrier
In this appendix we consider the role of quantum reflections in the tunneling
time delay for an electron wavepacket tunnelling through a one dimensional box
potential. This is perhaps the simplest example of tunnelling time delay since
the wavefunction is readily separable (and of analytic solution) in the
regions inside and outside the barrier, allowing the contributions of
reflections in to the time delay to be clearly identified.
It should be mentioned that this model differs from true strong field
ionization in that it considers the scattering of a wavepacket incident on a
potential barrier; in actual ionization the electron originates from within a
potential barrier. Be that as it may, this study provides a simple, intuitive
picture of tunnelling time delay, suitable even for the uninitiated.
### A.1 The Box Potential
Figure 10: Pictorial representation of the square barrier potential for an
incident monochromatic wave. The wavefunction is a piecewise solution of the
Schrödinger equation for the three regions shown, given by Equations
(37)-(38).
Consider a wave packet
$\Psi(x,t)=\int\mathop{}\\!\mathrm{d}p\>f(p-p_{0})\>\psi(p)\>e^{-i\>E(p)\;t}$
(36)
with energy $E(p)={p}^{2}/2$, incident on a potential barrier $V(x)=V_{0}$ for
$0\leq x\leq a$ and $0$ elsewhere, where $f(p-p_{0})$ is some distribution
peaked at $p_{0}$ (e.g. a Gaussian), as shown Fig. 10. Each $p$-component
wavefunction obeys the time-independent Schrödinger equation with the
piecewise solution
$\displaystyle\psi_{I}(x)$ $\displaystyle=$ $\displaystyle
e^{ip\,x}+Re^{-ip\,x},$ (37) $\displaystyle\psi_{II}(x)$ $\displaystyle=$
$\displaystyle C_{1}e^{qx}+C_{2}e^{-qx},$ (38) $\displaystyle\psi_{III}(x)$
$\displaystyle=$ $\displaystyle Te^{ipx},$ (39)
with momenta $p=\sqrt{2\,E}$, $q=\sqrt{2(V_{0}-E)}$. The amplitude of the
incoming wave has been set to unity, without loss of generality, and the co-
efficients $C_{1}$ and $C_{2}$ are the typical reflection and transmission
coefficients under the barrier, respectively.
Matching the above solutions and their derivatives at the boundaries yields
the coefficients
$\displaystyle C_{1}$
$\displaystyle=\dfrac{(-2i\chi)(1+i\chi)e^{-\xi}}{(1-i\chi)^{2}\>e^{\xi}-(1+i\chi)^{2}\>e^{-\xi}},$
(40) $\displaystyle C_{2}$
$\displaystyle=\dfrac{(-2i\chi)(1-i\chi)e^{\xi}}{(1-i\chi)^{2}\>e^{\xi}-(1+i\chi)^{2}\>e^{-\xi}},$
(41) $\displaystyle R$
$\displaystyle=\frac{(1+\chi^{2})\;\left(e^{-\xi}-e^{\xi}\right)}{(1-i\chi)^{2}\>e^{\xi}-(1+i\chi)^{2}\>e^{-\xi}},$
(42) and, $\displaystyle T$
$\displaystyle=\frac{(-4i\chi)e^{-ipa}}{(1-i\chi)^{2}\>e^{\xi}-(1+i\chi)^{2}\>e^{-\xi}}.$
(43)
We have introduced the dimensionless parameters $\chi=p/q$ and $\xi=q\,a$
which, loosely speaking, determine the relative length and height of the
barrier respectively.
### A.2 Time Delay
The wavepacket after transmission is of the form
$\Psi_{III}=\int\>\mathop{}\\!\mathrm{d}p\>|T(p)|\>\exp{[i\left(\varphi(p)+px-E(p)t\right)]},$
(44)
where $T=|T|e^{i\varphi}$. The maximum of this amplitude occurs when the phase
in Eq. (44) vanishes, that is when (after some re-arrangement):
$x=p_{0}\,t-\left[\frac{\partial\varphi}{\partial p}\right]_{p=p_{0}}$ (45)
In the absence of a potential barrier, the peak travels at the classical
velocity (in atomic units) $p_{0}$. Equation (45) shows that the barrier
causes a delay of the peak in reaching a given position $x$, a delay which is
given the by the energy derivative of the transmission phase $\varphi$. Thus,
$\tau=\frac{1}{p_{0}}\left[\frac{\partial\varphi}{\partial
p}\right]_{p=p_{0}}+\frac{a}{p_{0}},$ (46)
gives the time delay of the peak after crossing the barrier, i.e. the
tunneling time. For the box potential, one finds
$\tau=\frac{a}{p_{0}}\frac{\frac{1}{2\xi}\left(\chi+\frac{1}{\chi}\right)^{2}\tanh(\xi)+\frac{(1-\chi^{2})}{2}\mathop{}\\!\textnormal{sech}^{2}(\xi)}{1+\frac{1}{4}\left(\chi-\frac{1}{\chi}\right)^{2}\tanh^{2}(\xi)}.$
(47)
### A.3 Quantum Reflections and Time Delay
What is the origin of this time delay? For longer barriers, $\xi\gg 1$, the
tunnelling time $\tau$ becomes
$\lim_{\xi\gg 1}\tau\approx\frac{2\chi}{p_{0}^{2}},$ (48)
vanishing when $\chi\ll 1$. Thus, the time delay is vanishing for $\xi\gg 1$
and $\chi\ll 1$. With this knowledge, we can analyze the exact wave function
inside the barrier. The reflection coefficient $C_{1}$ in the given limiting
conditions, $\xi\gg 1$ and $\chi\ll 1$,
$C_{1}\approx-2i\chi e^{-2\xi},$ (49)
vanishes asymptotically. This suggests an intuitive link between the
reflection of wave components and the retardation of the wave packet.
In the box potential case, in contrast to the tunneling problem, there exists,
however, a second reflection: the reflection from incidence on the first
surface of the barrier, described by the coefficient $R$. Classically, $R=1$,
and there is no time delay at reflection. However, in the quantum case
$R-1\neq 0$, inducing a time delay of the electron wave packet from entry into
the barrier. In the limit $\xi\gg 1$, and $\chi\ll 1$, we have
$R\approx-(1+2i\chi)$, i.e.,
$|1-R|\approx\chi\ll 1$ (50)
Analyzing the exact wave function, we can conclude that the time delay
vanishes when the reflection from the barrier surface is classical, and when
the reflection inside the barrier is negligible.
One can deduce the time delay caused by reflection at the entry by applying
the Wigner delay formula, Eq. 1, at $x=0$. For $\xi\gg 1$,
$R\approx-(1+i\chi)/(1-i\chi)$ and
$\displaystyle\tau_{entry}$
$\displaystyle=\frac{-i}{p_{0}}\left.\,\frac{\partial}{\partial
p}\ln\left(\frac{\psi_{I}(x)}{|\psi_{I}(x)|}\right)\right|_{x=0}\approx\quad\frac{\chi}{p_{0}^{2}}=\quad\frac{1}{2}\;\lim_{\xi\gg
1}\,\tau.\;$ (51)
Thus, one can conclude the time taken for the peak of the electron wavepacket
to travel through the box potential is caused in equal parts by the
reflections of the wavepacket on the barrier entry ($x=0$) and exit ($x=a$)
surfaces.
## Appendix B Configurations of saddle points
Figure 11: Configurations of the saddle points of the argument
$\Phi(x,t,t^{\prime})$ of the wavefunction integral
$\psi_{i}=\int^{t}_{t_{0}}dt^{\prime}\exp(i\,\Phi)$ in complex $t^{\prime}$
plane, for parameter ranges $x=7,10,14$ and $\omega t=-\pi/20,0,+\pi,20$. The
dashed line corresponds to $t^{\prime}=0$ and each plot is centred around the
(scaled) observation time $\omega t$. The scale and colour coding are
identical to those of Fig. 6. As $x$ increases the two saddle points approach
vertically and, after a closest approach, separate horizontally. For the peak
of the pulse, $\omega t=0$, this closest approach is zero and the two saddle
points merge at the point $x_{t}$. Otherwise, varying $t$ skews the relative
orientation of the saddle points around the line
$\textnormal{Re}[t^{\prime}]=t$.
The configurations of the saddle points of the argument $\Phi(x,t,t^{\prime})$
of the wavefunction integral
$\psi_{i}=\int^{t}_{t_{0}}dt^{\prime}\exp(i\,\Phi)$ in complex $t^{\prime}$
plane, for a large $(x,t)$ parameter range are presented as a table in Fig.
11. The laser pulse evolves from left to right in this figure and mostly
shifts the reference line around which the saddle points are centred, namely
$\textnormal{Re}[t^{\prime}]=t$. It should be noted, by the definition in Eq.
17, a singularity is always to be observed at the point $t^{\prime}=t$.
As one moves down the table, configurations of the complex $t^{\prime}$ plane
are shown for spatial coordinates inside, neighbouring, and outside the
tunnelling barrier ($x=7,10,$ and $14$ a.u. respectively). There are marked
differences between each observation coordinate but the behaviour is
reminiscent of the complex plane configuration for the Airy integral,
displayed in Fig. 5. Indeed, the two pictures can easily be reconciled by a
$\nicefrac{{\pi}}{{2}}$ rad. clockwise rotation.
Inside the barrier, the saddle points are vertically aligned along the line
$\textnormal{Re}[t^{\prime}]=t$. As one increases the observation coordinate
and approaches the tunnel exit, $x\approx 10$, the two saddle points approach
each other vertically; as one exits the region neighbouring the tunnel exit,
$x\gg 10$, the saddles then separate from each other on the horizontal.
As one approaches the peak of the laser field, $t=0$, the distance of closest
approach around the tunnel exit shrinks. At the exact peak, this distance is
zero; that is, the saddle points merge. This phenomenon is directly comparable
to the merging of saddle points for the Airy integral at the classical tunnel
exit. Thus, we are able identify a new tunnel exit, $x_{t}$, from the complex
plane landscape, a topic discussed in Sec. IV.2.
The identification the exact value for the co-ordinate $x_{t}$ to a desired
precision is in principle achievable by a binary search or global minimization
of the distance function for the saddle points. However, such precision was
deemed unnecessary for the purposes of this work.
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|
# Noncommutative Henselizations
Masood Aryapoor
Division of Mathematics and Physics
Mälardalen University
Hamngatan 15, 632 17, Eskilstuna, Sweden
###### Abstract
In this paper, the familiar notion of a Henselian pair is extended to the
noncommutative case. Furthermore, the problem of Henselizations is studied in
the noncommutative context, and it is shown that every (not necessarily
commutative) pair which is Hausdorff with respect to a certain topology has a
left (and right) Henselization.
Keywords: Noncommutative Henselian pair, Noncommutative Henselization
## 1 Introduction
The notion of a Henselian ring, introduced by Azumaya in [3], is well-known in
the commutative case. This concept has been extended to the noncommutative
case, see [1]. However, the theory of noncommutative Henselian rings is not as
well-studied as its commutative counterpart, and many problems concerning
noncommutative Henselian rings are still open. One such problem, discussed in
[1], is the problem of noncommutative Henselizations. In the commutative case,
it is known that every local ring has a Henselization, see [8]. The aim of
this article is to investigate the notion of Henselization in the
noncommutative context. One of our results is that every noncommutative local
ring satisfying a kind of “commutativity” condition has a Henselization, see
Subsection 3.7.
In Section 2, we present the preliminaries. In particular, the notion of a
noncommutative Henselian ring, introduced in [1], is generalized in two
different ways. First, we introduce the notions of left Henselian rings and
right Henselian rings, which is more natural in the noncommutative context.
Second, this concept is generalized to pairs as done by Lafon in the
commutative case, see [6]. The final section is devoted to the notions of left
Henselizations and right Henselizations. It is shown that every perfect pair
has a left (and right) Henselization which is unique up to unique isomorphism.
## 2 Henselian pairs
In this section, we present the preliminaries, and in particular, the notions
of left Heneslian rings and right Henselian rings. In this paper, we shall
assume that all rings have a unit, and all ring homomorphisms are unit-
preserving.
### 2.1 The category of pairs
The category of pairs has been introduced in the commutative setting, see [6].
One can enlarge this category to include noncommutative rings. More precisely,
the objects of the category $\mathcal{P}$ of pairs are pairs $(A,I)$ where $A$
is a (unitary but not necessarily commutative) ring and $I$ is an ideal of
$A$. A morphism $\phi:(A,I)\to(B,J)$ of pairs is a ring homomorphism
$\phi:A\to B$ satisfying $\phi^{-1}(J)=I$. It is straightforward to verify
that any directed system in $\mathcal{P}$ has a direct limit. Likewise, any
inverse system in $\mathcal{P}$ has an inverse limit.
For a pair $(A,I)$, we denote the image of $a\in A$ in $A/I$ by $\bar{a}$.
Given a morphism $\phi:(A,I)\to(B,J)$ of pairs, we have a canonical
homomorphism $\bar{\phi}:A/I\to B/J$ induced by $\phi$. The proof of the
following lemma is straightforward and left to the reader.
###### Lemma 2.1.
Let $\phi:(A,I)\to(C,K)$ and $\psi:(B,J)\to(C,K)$ be morphisms of pairs. If
there is a ring homomorphism $\alpha:A\to B$ such that $\phi=\psi\circ\alpha$
as ring homomorphisms, then $\alpha:(A,I)\to(B,J)$ defines a morphism of
pairs.
### 2.2 Coprime polynomials
For a ring $A$, the ring of polynomials $F(x)=\sum_{i=0}^{n}a_{i}x^{i}$ over
$A$, where the indeterminate $x$ commutes with all $a\in A$, is denoted by
$A[x]$. For a subset $S\subset A$, the set of all polynomials
$\sum_{i=0}^{n}a_{i}x^{i}\in A[x]$, where $a_{0},...,a_{n}\in S$, is denoted
by $S[x]$.
Two polynomials $F_{1}(x),F_{2}(x)\in A[x]$ are called left coprime if
$A[x]F_{1}(x)+A[x]F_{2}(x)=A[x].$
The notion of right coprime polynomials is defined in the obvious way. It is
easy to see that polynomials $F_{1}(x),F_{2}(x)\in A[x]$ are left coprime if
and only if there are polynomials $G_{1}(x),G_{2}(x)\in A[x]$ such that
$G_{1}(x)F_{1}(x)+G_{2}(x)F_{2}(x)=1.$
For a pair $(A,I)$, we denote the canonical homomorphism $A[x]\to(A/I)[x]$ by
$F(x)=\sum_{i}a_{i}x^{i}\mapsto\bar{F}(x)=\sum_{i}\bar{a}_{i}x^{i}$.
### 2.3 Euclidean algorithm
Let $A$ be a ring. The following version of Euclidean algorithm holds in
$A[x]$.
###### Lemma 2.2.
Let $I$ be a right ideal of $A$. For every monic polynomial $F(x)\in A[x]$ and
every polynomial $G(x)\in I[x]$, there exist polynomials $Q(x),R(x)\in I[x]$
such that
$G(x)=Q(x)F(x)+R(x)\;\text{and}\;\deg(R(x))<\deg(F(x)).$
###### Proof.
We use induction on $\deg(G(x))$. If $\deg(G(x))<\deg(F(x))$, we set $Q(x)=0$
and $R(x)=G(x)$, so we are done. Let $G(x)=\sum_{i=0}^{m}b_{i}x^{i}$ and
$F(x)=a_{0}+a_{1}x+\cdots+a_{n-1}x^{n-1}+x^{n},$
where $m\geq n$. The polynomial $G_{1}(x)=G(x)-b_{m}x^{m-n}F(x)$ belongs to
$I$ because $b_{m}\in I$ and $I$ is a right ideal. Since
$\deg(G_{1}(x))<\deg(G(x)),$ by induction, there are polynomials
$Q_{1}(x),R(x)\in I[x]$ such that
$G_{1}(x)=Q_{1}(x)F(x)+R(x)\;\text{and}\;\deg(R(x))<\deg(F(x)).$
Setting $Q(x)=b_{m}x^{m-n}+Q_{1}(x)$, we obtain
$G(x)=Q(x)F(x)+R(x),$
where $Q(x),R(x)$ satisfy the desired conditions, and we are done. ∎
### 2.4 Jacobson pairs
A pair $(A,I)$ is called a Jacobson pair if $I\subset rad(A)$ where $rad(A)$
is the Jacobson radical of $A$. We have the following result concerning
Jacobson pairs.
###### Lemma 2.3.
Let $(A,I)$ be a Jacobson pair. Then, monic polynomials $F_{1}(x),F_{2}(x)\in
A[x]$ are left (resp. right) coprime if and only if the polynomials
$\bar{F}_{1}(x),\bar{F}_{2}(x)\in(A/I)[x]$ are left (resp. right) coprime.
###### Proof.
The “only if” part is trivial. To prove the other direction, suppose that
$\bar{F}_{1}(x),\bar{F}_{2}(x)\in(A/I)[x]$ are left coprime. It follows that
$A[x]F_{1}(x)+A[x]F_{2}(x)+I[x]=A[x]$
Considering
$M=\frac{A[x]}{A[x]F_{1}(x)}$
as a left $A$-module in the obvious way, we see that $M=N+IM$ where $N$ is the
submodule of $M$ generated by $F_{2}(x)$. The module $M$ is a finitely
generated $A$-module because $F_{1}(x)$ is monic. Since $I\subset rad(A)$,
Nakayama’s lemma implies that $N=M$, that is,
$A[x]F_{1}(x)+A[x]F_{2}(x)=A[x],$
and we are done. ∎
### 2.5 Local homomorphisms
A ring homomorphism $\phi:A\to B$ is called local if $\phi$ sends every
nonunit in $A$ to a nonunit in $B$. Here, we provide some facts concerning
local maps. The proof of the first lemma is easy and left to the reader.
###### Lemma 2.4.
Let $\phi:A\to B$ and $\psi:B\to C$ be ring homomorphisms. If $\psi\circ\phi$
is local, then $\phi$ is local too.
###### Lemma 2.5.
Let $\phi:A\to B$ be a local homomorphism. If $B$ is a local ring, then $A$ is
a local ring whose maximal ideal is $\phi^{-1}(I)$ where $I$ is the maximal
ideal of $B$.
###### Proof.
To show that $A$ is a local ring, we need to prove that if $a+b$ is invertible
in $A$ for some $a,b\in A$, then either $a$ or $b$ is invertible in $A$, see
Theorem 19.1 in [7]. If $a+b$ is invertible in $A$ for some $a,b\in A$, then
$\phi(a)+\phi(b)$ is invertible in $B$ which implies that $\phi(a)$ or
$\phi(b)$ is invertible in $B$, because $B$ is local. It follows that either
$a$ or $b$ is invertible in $A$ because $\phi$ is local. Therefore, $A$ is a
local ring. Since $\phi$ is a local homomorphism and $B$ is a local ring, we
see that $\phi^{-1}(I)$ consists of all nonunits in $A$, that is,
$\phi^{-1}(I)$ is the maximal ideal of $A$. ∎
### 2.6 Localizations
Let $\phi:A\to B$ be a ring homomorphism. In what follows, we introduce a ring
$A_{\phi}$, a ring homomorphism $\Lambda_{\phi}:A\to A_{\phi}$ and a local
homomorphism $\Psi_{\phi}:A_{\phi}\to B$ such that
$\phi=\Psi_{\phi}\circ\Lambda_{\phi}$. Let $S_{1}$ be the set of all elements
$s\in A$ such that $\psi_{1}(s)$ is invertible in $B$. Let $A_{1}=A_{S_{1}}$
be the localization of $A$ at $S_{1}$ and $\lambda_{1}:A\to A_{1}$ be the
canonical homomorphism, consult [4] for the definition and elementary
properties of localizations. It follows from the universal property of
$A_{S_{1}}$ that there exists a unique homomorphism $\psi_{1}:A_{1}\to B$ such
that $\phi=\psi_{1}\circ\lambda_{1}$. Let $S_{2}$ be the set of all elements
$s\in A_{1}$ such that $\phi_{1}(s)$ is invertible in $B$. Consider the
localization $A_{2}=(A_{1})_{S_{2}}$ of $A_{1}$ at $S_{2}$ and let
$\lambda_{2}:A_{1}\to A_{2}$ be the canonical homomorphism. It follows from
the universal property of $A_{2}$ that there exists a homomorphism
$\psi_{2}:A_{2}\to B$ such that $\psi_{1}=\psi_{2}\circ\lambda_{2}$.
Continuing this process, we obtain a sequence of rings
$A_{1},A_{2},A_{3},...$, homomorphisms $\lambda_{i}:A_{i-1}\to A_{i}$ and
$\psi_{i}:A_{i}\to B$ such that $\psi_{i-1}=\psi_{i}\circ\lambda_{i}$ for
$i=1,2,...,$ where $\psi_{0}=\phi$. We set $A_{\phi}=\varinjlim A_{i}$. We
have a canonical homomorphism $\Lambda_{\phi}:A\to A_{\phi}$ and a canonical
homomorphism $\Psi_{\phi}:A_{\phi}\to B$. In the following proposition, we
provide some facts about this construction, see Section 2 in [2] for a proof
of this proposition and a detailed discussion of the localization ring
$A_{\phi}$.
###### Proposition 2.6.
The ring homomorphism $\Psi_{\phi}:A_{\phi}\to B$ is local and satisfies the
equality $\phi=\Psi_{\phi}\circ\Lambda_{\phi}$.
### 2.7 Commutativity with respect to a filtration
Let $(A,I)$ be a pair. A descending sequence $\mathcal{F}$ of ideals
$I_{1}\supset I_{2}\supset...$
of $A$ is called a filtration on $(A,I)$ if $I_{1}=I$. The pair $(A,I)$ is
called commutative with respect to $\mathcal{F}$, or $\mathcal{F}$-commutative
for short, if $[A,I_{n}]\subset I_{n+1}$ for all $n=1,2,...$. The notation
$[S,T]$, where $S,T\subset A$, stands for the set of all elements of the form
$st-ts$ where $s\in S,t\in T$.
### 2.8 The commutator filtration
For every pair $(A,I)$, one can define a sequence $I^{(1)},I^{(2)},...$ of
ideals of $A$ as follows: $I^{(1)}=I$; for $n=1,2,...$, the ideal $I^{(n+1)}$
is the ideal generated by $[A,I^{(n)}]$. It is easy to see that the sequence
$I^{(1)},I^{(2)},...$ is a filtration on $(A,I)$ called the commutator
filtration of $(A,I)$. Clearly, $(A,I)$ is commutative with respect to its
commutator filtration. Note that if $\phi:(A,I)\to(B,J)$ is a morphism of
pairs then $\phi(I^{(n)})\subset J^{(n)}$ for all $n=1,2,...$.
### 2.9 Topologies defined by filtrations
Let $(A,I)$ be a pair and $\mathcal{F}:I_{1}=I,I_{2},...$ be a filtration on
$(A,I)$. The $\mathcal{F}$-topology on $(A,I)$ is the linear topology on $A$
for which the sets $I,I_{2},\dots$ form a fundamental system of neighborhoods
of $0$. The pair $(A,I)$ is called separated with respect to $\mathcal{F}$, or
$\mathcal{F}$-separated for short, if $A$ is Hausdorff with respect to the
$\mathcal{F}$-topology. Note that $(A,I)$ is $\mathcal{F}$-separated if and
only if $\cap_{n=1}^{\infty}I_{n}=\\{0\\}$. The pair $(A,I)$ is called
complete with respect to $\mathcal{F}$, or $\mathcal{F}$-complete for short,
if it is complete with respect to the $\mathcal{F}$-topology.
### 2.10 Unique factorization pairs
A pair $(A,I)$ is called a left unique factorization pair, or LUFP for short,
if for every factorization $\bar{F}(x)=f_{1}(x)f_{2}(x)$ of a monic polynomial
$F(x)\in A[x]$ over $A/I$, where $f_{1}(x)f_{2}(x)\in(A/I)[x]$ are left
coprime monic polynomials, there exists at most one factorization
$F(x)=F_{1}(x)F_{2}(x)$ such that $F_{1}(x),F_{2}(x)\in A[x]$ are monic
polynomials, and $\bar{F}_{1}(x)=f_{1}(x),\bar{F}_{2}(x)=f_{2}(x)$. The notion
of a right unique factorization pair (RUFP) is defined in a similar way. A
pair is called a unique factorization pair, or UFP for short, if it is both an
LUFP and an RUFP.
### 2.11 Henselian pairs
A pair $(A,I)$ is called left Henselian if $(A,I)$ is a Jacobson pair, and the
following version of Hensel’s lemma holds in A. For every monic polynomial
$F(x)\in A[x]$, if $\bar{F}(x)=f_{1}(x)f_{2}(x)$, where
$f_{1}(x),f_{2}(x)\in(A/I)[x]$ are left coprime monic polynomials, then there
exist unique monic polynomials $F_{1}(x),F_{2}(x)\in A[x]$ satisfying
$F(x)=F_{1}(x)F_{2}(x)$, $\bar{F}_{1}(x)=f_{1}(x)$ and $\bar{F}(x)=f_{2}(x)$.
We note that the polynomials $F_{1}(x)$ and $F_{2}(x)$ are left coprime, see
Lemma 2.3. The notion of a right Henselian pair is defined in a similar
fashion. A pair which is both left and right Henselian is called Henselian.
Obviously, every (resp. left or right) Henselian ring is a (resp. left or
right) UFP. We note that if $A/I$ is a commutative ring, then $(A,I)$ is left
Henselian if and only if it is right Henselian.
### 2.12 A class of left Henselian pairs
The following result generalizes Theorem 2.1 in [1].
###### Theorem 2.7.
Let $(A,I)$ be a pair and $\mathcal{F}:I_{1}=I,I_{2},...$ be a filtration on
$(A,I)$ such that $(A,I)$ is $\mathcal{F}$-commutative,
$\mathcal{F}$-separated and $\mathcal{F}$-complete. If $I_{n}[A,A]\subset
I_{n+1}$ and $I_{n}^{2}\subset I_{n+1}$ for all $n=1,2,...$, then $(A,I)$ is
left Henselian.
###### Proof.
First, we show that $(A,I)$ is a Jacobson pair. Let $a\in I$ be given. The
condition $I_{n}^{2}\subset I_{n+1}$ for all $n=1,2,...$, implies that the
geometric series
$1-a+a^{2}-\cdots$
converges to a unique limit in $A$ because $(A,I)$ is $\mathcal{F}$-separated
and $\mathcal{F}$-complete. The limit of this series is the inverse of $1+a$.
Therefore, every element in $1+I$ is invertible, which implies that $I\subset
rad(A)$, that is, $(A,I)$ is a Jacobson pair.
To show that $(A,I)$ is left Henselian, let $F(x)\in A[x]$ be a monic
polynomial such that $\bar{F}(x)=f_{1}(x)f_{2}(x)$ where
$f_{1}(x),f_{2}(x)\in(A/I)[x]$ are left coprime monic polynomials. We need to
show that we can lift this factorization to $A[x]$ in a unique way. First, we
show that there are sequences of monic polynomials
$F_{1,1}(x),F_{1,2}(x),...,F_{1,i}(x),...$
and
$F_{2,1}(x),F_{2,2}(x),...,F_{2,i}(x),...$
in $A[x]$ such that
$\bar{F}_{1,i}(x)=f_{1}(x),\;\bar{F}_{2,i}(x)=f_{2}(x),$
$F_{1,i+1}(x)-F_{1,i}(x)\in I_{i}[x],\;F_{2,i+1}(x)-F_{2,i}(x)\in I_{i}[x],$
$F(x)-F_{1,i}(x)F_{2,i}(x)\in I_{i}[x],$
for all $i\geq 1$. To construct these sequences, we use induction on $i$.
Since the canonical map $A\to A/I$ is onto, we can find monic polynomials
$F_{1,1}(x),F_{2,1}(x)\in A[x]$
such that $\bar{F}_{1,1}(x)=f_{1}(x),\bar{F}_{2,1}(x)=f_{2}(x)$. Clearly,
$F_{1,1}(x)$ and $F_{2,1}(x)$ satisfy the desired conditions. Having found
$F_{1,i}(x)$ and $F_{2,i}(x)$, we find $F_{1,i+1}(x)$ and $F_{2,i+1}(x)$ as
follows. Set
$G(x)=F(x)-F_{1,i}(x)F_{2,i}(x).$
I claim that there are polynomials $R_{1}(x),R_{2}(x)\in I_{i}[x]$ such that
$\deg(R_{1}(x))<\deg(F_{2,i}(x)),\deg(R_{2}(x))<\deg(F_{1,i}(x)),$
and
$R_{2}(x)F_{1,i}(x)+R_{1}(x)F_{2,i}(x)-G(x)\in I_{i+1}[x]$
By Lemma 2.3, there exit polynomials $H_{1}(x),H_{2}(x)\in A[x]$ such
$H_{1}(x)F_{1,i}(x)+H_{2}(x)F_{2,i}(x)=1.$
It follows that
$G_{1}(x)F_{1,i}(x)+G_{2}(x)F_{2,i}(x)=G(x),$
where both $G_{1}(x)=G(x)H_{1}(x)$ and $G_{2}(x)=G(x)H_{2}(x)$ belong to
$I_{i}[x]$. By Lemma 2.2, there are polynomials $Q(x),R_{1}(x)\in I_{i}[x]$
such that
$G_{1}(x)=Q(x)F_{2,i}(x)+R_{1}(x)\;\text{and}\;\deg(R_{1}(x))<\deg(F_{2,i}(x)).$
It follows that
$G(x)=R_{1}(x)F_{1,i}(x)+(G_{2}(x)+Q(x)F_{1,i}(x))F_{2,i}(x)+$
$Q(x)(F_{2,i}(x)F_{1,i}(x)-F_{1,i}(x)F_{2,i}(x))$
Using the condition $I_{i}[A,A]\subset I_{i+1}$, we see that
$R_{1}(x)F_{1,i}(x)+(G_{2}(x)+Q(x)F_{1,i}(x))F_{2,i}(x)-G(x)\in I_{i+1}[x].$
Let $G_{2}(x)+Q(x)F_{1,i}(x)=\sum_{i=0}^{m}b_{i}x^{i}.$ Assume that
$m\geq\deg(F_{1,i}(x))$. Since $F_{2,i}(x)$ is monic, and
$\deg(R_{1}(x)F_{1,i}(x))<\deg(F_{1,i}(x))+\deg(F_{2,i}(x)),$
$\deg(G(x))<\deg(F_{1,i}(x))+\deg(F_{2,i}(x)),$
the relation
$R_{1}(x)F_{1,i}(x)+(G_{2}(x)+Q(x)F_{1,i}(x))F_{2,i}(x)-G(x)\in I_{i+1}[x]$
implies that $b_{m}\in I_{i+1}$. Therefore, we have
$R_{1}(x)F_{1,i}(x)+(G_{2}(x)-b_{m}x^{m})F_{2,i}(x)-G(x)\in I_{i+1}[x].$
Using induction, we conclude that
$R_{1}(x)F_{1,i}(x)+R_{2}(x)F_{2,i}(x)-G(x)\in I_{i+1}[x],$
where $R_{2}(x)=\sum_{i<\deg(F_{1}(x))}b_{i}x^{i}$, proving the claim. We set
$F_{1,i+1}(x)=F_{1,i}(x)+R_{2}(x),$ $F_{2,i+1}(x)=F_{2,i}(x)+R_{1}(x).$
Clearly, we have
$\bar{F}_{1,i+1}(x)=f_{1}(x),\;\bar{F}_{2,i+1}(x)=f_{2}(x)$
$F_{1,i+1}(x)-F_{1,i}(x)\in I_{i}[x],\;F_{2,i+1}(x)-F_{2,i}(x)\in I_{i}[x]$
Moreover, we have
$F(x)-F_{1,i+1}(x)F_{2,i+1}(x)=$
$(F(x)-F_{1,i}(x)F_{2,i}(x))-F_{1,i}(x)R_{1}(x)-R_{2}(x)F_{2,i}(x)-R_{2}(x)R_{1}(x)=$
$(G(x)-R_{1}(x)F_{1,i}(x)-R_{2}(x)F_{2,i}(x))+$
$(R_{1}(x)F_{1,i}(x)-F_{1,i}(x)R_{1}(x))-R_{2}(x)R_{1}(x).$
Since $(A,I)$ is $\mathcal{F}$-commutative and $I_{i}^{2}\subset I_{i+1}$, we
deduce that
$F(x)-F_{1,i+1}(x)F_{2,i+1}(x)\in I_{i+1}[x].$
Having constructed the desired sequences, we proceed as follows. Since $A$ is
$\mathcal{F}$-complete, the limits
$F_{1}(x)=\lim_{i\to\infty}F_{1,i}(x)\,\text{and}\,F_{2}(x)=\lim_{i\to\infty}F_{2,i}(x)$
exist. Clearly, we have $\bar{F}_{1}(x)=f_{1}(x)$ and
$\bar{F}_{2}(x)=f_{2}(x)$. Since $A$ is $F$-Hausdorff, we have
$F(x)=F_{1}(x)F_{2}(x)$ and $F_{1},F_{2}$ are monic polynomials. Since $(A,I)$
is $\mathcal{F}$-commutative, it is easy to see that $I^{(n)}\subset I_{n}$
for all $n$. It follows that $(A,I)$ is also separated with respect to its
commutator filtration. Therefore, by Proposition 3.1, this factorization is
unique, and we are done. ∎
## 3 Noncommutative Henselizations
The notion of the Henselization of a pair has been introduced in the
commutative case, see [6]. It turns out that one can develop a similar theory
in the noncommutative case. However, we focus our attention on a special
subcategory $\mathcal{P}_{0}$ of the category $\mathcal{P}$ of pairs and prove
that every object in this subcategory has a left (and a right) Henselization
in $\mathcal{P}_{0}$, see Theorem 3.5. Our treatment of Henselization is
somewhat similar to the one given in [5].
### 3.1 The category of perfect pairs
A pair $(A,I)$ is called perfect if $(A,I)$ is separated with respect to its
commutator filtration, that is, $\cap_{n=1}^{\infty}I^{(n)}=\\{0\\}$. The full
subcategory of the category $\mathcal{P}$ consisting of perfect pairs is
denoted by $\mathcal{P}_{0}$. For any pair $(A,I)$, it is easy to see that the
pair
$\digamma{(A,I)}=(\frac{A}{\cap_{n=1}^{\infty}I^{(n)}},\frac{I}{\cap_{n=1}^{\infty}I^{(n)}})$
is a perfect pair. Furthermore, the assignment $(A,I)\mapsto\digamma{(A,I)}$
gives rise to a functor $\digamma:\mathcal{P}\to\mathcal{P}_{0}$. One can
easily check that $\digamma$ is a left adjoint of the inclusion function
$\iota:\mathcal{P}_{0}\to\mathcal{P}$.
### 3.2 Perfect Jacobson pairs
The reason for restricting our attention to $\mathcal{P}_{0}$ is in the
following result.
###### Proposition 3.1.
Every perfect Jacobson pair is a UFP.
###### Proof.
Let $(A,I)$ be a perfect Jacobson pair. We only show that $(A,I)$ is an LUFP.
The proof that A is an RUFP is similar. Assume, on the contrary, that there
are different factorizations
$F(x)=F_{1}(x)F_{2}(x)=G_{1}(x)G_{2}(x)$
of a monic polynomial $F(x)\in A[x]$ such that
$f_{1}(x)=\bar{F}_{1}(x)=\bar{G}_{1}(x)$ and
$f_{2}(x)=\bar{F}_{2}(x)=\bar{G}_{2}(x)$ are left coprime monic polynomials.
Without loss of generality, we may assume $F_{2}(x)\neq G_{2}(x)$. The facts
that $F_{2}(x)$ and $G_{2}(x)$ are monic polynomials, and
$\bar{F}_{2}(x)=\bar{G}_{2}(x)$, imply that there exists a polynomial $N(x)\in
I[x]$ such that
$F_{2}(x)=G_{2}(x)+N(x)\;\text{and}\;\deg(N(x))<\deg(G_{2}(x)).$
Since $\cap_{n=1}^{\infty}I^{(n)}=\\{0\\}$ and $N(x)\neq 0$, there exists
$d\geq 1$ such that
$N(x)\in I^{(d)}[x]\;\text{but}\;N(x)\notin I^{(d+1)}[x].$
Since $\bar{F}_{1}(x)$ and $\bar{G}_{2}(x)$ are left coprime, it follows from
Lemma 2.3 that there are polynomials $H_{1}(x),H_{2}(x)\in A[x]$ such that
$H_{1}(x)F_{1}(x)+H_{2}(x)G_{2}(x)=1.$
We can write
$F_{2}(x)=H_{1}(x)F_{1}(x)F_{2}(x)+H_{2}(x)G_{2}(x)F_{2}(x)=$
$(H_{1}(x)G_{1}(x)+H_{2}(x)F_{2}(x))G_{2}(x)+H_{2}(x)(G_{2}(x)N(x)-N(x)G_{2}(x)).$
Setting $K(x)=H_{1}(x)G_{1}(x)+H_{2}(x)F_{2}(x)$, we see that
$F_{2}(x)=K(x)G_{2}(x)$ as elements of $(A/I^{(d+1)})[x]$, because $(A,I)$ is
commutative with respect to its commutator filtration. Since $F_{2}(x)$ and
$G_{2}(x)$ are monic polynomials of the same degree, we conclude that
$F_{2}(x)=G_{2}(x)$ as elements of $(A/I^{(d+1)})[x]$, that is,
$N(x)=F_{2}(x)-G_{2}(x)\in I^{(d+1)}[x],$ a contradiction .
∎
### 3.3 Factorizations of polynomials over perfect pairs
In this part, we prove the following result.
###### Proposition 3.2.
Let $(A,I)$ be a perfect pair and $F(x)\in A[x]$ be a monic polynomial.
Suppose that $\bar{F}(x)$ has a factorization $\bar{F}(x)=f_{1}(x)f_{2}(x)$
over $A/I$ where $f_{1}(x),f_{2}(x)\in(A/I)[x]$ are left coprime monic
polynomials. Then, there exists a perfect Jacobson pair $(A\langle
F;f_{1},f_{2}\rangle,I\langle F;f_{1},f_{2}\rangle)$ and a morphism
$\Phi_{\langle F;f_{1},f_{2}\rangle}:(A,I)\to(A\langle
F;f_{1},f_{2}\rangle,I\langle F;f_{1},f_{2}\rangle)$
of pairs having the following universal property. For every morphism
$\phi:(A,I)\to(B,K)$
of pairs, where $(B,K)$ is a perfect Jacobson pair, if
$\phi(F(x))=G_{1}(x)G_{2}(x)$ for some monic polynomials $G_{1}(x),G_{2}(x)\in
B[x]$ such that
$\bar{G}_{1}(x)=\bar{\phi}(f_{1}(x)),\bar{G}_{2}(x)=\bar{\phi}(f_{2}(x)),$
then there exists a unique morphism
$\psi:(A\langle F;f_{1},f_{2}\rangle,I\langle F;f_{1},f_{2}\rangle)\to(B,K)$
of pairs such that $\phi=\psi\circ\Phi_{\langle F;f_{1},f_{2}\rangle}$.
###### Proof.
First, we give a construction of the pair $(A\langle
F;f_{1},f_{2}\rangle,I\langle F;f_{1},f_{2}\rangle)$ after which we prove its
universal property. Let
$F(x)=a_{0}+a_{1}x+\cdots+a_{d-1}x^{d-1}+x^{d},$
$f_{1}(x)=b_{0}+b_{1}x+\cdots+b_{d_{1}-1}x^{d_{1}-1}+x^{d_{1}},$
$f_{2}(x)=c_{0}+c_{1}x+\cdots+c_{d_{2}-1}x^{d_{2}-1}+x^{d_{2}}.$
We consider the free $A$-ring $A\langle
y_{0},...,y_{d_{1}-1},z_{0},...,z_{d_{2}-1}\rangle$ generated by the
noncommutating variables $y_{0},...,y_{d_{1}-1},z_{0},...,z_{d_{2}-1}$. We
have
$(y_{0}+y_{1}x+\cdots+y_{d_{1}-1}x^{d_{1}-1}+x^{d_{1}})(z_{0}+z_{1}x+\cdots+z_{d_{2}-1}x^{d_{2}-1}+x^{d_{2}})=$
$g_{0}+g_{1}x+\cdots+g_{d-1}x^{d-1}+x^{d}$
where $g_{0},...,g_{d-1}\in A\langle
y_{0},...,y_{d_{1}-1},z_{0},...,z_{d_{2}-1}\rangle$. Consider the ring
homomorphism
$\alpha:A\langle
y_{0},...,y_{d_{1}-1},z_{0},...,z_{d_{2}-1}\rangle\to\frac{A}{I}$
defined by
$\alpha(y_{0})=b_{0},...,\alpha(y_{d_{1}-1})=b_{d_{1}-1},\alpha(z_{0})=c_{0},...,\alpha(z_{d_{2}-1})=c_{d_{2}-1},$
$\alpha(a)=\bar{a}\,\;\text{where}\;a\in A.$
The ideal $\langle g_{0}-a_{0},...,g_{d-1}-a_{d-1}\rangle$ generated by
$g_{0}-a_{0},...,g_{d-1}-a_{d-1}$ is contained in the kernel of $\alpha$
because $\bar{F}(x)=f_{1}(x)f_{2}(x)$. Therefore, $\alpha$ gives rise to a
ring homomorphism
$\beta:\frac{A\langle
y_{0},...,y_{d_{1}-1},z_{0},...,z_{d_{2}-1}\rangle}{\langle
g_{0}-a_{0},...,g_{d-1}-a_{d-1}\rangle}\to\frac{A}{I}$
Consider the following localization ring (see subsection 2.6)
$R=\Big{(}\frac{A\langle
y_{0},...,y_{d_{1}-1},z_{0},...,z_{d_{2}-1}\rangle}{\langle
g_{0}-a_{0},...,g_{d-1}-a_{d-1}\rangle}\Big{)}_{\beta}.$
We have canonical ring homomorphisms $\gamma:R\to\frac{A}{I}$ and $\eta:A\to
R$ which satisfy $\gamma(\eta(a))=\bar{a}$ for every $a\in A$, see Proposition
2.6. We set $J=\ker(\gamma)$. By Proposition 2.6, $\gamma$ is local from which
it follows that $J\subset rad(A\langle F;f_{1},f_{2}\rangle)$, that is,
$(R,J)$ is a Jacobson pair. Since the quotient homomorphism $A\to A/I$ is a
morphism $(A,I)\to(A/I,0)$ of pairs, we can use Lemma 2.1 to deduce that
$\eta:(A,I)\to(R,J)$ is a morphism of pairs. Finally, we set
$(A\langle F;f_{1},f_{2}\rangle,I\langle
F;f_{1},f_{2}\rangle)=\digamma{(R,J)}$
The canonical morphism $\eta:(A,I)\to(R,J)$ composed with the quotient
morphism $(R,I)\to\digamma{(R,J)}$ gives a morphism
$\Phi_{\langle F;f_{1},f_{2}\rangle}:(A,I)\to(A\langle
F;f_{1},f_{2}\rangle,I\langle F;f_{1},f_{2}\rangle)$
of pairs.
To prove the universal property of $\Phi_{\langle F;f_{1},f_{2}\rangle}$, let
$\phi:(A,I)\to(B,K)$ be a morphism of pairs where $(B,K)$ is a perfect
Jacobson pair. Suppose that $\phi(F)(x)=G_{1}(x)G_{2}(x)$ where
$G_{1}(x),G_{2}(x)\in B[x]$ are monic polynomials and
$\bar{G}_{1}=\bar{\phi}(f_{1}),\bar{G}_{2}=\bar{\phi}(f_{2}).$
Let
$G_{1}(x)=e_{0}+e_{1}x+\cdots+e_{d_{1}-1}x^{d_{1}-1}+x^{d_{1}},$
$G_{2}(x)=f_{0}+f_{1}x+\cdots+f_{d_{2}-1}x^{d_{2}-1}+x^{d_{2}}.$
One can easily check that the assignments
$y_{0}\mapsto e_{0},...,y_{d_{1}-1}\mapsto e_{d_{1}-1},z_{0}\mapsto
f_{0},...,z_{d_{2}-1}\mapsto f_{d_{2}-1}$
yield a ring homomorphism
$\psi_{1}:\frac{A\langle
y_{0},...,y_{d_{1}-1},z_{0},...,z_{d_{2}-1}\rangle}{\langle
g_{0}-a_{0},...,g_{d-1}-a_{d-1}\rangle}\to B$
which extends the ring homomorphism $\phi$. Furthermore, since $K\subset
rad(B)$ and $(B,K)$ is perfect, one can extend $\psi_{1}$ to a ring
homomorphism
$\psi:A\langle F;f_{1},f_{2}\rangle\to B$
satisfying $\phi=\psi\circ\Phi_{\langle F;f_{1},f_{2}\rangle}$ as ring
homomorphisms. Since
$\bar{G}_{1}(x)=\bar{\phi}(f_{1}(x)),\bar{G}_{2}=\bar{\phi}(f_{2}(x))$, the
following diagram is commutative
$\begin{array}[]{ccc}A\langle F;f_{1},f_{2}\rangle&\xrightarrow{\psi}&B\\\
\downarrow{}&&\downarrow\\\ A/I&\xrightarrow{\bar{\phi}}&B/K\end{array}$
Using the commutativity of this diagram and Lemma 2.1, we conclude that
$\psi:(A\langle F;f_{1},f_{2}\rangle,I\langle F;f_{1},f_{2}\rangle)\to(B,K)$
is, in fact, a morphism of pairs. The uniqueness of $\psi$ follows from the
fact that $(B,K)$ is a UFP by Proposition 3.1. ∎
### 3.4 LF-extensions
Let $(A,I)$ be a perfect pair. A morphism $\phi:(A,I)\to(B,J)$ of pairs is
called a simple left factorization extension (or simple LF-extension for
short) of $(A,I)$ if $\phi=\Phi_{\langle F;f_{1},f_{2}\rangle}$ for some
polynomials $F(x)\in A[x]$, $f_{1}(x),f_{2}(x)\in(A/I)[x]$ satisfying the
conditions in Proposition 3.2. An LF-extension of $(A,I)$ is a morphism
$\phi:(A,I)\to(B,J)$ of pairs which is obtained by a finite sequence of simple
LF-extensions, that is, there are simple LF-extensions
$\phi_{i}:(A_{i},I_{i})\to(A_{i+1},I_{i+1}),\,\text{where}\;i=1,...,d,$
such that $(A_{1},I_{1})=(A,I)$, $(A_{d+1},I_{d+1})=(B,J)$ and
$\phi=\phi_{d}\circ\phi_{d-1}\circ\cdots\circ\phi_{1}.$
Obviously, the collection of all LF-extensions of a perfect pair $(A,I)$ is a
set which we denote by $LFext(A,I)$. Given morphisms
$\phi_{1}:(A,I)\to(B_{1},J_{1}),\;\text{and}\;\phi_{2}:(A,I)\to(B_{2},J_{2})$
in $LFext(A,I)$, we write $\phi_{1}\leq\phi_{2}$ if there exists a morphism
$\psi:(B_{1},J_{1})\to(B_{2},J_{2})$
of pairs such that $\phi_{2}=\psi\circ\phi_{1}$. Clearly, the relation $\leq$
defines a partial order on $LFext(A)$. Furthermore, we have the following
result.
###### Lemma 3.3.
Let $(A,I)$ be a perfect pair. (i) For all $\phi_{1}:(A,I)\to(B_{1},J_{1})$
and $\phi_{2}:(A,I)\to(B_{2},J_{2})$ in $LFext(A)$, there exists at most one
morphism $\psi:(B_{1},J_{1})\to(B_{2},J_{2})$ of pairs such that
$\phi_{2}=\psi\circ\phi_{1}$. (ii) The partial order $\leq$ on $LFext(A)$ is
directed.
###### Proof.
Part (i) can be proved using induction and the fact that $(B_{2},J_{2})$ is a
UFP. To prove (ii), we first assume that $\phi_{1}:(A,I)\to(B_{1},J_{1})$ is a
simple LF-extension. Therefore, $\phi=\Phi_{\langle F;f_{1},f_{2}\rangle}$ for
some polynomials $F(x)\in A[x]$, $f_{1}(x),f_{2}(x)\in(A/I)[x]$ satisfying the
conditions in Proposition 3.2. Let $G_{1}(x)=\phi_{2}(F(x))$,
$g_{1}(x)=\bar{\phi}_{2}(F(x))$ and $g_{2}(x)=\bar{\phi}_{2}(F(x))$. It is
easy to see that the polynomials $G_{1}(x),g_{1}(x),g_{2}(x)$ satisfy the
conditions in Proposition 3.2, giving rise to a morphism
$\Phi_{\langle G;g_{1},g_{2}\rangle}:(B_{2},J_{2})\to(B\langle
G;g_{1},g_{2}\rangle,I\langle G;g_{1},g_{2}\rangle)$
of pairs. Clearly
$\Phi_{\langle G;g_{1},g_{2}\rangle}\circ\phi_{2}:(A,I)\to(B\langle
G;g_{1},g_{2}\rangle,I\langle G;g_{1},g_{2}\rangle)$
is an LF-extension. By the universal property of $\phi=\Phi_{\langle
F;f_{1},f_{2}\rangle}$, we see that there exists a morphism
$\psi:(B_{1},J_{1})\to(B\langle G;g_{1},g_{2}\rangle,I\langle
G;g_{1},g_{2}\rangle)$
such that $\Phi_{\langle
G;g_{1},g_{2}\rangle}\circ\phi_{2}=\psi\circ\phi_{1}$. It follows that
$\phi_{1}\leq\Phi_{\langle
G;g_{1},g_{2}\rangle}\circ\phi_{2}\;\text{and}\;\phi_{2}\leq\Phi_{\langle
G;g_{1},g_{2}\rangle}\circ\phi_{2}.$
The general case is proved by induction. ∎
### 3.5 Left Henselizations
In this subsection, we prove the following result concerning the concept of
Henselization.
###### Theorem 3.4.
Let $(A,I)$ be a perfect pair. Then, there exists a left Henselian pair
$(A^{lh},I^{lh})$ and a morphism $\phi^{lh}:(A,I)\to(A^{lh},I^{lh})$ of pairs
having the following universal property. For every morphism
$\phi:(A,I)\to(B,J)$ of pairs from $(A,I)$ to a left Henselian prefect pair
$(B,J)$, there exists a unique morphism $\psi:(A^{lh},I^{lh})\to(B,J)$ of
pairs such that $\phi=\psi\circ\phi^{lh}$.
###### Proof.
By Lemma 3.3, the direct limit $(A^{lh},I^{lh})$ of elements in $LFext(A)$
exists. Moreover, it is a perfect pair. We also have a canonical morphism
$\phi^{lh}:(A,I)\to(A^{lh},I^{lh})$
of pairs. The universal property of $\phi^{h}$ follows from Proposition 3.2
and properties of direct limits. So, it remains to show that $(A^{lh},I^{lh})$
is left Henselian. Since $(A^{lh},I^{lh})$ is a direct limit of Jacobson
pairs, it is a Jacobson pair. Let a monic polynomial $F(x)\in A^{lh}[x]$ be
given such that $\bar{F}(x)=f_{1}(x)f_{2}(x)$ for some left coprime monic
polynomials $f_{1}(x),f_{2}(x)\in(A^{lh}/I^{lh})[x]$. Since $F(x)$ has only
finitely many (nonzero) coefficients, there exists an LF-extension
$\phi:(A,I)\to(B,J)$ such that $F(X)\in B[X]$. By Proposition 3.2, the
polynomial $F(X)$ has a factorization $F(x)=F_{1}(x)F_{2}(x)$ over
$B(F;f_{1},f_{2})$, hence over $A^{lh}$, such that $F_{1},F_{2}\in A^{lh}[X]$
are monic, and $\bar{F}_{1}(x)=f_{1}(x)$, $\bar{F}(x)=f_{2}(x)$. Since
$(A^{lh},I^{lh})$ is a perfect Jacobson ring, it is a UFP, see Proposition
3.1. Therefore, the factorization $F(x)=F_{1}(x)F_{2}(x)$ is unique. It
follows that $(A^{lh},I^{lh})$ is a left Henselian pair, and we are done. ∎
The pair $(A^{lh},I^{lh})$ is called the left Henselization of $(A,I)$. It is
easy to see that the pair $(A^{lh},I^{lh})$ is unique up to unique
isomorphism. Similarly, one can show that every perfect pair $(A,I)$ has a
right Henselizaiton
$\phi^{rh}:(A,I)\to(A^{rh},I^{rh})$
satisfying the corresponding universal property. We note that if $A/I$ is, in
addition, a commutative ring, then the right Henselization of $(A,I)$ is also
the left Henselization of $(A,I)$, and vice versa.
### 3.6 Commutative Henselizations
A pair $(A,I)$ is called commutative if $A$ is a commutative ring. It is known
that every commutative pair has a Henselization in the category
$\mathcal{P}_{c}$ of commutative pairs, see [6]. The Henselization of a
commutative pair $(A,I)$ in $\mathcal{P}_{c}$ is referred to as the
commutative Henselization of $(A,I)$, and is denoted by $(A^{ch},I^{ch})$.
Obviously, the category $\mathcal{P}_{c}$ is a full subcategory of
$\mathcal{P}_{0}$. The following result determines commutative Henselizations
in terms of left Henselizations.
###### Proposition 3.5.
Let $(A,I)$ be a commutative pair. Then, the ideal $J$ generated by
$[A^{lh},A^{lh}]$ is contained in $I^{lh}$. Moreover, the morphism
$q\circ\phi^{lh}:(A,I)\to(\frac{A}{J}^{lh},\frac{I}{J}^{lh}),$
where $q:(A^{lh},I^{lh})\to(A^{lh}/J,I^{lh}/J)$ is the quotient morphism, is
the commutative Henselization of $(A,I)$.
###### Proof.
Let $(A^{ch},I^{ch})$ be the commutative Henselization of $(A,I)$ and
$\phi^{ch}:(A,I)\to(A^{ch},I^{ch}),$
be the corresponding morphism of pairs. Since $(A^{ch},I^{ch})$ is left
Henselian, there exists a unique morphism
$\phi:(A^{lh},I^{lh})\to(A^{ch},I^{ch}),$
of pairs such that $\phi^{ch}=\phi\circ\phi^{lh}$. The fact that
$(A^{ch},I^{ch})$ is commutative implies that the ideal $J$ generated by
$[A^{lh},A^{lh}]$ is contained in the ideal
$\ker(\phi)=\phi^{-1}(0)\subset\phi^{-1}(I^{ch})=I^{lh}.$
Moreover, there exists a unique morphism
$\psi:(\frac{A}{J}^{lh},\frac{I}{J}^{lh})\to(A^{ch},I^{ch})$
such that $\phi=\psi\circ q$. Using the universal properties of $\phi^{lh}$
and $\phi^{ch}$, one can verify that $\psi$ is an isomorphism and the morphism
$q\circ\phi^{lh}:(A,I)\to(\frac{A}{J}^{lh},\frac{I}{J}^{lh})$
is the commutative Henselization of $(A,I)$. ∎
### 3.7 Henselizations of local rings
We conclude this article with a discussion of Henselizations of local rings. A
ring $A$ is called local if the set of all nonunits in $A$ form an ideal.
Every local ring $A$ has a unique maximal ideal $I$. A pair $(A,I)$ is called
a local pair if $A$ is a local ring and $I$ is its maximal ideal. We note that
any local pair is a Jacobson pair.
###### Proposition 3.6.
The left (right) Henselization of any perfect local pair is a local pair.
###### Proof.
Since the direct limit of local pairs is a local pair, it is enough to show
that any simple LF-extension of a perfect local pair is a local pair. Let
$\Phi_{\langle F;f_{1},f_{2}\rangle}:(A,I)\to(A\langle
F;f_{1},f_{2}\rangle,I\langle F;f_{1},f_{2}\rangle)$
be a simple LF-extension where $(A,I)$ is a local ring. Referring to the
notations used in Proposition 3.2, one can see that the ring homomorphism
$\gamma:R\to A/I$ is a local homomorphism. Since $A/I$ is a local ring, Lemma
2.5 implies that $R$ is a local ring whose maximal ideal is
$J=\gamma^{-1}(0)$. Using the relation
$(A\langle F;f_{1},f_{2}\rangle,I\langle
F;f_{1},f_{2}\rangle)=\digamma{(R,J)},$
we conclude that $(A\langle F;f_{1},f_{2}\rangle,I\langle
F;f_{1},f_{2}\rangle)$ is a local pair, and we are done. ∎
## References
* [1] Masood Aryapoor. Non-commutative henselian rings. Journal of Algebra, 322(6):2191–2198, 2009.
* [2] Masood Aryapoor. F-schemes. arXiv preprint arXiv:1001.1862, 2010.
* [3] Gorô Azumaya. On maximally central algebras. Nagoya Mathematical Journal, 2:119–150, 1951.
* [4] Paul Moritz. Cohn. Skew fields: theory of general division rings, 1995.
* [5] Silvio Greco. Henselization of a ring with respect to an ideal. Transactions of the American Mathematical Society, 144:43–65, 1969\.
* [6] J Lafon. Anneaux henséliens. Bulletin de la Société Mathématique de France, 91:77–107, 1963.
* [7] Tsit-Yuen Lam. A first course in noncommutative rings, volume 131. Springer Science & Business Media, 2013.
* [8] Masayoshi Nagata. Local rings. Interscience Tracts in Pure and Appl. Math., 1962.
|
# Comparison results for solutions to $p$-Laplace equations with Robin
boundary conditions
Vincenzo Amato, Andrea Gentile, Alba Lia Masiello
###### Abstract
In the last decades comparison results of Talenti type for Elliptic Problems
with Dirichlet boundary conditions have been widely investigated. In this
paper, we generalize the results obtained in [3] to the case of $p$-Laplace
operator with Robin boundary conditions.
The point-wise comparison, obtained in [3] only in the planar case, holds true
in any dimension if $p$ is sufficiently small.
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Universita‘ degli
Studi di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, via
Cintia - 80126 Napoli, Italy.
e-mail<EMAIL_ADDRESS>
e-mail<EMAIL_ADDRESS>
Mathematical and Physical Sciences for Advanced Materials and Technologies,
Scuola Superiore Meridionale, Largo San Marcellino 10, 80126 Napoli, Italy.
e-mail<EMAIL_ADDRESS>
000 2010 Mathematics Subject Classification: 35J92, 35P15. 000 Key words and
phrases: $p$-Lapacian, Robin boundary conditions.
## 1 Introduction
Let $\beta$ be a positive parameter and let $\Omega$ be a bounded open set of
$\operatorname{\mathbb{R}}^{n}$, $n\geq 2$, with Lipschitz boundary.
Let $f\in L^{p^{\prime}}(\Omega)$ be a non-negative function. We consider the
following problem
$\begin{cases}-\text{div}({\left|\nabla u\right|}^{p-2}\nabla u)=f&\text{ in
}\Omega\\\ {\left|\nabla u\right|}^{p-2}\displaystyle{\frac{\partial
u}{\partial\nu}}+\beta{\left|u\right|}^{p-2}u=0&\text{ on
}\partial\Omega.\end{cases}$ (1)
A function $u\in W^{1,p}(\Omega)$ is a weak solution to (1) if
$\int_{\Omega}{\left|\nabla u\right|}^{p-2}\nabla
u\nabla\varphi\,dx+\beta\int_{\partial\Omega}{\left|u\right|}^{p-2}u\varphi\,d\mathcal{H}^{n-1}(x)=\int_{\Omega}f\varphi\,dx\quad\forall\varphi\in
W^{1,p}(\Omega).$ (2)
We want to establish a comparison principle with the solution to the following
symmetrized problem
$\begin{cases}-\text{div}({\left|\nabla v\right|}^{p-2}\nabla
v)=f^{\sharp}&\text{ in }\Omega^{\sharp}\\\ {\left|\nabla
v\right|}^{p-2}\displaystyle{\frac{\partial
v}{\partial\nu}}+\beta{\left|v\right|}^{p-2}v=0&\text{ on
}\partial\Omega^{\sharp},\end{cases}$ (3)
where $\Omega^{\sharp}$ is the ball centered in the origin with the same
measure of $\Omega$ and $f^{\sharp}$ is the Schwarz rearrangement of $f$ (see
next section for its definition).
This kind of problems has been widely investigated in the last decades. The
first step is contained in [10], where Talenti proved a pointwise comparison
result between $u^{\sharp}$ and $v$ in the case of Dirichlet Laplacian. After
this, several papers generalized the result of Talenti: for instance, the one
by Talenti himself [11], in which the operator is a generic non-linear
operator in divergence form, or the one by Alvino, Lions and Trombetti [2] in
which the authors deal with both elliptic and parabolic cases: in both papers,
Dirichlet boundary conditions are considered.
Different kind of boundary conditions are considered by Alvino, Nitsch and
Trombetti in [3], where they establish a comparison between a suitable norm of
$u$ and $v$, respectively solution to
$\begin{cases}-\Delta u=f&\text{in}\,\Omega\\\ \displaystyle{\frac{\partial
u}{\partial\nu}}+\beta
u=0&\text{on}\,\partial\Omega.\end{cases}\quad\quad\begin{cases}-\Delta
v=f^{\sharp}&\text{in}\,\Omega^{\sharp}\\\ \displaystyle{\frac{\partial
u}{\partial\nu}}+\beta u=0&\text{on}\,\partial\Omega^{\sharp}.\end{cases}$
They found out that if $f$ is a non-negative function in $L^{2}(\Omega)$, then
$\displaystyle{\left\|u\right\|}_{L^{k,1}(\Omega)}\leq{\left\|v\right\|}_{L^{k,1}(\Omega^{\sharp})}\quad$
$\displaystyle\forall 0<k\leq\frac{n}{2n-2}$
$\displaystyle{\left\|u\right\|}_{L^{2k,2}(\Omega)}\leq{\left\|v\right\|}_{L^{2k,2}(\Omega^{\sharp})}\quad$
$\displaystyle\forall 0<k\leq\frac{n}{3n-4}$
where ${\left\|\cdot\right\|}_{L^{k,q}}$ is the so called _Lorentz norm_ ,
whose definition can be found in next section. Moreover, the authors in [3]
were able to establish a comparison á la Talenti,
$u^{\sharp}(x)\leq v(x),\quad\forall x\in\Omega^{\sharp}$
in the case $f\equiv 1$ and $n=2$. This will be the starting point of our
work: it will be clear that our results coincide with the one in [3] in the
case $p=2$.
For completeness sake, we cite that this wasn’t the first result in this
sense, indeed in [6] the authors study a comparison result for the
$p$-torsion, that is the case $f\equiv 1$, with a completely different
argument, obtaining
${\left\|u\right\|}_{L^{1}{\Omega}}\leq{\left\|v\right\|}_{L^{1}(\Omega^{\sharp})}.$
Another work that is worth to be mentioned is [1], where the authors obtained
similar results to [3] in the case of mixed Dirichlet and Robin boundary
conditions.
This paper is organized as follows. In the next section, we give some basic
notions about rearrangements of functions and Lorentz spaces. Moreover, we
list some properties of the solutions to problems (1) and (3). In section 3,
we prove the main results about comparison of the two solutions in terms of
the Lorentz norm.
In particular, we prove
###### Theorem 1.1.
Let $u$ and $v$ be the solutions to problem (1) and (3) respectively. Then we
have
${\left\|u\right\|}_{L^{k,1}(\Omega)}\,\leq{\left\|v\right\|}_{L^{k,1}(\Omega^{\sharp})}\,\;\forall\,0<k\leq\frac{n(p-1)}{(n-1)p},$
(4)
${\left\|u\right\|}_{L^{pk,p}(\Omega)}\,\leq{\left\|v\right\|}_{L^{pk,p}(\Omega^{\sharp})}\,\;\forall\,0<k\leq\frac{n(p-1)}{(n-2)p+n}.$
(5)
We observe that from Theorem 1.1, we have that, if $p\geq n$
${\left\|u\right\|}_{L^{1}(\Omega)}\leq{\left\|v\right\|}_{L^{1}(\Omega^{\sharp})}\quad\text{and}\quad{\left\|u\right\|}_{L^{p}(\Omega)}\leq{\left\|v\right\|}_{L^{p}(\Omega^{\sharp})}.$
###### Theorem 1.2.
Assume that $f\equiv 1$ and let $u$ and $v$ be the solutions to (1) and (3)
respectively.
1. $(i)$
If $\displaystyle{1\leq p\leq\frac{n}{n-1}}$ then
$u^{\sharp}(x)\leq v(x)\qquad x\in\Omega^{\sharp},$ (6)
2. $(ii)$
if $\displaystyle{p>\frac{n}{n-1}}$ and
$\displaystyle{0<k\leq\frac{n(p-1)}{n(p-1)-p}}$ , then
$\displaystyle{\left\|u\right\|}_{L^{k,1}(\Omega)}$
$\displaystyle\leq{\left\|v\right\|}_{L^{k,1}(\Omega^{\sharp})}$ (7)
$\displaystyle{\left\|u\right\|}_{L^{pk,p}(\Omega)}$
$\displaystyle\leq{\left\|v\right\|}_{L^{pk,p}(\Omega^{\sharp})}.$
Then we explicitly observe that in the case $f\equiv 1$ from Theorem 1.2 we
have that
${\left\|u\right\|}_{L^{1}(\Omega)}\leq{\left\|v\right\|}_{L^{1}(\Omega^{\sharp})}\quad\text{and}\quad{\left\|u\right\|}_{L^{p}(\Omega)}\leq{\left\|v\right\|}_{L^{p}(\Omega^{\sharp})}\qquad\text{
for }p>1.$
While we have the point-wise comparison only for
$\displaystyle{p\leq\frac{n}{n-1}}$.
In section 4, using tools from section 3, we give a new proof of the Faber-
Krahn inequality with Robin boundary conditions in the case $p\geq n$. This
topic was already studied in the papers by Bucur, Giacomini, Daners and
Trebeschi, [4], [5] , [6] and [7] where the authors proved the Faber-Krahn
inequality for the eigenvalues of the Laplacian, or of the $p$-Laplacian, with
Robin boundary conditions, for every $p>1$. Actually, the results in [4] are
more general, since they hold for every $p>1$, but they are obtained with
completely different tools than the ones contained in our paper.
Finally, in section 5, we provide some examples and open problems and we
discuss the optimality of our results.
## 2 Notions and preliminaries
###### Definition 2.1.
Let $u:\Omega\to\operatorname{\mathbb{R}}$ be a measurable function, the
_distribution function_ of $u$ is the function
$\mu:[0,+\infty[\,\to[0,+\infty[$ defined by
$\mu(t)={\left|\Set{x\in\Omega\,:\,{\left|u(x)\right|}>t}\right|}$
Here, and in the whole paper, ${\left|A\right|}$ stands for the
$n$-dimensional Lebesgue measure of the set $A$.
###### Definition 2.2.
Let $u:\Omega\to\operatorname{\mathbb{R}}$ be a measurable function, the
_decreasing rearrangement_ of $u$, denoted by $u^{\ast}$, is the distribution
function of $\mu$.
The _Schwarz rearrangement_ of $u$ is the function $u^{\sharp}$ whose level
sets are balls with the same measure as the level sets of $u$. The functions
$u^{\sharp}$ and $u^{*}$ are linked by the relation
$u^{\sharp}(x)=u^{*}(\omega_{n}{\left|x\right|}^{n})$
It is easily checked that $u$, $u^{*}$ e $u^{\sharp}$ are equi-distributed, so
it follows that
$\displaystyle{{\left\|u\right\|}_{L^{p}(\Omega)}={\left\|u^{*}\right\|}_{L^{p}(0,{\left|\Omega\right|})}=\lVert{u^{\sharp}}\rVert_{L^{p}(\Omega^{\sharp})}}.$
An important propriety of the decreasing rearrangement is the Hardy-
Littlewood inequaliy, that is
$\int_{\Omega}{\left|h(x)g(x)\right|}\,dx\leq\int_{0}^{{\left|\Omega\right|}}h^{*}(s)g^{*}(s)\,ds.$
So, by choosing $h=\chi_{\left\\{{\left|u\right|}>t\right\\}}$, one has
$\int_{{\left|u\right|}>t}{\left|h(x)\right|}\,dx\leq\int_{0}^{\mu(t)}h^{*}(s)\,ds.$
###### Definition 2.3.
Let $0<p<+\infty$ and $0<q\leq+\infty$. The Lorentz space $L^{p,q}(\Omega)$ is
the space of those functions such that the quantity:
${\left\|g\right\|}_{L^{p,q}}=\begin{cases}\displaystyle{p^{\frac{1}{q}}\left(\int_{0}^{\infty}t^{q}\mu(t)^{\frac{q}{p}}\,\frac{dt}{t}\right)^{\frac{1}{q}}}&0<q<\infty\\\
\displaystyle{\sup_{t>0}\,(t^{p}\mu(t))}&q=\infty\end{cases}$
is finite.
Let us observe that for $p=q$ the Lorentz space coincides with the $L^{p}$
space, as a consequence of the well known _Cavalieri’s Principle_
$\int_{\Omega}{\left|g\right|}^{p}=p\int_{0}^{+\infty}t^{p-1}\mu(t)\,dt.$
See [12] for more details on Lorentz space.
Let us consider the functional
$\mathfrak{F}(w)=\frac{1}{p}\int_{\Omega}{\left|\nabla
w\right|}^{p}\,dx+\frac{\beta}{p}\int_{\partial\Omega}{\left|w\right|}^{p}\,d\mathcal{H}^{n-1}(x)-\int_{\Omega}fw\,dx$
defined on $W^{1,p}(\Omega)$. This functional is well defined and its Euler-
Lagrange equation is exactly (1). If we show that the functional admits a
minimum, our problem will always have a solution.
1. $1)$
Let us show that the functional is bounded from below, indeed using the
parametric Young inequality, we have
$\displaystyle\mathfrak{F}(u)$
$\displaystyle\geq\frac{1}{p}\int_{\Omega}{\left|\nabla
u\right|}^{p}\,dx+\frac{\beta}{p}\int_{\partial\Omega}{\left|u\right|}^{p}\,d\mathcal{H}^{n-1}(x)-\frac{\varepsilon^{p}}{p}\int_{\Omega}{\left|u\right|}^{p}\,dx-\frac{1}{p^{\prime}\varepsilon^{p^{\prime}}}\int_{\Omega}{\left|f\right|}^{p^{\prime}}\,dx$
$\displaystyle\geq\frac{1}{p}\left(\int_{\Omega}{\left|\nabla
u\right|}^{p}\,dx+\beta\int_{\partial\Omega}{\left|u\right|}^{p}\,d\mathcal{H}^{n-1}(x)\right)-\frac{\varepsilon^{p}}{p}\int_{\Omega}{\left|u\right|}^{p}\,dx-\frac{1}{p^{\prime}\varepsilon^{p^{\prime}}}\int_{\Omega}{\left|f\right|}^{p^{\prime}}\,dx$
$\displaystyle\geq\frac{\lambda_{1,\beta}(\Omega)-\varepsilon^{p}}{p}\int_{\Omega}{\left|u\right|}^{p}\,dx-\frac{1}{p^{\prime}\varepsilon^{p^{\prime}}}\int_{\Omega}{\left|f\right|}^{p^{\prime}}\,dx$
In the last inequality we used the Sobolev inequality with trace term
$\int_{\Omega}{\left|\nabla
u\right|}^{p}+\beta\int_{\partial\Omega}{\left|u\right|}^{p}\geq\lambda_{1,\beta}(\Omega)\int_{\Omega}{\left|u\right|}^{p}.$
In general, the quantity $\lambda_{1,\beta}(\Omega)$ denotes the first
eigenvalue of the $p$-Laplacian with Robin boundary conditions, whose
definition is given in (28), which can be also seen as a trace constant of the
set $\Omega$.
If $\varepsilon$ is small enough, then the quantity
$\frac{\lambda_{1,\beta}(\Omega)-\varepsilon^{p}}{p}$
is non negative, and then
$\mathfrak{F}(u)\geq-\frac{1}{p^{\prime}\varepsilon^{p^{\prime}}}\int_{\Omega}{\left|f\right|}^{p^{\prime}}$
so
$m=\inf_{W^{1,p}}\mathfrak{F}(u)>-\infty.$
2. $2)$
Compactness and lower semicontinuity.
Let $\left\\{u_{i}\right\\}$ be a minimizing sequence. We can assume that
$\mathfrak{F}(u_{i})\leq m+1$, $\forall i$. Using again the Young inequality,
we have
$\displaystyle m+1$ $\displaystyle\geq\frac{1}{p}\int_{\Omega}{\left|\nabla
u_{i}\right|}^{p}\,dx+\frac{\beta}{p}\int_{\partial\Omega}{\left|u_{i}\right|}^{p}\,d\mathcal{H}^{n-1}(x)-\int_{\Omega}fu_{i}\,dx$
$\displaystyle\geq\frac{1}{p}\int_{\Omega}{\left|\nabla
u_{i}\right|}^{p}\,dx+\frac{\beta}{p}\int_{\partial\Omega}{\left|u_{i}\right|}^{p}\,d\mathcal{H}^{n-1}(x)-\frac{\varepsilon^{p}}{p}\int_{\Omega}{\left|u_{i}\right|}^{p}\,dx-\frac{1}{p^{\prime}\varepsilon^{p^{\prime}}}\int_{\Omega}{\left|f\right|}^{p^{\prime}}\,dx$
Then
$\displaystyle
m+1+\frac{1}{p^{\prime}\varepsilon^{p^{\prime}}}\int_{\Omega}{\left|f\right|}^{p^{\prime}}\,dx$
$\displaystyle\geq\frac{1}{2p}\left(\int_{\Omega}{\left|\nabla
u_{i}\right|}^{p}\,dx+\beta\int_{\partial\Omega}{\left|u_{i}\right|}^{p}\,d\mathcal{H}^{n-1}(x)\right)-\frac{\varepsilon^{p}}{p}\int_{\Omega}{\left|u_{i}\right|}^{p}\,dx$
$\displaystyle+\frac{1}{2p}\left(\int_{\Omega}{\left|\nabla
u_{i}\right|}^{p}\,dx+\beta\int_{\partial\Omega}{\left|u_{i}\right|}^{p}\,d\mathcal{H}^{n-1}(x)\right)$
$\displaystyle\geq\frac{1}{2p}\int_{\Omega}{\left|\nabla
u_{i}\right|}^{p}\,dx+\left(\frac{\lambda_{1,\beta}(\Omega)-2\varepsilon^{p}}{2p}\right)\int_{\Omega}{\left|u_{i}\right|}^{p}\,dx.$
Then, the minimizing sequence $\left\\{u_{i}\right\\}$ is bounded in
$W^{1,p}(\Omega)$, so there exists a subsequence $\Set{u_{i_{k}}}$ weakly
converging in $W^{1,p}(\Omega)$ and strongly in $L^{p}(\Omega)$ to a function
$u$. Let us show that $u$ is the minimum.
The function $t^{p}$ is strictly convex for $p>1$, so
$\displaystyle{\left|u_{i_{k}}\right|}^{p}\geq{\left|u\right|}^{p}+p{\left|u\right|}^{p-2}u(u_{i_{k}}-u)$
(8) $\displaystyle{\left|\nabla u_{i_{k}}\right|}^{p}\geq{\left|\nabla
u\right|}^{p}+p{\left|\nabla u\right|}^{p-2}\nabla u(\nabla u_{i_{k}}-\nabla
u)$ (9)
Putting (8) e (9) in $\mathfrak{F}(u_{i_{k}})$, we obtain
$\displaystyle\int_{\Omega}fu_{i_{k}}\,dx+\mathfrak{F}(u_{i_{k}})$
$\displaystyle\geq\frac{1}{p}\int_{\Omega}{\left|\nabla
u\right|}^{p}\,dx+\int_{\Omega}{\left|\nabla u\right|}^{p-2}\nabla u(\nabla
u_{i_{k}}-\nabla u)$
$\displaystyle+\frac{\beta}{p}\int_{\partial\Omega}{\left|u\right|}^{p}\,d\mathcal{H}^{n-1}(x)+\beta\int_{\partial\Omega}{\left|u\right|}^{p-2}u(u_{i_{k}}-u)\,d\mathcal{H}^{n-1}$
Passing to the limit for $k\to\infty$, by the weak convergence of
$\left\\{u_{i_{k}}\right\\}$ the integral over $\Omega$ on the right-hand side
goes to 0. The integral over $\partial\Omega$ goes to 0 as well. Indeed, the
space $W^{1,p}(\Omega)$ is compactly embedded in $L^{p}(\partial\Omega)$ (for
more details, see [9] 2.5), and $u_{i_{k}}-u\to 0$ in $L^{p}(\partial\Omega)$.
So, we obtain
$m\geq\mathfrak{F}(u).$
This ensures us that $u$ is the minimum of the functional.
The uniqueness of the minimum follows from the fact that $\mathfrak{F}(u)$ is
the sum of a strictly convex part and a linear part.
We observe that the solutions $u$ and $v$ to (1) and (3) respectively are both
$p$-superharmonic and then, by the strong maximum principle in [13], it
follows that they achieve their minima on the boundary. Denoting by $u_{m}$
and $v_{m}$ the minimum of $u$ and $v$ respectively, thanks to the
positiveness of $\beta$ and the Robin boundary conditions, we have that
$u_{m}\geq 0$ and $v_{m}\geq 0$. Hence $u$ and $v$ are strictly positive in
the interior of $\Omega$. Moreover we can observe that
$u_{m}=\min_{\Omega}u\leq\min_{\Omega^{\sharp}}v=v_{m},$ (10)
in fact, if we consider
$\begin{split}v_{m}^{p-1}\text{Per}(\Omega^{\sharp})&=\int_{\partial\Omega^{\sharp}}v(x)^{p-1}\,d\mathcal{H}^{n-1}(x)=\frac{1}{\beta}\int_{\Omega^{\sharp}}f^{\sharp}\,dx=\frac{1}{\beta}\int_{\Omega}f\,dx\\\
&=\int_{\partial\Omega}u(x)^{p-1}\,d\mathcal{H}^{n-1}(x)\\\ &\geq
u_{m}^{p-1}\text{Per}(\Omega)\geq{\left|u\right|}_{m}^{p-1}\text{Per}(\Omega^{\sharp}).\end{split}$
A consequence of (10) that will be used in what follows is that
$\mu(t)\leq\phi(t)={\left|\Omega\right|}\quad\forall t\leq v_{m}.$ (11)
### 2.1 Useful lemmas
Let $u$ be the solution to (1). For $t\geq 0$, we denote by
$U_{t}=\left\\{x\in\Omega:u(x)>t\right\\}\quad\partial U_{t}^{int}=\partial
U_{t}\cap\Omega,\quad\partial U_{t}^{ext}=\partial U_{t}\cap\partial\Omega,$
and by
$\mu(t)={\left|U_{t}\right|}\quad P_{u}(t)=Per(U_{t})$
where ${\left|\cdot\right|}$ is the Lebesgue measure on
$\operatorname{\mathbb{R}}^{n}$ and $Per(\cdot)$ is the perimeter.
If $v$ is the solution to (3), using the same notations, we set
$V_{t}=\left\\{x\in\Omega^{\sharp}:v(x)>t\right\\},\quad\phi(t)={\left|V_{t}\right|},\quad
P_{v}(t)=Per(V_{t}).$
Because of the invariance of the $p$-Laplacian and of the Schwarz
rearrangement of $f$ by rotation, there exists a radial solution to (3) and,
by uniqueness of solutions, this solution is $v$.
Since $v$ is radial, positive and decreasing along the radius then, for $0\leq
t\leq v_{m}$, $V_{t}=\Omega^{\sharp}$, while, for
$v_{m}<t<\max_{\Omega^{\sharp}}v$, $V_{t}$ is a ball, concentric to
$\Omega^{\sharp}$ and strictly contained in it.
###### Lemma 2.1 (Gronwall).
Let $\xi(t):[\tau_{0},+\infty[\,\to\operatorname{\mathbb{R}}$ be a continuous
and differentiable function satisfying, for some non negative constant $C$,
the following differential inequality
$\tau\xi^{\prime}(\tau)\leq(p-1)\xi(\tau)+C\quad\forall\tau\geq\tau_{0}>0.$
Then we have
* (i)
$\displaystyle{\xi(\tau)\leq\left(\xi(\tau_{0})+\frac{C}{p-1}\right)\left(\frac{\tau}{\tau_{0}}\right)^{p-1}-\frac{C}{p-1}\quad\forall\tau\geq\tau_{0}}$;
* (ii)
$\displaystyle{\xi^{\prime}(\tau)\leq\left(\frac{(p-1)\xi(\tau_{0})+C}{\tau_{0}}\right)\left(\frac{\tau}{\tau_{0}}\right)^{p-2}\quad\forall\tau\geq\tau_{0}}$.
###### Proof.
Dividing both sides of the differential inequality by $\tau^{p}$, we obtain
$\left(\frac{\xi^{\prime}(\tau)}{\tau^{p-1}}-(p-1)\frac{\xi(\tau)}{\tau^{p}}\right)=\left(\frac{\xi(\tau)}{\tau^{p-1}}\right)^{\prime}\leq\frac{C}{\tau^{p}}.$
Now, we integrate from $\tau_{0}$ to $\tau$ and we obtain
$\begin{split}\int_{\tau_{0}}^{\tau}\left(\frac{\xi(\tau)}{\tau^{p-1}}\right)^{\prime}\,d\tau&\leq\int_{\tau_{0}}^{\tau}\frac{C}{\tau^{p}}\,d\tau\\\
\implies\xi(\tau)&\leq\left(\xi(\tau_{0})+\frac{C}{p-1}\right)\left(\frac{\tau}{\tau_{0}}\right)^{p-1}-\frac{C}{p-1},\end{split}$
which gives _(i)_.
In order to obtain _(ii)_ , we just take into account _(i)_ in the
differential inequality. ∎
###### Lemma 2.2.
Let $u$ and $v$ be solutions to (1) and (3) respectively. Then for almost
every $t>0$ we have
$\gamma_{n}\mu(t)^{\left(1-\frac{1}{n}\right)\frac{p}{p-1}}\leq\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\left(-\mu^{\prime}(t)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{\text{ext}}}\frac{1}{u}\,d\mathcal{H}^{n-1}(x)\right),$ (12)
where
$\displaystyle{\gamma_{n}=\left(n\omega_{n}^{1/n}\right)^{\frac{p}{p-1}}}$.
And
$\gamma_{n}\phi(t)^{\left(1-\frac{1}{n}\right)\frac{p}{p-1}}=\left(\int_{0}^{\phi(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\left(-\phi^{\prime}(t)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
V_{t}^{\text{ext}}}\frac{1}{v}\,d\mathcal{H}^{n-1}(x)\right).$ (13)
###### Proof.
Let $t>0$ e $h>0$, we choose the test function
$\left.\varphi(x)=\right.\begin{cases}0&\text{ if }u<t\\\ u-t&\text{ if
}t<u<t+h\\\ h&\text{ if }u>t+h.\end{cases}$
Then,
$\begin{split}\int_{U_{t}\setminus U_{t+h}}{\left|\nabla
u\right|}^{p}\,dx&+\beta h\int_{\partial
U_{t+h}^{ext}}u^{p-1}\,d\mathcal{H}^{n-1}(x)+\beta\int_{\partial
U_{t}^{ext}\setminus\partial
U_{t+h}^{ext}}u^{p-1}(u-t)\,d\mathcal{H}^{n-1}(x)\\\ &=\int_{U_{t}\setminus
U_{t+h}}f(u-t)\,dx+h\int_{U_{t+h}}f\,dx.\end{split}$
Dividing by $h$, using coarea formula and letting $h$ go to 0, we have that
for a. e. $t>0$
$\int_{\partial U_{t}}g(x)\,d\mathcal{H}^{n-1}(x)=\int_{U_{t}}f\,dx,$
where
$\left.g(x)=\right.\begin{cases}{\left|\nabla u\right|}^{p-1}&\text{ if
}x\in\partial U_{t}^{int},\\\ \beta u^{p-1}&\text{ if }x\in\partial
U_{t}^{ext}.\end{cases}$
So, using the isoperimetric inequality, for a. e. $t>0$ we have
$\begin{split}n\omega_{n}^{1/n}\mu(t)^{\left(1-\frac{1}{n}\right)}&\leq
P_{u}(t)=\int_{\partial U_{t}}\,d\mathcal{H}^{n-1}(x)\leq\left(\int_{\partial
U_{t}}g\,d\mathcal{H}^{n-1}(x)\right)^{\frac{1}{p}}\left(\int_{\partial
U_{t}}\frac{1}{g^{\frac{1}{p-1}}}\,d\mathcal{H}^{n-1}(x)\right)^{1-\frac{1}{p}}\\\
&=\left(\int_{\partial
U_{t}}g\,d\mathcal{H}^{n-1}(x)\right)^{\frac{1}{p}}\left(\int_{\partial
U_{t}^{int}}\frac{1}{{\left|\nabla
u\right|}}\,d\mathcal{H}^{n-1}(x)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{ext}}\frac{1}{u}\,d\mathcal{H}^{n-1}(x)\right)^{1-\frac{1}{p}}\\\
&\leq\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p}}\left(-\mu^{\prime}(t)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{ext}}\frac{1}{u}\,d\mathcal{H}^{n-1}(x)\right)^{1-\frac{1}{p}}\quad
t\in[0,\max_{\Omega}u),\end{split}$
Then (12) follows. We notice that if $v$ is the solution to (3), than all the
inequalities are verified as equalities, so we have (13). ∎
###### Lemma 2.3.
For all $\tau\geq v_{m}$ we have
$\int_{0}^{\tau}t^{p-1}\left(\int_{\partial
U_{t}^{ext}}\frac{1}{u(x)}\,d\mathcal{H}^{n-1}(x)\right)\,dt\leq\frac{1}{p\beta}\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds.$
(14)
Moreover,
$\int_{0}^{\tau}t^{p-1}\left(\int_{\partial
V_{t}\cap\partial\Omega^{\sharp}}\frac{1}{v(x)}\,d\mathcal{H}^{n-1}(x)\right)\,dt=\frac{1}{p\beta}\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds,$
(15)
###### Proof.
If we integrate the quantity
$t^{p-1}\left(\int_{\partial
U_{t}^{ext}}\frac{1}{u(x)}\,d\mathcal{H}^{n-1}(x)\right),$
from 0 to $+\infty$, by Fubini theorem, we obtain
$\begin{split}\int_{0}^{\infty}\tau^{p-1}\left(\int_{\partial
U_{\tau}^{ext}}\frac{1}{u(x)}\,d\mathcal{H}^{n-1}(x)\right)\,d\tau&=\int_{\partial\Omega}\left(\int_{0}^{u(x)}\frac{\tau^{p-1}}{u(x)}\,d\tau\right)\,d\mathcal{H}^{n-1}(x)\\\
&=\frac{1}{p}\int_{\partial\Omega}u(x)^{p-1}\,d\mathcal{H}^{n-1}(x)\\\
&=\frac{1}{p\beta}\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds,\end{split}$
where the last equality follows from the fact that $u$ solves (1).
Analogously
$\int_{0}^{\infty}\tau^{p-1}\left(\int_{\partial
V_{\tau}\cap\partial\Omega^{\sharp}}\frac{1}{v(x)}\,d\mathcal{H}^{n-1}(x)\right)\,d\tau=\frac{1}{p\beta}\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds.$
Since $u$ is positive, we obtain, $\forall t\geq 0$,
$\int_{0}^{t}\tau^{p-1}\left(\int_{\partial
U_{\tau}^{ext}}\frac{1}{u(x)}\,d\mathcal{H}^{n-1}(x)\right)\,d\tau\leq\frac{1}{p\beta}\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds,$
on the other hand, since $\partial V_{t}\cap\partial\Omega^{\sharp}$ is empty
for $t\geq v_{m}$, we have
$\int_{0}^{t}\tau^{p-1}\left(\int_{\partial
V_{\tau}\cap\partial\Omega^{\sharp}}\frac{1}{v(x)}\,d\mathcal{H}^{n-1}(x)\right)\,d\tau=\frac{1}{p\beta}\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds.$
and the proof of lemma 2.3 is complete. ∎
###### Remark 2.1.
It can be observed that, since $\partial V_{t}\cap\partial\Omega^{\sharp}$ is
empty for $t\geq v_{m}$ and $\phi(t)={\left|\Omega\right|}$ for $t\leq v_{m}$,
for all $\delta>0$ and for all $t$, we have
$\begin{split}&\int_{0}^{t}\tau^{p-1}\phi(\tau)^{\delta}\left(\int_{\partial
V_{\tau}\cap\partial\Omega^{\sharp}}\frac{1}{v(x)}\,d\mathcal{H}^{n-1}(x)\right)\,d\tau=\\\
&\int_{0}^{v_{m}}\tau^{p-1}\phi(\tau)^{\delta}\left(\int_{\partial
V_{\tau}\cap\partial\Omega^{\sharp}}\frac{1}{v(x)}\,d\mathcal{H}^{n-1}(x)\right)\,d\tau=\\\
&\int_{0}^{+\infty}\tau^{p-1}\phi(\tau)^{\delta}\left(\int_{\partial
V_{\tau}\cap\partial\Omega^{\sharp}}\frac{1}{v(x)}\,d\mathcal{H}^{n-1}(x)\right)\,d\tau=\frac{{\left|\Omega\right|}^{\delta}}{p\beta}\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds.\end{split}$
## 3 Main results
Now we prove Theorem 1.1 and Theorem 1.2 .
###### Proof of Theorem 1.1.
Let $\displaystyle{0<k\leq\frac{n(p-1)}{p(n-1)}}$, so
$\displaystyle{\delta=\frac{1}{k}-\frac{(n-1)p}{n(p-1)}}$ is positive.
Multiplying (12) by $t^{p-1}\mu(t)^{\delta}$ and integrating from $0$ to
$\tau\geq v_{m}$, by the previous Lemma, we obtain
$\begin{split}\int_{0}^{\tau}\gamma_{n}t^{p-1}\mu(t)^{\frac{1}{k}}\,dt&\leq\int_{0}^{\tau}\left(-\mu^{\prime}(t)\right)t^{p-1}\mu(t)^{\delta}\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\,dt\\\
&+\frac{{\left|\Omega\right|}^{\delta}}{p\beta^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}.\end{split}$
(16)
Setting
$\displaystyle{F(l)=\int_{0}^{l}\omega^{\delta}\left(\int_{0}^{\omega}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\,d\omega}$,
we can integrate by parts both sides of the last inequality, getting
$\begin{split}\tau^{p-1}\left(\left(\int_{0}^{\tau}\gamma_{n}\mu(t)^{\frac{1}{k}}\,dt\right)+F(\mu(\tau))\right)&\leq(p-1)\int_{0}^{\tau}t^{p-2}\left(\left(\int_{0}^{t}\gamma_{n}\mu(s)^{\frac{1}{k}}\,ds\right)+F(\mu(t))\right)\,dt\\\
&+\frac{{\left|\Omega\right|}^{\delta}}{p\beta^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}.\end{split}$
Setting
$\displaystyle{\xi(\tau)=\int_{0}^{\tau}t^{p-2}\left(\int_{0}^{t}\gamma_{n}\mu(s)^{\frac{1}{k}}\,ds+F(\mu(t))\right)\,dt}$
and
$\displaystyle{C=\frac{{\left|\Omega\right|}^{\delta}}{p\beta^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}}$,
we are in the hypothesis of Lemma 2.1 (Gronwall), namely
$\tau\xi^{\prime}(\tau)\leq(p-1)\xi(\tau)+C,$
so, choosing $\tau_{0}=v_{m}$, we have
$\begin{multlined}\tau^{p-2}\left(\int_{0}^{\tau}\gamma_{n}\mu(s)^{\frac{1}{k}}\,ds+F(\mu(\tau))\right)\leq\left(\frac{(p-1)\xi(v_{m})+C}{v_{m}}\right)\left(\frac{\tau}{v_{m}}\right)^{p-2},\end{multlined}\tau^{p-2}\left(\int_{0}^{\tau}\gamma_{n}\mu(s)^{\frac{1}{k}}\,ds+F(\mu(\tau))\right)\leq\left(\frac{(p-1)\xi(v_{m})+C}{v_{m}}\right)\left(\frac{\tau}{v_{m}}\right)^{p-2},$
where
$\xi(v_{m})=\int_{0}^{v_{m}}t^{p-2}\left(\int_{0}^{t}\gamma_{n}\mu(s)^{\frac{1}{k}}\,ds+F(\mu(t))\right)\,dt.$
The previous inequality becomes an equality if we replace $\mu(t)$ with
$\phi(t)$. Since $\mu(t)\leq\phi(t)={\left|\Omega\right|},\quad\forall t\leq
v_{m}$, and $F(l)$ is monotone, we obtain
$\int_{0}^{v_{m}}t^{p-2}\left(\int_{0}^{t}\gamma_{n}\mu(s)^{\frac{1}{k}}\,ds+F(\mu(t))\right)\,dt\leq\int_{0}^{v_{m}}t^{p-2}\left(\int_{0}^{t}\gamma_{n}\phi(s)^{\frac{1}{k}}\,ds+F(\phi(t))\right)\,dt,$
hence
$\int_{0}^{\tau}\gamma_{n}\mu(s)^{\frac{1}{k}}\,ds+F(\mu(\tau))\leq\int_{0}^{\tau}\gamma_{n}\phi(s)^{\frac{1}{k}}\,ds+F(\phi(\tau)).$
Passing to the limit as $\tau\to\infty$, we get
$\int_{0}^{\infty}\mu(t)^{\frac{1}{k}}\,dt\leq\int_{0}^{\infty}\phi(t)^{\frac{1}{k}}\,dt,$
and hence
${\left\|u\right\|}_{L^{k,1}(\Omega)}\leq{\left\|v\right\|}_{L^{k,1}(\Omega^{\sharp})}\quad\forall\,0\,<k\leq\frac{n(p-1)}{p(n-1)}.$
To prove the inequality (5), it is enough to show that
$\int_{0}^{\infty}t^{p-1}\mu(t)^{\frac{1}{k}}\,dt\leq\int_{0}^{\infty}t^{p-1}\phi(t)^{\frac{1}{k}}\,dt.$
(17)
Let us consider equation (16), let us integrate by parts the first term on the
right-hand side from 0 to $\tau$ and then let us pass to the limit as
$\tau\to\infty$, we have
$\int_{0}^{\infty}\gamma_{n}t^{p-1}\mu(t)^{\frac{1}{k}}\,dt\leq(p-1)\int_{0}^{\infty}t^{p-2}F(\mu(t))\,dt+\frac{{\left|\Omega\right|}^{\delta}}{p\beta^{1+\frac{1}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}.$
Therefore, if we show that
$\int_{0}^{\infty}t^{p-2}F(\mu(t))\,dt\leq\int_{0}^{\infty}t^{p-2}F(\phi(t))\,dt$
(18)
we obtain the (17). To this aim, we multiply (12) by
$\displaystyle{t^{p-1}F(\mu(t))\mu(t)^{-\frac{(n-1)p}{n(p-1)}}}$ and
integrate. First, we observe that, by the choice
$\displaystyle{k\leq\frac{n(p-1)}{(n-2)p+n}}$, it follows that the function
$\displaystyle{h(l)=F(l)l^{-\frac{(n-1)p}{n(p-1)}}}$ is non decreasing. Hence,
we obtain
$\begin{split}\int_{0}^{\tau}\gamma_{n}t^{p-1}F(\mu(t))\,dt&\leq\int_{0}^{\tau}\Bigl{(}-\mu^{\prime}(t)\Bigr{)}t^{p-1}\mu(t)^{-\frac{(n-1)p}{n(p-1)}}F(\mu(t))\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\,dt\\\
&+F({\left|\Omega\right|})\frac{{\left|\Omega\right|}^{-\frac{(n-1)p}{n(p-1)}}}{p\beta^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}.\end{split}$
If we integrate by parts both sides of the last expression and set
$\displaystyle{C=F({\left|\Omega\right|})\frac{{\left|\Omega\right|}^{-\frac{p(n-1)}{n(p-1)}}}{p\beta^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}},$
we obtain
$\tau\int_{0}^{\tau}\gamma_{n}t^{p-2}F(\mu(t))\,dt+\tau
H_{\mu}(\tau)\leq\int_{0}^{\tau}\int_{0}^{t}r^{p-2}F(\mu(r))\,drdt+\int_{0}^{\tau}H_{\mu}(t)\,dt+C$
(19)
where
$H_{\mu}(\tau)=-\int_{\tau}^{+\infty}t^{p-2}\mu(t)^{-\frac{p(n-1)}{n(p-1)}}F(\mu(t))\biggl{(}\int_{0}^{\mu(t)}f^{*}(s)\,ds\biggr{)}^{\frac{1}{p-1}}\,d\mu(t).$
Setting
$\begin{multlined}\xi(\tau)=\int_{0}^{\tau}\int_{0}^{t}\gamma_{n}r^{p-2}F(\mu(r))\,dr+\int_{0}^{t}H_{\mu}(t)\,dt\end{multlined}\xi(\tau)=\int_{0}^{\tau}\int_{0}^{t}\gamma_{n}r^{p-2}F(\mu(r))\,dr+\int_{0}^{t}H_{\mu}(t)\,dt$
then (19) becomes
$\tau\xi^{\prime}(\tau)\leq\xi(\tau)+C.$
So lemma 2.1, with $\tau_{0}=v_{m}$, gives
$\int_{0}^{\tau}\gamma_{n}t^{p-2}F(\mu(t))\,dt+H_{\mu}(\tau)\leq\left(\frac{\displaystyle{(p-1)\int_{0}^{v_{m}}t^{p-2}F(\mu(t)\,dt+H_{\mu}(v_{m})+C}}{v_{m}}\right)\left(\frac{\tau}{v_{m}}\right)^{p-2}$
Of course, the inequality holds as an equality if we replace $\mu(t)$ with
$\phi(t)$, so we get, keeping in mind that
$\mu(t)\leq\phi(t)={\left|\Omega\right|}$ for $t\leq v_{m}$,
$\int_{0}^{\tau}\gamma_{n}t^{p-2}F(\mu(t)\,dt+H_{\mu}(\tau)\leq\int_{0}^{\tau}\gamma_{n}F(\phi(t))\,dt+H_{\phi}(\tau)$
Letting $\tau\to\infty$, one has
$\int_{0}^{\infty}t^{p-2}F(\mu(t))dt\leq\int_{0}^{\infty}t^{p-2}F(\phi(t))dt,$
as $H_{\mu}(\tau),H_{\phi}(\tau)\to 0$. This proves (18), and hence (5).
The fact that both $H_{\mu}$ and $H_{\phi}$ go to 0 as $\tau$ goes to infinity
can be easily deduced distinguishing the cases.
* •
If $p\geq 2$
$\displaystyle t^{p-2}\mu(t)$
$\displaystyle=\int_{u>t}t^{p-2}\,dx\leq\int_{u>t}u^{p-2}\,dx\leq{\left\|u\right\|}_{L^{p}}^{p-2}\mu(t)^{\frac{2}{p}}$
$\displaystyle\Rightarrow{\left|H_{\mu}(\tau)\right|}$
$\displaystyle=\int_{\tau}^{+\infty}t^{p-2}F(\mu(t))\mu(t)^{-\frac{p(n-1)}{n(p-1)}}\biggl{(}\int_{0}^{\mu(t)}f^{*}(s)\,ds\biggr{)}(-\mu^{\prime}(t))\,dt$
$\displaystyle\leq\biggl{(}\int_{0}^{{\left|\Omega\right|}}f^{*}(s)\,ds\biggr{)}{\left\|u\right\|}_{L^{p}}^{p-2}\int_{\tau}^{+\infty}F(\mu(t))\mu(t)^{\frac{2}{p}-\frac{p(n-1)}{n(p-1)}-1}(-\mu^{\prime}(t))\,dt\xrightarrow{\tau\to+\infty}0.$
* •
If $p<2$
$\displaystyle{\left|H_{\mu}(\tau)\right|}$
$\displaystyle=\int_{\tau}^{+\infty}t^{p-2}F(\mu(t))\mu(t)^{-\frac{p(n-1)}{n(p-1)}}\biggl{(}\int_{0}^{\mu(t)}f^{*}(s)\,s\biggr{)}(-\mu^{\prime}(t))\,dt$
$\displaystyle\leq\tau^{p-2}\int_{\tau}^{+\infty}F(\mu(t))\mu(t)^{-\frac{p(n-1)}{n(p-1)}}\biggl{(}\int_{0}^{\mu(t)}f^{*}(s)\,s\biggr{)}(-\mu^{\prime}(t))\,dt\xrightarrow{\tau\to+\infty}0.$
and analogously for $H_{\phi}$, which concludes the proof. ∎
###### Proof of Theorem 1.2.
.
1. $(i)$
Firstly, we observe that
$\displaystyle{\int_{0}^{\mu(t)}f^{\ast}(s)\,ds=\mu(t)}$, so (12) becomes
$\gamma_{n}\mu(t)^{\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}\leq-\mu^{\prime}(t)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{\text{ext}}}\frac{1}{u}\,d\mathcal{H}^{n-1}(x).$ (20)
Let us multiply both sides by $t^{p-1}\mu(t)^{\delta}$, where
$\delta=-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}$. We point out
that $\delta\geq 0$ for $p\leq\frac{n}{n-1}$. Hence, integrating from $0$ to
$\tau\geq v_{m}$, we have
$\begin{split}\int_{0}^{\tau}\gamma_{n}t^{p-1}&\leq\int_{0}^{\tau}t^{p-1}\mu(t)^{\delta}(-\mu^{\prime}(t))\,dt+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{0}^{\tau}t^{p-1}\mu(t)^{\delta}\int_{\partial
U_{t}^{\text{ext}}}\frac{1}{u}\,d\mathcal{H}^{n-1}(x)\\\
&\leq\int_{0}^{\tau}t^{p-1}\mu(t)^{\delta}(-\mu^{\prime}(t))\,dt+\frac{{\left|\Omega\right|}^{\delta+1}}{p\beta^{\frac{p}{p-1}}}\end{split}$
(21)
Taking into account remark 2.1, if we replace $\mu(t)$ with $\phi(t)$ the
previous inequality holds as equality.
Hence, we get
$\int_{0}^{\tau}t^{p-1}\mu(t)^{\delta}(-\mu^{\prime}(t))\,dt\geq\int_{0}^{\tau}t^{p-1}\phi(t)^{\delta}(-\phi^{\prime}(t))\,dt.$
Then an integration by parts gives
$-\tau^{p-1}\frac{\mu(\tau)^{\delta+1}}{\delta+1}+(p-1)\int_{0}^{\tau}t^{p-2}\frac{\mu(t)^{\delta+1}}{\delta+1}\,dt\geq-\tau^{p-1}\frac{\phi(\tau)^{\delta+1}}{\delta+1}+(p-1)\int_{0}^{\tau}t^{p-2}\frac{\phi(t)^{\delta+1}}{\delta+1}\,dt.$
Finally, using Gronwall’s Lemma with the function
$\displaystyle{\xi(\tau)=\int_{0}^{\tau}s^{p-2}\left(\frac{\mu(s)^{\delta+1}-\phi(s)^{\delta+1}}{\delta+1}\right)\,ds}$
we obtain
$\displaystyle{\tau^{p-2}\left(\frac{\mu^{\delta+1}(\tau)-\phi^{\delta+1}(\tau)}{\delta+1}\right)\leq(p-1)\frac{\tau^{p-2}}{v_{m}^{p-2}}\int_{0}^{v_{m}}s^{p-2}\left(\frac{\mu^{\delta+1}(s)-\phi^{\delta+1}(s)}{\delta+1}\right)\,ds.}$
The quantity on the right-hand side is non-positive, thanks to (10), so
$\mu(\tau)\leq\phi(\tau)\quad\forall\tau\geq v_{m}.$
and, remembering that,
$\mu(\tau)\leq\phi(\tau)={\left|\Omega\right|}\quad\forall\tau\leq v_{m},$
we get the point-wise inequality of the functions.
2. $(ii)$
Now we want to show that
${\left\|u\right\|}_{L^{k,1}(\Omega)}\leq{\left\|v\right\|}_{L^{k,1}(\Omega^{\sharp})}$
so it is enough to show
$\int_{0}^{+\infty}\mu(t)^{\frac{1}{k}}\,dt\leq\int_{0}^{+\infty}\phi(t)^{\frac{1}{k}}\,dt$
(22)
We multiply (20) by
$t^{p-1}\mu(t)^{\frac{1}{k}-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}$
and integrate from $0$ to $\tau\geq v_{m}$. Then using Lemma 2.3 and Remark
2.1, we obtain
$\int_{0}^{\tau}\gamma_{n}t^{p-1}\mu(t)^{\frac{1}{k}}\,dt\leq\int_{0}^{\tau}t^{p-1}\mu(t)^{\frac{1}{k}-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}(-\mu^{\prime}(t))\,dt+\frac{{\left|\Omega\right|}^{\frac{1}{k}-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}+1}}{p\beta^{\frac{p}{p-1}}}$
(23)
and equality holds if we replace $\mu$ with $\phi$. In order to be shorter, we
set
$\eta=\frac{1}{k}-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1},\quad
C=\frac{{\left|\Omega\right|}^{\eta+1}}{p\beta^{\frac{p}{p-1}}}.$
We point out that (23) follows by (21) if $\eta\geq 0$, namely
$0<k\leq\frac{n(p-1)}{n(p-1)-p}$
With these notations and keeping in mind that $\mu$ is a non increasing
function, we have from (23) taht
$\int_{0}^{\tau}\gamma_{n}t^{p-1}\mu(t)^{\frac{1}{k}}\,dt\leq\int_{0}^{\tau}-t^{p-1}\mu(t)^{\eta}\,d\mu(t)+C$
(24)
Let us set
$\displaystyle{G(\ell)=\int_{0}^{\ell}w^{\eta}\,dw=\frac{\ell^{\eta+1}}{\eta+1}}$,
let us integrate by parts both sides of (24) in order to obtain
$\begin{split}&\gamma_{n}\tau^{p-1}\int_{0}^{\tau}\mu(t)^{\frac{1}{k}}\,dt+\tau^{p-1}G(\mu(\tau))\\\
&\leq(p-1)\biggl{[}\int_{0}^{\tau}\gamma_{n}t^{p-2}\int_{0}^{t}\mu(r)^{\frac{1}{k}}\,drdt+\int_{0}^{\tau}t^{p-2}G(\mu(t))\,dt\biggr{]}+C\end{split}$
(25)
Setting
$\xi(\tau)=\int_{0}^{\tau}\left(\gamma_{n}t^{p-2}\int_{0}^{t}\mu(r)^{\frac{1}{k}}\,dr\right)\,dt+\int_{0}^{\tau}t^{p-2}G(\mu(t))\,dt$
(25) reads as follows
$\tau\xi^{\prime}(\tau)\leq(p-1)\xi(\tau)+C$
Hence, using Gronwall’s Lemma 2.1 with $\tau_{0}=v_{m}$, we get
$\gamma_{n}\tau^{p-2}\int_{0}^{\tau}\mu(t)^{\frac{1}{k}}\,dt+\tau^{p-2}G(\mu(\tau))\leq\biggl{(}\frac{(p-1)\xi(v_{m})+C}{v_{m}}\biggr{)}\biggl{(}\frac{\tau}{v_{m}}\biggr{)}^{p-2}$
where
$\displaystyle{\xi(v_{m})=\int_{0}^{v_{m}}\gamma_{n}t^{p-2}\int_{0}^{t}\mu(r)^{\frac{1}{k}}\,dr\,dt+\int_{0}^{v_{m}}t^{p-2}G(\mu(t))\,dt}$
Again, if we replace $\mu$ with $\phi$, the previous inequality holds as an
equality and $\xi(v_{m})$ is less or equal than the same quantity obtained by
replacing $\mu$ with $\phi$, as (10) holds. Keeping in mind (11), we have
$\tau^{p-2}\left(\gamma_{n}\int_{0}^{\tau}\mu(t)^{\frac{1}{k}}\,dt+G(\mu(\tau))\right)\leq\tau^{p-2}\left(\gamma_{n}\int_{0}^{\tau}\phi(t)^{\frac{1}{k}}\,dt+G(\phi(\tau))\right)$
Passing to the limit as $\tau\to+\infty$, we get
$\int_{0}^{+\infty}\mu(t)^{\frac{1}{k}}\,dt\leq\int_{0}^{+\infty}\phi(t)^{\frac{1}{k}}\,dt$
namely (22).
To conclude the proof, we have to show that
${\left\|u\right\|}_{L^{pk,p}(\Omega)}\leq{\left\|v\right\|}_{L^{pk,p}(\Omega^{\sharp})}\qquad\forall\,0<k\leq\frac{n(p-1)}{n(p-1)-p}$
that is to say
$\int_{0}^{+\infty}t^{p-1}\mu(t)^{\frac{1}{k}}\,dt\leq\int_{0}^{+\infty}t^{p-1}\phi(t)^{\frac{1}{k}}\,dt.$
We consider (24), pass to the limit as $\tau\to+\infty$ and integrate by parts
the first term on the right-hand side
$\int_{0}^{+\infty}\gamma_{n}t^{p-1}\mu(t)^{\frac{1}{k}}\,dt\leq(p-1)\int_{0}^{+\infty}t^{p-2}G(\mu(t))\,dt+C.$
Hence it is enough to show that
$\int_{0}^{+\infty}t^{p-2}G(\mu(t))\,dt\leq\int_{0}^{+\infty}t^{p-2}G(\phi(t))\,dt.$
To this aim, we multiply (20) by
$t^{p-1}G(\mu(t))\mu(t)^{-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}$
and integrate from $0$ to $\tau\geq v_{m}$
$\displaystyle\int_{0}^{\tau}\gamma_{n}t^{p-1}G(\mu(t))\,dt$
$\displaystyle\leq\int_{0}^{\tau}t^{p-1}G(\mu(t))\mu(t)^{-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}\,d\mu(t)$
$\displaystyle+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{0}^{\tau}t^{p-1}G(\mu(t))\mu(t)^{-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}\biggl{(}\int_{\partial
U_{t}^{ext}}\frac{1}{u}\,d\mathcal{H}^{n-1}\biggr{)}\,dt$
Since $\displaystyle{k\leq\frac{n(p-1)}{n(p-1)-p}}$, using Lemma 2.3 and the
fact that the function
$G(\ell)\ell^{-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}$ is non
decreasing, we obtain
$\int_{0}^{\tau}\gamma_{n}t^{p-1}G(\mu(t))\,dt\leq\int_{0}^{\tau}t^{p-1}G(\mu(t))\mu(t)^{-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}\,d\mu(t)+C$
(26)
with
$C=\frac{1}{p\beta^{\frac{p}{p-1}}}G({\left|\Omega\right|}){\left|\Omega\right|}^{-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}+1}$
If we replace $\mu$ with $\phi$ the previous inequality holds as an equality,
thanks to (2.1). Now, let us integrate by parts both sides of (26), obtaining
$\tau\int_{0}^{\tau}\gamma_{n}t^{p-2}G(\mu(t))\,dt+\tau
H(\tau)\leq\int_{0}^{\tau}\int_{0}^{t}\gamma_{n}t^{p-2}G(\mu(r))\,drdt+\int_{0}^{\tau}H_{\mu}(t)\,dt+C$
(27)
where
$H_{\mu}(\tau)=-\int_{\tau}^{+\infty}t^{p-2}G(\mu(t))\mu(t)^{-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}\,d\mu(t)$
Setting
$\xi(\tau)=\int_{0}^{\tau}\int_{0}^{t}\gamma_{n}t^{p-2}G(\mu(r))\,drdt+\int_{0}^{\tau}H_{\mu}(t)\,dt$
the (27) reads as follows
$\tau\xi^{\prime}(\tau)\leq\xi(\tau)+C$
Again, using Gronwall’s Lemma 2.1, we get
$\displaystyle{\int_{0}^{\tau}\gamma_{n}t^{p-2}G(\mu(t))\,dt+H_{\mu}(\tau)\leq\left(\frac{(p-1)\xi(v_{m})+C}{v_{m}}\right)\left(\frac{\tau}{v_{m}}\right)^{p-2}}$
with
$\xi(v_{m})=\int_{0}^{v_{m}}\int_{0}^{t}\gamma_{n}t^{p-2}G(\mu(r))\,drdt+\int_{0}^{v_{m}}H_{\mu}(t)\,dt.$
Keeping in mind that for $\phi$ the previous inequalities hold as equality and
the fact that $G$ is not decreasing, $\xi(v_{m})$ is less or equal to the same
quantity obtained by replacing $\mu$ with $\phi$. Hence, we obtain
$\displaystyle{\int_{0}^{\tau}\gamma_{n}t^{p-2}G(\mu(t))\,dt+H_{\mu}(\tau)\leq\int_{0}^{\tau}\gamma_{n}t^{p-2}G(\phi(t))\,dt+H_{\phi}(\tau)}$
and passing to the limit as $\tau\to+\infty$, we finally get
$\int_{0}^{+\infty}t^{p-2}G(\mu(t))\,dt\leq\int_{0}^{+\infty}t^{p-2}G(\phi(t))\,dt$
indeed, as in the proof of Theorem 1.1, $H_{\mu}(\tau)$ and $H_{\phi}(\tau)$
go to 0 as $\tau\to\infty$. That concludes the proof. ∎
###### Corollary 3.1.
Let $u$ and $v$ be the solutions to (1) and (3) respectively. Then, if $p\geq
n$, we have
${\left\|u\right\|}_{L^{1}(\Omega)}\leq{\left\|v\right\|}_{L^{1}(\Omega^{\sharp})}\quad\text{and}\quad{\left\|u\right\|}_{L^{p}(\Omega)}\leq{\left\|v\right\|}_{L^{p}(\Omega^{\sharp})}.$
Moreover in the case $f\equiv 1$, Theorem 1.2 gives
${\left\|u\right\|}_{L^{1}(\Omega)}\leq{\left\|v\right\|}_{L^{1}(\Omega^{\sharp})}\quad\text{and}\quad{\left\|u\right\|}_{L^{p}(\Omega)}\leq{\left\|v\right\|}_{L^{p}(\Omega^{\sharp})}\qquad\forall
p>1$
and the point-wise comparison for $\displaystyle{p\leq\frac{n}{n-1}}$.
###### Proof.
If $p\geq n$ the upper bounds of $k$, in both cases (4) e (5), are greater
than $1$ and so we can choose $k=1$. The assertion follows from the fact that
${\left\|\cdot\right\|}_{L^{p,p}(\Omega)}={\left\|\cdot\right\|}_{L^{p}(\Omega)}.$
Analogously if $f\equiv 1$. ∎
## 4 Faber–Krahn inequality
We recall that the first eigenvalue of $p$-Laplace operator with Robin
boundary conditions is obtained as the minimum of the Rayleigh quotients,
i.e.,
$\lambda_{1,\beta}(\Omega)=\min_{\begin{subarray}{c}\omega\in
W^{1,p}(\Omega)\\\ \omega\neq
0\end{subarray}}\frac{\displaystyle{\int_{\Omega}{\left|\nabla\omega\right|}^{p}\,dx+\beta\int_{\partial\Omega}{\left|\omega\right|}^{p}\,d\mathcal{H}^{n-1}(x)}}{\displaystyle{\int_{\Omega}{\left|\omega\right|}^{p}\,dx}}.$
(28)
We can observe that if $u$ achieves the minimum of Rayleigh quotients, so does
${\left|u\right|}$. From this we have that $u$ in non-negative. Furthermore,
we have if $u_{\lambda_{1}}\geq 0$, as a consequence of Harnack inequality,
$u_{\lambda_{1}}>0$.
Another important thing is that the eigenvalue is simple. Indeed, as shown in
[8], if $\Omega$ is smooth enough and $u$ and $v$ are to eigenfunctions
referred to the first eigenvalue, we can choose as test function
$\displaystyle{\varphi_{1}=\frac{u^{p}-v^{p}}{u^{p-1}}}$ in
$\int_{\Omega}{\left|\nabla u\right|}^{p-2}\nabla
u\nabla\varphi_{1}\,dx+\beta\int_{\partial\Omega}u^{p-1}\varphi_{1}\,d\mathcal{H}^{n-1}(x)=\int_{\Omega}\lambda_{1}u^{p-1}\varphi_{1}\,dx$
and $\displaystyle{\varphi_{2}=\frac{v^{p}-u^{p}}{v^{p-1}}}$ in
$\int_{\Omega}{\left|\nabla v\right|}^{p-2}\nabla
v\nabla\varphi_{2}\,dx+\beta\int_{\partial\Omega}v^{p-1}\varphi_{2}\,d\mathcal{H}^{n-1}(x)=\int_{\Omega}\lambda_{1}v^{p-1}\varphi_{2}\,dx.$
Summing the two equations, we have
$\begin{split}0&=\int_{\Omega}\left\\{1+(p-1)\left(\frac{v}{u}\right)^{p}\right\\}{\left|\nabla
u\right|}^{p}+\left\\{1+(p-1)\left(\frac{u}{v}\right)^{p}\right\\}{\left|\nabla
v\right|}^{p}\\\ &-\int_{\Omega}p\left(\frac{v}{u}\right)^{p-1}{\left|\nabla
u\right|}^{p-2}\nabla u\nabla v+p\left(\frac{u}{v}\right)^{p-1}{\left|\nabla
v\right|}^{p-2}\nabla v\nabla u\\\
&=\int_{\Omega}(u^{p}-v^{p})\left({\left|\nabla\log{u}\right|}^{p}-{\left|\nabla\log{v}\right|}^{p}\right)\\\
&-\int_{\Omega}pv^{p}{\left|\nabla\log{u}\right|}^{p-2}{\left|\nabla\log{u}\right|}\left(\nabla\log{v}-\nabla\log{u}\right)\\\
&-\int_{\Omega}pu^{p}{\left|\nabla\log{v}\right|}^{p-2}{\left|\nabla\log{v}\right|}\left(\nabla\log{u}-\nabla\log{v}\right)\end{split}$
Now, using the well known inequalities, which hold true for each $w_{1}$ and
$w_{2}\in\operatorname{\mathbb{R}}^{n}$ ,
$\begin{gathered}{\left|w_{2}\right|}^{p}\geq{\left|w_{1}\right|}^{p}+p{\left|w_{1}\right|}^{p-2}w_{1}\cdot(w_{2}-w_{1})+\frac{{\left|w_{2}-w_{1}\right|}^{p}}{2^{p-1}-1}\quad\text{if}\quad
p\geq 2\\\
{\left|w_{2}\right|}^{p}\geq{\left|w_{1}\right|}^{p}+p{\left|w_{1}\right|}^{p-2}w_{1}\cdot(w_{2}-w_{1})+C(p)\frac{{\left|w_{2}-w_{1}\right|}^{2}}{({\left|w_{1}\right|}+{\left|w_{2}\right|})^{2-p}}\quad\text{if}\quad
1<p<2,\end{gathered}$ (29)
if we consider the case $p\geq 2$, we choose $w_{2}=\nabla\log u$ and
$w_{1}=\nabla\log v$, we obtain
$\frac{1}{2^{p-1}-1}\int_{\Omega}\left(\frac{1}{v^{p}}+\frac{1}{u^{p}}\right){\left|v\nabla
u-u\nabla v\right|}^{p}=0.$
Hence, we obtain that $v\nabla u=u\nabla v$ a.e. in $\Omega$, and so there
exists a constant $K$ for which $u=Kv$. This means that $\lambda_{1}$ is
simple.
For the proof of (29), we refer to [8].
The following corollary of Theorem 1.1 holds true, that is Faber-Krahn
inequality.
###### Corollary 4.1.
Let $u$ and $v$ be the solutions to (1) and (3), respectively. Then , if
$p\geq n$, we have
$\lambda_{1,\beta}(\Omega)\geq\lambda_{1,\beta}(\Omega^{\sharp}).$
###### Proof.
Let $u$ an eigenfunction referred to the first eigenvalue of (1), then it
solves
$\begin{cases}-\Delta_{p}u=\lambda_{1,\beta}(\Omega)\,{\left|u\right|}^{p-2}u&\text{
in }\Omega\\\ {\left|\nabla u\right|}^{p-2}\displaystyle{\frac{\partial
u}{\partial\nu}}+\beta{\left|u\right|}^{p-2}u=0&\text{ on
}\partial\Omega.\end{cases}$
Now, let $z$ be a solution to the following problem
$\begin{cases}-\Delta_{p}z=\lambda_{1,\beta}(\Omega)\,\lvert
u^{\sharp}\rvert^{p-2}u^{\sharp}&\text{ in }\Omega^{\sharp}\\\ {\left|\nabla
z\right|}^{p-2}\displaystyle{\frac{\partial
z}{\partial\nu}}+\beta{\left|z\right|}^{p-2}z=0&\text{ on
}\partial\Omega^{\sharp}.\end{cases}$
In that case, corollary 3.1 gives
$\int_{\Omega}{\left|u\right|}^{p}\,dx=\int_{\Omega^{\sharp}}{\left|u^{\sharp}\right|}^{p}\,dx\leq\int_{\Omega^{\sharp}}{\left|z\right|}^{p}\,dx,$
and hence, by Hölder inequality
$\int_{\Omega^{\sharp}}(u^{\sharp})^{p-2}u^{\sharp}z\,dx\leq\left(\int_{\Omega^{\sharp}}{\left|u^{\sharp}\right|}^{p}\,dx\right)^{\frac{p-1}{p}}\left(\int_{\Omega^{\sharp}}z^{p}\,dx\right)^{\frac{1}{p}}\leq\int_{\Omega^{\sharp}}z^{p}\,dx.$
Therefore, observing that we can write the eigenvalue
$\lambda_{1,\beta}(\Omega)$ in the following way, we obtain
$\begin{split}\lambda_{1,\beta}(\Omega)&=\frac{\displaystyle{\int_{\Omega^{\sharp}}{\left|\nabla
z\right|}^{p}\,dx+\beta\int_{\partial\Omega^{\sharp}}z^{p}\,d\mathcal{H}^{n-1}(x)}}{\displaystyle{\int_{\Omega^{\sharp}}(u^{\sharp})^{p-2}u^{\sharp}z\,dx}}\\\
&\geq\frac{\displaystyle{\int_{\Omega^{\sharp}}{\left|\nabla
z\right|}^{p}\,dx+\beta\int_{\partial\Omega^{\sharp}}z^{p}\,d\mathcal{H}^{n-1}(x)}}{\displaystyle{\int_{\Omega^{\sharp}}z^{p}\,dx}}\geq\lambda_{1,\beta}(\Omega^{\sharp}).\end{split}$
∎
## 5 Conclusions
We have been able to extend the results obtained for the Laplacian to the
$p$-Laplacian. Many problems remain open, such as
Open Problem In the assumptions of Theorem 1.2, does the point-wise comparison
hold also for $p>\frac{n}{n-1}$?
We have already observed in the corollary 3.1 that if $p\geq n$ we have an
estimate on the $L^{p}$ norms of $u$ and $v$. Can we generalize this estimate
also for $q\neq p$? We know for sure that for $q=\infty$ this can’t be done,
as it can be seen in the following example.
###### Example 5.1.
Let $\Omega\subseteq\operatorname{\mathbb{R}}^{n}$ be the union of two
disjoint balls, $B_{1}$ and $B_{r}$ with radii 1 and $r$ respectively. We
choose $\displaystyle{\beta<\left(\frac{n-1}{p-1}\right)^{p-1}}$ with $p\neq
n$, and we fix $f=1$ on $B_{1}$ and $f=0$ on $B_{r}$. Both $u$ and $v$ can be
explicitly computed. We have
${\left\|u\right\|}_{\infty}-{\left\|v\right\|}_{\infty}=Cr^{n}+o(r^{n})$,
where $C$ is a positive constant.
_Proof._ We want an explicit expression of $u$ and $v$ respectively. Starting
from $u$, it is a solution to
$\begin{cases}-\text{div}({\left|\nabla u\right|}^{p-2}\nabla u)=f&\text{ in
}\Omega\\\ {\left|\nabla u\right|}^{p-2}\displaystyle{\frac{\partial
u}{\partial\nu}}+\beta{\left|u\right|}^{p-2}u=0&\text{ on
}\partial\Omega.\end{cases}$
with $f\rvert_{B_{1}}=1$ and $f\rvert_{B_{r}}=0$.
It’s clear that $u\rvert_{B_{r}}=0$ and $u(x)=u({\left|x\right|})$ it’s radial
on $B_{1}$.
So the equation (1) becomes
$s^{n-1}\Delta_{p}u(s)=\frac{d}{ds}\left(s^{n-1}{\left|u^{\prime}(s)\right|}^{p-2}u^{\prime}(s)\right)$
and then
$\displaystyle\frac{d}{ds}\left(s^{n-1}{\left|u^{\prime}(s)\right|}^{p-2}u^{\prime}(s)\right)$
$\displaystyle=s^{n-1}\Delta_{p}u(s)=-s^{n-1}$ $\displaystyle
s^{n-1}{\left|u^{\prime}(s)\right|}^{p-2}u^{\prime}(s)$
$\displaystyle=-\frac{s^{n}}{n}+c.$
We set $c=0$, in order to have a $C^{1}$-solution.
${\left|u^{\prime}(s)\right|}^{p-2}u^{\prime}(s)=-\frac{s}{n}\implies
u^{\prime}(s)=-\frac{s^{\frac{1}{p-1}}}{n^{\frac{1}{p-1}}}\qquad\alpha=\frac{1}{p-1}.$
If we integrate, we obtain
$u(s)=-\frac{p-1}{n^{\alpha}p}s^{\frac{p}{p-1}}+A.$
The Robin boundary conditions become
${\left|u^{\prime}(1)\right|}^{p-2}u^{\prime}(1)+\beta u(1)^{p-1}=0\quad(u\geq
0),$
now we can compute the value of $A$
$-\frac{1}{n}+\beta\left(-\frac{p-1}{n^{\alpha}p}+A\right)^{p-1}=0\implies
A=\frac{1}{(n\beta)^{\alpha}}+\frac{p-1}{n^{\alpha}p}.$
So
$u(s)=\frac{p-1}{n^{\alpha}p}\left(1-s^{\frac{p}{p-1}}\right)+\frac{1}{(n\beta)^{\alpha}}.$
As $u$ is decreasing, we have
${\left\|u\right\|}_{\infty}=u(0)=\frac{p-1}{n^{\alpha}p}+\frac{1}{(n\beta)^{\alpha}}.$
Now, let us compute $v(s)$. We will do this firstly for $s\in(0,1)$, then for
$s\in(1,\overline{r})$ where $\overline{r}=(1+r^{n})^{\frac{1}{n}}$ is
determined by the condition ${\left|\Omega\right|}=|\Omega^{\sharp}|$.
Let $s<1$
$\frac{d}{ds}\left(s^{n-1}{\left|v^{\prime}(s)\right|}^{p-2}v^{\prime}(s)\right)=-s^{n-1}$
${\left|v^{\prime}(s)\right|}^{p-2}v^{\prime}(s)=-\frac{s}{n}\implies
v^{\prime}(s)=-\frac{s^{\frac{1}{p-1}}}{n^{\alpha}}$
$v(s)=-\frac{p-1}{n^{\alpha}p}s^{\frac{p}{p-1}}+B.$
Now we can’t determine $B$ as before, as $v$ is not identically 0 in the
anulus $B_{\overline{r}}\backslash B_{1}$.
Let $s>1$ and $p\neq n$
$\frac{d}{ds}\left(s^{n-1}{\left|v^{\prime}(s)\right|}^{p-2}v^{\prime}(s)\right)=0$
${\left|v^{\prime}(s)\right|}^{p-2}v^{\prime}(s)=\frac{C}{s^{n-1}}$
by imposing the continuity of the derivative for $s=1$, we obtain that
$C=-1/n$
$\displaystyle v^{\prime}(s)$
$\displaystyle=-\frac{s^{-\frac{n-1}{p-1}}}{n^{\alpha}},$ $\displaystyle v(s)$
$\displaystyle=-\frac{p-1}{n^{\alpha}(p-n)}s^{\frac{p-n}{p-1}}+D,$
and by Robin conditions
$\displaystyle{\left|v^{\prime}(\overline{r})\right|}^{p-2}v^{\prime}(\overline{r})+\beta
v(\overline{r})^{p-1}=0,$
$\displaystyle-\frac{\overline{r}^{-n-1}}{n}+\beta\left(-\frac{p-1}{n^{\alpha}(p-n)}\overline{r}^{\frac{p-n}{p-1}}+D\right)^{(p-1)}=0,$
$\displaystyle
D=\frac{1}{(n\beta)^{\alpha}}\overline{r}^{-\frac{n-1}{p-1}}+\frac{p-1}{n^{\alpha}(p-n)}\overline{r}^{\frac{p-n}{p-1}}.$
By imposing the continuity of $v$ for $s=1$, we have
$B=\frac{p-1}{n^{\alpha}p}+\frac{\overline{r}^{-\frac{n-1}{p-1}}}{(n\beta)^{\alpha}}+\frac{p-1}{n^{\alpha}(p-n)}\left(\overline{r}^{\frac{p-n}{p-1}}-1\right)$
that is to say
$v(s)=\begin{cases}u(s)+\frac{1}{(n\beta)^{\alpha}}\left(\overline{r}^{-\frac{n-1}{p-1}}-1\right)+\frac{p-1}{n^{\alpha}(p-n)}\left(\overline{r}^{\frac{p-n}{p-1}}-1\right)&\text{
if }s<1\\\
\frac{1}{(n\beta)^{\alpha}}\overline{r}^{-\frac{n-1}{p-1}}+\frac{p-1}{n^{\alpha}(p-n)}\left(\overline{r}^{\frac{p-n}{p-1}}-s^{\frac{p-n}{p-1}}\right)&\text{
if }1<s<\overline{r}\end{cases}$
For convenience’s sake, we set
$\displaystyle{h=\frac{1}{(n\beta)^{\alpha}}\left(\overline{r}^{-\frac{n-1}{p-1}}-1\right)+\frac{p-1}{n^{\alpha}(p-n)}\left(\overline{r}^{\frac{p-n}{p-1}}-1\right)}$.
So we have
${\left\|v\right\|}_{L^{\infty}(\Omega^{\sharp})}={\left\|v\right\|}_{L^{\infty}(B_{1})}={\left\|u\right\|}_{L^{\infty}(\Omega)}+h=u(0)+h$
By using Taylor expansion of the function $(1+r^{n})^{\delta}$ we get
$h=\left(-\frac{1}{(n\beta)^{\alpha}}\frac{n-1}{n(p-1)}+\frac{1}{n^{\alpha+1}}\right)r^{n}+o(r^{n}),$
so, if we choose $\beta<\left(\frac{n-1}{p-1}\right)^{p-1}$, we get
${\left\|v\right\|}_{L^{\infty}(\Omega^{\sharp})}={\left\|u\right\|}_{L^{\infty}(\Omega)}-Cr^{n}+o(r^{n})\text{
where }C>0.$
Next example 5.2 is a counterexample to the corollary 3.1 in the case $n>p$.
###### Example 5.2.
Let $\Omega\subseteq\operatorname{\mathbb{R}}^{n}$, $p<n$ be the union of two
disjoint balls $B_{1}$ and $B_{r}$ with radii 1 and $r$ respectively. We
choose $\displaystyle{\beta\leq\left(\frac{n-p}{p(p-1)}\right)^{p-1}}$ and we
fix $f=1$ on $B_{1}$ and $f=0$ on $B_{r}$. Both $u$ and $v$ can be explicitly
computed. We have
${\left\|u\right\|}_{p}^{p}-{\left\|v\right\|}_{p}^{p}=Cr^{n}+o(r^{n})$, where
$C$ is a positive constant.
_Proof._ Let us consider the Taylor expansion of $(1+y)^{p}$, we get
${\left\|v\right\|}^{p}_{L^{p}(B_{1})}=\int_{B_{1}}(u+h)^{p}={\left\|u\right\|}^{p}_{L^{p}(B_{1})}+p{\left\|u\right\|}^{p-1}_{L^{p-1}(B_{1})}h+o(r^{n})$
Moreover
${\left\|v\right\|}^{p}_{L^{p}(B_{\overline{r}}\backslash
B_{1})}=\frac{\omega_{n}}{(n\beta)^{\alpha p}}r^{n}+o(r^{n})$
as if $1<s<\overline{r}$
$\frac{1}{(n\beta)^{\alpha}}\overline{r}^{-\frac{n-1}{p-1}}\leq
v(s)\leq\frac{1}{(n\beta)^{\alpha}}\overline{r}^{-\frac{n-1}{p-1}}+\frac{p-1}{n^{\alpha}(p-n)}\left(\overline{r}^{\frac{p-n}{p-1}}-1\right)$
thus
$v(s)=\frac{1}{(n\beta)^{\alpha}}+O(r^{n})$
and by integration we obtain the value of the norm in
$L^{p}(B_{\overline{r}}\backslash B_{1})$.
So
${\left\|v\right\|}^{p}_{L^{p}(\Omega^{\sharp})}={\left\|v\right\|}^{p}_{L^{p}(B_{1})}+{\left\|v\right\|}^{p}_{L^{p}(B_{\overline{r}}\backslash
B_{1})}={\left\|u\right\|}^{p}_{L^{p}(\Omega)}+p{\left\|u\right\|}^{p-1}_{L^{p-1}(B_{1})}h+\frac{\omega_{n}}{(n\beta)^{\alpha
p}}r^{n}+o(r^{n})$
and recalling that
$h=\left(-\frac{1}{(n\beta)^{\alpha}}\frac{n-1}{n(p-1)}+\frac{1}{n^{\alpha+1}}\right)r^{n}+o(r^{n})$
we get
${\left\|v\right\|}^{p}_{L^{p}(\Omega^{\sharp})}={\left\|u\right\|}^{p}_{L^{p}(\Omega)}+\left[p{\left\|u\right\|}^{p-1}_{L^{p-1}(B_{1})}\left(-\frac{1}{(n\beta)^{\alpha}}\frac{n-1}{n(p-1)}+\frac{1}{n^{\alpha+1}}\right)+\frac{\omega_{n}}{(n\beta)^{\alpha
p}}\right]r^{n}+o(r^{n}).$
We have to understand whether
$p{\left\|u\right\|}^{p-1}_{L^{p-1}(B_{1})}\left(-\frac{1}{(n\beta)^{\alpha}}\frac{n-1}{n(p-1)}+\frac{1}{n^{\alpha+1}}\right)+\frac{\omega_{n}}{(n\beta)^{\alpha
p}}<0.$ (30)
If we choose $\displaystyle{\beta<\left(\frac{n-1}{p-1}\right)^{p-1}}$ we have
$\displaystyle{-\frac{1}{(n\beta)^{\alpha}}\frac{n-1}{n(p-1)}+\frac{1}{n^{\alpha+1}}<0}$.
In order to have (30), we need
$\displaystyle{\left\|u\right\|}^{p-1}_{L^{p-1}(B_{1})}$
$\displaystyle>\frac{\omega_{n}}{(n\beta)^{\alpha
p}}\left[\frac{n(p-1)n^{\alpha}\beta^{\alpha}}{p(n-1)-p\beta^{\alpha}(p-1)}\right]$
$\displaystyle{\left\|u\right\|}^{p-1}_{L^{p-1}(B_{1})}$
$\displaystyle>\frac{\omega_{n}}{(n\beta)^{\alpha(p-1)}}\left[\frac{n(p-1)}{p(n-1)-p\beta^{\alpha}(p-1)}\right].$
If we show that
$\left[\frac{n(p-1)}{p(n-1)-p\beta^{\alpha}(p-1)}\right]\leq 1$ (31)
then
$u(s)>\frac{1}{(n\beta)^{\alpha}}\implies{\left\|u\right\|}^{p-1}_{L^{p-1}(B_{1})}>\frac{\omega_{n}}{(n\beta)^{\alpha(p-1)}}.$
We just have to verify (31)
$\begin{split}\left[\frac{n(p-1)}{p(n-1)-p\beta^{\alpha}(p-1)}\right]\leq
1&\iff n(p-1)\leq p(n-1)-p\beta^{\alpha}(p-1)\\\ &\iff
p-n\leq-p(p-1)\beta^{\alpha}<0\quad\text{ (if and only if $p<n!$)}\\\
&\iff\beta\leq\left(\frac{n-p}{p(p-1)}\right)^{p-1}\end{split}$
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# Joint quasiprobability distribution on the measurement outcomes of MUB-
driven operators
H S Smitha Rao1, Swarnamala Sirsi1 , Karthik Bharath2 1 Yuvaraja’s college,
University of Mysore, Mysuru, India 2 University of Nottingham, Nottingham,
U.K<EMAIL_ADDRESS>
## Abstract
We propose a method to define quasiprobability distributions for general
spin-$j$ systems of dimension $n=2j+1$, where $n$ is a prime or power of
prime. The method is based on a complete set of orthonormal commuting
operators related to Mutually Unbiased Bases which enable (i) a
parameterisation of the density matrix and (ii) construction of measurement
operators that can be physically realised. As a result we geometrically
characterise the set of states for which the quasiprobability distribution is
non-negative, and can be viewed as a joint distribution of classical random
variables assuming values in a finite set of outcomes. The set is an
$(n^{2}-1)$-dimensional convex polytope with $n+1$ vertices as the only pure
states, $n^{n+1}$ number of higher dimensional faces, and $n^{3}(n+1)/2$
edges.
## 1 Introduction
Expectation values of quantum mechanical observables can be studied on the
continuous phase space using quasiprobability distributions (QPDs). For spin
systems, prominent ones include the Wigner [1] and Margenau-Hill [2] QPDs; in
quantum optics, the Glauber and Sudarshan[3] QPD for quantum radiation is
used. These behave like probability density functions on the continuous phase
space, but, by their very definition, can assume negative values in certain
regions of the phase space. The issue also afflicts versions of the Wigner QPD
on discretised phase space[4, 5], tailored for finite-dimensional quantum
systems. Occurrence of negative values of QPDs, that can be attributed to non-
commutativity of quantum mechanical observables, is used as a signature of
non-classicality [6, 7, 8, 9] and is exploited profitably in various quantum
computational tasks[10, 11, 12, 13].
For finite-dimensional quantum systems, QPDs unrelated to the phase space view
have also been proposed [14, 8, 9] with a view towards ascribing joint
probabilities to a finite set of measurement outcomes using measurement
operators. The choice of measurement operators then plays an important role in
examining properties of the corresponding joint QPD: the form of the QPD can
be used to identify the subset of states for which the QPD behaves, and can be
interpreted, as a classical joint distribution. It is desirable to consider
measurement operators that are related to a complete set of Mutually Unbiased
Bases (MUBs)[15]; this is the approach adopted while defining the discrete
Wigner QPD [4, 5, 10]. For qubit or spin-1/2 systems, the Pauli operators can
be used since their eigenbases constiute a complete set of MUBs. In higher
prime or power-of-prime dimensions, the eigenbases of the generalised Pauli
operators[16] can be chosen as the MUB basis vectors, but the operators
themselves are not observables.
The two-fold purpose of this paper is: (i) to construct QPDs for
$n$-dimensional quantum systems, or equivalently spin-$j$ systems with
$n=2j+1$, where $n$ is a prime or power of a prime, using $n+1$ measurement
operators $M_{1},\ldots,M_{n+1}$ related to a complete set of MUBs; (ii)
obtain a geometric description of the set of states for which the constructed
QPD corresponds to a valid joint distribution on measurement outcomes. The two
objectives are achieved by employing an orthonormal operator basis given by
the complete set of commuting operators (CSCOs)
$\mathcal{A}=\\{\hat{\alpha}_{1},\ldots,\hat{\alpha}_{n^{2}-1}\\}$ proposed in
our earlier work[17]; the operators, through their relation to MUBs, can be
partitioned into $n+1$ disjoint subsets
$\mathcal{A}_{1},\ldots,\mathcal{A}_{n+1}$, such that each $\mathcal{A}_{i}$
contains exactly $n-1$ operators that commute and permit simultaneous
measurements. This enables us to construct the measurements $M_{i}$ as linear
combinations of the commuting operators in $\mathcal{A}_{i}$. A striking
feature of the constructed QPD is that when restricted to each
$\mathcal{A}_{i}$ it is a valid joint distribution of corresponding
measurement outcomes for _any_ state.
Our method of construction of the QPD uses a characteristic function defined
for a density matrix using a particular operator ordering of the CSCOs. As a
consequence, for states for which the QPD is non-negative it can effectively
be viewed as a joint distribution of classical random variables
$X_{1},\ldots,X_{n^{2}-1}$ assuming values on a finite set of outcomes.
Relatedly, bivariate and trivariate probability distributions for outcomes
from Pauli measurements corresponding to different definitions of
characteristic functions have been derived for qubits [18, 19], and
characteristic functions have also been used to derive trivariate moments for
arbitrary spin-$j$ systems[20, 21].
The use of CSCOs is well-motivated: unlike general $\mathfrak{su}(n)$
generators (e.g. Gell-Mann matrices) they are better suited for physical
implementation and interpretation owing to their relationship with MUBs; they
enable us to uncover the geometry of the set of states for which the QPD is
non-negative. The set of such states forms a regular convex polytope on
$n(n+1)$ vertices, $n^{n+1}$ number of higher dimensional faces, and
$n^{3}(n+1)/2$ edges with each vertex on the surface of the Bloch ball of
radius $\sqrt{n-1}$ which represents the set of all states. Interestingly, our
method of using MUBs results in the convex polytope which coincides with the
polytope identified by Galvão [10, 6] using probability coordinates obtained
from definition of the discrete Wigner QPD under a phase space description. We
briefly comment on this relationship (Section 5.1), and leave detailed
investigations for future work.
We first review quantum characteristic functions (Section 2). Then, starting
with a brief description of how the program is carried out for spin-1/2
systems (Section 3), we provide a detailed description of the QPD
construction, corresponding geometry and physical realisation for spin-1
systems (Sections 4 and 5), and provide some details for spin-3/2 systems
(Section 6). Inspection of these two cases will reveal how the methodology
extends to arbitrary spin-$j$ systems (Section 7).
## 2 Quantum characteristic functions
For a $k$-dimensional classical random vector $\vec{Y}$ with joint
distribution $p(\vec{y})=p(y_{1},\ldots,y_{k})$, the Fourier transform
$\phi(\vec{t})=\int
e^{i\vec{t}\cdot\vec{y}}p(\vec{y})\text{d}\vec{y}\quad\text{or}\quad\phi(\vec{t})=\sum_{\vec{y}}e^{i\vec{t}\cdot\vec{y}}p(\vec{y})$
is referred to as its characteristic function, depending on whether
$p(\vec{y})$ is a continuous or discrete distribution. A characteristic
function uniquely determines $p$ through its inverse Fourier transform. If we
view a vector $\vec{X}=(X_{1},\ldots,X_{k})$ of measurement operators $X_{k}$
as a quantum analogue of a classical random vector, noncommutativity implies
that there are multiple ways to define $e^{\vec{t}\cdot\vec{X}}$, and hence
the characteristic function $\phi$ [22]. This problem is typically addressed
using symmetrisation rules, popular amongst which are the Margenau-Hill [2]
rule, which, for example when $k=3$, proposes
$\displaystyle
e^{i\vec{t}\cdot\vec{X}}\longrightarrow\frac{1}{3!}\sum_{\pi\in\Pi_{3}}\Big{[}e^{it_{\pi(1)}{X_{\pi(1)}}}e^{it_{\pi(2)}{X_{\pi(2)}}}e^{it_{\pi(3)}{X_{\pi(3)}}}\Big{]},$
where $\Pi_{k}$ is the symmetric group of permutations of
$[k]=\\{1,\ldots,k\\}$ with bijections $\pi:[k]\to[k]$, and the Wigner-Weyl
[23] rule, which proposes $e^{i\vec{t}\cdot\vec{X}}\to
e^{i\vec{t}\cdot\vec{X}}$ for any fixed chosen ordering of
$X_{1},\ldots,X_{k}$. Note that $\vec{X}$ need not be a POVM for such a
definition of $\phi$. For a chosen symmetrisation rule the quantum
characteristic function associated with a state $\rho$ and operators $\vec{X}$
is then defined as
$\phi(\vec{t})=\Tr[\rho e^{i\vec{t}\cdot\vec{X}}].$
Irrespective of the symmetrisation rule, the map
$\vec{t}\mapsto\phi(\vec{t})$, unlike the situation with classical random
variables, is not guaranteed to be the Fourier transform of a joint
probability distribution $p(\vec{x})$ on measurement outcomes for every state
$\rho$111This is consequence of Bochner’s theorem: $\phi(\vec{t})$ is a valid
characteristic function if and only if for every $r$-tuple
$(\vec{t}_{1},\ldots,\vec{t}_{r})$ the $r\times r$ matrix with entries
$\phi(\vec{t}_{i}-\vec{t}_{j}),i,j=1,\dots,r$ is non-negative definite and
Hermitian. See Example 4.1 in [24] for a detailed discussion of the issue..
However, $\phi$ can be inverted to obtain a QPD on the measurement outcomes.
We will use the QPD arising from the Margenau-Hill symmetrisation rule using
measurement operators constructed using the CSCOs.
## 3 Spin-1/2 system
It is instructive to first describe our construction for the spin-1/2 case
with Pauli operators. The density matrix assumes the form
$\rho(\vec{\theta})=\frac{1}{2}(\mathbb{I}_{2}+\vec{\sigma}\cdot\vec{\theta})$
where $\vec{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ with $\sigma_{i}$,
$i=1,2,3$ denoting the well-known Pauli operators, and the components of Bloch
vector $\vec{\theta}$ are such that
$\theta_{i}=Tr(\rho(\vec{\theta})\sigma_{i})$. The constraint
$Tr[\rho(\vec{\theta})^{2}]\leq 1$ implies that
$\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}\leq 1$, with equality attained
only for pure states. The set of density matrices for qubits is then the Bloch
ball
$\mathcal{B}^{2}(\vec{\theta}):=\\{\vec{\theta}:\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}\leq
1\\}$ with the surface of the sphere corresponding to pure states.
It is known that the optimum measurement operators based on spin-1/2 MUBs can
be constructed and physically realized using the Stern-Gerlach experimental
setup. In this case, a particle having magnetic moment $\vec{\mu}$ is passed
through an inhomogeneous magnetic field $\vec{B}$. Here the potential energy
associated with the particle is $\mathcal{H}=-\vec{\mu}.\vec{B}$, where
$\vec{\mu}$ is proportional to spin. When the magnetic field is oriented along
the z-direction, one can measure the expectation value of $\sigma_{3}$. The
corresponding Hamiltonian is $\mathcal{H}=h_{0}\mathbb{I}+h_{3}\sigma_{3}$,
whose expectation value results in
$\langle\mathcal{H}\rangle=\Tr(\rho\mathcal{H})=h_{0}+\theta_{3}h_{3}$. Then,
the observable $\sigma_{1}$ can also be measured using the same apparatus if
its diagonal basis has the same form as $\sigma_{3}$. Experimentally this
corresponds to the application of magnetic field along x-direction. Similarly
is the measurement of $\sigma_{2}$. This results in the complete determination
of parameters characterizing the spin-1/2 density matrix.
### 3.1 Quasiprobability distribution and geometric description of non-
negative region
The eigenvalues of each of the three Pauli operators are $\pm 1$. Accordingly,
consider three classical random variables $X_{1},X_{2},X_{3}$ each of which
assumes values in $\mathcal{X}=\\{1,-1\\}$. Using the Margenau-Hill
symmetrisation rule on Pauli operators define
$\phi(t_{1},t_{2},t_{3})=\frac{1}{3!}\Tr[\rho(\vec{\theta})(\beta_{123}+\beta_{132}+\beta_{213}+\beta_{231}+\beta_{312}+\beta_{321})],$
(1)
where
$\beta_{abc}=e^{it_{a}\sigma_{a}}e^{it_{b}\sigma_{b}}e^{it_{c}\sigma_{c}}$
with $a,b,c=\\{1,2,3\\}$. Further simplification using
$\displaystyle e^{i\sigma_{k}t_{k}}$ $\displaystyle=I_{2}\cos
t_{k}+i\sigma_{k}\sin t_{k},\thickspace\sigma^{2}_{k}=\mathbb{I},\quad
k=1,2,3;$ (2) $\displaystyle\\{\sigma_{a},\sigma_{b}\\}$
$\displaystyle=\delta_{ab}\sigma_{c},\
[\sigma_{a},\sigma_{b}]=2i\epsilon_{abc}\sigma_{c},$
implies that $\phi$ can written as
$\phi(t_{1},t_{2},t_{3})=\cos t_{1}\cos t_{2}\cos t_{3}+i\theta_{1}\sin
t_{1}\cos t_{2}\cos t_{3}+\\\ i\theta_{2}\cos t_{1}\sin t_{2}\cos
t_{3}+i\theta_{3}\cos t_{1}\cos t_{2}\sin t_{3}.$
In [25] $\phi$ was inverted to obtain the QPD
$p(x_{1},x_{2},x_{3})=\frac{1}{8}(1+x_{1}\theta_{1}+x_{2}\theta_{2}+x_{3}\theta_{3}),\quad
x_{i}\in\mathcal{X},i=1,2,3.$
The QPD is non-negative only for those states
$\vec{\theta}=(\theta_{1},\theta_{2},\theta_{3})^{T}$ that, in addition to
belonging to the Bloch ball $\mathcal{B}^{2}(\vec{\theta})$, satisfy the
inequality $|\theta_{1}|+|\theta_{2}|+|\theta_{3}|\leq 1$. The inequality
characterises a octahedron in $\mathbb{R}^{3}$ with centre at $(0,0,0)$ within
the Bloch ball with six vertices $(\pm 1,0,0)$, $(0,\pm 1,0)$, $(0,0,\pm 1)$
on the surface of the ball. Figure 1 provides a graphical representation.
Thus for every state $\vec{\theta}=(\theta_{1},\theta_{2},\theta_{3})^{T}$
inside the octahedron the function $p(x_{1},x_{2},x_{3})$ is the joint
distribution of classical random variables $(X_{1},X_{2},X_{3})^{T}$, and we
can thus _prescribe_ and accordingly interpret $p(x_{1},x_{2},x_{3})$ as joint
probabilities of outcomes of non-commuting Pauli measurement operators.
Figure 1: Within the Bloch ball $\mathcal{B}^{2}(\vec{\theta})$, the
octahedron $|\theta_{1}|+|\theta_{2}|+|\theta_{3}|\leq 1$ with vertices on the
surface contains states for which the QPD corresponds to a joint probability
distribution on measurement outcomes.
## 4 Pauli-like complete set of commuting operators for spin-1 system
A spin-1 density matrix is characterized by eight independent parameters.
Extending the methodology used for spin-1/2 system requires the representation
of density matrix in a matrix basis which mimics the role played by Pauli-like
operators, whose eigenstates form a complete set of MUBs. For spin-1 and
higher-level spin systems it can be shown that using arbitrary
$\mathfrak{su}(n)$ Lie algebra generators do not necessarily lead to a
polytope similar to the spin-1/2 case. Moreover, their physical implementation
is not straightforward. We instead consider the MUB-driven operators
$\mathcal{A}=\\{\hat{\alpha}_{j},j=1,\ldots,8\\}$ proposed in [17]:
$\displaystyle\hat{{\alpha}}_{1}$
$\displaystyle={\sqrt{\frac{3}{2}}}\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&-1\\\
\end{array}\right),\qquad\hat{\alpha}_{2}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}1&0&0\\\
0&-2&0\\\ 0&0&1\\\ \end{array}\right),$ $\displaystyle\hat{\alpha}_{3}$
$\displaystyle={\frac{1}{\sqrt{2}}}\left(\begin{array}[]{ccc}0&-i\omega&i\omega^{2}\\\
i\omega^{2}&0&-i\omega\\\ -i\omega&i\omega^{2}&0\\\
\end{array}\right),\quad\hat{\alpha}_{4}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&-\omega&-\omega^{2}\\\
-\omega^{2}&0&-\omega\\\ -\omega&-\omega^{2}&0\\\ \end{array}\right),$
$\displaystyle\hat{\alpha}_{5}$
$\displaystyle={\frac{1}{\sqrt{2}}}\left(\begin{array}[]{ccc}0&-i&i\omega^{2}\\\
i&0&-i\omega^{2}\\\ -i\omega&i\omega&0\\\
\end{array}\right),\quad\hat{\alpha}_{6}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&-1&-\omega^{2}\\\
-1&0&-\omega^{2}\\\ -\omega&-\omega&0\\\ \end{array}\right),$
$\displaystyle\hat{\alpha}_{7}$
$\displaystyle={\frac{1}{\sqrt{2}}}\left(\begin{array}[]{ccc}0&-i\omega^{2}&i\omega^{2}\\\
i\omega&0&-i\\\ -i\omega&i&0\\\
\end{array}\right),\quad\hat{\alpha}_{8}={\frac{1}{\sqrt{2}}}\left(\begin{array}[]{ccc}0&-\omega^{2}&-\omega^{2}\\\
-\omega&0&-1\\\ -\omega&-1&0\\\ \end{array}\right).$
where $\omega=e^{2\pi i/3}$. Since
$\Tr(\hat{\alpha}_{i}\hat{\alpha}_{j})=3\delta_{ij}$, $\mathcal{A}$ is an
orthonormal operator basis for Hermitian matrices; it is a mutually disjoint,
maximally commuting set since it can be partitioned into four sets
$\mathcal{A}_{1}=\\{\hat{\alpha}_{1},\hat{\alpha}_{2}\\},\mathcal{A}_{2}=\\{\hat{\alpha}_{3},\hat{\alpha}_{4}\\},\mathcal{A}_{3}=\\{\hat{\alpha}_{5},\hat{\alpha}_{6}\\},\mathcal{A}_{4}=\\{\hat{\alpha}_{7},\hat{\alpha}_{8}\\}$
such that the operators within each $\mathcal{A}_{i}$
commute:$(\hat{\alpha}_{j},\hat{\alpha}_{j+1})$ commute for $j=1,3,5,7$. This
is the reason for them being referred to as a complete set of commuting
operators (CSCOs). They are Pauli-like in the sense that their eigenbases are
MUBs; we refer to [17] for details. The density matrix can be expressed as
$\rho(\vec{\theta})=\frac{1}{3}[\mathbb{I}_{3}+\sum_{j=1}^{8}\theta_{j}\hat{\alpha}_{j}],\quad\theta_{j}=\Tr[\rho(\vec{\theta})\hat{\alpha}_{j}],\thinspace
j=1,\ldots,8.$
The condition $\Tr[\rho(\vec{\theta})^{2}]\leq 1$ implies that
$\sum_{j=1}^{8}\theta^{2}_{j}\leq 2$, and a similar seven-dimensional Bloch
ball $\mathcal{B}^{7}(\vec{\theta})$ of radius $\sqrt{2}$ in eight dimensions
emerges. Bounds on the parameters are given by
$-\sqrt{\frac{3}{2}}\leq\theta_{j}\leq\sqrt{\frac{3}{2}}$ when $j=1,3,5,7$,
and $-{\sqrt{2}}\leq\theta_{j}\leq\frac{1}{\sqrt{2}}$ when $j=2,4,6,8$. Since
$\\{\hat{\alpha}_{j}\\}$ comprises of four sets of two commuting operators, we
have three pairs of eigenvalues as measurement outcomes shared between the
operators, denoted as tuples $z_{i}=(x_{i},y_{i}),i=1,2,3,4$, where
$z_{i}\in\mathcal{Z}=\left\\{\left(\sqrt{\frac{3}{2}},\sqrt{\frac{1}{2}}\right),\left(0,-\frac{2}{\sqrt{2}}\right),\left(-\sqrt{\frac{3}{2}},\sqrt{\frac{1}{2}}\right)\right\\}.$
## 5 Quasiprobabilities for spin-1 system
We consider four measurement operators $M_{i},i=1,2,3,4$, instead of
$n^{2}-1=8$, where each $M_{i}$ is defined using the two commuting operators
in $\mathcal{A}_{i}$. To our knowledge, such a construction is not possible
with other operator basis (for e.g. Gell-Mann matrices). For a fixed
$\vec{t}=(t_{1},\ldots,t_{8})$, let
$\displaystyle M_{1}$
$\displaystyle=\hat{\alpha}_{1}t_{1}+\hat{\alpha}_{2}t_{2},\quad
M_{2}=\hat{\alpha}_{3}t_{3}+\hat{\alpha}_{4}t_{4},$ $\displaystyle M_{3}$
$\displaystyle=\hat{\alpha}_{5}t_{5}+\hat{\alpha}_{6}t_{6},\quad
M_{4}=\hat{\alpha}_{7}t_{7}+\hat{\alpha}_{8}t_{8}.$
Unlike the case for spin-1/2 systems it is not straightforward to explicitly
compute the expression the characteristic function using the Margenau-Hill
symmetrization rule (or any rule for that matter) since a simplifying relation
such as (2) is unavailable.
Another advantage in using the CSCOs is that we can analyse the characteristic
function for the eight-dimensional QPD in a modular manner: characteristic
functions using pairs $M_{i},M_{j}$ can first be analysed and then for triples
$M_{i},M_{j},M_{k}$, which then enable a straightforward derivation of the
QPD. Relatedly, we can thus consider 4 bivariate random variables
$\vec{X}=((X_{1},Y_{1})^{T},(X_{2},Y_{2})^{T},(X_{3},Y_{3})^{T},(X_{4},Y_{4})^{T})^{T}$,
as opposed to a single 8-dimensional random vector. Each bivariate random
variable $(X_{i},Y_{i})^{T}$ assumes values
$z_{i}=(x_{i},y_{i})\in\mathcal{Z}$.
The operator $M_{1}$ is diagonal and Hermitian. Since the transformation from
$M_{i}$ to $M_{j}$ corresponds to a unitary transformation from one MUB basis
to another, we see that $M_{2},M_{3}$ and $M_{4}$ can also be diagonalised,
respectively, with unitary transformations
$\displaystyle U_{2}=\frac{1}{\sqrt{3}}\left(\begin{array}[]{ccc}1&1&1\\\
1&\omega&\omega^{2}\\\ 1&\omega^{2}&\omega\\\ \end{array}\right),\thinspace
U_{3}=\frac{1}{\sqrt{3}}\left(\begin{array}[]{ccc}1&\omega^{2}&1\\\
1&1&\omega^{2}\\\ 1&\omega&\omega\\\ \end{array}\right),\thinspace
U_{4}=\frac{1}{\sqrt{3}}\left(\begin{array}[]{ccc}1&\omega&1\\\
1&\omega^{2}&\omega^{2}\\\ 1&1&\omega\\\ \end{array}\right).$
For benefit of exposition we describe the construction in an incremental
fashion: aided by the decomposition of $\mathcal{A}$ into subsets containing
commuting operators and availability of explicit unitary transformations
between MUB bases, we detail how the four-dimensional marginal QPDs
(corresponding to any two $M_{i}$ and $M_{j}$) of the eight-dimensional QPD we
seek can be defined; the methodology extends to six-dimensional marginal QPDs
obtained using three operators. Consider the characteristic function
$\phi(t_{1},t_{2},t_{3},t_{4})=\frac{1}{2}\Tr[\rho(\vec{\theta})(e^{iM_{1}}e^{iM_{2}}+e^{iM_{2}}e^{iM_{1}})]$
defined only using $M_{1}$ and $M_{2}$. Since
$\hat{\alpha}_{3}=U^{\dagger}_{2}\hat{\alpha}_{1}U_{2}$ and
$\hat{\alpha}_{4}=U^{\dagger}_{2}\hat{\alpha}_{2}U_{2}$, we have
$\phi(t_{1},t_{2},t_{3},t_{4})=\frac{1}{2}\Tr[\rho(\vec{\theta})(e^{iM_{1}}U^{\dagger}_{2}e^{iM^{d}_{2}}U_{2}+U^{\dagger}_{2}e^{iM^{d}_{2}}U_{2}e^{iM_{1}})],$
(3)
where $M^{d}_{2}=\hat{\alpha}_{1}t_{3}+\hat{\alpha}_{2}t_{4}$. Denote the
joint eigenbases of $\hat{\alpha}_{1}$ and $\hat{\alpha}_{2}$ (since they
commute) in $\mathcal{A}_{1}$ as $|z_{1}\rangle$, where
$z_{1}=(x_{1},y_{1})\in\mathcal{Z}$ is the corresponding eigenvalue pair. That
is, $z_{1}$ can assume three values in $\mathcal{Z}$ and $|z_{1}\rangle$ thus
generically denotes any of the three eigenvectors that are common to both
$\hat{\alpha}_{1}$ and $\hat{\alpha}_{2}$. Similarly consider $|z_{2}\rangle$
from $\mathcal{A}_{2}$. Following the technique used in [14], we have
$\displaystyle\phi(t_{1},t_{2},t_{3},t_{4})=\sum_{z_{1},z_{2}}e^{i(x_{1}t_{1}+y_{1}t_{2}+x_{2}t_{3}+y_{2}t_{4})}\text{Re}\\{\langle
z_{2}|U_{2}|z_{1}\rangle\langle
z_{1}|\rho(\vec{\theta})U^{\dagger}_{2}|z_{2}\rangle\\},$
where $\text{R}e\\{y\\}$ denotes the real part of the complex number $y$.
Observing the right hand side of $\phi$, we can thus, without explicitly
inverting the characteristic function, prescribe the four-dimensional QPD
$\displaystyle p(z_{1},z_{2})$ $\displaystyle=\text{Re}\\{\langle
z_{2}|U_{2}|z_{1}\rangle\langle
z_{1}|\rho(\vec{\theta})U^{\dagger}_{2}|z_{2}\rangle\\}$
$\displaystyle=\frac{1}{9}\left[1+x_{1}{\theta_{1}}+y_{1}{\theta_{2}}+x_{2}{\theta_{3}}+y_{2}{\theta_{4}}\right]$
following some algebra. The QPD above is the four-dimensional marginal of the
eight-dimensional QPD of interest. Using the unitary operators
$U_{j},j=2,3,4$, the other four-dimensional QPDs $p(z_{k},z_{j}),j,k=2,3,4$
with $j\neq k$ can then be obtained as
$p(z_{j},z_{k})=\text{Re}\\{\langle
z_{k}|U_{k}U^{\dagger}_{j}|z_{j}\rangle\langle
z_{j}|U_{j}\rho(\vec{\theta})U^{\dagger}_{k}|z_{k}\rangle\\}.$
The above program can be carried out for operators in
$\mathcal{A}_{1},\mathcal{A}_{2},\mathcal{A}_{3}$ in almost identical fashion
using operators $M_{1},M_{2},M_{3}$ to obtain the six-dimensional marginal QPD
$\displaystyle p(z_{1},z_{2},z_{3})=\frac{1}{3^{3}}$
$\displaystyle\left[1+x_{1}{\theta_{1}}+y_{1}{\theta_{2}}+x_{2}{\theta_{3}}+y_{2}{\theta_{4}}+x_{3}{\theta_{5}}+y_{3}{\theta_{6}}\right],$
using which the other six-dimensional marginals QPD can be derived using the
corresponding unitary operators. Finally, using all four measurement operators
$M_{i}=1,2,3,4$, the characteristic function for the eight-dimensional QPD can
written as
$\phi(\vec{t})=\frac{1}{4!}\sum_{\pi\in\Pi_{4}}\Tr\left[\rho(\vec{\theta})\left(\beta(\pi(1)\pi(2)\pi(3)\pi(4))\right)\right],$
where
$\beta(abcd)=e^{i\hat{M}_{a}}e^{i\hat{M}_{b}}e^{i\hat{M}_{c}}e^{i\hat{M}_{d}}$
and $a,b,c,d\in\\{1,2,3,4\\}$. Following the above steps with unitaries
$U_{2},U_{3},U_{4}$ enables the definition of the required eight-dimensional
QPD
$\phi(\vec{t})=\sum_{z_{1},z_{2},z_{3},z_{4}\in\mathcal{Z}}e^{i(x_{1}t_{1}+y_{1}t_{2}+x_{2}t_{3}+y_{2}t_{4}+x_{3}t_{5}+y_{3}t_{6}+x_{4}t_{7}+y_{4}t_{8})}p(z_{1},z_{2},z_{3},z_{4}),$
with
$\displaystyle{}p(z_{1},z_{2},z_{3},z_{4})=\frac{1}{3^{4}}$
$\displaystyle\left[1+x_{1}{\theta_{1}}+y_{1}{\theta_{2}}+x_{2}{\theta_{3}}+y_{2}{\theta_{4}}+x_{3}{\theta_{5}}+y_{3}{\theta_{6}}+x_{4}{\theta_{7}}+y_{4}{\theta_{8}}\right].$
The form of the QPD for a spin-1 system is similar to that for spin-1/2 case.
This is a consequence of employing Pauli-like CSCOs.
### 5.1 Geometric description
The set of states within the Bloch ball $\mathcal{B}^{7}(\vec{\theta})$ for
which the QPD is a joint distribution on eight classical random variables is
characterised by the linear inequality
$x_{1}{\theta_{1}}+y_{1}{\theta_{2}}+x_{2}{\theta_{3}}+y_{2}{\theta_{4}}+x_{3}{\theta_{5}}+y_{3}{\theta_{6}}+x_{4}{\theta_{7}}+y_{4}{\theta_{8}}\geq-1.$
The set is convex polytope in $\mathbb{R}^{8}$. The use of $4\times 3=12$ MUB
basis vectors used to construct the operators $M_{1},M_{2},M_{3}$ leads to the
polytope to have 12 vertices on the surface of $\mathcal{B}^{7}(\vec{\theta})$
that are the only pure states given by $1/\sqrt{2}$ times the coordinates
$\displaystyle(\sqrt{3},1,\vec{0}_{6}),(-\sqrt{3},1,\vec{0}_{6}),(\vec{0}_{6},\sqrt{3},1),(\vec{0}_{4},-\sqrt{3},1,\vec{0}_{2}),$
$\displaystyle(\vec{0}_{6},-\sqrt{3},1),(0,-2,\vec{0}_{6}),(\vec{0}_{3},-2,\vec{0}_{4}),(\vec{0}_{5},-2,\vec{0}_{2}),$
$\displaystyle(\vec{0}_{7},-2),(\vec{0}_{2},\sqrt{3},1,\vec{0}_{4}),(\vec{0}_{2},-\sqrt{3},1,\vec{0}_{4}),(\vec{0}_{4},\sqrt{3},1,\vec{0}_{2}),$
where $\vec{0}_{r}$ denotes a vector of $r$ zeroes. We observe that each
vertex has two vertices with which it subtends an angle $2\pi/3$ at the
origin, and is orthogonal to the rest; the two vertices are the ‘diametrically
opposite’ points on the Bloch sphere and are linked by an $SU(3)$ rotation.
The three vertices comprise an equilateral triangle; consider for example, the
triplet
$(\sqrt{3},1,\vec{0}_{6}),(-\sqrt{3},1,\vec{0}_{6}),(0,-2,\vec{0}_{6})$.
Consequently, we note that each vertex is formed by four mutually orthogonal
equilateral triangular planes in the Bloch sphere, which implies that the
polytope has $3^{4}=81$ faces.
There are no edges between vertices that correspond to vectors from the same
basis set, and edges of equal length are formed with every vertex outside the
basis set. Recall that the MUB comprises 4 basis sets containing 3 vectors
each. Thus a vertex from the first MUB basis can share an edge with $3^{2}$
vertices that are orthogonal to it and do not belong to the same MUB basis.
This is true for each vertex in the first MUB set and there are hence $3^{3}$
edges involving vertices from the first MUB set. In similar fashion, there are
$3^{2}(3-1)=3^{2}\times 2$ edges involving vertices from the second MUB set,
and so on. The total number of edges of the polytope is hence
$3^{3}(3+1)/2=54$.
As a geometric object, the convex polytope matches the one described in [10]
using the discrete Wigner QPD and in [26] using the so-called probability
coordinates $\vec{p}$. They are related in the following manner: elements of
the Bloch vector $\vec{\theta}$ that coordinatises the Bloch ball are
expectation values $\theta_{i}=\Tr(\rho(\vec{\theta})\hat{\alpha}_{i})$ of
CSCOs $\hat{\alpha}_{i},i=1,\ldots,8$; if instead projection operators
$\Pi_{1},\ldots,\Pi_{8}$ (along with the identity operator) corresponding to
the MUB basis vectors are used, their expectation values
$p_{i}=\Tr(\rho(\vec{\theta})\Pi_{i})$ constitute the probability coordinates
$\vec{p}$ with $\sum_{i=1}^{8}p_{i}+(1-\sum_{i=1}^{8}p_{i})=1$. In effect,
this amount to a specific reparameterisation of the map
$\theta\mapsto\rho(\theta)$, whose image is thus preserved. The projectors
$\Pi_{i}$, however, do not form an orthonormal set and it is not possible to
provide Bloch vector-like coordinates to a density matrix; moreover, it is not
straightforward to physically realise projection operators. In contrast, the
CSCOs comprise an orthonormal operator basis with which the density matrix is
provided interpretable coordinates, and, as will be seen shortly, can be
physically realised.
### 5.2 Physical realization of measurement operators
The Hamiltonian associated with the first MUB of spin-1 system is a linear
combination of $\hat{\alpha}_{1}$ and $\hat{\alpha}_{2}$. For spin systems, it
is natural to consider the irreducible spherical tensor operator basis
$\tau^{k}_{q}$ rank $k$ in the $2j+1$ dimensional spin space with projection
$q$ along the axis of quantization in the real 3-dimensional space. The matrix
elements of $\tau^{k}_{q}$ are
$\langle
jm^{\prime}|\tau^{k}_{q}(\vec{J})|jm\rangle=\sqrt{2k+1}C(jkj;mqm^{\prime}),$
where $C(jkj;mqm^{\prime})$ are the Clebsch–Gordan coefficients.
$\tau^{k}_{q}$s satisfy
$\Tr({\tau^{k^{\dagger}}_{q}\tau^{k^{{}^{\prime}}}_{q^{{}^{\prime}}}})=(2j+1)\,\delta_{kk^{{}^{\prime}}}\delta_{qq^{{}^{\prime}}},\quad\tau^{k^{\dagger}}_{q}=(-1)^{q}\tau^{k}_{-q},$
where the normalization has been chosen so as to be in agreement with Madison
convention. Then, $\hat{\alpha}_{1}=\tau^{1}_{0}=\sqrt{\frac{3}{2}}J_{z}$,
$\hat{\alpha}_{2}=\tau^{2}_{0}=\frac{3J^{2}_{z}-J^{2}}{\sqrt{2}}$. The
expectation values of $\hat{\alpha}_{1}$ and $\hat{\alpha}_{2}$ are
respectively associated with the first and second order moments of $J_{z}$ and
constitute experimentally measurable parameters. The Hamiltonian associated
with second MUB is obtained from the Fourier transformation of the first
basis. Similarly, transition from second to third MUB is obtained from one-
axis twisting $e^{-iS^{2}_{z}t}$ for $t=2\pi/3$ and from second to fourth MUB
for $t=4\pi/3$ [27]. Thus, the complete state determination results in
determining the parameters $\theta_{i}$, $i=1,\ldots 8$, which is optimally
done using the complete set of commuting operators $\hat{\alpha}_{i}$.
Experimentally this corresponds to the application of external electric
quadrupole field in addition to the dipole magnetic field in the Stern-Gerlach
setup. The Hamiltonian is diagonal in the first MUB has the form
${\mathcal{H}}_{1}=h_{0}\mathbb{I}+h_{1}{\hat{\alpha}}_{1}+h_{2}{\hat{\alpha}}_{2}$;
alternatively, in terms of spherical tensors,
$\mathcal{H}_{1}=\sum_{k}^{2}h^{k}_{0}{\tau^{k}}^{\dagger}_{0}.$ What one
experimentally measures is the expectation value of the Hamiltonian
$\Tr(\rho\mathcal{H}_{1})=h_{0}+h_{1}\theta_{1}+h_{2}\theta_{2}.$
Unitary transformations connecting different MUBs from the canonical basis is
parametrized by a single parameter $\phi$,
${\hat{U}}_{i}=e^{i{\mathcal{H}_{i}}\phi_{i}}$, where ${\mathcal{H}}_{i}$ is
the Hamiltonian diagonal in the $i^{th}$ basis and
$\phi_{i}=2\pi/3,4\pi/3,2\pi$ for $i=2,3,4$ respectively.
## 6 Quasiprobabilities for spin-3/2 system
For spin-3/2 systems of dimension $n=4$, and $n^{2}-1=15$, we briefly describe
the construction of a QPD and the ensuing geometric picture of states along
the lines of what is done for spin-1 systems. Following the method proposed in
[17], the CSCOs for spin-3/2 system is explicitly given by
$\displaystyle{\hat{\beta}}_{1}=\frac{1}{\sqrt{5}}\left(\begin{array}[]{cccc}3&0&0&0\\\
0&1&0&0\\\ 0&0&-1&0\\\ 0&0&0&-3\\\
\end{array}\right),\quad{\hat{\beta}}_{2}=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&-1&0&0\\\ 0&0&-1&0\\\ 0&0&0&1\\\
\end{array}\right),\quad{\hat{\beta}}_{3}=\frac{1}{\sqrt{5}}\left(\begin{array}[]{cccc}1&0&0&0\\\
0&-3&0&0\\\ 0&0&3&0\\\ 0&0&0&-1\\\ \end{array}\right),$
$\displaystyle{\hat{\beta}}_{4}=\frac{1}{\sqrt{5}}\left(\begin{array}[]{cccc}0&1&2&0\\\
1&0&0&2\\\ 2&0&0&1\\\ 0&2&1&0\\\
\end{array}\right),\quad{\hat{\beta}}_{5}=\left(\begin{array}[]{cccc}0&0&0&1\\\
0&0&1&0\\\ 0&1&0&0\\\ 1&0&0&0\\\
\end{array}\right),\quad{\hat{\beta}}_{6}=\frac{1}{\sqrt{5}}\left(\begin{array}[]{cccc}0&2&-1&0\\\
2&0&0&-1\\\ -1&0&0&2\\\ 0&-1&2&0\\\ \end{array}\right),$
$\displaystyle{\hat{\beta}}_{7}=\frac{1}{\sqrt{5}}\left(\begin{array}[]{cccc}0&-i&-2i&0\\\
i&0&0&-2i\\\ 2i&0&0&-i\\\ 0&2i&i&0\\\
\end{array}\right),\quad{\hat{\beta}}_{8}=\left(\begin{array}[]{cccc}0&0&0&-1\\\
0&0&1&0\\\ 0&1&0&0\\\ -1&0&0&0\\\
\end{array}\right),\quad{\hat{\beta}}_{9}=\frac{1}{\sqrt{5}}\left(\begin{array}[]{cccc}0&-2i&i&0\\\
2i&0&0&i\\\ -i&0&0&-2i\\\ 0&-i&2i&0\\\ \end{array}\right),$
$\displaystyle{\hat{\beta}}_{10}=\frac{1}{\sqrt{5}}\left(\begin{array}[]{cccc}0&-i&2&0\\\
i&0&0&-2\\\ 2&0&0&i\\\ 0&-2&-i&0\\\
\end{array}\right),\quad{\hat{\beta}}_{11}=\left(\begin{array}[]{cccc}0&0&0&i\\\
0&0&i&0\\\ 0&-i&0&0\\\ -i&0&0&0\\\
\end{array}\right),\quad{\hat{\beta}}_{12}=\frac{1}{\sqrt{5}}\left(\begin{array}[]{cccc}0&-2i&-1&0\\\
2i&0&0&1\\\ -1&0&0&2i\\\ 0&1&-2i&0\\\ \end{array}\right),$
$\displaystyle{\hat{\beta}}_{13}=\frac{1}{\sqrt{5}}\left(\begin{array}[]{cccc}0&1&-2i&0\\\
1&0&0&2i\\\ 2i&0&0&-1\\\ 0&-2i&-1&0\\\
\end{array}\right),\quad{\hat{\beta}}_{14}=\left(\begin{array}[]{cccc}0&0&0&i\\\
0&0&-i&0\\\ 0&i&0&0\\\ -i&0&0&0\\\
\end{array}\right),\quad{\hat{\beta}}_{15}=\frac{1}{\sqrt{5}}\left(\begin{array}[]{cccc}0&2&i&0\\\
2&0&0&-i\\\ -i&0&0&-2\\\ 0&i&-2&0\\\ \end{array}\right),$
where $\Tr(\hat{\beta}_{i}\hat{\beta}_{j})=4\delta_{ij}$. The density matrix
can be expressed as
$\rho(\vec{\theta})=\frac{1}{4}[\mathbb{I}_{4}+\sum_{j=1}^{15}\theta_{j}\hat{\beta}_{j}],$
where $\theta_{j}=\Tr[\rho(\vec{\theta})\hat{\beta}_{j}]$. Denote by
$\mathcal{A}=\cup_{i=1}^{5}\mathcal{A}_{i}$ where
$\mathcal{A}_{1}=\\{\hat{\beta}_{1},\hat{\beta}_{2},\hat{\beta}_{3}\\},\mathcal{A}_{2}=\\{\hat{\beta}_{4},\hat{\beta}_{5},\hat{\beta}_{6}\\},\mathcal{A}_{3}=\\{\hat{\beta}_{7},\hat{\beta}_{8},\hat{\beta}_{9}\\},\mathcal{A}_{4}=\\{\hat{\beta}_{10},\hat{\beta}_{11},\hat{\beta}_{12}\\},\mathcal{A}_{5}=\\{\hat{\beta}_{13},\hat{\beta}_{14},\hat{\beta}_{15}\\}$;
we note that there are thus five sets $\mathcal{A}_{i}$ each consisting of
three commuting operators. Here, four eigenvalue triples are shared between
the operators, $z_{i}=(w_{i},x_{i},y_{i})$, $i=1,2,3,4,5$ and are given by,
$z_{i}\in\mathcal{Z}=\left\\{\left(\sqrt{\frac{9}{5}},1,\frac{1}{\sqrt{5}}\right),\left(\frac{1}{\sqrt{5}},-1,-\sqrt{\frac{9}{5}}\right),\left(-\frac{1}{\sqrt{5}},-1,\sqrt{\frac{9}{5}}\right),\left(-\sqrt{\frac{9}{5}},1,-\frac{1}{\sqrt{5}}\right)\right\\}.$
For a fixed $\vec{t}=(t_{1},\ldots,t_{15})$, define the measurement operators
$\displaystyle M_{1}$
$\displaystyle=\hat{\beta}_{1}t_{1}+\hat{\beta}_{2}t_{2}+\hat{\beta}_{3}t_{3},\quad
M_{2}=\hat{\beta}_{4}t_{4}+\hat{\beta}_{5}t_{5}+\hat{\beta}_{6}t_{6},\quad
M_{3}=\hat{\beta}_{7}t_{7}+\hat{\beta}_{8}t_{8}+\hat{\beta}_{9}t_{9},$
$\displaystyle M_{4}$
$\displaystyle=\hat{\beta}_{10}t_{10}+\hat{\beta}_{11}t_{11}+\hat{\beta}_{12}t_{12},\quad
M_{5}=\hat{\beta}_{13}t_{13}+\hat{\beta}_{14}t_{14}+\hat{\beta}_{15}t_{15}.$
The unitary transformations $\hat{U}_{j},j=2,3,4,5$ take $M_{2}$, $M_{3}$,
$M_{4}$, $M_{5}$ to their diagonal form are known [28]. Along the lines of
what has was done for spin-1 system, the resulting characteristic function for
spin-3/2 system is
$\phi(\vec{t})=\frac{1}{5!}\sum_{\pi\in\Pi_{4}}\Tr\left[\rho(\vec{\theta})\left(\zeta(\pi(1)\pi(2)\pi(3)\pi(4)\pi(5))\right)\right],$
where $\zeta(abcde)=e^{iM_{a}}e^{iM_{b}}e^{iM_{c}}e^{iM_{d}}e^{iM_{e}}$ and
$a,b,c,d,e\in\\{1,2,3,4,5\\}$. Let
$\vec{\theta}=(\theta_{1},\ldots,\theta_{15})^{T}$ with
$\vec{\theta}=(\vec{\theta}_{1},\vec{\theta}_{2},\ldots,\vec{\theta}_{5})^{T}$
where $\vec{\theta}_{1}=(\theta_{1},\theta_{2},\theta_{3})$, and so on. Recall
that the triples $z_{i}=(w_{i},x_{i},y_{i}),i=1,\ldots,5$ can be viewed as
three-dimensional vectors. Following the steps laid out for spin-1 results in
the fifteen-dimensional QPD
$p(z_{1},z_{2},z_{3},z_{4},z_{5})=\frac{1}{1024}\left[1+z_{1}\cdot\vec{\theta}_{1}+\cdots+z_{5}\cdot\vec{\theta}_{5}\right],\quad
z_{i}\in\mathcal{Z},$
where $a\cdot b$ is the dot product between vectors $a$ and $b$. Each
measurement operator $M_{i},i=1,\ldots,5$ will engender a three-dimensional
marginal of the joint QPD.
The convex polytope within which the QPD is a valid joint distribution on
classical random variables is given by the inequality
$z_{1}\cdot\vec{\theta}_{1}+\cdots+z_{5}\cdot\vec{\theta}_{5}\geq-1$. Along
the line of reasoning used for the spin-1 case, we note that the polytope has
$5\times 4=20$ vertices; each vertex is equidistant from the 3 other vertices
within the MUB set, but does not share an edge with any of them; it shares an
edge, and is orthogonal, with the rest of the vertices. The polytope thus has
$4^{5}=1024$ faces, and $4^{3}(4+1)/2=160$ edges.
## 7 Higher-order spin systems
Method of construction of the QPD described can be used for higher-order
spin-$j$ systems if the corresponding set of CSCOs are available. The CSCOs,
in principle, can be constructed since a complete set of MUBs is known to
exist when the dimension $n=2j+1$ is a prime or power of a prime. For such
$n$, we can consider the irreducible tensor operators $\tau^{k}_{q}$ discussed
above, with $\tau^{0}_{0}=\mathbb{I}_{n}$ being the identity operator. The
matrix elements of diagonal operators are $\langle
jm^{\prime}|\tau^{k}_{0}|jm\rangle=\sqrt{2k+1}C(jkj;m0m^{\prime})$.
Thus in the canonical basis of Hilbert space of dimension $n=2j+1$, we know
the $2j$ diagonal matrices. It is possible to generate CSCOs from the unitary
transformations connecting different MUB sets[17], resulting in $n+1$ sets
$\mathcal{A}_{i},i=1,\ldots,n+1$ of operators, where each $\mathcal{A}_{i}$
contains $n-1$ commuting operators. The $n+1$ measurement operators
$M_{i},i=1,\ldots,n+1$ are then constructed taking linear combinations of the
operators within each $\mathcal{A}_{i}$. The CSCOs determine the Bloch vector
$\vec{\theta}=(\theta_{1},\ldots,\theta_{n^{2}-1})$ through the corresponding
density matrix representation. Inspection of our method reveals that the
requirement to extend this to an arbitrary finite dimensional system is that a
complete set of MUBs is known to exist. For such systems, physical realization
amounts to the identification of a suitable Hamiltonian which plays a role
similar to the multipole fields used for spin-$j$ systems.
With $\vec{v}_{i}=(v_{1i},\ldots,v_{(n-1)i}),i=1,\ldots,n+1$ and a
commensurate partitioning of the vector $\vec{\theta}$ as
$\vec{\theta}=(\vec{\theta}_{1},\ldots,\vec{\theta}_{n+1})^{T}$, the general
form of the $(n^{2}-1)$-dimensional QPD for an $n$-dimensional system then is
$p(\vec{v}_{1},\ldots,\vec{v}_{n+1})=\frac{1}{n^{n+1}}\left[1+\vec{v}_{1}\cdot\vec{\theta}_{1}+\cdots+\vec{v}_{n+1}\cdot\vec{\theta}_{n+1}\right],$
where the $\vec{v}_{i}$ assume values in a set consisting of $n$ elements,
where each element is an $(n-1)$-dimensional vector. The set of states within
the Bloch ball $\mathcal{B}^{n^{2}-2}(\vec{\theta})$ for which the QPD is non-
negative is given by the inequality
$\vec{v}_{1}\cdot\vec{\theta}_{1}+\cdots+\vec{v}_{n+1}\cdot\vec{\theta}_{n+1}\geq-1$.
The convex polytope has $n(n+1)$ vertices, $n^{n+1}$ faces, and has
$n^{3}(n+1)/2$ edges.
## 8 Acknowledgements
HSS thanks the Department of Science and Technology (DST), India for the grant
of INSPIRE Fellowship. KB acknowledges partial support from grants NSF DMS
grants 1613054, 2015374 and NIH R37-CA214955.
## References
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|
# Polynomial interpolation and residue currents
Jimmy Johansson Department of Mathematical Sciences, Chalmers University of
Technology and the University of Gothenburg, Gothenburg SE-412 96, Sweden
<EMAIL_ADDRESS>
(Date: January 20, 2021)
###### Abstract.
We show that a global holomorphic section of $\mathscr{O}(d)$ restricted to a
closed complex subspace $X\subset\mathbb{P}^{n}$ has an interpolant if and
only if it satisfies a set of moment conditions that involves a residue
current associated with a locally free resolution of $\mathscr{O}_{X}$. When
$X$ is a finite set of points in $\mathbb{C}^{n}\subset\mathbb{P}^{n}$ this
can be interpreted as a set of linear conditions that a function on $X$ has to
satisfy in order to have a polynomial interpolant of degree at most $d$.
###### 2010 Mathematics Subject Classification:
32A27, 32C30
## 1\. Introduction
Let $i:X\hookrightarrow\mathbb{C}^{n}$ be a subvariety or complex subspace
whose underlying space, $X_{\textnormal{red}}$, is a finite set of points
$\\{p_{0},\dots,p_{r}\\}\subset\mathbb{C}^{n}$. Let $g$ be a holomorphic
function on $X$, i.e., a global holomorphic section of $\mathscr{O}_{X}$, and
let $G\in\mathbb{C}[\zeta_{1},\dots,\zeta_{n}]$ be a polynomial. We say that
$G$ interpolates $g$ if the pull-back of $G$ to $X$ equals $g$, i.e.,
$i^{*}G=g$.
If $X$ is reduced, then a holomorphic function $g$ on $X$ is just a function
from $X_{\textnormal{red}}=\\{p_{0},\dots,p_{r}\\}$ to $\mathbb{C}$, and we
have that $G$ interpolates $g$ if $G(p_{j})=g(p_{j})$ for each $j=0,\dots,r$.
In the univariate case this is referred to as Lagrange interpolation. If $X$
is not reduced, then at each point $p_{j}$, $G$ also has to satisfy some
conditions on its derivatives. In the univariate case this is referred to as
Hermite interpolation, see Example 4.3.
The motivating question for this note is the following. What are the necessary
and sufficient conditions on $g$ for the existence of an interpolant of degree
at most $d$?
Let $A_{X}$ denote the vector space of holomorphic functions on $X$, i.e.,
$A_{X}=H^{0}(\mathbb{C}^{n},\mathscr{O}_{X})$. Since the set of holomorphic
functions on $X$ that have an interpolant of degree at most $d$ is a linear
subspace of $A_{X}$, we have that a function $g\in A_{X}$ has an interpolant
of degree at most $d$ if and only if it satisfies a finite set of linear
conditions. In this note we will show how these linear conditions can be
explicitly realized as a set of moment conditions that involves a so-called
residue current associated with a locally free resolution of
$\mathscr{O}_{X}$.
Recall that since $X_{\textnormal{red}}$ is a finite set of points, $X$ can be
viewed as a closed complex subspace of $\mathbb{P}^{n}$, and we have that
polynomials of degree at most $d$ on $\mathbb{C}^{n}$ naturally correspond to
global holomorphic sections of the line bundle
$\mathscr{O}(d)\rightarrow\mathbb{P}^{n}$ via $d$-homogenization. This
motivates the following more general notion of interpolation that we shall
consider in this note. Let $i:X\hookrightarrow\mathbb{P}^{n}$ be a closed
complex subspace of arbitrary dimension. Let $\Phi$ and $\varphi$ be global
holomorphic sections of $\mathscr{O}(d)$ and
$\mathscr{O}_{X}(d)=i^{*}\mathscr{O}(d)$, respectively. We say that $\Phi$
interpolates $\varphi$ if $i^{*}\Phi=\varphi$.
From a minimal graded free resolution of the homogeneous coordinate ring of
$X$, $S_{X}$, we obtain a locally free resolution of $\mathscr{O}_{X}$ of the
form
$0\longrightarrow\mathscr{O}(E_{n})\overset{f_{n}}{\longrightarrow}\dots\overset{f_{2}}{\longrightarrow}\mathscr{O}(E_{1})\overset{f_{1}}{\longrightarrow}\mathscr{O}_{\mathbb{P}^{n}}\longrightarrow\mathscr{O}_{X}\longrightarrow
0$ (1.1)
where $E_{k}=\bigoplus_{\ell}\mathscr{O}(-\ell)^{\beta_{k,\ell}}$, see
[eisenbud] and [aw]*Section 6. The $\beta_{k,\ell}$ are referred to as the
_graded Betti numbers_ of $S_{X}$. We equip the $E_{k}$ with the natural
Hermitian metrics. In [aw], Andersson and Wulcan showed that with (1.1), one
can associate a residue current $R$ that generalizes the classical
Coleff–Herrera product [ch], see Section 2. It can be written as
$R=\sum_{k,\ell}R_{k,\ell}$, where each $R_{k,\ell}$ is an
$\mathscr{O}(-\ell)^{\beta_{k,\ell}}$-valued $(0,k)$-current. In [aw2], the
same authors proved a result which as a special case gives a cohomological
condition in terms of the current $R$ for when $\Phi$ interpolates $\varphi$.
In Section 3 we will show that in our setting this condition amounts to the
following set of moment conditions.
###### Theorem 1.1.
Let $X\subset\mathbb{P}^{n}$ be a closed complex subspace, and let $R$ be the
residue current associated with (1.1). Moreover, let $\omega$ be a
nonvanishing holomorphic $\mathscr{O}(n+1)$-valued $n$-form. A global
holomorphic section $\varphi$ of $\mathscr{O}_{X}(d)$ has an interpolant if
and only if for each $\ell$ it holds that
$\int_{\mathbb{P}^{n}}R_{n,\ell}\varphi\wedge h\omega=0$ (1.2)
for all global holomorphic sections $h$ of $\mathscr{O}(\ell-d-n-1)$.
Recall that the _interpolation degree_ of $X$ is defined as
$\inf\\{d:\text{all global holomorphic sections of $\mathscr{O}_{X}(d)$ has an
interpolant}\\}.$
In particular, if $X_{\textnormal{red}}$ is a finite set of points in
$\mathbb{C}^{n}$, then the interpolation degree of $X$ is the smallest number
$d$ such that any $g\in A_{X}$ has an interpolant of degree at most $d$.
Define
$t_{k}(S_{X})=\sup\\{\ell:\beta_{k,\ell}\neq 0\\}.$ (1.3)
(We use the convention that the supremum of the empty set is $-\infty$.) As a
consequence of Theorem 1.1 we get the following bound of the interpolation
degree.
###### Corollary 1.2.
Let $X\subset\mathbb{P}^{n}$ be a closed complex subspace with homogeneous
coordinate ring $S_{X}$. The interpolation degree of $X$ is less than or equal
to $t_{n}(S_{X})-n$.
It can be shown by purely algebraic means that the interpolation degree of $X$
is in fact equal to $t_{n}(S_{X})-n$, see e.g. [johansson]*Corollary 1.6. If
$X_{\textnormal{red}}$ consists of a finite set of points, then it can be
shown that $t_{n}(S_{X})-n$ is equal to the Castelnuovo–Mumford regularity of
$S_{X}$, see [eisenbud]*Exercise 4E.5, and the statement in this case is
Theorem 4.1 in [eisenbud].
In Section 4 we will consider the case when $X_{\textnormal{red}}$ is a finite
set of points in $\mathbb{C}^{n}$. In this case the corresponding versions of
Theorem 1.1 and Corollary 1.2 first appeared in [standar]. We will consider
some examples where we explicitly write down the conditions for when $g\in
A_{X}$ has an interpolant of degree at most $d$. In particular, we will obtain
the precise conditions for when the Hermite interpolation problem has a
solution.
## 2\. Residue currents
Let $f$ be a holomorphic function in an open set in $\mathbb{C}^{n}$. Let
$\xi$ be a smooth $(n,n)$-form with compact support. In [hl], using Hironaka’s
desingularization theorem, Herrera and Lieberman proved that the limit
$\lim_{\epsilon\rightarrow 0}\int_{|f|>\epsilon}\frac{\xi}{f}$ (2.1)
exists. Thus (2.1) defines a current known as the principal value current,
which is denoted by $[1/f]$. The residue current $R^{f}$ of $f$ is the
$(0,1)$-current $\bar{\partial}[1/f]$. It is easy to see that $R^{f}$ has its
support on $V(f)=f^{-1}(0)$, and that it satisfies the following _duality
principle_ : A holomorphic function $\Phi$ belongs to the ideal $(f)$ if and
only if $R^{f}\Phi=0$.
###### Example 2.1.
Let $\zeta_{0}\in\mathbb{C}$. We have that the action of
$\bar{\partial}[1/(\zeta-\zeta_{0})]$ on a test form $\xi(\zeta)d\zeta$ is
given by
$\left\langle\bar{\partial}\left[\frac{1}{\zeta-\zeta_{0}}\right],\xi(\zeta)d\zeta\right\rangle=2\pi
i\xi(\zeta_{0}).$ (2.2)
∎
### 2.1. Residue currents associated with generically exact complexes
We will now consider a generalization of the above construction due to
Andersson and Wulcan. Consider a generically exact complex of Hermitian
holomorphic vector bundles over a complex manifold $Y$ of dimension $n$,
$0\longrightarrow
E_{n}\overset{f_{n}}{\longrightarrow}\dots\overset{f_{2}}{\longrightarrow}E_{1}\overset{f_{1}}{\longrightarrow}E_{0}\longrightarrow
0,$ (2.3)
i.e., a complex that is exact outside an analytic variety $Z\subset Y$ of
positive codimension. The vector bundle $E=\bigoplus_{k}E_{k}$ has a natural
superbundle structure, i.e., a $\mathbb{Z}_{2}$-grading, $E=E_{+}\oplus
E_{-}$, where $E_{+}=\bigoplus_{k}E_{2k}$ and $E_{-}=\bigoplus_{k}E_{2k+1}$,
which we shall refer to as the subspaces of even and odd elements,
respectively. This induces a $\mathbb{Z}_{2}$-grading on the sheaf of
$E$-valued currents $\mathscr{C}(E)$; if $\omega\otimes\xi$ is an $E$-valued
current, where $\omega$ is a current and $\xi$ is a smooth section of $E$,
then the degree of $\omega\otimes\xi$ is the sum of the degree of $\xi$ and
the current degree of $\omega$ modulo 2.
We say that an endomorphism on $E$ is even (resp. odd) if it preserves (resp.
switches) the degree. If $\alpha$ is a smooth section of
$\operatorname{End}E$, then it defines a map on $\mathscr{C}(E)$ via
$\alpha(\omega\otimes\xi)=(-1)^{(\deg\alpha)(\deg\omega)}\omega\otimes\alpha(\xi),$
where $\omega$ is a current and $\xi$ is a smooth section of $E$. In
particular, the map $f=\sum_{k=1}^{n}f_{k}$ defines an odd map on
$\mathscr{C}(E)$. We define an odd map on $\mathscr{C}(E)$,
$\nabla=f-\bar{\partial}$, which, since $f$ and $\bar{\partial}$ anti-commute,
satisfies $\nabla^{2}=0$. The map $\nabla$ extends to an odd map on
$\mathscr{C}(\operatorname{End}E)$ via Leibniz’s rule,
$\nabla(\alpha\xi)=(\nabla\alpha)\xi+(-1)^{\deg\alpha}\alpha\nabla\xi.$
In [aw], Andersson and Wulcan constructed $\operatorname{End}E$-valued
currents
$U=\sum_{\ell}U^{\ell}=\sum_{\ell}\sum_{k\geq\ell+1}U_{k}^{\ell},$
and
$R=\sum_{\ell}R^{\ell}=\sum_{\ell}\sum_{k\geq\ell+1}R_{k}^{\ell},$
where $U_{k}^{\ell}$ and $R_{k}^{\ell}$ are
$\operatorname{Hom}(E_{\ell},E_{k})$-valued currents of bidegree
$(0,k-\ell-1)$ and $(0,k-\ell)$, respectively, which satisfy
$\nabla U=\operatorname{id}_{E}-R,\quad\nabla R=0.$ (2.4)
The current $R$ is referred to as the residue current associated with (2.3)
and it has its support on $Z$.
Suppose that the complex of locally free sheaves corresponding to (2.3),
$0\longrightarrow\mathscr{O}(E_{n})\overset{f_{n}}{\longrightarrow}\dots\overset{f_{2}}{\longrightarrow}\mathscr{O}(E_{1})\overset{f_{1}}{\longrightarrow}\mathscr{O}(E_{0}),$
(2.5)
is exact. When the $E_{k}$ are equipped with Hermitian metrics, we shall refer
to (2.5) as a _Hermitian resolution_ of the sheaf
$\mathscr{O}(E_{0})/\operatorname{im}f_{1}$. In this case it holds that
$R^{\ell}=0$ if $\ell\geq 1$, and henceforth we shall write $R_{k}$ for
$R_{k}^{0}$. Moreover, we have that $R$ satisfies the following properties:
_Duality principle_ : A holomorphic section $\Phi$ of $E_{0}$ belongs to
$\operatorname{im}f_{1}$ if and only if $R\Phi=0$.
_Dimension principle_ : If $\operatorname{codim}Z>k$, then $R_{k}=0$.
Note that the second equality in (2.4) is equivalent to
$\displaystyle f_{1}R_{1}$ $\displaystyle=0,$ (2.6) $\displaystyle
f_{k+1}R_{k+1}-\bar{\partial}R_{k}$ $\displaystyle=0,\quad 1\leq k\leq n-1,$
(2.7) $\displaystyle\bar{\partial}R_{n}$ $\displaystyle=0.$ (2.8)
Let $i:X\hookrightarrow Y$ be a closed complex subspace with ideal sheaf
$\mathscr{I}_{X}$, and suppose that $\mathscr{O}_{X}=i^{*}\mathscr{O}_{Y}$,
which we identify with $\mathscr{O}_{Y}/\mathscr{I}_{X}$, has a Hermitian
resolution of the form
$0\longrightarrow\mathscr{O}(E_{n})\overset{f_{n}}{\longrightarrow}\dots\overset{f_{2}}{\longrightarrow}\mathscr{O}(E_{1})\overset{f_{1}}{\longrightarrow}\mathscr{O}_{Y}\longrightarrow\mathscr{O}_{X}\longrightarrow
0,$ (2.9)
cf. (2.5) where $E_{0}$ is the trivial line bundle. For the associated residue
current $R=R_{1}+\dots+R_{n}$, we can view each $R_{k}$ as an $E_{k}$-valued
$(0,k)$-current. Since $\operatorname{im}f_{1}=\mathscr{I}_{X}$, we have that
$i^{*}\Phi=0$ if and only if $R\Phi=0$ by the duality principle. More
generally, let $L\rightarrow Y$ be a holomorphic line bundle. If we equip $L$
with a Hermitian metric, then we obtain a Hermitian resolution of
$i^{*}L=\mathscr{O}_{X}\otimes L$ by tensoring (2.9) with $L$, and we have
that $R$ is the associated residue current with this resolution as well.
### 2.2. The Coleff–Herrera product
Let $f=(f_{1},\dots,f_{p}):\mathbb{C}^{n}\rightarrow\mathbb{C}^{p}$ be a
holomorphic mapping such that $V(f)=f^{-1}(0)$ has codimension $p$. In [ch]
Coleff and Herrera gave meaning to the product
$\mu^{f}=\bar{\partial}\left[\frac{1}{f_{1}}\right]\wedge\dots\wedge\bar{\partial}\left[\frac{1}{f_{p}}\right],$
(2.10)
which is known as the _Coleff–Herrera product_. In particular, if each $f_{j}$
only depends on $\zeta_{j}$, then (2.10) is just the tensor product of the
one-variable currents $\bar{\partial}[1/f_{j}]$ described above. The current
$\mu^{f}$ is $\bar{\partial}$-closed, has support $V(f)$, and is anti-
commuting in the $f_{j}$. Moreover, $\mu^{f}$ satisfies the duality principle,
i.e., $\mu^{f}\Phi=0$ if and only if $\Phi\in\mathscr{I}(f)$, where
$\mathscr{I}(f)$ is the ideal sheaf generated by $f$.
Let $H\rightarrow Y$ be a holomorphic Hermitian vector bundle of rank $p$, and
let $f$ be a holomorphic section of the dual bundle $H^{*}$. Let
$E_{k}=\bigwedge^{k}H$, and define $\delta_{k}:E_{k}\rightarrow E_{k-1}$ as
interior multiplication by $f$. This gives a generically exact complex (2.3).
Suppose $f=f_{1}e_{1}^{*}+\dots+f_{p}e_{p}^{*}$ in some local holomorphic
frame $e_{j}^{*}$ for $H^{*}$. If $\operatorname{codim}f^{-1}(0)=p$, then the
corresponding complex of sheaves is a Hermitian resolution of
$\mathscr{O}_{Y}/\mathscr{I}(f)$ known as the _Koszul complex_ , and it was
proven in [mats] that the associated residue current is given by
$R=R_{p}=\mu^{f}e_{1}\wedge\dots\wedge e_{p}$.
### 2.3. A comparison formula for residue currents
We have the following _comparison formula_ for residue currents, see Theorem
1.3 and Corollary 4.7 in [larkang]. Let $X\subset X^{\prime}$ be complex
subspaces of codimension $p$ of $Y$. Suppose that there exist Hermitian
resolutions of length $p$ of $\mathscr{O}_{X}$ and $\mathscr{O}_{X^{\prime}}$,
respectively, and let $R$ and $R^{\prime}$ be the associated residue currents.
Moreover, suppose that there exists a map of complexes
${0}$${\mathscr{O}(E^{\prime}_{p})}$${\cdots}$${\mathscr{O}(E^{\prime}_{1})}$${\mathscr{O}_{Y}}$${\mathscr{O}_{X^{\prime}}}$${0}$${0}$${\mathscr{O}(E_{p})}$${\cdots}$${\mathscr{O}(E_{1})}$${\mathscr{O}_{Y}}$${\mathscr{O}_{X}}$${0}$$\scriptstyle{\psi_{p}}$$\scriptstyle{f^{\prime}_{p}}$$\scriptstyle{f^{\prime}_{2}}$$\scriptstyle{\psi_{1}}$$\scriptstyle{f^{\prime}_{1}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{f_{p}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$
Then $R_{p}=\psi_{p}R^{\prime}_{p}$.
## 3\. Interpolation and residue currents
Let $Y$ be a complex manifold of dimension $n$, and let $i:X\hookrightarrow Y$
be a closed complex subspace. Let $L\rightarrow Y$ be a holomorphic line
bundle, and let $\Phi$ and $\varphi$ be global holomorphic sections of $L$ and
$i^{*}L$, respectively. We say that $\Phi$ interpolates $\varphi$ if
$i^{*}\Phi=\varphi$.
Suppose that there exists a Hermitian resolution of $\mathscr{O}_{X}$ of the
form (2.9), and let $R$ denote the associated residue current. For each point
$x\in Y$ there is a neighborhood $\mathscr{U}$ and a holomorphic section
$\widetilde{\varphi}$ of $L$ such that $i^{*}\widetilde{\varphi}=\varphi$ on
$\mathscr{U}$. We define the current $R\varphi$ on $Y$ locally as
$R\widetilde{\varphi}$. This is well-defined since if
$\widetilde{\varphi}^{\prime}$ is another section such that
$i^{*}\widetilde{\varphi}^{\prime}=\varphi$, then
$R(\widetilde{\varphi}-\widetilde{\varphi}^{\prime})=0$ by the duality
principle.
We have the following result which follows immediately as a special case of
Lemma 4.5 (ii) in [aw2].
###### Lemma 3.1.
Let $\Phi$ and $\varphi$ be global holomorphic sections of $L$ and $i^{*}L$,
respectively. Then $\Phi$ interpolates $\varphi$ if and only if there exists a
current $w$ such that $\Phi-R\varphi=\nabla w$.
In other words, $\varphi$ has an interpolant if and only if there exist
currents $w_{1},\dots,w_{n}$ such that $\bar{\partial}w_{n}=R_{n}\varphi$, and
$\bar{\partial}w_{k}=f_{k+1}w_{k+1}+R_{k}\varphi,\quad 1\leq k\leq n-1.$ (3.1)
Moreover, in this case an interpolant of $\varphi$ is given by
$\Phi=f_{1}w_{1}$. Note that $\Phi$ is holomorphic since
$\bar{\partial}\Phi=-f_{1}\bar{\partial}w_{1}=-f_{1}(f_{2}w_{2}+R_{1}\varphi)=-(f_{1}R_{1})\varphi=0$.
Here the last equality follows from (2.6).
Let us now consider interpolation on $Y=\mathbb{P}^{n}$ with respect to the
line bundle $L=\mathscr{O}(d)$. Recall that there is a Hermitian resolution of
$\mathscr{O}_{X}$ of the form (1.1). We write $R_{k,\ell}$ for the
$\mathscr{O}(-\ell)^{\beta_{k,\ell}}$-valued component of $R_{k}$.
If $R_{n}\varphi=\bar{\partial}w_{n}$ for some current $w_{n}$, then one can
successively find currents $w_{n-1},\dots,w_{1}$ such that (3.1) holds since,
in view of (2.7),
$\bar{\partial}(f_{k+1}w_{k+1}+R_{k}\varphi)=-f_{k+1}\bar{\partial}w_{k+1}+(\bar{\partial}R_{k})\varphi=-(f_{k+1}R_{k+1}-\bar{\partial}R_{k})\varphi=0,$
and it follows from, e.g., [demailly]*Theorem 10.7 that
$H^{0,k}(\mathbb{P}^{n},E_{k}\otimes\mathscr{O}(d))=0,\quad 1\leq k\leq n-1.$
We thus have the following condition for the existence of an interpolant.
###### Lemma 3.2.
A global holomorphic section $\varphi$ of $\mathscr{O}_{X}$ has an interpolant
if and only if $R_{n}\varphi$ is $\bar{\partial}$-exact, i.e., there exists a
current $\eta$ such that $R_{n}\varphi=\bar{\partial}\eta$.
###### Proof of Theorem 1.1.
By Serre duality we have that $R_{n,\ell}\varphi$ is $\bar{\partial}$-exact if
and only if
$\int_{\mathbb{P}^{n}}R_{n,\ell}\varphi\wedge\eta=0$
for all global $\bar{\partial}$-closed $\mathscr{O}(\ell-d)$-valued
$(n,0)$-forms $\eta$. Note that each such form is of the form $h\omega$ for
some global holomorphic section $h$ of $\mathscr{O}(\ell-d-n-1)$. Since
$R_{n}\varphi$ is $\bar{\partial}$-exact if and only if each component
$R_{n,\ell}\varphi$ is, the statement follows from Lemma 3.2. ∎
###### Proof of Corollary 1.2.
Let $d\geq t_{n}(S_{X})-n$, see (1.3), and let $\varphi$ be a global
holomorphic section of $\mathscr{O}_{X}(d)$. We have for each $\ell$ that
$\int_{\mathbb{P}^{n}}R_{n,\ell}\varphi\wedge h\omega=0$
for all global holomorphic sections $h$ of $\mathscr{O}(\ell-d-n-1)$. Indeed,
if $\ell\leq d+n$, the only such $h$ is the zero section, and if $\ell>d+n$,
then $R_{n,\ell}=0$ since $\beta_{n,\ell}=0$. Therefore $\varphi$ has an
interpolant by Theorem 1.1. ∎
## 4\. Polynomial interpolation
Let us now return to the topic of polynomial interpolation. Recall that the
setting is that $X$ is a complex subspace of $\mathbb{C}^{n}$ such that
$X_{\textnormal{red}}$ is a finite set of points. The aim of this section is
to give some examples where we explicitly compute the residue current $R$
associated with a Hermitian resolution of $\mathscr{O}_{X}$ and write down the
moment conditions that Theorem 1.1 imposes on a function $g\in A_{X}$ for the
existence of an interpolant of degree at most $d$. We do this by identifying
$g$ with a global holomorphic section $\varphi$ of $\mathscr{O}_{X}(d)$ and
use the fact that $g$ has an interpolant of degree at most $d$ if and only if
$\varphi$ has an interpolant. More precisely, we let
$[z]=[z_{0}\,{:}\,\dots\,{:}\,z_{n}]$ denote homogeneous coordinates on
$\mathbb{P}^{n}$, and we view $\mathbb{C}^{n}$ as an open complex subspace of
$\mathbb{P}^{n}$ via the embedding
$(\zeta_{1},\dots,\zeta_{n})\mapsto[1\,{:}\,\zeta_{1}\,{:}\,\dots\,{:}\,\zeta_{n}]$.
Recall that on $\mathbb{C}^{n}$ there is a frame $e$ for $\mathscr{O}(1)$ such
that a global holomorphic section $\Phi$ of $\mathscr{O}(d)$ is given by
$\Phi(\zeta_{1},\dots,\zeta_{n})=G(\zeta_{1},\dots,\zeta_{n})e(\zeta_{1},\dots,\zeta_{n})^{\otimes
d},$
where $G$ is a polynomial of degree at most $d$ on $\mathbb{C}^{n}$.
Throughout this section we shall let $\omega$ in Theorem 1.1 be the
nonvanishing holomorphic $\mathscr{O}(n+1)$-valued $n$-form on
$\mathbb{P}^{n}$ such that
$\omega=d\zeta_{1}\wedge\dots\wedge d\zeta_{n}\otimes e^{\otimes(n+1)}$
on $\mathbb{C}^{n}\subset\mathbb{P}^{n}$.
Note that the dimension principle gives that $R=R_{n}$, and throughout this
section we write $R_{\ell}$ rather than $R_{n,\ell}$ for the
$\mathscr{O}(-\ell)^{\beta_{n,\ell}}$-valued component of $R_{n}$.
###### Example 4.1.
Let
$X=\\{(0,0),(1,0),(0,1),(1,1)\\}\subset\mathbb{C}^{2}\subset\mathbb{P}^{2}$.
We have that $X$ is defined by the homogeneous ideal $I_{X}=(f_{1},f_{2})$,
where $f_{1}=z_{1}(z_{1}-z_{0})$ and $f_{2}=z_{2}(z_{2}-z_{0})$. A Hermitian
resolution of $\mathscr{O}_{X}$ is given by the Koszul complex, see Section
2.2, where we interpret $(f_{1},f_{2})$ as a global holomorphic section of
$\mathscr{O}(2)^{2}$. Thus the associated residue current takes values in
$\mathscr{O}(-4)$, and is given by the Coleff–Herrera product, see Section
2.2,
$R=R_{4}=\bar{\partial}\left[\frac{1}{\zeta_{1}(\zeta_{1}-1)}\right]\wedge\bar{\partial}\left[\frac{1}{\zeta_{2}(\zeta_{2}-1)}\right]e^{\otimes(-4)}.$
By a straightforward computation, cf. (2.1), we get
$\displaystyle R_{4}$
$\displaystyle=\left(\bar{\partial}\left[\frac{1}{\zeta_{1}}\right]\wedge\bar{\partial}\left[\frac{1}{\zeta_{2}}\right]-\bar{\partial}\left[\frac{1}{\zeta_{1}-1}\right]\wedge\bar{\partial}\left[\frac{1}{\zeta_{2}}\right]\right.$
$\displaystyle-\left.\bar{\partial}\left[\frac{1}{\zeta_{1}}\right]\wedge\bar{\partial}\left[\frac{1}{\zeta_{2}-1}\right]+\bar{\partial}\left[\frac{1}{\zeta_{1}-1}\right]\wedge\bar{\partial}\left[\frac{1}{\zeta_{2}-1}\right]\right)e^{\otimes(-4)}.$
By Theorem 1.1, we now get the following. Since $R_{\ell}=0$ for $\ell\geq 5$,
we have that any $g\in A_{X}$ has an interpolant of degree at most $2$.
Moreover, $g$ has an interpolant of degree at most 1 if and only if (1.2)
holds when $\ell=4$ and $h=1$. In view of (2.2) this amounts to
$g(0,0)-g(1,0)-g(0,1)+g(1,1)=0,$ (4.1)
which is expected since the values of $g$ at $(0,0)$, $(1,0)$, and $(0,1)$
uniquely determines a polynomial of degree at most 1 that takes the value
$g(1,0)+g(0,1)-g(0,0)$ at $(1,1)$. Note that this gives that the interpolation
degree of $X$ is 2.
We have that $g$ has a constant interpolant if and only if (1.2) holds for all
global holomorphic $h$ of $\mathscr{O}(1)$. By linearity we only need to check
$h=z_{0},z_{1},z_{2}$, which amounts to (4.1), $g(1,1)-g(1,0)=0$, and
$g(1,1)-g(0,1)=0$. This amounts to
$g(0,0)=g(1,0)=g(0,1)=g(1,1)$
as expected. ∎
###### Example 4.2.
Let
$X=\\{(0,0),(1,0),(0,1),(0,2)\\}\subset\mathbb{C}^{2}\subset\mathbb{P}^{2}$.
We have that $X$ is defined by the homogeneous ideal
$I_{X}=(z_{1}a_{1},z_{1}z_{2},z_{2}a_{2})$, where $a_{1}=z_{1}-z_{0}$ and
$a_{2}=(z_{2}-z_{0})(z_{2}-2z_{0})$. We have Hermitian resolutions of
$\mathscr{O}_{\mathbb{P}^{2}}/\mathscr{I}(z_{1}a_{1},z_{2}a_{2})$ and
$\mathscr{O}_{X}$ and a map of complexes:
${\mathscr{O}(-5)}$${\mathscr{O}(-2)\oplus\mathscr{O}(-3)}$${\mathscr{O}_{\mathbb{P}^{2}}}$${\mathscr{O}_{\mathbb{P}^{2}}/\mathscr{I}(z_{1}a_{1},z_{2}a_{2})}$${\mathscr{O}(-3)\oplus\mathscr{O}(-4)}$${\mathscr{O}(-2)^{2}\oplus\mathscr{O}(-3)}$${\mathscr{O}_{\mathbb{P}^{2}}}$${\mathscr{O}_{X}}$$\scriptstyle{\psi_{2}}$$\scriptstyle{\delta_{2}}$$\scriptstyle{\psi_{1}}$$\scriptstyle{\delta_{1}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$
where the upper complex is the Koszul complex, see Section 2.2. Moreover,
$f_{1}=\begin{bmatrix}z_{1}a_{1}&z_{1}z_{2}&z_{2}a_{2}\end{bmatrix},\quad
f_{2}=\begin{bmatrix}-z_{2}&0\\\ a_{1}&-a_{2}\\\ 0&z_{1}\end{bmatrix},$
and
$\psi_{1}=\begin{bmatrix}1&0\\\ 0&0\\\
0&1\end{bmatrix},\quad\psi_{2}=\begin{bmatrix}a_{2}\\\ a_{1}\end{bmatrix}.$
Let $R$ and $R^{\prime}$ denote the residue currents associated with the
resolutions of $\mathscr{O}_{X}$ and
$\mathscr{O}/\mathscr{I}(z_{1}a_{1},z_{2}a_{2})$, respectively. We have that
$R^{\prime}=R^{\prime}_{5}$ takes values in $\mathscr{O}(-5)$ and is given by
the Coleff–Herrera product, see Section 2.2,
$R^{\prime}_{5}=\bar{\partial}\left[\frac{1}{\zeta_{1}(\zeta_{1}-1)}\right]\wedge\bar{\partial}\left[\frac{1}{\zeta_{2}(\zeta_{2}-1)(\zeta_{2}-2)}\right]e^{\otimes(-5)}.$
Thus by the comparison formula, see Section 2.3,
$R=\psi_{2}R^{\prime}=R_{3}\oplus R_{4}$. A straightforward computation gives
that
$\displaystyle R_{3}$
$\displaystyle=\bar{\partial}\left[\frac{1}{\zeta_{1}(\zeta_{1}-1)}\right]\wedge\bar{\partial}\left[\frac{1}{\zeta_{2}}\right]e^{\otimes(-3)}$
$\displaystyle=\left(-\bar{\partial}\left[\frac{1}{\zeta_{1}}\right]\wedge\bar{\partial}\left[\frac{1}{\zeta_{2}}\right]+\bar{\partial}\left[\frac{1}{\zeta_{1}-1}\right]\wedge\bar{\partial}\left[\frac{1}{\zeta_{2}}\right]\right)e^{\otimes(-3)},$
and
$\displaystyle R_{4}$
$\displaystyle=\bar{\partial}\left[\frac{1}{\zeta_{1}}\right]\wedge\bar{\partial}\left[\frac{1}{\zeta_{2}(\zeta_{2}-1)(\zeta_{2}-2)}\right]e^{\otimes(-4)}$
$\displaystyle=\left(\frac{1}{2}\bar{\partial}\left[\frac{1}{\zeta_{1}}\right]\wedge\bar{\partial}\left[\frac{1}{\zeta_{2}}\right]-\bar{\partial}\left[\frac{1}{\zeta_{1}}\right]\wedge\bar{\partial}\left[\frac{1}{\zeta_{2}-1}\right]+\frac{1}{2}\bar{\partial}\left[\frac{1}{\zeta_{1}}\right]\wedge\bar{\partial}\left[\frac{1}{\zeta_{2}-2}\right]\right)e^{\otimes(-4)}.$
By Theorem 1.1, we now get the following. Since $R_{\ell}=0$ for $\ell\geq 5$,
we have that any $g\in A_{X}$ has an interpolant of degree at most $2$.
Moreover, $g$ has an interpolant of degree at most 1 if and only if (1.2)
holds when $\ell=4$ and $h=1$. (Note that there is no condition involving
$R_{3}$ since $\ell-d-n-1<0$ in this case.) In view of (2.2) we get the
condition
$\frac{1}{2}g(0,0)-g(0,1)+\frac{1}{2}g(0,2)=0,$ (4.2)
which is expected since $g$ has an interpolant of degree at most 1 if and only
if $g(0,1)$ is the average of $g(0,0)$ and $g(0,2)$. Note that this gives that
the interpolation degree of $X$ is 2.
We get that $g$ has a constant interpolant if and only if (1.2) holds when
$\ell=4$ for $h=z_{0},z_{1},z_{2}$, and when $\ell=3$ and $h=1$. Since $z_{1}$
vanishes on the support of $R_{4}$, this amounts to the equations (4.2),
$g(0,2)-g(0,1)=0$ and $g(1,0)-g(0,0)=0$. This amounts to
$g(0,0)=g(1,0)=g(0,1)=g(0,2)$
as expected. ∎
We end this note by considering Hermite interpolation. We refer to, e.g.,
[calvi, spitz, traub], and references therein for a classical survey of this
topic.
###### Example 4.3.
Let $p_{0},\dots,p_{r}\in\mathbb{C}$, and let $g$ be a holomorphic function on
the complex subspace $X\subset\mathbb{C}$ defined by the ideal generated by
$\prod_{j=0}^{r}(\zeta-p_{j})$. Here we allow for the possibility that
$p_{i}=p_{j}$ for some $i,j$, so that $X$ is nonreduced in general, and we
denote the number of times that $p_{j}$ occurs by $m_{j}$. We have that a
polynomial $G$ interpolates $g$ if and only if, for each $j=0,\dots,r$,
$G^{(k)}(p_{j})=g^{(k)}(p_{j}),\quad k=0,\dots,m_{j}-1.$
We say that a polynomial interpolates $g$ with respect to $p_{0},\dots,p_{k}$,
$k\leq r$, if it interpolates the pull-back of $g$ to the complex subspace
defined by the ideal generated by $\prod_{j=0}^{k}(\zeta-p_{j})$. We denote
the unique polynomial that interpolates $g$ with respect to
$p_{0},\dots,p_{k}$ by $H[g;p_{0},\dots,p_{k}]$. The coefficient of its
$\zeta^{k}$-term is referred to as the $k$th _divided difference_ of $g$ and
we denote it by $g[p_{0},\dots,p_{k}]$. By induction it is not difficult to
see that
$H[g;p_{0},\dots,p_{r}](\zeta)=\sum_{k=0}^{r}g[p_{0},\dots,p_{k}]\prod_{j=0}^{k-1}(\zeta-
p_{j}),$
see [calvi]*Theorem 1.8. This is referred to as Newton’s formula.
We claim that $g\in A_{X}$ has an interpolant of degree at most $d$ if and
only if $(gh)[p_{0},\dots,p_{r}]=0$ for all polynomials $h$ of degree at most
$r-d-1$. Let us show how this condition follows from Theorem 1.1. Since the
ideal is generated by a single element, we have that the associated residue
current is given by
$R_{r+1}=\bar{\partial}\left[\frac{1}{\prod_{j=0}^{r}(\zeta-
p_{j})}\right]e^{\otimes(-r-1)}.$
Theorem 1.1 together with Stokes’ formula gives that $g$ has an interpolant if
and only if
$\int_{\mathbb{C}}\bar{\partial}\left[\frac{1}{\prod_{j=0}^{r}(\zeta-
p_{j})}\right]gh\wedge
d\zeta=\int_{C_{R}}\frac{H[gh;p_{0},\dots,p_{r}](\zeta)}{\prod_{j=0}^{r}(\zeta-
p_{j})}\,d\zeta=0$
for all polynomials $h$ of degree at most $r-d-1$, where $C_{R}$ is a circle
of radius $R\gg 0$. Here we have used the fact that $H[gh;p_{0},\dots,p_{r}]$
interpolates $gh$. By letting $R\rightarrow\infty$, a direct calculation gives
that the second integral is equal to $2\pi i\cdot(gh)[p_{0},\dots,p_{r}]$, and
hence the claim follows.
## Acknowledgments
I would like to thank Elizabeth Wulcan and Mats Andersson for helpful
discussions and comments on preliminary versions of this note.
## References
|
# Effective connectivity determines the critical dynamics of biochemical
networks
Santosh Manicka1,2,+ Manuel Marques-Pita1,2,3,+††+ These authors contributed
equally to this work. Luis M. Rocha1,2
( 1Center for Social and Biomedical Complexity, Luddy School of Informatics,
Computing & Engineering, Indiana University, Bloomington IN, USA
2Instituto Gulbenkian de Ciência, 2780-156, Oeiras, Portugal
3Universidade Lusófona, COPELABS. 1700-097, Lisbon, Portugal
<EMAIL_ADDRESS>
)
###### Abstract
Living systems operate in a critical dynamical regime—between order and
chaos—where they are both resilient to perturbation, and flexible enough to
evolve. To characterize such critical dynamics, the established structural
theory of criticality uses automata network connectivity and node bias (to be
on or off) as tuning parameters. This parsimony in the number of parameters
needed sometimes leads to uncertain predictions about the dynamical regime of
both random and systems biology models of biochemical regulation. We derive a
more accurate theory of criticality by accounting for canalization, the
existence of redundancy that buffers automata response to inputs. The new
canalization theory of criticality is based on a measure of effective
connectivity. It contributes to resolving the problem of finding precise ways
to design or control network models of biochemical regulation for desired
dynamical behavior. Our analyses reveal that effective connectivity
significantly improves the prediction of critical behavior in random automata
network ensembles. We also show that the average effective connectivity of a
large battery of systems biology models is much lower than the connectivity of
their original interaction structure. This suggests that canalization has been
selected to dynamically reduce and homogenize the seemingly heterogeneous
connectivity of biochemical networks.
## Introduction
In the study of biological, social, and technological systems, network models
have become important tools. [10] Network model structure is defined by a
graph $\mathcal{G}\equiv(X,E)$, where actors (e.g., biochemical species and
environmental factors) are represented as a set of _nodes_ , $X$, and
interactions between pairs of nodes as a set of _edges_ , $E$. Several
dynamical properties of networks can be inferred from their structure alone.
[11] The structural properties of networks yield insights into the
organization of living systems and societies. [10, 9] Yet, the rules of
interaction between nodes must be considered in order to study network
dynamics. For biochemical systems, understanding the precise inference of
interaction rules is a difficult task because vast amounts of data are
required to estimate the kinetic parameters governing molecular concentration
rates. In response, a growing number of successful modelers have overcome the
need for precise parameter estimation by relying on coarse-grained qualitative
approaches to modeling interactions between nodes. [59, 1, 50, 32, 2]
In 1969 Kauffman introduced the simplest qualitative model of biochemical
regulation and signaling, the Boolean network (BN). [37] Nodes in a Kauffman
network are defined as simple Boolean automata, and consequently, their
interactions are described as logical state transition rules. The state
transition rule of a node, $x_{i},$ incorporates $k_{i}$ inputs—typically the
states of other nodes, or external signals. Network dynamics ensue as the
state of every node $x_{i}\in X$ is updated synchronously in discrete time
steps. As the dynamics unfold from an initial configuration, the network
eventually settles into an _attractor_ configuration. An attractor can be a
stable fixed-point—a network configuration that leads to itself in the next
time step—or a sequence of configurations that repeat periodically. Kauffman
observed that hard-to-predict behaviors of real-world biochemical systems are
also exhibited by these canonical networks, and they have since been widely
used to model genetic regulation and signaling. [39, 2, 61, 12, 57, 3, 32]
State transition rules in these models are derived from molecular data and
used to capture the characteristic combinatorial regulation pervasive in
biochemical networks. [28, 44, 21, 3, 17, 5] Attractors then correspond to
stable states of real systems, such as those that determine cell fate, as
evidenced in a large number of models. [39, 44, 21, 13, 31]
Important discoveries in biology have been made using automata models even
though they are built from coarse qualitative representations of biochemical
entities and interactions and they use state-transition rules that often
ignore the precise specification of interaction timings. [3, 12, 63, 44] For
instance, the BN model of the yeast cell cycle reproduced the complete
dynamical trajectories originating from known initial conditions to known
attractor configurations. [22] For another instance, a BN model of intra-
cellular signal transduction in breast cancer reproduced known drug resistance
mechanisms and uncovered new and effective drug interventions. [64] A third,
striking instance is a prescription obtained from a BN model for how to
reprogram already differentiated cells. [16] While most existing models of
biochemical regulation are Boolean, an increasing number of models have
considered automata with more than two states. [3, 65] In addition, automata
networks have been extended to include various sources of stochasticity. [12,
46] BNs have become important conceptual frameworks to study a number of
general principles in theoretical biology, two of which, _canalization_ [62,
46] and _criticality_ , [37, 39, 42, 41] are at the core of the research
presented here.
### Critical dynamics in complex networks
The notion of criticality emerged from the observation that some dynamical
systems can be in a state of thermodynamic equilibrium that depends on some
_critical_ parameter. Tuning this parameter makes the system undergo phase
transitions. In an _ordered_ phase, the system becomes insensitive to
perturbations and changes in initial conditions. Conversely, in a _chaotic_
phase, dynamic trajectories within the system vary vastly as a result of small
perturbations or minute differences in initial conditions. In the _critical_
phase—the one between order and chaos—the system is robust to most small
perturbations, yet sensitive to some, making it flexible enough to respond
differentially to a range of input signals. In this phase, small changes in
initial conditions do not lead to completely different dynamic trajectories.
Though other notions of criticality exist, this is the focus of the research
presented here. In theory, complex networks in the critical phase can perform
collective information processing, which may be a key aspect in complex life
processes, such as genetic regulation. [49, 19, 41, 38]
### Criticality in living systems
Kauffman [37] not only introduced random BNs (RBNs) to model genetic
regulation, but also presented one of the first intuitions about criticality
in living systems. He suggested that biologically plausible regulatory
networks that exhibit the kinds of stable dynamics seen in biology, must have,
on average, _low_ connectivity. Later, Kauffman elaborated on this intuition
by proposing the hypothesis that biological systems operate in a _critical
dynamical regime_ , between order and chaos, and that a critical tuning
parameter is the network connectivity. Kauffman hypothesized that each
biochemical species in a given regulatory network should have two regulators
on average—that is, a regular network connectivity where every state
transition rule incorporates $k\approx 2$ inputs. [38, 39] Fifty years after
the publication of the original RBN paper, Kauffman and Bornholdt revisited
the main claims. [13] They noted that the _attractor hypothesis_ —network
attractors correspond to cell types in genetic networks—had become an accepted
fact. Kauffman and Bornholdt examined the research on the biochemical
criticality hypothesis and highlighted the following supporting evidence: (a)
the distribution of genes damaged by the spreading effects of deleting
selected genes in yeast mutant has a power law distribution, which indicates
criticality; [51, 55] (b) similar, biologically-plausible initial
configurations in global gene expression data obtained from macrophages follow
somewhat parallel trajectories to attractors. These trajectories are neither
identical, which would indicate order, nor divergent, which would indicate
chaos; [48] (c) a large battery of sixty-seven Boolean models of real
biochemical networks operate in the critical regime based on the analysis of
their predicted structural and dynamical properties. [20] Indeed, it is now
widely accepted that biochemical complex networks are critical. [33, 40, 30,
8, 57] See Roli et al.[53] and Muñoz[47] for recent reviews that explore
further evidence of criticality in living systems.
### The parameters and quantification of criticality
Several methods have been proposed for quantifying criticality in complex
networks. Early developments were grounded in physics and dynamical systems
theory. For example, Bak [6] showed that regular, spatially-extended,
dissipative dynamical systems can evolve to a self-organized critical state
with spatial and temporal power-law scaling behavior. Around the same period,
Langton studied network dynamics using computer simulations of cellular
automata (CA)—a canonical discrete idealization of spatially extended systems.
[42, 41] Langton made a very important connection between the local behavior
of CA state transition rules and the collective phenomenon of criticality. He
introduced a (local) parameter, $\lambda$, to measure the proportion of state
transitions in a given CA rule that do not go to a basal (quiescent) state. CA
with different values of $\lambda$ have collective behaviors that closely
match distinct classes. In Langton’s account, the transition between order and
chaos takes place at $\lambda\approx 0.5$ where he observed the properties of
a second-order phase transition, such as in power-law distributed transients,
and the maximization of average mutual information between cells.
In the same period, Derrida and collaborators made very similar connections
between network parameters and collective critical behavior, but instead of
looking at spatially extended cellular automata—characterized by having fixed
and regular network structure—they focused on Kauffman’s RBNs, where nodes
have distinct state-transition functions, and networks have heterogeneous
structure. [24, 23] Indeed, Derrida and Pomeau defined what we refer to as the
_structural theory_ (ST) of criticality for BNs. [23] According to this
theory, if a BN has homogeneous in-degree, $k,$ and fixed bias, $p$, then the
critical boundary between ordered and chaotic network dynamics is given by,
$2kp(1-p)=1.$ (1)
The ST, as defined originally, holds for fixed connectivity and fixed bias. It
has since been shown that the same theory holds when connectivity is not
fixed, but normally distributed around a characteristic mean value. The same
applies to the bias. [4] While Eq. 1 is theoretically well-founded, it is not
an accurate predictor of dynamical regime, particularly if the BN dynamics are
near the critical edge. This is the case even for BNs that abide the most
strict, fixed in-degree and bias assumptions. We elaborate on this in the
following sections.
While Langton determined dynamical regime using properties such as transient
lengths, number of attractors, and attractor sizes in CA, Derrida and
colleagues formalized a quantitative collective measure known as the _Derrida
parameter_ , $\zeta$. [24] This parameter is derived from the Derrida plot: a
curve that shows the degree to which small perturbations in pairs of otherwise
identical initial configurations diverge in their dynamical trajectories. This
divergence is measured as the average number of different node-states (Hamming
distance) that separate the two initial trajectories after a number of time-
steps. The $\zeta$ parameter is the slope of the Derrida plot at the origin.
If $\zeta<1$, the BN is classified in the ordered regime. Conversely, if
$\zeta>1$, it is classified as chaotic. Thus a value $\zeta\approx 1$
indicates criticality. We use $\zeta$ to determine the dynamical regime of BN
ensembles in this research. (See § Methods for details.)
### Canalization
_Canalization_ [62] is used to characterize the buffering of genetic and
epigenetic perturbations that lead to the stability of phenotypic traits. [58,
60] Gene regulatory networks, for example, have the remarkable feature that
they tend to be made of highly canalizing regulatory interactions. [39, 30, 8,
20] Canalization has been studied by characterizing redundancy in the state
transition rules of automata. [38, 39, 52, 36, 46] It is observed when an
automaton’s state transition can be determined from the known state of a
subset of its inputs, which means the remaining inputs are contextually
irrelevant or redundant. [52] Canalization influences the dynamical behavior
of automata networks, contributes to their stability, [35, 8, 36, 43, 56]
increases the likelihood of modular attractors, and makes canalizing networks
more controllable by tuning external signals. [46, 27] Yet, canalization has
not previously been considered as a tuning parameter to define the structure
and dynamics of critical networks or to find the most biologically plausible
models of biochemical systems by targeting their stability and
criticality.[20]
Previous studies of the effects of canalization on network stability and
criticality have focused on the so-called _strictly canalizing_ state-
transition rules. [52] Such rules always have one input that, in at least one
of its possible states, is sufficient to determine the automaton’s state
transition. The idea behind these studies was to build BNs with strictly
canalizing state-transition rules, measure the _average sensitivity_ of their
nodes, and quantify the propensity of an automaton to change its state as the
result of perturbation to one of its inputs. [56] This node measure was then
extended to quantify sensitivity at the network level. Notably, the average
_network_ sensitivity is equivalent to the ST defined in Eq. 1 for predicting
criticality. [20] However, canalization is a much more frequent phenomenon
when we consider, not only strict canalization, but also _collective_
canalization. [52]
In Boolean automata, collective canalization is observed when a subset of
inputs, in some state combination, jointly determines an automaton’s state
transition. An automaton’s _effective connectivity_ , $k_{e}$, introduced by
authors Marques-Pita and Rocha,[46] is a measure of the expected minimal
number of inputs that are necessary to determine its state transitions. It
accounts for the existence of both strictly and collectively canalizing
inputs. If we consider how the original connectivity structure of a BN is
affected by canalization, it becomes clear that it is not a useful
representation of how control signals propagate in network dynamics because
canalizing rules make some edges in the original structure contextually
redundant. The roles of some edges in transmitting control signals vary,
however, in that some edges become completely redundant, or conversely,
essential, in different dynamical trajectories and attractors. There are many
possible _effective structures_ with very distinct dynamical behaviors for any
given in-degree network structure. [27] This must be considered in order to
understand how structure and dynamics account for control and criticality
together. [20, 27] Effective connectivity can easily be extended to a network
measure by computing the mean in-degree of the effective structure of a given
automata network. The effective _network_ connectivity characterizes both the
interaction structure and the canalization in one parameter.
We have addressed the limitations of the ST of criticality in automata
networks [20] in Eq. 1 by including the _effective network connectivity_ as a
tuning parameter to account for canalization. To test our main hypothesis that
this parameter is a better predictor of criticality than the in-degree
connectivity, $k$, we frame the prediction of dynamical stability as a binary
classification problem. We have produced a large dataset of random Boolean-
network ensembles with nearly 300K distinct networks generated under the same
assumptions made by ST concerning network connectivity, $k,$ and bias, $p$. To
include effective network connectivity as a tuning parameter, we have produced
a catalog of state-transition rules for each $k_{e}$ value across the values
of $k$ and $p$. Furthermore, we have analyzed a large set of 63 models of
biochemical regulation and signaling obtained from the Cell Collective
repository. [32, 31] For potential application areas, note that this
repository includes automata models on, for example, lac operon interaction,
T-cell receptor signaling, yeast cell cycle and apoptosis, cholesterol
regulation, Influenza A replication, Drosophila body segmentation, lymphocyte
differentiation, and cortical area development.
## Results
We define the effective connectivity of a given homogeneous BN as the mean
effective connectivity of its nodes, all of which have been sampled from a
small interval of size $\Delta k_{e}=0.5$. The characteristic mean value of
every such interval is denoted by $\langle k_{e}\rangle$. In addition, based
the principle of bias symmetry in logical rules, the compound term $p(1-p)$ is
set as an independent variable that represents the bias parameter. The
dynamical regime of every BN in our ensemble data is characterized by a binary
transformation of its Derrida parameter, $\zeta$, whereby $\zeta>1$ represents
the chaotic regime and $\zeta\leq 1$ represents the ordered or critical
dynamical regimes (considered together for classification purposes since
critical networks are found in the boundary between ordered and chaotic
networks). See § Methods for further details.
Figure 1: Dynamical regimes in the $(k,\langle k_{e}\rangle)$ parameter space.
Dynamical regime (ordered or chaotic) was determined using the Derrida
parameter $\zeta$ computed for the large RBN ensemble we generated (see main
text). Pie charts depict the dynamical regimes of RBN aggregates for the
possible $(k,\langle k_{e}\rangle)$ pairs in our RBN ensembles. Blue and red
areas indicate the proportions of networks with stable and chaotic dynamics,
respectively. The black dashed line corresponds to the critical $\langle
k_{e}\rangle$, as described in the main text. The critical boundary equation,
derived using binary symbolic regression in a linear model with an interaction
term, is $0.1k+0.7\langle k_{e}\rangle-0.1k\langle k_{e}\rangle=1$. Out of the
266,400 BNs in our RBN ensembles, 224,083 (approx. $84\%)$ are classified as
chaotic.
### The canalization theory of criticality
We search and optimize binary classifiers to predict the dynamical regime of
the RBNs in our ensemble dataset using six specific model classes of
increasing complexity. For each class, there is a model instance that
considers the original connectivity, $k$, and another that considers $\langle
k_{e}\rangle$ instead. All other elements of a given class are kept identical
in both instances; see § Methods for details. Figure 1 depicts the proportions
of chaotic and stable BNs in our ensemble dataset for the values of $k$ and
$\langle k_{e}\rangle$, as well as the best criticality decision boundary we
obtained when considering $k$ and $\langle k_{e}\rangle$ as tuning parameters.
The majority of BNs in our ensembles are classified as chaotic $(84\%)$, based
on the Derrida parameter. Therefore, cross-validation prediction performance
is best captured by measures tailored for unbalanced classification scenarios
such as the Matthews Correlation Coefficient (MCC). [7] We also show results
for McFadden’s $R^{2}$ since we are performing logistic regression, as well as
the Area Under the Curve (AUC) for ranking performance; see §Methods for
details.
Model class (1) has the lowest complexity, and serves to compare the
predictive power of the original network connectivity, $k$, with that of the
effective connectivity, $\langle k_{e}\rangle$, disregarding the bias
parameter. It yields the following decision boundaries: $-0.09k=1,$ and
$0.63\langle k_{e}\rangle=1$. The corresponding critical values for the tuning
parameters are $k=-11.11$, and $\langle k_{e}\rangle=1.59$. The prediction
superiority of $\langle k_{e}\rangle$ over $k$ is clear in this model class,
since the model instance based on, $k,$ classifies every BN as chaotic,
whereas the instance based on $\langle k_{e}\rangle$ partitions the data into
two reasonably correct dynamical regimes. Indeed, as shown in the first column
on the left in Figure 2, while MCC$(k)$ $\approx 0$, MCC$(\langle
k_{e}\rangle)\approx 0.49$, with similar behavior for $R^{2}$. Moreover,
AUC$(k)\approx 0.5$, while AUC$(\langle k_{e}\rangle)\approx 0.88$. Thus, the
best classifier based solely on in-degree $k$ is equivalent to a random coin
toss, while the best classifier based solely on effective connectivity $k_{e}$
yields reasonably good performancebbbTo further ascertain whether $k$ and
$k_{e}$ interact synergistically to predict criticality, we perform binary
logistic regression considering the linear effects of $k$ and $\langle
k_{e}\rangle$, as well as their interaction. The critical decision boundary of
the best such classifier is $0.1k+0.7\langle k_{e}\rangle-0.1k\langle
k_{e}\rangle=1$. However, the MCC $\approx 0.49$ is essentially the same as
$\langle k_{e}\rangle$ instance of model 1 , which demonstrates that adding
$k$ does not increase the classification performance of using $k_{e}$ alone.
This is also clear from the best interaction model where the coefficient of
$\langle k_{e}\rangle$ is seven times larger than the others.. The study of
model class (1), as well as the lack of synergy between $k$ and $k_{e}$,
demonstrates that the original network connectivity on its own carries no
information about criticality, but effective connectivity, on its own, yields
a reasonable prediction of criticality. This result strongly suggests that
dynamical canalization alone is an important factor in criticality.
Model class (2) is defined by the interaction between the term for the bias
parameter $p(1-p)$ and the term for either $k$ or $\langle k_{e}\rangle$ (see
§Methods). The optimal decision boundaries obtained are: $1.49kp(1-p)=1$, and
$3.93\langle k_{e}\rangle p(1-p)=1$. The corresponding performance metrics are
shown in the second column of Figure 2. Remarkably, all classification
performance measures for model instance $\langle k_{e}\rangle$ in class (2)
are very high, with near-perfect MCC and $R^{2}$ scores, and perfect ranking
performance measured by AUC, as detailed below. In contrast, the
classification performance for the model instance based on $k$ in the same
class is substantially, and significantly, lower (see also Figure 5). To
better understand the performance difference for models in class (2) consider
Figure 3. First, a very crisp boundary exists between stable and chaotic
dynamics in the space $(\langle k_{e}\rangle,p)$; the two regimes are more
neatly organized with almost no misclassifications beyond the critical
boundary. This is in sharp contrast with the less distinct boundary observed
in the $(k,p)$ space around the critical boundaries, predicted by both the ST
and the optimized the instance of the model based on $k$. Indeed, in these
cases substantial misclassifcations occur, whereby stable networks are
observed well into the predicted chaotic regime, and vice versa, especially
for the ST boundarycccNotice that the ST is defined by a slightly different
critical boundary (eq. 1) than what we obtained by optimizing model 2 $(k)$
against random ensemble data. This is likely because the ST was derived
theoretically while model 2 was derived from empirical data circumscribed to a
finite $k$ range. In any case, the ST is not optimal on this range and leads
to slightly worse classification performance, MCC$=0.73$ and $R^{2}=0.28$,
than model 2 $(k)$.. Notice, for instance, that BNs with stable dynamics are
observed for most values of $k$ when $p=0.5$ (the most adverse value of bias
for stability). The ST predicts most of these networks to be chaotic, in
Figure 3(A), but in Figure 3(B), in the $(\langle k_{e}\rangle,p)$ space,
these networks neatly cluster at $\langle k_{e}\rangle=1$, right on the
critical boundary. Similar behavior occurs for all bias values.
The classification performance together with the observation of the
arrangement of dynamical regimes around the critical boundaries in Figure 3
demonstrate that using effective connectivity ($\langle k_{e}\rangle$) instead
of the original connectivity ($k$) in RBNs leads to a much more accurate,
near-perfect prediction of the critical boundary that separates stable and
chaotic dynamics, as well as a more organized characterization of both
regimes. In other words, accounting for canalization and interaction bias at
the node- or micro-level, leads to optimal prediction of macro-level dynamics.
Indeed, model 2 $(\langle k_{e}\rangle)$ shows the most accurate decision
boundary for the critical boundary, with more complex models yielding no
increase in classification performance. We refer to this model as the
_canalization theory_ (CT) of criticality in BNs.
Figure 2: Performance scores for the regression models used to find the
optimal critical boundary. Each model belongs to one of six model
classes—labeled in increasing order of complexity. For each model class,
orange illustrates $k$ as a tuning parameter, and red, $\langle k_{e}\rangle$
instead. In every class, $\langle k_{e}\rangle$ is a better tuning parameter
for criticality than $k$. See §Methods for further details about classes and
performance measures. Figure 3: Dynamical regimes in the $(k,p)$ and $(\langle
k_{e}\rangle,p)$ parameter spaces. Dynamical regime (ordered, or chaotic) was
determined using the Derrida parameter $\zeta$ computed for the large RBN
ensemble we generated (see main text). Blue areas indicate proportions of
networks with ordered dynamics, and the red areas indicate the proportions
that were found to be chaotic. Panel (A) depicts the $(k,p)$ space, while
panel (B), the $(\langle k_{e}\rangle,p)$ space. The black dashed curves
represent criticality models in model class (2) described in the main text.
The dashed blue curve shown in (A) corresponds to the current criticality
model per the ST, shown for reference.
### The Canalization Theory optimizes complexity and classification
performance
We use a Pareto front method to optimize for decision boundaries that best
balance the trade-off between model complexity and classification performance.
This method relies on the graphical representation shown in Figure 4, which
depicts the performance of decision boundaries obtained from the different
model classes considered. A given boundary is marked with an arrow if and only
if its performance is greater than that of all models of lower complexity. In
short, performance increases substantially when passing from model class (1)
to (2), but not by using more complex model classes (3) and beyond. Indeed, in
model 2 $(\langle k_{e}\rangle)$, the CT, achieves near-perfect classification
performance with $\text{MCC}=0.96$ and $R^{2}=0.94$, and perfect ranking AUC
$\approx 1$. Therefore, more complex model classes could not improve much at
all over such performance. Interestingly, models based on $k$ do not show much
improvement in performance beyond model complexity class (2), even though the
performance of model 2 ($k$) is much smaller than that of 2 $(\langle
k_{e}\rangle)$ with MCC $=0.73$ and $R^{2}=0.58$—the former is 32% and the
latter is 64% smaller than the respective values for the model 2 instance
based on $\langle k_{e}\rangle$. In other words, even though there is much
room to improve, increasing the complexity of the models based on the original
connectivity, $k,$ does not yield performance gains. This implies that unless
canalization is factored in, as in the model instances based on $\langle
k_{e}\rangle$, no increase in performance is gained over the ST. We thus
conclude that model class (2) is optimal in terms of simplicity and
performance for both instances, but the instance that uses $\langle
k_{e}\rangle$ is considerably (and significantly as shown below) better at
predicting the dynamical regime of BNs.
Figure 4: Pareto front analysis of model complexity vs. performance for the
six model classes fit to RBN ensembles. Models are in increasing order of
complexity from class 1 to class 6. A model class is labeled on the axis only
if its performance is greater than the performances of all models of lower
complexity. For each model class, orange illustrates $k$ as a tuning
parameter, and red, $\langle k_{e}\rangle$ instead. Arrows mark the
performance of the optimal model class, characterized by a substantial rise
followed by very little gain afterward. Notice that for all performance
measures, model class 2 with $\langle k_{e}\rangle$ has the best Pareto front
performance.
### The classification performance of the CT is significantly better than that
of the ST
To estimate the statistical significance of the increased performance of the
CT, as well as to ensure that it does not derive from over-fitting the data,
we compare both instances of models in every class under cross validation
(details in §Methods). The statistical significance results for class (2) are
shown in Figure 5. All performance measures for the CT are significantly
better than for the model instance using $k$, based on paired-sample t-tests
$(P<0.001$). In addition, Vuong and Clarke tests and indicate similar results.
These cross-validation results also demonstrate that the performance of the CT
generalizes very well to out-of-sample data. All together, this analysis
supports the assertion that the CT predicts criticality in BNs significantly
better than does the ST. See S1 in the SI appendix for details.
Figure 5: Classification performance of models in class 2 under nested 4-fold
cross-validation. Significant differences ($P<0.001$) are indicated with
’***’. We use a one-sided paired-sample t-test to account for the alternative
hypothesis that the mean score of the CT (model with $k_{e}$) is _greater_
than that of the best instance of model class 2 with $(k)$—a class that
includes the ST.
### The CT via unconstrained symbolic regression
We performed an unconstrained search using Symbolic Regression [54] alongside
the constrained search reported previously in this section. We used the
symmetric effect of biases $p$ and $1-p$ on the Derrida parameter $\zeta$ to
justify using only $0<p\leq 0.5$ rather than using the compound term $p(1-p)$.
This type of unconstrained search works in a much larger space of model
classes, so finding an optimal model that also guarantees minimal class
complexity can be hard. Interpreting the fitting functions and coefficients
can also be difficult, as stochastic searches sometimes introduce artifacts in
the classifier. Despite these potentially limiting aspects inherent to
stochastic search algorithms, we obtained a high-performance classifier that
belongs to the same model class as the ST and the CT. The decision boundary
for this classifier is the function $3.125\langle k_{e}\rangle p=1$. The
performance of this classifier is $R^{2}=0.88$, and $\text{MCC}=0.93$, values
very similar to those of the CT. Additional information about the top
classifiers produced by symbolic regression is shown in appendix S2.
### The dynamics of systems biology models is very canalized and better
characterized by the CT
We analyze 63 Boolean models of biochemical networks that have been
experimentally validated. These models are from the Cell Collective
repository. [31] We refer to the set $2,979$ automata in these models that are
neither tautological nor contradictory as ${C}$. Approximately $48\%$ of the
automata in ${C}$ have one input $(k=1)$ so the connectivity of these cannot
be further reduced by computing $k_{e}$. Automata with $2\leq k\leq 9$ inputs
account for $50\%$ of $C$, and the remaining $2\%$ have $10\leq k\leq 15$
inputs, see Figure 6(A). We excluded automata with $k=1$, since they cannot be
reduced, and the very few automata with $k\geq 10$, were merged into a set
${C}^{*}$. The in-degree distribution $(k)$ of the automata in both sets $C$
and $C^{*}$ is highly right-skewed with skewness $\approx 2$. In addition,
both distributions are leptokurtic, with normalized $\text{kurtosis}\geq 5$.
We thus report the median and interquartile range as measures of central
tendency and dispersion: Med$({C}_{k})=2$, Med$({C}_{k}^{*})=3$ and
IQR$({C}_{k})$ = IQR$({C}_{k}^{*})=3-1$. The $k_{e}$ distributions for
automata in ${C}$ and ${C}^{*}$ are also heavily right-skewed with
approximately the same skewness $\approx 2$, and leptokurtic, with normalized
$\text{kurtosis}\geq 6$ in both cases. The medians and interquartile ranges
for $k_{e}$ are: Med$({C}_{k_{e}})=1.125$, Med$({C}_{k_{e}}^{*})=1.25$,
IQR$({C}_{k_{e}})=1.25-1$, and IQR$({C}_{k_{e}}^{*})=1.43-1.25$. Figure 6
shows the $k_{e}$ box plots of the automata in ${C}_{k_{e}}^{*}$, one for each
value of $k$. Being so heavily leptokurtic, most of the automata in ${C}^{*}$
have both in-degree $k$, and effective connectivity $k_{e}$, very close to the
respective central tendency, namely $k=3$, and $k_{e}\approx 1.25$. However,
the wider dispersion for $k$ suggests that effective connectivity _flattened_
the original in-degree distributions of the BN models considered and shows
that canalization is both very high and pervasive across different systems
biology models. See appendix S4 for additional details.
The dynamical regime of the Cell Collective models can be inferred from their
Derrida Parameter, $\zeta$, (§Methods), which varies very little:
$\text{IQR}(\zeta)=0.976-0.9$ and Min/Max range $\zeta\in[0.65,1.15])$. Only
eleven (out of 63) models have $\zeta>1$. The other 52 models have $\zeta$
values slightly below $\zeta=1$. The low dispersion, $\zeta\approx 1$, is a
strong indication that the Cell Collective models are in, or very close to,
the critical regime, validating what is known about them[20]. In Figure 7, we
show that the near-critical status of these models is not clear in the
$(\langle k\rangle,\langle p\rangle)$ space of the ST, but is quite clearly
revealed in the $(\langle k_{e}\rangle,\langle p\rangle)$ space of the CT. The
critical boundary curves are derived by fitting class-2 models representing
the ST and the CT to maximize the MCC score. While the networks are dispersed
mostly far from the boundary curve in the ST space, they cluster very near the
boundary in the CT space. Thus, the latter better characterizes the known
dynamics of these models, which are mostly near critical. Indeed, looking at
the AUC ranking measure, we have $\text{AUC (ST)}=0.54$, which is only
marginally better than a random toss, while $\text{AUC (CT)}=0.81$. In other
words, The ranking (by distance to boundary) is far superior for the CT. The
classification performance is also superior for the CT, even though the many
near-critical (and few chaotic) models make classification performance less
relevant: $\text{MCC (ST)}=0.44$, $\text{MCC (CT)}=0.58$.
A caveat to this analysis of the Cell Collective models should be noted. We
have developed the CT for homogeneous networks with fixed $k$ and $p$, but the
Cell Collective networks are heterogeneous. Therefore, we use the mean values
of these quantities in our analysis, as shown in Figure 7. While the CT can be
properly developed for heterogeneous networks in the future (see §Discussion),
here we derive new critical boundary curves by re-fitting both variants of
model class 2 to the heterogeneous Cell Collective data. Still, $c$
coefficients of the new curves are not very different from the optima found
for the homogeneous case (Figure 3): in the ($\langle k\rangle,\langle
p\rangle$) space, the new $c=1.03$ (was $c=1.49$ for homogeneous case), and in
the ($\langle k_{e}\rangle,\langle p\rangle$) space the new $c=3.2$ (for the
homogeneous case, it was $c=3.93$). The change in $c$ results in shifting the
boundary curves slightly to the right in the case of the heterogeneous
networks of the Cell Collective, thus increasing the area of the stable
regime. This is an expected result, since we know that heterogeneous
connectivity leads to more stable BN dynamics.[4] In summary, it is clear that
including canalizing dynamics in a model of criticality yields a substantially
better characterization (cf. AUC score) and prediction (cf. MCC score) of the
dynamics of systems biology automata network models.
Figure 6: Characterization of $k$ and $k_{e}$ for the automata in the 63 Cell
Collective BN models analyzed. Depicted is the subset of automata that have
two or more inputs $(52\%ofthetotal)$, denoted in the main text by $C^{*}$.
The median value for $k$ is Med$(C_{k}^{*})=3$, while for $k_{e}$,
Med$(C_{k_{e}}^{*})=1.25$. The low median values (and low dispersion for
$k_{e}$; see main text) indicate, not only that there is a pervasive
canalization in validated BN models of biochemical systems, but also that
effective connectivity ‘flattens’ the original degree distributions. On
average, knowing the state of 1.25 inputs is sufficient to determine the state
transitions of these automata. Figure 7: Predicted dynamic regimes of Cell
Collective BNs by ST (panel A) and CT (panel B). Blue dots denote stable
models ($\zeta<1$), and red dots denote chaotic models ($\zeta>1$). The axes
are labeled with the mean value of the relevant tuning parameters for each of
the 63 BN models considered. The critical boundary curves are shown in blue
and have been derived by fitting class-2 models to maximize the MCC score.
## Discussion
### The CT is more accurate in predicting criticality than the ST and belongs
to the same model class
Previous studies of criticality in automata networks have relied on the ST,
which characterizes networks and their critical boundary in the $(k,p)$ space.
The CT introduced here includes the effects of node-level canalization and
characterizes networks and their critical boundary in the $(\langle
k_{e}\rangle,p)$ space instead. In this new space, the criticality boundary
leads to much more accurate predictions (Figs. 2, 4, & 5), and also reveals a
much more organized dynamical regime space in both random ensembles (Fig. 3)
and systems biology models (Fig. 7). Notably, the CT belongs to the same model
class as the STdddWe pursued both class-constrained and unconstrained
regression analysis, leading to almost identical critical boundaries in the
same model class. The Pareto-optimal model class is of the form $c\kappa
p(1-p)$, where the network connectivity term $\kappa$ is the original in-
degree ($k$) in the ST or the effective connectivity ($k_{e}$) in our new CT
(See §Methods). The bias of state transition rules in the network is denoted
$p$, and coefficient $c$ defines where the curve is positioned in the relevant
parameter space (the smallest value of $\kappa$ when $p=1/2$). Thus, in both
theories, the tuning of criticality depends on interaction between the
connectivity and bias parameters. However, our work reveals that a correct
measure of connectivity needs to include the influence from canalization that
derives from node (state-transition) dynamics. Canalization at the micro-level
of node dynamics defines the true connectivity of automata networks and thus
ultimately their macro-level dynamical regime. Importantly, the prediction
performance of the CT vis a vis that of the ST demonstrates that criticality
depends not only on structural connectivity and bias, but also very
significantly on canalizing dynamics. Indeed, a prediction of criticality
without bias (model class 1 in §Results) shows that effective connectivity
alone yields a reasonable prediction performance, but in-degree alone does not
(§Results and Figs. 1, 2, & 4).
### Effective connectivity captures characteristic properties of dynamical
regime
In the space of $2^{2^{k}}$ possible logical rules for a given $k$, there are
only $(2^{k})-2$ distinct values of $p$ when tautologies and contradictions
are ignored, and this number is halved when taking into account the principle
of bias symmetry in Boolean functions. The ST implicitly assumes that all
functions of same $k$ and $p$ contribute in the same way to dynamical regime.
We demonstrate, however, that the finer characterization of the canalized
logic of individual automata is necessary to accurately predict the dynamical
regime of automata networks. In Figure 3(A), homogeneous networks of the same
size whose nodes are automata with the exact same $k$ and $p$ are shown to
have opposite dynamical regimes, even far from the critical boundary of the
ST. In contrast, when we transform the critical phase transition space to the
finer characterization enabled by $k_{e}$, as in Figure 3(B), networks with
the same $p$ and $\langle k_{e}\rangle$ almost always display the same
dynamical regime—except very near the CT critical boundary—as demonstrated by
a near-perfect MCC score (§Results). Notice further that in this latter case,
networks are not homogeneous in $k_{e}$ and are thus grouped by $\langle
k_{e}\rangle$. Therefore, some variation in dynamical regime for the same $p$
and $\langle k_{e}\rangle$ is expected. Even so, such variation is only
observed near the critical boundary, which demonstrates that $k_{e}$ (and its
mean value in the BN) is very characteristic of the dynamical regime. Finally,
note that $k_{e}$ includes the contribution of collective canalization, while
other measures of canalization such as sensitivity do not (§Methods). This
means that the nonlinear effects of collective canalization are included and
contribute to the finer characterization of criticality that the CT provides.
### Effective structure is more homogeneous than original structure
While we are aware that the ST has been extended to consider heterogeneous
BNs—with, for example, power-law distributions [4, 25]—we have not yet
considered such an extension for the CT. One reason is that the BN models of
biochemical regulation and random ensembles used here are not large enough to
properly distinguish heterogeneous degree distributions. [14] Another
important reason, as this study reveals, is that the original interaction
structure of the BN is replaced by canalized dynamics that instantiates a more
homogeneous effective structure with low-degree distributions. Indeed, a
consistent observation in our results is that $\langle k_{e}\rangle\ll k$ for
most automata both in the random BN ensembles and in the 63 heterogeneous
Boolean models of biochemical regulation and signaling that we
analyzed—$k_{e}=k$ only for the two parity functions for each $k$.[46]
Furthermore, $k_{e}$ is significantly smaller in Cell Collective automata than
for same size and bias random automata.[26] Therefore, the ubiquitous
canalization (redundancy) present in automata nodes can dramatically alter the
original interaction structure of a network, revealing a truer effective
structure that takes canalizing dynamics into account.
It is known that such effective structure affects the dynamics and
controlability of BNs.[46, 27] While effective structure can be easily
computed [18] and used to uncover control pathways in biochemical regulation
and signaling, [26, 46] we do not yet know how its topology is organized
across random and real-world networks. The evidence presented here for the
systems biology models in the Cell Collective indicates that the effective
structure is much more homogeneous than the original interaction structures,
as demonstrated by the small dispersion of $k_{e}$ values in comparison to the
dispersion of $k$ (§Results). This suggests that very heterogeneous biological
regulation and signaling networks (lognormal or asymptotic power-law degree
distributions) may effectively function dynamically with more homogeneous and
low-degree distributions. An exhaustive study of the topology of effective
structure is still needed to investigate this hypothesis. The present
research, however, offers much evidence that the canalizing dynamics that
defines an underlying effective structure is an important factor in
determining critical dynamics in random and biochemical networks.
### Beyond criticality: harnessing canalization in complex systems
The theoretical development and experimental results we present provide a new
theory of criticality that accounts for canalization, the CT. Based on the
same class of functions, the new theory does not increase the complexity of
the current theory, but increases substantially and significantly the ability
to accurately predict the dynamical regime of automata networks. Given that
automata networks are canonical examples of complex multivariate dynamical
systems, the high classification accuracy of the new theory strongly suggests
that canalization is a prime mechanism for tuning the dynamical regime of
complex systems. Indeed, our results with systems biology models suggest that
canalization plays a fundamental role in the dynamics of biochemical
regulation and signaling, which is missed by studying the structure of
biochemical interactions alone. Therefore, beyond the study of criticality, a
precise characterization of canalization is likely to enable the tailoring of
interventions in complex systems towards desirable dynamical behavior.[46, 27]
The concept of effective connectivity underlying the CT integrates information
about the structure and dynamics of multivariate interactions—in-degree
connectivity and input redundancy in state transitions, respectively. It
implies that the behavior and function of complex systems is dictated by an
effective structure that is revealed only after removal of causal redundancy
in the logic of how variables integrate input signals. This truer structure of
interactions is a more accurate portrait of causal multivariate dynamics,
which is more canalized than the original structure of interactions implies.
This is why we find stable (or critical) dynamics in networks whose structure
would be predicted by the current ST to be chaotic, and vice versa (see
Figures 3 & 7). In this sense, canalization is a network-level mechanism that
can be tailored by evolution. Going forward, the methodology can provide
powerful analytical tools to uncover the causal pathways that determine
control and resilience to interventions in various complex systems,[18] such
as genetic regulation in biological development,[46] and treatment strategies
in cancer and other diseases.[26]
## Methods
### Boolean automata definitions and notation
A _Boolean automaton_ is a binary variable, $x\in\\{0,1\\}$, where state 0 is
interpreted as _false_ (_off_ or _unexpressed_), and state 1 as _true_ (_on_
or _expressed_). The states of $x$ are updated in discrete time-steps, $t$,
according to a _Boolean state transition rule_ of $k$ inputs:
$x^{t+1}=f\left(i_{1}^{t},...,i_{k}^{t}\right)$. Therefore
$f:\\{0,1\\}^{k}\rightarrow\\{0,1\\}$. Such a rule can be defined by a
_Boolean logic formula_ or by a _look-up (truth) table_ (LUT) with $2^{k}$
entries. Each LUT entry of an automaton $x$, $f_{\alpha}$, is defined by (1) a
specific _condition_ , which is a conjunction of $k$ inputs represented as a
unique $k$-tuple of input-variable (Boolean) states, and (2) the automaton’s
_next state_ (transition) $x^{t+1}$, given the condition. We denote the entire
state transition rule of an automaton $x$ in its LUT representation as
$F\equiv\\{f_{\alpha}:\alpha=1,...,2^{k}\\}$.
### Boolean networks
A _Boolean Network_ (BN) is a graph $\mathcal{B}\equiv(X,E)$, where $X$ is a
set of $n$ Boolean automata _nodes_ $x_{i}\in X,i=1,...,n$, and $E$ is a set
of directed edges $e_{ji}\in E:x_{i},x_{j}\in X$. If $e_{ji}\in E$, then
automaton $x_{j}$ is an input to automaton $x_{i}$, as computed by $F_{i}$.
$X_{i}=\\{x_{j}\in X:e_{ji}\in E\\}$ which denotes the set of input automata
of $x_{i}$. Its cardinality, $k_{i}=|X_{i}|$, is the _in-degree_ of node
$x_{i}$, which determines the size of its LUT, $|F_{i}|=2^{k{{}_{i}}}$. We
refer to each entry of $F_{i}$ as $f_{i:\alpha},\alpha=1...2^{k{{}_{i}}}$. At
any given time $t$, $\mathcal{B}$ is in a specific _configuration_ of node
states, $\boldsymbol{x}^{t}=[x_{1},x_{2},...,x_{n}]$. We use the terms _state_
for individual automata $(x)$ and _configuration_ $(\boldsymbol{x})$ for the
collection of states of the set of automata of $\mathcal{B}$, i.e., the
collective network state. Starting from an initial configuration,
$\boldsymbol{x}^{0}$, the nodes of a BN are updated with a _synchronous_ or
_asynchronous_ policy. The _dynamics_ of $\mathcal{B}$ is thus defined by the
temporal sequence of the $2^{n}$ possible configurations that ensue. The
transitions between configurations can be represented as a _state transition
graph_ , STG, where each vertex is a configuration, and each directed edge
denotes a transition from $\boldsymbol{x}^{t}$ to $\boldsymbol{x}^{t+1}$. The
STG of $\mathcal{B}$ thus encodes the network’s entire _dynamical landscape_.
Under the synchronous updating scheme (used in the studies reported in this
paper) configurations that repeat, such that
$\boldsymbol{x}^{t+\mu}=\boldsymbol{x}^{t}$, are known as _attractors_ ;
_fixed point_ when $\mu=1$, and _limit cycle_ , with period $\mu$, when
$\mu>1$. The disconnected subgraphs of a STG that lead to an attractor are
known as _basins of attraction_. A BN $\mathcal{B}$ has a finite number of
attractors, $b$, each denoted by $\mathcal{A}_{i}:i=1,...,b$.
### Effective Connectivity
The effective connectivity ($k_{e}$) tallies the expected number of inputs of
an automaton $x_{i}$ that are _minimally sufficient_ to determine an its state
transitions. When a subset of such minimal inputs is in a certain state
combination, the remaining inputs are effectively redundant—they can be in any
state with no effect on the transition of $x_{i}$. These effective inputs, or
_enputs_ for short, can be identified using the schema redescription
methodology introduced by Marques-Pita and Rocha, [46] which we illustrate
next. The formula for the logic rule OR with two inputs can be written as
$x=i_{1}\lor i_{2}$. The Truth Table for this expression can be redescribed as
wildcard schemata as follows: $F^{\prime}_{1}=\\{(1,\\#),(\\#,1)\\}$ and
$F^{\prime}_{0}=\\{(0,0)\\}$, where $F^{\prime}_{1}$ denotes the set of
wildcard schemata that prescribe transitions to 1 (ON), and conversely,
$F^{\prime}_{0}$ denotes the wildcard schemata prescribing transitions to 0
(OFF), a set that contains only one schema in this case. The wildcard symbol
‘#’ in a schema denotes a redundant input state. For example, $(1,\\#)$ is
interpreted as follows: given $i_{1}=1$, then the transition $x^{t+1}=1$ is
guaranteed, regardless of the state of $i_{2}$. A closer look at
$F^{\prime}_{1}$ reveals that only one input is necessary to settle
transitions to 1 (ON) in this example, and this is the case for the OR rule
with any number of inputs. The entire set of schemata for a given automaton
can be used to determine its effective connectivity. This requires the
computation of the average _minimal_ number of enputs necessary to determine
its state transition. Effective connectivity is computed from the upper bound
on _input redundancy_ , [46] yielding a sum of the minimal number of enputs
required to settle each of the possible $2^{k}$ state transitions specified in
the automaton’s LUT. This value is then divided by $2^{k}$ to obtain $k_{e}$.
For this computation we iterate over the entire LUT of the automaton; for each
LUT entry we accumulate the number of enputs of the wildcard schema matched,
with the largest number of wildcard symbols; once all LUT entries have been
processed, the final accumulated sum is divided the the LUT size. In our
example $k_{e}=1.25$. This is the case since _three_ of the _four_ look-up
entries in the LUT have _one_ of the inputs in the _on_ state, which is
sufficient to settle the transition, while one of the entries requires _two_
$(i_{1}=0,i_{2}=0)$, so in this case $k_{e}=[(3\times 1)+(1\times 2)]/4$, see
[46] for details. Note that $k_{e}\leq k$ and that the higher the difference
between $k_{e}$ and $k$, the more canalization there is in the automaton rule,
and also, the lower the effective connectivity the automaton will have as a
node in a BN.
Other measures of canalization in Boolean automata exist and have been linked
to criticality, such as _average sensitivity_ , [56] and the more general _c-
sensitivity_.[34] Effective connectivity presents several advantages over
these measures. First and foremost, it is designed to capture collective
canalization,[46] a very common non-linear phenomenon in automata whereby a
subset of inputs jointly determine the state of an automaton, while rendering
redundant the complement subset of inputs.[52] In contrast, sensitivity
independently aggregates the influence (activity) of each individual input to
an automaton. It is thus a linear measure of canalization. This means that
effective connectivity provides a more nuanced and realistic measurement of
canalization that includes non-linear effects.[45, 26] For instance, even for
automata of $k=2$, sensitivity does not discriminate between such common
Boolean functions as conjunction/disjunction and proposition/negation:
$s(x_{1}\land x_{2})=s(x_{1}\lor x_{2})=s(x_{1})=s(\lnot x_{1})=1$. Effective
connectivity, on the other hand, correctly accounts for the additional
collective canalization that is present in the conjunction/disjunction (and
other) functions: $k_{e}(x_{1}\land x_{2})=k_{e}(x_{1}\lor x_{2})=5/4=1.25$,
while $k_{e}(x_{1})=k_{e}(\lnot x_{1})=1$. Since non-linear, collective
canalization increases with $k$,[52, 26] the finer characterization of the
phenomenon provided by effective connectivity becomes more relevant as well.
Interestingly, both sensitivity and effective connectivity can be easily
computed from our schema description methodology,[26] which is available in
the CANA Python package. [18] Finally, ‘$c$-sensitivity’ [34] extends
sensitivity to subsets of $c$ inputs, but it results in a vector of $k$ values
for each $c$, which is much less amenable to the regression analysis of
criticality boundaries we pursue in this study than is the scalar value
measured by $k_{e}$.
### Generation of RBN ensembles
Each of the ensembles of RBNs that we produced for this study is characterized
by a set of tuning parameters, namely $(k,k_{e},p)$. The network connectivity
$k$ is a fixed (homogeneous) variable. This means that in our ensembles every
node $x_{i}$ is connected to $k$ nodes. The effective connectivity is the mean
value in a small interval (bin), and the bias is also fixed (homogeneous).
Note that the values of these parameters are always homogeneously distributed,
in alignment with the assumptions made by the ST in Eq. 1. For a given value
combination of $(k,k_{e},p)$ a single random BN is generated by choosing: (1)
for each constituent node, a random set of $k$ input nodes; and (2) a random
Boolean automaton with $k$ inputs, output-bias $p$, and effective connectivity
in a small range $k_{e}\pm\epsilon$ from an existing catalog. The reason for
binning $k_{e}$ is that the possible values for this parameter vary
significantly for each combination of $k$ and $p$, which leads to a sparse
matrix of viable ensembles $(k,k_{e},p)$, where viability is determined by the
existence of Boolean state transition rules that satisfy specific combinations
of the parameter values (see appendix S3 for further details). Thus, without
loss of information, we bin $k_{e}$ using a small bin size $\epsilon=0.25$
leading to $k_{e}$ being homogeneously distributed in regular intervals of
size $\Delta k_{e}=0.5$, and to a more dense matrix of viable ensembles.
Because the values of $k_{e}$ are binned, we refer to the $k_{e}$ tuning
parameter as $\langle k_{e}\rangle$. Producing a random Boolean automaton with
a given $(k,p)$ is simple: (1) generate an all-zeroes vector of length
$2^{k}$; (2) assign the state one (_on_) to $(2^{k})p$ LUT random entries in
the resulting vector; and (3) assume the updated vector represents the state
transitions of the automaton in the lexicographic order of input combinations.
To control for $k_{e}$, we generate a catalog of Boolean automata with a large
number of $(k,k_{e},p)$ value combinations, from which automata with the
appropriate parameter values are picked during the generation of the RBN
ensembles. The catalogs for Boolean rules of $k={2,3,4}$ are exhaustive. For
larger $k$, automata are first obtained by random generation for a given $k$
and $p$, with their $k_{e}$ subsequently computed. The number of possible
automata for a given $k$ and $p$ is $\binom{2^{k}}{p(2^{k})}$. Thus, for
$k>4$, the catalogs contain a random sample of $10^{4}$ Boolean rules for each
$(k,p)$ if the total number possible is greater than $10^{4}$, and all the
Boolean state transition rules otherwise. Additionally, to obtain automata
with $k_{e}$ in ranges essentially inaccessible to random generation via $k$
and $p$ alone, we use a genetic algorithm. We refer the interested reader to
appendix S3 for details. We have considered the following ranges for our
tuning parameters: the number of nodes per network $N=100$,
$k\in\\{2,3,4,6,8\\}$, $p=[0.01,0.5]$ with $\Delta p=1/2^{k}$, and $\langle
k_{e}\rangle=[1,k]$ with $\Delta k_{e}=0.5$. By sweeping the space of values
for our ensemble parameters we have generated a total of 266.4K RBNs.
### Computation of the Derrida parameter
For a given BN, we compute the $\zeta$ parameter [24, 23, 39] by first
generating $I=250$ random initial configurations, and producing an almost
identical copy for each, where the copy differs only in the state of a small
number $m$ of states that have been perturbed (flipped). We set this value to
be a random integer $m\in[1,..,N/10]$. Second, allowing the BN to advance each
pair of initial configurations (original and perturbed) for $t$ time steps; we
set $t=1$. Third, computing the Hamming distance between the two resulting
configurations. Fourth, for each value of $m$, averaging the Hamming distances
obtained in the previous step and and plotting them against $m$ to produce the
Derrida plot. Finally, fifth, calculating $\zeta$ as the slope of the Derrida
plot at the origin. A value of $\zeta=1$ indicates criticality. A value above
(below) this is interpreted as meaning the BN is in the chaotic (stable)
dynamical regime.
### Constrained search for decision boundaries
The dataset we produce contains individual RBNs, each characterized by the
independent variables $k,p$, and $k_{e}$, and with one dependent variable with
value one (1) if $\zeta>1$ (chaos), and zero (0) otherwise. We perform binary
logistic regression to identify the decision boundary separating dynamic
regimes using a set of predefined model classes. The general form of all
models in every class is: R=step(logistic(model)), where the output of the
logistic function is the probability that the dependent variable has value one
(chaotic regime). The output of the step function is the predicted binary
value of the dependent variable given a threshold $\tau=0.5$. If the output of
the step function for the BN variables in a given model is greater than $\tau$
then the classifier predicts that BN to be in the chaotic regime, and
critical/stable otherwise. Each model tested belongs to one of the following
model classes, where $\kappa$ is the in-degree $k$ in the ST or the mean
effective connectivity $\langle k_{e}\rangle$ in the CT, listed in increasing
order of model complexity. Model complexity is defined by the number of terms
and the number of predictors in each term (in that order):
1. 1.
$c_{1}\kappa;$
2. 2.
$c_{1}\kappa p(1-p);$
3. 3.
$c_{1}\kappa+c_{2}p(1-p);$
4. 4.
$c_{1}\kappa+c_{2}\kappa p(1-p);$
5. 5.
$c_{1}\kappa p(1-p)+c_{2}p(1-p);$
6. 6.
$c_{1}\kappa+c_{2}\kappa p(1-p)+c_{3}p(1-p);$
In our binary logistic regression we use the $p(1-p)$ as a single independent
variable accounting for the bias, rather than just $p$ due to the principle of
duality in Boolean logic. The coefficients derived for each criticality model
are used to construct a decision surface. For this, the resulting equations
have been manipulated so that the independent variables and their coefficients
are on the left-hand side and the value (1) on the right-hand side, thus
facilitating comparisons with the ST.
### Performance measures
Mc-Fadden’s $\text{R}^{2}$ is a standard goodness-of-fit measure used for
logistic regression models. It is computed as one minus the ratio of the log-
likelihood of the model to that of the intercept-only model. [15] The maximum
value of this pseudo $\text{R}^{2}$ is 1. The MCC is ideal for computing
classification performance in unbalanced scenarios, [7] such as the one
studied here, whereby there are many more instances of chaotic automata
networks in the random ensembles than instances of stable network dynamics.
Computed for the classifier using model predictions and test data, it is
defined as a function of the number of true positives (TP), false positives
(FP), true negatives (TN) and false negatives (FN): $MCC=\frac{TP\times TN-
FP\times FN}{\sqrt{(TP+FP)(TP+FN)(TN+FP)(TN+FN)}}$. [7] The MCC ranges between
-1 and 1, where -1 indicates perfect opposite classification, 1 indicates
perfect classification, and 0 indicates random classification. Here, the
positive label is associated with the chaotic dynamical regime $R=1$, and the
negative label with the stable (stable/critical) regime $R=0$. The AUC is
defined as a function of the true positive rate (TPR), the proportion of true
positives in the total number of positive instances, false positive rate
(FPR), and the proportion of false positives in the total number of negative
instances, as follows: $AUC=\int_{1}^{0}TPR(T)FPR^{\prime}(T)dT$. The AUC
ranges between 0 and 1, for perfectly incorrect and correct classification at
the endpoints, respectively. A random classifier yields a value of $0.5$. It
is interpreted as the probability with which the classifier ranks positive
instances (label 1) higher than negative instances (label 0). [29]
### Cross-validation
The full dataset was randomly split into 4 non-overlapping equally sized
partitions ($75\%-25\%$ training and testing splits). This was repeated 4
times, thus yielding _outer foldings_. A similar procedure was followed on
each of the training splits, yielding a total of 16 training-testing pairs
(see appendix S1 for further details). Measures of classification and
regression performance (as with the full dataset) on the testing splits were
collected. The 16 sets of performance scores were averaged to produce an
estimate of generalization performance score for each measure. Between-model
comparisons were made using pair-sample t-tests because the two models were
evaluated on the same set of sixteen test folds. The paired t-tests were one-
sided with the alternative hypothesis that the mean score of model 2 $(\langle
k_{e}\rangle)$ is greater than that of model 2 $(k)$.
### Symbolic regression
A supplemental study was performed using a different curve fitting method to
find the critical decision surface. We used symbolic regression (a type of
unconstrained search), which is, in essence, a genetic programming algorithm.
[54] The symmetric effect of the biases $p$ and $1-p$ on the Derrida parameter
was used to prune the search space by considering $0<p\leq 0.5$ only. Note
that symbolic regression works in a much larger space of many function classes
than the space of six model classes considered in our main methodology.
Because of this, it can be hard to find an optimal function that is both
consistent and guarantees minimal complexity. Furthermore, the obtained
classifiers and coefficients can be hard to interpret in some cases. One of
the relevant uses of this kind of method is to find different models for a
given classification problem, for example, and compare them. One of the
benefits of this is to help in determining suitable function classes to
describe a classification decision boundary.
Symbolic regression was performed on our dataset from different (random) seeds
eight times. We allowed for any formula in evolving populations that included
basic arithmetic operators, coefficients, exponents, the sine, cosine, and
logarithmic functions. In every execution of the algorithm we consistently
obtained a classifier with the same function form based on an interaction
between $k_{e}$ and $p$ with a coefficient that varied slightly in different
runs. The ensembles were defined in the same way as in the main methods with
the only difference that we used networks of size $N=48$ instead of $N=100$.
The best classifier found was the function $3.125\langle k_{e}\rangle p=1$,
with performance values very close to those of the CT. See appendix S2 for
further details.
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## Acknowledgements
MMP Acknowledges input and discussions about the original ideas with Prof.
Christof Teuscher and Prof. Melanie Mitchell (Portland State University, USA),
as well as funding provided by Fundação para a Ciência e a Tecnologia
(Portugal) grant PTDC/EIA-CCO/114108/2009. The authors thank Deborah Rocha for
thorough line editing. LMR was partially funded by the National Institutes of
Health, National Library of Medicine Program, grant 01LM011945-01, by a
Fulbright Commission fellowship, by NSF-NRT grant 1735095 “Interdisciplinary
Training in Complex Networks and Systems,” and by Fundação para a Ciência e a
Tecnologia (Portugal) grant PTDC/EIA-CCO/114108/2009
## Author contributions statement
MMP and LMR conceived hypothesis and research rationale. SM, MMP and LMR
designed and executed the experiments, analyzed the data, and wrote the paper
## Additional information
Competing interests The authors do not declare any conflicts of interest.
|
# Fundamental measure theory of inhomogeneous two-body correlation functions
S. M. Tschopp J. M. Brader Department of Physics, University of Fribourg,
CH-1700 Fribourg, Switzerland
###### Abstract
For the three-dimensional hard-sphere model we investigate the inhomogeneous
two-body correlations predicted by Rosenfeld’s fundamental measure theory. For
the special cases in which the density has either planar or spherical symmetry
we provide analytic formulae for the Hankel and Legendre transforms,
respectively, of the inhomogeneous two-body direct correlation function as
explicit functionals of the density. When combined with the Ornstein-Zernike
relation our analytical results allow for rapid calculation of inhomogeneous
hard-sphere density correlations in real-space. These provide not only
information about the packing structures of the hard-sphere system, but also
form an essential building-block for constructing perturbation theories of
more realistic models.
## I Introduction
Two-body correlation functions give important information about the
microstructural particle arrangement in a classical fluid. In the presence of
an external field the density becomes nonuniform and the corresponding
inhomogeneous two-body correlations can deviate significantly from those in
bulk, e.g. for fluids at interfaces or under spatial confinement. For systems
interacting via a pair potential these deviations provide direct access to
interfacial thermodynamic quantities, such as the surface tension between
coexisting phases [1, 2, 3].
The most familiar theoretical approach to calculating two-body correlations is
the method of integral equations, based on closures of the Ornstein-Zernike
(OZ) equation [4]. Approximations such as the Percus-Yevick (PY) or the
hypernetted-chain have been widely used to study thermodynamics and two-body
correlations in bulk [5], where translational invariance enables fast Fourier
transform methods to be employed to great advantage in numerical calculations.
For inhomogeneous systems this luxury is absent; the two-body correlation
functions generally depend upon two vector arguments. However, for systems in
which the density has a simple geometry (usually planar or spherical) the OZ
equation can be reduced to a more manageable form. In such cases,
generalizations of the bulk closure approximations have been used to calculate
the inhomogeneous two-body correlations (see Refs.[6, 7, 8, 9, 10, 11, 12, 13,
14] for examples and chapter four in Ref.[2] for an overview).
An alternative approach is to use classical density functional theory (DFT).
Within the DFT framework, correlation functions are generated by successive
functional differentiation of the excess Helmholtz free energy functional with
respect to the density. The functional contains complete statistical
information about the system and can thus be used to calculate correlation
functions of any order. Calculation of the inhomogeneous two-body correlations
proceeds in the following way: (i) minimize the grand potential functional to
obtain the equilibrium one-body density, (ii) evaluate the two-body direct
correlation function (generated by taking two functional derivatives of the
excess Helmholtz free energy) at the equilibrium density and then solve the OZ
equation for the two-body total correlation function. No closure is required,
as the direct correlation function is uniquely specified by the generating
functional. This two-step scheme, sometimes referred to as the ‘Ornstein-
Zernike route’ is often used to obtain bulk two-body correlations (in which
case the equilibrium density is a trivial constant), but is more rarely
exploited to address inhomogeneous systems (for some examples see, e.g.
Refs.[15, 16, 17]).
The most well-studied model in liquid-state theory is the hard-sphere system.
In a classic 1989 paper Rosenfeld introduced a geometrically-based fundamental
measure theory (FMT) density functional for hard-spheres [18]. The predictions
of Rosenfeld FMT for the one-body density profile were found to be in
excellent agreement with computer simulation data for a wide variety of
external fields [19]. Although the original FMT encountered difficulties for
strongly confined fluids and ordered states, subsequent versions of hard-
sphere FMT corrected these shortcomings. The FMT, in common with other DFT
approximations, is usually employed to obtain the equilibrium one-body density
profile in a given external potential. Higher-body correlation functions are
typically only evaluated in bulk [18] and inhomogeneous pair and higher-order
correlation functions from FMT remain largely unexplored. This is perhaps
surprising, given that the analytic formulae for the direct correlation
functions present an obvious (and computationally advantageous) alternative to
the inhomogeneous integral equation closures mentioned above. A deeper
investigation of higher-order FMT correlation functions would not only provide
insight into the structure of hard-sphere FMT, possibly suggesting
improvements, but is also needed for the construction of perturbation theories
aiming to describe more realistic inhomogeneous fluids.
In this paper we will address these issues and analyze in detail the
inhomogeneous two-body correlations generated by FMT. We focus on situations
for which the one-body density exhibits either planar or spherical symmetry
and derive analytic formulae for the Hankel (planar geometry) and Legendre
(spherical geometry) transforms of the inhomogeneous two-body FMT direct
correlation function. These explicit functionals of the (one-dimensional)
density profiles then provide rapid access to the direct and total pair
correlation functions in real-space. Our results for hard-spheres will be
tested against the inhomogeneous PY integral equation theory and existing
Monte-Carlo data. Once the quality of the FMT correlations has thus been
established we will show how these can be exploited as input to a recently
developed perturbative density functional theory for treating systems with
attractive interactions [20].
The paper will be structured as follows: In Section II we briefly outline
relevant aspects of classical DFT. In Section III we introduce the FMT and
give explicit formulae for the Hankel and Legendre transforms of the two-body
direct correlation function in planar and spherical geometry, respectively. In
Section IV we present numerical results for the inhomogeneous total
correlation function of hard-spheres confined between planar walls and in the
presence of a fixed test particle. In Section V we show how our results for
hard-spheres can be used as input to a perturbation theory of attractive
interactions. In Section VI we give results obtained using this perturbation
theory for the well-known hard-core Yukawa model. Finally, in Section VII, we
discuss our findings and give an outlook for future investigations.
## II Density functional theory
DFT is an exact formalism for the study of classical many-body systems in
external fields [1, 2, 3]. The central object of interest is the grand
potential functional
$\displaystyle\Omega[\,\rho\,]=F^{\rm id}[\,\rho\,]+F^{\rm
exc}[\,\rho\,]-\int\\!d{\bf r}\big{(}\mu-V_{\rm ext}({\bf r})\big{)}\rho({\bf
r}),$ (1)
where $\mu$ is the chemical potential, $V_{\rm ext}({\bf r})$ is the external
potential and $\rho({\bf r})$ is the one-body ensemble averaged density. The
square brackets indicate a functional dependence. The Helmholtz free energy of
the ideal gas is exactly given by
$\displaystyle F^{\rm id}[\,\rho\,]=k_{B}T\int\\!d{\bf r}\,\rho({\bf
r})\left(\,\ln(\rho({\bf r}))-1\,\right),$ (2)
where $k_{B}$ is the Boltzmann constant, $T$ is the temperature and we have
set the thermal wavelength equal to unity. The excess Helmholtz free energy,
$F^{\rm exc}$, includes all information regarding the interparticle
interactions and usually has to be approximated. The grand potential satisfies
the variational condition
$\displaystyle\frac{\delta\Omega[\rho\,]}{\delta\rho({\bf r})}=0.$ (3)
This yields the following Euler-Lagrange equation for the equilibrium one-body
density
$\displaystyle\rho({\bf r})=e^{-\beta\left(V_{\rm ext}({\bf r})-\mu-
k_{B}Tc^{(1)}({\bf r})\right)},$ (4)
where the one-body direct correlation function is generated from the excess
Helmholtz free energy by a functional derivative
$\displaystyle c^{(1)}({\bf r})=-\frac{\delta\beta F^{\rm
exc}}{\delta\rho({\bf r})}.$ (5)
Substitution of the solution of (4) into (1) yields the equilibrium grand
potential, thus providing access to all thermodynamic properties of the
system.
Information about the two-body correlations in the inhomogeneous fluid can be
obtained from a second functional derivative of the free energy
$\displaystyle c^{(2)}({\bf r}_{1},{\bf r}_{2})=-\frac{\delta^{2}\beta F^{\rm
exc}}{\delta\rho({\bf r}_{1})\delta\rho({\bf r}_{2})},$ (6)
where $c^{(2)}$ is the two-body direct correlation function. The connection
between $c^{(2)}$ and the total correlation function, $h$, can be established
by considering the functional derivative of equation (3) with respect to the
external field
$\displaystyle\frac{\delta^{2}\Omega}{\delta V_{\rm ext}({\bf
r}_{1})\delta\rho({\bf r}_{2})}=0.$ (7)
While the vanishing of this mixed derivative is a trivial consequence of (3),
it is nevertheless a useful result. Explicit calculation of the left-hand side
of (7) yields the OZ equation
$\displaystyle h({\bf r}_{1},{\bf r}_{2})=c^{(2)}({\bf r}_{1},{\bf
r}_{2})\\!+\\!\int\\!d{\bf r}_{3}\,h({\bf r}_{1},{\bf r}_{3})\rho({\bf
r}_{3})c^{(2)}({\bf r}_{3},{\bf r}_{2}),$ (8)
Note that the external potential does not appear explicitly in the OZ
equation, this information is implicitly contained within the one-body
density.
## III Hard-spheres
### Fundamental measure theory
We now focus our attention on a system of three-dimensional hard-spheres of
radius $R$. Within FMT the excess Helmholtz free energy is approximated by an
integral over a function of weighted densities [18]
$\displaystyle\beta F^{\rm exc}_{\rm hs}[\,\rho\,]=\int d{\bf
r}_{1}\;\Phi\left(\left\\{n_{\alpha}({\bf r}_{1})\right\\}\right).$ (9)
The original Rosenfeld formulation of FMT employs the following reduced excess
free energy density
$\Phi=-n_{0}\ln(1-n_{3})+\frac{n_{1}n_{2}-{\bf n}_{1}\cdot{\bf
n}_{2}}{1-n_{3}}+\frac{n_{2}^{3}-3n_{2}{\bf n}_{2}\cdot{\bf
n}_{2}}{24\pi(1-n_{3})^{2}}.$ (10)
The weighted densities are generated by convolution
$n_{\alpha}({\bf r}_{1})=\int d{\bf r}_{2}\;\rho({\bf
r}_{2})\,\omega_{\alpha}({\bf r}_{1}-{\bf r}_{2}),$ (11)
where the weight functions, $\omega_{\alpha}$, are characteristic of the
geometry of the spheres. Of the six weight functions, four are scalars
$\displaystyle\omega_{3}({\bf r})$ $\displaystyle=\Theta(R-r),\hskip
14.22636pt\omega_{2}({\bf r})=\delta(R-r),$ $\displaystyle\omega_{1}({\bf r})$
$\displaystyle=\frac{\delta(R-r)}{4\pi R},\hskip 14.51074pt\omega_{0}({\bf
r})=\frac{\delta(R-r)}{4\pi R^{2}},$
and two are vectors
$\displaystyle\omega_{\mathbb{2}}({\bf r})$
$\displaystyle=\mathrm{\mathbf{e}}_{{\bf r}}\,\delta(R-r),\hskip
14.22636pt\omega_{\mathbb{1}}({\bf r})=\mathrm{\mathbf{e}}_{{\bf
r}}\frac{\delta(R-r)}{4\pi R},$
where $\mathrm{\mathbf{e}}_{{\bf r}}={\bf r}/r$ is a unit vector. The presence
of summations over both vector and scalar weights in many FMT expressions
presents some notational difficulty and the analytical calculations below
demand clarity regarding the scalar or vector character of various functions.
We have thus chosen to employ the symbol $\omega$ for all weight functions,
both scalar and vector, where the latter will be distinguished by employing a
bold font index. This choice also enables us to use the convenient notation
$\omega_{|{\mathbb{2}}|}\\!=\\!\omega_{2}$ and
$\omega_{|{\mathbb{1}}|}\\!=\\!\omega_{1}$.
Applying the definition (5) to the free energy (9) generates the following
approximate form for the one-body direct correlation function
$\displaystyle c^{(1)}_{\rm hs}({\bf r}_{1})=-\sum_{\alpha}\int d{\bf
r}_{2}\,\Phi^{{}^{\prime}}_{\alpha}({\bf r}_{2})\,\omega_{\alpha}({\bf
r}_{21}),$ (12)
where $\Phi^{{}^{\prime}}_{\alpha}\\!=\\!\partial\Phi/\partial n_{\alpha}$,
${\bf r}_{21}={\bf r}_{2}-{\bf r}_{1}$ and the summation runs over all scalar
and vector indices. Consistent with our established notation the function
$\Phi^{\prime}_{\alpha}$ is a vector quantity when $\alpha$ takes the value
${\mathbb{1}}$ or ${\mathbb{2}}$, in which case a scalar product is implied in
equation (12). While equation (12) makes an appearance in practically all FMT
studies, the FMT approximation to the two-body direct correlation function is
less frequently encountered and its general structure thus deserves some
careful attention. Taking two functional derivatives of the free energy (9)
generates the following expression
$\displaystyle c^{(2)}_{\rm hs}({\bf r}_{1},{\bf r}_{2})$
$\displaystyle=-\\!\sum_{\alpha\beta}\int\\!d{\bf r}_{3}\,\omega_{\alpha}({\bf
r}_{31})\,\Phi^{{}^{\prime\prime}}_{\alpha\beta}({\bf
r}_{3})\,\omega_{\beta}({\bf r}_{32}),$
$\displaystyle\equiv-\sum_{\alpha\beta}c_{\alpha\beta}({\bf r}_{1},{\bf
r}_{2})$ (13)
where $\Phi^{{}^{\prime\prime}}_{\alpha\beta}\\!=\\!\partial^{2}\Phi/\partial
n_{\alpha}\partial n_{\beta}$. The terms, $c_{\alpha\beta}({\bf r}_{1},{\bf
r}_{2})$, contributing to the sum in (13), can be separated into three
distinct classes according to the values of the $\alpha$ and $\beta$ indices:
1\. both scalars, 2. one scalar and one vector, and 3. both vectors. Each of
these three classes involves a function $\Phi^{\prime\prime}_{\alpha\beta}$ of
different tensorial rank (for convenience all first and second derivatives of
$\Phi$ are given explicitly in Appendix A).
For terms belonging to class 1, the products in equation (13) are self-
explanatory, as the second derivative $\Phi^{\prime\prime}_{\alpha\beta}$ and
the weight functions are all scalars. For class 2 terms, we have one scalar
weight, one vector weight and a vectorial second derivative function. A scalar
product between the vector weight and $\Phi^{\prime\prime}_{\alpha\beta}$ is
thus implied. For example, if $\alpha\\!=\\!2$ and $\beta\\!=\\!{\mathbb{2}}$
then the corresponding term in the sum (13) is given by
$c_{2{\mathbb{2}}}({\bf r}_{1},{\bf r}_{2})=\\!\int d{\bf
r}_{3}\,\omega_{2}({\bf
r}_{31})\,\Phi^{{}^{\prime\prime}}_{2{\mathbb{2}}}({\bf
r}_{3})\cdot\omega_{\mathbb{2}}({\bf r}_{32}).$ (14)
For terms in class 3, we have two vector weights and
$\Phi^{\prime\prime}_{\alpha\beta}$ is a second rank tensor. For example, the
term with $\alpha\\!=\\!{\mathbb{2}}$ and $\beta\\!=\\!{\mathbb{2}}$ is given
by a quadratic form
$c_{{\mathbb{2}}{\mathbb{2}}}({\bf r}_{1},{\bf r}_{2})=\\!\int d{\bf
r}_{3}\,\omega_{\mathbb{2}}({\bf
r}_{31})\cdot\Phi^{{}^{\prime\prime}}_{{\mathbb{2}}{\mathbb{2}}}({\bf
r}_{3})\cdot\omega_{\mathbb{2}}({\bf r}_{32}).$ (15)
The Helmholtz free energy density of the Rosenfeld FMT (10) is quadratic in
the vector weighted densities. This has the simplifying consequence that
$\Phi^{\prime\prime}_{\alpha\beta}$ for class 3 terms is proportional to the
unit tensor. We note that this would generally not be the case for FMT
approximations involving an extended set of weight functions (e.g. the
Tarazona FMT [21]).
### Inhomogeneous Percus-Yevick closure
An alternative approach to obtaining the inhomogeneous pair correlations is to
supplement the OZ equation (8) by a second (usually local) closure relation
between the pair direct correlation function and the total correlation
function, and then to solve self-consistently the two coupled equations. A
closure which is known to work well for hard-spheres is the PY approximation
[4, 22, 7]
$\displaystyle h_{\rm hs}({\bf r}_{1},{\bf r}_{2})$
$\displaystyle=-1\;\;\;\text{for}\;|{\bf r}_{1}-{\bf r}_{2}|<2R,$
$\displaystyle c_{\rm hs}^{(2)}({\bf r}_{1},{\bf r}_{2})$
$\displaystyle=0\;\;\;\;\;\;\text{for}\;|{\bf r}_{1}-{\bf r}_{2}|>2R.$ (16)
The first of these relations is the exact ‘core condition’ reflecting the
impossibility of hard-sphere overlaps. The PY theory can be solved exactly in
bulk and yields an expression for the pair direct correlation function
identical to that generated by the Rosenfeld FMT. However, as this agreement
occurs only in bulk, care should be taken not to label the Rosenfeld FMT as
the ‘PY functional’. For inhomogeneous situations the predictions of equation
(13) for any given density profile will differ from the solution of the
coupled equations (8) and (III). In particular, the FMT expression (13) will
not satisfy exactly the core condition, although it may provide a good
approximation. The PY theory has been shown to perform well in a variety of
inhomogeneous situations [7, 11, 10] and we will thus use it as a benchmark
for assessing the quality of the pair correlations generated by FMT.
### FMT in planar geometry
When the external field has planar symmetry the density only varies as a
function of a single cartesian coordinate, which we take to be the $z$-axis.
The inhomogeneous pair correlations thus exhibit cylindrical symmetry and
require as input two coordinates, $z_{1}$ and $z_{2}$, and a cylindrical
radial distance, $r$, separating them (see Fig.1). In this case the OZ
equation (8) can be simplified using a Hankel transform (two dimensional
Fourier transform) in the plane orthogonal to $z$. The Hankel transform of the
pair direct correlation function is given by
$\displaystyle\overline{c}^{\,(2)}_{\rm
hs}(z_{1},z_{2},k)=2\pi\int_{0}^{\infty}\\!dr\,rJ_{0}(kr)c^{(2)}_{\rm
hs}(z_{1},z_{2},r),$ (17)
where $k$ is the absolute value of the two-dimensional wavevector
$\mathrm{\mathbf{k}}$ and $J_{0}$ is a Bessel function. The back-transform is
given by
$\displaystyle c^{(2)}_{\rm
hs}(z_{1},z_{2},r)=\frac{1}{2\pi}\int_{0}^{\infty}\\!dk\,kJ_{0}(kr)\,\overline{c}^{\,(2)}_{\rm
hs}(z_{1},z_{2},k).$ (18)
Analogous expressions can be written for the total correlation function.
Hankel transform of the OZ equation (8) yields (see Appendix B)
$\displaystyle\overline{h}_{\rm hs}(z_{1},z_{2},k)$
$\displaystyle=\overline{c}^{\,(2)}_{\rm hs}(z_{1},z_{2},k)$ (19)
$\displaystyle+\int_{-\infty}^{\infty}\\!dz_{3}\;\overline{h}_{\rm
hs}(z_{1},z_{3},k)\,\rho(z_{3})\,\overline{c}^{\,(2)}_{\rm
hs}(z_{3},z_{2},k).$
If $\rho$ and $\overline{c}^{\,(2)}_{\rm hs}\\!$ are known functions, then
(19) becomes a linear integral equation for the remaining unknown
$\overline{h}_{\rm hs}$.
12$r_{sph}$$r$$z_{1}$$z_{2}$
Figure 1: Sketch of the planar geometry.
Equation (13) gives the general FMT approximation to the pair direct
correlation function as a functional of the three-dimensional density, but is
not in a form suitable for numerical implementation. This is probably the
reason why (13) has not been exploited for the development of liquid-state
theory. In the following we will show that the Hankel transform of equation
(13) can be reduced to an expression which allows for rapid and precise
numerical evaluation of the pair correlations for any given planar density
profile.
This ‘FMT route’ to the hard-sphere pair correlations is computationally
efficient for a number of reasons: (i) The iterative solution of the linear
integral equation (19) is both rapid and stable. (ii) The equations can be
solved entirely in Hankel space with no need to back-transform to real-space
during the iteration loop. (iii) The inhomogeneous pair correlation functions
can be determined for a given value of $k$, independently of all other
wavevectors. Calculations can thus be performed in parallel for different
$k$-values. It is worth to compare this comfortable situation with the demands
of solving numerically the nonlinear PY integral equation theory (equations
(8) and (III)) where we observe: (i) The iterative convergence rate is very
slow at high densities and small Broyles mixing parameters must be employed to
maintain stability [4, 19]. (ii) The OZ equation (19) is treated in Hankel
space, whereas the closure (III) can only be implemented in real-space. This
prevents parallel computation and demands an expensive back-and-forth Hankel
transformation at each iteration step.
Hankel transform of the two-body direct correlation function (13) generates a
sum of terms
$\overline{c}^{(2)}_{\rm
hs}(z_{1},z_{2},k)=-\sum_{\alpha\beta}\overline{c}_{\alpha\beta}(z_{1},z_{2},k).$
(20)
The main building blocks for each of the terms in (20) are the Hankel
transformed scalar weight functions
$\displaystyle\overline{\omega}_{3}(z_{1},z_{2},k)$
$\displaystyle=2\pi\frac{R_{12}}{k}\Theta_{12}J_{1}\left(kR_{12}\right),$
$\displaystyle\overline{\omega}_{2}(z_{1},z_{2},k)$ $\displaystyle=2\pi
R\Theta_{12}J_{0}\left(kR_{12}\right),$
$\displaystyle\overline{\omega}_{1}(z_{1},z_{2},k)$
$\displaystyle=\frac{\overline{\omega}_{2}(z_{1},z_{2},k)}{4\pi R},$
$\displaystyle\overline{\omega}_{0}(z_{1},z_{2},k)$
$\displaystyle=\frac{\overline{\omega}_{2}(z_{1},z_{2},k)}{4\pi R^{2}},$ (21)
where $\Theta_{12}\\!\equiv\\!\Theta\left(R-|z_{12}|\right)$ is the Heaviside
step function, $R_{12}^{2}\\!\equiv\\!R^{2}-z_{12}^{2}$ and
$z_{12}=z_{1}-z_{2}$.
In the discussion below equation (13) we identified three classes of terms
appearing in the sum, grouped according to the values of the pair of indices
$\alpha$ and $\beta$. For terms belonging to class 1 the steps involved in
transforming $c_{\alpha\beta}$ are identical to those required to transform
the OZ equation (see Appendix B). This yields for
$\alpha,\beta\in\\{0,1,2,3\\}$ the following one-dimensional integral
$\displaystyle\overline{c}_{\alpha\beta}(z_{1},z_{2},k)=\\!\int_{a}^{\,b}\\!\\!dz_{3}\;\overline{\omega}_{\alpha}(z_{3},z_{1},k)\Phi^{{}^{\prime\prime}}_{\alpha\beta}(z_{3})\,\overline{\omega}_{\beta}(z_{3},z_{2},k),$
(22)
for $|z_{12}|\leq 2R$ and zero otherwise. The integration limits are a
consequence of the finite range of the weight functions and are given by
$a=\max(z_{1},z_{2})-R$ and $b=\min(z_{1},z_{2})+R$, respectively.
The mixed terms belonging to class 2 have one scalar and one vector index. For
example, if we have $\alpha\in\\{0,1,2,3\\}$ and
$\beta\in\\{{\mathbb{1}},{\mathbb{2}}\\}$, then we must consider the following
scalar product
$\displaystyle\Phi^{{}^{\prime\prime}}_{\alpha\beta}(z_{3})\cdot\omega_{\beta}({\bf
r}_{32})=|\Phi^{{}^{\prime\prime}}_{\alpha\beta}(z_{3})|\,\omega_{|\beta|}({\bf
r}_{32})\,\mathrm{\mathbf{e}}_{z_{3}}\cdot\mathrm{\mathbf{e}}^{\,\rm
shell}_{32},$
where we define an (outwards pointing) unit vector, orthogonal to the surface
of the spherical delta-shell centered at ${\bf r}_{2}$:
$\displaystyle\mathrm{\mathbf{e}}^{\,\rm shell}_{32}=\begin{cases}\frac{{\bf
r}_{3}-{\bf r}_{2}}{R}&|{\bf r}_{3}-{\bf r}_{2}|=R,\vspace*{0.2cm}\\\ \hskip
10.81218pt0&{\rm otherwise}.\end{cases}$ (23)
The scalar product is obtained by simple trigonometry,
$\mathrm{\mathbf{e}}_{z_{3}}\\!\cdot\,\mathrm{\mathbf{e}}^{\,\rm
shell}_{32}=z_{32}/R$. As this result depends only on $z_{3}$ and $z_{2}$ its
presence in the integrand of (13) does not interfere with the Hankel
transformation and the standard procedure outlined in Appendix B can be
applied without modification. We thus obtain the following expression
$\displaystyle\overline{c}_{\alpha\beta}(z_{1},z_{2},k)=\frac{1}{R}\int_{a}^{\,b}\\!dz_{3}$
$\displaystyle\;\overline{\omega}_{\alpha}(z_{3},z_{1},k)\,|\Phi^{{}^{\prime\prime}}_{\alpha\beta}(z_{3})|$
$\displaystyle\times z_{32}\;\overline{\omega}_{|\beta|}(z_{3},z_{2},k),$ (24)
for $|z_{12}|\leq 2R$ and zero otherwise. The same considerations apply when
the rank of the indices is exchanged,
$\alpha\in\\{{\mathbb{1}},{\mathbb{2}}\\}$ and $\beta\in\\{0,1,2,3\\}$. This
yields
$\displaystyle\overline{c}_{\alpha\beta}(z_{1},z_{2},k)=\frac{1}{R}\int_{a}^{\,b}\\!dz_{3}$
$\displaystyle\;\overline{\omega}_{|\alpha|}(z_{3},z_{1},k)\,z_{31}$
$\displaystyle\times\;|\Phi^{{}^{\prime\prime}}_{\alpha\beta}(z_{3})|\;\overline{\omega}_{\beta}(z_{3},z_{2},k),$
(25)
for $|z_{12}|\leq 2R$ and zero otherwise.
$r_{sph}$$r_{2}$$|z_{1}-z_{2}|$$\bullet$$\bullet$$\bullet$$1$$2$$3$$\gamma$
Figure 2: Geometrical sketch for evaluation of the scalar product given in
(27).
Terms in class 3 have a product of two vector weight functions,
$\alpha,\beta\in\\{{\bf 1},{\bf 2}\\}$, and are more difficult to deal with.
For the original FMT used in this work the second derivative tensor can be
expressed as
$\Phi^{{}^{\prime\prime}}_{\alpha\beta}\equiv\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}{\mathbb{1}}$,
where $\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}$ is a scalar
function and ${\mathbb{1}}$ is the unit tensor. This enables us to simplify
the quadratic form in the integrand of $\overline{c}_{\alpha\beta}$ to a
scalar product between unit vectors
$\displaystyle\omega_{\alpha}({\bf
r}_{31})\cdot\Phi^{{}^{\prime\prime}}_{\alpha\beta}(z_{3})\cdot\omega_{\beta}({\bf
r}_{32})=$ (26)
$\displaystyle\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}(z_{3})\,\omega_{|\alpha|}({\bf
r}_{31})\,\omega_{|\beta|}({\bf r}_{32})\;\mathrm{\mathbf{e}}^{\,\rm
shell}_{31}\cdot\mathrm{\mathbf{e}}^{\,\rm shell}_{32}.$
In Fig.2 we sketch the intersection of a pair of delta-shells centered at
${\bf r}_{1}$ and ${\bf r}_{2}$, representing a product of weight functions
$\omega_{\alpha}({\bf r}_{31})\,\omega_{\beta}({\bf r}_{32})$ for
$\alpha,\beta\in\\{1,2\\}$. The values of the integration variable ${\bf
r}_{3}$ which yield a nonzero contribution to (26) lie on the intersection
circle of the delta-shells. For such cases the points ${\bf r}_{1}$, ${\bf
r}_{2}$ and ${\bf r}_{3}$ define an isosceles triangle with fixed angles. If
we choose $z_{1}$ as the axis of our cylindrical coordinate system then it is
a straightforward geometrical exercise to show that for ${\bf r}_{3}$ anywhere
on the intersection circle
$\displaystyle\mathrm{\mathbf{e}}^{\,\rm
shell}_{31}\cdot\mathrm{\mathbf{e}}^{\,\rm
shell}_{32}\equiv\cos(\gamma)=1-\frac{z_{12}^{2}}{2R^{2}}-\frac{r_{2}^{2}}{2R^{2}},$
(27)
where $\gamma$ is defined in Fig.2. Due to our identification of $z_{1}$ with
the cylindrical coordinate axis the variable $r_{1}$ does not appear in (27).
We thus seek to evaluate the Hankel transform of
$\displaystyle
c_{\alpha\beta}(z_{1},z_{2},r_{2})=c_{\alpha\beta}^{A}(z_{1},z_{2},r_{2})+c_{\alpha\beta}^{B}(z_{1},z_{2},r_{2})$
(28)
where the two contributions are given by
$\displaystyle c_{\alpha\beta}^{A}(z_{1},z_{2},r_{2})=\\!\\!\int\\!\\!d{\bf
r}_{3}\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}(z_{3})\omega_{|\alpha|}({\bf
r}_{31})\,\omega_{|\beta|}({\bf
r}_{32})\\!\\!\left(\\!1\\!-\\!\frac{z_{12}^{2}}{2R^{2}}\\!\right)\\!,$
$\displaystyle c_{\alpha\beta}^{B}(z_{1},z_{2},r_{2})=-\\!\\!\int\\!\\!d{\bf
r}_{3}\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}(z_{3})\omega_{|\alpha|}({\bf
r}_{31})\,\omega_{|\beta|}({\bf
r}_{32})\\!\left(\frac{r_{2}^{2}}{2R^{2}}\right)\\!.$
The factor $1-z_{12}^{2}/2R^{2}$ appearing in the first of these contributions
is independent of the radial coordinate. The Hankel transformation of
$c_{\alpha\beta}^{A}$ thus proceeds in the same way as for the OZ equation
(see Appendix B) and yields
$\displaystyle\overline{c}^{\,A}_{\alpha\beta}(z_{1},z_{2},k)=\left(1-\frac{z_{12}^{2}}{2R^{2}}\right)\mathcal{A}_{\alpha\beta}(z_{1},z_{2},k),$
(29)
for $|z_{12}|\leq 2R$ and zero otherwise, where the function
${\mathcal{A}}_{\alpha\beta}$ is given by
${\mathcal{A}}_{\alpha\beta}(z_{1},z_{2},k)\\!=\\!\\!\int_{a}^{\,b}\\!\\!\\!dz_{3}\,\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}(z_{3})\overline{\omega}_{|\alpha|}(z_{3},z_{1},k)\,\overline{\omega}_{|\beta|}(z_{3},z_{2},k).$
Hankel transform of $c_{\alpha\beta}^{B}$ is complicated by the presence of
the factor $r_{2}^{2}$. Following again the procedure outlined in Appendix B,
we find that the first step of the calculation can be carried through easily,
leading to
$\displaystyle c^{\,B}_{\alpha\beta}(z_{1},z_{2},r_{2})$
$\displaystyle=-\frac{r_{2}^{2}}{4\pi
R^{2}}\int_{a}^{\,b}\\!\\!dz_{3}\,\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}(z_{3})$
(30)
$\displaystyle\times\int_{0}^{\,\infty}\\!\\!dk^{\prime}k^{\prime}\,\overline{\omega}_{\alpha}(z_{3},z_{1},k^{\prime})\,\overline{\omega}_{\beta}(z_{3},z_{2},k^{\prime})\,J_{0}(k^{\prime}r_{2}).$
It is the second step of the calculation (Hankel transformation with respect
to the external coordinate $r_{2}$) which presents difficulties. Applying the
integral operator $2\pi\\!\int_{0}^{\infty}dr_{2}\,r_{2}J_{0}(kr_{2})$ to (30)
yields the Hankel transformation
$\displaystyle\\!\overline{c}^{\,B}_{\alpha\beta}(z_{1},z_{2},k)\\!=\\!\frac{1}{4\pi
R^{2}}\\!\int_{a}^{\,b}\\!\\!\\!dz_{3}\,\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}(z_{3})\\!\\!\int_{0}^{\,\infty}\\!\\!\\!dk^{\prime}k^{\prime}\,\overline{\omega}_{\alpha}(z_{3},z_{1},k^{\prime})$
$\displaystyle\times\overline{\omega}_{\beta}(z_{3},z_{2},k^{\prime})\left[2\pi\\!\\!\int_{0}^{\,\infty}dr_{2}\,r_{2}J_{0}(kr_{2})\left(-r_{2}^{2}J_{0}(k^{\prime}r_{2})\right)\right]\\!.$
(31)
To make progress we must evaluate the integral in square brackets; the Hankel
transform of $-r_{2}^{2}J_{0}(k^{\prime}r_{2})$. Given a test function,
$f(r)$, which vanishes sufficiently rapidly as $r\\!\rightarrow\\!\infty$, it
can be shown that the Hankel transform of $-r^{2}f(r)$ is given by
$\displaystyle\overline{-r^{2}f}(r)$
$\displaystyle=\frac{d^{2}\overline{f}(k)}{dk^{2}}+\frac{1}{k}\frac{d\overline{f}(k)}{dk}.$
(32)
Setting $f(r)=J_{0}(k^{\prime}r)$ and using the result (81) yields
$\displaystyle\overline{-r_{2}^{2}J_{0}}(k^{\prime}r_{2})=(2\pi)^{2}\left(\delta^{\prime\prime}(\mathrm{\mathbf{k}}-\mathrm{\mathbf{k^{\prime}}})+\frac{1}{k}\delta^{\prime}(\mathrm{\mathbf{k}}-\mathrm{\mathbf{k^{\prime}}})\right),$
(33)
where the prime(s) on the $\delta$-functions express the first(second)
derivative with respect to $k$. Using the known properties of delta-function
derivatives the expression (III) thus becomes
$\displaystyle\overline{c}^{\,B}_{\alpha\beta}(z_{1},z_{2},k)$
$\displaystyle=\frac{2\pi^{2}}{(4\pi
R)^{\delta_{1|\alpha|}+\delta_{1|\beta|}}}\int_{a}^{\,b}\\!\\!dz_{3}\,\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}(z_{3})$
(34) $\displaystyle\times\left(\frac{\partial^{2}}{\partial
k^{2}}+\frac{1}{k}\frac{\partial}{\partial
k}\right)\Big{(}J_{0}\left(kR_{13}\right)J_{0}\left(kR_{23}\right)\Big{)},$
where we have used the explicit expressions for the transformed weight
functions (21) to introduce the Bessel functions. The derivatives of the
Bessel function product yield the following expression
$\displaystyle\left(\frac{\partial^{2}}{\partial
k^{2}}+\frac{1}{k}\frac{\partial}{\partial
k}\right)\Big{(}J_{0}\left(kR_{13}\right)J_{0}\left(kR_{23}\right)\Big{)}=$
(35)
$\displaystyle-\frac{1}{2}\left(R_{13}^{2}+R_{23}^{2}\right)J_{0}\left(kR_{13}\right)J_{0}\left(kR_{23}\right)$
$\displaystyle+2R_{13}R_{23}J_{1}\left(kR_{13}\right)J_{1}\left(kR_{23}\right)$
$\displaystyle+\frac{1}{2}R_{13}^{2}J_{2}\left(kR_{13}\right)J_{0}\left(kR_{23}\right)+\frac{1}{2}R_{23}^{2}J_{0}\left(kR_{13}\right)J_{2}\left(kR_{23}\right)$
$\displaystyle-\frac{1}{k}\Bigl{(}R_{13}J_{1}\left(kR_{13}\right)J_{0}\left(kR_{23}\right)+R_{23}J_{0}\left(kR_{13}\right)J_{1}\left(kR_{23}\right)\Bigr{)}.$
Putting the results (34) and (35) together with (28) and (29) yields the final
result for the Hankel transform of the class 3 contributions with
$\alpha,\beta\in\\{{\bf 1},{\bf 2}\\}$
$\overline{c}_{\alpha\beta}(z_{1},z_{2},k)=\left(1-\frac{z_{12}^{2}}{2R^{2}}\right)\mathcal{A}_{\alpha\beta}(z_{1},z_{2},k)\\\
+\frac{2\pi^{2}}{(4\pi
R)^{\delta_{1|\alpha|}+\delta_{1|\beta|}}}{\mathcal{B}}_{\alpha\beta}(z_{1},z_{2},k),$
(36)
for $|z_{12}|\leq 2R$ and zero otherwise, where
$\displaystyle{\mathcal{B}}_{\alpha\beta}(z_{1},z_{2},k)=\int_{a}^{\,b}\\!\\!dz_{3}\,\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}(z_{3})$
$\displaystyle\times\bigg{\\{}-\frac{R_{13}^{2}+R_{23}^{2}}{2}J_{0}\left(kR_{13}\right)J_{0}\left(kR_{23}\right)$
$\displaystyle+2R_{13}R_{23}J_{1}\left(kR_{13}\right)J_{1}\left(kR_{23}\right)+\frac{R_{13}^{2}}{2}J_{2}\left(kR_{13}\right)J_{0}\left(kR_{23}\right)$
$\displaystyle+\frac{R_{23}^{2}}{2}J_{0}\left(kR_{13}\right)J_{2}\left(kR_{23}\right)-\frac{1}{k}\Bigl{(}R_{13}J_{1}\left(kR_{13}\right)J_{0}\left(kR_{23}\right)$
$\displaystyle+R_{23}J_{0}\left(kR_{13}\right)J_{1}\left(kR_{23}\right)\Bigr{)}\bigg{\\}}.$
To summarize, equations (20), (22), (III), (III), and (36) provide the Hankel
transform of the FMT pair direct correlation function as an explicit
functional of the one-dimensional planar density profile. Given
$\overline{c}^{(2)}_{\rm hs}$ the total correlation function,
$\overline{h}_{\rm hs}$, can be calculated using the OZ relation (19). The
pair correlations in real-space can then be obtained via (numerical) Hankel
back-transformation using (18).
### FMT in spherical geometry
When the external field has spherical symmetry the density only varies as a
function of the distance from the origin. The inhomogeneous pair correlations
thus require as input the two radial distances, $r_{1}$ and $r_{2}$, and the
cosine of the angle between them, $x_{12}\\!=\\!\cos(\,\theta_{12})$ (see
Fig.3). In planar geometry the transformed weight functions (21) depend only
on the separation $z_{12}$, whereas the analogous expressions in spherical
geometry depend on both arguments $r_{1}$ and $r_{2}$. In addition, the
transformed weight functions in spherical geometry change their functional
forms whenever one (or both) of these arguments approaches the origin to
within a distance $R$. With the intention of sparing the reader technical
overload, the results to be presented below will be restricted to cases
satisfying both $r_{1}\\!\geq\\!R$ and $r_{2}\\!\geq\\!R$, If other situations
arise, as would be the case for e.g. hard-spheres confined to a spherical
cavity, then the methods to be discussed below could be easily generalized. In
the following we will consider only test-particle calculations, for which the
density, $\rho(r)$, is zero for $r\\!<\\!R$.
12$r_{1}$$r_{2}$$\theta_{12}$
Figure 3: Sketch of the spherical geometry.
The Legendre transform of the pair direct correlation function is given by
$\displaystyle\hat{c}^{\,(2)}_{\rm
hs}(r_{1},r_{2},n)=\frac{2n+1}{2}\int_{-1}^{1}\\!dx_{12}\,P_{n}(x_{12})c^{(2)}_{\rm
hs}(r_{1},r_{2},x_{12}),$ (37)
where $P_{n}(x_{12})$ is a Legendre polynomial. The back-transform is given by
$\displaystyle c^{(2)}_{\rm
hs}(r_{1},r_{2},x_{12})=\sum_{n=0}^{\infty}P_{n}(x_{12})\,\hat{c}^{\,(2)}_{\rm
hs}(r_{1},r_{2},n).$ (38)
Legendre transform of the OZ equation (8) simplifies the three-dimensional
integral and yields the following equation for the transforms (see Appendix C)
$\hat{h}_{\rm hs}(r_{1},r_{2},n)=\hat{c}^{\,(2)}_{\rm hs}(r_{1},r_{2},n)\\\
+\frac{4\pi}{2n+1}\int_{0}^{\infty}\\!dr_{3}\;r_{3}^{2}\,\hat{h}_{\rm
hs}(r_{1},r_{3},n)\,\rho(r_{3})\,\hat{c}^{\,(2)}_{\rm hs}(r_{3},r_{2},n).$
(39)
The Legendre transform of the two-body direct correlation function (13) can be
expressed as the following sum
$\hat{c}^{(2)}_{\rm
hs}(r_{1},r_{2},n)=-\sum_{\alpha\beta}\hat{c}_{\alpha\beta}(r_{1},r_{2},n),$
(40)
where the terms in (40) are constructed using the Legendre transformed scalar
weight functions
$\displaystyle\hat{\omega}_{3}(r_{1},r_{2},n)$
$\displaystyle\\!=\\!\begin{cases}\frac{1}{2}\\!\left(1-x_{12}\right)\\!\Theta_{12}&\\!\\!n\\!=\\!0,\\\
\frac{3}{4}\\!\left(1-x_{12}^{2}\right)\\!\Theta_{12}&\\!\\!n\\!=\\!1,\\\
\frac{2n+1}{2n}\\!\big{(}x_{12}P_{n}(x_{12})\\!-\\!P_{n+1}(x_{12})\big{)}\Theta_{12}&\\!\\!n\\!\geq\\!2,\end{cases}$
$\displaystyle\hat{\omega}_{2}(r_{1},r_{2},n)$
$\displaystyle\\!=\\!\frac{2n+1}{2}\frac{R}{r_{1}r_{2}}P_{n}(x_{12})\Theta_{12},$
$\displaystyle\hat{\omega}_{1}(r_{1},r_{2},n)$
$\displaystyle\\!=\\!\frac{\hat{\omega}_{2}(r_{1},r_{2},n)}{4\pi R},$
$\displaystyle\hat{\omega}_{0}(r_{1},r_{2},n)$
$\displaystyle\\!=\\!\frac{\hat{\omega}_{2}(r_{1},r_{2},n)}{4\pi R^{2}},$ (41)
where $x_{12}=(r_{1}^{2}+r_{2}^{2}-R^{2})/(2r_{1}r_{2})$,
$\Theta_{12}\\!\equiv\\!\Theta\left(R-|r_{12}|\right)$ is the Heaviside step
function and $r_{12}=r_{1}-r_{2}$.
In analogy with our treatment of planar geometry we consider separately the
three classes of terms contributing to the sum (40). For terms belonging to
class 1 the steps involved in transforming $c_{\alpha\beta}$ are identical to
those required to transform the OZ equation (see Appendix C). For
$\alpha,\beta\in\\{0,1,2,3\\}$ this yields
$\displaystyle\hat{c}_{\alpha\beta}(r_{1},r_{2},n)$
$\displaystyle=\frac{4\pi}{2n+1}\\!\int_{d}^{e}\\!\\!dr_{3}\;r_{3}^{2}$ (42)
$\displaystyle\quad\times\hat{\omega}_{\alpha}(r_{3},r_{1},n)\Phi^{{}^{\prime\prime}}_{\alpha\beta}(r_{3})\,\hat{\omega}_{\beta}(r_{3},r_{2},n),$
for $|r_{12}|\leq 2R$ and zero otherwise. For the restricted ranges of $r_{1}$
and $r_{2}$ under consideration the integration limits are
$d=\max(r_{1},r_{2})\\!-\\!R$ and $e=\min(r_{1},r_{2})\\!+\\!R$.
For mixed terms with $\alpha\in\\{0,1,2,3\\}$ and
$\beta\in\\{{\mathbb{1}},{\mathbb{2}}\\}$, we consider the following scalar
product
$\displaystyle\Phi^{{}^{\prime\prime}}_{\alpha\beta}(r_{3})\cdot\omega_{\beta}({\bf
r}_{32})=|\Phi^{{}^{\prime\prime}}_{\alpha\beta}(r_{3})|\,\omega_{|\beta|}({\bf
r}_{32})\,\mathrm{\mathbf{e}}_{r_{3}}\cdot\mathrm{\mathbf{e}}^{\,\rm
shell}_{32},$
where $\mathrm{\mathbf{e}}_{r_{3}}\\!\cdot\,\mathrm{\mathbf{e}}^{\,\rm
shell}_{32}=(R^{2}+r_{3}^{2}-r_{2}^{2})/(2r_{3}R)$. This result depends only
on $r_{3}$ and $r_{2}$ and so its presence in the integrand does not interfere
with the Legendre transformation and the standard procedure given in Appendix
C can be applied without modification. We thus obtain the following expression
$\displaystyle\hat{c}_{\alpha\beta}(r_{1},r_{2},n)=$
$\displaystyle\frac{4\pi}{2n+1}\int_{d}^{e}\\!\\!dr_{3}\,r_{3}^{2}\,\hat{\omega}_{\alpha}(r_{3},r_{1},n)\,|\Phi^{{}^{\prime\prime}}_{\alpha\beta}(r_{3})|$
$\displaystyle\times\left(\frac{R^{2}+r_{3}^{2}-r_{2}^{2}}{2r_{3}R}\right)\hat{\omega}_{|\beta|}(r_{3},r_{2},n),$
(43)
for $|r_{12}|\leq 2R$ and zero otherwise. Similarly, when the rank of the
indices is exchanged, $\alpha\in\\{{\mathbb{1}},{\mathbb{2}}\\}$ and
$\beta\in\\{0,1,2,3\\}$, we find
$\displaystyle\hat{c}_{\alpha\beta}(r_{1},r_{2},n)=$
$\displaystyle\frac{4\pi}{2n+1}\int_{d}^{e}\\!dr_{3}\;r_{3}^{2}\,\hat{\omega}_{|\alpha|}(r_{3},r_{1},n)$
(44)
$\displaystyle\times\left(\frac{R^{2}+r_{3}^{2}-r_{1}^{2}}{2r_{3}R}\right)|\Phi^{{}^{\prime\prime}}_{\alpha\beta}(r_{3})|\;\hat{\omega}_{\beta}(r_{3},r_{2},n),$
for $|r_{12}|\leq 2R$ and zero otherwise.
$\bullet$$\bullet$$\bullet$$\bullet$$O$$1$$2$$3$$\mathrm{\mathbf{r}}_{1}$$\mathrm{\mathbf{r}}_{2}$$\mathrm{\mathbf{r}}_{3}$$R$$R$$\alpha$$\theta_{2}$
Figure 4: Geometrical sketch for evaluation of the scalar product given in
(46). We orient our spherical coordinate system such that the $z$-axis lies
along $\mathrm{\mathbf{r}}_{1}$.
As in the planar case, the terms in class 3 are more difficult to deal with
and we must consider the following scalar product
$\displaystyle\omega_{\alpha}({\bf
r}_{31})\cdot\Phi^{{}^{\prime\prime}}_{\alpha\beta}(r_{3})\cdot\omega_{\beta}({\bf
r}_{32})=$ (45)
$\displaystyle\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}(r_{3})\,\omega_{|\alpha|}({\bf
r}_{31})\,\omega_{|\beta|}({\bf r}_{32})\;\mathrm{\mathbf{e}}^{\,\rm
shell}_{31}\cdot\mathrm{\mathbf{e}}^{\,\rm shell}_{32}.$
In Fig.4 we sketch the three vectors ${\bf r}_{1}$, ${\bf r}_{2}$ and ${\bf
r}_{3}$. The values of ${\bf r}_{3}$ which yield a nonzero contribution to
(45) lie on the intersection circle of delta-shells centered at ${\bf r}_{1}$
and ${\bf r}_{2}$. If we choose ${\bf r}_{1}$ along the $z$-axis of our
spherical coordinate system, then for ${\bf r}_{3}$ anywhere on the
intersection circle we find
$\displaystyle\mathrm{\mathbf{e}}^{\,\rm
shell}_{31}\cdot\mathrm{\mathbf{e}}^{\,\rm
shell}_{32}\equiv\cos(\alpha)=1-\frac{r_{1}^{2}+r_{2}^{2}}{2R^{2}}+\frac{r_{1}r_{2}}{R^{2}}P_{1}(x_{2}),$
(46)
where the angle $\alpha$ is defined in Fig.4 and $x_{2}=\cos(\theta_{2})$. We
thus seek to evaluate the Legendre transform of
$\displaystyle
c_{\alpha\beta}(r_{1},r_{2},x_{2})=c_{\alpha\beta}^{\,D}(r_{1},r_{2},x_{2})+c_{\alpha\beta}^{\,E}(r_{1},r_{2},x_{2}),$
(47)
where the two contributions are given by
$\displaystyle c_{\alpha\beta}^{\,D}(r_{1},r_{2},x_{2})=\\!\\!\int$
$\displaystyle d{\bf
r}_{3}\,\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}(r_{3})$
$\displaystyle\times\omega_{|\alpha|}({\bf r}_{31})\,\omega_{|\beta|}({\bf
r}_{32})\\!\\!\left(\\!1\\!-\\!\frac{r_{1}^{2}+r_{2}^{2}}{2R^{2}}\\!\right)\\!,$
$\displaystyle c_{\alpha\beta}^{\,E}(r_{1},r_{2},x_{2})=\\!\\!\int$
$\displaystyle d{\bf
r}_{3}\,\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}(r_{3})$
$\displaystyle\times\omega_{|\alpha|}({\bf r}_{31})\,\omega_{|\beta|}({\bf
r}_{32})\\!\left(\frac{r_{1}r_{2}}{R^{2}}P_{1}(x_{2})\right)\\!.$
The factor $1-(r_{1}^{2}+r_{2}^{2})/(2R^{2})$ is independent of $x_{2}$ and
Legendre transformation thus proceeds in the same way as for the OZ equation
(see Appendix C). This yields
$\displaystyle\hat{c}^{\,D}_{\alpha\beta}(r_{1},r_{2},n)=\left(\\!1\\!-\\!\frac{r_{1}^{2}+r_{2}^{2}}{2R^{2}}\right)\mathcal{D}_{\alpha\beta}(r_{1},r_{2},n),$
(48)
for $|r_{12}|\leq 2R$ and zero otherwise, where the function
${\mathcal{D}}_{\alpha\beta}$ is given by
$\displaystyle{\mathcal{D}}_{\alpha\beta}(r_{1},r_{2},n)\\!$
$\displaystyle=\\!\frac{4\pi}{2n+1}\int_{0}^{\infty}\\!\\!\\!dr_{3}\,r_{3}^{2}$
$\displaystyle\quad\times\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}(r_{3})\hat{\omega}_{|\alpha|}(r_{3},r_{1},n)\,\hat{\omega}_{|\beta|}(r_{3},r_{2},n).$
Legendre transform of $c_{\alpha\beta}^{E}$ is more difficult due to the
presence of the factor $P_{1}(x_{2})$. If we follow the procedure of Appendix
C we find that the first step of the calculation can be carried through easily
to obtain
$\displaystyle
c^{\,E}_{\alpha\beta}(r_{1},r_{2},x_{2})=2\pi\int_{0}^{\infty}\\!dr_{3}\;r_{3}^{2}\,\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}(r_{3})$
(49)
$\displaystyle\\!\times\sum_{i=0}^{\infty}\frac{2}{2i+1}\hat{\omega}_{|\alpha|}(r_{3},r_{1},i)\,\hat{\omega}_{|\beta|}(r_{3},r_{2},i)\frac{r_{1}r_{2}}{R^{2}}P_{1}(x_{2})P_{i}(x_{2}).$
Applying the operator $\frac{2n+1}{2}\\!\int_{-1}^{\,1}dx_{2}\,P_{n}(x_{2})$
to (49) then yields the Legendre transformation
$\displaystyle\hat{c}^{\,E}_{\alpha\beta}(r_{1},r_{2},n)\\!$
$\displaystyle=\\!2\pi\frac{2n+1}{2}\\!\int_{0}^{\infty}\\!\\!\\!dr_{3}\,r_{3}^{2}\,\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}(r_{3})$
$\displaystyle\times\sum_{i=0}^{\infty}\frac{2}{2i+1}\hat{\omega}_{|\alpha|}(r_{3},r_{1},i)\,\hat{\omega}_{|\beta|}(r_{3},r_{2},i)$
$\displaystyle\times\frac{r_{1}r_{2}}{R^{2}}\\!\\!\int_{-1}^{\,1}\\!\\!\\!dx_{2}\,P_{1}(x_{2})P_{n}(x_{2})P_{i}(x_{2}).$
(50)
The extra complication here is caused by the integral of a triple product of
Legendre polynomials. Fortunately, this integration has been well-studied in
the context of quantum mechanics and can be reexpressed using the Wigner $3j$
notation (see e.g. [23])
$\displaystyle\int_{-1}^{\,1}\\!\\!\\!dx_{2}\,P_{l}(x_{2})P_{n}(x_{2})P_{i}(x_{2})=2{\begin{pmatrix}l&n&i\\\
0&0&0\end{pmatrix}}^{2}.$ (51)
For the case of interest here, $l\\!=\\!1$, there are only two terms in the
sum over $i$ appearing in (III). This leads to the result
$\displaystyle\hat{c}^{\,E}_{\alpha\beta}(r_{1},r_{2},n)=\frac{r_{1}r_{2}}{R^{2}}{\mathcal{E}}_{\alpha\beta}(r_{1},r_{2},n),$
(52)
for $|r_{12}|\leq 2R$ and zero otherwise, where
$\displaystyle{\mathcal{E}}_{\alpha\beta}(r_{1},r_{2},n)=\frac{4\pi}{2n+1}\\!\int_{0}^{\infty}\\!\\!\\!dr_{3}\,r_{3}^{2}\,\widetilde{\Phi}^{{}^{\prime\prime}}_{\alpha\beta}(r_{3})$
(53)
$\displaystyle\times\bigg{(}\frac{n(2n+1)}{(2n-1)^{2}}\hat{\omega}_{|\alpha|}(r_{3},r_{1},n\\!-\\!1)\,\hat{\omega}_{|\beta|}(r_{3},r_{2},n\\!-\\!1)\bigg{.}\Biggr{.}$
$\displaystyle\Biggl{.}\bigg{.}$
$\displaystyle+\frac{(n+1)(2n+1)}{(2n+3)^{2}}\hat{\omega}_{|\alpha|}(r_{3},r_{1},n\\!+\\!1)\,\hat{\omega}_{|\beta|}(r_{3},r_{2},n\\!+\\!1)\bigg{)}.$
Putting together (47), (48) and (52) yields the final result for the Legendre
transform of the class 3 contributions with $\alpha,\beta\in\\{{\bf 1},{\bf
2}\\}$
$\hat{c}_{\alpha\beta}(r_{1},r_{2},n)=\left(\\!1\\!-\\!\frac{r_{1}^{2}+r_{2}^{2}}{2R^{2}}\right)\mathcal{D}_{\alpha\beta}(r_{1},r_{2},n)\\\
+\frac{r_{1}r_{2}}{R^{2}}{\mathcal{E}}_{\alpha\beta}(r_{1},r_{2},n),$ (54)
for $|r_{12}|\leq 2R$ and zero otherwise. In summary, equations (40), (42),
(III), (44), and (54) provide the Legendre transform of the FMT pair direct
correlation function, $\hat{c}^{(2)}_{\rm hs}$, as an explicit functional of
the one-dimensional spherical density profile, $\rho(r)$.
### Numerical consistency checks
The formulae presented in the previous subsections for the Hankel and Legendre
transforms of the pair direct correlation functions are, in principle,
straightforward to implement. However, when developing numerics to evaluate
the remaining one-dimensional integrals it is useful to have some checks and
limiting cases to help eliminate possible coding errors. In bulk there are two
helpful benchmarks: (i) In the low density limit the pair direct correlation
function reduces to the Mayer function [4] for which both the Hankel and
Legendre transforms are known exactly. (ii) At finite density the (real-space)
analytic expression for the PY pair direct correlation function [4] can be
numerically Hankel/Legendre transformed using (17) and (37), respectively. The
result thus obtained should agree with the predictions of our analytical
expressions. Contributions arising from class 2 terms (scalar-vector
combinations) vanish in bulk and can thus only be tested by considering
inhomogeneous density profiles. A useful check is the following relation
between the one- and two-body direct correlation functions
$\nabla_{1}c^{(1)}(\mathrm{\mathbf{r}}_{1})=\int
d\mathrm{\mathbf{r}}_{2}\,c^{(2)}(\mathrm{\mathbf{r}}_{1},\mathrm{\mathbf{r}}_{2})\,\nabla_{2}\rho(\mathrm{\mathbf{r}}_{2}),$
(55)
known as the Lovett-Mou-Buff-Wertheim sum-rule [3].
Figure 5: Hard-spheres at a hard-wall. FMT density profile for $\mu=2$ (solid
line) with its mid-point value indicated (horizontal dotted line). Points
indicate the $z$-coordinates for which we show the total correlation function
in Fig.6.
In planar geometry this reduces to
$\frac{\partial c^{(1)}(z_{1})}{\partial
z_{1}}=\int_{-\infty}^{\infty}dz_{2}\,\frac{\partial\rho(z_{2})}{\partial
z_{2}}\,\overline{c}^{(2)}(z_{1},z_{2},k\\!=\\!0),$ (56)
where both $c^{(1)}$ and $c^{(2)}$ are evaluated at the equilibrium density,
$\rho$. In spherical geometry equation (55) becomes
$\frac{\partial\,\hat{c}^{(1)}(r_{1})}{\partial
r_{1}}=\frac{4\pi}{3}\\!\int_{0}^{\infty}\\!\\!dr_{2}\,r_{2}^{2}\;\frac{\partial\rho(r_{2})}{\partial
r_{2}}\,\hat{c}(r_{1},r_{2},n\\!=\\!1).$ (57)
Finally, in planar geometry a transverse structure factor can be defined as
[1]
$\displaystyle H(z_{1},k)$ $\displaystyle\equiv 1+\int
dz_{2}\,\overline{h}(z_{1},z_{2},k)\,\rho(z_{2})$ (58) $\displaystyle=1+\int
dz_{2}\,H(z_{2},k)\,\rho(z_{2})\,\overline{c}^{(2)}(z_{2},z_{1},k),$ (59)
where the second equality is an integral equation requiring iterative
solution. The transverse structure factor is related to the local
compressibility according to
$\displaystyle
H(z,k\\!=\\!0)=\frac{1}{\beta\rho(z)}\,\frac{\partial\rho(z)}{\partial\mu}.$
(60)
Satisfying equations (58) and (60) provides an additional check that the
numerical solution of the OZ equation (19) for $\bar{h}_{\rm hs}$ has been
performed correctly.
## IV Results for hard-spheres
### Planar geometry
In Fig.5 we show a FMT density profile for hard-spheres at a hard-wall. The
chosen value of the chemical potential ($\mu\\!=\\!2$) corresponds to a
liquid-state of intermediate bulk density. In the vicinity of the wall we
observe the familiar packing oscillations which then decay rapidly into the
bulk. The four points marked on the curve indicate the positions at which we
will investigate the inhomogeneous two-body correlations.
Figure 6: Hard-spheres at a hard-wall. Total correlation function for equal
values of the two $z$-arguments, $h_{\rm hs}(z,z,r)$, corresponding to the
$z$-positions indicated on the density profile shown in Fig.5. FMT (full
orange line), inhomogeneous PY (dashed blue line) and PY bulk solution (grey
dotted line). At points A and B all three curves remain essentially identical.
At points C and D there are deviations from bulk, but excellent agreement
between FMT and PY theory.
When analyzing two-body correlations with planar symmetry we are faced with a
function of three independent scalar arguments. This naturally presents many
alternatives for graphical representation of the data. Following a quite
extensive study of these various possibilities we have come to the conclusion
that simple one-dimensional plots showing the variation of the correlation
functions as a function of $r$ for equal values of the $z$-coordinates
provides a reasonable way to compare different theories. Similar plots for
fixed, but distinct, values of the $z$-coordinates were not found to offer any
greater insight.
In Fig.6, we show the total correlation function for
$z_{1}\\!=\\!z_{2}\\!=\\!z$ as a function of the cylindrical radial
coordinate, $r$. In each panel we indicate the bulk function, $h_{\rm
hs}(z\\!\rightarrow\\!\infty,z\\!\rightarrow\\!\infty,r)$, as a visual
reference. Moving through the panels from A to D we observe increasing
deviation of the inhomogeneous total correlation function from its bulk form.
At all of the considered points the FMT prediction stays very close to that of
the inhomogeneous PY theory, even when the density is strongly varying. This
good level of agreement between the PY theory and FMT gives us some confidence
in the quality of FMT at the two-body level, at least for these intermediate
densities. We note that the PY total correlation function of hard-spheres is
unique, in the sense that it is generated by a strictly truncated direct
correlation for $r_{12}>1$ while still satisfying the core condition (see
(III)). Due to the finite range of the weight functions the FMT direct
correlation function automatically satisfies the first of these conditions,
but not the second (except in the low density limit). It thus follows that any
deviation of the FMT total correlation function from the PY theory is a
consequence of core condition violation. The good level of agreement shown in
Fig.6 can be therefore taken as an indirect indication that the core condition
is well approximated by FMT for the considered density.
Figure 7: Confined hard-sphere system. FMT density profile for $\mu=5$ (solid
line). Points indicate the $z$-coordinates for which we show the total
correlation function in Fig.8.
We next consider a more demanding case: densely packed hard-spheres confined
between two parallel hard-walls separated by four particle diameters. In Fig.7
we show the density calculated at $\mu=5$, which generates a strongly
inhomogeneous profile. The points label the positions at which we will
investigate the inhomogeneous total correlation function. In Fig.8 we show
$h_{\rm hs}$ as a function of $r$ for $z\\!=\\!1$ and $z\\!=\\!0.5$ (the
positions labelled C and D in Fig.7). At the point C, close to the first
minimum of the profile, we find very close agreement between the PY theory and
FMT, with only slight deviation at around $r\\!=\\!1.75$. At point D,
corresponding to the contact peak of the profile, we find more substantial
differences between the two approaches. The amplitude of the oscillations
predicted by the FMT are somewhat larger than those from the PY theory, but
the overall level of agreement remains satisfactory. For separations $r>2.75$
the predictions of PY theory and the FMT become very similar.
### Spherical geometry
As a test of our analytic FMT formulae in spherical geometry we will use the
inhomogeneous total correlation function to calculate the three-body
correlations of the bulk fluid. This can be achieved by extending the test-
particle idea of Percus [2] to the two-body level. If we specify the external
field to represent a hard-sphere fixed at the coordinate origin, then the
inhomogeneous correlation function $g^{tp}_{\rm hs}({\bf r}_{1},{\bf
r}_{2})\equiv h^{tp}_{\rm hs}({\bf r}_{1},{\bf r}_{2})+1$ is related to the
bulk triplet correlation function according to
$\displaystyle
g^{(3)}(r_{1},r_{2},r_{12})=\frac{\rho^{tp}(r_{1})\rho^{tp}(r_{2})g^{tp}({\bf
r}_{1},{\bf r}_{2})}{\rho^{2}_{b}},$ (61)
where we employ the superscript $tp$ to indicate functions calculated in the
presence of a test-particle at the origin. Experience with triplet
correlations has shown that direct analysis of $g^{(3)}$ is not the best
choice when seeking to assess the quality of a given approximation. A better
option is the following function
$\displaystyle\Gamma(r_{1},r_{2},r_{12})=\frac{\rho_{b}^{3}\,g^{(3)}\\!(r_{1},r_{2},r_{12})}{\rho^{tp}(r_{1})\,\rho^{tp}(r_{2})\,\rho^{tp}(r_{12})},$
(62)
which scales the triplet correlation function by the well-known Kirkwood
superposition approximation [24]. Deviations of $\Gamma$ from unity thus
provide a sensitive measure of nontrivial contributions to the three-body
correlations.
Figure 8: Confined hard-sphere system. Two-body total correlation function
corresponding to the density profile shown in Fig.7. Inhomogeneous PY
solutions (solid black lines) and FMT for $z=0.5$ (dashed blue line) and for
$z=1$ (dotted orange line). We observe close agreement between FMT and PY
theory at point C, but deviations between the two theories emerge at point D.
In Fig.9 we show the function $\Gamma$ generated by FMT for ‘rolling contact’
configurations at bulk densities $\rho_{b}\\!=\\!0.3,0.5$ and $0.7$ (marked A,
B and C, respectively, in the figure). These configurations are where the
Kirwood superposition (independent probability) approximation is most severely
tested, but are also of central importance in kinetic theories for the
transport properties of hard-spheres (see e.g. [25, 26]). The FMT predictions
are compared with Monte-Carlo simulation data taken from Refs. [27] and [28].
For the two lower bulk densities considered (points A and B) we find a good
level of agreement between FMT and simulation. However, at
$\rho_{b}\\!=\\!0.7$ discrepancies emerge and the FMT prediction for the
amplitude and position of the peak is less accurate. This suggests that we are
approaching the limit at which the FMT two-body correlations can be considered
reliable.
To investigate further this breakdown at high densities we compare in Fig.10
the predictions of FMT with simulation data for a more varied selection of
configurations at the even higher density, $\rho_{b}\\!=\\!0.8$. Panel A shows
the variation for a rolling contact configuration. An unphysical ‘shoulder’,
already visible in panel C of Fig.9, becomes more pronounced, although one
could argue that the overall description remains acceptable. This shoulder
feature becomes more prominent when considering a rolling configuration with
slightly more separation between the particles, shown in panel B of Fig.10.
Despite showing reasonable behaviour at larger separations, for $r\\!<\\!2$
the description of the simulation data is rather poor. Panels C and D focus on
stretched isoceles triangle configurations, which also serve to expose
deficiencies of the FMT. While it is apparent that the general trends of the
simulation data are roughly captured, the amplitude of oscillation is
significantly overestimated. It would appear that the FMT performs best for
rolling contact situations but leaves much to be desired at intermediate
particle separations, at least for densities $\rho_{b}>0.7$. It seems to us
that the overall level of agreement of the FMT predictions with the Monte-
Carlo data is on a similar level to that of earlier theories of the triplet
correlation, such as those of Haymet et al. [29] and Barrat et al. [30]. For
some examples of this we refer the reader to Fig.9 of the paper by Bilstein
and Kahl [28].
$\bullet$$\bullet$$\bullet$
Figure 9: Hard-sphere triplet correlations. Comparison of FMT (lines) with
simulation data (points) [27, 28] for the quantity $\Gamma(1,1,r)$ at
$\rho_{b}\\!=\\!0.3$ (A), $0.5$ (B) and $0.7$ (C). We consider rolling contact
configurations for which the separation $r$ is indicated by an arrow in the
sketch.
## V Perturbation theory
The hard-sphere model is not sufficient to capture all of the phenomena
exhibited by real fluids. An improved description can be achieved if we
supplement the hard-sphere repulsion with an attractive component to the
interaction potential, $u=u_{\rm hs}+u_{\rm att}$. If the attraction if
sufficiently weak and long ranged, then the following first-order perturbation
theory provides a good approximation to the Helmholtz free energy
$\displaystyle F_{\rm BH}[\,\rho\,]$ $\displaystyle=F_{\rm hs}[\,\rho\,]$ (63)
$\displaystyle+\frac{1}{2}\\!\int\\!d{\bf r}_{1}\\!\int\\!d{\bf
r}_{2}\,\rho({\bf r}_{1})\rho({\bf r}_{2})u^{\rm att}(r_{12})\big{(}1+h_{\rm
hs}({\bf r}_{1},{\bf r}_{2};[\,\rho\,])\big{)},$
where the first term is the Helmholtz free energy of hard-spheres, including
the ideal gas contribution. The density enters equation (63) both explicitly,
via the quadratic product in the integrand, and implicitly, via the functional
dependence of the hard-sphere Helmholtz free energy and total correlation
function. In bulk, equation (63) reduces to the well-known first-order
perturbation theory of Barker and Henderson [31, 32, 33]. For this reason the
approximation (63) has been called the Barker-Henderson (BH) functional.
In Ref.[20] we investigated the density obtained from numerical minimization
of the BH grand potential (using equations (1), (3) and (63)) and found
excellent agreement with simulation data for several inhomogeneous situations.
Our findings suggest that the BH functional provides a quantitatively accurate
description of inhomogeneous fluids with hard-core repulsion and weak
attraction. Despite these promising results, widespread application of the BH
functional, as implemented in [20], is likely to be hindered by the numerical
effort required to minimize the grand potential. The strategy adopted, which
we will henceforth refer to as the BH-PY approach, was to use FMT to
approximate the first (reference) term in (63) and to obtain $h_{\rm hs}$ by
iteratively solving the inhomogeneous PY theory (equations (8) and (III)).
Clearly, using the numerical solution of an inhomogeneous integral equation
theory as part of a self-consistent minimization scheme is computationally
expensive, particularly when larger systems are required (e.g. studies of the
liquid-vapour interface). However, we are now in a position to improve this
situation by incorporating the FMT total correlation function, rather than
that from the PY approximation, into (63). This BH-FMT approach yields a huge
reduction in computation time and thus opens the door to applications which
would be practically impossible using BH-PY.
$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$
Figure 10: Hard-sphere triplet correlations. Comparison of the FMT (lines)
with simulation data (points) [27] for the quantity $\Gamma(s,t,r)$ at
$\rho_{b}=0.8$. Configurations are sketched in each figure. The separation $r$
is indicated by the bold arrow. A and B are rolling geometries at $s=t=1.0$
and $s=t=1.1$, respectively. C and D are isosceles triangle configurations
with $s=r$ and the base length of the triangle fixed at $t=1.1$ and $1.3$,
respectively.
We now briefly summarize the steps required to minimize the BH-FMT grand
potential. Although this proceeds in much the same way as discussed in [20],
there is a subtle but very important difference to be observed when evaluating
the derivative contribution to the one-body direct correlation function. To
limit the length of the presentation we restrict attention to the case of
planar geometry, although similar calculations in spherical geometry would
pose no greater difficulty. To solve the Euler-Lagrange equation (4), we
require the one-body direct correlation function (5). Using (63) to evaluate
the derivative yields
$\displaystyle c^{(1)}=c_{\rm hs}^{(1)}+c_{\rm smf}^{(1)}+c_{\rm
corr}^{(1)}+c_{\rm der}^{(1)},$ (64)
where $c^{(1)}_{\rm hs}$ is given by (12). The remaining terms involve
integrals over the attractive interaction:
$\displaystyle c_{\rm smf}^{(1)}({\bf r}_{1})$
$\displaystyle=-\\!\int\\!\\!d{\bf r}_{2}\,\rho({\bf r}_{2})\beta u^{\rm
att}(r_{12}),$ (65) $\displaystyle c_{\rm corr}^{(1)}({\bf r}_{1})$
$\displaystyle=-\\!\int\\!\\!d{\bf r}_{2}\,\rho({\bf r}_{2})\beta u^{\rm
att}(r_{12})h_{\rm hs}({\bf r}_{1},{\bf r}_{2}),$ (66) $\displaystyle c_{\rm
der}^{(1)}({\bf r}_{1})$ $\displaystyle=-\\!\int\\!\\!d{\bf
r}_{2}\\!\int\\!\\!d{\bf r}_{3}\frac{\rho({\bf r}_{2})\rho({\bf r}_{3})\beta
u^{\rm att}(r_{23})}{2}\,\frac{\delta h_{\rm hs}({\bf r}_{2},{\bf
r}_{3})}{\delta\rho({\bf r}_{1})}.$ (67)
The first of these is easy to calculate. The second contribution can be
evaluated by using our analytic formulae for $\bar{c}_{\rm hs}$ as input to
the OZ equation (19) and then transforming the resulting $\bar{h}_{\rm hs}$
back to real-space.
The functional derivative in (67) can be reexpressed in terms of a derivative
with respect to the one-dimensional density profile
$\displaystyle\frac{\delta h_{\rm hs}({\bf r}_{1},{\bf
r}_{2})}{\delta\rho({\bf r})}=\frac{1}{A}\frac{\delta h_{\rm
hs}(z_{1},z_{2},r)}{\delta\rho(z)},$ (68)
where $A$ is an (arbitrary) area perpendicular to the $z$-axis (eliminated
when performing the integrals in (67)). Using finite differences the
functional derivative becomes
$\displaystyle\frac{\delta h_{\rm hs}({\bf r}_{1},{\bf
r}_{2})}{\delta\rho({\bf r})}=\lim_{\varepsilon\rightarrow
0}\frac{h^{\varepsilon z\\!}_{\rm hs}(z_{1},z_{2},r)-h_{\rm
hs}(z_{1},z_{2},r)}{A\varepsilon},$ (69)
where $h^{\varepsilon z}_{\rm hs}$ is the total correlation function evaluated
at the perturbed density
$\rho_{z}(z_{3})=\rho(z_{3})+\varepsilon\,\delta(z_{3}-z)$. Substitution of
the perturbed density into the transformed OZ equation (19) yields
$\displaystyle\\!\\!\\!\overline{h}^{\,\varepsilon z\\!}_{\rm
hs}(z_{1},z_{2},k)=\int_{-\infty}^{\infty}\\!\\!\\!dz_{3}\,\overline{h}^{\,\varepsilon
z\\!}_{\rm hs}(z_{1},z_{3},k)\rho(z_{3})\overline{c}^{\,(2),\varepsilon
z\\!}_{\rm hs}(z_{3},z_{2},k)$
$\displaystyle+\;\overline{c}^{\,(2),\varepsilon z\\!}_{\rm
hs}(z_{1},z_{2},k)\,+\,\varepsilon\,\overline{h}^{\,\varepsilon z\\!}_{\rm
hs}(z_{1},z,k)\overline{c}^{\,(2),\varepsilon z\\!}_{\rm hs}(z,z_{2},k).$ (70)
Before solving (V) for $\overline{h}^{\,\varepsilon z\\!}_{\rm hs}$ we
evaluate the perturbed function $\overline{c}^{\,(2),\varepsilon z\\!}_{\rm
hs}$ using our analytical results. Since the Hankel transformed two-body
direct correlation function depends on the density only through the weighted
densities in $\Phi^{{}^{\prime\prime}}_{\alpha\beta}$, the perturbed pair
direct correlation function can be obtained simply by substituting perturbed
weighted densities, $n^{\varepsilon z\\!}_{\alpha}$, into
$\Phi^{{}^{\prime\prime}}_{\alpha\beta}$. In planar geometry the weighted
densities can be expressed as
$n_{\alpha}(z_{1})=\int dz_{2}\;\rho(z_{2})\,\omega_{\alpha}(z_{1}-z_{2}),$
(71)
where the one-dimensional weight functions are given by
$\displaystyle\omega_{3}(z)$
$\displaystyle\\!=\\!\pi\left(R^{2}\\!-\\!z^{2}\right)\Theta\left(R\\!-\\!|z|\right),\hskip
1.42271pt\omega_{2}(z)\\!=\\!2\pi R\Theta\left(R-|z|\right),$
$\displaystyle\omega_{1}(z)$ $\displaystyle\\!=\\!\frac{\omega_{2}(z)}{4\pi
R},\hskip 71.13188pt\omega_{0}(z)\\!=\\!\frac{\omega_{2}(z)}{4\pi R^{2}},$
$\displaystyle\omega_{\mathbb{2}}(z)$ $\displaystyle\\!=\\!2\pi
z\,\mathrm{\mathbf{e}}_{z}\Theta\left(R-|z|\right),\hskip
19.91684pt\omega_{\mathbb{1}}(z)\\!=\\!\frac{\omega_{\mathbb{2}}(z)}{4\pi R}.$
The perturbed weighted densities are therefore given by
$\displaystyle n^{\varepsilon
z\\!}_{\alpha}(z_{1})=n_{\alpha}(z_{1})+\varepsilon\,\omega_{\alpha}(z_{1}-z).$
(72)
In [20] equation (V) was solved using the PY closure (III) on the perturbed
functions $h^{\varepsilon z}_{\rm hs}$ and $c^{\varepsilon z}_{\rm hs}$ and
solving the resulting nonlinear integral equation. This is much more demanding
than the FMT route proposed here.
Figure 11: Attractive hard-core Yukawa system under confinement. Profiles for
$\mu=-2.00$, $-1.50$ and $-1.00$, respectively, at parameter values
$\kappa=0.75$ and $\alpha=1.8$. All profiles here are thus super-critical,
recalling that $\kappa$ plays the role of an inverse temperature. Simulation
data (solid black lines) [20], BH-PY (dashed blue lines) and BH-FMT (dotted
orange lines). The inset shows the phase diagram for $\alpha=1.8$, where we
indicate the mid-point density of the calculated density profiles (black
points) for fixed parameter $\kappa=0.75$ (dashed silver line).
## VI Results for the hard-core Yukawa model
In Fig.11 we show density profiles for the attractive hard-core Yukawa (HCY)
model confined between two planar walls separated by a distance of ten
particle diameters. In addition to a hard-core repulsion the pair interaction
potential of the HCY model has an attractive contribution given by
$\displaystyle u_{\rm
att}(r_{12})=-\kappa\,\frac{e^{-\alpha(r_{12}-1)}}{r_{12}}\hskip
14.22636ptr_{12}\geq 1,$ (73)
where $\kappa$ and $\alpha$ are positive constants.
Fig.11 shows profiles calculated using both the BH-PY and BH-FMT for three
different chemical potentials. The theoretical predictions are compared with
Monte-Carlo data taken from [20]. The inset to Fig.11 shows the bulk binodal,
where the points indicate the mid-point density of each of the considered
profiles. For all three chemical potentials the BH-PY and BH-FMT are in very
close agreement and both provide a good description of the simulation data. We
wish to emphasize that this close agreement between BH-FMT and BH-PY is a key
result of the present work and central to the ultimate success of the method.
Indeed, establishing this agreement provided much of the motivation for the
present study of the two-body inhomogeneous correlation functions of hard-
spheres. The fact that the accurate first-principles predictions of the BH-PY
theory can be essentially reproduced, but with greatly reduced computational
effort, by the BH-FMT is a significant step in turning the BH perturbation
theory into a practically viable method for predicting the properties of
realistic inhomogeneous fluids. While it remains to be seen whether the high
level of agreement between BH-FMT and BH-PY remains in other situations, the
data shown in Fig.11 seem to us to be very promising. To give the reader some
feeling for the demands involved and the time saved by employing parallel
computation - each of the BH-FMT profiles shown in Fig.11 required around four
hours of computation time on a standard eight-core desktop machine (runtime
effectively scales with the number of cores), whereas the corresponding BH-PY
results each required several days to converge.
## VII Discussion
In this paper we have provided a detailed analysis of the inhomogeneous two-
body correlation functions generated by FMT. Our formulae for the Hankel and
Legendre transforms of the two-body direct correlation function enable rapid
numerical evaluation of the real-space total correlation function and
circumvent many of the usual numerical difficulties associated with iterative
solution of inhomogeneous integral equation closures.
Considering hard-spheres, our developments both facilitate the study of
inhomogeneous microstructure and provide a fresh line of enquiry when
analyzing the FMT. Past optimization strategies have focussed on thermodynamic
(zero-body) and one-body quantities. It is our view that explicit
consideration of the two-body correlation functions could lead to new insight
into FMT, yielding both quantitative criteria for the assessment of existing
approximations as well as suggesting possible improvements. For example, it
would be interesting to know the influence of either improved thermodynamics
or tensorial weight functions on the predictions for the bulk triplet
correlation function [19].
Using the FMT total correlation function as input to the BH perturbation
theory (63) yields a computationally viable approach for models of fluids with
attractive interparticle interactions. The key advantage is that within FMT
the inhomogeneous two-body correlation functions can be obtained using
parallel computation. This is an essential feature if density functional
theory beyond the one-body level is ever to become a practical tool for the
investigation of relevant and interesting phenomena. The BH-FMT approach is
undoubtedly much more efficient to implement than the BH-PY theory [20] and
does not appear to lead to any significant reduction in accuracy (see Fig.11),
even at high density.
At first sight, this conclusion might seem to be in contradiction to the
triplet correlation data presented in Figs.9 and 10, where we find generally
unsatisfactory performance of FMT at higher densities. We thus make the the
following observations: (i) The integrand appearing in the BH free energy
functional (63) is weighted by the attractive part of the pair interaction
potential, which lends particular importance to configurations around
$r_{12}\\!\approx\\!1$. (ii) The integral in (63) runs over both ${\bf r}_{1}$
and ${\bf r}_{2}$ and represents a complicated average over the inhomogeneous
total correlation function. These two features apparently lend the BH-FMT
approach a certain robustness with respect to errors in the FMT hard-sphere
two-body correlations and therefore open the door to many applications of the
BH-FMT approach. However, only a more extensive investigation for different
external fields and values of the model parameters will reveal under which
conditions BH-PY and BH-FMT remain in such good agreement.
## Appendix A
The nonzero derivatives of the free energy density,
$\Phi^{{}^{\prime}}_{\alpha}\\!=\\!\partial\Phi/\partial n_{\alpha}$, required
for calculation of the FMT one-body direct correlation function are given by
$\displaystyle\Phi^{{}^{\prime}}_{0}$
$\displaystyle=-\ln(1-n_{3}),\;\;\;\Phi^{{}^{\prime}}_{1}=\frac{n_{2}}{1-n_{3}},$
$\displaystyle\Phi^{{}^{\prime}}_{2}$
$\displaystyle=\frac{n_{1}}{1-n_{3}}+\frac{3n_{2}^{2}-3\mathrm{\mathbf{n}}_{2}\cdot\mathrm{\mathbf{n}}_{2}}{24\pi(1-n_{3})^{2}},$
$\displaystyle\Phi^{{}^{\prime}}_{3}$
$\displaystyle=\frac{n_{0}}{1-n_{3}}+\frac{n_{1}n_{2}-\mathrm{\mathbf{n}}_{1}\cdot\mathrm{\mathbf{n}}_{2}}{(1-n_{3})^{2}}+\frac{n_{2}^{3}-3n_{2}\mathrm{\mathbf{n}}_{2}\cdot\mathrm{\mathbf{n}}_{2}}{12\pi(1-n_{3})^{3}},$
$\displaystyle\Phi^{{}^{\prime}}_{\mathbb{1}}$
$\displaystyle=-\frac{\mathrm{\mathbf{n}}_{2}}{1-n_{3}},\;\;\;\;\;\Phi^{{}^{\prime}}_{\mathbb{2}}=-\frac{\mathrm{\mathbf{n}}_{1}}{1-n_{3}}-\frac{n_{2}\mathrm{\mathbf{n}}_{2}}{4\pi(1-n_{3})^{2}}.$
The nonzero second derivatives,
$\Phi^{{}^{\prime\prime}}_{\alpha\beta}\\!=\\!\partial^{2}\Phi/\partial
n_{\alpha}\partial n_{\beta}$, required to calculate the two-body direct
correlation function are given by
$\displaystyle\Phi^{{}^{\prime\prime}}_{03}$
$\displaystyle=\Phi^{{}^{\prime\prime}}_{30}=\frac{1}{1-n_{3}},\;\;\;\;\;\;\;\;\;\;\;\;\;\Phi^{{}^{\prime\prime}}_{12}=\Phi^{{}^{\prime\prime}}_{21}=\frac{1}{1-n_{3}},$
$\displaystyle\Phi^{{}^{\prime\prime}}_{13}$
$\displaystyle=\Phi^{{}^{\prime\prime}}_{31}=\frac{n_{2}}{(1-n_{3})^{2}},\;\;\;\;\;\;\;\;\,\Phi^{{}^{\prime\prime}}_{22}=\frac{n_{2}}{4\pi(1-n_{3})^{2}},$
$\displaystyle\Phi^{{}^{\prime\prime}}_{23}$
$\displaystyle=\Phi^{{}^{\prime\prime}}_{32}=\frac{n_{1}}{(1-n_{3})^{2}}+\frac{n_{2}^{2}-\mathrm{\mathbf{n}}_{2}\cdot\mathrm{\mathbf{n}}_{2}}{4\pi(1-n_{3})^{3}},$
$\displaystyle\Phi^{{}^{\prime\prime}}_{2{\mathbb{2}}}$
$\displaystyle=\Phi^{{}^{\prime\prime}}_{{\mathbb{2}}2}=-\frac{\mathrm{\mathbf{n}}_{2}}{4\pi(1-n_{3})^{2}},$
$\displaystyle\Phi^{{}^{\prime\prime}}_{33}$
$\displaystyle=\frac{n_{0}}{(1-n_{3})^{2}}+\frac{2(n_{1}n_{2}-\mathrm{\mathbf{n}}_{1}\cdot\mathrm{\mathbf{n}}_{2})}{(1-n_{3})^{3}}+\frac{n_{2}^{3}-3n_{2}\mathrm{\mathbf{n}}_{2}\cdot\mathrm{\mathbf{n}}_{2}}{4\pi(1-n_{3})^{4}},$
$\displaystyle\Phi^{{}^{\prime\prime}}_{3{\mathbb{1}}}$
$\displaystyle=\Phi^{{}^{\prime\prime}}_{{\mathbb{1}}3}=-\frac{\mathrm{\mathbf{n}}_{2}}{(1-n_{3})^{2}},$
$\displaystyle\Phi^{{}^{\prime\prime}}_{3{\mathbb{2}}}$
$\displaystyle=\Phi^{{}^{\prime\prime}}_{{\mathbb{2}}3}=-\frac{\mathrm{\mathbf{n}}_{1}}{(1-n_{3})^{2}}-\frac{n_{2}\mathrm{\mathbf{n}}_{2}}{2\pi(1-n_{3})^{3}},$
$\displaystyle\Phi^{{}^{\prime\prime}}_{{\mathbb{1}}{\mathbb{2}}}$
$\displaystyle=\Phi^{{}^{\prime\prime}}_{{\mathbb{2}}{\mathbb{1}}}=-\frac{1}{1-n_{3}}{\mathbb{1}},\;\;\;\;\;\;\;\Phi^{{}^{\prime\prime}}_{{\mathbb{2}}{\mathbb{2}}}=-\frac{n_{2}}{4\pi(1-n_{3})^{2}}{\mathbb{1}},$
where ${\mathbb{1}}$ is the unit tensor.
## Appendix B
We show how to obtain the Hankel transformed OZ equation for planar geometry
(19) starting from the general expression (8). To clarify the treatment of
‘class 3’ terms in the main text we find it convenient to break the
calculation into two steps.
$z$$O$$z_{2}$$z_{3}$$r_{1}=0$$r_{2}$$r_{3}$$r_{32}$$\bullet$$\bullet$$\bullet$$\theta$$\bullet$$\circ$
Figure 12: Sketch of the geometry used to rewrite the OZ equation in planar
geometry. The point $O$ indicates the origin of the cylindrical coordinate
system ($z_{1}=0,r_{1}=0$).
Step 1: When the density has a planar geometry we can express the OZ equation
(8) in the cylindrical coordinate system
$\displaystyle
h(z_{1},z_{2},r_{2})=c^{(2)}(z_{1},z_{2},r_{2})+\int_{-\infty}^{\infty}dz_{3}\int_{0}^{2\pi}\\!\\!d\theta$
(74)
$\displaystyle\times\int_{0}^{\infty}dr_{3}r_{3}\,h(z_{1},z_{3},r_{3})\,\rho(z_{3})\,c^{(2)}(z_{3},z_{2},r_{32}).$
In Fig.12 we specify the geometry. If we choose $z_{1}$ as the axis of our
cylindrical coordinates, then the $z$-projected separation between the points
at ${\bf r}_{2}$ and ${\bf r}_{3}$ is given by
$r_{32}=\sqrt{r_{2}^{2}+r_{3}^{2}-2r_{2}r_{3}\cos(\theta)}$. The Hankel back-
transform of the pair direct correlation function (18) and the total
correlation function can be expressed as
$\displaystyle c^{(2)}(z_{1},z_{2},r)$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int\\!d\mathrm{\mathbf{k}}\,J_{0}(kr)\,\overline{c}^{\,(2)}(z_{1},z_{2},k),$
(75) $\displaystyle h(z_{1},z_{2},r)$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int\\!d\mathrm{\mathbf{k}}\,J_{0}(kr)\,\overline{h}(z_{1},z_{2},k),$
(76)
where $d\mathrm{\mathbf{k}}$ is an area element in the plane othogonal to the
$z$-axis. Substitution of (75) and (76) into (74) yields
$\displaystyle h(z_{1},z_{2},r_{2})=$ (77) $\displaystyle
c^{(2)}(z_{1},z_{2},r_{2})+\frac{1}{(2\pi)^{4}}\\!\\!\int_{-\infty}^{\infty}\\!\\!\\!dz_{3}\,\rho(z_{3})\int_{0}^{\infty}\\!\\!\\!dr_{3}\,r_{3}\int\\!d\mathrm{\mathbf{k^{\prime}}}\\!\\!\int\\!d\mathrm{\mathbf{k^{\prime\prime}}}$
$\displaystyle\times\overline{h}(z_{1},z_{3},k^{\prime})\,\overline{c}^{\,(2)}(z_{3},z_{2},k^{\prime\prime})\,J_{0}(k^{\prime}r_{3})\\!\\!\int_{0}^{2\pi}\\!\\!d\theta
J_{0}(k^{\prime\prime}r_{32}),$
where we note that the separation $r_{32}$ is a function of $\theta$. Graf’s
addition theorem for Bessel functions states that
$J_{0}(r_{23})=\sum_{n=-\infty}^{\infty}J_{n}(r_{2})J_{n}(r_{3})e^{in\theta},$
(78)
which implies the useful result
$\int_{0}^{2\pi}d\theta\;J_{0}(r_{23})=2\pi J_{0}(r_{2})J_{0}(r_{3}).$ (79)
Using (79) to perform the $\theta$-integral in (77) yields
$\displaystyle
h(z_{1},z_{2},r_{2})=c^{(2)}(z_{1},z_{2},r_{2})+\int_{-\infty}^{\infty}\\!\\!dz_{3}\,\rho(z_{3})$
$\displaystyle\times\\!\int
d\mathrm{\mathbf{k^{\prime}}}\\!\\!\int\\!d\mathrm{\mathbf{k^{\prime\prime}}}\,\overline{h}(z_{1},z_{3},k^{\prime})\,\overline{c}^{(2)}(z_{3},z_{2},k^{\prime\prime})J_{0}(k^{\prime\prime}r_{2})$
$\displaystyle\times\frac{1}{(2\pi)^{3}}\int_{0}^{\infty}\\!\\!dr_{3}\,r_{3}J_{0}(k^{\prime}r_{3})J_{0}(k^{\prime\prime}r_{3}).$
(80)
Bessel functions obey the orthogonality relation
$\displaystyle
2\pi\\!\int_{0}^{\infty}\\!\\!dr_{3}\;r_{3}J_{0}(kr_{3})J_{0}(k^{\prime}r_{3})=(2\pi)^{2}\delta(\mathrm{\mathbf{k}}-\mathrm{\mathbf{k^{\prime}}}),$
(81)
which could also be viewed as the Hankel transform of the zero-order Bessel
function. Using this in (80) yields
$\displaystyle h(z_{1},$ $\displaystyle
z_{2},r_{2})=c^{(2)}(z_{1},z_{2},r_{2})+\int_{-\infty}^{\infty}dz_{3}\,\rho(z_{3})$
(82) $\displaystyle\times\frac{1}{(2\pi)^{2}}\\!\int
d\mathrm{\mathbf{k^{\prime}}}\,\overline{h}(z_{1},z_{3},k^{\prime})\,\overline{c}^{(2)}(z_{3},z_{2},k^{\prime})J_{0}(k^{\prime}r_{2}).$
Step 2: Now that we have reexpressed the integration over the internal
coordinate ${\bf r}_{3}$ we will Hankel transform (82) with respect to the
external coordinate $r_{2}$. Applying the operator
$2\pi\int_{0}^{\infty}dr_{2}\,r_{2}J_{0}(kr_{2})$ to both sides of the
equation yields
$\displaystyle\overline{h}(z_{1},z_{2},k)=\overline{c}^{(2)}(z_{1},z_{2},k)+\frac{1}{2\pi}\int_{-\infty}^{\infty}dz_{3}\,\rho(z_{3})$
(83) $\displaystyle\times\\!\\!\\!\int
d\mathrm{\mathbf{k^{\prime}}}\,\overline{h}(z_{1},z_{3},k^{\prime})\,\overline{c}^{(2)}(z_{3},z_{2},k^{\prime})\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!dr_{2}\,r_{2}J_{0}(kr_{2})J_{0}(k^{\prime}r_{2}).$
Using once more the orthogonality relation (81) then leads directly to the
Hankel transformed OZ equation (19) in the main text.
## Appendix C
We show here the calculation analogous to that in the preceding Appendix, but
now for spherical geometry. Starting from the general expression (8) we obtain
the Legendre transformed OZ equation (39) (closely following the presentation
of Refs.[7] and [22]). To clarify the treatment of ‘class 3’ terms in the main
text we break the calculation into two steps.
Step 1: The OZ equation (8) can be rewritten as
$\displaystyle h(r_{1},r_{2},x_{2})$
$\displaystyle=c^{(2)}(r_{1},r_{2},x_{2})+\int_{0}^{\infty}\\!dr_{3}\;r_{3}^{2}\,\rho(r_{3})$
(84)
$\displaystyle\times\int_{0}^{2\pi}\\!d\phi_{3}\int_{-1}^{1}\\!dx_{3}\,h(r_{1},r_{3},x_{3})\,c^{(2)}(r_{3},r_{2},x_{32}),$
where we have chosen the $z$-axis of the spherical coordinate system to
coincide with the vector ${\bf r}_{1}$, which implies
$\theta_{1}\\!=\\!\phi_{1}\\!=\\!0$. Without loss of generality we can also
orient the coordinates such that ${\bf r}_{2}$ lies in the $xz$-plane, such
that $\phi_{2}=0$. Using the back-transform (38) to represent both the pair
direct and total correlation functions yields
$\displaystyle
h(r_{1},r_{2},x_{2})=c^{(2)}(r_{1},r_{2},x_{2})+\sum_{i,j=0}^{\infty}\int_{0}^{\infty}\\!dr_{3}\;r_{3}^{2}\,\rho(r_{3})$
$\displaystyle\times\int_{0}^{2\pi}\\!d\phi_{3}\int_{-1}^{1}\\!dx_{3}\,\hat{h}(r_{1},r_{3},i)\,\hat{c}^{(2)}(r_{3},r_{2},j)P_{i}(x_{3})P_{j}(x_{32}).$
(85)
For our chosen orientation of coordinate system the addition theorem for
spherical harmonics states that
$\displaystyle P_{j}(x_{32})$ $\displaystyle=P_{j}(x_{3})P_{j}(x_{2})$ (86)
$\displaystyle\quad+2\sum_{m=1}^{j}\frac{(i-m)!}{(j+m)!}P_{j}^{m}(x_{3})P_{j}^{m}(x_{2})\cos(m\phi_{3}).$
Substitution of (86) into (C) and performing the integration over $\phi_{3}$
yields
$\displaystyle
h(r_{1},r_{2},x_{2})=c^{(2)}(r_{1},r_{2},x_{2})+\sum_{i,j=0}^{\infty}\int_{0}^{\infty}\\!dr_{3}\;r_{3}^{2}\,\rho(r_{3})$
$\displaystyle\times
2\pi\int_{-1}^{1}\\!dx_{3}\,\hat{h}(r_{1},r_{3},i)\,\hat{c}^{(2)}(r_{3},r_{2},j)P_{i}(x_{3})P_{j}(x_{3})P_{j}(x_{2}).$
(87)
Legendre polynomials obey the orthogonality relation
$\int_{-1}^{\,1}\\!dx\;P_{i}(x)P_{j}(x)=\frac{2}{2i+1}\delta_{ij},$ (88)
where $\delta_{ij}$ is the Kronecker delta. Using this in (C) yields
$\displaystyle
h(r_{1},r_{2},x_{2})=c^{(2)}(r_{1},r_{2},x_{2})+\sum_{j=0}^{\infty}\int_{0}^{\infty}\\!dr_{3}\;r_{3}^{2}\,\rho(r_{3})$
$\displaystyle\times\frac{4\pi}{2j+1}\hat{h}(r_{1},r_{3},j)\,\hat{c}^{(2)}(r_{3},r_{2},j)P_{j}(x_{2}).$
(89)
Step 2: Legendre transform (C) with respect to the external coordinate
$x_{2}$. Applying the operator
$\frac{2n+1}{2}\\!\int_{-1}^{\,1}dx_{2}\,P_{n}(x_{2})$ to both sides of the
equation yields
$\displaystyle\hat{h}(r_{1},r_{2},n)=\hat{c}^{(2)}(r_{1},r_{2},n)+\sum_{j=0}^{\infty}\int_{0}^{\infty}\\!dr_{3}\;r_{3}^{2}\,\rho(r_{3})\,2\pi$
(90)
$\displaystyle\times\frac{2n+1}{2j+1}\,\hat{h}(r_{1},r_{3},j)\,\hat{c}^{(2)}(r_{3},r_{2},j)\int_{-1}^{\,1}dx_{2}\,P_{n}(x_{2})P_{j}(x_{2}).$
Using once more the orthogonality relation (88) then leads directly to the
Legendre transformed OZ equation (39) in the main text.
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|
111Lei Zhang is partially supported by a Simons Foundation Collaboration Grant
# On Liouville systems at critical parameters, Part 2: Multiple bubbles
Hsin-Yuan Huang Department of Applied Mathematics
National Chiao-Tung University
Hsinchu, Taiwan<EMAIL_ADDRESS>and Lei Zhang Department of
Mathematics
University of Florida
358 Little Hall P.O.Box 118105
Gainesville FL 32611-8105<EMAIL_ADDRESS>
###### Abstract.
In this paper, we continue to consider the generalized Liouville system:
$\Delta_{g}u_{i}+\sum_{j=1}^{n}a_{ij}\rho_{j}\left(\frac{h_{j}e^{u_{j}}}{\int
h_{j}e^{u_{j}}}-{1}\right)=0\quad\text{in \,}M,\quad i\in I=\\{1,\cdots,n\\},$
where $(M,g)$ is a Riemann surface $M$ with volume $1$, $h_{1},..,h_{n}$ are
positive smooth functions and $\rho_{j}\in\mathbb{R}^{+}$($j\in I$). In
previous works Lin-Zhang identified a family of hyper-surfaces $\Gamma_{N}$
and proved a priori estimates for $\rho=(\rho_{1},..,\rho_{n})$ in areas
separated by $\Gamma_{N}$. Later Lin-Zhang also calculated the leading term of
$\rho^{k}-\rho$ where $\rho\in\Gamma_{1}$ is the limit of $\rho^{k}$ on
$\Gamma_{1}$ and $\rho^{k}$ is the parameter of a bubbling sequence. This
leading term is particularly important for applications but it is very hard to
be identified if $\rho^{k}$ tends to a higher order hypersurface $\Gamma_{N}$
($N>1$). Over the years numerous attempts have failed but in this article we
overcome all the stumbling blocks and completely solve the problem under the
most general context: We not only capture the leading terms of
$\rho^{k}-\rho\in\Gamma_{N}$, but also reveal new robustness relations of
coefficient functions at different blowup points.
###### Key words and phrases:
Liouville system, asymptotic analysis, a priori estimate, classification of
solutions, method of unique continuation, Pohozaev identity, blowup phenomenon
###### 1991 Mathematics Subject Classification:
35J60, 35J55
## 1\. Introduction
Let $(M,g)$ be a compact Riemann surface with volume $1$, $h_{1},...,h_{n}$ be
positive $C^{3}$ functions on $M$, $\rho_{1},..,\rho_{n}$ be nonnegative
constants. In this article we continue our study of the following Liouville
system defined on $(M,g)$:
(1.1)
$\Delta_{g}u_{i}+\sum_{j=1}^{n}\rho_{j}a_{ij}\left(\frac{h_{j}e^{u_{j}}}{\int_{M}h_{j}e^{u_{j}}dV_{g}}-1\right)=0,\quad
i\in I:=\\{1,..,n\\}$
where $dV_{g}$ is the volume form, $\Delta_{g}$ is the Laplace-Beltrami
operator $\Delta_{g}\leq 0$. When $n=1$, equation (1.1) is the mean field
equation of the Liouville type:
(1.2)
$\Delta_{g}u+\rho\left(\frac{he^{u}}{\int_{M}he^{u}dV_{g}}-{1}\right)=0\quad\text{in
\,}M$
when $a_{11}=1$. Therefore, the Liouville system (1.1) is a natural extension
of the classical Liouville equation, which has been extensively studied for
decades because of its profound connections with various fields in geometry
and physics. Since the general form of Liouville systems includes many models
from Biology, Physics and other disciplines of sciences, it is very desirable
to study generical Liouville systems and derive common features. Recently, the
Liouville system has drawn a lot of attention because it also arises from the
stationery solutions of multi-species Patlak-Keller-Segel system[41] and self-
dual condensate solutions of Ablian Chern-Simons model with $N$ Higgs
particles[38, 31] when certain parameter tends to zero. In particular, these
two examples exhibit the bubbling phenomenon. The study of bubbling solutions
represents an essential difficulty of Liouville system and it not only impacts
the immediately related fields but also depends on the development of them.
The readers may look into the following references for closely related
discussions [1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 14, 17, 18, 19, 20, 23, 25, 26,
27, 28, 30, 35, 36, 37, 39, 40, 42, 43, 44].
For system (1.1) the Sobolev spaces for solutions are
${}^{\text{\r{}}}H^{1,n}=\,^{\text{\r{}}}H^{1}(M)\times\cdots\times\,^{\text{\r{}}}H^{1}(M)$
where
${}^{\text{\r{}}}H^{1}(M)=\left\\{u\in L^{2}(M)\,\Big{|}\,|\nabla_{g}u|\in
L^{2}(M)\text{ and }\int_{M}u\,dV_{g}=0\right\\}.$
For any $\rho=(\rho_{1},\cdots,\rho_{n})$, $\rho_{i}>0$, let $\varPhi_{\rho}$
be a nonlinear functional defined in ${}^{\text{\r{}}}H^{1,n}$ by
$\varPhi_{\rho}(u)=\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}a^{ij}\int_{M}\nabla_{g}u_{i}\cdot\nabla_{g}u_{j}dV_{g}-\sum_{j=1}^{n}\rho_{j}\log\int_{M}h_{j}e^{u_{j}}dV_{g}$
where $(a^{ij})_{n\times n}$ is the inverse of $A=(a_{ij})_{n\times n}$, $I$
is the set of indexes: $I=\\{1,...,n\\}$. It is easy to see that equation
(1.1) is the Euler-Lagrangian equation of $\varPhi_{\rho}$.
In [32, 33], Lin and the second author completed a degree counting program for
(1.2) under the following two assumptions on the matrix A:
$\displaystyle(H1):\quad A\mbox{ is symmetric, nonnegative, irreducible and
invertible.}$ $\displaystyle(H2):\quad a^{ii}\leq 0,\,\,\forall i\in I,\quad
a^{ij}\geq 0\,\,\forall i\neq j\in I,\quad\sum_{j=1}^{n}a^{ij}\geq
0\,\,\forall i\in I.$
Roughly speaking $(H1)$ is a rather standard assumption for Liouville systems,
$(H2)$ says the interaction between equations has to be strong. In [33] Lin-
Zhang identified a family of hypersurfaces
$\Gamma_{N}=\left\\{\rho\,\big{|}\,\rho_{i}>0,i\in
I;\,\,\Lambda_{I}(\rho)=0,\quad\Lambda_{J}(\rho)>0,\,\,\forall\,\,\emptyset\neq
J\subsetneq I,\,\,\right\\}.$
where
$\Lambda_{I}(\rho)=4\sum_{i=1}^{n}\frac{\rho_{i}}{2\pi
N}-\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}\frac{\rho_{i}}{2\pi
N}\frac{\rho_{j}}{2\pi N}$
($\Lambda_{J}(\rho)$ is understood similarly). Furthermore they proved that if
$\rho=(\rho_{1},...,\rho_{n})$ stays in a regions bounded by these critical
hypersurfaces, a priori estimate holds. Based on the a priori estimate, Lin-
Zhang proved the following degree counting formula which implies existence of
solution if the degree is not zero:
(1.3) $d_{\rho}=\left\\{\begin{array}[]{ll}1\quad\text{if
}\rho\in\mathcal{O}_{0}\\\
\frac{1}{N!}\bigg{(}(-\chi_{M}+1)...(-\chi_{M}+N)\bigg{)}\quad\text{if
}\rho\in\mathcal{O}_{N}.\end{array}\right.$
where $\chi_{M}$ is the Euler characteristic of $M$, $\mathcal{O}_{N}$ is the
region between $\Gamma_{N}$ and $\Gamma_{N+1}$. The jump of the degree from
$\mathcal{O}_{N-1}$ to $\mathcal{O}_{N}$ is contributed by the blowup
solutions to (1.1).
In this article we consider the case $\rho\to\Gamma_{N}$ from
$\mathcal{O}_{N-1}$ or $\mathcal{O}_{N}$: suppose
$\rho=(\rho_{1},..,\rho_{n})\in\Gamma_{N}$ is a limit point,
$\rho^{k}=(\rho_{1}^{k},..,\rho_{n}^{k})$ is a sequence of parameters
corresponding to bubbling solutions, the aim of this article is to identity
the leading term of $\rho^{k}-\rho$ as $k\to\infty$. Since the normal vector
at $\rho$ is proportional to
$(\sum_{j=1}^{n}a_{1j}\frac{\rho_{j}}{2\pi
N}-2,...,\sum_{j=1}^{n}a_{nj}\frac{\rho_{j}}{2\pi N}-2),$
which has all components positive (implied by $\Lambda_{J}>0$ in the
definition of $\Gamma_{N}$, see [33]), we assume
(1.4) $\frac{\rho_{i}^{k}-\rho_{i}}{\rho_{j}^{k}-\rho_{j}}\sim
1\quad\forall\quad i\neq j\in I.$
Note that $A_{k}\sim B_{k}$ means $CA_{k}\leq B_{k}\leq C_{1}A_{k}$ for some
$C,C_{1}>0$ independent of $k$. It is established in [33] that when
$\rho^{k}=(\rho_{1}^{k},..,\rho_{n}^{k})$ tends to $\rho\in\Gamma_{N}$, blowup
solutions have exactly $N$ disjoint blowup points: $p_{1},...,p_{N}$. In [34]
Lin-Zhang derived the leading term when $N=1$. It is interesting to observe in
[34] that there is one particular point $Q\in\Gamma_{1}$ such that if $\rho\to
Q$ the leading term contains local curvature at $Q$ only, but if $\rho$ tends
to any other point, the leading term is involved with global integration of
the whole manifold.
The main purpose of this article is to extend the result in [34] to
$\rho^{k}\to\Gamma_{N}$ when $N>1$. Among other things, we obtain the leading
term of $\Lambda_{I}(\rho^{k})$ as $\rho^{k}\to\Gamma_{N}$ which gives us the
sufficient conditions of a uniform bound of solutions as
$\rho^{k}\to\Gamma_{N}$. Let $p_{1}$,…,$p_{N}$ be $N$ disjoint blowup points,
which means for each $p_{t}$, there exist $p_{t}^{k}\to p_{t}$ such that
$\max_{i\in I}u_{i}^{k}(p_{t}^{k})\to\infty$, $t\in\\{1,\cdots,N\\}$. For
$h_{i}^{k}$ we assume that they are uniformly bounded by positive constants:
(1.5) $\frac{1}{C}\leq h_{i}^{k}(x)\leq C,\quad\|h_{i}^{k}\|_{C^{3}(M)}\leq C$
for all $i$ and a $C>0$ independent of $k$. Throughout the paper we use
$u^{k}=(u_{1}^{k},...,u_{n}^{k})$ to denote blowup solutions and $M_{k,t}$ to
denote the magnitude of $u^{k}$ near the blowup point $p_{t}$ and use
$\epsilon_{k,t}=e^{-\frac{1}{2}M_{k,t}}$ to measure the errors:
(1.6) $M_{k,t}=\max_{i\in I}\max_{x\in
B(p_{t},\delta_{0})}\\{u_{i}^{k}(x)-\log\int_{M}h_{i}^{k}e^{u_{i}^{k}}dV_{g},\\},\quad\epsilon_{k,t}=e^{-\frac{1}{2}M_{k,t}},$
Here we require $\delta_{0}$ to be small enough so that
$B(p_{t_{1}},\delta_{0})\cap B(p_{t_{2}},\delta_{0})=\emptyset$ for all
$t_{1}\neq t_{2}$. Let $p_{t}^{k}$ be the point where
$\max_{i\in I}\max_{x\in
B(p_{t},\delta_{0})}u_{i}^{k}(x)-\log\int_{M}h_{i}^{k}e^{u_{i}^{k}}dV_{g}$
is taken in $B(p_{t},\delta_{0})$. Under the assumptions $(H1)$ and $(H2)$, we
have a full blow-up picture in all balls (see [33]).
To understand more precise information for the blowup phenomenon to (1.1), we
shall study the convergence rate of $\Lambda_{I}(\rho^{k})$ as
$\rho^{k}\to\rho$ in terms of the magnitude of $u^{k}$. The following two
fundamental questions will be answered in this paper:
* (1)
Are the magnitude of $u^{k}$ at different blowup points comparable to each
other?
* (2)
What is the convergence rate of the difference of the local masses
$\sigma_{i,s}^{k}-\sigma_{i,t}^{k}$ where
$\sigma_{i,s}^{k}=\int_{B(p_{s},\delta_{0})}e^{u_{i}^{k}}dV_{g}$ and
$\sigma_{i,t}^{k}=\int_{B(p_{t},\delta_{0})}e^{u_{i}^{k}}dV_{g}$?
The first main result is to answer the first question.
###### Theorem 1.1.
Let $u^{k}\in\,\,^{\text{\r{}}}H^{1,n}(M)$ be a sequence of blowup solutions
of (1.1). Suppose $(H1),(H2)$ holds for $A$, (1.4) holds for $\rho^{k}$ and
(1.5) holds for $h_{i}^{k}$. Then
(1.7) $|M_{k,s}-M_{k,t}|=O(1),\quad\text{ for
}\,\,s\not=t,\,\,s,t\in\\{1,\cdots,N\\}.$
Here, $O(1)$ is independent of $k$.
Note that without knowing $|M_{k,s}-M_{k,t}|=O(1)$ we do not even know if
$O(\epsilon_{k,s})=O(\epsilon_{k,t})$. We thus can use
$\epsilon_{k}=e^{-\frac{1}{2}M_{k}}$ to measure the errors, where
$M_{k}=\max_{i\in I}\max_{x\in
M}\\{u_{i}^{k}(x)-\log\int_{M}h_{i}^{k}e^{u_{i}^{k}}dV_{g},\\}.$
Let
(1.8) $m_{i}=\sum_{j=1}^{n}a_{ij}\frac{\rho_{j}}{2\pi N},\quad i=1,..,n,\quad
m=\min_{i\in I}m_{i}.$
Here we note that either $2<m<4$ or all $m_{i}=4$ for all $m_{i}$ (see
(2.15)). Now we define a special point $Q_{N}=(q_{1},...,q_{n})$ on
$\Gamma_{N}$, which satisfies
$\sum_{j=1}^{n}a_{ij}q_{j}=8\pi N\quad\forall i\in I.$
The second result is showing the tightness of the local masses.
###### Theorem 1.2.
Under the same assumptions in Theorem 1.1.
* (1)
If $\rho^{k}\to\rho\in\Gamma_{N}(\rho\neq Q_{N})$ from $\mathcal{O}_{N-1}$ or
$\mathcal{O}_{N}$, then
(1.9) $|\sigma_{i,s}^{k}-\sigma_{i,t}^{k}|=O(\epsilon_{k}^{m-2}),\quad\text{
for }s\not=t,\,\,s,t\in\\{1,\cdots,N\\},\,\,i\in I.$
* (2)
If $\rho^{k}\to Q_{N}$ from $\mathcal{O}_{N-1}$ or $\mathcal{O}_{N}$, then
(1.10)
$|\sigma_{i,s}^{k}-\sigma_{i,t}^{k}|=O(\epsilon_{k}^{2}\log\frac{1}{\epsilon_{k}}),\quad\text{
for }s\not=t,\,\,s,t\in\\{1,\cdots,N\\},\,\,i\in I.$
We remark here that the techniques developed in the proof of Theorems 1.1 and
1.2 also play the key roles in the study of the local uniqueness of the
bubbling solutions in Liouville systems[24]. For the sake of contradiction,
one also need to compare two sequence of the bubbling solutions at the blowup
points with the same limit $\rho$.
The leading terms of $\Lambda_{I}(\rho^{k})$ are different in two cases.
Before we state the results, we give some notations:
We define $N$ open sets $\Omega_{t,\delta_{0}}$ such that they are mutually
disjoint, each of them contains a bubbling disk and their union is $M$:
(1.11)
$B(p_{t}^{k},\delta_{0})\subset\Omega_{t,\delta_{0}},\quad\cup_{t=1}^{N}\overline{\Omega_{t,\delta_{0}}}=M,\quad\Omega_{t,\delta_{0}}\cap\Omega_{s,\delta_{0}}=\emptyset,\,\,\forall
t\neq s.$
Let
$I_{1}=\\{i\in I;\quad\lim_{k\to\infty}m_{i}^{k}=m.\quad\\}.$
and $G$ be the Green’s function defined by
$-\Delta_{g}G(x,\cdot)=\delta_{p}-1,\quad\int_{M}G(x,\eta)dV_{g}(\eta)=0,$
and $\gamma$ is the regular part of the Green’s function. Note that in local
coordinates of a point, say $\eta$, $G$ is of the form
$G(x,\eta)=-\frac{1}{2\pi}\log|x-\eta|+\gamma(x,\eta).$
We also define
(1.12)
$G^{*}(p_{t}^{k},p_{s}^{k})=\left\\{\begin{array}[]{ll}\gamma(p_{t}^{k},p_{t}^{k}),\quad
s=t,\\\ G(p_{t}^{k},p_{s}^{k}),\quad s\neq t.\end{array}\right.\qquad
s,t=1,...,N.$
The third result is the leading terms of $\Lambda_{I}(\rho^{k})$ of the first
case.
###### Theorem 1.3.
Under the same assumptions in Theorem 1.1. If
$\rho^{k}\to\rho\in\Gamma_{N}(\rho\neq Q_{N})$ from $\mathcal{O}_{N-1}$ or
$\mathcal{O}_{N}$, then
(1.13) $\Lambda_{I}(\rho^{k})=(D+o(1))\frac{\epsilon_{k}^{m-2}}{N}.$
Here, The quantity $D$ is defined as follows
(1.14) $D=\sum_{i\in
I_{1}}e^{D_{i}-\alpha_{i}}\sum_{t=1}^{N}c_{t}\lim_{\delta_{0}\to
0}\bigg{(}\delta_{0}^{2-m}-\frac{(m-2)}{2\pi}\int_{\hat{\Omega}_{t,\delta_{0}}}(\frac{h_{i}^{k}(x)}{h_{i}^{k}(p_{t}^{k})}e^{2\pi
m\sum_{l=1}^{N}(G(x,p_{l}^{k})-G^{*}(p_{t}^{k},p_{l}^{k}))}dV_{g}\bigg{)}.$
where $\hat{\Omega}_{t,\delta_{0}}=\Omega_{t,\delta_{0}}\setminus
B(p_{t}^{k},\delta_{0})$, $c_{t}=\frac{h_{i}^{k}(p_{t}^{k})e^{2\pi
m\sum_{l=1}^{N}G^{*}(p_{t}^{k},p_{l}^{k})}}{h_{i}^{k}(p_{1}^{k})e^{2\pi
m\sum_{l=1}^{N}G^{*}(p_{1}^{k},p_{l}^{k})}}$ and $D_{i}$, $\alpha_{i}$ are
constants defined in (3.14).
We remark here that $\Lambda_{I}(\rho^{k})>0$ for
$\rho^{k}\in\mathcal{O}_{N-1}$ and $\Lambda_{I}(\rho^{k})<0$ for
$\rho^{k}\in\mathcal{O}_{N}$. Thus, if $D\not=0$, then the blowup solutions
with $\rho^{k}$ occur as $\rho^{k}\to\rho\in\Gamma_{N}(\rho\not=Q_{N})$ only
from one side of $\Gamma_{N}$. Furthermore, it yields a uniform bound of
solutions as $\rho^{k}$ converges to $\Gamma_{N}(\rho\not=Q_{N})$ from
$\mathcal{O}_{N-1}$ provided that $D<0$. It is easy to see that for fixed $k$
the following limit exists:
(1.15) $\lim_{\delta_{0}\to
0}\bigg{(}\delta_{0}^{2-m}-\frac{m-2}{2\pi}\int_{\hat{\Omega}_{t,\delta_{0}}}\frac{h_{i}^{k}(x)}{h_{i}^{k}(p_{t}^{k})}e^{2\pi
m(\sum_{l=1}^{N}G(x,p_{l}^{k})-G^{*}(p_{t}^{k},p_{l}^{k}))}dV_{g}\bigg{)}$
because the leading terms from the Green’s function is
$-\frac{1}{2\pi}\log|x-p_{t}^{k}|$. The study of the sign of $D$ is another
interesting fundamental question. However, it is out of the scope of of the
present article. We will come back to this issue in the further study.
The fourth major result is concerned with the leading term of
$\Lambda_{I}(\rho^{k})$ when $\rho^{k}\to Q_{N}$ and the major difference is
that the leading term only depends on the curvature and coefficient functions
at blowup points: $p_{1}^{k}$, …, $p_{N}^{k}$:
###### Theorem 1.4.
Under the same assumptions in Theorem 1.1. If $\rho^{k}\to Q_{N}$ from
$\mathcal{O}_{N-1}$ or $\mathcal{O}_{N}$, then
(1.16)
$\Lambda_{I}(\rho^{k})=-4\sum_{i=1}^{n}\sum_{t=1}^{N}b_{it}^{k}\epsilon_{k}^{2}\log\epsilon_{k}^{-1}+O(\epsilon_{k}^{2})$
where
(1.17) $\displaystyle b_{it}^{k}$
$\displaystyle=e^{D_{i}-\alpha_{i}}\bigg{(}\frac{1}{4}\frac{\Delta
h_{i}^{k}(p_{t}^{k})}{h_{i}^{k}(p_{t}^{k})}-K(p_{t}^{k})+4\pi N$
$\displaystyle+4\pi\frac{\nabla
h_{i}^{k}(p_{t}^{k})}{h_{i}^{k}(p_{t}^{k})}\cdot\sum_{l=1}^{N}\nabla_{1}G^{*}(p_{t}^{k},p_{l}^{k})+16\pi^{2}|\sum_{l=1}^{N}\nabla_{1}G^{*}(p_{t}^{k},p_{l}^{k})|^{2}\bigg{)}.$
and $K$ is the Gaussian curvature.
As an application of the formula (1.16), we obtain the uniform bound of
solutions as $\rho^{k}$ converges to $Q_{N}$ from $\mathcal{O}_{N-1}$ provided
$b_{it}^{k}>0$ for $t=1,\cdots,N.$
The fifth result is about the locations of the blowup points and the mutual
relation of coefficient functions.
###### Theorem 1.5.
If $\rho^{k}\to\rho\neq Q$ as in (1.4), then for $t=1,...,N$
(1.18) $\sum_{i=1}^{n}\big{(}\nabla(\log h_{i}^{k})(p_{t}^{k})+2\pi
m_{i}\sum_{s=1}^{N}\nabla_{1}G^{*}(p_{t}^{k},p_{s}^{k})\big{)}=O(\epsilon_{k}^{m-2}),\,\,t=1,..,N$
where $\nabla_{1}$ means the differentiation with respect to the first
component. If $\rho^{k}\to Q$ as in (1.4),
(1.19) $\sum_{i=1}^{n}\big{(}\nabla(\log
h_{i}^{k})(p_{t}^{k})+8\pi\sum_{s=1}^{N}\nabla_{1}G^{*}(p_{t}^{k},p_{s}^{k})\big{)}=O(\epsilon_{k}^{2}\log\epsilon_{k}^{-1}),\,\,t=1,..,N$
Besides the location of blowup points, we also reveal new information about
coefficient functions in the next theorem. For convenience we set
(1.20) $H_{i,t}^{k}:=\log h_{i}^{k}(p_{t}^{k})+2\pi
m_{i}\sum_{l=1}^{N}G^{*}(p_{t}^{k},p_{l}^{k}).$
###### Theorem 1.6.
Let $H_{i,t}^{k}$ be defined as in (1.20), then we have
(1.21) $H_{i,t}^{k}=H_{i,s}^{k}+O(\epsilon_{k}^{m-2}),\quad\forall i\in
I,\quad\forall t\neq s,\quad\mbox{if}\quad m<4,$ (1.22)
$H_{i,t}^{k}=H_{i,s}^{k}+O(\epsilon_{k}^{2}\log\frac{1}{\epsilon_{k}}),\quad\forall
i\in I,\quad t\neq s,\quad\mbox{if}\quad m=4.$
The tightness of the coefficient functions is not seen in the case of $N=1$.
This show that the occurrence of multi bubbles forces the coefficient
functions at the blowup points to be the same. In construction of bubbling
solutions one needs to know the precise information about bubbling
interactions, exact location of blowup points, accurate vanishing rate of
coefficient functions and specific leading terms in asymptotic expansions. All
these have been covered in the main results of this article. Until now the
construction of bubbling solutions for Liouville systems is still in the early
stage of development, as the constructions so far are still restricted to
single blowup point situations [22].
In all these main results the readers can see that sharp estimates are
obtained for all the error terms. This is why we think these results will play
as a central role in applications. We expect the theorems of this article to
serve as a benchmark for more sophisticated discussions in the near future.
Also Gu-Zhang [21] completed a degree counting program for singular Liouville
systems, the corresponding discussion for leading terms of approximating
critical hyper-surfaces of singular Liouville systems is another exciting
unconquered land to explore. Besides these immediate impacts to closely
related fields, the idea of the proof in this article, the way to overcome
major difficulties in bubble interaction could lead to major advance in Chern-
Simons type equations (see [23]) as well.
The main difficulty in the proof of the main theorems is on the interaction of
bubbling solutions, which has a large to do with the nature of Liouville
systems. For global solutions defined in $\mathbb{R}^{2}$, it is established
in [15, 32] that total integrations of all components form a $n-1$ dimensional
hyper-surface similar to $\Gamma_{1}$. This continuum of energy brings great
difficulty for bubbling interaction: If we only have two bubbling disks, the
energy in each disk is very close to $\Gamma_{1}$, but to identify the leading
term of $\rho^{k}-\rho\in\Gamma_{2}$ one has to prove that they are both
tending the _same point_ on $\Gamma_{1}$. It does not help to use the fact
that they have the same limit, because the energy sequence may tend to its
limit position very slowly. This difficulty does not exist for Toda systems,
because the energy set for Toda systems is discrete. When we have only one
bubble, this bubble interaction situation can be avoided ( see [34]). So the
main contribution in this article is to prove that the bubbling solutions have
almost the same energy in each bubbling disk. The key idea is as follows:
Around each blowup point, we first use an approximation theorem of Lin-Zhang
[34] to have an initial expansion of the bubbling solution. The first term of
this expansion is a sequence of global solutions. In this article we identify
what the global sequence is around each bubbling point and compare them using
a key idea of Lin-Zhang [32] in their proof of the classification theorem for
Liouville systems. It turns out that after scaling, the global sequences are
extremely close to one another. Then we further prove that the energy of the
global sequence is not too much different from the bubbling solutions in each
bubbling disk. All the error terms must be carefully identified in order to
single out the leading terms in the main theorems.
The organization of this article is as follows: In section two we deploy the
basic setting for all the topics in this article and invoke a few
approximation results in previous works of Lin-Zhang. In section three we
prove the closeness of bubbling solutions around different blowup points, in
which Theorems 1.1, 1.2 and 1.6 will be proved. The proof is set-up in two
stages as the approximation becomes better in the second stage. Finally in
section four all other main theorems will be proved based on the precise
estimates in section three.
## 2\. Approximation around a blowup point
First we claim that we can assume $u^{k}=(u_{1}^{k},...,u_{n}^{k})$ to satisfy
(2.1) $\int_{M}h_{i}^{k}e^{u_{i}^{k}}dV_{g}=1,\quad Vol(M)=1.$
because otherwise we just consider
(2.2) $\Theta_{i}^{k}=u_{i}^{k}-\log\int_{M}h_{i}^{k}e^{u_{i}^{k}}dV_{g},\quad
i\in I.$
Then we have
(2.3)
$-\Delta_{g}\Theta_{i}^{k}=\sum_{j=1}^{n}a_{ij}\rho_{j}^{k}(h_{j}^{k}e^{\Theta_{j}^{k}}-1).$
where $\Theta^{k}$ satisfies (2.1). Here we first set up preliminary
discussions about the profile of $u^{k}$ near a blowup point. Suppose $p$ is a
blowup point and in $B(p,\delta)$ there is only one blowup point of $u^{k}$.
Let
$\tilde{M}_{k}=\max_{i\in I,x\in
B(p,\delta)}u_{i}^{k}(x)+\log(\rho_{i}^{k}h_{i}^{k}(\tilde{p}_{k}))\quad\mbox{and
}\quad\tilde{\epsilon}_{k}=e^{-\frac{1}{2}\tilde{M}_{k}},$
and $\tilde{p}_{k}$ be where $\tilde{M}_{k}$ is attained ( $\tilde{p}_{k}\to
p$). Then the functions
$\tilde{v}_{i}^{k}(y)=u_{i}^{k}(\tilde{p}_{k}+\tilde{\epsilon}_{k}y)+2\log\tilde{\epsilon}_{k}$
converge in $C^{2}_{loc}(\mathbb{R}^{2})$ to the limit function
$v=(v_{1},...,v_{n})$ which is a global solution of
(2.4) $\left\\{\begin{array}[]{ll}-\Delta
v_{i}=\sum_{j}a_{ij}e^{v_{j}},\quad\mathbb{R}^{2},\quad i=1,..,n\\\ \\\
\int_{\mathbb{R}^{2}}e^{v_{i}}<\infty,\quad i=1,..,n,\quad\max_{i\in
I}v_{i}(0)=0.\end{array}\right.$
Here we note that it is established in [33] that with assumptions (H1), (H2)
all the bubbling solutions are fully bubbling: the limit must have $n$
equations and no component is lost in the limiting taking process. The
classification of all global solutions of (2.4) was completed in the work of
Chipot-Shafrir-Wolansky [15] and Lin-Zhang [32]. All components of
$v=(v_{1},...,v_{n})$ have one common point of symmetric symmetry. In this
context, this common point is the origin.
To state more precise approximation results we write the equation in local
coordinates around $\tilde{p}_{k}$. In this coordinate $ds^{2}$ has the form
$e^{\phi(y_{\tilde{p}_{k}})}(dy_{1}^{2}+dy_{2}^{2})$ where
(2.5)
$|\nabla\phi(0)|=0,\quad\phi(0)=0,\quad\Delta_{y_{\tilde{p}_{k}}}\phi=-2Ke^{\phi}$
where $K$ is the Gauss curvature. In local coordinates, (2.2) becomes
(2.6) $-\Delta
u_{i}^{k}=\sum_{j=1}^{n}a_{ij}\rho_{j}^{k}e^{\phi}(h_{j}^{k}e^{u_{j}^{k}}-1),\quad\mbox{in}\quad
B(0,\delta).$
Let $f_{i}^{k}$ be defined by
(2.7) $\left\\{\begin{array}[]{ll}\Delta
f_{i}^{k}=\sum_{j=1}^{n}a_{ij}\rho_{j}^{k}e^{\phi},\quad\mbox{in}\quad
B(0,\delta).\\\ \\\ f_{i}^{k}(0)=0,\quad\nabla
f_{i}^{k}(0)=0.\end{array}\right.$
Then we have
(2.8)
$-\Delta(u_{i}^{k}-f_{i}^{k})=\sum_{j=1}^{n}a_{ij}\rho_{j}^{k}h_{j}^{k}e^{u_{j}^{k}-f_{j}^{k}}e^{f_{j}^{k}}e^{\phi},\quad\mbox{in}\quad
B(0,\delta).$
If we set
$\tilde{h}_{i}^{k}(x)=\frac{h_{i}^{k}(x)}{h_{i}^{k}(\tilde{p}_{k})}e^{\phi+f_{i}^{k}},$
we have $\tilde{h}_{i}^{k}(0)=1$,
(2.9) $\nabla(\log\tilde{h}_{i}^{k})(0)=\nabla(\log
h_{i}^{k})(\tilde{p}_{k}).$ (2.10)
$\Delta(\log\tilde{h}_{i}^{k})(0)=\Delta(\log
h_{i}^{k})(\tilde{p}_{k})-2K(\tilde{p}_{k})+\sum_{j=1}^{n}a_{ij}\rho_{j}^{k}.$
Thus we set
(2.11) $\tilde{u}_{i}^{k}=u_{i}^{k}+\log\rho_{i}^{k}+\log
h_{i}^{k}(\tilde{p}_{k})-f_{i}^{k},$
and write the equation of $\tilde{u}_{i}^{k}$ as
(2.12)
$\Delta\tilde{u}_{i}^{k}+\sum_{j=1}^{n}a_{ij}\tilde{h}_{j}^{k}e^{\tilde{u}_{j}^{k}}=0,\quad\mbox{
in}\quad B(0,\delta).$
Now we introduce $\phi_{i}^{k}$ to be a harmonic function defined by the
oscillation of $\tilde{u}_{i}^{k}$ on $B(\tilde{p}_{k},\delta)$:
(2.13) $\left\\{\begin{array}[]{ll}-\Delta\phi_{i}^{k}=0,\quad\mbox{in}\quad
B(0,\delta),\\\ \\\
\phi_{i}^{k}=\tilde{u}_{i}^{k}-\frac{1}{2\pi\delta}\int_{\partial
B(0,\delta)}\tilde{u}_{i}^{k},\quad\mbox{on}\quad\partial
B(0,\delta).\end{array}\right.$
Obviously $\phi_{i}^{k}(0)=0$ by the mean value theorem and $\phi_{i}^{k}$ is
uniformly bounded on $B(0,\delta/2)$ because $u_{i}^{k}$ has finite
oscillation away from blowup points. It is a standard fact (see [32, 33]) that
the location of $\max_{i\in I}\max_{x\in
B(0,\delta)}\tilde{u}_{i}^{k}(x)-\phi_{i}^{k}(x)$ is $O(\epsilon_{k}^{2})$
(roughly speaking, the reason is $0$ is a non-degenerate maximum of
$u_{i}^{k}$ and $\phi_{i}^{k}(0)=0$).
Going back to the original coordinate system, we call the maximum point after
perturbation $p_{k}$. Now we set
$M_{k}=\max_{i\in I}\max_{x\in
B(0,\delta)}u_{i}^{k}(x)+\log(\rho_{i}^{k}h_{i}^{k}(p_{k}))-\phi_{i}^{k}(x),\quad\epsilon_{k}=e^{-\frac{1}{2}M_{k}},$
and we let $V^{k}=(V_{1}^{k},..,V_{n}^{k})$ be the radial solutions of
(2.14) $\left\\{\begin{array}[]{ll}-\Delta
V_{i}^{k}=\sum_{j=1}^{n}a_{ij}e^{V_{j}^{k}}\quad\mbox{in}\quad\mathbb{R}^{2},\quad
i\in I,\\\ \\\
V_{i}^{k}(0)=u_{i}^{k}(p_{k})+\log(\rho_{i}^{k}h_{i}^{k}(p_{k}))-\phi_{i}^{k}(p_{k}),\quad
i\in I.\end{array}\right.$
Note that since $p_{k}=\tilde{p}_{k}+O(\epsilon_{k}^{2})$, it is easy to
obtain that the oscillation of $V_{i}^{k}$ on $\partial B(p_{k},\delta)$ is
$O(\epsilon_{k}^{2})$. The sequence of function
$V^{k}=(V_{1}^{k},...,V_{n}^{k})$, which agrees with
$u_{i}^{k}(x)+\log(\rho_{i}^{k}h_{i}^{k}(p_{k}))-\phi_{i}^{k}(x)$ at $p_{k}$,
gives the first term in the approximation of $u_{i}^{k}$ near $p$. To state
more precise approximation terms, we use
$v_{i}^{k}(y)=u_{i}^{k}(p_{k}+\epsilon_{k}y)+\log(\rho_{i}^{k}h_{i}^{k}(p_{k}))-\phi_{i}^{k}(\epsilon_{k}y)+2\log\epsilon_{k},\quad|y|<\frac{\delta}{2}\epsilon_{k}^{-1}$
and the following rough approximation theorem is established in [34]:
###### Remark 2.1.
The notations $M_{k}$, $\epsilon_{k}$ are the same as those in the
introduction. It is confusing at this moment, later in the multiple bubbling
situation we will use $M_{k,t}$ and $\epsilon_{k,t}$ to denote the maximum of
bubbling solutions and decay rate in each bubbling disk
$B(p_{t}^{k},\delta_{0})$. $M_{k}$ is the maximum of $M_{k,t}$. But our
analysis will show that we can replace $M_{k}$ by $M_{k,t}$, $\epsilon_{k}$ by
$\epsilon_{k,t}$ for any $t$ and the nature of the proof does not change.
Before citing the approximation theorems in [34] we mention one simple fact
implied by the Pohozaev identity. Let
$\sigma_{i}^{k}=\frac{1}{2\pi}\int_{B(p_{k},\delta)}h_{i}^{k}e^{u_{i}^{k}}$
and $m_{i}^{k}=\sum_{j=1}^{n}a_{ij}\sigma_{j}^{k}$. Let
$\sigma_{i}=\lim_{k\to\infty}\sigma_{i}^{k}$ and
$m_{i}=\lim_{k\to\infty}m_{i}^{k}$. As usual we set
$m=\min\\{m_{1},...,m_{n}\\}$. Then it is established in [32] that each
$m_{i}>2$ and
$\sum_{i=1}^{n}\sigma_{i}(m_{i}-4)=0.$
Since each $\sigma_{i}>0$ it is easy to see that either
(2.15) $m<4,\quad\mbox{or}\quad m_{i}=4\,\,\forall i\in I.$
The first approximation theorem established in [34] is a rough one that does
not distinguish $m<4$ or $m=4$.
###### Theorem 2.1.
Given $\delta>0$, there exist $C(\delta)>0$, $k_{0}(\delta)>1$ such that for
$|y|\leq\frac{\delta}{2}\epsilon_{k}^{-1}$ and $|\alpha|=0,1$, the following
holds for all $k\geq k_{0}$
(2.16)
$|D^{\alpha}(v_{i}^{k}(y)-V_{i}^{k}(\epsilon_{k}y)-2\log\epsilon_{k}-\Phi_{i}^{k}(y))|\leq
C\epsilon_{k}^{2}(1+|y|)^{4-m-|\alpha|+\delta}.$
where
$\Phi_{i}^{k}(y)=\epsilon_{k}(G_{1,i}^{k}(r)\cos\theta+G_{2,i}^{k}(r)\sin\theta)$
with
(2.17) $|G_{t,i}^{k}(r)|\leq Cr(1+r)^{2-m+\delta}\quad t=1,2.$
Note that $\Phi^{k}=(\Phi_{1}^{k},..,\Phi_{n}^{k})$ denotes the projection of
$v_{i}^{k}$ onto $span\\{\sin\theta,\cos\theta\\}$. i.e.
(2.18)
$\Phi^{k}_{i}(r\cos\theta,r\sin\theta)=\epsilon_{k}(G_{1,i}^{k}(r)\cos\theta+G_{2,i}^{k}(r)\sin\theta),\quad
i\in I$
with $G_{t,i}^{k}(r)$ ($t=1,2$) satisfying some ordinary differential
equations to be specified later. The estimate for $|\alpha|=1$ follows from
standard gradient estimate for elliptic equations.
Theorem 2.1 does not distinguish $m<4$ or $m=4$. But using Theorem 2.1 in more
careful computation for $m<4$ and $m=4$ gives rise to more accurate results as
follows: Here it is important to observe that $m-2<2$ if $m<4$. If we use
$m_{i}^{k}=\frac{1}{2\pi}\int_{B(p,\delta)}\rho_{i}^{k}h_{i}^{k}(x)e^{u_{i}^{k}}dV_{g},\quad
m_{k}=\min\\{m_{1}^{k},...,m_{n}^{k}\\}.$
Clearly $m_{k}\to m\in(2,4)$.
###### Theorem 2.2.
Suppose $m<4$, then for $|y|\leq\frac{\delta_{0}}{2}\epsilon_{k}^{-1}$ and
$i\in I$
$\displaystyle|D^{\alpha}\big{(}v_{i}^{k}(y)-(V_{i}^{k}(\epsilon_{k}y)+2\log\epsilon_{k})-\Phi^{k}_{i}(y)\big{)}|$
$\displaystyle\leq$ $\displaystyle
C\epsilon_{k}^{2}(1+|y|)^{4-m_{k}-|\alpha|}\log(2+|y|).\quad|\alpha|=0,1$
Moreover $G_{t,i}^{k}$ ($t=1,2,i\in I$) satisfy
(2.20) $|G_{t,i}^{k}(r)|\leq Cr(1+r)^{2-m_{k}}\quad 0<r<\epsilon_{k}^{-1}.$
Note that Theorem 2.2 is slightly stronger than Theorem 4.2 of [34] because
the latter has a logarithmic term. The reason is in the context of Theorem
2.2, the function $v_{i}^{k}$ agrees with its approximation at the origin.
Theorem 2.2 can be proved just like Theorem 4.2 in [34].
###### Theorem 2.3.
If $m=4$ and $|m_{i}^{k}-4|\leq C/\log\epsilon_{k}^{-1}$ for all $i\in I$,
then we have, for $|y|\leq\frac{\delta_{0}}{2}\epsilon_{k}^{-1}$ and $i\in I$
$\displaystyle|D^{\alpha}(v_{i}^{k}(y)-(V_{i}^{k}(\epsilon_{k}y)+2\log\epsilon_{k})-\Phi_{i}^{k}(y)\big{)}|$
$\displaystyle\leq$ $\displaystyle
C\epsilon_{k}^{2}(1+|y|)^{-|\alpha|}(\log(2+|y|))^{2}.\quad|\alpha|=0,1,$
where $\Phi^{k}$ is of the form stated in (2.18) with $G_{t,i}^{k}$ ($t=1,2$)
satisfying
(2.22) $|G_{t,i}^{k}(r)|\leq Cr(1+r)^{-2},\quad 0<r<\epsilon_{k}^{-1},\quad
i\in I.$
## 3\. Rough estimates about bubbling magnitudes
In this section, we will prove Theorems 1.1, 1.2 and 1.6. First in this
section for simplicity we assume there are only two blowup points $p$ and $q$.
The nature of analysis does not change if we have more blowup points.
Now we use Green’s representation to describe the neighborhood of $p_{k}$. The
expression of $u_{i}^{k}$ is
$\displaystyle u_{i}^{k}(x)$
$\displaystyle=\bar{u}_{i}^{k}+\int_{M}G(x,\eta)\sum_{j=1}^{n}a_{ij}\rho_{j}^{k}h_{j}^{k}e^{u_{j}^{k}}dV_{g}$
$\displaystyle=\bar{u}_{i}^{k}+\left(\int_{B(p_{k},\delta)}+\int_{B(q_{k},\delta)}+\int_{M\setminus(B(p_{k},\delta)\cup
B(q_{k},\delta))}\right)G(x,\eta)\sum_{j=1}^{n}a_{ij}\rho_{j}^{k}h_{j}^{k}e^{u_{j}^{k}}dV_{g}$
$\displaystyle=\bar{u}_{i}^{k}+I+II+III.$
Here $\bar{u}_{i}^{k}=\int_{M}u_{i}^{k}dV_{g}$ and we use
$E=O(\epsilon_{k}^{m-2+\delta})$ to denote a very rough order of the error. If
we use Theorem 2.1 in the evaluation, after cancellation we have
$\displaystyle I$
$\displaystyle=\int_{B_{\delta}}(-\frac{1}{2\pi}\log|\eta|+\gamma(x,\eta))\sum_{j=1}^{n}a_{ij}\tilde{h}_{j}^{k}e^{\tilde{u}_{j}^{k}}d\eta$
$\displaystyle=-m_{i}^{k}\log|x|+2\pi m_{i}^{k}\gamma(x,p_{k})+E.$
where $m_{i}^{k}=\frac{1}{2\pi}\sum_{j=1}^{n}a_{ij}\sigma_{j}^{k}$ and
$\sigma_{i}^{k}=\int_{B(p,\delta)}h_{i}^{k}e^{u_{i}^{k}}dV_{g}$, we use
$\bar{\sigma}_{i}^{k}$, $\bar{m}_{i}^{k}$ to denote the integrations in
$B(q_{k},\delta)$. Here we note that in the evaluation of the integrals, the
$\Phi_{i}^{k}$ terms disappear because of the symmetry of the domain.
Similarly the integration around $q_{k}$ gives
$II=2\pi\bar{m}_{i}^{k}G(x,q_{k})+E,\quad III=E.$
Thus in the neighborhood of $p_{k}$ we have
(3.1) $u_{i}^{k}(x)=\bar{u}_{i}^{k}-m_{i}^{k}\log|x|+2\pi
m_{i}^{k}\gamma(x,p_{k})+2\pi\bar{m}_{i}^{k}G(x,q_{k})+E$
in, say $B(p_{k},2\delta)\setminus B(p_{k},\delta/2)$. If we use the
approximation theorems to evaluate $u_{i}^{k}$ at $p_{t}^{k}$, it is easy to
obtain
(3.2) $\bar{u}_{i}^{k}=(1-\frac{m_{i}^{k}}{2})M_{k}+O(1).$
Before more advanced estimates we first establish an elementary one:
Proof of Theorem 1.1: From (3.2) we have
(3.3)
$(1-\frac{m_{i}^{k}}{2})M_{k}=(1-\frac{\bar{m}_{i}^{k}}{2})\bar{M}_{k}+O(1).$
Let
$\lambda_{k}=M_{k}/\bar{M}_{k},\quad\delta_{i}^{k}=O(1)/\bar{M}_{k},$
we have
(3.4)
$(\frac{m_{i}^{k}}{2}-1)\lambda_{k}+\delta_{k}=(\frac{\bar{m}_{i}^{k}}{2}-1)+o(\epsilon_{k}^{\delta}),\quad\mbox{for
some }\delta>0.$
It is established in [32, 34] that $m^{k}=(m_{1}^{k},...,m_{n}^{k})$ satisfies
(3.5)
$\sum_{i=1}^{n}\sum_{j=1}^{n}a^{ij}(\frac{m_{i}^{k}-2}{2})(\frac{m_{j}^{k}-2}{2})=\sum_{i=1}^{n}\sum_{j=1}^{n}a^{ij}+E.$
and (3.5) also holds for $\bar{m}^{k}=(\bar{m}_{1}^{k},...,\bar{m}_{n}^{k})$.
Thus using (3.4) in (3.5) for $\bar{m}_{k}$, we have
$\sum_{i=1}^{n}\sum_{j=1}^{n}a^{ij}(\frac{m_{i}^{k}-2}{2}\lambda_{k}+\delta_{i}^{k})(\frac{m_{j}^{k}-2}{2}\lambda_{k}+\delta_{j}^{k})=\sum_{i=1}^{n}\sum_{j=1}^{n}a^{ij}+E,$
which can be written as a quadratic expression of $\lambda_{k}$:
$\displaystyle\lambda_{k}^{2}\sum_{i=1}^{n}\sum_{j=1}^{n}a^{ij}(\frac{m_{i}^{k}-2}{2})(\frac{m_{j}^{k}-2}{2})+2\lambda_{k}\sum_{i=1}^{n}\sum_{j=1}^{n}a^{ij}(\frac{m_{i}^{k}-2}{2})\delta_{j}^{k}+\sum_{i=1}^{n}\sum_{j=1}^{n}a^{ij}\delta_{i}^{k}\delta_{j}^{k}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\sum_{j=1}^{n}a^{ij}+E/\bar{M}_{k}.$
Let
$B_{k}=\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}a^{ij}(m_{i}^{k}-2)\delta_{j}^{k}}{\sum_{i=1}^{n}\sum_{j=1}^{n}a^{ij}},\quad
C_{k}=\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}a^{ij}\delta_{i}^{k}\delta_{j}^{k}}{\sum_{i=1}^{n}\sum_{j=1}^{n}a^{ij}}.$
Here we note that $\sum_{i=1}^{n}\sum_{j=1}^{n}a^{ij}>0$ because $(H2)$
requires $A$ to be invertible and $\sum_{j=1}^{n}a^{ij}\geq 0$ for all $i$.
Then
$\lambda_{k}^{2}+B_{k}\lambda_{k}+C_{k}=1+E/\bar{M}_{k}.$
First it is obvious to observe that $\lim_{k\to\infty}\lambda_{k}=1$. Thus by
$C_{k}=O(\bar{M}_{k}^{-2})$ and $B_{k}=O(\bar{M}_{k}^{-1})$
$\lambda_{k}=\sqrt{1-C_{k}+\frac{B_{k}^{2}}{4}}-\frac{B_{k}}{2}=1-\frac{B_{k}}{2}-\frac{1}{2}(C_{k}-\frac{B_{k}^{2}}{4})+O(\bar{M}_{k}^{-4}).$
This verifies that $M_{k}-\bar{M}_{k}=O(1)$ which justifies
$O(\epsilon_{k}^{m_{k}-2})=O(\bar{\epsilon}_{k}^{\bar{m}_{k}-2})$. Theorem 1.1
is established. $\Box$
By Theorem 1.1, all the error terms above can be improved to
$E=O(\epsilon_{k}^{m_{k}-2})$. Note that it is not $O(\epsilon_{k}^{m-2})$
yet, because the closeness of $m_{k}$ and $m$ is not derived yet. Another
consequence of Theorem 1.1 is that
(3.6)
$\frac{\sum_{j=1}^{n}a_{ij}\rho_{i}^{k}}{2\pi}-m_{i}^{k}-\bar{m}_{i}^{k}=O(\epsilon_{k}^{m_{k}-2}).$
Indeed, integrating the equation for $u_{i}^{k}$ (which is (1.1)), we have
$\sum_{j=1}^{n}\int_{M}a_{ij}\rho_{j}^{k}h_{j}^{k}e^{u_{j}^{k}}=\sum_{j=1}^{n}a_{ij}\rho_{j}^{k}.$
The integration of the left in $B(p,\delta)$ and $B(q,\delta)$ gives
$m_{i}^{k}+\bar{m}_{i}^{k}+O(\epsilon_{k}^{m_{k}-2})=\sum_{j=1}^{n}a_{ij}\rho_{j}^{k}/(2\pi).$
Thus (3.6) is verified.
Here we recall a theorem in [34] about location of blowup points:
Let $p_{t}^{k}$ be blowup points described as before. Then at each blowup
point $p_{t}^{k}$, let $\phi_{it}^{k}$ be the harmonic function that
eliminates the oscillation of $\tilde{u}_{i}^{k}$ on $\partial
B(\tilde{p}_{t}^{k},\delta)$ for $\delta>0$ small. Then it is proved in [34]
that
###### Theorem 3.1.
If $m<4$
(3.7) $|\sum_{i=1}^{n}\bigg{(}\partial_{l}(\log
h_{i}^{k})(p_{t}^{k})+\partial_{l}\phi_{it}^{k}(p_{t}^{k})\bigg{)}\rho_{it}^{k}|\leq
C\epsilon_{k}^{m_{k}-2},\quad l=1,2,$
where $C$ is independent of $k$. On the other hand, if $m=4$
(3.8) $|\sum_{i=1}^{n}\bigg{(}\partial_{l}(\log
h_{i}^{k})(p_{t}^{k})+\partial_{l}\phi_{i}^{k}(p_{t}^{k})\bigg{)}\rho_{it}^{k}|\leq
C\epsilon_{k}^{2}\log\epsilon_{k}^{-1},\quad l=1,2,$
where
$\rho_{it}^{k}=\int_{B(p_{t}^{k},\delta)}\rho_{i}^{k}h_{i}^{k}e^{u_{i}^{k}}dV_{g}$.
In $B(p_{k},\delta)$, by the definition of $\tilde{u}_{i}^{k}$ in (2.11) and
the estimate of $u_{i}^{k}$ in (3.1) we now have
(3.9) $\displaystyle\tilde{u}_{i}^{k}(x)=\bar{u}_{i}^{k}-m_{i}^{k}\log|x|+2\pi
m_{i}^{k}\gamma(x,p_{k})+2\pi\bar{m}_{i}^{k}G(x,q_{k})$
$\displaystyle+\log\rho_{i}^{k}+\log
h_{i}^{k}(p_{k})-f_{i}^{k}(x)+O(\epsilon_{k}^{m_{k}-2}).$
In this neighborhood, $\tilde{u}_{i}^{k}$ is of the form
$\tilde{u}_{i}^{k}(x)=V_{i}^{k}(x)+\phi_{i}^{k}(x)+O(\epsilon_{k}^{m_{k}-2}),\quad
x\in B(0,\delta/2)\setminus B(0,\delta/8)$
where $\phi_{i}^{k}$ be the harmonic function on $B(p_{k},\delta)$ that
eliminates the oscillation of $\tilde{u}_{i}^{k}$:
$\Delta\phi_{i}^{k}(x)=0,\quad\mbox{in}\quad
B(p_{k},\delta),\quad\phi_{i}^{k}(x)=\tilde{u}_{i}^{k}(x)-\frac{1}{2\pi\delta}\int_{\partial
B(p_{k},\delta)}\tilde{u}_{i}^{k}.$
Note that we have used the fact that the first order terms
$\Phi_{i}^{k}(x)=O(\epsilon_{k}^{m_{k}-2})$ when $x\sim 1$. Notice that by
(3.6) and the fact that $q_{k}$ is not in $B(p_{k},\delta)$,
$\displaystyle\Delta_{g}(2\pi
m_{i}^{k}\gamma(x,p_{k})+2\pi\bar{m}_{i}^{k}G(x,q_{k})-f_{i}^{k})$
$\displaystyle=$ $\displaystyle 2\pi
m_{i}^{k}+2\pi\bar{m}_{i}^{k}-\sum_{j=1}^{n}a_{ij}\rho_{j}^{k}=O(\epsilon_{k}^{m_{k}-2})$
Thus
(3.10) $\Delta(2\pi
m_{i}^{k}\gamma(x,p_{k})+2\pi\bar{m}_{i}^{k}G(x,q_{k})-f_{i}^{k})=O(\epsilon_{k}^{m_{k}-2}).$
By the definition of $\tilde{u}_{i}^{k}$ in (2.11) and the comparison of (3.1)
and (3.10) we have
(3.11) $\displaystyle\phi_{i}^{k}(x)=2\pi
m_{i}^{k}(\gamma(x,p_{k})-\gamma(p_{k},p_{k}))+2\pi\bar{m}_{i}^{k}(G(x,q_{k})-G(p_{k},q_{k}))$
$\displaystyle-f_{i}^{k}+O(\epsilon_{k}^{m_{k}-2}).$
So if we rewrite the expression of $\tilde{u}_{i}^{k}$ as
$\displaystyle\tilde{u}_{i}^{k}(x)$
$\displaystyle=\bar{u}_{i}^{k}-m_{i}^{k}\log|x|+\phi_{i}^{k}(x)$
$\displaystyle+2\pi
m_{i}^{k}\gamma(p_{k},p_{k})+2\pi\bar{m}_{i}^{k}(G(p_{k},q_{k}))+\log(\rho_{i}^{k}h_{i}^{k}(p_{k}))+O(\epsilon_{k}^{m_{k}-2}).$
then we see that for $x\in B(0,\delta)\setminus B(0,\delta/8)$,
(3.12) $\displaystyle V_{i}^{k}(x)=-m_{i}^{k}\log|x|+\bar{u}_{i}^{k}$
$\displaystyle+2\pi
m_{i}^{k}\gamma(p_{k},p_{k})+2\pi\bar{m}_{i}^{k}G(p_{k},q_{k})+\log(\rho_{i}^{k}h_{i}^{k}(p_{k}))+O(\epsilon_{k}^{m_{k}-2}).$
Similarly around $q_{k}$ we have
(3.13)
$\displaystyle\tilde{V}_{i}^{k}(x)=-\bar{m}_{i}^{k}\log|x|+\bar{u}_{i}^{k}$
$\displaystyle+2\pi\bar{m}_{i}^{k}\gamma(q_{k},q_{k})+2\pi
m_{i}^{k}G(q_{k},p_{k})+\log(\rho_{i}^{k}h_{i}^{k}(q_{k}))+O(\epsilon_{k}^{m_{k}-2}).$
Let
$M_{k}=\max_{i\in I}\max_{x}\tilde{u}_{i}^{k}(x),\quad\mbox{in}\quad
B(p_{k},\delta),$
and $\bar{M}_{k}$ be the corresponding maximum in $B(q_{k},\delta)$. Then it
is proved in [33] that
$M_{k}-\tilde{u}_{i}^{k}(p_{k})=O(1).$
We shall use the following notations:
$D_{i}^{k}=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\sum_{j=1}^{n}a_{ij}e^{V_{j}^{k}},\quad\alpha_{i}^{k}=M_{k}-\tilde{u}_{i}^{k}(p_{k}).$
$\bar{D}_{i}^{k}$, $\bar{\alpha}_{i}^{k}$ can be understood similarly.
In order to obtain accurate estimate of $|m_{i}^{k}-\bar{m}_{i}^{k}|$, we
first derive a simple fact about global solutions of Liouville system.
###### Lemma 3.1.
Let $U=(U_{1},...,U_{n})$ be global solution of
$\Delta
U_{i}+\sum_{j=1}^{n}a_{ij}e^{U_{j}}=0,\quad\mbox{in}\quad\mathbb{R}^{2},\quad\int_{\mathbb{R}^{2}}e^{U_{i}}<\infty,\quad
U_{i}\,\,\,\mbox{is radial}$
and suppose $\max_{i\in I}U_{i}(0)=0$. Then
(3.14) $U_{i}(r)=-m_{i}\log
r+D_{i}-\alpha_{i}-\sum_{j=1}^{n}\frac{a_{ij}}{(m_{j}-2)^{2}}e^{D_{j}-\alpha_{j}}r^{2-m_{j}}+O(r^{2-m-\delta}).$
where
$m_{i}=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\sum_{j=1}^{n}a_{ij}e^{U_{j}(x)}dx$,
$\alpha_{i}=-U_{i}(0)$ and $D_{i}=\int_{0}^{\infty}\log
r\sum_{j=1}^{n}a_{ij}e^{U_{j}(r)}rdr.$.
Proof of Lemma 3.1:
It is easy to see that
$U_{i}(x)=-\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\log|x-y|\sum_{j=1}^{n}a_{ij}e^{U_{j}}dy+c_{i}.$
Recall that $U_{i}(0)=-\alpha_{i}$, then
$-\alpha_{i}=-\int_{0}^{\infty}\log
r\sum_{j=1}^{n}a_{ij}e^{U_{j}(r)}rdr+c_{i}.$
and
(3.15) $\displaystyle U_{i}(x)$
$\displaystyle=-\frac{1}{2\pi}\int_{\mathbb{R}^{2}}(\log|x-y|-\log|x|)\sum_{j=1}^{n}a_{ij}e^{U_{j}(y)}dy+D_{i}-\alpha_{i}-m_{i}\log|x|$
$\displaystyle=-m_{i}\log|x|+D_{i}-\alpha_{i}+O(|x|^{-\delta}).$
for some $\delta>0$. This expression gives
$e^{U_{i}(r)}=r^{-m_{i}}e^{D_{i}-\alpha_{i}}+O(r^{-m-\delta}).$
Then we use this in the ode that $U_{i}$ satisfies:
$U_{i}^{\prime\prime}(r)+\frac{1}{r}U_{i}^{\prime}(r)=-\sum_{j=1}^{n}a_{ij}e^{U_{j}(r)},\quad
0<r<\infty.$
Here we use the fact that
$\lim_{r\to\infty}rU_{i}^{\prime}(r)=-m_{i}.$
Thus
$\displaystyle-
m_{i}-rU_{i}^{\prime}(r)=-\sum_{j=1}^{n}a_{ij}\int_{r}^{\infty}se^{U_{j}(s)}ds$
$\displaystyle=-\sum_{j=1}^{n}a_{ij}\frac{e^{D_{j}-\alpha_{j}}}{m_{j}-2}r^{2-m_{j}}+O(r^{2-m-\delta}).$
Then we have
$U_{i}^{\prime}(r)=-\frac{m_{i}}{r}+\sum_{j=1}^{n}\frac{a_{ij}}{m_{j}-2}e^{D_{j}-\alpha_{j}}r^{1-m_{j}}+O(r^{1-m-\delta}).$
After integration and using (3.15) we see that (3.14) holds. Lemma 3.1 is
established. $\Box$
The main result in this section is:
###### Proposition 3.1.
If $m<4$,
$|m_{i}^{k}-\bar{m}_{i}^{k}|\leq C\delta_{0}^{4-m}\epsilon_{k}^{m_{k}-2},\quad
i\in I.$
Proof of Proposition 3.1: Let $V_{i}^{k}$ be the sequence of global solutions
that approximate $\tilde{u}_{i}^{k}$ around $p_{k}$, and in a neighborhood
centered at $p_{k}$, $V_{i}^{k}(0)$ agrees with $\tilde{u}_{i}^{k}$ at
$p_{k}$. $\bar{V}_{i}^{k}$ is understood as the first approximation around
$q_{k}$ and we have We use $m_{iv}^{k}$ to denote
$m_{iv}^{k}=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\sum_{j=1}^{n}a_{ij}e^{V_{j}^{k}}dx,\quad
i=1,..,n,$
and $\bar{m}_{iv}^{k}$ is for $\bar{V}_{i}^{k}$. Note that a rough estimate of
$m_{i}^{k}$ based on Theorem 2.2 gives
$m_{i}^{k}-m_{iv}^{k}=O(\epsilon_{k}^{m_{k}-2}).$
If we use
$U_{i}(y)=V_{i}^{k}(x)+2\log\epsilon_{k},\quad\epsilon_{k}=e^{-\frac{1}{2}M_{k}},\quad
x=\epsilon_{k}y.$
Then the expansion of $V_{i}^{k}$ for $|x|$ is
(3.16) $\displaystyle V_{i}^{k}(x)$
$\displaystyle=-m_{iv}^{k}\log|x|-\frac{m_{iv}^{k}-2}{2}M_{k}+D_{i}-\alpha_{i}$
$\displaystyle-\sum_{j=1}^{n}\frac{a_{ij}}{(m_{jv}^{k}-2)^{2}}e^{D_{j}-\alpha_{j}}\epsilon_{k}^{m_{jv}^{k}-2}|x|^{2-m_{jv}^{k}}+O(\epsilon_{k}^{m-2+\delta}).$
$V^{k}=(V_{i}^{k},....,V_{n}^{k})$ is the sequence of global functions that
serves as the first term in the expansion of $\tilde{u}^{k}$ around $p_{k}$.
Similarly
$\bar{V}_{i}^{k}(x)=-\bar{m}_{iv}^{k}\log|x|-\frac{\bar{m}_{iv}^{k}-2}{2}\bar{M}_{k}+\bar{D}_{i}-\bar{\alpha}_{i}+O(\epsilon_{k}^{m_{k}-2})|x|^{2-\bar{m}_{iv}^{k}}$
Since both $V^{k}=(V_{1}^{k},...,V_{n}^{k})$ and
$\bar{V}^{k}=(\bar{V}_{1}^{k},...,\bar{V}_{n}^{k})$ are radial and satisfy the
same Liouville system. The dependence on initial condition gives
(3.17) $|V_{i}^{k}(x)-(\bar{V}_{i}^{k}(\eta x)+2\log\eta)|\leq
C\sum_{i=1}^{n}|\alpha_{i}-\bar{\alpha}_{i}|\quad\mbox{in}\quad B(0,R).$
where $R>0$ is a constant, $\eta$ is chosen to make the heights equal, in this
case $2\log\eta=M_{k}-\bar{M}_{k}$. We also note that one of
$\alpha_{i}=\bar{\alpha}_{i}$. Here we invoke an important result in [32].
Suppose $\alpha_{1}=0$, the mapping from $(\alpha_{2},...,\alpha_{n})$ to
$(\sigma_{2},...,\sigma_{n})$ is invertible. In fact the following matrix
$\mathbb{M}=\left(\begin{array}[]{ccc}\partial_{\alpha_{2}}\sigma_{2}&...&\partial_{\alpha_{n}}\sigma_{2}\\\
\vdots&\vdots&\vdots\\\
\partial_{\alpha_{2}}\sigma_{n}&...&\partial_{\alpha_{n}}\sigma_{n}\end{array}\right)$
is invertible. Note that in this proposition we use
$\sigma_{i}^{k}=\sum_{j=1}^{n}a^{ij}m_{jv}^{k},\quad\bar{\sigma}_{i}^{k}=\sum_{j=1}^{n}a^{ij}\bar{m}_{jv}^{k}.$
Thus (3.17) can be written as
(3.18) $|V_{i}^{k}(x)-(\bar{V}_{i}^{k}(\eta x)+2\log\eta)|\leq
C\sum_{i=2}^{n}|\sigma_{i}^{k}-\bar{\sigma}_{i}^{k}|.$
By the expansion of $\bar{V}_{i}^{k}$, we find that
(3.19) $\bar{V}_{i}^{k}(\eta
x)+2\log\eta=-\frac{\bar{m}_{iv}^{k}-2}{2}M_{k}-\bar{m}_{iv}^{k}\log|x|+\bar{D}_{i}-\bar{\alpha}_{i}+O(\epsilon_{k}^{m_{k}-2}).$
The difference between $V_{i}^{k}$ and $\bar{V}_{i}^{k}(\eta x)+2\log\eta$
gives
$\displaystyle V_{i}^{k}(x)-(\bar{V}_{i}^{k}(\eta x)+2\log\eta)$
$\displaystyle=$
$\displaystyle(\bar{m}_{iv}^{k}-m_{iv}^{k})\log|x|+\frac{\bar{m}_{iv}^{k}-m_{iv}^{k}}{2}M_{k}+D_{i}-\bar{D}_{i}+\bar{\alpha}_{i}-\alpha_{i}+O(\epsilon_{k}^{m_{k}-2}).$
By the dependence of initial condition and fast decay of $V^{k}$ and
$\bar{V}^{k}$, we have
$|D_{i}-\bar{D}_{i}|\leq C\sum_{i=1}^{n}|\alpha_{i}-\bar{\alpha}_{i}|\leq
C\sum_{i=2}^{n}|\sigma_{i}^{k}-\bar{\sigma}_{i}^{k}|.$
And by the one to one correspondence between $(\alpha_{1},...,\alpha_{n})$ to
$(\sigma_{2},...,\sigma_{n})$ and the smoothness of the mapping we also obtain
$\sum_{i=1}^{n}|\alpha_{i}-\bar{\alpha}_{i}|\leq
C\sum_{i=2}^{n}|\sigma_{i}^{k}-\bar{\sigma}_{i}^{k}|.$
Combining these estimates we have
(3.20) $|\frac{\bar{m}_{iv}^{k}-m_{iv}^{k}}{2}M_{k}|\\\ \leq
C\sum_{j=2}^{n}|\sigma_{j}^{k}-\bar{\sigma}_{j}^{k}|+O(\epsilon_{k}^{m_{k}-2})\quad
i=1,...,n.$
After that we multiply $a^{ij}$ with summation on $i$ and take summation on
$j$, then we have
(3.21)
$\sum_{j=1}^{n}|\bar{\sigma}_{j}^{k}-\sigma_{j}^{k}|\leq\frac{C}{M_{k}}\sum_{l=2}^{n}|\sigma_{l}^{k}-\bar{\sigma}_{l}^{k}|+O(\epsilon_{k}^{m_{k}-2}).$
We thus obtain the following important closeness result:
(3.22)
$\sigma_{i}^{k}-\bar{\sigma}_{i}^{k}=O(\epsilon_{k}^{m_{k}-2})/M_{k},\quad
i=1,...,n.$
Thus we have proved that
$m_{iv}^{k}-\bar{m}_{iv}^{k}=O(\epsilon_{k}^{m_{k}-2}/M_{k})$, using the
expansion of $\tilde{u}_{i}^{k}$ in the calculation of
$\int_{B(p_{k},\delta_{0})}\rho_{i}^{k}h_{i}^{k}e^{u_{i}^{k}}dV_{g}$. It is
easy to see that
$|m_{i}^{k}-m_{iv}^{k}|\leq
C\delta_{0}^{m_{k}-4}\epsilon_{k}^{m_{k}-2},\quad|\bar{m}_{i}^{k}-\bar{m}_{iv}^{k}|\leq
C\delta_{0}^{4-m}\epsilon_{k}^{m_{k}-2}.$
Thus
$|m_{i}^{k}-\bar{m}_{i}^{k}|\leq C\delta_{0}^{4-m}\epsilon_{k}^{m_{k}-2}.$
Proposition 3.1 is established. $\Box$
###### Remark 3.1.
By the same argument for the case $m=4$, we also have
(3.23) $m_{iv}^{k}-\bar{m}_{iv}^{k}=O(\epsilon_{k}^{m_{k}-2}/M_{k}).$
Here $m_{k}\to 4$. Later we shall see $m_{k}$ can be replaced by $4$.
Correspondingly
(3.24)
$m_{i}^{k}-\bar{m}_{i}^{k}=O(\epsilon_{k}^{2}\log\frac{1}{\epsilon_{k}}),$
but the leading term (of the order
$O(\epsilon_{k}^{2}\log\frac{1}{\epsilon_{k}})$ ) can be identified as a local
term that involves curvature at the blowup point.
In the more general situation of $N$ bubbles, if we use $m_{it}^{k}$ to denote
the energy in $B(p_{t}^{k},\delta_{0})$, $m_{itv}^{k}$ to denote the energy of
the global sequence as the first term in the approximation, we have, for
$m<4$:
(3.25) $m_{it}^{k}-m_{is}^{k}=O(\epsilon_{k}^{m_{k}-2}),\quad s\neq t,\quad
m<4$ (3.26)
$m_{itv}^{k}-m_{isv}^{k}=O(\epsilon_{k}^{m_{k}-2}/\log\frac{1}{\epsilon_{k}}),\quad
s\neq t,\quad m<4.$
For $m=4$, (3.26) also holds, and
$m_{it}^{k}-m_{is}^{k}=O(\epsilon_{k}^{m_{k}-2}\log\epsilon_{k}^{-1})$. The
following lemma gives an estimate of $\rho_{i}^{k}-\rho$, which determines as
a consequence that
$O(\epsilon_{k}^{m_{k}-2})=O(\epsilon_{k}^{m-2})\quad\mbox{if}\quad m\in(2,4)$
$O(\epsilon_{k}^{m_{k}-2}\log\frac{1}{\epsilon_{k}})=O(\epsilon_{k}^{2}\log\frac{1}{\epsilon_{k}}),\quad\mbox{if}\quad
m=4.$
###### Lemma 3.2.
If $m<4$
$\rho_{i}^{k}-\rho_{i}=O(\epsilon_{k}^{m-2}),\quad i\in I.$
Proof of Lemma 3.2: Recall that $\rho\in\Gamma_{N}$. Let
$\rho_{it}^{k}=\int_{B(p_{t},\delta)}\rho_{i}h_{i}^{k}e^{u_{i}^{k}}dV_{g},\quad
t=1,...,N,\quad E_{i}^{k}=\rho_{i}^{k}-\sum_{t=1}^{N}\rho_{it}^{k}.$
Here $\delta>0$ is small so that bubbling disks are disjoint. Clearly from
$\int_{M}h_{i}^{k}e^{u_{i}^{k}}=1$ and (3.28) we have
$E_{i}^{k}=O(\epsilon_{k}^{m_{k}-2}),\quad\rho_{i}^{k}=\sum_{i=1}^{N}\rho_{it}^{k}+E_{i}^{k}.$
Let $\sigma_{i}=\frac{\rho_{i}}{2\pi N}$ and
$\sigma_{it}^{k}=\frac{\rho_{it}^{k}}{2\pi}$. Then we write
$\sigma_{i}=\sigma_{it}+s_{it}^{k}$. We have known that $s_{it}^{k}=o(1)$ as
$k\to\infty$. Since $\rho\in\Gamma_{N}$ we know
$\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}\sigma_{i}\sigma_{j}-4\sum_{i=1}^{n}\sigma_{i}=0.$
On the other hand the Pohozaev identity around $p_{t}$ gives
$\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}\sigma_{it}^{k}\sigma_{jt}^{k}-4\sum_{i=1}^{n}\sigma_{it}^{k}=O(\epsilon_{k}^{m_{k}-2}).$
The difference between these two equations gives
(3.27)
$\sum_{i=1}^{n}2(m_{i}-2)s_{it}^{k}+\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}s_{it}^{k}s_{jt}^{k}=O(\epsilon_{k}^{m_{k}-2}),\quad
t=1,...,N.$
Taking the sum of $N$ equations we have
$\sum_{i=1}^{n}2(m_{i}-2)(\sum_{t=1}^{N}s_{it}^{k})+\sum_{t=1}^{N}a_{ij}s_{it}^{k}s_{jt}^{k}=O(\epsilon_{k}^{m_{k}-2}).$
From Proposition 3.1 we know that the difference between any two $s_{it}^{k}$
is $O(\epsilon_{k}^{m_{k}-2})$,
$\sum_{t=1}^{N}s_{it}^{N}=Ns_{i1}^{k}+O(\epsilon_{k}^{m_{k}-2})$. Thus we have
$N\sum_{i=1}^{n}s_{i1}^{k}+\sum_{t=1}^{N}\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}s_{it}^{k}s_{jt}^{k}=O(\epsilon_{k}^{m_{k}-2}).$
By the assumption $\frac{\rho_{i}^{k}-\rho_{i}}{\rho_{j}^{k}-\rho_{j}}\sim 1$
we have $s_{i1}^{k}/s_{j1}^{k}\sim 1$ for all $i,j$ because
$\sum_{t=1}^{N}s_{it}^{k}=\frac{1}{2\pi}(\rho_{i}^{k}-\rho_{i})$. Thus
$s_{it}^{k}=O(\epsilon_{k}^{m_{k}-2}),\quad t=1,...,N.$
Thus Lemma 3.2 is established. $\Box$
Now we are ready to prove Theorem 1.2.
Proof of Theorem 1.2: Due to the closeness between $m_{i}^{k}$ and $m$ (which
is of the order $O(\epsilon_{k}^{m_{k}-2})$ we shall use
$O(\epsilon_{k}^{m-2})$ instead of $O(\epsilon_{k}^{m_{k}-2})$ in Proposition
3.1. By a similar argument, based on (3.24) we clearly have
$\mbox{ For
}m=4,\quad\rho_{i}^{k}-\rho_{i}=O(\epsilon_{k}^{2}\log\frac{1}{\epsilon_{k}}),$
which gives $O(\epsilon_{k}^{m_{k}-2})=O(\epsilon_{k}^{2})$. We thus have
$m_{i,t}^{k}-m_{i,s}^{k}=O(\varepsilon_{k}^{2}\log\frac{1}{\varepsilon_{k}}),\,\,t\not=s,\,\,s,t\in\\{1,\cdots,N\\},\,\,i\in
I.$
Recall that $\sigma_{i,t}^{k}=2\pi\sum_{j\in I}a^{ij}m_{i,t}^{k}$ and
$A=(a_{ij})$ is invertible. This theorem is proved. $\Box$
Because of (3.25) the evaluation of $u_{i}^{k}(x)$ away from bubbling disks
now becomes:
$u_{i}^{k}(x)=\bar{u}_{i}^{k}+2\pi
m_{i}^{k}\sum_{l=1}^{N}G^{*}(x,p_{l}^{k})+O(\epsilon_{k}^{m-2}),\quad x\in
M\setminus(\cup B(p_{t}^{k},\delta_{0})),\quad m<4,$
$u_{i}^{k}(x)=\bar{u}_{i}^{k}+2\pi
m_{i}^{k}\sum_{l=1}^{N}G^{*}(x,p_{l}^{k})+O(\epsilon_{k}^{2}\log\frac{1}{\epsilon_{k}}),\quad
x\in M\setminus(\cup B(p_{t}^{k},\delta_{0})),\quad m=4,$
for $\delta_{0}>0$ small. We recall the notation
$G^{*}(p_{t}^{k},p_{l}^{k})=\left\\{\begin{array}[]{ll}\gamma(p_{t}^{k},p_{t}^{k}),\quad\mbox{if}\quad
t=l,\\\ G(p_{t}^{k},p_{l}^{k}),\quad\mbox{if}\quad t\neq l.\end{array}\right.$
With the information available we are in the position to prove Theorem 1.6.
Proof of Theorem 1.6: In $B(p_{t}^{k},\delta_{0})$ we use
$V_{t}^{k}=(V_{1t}^{k},...,V_{nt}^{k})$ to denote the sequence of global
solutions as the first term in the approximation. In the context of multiple
bubbles, $V_{it}^{k}$ has two expressions:
$V_{it}^{k}=-m_{it}^{k}\log|x|+\bar{u}_{i}^{k}+2\pi
m_{i}^{k}\sum_{l=1}^{N}G^{*}(p_{t}^{k},p_{l}^{k})+\log(\rho_{i}^{k}h_{i}^{k}(p_{t}^{k}))+O(\epsilon_{k}^{m-2}),$
and
$V_{it}^{k}=-m_{itv}^{k}\log|x|-\frac{m_{itv}^{k}-2}{2}M_{k,t}+D_{it}^{k}-\alpha_{it}^{k}+O(\epsilon_{k}^{m-2}),$
where $m_{itv}^{k}=\sum_{j=1}^{n}a_{ij}\sigma_{itv}^{k}$,
$\sigma_{itv}^{k}=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}e^{V_{it}^{k}}$.
By comparing the two expressions of $V_{it}^{k}$ we have
$\displaystyle-m_{it}^{k}\log|x|+\bar{u}_{i}^{k}+2\pi
m_{it}^{k}\sum_{l=1}^{N}G^{*}(p_{t}^{k},p_{l}^{k})+\log(h_{i}^{k}(p_{t}^{k})\rho_{i}^{k})$
$\displaystyle=-m_{itv}^{k}\log|x|-\frac{m_{itv}^{k}-2}{2}M_{k,t}+D_{it}^{k}-\alpha_{it}^{k}+O(\epsilon_{k}^{m-2}).$
where all the notations are clearly understandable under this context. Thus,
for $t\neq s$, based on the two different expression of $\bar{u}_{i}^{k}$, we
have
$\sum_{l=1}^{N}2\pi m_{i}^{k}G^{*}(p_{t}^{k},p_{l}^{k})+\log
h_{i}^{k}(p_{t}^{k})=\sum_{l=1}^{N}2\pi
m_{i}^{k}G^{*}(p_{s}^{k},p_{l}^{k})+\log
h_{i}^{k}(p_{s}^{k})+O(\epsilon_{k}^{m-2}),$
where we have used
$m_{ivt}^{k}-m_{ivs}^{k}=O(\epsilon_{k}^{m-2}/\log\frac{1}{\epsilon_{k}})$ and
used $m_{i}^{k}$ to replace all $m_{it}^{k}$. (1.21) is verified. (1.22) can
be verified similarly. Theorem 1.6 is established. $\Box$
The following expression of $\bar{u}_{i}^{k}$ will be used
(3.28) $\displaystyle\bar{u}_{i}^{k}$
$\displaystyle=(1-\frac{m_{i}^{k}}{2})M_{k,t}-\log(\rho_{i}^{k}h_{i}^{k}(p_{t}^{k}))-2\pi
m_{i}^{k}\sum_{l=1}^{N}G^{*}(p_{t}^{k},p_{l}^{k})$
$\displaystyle+D_{i}^{k}-\alpha_{i}^{k}+O(\epsilon_{k}^{m_{k}-2}),\quad
t=1,...,N,$
where we used $\alpha_{i}^{k}$ and $D_{i}^{k}$ to replace any $D_{it}^{k}$ and
$\alpha_{it}^{k}$ for obvious reasons. One of $M_{k,t}$ is $M_{k}$ and the
difference between any two of these is bounded. In fact, for $t\neq s$, the
difference on equations (3.28) for $t$ and $s$ gives
$2\pi
m_{i}(\sum_{l=1}^{N}(G^{*}(p_{t}^{k},p_{l}^{k})-G^{*}(p_{s}^{k},p_{l}^{k}))+\log\frac{h_{i}^{k}(p_{t}^{k})}{h_{i}^{k}(p_{s}^{k})}=-\frac{m_{i}-2}{2}(M_{k,t}-M_{k,s})+O(\epsilon_{k}^{m-2}).$
Equivalent form is
(3.29) $exp(2\pi
m_{i}\sum_{l=1}^{N}(G^{*}(p_{t}^{k},p_{l}^{k})-G^{*}(p_{s}^{k},p_{l}^{k})))\,\frac{h_{i}^{k}(p_{t}^{k})}{h_{i}^{k}(p_{s}^{k})}=\frac{\epsilon_{k,t}^{m_{i}-2}}{\epsilon_{k,s}^{m_{i}-2}}+O(\epsilon_{k}^{m-2}),$
where $\epsilon_{k,t}=e^{-\frac{1}{2}M_{k,t}}$. Also, (3.28) gives
(3.30)
$e^{\bar{u}_{i}^{k}}=\epsilon_{k,t}^{m_{i}-2}\frac{e^{D_{i}^{k}-\alpha_{i}^{k}}}{\rho_{i}^{k}h_{i}^{k}(p_{t}^{k})}e^{-2\pi
m_{i}\sum_{l=1}^{N}G^{*}(p_{t}^{k},p_{l}^{k})}+O(\epsilon_{k}^{m-2+\delta}).$
## 4\. Proof of leading terms
Now we are in the position to prove Theorem 1.3.
Proof of Theorem 1.3:
Since $\int_{M}h_{i}^{k}e^{u_{i}^{k}}dV_{g}=1$ we write
$\rho_{i}^{k}=\sum_{t=1}^{N}\int_{B(p_{t}^{k},\delta_{0})}\rho_{i}^{k}h_{i}^{k}e^{u_{i}^{k}}dV_{g}+\int_{M\setminus\cup_{t}B(p_{t}^{k},\delta_{0})}\rho_{i}^{k}h_{i}^{k}e^{u_{i}^{k}}dV_{g}=\sum_{t=1}^{N}\rho_{it}^{k}+\rho_{ib}^{k}$
where in local coordinates
$\rho_{it}^{k}=\int_{B(0,\delta_{0})}\tilde{h}_{i}^{k}e^{\tilde{u}_{i}^{k}}d\eta.$
Let
$I_{1}=\\{i\in I;\quad\lim_{k\to\infty}m_{i}^{k}=m.\quad\\}.$
Based on Proposition 3.1 for each $i\in I_{1}$,
$m_{i}^{k}-m=O(\epsilon_{k}^{m-2})$. If we use $V_{i}^{k}$ to be the leading
term in the approximation of $\tilde{u}_{i}^{k}$ and $U_{i}^{k}$ be the scaled
version of $V_{i}^{k}$, by (3.16) we have (since $m<4$)
(4.1) $\displaystyle\frac{1}{2\pi}\rho_{it}^{k}$
$\displaystyle=\frac{1}{2\pi}\int_{B(0,\delta_{0}\epsilon_{k}^{-1})}\tilde{h}_{i}^{k}(0)e^{V_{i}^{k}(y)}dy+o(\delta_{0})\epsilon_{k}^{m-2},$
$\displaystyle=\sigma_{iv}^{k}-\frac{e^{D_{i}-\alpha_{i}}}{m-2}\epsilon_{k,t}^{m-2}\delta_{0}^{2-m}+E_{\delta_{0}},\quad
i\in I_{1}.$
where we use
$\sigma_{iv}^{k}=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}e^{V_{it}^{k}}$ and
$E_{\delta_{0}}$ to denote $o(\delta_{0})\epsilon_{k}^{m-2}$,
$\epsilon_{k,t}=e^{-\frac{M_{k,t}}{2}}$. We did not use $t$ in
$\sigma_{iv}^{k}$ because the difference between any two of them is
$O(\epsilon_{k}^{m-2}/M_{k})$. Now for $i\not\in I_{1}$ we have
(4.2) $\frac{1}{2\pi}\rho_{it}^{k}=\sigma_{iv}^{k}+E_{\delta_{0}},\quad
i\not\in I_{1}$
and
(4.3) $|\rho_{ib}^{k}|=E_{\delta_{0}},\quad i\not\in I_{1}.$
Combining (4.1), (4.2), (4.3) we have
$\displaystyle\sum_{i=1}^{n}\frac{4}{2\pi}\rho_{it}^{k}-\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}\frac{\rho_{it}^{k}}{2\pi}\frac{\rho_{jt}^{k}}{2\pi}$
$\displaystyle=$ $\displaystyle
4\sum_{i=1}^{n}(\sigma_{iv}-\frac{e^{D_{i}-\alpha_{i}}}{m_{i}-2}\delta^{2-m_{i}}\epsilon_{k,t}^{m_{i}-2})$
$\displaystyle-\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}(\sigma_{iv}^{k}-\frac{e^{D_{i}-\alpha_{i}}}{m_{i}-2}\delta^{2-m_{i}}\epsilon_{k,t}^{m_{i}-2})(\sigma_{jv}^{k}-\frac{e^{D_{j}-\alpha_{j}}}{m_{j}-2}\delta^{2-m_{j}}\epsilon_{k,t}^{m_{j}-2})+E_{\delta_{0}}$
$\displaystyle=$ $\displaystyle-\frac{4}{m-2}\sum_{i\in
I_{1}}e^{D_{i}-\alpha_{i}}\delta_{0}^{2-m}\epsilon_{k,t}^{m-2}+2\sum_{i\in
I_{1}}\sum_{j=1}^{n}a_{ij}\sigma_{jv}^{k}\frac{e^{D_{j}-\alpha_{j}}}{m_{j}-2}\delta_{0}^{2-m_{j}}\epsilon_{k,t}^{m_{j}-2}+E_{\delta_{0}}$
$\displaystyle=$ $\displaystyle
2\delta_{0}^{2-m}\epsilon_{k,t}^{m-2}\sum_{i\in
I_{1}}e^{D_{i}-\alpha_{i}}+E_{\delta_{0}}.$
Note that
$m_{i}=\sum_{j=1}^{n}a_{ij}\sigma_{jv}^{k}+O(\epsilon_{k}^{m-2}/M_{k})$. For
$i\in I_{1}$, using (3.28) we have
$\displaystyle\rho_{ib}^{k}$
$\displaystyle=\int_{M\setminus(\cup_{t}B(p_{t}^{k},\delta_{0}))}\rho_{i}^{k}h_{i}^{k}e^{u_{i}^{k}}dV_{g}$
$\displaystyle=$
$\displaystyle\rho_{i}^{k}e^{\bar{u}_{i}^{k}}\int_{M\setminus\cup_{t}B(p_{t}^{k},\delta_{0})}h_{i}^{k}e^{2\pi
m\sum_{t=1}^{N}G(x,p_{t}^{k})}+E_{\delta_{0}},\quad i\in I_{1}.$
Now we define $N$ open sets $\Omega_{t,\delta_{0}}$ such that they are
mutually disjoint, each of them contains a bubbling disk and their union is
$M$:
$B(p_{t}^{k},\delta_{0})\subset\Omega_{t,\delta_{0}},\quad\cup_{t=1}^{N}\overline{\Omega_{t,\delta_{0}}}=M,\quad\Omega_{t,\delta_{0}}\cap\Omega_{s,\delta_{0}}=\emptyset,\,\,\forall
t\neq s.$
In each $\Omega_{t,\delta_{0}}$ we use (3.30)) to write $\rho_{ib}^{k}$ as
(for $i\in I_{1}$)
(4.4)
$\displaystyle\rho_{ib}^{k}=\rho_{i}^{k}e^{\bar{u}_{i}^{k}}\sum_{t=1}^{N}\int_{\hat{\Omega}_{t,\delta_{0}}}h_{i}^{k}e^{2\pi
m\sum_{l=1}^{N}G(x,p_{l}^{k})}$
$\displaystyle=\sum_{t=1}^{N}\int_{\hat{\Omega}_{t,\delta_{0}}}\epsilon_{k,t}^{m-2}\frac{h_{i}^{k}(x)}{h_{i}^{k}(p_{t}^{k})}e^{D_{i}-\alpha_{i}}e^{2\pi
m\sum_{l=1}^{N}(G(x,p_{l}^{k})-G^{*}(p_{t}^{k},p_{l}^{k}))}dV_{g}$
$\displaystyle+E_{\delta_{0}},\quad i\in I_{1},$
where $\hat{\Omega}_{t,\delta_{0}}=\Omega_{t,\delta_{0}}\setminus
B(p_{t}^{k},\delta_{0})$. Now we put estimates together to have
$\displaystyle 4\sum_{i=1}^{n}\frac{\rho_{i}^{k}}{2\pi
N}-\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}\frac{\rho_{i}^{k}}{2\pi
N}\frac{\rho_{j}^{k}}{2\pi N}$ $\displaystyle=$ $\displaystyle
4\sum_{i=1}^{n}(\sum_{t=1}^{N}\frac{\rho_{it}^{k}}{2\pi
N}+\frac{\rho_{ib}}{2\pi
N})-\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}(\sum_{t=1}^{N}\frac{\rho_{it}^{k}}{2\pi
N}+\frac{\rho_{ib}^{k}}{2\pi N})(\sum_{s=1}^{N}\frac{\rho_{js}^{k}}{2\pi
N}+\frac{\rho_{jb}^{k}}{2\pi N})$ $\displaystyle=$
$\displaystyle\frac{2}{N}\sum_{t=1}^{N}\sum_{i\in
I_{1}}\delta_{0}^{2-m}\epsilon_{k,t}^{m-2}e^{D_{i}-\alpha_{i}}-2(m-2)\sum_{i\in
I_{1}}\frac{\rho_{ib}^{k}}{2\pi N}+E_{\delta_{0}}.$
Using (4.4) in the expression above we have
(4.5) $\displaystyle\qquad 4\sum_{i=1}^{n}\frac{\rho_{i}^{k}}{2\pi
N}-\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}\frac{\rho_{i}^{k}}{2\pi
N}\frac{\rho_{j}^{k}}{2\pi N}$ $\displaystyle=\frac{2}{N}\sum_{i\in
I_{1}}e^{D_{i}-\alpha_{i}}\sum_{t=1}^{N}\epsilon_{k,t}^{m-2}\bigg{(}\delta_{0}^{2-m}-\frac{(m-2)}{2\pi}\int_{\hat{\Omega}_{t,\delta_{0}}}(\frac{h_{i}^{k}(x)}{h_{i}^{k}(p_{t}^{k})}e^{2\pi
m\sum_{l=1}^{N}(G(x,p_{l}^{k})-G^{*}(p_{t}^{k},p_{l}^{k}))}dV_{g}\bigg{)}$
$\displaystyle+E_{\delta_{0}}$
Since one of $\epsilon_{k,t}$ is $\epsilon_{k}$, say
$\epsilon_{1,t}=\epsilon_{k}$, based on (3.29) we use
(4.6)
$c_{t}=\epsilon_{k,t}^{m-2}/\epsilon_{k}^{m-2}=\frac{h_{i}^{k}(p_{t}^{k})e^{2\pi
m\sum_{l=1}^{N}G^{*}(p_{t}^{k},p_{l}^{k})}}{h_{i}^{k}(p_{1}^{k})e^{2\pi
m\sum_{l=1}^{N}G^{*}(p_{1}^{k},p_{l}^{k})}},$
Here we note that (3.29) implies that for $i\in I_{1}$, $c_{t}$ is independent
of $i\in I_{1}$. Then we use $D+o(1)$ to represent
(4.7) $\displaystyle D+o(1):=$ $\displaystyle\sum_{i\in
I_{1}}e^{D_{i}-\alpha_{i}}\sum_{t=1}^{N}c_{t}\bigg{(}\delta_{0}^{2-m}-\frac{(m-2)}{2\pi}\int_{\hat{\Omega}_{t,\delta_{0}}}(\frac{h_{i}^{k}(x)}{h_{i}^{k}(p_{t}^{k})}e^{2\pi
m\sum_{l=1}^{N}(G(x,p_{l}^{k})-G^{*}(p_{t}^{k},p_{l}^{k}))}dV_{g}\bigg{)}.$
and the leading term of $4\sum_{i=1}^{n}\frac{\rho_{i}^{k}}{2\pi
N}-\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{\rho_{i}^{k}}{2\pi
N}\frac{\rho_{j}^{k}}{2\pi N}$ is written as
$4\sum_{i=1}^{n}\frac{\rho_{i}^{k}}{2\pi
N}-\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{\rho_{i}^{k}}{2\pi
N}\frac{\rho_{j}^{k}}{2\pi N}=\frac{2}{N}\epsilon_{k}^{m-2}(D+o(1)),$
where $o(1)$ stands for the infinitesimal quantity when $\delta_{0}\to 0$.
Here it is important to observe that $D$ is involved with integration on the
whole manifold. Theorem 1.3 is established. $\Box$
Proof of Theorem 1.5:
Around each $p_{t}^{k}$, an extension of (3.11) can be easily determined to be
(4.8) $\displaystyle\phi_{i}^{k}(x)$ $\displaystyle=2\pi
m_{i}(\gamma(x,p_{t}^{k})-\gamma(p_{t}^{k},p_{t}^{k}))$
$\displaystyle+\sum_{l\neq t}2\pi
m_{i}(G^{*}(x,p_{l}^{k})-G^{*}(p_{t}^{k},p_{l}^{k}))-f_{i}^{k}(x)+E_{1}$
where $E_{1}=O(\epsilon_{k}^{m-2})$ if $m<4$ and is
$O(\epsilon_{k}^{2}\log\frac{1}{\epsilon_{k}})$ if $m=4$. Correspondingly,
(4.9) $\nabla\phi_{i}^{k}(p_{t}^{k})=2\pi
m_{i}\sum_{l=1}^{N}\nabla_{1}G^{*}(p_{t}^{k},p_{l}^{k})+E_{1}.$
With these notations, (1.18) and (1.19) follow immediately. Theorem 1.5 is
established. $\Box$
Finally we prove Theorem 1.4.
Proof of Theorem 1.4:
$\rho_{i}^{k}=\sum_{t=1}^{N}\int_{B(p_{t}^{k},\delta_{0})}\rho_{i}^{k}h_{i}^{k}e^{u_{i}^{k}}dV_{g}+\int_{M\setminus\cup_{t}B(p_{t}^{k},\delta_{0})}\rho_{i}^{k}h_{i}^{k}e^{u_{i}^{k}}dV_{g}.$
We continue to use the notation $\rho_{it}^{k}$ and $\rho_{ib}^{k}$. By (3.2)
and Theorem 2.3 the second integral is $O(\epsilon_{k}^{2})$, this is the same
as the computation for the single equation [12].
Now we use the expansion of bubbles to compute each $\rho_{it}^{k}$. By the
expansion of $\tilde{u}_{i}^{k}$ around $p_{t}^{k}$, we have
$\displaystyle\rho_{it}^{k}=\int_{B(p_{t}^{k},\delta_{0})}\rho_{i}^{k}h_{i}^{k}e^{u_{i}^{k}}dV_{g}=\int_{B(0,\delta_{0})}\tilde{h}_{i}^{k}e^{\phi_{i}^{k}}e^{\tilde{u}_{i}^{k}-\phi_{i}^{k}}d\eta$
(4.10)
$\displaystyle=\int_{B(0,\delta_{0}\epsilon_{k,t}^{-1})}\rho_{i}^{k}h_{i}^{k}(p_{t}^{k})e^{U_{i}^{k}(\eta)}d\eta+O(\epsilon_{k}^{2})$
$\displaystyle+\int_{B(0,\delta_{0}\epsilon_{k,t}^{-1})}\epsilon_{k,t}^{2}(\frac{1}{4}\Delta\tilde{h}_{i}^{k}(0)+\frac{1}{2}\nabla\tilde{h}_{i}^{k}(0)\cdot\nabla\phi_{i}^{k}(0)+\frac{1}{4}|\nabla\phi_{i}^{k}(0)|^{2})|y|^{2}e^{U_{i}^{k}}d\eta.$
The first integral on the right hand side of the above is
$O(\epsilon_{k}^{2})$ different from the global solution in the approximation
of $\tilde{u}_{i}^{k}$ around $p_{t}^{k}$. So we use $\sigma_{ivt}^{k}$ to
denote it. For $t\neq s$, from (3.23) we see that
$\sigma_{ivt}^{k}-\sigma_{ivs}^{k}=O(\epsilon_{k}^{2}/\log\frac{1}{\epsilon_{k}}).$
To evaluate the last term, we first use the definition of the
$\tilde{h}_{i}^{k}$ to have
$\nabla\tilde{h}_{i}^{k}(0)\cdot\nabla\phi_{i}^{k}(0)=2\pi m_{i}\frac{\nabla
h_{i}^{k}(p_{t}^{k})}{h_{i}^{k}(p_{t}^{k})}\cdot\sum_{l=1}^{N}\nabla_{1}G^{*}(p_{t}^{k},p_{l}^{k})+O(\epsilon_{k}^{2}\log\frac{1}{\epsilon_{k}}),$
and
$\displaystyle\Delta\tilde{h}_{i}^{k}(0)=\frac{\Delta
h_{i}^{k}(p_{t}^{k})}{h_{i}^{k}(p_{t}^{k})}-2K(p_{t}^{k})+\sum_{j=1}^{n}a_{ij}\rho_{j}+O(\epsilon_{k}^{2}\log\frac{1}{\epsilon_{k}})$
(4.11) $\displaystyle=\frac{\Delta
h_{i}^{k}(p_{t}^{k})}{h_{i}^{k}(p_{t}^{k})}-2K(p_{t}^{k})+8\pi
N+O(\epsilon_{k}^{2}\log\frac{1}{\epsilon_{k}}).$
Then we define $b_{it}^{k}$ as
(4.12) $\displaystyle b_{it}^{k}$
$\displaystyle=e^{D_{i}-\alpha_{i}}\bigg{(}\frac{1}{4}\frac{\Delta
h_{i}^{k}(p_{t}^{k})}{h_{i}^{k}(p_{t}^{k})}-K(p_{t}^{k})+4\pi N$
$\displaystyle+4\pi\frac{\nabla
h_{i}^{k}(p_{t}^{k})}{h_{i}^{k}(p_{t}^{k})}\cdot\sum_{l=1}^{N}\nabla_{1}G^{*}(p_{t}^{k},p_{l}^{k})+16\pi^{2}|\sum_{l=1}^{N}\nabla_{1}G^{*}(p_{t}^{k},p_{l}^{k})|^{2}\bigg{)}.$
With this $b_{it}^{k}$ we have
$\frac{\rho_{it}^{k}}{2\pi}=\sigma_{iv}^{k}+b_{it}^{k}\epsilon_{k,t}^{2}\log\frac{1}{\epsilon_{k}}+O(\epsilon_{k}^{2}).$
From (3.29) we see that $\epsilon_{k,t}$ can be replaced by $\epsilon_{k}$.
Consequently,
(4.13) $\displaystyle 4\sum_{i=1}^{n}\frac{\rho_{i}^{k}}{2\pi
N}-\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}\frac{\rho_{i}^{k}}{2\pi
N}\frac{\rho_{j}^{k}}{2\pi N}$ $\displaystyle=$ $\displaystyle
4\sum_{i=1}^{n}\sum_{t=1}^{N}\frac{\rho_{it}^{k}}{2N\pi}-\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}(\sum_{t=1}^{N}\frac{\rho_{it}}{2\pi
N})(\sum_{s=1}^{N}\frac{\rho_{js}}{2\pi N})$ $\displaystyle=$ $\displaystyle
4\sum_{i=1}^{n}\sum_{t=1}^{N}(\frac{\sigma_{ivt}}{N}+\tilde{\epsilon}_{k}\frac{b_{it}^{k}}{N})-\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{t=1}^{N}\sum_{s=1}^{N}a_{ij}(\frac{\sigma_{ivt}}{N}+\tilde{\epsilon}_{k}\frac{b^{k}_{it}}{N})(\frac{\sigma_{jv}}{N}+\tilde{\epsilon}_{k}\frac{b^{k}_{js}}{N})$
$\displaystyle=$ $\displaystyle
4\sum_{i=1}^{n}\frac{\sigma_{iv}}{N}-\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}\frac{\sigma_{iv}\sigma_{jv}}{N}+4\sum_{i=1}^{n}\sum_{t=1}^{N}\tilde{\epsilon}_{k}\frac{b^{k}_{it}}{N}-2\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{t=1}^{N}a_{ij}\sigma_{iv}\tilde{\epsilon}_{k}\frac{b^{k}_{jt}}{N}+O(\epsilon_{k}^{2})$
$\displaystyle=$
$\displaystyle-4\epsilon_{k}^{2}\log\epsilon_{k}^{-1}\sum_{i=1}^{n}\sum_{t=1}^{N}b_{it}^{k}+O(\epsilon_{k}^{2}).$
where $\tilde{\epsilon}_{k}$ stands for
$\epsilon_{k}^{2}\log\frac{1}{\epsilon_{k}}$. Theorem 1.4 is established.
$\Box$
## Acknowledgment
The authors are grateful to Professor C.-S. Lin for his helpful discussions
and comments on the article.
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|
# Core Fragmentation in Simplest Superfluid Dark Matter Scenario
Lasha Berezhiani Max-Planck-Institut für Physik, Föhringer Ring 6, 80805
München, Germany Arnold Sommerfeld Center, Ludwig-Maximilians-Universität,
Theresienstraße 37, 80333 München, Germany Giordano Cintia Max-Planck-
Institut für Physik, Föhringer Ring 6, 80805 München, Germany Max Warkentin
Arnold Sommerfeld Center, Ludwig-Maximilians-Universität, Theresienstraße 37,
80333 München, Germany
###### Abstract
We study the structure of galactic halos within a scalar dark matter model,
endowed with a repulsive quartic self-interaction, capable of undergoing the
superfluid phase transition in high-density regions. We demonstrate that the
thermalized cores are prone to fragmentation into superfluid droplets due to
the Jeans instability. Furthermore, since cores of astrophysical size may be
generated only when most of the particles comprising the halo reside in a
highly degenerate phase-space, the well-known bound on the dark matter self-
interaction cross section inferred from the collision of clusters needs to be
revised, accounting for the enhancement of the interaction rate due to
degeneracy. As a result, generation of kpc-size superfluid solitons, within
the parameter subspace consistent with the Bullet Cluster bound, requires dark
matter particles to be ultra-light.
## I I. Introduction
The cold dark matter (CDM) paradigm, in which dark matter is incorporated as a
non-relativistic fluid composed of collisionless particles, is in spectacular
agreement with observations at large scales. However, it presents several
challenges when it comes to galactic scale phenomena Bullock:2017xww . Among
those, the mismatch between the Navarro-Frank-White (NFW) density profile in
inner regions of galaxies and clusters obtained from CDM simulations
Navarro:1995iw and the density profile inferred from observations, known in
the literature as ’the core-cusp problem’, appears to be a central one and its
resolution will most likely affect the status of some of the other puzzles as
well. In particular, there have been claims regarding the excess of the
dynamical friction predicted by CDM Debattista:1997bi ; Sellwood:2016 ;
Fornaxold ; Tremaine , which is a sensitive function of the dark matter
density-profile (a potentially ameliorating discussion within CDM, in some
cases, can be found in Cole ; SanchezSalcedo:2006fa ). Furthermore,
observations seem to indicate a significant correlation between the
gravitational acceleration of baryons and their distribution in disk galaxies
McGaugh:2016leg ; Lelli:2017vgz . This is usually referred to as the mass
discrepancy acceleration relation, which is the generalized version of the
baryonic Tully-Fisher relation McGaugh:2011ac ; LelliBTFR , see also
Famaey:2011kh and references therein. These are well accounted for by
Milgrom’s empirical law Milgrom:1983ca ; Milgrom:1983pn ; Milgrom:1983zz ,
which was originally suggested as a modification of gravity. The origin of
these observed relations is not yet understood, but seems to be indicative of
some form of interaction between baryons and the dark sector. However, it must
be pointed out that depending on the mass-range for dark matter particles, the
cosmological constraints on such interactions could be significant
Dvorkin:2013cea . Although it is still unclear whether a purely gravitational
interaction present in CDM, without invoking self- or cross-species
interactions, is sufficient for generating such correlations, it seems to be
an unlikely scenario Oman:2015xda (the possibility of reconciliation has been
argued Read:2016 ).
These and other related observations have motivated extensions of the standard
paradigm, by giving dark matter some additional properties that could be
reflected in a more desirable galactic scale behaviour. One of the directions
that has attracted much attention recently concerns self-interacting dark
matter Spergel:1999mh . If strong enough, interactions could affect the
density distribution of central regions of the halo significantly, due to a
higher number density of dark matter particles. In particular, numerical
simulations have revealed that the density profile starts deviating from the
NFW profile in regions where particles had the chance to interact at least
once throughout the lifetime of the halo Kaplinghat:2015aga . The main effect
of those interactions is the dark matter redistribution, alleviating the
central cusp of the NFW profile. At large distances from the centre of the
halo, where dark matter particles have not had a chance to interact yet due to
low densities, the density profile is similar to the one in CDM.
An interesting version of self-interacting dark matter was considered in
Goodman:2000tg ; Slepian:2011ev 111See also Boehmer:2007um for the discussion
of the core profile and galactic rotation curves within the Bose-Einstein
condensate dark matter scenario. in the form of a repulsive, sub-eV, scalar
field exhibiting superfluidity in galaxies, motivated by theoretical
simplicity and the capacity for an additional suppression of the dynamical
friction. The considered structure of the dark matter halo had a superfluid
core enclosed by an isothermal envelope. In that scenario, dark matter was
assumed to be in thermal equilibrium which led to the formation of the
superfluid core in the central region, where the temperature of the substance
is subcritical (due to high densities). The presence of repulsive self-
interaction was vital for the stability of the core, as well as for
superfluidity, as we will discuss shortly. The envelope, on the other hand,
was stabilized by the thermal pressure of dark matter particles, with the
junction condition determined in terms of the finite-temperature equation of
state. Imposing the Bullet Cluster constraint and the reproduction of the
observed features of rotational curves, the model was claimed to be ruled out
in Slepian:2011ev . See Sharma:2018ydn for a more recent discussion of a
spherically symmetric density profile for a thermalized halo, derived from an
improved finite-temperature equation of state. The core profile and galactic
rotation curves within the Bose-Einstein condensate dark matter scenario have
been discussed in Boehmer:2007um as well. The same scalar dark matter model
has been also considered in the context of its gravitational production at the
end of inflation Peebles:1999fz ; Peebles:1999se .
For attractive bosons, which is the case for axions, one lacks positive
pressure and consequently the condensate formed at subcritical temperatures is
prone to fragmentation Guth:2014hsa . The only possibility for sustaining the
macroscopic homogeneous core is to consider the case of ultra-light particles.
Achieving macroscopic homogeneity in such a way, in the core of the dark
matter halo, was first suggested by Hu:2000ke , and was coined as Fuzzy Dark
Matter. There, the scale of homogeneity is set by the de Broglie wavelength,
and the presence of a kpc-size core requires particles lighter than
$10^{-21}{\rm eV}$. The detailed analysis of phenomenological implications of
such a scenario was performed in Hui:2016ltb (see May:2021wwp for recent
numerical simulations of the structure formation). The idea of substructure
formation has been discussed in the context of the superfluid dark matter
scenario as well, see Schiappacasse:2017ham ; Alexander:2019qsh and
references therein.
Recently, the idea of dark matter superfluidity was revitalized in
Berezhiani:2015pia ; Berezhiani:2015bqa as a novel mechanism that could be
behind the above-mentioned long-range correlations in galactic dynamics. The
idea is to have a superfluid dark matter core in galaxies and utilize phonons
to mediate an emergent long-range interaction between baryons submerged within
this quantum liquid222The idea of emergent long-range interactions within an
ideal Bose-Einstein condensate and the cosmological implications for the
scalar dark matter scenario was first considered in Ferrer:2000hm ;
Ferrer:2004xj . Recently, the idea was revisited for superfluids
Berezhiani:2018oxf . It was demonstrated that superfluidity shortens the
range, but still keeping it much longer than the Compton wavelength of dark
matter particles. The potential significance of the mechanism for the galactic
dynamics, in the context of the ultra-light dark matter scenario (like Fuzzy
Dark Matter), was studied as well.. It was demonstrated in Berezhiani:2017tth
that in this incarnation of the superfluid dark matter scenario it was more
natural to give up global thermal equilibrium and instead to require
thermalization within the central neighbourhood of a halo. However, there much
freedom was given in specifying the profile by keeping the superfluid pressure
not very closely related to the $2\rightarrow 2$ scattering cross
section.333The reason behind this was the assumption that the thermalization
could have been governed by two-body interactions (i.e. in the disordered
phase the $2\rightarrow 2$ scattering was considered to be dominant), while
the superfluid pressure was considered to be determined by three-body
interactions.
In this work, we study the structure of galactic halos within the simplest
model of dark matter superfluidity combining ideas of Goodman:2000tg ;
Guth:2014hsa ; Schiappacasse:2017ham and Berezhiani:2017tth together. The
additional ingredient added to the pre-existing ideas being the proper
application of the Bullet Cluster bound to sub-eV particles, by including the
Bose-enhancement factor in the calculation of scattering rates. The result is
the inevitable fragmentation of the thermalized core into superfluid islands
due to the Jeans instability. It is demonstrated that the extremely light
values for the mass are required to give a kpc-size coherence length.
One of the theoretically simplest interacting dark matter models can be
introduced as a massive scalar field, with quartic self-interaction and
minimally coupled to gravity, with the following action
$\displaystyle S=\int{d^{4}x\sqrt{-g}\left(\frac{1}{16\pi
G}R-|\partial\Phi|^{2}-m^{2}|\Phi|^{2}-\frac{\lambda}{2}|\Phi|^{4}\right)}\,.$
(1)
Here, we have chosen to work with a complex ($U(1)$-invariant) field, due to
manifest particle number conservation. We could have begun with a real scalar,
but considering that we are interested in a non-relativistic substance, the
net result would have been identical; in this limit, both theories flow to the
nonlinear Schrödinger’s action.
Before diving into the description of the superfluid scenario, let us begin by
recapping the Bullet Cluster constraint Markevitch:2003at for such a theory.
It is usually invoked as the following bound on the scattering cross section
for dark matter particles
$\displaystyle\frac{\sigma}{m}\lesssim 1{\text{ cm}^{2}}/{\text{g}}\,.$ (2)
Strictly speaking, the bound obtained by Markevitch:2003at is for the
scattering rate, which for the non-degenerate phase-space translates into (2).
Although the latter has been widely applied to various dark matter models,
including the sub-eV mass-range, we will argue that it may not be necessarily
legitimate. Having said this, let us put this caveat aside for a moment by
focussing on $m\gg{\rm eV}$ and translate (2) into a bound on the coupling
constant
$\displaystyle\lambda\lesssim\left(\frac{m}{10~{}{\rm MeV}}\right)^{3/2}\,.$
(3)
In other words, for dark matter heavier than few MeV, the theory has to be
strongly coupled in order to violate the Bullet Cluster bound.
For the superfluid scenario one is mostly interested in sub-eV particles. As
it is well known, such a candidate must be a non-thermal relict and should be
produced via the axion-like vacuum misalignment mechanism. Consequently, it is
expected to be in the form of the condensate on cosmological scales. Requiring
the equation of state due to interaction-pressure to be the one for a non-
relativistic fluid, one arrives at the following bound for the scattering
cross section
$\displaystyle\frac{P}{\rho}\Big{|}_{\text{equality}}=\frac{\lambda\rho\rvert_{\text{equality}}}{8m^{4}}\ll
1\,,\quad\Rightarrow\quad\frac{\sigma}{m}\ll\left(\frac{m}{2\times
10^{-5}\text{ eV}}\right)^{5}\frac{\text{cm}^{2}}{\text{g}}\,.$ (4)
We have used $\rho\rvert_{\text{equality}}\simeq 0.4\text{ eV}^{4}$ as a dark
matter density at matter-radiation equality. Notice that if a fraction of dark
matter is produced by a mechanism other than the vacuum misalignment, an
additional statistical contribution to the pressure would be generated that
could tighten the bound (4). Interestingly, for light-enough particles (4)
seems to compete with the merger constraint. However, the real Bullet Cluster
bound turns out to be even more restrictive than the naively obtained
inequality (2). Due to phase-space degeneracy for sub-eV particles, the
interaction rate is boosted by the bosonic enhancement factor. In particular,
denoting the average degeneracy factor by $\mathcal{N}$, the improved version
of (2) takes the following form for sub-eV dark matter particles of interest
$\displaystyle\frac{\sigma}{m}\mathcal{N}\lesssim 1{\text{
cm}^{2}}/{\text{g}}\,,\quad\Rightarrow\quad\frac{\sigma}{m}\lesssim
10^{-2}\left(\frac{m}{\text{eV}}\right)^{4}\frac{\text{cm}^{2}}{\text{g}}\,.$
(5)
Moreover, this improved bound results in the exclusion of a significant part
of the parameter space, as we are about to show.
The paper is organized as follows. In section II, we overview some of the key
properties of the condensate of interacting particles governed by (1). In
section III, we study the conditions that lead to a superfluid phase
transition in galaxies and clusters. We show that, in order to have kpc-size
superfluid cores, we have to consider a scenario with highly degenerate phase-
space, implying an ultra-light mass range for dark matter particles. In
section IV, we revise the well known bound on dark matter self-interaction
cross section inferred from merging clusters accordingly. Section V is devoted
to the detailed analysis of the full parameter space for a Milky Way-like
halo. We summarize the results in section VI.
## II II. Superfluid Properties
In a theory of self-interacting bosons, superfluidity may be achieved through
Bose-Einstein condensation. Within the theory given by (1), the condensate of
$\Phi$ particles can be well described by a homogeneous classical field
configuration with finite number density, which spontaneously breaks the U(1)
global symmetry of the Lagrangian. As it is well known, any homogeneous fluid
is susceptible to the gravitational Jeans instability above a certain length-
scale. For the theory at hand, the low-energy spectrum of excitations around
the homogeneous condensate (phonon spectrum) is given by
$\displaystyle\omega_{k}^{2}=-4\pi
G\rho+c_{s}^{2}k^{2}+\frac{k^{4}}{4m^{2}}\,,\qquad
c_{s}^{2}\equiv\frac{\lambda\rho}{4m^{4}}\;,$ (6)
with $G$ standing for the gravitational constant and $\rho$ denoting the
superfluid density444See, e.g., Berezhiani:2019pzd for the derivation.. The
first term in (6) is a tachyonic contribution responsible for the Jeans
instability, the second term describes the energy cost for exciting the sound
waves and the last one is the kinetic energy of a massive constituent; in
other words, the last term indicates that in order to create an excitation, we
need to make a massive constituent of the superfluid mobile along the way. It
is straightforward to see that, in order to have stable sound waves, $\lambda$
needs to be positive which corresponds to the case of repulsive bosons555In
the opposite case ($\lambda\leq 0$), the only stabilizing contribution to (6)
would have been the last term., which is essential for superfluidity and will
be assumed throughout this work.
As one can easily deduce from (6), for the homogeneous condensate, the modes
softer than the critical momentum-scale $k_{*}$ are unstable; with
$\displaystyle k_{*}^{2}\equiv 2m^{2}c_{s}^{2}\left(-1+\sqrt{1+\frac{4\pi
G\rho}{m^{2}c_{s}^{4}}}\right)\,.$ (7)
The existence of this Jeans scale implies that due to gravity there is an
upper bound on the coherence length for the homogeneous superfluid
configuration, which is the result of the equilibrium between the
gravitational attraction and either the repulsive self-interaction (giving
rise to the sound-speed) or the quantum pressure (à la Fuzzy Dark Matter
scenario).
It is easy to see that, depending on the value of $c_{s}$, which in turn is
determined by the self-interaction strength $\lambda$ (or equivalently by the
scattering cross section $\sigma$), the Jeans momentum has two interesting
limits depending on the magnitude of the dimensionless quantity
$\displaystyle\xi\equiv\frac{m^{2}c_{s}^{4}}{4\pi G\rho}\,.$ (8)
In particular, in the non-interacting limit $\xi\ll 1$ we get
$\displaystyle\lim_{\xi\ll 1}k_{*}^{2}=\sqrt{16\pi G\rho m^{2}}\,,$ (9)
which is the Jeans scale stabilized by the quantum pressure. It corresponds to
the Fuzzy Dark Matter scenario Hu:2000ke ; Hui:2016ltb and we refer to it as
the degeneracy pressure case. For completeness, let us stress that in this
limit one cannot talk about sound waves anymore, since the dispersion relation
of phonons with wavelength well within the homogeneity domain are highly
dominated by the last term of (6). Although subdominant, the presence of the
sound-term can still give rise to superfluidity by providing an additional
energy cost for excitations.
For the opposite case, with $\xi\gg 1$, we get
$\displaystyle\lim_{\xi\gg 1}k_{*}^{2}=\frac{4\pi G\rho}{c_{s}^{2}}\,,$ (10)
which is the standard result for the Jeans scale for the superfluid. In this
case, the gravitational instability is counteracted and balanced by the
repulsive interactions (regular positive pressure). This is the case we are
interested in, and we refer to it as the interaction pressure case.
Since the magnitude of the parameter $\xi$ defines the nature of the pressure
that sustains the condensate, let us evaluate it at typical galactic
densities,
$\displaystyle\xi=\frac{4\pi M_{\rm pl}^{2}\rho\sigma}{m^{4}}\simeq
10^{27}\left(\frac{\sigma/m}{{\rm cm}^{2}/{\rm g}}\right)\left(\frac{m}{\rm
eV}\right)^{-3}\,,$ (11)
where we estimated $\rho\simeq 10^{-25}{\rm g/cm}^{3}$, which is the average
dark matter density of inner regions of the Milky Way. This is justified as we
are after the scenario in which the dark matter density profile is altered
significantly only at short scales. In deriving (11), we have used the
relation between the sound speed and the scattering cross section
$\displaystyle c_{s}^{2}=\frac{\rho}{m^{4}}\sqrt{2\pi m^{2}\sigma}\,.$ (12)
As it can be seen from (11), keeping in mind that we do not wish $m$ to be
significantly heavier than eV, the only way one could end up with $\xi\ll 1$
would be to take an extremely small scattering cross section (per mass)
$\sigma/m$.
For $\xi\gg 1$, it is straightforward to derive the self-sustained spherical
density profile of a zero-temperature superfluid soliton by solving the
equation for the hydrostatic equilibrium and Poisson’s equation. For the
quartic interaction at hand, the superfluid equation of state is
$P=\lambda\rho^{2}/8m^{4}$, which leads to the following analytic expression
for the self-sustained density profile chandrabook
$\rho(r)=\rho_{0}\frac{\sin\left({2\pi r/\ell}\right)}{2\pi r/\ell},$ (13)
where $\ell\equiv 2\pi/k_{*}=\sqrt{\frac{\pi\lambda}{4Gm^{4}}}$ is the Jeans
length in the $\xi\gg 1$ limit and $\rho_{0}$ is the central density of the
soliton. Equation (13) shows the equivalence between the Jeans length $\ell$
and the size of the soliton diameter. It must be noted that the density-
independence of $\ell$ is tightly connected with the type of self-interaction
present for the dark matter field; it would not have been the case for any
other form of the potential. It has proven to be convenient to express the
size of the soliton in terms of the cross section Slepian:2011ev
$\displaystyle\ell=2\pi\left(\frac{8\pi M_{\rm
pl}^{4}}{m^{5}}\frac{\sigma}{m}\right)^{1/4}\simeq 2~{}{\rm
kpc}\left(\frac{\sigma/m}{{\rm cm}^{2}/{\rm
g}}\right)^{1/4}\left(\frac{m}{{\rm meV}}\right)^{-5/4}\,.$ (14)
This way, one can get an idea of what it takes to have a macroscopic core.
Therefore, if we take a nearly-zero-temperature homogeneous superfluid of
$\Phi$s (that would have been stable in the absence of gravity), it will break
into superfluid islands of size $\ell$ and the density profile given by (13),
similar to the formation of the cluster of stars from a baryonic cloud.
We would like to finish this section by pointing out that the above discussion
applies to a zero-temperature superfluid. In the dark matter context, the
initial thermalized region has a finite temperature Slepian:2011ev ;
Berezhiani:2015bqa . Therefore, each superfluid soliton will be dressed in an
envelope of normal dark matter particles. The transition area is expected to
be located where the density drops below the critical one; i.e. the density
for which the de Broglie wavelength becomes shorter than the inter-particle
separation. Within the superfluid soliton the densities will be of order of
the galactic values (maybe somewhat larger, as they would be the result of a
collapse), while the transition density can be roughly estimated as
$\rho_{c}\sim m^{4}v^{3}$; with $v$ denoting the characteristic galactic
velocity determining the temperature. Assuming $v\sim 10^{-3}$, one gets
$\rho_{c}\sim 10^{-28}{\rm g/cm}^{3}(m/{\rm eV})^{4}$. In other words, if the
dark matter mass is significantly sub-eV (which will be the range of our
interest, as we will show), then the envelope begins at the distance from the
centre of the soliton where the densities have dropped by more than few orders
of magnitude compared to the core. Taking this into account, it is safe to
assume that the significant fraction of the dark matter within the thermalized
region will be in the form of the superfluid islands. The rest of the matter
will be split between the envelopes (gravitationally bound to solitons) and
the inter-soliton gas.
## III III. Thermalization and Superfluid Formation
Let us investigate what are the conditions that lead to the formation of a
superfluid in the galactic medium. We would like to begin by mentioning that
it is possible to have an effective condensate without a local thermal
equilibrium, as long as the number of particles within the de Broglie volume
is large. In other words, the high degeneracy enables us to describe the
quantum state by a homogeneous field configuration, the perturbations around
which obey (6). Therefore, the coherence length can be estimated as
Guth:2014hsa
$\displaystyle\ell\simeq{\rm min}\left(2\pi/k_{*},\lambda_{\rm dB}\right)\,,$
(15)
with $\lambda_{\rm dB}$ denoting the de Broglie wavelength. If equilibrium is
not reached, then $\lambda_{\rm dB}$ is determined by characteristic dark
matter velocities obtained from N-body simulations. Using the value of the
virial velocity for a typical galactic halo, it is straightforward to verify
that (15) will always reduce to $\lambda_{\rm dB}$, unless one considers
particle masses even lighter than the one for the Fuzzy Dark Matter scenario.
Therefore, in practice, one needs to invoke thermalization in order to even
hope to get a macroscopic (kpc-size) core for moderately sub-eV particles.
Following what we said, the Bose-Einstein condensation for weakly interacting
bosons sets in if two conditions are satisfied:
* •
First, the system must reach the equilibrium. This is achieved after particles
had sufficient time to interact and reach the (nearly) maximum entropy state,
since otherwise the applicability of the Bose-Einstein statistics would be
questionable. The time it takes to reach the equilibrium can be estimated as
$t_{\rm eq}>t_{1}$, with $t_{1}$ denoting the time it takes each particle to
scatter at least once. The longer one waits, compared to $t_{1}$, the more
certain one can be for being close to equilibrium. A more precise statement is
beyond the scope of this paper.
* •
Second, the de Broglie wavelengths of particles must overlap. This corresponds
to the system being colder than the critical temperature $T_{c}\sim
n^{2/3}/m$; with $n$ denoting the particle number density. Physically, what
happens is that at high temperatures, for which the de Broglie wavelength is
shorter than the inter-particle separation, the gas of weekly interacting
particles behaves as a classical system and the Bose-Einstein distribution is
well-approximated by the Boltzmann distribution. Below the critical
temperature, on the other hand, the latter fails to adequately describe the
system, because indistinguishable particles with overlapping wave-packets
start to populate the zero momentum state (in compliance with the Bose-
Einstein statistics). In fact, for $T\ll T_{c}$ almost all particles of the
gas are in the ground state.
Therefore, we expect the dark matter halo to possess few relevant length
scales which are not in a one-to-one correspondence: the first one, the
thermal radius $R_{T}$, identifies the region where interactions are efficient
enough to allow thermal equilibrium. The second one is the degeneracy radius
$R_{\rm deg}$ within which the de Broglie wavelength exceeds the inter-
particle separation. The shortest of these identifies the region where the
phase transition is expected to take place. The last, but not the least, is
$\ell$ connected to the scale below which the condensate is stable.
Understanding the hierarchy between these scales is vital in order to
understand whether the fragmentation takes place or not, namely if galaxies
present a single superfluid core or a collection of superfluid substructures.
The formation of a dark matter halo is a non-linear process and as such it is
challenging (if not impossible) to establish a precise density profile
analytically. For purely gravitationally interacting (standard) dark matter
models, $N$-body simulations reveal the more or less universal density
distribution, known as the NFW profile
$\displaystyle\rho(r)=\frac{\rho_{0}}{\frac{r}{r_{s}}\left(1+\frac{r}{r_{s}}\right)^{2}}\,.$
(16)
The characteristic density $\rho_{0}$ and the scale radius $r_{s}$ vary from
halo to halo. However, according to simulations, there exists a tight relation
between these two parameters, known as the mass-concentration relation
Dutton:2014xda . The size of the halo itself is conventionally defined by the
virial radius $R_{V}$, which represents the radius within which the average
density of the halo (denoted as $\rho_{200}$) is about 200 times the critical
density. For our qualitative analysis we begin with the NFW profile and
examine under what conditions a significant modification of the profile,
followed by a superfluid formation, takes place. As we have already pointed
out, one should expect the aforementioned thermalization and degeneracy
requirements to be more easily satisfied in central (high density) regions.
Depending on the parameters of the model $m$ and $\lambda$, dark matter
particles could become degenerate at densities lower than the ones at which
the equilibrium can be reached. In that case, the interaction rate,
responsible for thermalization, will be assisted by the degeneracy factor that
roughly counts the number of particles in the de Broglie volume. In general,
the relaxation rate for highly degenerate particles can be estimated as
Sikivie:2009qn ; Erken:2011dz
$\displaystyle\Gamma=\frac{\sigma}{m}\rho
v\mathcal{N}\,,\qquad\mathcal{N}={\rm
max}\left\\{1,~{}\frac{\rho}{m}\left(\frac{2\pi}{mv}\right)^{3}\right\\}\,,$
(17)
where $v(r)=\sqrt{\frac{GM(r)}{r}}$ stands for the orbital velocity of dark
matter particles and the velocity dispersion has been assumed to be of order
$v$, while $M(r)$ is the mass of the halo enclosed in an orbit of radius $r$.
Following our earlier discussion, we estimate the thermal radius $R_{T}$ as
the one within which the particles had a chance to scatter at least once
throughout the lifetime of the galaxy; i.e. $R_{T}$ is a radius within which
we have $\Gamma t_{\text{g}}>1$, with $t_{\text{g}}\approx 13~{}{\rm Gyrs}$
being the age of the galaxy.
For completeness, let us note that using $t_{\text{g}}$ as the time-scale for
thermalization implies the assumption that it is possible to ignore the phase-
space reshuffling of $\Phi$ due to dynamical effects within the galaxy. If
this is not justified, the dynamical time $t_{\text{dyn}}=r/v$ is more
appropriate to determine the thermal radius. We will demonstrate in the
appendix that the utilization of $t_{\text{dyn}}$ results in a reduction of
the thermalization radius $R_{T}$ by a factor of few.
Depending on whether the equilibrium is reached while $\mathcal{N}\gg 1$ or
not, there could be two qualitatively distinct cases to consider:
* (i)
If thermalization is reached at radius $R_{T}$ while $\mathcal{N}\simeq 1$,
then we would have a non-degenerate classical gas of weakly interacting
particles at distances $r>R_{T}$ from the centre of the halo. In other words,
for $r>R_{T}>R_{\rm deg}$ particles are not aware of interactions, for
$R_{T}>r>R_{\rm deg}$ particles had the chance to experience interactions and
as such the distribution will be more fuzzed out compared to the non-
interacting case. Since the particles are non-degenerate in this region, the
profile would resemble the profile one obtains in self-interacting dark matter
models. At $r<R_{\rm deg}<R_{T}$, on the other hand, the Bose-Einstein
condensation would take place and we would expect to see the presence of
superfluid islands of size $\ell$.
* (ii)
An alternative scenario would be that the high degeneracy is reached at
distances larger than the thermal radius ($R_{\rm deg}>R_{T}$). In this case,
the halo would have a simpler structure. In particular, at $r>R_{T}$ the
density profile would be similar to the one for the non-self-interacting dark
matter, i.e. like NFW, with the possibility of a BEC-like sub-structure at
scales shorter than the de Broglie wavelength. Then we would expect the
superfluid phase transition directly at $r<R_{T}$, populating the
corresponding volume with the aforementioned superfluid islands of size
$\ell$.
Now, we are going to study those two scenarios separately. For definiteness,
the numerical estimates will be performed for a Milky Way-like galaxy with the
total mass $M_{\rm DM}=10^{12}M_{\odot}$ and the concentration parameter
$c=R_{\rm V}/r_{s}=6$.
## Case (i): Non-degenerate Thermalization
Let us focus on the case where the thermalization of $\Phi$s happens in a non-
degenerate setting. The thermal radius $R_{T}$ can be estimated as
$\displaystyle\Gamma
t_{\text{g}}=\frac{\sigma}{m}\rho(R_{T})v(R_{T})t_{\text{g}}=1\,.$ (18)
For a given density profile, this equality gives $R_{T}$ as a function of
$\sigma/m$. Concerning the scaling of (18) with the parameters of the NFW
profile, this is a cumbersome function of $R_{T}$, $\rho_{0}$ and $r_{s}$.
However, the behaviour is simple in limiting cases
$R_{T}\simeq{}r_{s}\Big{(}\rho_{0}r_{s}\sqrt{2\pi
G\rho_{0}}\frac{\sigma}{m}t_{\rm g}\Big{)}^{\gamma}\,,\quad{\rm
with}\quad\gamma=\begin{cases}{{2}},&\mbox{for }R_{T}\ll r_{s}\\\
2/7,&\mbox{for }R_{T}\gg r_{s}\end{cases}$ (19)
Not surprisingly, the overall result for $R_{T}$ is a monotonically increasing
function of the cross section. In other words, for larger $\sigma/m$ the dark
matter particles manage to reach equilibrium at larger radii (i.e. lower
densities). It is easy to find that for the Milky Way-like halo at hand, using
$\rho\simeq 10^{-25}{\rm g/cm}^{3}$ , one can conveniently express the
thermalization radius as:
$\displaystyle R_{T}^{\rm MW}\simeq r_{s}\left(\frac{\sigma/m}{{\rm
cm}^{2}/{\rm g}}\right)^{\gamma}$ (20)
Let us stress that the case $R_{T}\ll r_{s}$ is sensitive to the specific
values of $r_{s}$ and $\rho_{0}$ and could have strongly been affected by
different fits. Moreover, we may see how the strength of the self-interactions
affects more mildly $R_{T}$ in outer regions of the halo: since the density
scales as $r^{-3}$, stronger self-interactions are needed to overcome the
density drop.
So far, we have not said anything about the mass of the dark matter particle.
The equality (18) will successfully provide us with the estimate for $R_{T}$,
as long as the degeneracy factor is small. Examining the expression for this
factor (see (17)) it is easy to see that to have $\mathcal{N}<1$ at the
distance $R_{T}\ll r_{s}$ from the centre,
$\displaystyle m\gtrsim 20~{}{\rm eV}\cdot\left(\frac{\sigma/m}{{\rm
cm}^{2}/{\rm g}}\right)^{-5/4}\,,$ (21)
implying masses significantly greater than eV. One gets a similar constraint
for $R_{T}\gg r_{s}$, albeit with a different power law. The important message
is that this scenario requires masses above eV, if the interaction strength
satisfies the Bullet cluster constraint. This observation together with (14),
and using $\sigma/m\lesssim{\rm cm}^{2}/{\rm g}$, gives us an absolute upper
bound on the size of the superfluid soliton
$\displaystyle\ell\lesssim 5\cdot 10^{-2}~{}{\rm pc}\,.$ (22)
Interestingly enough, the size of the superfluid patches would be an order of
magnitude or so larger than the solar system scale if one were to saturate
this, taking the density of the order of the NFW density at our location.
However, the density could be somewhat larger as these solitons result from a
fragmentation of a locally thermalized dark matter distribution.
## Case (ii): Degenerate Thermalization
It seems that, as long as thermalization happens in a non-degenerate setting,
particles cannot rely on superfluidity to generate kpc-size solitons. This
indicates that we have to explore the possibility of $R_{\rm deg}>R_{T}$, thus
violating (21). In this case we have to replace (18) with its degenerate
counterpart
$\displaystyle\Gamma
t_{\text{g}}=\frac{\sigma}{m}\rho(R_{T})v(R_{T})\mathcal{N}t_{\text{g}}=1\,,\quad\text{with}\quad\mathcal{N}=\frac{\rho}{m}\left(\frac{2\pi}{mv}\right)^{3}\gg
1\,.$ (23)
Here too, the exact expression for $R_{T}$ is cumbersome. However, similar to
the previous case, it can be nicely presented in limiting cases
$R_{T}\simeq
r_{s}\left(\frac{4\pi^{2}\rho_{0}}{Gm^{4}r_{s}^{2}}\frac{\sigma}{m}t_{\rm
g}\right)^{\delta}\,,\quad{\rm with}\quad\delta=\begin{cases}1/3,&\mbox{for
}R_{T}\ll r_{s}\\\ 1/5,&\mbox{for }R_{T}\gg r_{s}\end{cases}$ (24)
For a Milky Way-like halo, the expression reduces to
$\displaystyle R_{T}^{\rm MW}\simeq\begin{cases}60\cdot
r_{s}\left[\frac{\sigma/m}{{\rm cm}^{2}/{\rm g}}\left(\frac{m}{\rm
eV}\right)^{-4}\right]^{1/3},&\mbox{for }R_{T}\ll r_{s}\\\ 10\cdot
r_{s}\left[\frac{\sigma/m}{{\rm cm}^{2}/{\rm g}}\left(\frac{m}{\rm
eV}\right)^{-4}\right]^{1/5},&\mbox{for }R_{T}\gg r_{s}\end{cases}$ (25)
Unlike the non-degenerate case, here the size of the thermalized region is
determined by $\left({\sigma}/{m}\right)m^{-4}$, as it was previously
demonstrated in Berezhiani:2017tth . For instance, as one can see from (25),
for the galactic halo in question
$\displaystyle R_{T}\gtrless
r_{s}\,,\quad\Leftrightarrow\quad\frac{\sigma/m}{{\rm cm}^{2}/{\rm
g}}\left(\frac{m}{\rm eV}\right)^{-4}\gtrless 10^{-5}\,.$ (26)
Moreover, since (24) depends only mildly on $r_{s}$ and $\rho_{0}$, this
result is not very sensitive to a fine tuning of the NFW-parameters.
Therefore, it seems there exists a wide range of parameters, for which a
significant fraction of the halo has had enough time to have reached
equilibrium; the parameter space here is even larger than in case (i), since a
specific value of the thermal radius is now generated by different
combinations of $m$ and $\sigma/m$ due to the introduction of $\mathcal{N}$.
## IV IV. Bullet Cluster Constraint
In this section, we would like to revisit the Bullet Cluster constraint
Markevitch:2003at for the ultra-light bosonic dark matter candidate. The
system in question is a merger of two clusters, in which the dark matter
component is offset with respect to the gas component. As it is well known,
the comparison of the observed mass distribution with the simulated one does
not seem to indicate the presence of any other dark matter interaction besides
the gravitational one. In other words, the average number of scatterings
experienced by a given dark matter particle from the bullet cluster, while
crossing the target cluster, seems to be less than one
$\langle n_{sc}\rangle<1.$ (27)
The value $\langle n_{sc}\rangle$ is averaged over all the bullet cluster
particles and could be estimated as the product of the interaction rate
$\Gamma$ and the crossing time
$\langle n_{sc}\rangle=\Gamma\frac{2R_{V}}{v_{\text{in-fall}}},$ (28)
where $v_{\text{in-fall}}\simeq 10^{-2}$ is the in-fall velocity and
$R_{\text{V}}$ is the virial radius of the target cluster. Clearly, the bound
on the scattering cross section that can be extracted from (27) and (28)
depends on the nature of dark matter.
For example, assuming non-degenerate dark matter that interacts through 2-body
interactions
$\langle n_{sc}\rangle=2\frac{\sigma}{m}R_{V}\rho.$ (29)
Using $\rho\simeq 10^{-25}{\text{ g}}/{\text{cm}^{3}}$ as an average dark
matter density and $R_{V}\simeq 2$ Mpc, one obtains
$\frac{\sigma}{m}\lesssim 1{\text{ cm}^{2}}/{\text{g}}.$ (30)
The value of the density that we have chosen represents the average density of
the Target Cluster within 500 kpc, obtained fitting the matter distribution
using an NFW profile Clowe:2003tk 666At this distance, the density profile
changes from $1/r^{3}$ to $1/r$: therefore, we are averaging over the region
where the mass scales as $r^{2}$ but not in the part where the mass scales
logarithmically with $r$..
As we have already discussed from the point of view of dark matter
thermalization in galaxies, if particles have a degenerate phase-space, then
the interaction rate is enhanced by $\mathcal{N}$ (17). The same goes for
mergers, if colliding halos are significantly degenerate, then (30) needs to
be reconsidered. It is straightforward to see that for the NFW profile of a
cluster (with $R_{V}\simeq 2~{}{\rm Mpc}$, $R_{V}/r_{s}\simeq 4$ and
$\rho_{0}\simeq 10^{-25}{\text{ g}}/{\text{cm}^{3}}$) the degeneracy factor
exceeds unity everywhere inside the virial radius if $m\ll{\rm eV}$. Notice
that $\mathcal{N}_{\text{cluster}}\simeq 10^{-3}\mathcal{N}_{\text{galaxy}}$,
since the velocities are an order of magnitude or so higher in clusters.
Therefore, if galaxies are strongly degenerate, which is the case for the
parameters of interest, then the scattering rates in clusters are expected to
be enhanced as well.
We can estimate the improved Bullet Cluster bound on the self-interaction
cross section by boosting the interaction rate by a typical (average) value of
the degeneracy factor for a cluster, resulting in
$\frac{\sigma}{m}\lesssim
10^{-2}\left(\frac{m}{\text{eV}}\right)^{4}\frac{\text{cm}^{2}}{\text{g}}\,.$
(31)
We would like to stress that this relation applies if the halos are in the
form of a gaseous medium of particles with $m\ll{\rm eV}$ and the significant
velocity dispersion. If the entire halo of the cluster were to thermalize and
undergo fragmentation into superfluid solitons, without significant leftover
in the form of a dispersed gas, then (31) would need to be ameliorated: in
this case, almost all particles would lie in the ground state and transitions
to excited states would not be enhanced by degeneracy. However, it seems
unlikely not to end up with a significant fraction of particles to remain un-
condensed in the process of thermalization and fragmentation. Here, we take
this qualitative statement for granted and leave more detailed analysis to
future work.
The revised Bullet Cluster constraint (31) leads to the following upper bound
on the thermal radius of the Milky Way
$\displaystyle\left(R_{T}\right)_{\text{Milky Way}}\lesssim 125\text{ kpc}$
(32)
Moreover, it is straightforward to see that the thermal radii of the target
and bullet clusters cannot exceed $\approx 0.5$ Mpc.
Therefore, even if the estimated inequality (31) is saturated, only inner
regions of the dark matter halo are capable of reaching equilibrium; with the
outskirts being unaffected by the presence of self-interactions.
## V V. Relative Size of thermalized region and coherence length
In this section, we focus on a relation between the Jeans scale $\ell$ and the
thermal radius $R_{T}$. Since we are interested in the superfluid regime of
the theory, let us begin by pointing out that for interaction-pressure
dominance (large $\xi$) $R_{T}$ and $\ell$ (given by (14)) are controlled by
different combinations of $m$ and $\sigma$. Therefore, it may seem possible to
pick the parameter values in such a way as to have kpc-size superfluid islands
while $R_{T}$ could vary from values lower than $\ell$ up to $R_{V}$. We will
demonstrate that the dark matter masses required for $\ell\geq R_{T}$ are so
low that one enters the Fuzzy Dark Matter parameter space. This is equivalent
to a transition from a superfluid whose degrees of freedom are collective
modes (phonons with a linear dispersion relation) to a condensate whose
dynamics is described by almost-free constituents (quadratic dispersion
relation).
It is straightforward to see that $\ell$, given by (14), can be expressed in
terms of $R_{T}$ using (23) as
$\left(\frac{\ell}{2\text{kpc}}\right)\simeq\left[\mathcal{F}\left({R_{T}}/{\text{kpc}}\right)\cdot{\left(\frac{m}{\text{meV}}\right)^{-1}}\right]^{1/4}\
,$ (33)
where $\mathcal{F}$ is determined by the density profile of a halo. The
explicit form of $\mathcal{F}$ may be easily deduced from (23). This equation
shows that for a given $R_{T}$ the Jeans scale depends only mildly on the mass
of the dark matter particle. Because of this, boosting $\ell$ to sufficiently
large values may require lowering $m$ so much that we may end up leaving the
interaction-pressure domination regime and enter the quantum pressure
dominance. In other words, $\ell$, $\xi$ and $R_{T}$ depend on three different
combinations of $m$ and $\sigma/m$. At the same time, the superfluidity
requires $\xi\gg 1$, which is not satisfied by a generic choice of $\ell$ and
$R_{T}$, due to the mild mass-dependence of (33).
Figure 1: Black lines show the slices of the [m,$\frac{\sigma}{m}$]-parameter
space that generate a Jeans scale of $0.1$ kpc/ $2$ kpc/ $6$ kpc in degenerate
regions of the Milky Way dark matter halo, assuming a particle species
governed by the Lagrangian (1). They are obtained evaluating (7) numerically.
The sloped part of the curve represents the superfluid regime of the theory
($\xi\gg 1$) while the flat part ($\xi\ll 1$) represents the region where
particles self-interact so little as to be considered a non-interacting
species. To evaluate $\ell$ we assumed $\rho\simeq 10^{-25}\text{g/cm}^{-3}$,
which is the expected average density of the thermal core before
fragmentation. This value influences only the $\xi\leq 1$ part of the Jeans
scale, which is favoured by small $R_{T}$. Blue/Orange/Red dashed curves
correspond to the parameter space that generates a thermal radius of 0.1 kpc/2
kpc/6 kpc, estimating $R_{T}$ as the radius within which the particles had the
chance to scatter at least once throughout the lifetime of the halo. On the
right of the gray line, the Milky Way dark matter halo is in global thermal
equilibrium. In order to stress that we do not really know what is the
sufficient number of scattering events for reaching equilibrium, coloured
regions show how $R_{T}$ changes varying the sufficient mean number of
scattering events in the interval 1-10. We see that it is impossible to have
an interaction pressure dominated core with $\ell\gtrsim R_{T}$.
Having made those observations, let us focus on narrowing down the parameter
space for which we can get the macroscopic superfluid patches within the
thermalization radius. For this we use the expression for the Jeans scale (7)
that covers both the superfluid and Fuzzy Dark Matter regimes. The result is
given by Fig.1 for a Milky Way-like halo, where different values of the Jeans
scale and the thermal radius are shown. Let us stress that, even if $\ell$ is
density independent in the interaction pressure case (in case of the
$\lambda\Phi^{4}$-potential), this is not true for $\xi\lesssim 1$. Thus,
while the sloped part of solid curves is the same in every astrophysical
structure, this is not true for the turning point and the flat part.
Now, Fig.1 highlights the impossibility of $\ell\gtrsim R_{T}$ for halos
sustained by the interaction pressure: lines corresponding to a given $\ell$
do not intersect lines describing thermal radii with a smaller relative
magnitude in the interaction pressure limit. Therefore, if the dark matter is
a scalar particle with a $\lambda\Phi^{4}$-potential which forms an
interaction-pressure-supported superfluid core, we have
$\ell<R_{T}\ .$ (34)
To show that (34) is not an artefact of our specific definition of the thermal
radius, coloured regions represent the parameter space that generates specific
values of the thermal radius when changing the sufficient number of scattering
events to equilibrate the dark matter halo in the interval 1-10.
## VI VI. Summary
Let us conclude by reiterating the qualitative tale of dark matter
superfluidity discussed in this work. As dark matter particles begin to clump
to form a halo, they start similarly to CDM. Up until the point when particles
begin to scatter from each other, they are striving towards an NFW density
profile. In regions where densities increase sufficiently for particles to
start experiencing collisions, the evolution of the density profile starts to
depart from its collisionless counterpart. In fact, the regions within which
each dark matter particle has had a chance to scatter few times should be
close to equilibrium. Although efficient interactions tend to homogenize the
density profile, the phase transition and the formation of superfluid droplets
may take place if the de Broglie wavelengths begin to overlap inside
thermalized regions. Both equilibration and the overlapping wave-functions
favour high densities and are hence easier to achieve within central regions.
Denoting the radii of the corresponding boundaries as $R_{T}$ and $R_{\rm
deg}$ respectively, we have demonstrated that a presence of $\sim$kpc-size
superfluid patches enforces the parameter space for which $R_{\rm deg}>R_{T}$.
Within $R_{T}$ the core breaks into superfluid islands of size $\ell$
(determined by the parameters of the model) due to the gravitational Jeans
instability. The dynamics of the droplets plays an important role to shape the
density distribution of the thermal core. As we have shown in the first
section, each island is a superfluid soliton (non-topological) with a
practically homogeneous core.777This is similar to the Bose-Einstein
condensation of bosonic cold-atoms with attractive contact interactions. As a
result, the homogeneous condensate is unstable and fragments into solitons.
For $\ell\ll R_{T}$, the course-grained (over scales larger than $\ell$)
density profile should resemble the one in CDM as superfluid solitons are
expected to behave as weakly interacting effective particles.
In Fig. 2 we combine the limits discussed in this work to understand what is
the parameter space of a dark matter candidate governed by the Lagrangian (1)
that could generate kpc-size solitons in partially thermalized clusters. The
pink region is excluded by the analysis of the collision of degenerate
clusters (leading to an upper limit of the self-interaction strength). For
completeness, let us stress that the results of our qualitative analysis are
sensitive to the pre-thermalization shape of the density profile and in
particular to the values for the parameters of the NFW distribution. Also, the
value we used to estimate the mean density of the thermal core would drop by
one order of magnitude if thermalization is strong enough to affect outer
regions of the galactic halo. However, this would neither change the behaviour
of the theory in the interaction pressure regime, which is density
independent, nor the conclusion on the fragmentation of the halo. In fact, the
transition point between degeneracy and interaction pressure highlighted by
Fig. 1 would move to the right, since stronger self-interactions are necessary
to compensate the smaller mean density of the thermal core.
Moreover, as we pointed out in Section 4, the bound (31) applies only if a
significant fraction of the bullet cluster halos is in the gaseous form. This
constraint would have been downgraded to (30) if both the bullet and the
target clusters were to thermalize completely and undergo the fragmentation
into superfluid solitons, without a significant leftover in the dispersed
phase. On the off chance that this could happen, we have highlighted the
region of the parameter space in Fig. 2 where both the target and bullet
cluster are expected to have thermalized in their entirety. In this case the
density profile of clusters would deviate from the usual NFW profile, and the
analysis of the profile and the substructure of clusters could lead to
constraints on this scenario. The detailed analysis of this scenario being
beyond the scope of this work, we have assumed such a fragmentation to result
in a significant portion of the cluster mass to have remained in the form of a
gaseous medium.
Figure 2: Parameter space for $\Phi$. We focussed on the superfluid regime and
we excluded halos sustained by the quantum pressure (green region). The pink
region is excluded by the bullet cluster constraint while the purple one
represents condensates which were relativistic at matter-radiation equality.
The striped region highlights the parameter space where both the bullet and
target cluster are in global thermal equilibrium. In the left panel, Black
solid lines identify the parameter space that generates a Jeans scale of 0.5
kpc/2 kpc/6 kpc. In the right panel, we plotted the whole parameter space,
highlighting $\ell=10^{-3}$ pc/ 1 pc/6 kpc for reference.
Finally, let us comment on the magnitude of the coupling constant $\lambda$.
Fig. 2 illustrates that the generation of solitons with a diameter
$\ell\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}0.5$ kpc is compatible with the revised bullet cluster bound
only for cross sections $\sigma/m\lesssim 10^{-63}\text{ cm}^{2}/$g and masses
$m\lesssim 10^{-15}$ eV. It is straightforward to verify that this means
considering $\lambda\lesssim 10^{-65}$: the smallness of the coupling constant
involved is a consequence of the extreme enhancement of the interaction rate
by the sub-eV mass of $\Phi$, both through $\mathcal{N}$ and through the mass
dependence of $\sigma/m$.
## Achnowledgements
We would like to thank Justin Khoury for valuable discussions and comments.
## Appendix: Dynamical time as the time-scale for thermalization
Let us show the differences emerging by using dynamical time $t_{\text{dyn}}$
to describe the time-scale in which dark matter thermalizes. Indeed, if
galactic dynamical effects are efficient enough to reshuffle the phase-space
of $\Phi$ significantly, the correct time-scale involved would be
$t_{\text{dyn}}=r/v$. Therefore, focussing on the degenerate case, particles
had the time to interact only if
$\Gamma t_{\text{dyn}}=\mathcal{N}\frac{\sigma}{m}\rho(R_{T})R_{T}=1\ .$ (35)
In this case, $\rho_{0}$, $R_{T}$ and $r_{s}$ enter in equation (35) through
the combination $\rho^{2}R_{T}/v^{3}$. Again, we may extract the following
limits:
$R_{T}\simeq
r_{s}\left(\sqrt{\frac{8\pi^{3}\rho_{0}}{G^{3}}}\frac{\sigma/m}{m^{4}r_{s}^{2}}\right)^{\delta}\,,\quad{\rm
with}\quad\delta=\begin{cases}2/5,&\mbox{for }R_{T}\ll r_{s}\\\ 2/7,&\mbox{for
}R_{T}\gg r_{s}\end{cases}$ (36)
For a Milky Way-like halo, the expression reduces to
$\displaystyle R_{\rm T,dyn}^{\rm MW}\simeq\begin{cases}30\cdot
r_{s}\left[\frac{\sigma/m}{{\rm cm}^{2}/{\rm g}}\left(\frac{m}{\rm
eV}\right)^{-4}\right]^{2/5},&\mbox{for }R_{T}\ll r_{s}\\\ 10\cdot
r_{s}\left[\frac{\sigma/m}{{\rm cm}^{2}/{\rm g}}\left(\frac{m}{\rm
eV}\right)^{-4}\right]^{2/7},&\mbox{for }R_{T}\gg r_{s}\end{cases}$ (37)
We may now compare (25) and (37):
$\displaystyle\frac{R_{\rm T,dyn}^{\rm MW}}{R_{\rm T,g}^{\rm
MW}}\simeq\begin{cases}1/2\left[\frac{\sigma/m}{{\rm cm}^{2}/{\rm
g}}\left(\frac{m}{\rm eV}\right)^{-4}\right]^{1/15},&\mbox{for }R_{T}\ll
r_{s}\\\ \left[\frac{\sigma/m}{{\rm cm}^{2}/{\rm g}}\left(\frac{m}{\rm
eV}\right)^{-4}\right]^{3/35},&\mbox{for }R_{T}\gg r_{s}\end{cases}$ (38)
One can see that $R_{T}$ gets reduced by a factor 2 at most (the $\sigma$ and
$m$ dependence is too mild to affect the ratio by an order one contribution).
In this case, the analogue of Fig. 2 would reveal a similar picture of what we
obtained using $t_{\text{g}}$.
Finally, we report the upper bound from the bullet cluster on the thermal
radius of the Milky Way using $t_{\text{dyn}}$:
$\displaystyle(R^{\text{dyn}}_{T})_{\text{Milky Way}}\lesssim 60\text{ kpc}.$
(39)
As expected, the bound on the thermal radius is tighter than (32) when we use
$t_{\text{dyn}}$ as the time scale for thermalization.
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|
# A short proof on the rate of convergence of the empirical measure for the
Wasserstein distance
Vincent Divol<EMAIL_ADDRESS>
###### Abstract
We provide a short proof that the Wasserstein distance between the empirical
measure of a $n$-sample and the estimated measure is of order $n^{-1/d}$, if
the measure has a lower and upper bounded density on the $d$-dimensional flat
torus.
[1]organization=Université Paris-Saclay and Inria Saclay, country=France
For $1\leq p<\infty$, let $W_{p}$ be the $p$-Wasserstein distance between
measures, defined for two probability measures $\mu,\nu$ with finite $p$th
moments supported on a metric space $(\Omega,\rho)$ by
$W_{p}(\mu,\nu)\vcentcolon=\inf_{\pi\in\Pi(\mu,\nu)}C_{p}(\pi)^{1/p},$ (1)
where $\Pi(\mu,\nu)$ is the set of transport plans between $\mu$ and $\nu$,
that is the set of probability measures on $\Omega\times\Omega$, with first
marginal $\mu$ and second marginal $\nu$, and
$C_{p}(\pi)=\iint\rho(x,y)^{p}\mathrm{d}\pi(x,y)$ is the cost of the plan
$\pi$. We define the distance $W_{\infty}$ by replacing the quantity
$C_{p}(\pi)^{1/p}$ by the $\pi$-essential supremum of $\rho$.
Let $\mu$ be a probability measure on some metric space $(\Omega,\rho)$, and
let $\mu_{n}$ be the empirical measure associated to a $n$-sample
$X_{1},\dots,X_{n}$ of law $\mu$. The question of studying rates of
convergence between $\mu$ and $\mu_{n}$ for Wasserstein distances $W_{p}$ has
attracted a lot of attention over recent years (see e.g. [5, 6]). If no bounds
on the density are assumed, then the quantity $\mathbb{E}W_{p}(\mu_{n},\mu)$
is known to be bounded by a quantity of order
$n^{-\frac{1}{2p}}+n^{-\frac{1}{d}}$ when $\Omega$ is a $d$-dimensional
domain, and this bound is tight (see e.g [5]). For $p=\infty$, Nicolás García
Trillos and Dejan Slepčev [6] have shown that
$\mathbb{E}W_{\infty}(\mu_{n},\mu)$ is of order $(\log n/n)^{1/d}$ (for $d\geq
3)$ in the case where $\mu$ has a density $f$ which is lower bounded and upper
bounded on some convex domain $\Omega$. As $W_{p}\leq W_{\infty}$, the same
rate also holds for any $1\leq p\leq\infty$. This exhibits the following
phenomenon: when $2p>d$, the problem of reconstructing $\mu$ for the
Wasserstein distance is strictly harder if no bounds on the underlying density
are assumed.
In this note, we propose to give a short proof of the fact that
$\mathbb{E}W_{p}(\mu_{n},\mu)\lesssim n^{-1/d}$ (for $d\geq 3$) for bounded
densities. We restrict to the case where $\Omega$ is the $d$-dimensional flat
torus $\Omega$ in order to avoid complications due to boundary effects. Let
$\mathcal{P}_{0}$ be the set of probability distributions on $\Omega$, having
a density $f$ satisfying $f_{\min}\leq f\leq f_{\max}$ for some $f_{\max}\geq
f_{\min}>0$.
###### Theorem.
Let $\mu\in\mathcal{P}_{0}$ and $1\leq p<\infty$. Then, there exists a
constant $C$ such that
$\mathbb{E}W_{p}(\mu_{n},\mu)\leq C\begin{cases}n^{-1/d}&\text{ if }d\geq
3,\\\ (\log n)^{1/2}n^{-1/2}&\text{ if }d=2,\\\ n^{-1/2}&\text{ if
}d=1.\end{cases}$ (2)
The standard approach for bounding the distance $W_{p}(\mu_{n},\mu)$ consists
in precisely assessing the masses given by the measures $\mu_{n}$ and $\mu$ on
dyadic partitions of the domain $\Omega$ (see e.g. [6]). We propose to take a
different route by relying on a result from [3] which asserts that the
Wasserstein distance is controlled by the pointed negative Sobolev distance
when comparing measures having lower bounded densities. The proof is then
completed by using tools from Fourier analysis.
We also note that minimax results from [7] (proven for measures on the cube)
can be straightforwardly adapted to the setting of the flat torus. In
particular, those results imply that the rates exhibited in the theorem are
optimal on the class $\mathcal{P}_{0}$ (up to a logarithmic factor for $d=2$).
## The proof
As $W_{p}\geq W_{q}$ if $p\geq q$, we may assume that $p\geq 2$. The proof of
the theorem is heavily based on the following result of optimal transport
theory, appearing in [3, 2]. Let $p^{*}$ be the conjugate exponent of $p$. For
$\phi\in L_{p}$ with $\int\phi=0$, introduce the pointed negative Sobolev norm
$\|\phi\|_{\dot{H}_{p}^{-1}}\vcentcolon=\sup\left\\{\int\phi\psi,\
\|\nabla\psi\|_{L_{p^{*}}}\leq 1\right\\},$ (3)
where the supremum is taken over all smooth functions $\psi$ defined on
$\Omega$.
###### Lemma 1.
Let $\mu,\nu$ be two measures on $\Omega$ having densities $f,g$. Assume that
$f\geq f_{\min}$. Then,
$W_{p}(\mu,\nu)\leq pf_{\min}^{1/p-1}\|f-g\|_{\dot{H}_{p}^{-1}}.$ (4)
Let $K$ be a smooth radial nonnegative function with $\int K=1$, supported on
the unit ball and, for $h>0$, let $K_{h}=h^{-d}K(\cdot/h)$. Let $\mu_{n,h}$ be
the measure having density $K_{h}*\mu_{n}$ on $\Omega$, i.e. the density at a
point $x\in\Omega$ is given by
$f_{n,h}(x)\vcentcolon=\sum_{j=1}^{n}K_{h}(x-X_{j})/n$.
###### Lemma 2.
We have $W_{p}(\mu_{n},\mu_{n,h})\leq C_{0}h$, where
$C_{0}=\left(\int|x|^{p}K(x)\mathrm{d}x\right)^{1/p}$.
###### Proof.
Consider the unique transport plan $\pi_{j}$ between $K_{h}*\delta_{X_{j}}$
and $\delta_{X_{j}}$. The cost of $\pi_{j}$ is equal to
$\int|x-X_{j}|^{p}K_{h}(x-X_{j})\mathrm{d}x=h^{p}\int|x|^{p}K(x)\mathrm{d}x$.
The measure $\frac{1}{n}\sum_{j=1}^{n}\pi_{j}$ is a transport plan between
$\mu_{n,h}$ and $\mu_{n}$, with associated cost equal to
$h^{p}\int|x|^{p}K(x)\mathrm{d}x$. ∎
By Lemmas 1 and 2,
$\begin{split}\mathbb{E}W_{p}(\mu_{n},\mu)&\leq\mathbb{E}W_{p}(\mu_{n},\mu_{n,h})+\mathbb{E}W_{p}(\mu_{n,h},\mu)\\\
&\leq
C_{0}h+pf_{\min}^{1/p-1}\mathbb{E}\|f_{n,h}-f\|_{\dot{H}_{p}^{-1}}.\end{split}$
(5)
To further bound this quantity, we use the following relation between the
negative Sobolev norm and the Fourier decomposition of a signal. Given
$\phi\in L_{p}$, we let $\hat{\phi}$ be the sequence of Fourier coefficients
of $\phi$ (indexed by $\mathbb{Z}^{d}$) and denote by ∨ the inverse Fourier
transform. Let $|x|\vcentcolon=\sum_{i=1}^{d}|x_{i}|$ for
$x\in\mathbb{R}^{d}$. A multiplier $s$ is a bounded sequence indexed by
$\mathbb{Z}^{d}$ such that the operator $\phi\in
L_{p}\mapsto(s\hat{\phi})^{\vee}\in L_{p}$ is bounded. A sufficient condition
for a sequence to be a multiplier is given by Mikhlin multiplier theorem [1,
Theorem 3.6.7, Theorem 5.2.7].
###### Lemma 3.
Let $s:\mathbb{R}^{d}\to\mathbb{R}$ be a smooth function such that
$|\partial^{\alpha}s(\xi)|\leq B|\xi|^{-|\alpha|}$ for every multiindex
$\alpha$ with $|\alpha|\leq d/2+1$. Then, the sequence
$(s(m))_{m\in\mathbb{Z}^{d}}$ is a multiplier with corresponding operator of
norm smaller than $C_{p,d}B$.
Let $a:\mathbb{R}^{d}\to\mathbb{R}$ be a smooth function with $a(\xi)=1/|\xi|$
for $|\xi|\geq 1$ and $a(0)=0$. Let $\mathcal{A}$ be the associated multiplier
operator (by Lemma 3) defined by $\mathcal{A}(\phi)=(a\hat{\phi})^{\vee}$.
###### Lemma 4.
Let $\phi\in L_{p}$ with $\int\phi=0$. Then, $\|\phi\|_{\dot{H}_{p}^{-1}}\leq
C_{5}\|\mathcal{A}(\phi)\|_{L_{p}}$.
###### Proof.
Let $\psi:\Omega\to\mathbb{R}$ be a smooth function with
$\|\nabla\psi\|_{L_{p^{*}}}\leq 1$. As $\hat{\phi}(0)=0$, we have
$\int\phi\psi=\sum_{m\in\mathbb{Z}^{d}}\hat{\phi}(m)\hat{\psi}(m)=\sum_{m\in\mathbb{Z}^{d}}a(m)\hat{\phi}(m)|m|\hat{\psi}(m)\leq\|\mathcal{A}(\phi)\|_{L_{p}}\|(|\cdot|\hat{\psi})^{\vee}\|_{L_{p^{*}}}.$
Note that $|\cdot|=\sum_{i=1}^{d}\varepsilon_{i}e_{i}$, where $e_{i}(m)=m_{i}$
and $\varepsilon_{i}(m)$ is the sign of $m_{i}$. As $\varepsilon_{i}$ is a
multiplier (by Lemma 3), we have
$\|(|\cdot|\hat{\psi})^{\vee}\|_{L_{p^{*}}}\leq
c\sum_{i=1}^{d}\|(e_{i}\hat{\psi})^{\vee}\|_{L_{p^{*}}}=c\sum_{i=1}^{d}\|\partial_{i}\psi\|_{L_{p^{*}}}\leq
C_{5}$. ∎
Hence, to conclude, it suffices to bound
$\mathbb{E}\|\mathcal{A}(f_{n,h}-f)\|_{L_{p}}\leq\|\mathcal{A}(f_{h}-f)\|_{L_{p}}+\mathbb{E}\|\mathcal{A}(f_{n,h}-f_{h})\|_{L_{p}}.$
### Bound of the bias
Let $\kappa$ be the Fourier transform of $K$. As $K$ is smooth and compactly
supported, $\kappa$ is a multiplier by Lemma 3. Also, the function
$M=a\cdot(\kappa-1)$ is a multiplier as a product of multiplier. Remark that
$\hat{f}_{h}-\hat{f}=(\kappa(h\hskip 1.42271pt\cdot\hskip
1.42271pt)-1)\hat{f}$, so that $\mathcal{A}(f_{h}-f)=h(M(h\hskip
1.42271pt\cdot\hskip 1.42271pt)\hat{f})^{\vee}$. As the multiplier norms of
$M$ and $M(h\hskip 1.42271pt\cdot\hskip 1.42271pt)$ are equal [1, Theorem
3.6.7], we have
$\|\mathcal{A}(f_{h}-f)\|_{L_{p}}\leq hC_{6}\|f\|_{L_{p}}\leq hC_{6}f_{\max}.$
(6)
### Bound of the fluctuations
Eventually, we bound
$\mathbb{E}\|\mathcal{A}(f_{n,h}-f_{h})\|_{L_{p}}\leq\mathbb{E}\left[\|\mathcal{A}(f_{n,h}-f_{h})\|_{L_{p}}^{p}\right]^{1/p}.$
(7)
The random variable $\mathcal{A}(f_{n,h})$ is equal to
$n^{-1}\sum_{j=1}^{n}U_{j}$, where
$U_{j}\vcentcolon=\mathcal{A}(K_{h}*\delta_{X_{j}})=\mathcal{A}(K_{h})(\hskip
1.42271pt\cdot-X_{j})$ and $\mathbb{E}U_{j}=\mathcal{A}(f_{h})$. We control
the expectation of the $L_{p}$-norm of the sum of i.i.d. centered functions
thanks to the next lemma, which is a direct consequence of Rosenthal
inequality [4].
###### Lemma 5.
Let $U_{1},\dots,U_{n}$ be i.i.d. functions on $L_{p}$. Then, the expectation
$\mathbb{E}\left\|\frac{1}{n}\sum_{i=1}^{n}(U_{i}-\mathbb{E}U_{i})\right\|_{L_{p}}^{p}$
is smaller than
$C_{p}n^{-p/2}\int\left(\mathbb{E}|U_{1}(x)|^{2}\right)^{p/2}\mathrm{d}x+C_{p}n^{1-p}\int\mathbb{E}\left[|U_{1}(x)|^{p}\right]\mathrm{d}x.$
(8)
Let $v_{h}$ be the sequence in $\ell_{p^{*}}(\mathbb{Z}^{d})$ defined by
$v_{h}(m)=a(m)\kappa(hm)$ for $m\in\mathbb{Z}^{d}$. By a change of variable,
we obtain
$\mathbb{E}\left[|U_{1}(x)|^{p}\right]=\int
f(y)|\mathcal{A}(K_{h})(x-y)|^{p}\mathrm{d}y\leq
f_{\max}\|\mathcal{A}(K_{h})\|_{L_{p}}^{p}\leq
f_{\max}\|v_{h}\|_{\ell_{p^{*}}}^{p},$ (9)
where, at the last line, we applied Hausdorff-Young inequality [8, Section
XII.2]. The last step consists in bounding $\|v_{h}\|_{\ell_{p^{*}}}^{p^{*}}$.
We separate this quantity into two parts: $S_{0}=\sum_{|hm|\leq
1}|v_{h}(m)|^{p^{*}}$ and $S_{1}=\sum_{|hm|>1}|v_{h}(m)|^{p^{*}}$. To bound
$S_{0}$, we use that $\kappa$ is bounded on the unit ball, so that $S_{0}$ is
of the order
$\sum_{|hm|\leq 1}|m|^{-p^{*}}\lesssim\begin{cases}h^{p^{*}-d}&\text{ if
}d\geq 3\text{ or }(d=2\text{ and }p>2)\\\ -\log h&\text{ if }p=d=2\\\
1&\text{ if }d=1.\end{cases}$ (10)
To bound $S_{1}$, we use that $|\kappa(hm)|\leq C_{\gamma}|hm|^{-\gamma}$ for
any $\gamma>0$. Choosing $\gamma$ such that $\gamma p^{*}+p^{*}>d$, we obtain
that $S_{1}$ is of the order
$h^{-\gamma p^{*}}\sum_{|hm|>1}|m|^{-\gamma p^{*}-p^{*}}\lesssim h^{p^{*}-d}.$
(11)
Putting together inequalities (8), (10) and (11) yields that, for $h$ of the
order $n^{-1/d}$, the expectation
$\mathbb{E}\|\mathcal{A}(f_{n,h}-f_{h})\|_{L_{p}}$ is of the order
$\begin{cases}h/\sqrt{nh^{d}}\lesssim n^{-1/d}&\text{ if }d\geq 3,\\\
\sqrt{(-\log h)/n}\lesssim(\log n)^{1/2}n^{-1/2}&\text{ if }d=2,\\\
n^{-1/2}&\text{ if }d=1.\end{cases}$ (12)
We conclude the proof by putting together the estimates (5), (6) and (12).
### Remark 1
For $p=2$, Mikhlin multiplier theorem can be replaced by Parseval’s theorem,
further simplifying the proof.
### Remark 2
A similar proof shows that the risk of the measure $\mu_{n,h}$ satisfies
$\mathbb{E}W_{p}(\mu_{n,h},\mu)\lesssim n^{-(s+1)/(2s+d)}$ if $f$ is assumed
to be of regularity $s$. Indeed, we can exploit the regularity of $s$ to show
that, if $\kappa$ has sufficiently many zero derivatives at $0$, then the bias
term is of order $h^{s+1}$, while the fluctuation terms is bounded in the same
way. We then obtain the desired rate by choosing $h$ of the order
$n^{-1/(2s+d)}$. This rate is in accordance with the minimax result of [7],
where a modified wavelet density estimator is shown to attain the same rate of
convergence.
## References
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* [2] Sloan Nietert, Ziv Goldfeld, and Kengo Kato. From smooth Wasserstein distance to dual Sobolev norm: Empirical approximation and statistical applications, 2021.
* [3] Rémi Peyre. Comparison between $W_{2}$ distance and $\dot{H}_{-1}$ norm, and localization of Wasserstein distance. ESAIM. Control, Optimisation and Calculus of Variations, 24(4), 2018\.
* [4] Haskell P Rosenthal. On the subspaces of $L_{p}$ ($p>2$) spanned by sequences of independent random variables. Israel Journal of Mathematics, 8(3):273–303, 1970.
* [5] Shashank Singh and Barnabás Póczos. Minimax distribution estimation in Wasserstein distance. arXiv preprint arXiv:1802.08855, 2018.
* [6] Nicolás Garcia Trillos and Dejan Slepčev. On the rate of convergence of empirical measures in $\infty$-transportation distance. Canadian Journal of Mathematics, 67(6):1358–1383, 2015.
* [7] Jonathan Weed and Quentin Berthet. Estimation of smooth densities in Wasserstein distance. In Conference on Learning Theory, pages 3118–3119, 2019.
* [8] A. Zygmund and R. Fefferman. Trigonometric Series. Cambridge Mathematical Library. Cambridge University Press, 2003.
|
Pub. in Lett. in High En. Phys., LHEP-199, 2021.
http://journals.andromedapublisher.com/index.php/LHEP/article/view/199
# 5 reasons to expect an 8 MeV line in the SN 1987A neutrino spectrum
Robert Ehrlich George Mason University, Fairfax, VA 22030<EMAIL_ADDRESS>
###### Abstract
Evidence was previously reported for an 8 MeV neutrino line associated with SN
1987A based on an analysis of 997 events recorded in the Kamiokande-II
detector on the day of the supernova. That claimed line, however, occurred at
the peak of the background spectrum, and both had a similar shape, making the
claim tenuous at best. Here the claim is buttressed by providing five reasons
to expect such an 8 MeV neutrino line. A final section of the paper concerns
the ongoing KATRIN experiment to find the neutrino mass, which might provide
additional support for the line, should it validate a controversial $3+3$
model of the neutrino masses, including a tachyonic ($m^{2}<0$) mass.
ArXiv/2101.08128
keywords: neutrino line, monochromatic neutrinos, SN 1987A, supernova, dark
matter, galactic center, KATRIN experiment
## I Introduction
Elsewhere evidence was presented for an 8 MeV neutrino line in the SN 1987A
spectrum – see appendix in ref. (Eh2018, ). The line had the right shape and
width, and its background was independently derived. One potentially fatal
problem, however, was that the line occurred at the peak of the background and
had a similar shape, making its existence uncertain. This paper buttresses the
claim by providing five reasons to expect such an 8 MeV line from SN 1987A. In
addition, it presents other new evidence that makes the case for the line
stronger, and discusses how potential contradictions can be satisfactorily
addressed. Finally, two specific tests are proposed for such a neutrino line.
One test involves searches for diffuse supernovae using a novel approach with
existing data, and the second involves the ongoing KATRIN experiment to
measure the electron neutrino mass.
The data in support of an 8 MeV neutrino line are the 997 events recorded on
the date of SN 1987A by the Kamiokande-II detector (Hi1988, ). Before
reviewing that data, let us first explain three of the five reasons to expect
such a supernova neutrino line. They all involve the possible existence of
cold dark matter particles of mass $m_{X}\sim 10MeV.$
## II 1st Reason: Galactic center $\gamma$-rays
The galactic center (GC) has been considered as a possible place for dark
matter (DM) annihilations to occur, in which case the $\gamma-$rays from the
GC could be the result of $XX\rightarrow e^{+}e^{-}$ followed by
$e^{+}e^{-}\rightarrow 2\gamma.$ We can, therefore, learn about the possible
presence of DM near the GC by examining the spectrum of $\gamma-$rays from
that source. There is also a direct connection between $XX$ annihilations and
neutrino lines. Thus, if there were evidence for cold dark matter particles of
mass $m_{X},$ their annihilation $XX\rightarrow\nu\bar{\nu},$ would give rise
to be nearly monochromatic $\nu,\bar{\nu}$ having $E=m_{X}$.
Fig. 1 shows the predicted enhancement above background for the GC $\gamma-$
ray spectrum due to dark matter annihilation. The four curves correspond to
different $m_{X}$ values. These curves are found by assuming that the $e^{+}$
are created in $XX\rightarrow e^{+}e^{-}$ with an initial energy
$E_{0}=m_{X}.$ Of those $e^{+},$ we assume $97\%$ will annihilate at rest
yielding the 511 keV line, while the remaining $3\%$ propagate in a neutral
medium before annihilating in flight (Je2006, ). Also shown in the figure is
the GC $\gamma-$ ray flux data from four instruments. Note that most of the
data and three of the four enhancement curves previously appeared in refs.
(Si2006, ; Pr2011, ), but the author has added the 8.3 MeV enhancement curve
and the 7 OSSE points from Ref. (Ki2001, ).
The data in Fig. 1 can be seen to be consistent with $m_{X}=10$ MeV (black
curve), with $\chi^{2}=7.3,$ $p=89\%,dof=13.$ In contrast, the fit to the null
hypothesis, i.e., the dashed line power law, is completely unacceptable:
$\chi^{2}=960.$ Acceptable fits to the data in Fig. 1 can only be found for
the range: $m_{X}=10^{+5}_{-1.7}$ MeV. Thus, the null hypothesis is excluded
by $N=10/1.7\sim 6$ standard deviations. This rejection of the null hypothesis
is in marked contrast to the conclusion in refs. (Si2006, ; Pr2011, ), which
failed to include the OSSE data. The key role of the OSSE data here arises
from their very small error bars, which is discussed in section 3.1.3 of ref.
(Eh2018, ).
We now consider other data that strengthens the case for $m_{X}\sim 10MeV,$
and hence a neutrino line with this energy. Incidentally, an $m_{X}\sim 10MeV$
for a DM particle is just within BBN and CMB cosmological constraints, which
excludes a thermal dark matter particle with a mass $m_{X}<7–10MeV.$ (De2019,
).
Figure 1: Flux, i.e., $E\times\frac{dF}{dE}(cm^{-2}s^{-1})$ versus energy for
$\gamma-$rays from the inner galaxy, as measured by: SPI(open circle), COMPTEL
(open squares), EGRET (filled circles), and OSSE (filled triangles). All but
the OSSE data (from Ref.(Ki2001, )) are from ref. (Si2006, ). The computed
enhancements above the straight line are for positrons injected into a neutral
medium at initial energies $E_{0}=m_{X}=5,8.3,10,50$ MeV displayed
respectively as: lower grey, dotted, black, and upper grey. The sloped line is
a power law (index 1.55) fit to data at high and low energies.
## III 2nd reason: the $Z^{\prime}$ Boson
If cold dark matter $X$ particles having $m_{X}\sim 10MeV$ exist and their
annihilation yields monochromatic $e^{+}e^{-}$ pairs as suggested in the
previous section, it is reasonable to suppose that the reaction proceeds via
some mediator particle Z’ as in $XX\rightarrow Z^{\prime}\rightarrow
e^{+}e^{-},$ whose mass is $m_{Z^{\prime}}=2m_{X},$ by energy conservation.
The natural place to look for such a Z’ would be in a nuclear physics
experiment, where the decay of some nuclear excited state $N^{*}$ produced
$e^{+}e^{-}$ pairs via $N^{*}\rightarrow N+Z^{\prime}\rightarrow
N+e^{+}e^{-}.$ Of course, most of the time when $e^{+}e^{-}$ pairs are
observed it would be when the mediator particle is a photon, so the existence
of such a Z’ would be revealed by an enhancement to that reaction, i.e, an
excess of $e^{+}e^{-}$ pairs having a specific opening angle, corresponding to
$m_{Z^{\prime}}.$ In 2016 exactly such an enhancement was reported by
Krasznahorkay et al. (the Atomki group) for $e^{+}e^{-}$ emissions in the
reaction ${}^{7}Li(p,\gamma)Be^{8}.$ (Kr2016, ) Their result implied an
intermediate short-lived Z’ particle (sometimes called X17) with mass
$m=16.7\pm 0.6$ MeV appearing in the two step decay process of the excited
${}^{8}Be,$ i.e.: ${}^{8}Be^{*}\rightarrow^{8}BeZ^{\prime},$ followed by
$Z^{\prime}\rightarrow e^{+}e^{-}.$
In 2020 the Atomki group has reported the same anomaly in the decay of excited
helium atoms in the reaction ${}^{3}H(p,\gamma)^{4}He$ as they earlier
observed in ${}^{8}Be$. (Kr2020, ). A particle having a 16.7 MeV mass would
also be expected to be found in some accelerator experiments. However, the
NA64 experiment (and others) at the CERN SPS have not observed it. (De2020, ).
On the other hand, these negative results do not contradict those in refs.
(Kr2016, ; Kr2020, ) because they were not sensitive to a small range of
particle lifetimes consistent with that reported in ref. (Kr2016, ) – see Fig.
1 in ref. (De2020, ).
Various discrepancies from standard model predictions also support the Z’
interpretation of the Atomki anomaly. Thus, ref. (Ki2020, ) explains how the
$(g-2)_{\mu}$ anomaly $(3.7\sigma)$can be explained based on an extension of
the Standard Model, including a light Z’ boson as observed by the Atomki
group. In addition, the neutron lifetime puzzle can be explained by assuming a
virtual $Z^{\prime}$ exchange into a neutrino and its Kaluza-Klein
sibling.(Du2020, ) Finally, such a particle could account for the $\sim
2-3\sigma$ deviations from standard model predictions seen in the leptonic
decays of the $\pi$ (Al2020, ) and $B$ (Gr2019, ) mesons.
However, the $Z^{\prime}$ boson interpretation of the Atomki anomaly has also
been challenged. For example, Aleksejevs etal. (Al2021, ) and Koch (Ko2020, ),
claim that the observations can be explained within the standard model by (1)
adding the full set of second-order corrections and the interference terms to
the Born-level decay amplitudes, and (2) accounting for detector and analysis
bias. Similarly, Zhang and Miller also provide an alternate standard model
explanation of the Atomki anomaly. (Zh2021, ). However, refs. (Al2021, ;
Ko2020, ; Zh2021, ) only apply to the ${}^{8}B$ data and they did not consider
the similar anomaly seen in the helium case. Furthermore, the analysis in ref.
(Zh2021, ) only rules out a new $Z^{\prime}$ _vector_ boson as the explanation
of the Atomki anomaly, not a scalar.
If the anomalies observed in refs. (Kr2016, ; Kr2020, ) really are due to a
new $Z^{\prime}$ boson, this particle would be the mediator of a fifth force.
(Fe2016, ) Moreover, as already noted, for cold DM $X$ particles, the end
product of $XX\rightarrow Z^{\prime}\rightarrow\nu\bar{\nu}$ (which is the
only other Z’ decay mode according to ref. (Ch2016, )) would be nearly
monochromatic $\nu$ and $\bar{\nu}$ pairs having $E_{\nu}=8.4\pm 0.3$ MeV.
## IV 3rd reason: DM in supernovae
Many researchers have suggested that dark matter might collect in the core of
some stars. (Ba2008, ) DM annihilation triggering a supernova is plausible
because without such an “extra” energy, shock wave stalling has been a
difficulty with most supernova models, the best of which have elements that
can still only be understood in qualitative terms. (Ja2017b, ) It may be true
that as of 2020 a self-consistent 3D simulation with detailed neutrino
transport has finally achieved a neutrino-driven explosion with properties
similar to SN 1987A without dark matter. (Bo2020, ) However, even if DM may
not be required to trigger a neutrino-driven explosion, the presence of large
amounts of DM in the stellar core could still play a role in the explosion and
be the source of significant long-lasting monochromatic neutrino emissions. In
fact, Fayet et al. (Fa2006, ) have shown that $m_{X}\sim 1-30MeV$ dark matter
particles can play a significant role in core-collapse supernovae. They also
note that if the DM particles have relatively large annihilation and
scattering cross sections, and have $m_{X}<10MeV,$ the DM would cool on a time
scale perhaps $>100$ times that in the standard scenario, (Fa2006, ) as would
be implied by an analysis of the Kamiokande data now described.
## V Analysis of Kamiokande data
The largest of the four detectors operating at the time of the SN 1987A
observation was Kamiokande-II. (Hi1988, ) This detector recorded neutrino
arrival times and their energies, which could be deduced from the “visible”
energies, $E_{vis}$ based on $E_{\nu}=E_{vis}+1.3$ MeV, assuming the dominant
reaction to be $\bar{\nu}_{e}+p\rightarrow n+e^{+}.$ In addition to observing
the main 12-event burst, Kamiokande-II also recorded 997 events occurring
during eight 17-min long intervals during several hours before and after the
burst. Figs. 4 (a)-(h) of Ref. (Hi1988, ) show scatter plots for each event
displaying the number of “hits,” $N_{hit},$ (PMT’s activated) versus the event
occurrence time, $t,$ during that $\Delta t=8\times 17\rm{min}=0.094$ day time
interval.
Those eight plots of $N_{hit}$ vs $t$ were digitized by the author who then
counted the number of times various $N_{hit}$ values occurred. The $N_{hit}$
frequency distribution is shown in Fig. 2. Note that the $N_{hit}$ values are
found to be proportional to the visible energy, i.e., $E_{vis}=cN_{hit}MeV$
with $c=0.363\pm 5\%,$ as shown in Fig. 4(a) of ref. (Eh2018, ). Thus, Fig. 2
is actually a spectrum for the events observed over several hours, with the
peak at 17 hits corresponding to $E_{\nu}=7.5\pm 0.5MeV$.
Figure 2: Kamiokande II neutrino data on Feb. 24, 1987. Histogram of $N_{hit}$
values for 997 events in Fig. 4 in ref. (Hi1988, ) The solid and dashed curves
are two versions of the background for the detector. The dashed one was
extracted from published data on a search for ${}^{8}B$ solar neutrinos – see
Appendix A in ref. (Eh2018, ) for details. The solid background curve is based
on a fit to the data using a Gaussian for the ten data points with
$N_{hit}<14$ and $N_{hit}>22.$ The value $N_{hit}=17$ corresponds to $7.5\pm
0.5$ MeV.
### V.1 Finding the background spectrum
Clearly, any claim of a peak above background in Fig. 2 depends critically on
how the background is determined. If the background is correctly represented
here the peak would have very high statistical significance $(\sim 30\sigma),$
but no such claim can be made here given the strong similarity between the
shape of the signal and background. Two versions of the background are
depicted in Fig. 2 – one dashed and on solid. The dashed background has been
found based on a 1989 publication by the Kamiokande-II Collaboration (K-II) on
a search for solar neutrinos from the reaction
${{}^{8}}B\rightarrow{{}^{8}Be^{*}}+e^{+}+\nu_{e}.$ (Hi1989, ), as explained
in Appendix A of ref. (Eh2018, ). The beginning of the 450 day data-taking
period preceded the date of SN 1987A, but most of it was many months
afterwards. The excess counts above the dashed background in Fig. 2 can be
well-fit by a Gaussian curve centered on $E_{max}=7.5\pm 0.4$ MeV, with a
width consistent with the expected $25\%$ energy resolution based on $\Delta
E/E=22\%/\sqrt{E/10},$ (Hi1988, ) consistent with what would be expected for
an 8 MeV neutrino line.
The most obvious flaw with using the data from the ${}^{8}B$ neutrino search
for the background on the day of SN 1987A would be if the background count
rate in the detector were significantly time dependent. It could be risky to
assume a constant background over time given various improvements made to the
detector. (Hi1991, ) Therefore, the solid background curve in Fig. 2 has been
found using the data themselves by the procedure outlined in the caption. Of
course, normally, when this technique is used to find evidence for a signal
above an unknown background, the signal and background curves have very
different shapes, and the evidence for the spectral line’s existence would be
much more certain than in the present case.
## VI Possible contradictions
Here we discuss three possible contradictions to there being an 8 MeV neutrino
line from SN 1987A, and show how they can be addressed.
### VI.1 Lack of time variation in line prominence
If the 8 MeV line is real, one might have expected that of the eight 17-min
time intervals for which ref. (Hi1988, ) provided data, those time intervals
closest to the 12-event burst would show a greater excess above background,
whereas no such variation is found in the data. Unfortunately, there is no
surviving record of the background on the days preceding and following the
supernova. The lack of any such time variation in the strength of the line is
worrisome, but it would be consistent with the long-lasting luminosity
expected for neutrinos emitted via light dark matter annihilation. (Fa2006, )
If the DM annihilation in supernovae were strongly coupled to neutrinos, with
neutrinos and light DM particles decoupling at $<\sim 3.3MeV$ rather than the
usual 8 MeV for neutrinos, the supernova cooling time scale would be larger by
a factor of $\sim 20$ than in the standard scenario. (Fa2006, ) Furthermore,
in the interior of the interior of the proto-neutron star, the cooling time
scale would be a factor $\geq 100$ larger (Fa2006, ). Such long cooling times
would be expected to be matched by long heating times for the dark matter, so
it would not be surprising to find that the emissions from DM annihilation are
not simply confined to times after the main 12 event burst. The neutrino
emission of massive stars before a supernova has been studied by many authors
and they all agree on significant, potentially detectable neutrino
luminosities, see the review in Kato et al. (Ka2017, )
### VI.2 Non-observation in diffuse supernova searches
The diffuse supernova neutrino background is a theoretical population of
neutrinos cumulatively originating from all of the supernovae events which
have occurred throughout the Universe for which only upper limits currently
exist. Given the large number of excess events above background comprising the
8 MeV neutrino line, one might expect it would have been detected in previous
searches for diffuse supernovae. However, those searches either focused on
neutrinos having energies well above 8 MeV (SK2012, ; Ah2020, ) or
alternatively neutrinos emitted in seconds-long bursts. (No2017, ) Thus,
despite those negative results, it might be possible to see such emissions
even with today’s neutrino detectors, especially those which have operated
over many years, and have a low energy threshold. In doing such a search one
might look for a significant excess of counts in any day-long time interval
for events in an energy band centered on $E_{\nu}=8MeV.$ One day would be
about the optimum size search window to use, because if the duration of the
monochromatic emissions was much shorter than that, a time variation in the
strength of the 8 MeV neutrino line would have been seen in the Kamiokande-II
data, and if the monochromatic emissions lasted much longer than a day, the
precursor star to SN 1987A would have to contain an impossible amount of dark
matter – see next section.
### VI.3 Impossible number of 8 MeV neutrinos
A third problematic aspect of the claimed 8 MeV neutrino line from SN 1987A is
the sheer magnitude of the excess events in Fig. 2 being $\sim 700$ above
background. Recall that these data were from eight 17 minute long time
intervals, i.e., roughly two hours duration chosen at random from a total time
of ten hours. Therefore, it is reasonable that given the lack of time
variation seen over those ten hours, the total time during which monochromatic
neutrinos were emitted by SN 1987A would need to be at least 12 hours
duration. In that case, one would estimate the number of monochromatic
neutrinos in those 12 hours to be $700\times 12/2\sim 4,000,$ as compared to
12 events seen in the main burst. In other words if we take the usual estimate
of $10^{58}$ neutrinos emitted from SN 1987A, the number associated with DM
annihilation would be $\sim 400$ times more or $4\times 10^{60}$. Let us now
see if such a value is even remotely possible.
The star giving rise to SN 1987A was estimated to have 18 solar masses. Let us
assume it consisted of half DM, and that, as was assumed earlier, the DM X
particles, had a mass of 8 MeV, yielding in total $\sim 10^{61}$ $X$ particles
in the star. We, therefore, find that at least a half the DM in the star would
need to annihilate producing 8 MeV neutrinos to match the number $\sim 4\times
10^{60},$ of emitted monochromatic in an emission duration of half a day.
Thus, under our assumptions, finding as many as $\sim 700$ neutrinos
constituting an observed 8 MeV spectral line in the Kamiokande II data is not
impossible. Lending further credence to the notion of such a large
monochromatic neutrino emission, Brito et al. (Br2015, ) have shown that
stellar cores do not necessarily collapse when they grow beyond their
Chandrasekhar limit, and that large fractions of DM in stellar cores are not
ruled out. (Br2016, )
## VII 4th Reason: Mont Blanc Burst
There are two more pieces of evidence in support of an 8 MeV neutrino line
from SN 1987A in addition to the three previously noted. First, the
controversial Mont Blanc neutrino burst on the day of SN 1987A has been
disregarded by most physicists with some exceptions (Gi1999, ; St1987, ;
Vo1987, ), owing to its 5 hour early arrival and its absence in the Kamiokande
II detector. Unlike the bursts seen in the other three detectors then
operating which all had much higher energies, the five Mont Blanc neutrinos
all have energies strangely consistent with the value 8 MeV within
uncertainties, (Aga1988, ; Agb1987, ) in which case they could be the result
of the 8 MeV neutrino line. The absence of such a 5 hour early burst in the
Kamiokande II data can be explained by the higher energy threshold of that
detector ($20\%$ efficient at $E_{\nu}=8MeV$), (Hi1988, ) and the
synchronization of the two detectors being no better than $\sim\pm 1$ minute.
Aglietta et al. gives further reasons why the Kamiokande II detector would
have probably seen only 1.5 neutrino events at the time of the Mont Blanc
early burst. (Agc1987, ).
Figure 3: Computed difference between the integral beta spectra for the $3+3$
model of the neutrino masses and the standard model of a small single
effective mass. For the $3+3$ model three values of the contribution for the
4.0 eV mass are shown: $92\%,94\%$ and $96\%.$ Only for the $94\%$ case are
the two integral spectra everywhere within $\pm 0.8\%$ of each other. The
graph shows the difference in the two integral spectra, because the integral
spectrum is what KATRIN measures.
## VIII 5th Reason: $3+3$ model and KATRIN
A final reason to expect there to be an 8 MeV neutrino line from SN 1987A is
provided by the author’s exotic $3+3$ neutrino model. The model postulates
that the neutrino flavor states are each comprised of three active-sterile
doublets having these masses: $m_{1}=4.0\pm 0.5eV,$ $m_{2}=21.4\pm 1.2eV,$ and
$m_{3}^{2}\sim-0.2keV^{2}.$ (Eh2013, ) Although the model is highly exotic,
and involves a “tachyonic” mass state, it has satisfied a number of empirical
tests. (Eh2019a, ) This controversial model requires any 5-hour early neutrino
burst like that observed in the Mont Blanc detector to be monochromatic, with
an energy roughly 8 MeV, something the author initially believed to be
“inconceivable.” (Eh2013, )
The $3+3$ model can be tested in the KATRIN experiment, which is measuring the
electron neutrino effective mass based on fitting the shape of the beta decay
integral spectrum of tritium near its endpoint, $E_{0}.$ KATRIN’s first
results published in 2019 (Ak2019, ) have been shown to yield a slightly
better fit to the model than to a single small effective mass with
$m_{\nu}<1eV.$ (Eh2019, ) However, a good fit to the model only is found for a
narrow range of contributions from the 4.0 eV mass, specifically for
$\alpha=|U_{i,1}|^{2}=0.94\pm 0.02.$(Eh2019, ) The need for a narrow range in
$\alpha$ is illustrated in Fig. 3, which shows the computed difference in the
integral spectra for the two models. To find those curves, the differential
spectrum for each mass $m_{i}$ is first found from the square of the Kurie
function:
$K_{i}^{2}(E)=(E_{0}-E)\sqrt{R[(E_{0}-E)^{2}-m_{i}^{2}]}$ (1)
where $R(x)$ is the ramp function ($R(x)=x$ for $x>0$ and $R(x)=0$ otherwise).
The respective integral spectra for a given mass is then found by a numerical
integration of Eq. 1 from $E$ to $E_{0},$ and the spectra for both signs of
$m^{2}$ are assumed to vanish for $E-E_{0}>0.$
A consequence of Fig. 3 is that if the KATRIN data are well described by the
$3+3$ model, but they are fit to a single small effective mass spectrum, then
one would expect to see two “bumps,” that is, excess numbers of events
($O(0.5\%)$) at a distance from $E_{0}$ close to the two $m^{2}>0$ masses in
the model. The initial KATRIN results from 2019 were such that the statistical
error bars were not small enough to clearly show whether these excesses were
present, although the residuals for their best fit did show an indication they
were. In fact, the ninth residual in their fit located at $E-E_{0}=-23eV$,
which was consistent with the position of the left peak in Fig. 3, fell
$+2.5\sigma$ above their best four free parameter fit to a single effective
mass.(Eh2019, ). However, the most recent KATRIN experiment fit to the
spectrum released in May 2021 has used not four free parameters to fit the
data, but rather 37. (Ak2021, ). One of those parameters is the value of
$m^{2}$ and the other 36 are the signal amplitude, $S_{i}$, background
amplitude, $B_{i},$ and endpoint energy, $E_{0,i},$ for the data recorded in
each of 12 concentric rings on the detector. While the assumption of a radial
dependence of signal and background is reasonable in one sense, it was not
done previously. Moreover, the use of so many free parameters can mask the
$O(0.5\%)$ departures from the single mass spectrum that would occur if the
$3+3$ model were a valid description of the data – see Fig. 3. In fact, the
number of adjustable parameters KATRIN uses in its fits is even greater than
37 if one counts some unspecified number of “pull terms” corresponding to
constrained parameters that can vary within limits.
The mathematician John Von Neumann famously once said: “With four parameters I
can fit an elephant, and with five I can make him wiggle his trunk.” (Dy2004,
) This quote was his humorous way of telling us to be suspicious of using too
many free parameters in doing a fit. Despite Von Neumann’s caution, the use of
four parameters by KATRIN to fit its initial data was exactly appropriate, but
37-plus parameters it now uses would seem to be excessive. Whether the
departures from the single effective mass spectrum indicated in Fig. 3 are in
fact present in the KATRIN data remains uncertain at present, and will
probably remain so, unless the number of free and partly free parameters used
to fit the data is significantly reduced. If four parameters are enough to fit
an elephant, 37-plus are enough to fit (or here hide) a whole herd of
elephants, all with their trunks wiggling.
## Conflicts of interest
I have no competing conflicts of interest.
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|
# Machine learning for rapid discovery of laminar flow channel wall
modifications that enhance heat transfer
Matthias Schniewind Institute of Theoretical Informatics, Karlsruhe Institute
of Technology, Karlsruhe, Germany<EMAIL_ADDRESS>Alexander Stroh
Institute of Fluid Mechanics, Karlsruhe Institute of Technology, Karlsruhe,
Germany<EMAIL_ADDRESS>Bradley P. Ladewig Institute for Micro
Process Engineering, Karlsruhe Institute of Technology, Karlsruhe, Germany
<EMAIL_ADDRESS>Pascal Friederich Institute of Theoretical
Informatics, Karlsruhe Institute of Technology, Karlsruhe, Germany Institute
of Nanotechnology, Karlsruhe Institute of Technology, Karlsruhe, Germany
<EMAIL_ADDRESS>
###### Abstract
The calculation of heat transfer in fluid flow in simple flat channels is a
relatively easy task for various simulations methods. However, once the
channel geometry becomes more complex, numerical simulations become a
bottleneck in optimizing wall geometries. We present a combination of accurate
numerical simulations of arbitrary, non-flat channels and machine learning
models predicting drag coefficient and Stanton number. We show that
convolutional neural networks can accurately predict the target properties at
a fraction of the time of numerical simulations. We use the CNN models in a
virtual high-throughput screening approach to explore a large number of
possible, randomly generated wall architectures. We find that S-shaped channel
geometries are Pareto-optimal, a result which seems intuitive, but was not
obvious before analysing the data. The general approach is not only applicable
to simple flow setups as presented here, but can be extended to more complex
tasks, such as multiphase or even reactive unit operations in chemical
engineering.
1.75pt 28
*
## 1 Introduction
Heat transfer in fluid flow is an important physical phenomenon, with
relevance across all areas of science and engineering ranging from
microfluidic devices in chemical engineering and biomedical implants, all the
way to high temperature physics and cosmology. In this proof-of-concept study
we explore an interesting engineering question, which could be posed as ”is it
possible to introduce structural changes to the wall of a pipe that increase
heat transfer, without a corresponding increase in the pressure drop?”. This
fundamental question linked to the ultimate goal of dissimilar flow control or
dissimilar heat transfer enhancement has been asked for decades by various
research groups in different application fields.
Dissimilar heat transfer enhancement is proven to be extremely challenging due
to similarity in the mechanisms of momentum and heat transfer [3].
Investigations of various surfaces including specially designed fins [17, 14],
dimples [7] or vortex generators [8] report that an increase in heat transfer
(described by Stanton number $St$) is always accompanied by inevitable
manifold increase in the drag coefficient $C_{f}$, which eventually results in
a decrease of the Reynolds analogy factor $RA=2St/C_{f}$ [21] in comparison to
a flat wall configuration. It is, however, known that a dissimilar
modification of momentum and heat transfer is possible, when more
sophisticated flow control methods are applied. Those control methods are
based on introduction of flow perturbations or optimally distributed
blowing/suction profiles from the wall surface [12, 11, 19, 13]. These studies
confirm, that a significant enhancement of Reynolds analogy factor (tripling
$RA$ in comparison to the uncontrolled channel flow) is possible when an
appropriate flow manipulation is created. It is found that an introduction of
large-scale spanwise rolls significantly promotes the heat transfer while the
drag coefficient remains less affected. This concept has also been
successfully tested in the framework of turbulent channel flows, where $RA>2$
can be achieved instead of $RA=1$ in an uncontrolled flow configuration [10,
23]. Recent studies in turbulent flows also report a possibility of $RA$
modification using streamwise elongated structures leading to formation of
turbulence-driven secondary motions [22]. The modification of $RA$ is however
limited in this case to several percent due to the increase of the wetted area
and corresponding increase in $C_{f}$.
To simplify the scenario, we consider a two-dimensional channel with laminar
flow, heat transfer and immersed boundary method for introduction of surface
structuring. This allows for the use of a quickly executed direct numerical
simulation (DNS), where a large set of arbitrary generated surfaces can be
investigated. In this proof-of-principle study, we present a workflow
consisting of numerical simulations (Section 2.1 and Section 2.2), generation
of a dataset of simulation data, and training of machine learning (ML) models
(Section 2.3). We show that the ML model can predict fluid flow and heat
transfer characteristics with a large speedup compared to numerical
simulations and with an accuracy which is high enough to screen a large
database of possible channel geometries to find Pareto optimal structures
(Section 3).
## 2 Methods
### 2.1 Numerical procedure
white$u$white$\theta$$\delta_{l}$$\delta_{u}$$L_{x}=10\delta$$x$$y$$h_{l}$$h_{u}$centerline$L_{y}=2\delta$white$\theta_{u}=1$white$\theta_{l}=0$
Figure 1: Laminar channel flow with imposed wall structuring.
For the problem setup we consider a laminar channel flow with arbitrary wall
structuring. The coordinate system of the numerical domain and its geometry
($L_{x}\times L_{y}=10\delta\times 2\delta$ with $\delta$ being the half
channel height) are illustrated in Figure 1, where ($x$, $y$) = ($x_{1}$,
$x_{2}$) correspond to the streamwise and wall-normal directions respectively.
The velocity components in the three directions are denoted by ($u$, $v$) =
($u_{1}$, $u_{2}$). The analysis is carried out using flow and temperature
fields produced by a direct numerical simulation (DNS) in a channel flow
driven at constant flow rate (CFR). Assuming an incompressible flow, the
velocity field is required to satisfy continuity:
${\frac{\partial u_{i}}{\partial x_{i}}=0,}$ (1)
and the Navier-Stokes equations for a constant property Newtonian fluid:
$\frac{\partial u_{i}}{\partial t}+\frac{\partial u_{i}u_{j}}{\partial
x_{j}}=\frac{1}{\rho}P_{x}\delta_{i1}-\frac{1}{\rho}\frac{\partial p}{\partial
x_{i}}+\nu\frac{\partial^{2}u_{i}}{\partial{x_{j}}\partial{x_{j}}}+F_{\textrm{IBM},i},$
(2)
where $p$ is the fluctuating pressure part, $\rho$ is density, $\nu$ is the
kinematic viscosity and $F_{{IBM},i}$ represents the external volume force per
unit mass required for the immersed boundary method (IBM) with which the
structured surface is introduced into the flow domain [9]. In the present
configuration $F_{\textrm{IBM},i}$ corresponds to the frictional drag between
the flow and the part of the surface reproduced by the immersed boundary
method, i.e. the elevated surface. $P_{x}$ is the absolute value of the mean
streamwise pressure gradient added to the equation in order to drive the flow
through the channel.
Due to the CFR approach the bulk Reynolds number is fixed to
$\mathrm{Re}_{b}=2U_{b}\delta/\nu=200$ for all considered simulations, which
means that any modification of the flow is translated into an alteration of
the resulting mean streamwise pressure gradient $P_{x}$ required to maintain
the chosen flow rate. Periodic boundary conditions are applied in the
streamwise directions while the wall-normal extension of the flow domain is
bounded by no-slip boundary conditions at the lower and upper domain wall
($y=0,2\delta$). Subscript $l$ and $u$ are used throughout the manuscript to
denote quantities on lower and upper wall, respectively.
Temperature $T$ is treated as passive scalar:
$\frac{\partial T}{\partial t}+\frac{\partial u_{j}T}{\partial
x_{j}}=\alpha\frac{\partial^{2}T}{\partial{x_{j}}^{2}}+Q_{\textrm{IBM}}\mbox{,}$
(3)
where $\alpha$ denotes the thermal diffusivity. Periodic boundary conditions
are applied for the thermal field in $x$-direction, while constant
temperatures on lower and upper wall of the flow domain are prescribed. The
non-dimensionalized temperature is defined as $\theta=(T-T_{l})/\Delta T_{w}$
with $\Delta T_{w}=T_{u}-T_{l}$, such that $\theta_{l}=0$ and $\theta_{u}=1$.
The Prandtl number is chosen to be $\mathrm{Pr}=\nu/\alpha=1$.
$Q_{\textrm{IBM}}$ is proportional to the heat transfer rate between the flow
and the ridges and can be considered as a counterpart to the volume force
$F_{\textrm{IBM},i}$ in the momentum equation. This term is adjusted to
fulfill the temperature boundary condition on the elevated ridges. Due to use
of periodic boundary condition for temperature, the absolute value of heat
transfer rate on the two walls should be identical once the solution reaches
the thermal equilibrium. For the same reason, the mean heat flux in wall-
normal direction is constant in the channel. The present thermal boundary
condition is chosen following other studies of heat transfer above structured
walls [16, 18, 20].
The solver implementation is based on a spectral solver for incompressible
boundary layer flows [5]. The Navier-Stokes equations are numerically
integrated using the velocity-vorticity formulation by a spectral method with
Fourier decomposition in the horizontal directions and Chebyshev
discretization in the wall-normal direction. For temporal advancement, the
convection and viscous terms are discretized using the third-order Runge-Kutta
and Crank-Nicolson methods, respectively. The flow domain is discretized with
$N_{x}\times N_{y}=256\times 129$ grid nodes, while the immersed boundary
method is applied on the dealiased grid (3/2 rule) with $384\times 129$ grid
nodes.
### 2.2 Performance indices
Contrary to the laminar flow in a flat channel (see supplementary materials),
no universal analytical solution can be derived for a channel with arbitrary
structuring at both channel walls. Utilizing the melt-down heights of the
imposed structure for both walls ($h_{u}$, $h_{l}$) and splitting the flow
into two halves based on the position of the maximal spatially averaged
velocity denoted with $y_{c}$ (Fig. 1), the balance between pressure drop
$P_{x}$ and the average effective wall shear stress $\tau_{eff}$ is given by
$\tau_{eff}=\frac{(\delta_{l}+\delta_{u})}{2}P_{x},$ (4)
where $\delta_{u}$ and $\delta_{l}$ define the upper and lower effective
channel half heights in respect to $y_{c}$. Based on the wall shear stress the
mean drag coefficient is given as
$C_{f}=\frac{2\tau_{eff}}{\rho{U^{eff}_{b}}^{2}}\quad\textrm{with}\quad
U_{b}^{eff}=\frac{1}{(\delta_{u}+\delta_{l})}\int_{0}^{2\delta}\left<u\right>\mathrm{d}y=\frac{2\delta}{(\delta_{u}+\delta_{l})}U_{b}.$
(5)
The brackets $\left<\right>$ denote a quantity averaged in $x$-direction so a
split-up into the mean part $\left<\phi\right>(y)$ and spatial fluctuation
part $\phi^{\prime}(x,y)$ can be performed for any quantity $\phi(x,y)$:
$\phi(x,y)=\left<\phi\right>(y)+\phi^{\prime}(x,y).$ (6)
Due to the asymmetry in the temperature boundary condition the heat transfer
properties have to be separately evaluated for each wall. Hence, the hydraulic
diameter is defined for upper and lower wall as
$D_{h,u/l}=4\delta_{u/l}.$ (7)
The Nusselt number for both walls can be estimated with
$Nu_{u/l}=\frac{4\delta_{u/l}q_{tot}}{\lambda\Delta\theta_{b,u}},$ (8)
where $q_{tot}$ denotes the total heat flux and the bulk mean temperature
differences are defined as
$\Delta\theta_{b,l}=\frac{1}{\delta_{l}U_{b,l}^{eff}}\int_{0}^{y_{c}}\left<u\right>\left<\theta\right>\mathrm{d}y,$
(9)
and
$\Delta\theta_{b,u}=\frac{1}{\delta_{u}U_{b,u}^{eff}}\int_{y_{c}}^{2\delta}\left<u\right>(1-\left<\theta\right>)\mathrm{d}y.$
(10)
The average of $\mathrm{Nu}_{l}$ and $\mathrm{Nu}_{u}$ is computed to
determine the resultant Nusselt number of a particular case. The effective
bulk mean velocity for each channel half is given by
$U_{b,l}^{eff}=\frac{1}{\delta_{l}}\int_{0}^{y_{c}}\left<u\right>\mathrm{d}y\quad\textrm{or}\quad
U_{b,u}^{eff}=\frac{1}{\delta_{u}}\int_{y_{c}}^{2\delta}\left<u\right>\mathrm{d}y.$
(11)
The total heat flux $q_{tot}$ is estimated as the sum of the viscous and
spatial fluctuation contributions at the position of maximum velocity $y_{c}$
in the channel:
$q_{tot}=\lambda\left.\frac{\mathrm{d}\left<{\theta}\right>}{\mathrm{d}y}\right|_{y_{c}}-\rho
c_{p}\left.\left<{v^{\prime}\theta^{\prime}}\right>\right|_{y_{c}}.$ (12)
Here $c_{p}$ denotes the specific heat capacity. Finally, the Stanton number
is defined based on $Re_{dh}=2(\delta_{l}+\delta_{u})U_{b}^{eff}/\nu$ and
Prandtl number $Pr$:
$St=\frac{Nu}{Re_{dh}Pr}.$ (13)
Reynolds analogy factor $RA$ relates Stanton number to the drag coefficient
$RA=\frac{2St}{C_{f}},$ (14)
and is used to evaluate the similarity between drag coefficient and heat
transfer [4]. An increase in $RA$ highlights a stronger enhancement in heat
transfer than in the drag coefficient and hence is desirable in a design of a
fluidic system. It has to be noted that for the chosen boundary conditions
$RA=0.533$ with $St=0.016$ and $C_{f}=0.06$ in the flat channel configuration
(see derivation in supplementary materials).
### 2.3 Dataset and machine learning model
To generate a diverse dataset of wall structuring, we used a random walk
algorithm combined with spline interpolation and discretization on the
simulation grid. For each generated wall structure, we calculate the drag
coefficient $C_{f}$ and Stanton number $\mathrm{St}$ using the simulation
method described above. An initial training set for the machine learning model
was collected by generating $13,824$ random channel geometries and
corresponding $C_{f}$ and $\mathrm{St}$ values. From that, $13,029$ passed a
set of filters regarding temperature convergence and geometric validity.
We used varying fractions of the dataset to train convolutional neural
networks (CNNs) with 3 convolutional layers (feature maps: 32, 64, 128), one
non-linear fully-connected layer, and a linear output layer with two neurons.
The input for the CNN are the $384\times 129$ binary images representing a
cross section of the channel geometries, i.e. exactly the same input which is
also used in the numerical simulations. Each convolutional layer consists of
$3\times 3$ filters, followed by relu-activations and max-pooling layers (size
2, stride 2). Additionally, a dropout of 25% was used for regularization. The
convolution-outputs are flattened and followed by one densely connected layer
with size 256, relu-activation and 50% dropout. The model was implemented
using TensorFlow [2] and Keras [6], and training was done using the Adam
optimizer [15]. We used MLFlow [1] for model tracking and storage.
## 3 Results
We split the dataset in a test set (1000 data points), a training set (90% of
the remaining data points, i.e. $\approx 11,700$ data points) and a validation
set ($\approx 1300$ data points). We trained the CNN model described above and
obtained mean absolute errors (and r2 scores) of $MAE=1.05\cdot 10^{-3}$
($r^{2}=0.900$) and $MAE=4.82\cdot 10^{-5}$ ($r^{2}=0.805$) for predictions of
$c_{f}$ and $\mathrm{St}$. A comparison of CNN predictions and simulated
ground truth on the test set are shown in Figure 2.
Figure 2: Predictions of the CNN model of a) drag coefficient $c_{f}$ and b)
Stanton number $\mathrm{St}$ compared to the ground truth on a validation set.
In order to evaluate how well the CNN performs on smaller datasets, we
generated learning curves (see Figure 3), where we observe the mean absolute
error in $c_{f}$ and $\mathrm{St}$ as a function of the training set size. The
hyperparameters were kept constant, the amount of training data was varied
from $5~{}\%$ to $90~{}\%$ of the available training and validation data, and
at each training set size, four independent models with random initialization
were trained and their test set performance was averaged. We observe an
exponential decrease of the mean absolute error with the training set size.
Figure 3: Learning curve, i.e. mean absolute error as a function of the
training set size of a CNN model for a) drag coefficient $C_{f}$ and b)
Stanton number $\mathrm{St}$.
A scatter plot of the total dataset of $13,029$ labeled data points is shown
in Figure 2a. We find a strong correlation between $C_{f}$ and $\mathrm{St}$.
Deviations from a flat channel ($C_{f}=0.06$ and $\mathrm{St}=0.016$)
necessarily lead to an increase in drag coefficient and a simultaneous
increase in Stanton number, which leaves limited possibilities for
optimization. However, it is a non-trivial task to predict which channel
geometries yield an optimal $\mathrm{St}$ at a given fixed value of $C_{f}$.
We therefore exploited the speed up of the surrogate machine learning model
($<100~{}\mathrm{ms}$ per channel) compared to the numerical simulation
($\approx 20-30~{}\mathrm{min}$ per channel) to explore a much larger set of
unlabeled channel geometries (Figure 4b). The best channel geometry with a
Stanton number of $\mathrm{St}\approx 0.017$ at $C_{f}=0.07$ is shown in
Figure 4c. This modification results in $6.25$% and $16.6$% increase in $St$
and $C_{f}$ compared to the flat channel configuration. The Reynolds analogy
factor is reduced from $0.533$ to $0.486$ highlighting the prevalence of drag
increase over the heat transfer enhancement. We find a large-scale S-shaped
channel geometry where the fluid flow is mainly redirected from one wall to
another without introduction of contractions for the fluid flow and hence
avoiding regions of flow with acceleration and deceleration.
Figure 4: a) Training data distribution (drag coefficients $C_{f}$ and Stanton
numbers St) of the $\approx 13,000$ channel geometries used for training and
validation of the ML model, b) ML predictions of $C_{f}$ and $\mathrm{St}$ on
a larger dataset of 50,000 channel geometries, c) Channel geometry with
highest predicted St at a given $C_{f}\approx 0.07$.
## 4 Conclusions and outlook
We presented a combination of accurate numerical simulations of fluid flow and
heat transfer in arbitrary, non-flat channels and machine learning models
predicting drag coefficient and Stanton number. We show that the CNNs predict
the target properties at a fraction of the time of numerical simulations which
can be exploited for exploration and optimization tasks. The general approach
is not only applicable to simple flow setups as presented here, but can be
extended to more complex tasks, such as three-dimensional multiphase or even
reactive unit operations in chemical engineering. The only limitation is the
availability of data or the associated computational cost of the underlying
simulations.
In order to further exploit ML models in general and CNNs in particular for
the design of chemical engineering unit operations, we plan to implement
active learning approaches and generative models to reliably explore the
possible design space of channel structures and directly solve the inverse
problem, i.e. the suggestion of channel architectures given desired target
properties.
## Code and data availablity
The code to train the CNNs can be found on https://github.com/aimat-
lab/ChemEngML. The data that support the findings of this study are available
upon reasonable request from the authors.
## References
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|
# Equivalence between condensation and boiling in a Lennard Jones fluid
I. Sanchez-Burgos1, P. Montero de Hijes1, P. Rosales-Pelaez1, C. Vega1 and E.
Sanz1 1Departamento de Química Física, Facultad de Ciencias Químicas,
Universidad Complutense de Madrid, 28040 Madrid, Spain
###### Abstract
Condensation and boiling are phase transitions highly relevant to industry,
geology or atmospheric science. These phase transitions are initiated by the
nucleation of a drop in a supersaturated vapor and of a bubble in an
overstretched liquid respectively. The surface tension between both phases,
liquid and vapor, is a key parameter in the development of such nucleation
stage. Whereas the surface tension can be readily measured for a flat
interface, there are technical and conceptual limitations to obtain it for the
curved interface of the nucleus. On the technical side, it is quite difficult
to observe a critical nucleus in experiments. From a conceptual point of view,
the interfacial free energy depends on the choice of the dividing surface,
being the surface of tension the one relevant for nucleation. We bypass the
technical limitation by performing simulations of a Lennard Jones fluid where
we equilibrate critical nuclei (both drops and bubbles). Regarding the
conceptual hurdle, we find the relevant cluster size by searching the radius
that correctly predicts nucleation rates and nucleation free energy barriers
when combined with Classical Nucleation Theory. With such definition of the
cluster size we find the same value of the surface tension for drops and
bubbles of a given radius. Thus, condensation and boiling can be viewed as two
sides of the same coin. Finally, we combine the data coming from drops and
bubbles to obtain, via two different routes, estimates of the Tolman length, a
parameter that allows describing the curvature dependence of the surface
tension in a theoretical framework.
## I Introduction
Understanding first order phase transitions is of great importance to many
fields, ranging from biology cryopres , to atmospheric science
cantrell2005production , physics debenedettibook , geology icemars2018 or
industry foodcrystallization ; brittain .
In the absence of impurities or external surfaces, first order phase
transitions start with the emergence of a nucleus of the stable phase in the
bulk of the parent metastable phase kelton ; skripov1974metastable . A nucleus
is “critical” if it is big enough so that it has 50 per cent chances to either
grow or redissolve.
Although the emerging phase is more stable, the presence of an unfavourable
interface between the nucleus and the parent phase can delay to a great extent
the phase transition. Thus, for instance, alkane vapors can be saturated
thousands of times over their vapor pressure before condensation takes place
wyslouzil2016overview , alkane liquids can be substantially superheated above
the boiling temperature skripov2010phenomenon ; lipnyagov2018going ;
skripov1992metastable or liquid water can be supercooled up to $\sim 60$ K
below melting until it freezes manka2012 ; amaya2018ice ; hu2008water ;
bhabheJPCA2013 .
Therefore, the surface tension, $\gamma$, or the free energy per unit area
between both phases, plays a key role in the development of first order phase
transitions. Whereas $\gamma$ can be readily measured for a flat interface at
equilibrium –at least between fluid phases ickespccp2015 – it cannot be
directly probed for curved interfaces, which is the relevant case for
nucleation. Moreover, the fact that critical nuclei are nanoscopic objects
makes it very difficult to observe them in experiments, let alone measuring
their $\gamma$. The usual strategy is to infer $\gamma$ by combining a
theoretical description of nucleation with measurements of the nucleation rate
(the number of nuclei that appear per unit of time and volume) kelton ;
ickespccp2015 ; caupinPRL2016curvature ; granasy:6157 . This approach relies
on the validity of theoretical approximations that are difficult to assess.
Computer simulations do have access to the time and length scales relevant for
the observation of critical nuclei. However, whereas the methodology and
theoretical framework for computing $\gamma$ for flat interfaces is very well
established kirkwood1949statistical ; JCP_2005_123_134703 ; broughton:5759 ;
PhysRevLett.86.5530 ; PNAS_2002_99_12562 ; ubertiPRB2010 ; verrocchioPRL2012 ;
PhysRevLett.94.176105 ; Nature_2001_409_1020 ; espinosaJCP2014_2 , that for
curved interfaces is still under development lau2015surface ;
ceriottitolman2018 ; binder2012beyond . One of the key issues is that $\gamma$
for curved interfaces depends on the location of the interface, that can be
defined in different ways rowlinson2013molecular ; binder2012beyond . The
current situation is that the dependence of $\gamma$ with the curvature of the
interface is contradictory between different groups blokhuis2006thermodynamic
; block2010curvature ; sampayo ; malijevsky2012perspective ; binder2012beyond
; size-dependendentgamma2005 ; ceriottitolman2018 ; JCP_1984_81_00530 ;
lau2015surface ; vrabec2006comprehensive ; wilhelmsen2015tolman ;
joswiak2016energetic ; schmelzer2019entropy ; richard2018crystallization ;
caupinPRL2016curvature
In this work we address fundamental questions regarding the liquid-vapor
interface with computer simulations. It has been shown in different simulation
works that spherical nuclei can be equilibrated at constant volume and
temperature in finite systems block2010curvature ; schrader2009simulation ;
matsumoto2008nano ; troster2012numerical ; macdowell2006nucleation ;
richard2018crystallization ; statt2015finite ; koss2018phase ;
gunawardana2018theoretical ; zierenberg2015exploring ; zierenberg2017canonical
. Recently, we showed with simulations of bubbles seedingNVT and crystals
montero2020interfacial that nuclei thus equilibrated are critical, in
agreement with Density Functional Theory (DFT) predictions lutsko2018classical
; lutsko2019crystals . On the other hand, we have extensively developed in the
past years the so-called Seeding method jacs2013 ; knottJACS2012 ; baiJCP2006
; seedingvienes to study nucleation phenomena. This method consists in
obtaining with simulations the properties of critical clusters and “plug” them
in the Classical Nucleation Theory (CNT) formalism ZPC_1926_119_277_nolotengo
; becker-doring ; gibbsCNT1 ; gibbsCNT2 to obtain predictions of the
nucleaiton rate and of the $\gamma$ curvature dependence. This approach has
been successful for a wide range of systems knottJACS2012 ; jacs2013 ;
seedingvienes ; zaragozaJCP2015 ; espinosaPRL2016 ; espinosaJPCL2017 ;
espinosa2018homogeneous ; seedingNpT ; seedingNVT and we use it here for the
first time to study condensation. In particular, we apply Seeding at constant
volume both to condensation and to cavitation for a Lennard Jones model.
Since Seeding relies on CNT, it is necessary to validate it by comparing its
predictions with independent calculations that do not rely on such framework.
We do so by computing nucleation rates via Umbrella Sampling (US)
torrie1974monte ; JCP_1992_96_4655 and direct brute force simulations
filion:244115 as well as by testing the consistency of the $\gamma$-curvature
dependence obtained via Seeding with the value for a flat interface.
All consistency tests are successfully passed for our Seeding simulations
provided that the nucleus surface is identified with that where the density is
the average between the density of both phases (“equi-density” surface).
Therefore, we identify the equidensity surface with the surface of tension. On
the other hand, we directly compare the condensation of liquid drops in a
supersaturated vapor with the cavitation of vapor bubbles in an overstretched
liquid. We find that, for a given temperature, drops and bubbles of the same
radius have the same $\gamma$ when using the equidensity definition of the
surface of tension. Finally, we estimate the Tolman lenght tolman1949effect ,
a parameter useful to predict the $\gamma$-curvature dependence, via two
different routes, as recently proposed in montero2020interfacial .
## II Simulation details
The Lennard Jones model potential, as well as the simulation details, are the
same as in our previous work seedingNpT ; seedingNVT . In particular, we study
the truncated and force-shifted Lennard-Jones (TSF-LJ) potential
wang2008homogeneous , a model for which the vapor-liquid transition has been
previously investigated wang2008homogeneous ; tanaka2015simple ;
meadley2012thermodynamics ; seedingNpT :
$U_{TSF-LJ}(r)=U_{LJ}(r)-U_{LJ}(r_{c})-(r-r_{c})U^{\prime}_{LJ}(r_{c}),$ (1)
where $U_{LJ}(r)$ is the standard 12-6 Lennard-Jones potential and
$U^{\prime}_{LJ}(r)$ is its first derivative. The interaction potential is
truncated and shifted at $r_{c}=2.5\sigma$, being $\sigma$ the particle’s
diameter and $\epsilon$ the depth of the un-truncated Lennard-Jones potential.
Unless otherwise specified, all magnitudes in this work are given in Lennard-
Jones reduced units seedingNVT . Thus, the reported temperatures are reduced
by $\epsilon/k_{B}$, distances by $\sigma$, densities by $\sigma^{-3}$,
pressures by $\epsilon/\sigma^{3}$, times by
$\tau=\sqrt{m\sigma^{2}/\epsilon}$ (being $m$ the particle mass), interfacial
free energies by $\epsilon/\sigma^{2}$ and nucleation rates by
$1/(\tau\sigma^{3})$.
We use cubic boxes with periodic boundary condition and the Molecular Dynamics
(MD) LAMMPS package lammps_program to perform all simulations of this work.
The equations of motion are integrated with a leap-frog algorithm leapfrog .
In the MD Seeding simulations we used a time-step of $0.0012$. The system was
kept at constant temperature using the Nosé-Hover thermostat JCP_1984_81_00511
with a relaxation time of 0.46.
For the MD simulations used within the Umbrella Sampling scheme we set the
time step for the integration of the motion equations to $0.0012$. The
relaxation times for the Nose-Hover thermostat and barostat were 0.46 and 1.6
respectively.
All simulations are carried out at T=0.785. The coexistence pressure at such
temperature for the model is $p_{coex}$=0.0267. We determined this value,
refined with respect to that of 0.026 previously published wang2008homogeneous
, by running long ($4\cdot 10^{5}$ $\tau$) MD NVT simulations with an
elongated box (50 x 17 x 17) where the vapor and the liquid were put at
contact at the temperature of interest. The average pressure normal to the
interface in such simulation corresponds to $p_{coex}$.
## III Seeding of condensation
This work is based on a recent publication where we demonstrate how to compute
bubble nucleation rates in an overstretched Lennard-Jones fluid by
equilibrating critical bubbles in the NVT ensemble, an approach we call “NVT-
Seeding” seedingNVT . The Seeding method, originally developed to study
crystal nucleation jacs2013 ; knottJACS2012 ; baiJCP2006 ; seedingvienes , and
more recently applied to vapor cavitation seedingNpT ; seedingNVT ;
baidakov2020molecular , consists in combining CNT ZPC_1926_119_277_nolotengo ;
becker-doring ; gibbsCNT1 ; gibbsCNT2 with computer simulations to estimate
nucleation free energy barrier heights, $\Delta G_{c}$, interfacial free
energies, $\gamma$, and, most importantly, nucleation rates, $J$.
According to CNT, the Gibbs free energy barrier for the nucleation of a
spherical liquid drop is given by the following expression:
$\Delta G=\gamma A-V\Delta p$ (2)
Where $V$ and $A$ are the volume and the area of the drop respectively. By
maximizing Eq. 2 assuming a spherical drop shape one obtains both the height
of the nucleation free energy barrier,
$\Delta G_{c}=\frac{2\pi R_{c}^{3}\Delta p}{3},$ (3)
where $R_{c}$ is the critical droplet radius and $\Delta p$ is the pressure
difference between the interior of the drop and the surrounding vapor, and the
number of particles in the critical drop,
$N_{c}=(32\pi\rho_{l}\gamma^{3})/(3\Delta p^{3}),$ (4)
where $\rho_{l}$ is the critical drop number density and $\gamma$ is the
liquid-vapor surface tension. By substituting in the equation above $N_{c}$ by
the droplet volume ($4/3\pi R_{c}^{3}$) times $\rho_{l}$ one recovers the
Laplace equation:
$\Delta p=\frac{2\gamma}{R_{c}}.$ (5)
This derivation shows that the Laplace equation, which is valid when the
droplet surface is located at the the surface of tension, is implicit in CNT.
Consequently, $R_{c}$ should be identified with the radius of tension,
$R_{s}$. This is an important point that we will use later on in the paper.
The CNT prediction for the nucleation rate of drops is given by kelton :
$J=A_{0}\rho_{vap}\exp\left(-\frac{\Delta G_{c}}{k_{B}T}\right),$ (6)
where $k_{B}$ is the Boltzmann constant, $\rho_{vap}$ is the density of the
vapor phase that multiplied by $\exp\left(-\frac{\Delta G_{c}}{k_{B}T}\right)$
gives the number density of critical clusters and $A_{0}$ is a kinetic pre-
factor.
$A_{0}$ is computed as the product of the Zeldovich factor, $Z$, and the rate
of attachment to the critical nucleus, $f^{+}$ becker-doring ; kelton :
$A_{0}=Zf^{+}.$ (7)
$Z$ takes into account the establishment of a steady state and, according to
CNT, is given by kelton ; ZPC_1926_119_277_nolotengo ; becker-doring :
$Z=\sqrt{\frac{|\Delta G(N)^{{}^{\prime\prime}}|_{N_{c}}}{2\pi
k_{B}T}}=\sqrt{\frac{\Delta p}{6\pi k_{B}T\rho_{l}N_{c}}}=\sqrt{\frac{\Delta
p}{8\pi^{2}k_{B}T\rho_{l}^{2}\cdot R_{c}^{3}}}$ (8)
where $N_{c}$ is the number of particles in the drop and $|\Delta
G_{c}(N)^{{}^{\prime\prime}}|_{N_{c}}$ is the curvature of $\Delta G(N)$
evaluated at the barrier top.
The attachment rate, $f^{+}$, can be estimated by multiplying the collision
frequency of the vapor per unit of wall area given by the kinetic theory of
gasses (ktg) by the area of the critical bubble:
$f^{+}_{ktg}=\sqrt{\frac{k_{B}T}{2\pi
m}}\left(\frac{6\sqrt{\pi}N_{c}}{\rho_{l}}\right)^{2/3}$ (9)
where the subscript “$ktg$” stresses the fact that this expression of the
attachment rate is based on the kinetic theory of gasses.
Combining this equation with 8, 4, and 5, the following kinetic pre-factor is
obtained:
$A^{ktg}_{0}=\sqrt{\frac{\Delta pR_{c}}{\pi m}}\frac{\rho_{vap}}{\rho_{l}}.$
(10)
The equations above are quite powerful, because only $R_{c}$, $\Delta p$ and
the density of both phases are required to obtain key nucleation parameters as
free energy barriers, interfacial free energies and ınucleation rates. The
Seeding method consists in performing simulations of a cluster of the stable
phase surrounded by the mestastable phase (a liquid drop surrounded by
supersaturated vapor in our case as shown in Fig. 1) to compute $R_{c}$,
$\Delta p$, $\rho_{l}$ and $\rho_{vap}$ in order to get “cheap” estimates of
$\Delta G_{c}$, $\gamma$ and, most importantly, $J$ through the expressions
above.
The main drawback of Seeding is that the definition of $R_{c}$ is not unique.
Therefore, the resulting free energy barrier depends on the specific
definition of $R_{c}$. This contrasts with rigorous simulation methods like
Umbrella Sampling JCP_2005_122_194501 ; filion:244115 or with theoretical
approaches like DFT shen2001density ; gallo2020nucleation ; lutsko2008density
where the nucleation free energy does not depend on the criterion chosen to
measure the nucleus size, which can be estimated _a posteriori_ via, e. g. the
nucleation theorem JCP_1982_76_05098 ; kashchiev2006forms ; wedekind2007best
(although in the particular case of DFT an approximate functional needs to be
proposed so that the results do also contain approximations). To assess the
suitability of our choice to compute $R_{c}$ we complement Seeding with
Umbrella Sampling simulations.
Figure 1: Snapshot of a critical drop equilibrated in the NVT ensemble at
T=0.785 surrounded by supersaturated vapor. The droplet radius is about 6.8
and the density of the surrounding vapor 0.0550.
### III.1 $R_{c}$, $\Delta p$, $\rho_{l}$ and $\rho_{vap}$
We use the $NVT$ ensemble to run the simulations of the drops given that in
such ensemble critical nuclei are naturally equilibrated and stabilised for
long times seedingNVT ; montero2020interfacial . We equilibrate drops in 10
different systems. The edge of the cubic simulation box, $L$, and the total
number of particles in each system, $N_{T}$, are reported in table 1. A large
number of particles is used to minimize finite size effects
zierenberg2015exploring ; marchio2018pressure ; seedingNVT . Each system was
simulated for about $10^{3}$ Lennard Jones times of equilibration and $2\cdot
10^{5}$ of production.
To prepare the initial configuration we cut a spherical liquid drop from a
bulk liquid simulation and insert it in a bulk vapor box removing the
overlapping vapor particles. The liquid drop is cut with a certain tentative
radius, but the precise number of particles in each phase is not crucial given
that equilibrium is reached along the course of the $NVT$ simulation.
From a simulation of a drop surrounded by supersaturated vapor one can obtain
an average radial density profile starting from the center of the drop as that
shown in Fig. 2 (to find the drop centre in each configuration we use a
similar strategy to that described in our previous work seedingNVT consisting
in this case in identifying the maxima of density profiles computed along each
cartesian coordinate). Following Refs. seedingNpT ; seedingNVT , we obtain
$R_{c}$ from such density profile as the distance at which the density is
average between the liquid and the vapor plateaux. This is indicated with a
vertical dashed line in Fig. 2. We refer to this way of obtaining $R_{c}$ as
the “equi-density” criterion. The $R_{c}$’s thus obtained in our NVT-Seeding
simulations are also reported in Table 1. Other definitions of $R_{c}$ are in
principle as valid as the equi-density criterion seedingNpT ;
lutsko2008density ; gallo2020nucleation . We argue later on in the paper that
our $R_{c}$ definition is a good one because it makes Seeding predictions
consistent with independent calculations of $\gamma$, $J$ or $\Delta G_{c}$.
Figure 2: Density profile of a critical drop equilibrated in the NVT ensemble
at T=0.785 surrounded by supersaturated vapor. The droplet radius, indicated
by a red vertical line in the figure, is given by the point at which $\rho(r)$
takes an average value between both plateaux (equi-density criterion). The
density profile corresponds to the system labelled as IV in table 1
.
Figure 3: Box volume versus time in NpT simulations starting from 40
configurations taken from the NVT-Seeding simulation labelled as IX in Table
1. The imposed pressure is the average viral pressure of the NVT-Seeding run.
(a)
(b)
(c)
Figure 4: (a) $\gamma$ vs vapor pressure obtained from NVT-Seeding data of
droplets surrounded by supersaturated vapor. The surface tension at
coexistence (p=0.0267) is included seedingNpT . (b) $\Delta G_{c}$ vs vapor
pressure. NVT-Seeding and US data are compared. Empty black symbols correspond
to Seeding predictions when the Gibbs dividing (equi-molar) –instead of the
equi-density– surface is employed to identify the cluster radius. (c)
Nucleation rate versus vapor pressure as obtained from NVT-Seeding, US and
spontaneous nucleation.
To get $\Delta p=p_{l}-p_{vap}$ we obtain first the vapor density,
$\rho_{vap}$, by counting the number of particles outside a sphere concentric
with the drop but with a larger radius (we use a sphere radius 7$\sigma$
larger than that of the drop, but we have checked for a few selected cases
that any value beyond $\sim 5\sigma$ gives the same result). $\rho_{vap}$ is
given by the number of particles outside the sphere divided by the $L^{3}$
minus the sphere volume. We then use the bulk vapor equation of state to infer
$p_{vap}$ from $\rho_{vap}$. We report $p_{vap}$ and $\rho_{vap}$ in table 1.
We have checked for all studied systems that $p_{vap}$ coincides with the
overall virial pressure of the system. On the other hand, $p_{l}$ is obtained,
as in our previous work seedingNpT ; seedingNVT , by assuming equal chemical
potential between the critical drop and the surrounding vapor:
$\int^{p_{vap}}_{p_{coex}}\frac{1}{\rho_{vap}(p)}dp=\int^{p_{l}}_{p_{coex}}\frac{1}{\rho_{l}(p)}dp$
(11)
where $p_{coex}$ is the coexistence pressure and $\rho_{vap}(p)$ and
$\rho_{l}(p)$ are the bulk vapor and bulk liquid number densities at pressure
$p$. In table 1 we report $p_{l}$ and $\Delta p$ for all studied systems. Once
$p_{l}$ is known, $\rho_{l}$, also reported in the table, can be easily
computed from the bulk liquid equation of state. In all cases, this
computation of $\rho_{l}$, based on the equality of chemical potential between
both phases, is consistent with that obtained from the density profiles. For
instance, for system IV we get $\rho_{l}=0.0680$, which is fully consistent
with the first plateau observed in the density profile shown in Fig. 2. This
means that the mechanical pressure and the thermodynamic pressure inside the
drop coincide, a matter of current debate for solid-liquid nucleation
presiontermod .
There has been much simulation, theoretical and experimental work devoted to
study the formation of nuclei confined at constant volume vincent2017statics ;
marti2012effect ; macdowell2006nucleation ; finitesizenucreguera ;
reguera2004fusion ; JCP_2003_118_00340 ; binder2012beyond ; bindernuccryst2017
; richard2018crystallization ; yang1985thermodynamical ; gallo2018thermally ;
gallo2020heterogeneous . In Refs. seedingNVT ; montero2020interfacial we
showed with simulations that nuclei equilibrated in the NVT ensemble are
critical because they have equal chances to grow or shrink when simulated in
the NpT ensemble at the same temperature and at the average pressure along the
NVT run. Based on this result, we opted to study here drop nucleation in the
NVT ensemble, where statistics is better because clusters remain stable for
very long times seedingNVT . Stabilising nuclei to gain time to study their
properties is something quite desirable. An alternative strategy to the use of
constant volume simulations is to pin the nucleus to a heterogeneous solid
substrate xiao2017experiments .
Despite having already shown the equivalence between stable (NVT) and critical
(NpT) nuclei for cavitation seedingNVT , we check here for one of the NVT-
Seeding simulations if the drops equilibrated at constant volume and
temperature do correspond to a Gibbs free energy maximum. In Fig. 3 we show
the evolution of the box volume in NpT simulations started from 40 independent
configurations gathered along the NVT-Seeding trajectory labelled as IX in
Table 1. The imposed pressure is the average virial pressure along the NVT-
Seeding run. Roughly, in 50 per cent of the cases the box expands (the drop
dissolves) and in the other half of the cases the box shrinks (the drop
grows). This result supports the use of NVT to study drop condensation in the
same manner we did for bubble cavitation and cystal nucleation seedingNVT ;
montero2020interfacial . Furthermore, the equivalence between clusters
equilibrated at constant volume and critical nuclei has been recently proven
with DFT theoretical arguments for crystallization (see supplementary material
of Ref. lutsko2019crystals ).
Label | $L$ | $N_{T}$ | $\rho_{l}$ | $\rho_{vap}$ | $p_{l}$ | $p_{vap}$ | $\Delta p$ | $R_{c}$ | $\gamma$ | $\Delta G_{c}/(k_{B}T)$ | $A_{0}^{ktg}$ | $A_{0}^{af}$ | $\log_{10}(J)$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
I | 38.019 | 3774 | 0.6833 | 0.05690 | 0.09279 | 0.03161 | 0.0612 | 5.86 | 0.1792 | 32.8 | 0.028 | | -17.1
II | 39.160 | 4291 | 0.6811 | 0.05496 | 0.0846 | 0.03092 | 0.0537 | 6.82 | 0.1831 | 45.5 | 0.028 | | -22.6
III | 39.160 | 4373 | 0.6801 | 0.05414 | 0.08095 | 0.03063 | 0.0503 | 7.15 | 0.1800 | 49.2 | 0.027 | | -24.2
IV | 39.160 | 4510 | 0.6797 | 0.05384 | 0.07961 | 0.03052 | 0.0491 | 7.48 | 0.1835 | 54.8 | 0.027 | 0.030 | -26.6
V | 39.160 | 4623 | 0.6792 | 0.05342 | 0.07769 | 0.03037 | 0.0473 | 7.79 | 0.1843 | 59.7 | 0.027 | | -28.8
VI | 39.160 | 4796 | 0.6782 | 0.05277 | 0.07468 | 0.03013 | 0.0446 | 8.20 | 0.1827 | 65.5 | 0.027 | 0.036 | -31.3
VII | 39.160 | 4964 | 0.6783 | 0.05269 | 0.07432 | 0.03010 | 0.0442 | 8.49 | 0.1878 | 72.3 | 0.027 | | -34.2
VIII | 39.160 | 5163 | 0.6776 | 0.05216 | 0.07181 | 0.02991 | 0.0419 | 8.86 | 0.1856 | 77.8 | 0.027 | | -36.6
IX | 39.160 | 5435 | 0.6771 | 0.05176 | 0.06989 | 0.02976 | 0.0401 | 9.30 | 0.1865 | 86.0 | 0.026 | | -40.2
X | 85.264 | 34519 | 0.6761 | 0.05105 | 0.06638 | 0.02949 | 0.0369 | 9.99 | 0.1843 | 98.2 | 0.026 | | -45.5
Table 1: NVT-Seeding data for the different drops studied in this work at
T=0.785.
Having computed $R_{c}$, $\Delta p$ and $\rho_{vap}$ and $\rho_{l}$ we have
everything needed to obtain $\gamma$, $\Delta G_{c}$ and $J$ according to the
equations presented in section III. We report the values for these variables
in table 1 and plot them in fig. 4(a)-(c) versus the vapor pressure with black
dots. In the following section we comment each of these graphs.
### III.2 $\gamma$, $\Delta G_{c}$ and $J$
#### III.2.1 $\gamma$
As shown in Fig. 4 (a) the prediction we obtain from Seeding is that $\gamma$
decreases as the vapor supersaturation increases. This trend is in agreement
with previous work block2010curvature ; troster2012numerical . Accordingly,
using the capillarity approximation (i. e., that $\gamma$ is pressure
independent) would be erroneous. The green square in Fig. 4(a) corresponds to
the surface tension at coexistence seedingNpT obtained through the pressure
tensor walton1983pressure in an NVT simulation of a liquid and a vapor at
contact. The trend of the Seeding data is fully consistent with the
coexistence value, as shown by the linear fit in the figure. This is a good
consistency test, although the $\gamma$ values provided by Seeding could still
be incorrect despite the fact that they extrapolate correctly to coexistence.
Therefore, a test for Seeding predictions away from coexistence is needed.
#### III.2.2 $\Delta G_{c}$
To further test our Seeding results we compare $\Delta G_{c}$ obtained with
Seeding with that computed via Umbrella Sampling. In Fig. 4(b), where we plot
$\Delta G_{c}$ versus the vapor pressure, black solid dots correspond to
Seeding and red ones to US (details on US calculations are described in
section IV). Whereas Seeding predictions rely on the validity of CNT and on a
proper definition of $R_{c}$, Umbrella Sampling calculations are rigorous and
independent on the criterion to identify the nucleus size filion:244115 . On
the other hand Seeding is much “cheaper” than US from a computational point of
view. As a matter of fact, Seeding has access to much higher nucleation
barriers than US. The accordance between Seeding and US shown in Fig. 4 is
excellent, which gives us great confidence on Seeding predictions. The choice
of the equi-density surface to identify the drop radius has proven correct. If
we use another criterion, such as the Gibbs (equi-molar) dividing surface, the
agreement between Seeding and Umbrella Sampling deteriorates (empty black
symbols in Fig. 4 (b)). To compute $R_{e}$, the radius associated to the Gibbs
dividing surface, we use $N_{T}=N_{l}+N_{vap}$ where $N_{l}=4/3\pi
R_{e}^{3}\rho_{l}$ and $N_{vap}=[V_{T}-(4/3\pi R_{e}^{3})]\rho_{vap}$, where
$V_{T}$ is the volume of the simulation box and the densities $\rho_{l}$ and
$\rho_{vap}$ are obtained as described in Sec. III.
In a recent publication on cavitation (nucleation of bubbles instead of drops)
we compared the performance of different criteria to identify the cluster
radius and found that the equi-density criterion also made Seeding predictions
consistent with other rigorous calculations seedingNpT . Therefore,
identifying the critical drop radius with the equi-density distance seems to
be quite general for condensation-evaporation transitions.
#### III.2.3 J
Once $\Delta G_{c}$ is known computing $J$ via Eq. 6 is quite straight
forward. The kinetic pre-factor $A_{0}$ given by the kinetic theory of gases,
Eq. 10, depends on parameters we already have under control: $\Delta p$,
$R_{c}$ and the density of both phases. The values of $A_{0}$ computed via Eq.
10, $A_{0}^{ktg}$, are reported in Table 1.
These $A_{0}$ values are approximate since they rely on the validity of the
kinetic theory of gasses to estimate the attachment rate (see section III). We
therefore have to check $A_{0}^{ktg}$ by computing the attachment rate with an
alternative approach. Following the work by Auer and Frenkel auerJCP2004 , the
attachment rate can be computed from the diffusion of $N$, the number of
particles in the liquid drop, around the critical drop auerJCP2004 :
$f^{+}_{af}=\frac{\left<(N(t)-N(0))^{2}\right>_{Nc}}{2t},$ (12)
where the average is performed over several trajectories starting from a
critical drop configuration. The $af$ subscript stresses the fact that this
expression of the attachment rate is based on the work by Auer and Frenkel.
To compute $N$ we follow JCP_1998_109_09901 . We count as neighbors all
particles within a 1.625 distance of a tagged particle. Particles with 8 or
more neighbors are labelled as “liquid”. Two liquid particles belong to the
same drop if their mutual distance is less than 1.625. An example of the
calculation of $f^{+}$ according to equation 12 is illustrated in Fig. 5.
Typically, $\left<N(t)-N(0)\right>_{N_{c}}$ is obtained by averaging 20 NpT
runs started from independent configurations of the critical drop, coming
either from NVT-Seeding or from Umbrella Sampling simulations (see section
IV). In these runs, the pressure is fixed to the virial value of the
simulations were the critical clusters were previously equilibrated. According
to Eq. 12, the slope of Fig. 5 divided by 2 gives $f^{+}$. Multiplying such
$f^{+}$ by the Zeldovich factor we get an estimate of the kinetic pre-factor,
$A_{0}^{af}$, that does not rely on the kinetic theory of gases. $A_{0}^{af}$
is reported it in table 1 for a couple of critical clusters generated with
NVT-Seeding (systems IV and VI). $A_{0}^{af}$ is very close to $A_{0}^{ktg}$.
This agreement suggests the validity of the kinetic theory of gases to
estimate the attachment rate and makes the theoretical framework that supports
the Seeding technique quite powerful given that, since $A_{0}^{ktg}$ can be
used, only $R_{c}$, $\Delta p$ and the density of both phases are required to
get accurate estimates of $J$ in a wide range of orders of magnitude. Note in
Fig. 4(c) that Seeding (black dots) has access to $J$ values many orders of
magnitude lower than US (red dots).
Label | $N_{T}$ | $\langle V\rangle$ | $p_{v}$ | $\rho_{v}$ | $\log_{10}(J)$
---|---|---|---|---|---
BF-1 | 4000 | 57145 | 0.035 | 0.0700 | -9.235
BF-2 | 4000 | 53456 | 0.036 | 0.0748 | -8.287
Table 2: Data corresponding to the brute force calculations.
The green dots in Fig. 4(c) correspond to rate estimates obtained in brute
force NPT molecular simulation runs performed at high supersaturations where
condensation occurs spontaneously from an unseeded vapor. In such cases the
nucleation rate can be estimated as $J=1/(t<V>)$, where $<V>$ is the average
volume before nucleation and $t$ is the nucleation time averaged over a number
of independent trajectories (typically 20 in our case). $N_{T}$, $V$, the
vapor pressure and density, and $J$ for the two states where we studied
spontaneous condensation are reported in Table 2. In Fig. 4(c) we show that
$J$ estimates from Seeding and from spontaneous nucleation are consistent with
each other, which further indicates the ability of Seeding to predict
nucleation rates. It is worth mentioning here that NVT-Seeding and spontaneous
nucleation are complementary techniques. On the one hand, the former does not
have access to such high supersaturations given the difficulty to equilibrate
small clusters in the NVT ensemble seedingNVT ; montero2020interfacial . That
said, it would be nonsense using Seeding where nucleation occurs spontaneously
in a straightforward manner. On the other hand, spontaneous nucleation is
limited to a narrow window of nucleation rates (that enabled by computational
time) whereas Seeding has access to extremely low rates.
We would like to end this section by discussing finite size effects, which
could be present if a nucleus sees its replica through periodic boundary
conditions. On the one hand, we made sure that the density of the outer phase
reaches a plateau before L/2 by looking at radial density profiles such as
that shown in Fig. 2. On the other hand, we note that the box side of system X
is more than twice than those of the other systems. By looking at Figs. 4 (a),
(b) and (c) one can see that the results from system X are fully consistent
with those inferred from the other systems, which strongly supports the
absence of noticeable finite size effects in our simulations.
Figure 5: Time dependence of the mean squared deviation of the number of
particles in the critical drop for system VI in table 1. Half the slope of
this plot gives the attachment rate according to Eq. 12
.
Label | $L$ | $N_{T}$ | $\Delta G_{c}/(k_{B}T)$ | $A_{0}^{af}$ | $\log_{10}(J)$
---|---|---|---|---|---
US-1 | 39.112 | 4000 | 17.7 | 0.041 | -10.3
US-2 | 38.501 | 4000 | 40.7 | 0.039 | -20.3
Table 3: Data corresponding to the US calculations.
## IV Umbrella Sampling
As previously indicated, to validate the Seeding results we used the US
technique. We followed Refs. JCP_1998_109_09901 ; gonzalezPCCP2014 to compute
$\Delta G_{c}$ for two different vapor pressures: p=0.031 and p=0.033. Details
on the simulation box size and number of particles in the systems used to
perform the US calculations are given in table 3.
The free energy associated to the formation of an $N$ particle cluster drop
can be obtained from:
$\Delta G(N)=-k_{B}T\ln[P(N)],$ (13)
where $P(N)$ is the probability distribution of $N$. Our criterion to compute
$N$ is described in Section III.2.3. It is important to note that even though
different criteria may give different $N$ for a given configuration, the
height of an US free energy barrier does not depend on the criterion to
determine the cluster size filion:244115 . Therefore, contrary to what happens
in Seeding, the US method does not depend on the specific criterion to
determine the nucleus size. This is why it is important to validate the
Seeding method with other tecniques such us US.
With conventional NpT simulations at the selected pressures $P(N)$ can only be
sampled up to $N\sim 40$ while the critical cluster is much larger in this
regime. To sample the rest of the free energy barrier a biasing potential,
$U_{bias}$, is added to the original hamiltonian:
$U_{bias}=\frac{1}{2}k_{bias}\left(N-N_{0}\right)^{2},$ (14)
where $N_{0}$ controls the cluster size around which the sampling will be
centred and $k$ the width of such sampling. Tens of overlapping sampling
“windows” centered at different $N_{0}$ values are required to reconstruct the
whole free energy barrier. The effect of the bias potential on the calculation
of the free energy barrier is removed as follows torrie1974monte :
$\Delta G(N)=-k_{B}T\ln\left<\frac{\chi_{N}}{e^{-U_{bias}/(k_{B}T)}}\right>+C$
(15)
where $\chi_{N}$ is the fraction of clusters with $N$ particles that appear
within a certain window and $C$ is a constant. The constant is obtained by
gluing together the first part of the energy barrier evaluated without the
biasing potential (Eq. 13 ) with the rest of the windows. The result is the
whole free energy barrier.
To compute each window we use the hybrid Molecular Dynamics-Monte Carlo scheme
labelled as HMC(nM-NpT)/US in Ref. gonzalezPCCP2014 . From the starting
configuration, random velocities are assigned to every particle according to a
Maxwell-Boltzmann distribution and a short ($\Delta t$ 19.2 Lennard Jones
times) MD simulation is run for generating a new configuration, which is
accepted with probability $\min[1,\exp[-(U_{bias}(\Delta
t)-U_{bias}(0))/(k_{B}T)]]$. Either in case of acceptance or rejection new
random velocities are assigned at the beginning of each short MD cycle. For
each window, 10000 of such cycles were performed for equlibration and 60000 to
obtain the free energy barrier. We used $k_{bias}=0.04k_{B}T$ in the biasing
potential (Eq. 14), which gives an acceptance rate of $\sim 25\%$.
In figure 6 we plot both free energy barriers, being $\Delta G_{c}=17.7k_{B}T$
for p=0.033 and $\Delta G_{c}=40.7k_{B}T$ for p=0.031 (also reported in table
3). As already discussed, the agreement between US and Seeding is excellent
(see Fig. 4(b)).
Additionally, we compute the kinetic pre-factor $A_{0}^{af}$ (Eq. 7) to obtain
the nucleation rate (Eq. 6). To do that, we launch tens of unbiased
trajectories from independent configurations at the barrier top in order to
compute the attachment rate via Eq. 12. The Zeldovich factor (Eq. 8) can be
obtained by numerically calculating the curvature of $\Delta G(N)$ at the
barrier top. We report $A_{0}^{af}$ thus calculated and the corresponding $J$
in table 3. As previously discussed, $J$ from US is fully consistent with that
coming from Seeding (see Fig. 4 (c)).
In summary, we have compared Seeding, that relies on the theoretical
assumptions by CNT and $ktg$ and depends on the criterion employed to
determine the cluster size, with US, that does not have these limitations. We
have obtained an excellent agreement between both techniques. This is very
good news because Seeding is much more efficient than US and has access to
much lower values of the nucleation rate.
Figure 6: Free energy for two different pressures (p=0.031 and p=0.033) versus the number of particles in the drop as obtained from US calculations. The different colours represent the different windows performed. Label | $L$ | $N_{T}$ | $\rho_{vap}$ | $\rho_{l}$ | $p_{vap}$ | $p_{l}$ | $\Delta p$ | $R_{c}$ | $\gamma$ | $\Delta G_{c}/(k_{B}T)$ | $A_{0}^{BK}$ | $\log_{10}(J)$
---|---|---|---|---|---|---|---|---|---|---|---|---
I | 36.731 | 30795 | 0.03765 | 0.6453 | 0.02365 | -0.02601 | 0.0497 | 7.35 | 0.1826 | 52.7 | 0.341 | -23.5
II | 36.731 | 30342 | 0.03834 | 0.6484 | 0.02398 | -0.01914 | 0.0431 | 8.50 | 0.1832 | 70.6 | 0.342 | -31.3
III | 36.731 | 29760 | 0.03875 | 0.6501 | 0.02419 | -0.01503 | 0.0392 | 9.53 | 0.1869 | 90.6 | 0.345 | -40.0
IV | 36.731 | 29034 | 0.03907 | 0.6514 | 0.02433 | -0.01191 | 0.0362 | 10.52 | 0.1906 | 112.5 | 0.348 | -49.5
V | 36.731 | 28147 | 0.03949 | 0.6530 | 0.02453 | -0.00776 | 0.0323 | 11.50 | 0.1857 | 131.1 | 0.344 | -57.6
VI | 36.731 | 27082 | 0.03972 | 0.6539 | 0.02464 | -0.00558 | 0.0302 | 12.45 | 0.1881 | 155.7 | 0.346 | -68.3
Table 4: NVT-Seeding data for the different bubbles studied in this work at
T=0.785.
(a) (b)
(c) (d)
Figure 7: (a) $\Delta p$ vs. $1/R_{c}$, (b) $\gamma$ vs. $1/R_{c}$, (c)
$\Delta G_{c}$ vs. $R_{c}$, and (d) $log_{10}J$ vs. $1/R_{c}$ for droplets
(black symbols) and bubbles (red symbols) as obtained from NVT-Seeding.
## V Condensation vs boiling
### V.1 Comparison for a given $R_{c}$
We have studied quite recently the nucleation of bubbles for the same Lennard
Jones model employed here seedingNVT . Since the study was performed at the
same temperature, the question that naturally arises is whether bubbles and
drops with the same radius have the same interfacial properties. To establish
the comparison we have repeated the analysis performed in Ref. seedingNVT
because in such work we used 0.026 as the coexistence pressure instead of
0.0267. We took the 0.026 value from a paper published more than a decade ago
wang2008homogeneous . However, we have recomputed more carefully the
coexistence pressure at $T=0.785$ and obtain $p=0.0267$ instead, which is the
value we use in this work. The difference is subtle, but given that the
pressure inside the nucleating phase is obtained by integrating from the
coexistence pressure (see Eq. 11) it is very important to use an accurate
value for the latter.
The simulation data for different bubbles equilibrated at T=0.785 in the NVT
ensemble are reported in table 4. The values of $R_{c}$ corresponding to each
system (obtained with the equi-density criterion as discussed in section III.1
and in Ref. seedingNVT ) are very close to those recently reported by
ourselves seedingNVT . However, the values of $\Delta p$ here reported are not
identical to those of Ref.seedingNVT due to the coexistence pressure issue
discussed above. In Fig.7(a) we plot $\Delta p$ vs $1/R_{c}$ for bubbles and
and drops at T=0.785. Drops and bubbles of the same size have the same $\Delta
p$, which is perhaps the most important result of the paper. Note that, for a
given $R_{c}$, the pressures of the external and the internal phases are not
the same if one compares cavitation and condensation. What is the same is the
pressure difference between the internal and the external phases. For
instance, let’s focus on the case of drop VII and bubble II, both with
$R_{c}\approx 8.5$. In Fig. 8 we compare their radial density profiles. The
density of the liquid inside the drop is different from that of the liquid
outside the bubble. Also, the density of the vapor inside the bubble is
different from that of the vapor outside the drop. The bubble is surrounded by
a liquid of pressure -0.01914 whereas the drop by a vapor of pressure 0.0301:
the pressures of the external phases do not even have the same sign. The
bubble and the drop also have very different pressures: 0.02398 and 0.07432
respectively. Despite the fact that the external and the internal pressures
are very different, $\Delta p$ is not: 0.043 and 0.044 for the bubble and the
drop respectively.
Figure 8: Radial density profiles of drop VII and bubble II, compared. They
have almost identical radius, $R_{c}$.
According to the Laplace equation, that $\Delta p(R_{c})$ is the same for
drops and bubbles, implies that $\gamma$ must be the also same regardless the
identity of the internal and the external phases. In Fig. 7 (b) we plot
$\gamma$ vs $1/R_{c}$ for bubbles and drops and find that, indeed, they have
the same $\gamma$ within our statistical noise. Of course, attending to Eq. 3,
$\Delta G_{c}$, that only depends on $R_{c}$ and $\Delta p$, is also the same
for a given $R_{c}$, as illustrated in Fig. 7(c).
The nucleation rate for bubbles with a given $R_{c}$ is close to the
corresponding drop, but is not exactly the same, given that the kinetic pre-
factor is not identical. In the case of bubble nucleation we have carefully
assessed seedingNpT that the following expression by Blander and Katz (BK)
provides a good approximation for $A_{0}$:
$A_{0}^{BK}=\sqrt{\frac{\Delta pR_{c}}{\pi m}},$ (16)
which is very similar, but with a missing $(\rho_{vap}/\rho_{l})$ factor with
respect to the $ktg$ expression we use for drop condensation (Eq. 10). The
$A_{0}^{BK}$ values we use in our Seeding predictions of bubble cavitation are
reported in table 4 alongside the resulting values of $J$ obtained as
$J=\rho_{l}A_{0}^{BK}\exp[{-\Delta G_{c}/(k_{B}T)}$]. As it can be seen in
Fig. 7(d), the rate for bubbles and drops for a given $R_{c}$ is quite
similar, although it is systematically lower for the latter due to the
$\rho_{vap}/\rho_{l}$ factor previously mentioned.
Condensation and cavitation have already been compared in the literature
guermeur1985density ; schmelzer2019entropy ; macdowell2006nucleation ;
JCP_2004_120_05293 ; block2010curvature ; binder2012beyond ; caupin2015 .
However, there are only few cases in which $\gamma$ has been compared for a
given temperature as a function of the droplet/bubble size block2010curvature
; binder2012beyond as we do in this work. Establishing such comparison in
experiments is difficult because it is not possible to detect the critical
nucleus. In simulations the nucleus can be visualized, but computing $\gamma$
is a hard task. It requires either computing the free energy of a system with
the nucleus inside blokhuis2006thermodynamic ; binder2012beyond or, more
easily, computing the nucleus size and using a theory to infer $\gamma$
baiJCP2006 ; carignano as we do in this work. In either approach, one has to
deal with the arbitrariness of establishing a location for the interface.
In our case, we found in a recent work by “trial and error” that the equi-
density surface gives good results for cavitation seedingNpT . By “good
results” we mean that Seeding predictions of nucleation are consistent with
those coming from independent methods that do not rely on a precise definition
of the nucleus size. In this work we have demonstrated that the same criterion
to locate the interface is successful in condensation. Therefore, one of our
main findings is that the equi-density surface is the one that provides good
predictions when CNT is used both for cavitation and for condensation. This
means that the equi-density surface can be identified with the surface of
tension, which is the one for which CNT works and the Laplace equation holds
(see section III) kashchievbook ; montero2020interfacial ;
troster2012numerical ; statt2015finite . We believe that identifying the
surface of tension with the equi-density surface both for cavitation and
condensation is an important finding of our work. This leads to the relevant
conclusion that condensation and cavitation are two sides of the same coin in
the sense that they share the same surface tension.
In Ref. block2010curvature $\gamma$ was found to be quite different for both
phenomena, but the comparison was not established for the surface of tension
but for the equimolar surface. In Ref. binder2012beyond , however, the
comparison was established for the first time for the surface of tension and,
although $\gamma$ was similar for condensation and cavitation, there were
significant differences that need to be further investigated in order to match
our work with that of Ref. binder2012beyond .
### V.2 Comparison for a given metastability degree
In Ref. shen2001density it was proposed in a DFT study that the work of
formation of critical bubbles studied at different temperatures collapse when
plotted against the metastability degree, $X_{m}$, quantified as:
$X_{m}=\frac{\mu_{nuc}-\mu_{coex}}{\mu_{spinodal}-\mu_{coex}}$ (17)
where $\mu_{nuc}$ is the chemical potential of the parent phase at the
conditions where nucleation is studied, $\mu_{coex}$ is the coexistence
chemical potential at the same temperature and at coexistence pressure, and
$\mu_{spinodal}$ is the chemical potential at the same temperature but at the
pressure where spinodal decomposition takes place. To estimate the spinodal
pressure we run NpT simulations of the bulk liquid and vapor phases with 4000
particles. We estimate the spinodal decomposition pressure as that for which
we the system undergoes a phase transition without any induction period, right
after the start of the simulation. Both chemical potential differences in Eq.
17 can be easily obtained by numerically integrating the molar volume along
pressure at constant temperature. The denominator is the maximum possible
metastability whereas the numerator is the actual metastability of the state
where nucleation is studied. Therefore, $X_{m}$ varies from 0 at coexistence,
to 1 at spinodal decomposition. The mestastability degree above described can
be computed for drop as well as for bubble nucleation. Therefore, we have the
chance to compare nucleation free energy barriers for drops and bubbles as a
function of $X_{m}$. The comparison, shown in Fig. 9, reveals the interesting
conclusion that $\Delta G_{c}$ for bubble and drop nucleation is the same for
a given metastbility degree. Therefore, not only nucleation barriers at
different temperatures can be collapsed via the metastability degree as
proposed in Ref. shen2001density , but also bubble and drop nucleation data
match for a given metastability degree.
Figure 9: Nucleation free energy barrier for drops and bubbles (see legend) as
a function of the metastability degree, $X_{m}$, defined in Eq. 17.
### V.3 Tolman Legth
Since bubbles and drops of the same radius have the same interfacial
properties, we can use the data coming from both systems altogether in order
to compute the Tolmann Lenght, $\delta_{T}$, which is defined as
tolman1949effect ; blokhuis2006thermodynamic :
$\delta_{Tolman}=\lim_{R_{s}\rightarrow\infty}(R_{e}-R_{s})$ (18)
where $R_{e}$ is the Gibbs equi-molar radius and $R_{s}$ is the radius of the
surface of tension. We identify $R_{s}$ with $R_{c}$ (the equi-density radius)
given that (i) we obtain good predictions of nucleation when we use $R_{c}$
and (ii) $R_{s}$ is the radius that enters CNT kashchievbook ;
montero2020interfacial ; troster2012numerical ; statt2015finite . To underline
the fact that we identify $R_{c}$ with $R_{s}$ we label $R_{c}$ as $R_{s=c}$
in the following figures. $R_{e}$ can be easily computed from the radial
density profiles seedingNpT ; montero2020interfacial . In Fig. 10(a) we show
$R_{e}-R_{s=c}$ versus 1/$R_{s=c}$ for all data (either bubbles or drops)
coming from this work. The extrapolation to $1/R_{s=c}=0$ provides an estimate
of $\delta_{Tolman}$, indicated with an empty blue dot in the figure. We
obtain $\delta_{Tolman}=0.15\pm 0.02$. We showed in a recent paper, in which
we analysed spherical hard sphere crystals in equilibrium with the fluid, that
$\delta_{Tolman}$ can be also estimated by fitting $\gamma$ to the following
expression:
$\gamma=\gamma_{0}\left(1-2\frac{\delta_{T}}{R_{s}}\right),$ (19)
where $\gamma_{0}$ is the value of $\gamma$ at coexistence at the temperature
of interest and $\delta_{T}$ is the fitting parameter that serves as an
estimate for $\delta_{Tolman}$ montero2020interfacial . This approach is
similar in spirit to those that include $\gamma$ given by Eq. 19 in CNT to fit
free energy barriers obtained by rare event methods gallo2020nucleation ;
menzl2016molecular . Again, we identify here $R_{s}$ with $R_{c}$.
Consequently, we use the $\gamma$ data coming from such radius (that reported
in tables 1 and 4) to obtain an estimate of $\delta_{T}$ with the expression
above. The data of $\gamma$ vs $1/R_{s=c}$ are shown in green in Fig. 10(b).
The solid line is a linear fit of $\gamma$ vs $1/R_{s=c}$ which includes
$\gamma_{0}$ (the green square in the figure). The $\delta_{T}$ value coming
from such fit, $\delta_{T}=0.21\pm 0.03$, is shown with a red dot in Fig.
10(a). Both values, $\delta_{Tolman}$ obtained via Eq. 18 (blue dot in Fig.
10(a)) and $\delta_{T}$ coming from Eq. 19 (red dot the same figure), are
consistent with each other within the statistical uncertainty of our
estimates. This corroborates the idea, recently checked for the first time for
hard sphere crystals montero2020interfacial , that the Tolman lenght can be
obtained either from Eq. 19 or from Eq. 18. Hence, this idea seems to be a
general one pertaining not only to the crystal-fuid equilibrium but also to
the liquid-vapor one.
This study may shed some light in the intense literature debate about the
magnitude and sign of the Tolman length blokhuis2006thermodynamic ;
block2010curvature ; sampayo ; malijevsky2012perspective ; binder2012beyond ;
size-dependendentgamma2005 ; ceriottitolman2018 ; JCP_1984_81_00530 ;
lau2015surface ; vrabec2006comprehensive ; wilhelmsen2015tolman ;
joswiak2016energetic ; schmelzer2019entropy ; richard2018crystallization ;
caupinPRL2016curvature . We obtain a Tolman length of about twenty per cent
the particle diameter. Its sign is positive, which means that $\gamma$
decreases when one moves away from coexistence at constant temperature.
(a)
(b)
Figure 10: (a) $R_{e}-R_{s=c}$ and (b) $\gamma$ vs. $1/R_{s=c}$ for drops and
bubbles together.
## VI Conclusions
The main conclusions we draw from our work are the following:
1. 1.
We have used NVT-Seeding to investigate droplet nucleation in a supersaturated
Lennard-Jones vapor. The results obtained from this technique are consistent
with: (i) independent calculations of the nucleation free energy barrier
performed with Umbrella Sampling (ii) the surface tension of a flat interface
obtained from the pressure tensor in a vapor-liquid coexistence simulation
(iii) the drop nucleation rate obtained both with US and in brute force
spontaneous nucleation simulations.
2. 2.
NVT-Seeding requires defining the radius of a droplet equilibrated in the NVT
ensemble. The radius definition that passes the consistency tests mentioned in
the previous paragraph is that given by the surface where the density is
average between that of the interior and that of the exterior phases. Such
radius definition was also successful in our earlier studies of bubble
nucleation seedingNpT ; seedingNVT . Therefore, we identify this “equi-
density” radius with the radius of tension, $R_{s}$.
3. 3.
The good performance of Seeding strongly supports the use of CNT to describe
nucleation. However, the capillarity approximation (that $\gamma$ is curvature
independent) does not provide good results. A $\gamma$ dependent on the
curvature of the critical nucleus must be plugged into the the theory.
Therefore, the theory, although powerful, requires the involvement of
simulations given that the $\gamma$-curvature dependence is obtained by
computing the size of the critical cluster at different pressures.
4. 4.
The kinetic theory of gases provides very good estimates of the kinetic pre-
factor of the condensation nucleation rate. This makes the theoretical
framework very powerful given that only the size of the critical cluster, the
density of the external phase and the bulk phases equations of state are
needed to estimate nucleation rates.
5. 5.
We compare NVT-Seeding results of droplets with those obtained for bubbles and
find that, for a given temperature, bubbles and droplets of the same radius
have, within the accuracy of our method, the same pressure difference with the
surrounding medium. Therefore, bubbles and droplets of the same size have the
same surface tension and the same nucleation free energy barrier. In this
respect, condensation and boiling can be seen as two sides of the same coin.
Such duality is only verified if the size of the critical nucleus (either a
bubble or a drop) is determined with the equi-density radius (our empirical
definition of the surface of tension).
6. 6.
We estimate the Tolman length, $\delta_{T}$, by extrapolating to infinite-size
drops/bubbles the difference between the equimolar radius, $R_{e}$, and
$R_{s}$. Such $\delta_{T}$ is consistent with that obtained by linearly
fitting $\gamma(1/R_{c})$, in accordance with our recent work of hard sphere
crystals montero2020interfacial .
## VII Acknowledgments
This work was funded by grants FIS2016/78117-P and PID2019-105898GB-C21 of the
MEC. The authors acknowledge the computer resources and technical assistance
provided by the RES. P. R. thanks a doctoral grant from UCM. P. M. H.
acknowledges financial support from the FPI grant no. BES- 671 2017-080074. We
also thank the reviewers of this work for their suggestions.
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# Relativistic quantum information as radiation reaction: entanglement entropy
and self-force of a moving mirror analog to the CGHS black hole
Aizhan Myrzakul1<EMAIL_ADDRESS>Chi Xiong2,3<EMAIL_ADDRESS>Michael R.R. Good1<EMAIL_ADDRESS>1Physics Department & Energetic
Cosmos Laboratory, Nazarbayev University,
53 Kabanbay Batyr, Nur-Sultan, 010000, Kazakhstan
2School of Mathematical and Physical Sciences, Nanyang Technological
University,
50 Nanyang Avenue, 639798, Singapore
3Department of Mathematics and Applied Mathematics,
Xiamen University Malaysia,
Selangor, 43900, Malaysia
###### Abstract
The CGHS black hole has a spectrum and temperature that corresponds to an
accelerated reflecting boundary condition in flat spacetime. The beta
coefficients are identical to a moving mirror model where the acceleration is
exponential in laboratory time. The center and the event horizon of the black
hole are at the same location modeled by the perfectly reflecting regularity
condition that red-shifts the field modes. In addition to computing the energy
flux, we find the corresponding parameter associated with the black hole mass
and the cosmological constant in the gravitational analog system. Generalized
to any mirror trajectory we derive the self-force (Lorentz-Abraham-Dirac) and
express it and the power (Larmor) in connection with entanglement entropy,
inviting an interpretation of acceleration radiation in terms of information
flow. The mirror self-force and radiative power are applied to the particular
CGHS black hole analog moving mirror which reveals the physics of information
at the horizon during asymptotic approach to thermal equilibrium.
## I Introduction
Three decades ago, several (1+1)-dimensional black hole models were introduced
to gain insight into the quantum nature of black hole radiation, with one of
the most prominent and physically interesting models being the Callan-
Giddings-Harvey-Strominger (CGHS) system Callan _et al._ (1992). Simplified
CGHS models, albeit with certain limitations, are exactly soluble and lead to
many associated discoveries. New surprises related to complexity,
temperatures, and entropy are still being found Strominger (1994); Giddings
(1994); Thorlacius (1995); Fabbri and Navarro-Salas (2005); Ashtekar _et al._
(2011).
Moving mirrors are accelerated boundaries that create energy, particles, and
entropy. They are simplified (1+1)-dimensional versions of the dynamical
Casimir effect Moore (1970); DeWitt (1975). Interesting in their own right,
they also act as toy models for black hole evaporation Hawking (1975); Unruh
(1976); Fulling and Davies (1976); Davies and Fulling (1977); Juárez-Aubry and
Louko (2018); Cong _et al._ (2020); Wilson _et al._ (2020). The general and
physically relevant connections of moving mirrors to black hole physics can be
found in canonical textbooks Fabbri and Navarro-Salas (2005); Birrell and
Davies (1984) and also in recent works, e.g. Good _et al._ (2017a); Myrzakul
and Good (2018); Good _et al._ (2019); Lin _et al._ (2020).
There have been a number of studies that relate different specific black hole
models (e.g. the Schwarzschild Good _et al._ (2016) case) and their analog
moving mirrors, including the extremal Reissner-Nordström Liberati _et al._
(2000); Good (2020a), extremal Kerr Rothman (2000), Reissner-Nordström Good
and Ong (2020), Taub-NUT Foo _et al._ (2020) and Kerr Good _et al._ (2020a)
black holes. In addition, de Sitter and anti-de Sitter cosmologies Good _et
al._ (2020b) are also modeled by moving mirror trajectories. For appropriately
chosen trajectories Good and Linder (2019), close comparisons can be made with
the radiation emitted from dynamic spacetimes Wilczek (1993); Good and Linder
(2017). Such an equivalence between a mirror and a curved spacetime is called
an accelerated boundary correspondence (ABC).
Our motivation in this paper is to synthesize and strongly link the well-known
and important CGHS black hole model with its analog moving mirror counterpart.
In the process we want to derive the spectrum of particle production exactly
and analytically, drawing close parallels between the two systems via the
temperature, horizons and parameter analogs associated with CGHS black hole
mass and cosmological constant. Furthermore, we aim to initiate an
investigation into the entanglement entropy of a generalized mirror system and
its relationship to the self-force on the mirror and power of the emitted
vacuum radiation. As we shall see, this link of inquiry reveals a close
connection between the seemingly distinct concepts of self-force and
information. Application of the results for the CGHS mirror reveals the
divergent self-force is directly a consequence of information loss.
Additionally, we push the correspondence further, by considering the close
connection to classical electrodynamic analogies. The moving mirror is found
to behave almost like a neutral particle coupled to the massless scalar field
(similar to a charged particle coupled to the electromagnetic field). Hence
some of the familiar radiation results in classical electrodynamics has direct
correspondence in the mirror case.
The paper is organized in the following: in Sec. II we briefly review the CGHS
action, the corresponding field equations of motion, and the formation of the
CGHS black hole. In Sec. III, the details of the CGHS metric and the
transformation of this coordinate system to an accelerated mirror trajectory
are investigated. In Sec. IV, we derive the particle flux radiated from the
exponentially accelerated mirror in laboratory time and demonstrate its
thermal characters for late-times. In Sec. V, the particle flux radiated from
the CGHS black hole is reviewed. Sec. VI and VII are dedicated to an overview
of the quantum stress tensor and mirror entanglement entropy, respectively. In
Sec. VIII, the moving mirror Larmor formula and Lorentz-Abraham-Dirac (LAD)
force analogs are derived with an emphasis on entanglement entropy. In Sec.
IX, the formulas derived in the previous section are used to find the
radiative power and radiation reaction force for our particular moving mirror
(CGHS). In the last section, Sec. X, the properties of the correspondence
between the CGHS black hole and the exponentially accelerated mirror as well
as the relation between the moving mirror and electrodynamics are summarized.
In addition, some insight into future directions are provided. We use
$\hbar=c=1$ except in the results of Eq. (111) and Eq. (112) of Sec. IX.
## II Action & Field Equation
In this section we will briefly summarize the action and field equations of
the Callan–Giddings–Harvey–Strominger model in which a linear dilaton vacuum
evolves into a black hole from matter injection. The CGHS action reads as,
$S=\frac{1}{2}\int
d^{2}x\sqrt{-g}\left[e^{-2\phi}\left(R+4(\nabla\phi)^{2}+4\Lambda^{2}\right)-|\nabla\chi|^{2}\right],$
(1)
where $g$ is the metric tensor, $\phi$ is the dilaton field, $\Lambda$ is the
cosmological constant, and $\chi$ are the matter fields. To obtain the
equations of motion one may vary the action, Eq. (1), with respect to the
metric $g^{ab}$ and the dilaton field $\phi$, respectively,
$\displaystyle
2e^{-2\phi}\left[\nabla_{a}\nabla_{b}\phi+g_{ab}((\nabla\phi)^{2}-\nabla^{2}\phi-\Lambda^{2})\right]$
$\displaystyle=$ $\displaystyle T_{ab},$ (2) $\displaystyle
e^{-2\phi}\left[-R+4(\nabla\phi)^{2}-4\nabla^{2}\phi-4\Lambda^{2})\right]$
$\displaystyle=$ $\displaystyle 0.$ (3)
where
$T_{ab}\equiv\nabla_{a}\chi\nabla_{b}\chi-\frac{1}{2}g_{ab}(\nabla\chi)^{2}$.
Following Fabbri and Navarro-Salas (2005), one can readily solve Eq. (2) in
the conformal gauge, $ds^{2}=-e^{2\rho}dx^{+}dx^{-}$. The solution is
$\rho=\phi$ (gauge fixing) and
$e^{-2\rho}=\frac{M(x^{+})}{\Lambda}-\Lambda^{2}x^{+}\left(x^{-}+\frac{\mathcal{C}(x^{+})}{\Lambda^{2}}\right),$
(4)
where the functions $M(x^{+})$ and $\mathcal{C}(x^{+})$ are integrals
depending on the stress-energy tensor of the matter field, connected to the
mass of CGHS black hole and the event horizon, respectively. Assume that we
start with a linear dilaton vacuum, then turn on the matter flux injected to
the system at some time $x^{+}_{i}$ and turn it off after the time
$x^{+}_{f}$. Then when $x^{+}>x^{+}_{f}$ the geometry of the system will
approach and finally settle down to the static CGHS black hole background. The
value $M(x^{+}_{f})$ becomes the mass of the black hole and
$\mathcal{C}(x^{+}_{f})$ gives the curve of the event horizon. Therefore one
can observe how the linear dilaton vacuum (before the time $x^{+}_{i}$)
evolves into a CGHS black hole (after the time $x^{+}_{f}$) due to the matter
injection.
## III CGHS black hole and matching condition
In this section we concentrate on the CGHS black hole solution and some of the
significant observable quantities within this model. The relevant metric for
the CGHS black hole system can be cast in the form Giddings (2016) (c.f.
Schwarzschild gauge Fabbri and Navarro-Salas (2005)),
$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2},$ (5)
where
$f(r)=1-\frac{M}{\Lambda}e^{-2\Lambda r},$ (6)
with $\Lambda>0$ as the cosmological constant parameterization scale of the
spacetime and $M>0$ as the mass of the CGHS black hole. The curve of the event
horizon function $\mathcal{C}(x^{+})$ is set to zero in Eq. (4) in order to
obtain the metric for stationary CGHS black hole, Eq. (5). Notice that when
$M=\Lambda$, Eq. (5) possesses a singularity at the radial coordinate $r=0$
which also reduces to the event horizon, in contrast to the coordinate
singularity of the Schwarzschild black hole event horizon (see e.g. the
Schwarzschild mirror Good _et al._ (2016); Good (2017); Anderson _et al._
(2017); Good _et al._ (2017b)). For general $M$ and $\Lambda$, the horizon is
at $r_{H}=\frac{1}{2\Lambda}\ln\frac{M}{\Lambda}$. The surface gravity of the
CGHS black hole can be obtained as Derek and Edwin (2009),
$\displaystyle\kappa$
$\displaystyle=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}r}f(r)\Big{|}_{r=r_{H}}=\Lambda,$
(7)
where $r_{H}=0$, so that consistency with the laws of black hole
thermodynamics dictates the temperature is $T=\Lambda/(2\pi)$. For a double
null coordinate system ($u,v$) with $u=t-r^{\star}$ and $v=t+r^{\star}$, the
associated tortoise coordinate $r^{\star}$ can be obtained in the usual way
Fabbri and Navarro-Salas (2005), via
$\displaystyle r^{\star}$ $\displaystyle=\int\frac{\mathrm{d}r}{f(r)},$ (8)
which yields
$r^{*}=\frac{1}{2\Lambda}\ln\left|\frac{M}{\Lambda}-e^{2\Lambda r}\right|.$
(9)
The absolute brackets are critical for real coordinate values.
Following Wilczek Wilczek (1993), let us coincide the inner and outer regions
of a collapsing null shell to form a black hole, where the exterior background
is given by the CGHS metric,
$ds^{2}=\begin{cases}-dt_{in}^{2}+dr^{2},&\text{for $t_{in}+r\leq v_{0}$},\\\
-fdt_{out}^{2}+f^{-1}dr^{2},&\text{for $t_{out}+r\geq v_{0}$}.\end{cases}$
(10)
and $v_{0}$ is a light-like shell. In null coordinates, the system Eq. (10)
can be rewritten as,
$ds^{2}=\begin{cases}-dUdV,&\text{where $U=t_{in}-r$, $V=t_{in}+r$},\\\
-fdudv,&\text{where $u=t_{out}-r_{*}$, $v=t_{out}+r_{*}$}.\end{cases}$ (11)
So, the metric for the geometry describing the outside region of a collapsing
shell takes the simplified form, $\mathrm{d}s^{2}=-fdudv$.
The matching condition (see Fabbri and Navarro-Salas (2005); Wilczek (1993);
Good _et al._ (2020a)) with the flat interior geometry, described by the
interior coordinates $(U,V)$ is the trajectory corresponding to $r=0$,
expressed in terms of the exterior function $u(U)$. We can obtain this
matching via the association $r=r^{\star}$, and taking
$r^{\star}(r=(v_{0}-U)/2)=(v_{0}-u)/2$ along the light ray, $v_{0}$. We set
$v_{0}=0$ for simplicity without loss of generality. This matching condition,
$u(U)=-\frac{1}{\Lambda}\ln\left|\frac{M}{\Lambda}-e^{-\Lambda U}\right|,$
(12)
is the outside $u$ trajectory of the origin as a function of the inside
coordinate $U$. We can write this as,
$u(U)=U-\frac{1}{\Lambda}\ln\left(1-\frac{M}{\Lambda}e^{\Lambda U}\right),$
(13)
where $U<0$ and $\Lambda>0$. The regularity condition of the modes requires
that they vanish at $r=0$, which acts as a reflecting boundary in the black
hole system. In the accelerated boundary correspondence (ABC) of the mirror
system, the origin of the black hole functions as the mirror trajectory in
flat spacetime. The position of the origin is a dynamic function $u$ with
independent variable $U$. Since the field vanishes (does not exist for $r<0$),
the form of the field modes can be determined, allowing for the identification
$U\Leftrightarrow v$ (where $v$ is the flat spacetime advanced time in the
moving mirror model) for the Doppler-shifted field modes. In the next section
we will define the analog mirror trajectory for the CGHS spacetime by making
the identification $u(U)\Leftrightarrow f(v)$, which is a known function of
the advanced time $v$.
## IV Exponentially Accelerated Mirror
In this section we focus on the trajectory and particle flux radiation of the
exponentially accelerated mirror in coordinate time to demonstrate their
equivalence with the corresponding quantities in the CGHS black hole model.
In line with previous accelerated boundary correspondences, consider the
exponentially accelerated mirror trajectory with proper acceleration Good and
Linder (2018):
$\alpha(t)=-\frac{\kappa}{2}e^{\kappa(t-v_{H})},$ (14)
where $\kappa>0$ is a parameter of the acceleration and $v_{H}$ is the horizon
in advanced time, $v=t+x$. This $x$ and $t$ are the usual lab coordinates of
flat (1+1)-dimensional Minkowski spacetime. The trajectory in light cone
coordinates as a function of advanced time is
$f(v)=v-\frac{1}{\kappa}\ln\left(1-e^{\kappa(v-v_{H})}\right),$ (15)
where, identifying Eq. (15) with Eq. (13) as usual (see prior ABCs), the
associated parameters in the CGHS system define the moving mirror’s null-ray
horizon,
$v_{H}=\frac{1}{\Lambda}\ln\left(\frac{\Lambda}{M}\right),$ (16)
which is the location that the last incoming left-moving ray reflects off the
mirror. Past this position, there is no more reflection and left-mover modes
never make it to an observer at right null-infinity, $\mathscr{I}^{+}_{R}$.
The mirror horizon couples the parameters $\Lambda$ and $M$, which are the
cosmological constant and mass of the black hole, respectively, in the CGHS
system. The fact that $v_{H}$, which is the finite $v$ for the mirror horizon,
is also closely related to the CGHS black hole horizon, through
$2r_{H}=-v_{H}$, further corroborates a correspondence between the CGHS black
hole and the exponentially accelerated mirror. A spacetime plot of this
asymptotic light-like moving mirror is given in Fig. 1. A Penrose conformal
diagram is given in Fig. 2. Notice when $v_{H}=0$, Eq. (15) is Eq. (13), i.e.
$u(U)\Leftrightarrow f(v)$, when $\Lambda=M$.
It is interesting to note that in Schwarzschild black hole case the
singularity is located at the center, that corresponds to the mirror, and the
event horizon is located at $r=2M$. Unlike the Schwarzschild black hole, for
the CGHS black hole the singularity happens at $r=0$ which is the location of
event horizon as well when $\Lambda=M$. So, the mirror mimics both the event
horizon and the center of the black hole, simplifying physical interpretation
and giving a straightforward answer to the origin of particle creation in the
CGHS system.
Now we will derive the thermal Planck distribution of the exponentially
accelerated moving mirror particles by use of the beta Bogolubov coefficient.
The beta coefficient can be found via an integration Good _et al._ (2017a) by
parts where we ignore non-contributing surface terms,
$\beta_{\omega\omega^{\prime}}=\frac{1}{2\pi}\sqrt{\frac{\omega^{\prime}}{\omega}}\int_{-\infty}^{v_{H}=0}dv\;e^{-i\omega^{\prime}v}e^{-i\omega
f(v)},$ (17)
to get
$\beta_{\omega\omega^{\prime}}=\frac{1}{2\pi\kappa}\sqrt{\frac{\omega^{\prime}}{\omega}}B\left[-\frac{i\omega_{+}}{\kappa},1+\frac{i\omega}{\kappa}\right],$
(18)
where we utilize the Euler integral of the first kind as a Beta function,
$B(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$, and
$\omega_{+}=\omega+\omega^{\prime}$. Multiplying by its complex conjugate
gives the particle count per $\omega^{\prime}$ mode, per $\omega$ mode:
$|\beta_{\omega\omega^{\prime}}|^{2}=\frac{1}{4\pi^{2}\kappa^{2}}\frac{\omega^{\prime}}{\omega}\left|B\left[\frac{i\omega_{+}}{\kappa},1-\frac{i\omega}{\kappa}\right]\right|^{2},$
(19)
or, equivalently expressed,
$|\beta_{\omega\omega^{\prime}}|^{2}=\frac{e^{\frac{2\pi\omega}{\kappa}}\left(e^{\frac{2\pi\omega^{\prime}}{\kappa}}-1\right)}{2\pi\kappa\omega_{+}\left(e^{\frac{2\pi\omega}{\kappa}}-1\right)\left(e^{\frac{2\pi\omega_{+}}{\kappa}}-1\right)}.$
(20)
Thermal character results in the high frequency limit
$\omega^{\prime}\gg\omega$ approximation (a good explanation for how this
corresponds to late times is given by Hawking Hawking (1975)). We can see by
inspection that,
$|\beta_{\omega\omega^{\prime}}|^{2}\approx\frac{1}{2\pi\kappa\omega^{\prime}}\frac{1}{e^{2\pi\omega/\kappa}-1}\quad\text{for}\;\omega^{\prime}\gg\omega$
(21)
so that $T=\kappa/(2\pi)$ at late times.
Figure 1: Spacetime diagram with $M=\Lambda=2$, of Eq. (15), $f(v)$. The
mirror starts asymptotically timelike and finishes asymptotically lightlike
with infinite acceleration along the horizon $v_{H}=0$. Figure 2: Penrose
diagram with Blue, Green, Red, Black, parametrized respectively with
$\kappa=1/2,1,2,4$ of Eq. (15), $f(v)$. As in Figure 1 (the Red line is the
same trajectory) the mirror starts asymptotically timelike and finishes
asymptotically lightlike with infinite acceleration along the horizon
$v_{H}=0$.
## V CGHS particle radiation
In this section we calculate the particle flux radiated by the CGHS black hole
and the corresponding beta Bogolubov coefficients, following the standard
procedure (see e.g. Fabbri and Navarro-Salas (2005)). Remarkably, the beta
Bogolubov coefficients match those corresponding ones obtained from the
particle radiation of the exponentially accelerating mirror (Section IV).
The standard Bogolubov procedure for calculating the Hawking radiation
considers two relevant regions as mentioned in Sec. II: the linear dilaton
vacuum and the CGHS black hole, described by the corresponding “in” and “out”
coordinates which can be connected via the Kruskal coordinates (see Fabbri and
Navarro-Salas (2005) for details and elaboration on the standard notation).
One then identifies plane wave modes for ingoing and outgoing sectors using
null Minkowski coordinates $(\sigma_{in}^{\pm},\sigma_{out}^{\pm})$. They are
related as,
$\sigma_{in}^{+}=\sigma_{out}^{+}=\sigma^{+},$ (22)
$\sigma_{in}^{-}=-\frac{1}{\Lambda}\ln\left(e^{-\Lambda\sigma_{out}^{-}}+\frac{\mathcal{C}(x_{f}^{+})}{\Lambda}\right).$
(23)
So, using $\sigma^{-}$ sector the plane wave modes have the following forms,
$g_{\omega^{\prime}}^{in}=\frac{1}{\sqrt{4\pi\omega^{\prime}}}e^{-i\omega^{\prime}\sigma_{in}^{-}},$
(24)
$g_{\omega}^{out}=\frac{1}{\sqrt{4\pi\omega}}e^{-i\omega\sigma_{out}^{-}},$
(25)
designated by $g$ (sometimes $u$ is used but here $u$ is already a retarded
time null coordinate). The next step is to evaluate the beta Bogolubov
coefficients by calculating scalar product between the plane wave modes. For
our particular case the corresponding integral is,
$\beta_{\omega\omega^{\prime}}=(g_{\omega}^{out},g_{\omega^{\prime}}^{in*})=2i\int_{-\infty}^{+\infty}d\sigma_{in}^{-}g_{\omega}^{out}\frac{\partial
g_{\omega^{\prime}}^{in}}{\partial\sigma_{in}^{-}}.$ (26)
Substituting Eq. (24) and Eq. (25) into Eq. (26), and also using Eq. (23), the
above integral becomes,
$\beta_{\omega\omega^{\prime}}=\frac{1}{2\pi}\sqrt{\frac{\omega^{\prime}}{\omega}}\int_{-\infty}^{\sigma_{in,H}^{-}}d\sigma_{in}^{-}\frac{e^{\frac{i\omega}{\Lambda}\ln\left(e^{-\Lambda\sigma_{in}^{-}-\frac{\mathcal{C}(x_{f}^{+})}{\Lambda}}\right)}}{e^{i\omega^{\prime}\sigma_{in}^{-}}},$
(27)
where
$\sigma_{in,H}^{-}=\frac{1}{\Lambda}\ln\frac{\Lambda}{\mathcal{C}(x_{f}^{+})}$
(28)
is the black hole event horizon location that is formed when injecting matter
into linear dilaton vacuum. Note that Eq. (28) is similar to Eq. (16), i.e.
$v_{H}\Leftrightarrow\sigma_{in,H}^{-}$. Calculation of Eq. (27) gives,
$\beta_{\omega\omega^{\prime}}=\frac{1}{2\pi\Lambda}\sqrt{\frac{\omega^{\prime}}{\omega}}\left(\frac{\mathcal{C}(x_{f}^{+})}{\Lambda}\right)^{\frac{i\omega_{+}}{\Lambda}}B\left[-\frac{i\omega_{+}}{\Lambda},1+\frac{i\omega}{\Lambda}\right].$
(29)
The complex conjugate squared of Eq. (29) yields,
$|\beta_{\omega\omega^{\prime}}|^{2}=\frac{1}{4\pi^{2}\Lambda^{2}}\frac{\omega^{\prime}}{\omega}\left|B\left[\frac{i\omega_{+}}{\Lambda},1-\frac{i\omega}{\Lambda}\right]\right|^{2},$
(30)
which is exactly Eq. (19) given that $|\Lambda|=|\kappa|$. Interestingly, the
mass of the CGHS black hole, $M(x^{+}_{f})$, does not appear in
$\beta_{\omega\omega^{\prime}}$, and $\mathcal{C}(x^{+}_{f})$ disappears when
calculating $|\beta_{\omega\omega^{\prime}}|^{2}$. This is because the
spectrum of the CGHS black hole does not explicitly depend on its mass. The
correspondence between the black hole mass and the curve of the event horizon,
$\mathcal{C}(x_{f}^{+})$, can be seen from the comparison of Eq. (16) and Eq.
(28). The curve of the event horizon is defined by the so-called “apparent
horizon”, $x^{-}=-\mathcal{C}(x^{+})/\Lambda^{2}$, which is spacelike or null
and coincides with the event horizon after the matter injection has finished
at time $x^{+}_{f}$, i.e. after time $x^{+}_{f}$ when the geometry of the CGHS
black hole is settled and the apparent horizon becomes the event horizon.
Hereinafter, due to the identical particle production Eq. (19) between the
exponentially accelerated mirror in coordinate laboratory time and the CGHS
system Eq. (30), for short, we refer to this specific perfectly reflecting
boundary trajectory, Eq. (15), as the CGHS mirror.
## VI Quantum stress-tensor
In this section we will briefly review the quantum stress tensor of the moving
mirror following Davies and Fulling Fulling and Davies (1976); Davies and
Fulling (1977). We will need this to specialize to the CGHS mirror and compute
its energy flux which we do in Section IX. In (1+1)-dimensional flat spacetime
the energy-momentum tensor is determined by the following $2\times 2$ matrix,
$T_{\mu\nu}=\frac{1}{2}\begin{bmatrix}\left(\frac{\partial\phi}{\partial
t}\right)^{2}+\left(\frac{\partial\phi}{\partial
x}\right)^{2}&\frac{\partial\phi}{\partial x}\frac{\partial\phi}{\partial
t}+\frac{\partial\phi}{\partial t}\frac{\partial\phi}{\partial x}\\\
\frac{\partial\phi}{\partial t}\frac{\partial\phi}{\partial
x}+\frac{\partial\phi}{\partial x}\frac{\partial\phi}{\partial
t}&\left(\frac{\partial\phi}{\partial
t}\right)^{2}+\left(\frac{\partial\phi}{\partial x}\right)^{2}\end{bmatrix}.$
(31)
Here $\phi(t,x)$ is technically a free field that obeys the massless scalar
wave equation,
$\frac{\partial^{2}\phi}{\partial t^{2}}-\frac{\partial^{2}\phi}{\partial
x^{2}}=0,$ (32)
but there are effective interactions with the boundary condition,
$\phi(t,x)|_{z}=0,$ (33)
which is imposed on the field equation of motion, where $x=z(t)$ is the
trajectory of the moving mirror. In this two dimensional case, the mirror is a
perfectly reflecting point moving along a timelike worldline $z(t)$.
In the usual quantum field theory, $\phi(t,x)$ is an operator defined by field
modes as,
$\phi(t,x)=\int_{0}^{\infty}\left[\hat{a}_{\omega}^{in}\phi_{\omega}+\hat{a}_{\omega}^{in\dagger}\phi_{\omega}^{*}\right]d\omega,$
(34)
where $\hat{a}_{\omega}$ and $\hat{a}_{\omega}^{\dagger}$ are ladder
operators, and $\phi_{\omega}^{*}$ is a complex conjugate of $\phi$. After
inserting Eq. (34) into the matrix Eq. (31), the stress-tensor can be written
as,
$T_{\mu\nu}=:T_{\mu\nu}:+\braket{T_{\mu\nu}},$ (35)
where the first term has normal ordering of the ladder operators, i.e.
$:\hat{a}_{\omega}^{in}\hat{a}_{\omega}^{in\dagger}:=\hat{a}_{\omega}^{in\dagger}\hat{a}_{\omega}^{in}.$
(36)
The second term in Eq. (35) is the expectation value of the operator in
vacuum, which is defined as,
$\braket{T_{\mu\nu}}=\int_{0}^{\infty}T_{\mu\nu}(\phi_{\omega},\phi_{\omega}^{*})d\omega.$
(37)
The stress tensor here, as it stands, is of significant interest, but
unfortunately the above integral is divergent. In order to make the integral
in Eq. (37) finite and extract useful information about the emitted radiation,
point-splitting regularization is utilized. The key idea is to evaluate the
field modes at different times: $\phi$ at $(t,x)$ and $\phi^{*}$ at
$(t+\epsilon,x)$, respectively, where $\epsilon$ is an infinitesimally small
quantity. So, the field modes at corresponding points are:
$\begin{cases}\begin{matrix}\frac{\partial\phi_{\omega}}{\partial t}\\\
\frac{\partial\phi_{\omega}}{\partial
x}\end{matrix}\end{cases}=\sqrt{\frac{\omega}{4\pi}}\left[e^{-i\omega v}\mp
p^{\prime}(u)e^{-i\omega p(u)}\right],$ (38)
$\begin{cases}\begin{matrix}\frac{\partial\phi_{\omega}^{*}}{\partial t}\\\
\frac{\partial\phi_{\omega}^{*}}{\partial
x}\end{matrix}\end{cases}=\sqrt{\frac{\omega}{4\pi}}\left[e^{i\omega(v+\epsilon)}\mp
p^{\prime}(u+\epsilon)e^{i\omega p(u+\epsilon)}\right],$ (39)
where $p(u)\equiv 2t(u)-u$ and $u\equiv t-z(t)$. Inserting Eqs. (38) and (39)
into Eq. (37) leads to,
$\begin{cases}\begin{matrix}\braket{T_{00}}=\braket{T_{11}}\\\
\braket{T_{10}}=\braket{T_{01}}\end{matrix}=\frac{1}{4\pi}\int_{0}^{\infty}\omega\left[e^{i\omega\epsilon}\pm\frac{p^{\prime}(u)p^{\prime}(u+\epsilon)}{e^{-i\omega(p(u+\epsilon)-p(u))}}\right]d\omega.\end{cases}$
(40)
Calculation of the above integrals results in,
$\begin{cases}\begin{matrix}\braket{T_{00}}\\\
\braket{T_{01}}\end{matrix}=-\frac{1}{4\pi\epsilon^{2}}\mp\frac{1}{4\pi}\frac{p^{\prime}(u)p^{\prime}(u+\epsilon)}{[p(u)-p(u+\epsilon)]^{2}}.\end{cases}$
(41)
Consequently,
$\braket{T_{00}}=-\frac{1}{2\pi\epsilon^{2}}-\braket{T_{01}}.$ (42)
The result for $\braket{T_{01}}$ in Eq. (41) is Taylor expanded in $\epsilon$
to give,
$\braket{T_{01}}=\frac{1}{24\pi}\left[\frac{p^{\prime\prime\prime}}{p^{\prime}}-\frac{3}{2}\left(\frac{p^{\prime\prime}}{p^{\prime}}\right)^{2}\right]+O(\epsilon).$
(43)
In the limit $\epsilon\rightarrow 0$, the first term in Eq. (42) becomes
divergent. Since the second term is independent of $\epsilon$, it does not
vanish like the higher order terms. This second term is of particular physical
interest, corresponding to the energy flux radiated by the mirror,
$\braket{T_{00}}=\mathcal{F}(u)=-\frac{1}{24\pi}\left[\frac{p^{\prime\prime\prime}}{p^{\prime}}-\frac{3}{2}\left(\frac{p^{\prime\prime}}{p^{\prime}}\right)^{2}\right].$
(44)
The first component of the renormalized stress tensor expectation value gives
energy flux radiated by the mirror into the vacuum, characterizing the
amplified quantum fluctuations due to the presence of the accelerating
boundary.
Using the relations between spacetime $(t,x)$ and null $(u,v)$ coordinates as,
$u\equiv t-z(t),~{}~{}~{}\quad v\equiv t+z(t),$ (45)
and
$p(u)=2t(u)-u,~{}~{}~{}\quad f(v)=2t(v)-v,$ (46)
the energy flux can also be expressed with care, using straightforward
differential algebra,
$\mathcal{F}(t)=\dfrac{\dddot{z}(\dot{z}^{2}-1)-3\dot{z}\ddot{z}^{2}}{12\pi(\dot{z}+1)^{2}(\dot{z}-1)^{4}},$
(47) $\mathcal{F}(x)=\dfrac{t^{\prime\prime\prime}(t^{\prime
2}-1)-3t^{\prime}t^{\prime\prime
2}}{12\pi(t^{\prime}+1)^{2}(t^{\prime}-1)^{4}},$ (48)
$\mathcal{F}(v)=\dfrac{1}{24\pi}\bigg{[}\dfrac{f^{\prime\prime\prime}}{f^{\prime}}-\dfrac{3}{2}\bigg{(}\dfrac{f^{\prime\prime}}{f^{\prime}}\bigg{)}^{2}\bigg{]}\dfrac{1}{f^{\prime
2}},$ (49)
where dots and primes denote derivatives with respect to arguments $t$, $x$
and $v$, respectively. Eq. (48) will be used to find the flux of the CGHS
moving mirror in Sec. IX.
## VII Mirror Entanglement Entropy
In this section we will review the derivation of (1+1)-dimensional
entanglement (geometric) entropy in conformal field theory (CFT) and its
connection to the rapidity of the moving mirror (see other derivations e.g.
Good (2020b); Fitkevich _et al._ (2020)).
Consider the entropy of a system in (1+1)-D CFT Holzhey _et al._ (1994),
$S=\frac{1}{6}\ln\frac{L}{\epsilon},$ (50)
where $L$ is the size of the system in general (and in our case it is the
mirror trajectory which measures the size of the system by the spacetime
traversed accessible to the quantum field), and $\epsilon$ is a UV cut-off.
For a general arbitrary moving mirror,
$L\equiv p(u)-p(u_{0}),$ (51)
where $u$ and $u_{0}$ are null coordinates that form the region in the system
which we are considering, and $\epsilon$ is asymmetrically smeared, i.e.
$\epsilon^{2}\equiv\epsilon_{p}\epsilon_{p_{0}}$. Here $p(u)$ is the
trajectory of the mirror in null coordinates (it is a function of retarded
time $u$). The smearing and dynamics of the mirror are related as,
$\epsilon_{p}=p^{\prime}(u)\epsilon_{u},\qquad\epsilon_{p_{0}}=p^{\prime}(u_{0})\epsilon_{u_{0}}.$
(52)
Substituting Eqs. (52) into Eq. (50) yields the bare entropy of the system,
$S_{bare}=\frac{1}{12}\ln\frac{\left[p(u)-p(u_{0})\right]^{2}}{p^{\prime}(u)p^{\prime}(u_{0})\epsilon_{u}\epsilon_{u_{0}}}.$
(53)
The vacuum entropy of the system can be found by considering a static mirror
where $L=u-u_{0}$ and $\epsilon^{2}=\epsilon_{u}\epsilon_{u_{0}}$. Thus,
$S_{vac}=\frac{1}{12}\ln\frac{(u-u_{0})^{2}}{\epsilon_{u}\epsilon_{u_{0}}}.$
(54)
Even though the entropies above are defined in terms of smearing, this
dependence can be removed by an intuitive renormalization via,
$S_{ren}=S_{bare}-S_{vac}=\frac{1}{12}\ln\frac{[p(u)-p(u_{0})]^{2}}{p^{\prime}(u)p^{\prime}(u_{0})(u-u_{0})^{2}}.$
(55)
Further simplification proceeds by a Taylor expansion of our arbitrary
function $p(u)$ around $u=u_{0}$ up to first order, that is,
$p(u)=p(u_{0})+p^{\prime}(u_{0})(u-u_{0})+\mathcal{O}(u-u_{0})^{2}.$ (56)
Substituting Eq. (56) into Eq. (55) brings us to,
$S_{ren}=\frac{1}{12}\ln\frac{p^{\prime}(u_{0})}{p^{\prime}(u)}.$ (57)
Moreover, for a static mirror $p(u)=u$ and $p(u_{0})=u_{0}$, therefore
$p^{\prime}(u_{0})=1$. As a result, Eq. (57) reduces to $S_{ren}\rightarrow
S(u)$ where,
$S(u)=-\frac{1}{12}\ln p^{\prime}(u).$ (58)
Eq. (58) is valid for any moving mirror that starts asymptotically static
(zero velocity). Notable exceptions are the eternally thermal Carlitz-Willey
mirror Carlitz and Willey (1987) and the eternally uniformly accelerated
mirror Birrell and Davies (1984); however, most of the solved mirrors in the
literature, by construction, do start static as they are often used to model
gravitational collapse. The CGHS mirror is no exception.
Eq. (58) is more intuitively written in spacetime coordinates using the
relation between null and spacetime trajectories of the mirror as,
$p^{\prime}(u)=\frac{1+\dot{z}(t)}{1-\dot{z}(t)}.$ (59)
Applying this relation into the Eq. (58) yields,
$S(t)=-\frac{1}{6}\tanh^{-1}[\dot{z}(t)]=-\frac{1}{6}\eta(t),$ (60)
where $\eta(t)\equiv\tanh^{-1}[\dot{z}(t)]$ is the time-dependent rapidity. It
is simple to see that the magnitude of the entropy increases as the mirror
moves faster. Unitarity in this context Bianchi and Smerlak (2014a) strictly
requires that the entropy must achieve a constant value in the far past and
far future.
This von Neumann entropy measure of the degree of quantum entanglement between
the two subsystems (past & future) constitutes a two-part composite quantum
system. It explicitly reveals the connection of information of entanglement to
the dynamics (rapidity) of the moving mirror system. Allow us to speculate a
thermodynamic treatment of the system, and the corresponding macroscopic state
of the entanglement is characterized by a distribution of its microstates,
then it may be appropriate to coincide the Boltzmann entropy with the von
Neumann entropy. In this conjectural case, a discrete speed based on the basic
smallest unit of an operable binary digit of information results.111This
discreteness necessarily leads to a smallest non-zero speed for the moving
mirror. In SI units, this is $17.2\;\textrm{fm/s}$, from $v=c\tanh(6k_{B}\ln
2)$ which is about the diameter of a gold nucleus in one second, or about 5
centimeters in 100,000 years.
In the next section we will ultimately apply the above entanglement-rapidity
relationship, Eq. (60), which can be expressed independently of coordinates or
its argument, $-6S=\eta$, to gain insight into the self-force and Larmor power
by reformulating them in terms of entropy.
## VIII Mirror Larmor formula and LAD force
### VIII.1 Quantum relativistic Larmor formula
In this subsection we derive the quantum relativistic power radiated by the
moving mirror and find it has the same form as the classical relativistic
Larmor formula of electrodynamics. We account for the power radiated to both
sides of the moving mirror, utilizing the quantum stress tensor derived in
Section VI.
Let us consider the total energy radiated as derived from the Davies-Fulling
quantum stress tensor, to the right side of the mirror, expressed as Equation
(2.34) of Good _et al._ (2013),
$E^{R}=\frac{1}{12\pi}\int_{-\infty}^{\infty}\alpha^{2}(1+\dot{z})\,\operatorname{d}\\!{t}.$
(61)
An observer at $\mathscr{I}^{+}_{R}$ measures $E^{R}$ energy emitted, but the
mirror also radiates energy to $\mathscr{I}^{+}_{L}$ for an observer on the
left. The energy radiated to the left, $E^{L}$, is found by the same
expression but with a parity flip, $\dot{z}\rightarrow-\dot{z}$, so that the
total radiated energy is,
$E=E^{R}+E^{L}=\frac{1}{6\pi}\int_{-\infty}^{\infty}\alpha^{2}\,\operatorname{d}\\!{t}.$
(62)
We define the quantity, $E$, without an average giving us a measure of the
total radiation, independent of observer. This allows us to identify and
define a quantum relativistic Larmor power analog for the moving mirror,
$P=\operatorname{d}\\!{E}/\operatorname{d}\\!{t}$,
$E=\int_{-\infty}^{\infty}\frac{dE}{dt}\operatorname{d}\\!{t}\equiv\int_{-\infty}^{\infty}P\operatorname{d}\\!{t},$
(63)
which gives the familiar relativistic Larmor scaling for proper acceleration:
$P=\frac{\alpha^{2}}{6\pi}.$ (64)
The quantum power radiated by the mirror takes the same form as that of a
classical point charge in electrodynamics (see Zhakenuly _et al._ (2020) for
the derivation and distribution in (3+1) dimensions). Recall here that
$\alpha$ is the scalar invariant which is defined as the proper time
derivative of rapidity, $\alpha=\eta^{\prime}(\tau)$, even though the integral
in which we define the power with respect to uses ordinary coordinate time.
Eq. (64) is in harmony with Ford-Vilenkin Ford and Vilenkin (1982) who found
that the self-force of the moving mirror also has the same form as radiation-
reaction of a point charge in classical electrodynamics.
### VIII.2 Relativistic entanglement-power
In this subsection we apply the entanglement-rapidity relationship, Eq. (60),
to the power derived in the previous subsection, Eq. (64). Motivated to
understand the radiated emission in terms of information flow, we find the
quantum power is expressed as the square of the first derivative of the
entanglement.
The covariant Larmor power $P$ is a Lorentz scalar invariant since it is
proportional to the square of the proper acceleration,
$P=\frac{\alpha^{2}}{6\pi}=\frac{\eta^{\prime}(\tau)^{2}}{6\pi}.$ (65)
The rapidity-entanglement relationship, given by $\eta=-6S$ (see also Section
VII and further references Good _et al._ (2017a); Bianchi and Smerlak
(2014b); Good (2020a)), can be expressed with independent variable as proper
time, $\eta^{\prime}(\tau)=-6S^{\prime}(\tau)$, and thus the power is
formulated as,
$P=\frac{6}{\pi}S^{\prime}(\tau)^{2}.$ (66)
This entanglement-power relationship characterizes one-dimensional
transmission of entropy or information for non-thermal radiation. At thermal
equilibrium, this can be compared to Pendry’s maximum entropy rate for power,
$\dot{S}=\sqrt{\pi P/3}$, also called the noiseless quantum channel capacity,
investigated by Bekenstein-Mayo in the context of black holes information flow
as (1+1)-dimensional Bekenstein and Mayo (2001). The factor of 2 is accounted
for by a uni-directional flow to a single observer, conventionally taken to be
situated at $\mathscr{I}^{+}_{R}$ future null right infinity. The result Eq.
(66) compliments the celebrated Bianchi-Smerlak formula Bianchi and Smerlak
(2014b) revealing the energy-entanglement connection,
$2\pi\mathcal{F}(u)=6S^{\prime}(u)^{2}+S^{\prime\prime}(u),$ (67)
which implies that the outgoing flux is completely determined by the structure
of entanglement at future null infinity, and vice versa, the entanglement
entropy is completely determined by the flux through the second order
differential equation from the values of $S(u)$ and $S^{\prime}(u)$ at a
single point at $\mathscr{I}^{+}_{R}$. Eq. (67) involves derivatives with
respect to retarded time $u$ rather than proper time $\tau$ as is the case in
Eq. (66).
### VIII.3 Averaging Radiation Reaction
Turning our attention to the self-force in this subsection, we derive the
formula for radiation reaction from our previously derived mirror Larmor
power, Eq. (64), using an average over proper time. Our results in this
subsection confirm those of Ford-Vilenkin Ford and Vilenkin (1982), building
confidence in the overall theme by connecting the self-force to the result for
Larmor power.
Energy lost by radiation for an accelerating point charge tends to slow it
down. This is because there is a backreaction of the radiation on the particle
itself. In the case of a moving mirror this is in practise not so, because the
trajectory is usually just assumed apriori. Allow us to try to find this
radiation reaction on the mirror without assuming locally any trajectory and
by use of an averaging over proper time. Starting with the Larmor mirror
power, Eq. (64), expressed in rapidity Eq. (65),
$P=\frac{\eta^{\prime}(\tau)^{2}}{6\pi},$ (68)
the reaction must be, averaged over proper time, the work done on the mirror
equal to the negative of the energy lost to the vacuum radiation:
$\overline{F\eta}=-\frac{\overline{\eta^{\prime}(\tau)^{2}}}{6\pi}.$ (69)
We are still in natural units, $c=1$, and so $\overline{F\eta}$ is an average
radiation reaction power linearly and proportionally dependent on the
rapidity, $\eta$, of the mirror. Writing the proper acceleration as,
$\eta^{\prime}(\tau)^{2}=\frac{d}{d\tau}(\eta\eta^{\prime})-\eta\eta^{\prime\prime},$
(70)
where the total derivative with respect to proper time vanishes due to our
averaging procedure (or in the case of assuming global asymptotic inertial
trajectory222This is also equivalently accomplished by assuming periodicity.),
we then have
$\overline{F\eta}=+\frac{\overline{\eta\eta^{\prime\prime}(\tau)}}{6\pi}.$
(71)
This allows us to identify the radiation reaction force as,
$F=\frac{\eta^{\prime\prime}(\tau)}{6\pi}.$ (72)
We will show that this result is in agreement with the magnitude, Eq. (88), of
the covariant LAD 4-force Eq. (78), derived using the Davies-Fulling stress
tensor, in the next subsection.
### VIII.4 Confirmation of LAD magnitude
A rigorous non-averaging derivation of Eq. (72) is accomplished by
relativistic covariance. In this subsection we derive the mirror LAD force
using electromagnetic 4-vector formulation and quantities known in special
relativity, e.g. proper acceleration, celerity and rapidity.
Before turning to the derivation of the force, let us briefly review some
known formulations. The point charge in SI units to moving mirror natural
units ($\mu_{0}=\epsilon_{0}=1$) has a coupling which can be expressed by the
substitution:
$\frac{2}{3}\left(\frac{q^{2}}{4\pi\epsilon_{0}c^{3}}\right)=\frac{q^{2}}{6\pi\epsilon_{0}c^{3}}=\frac{\mu_{0}q^{2}}{6\pi
c}\quad\Rightarrow\quad\frac{1}{6\pi}.$ (73)
Notice Gaussian units are $4\pi\epsilon_{0}=1$ and $\mu_{0}=4\pi$. We will
need the proper acceleration, $\alpha$, that is a Lorentz invariant, defined
by:
$\alpha^{2}\equiv-\frac{d^{2}x^{\mu}}{d\tau^{2}}\frac{d^{2}x_{\mu}}{d\tau^{2}}.$
(74)
It will also be helpful to have the velocity $v$, the Lorentz factor $\gamma$,
and the celerity $w$, which are defined through the rapidity $\eta$,
$v=\tanh\eta,\quad\gamma=\cosh\eta,\quad w=\sinh\eta,$ (75)
or similarly,
$v=\frac{dx}{dt},\qquad\gamma=\frac{dt}{d\tau},\qquad w=\frac{dx}{d\tau}.$
(76)
The self-force scalar invariant, $F$, can be written via 4-vectors as,
$F^{2}\equiv-F^{\mu}F_{\mu},$ (77)
which will be the final object that we obtain in this subsection.
Let us now confirm the mirror self-force by substituting Eqs. (73)-(77) into
the radiative force introduced by Ford-Vilenkin Ford and Vilenkin (1982),
$6\pi
F^{\mu}=\frac{d^{3}x^{\mu}}{d\tau^{3}}-\alpha^{2}\frac{dx^{\mu}}{d\tau}.$ (78)
Here the 4-vector has time $F^{0}=F^{t}$ and space $F^{1}=F^{x}$ components
respectively for $6\pi F^{\mu}$:
$\gamma^{\prime\prime}(\tau)-\alpha^{2}\gamma=\alpha^{\prime}(\tau)w,$ (79)
$w^{\prime\prime}(\tau)-\alpha^{2}w=\alpha^{\prime}(\tau)\gamma,$ (80)
where $\alpha=\eta^{\prime}(\tau)$. So, the time and space components are
expressed in rapidity as:
$6\pi F^{t}=\eta^{\prime\prime}(\tau)\sinh\eta,$ (81) $6\pi
F^{x}=\eta^{\prime\prime}(\tau)\cosh\eta.$ (82)
With signature $(+,-,-,-)$ or just $(+,-)$ for our (1+1)-dimensional context,
the magnitude can be found by,
$\displaystyle F$ $\displaystyle=$ $\displaystyle\sqrt{-F^{\mu}F_{\mu}}$ (83)
$\displaystyle=$ $\displaystyle\sqrt{-(|F^{0}|^{2}-|F^{1}|^{2})}$ (84)
$\displaystyle=$ $\displaystyle\sqrt{|F^{x}|^{2}-|F^{t}|^{2}}$ (85)
$\displaystyle=$
$\displaystyle\frac{\eta^{\prime\prime}}{6\pi}\sqrt{\cosh^{2}\eta-\sinh^{2}\eta}$
(86) $\displaystyle=$ $\displaystyle\frac{1}{6\pi}\eta^{\prime\prime}(\tau),$
(87)
giving us the simple relationship for jerk,
$\alpha^{\prime}(\tau)=6\pi F,$ (88)
which coincides with the Eq. (72) taking into account the change of rapidity
with respect to proper time, $\alpha=\eta^{\prime}(\tau)$.
### VIII.5 Derivation of LAD formula
Let us now move to the derivation of the self-force from the moving mirror
point of view explicitly. Consider the total energy-momentum emitted to the
right of the mirror,
$E^{R}=\int_{-\infty}^{\infty}\mathcal{F}^{R}\operatorname{d}\\!{u},$ (89)
where the Schwarzian derivative defines the quantum stress tensor Fulling and
Davies (1976),
$\mathcal{F}^{R}(u)=-\frac{1}{24\pi}\\{p(u),u\\},$ (90)
which we convert to proper time Good and Linder (2018),
$\mathcal{F}^{R}(\tau)=-\frac{1}{12\pi}\eta^{\prime\prime}(\tau)e^{+2\eta(\tau)},$
(91)
so that with Jacobian $du=e^{-\eta}d\tau$,
$\displaystyle E^{R}$ $\displaystyle=$
$\displaystyle-\frac{1}{12\pi}\int_{-\infty}^{\infty}\eta^{\prime\prime}e^{+2\eta}(e^{-\eta}\operatorname{d}\\!{\tau}),$
(92) $\displaystyle=$
$\displaystyle-\frac{1}{12\pi}\int_{-\infty}^{\infty}\eta^{\prime\prime}e^{+\eta}\operatorname{d}\\!{\tau}.$
(93)
Similarly, with a parity flip, the energy-momentum emitted to left of the
mirror is,
$E^{L}=+\frac{1}{12\pi}\int_{-\infty}^{\infty}\eta^{\prime\prime}e^{-\eta}\operatorname{d}\\!{\tau}.$
(94)
We are looking for the difference in energy-momentum between the left and
right sides of the mirror to construct the 4-vector radiation reaction self-
force. The time component, $F^{t}=Fw$, is constructed from the energy,
$\Delta U=\int F\operatorname{d}\\!{x}=\int Fv\operatorname{d}\\!{t}=\int
Fv\gamma\operatorname{d}\\!{\tau}=\int Fw\operatorname{d}\\!{\tau}.$ (95)
On the other hand, the energy is defined as,
$\Delta
U=U^{L}-U^{R}=\int_{-\infty}^{\infty}\frac{dU}{d\tau}\operatorname{d}\\!{\tau}=\int_{-\infty}^{\infty}F^{t}\operatorname{d}\\!{\tau},$
(96)
where
$F^{t}=\frac{dU}{d\tau}=\frac{dU}{dx}\frac{dx}{d\tau}=F\sinh\eta=Fw.$ (97)
The space component of the force, $F^{x}=F\gamma$, is constructed from the
momentum,
$\Delta\mathcal{P}=\int F\operatorname{d}\\!{t}=\int
F\gamma\operatorname{d}\\!{\tau}.$ (98)
The difference in momentum radiated between the two sides is expressed as,
$\Delta\mathcal{P}=\mathcal{P}^{L}-\mathcal{P}^{R}=\int_{-\infty}^{\infty}\frac{d\mathcal{P}}{d\tau}\operatorname{d}\\!{\tau}=\int_{-\infty}^{\infty}F^{x}\operatorname{d}\\!{\tau},$
(99)
which defines the space component piece of the radiation reaction on the
mirror. This is explicitly,
$F^{x}=\frac{\eta^{\prime\prime}}{12\pi}[e^{-\eta}-(-e^{+\eta})]=\frac{\eta^{\prime\prime}}{6\pi}\cosh\eta=F\gamma,$
(100)
where one can already see that $F=\eta^{\prime\prime}/6\pi$. The radiation
reaction force,
$F^{\mu}=(F^{t},F^{x})=\gamma(Fv,F)=(wF,\gamma F),$ (101)
or, equivalently written in covariant notation as in Eq. (78), has a Lorentz
scalar invariant jerk,
$F=\frac{\eta^{\prime\prime}(\tau)}{6\pi}.$ (102)
While our derivation in this subsection for the radiation-reaction force
ostensibly relies on the expression for finite conserved energy, the
integration is not explicitly taken. For finite energy one requires the
acceleration to vanish asymptotically. It is safe to assume the self-force
does not actually require this constraint. This is congruent with the usual
LAD expression in electrodynamics which holds even in the mathematical case
where a charged point is accelerated in both asymptotic limits.
We have derived the moving mirror LAD formula for the radiation reaction using
conservation of energy (the difference in energy-momentum between the right
and left sides of the mirror), but made no effort to identify the mechanism
responsible for the force. In the case of a point charge, one imagines the
force as the recoil effect of the particle’s own field acting back on the
charge, but in the case of a mirror we see it is not the source of a field
whatsoever.
In the electromagnetic case, one has the problem of the field blowing up right
at the point charge. But in the mirror model we know the field is identically
zero at the mirror. So what then, is the mechanism? The accepted answer in the
case of charge is that an extended charge distribution divided into
infinitesimal pieces gives rise to a net force of the charge on itself - the
self-force - as a consequence of the breakdown of Newton’s third law within
the structure of the particle. Perhaps a distributed boundary condition
calculation could also give rise to a self-force on the mirror. In other
words, the net force exerted by the scalar field generated by different pieces
of the distributed boundary condition acts on each other to produce a mirror
self-force. Such a calculation is beyond the scope of this paper.
### VIII.6 Entanglement & Radiative Force
In the relativistic entanglement-power Section VIII.2, we applied the
entanglement-rapidity relationship to the power. This result gave us insight
into the information flow distributed by the radiation. In this subsection we
apply the entanglement-rapidity relationship to the radiation reaction itself.
We find the self-force is proportional to the second derivative of the von
Neumann entanglement entropy in a simple entanglement-force relationship.
Using the Davies-Fulling exact relativistic quantum stress tensor, expressed
in proper time $\tau$ as $T_{00}=\mathcal{F}(\tau)$ Good and Linder (2018)
where,
$12\pi\mathcal{F}(\tau)=-\eta^{\prime\prime}(\tau)e^{2\eta(\tau)},$ (103)
as well as the Lorentz-Abraham-Dirac 4-force in one dimension, we have
demonstrated that $F^{t}=Fw$, where the celerity is $w=dx/d\tau$, and
$F^{x}=F\gamma$, where the Lorentz factor is $\gamma=dt/d\tau$, (see the
closely related results of Higuchi-Martin Higuchi and Martin (2005)). These
results ultimately led to,
$6\pi F(\tau)=\alpha^{\prime}(\tau).$ (104)
The derivation reveals Eq. (104) as the most simple interpretation of the LAD
force, with magnitude $F$ of Eq. (78) as the jerk of the mirror, i.e. the
proper time derivative of the proper acceleration. The source of this force is
the reaction of the scalar field to the presence of the accelerating mirror in
vacuum. The proper time derivative of the proper acceleration determines the
force and can be nonzero even when the acceleration itself of the mirror is
instantaneously zero, and the mirror is not radiating particles. The
disturbing implications of the Lorentz-Abraham-Dirac formula which are still
not entirely understood in classical electrodynamics (see e.g. Rohrlich
(1997); Yaghjian (2006)), carry over in analog, to the quantum scalar field of
the moving mirror model.
Using the rapidity-entanglement relationship, $\eta=-6S$ in Eq. (102), it is
easy to find the von Nuemann entanglement entropy in terms of the radiative
reaction force,
$S^{\prime\prime}(\tau)=-\pi F(\tau).$ (105)
This relationship connects information flow in the system to the self-force on
the mirror. In a similar vein to the interesting features like negativity and
thermodynamic interpretations of entropic forces Visser (2011), this
entanglement self-force also assumes negative values and demonstrates an
information interpretation of the radiative reaction force: it is the second
proper time derivative of the von Neumann entanglement entropy.333The sign in
$S^{\prime\prime}=-\pi F$ tells us that when the force on the mirror is to the
left, away from the observer at $\mathscr{I}^{+}_{R}$, then $S^{\prime\prime}$
is positive. The sign is by convention because the observer is chosen to be
located on the right.
It would be interesting to know whether Eq. (105) holds outside the moving
mirror model considering the closely related accelerated boundary
correspondences (ABCs) with cosmologies and black holes. Regardless, advances
in general relativity, like the maximum force conjecture, may play a role in
better understanding the moving mirror model and associated entanglement
entropy. There have been a number of works in the last decades indicating that
$F_{\max}=c^{4}/(4G)$ is the limiting force (see Barrow and Gibbons (2015);
Good and Ong (2015) for more references) in general relativity. Taking into
account entropy-force relation of Eq. (105), this implies a constraint on the
rate of change of the entanglement entropy, i.e. if the force, or jerk
$\alpha^{\prime}(\tau)$, has this maximum, then the second derivative of
entropy has a minimum possible value,
$S^{\prime\prime}_{\textrm{min}}=-\pi/4$.
## IX CGHS Larmor power and self-force
Having derived the power and self-force for any moving mirror in general, we
now specialize to the exponentially accelerated mirror that has particle
production which corresponds to the CGHS system. We apply the Larmor power and
LAD force derived in previous sections to our particular CGHS mirror and find
that a simple entanglement-over-distance relationship is revealed connecting
the entanglement-rapidity relationship to the space traversed. In addition,
the loss of unitarity is explicitly manifest in the divergence of the power
and self-force at the time the horizon forms in the proper frame.
Let us start from the trajectory of the CGHS mirror in spacetime coordinates
Good and Linder (2018),
$z(t)=-\frac{1}{\kappa}\sinh^{-1}\left(\frac{e^{\kappa(t-v_{H})}}{2}\right).$
(106)
The Larmor power and self-force for the CGHS mirror are found using Eq. (64)
and Eq. (88), where $\alpha(\tau)$ is the acceleration in proper time. The
procedure of defining and deriving $\alpha(\tau)$ is given in Good and Linder
(2018). Let us start from the connection between proper and coordinate times.
For CGHS mirror it is obtained to be,
$\tau(t)=\int\frac{dt}{\gamma(t)}=\frac{1}{2\kappa}\ln\left|\frac{\sqrt{4+e^{2\kappa(t-v_{H})}}-2}{\sqrt{4+e^{2\kappa(t-v_{H})}}+2}\right|.$
(107)
The inverse of Eq. (107) yields,
$t(\tau)=v_{H}+\frac{1}{2\kappa}\ln\left|\frac{16e^{2\kappa\tau}}{(1-e^{2\kappa\tau})^{2}}\right|.$
(108)
Applying it into Eq. (106) leads to the trajectory in proper time,
$z(\tau)=\frac{1}{\kappa}\sinh^{-1}\left(\csch(\kappa\tau)\right).$ (109)
The next step to obtain $\alpha(\tau)$ is to find celerity and then rapidity.
Using the rapidity, the proper acceleration is found to be,
$\alpha(\tau)=\frac{d\eta(\tau)}{d\tau}=\kappa\csch(\kappa\tau).$ (110)
This result, Eq. (110), is found using a different method in Juárez-Aubry
Juárez-Aubry (2017) and is in agreement. Substituting Eq. (110) into Eq. (64)
and Eq. (88), we obtain corresponding Larmor power and radiation reaction
force for the CGHS mirror as,
$P=\frac{\alpha^{2}}{6\pi}=\frac{\hbar}{c^{2}}\frac{\kappa^{2}\csch^{2}(\frac{\kappa\tau}{c})}{6\pi},$
(111)
and
$F=\frac{\alpha^{\prime}}{6\pi}=-\frac{\hbar}{c^{3}}\frac{\kappa^{2}}{6\pi}\coth(\frac{\kappa\tau}{c})\csch(\frac{\kappa\tau}{c}).$
(112)
The terms on the right of Eq. (111) and Eq. (112) have reinstated $\hbar$ and
$c$, noting that $\kappa$ has units of an acceleration, in order to emphasize
they are a quantum Larmor power and quantum self-force, respectively. The
dependences of the CGHS mirror Larmor power and self-force on proper time and
$\kappa$, Eq. (111) and Eq. (112), are demonstrated graphically in Figs. 3 and
4.
Figure 3: Larmor power for CGHS mirror, Eq. (111), where plots from left to
the right correspond to $\kappa=1,2,3,4,5$ cases, respectively, $\hbar=c=1$,
and power is normalised by $10$. The power increases asymptotically as time
approaches $\tau=0$. The key takeaway is this divergence at a finite proper
time when the horizon forms. Figure 4: Radiative reaction or the self-force
for CGHS mirror, Eq. (112), where blue, orange, green, red and purple lines
correspond to $\kappa=1,2,3,4,5$ cases, respectively, $\hbar=c=1$, and force
is normalised by $10^{2}$. The self-force has opposite direction with respect
to the Larmor power and demonstrates left-hand side trend as the mirror is
moving to the left.
Fig. 4 has lines corresponding to different values of $\kappa$ which
intersect. This is explained by the fact that the dependence of the CGHS self-
force, Eq. (112), on the single parameter of the system $\kappa$ is non-
trivially different from the dependence of the power, Eq. (111).
Let us now consider the timespace trajectory of the mirror in natural units,
$t(x)=v_{H}+\frac{1}{\kappa}\ln[-2\sinh(\kappa x)].$ (113)
Using this form of the trajectory we find rapidity in terms of space
coordinate $x$,
$\eta(x)=\kappa x.$ (114)
So, the rapidity, or the information defining dynamical quantity, in terms of
$x$ has surprisingly simple form: it linearly depends on the space coordinate.
Eq. (114) is the simplest way to express the trajectory of the CGHS mirror.
The last interesting quantity we compute is the energy flux in terms of $x$.
Using Eq. (48), the CGHS mirror flux is found to be,
$\mathcal{F}(x)=\frac{\kappa^{2}}{48\pi}(1-e^{4\kappa x}).$ (115)
This form immediately clarifies that at late times (far-left positions,
$x\to-\infty$), the energy flux is a constant associated with thermal emission
that is in agreement with the thermal behaviour of the CGHS black hole
radiation, $\mathcal{F}=\kappa^{2}/(48\pi)$. The graphical illustration of
this flux is shown in Fig. 5.
Figure 5: This graph should be read from right to left mapping the initial
motion of the mirror in the past to the future. This is the energy flux for
the CGHS mirror, Eq. (115), with $\kappa=\sqrt{48\pi}$ for illustration so
that at thermality the flux is equal to one. The mirror moves to the left,
while the flux ascends from $x=0$, then monotonically approaches constant
thermal emission at late positions, $x=-\infty$.
## X Conclusions & Future Work
Overall, the equivalence between the CGHS black hole and the exponentially
accelerating moving mirror in lab time can be seen from several explicit
matching quantities: the matching condition for the CGHS black hole and the
trajectory of the mirror in null coordinates, the spectra, and consequently,
the temperatures. A one-to-one correspondence is ensured as long as
$|\Lambda|=|\kappa|$ requirement is met. The correspondence is summarized in
Table 1.
An important physical result of this correspondence is the demonstration that
the moving mirror horizon corresponds in the black hole system to a quantity
determined by the black hole parameters $\Lambda$ and $M$. This is understood
by seeing that the horizon location of the mirror is associated with the CGHS
black hole spacetime geometry as determined by the global non-zero curvature.
This geometry, in turn, is determined by the cosmological constant and
curvature caused by the black hole mass. Interestingly, it has been found that
unlike Schwarzschild black hole case, where the singularity and the event
horizon are located at different positions, the CGHS mirror mimics both the
event horizon and the center of the CGHS black hole as they happen at the same
location, $r=0$. This overlap is almost assuredly responsible for the
particular simplicity of the mathematics in the mirror case and the ease and
utility in describing this exact spectrum via a simple accelerating trajectory
in a flat-spacetime background. It also highlights no conflict between the
origin singularity and event horizon as the location of particle production,
since they are both one and the same.
More general considerations have given us the Larmor power radiated by an
arbitrary moving mirror and the LAD formula for the radiation reaction. The
derivations utilize general dynamics of the mirror expressed in terms of
proper acceleration and rapidity, and lead naturally to an information
interpretation by expressing the rapidity in terms of entanglement entropy.
The power and force are found to have the same dynamic form as that in
classical electrodynamics for a moving point charge. In terms of information,
the entanglement power and entanglement self-force are interpreted in terms of
first and second derivatives of the von-Neuman entanglement entropy,
respectively.
Specializing to our particular CGHS moving mirror, the Larmor power is found
to diverge as $\tau\to 0^{-}$. As proper time ticks to $\tau=0$, the mirror is
infinitely accelerating, reaching the speed of light. Consistently, the
direction of the self-force is opposite the direction of the radiated Larmor
power. It is worth emphasizing that as a guide, in SI units both the Larmor
power and LAD force for the CGHS mirror are proportional to $\hbar$,
underscoring the fact that the power and self-force are quantum (not
classical) measures.
Lastly, the CGHS mirror has two simplifying results when expressed in terms of
space rather than time or light-cone coordinates: the trajectory rapidity is
simply proportional to the distance travelled, $\eta=\kappa x$ and the
radiative flux emitted by the CGHS mirror is seen by eye as thermal (constant
emission) at far-left positions (late times). In summary list-form, the
salient features of this work are:
* •
CGHS mirror $\Leftrightarrow$ EH & BH center; IV
* •
Mirror Larmor power; $P\sim\alpha^{2}(\tau)$; VIII.1
* •
Entanglement-power; $P\sim S^{\prime}(\tau)^{2}$; VIII.2
* •
Larmor to LAD Averaging; $P\rightarrow F$; VIII.3
* •
Mirror LAD self-force; $F\sim\alpha^{\prime}(\tau)$; VIII.4 & VIII.5
* •
Entanglement-force; $F\sim S^{\prime\prime}(\tau)$; VIII.6
* •
CGHS self-force and CGHS power; IX
Future extensions of this work are foreseen. Hawking Hawking (1975) pointed
out that at very early times of gravitational collapse, a star cannot be
described by the no-hair theorem. So in this context, a variety of different
collapse situations corresponds to different mirror trajectories. It is likely
to be fruitful to consider modifications to the mirror trajectory (one of
which has already been done to CGHS e.g. Good _et al._ (2013)) made to
provide different early time approaches to a thermal distribution,
particularly those modifications that can afford unitarity and finite
evaporation energy, modeling more realistic situations congruent with finite
mass black holes and quantum purity.
The modifications that can take into account energy conservation like those of
the dilaton gravity models and moving mirror models have had significant
success as a laboratory for studying black hole evaporation. The physical
problem in (1+1) dilaton gravity of the evaporating black hole and its
modified emission extends to complete evaporation for the Russo, Susskind, and
Thorlacius (RST) model Russo _et al._ (1992) and to partial evaporation
leaving a remnant for the Bose, Parker, and Peleg (BPP) model Bose _et al._
(1995). The two-dimensional RST model for evaporating black holes is locally
equivalent - at the full quantum level - to Jackiw-Teitelboim (JT) gravity
that was recently shown to be unitary Fitkevich _et al._ (2020).
The similarity of drifting moving mirrors (see e.g. Good _et al._ (2017a)) to
the BPP model is striking in several qualitative aspects: NEF emission as a
thunderpop (NEF emission from evaporating black holes, at least in the case of
a (1+1)-dimensional dilaton gravity, has already been known in the literature
for over 20 years Bose _et al._ (1996)), a left over remnant, and finite
total energy emission. It is also interesting that the mass of the remnant in
the BPP model is independent of the mass $M$ of the infalling matter, since
with respect to the issue of energy conservation, there is no known physical
analog for $M=1/(4\kappa)$, the initial mass of the shockwave, in the mirror
model.
We hope this work offers insight for future direction, using the formulas for
mirror radiative power and radiation reaction force. Investigations of
behavior of the Larmor power and LAD force of other existing moving mirror
models will be used to compare results to better understand the specific
physics of moving mirrors and the general physics of acceleration radiation
from the quantum vacuum.
Table 1: Matching quantities for CGHS black hole and exponentially accelerating mirror. Quantity | black hole | moving mirror
---|---|---
Trajectory | $u(U)=U-\frac{1}{\Lambda}\ln\left(1-e^{\Lambda(U-v_{0})}\right)$ | $f(v)=v-\frac{1}{\kappa}\ln\left(1-e^{\kappa(v-v_{H})}\right)$
Spectrum | $|\beta_{\omega\omega^{\prime}}|^{2}=\frac{1}{4\pi^{2}\Lambda^{2}}\frac{\omega^{\prime}}{\omega}\left|B\left[\frac{i\omega_{+}}{\Lambda},1-\frac{i\omega}{\Lambda}\right]\right|^{2}$ | $|\beta_{\omega\omega^{\prime}}|^{2}=\frac{1}{4\pi^{2}\kappa^{2}}\frac{\omega^{\prime}}{\omega}\left|B\left[\frac{i\omega_{+}}{\kappa},1-\frac{i\omega}{\kappa}\right]\right|^{2}$
Temperature | $T=\frac{\Lambda}{2\pi}$ | $T=\frac{\kappa}{2\pi}$
###### Acknowledgements.
Funding from state-targeted program “Center of Excellence for Fundamental and
Applied Physics” (BR05236454) by the Ministry of Education and Science of the
Republic of Kazakhstan is acknowledged. MG is also funded by the
FY2018-SGP-1-STMM Faculty Development Competitive Research Grant No.
090118FD5350 at Nazarbaev University.
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|
# The effects of $q$-statistics on cosmology
M. Senay
Naval Academy, National Defence University, 34940, Istanbul, Turkey
<EMAIL_ADDRESS>
(April 04, 2021)
###### Abstract
Considering Verlinde’s entropic gravity proposal, we focus the effects of
fermionic $q$ deformation on the Einstein’s field equations and Friedmann
equations. For this purpose, we represent the thermodynamical properties of
the deformed fermion gas model in two-dimensional space. To describe the
behavior of the quantum black holes, deformed Einstein field equations are
derived with the help of deformed entropy function. Moreover, deformed
Friedmann equations are investigated by using Friedmann-Robertson-Walker (FRW)
metric. We also present the effective ones of the energy density, the
pressure, and the equation of state for the dark energy. Lastly, we derive an
analytical expression of the effective density parameter of dark energy.
Keywords: $q$-deformed, gravity, dark energy
## I Introduction
With the discovery of thermodynamic properties of black holes, there has been
found out a deep connection between gravity and thermodynamics. In 1971,
Hawking Hawking1971 demonstrated that the area of black hole event horizon
cannot be reduced by any process and when two black holes merged, the
generated new black hole area cannot be smaller than the sum of the initial
black hole areas. In 1973, Bekeinstein Bekeinstein1973 showed that the
surface of the event horizon of the black hole and its entropy were related
with each other. In 1995, Jacobson Jacobson1995 indicated that Eisntein’s
field equations can be derived by using the first law of thermodynamics.
In 2011, Verlinde Verlinde2011 suggested a new remarkable perspective on the
relation between gravity and thermodynamics. According to Verlinde’s
suggestion, gravity can be explained as an entropic force when taken into
holographic viewpoint of gravity. Using the entropic force together with the
holographic principle and the equipartition law of energy, he derived Newton’s
law of gravitation, and Einstein’s field equations.
The holographic approach on gravity has enabled to conduct many studies in the
literature Sheykhi2012 ; Moradpour2015 ; Moradpour2018 ; Sefiedgar2017 ;
Feng2016 ; Abreu2018 ; Abbasi2020 ; Saridakis2020 . For example, in the Ref
Sheykhi2012 , the equipartition law of energy theorem was modified with Debye
function and MOND theory and Einstein’s field equations were derived under
Debye entropic gravity scenario. In the Ref Saridakis2020 , modifed Friedmann
equations were obtained by using Barrow entropy instead of usual Bekeinstein-
Hawking entropy.
On the other hand, the mass of a charged black hole decreases via Hawking
radiation until it reaches a minimum mass proportional to it’s charge
Strominger1993 . The black hole having a minimum mass is assumed as an
extremal black hole. The extremal black holes can be used to find the solution
to the quantum puzzles of black holes. For instance, Strominger Strominger1993
considered the scattering of two extremal black holes for solving the quantum
black hole puzzles. He looked to answer the question of whether do extremal
black holes scatter as bosons, fermions or something else? According to the
results from his study, the extremal black holes obey the deformed statistics
instead of usual statistics. Hence, they can be considered as deformed bosons
or fermions. In recent years, several authors have intensively examined both
the quantum mechanical and thermodynamical properties of such deformed bosons
and fermions Lavagno2002 ; Gavrilik2011 ; Algin2016a , and their applications
on different systems Dil2015 ; Dil2017 ; Senay2018 ; Kibaroglu2019 .
In the view of the above motivations, in this study, based on Strominger’s
idea we specially consider extremal black holes as $q$-deformed fermions.
Thermodynamical properties of these $q$-deformed fermions were investigated in
Ref. Algin2016b . The present paper is aimed to investigate the effects of
fermionic _q_ -deformation on the Einstein’s field equations and Friedmann
equations. Hence, some thermodynamical properties of deformed fermion gas
model in two dimensional space is briefly introduced in Section II. With the
help of deformed entropy function, deformed Einstein’s field equations are
found in Section III. The deformed Friedman equations are derived in Section
IV. The cosmological implications are given in Section V. The conclusions are
given in the last section.
## II Deformed fermion statistics in two dimensions
The non-symmetric deformed fermion oscillators algebra is defined in terms of
the creation operator $f^{*}$ and annihilation operator $f$ in the following
form Parthasarathy1991 ; Viswanathan1992 ; Chaichian1993 ; Algin2016b ,
$ff^{*}+qf^{*}f=1,$ (1) $[\hat{N},f^{*}]=f^{*},\,\,\,\,[\hat{N},f]=-f,$ (2)
where $\hat{N}$ is the fermionic number operator and q is the real positive
deformation parameter which takes values in the interval $0<\emph{q}<1$. The
basic deformed quantum number is given as
$[n]=\frac{1-(-1)^{n}q^{n}}{1+q}.$ (3)
To investigate the thermostatistical properties of the deformed fermion gas
model, the fermionic Jackson derivative operator is used instead of the
standard derivative operator
$D_{x}^{q}f(x)=\frac{1}{x}\left[\frac{f(x)-f(-qx)}{1+q}\right],$ (4)
for any function $f(x)$. The mean occupation number of the deformed fermion
gas model is expressed as
$n_{i}=\frac{1}{|lnq|}\left|ln\left(\frac{|1-ze^{\beta\epsilon_{i}}|}{1+qze^{\beta\epsilon_{i}}}\right)\right|,$
(5)
where $\epsilon_{i}$ is the kinetic energy of a particle in the one-particle
energy state, $\beta=1/k_{B}T$, $k_{B}$ is the Boltzmann constant, $T$ is the
temperature of the system and the fugacity $z=exp(\mu/k_{B}T)$ has the
standard form.
Using the Eq. (5), the total number of particles and the energy of the system
can be, respectively, expressed as
$N=\sum_{i}\frac{1}{|lnq|}\left|ln\left(\frac{|1-ze^{\beta\epsilon_{i}}|}{1+qze^{\beta\epsilon_{i}}}\right)\right|,$
(6)
$U=\sum_{i}\frac{\epsilon_{i}}{|lnq|}\left|ln\left(\frac{|1-ze^{\beta\epsilon_{i}}|}{1+qze^{\beta\epsilon_{i}}}\right)\right|.$
(7)
For a large volume and a large number of particles, we can replace the sums
over states by integrals. Using the density of state in two dimensions
$g(\epsilon)=(2{\pi}{mA}/h^{2})$, one can easily reach the following relations
$\frac{N}{A}=\frac{1}{\lambda^{2}}f_{1}(z,q),$ (8)
$\frac{U}{A}=\frac{1}{\lambda^{2}}k_{B}Tf_{2}(z,q),$ (9)
where $\lambda=h/(2{\pi}{mk_{B}T})^{1/2}$ is the thermal wavelength and the
generalized Fermi-Dirac function $f_{n}(z,q)$ is defined as
$\displaystyle f_{n}(z,q)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(n)}\int_{0}^{\infty}{\frac{x^{n-1}}{|lnq|}}\left|ln\left(\frac{|1-ze^{-x}|}{1+qze^{-x}}\right)\right|dx$
(10) $\displaystyle=$
$\displaystyle\frac{1}{|lnq|}\left[\sum_{l=1}^{\infty}\frac{(zq)^{l}}{l^{n+1}}-\sum_{l=1}^{\infty}\frac{z^{l}}{l^{n+1}}\right],$
where $x=\beta\epsilon$. From the thermodynamic relation $F=\mu{N}-PA$, the
Helmholtz free energy in two dimensions can be found as
$F=\frac{k_{B}TA}{\lambda^{2}}\left[{f_{1}(z,q)lnz-f_{2}(z,q)}\right].$ (11)
The entropy function of the model can be obtained by the relation $S=(U-F)/T$
as
$S=\frac{k_{B}A}{\lambda^{2}}\left[{2f_{2}(z,q)-f_{1}(z,q)lnz}\right].$ (12)
If we take the one particle kinetic energy $E=k_{B}T$ the Eq. (12) can be re-
expressed as
$S=\frac{2{\pi}mA}{h^{2}T}E^{2}F(z,q),$ (13)
where
$F(z,q)=2f_{2}(z,q)-f_{1}(z,q)lnz.$ (14)
## III The Einstein field equations due to deformed entropic gravity
In this section, we investigate the fermionic $q$-deformation effect on
Einstein’s field equations under Verlinde’s approach. Acoording to Verlinde’s
proposal Verlinde2011 , when a test particle approaches a holographic screen,
the entropic force of the system is expressed as
$\mathcal{F}=T\frac{\Delta{S}}{\Delta{x}},$ (15)
where $\Delta{S}$ is the change of the entropy on holographic screen and
$\Delta{x}$ is the displacement of the particle from holographic screen. The
holographic screen can be thought as maximal storage space for information.
The total number of bits $N$ is assumed to be proportional to the area $A$ of
the holographic screen and it is given as
$N=\frac{Ac^{3}}{2G\hbar}.$ (16)
When the entropic force is equal to the force increasing the entropy, the
total entropy of the system remains constant. Therefore, the variation of the
entropy goes to zero when the system has statistical equilibrium
$\frac{d}{dx^{a}}S(E,x^{a})=0,$ (17)
and it can be re-expressed as
$\frac{\partial{S}}{\partial{E}}{\frac{\partial{E}}{\partial{x^{a}}}+\frac{\partial{S}}{\partial{x^{a}}}}=0,$
(18)
where $\frac{\partial{E}}{\partial{x^{a}}}=-F_{a}$ and
$\frac{\partial{S}}{\partial{x^{a}}}=\nabla_{a}{S}$. Substituting Eq. (13)
into Eq. (18), the deformed temperature can be derived as
$T=\frac{mAE}{\pi{h}}F(z,q)e^{\phi}{N^{a}}{\nabla_{a}{\phi}},$ (19)
where we have used $F_{a}=me^{\phi}{\nabla_{a}}\phi$ and
$\nabla_{a}{S}=-2\pi{mN_{a}}/\hbar$ in the last equation. Also, $e^{\phi}$
indicates the redshift factor and $N^{a}$ is the unit outward pointing vector.
The above temperature equation is re-expressed as
$T=2\tilde{\alpha}({z,q})T_{U}$ (20)
where $T_{U}=\frac{\hbar}{2\pi}e^{\phi}{N^{a}}{\nabla_{a}{\phi}}$ is the Unruh
temperature Unruh1976 ; Verlinde2011 and $\tilde{\alpha}(z,q)$ carries the
information of deformed fermion system and defined as
$\tilde{\alpha}(z,q)=\frac{2\pi{mAE}}{h^{2}}F(z,q).$ (21)
Using the relation between mass-energy $M=N{T}/2$, the total mass can be
written as
$M=\frac{\tilde{\alpha}(z,q)}{4\pi{G}}\int_{\mathcal{S}}e^{\phi}{\nabla{\phi}}dA,$
(22)
where $\mathcal{S}$ is the holographic screen. The integral on the right hand
side relates with the modified Komar mass. Hence, the Eq. (22) can be
considered as the modified Gauss’s law in general relativity. Using the Stokes
theorem and following same procedure in Ref. Senay2018 ; Kibaroglu2019 , the
Eq. (22) can be written in terms of the Killing vector $\xi^{a}$ and Ricci
curvature tensor $R_{ab}$ as
$M=\frac{\tilde{\alpha}(z,q)}{4\pi{G}}\int_{V}R_{ab}n^{a}\xi^{b}dV,$ (23)
where $V$ is a spacelike hypersurface demonstrating the space volume and
$n_{a}$ is the unit vector normal to $V$. Moreover, the Komar mass can be
expressed in terms of the stress-energy tensor $T_{ab}$ Wald1984 as
$\mathcal{M}=2\int_{V}\left(T_{ab}-\frac{1}{2}g_{ab}T+\frac{\Lambda}{8\pi{G}}g_{ab}\right){n^{a}\xi^{b}}dV,$
(24)
where $g_{ab}$ is metric tensor. Equating Eq. (23) and Eq. (24), we obtain
$\tilde{\alpha}(z,q){R_{ab}}=8\pi{G}\left(T_{ab}-\frac{1}{2}g_{ab}T+\frac{\Lambda}{8\pi{G}}g_{ab}\right).$
(25)
If we take the trace of the last equation, we reach
$R_{ab}-{\frac{1}{2}g_{ab}R+{\frac{\Lambda}{\tilde{\alpha}(z,q)}}g_{ab}=\frac{8\pi{G}}{\tilde{\alpha}(z,q)}T_{ab}},$
(26)
Thus, we get a modification of Einstein’s field equations resulting from
deformed fermion theory under Verlinde’s entropic gravity scenario.
## IV Friedmann equation due to deformed entropic gravity
In this section, we want to derive modifed version of Friedmann equations. In
the homogeneous and isotropic space time, the FRW universe is given by the
line element
$ds^{2}=c^{2}dt^{2}-a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right),$
(27)
where $a(t)$ is the scale fator and $k$ is constant relating curvature of the
universe with $k=0,1,-1$ corresponding to flat, closed, and open universes,
respectively. The dynamical apparent horizon for the FRW univers can be
defined as
$\tilde{r}=ar=\frac{1}{\sqrt{H^{2}+k/a^{2}}},$ (28)
where $H=\dot{a}/a$ is the Hubble parameter and the dots represents the time
derivative. Now, assuming that the matter source in the FRW universe can be
expressed as a perfect fluid with the following stress-energy tensor
$T_{ab}=\left(\rho+p\right)u_{a}u_{b}-pg_{ab},$ (29)
where $\rho$ is the mass-energy density, $p$ is the pressure of the fluid, and
$u_{a}$ is the fluid’s four vector. The conservation law of energy-momentum
leads to the following continuity equation
$\dot{\rho}+3H\left(\rho+p\right)=0.$ (30)
Now, we consider a compact spatial region $V=\frac{4}{3}\pi\tilde{r}^{3}$ with
a compact boundary $\mathcal{S}=4\pi\tilde{r}^{2}$. The acceleration of radial
observe is given as
$a_{r}=-\ddot{a}r,$ (31)
and it leads to take the following form for Unruh temperature
$T_{U}=\frac{\hbar{a_{r}}}{2\pi}.$ (32)
The total physical mass $M$ inside the volume $V$ can be defined as
$M=\int_{V}{dV(T_{ab}u^{a}u^{b})}=\frac{4}{3}\pi\tilde{r}^{3}\rho.$ (33)
Using the Eqs. (16), (20), (31)-(33), one can easily find
$\frac{\ddot{a}}{a}=-\frac{4\pi{G}}{3\tilde{\alpha}(z,q)}\rho$ (34)
which is modified version of the dynamical equations for the Newtonian
cosmology in the deformed case. Now, we want to derive the Friedmann equations
of FRW universe. For this purpose, we need to use the active gravitational
mass $\mathcal{M}$ instead of the total mass $M$, since the active
gravitational mass produces the acceleration in the dynamical background. In
our context, the active gravitational mass can defined as
$\mathcal{M}=\left(\rho+3p+\frac{\Lambda}{4\pi{G}}\right)V.$ (35)
Replacing the total mass $M$ with active gravitational mass $\mathcal{M}$ in
the Eq. (33), we reach,
$\frac{\ddot{a}}{a}=-\frac{4\pi{G}}{3\tilde{\alpha}(z,q)}(\rho+3p)-\frac{\Lambda}{3\tilde{\alpha}(z,q)},$
(36)
which is the deformed acceleration equation for the dynamical evolution of the
FRW universe. Using the Eq. (30) and multiplying both sides of Eq. (36) with
$a\dot{a}$ and then integrating it, we have
$H^{2}+\frac{k}{a^{2}}=\frac{8\pi{G}}{3}\left(\frac{\rho}{\tilde{\alpha}(z,q)}-\frac{\Lambda}{8\pi{G}\tilde{\alpha}(z,q)}\right).$
(37)
For simplicity we focus on the flat case $k=0$, the last equation can re-
express as
$H^{2}=\frac{8\pi{G}}{3}(\rho+\rho_{DE}),$ (38)
where $\rho_{DE}$ is the effective energy density of the dark energy
$\rho_{DE}=\left(\frac{1}{\tilde{\alpha}(z,q)}-1\right)\rho-\frac{\Lambda}{8\pi{G}\tilde{\alpha}(z,q)}.$
(39)
Moreover, using the relation $\dot{H}=\ddot{a}/a-H^{2}$ together with Eqs.
(36)-(39), one can easily obtain the following expression
$\dot{H}=-4\pi{G}(\rho+p+\rho_{DE}+p_{DE}),$ (40)
where $p_{DE}$ is the effective pressure of the dark energy
$p_{DE}=\left(\frac{1}{\tilde{\alpha}(z,q)}-1\right)p+\frac{\Lambda}{8\pi{G}\tilde{\alpha}(z,q)}.$
(41)
Moreover, the effective equation of state can be easily found in the following
form:
$w_{DE}=\frac{p_{DE}}{\rho_{DE}}=\frac{\left(\frac{1}{\tilde{\alpha}(z,q)}-1\right)p+\frac{\Lambda}{8\pi{G}\tilde{\alpha}(z,q)}}{\left(\frac{1}{\tilde{\alpha}(z,q)}-1\right)\rho-\frac{\Lambda}{8\pi{G}\tilde{\alpha}(z,q)}}.$
(42)
Before we close this section, we want to remark that we considered a static
background by following Verlinde’s way. In order to obtain Einstein’s
equation, he uses a timelike Killing vector which exists for static or
stationary spacetime Cai2010 . In case of the FRW spacetime, this can be
possible only if the metric Eq. (27) reduces to the de Sitter or Minkowski
spacetime Tower2014 . By considering this assumption, we have obtained the
modified version of the dynamical equations governing the evolution of the FRW
universe.
## V Cosmological Results
In this section, we briefly investigate the comological evolution in the
scenario of $q$-deformed dark energy model. Now we consider the case of dust
matter (${p}\approx{0}$) in which case Eq. (30) leads to $\rho=\rho_{0}a^{-3}$
Roos2015 , where $\rho_{0}$ is the current value of the dust density.
Furthermore, the density parameters of the matter and the effective dark
energy can be defined in the following form, respectively
$\Omega_{m}=\frac{8\pi{G}}{3H^{2}}\rho,$ (43)
$\Omega_{m}=\frac{8\pi{G}}{3H^{2}}\rho_{DE}.$ (44)
From Eq. (43) we reach $\Omega_{m}=(\Omega_{m0}{H^{2}_{0}})/(a^{3}H^{2})$ and
using the definition
$\Omega_{m}+\Omega_{DE}=1,$ (45)
we find
$H=\frac{\sqrt{\Omega_{m0}}{H_{0}}}{\sqrt{a^{3}(1-\Omega_{DE})}}.$ (46)
Differentiating last equation and using redsift $z_{R}=a^{-1}-1$, we obtain
$\dot{H}=-\frac{H^{2}}{2(1-\Omega_{DE})}\left(3(1-\Omega_{DE})+(1+z_{R})\Omega^{\prime}_{DE}\right)$
(47)
where the prime represents $z_{R}$-derivative. Inserting Eq. (39) into Eq.
(44) and using Eq. (46), we can easily reach
$\Omega_{DE}=1-\frac{1}{1+\frac{8\pi{G}}{3\Omega_{m0}H_{0}^{2}(1+z_{R})^{3}}\left[\left(\frac{1}{\tilde{\alpha}(z,q)}-1\right)\rho-\frac{\Lambda}{8\pi{G}\tilde{\alpha}(z,q)}\right]},$
(48)
which is the analytical solution of the dark energy density parameter for the
dust matter case in a flat universe.
## VI Conlusion
In this paper, we introduced some of the high temperature thermodynamical
properties of $q$-deformed fermion gas model in two-dimensional space, and
particularly discussed its effects on the Einstein field equations, Friedmann
equations and dark energy. For this aim, we considered quantum black holes as
a deformed fermion particles proposed by StromingerStrominger1993 and used
entropic gravity assumption proposed by Verlinde Verlinde2011 .
By the help of the $q$-deformed entropy function in Eq. (12), we obtained
$q$-deformed temperature function in Eq. (19). Using it in the mass-energy
relation and Komar mass definition, we derived $q$-deformed Einstein field
equations in Eq. (26). The factor $\tilde{\alpha}(z,q)$ in the Eq. (26)
carries the information of $q$-deformed fermion gas model. This factor does
not reduce to unity in the limit $q\rightarrow 1$. Therefore, the $q$-deformed
Einstein field equations do not reduce to standard ones when considered
$q\rightarrow 1$.
Moreover, it can be seen in Eq. (26) that the field equations remain intant in
their original form while the $q$-deformation brings a certain kind of re-
scaling to the two constant, namely the Newton’s gravitational constant and
cosmological constant. For a more novel approach, the field equations ought to
get correction to the curvature or the matter sector rather a mere re-scaling
of certain constants. Although $q$ is a free parameter which varies between 0
and 1, both $G$ and $\Lambda$ might be varying constant. The $q$-formation to
field equations would not effect any physical processes under consideration.
Considering FRW metric, we could obtain $q$-deformed version of Friedmann
equations in Eqs. (36) and (37). These equations do not coincide with
$\Lambda{CDM}$ paradigm due to the factor $\tilde{\alpha}(z,q)$. However,
these may be considered as the generalized version of $\Lambda{CDM}$ model.
Also, taking into account flat case $k=0$, we determined the effective energy
density, the effective pressure, and the effective equation of state for the
dark energy sector. Moreover, supposing a background filled by a pressureless
source, we derived analytical expression of the effective density parameter of
dark energy in Eq. (48).
Consequently, the $q$-deformed fermion theory plays a crucial role on the
investigation of cosmological evolution under gravity-thermodynamic
connection. We hope that the results obtained this study may be used to
understand the properties of the early universe and dark energy.
The investigation of the present $q$-deformed fermion model on the Einstein
field equations and cosmology in the low-temperature limit is one of the open
problems to study in the near future.
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|
# Positive spectrahedra:
Invariance principles and Pseudorandom generators
Srinivasan Arunachalam
IBM Quantum.
IBM T.J. Watson Research Center
Yorktown Heights, USA
<EMAIL_ADDRESS>Penghui Yao
State Key Laboratory for
Novel Software Technology,
Nanjing University
<EMAIL_ADDRESS>
###### Abstract
In a recent work, O’Donnell, Servedio and Tan (STOC 2019) gave explicit
pseudorandom generators ($\mathsf{PRG}$s) for arbitrary $m$-facet polytopes in
$n$ variables with seed length poly-logarithmic in $m,n$, concluding a
sequence of works in the last decade, that was started by Diakonikolas,
Gopalan, Jaiswal, Servedio, Viola (SICOMP 2010) and Meka, Zuckerman (SICOMP
2013) for fooling linear and polynomial threshold functions, respectively. In
this work, we consider a natural extension of $\mathsf{PRG}$s for
intersections of positive spectrahedra. A positive spectrahedron is a Boolean
function $f(x)=[x_{1}A^{1}+\cdots+x_{n}A^{n}\preceq B]$ where the $A^{i}$s are
$k\times k$ positive semidefinite matrices. We construct explicit
$\mathsf{PRG}$s that $\delta$-fool “regular” width-$M$ positive spectrahedra
(i.e., when none of the $A^{i}$s are dominant) over the Boolean space with
seed length $\operatorname{poly}(\log k,\log n,M,1/\delta)$.
Our main technical contributions are the following: We first prove an
invariance principle for positive spectrahedra via the well-known Lindeberg
method. As far as we are aware such a generalization of the Lindeberg method
was unknown. Second, we prove an upper bound on noise sensitivity and a
Littlewood-Offord theorem for positive spectrahedra. Using these results, we
give applications for constructing $\mathsf{PRG}$s for positive spectrahedra,
learning theory, discrepancy sets for positive spectrahedra (over the Boolean
cube) and $\mathsf{PRG}$s for intersections of structured polynomial threshold
functions.
###### Contents
1. 1 Introduction
1. 1.1 Prior work and conceptual challenges
2. 1.2 Our main result
3. 1.3 Sketch of the [HKM13] invariance principle for polytopes
4. 1.4 First contribution: Invariance principle for Bentkus mollifier
5. 1.5 Second contribution: Geometric properties of positive spectrahedra
6. 1.6 Applications
7. 1.7 Future work
2. 2 Preliminaries
1. 2.1 Derivatives and multidimensional Taylor expansion
2. 2.2 Combinatorial properties of Boolean functions
3. 2.3 Matrix analysis and Random matrices
4. 2.4 Matrix functions, spectral functions and Fréchet derivatives
5. 2.5 spectrahedra and Positive spectrahedra
6. 2.6 Pseudorandomness
7. 2.7 Tensors
3. 3 Bentkus mollifier
1. 3.1 Properties of the mollifier and its derivatives
2. 3.2 Properties of the spectral norm of the mollifier
4. 4 Computing spectral derivatives
1. 4.1 Formulas for spectral derivatives
2. 4.2 Third order Fréchet derivatives of smooth functions
3. 4.3 Main theorem: Fréchet derivatives of Bentkus function
4. 4.4 Bounding terms ($1$)-($5$) in Theorem 27 for Bentkus function
5. 4.5 Bounding terms $(6,7)$ in Theorem 27 for Bentkus function
5. 5 Properties of positive spectrahedra
1. 5.1 Average sensitivity and Noise sensitivity
2. 5.2 Boolean Anti-concentration: Littlewood Offord for spectrahedra
6. 6 Invariance principle for positive spectrahedra
1. 6.1 Invariance principle for the spectral Bentkus mollifier
2. 6.2 Invariance principle for positive spectrahedra
3. 6.3 Application: Pseudorandom generators for positive spectrahedra.
7. A Proof of Lemma 34: Case 2
1. A.1 Upper bounding first term in Eq. (58)
2. A.2 Upper bounding $(\dagger)$ first term in $(\star)$ in Eq. (59)
3. A.3 Upper bounding first term in $(\dagger\dagger)$ in Eq. (60)
4. A.4 Upper bounding the second term ($\P$) in Eq. (60)
## 1 Introduction
Constructing explicit pseudorandom generators $(\mathsf{PRG})$ for a class of
interesting Boolean functions has received tremendous attention in the last
few decades. One particular class of functions that has seen a flurry of works
is the class of halfspaces. A _halfspace_ is a Boolean function
$f:\\{-1,1\\}^{n}\rightarrow\left\\{0,1\right\\}$ that can be expressed as
$f(x)=\operatorname{sign}(a_{1}x_{1}+\cdots+a_{n}x_{n}-b)$ for some real
values $a_{1},\ldots,a_{n},b\in~{}\mathbb{R}$. Halfspaces arise naturally in
many areas of theoretical computer science including machine learning,
communication complexity, circuit complexity and pseudorandomness. A
successful line of work [Ser06, DHK+10, MZ13, KM15, GKM18] resulted in
$\mathsf{PRG}$s that $\varepsilon$-fool halfspaces with seed length poly-
logarithmic in $(n/\varepsilon)$ over the Boolean space.
Given the success in designing $\mathsf{PRG}$s for single halfspaces (or
linear threshold function), two alternate lines of work received a lot of
attention, _polynomial threshold functions_ and _intersections_ of halfspaces.
A degree-$d$ polynomial threshold function ($\mathsf{PTF}$) is simply a
function $f(x)=\operatorname{sign}(p(x))$ where $p$ is a degree-$d$
polynomial. In this direction, there have been a sequence of works [DGJ+10,
DHK+10, Kan10, Kan11a, Kan11b, Kan11c, Kan14b, OST20] that produced
$\mathsf{PRG}$s with seed length exponential in $d$ over the Boolean space and
quasi-polynomial in $d$ over the Gaussian space. Alternatively, another line
of work considered _intersections_ of halfspaces (i.e., a polytope). In this
direction, a sequence of works [GOWZ10, HKM13, ST17, CDS19, OST19] produced a
$\mathsf{PRG}$ for $m$-facet polytopes in $n$ variables with seed length poly-
logarithmic in $m,n$.
In this work, we initiate the construction of $\mathsf{PRG}$s for
spectrahedra: a natural generalization of halfspaces, polytopes and
$\mathsf{PTF}$s in one framework. A _spectrahedron_ $S\subseteq\mathbb{R}^{n}$
is a feasible region of a _semidefinite program_. Namely,
$S=\left\\{x\in\mathbb{R}^{n}:\sum_{i}x_{i}A^{i}\preceq B\right\\}$
for some $k\times k$ symmetric matrices $A^{1},\ldots,A^{n},B$, where
$\preceq$ is the standard Löwner ordering.111In this ordering, we say
$A\preceq B$ if $B-A$ is positive semidefinite, i.e., all the eigenvalues of
$B-A$ are non-negative. We say $S$ is a _positive spectrahedron_ if either all
$A^{i}$s are positive semidefinite ($\mathsf{PSD}$) or all $A^{i}$s are
negative semidefinite. spectrahedra are important basic objects in polynomial
optimization and algebraic geometry [BPT12, Sch18]. Mathematically,
spectrahedra have rich and complicated structures and include well-known
geometric objects like polytopes, cylinders, polyhedrons, elliptopes.
Computationally, semidefinite programming has found many applications in
theoretical computer science in the field of optimization [AK07],
approximation theory [GW95, GM12], algorithms [AHK05, JLL+20], SoS hierarchy
[BHK+19], extension complexity [FMP+15, LRS15]. The class of semidefinite
programs that consists of only $\mathsf{PSD}$ matrices is an important class
of SDPs, termed as positive semidefinite programs, which has been used to
characterize various quantum interactive proof systems [JUW09, JJUW11, GW13].
Their computational complexity has also received a lot of attention in the
past decade [JY11, PT12, AZLO16, JLL+20]. But in several ways, our
understanding of spectrahedra is at an early stage and seriously lags behind
our understanding of polytopes. Many basic geometric properties of
spetrahedrons, such as dimensions, numbers of connected components, matrix
ranks [Viz17] are not well understood, even basic properties such as proving
the membership of spetrahedrons for some geometric objects is highly non-
trivial [NPS08].
Our main result in this work is $\mathsf{PRG}$s for regular positive
spectrahedra with seed length poly-logarithmic in $n$ and $k$, which we define
in Section 1.2. Before stating our main results, we briefly discuss the
techniques developed by prior works to construct $\mathsf{PRG}$s for polytopes
before discussing the challenges we need to handle here.
### 1.1 Prior work and conceptual challenges
#### 1.1.1 Prior work
One of the earliest works that considered fooling threshold functions was by
Meka-Zuckerman [MZ13] and [DGJ+10]. A powerful technique that Meka-Zuckerman
introduced was a general recipe to construct $\mathsf{PRG}$s for functions $f$
via _invariance principles_. Roughly speaking, an invariance principle for a
function $f:\\{-1,1\\}^{n}\rightarrow\\{0,1\\}$ states that, the expected
value of $f(\mathcal{U}^{n})$ (where the input is uniformly random in
$\\{-1,1\\}^{n}$) is close to the expected value of $f(\mathcal{G}^{n})$
(where the input is a standard $\mathcal{G}^{n}=\mathcal{N}(0,1)^{n}$
Gaussian). Invariance theorems are generalizations of the classic Berry-Esseen
central limit theorem, proven using the well-known Lindeberg method [Lin22].
The versatile framework of [MZ13] allows one to use invariance principles
along with a few more ingredients to construct $\mathsf{PRG}$s, so the
technical challenge is in establishing invariance principles.
Using this framework, Harsha, Klivans and Meka [HKM13] proved an _invariance
principle for regular polytopes_ (i.e., when the coefficients in (all) the
halfspaces are “regular”). The main novelty in their work was the _poly-
logarithmic_ (in the input parameters) error dependence. In order to prove
this, they first proved a general invariance principle for smooth functions
(over polytopes). Subsequently they instantiate their invariance principle for
the so-called Bentkus mollifier [Ben90],222The Bentkus _mollifier_ is a
function which provides a “smooth” continuous approximation to the the
discrete multivariate indicator function (also referred to as _orthant
functions_). We discuss this further below. crucially relying on the fact that
the mollifier has derivatives that scale poly-logarithmic in the input size.
Finally in order to go from invariance principles (for the mollifier) to
fooling regular polytopes, they need to prove an _anti-concentration of
polytopes_ in the Gaussian space. For this, they use (as a black-box) a well-
known result of Nazarov [Naz03, KOS08], which bounds the _Gaussian surface
area_ $(\mathsf{GSA})$ of polytopes. Putting together the invariance principle
for smooth functions, Bentkus mollifier and Nazarov’s bound on $\mathsf{GSA}$,
[HKM13] obtained their main results for regular polytopes. We discuss this
proof idea in more detail in Section 1.3.
Subsequently, Servedio and Tan [ST17] improved the results of [HKM13] by
considering “low-weight” polytopes, which removes the regularity condition
(albeit, with the seed length of the $\mathsf{PRG}$ in [ST17] depending on the
weight). Finally, O’Donnell, Servedio and Tan [OST19] showed how to fool
arbitrary polytopes. In [OST19] they bypass the entire Gaussian space (in fact
it is a _necessity_ to avoid this Gaussian space since standard invariance
principles do not hold for non-regular polytopes) and proved a “Boolean-
invariance principle” for the Bentkus mollifier. Although they bypass the
Gaussian intermediate (which is standard in invariance principles), their
proof techniques still use the Lindeberg method. Additionally, a crucial tool
introduced by them was the Boolean anti-concentration of polytopes, since they
can no longer use the $\mathsf{GSA}$ bound of Nazarov which used by [HKM13,
ST17, CDS19] for _Gaussian_ anti-concentration.
#### 1.1.2 PRGs for spectrahedra: Conceptual challenges
There are two straightforward approaches to constructing $\mathsf{PRG}$s for
positive spectrahedra. The first is to write a spectrahedron as a linear
program. Naturally one can approximate a positive-semidefinite constraint
$X\succeq 0$ of a $k\times k$ symmetric matrix with exponentially many
constraints $z^{T}\hskip 1.42262ptX\hskip 1.42262ptz\geq 0$ for
$z\in~{}\mathbb{R}^{k}$. However the results of [HKM13, OST19] would be moot
here since the seed-lengths of their $\mathsf{PRG}$s are poly-logarithmic in
the number of constraints, which is polynomial in the dimension $k$, while our
goal it to have a seed length _poly-logarithmic_ in $k$. The second approach
is to use Sylvester’s criterion to write out $k$ polynomials of degree at most
$k$ (corresponding to the $k$ determinantal representation of the $k$ minors)
and one could use $\mathsf{PRG}$s for polynomial threshold functions
($\mathsf{PTF}$). However, finding optimal $\mathsf{PRG}$s for $\mathsf{PTF}$s
has remains open and the best-known $\mathsf{PRG}$s we have for degree-$k$
$\mathsf{PTF}$s over the Boolean space depends _exponentially_ in $k$ [MZ13].
This naturally motivates us to use the “eigenstructure” of $X\succeq 0$
crucially in understanding spectrahedra. The next line of approach is to use
the existing invariance-principle framework of [MZ13] which we overviewed in
the previous section, but this opens up a few challenges:
1. 1.
Invariance principles: Since a spectrahedron naturally deals with eigenvalues
of matrices, it is unclear if we could use known invariance principles for
spectrahedra. In fact, we are not even aware of a generalization of the
Lindeberg-type argument to show an invariance principle for _spectral
functions_ (i.e., functions that act on the eigenspectra of matrices).
2. 2.
Geometric properties: Prior works of [KOS08, HKM13, ST17, CDS19] crucially
used the work of Nazarov [Naz03] which bounds the Gaussian surface area of
polytopes in order to prove their anti-concentration. However, spectrahedra
are very poorly understood, and even more basic questions about their average
sensitivity, noise sensitivity, surface area are unknown.
3. 3.
Anti-concentration: An important technique for constructing $\mathsf{PRG}$s
using invariance principles requires one to prove _anti-concentration_ , i.e.,
when moving from the smooth mollifiers to the orthant functions a crucial
ingredient is anti-concentration. It is far from clear if spectrahedra enjoy
such nice properties in either Boolean spaces or Gaussian spaces.
As far as we are aware, none of these questions have been considered for any
class of spectrahedra except polytopes. Our main contribution is to make
significant progress in all these questions for the class of positive
spectrahedra.
### 1.2 Our main result
In order to state our main result we first define $\mathsf{PRG}$s and
$(\tau,M)$-regular spectrahedra. A _pseudorandom generator_ is a function
$G:\\{-1,1\\}^{r}\rightarrow\\{-1,1\\}^{n}$ and is said to $\varepsilon$-fool
a class of functions
$\mathcal{F}\subseteq\\{f:\\{-1,1\\}^{n}\rightarrow\\{0,1\\}\\}$ with _seed
length_ $r$ if it satisfies the following: for every $f\in\mathcal{F}$, we
have
$\left|\Pr_{\bm{x}\sim\mathcal{U}_{n}}[f(\bm{x})=1]-\Pr_{\bm{y}\sim\mathcal{U}_{r}}[f\left(G(\bm{y})\right)=1]\right|\leq\varepsilon,$
where $\mathcal{U}_{n}$ (resp. $\mathcal{U}_{r}$) corresponds to uniform
distribution over $\\{-1,1\\}^{n}$ (resp. $\\{-1,1\\}^{r}$). We next define
the class of regular positive spectrahedra. Given $\tau,M>0$, we say a
sequence of $k\times k$ positive semidefinite matrices
$\left(A^{1},\ldots,A^{n}\right)$ is _$\left(\tau,M\right)$ -regular_ if
$\mathbb{I}\preceq\sum_{i=1}^{n}\left(A^{i}\right)^{2}\preceq
M\cdot\mathbb{I}\hskip 5.69054pt~{}\mbox{and}~{}\hskip
5.69054ptA^{i}\preceq\tau\cdot\mathbb{I}\text{ for every }i\in[n].$ (1)
This regularity assumption is a very natural assumption, it says that the
_width_ of a semidefinite program defined by these matrices is bounded. We
remark that our regularity condition naturally extends (and is in fact _less_
restrictive) the regularity condition that was used in prior works on fooling
halfspaces and polytopes [GOWZ10, DGJ+10, MZ13, HKM13]. In Section 1.5.2 we
discuss more about why this notion of regularity is necessary and sufficient
for our proof techniques.
A _spectrahedron_ $S\subseteq{\mathbb{R}}^{n}$ is a feasible region of the
convex set $S=\left\\{x\in\mathbb{R}^{n}:\sum_{i}x_{i}A^{i}\preceq
B\right\\}$.333For simplicity in exposition, we assume here that $\|B\|\leq M$
(our main theorems depend on the norm of $B$). We say $S$ is a _positive
spectreheron_ if either all $A^{i}$s are positive semidefinite
($\mathsf{PSD})$ or all $A^{i}$s are negative semidefinite. We say $S$ is a
_$(\tau,M)$ -regular positive spectrahedron_ if $(A^{1},\ldots,A^{n})$ are
$(\tau,M)$ regular. It is also natural to consider an _intersection_ of
positive spectrahedra $S_{1},\ldots,S_{t}$. However, without loss of
generality one can assume that $t=2$ since one can “pack” all the $S_{i}$s
with $\mathsf{PSD}$ matrices into a larger block diagonal matrix with
dimension $t\cdot k$ and similarly all the negative semidefinite matrices, so
we can always assume we are working with an intersection of two positive
spectrahedra.444Crucially we remark that the seed length of our $\mathsf{PRG}$
has dependence only logarithmic in $k$, so even with an intersection of $t$
positive spectrahedra, the dependence would be logarithmic in $t$ as well. For
simplicity, in the introduction we assume that we are working with a single
regular positive spectrahedron here and state our main theorem.
###### Result 1 (PRG for positive spectrahedra).
There exists a $\mathsf{PRG}$ $G:\\{0,1\\}^{r}\rightarrow\\{-1,1\\}^{n}$ with
seed length
$r=O(\log n\cdot\log k\cdot M\cdot 1/\delta)$
that $\delta$-fools $(\tau,M)$-regular positive spectrahedra for
$\tau\leq\operatorname{poly}(\delta/(M\cdot\log k))$.
Typically, handling the “regular case” is the first step towards obtaining
optimal results in pseudorandom generators for geometric objects and we have
accomplished that here for the first time. To prove this theorem, we follow
the well-known three-step approach and prove the following:
1. 1.
An invariance principle for the Bentkus mollifier of arbitrary regular
spectrahedra.
2. 2.
Boolean and Gaussian anti-concentration for _positive_ regular spectrahedra.
3. 3.
An invariance principle for _positive_ regular spectrahedra
Before proving these statements, we first overview the [HKM13, OST19] approach
to proving invariance principles (since our high-level ideas are inspired by
their works).
### 1.3 Sketch of the [HKM13] invariance principle for polytopes
First recall that a polytope is the feasible region of the set
$\\{x\in\mathbb{R}^{n}:Wx\leq b\\}$ for a fixed $W\in\mathbb{R}^{n\times
n},b\in\mathbb{R}^{n}$.555For simplicity, we assume that the number of
constraints and variables are equal. Their analysis is more general. We say a
polytope is $\tau$-regular if each row $W^{i}$ satisfies $\|W^{i}\|_{2}=1$ and
$\|W^{i}\|_{4}\leq\tau$. At a high-level the [HKM13] invariance principle
states the following:
$\displaystyle\left|\Pr_{\bm{x}\sim\mathcal{U}_{n}}[W\bm{x}\leq
b]-\Pr_{\bm{g}\sim\mathcal{G}^{n}}[W\bm{g}\leq
b]\right|\leq\operatorname{poly}(\log n,\tau).$ (2)
To show this, they first express the _orthant_ function above (which we denote
$\mathcal{O}:\mathbb{R}^{n}~{}\rightarrow~{}\\{0,1\\})$, as $[W\bm{x}\leq
b]=[W^{1}\bm{x}\leq b_{1}]\cdots[W^{n}\bm{x}\leq b_{n}]$. Given this
structure, they now use the well-known Lindeberg method [Lin22] (see [O’D14,
Tao10] for a detailed exposition) to move from the uniform distribution over a
Boolean space to the Gaussian space. To establish Eq. (2), they follow a
three-step approach: (1) First, they prove a version of Eq. (2) for _smooth_
functions $\widetilde{\mathcal{O}}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ (i.e.,
functions who have bounded multivariate derivatives). In particular, they use
the Lindeberg method to show that the expected value of
$\widetilde{\mathcal{O}}(W\bm{x})$ for $x\sim\mathcal{U}_{n}$, is “close” to
the expected value of $\widetilde{\mathcal{O}}(W\bm{g})$ for
$\bm{g}\sim\mathcal{G}^{n}$. To understand this closeness, they write out
$\widetilde{\mathcal{O}}(W\bm{z})$ using the standard multivariate Taylor
expansion and bound the distance between $\widetilde{\mathcal{O}}(W\bm{x})$
and $\widetilde{\mathcal{O}}(W\bm{g})$ by the higher-order derivatives of the
smooth function $\widetilde{\mathcal{O}}$. (2) Second, they observe that a
result of Bentkus [Ben90] provides exactly an approximator
$\widetilde{\mathcal{O}}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ (which we refer
to as the _Bentkus mollifier_) which serves as a _smooth_ approximation to the
$\\{0,1\\}$-valued orthant function $\mathcal{O}(x)=~{}[W\bm{x}\leq b]$.
Additionally this mollifier _crucially_ satisfies the property that
$\|\widetilde{\mathcal{O}}^{(\ell)}\|_{1}\leq
O\left(\log^{\ell}n\right)$.666Here $\|f^{(\ell)}\|_{1}$ is the $1$-norm of
the coefficients in the $\ell$-th derivative. In [HKM13], they care about
$\|f^{(4)}\|_{1}=\max_{x}\sum_{p,q,r,s}\left|\partial_{p}\partial_{q}\partial_{r}\partial_{s}f(x)\right|$.
(3) So far they established that the Bentkus mollifier (which served as a
proxy for $[W\bm{x}\leq b]$) satisfies an approximate version of Eq. (2). In
order to go from being close with respect to this Bentkus mollifier to
multidimensional CDF closeness, they prove _Gaussian_ anti-concentration of
polytopes. For this, they use a result of Nazarov [Naz03] (as a black-box)
which shows that the Gaussian surface area of a polytope is $O(\sqrt{\log
n})$. These three steps allow them to prove Eq. (2).
### 1.4 First contribution: Invariance principle for Bentkus mollifier
We begin by defining spectral functions. Let
$f:\mathbb{R}^{k}\rightarrow\mathbb{R}$, we say
$\psi:\mathsf{Sym}_{k}\rightarrow\mathbb{R}$ is a _spectral function_ if
$\psi(M)=f(\lambda(M))$ for all $M\in\mathsf{Sym}_{k}$ where
$\lambda(M)=(\lambda_{1},\ldots,\lambda_{k})$ are the $k$ eigenvalues of $M$.
In other words, a spectral function $\psi(\cdot)$ depends on a function $\psi$
applied to the eigenvalues of its argument. We say $f$ satisfies an invariance
principle if
$\operatorname*{\mathbb{E}}_{\bm{x}\sim\mathcal{U}_{n}}\left[\psi\left(\sum_{i}\bm{x}_{i}A^{i}-B\right)\right]\approx_{\varepsilon}\operatorname*{\mathbb{E}}_{\bm{g}\sim\mathcal{G}^{n}}\left[\psi\left(\sum_{i}\bm{g}_{i}A^{i}-B\right)\right],$
for symmetric matrices $A_{1},\ldots,A_{n},B$. A conceptual challenge in
proving an invariance principle even for smooth spectral functions is that
standard Lindeberg-style proofs of invariance theorems use multivariate Taylor
series of the mollifier function cannot be used here, since our functions act
on the _eigenvalues_ of matrices. In the past, there have been various
invariance principles [MOO05, Mos08, IM12, HKM13, Yao19] but none of them
apply here; as far as we are aware invariance principles with non-diagonal
$A^{i},B$ have not been studied. In this work, we overcome this challenge and
adapt the Lindeberg-style proofs of probabilistic invariance principles to
prove its analogue for spectral functions.
To this end, recall that we are concerned with spectrahedra whose feasible
regions are given by $\\{x\in\mathbb{R}^{n}:\sum_{i}x_{i}A^{i}\preceq B\\}$,
which can alternatively be written as
$\\{x:\lambda_{\max}\left(\sum_{i}x_{i}A^{i}-B\right)\leq 0\\}$. So we let our
spectral function $f:\mathbb{R}^{k}\rightarrow\mathbb{R}$ to be
$f(\lambda)=[\max_{i}\lambda_{i}\leq 0]$ (recall that although our
spectrahedron acts on $n$ bits on which we want to prove an invariance
principle, our spectral function acts only on the $k$ eigenvalues). For this
function, we can still use the Bentkus mollifier
$\widetilde{\mathcal{O}}:\mathbb{R}^{k}\rightarrow\mathbb{R}$ as a smooth
approximation to $f$.777In fact our analysis can allow arbitrary orthant
functions which can be approximated by a Bentkus mollifier. So our first main
contribution is to prove an invariance principle for the Bentkus mollifier
applied to the spectra of matrices. We remark that in contrast to [HKM13], we
do not prove a general invariance principle for spectral functions, instead
our spectral function is tailored for the Bentkus mollifier (which is also the
case for [OST19]).
##### Fréchet derivatives.
Since our Bentkus mollifier is acting on the eigenspectra of matrices, instead
of multivariate Taylor expansion, we adopt _Fréchet derivatives_ , a notion of
derivatives that is studied in Banach spaces. Unfortunately, Fréchet series
(in contrast to standard multivariate series) are still not well understood.
In fact even basic properties such as continuity, Lipschitz continuity,
differentiability, continuous differentiability, were only proven in the last
three decades [BSS98, Lew96, BS99, CQT03], which have been well-known for
centuries in standard calculus. In particular, even a succinct representation
of high-order Fréchet derivatives [Sen07, AS10, AS12, AS16] for _spectral_
functions only appeared in the last decade.
Fortunately for us, Sendov [Sen07] provided a tensorial representation of
high-order Fréchet series for spectral functions which we employ to analyze
the Fréchet derivatives of the Bentkus mollifier. The challenge is in bounding
the 3-tensors that appears in Sendov’s theorem, which produce $7$ terms
corresponding to different permutations of the tensors after simplification.
Three of these $7$ terms can simply be upper bounded by
$\|\widetilde{\mathcal{O}}^{(3)}\|_{1}$ which we know to be small for the
Bentkus mollifier. We remark that these are exactly, and the only, terms that
appear in the standard invariance principle proofs for linear forms.
Intuitively this is not surprising since the first three terms simply
correspond to the case when the $A^{i},B$ are _diagonal_ which reduces a
spectrahedron to a polytope. However, bounding the remaining terms is highly
non-trivial and one of our technical contributions is in showing these
remaining terms are bounded for the Bentkus mollifier.
##### Bounding derivatives and obtaining invariance principle.
Bounding these last three terms of the $3$-tensors significantly deviates from
the analysis of [HKM13] since we need to deal with off-diagonal entries of
matrices which is unique to the matrix-spectrahedron case and is not faced in
[HKM13, ST17, OST19]. To bound this, we use several properties of Fréchet
derivatives such as, mean value theorems for Fréchet derivatives, divided
differences representations of Fréchet derivatives [BLZ05], and Dyson’s
theorem [Bha13] which provides a useful integral expression for Fréchet
derivatives (using the structure of the mollifier). More importantly, since we
work with the Bentkus mollifier [Ben90], we completely open up the Bentkus
black-box and show various analytic properties of this mollifier
$\widetilde{\mathcal{O}}$ in order to prove that our Fréchet derivatives are
bounded.
In order to go from bounded third-order Fréchet derivatives to a final
invariance principle, we still need to borrow some results from random matrix
theory to upper bound the moments of $\sum_{i}\bm{x}_{i}A^{i}$. Although, the
concentration of $\sum_{i}\bm{x}_{i}A^{i}$ for uniformly random
$\bm{x}\sim\mathcal{U}_{n}$ is well-studied by standard matrix Chernoff bounds
[Tro15], we need better concentration of this random matrix variable at higher
Schatten norms. For the diagonal polytope case [HKM13] used the standard
hypercontractivity and [OST19] used Rosenthal’s inequality. Fortunately for
us, a matrix-version of Rosenthal’s inequality [MJC+14] was proven a few years
back and we use it to conclude our proof (in fact we also crucially rely on
this inequality to construct our $\mathsf{PRG}$). Putting everything together,
for arbitrarily small $\tau>0$, we obtain our main invariance principle for
the Bentkus mollifier applied as a spectral function
$\displaystyle\left|\operatorname*{\mathbb{E}}_{\bm{x}\sim\mathcal{U}_{n}}\left[\widetilde{\mathcal{O}}\left(\sum_{i=1}^{n}\bm{x}_{i}A^{i}-B\right)\right]-\operatorname*{\mathbb{E}}_{\bm{g}\sim\mathcal{G}^{n}}\left[\widetilde{\mathcal{O}}\left(\sum_{i=1}^{n}\bm{g}_{i}A^{i}-B\right)\right]\right|\leq\operatorname{poly}(\log
k,M,\tau).$ (3)
We remark that the invariance principle above does not assume the positivity
of the matrices. We believe this is a necessity for future work on fooling
arbitrary spectrahedra.
### 1.5 Second contribution: Geometric properties of positive spectrahedra
Even with an invariance principle in hand, we are faced with the same
challenges as [HKM13, ST17, OST19] to show an anti-concentration statement.
Recall that our goal is to show that for a $(\tau,M)$-regular positive
spectrahedron $S$, the expected value of the _indicator function_ $[x\in S]$
for $x\sim\mathcal{U}_{n}$ is close to the expected value of $[\bm{g}\in S]$
for $\bm{g}\sim\mathcal{G}^{n}$. This is “almost” what we showed in the
previous section except that the Bentkus mollifier $\widetilde{\mathcal{O}}$
in Eq. (3) is replaced by the orthant indicator function
$f(x)=[\max_{i}x_{i}\leq 0]$. In order to move from the smooth function
distance to CDF distance, one particular approach taken by [HKM13, ST17,
CDS19] is to use geometric properties of polytopes, and as far as we are aware
this is widely open for spectrahedra.
#### 1.5.1 Properties of positive spectrahedron
Understanding average sensitivity and noise sensitivity of geometric objects
has been an important area in theoretical computer science. For the class of
halfspaces, we have several results that upper bound these properties [Per04,
HKM13, DGJ+10, Kan14a], however upper bounds on these properties are poorly
understood for the case of spectrahedra. Below, we prove upper bounds on these
quantities.
###### Result 2 (Geometric properties of positive spectrahedra).
Let $S$ be a positive spectrahedron and consider
$F:\\{-1,1\\}^{n}\rightarrow\\{0,1\\}$ defined as $F(x)=[x\in S]$. The average
sensitivity of $F$ is $O(\sqrt{n})$, the $\varepsilon$-Boolean noise
sensitivity of $F$ is $O(\sqrt{\varepsilon})$.
We remark that the noise-sensitivity statement we have above can be viewed as
a “positive-matrix-analogue” version of the well-known Peres’s theorem
[Per04]. In order to prove this statement, we first observe that the average
sensitivity of $F$ being $O(\sqrt{n})$ immediately follows by the observation
that positive spectrahedra correspond to _unate functions_ and Kane [Kan14a]
showed $\mathsf{AS}(f)\leq O(\sqrt{n})$ if $f$ is unate (and a similar
statement is known to be false for noise sensitivity). One issue we need to
handle when translating between noise sensitivity and average sensitivity is
the following: in the standard technique of [Per04, DGJ+10, Kan14a], one upper
bounds the $\varepsilon$-noise sensitivity of a function $f$ by “bucketing”
the input variables into $m=O(1/\varepsilon)$ buckets $B_{1},\ldots,B_{m}$ and
reduces the function $f:\\{-1,1\\}^{n}\rightarrow\\{-1,1\\}$ to a function
$g:\\{-1,1\\}^{m}\rightarrow\\{-1,1\\}$ defined as
$g(b)=\sum_{\ell=1}^{m}b_{i}\sum_{i\in B_{\ell}}z_{i}A^{i}$ (for uniformly
random $z$). One then upper bounds $\mathsf{NS}_{\varepsilon}(f)$ using
$\mathsf{AS}(g)$ (up to a factor $\varepsilon$). Clearly when using this
technique to bound $\varepsilon$-noise sensitivity of halfspaces, both $f,g$
are intersections of halfspaces and one can upper bound the average
sensitivity of $g$ using Kane’s result [Kan14a] to be $O(\sqrt{m})$. However
in our setting if $f$ is an indicator of a _positive_ spectrahedron, then $g$
no longer needs to be an indicator of a _positive_ spectrahedron since
$\sum_{i\in B_{\ell}}z_{i}A^{i}$ need not even be either a positive
semidefinite matrix or a negative semidefinite matrix. We overcome this by
modifying the bucketing procedure of [DGJ+10] to ensure $g$ is an indicator of
a unate function. However, in the process case we end up upper bounding
$\mathsf{NS}_{\varepsilon}(f)$ by the “average $2$-sensitivity” of $g$. We
extend the results of Kane [Kan14a] by showing that even the “average
$2$-sensitivity” of $g$ is small for our setting.
#### 1.5.2 Boolean anti-concentration
For polytopes, Gaussian anti-concentration immediately follows from the fact
that the Gaussian surface area of polytopes is bounded since its surface has
only finite normed vectors. This is crucially used in [HKM13, ST17, CDS19].
However, it is not clear how to upper bound the $\mathsf{GSA}$ of positive
spectrahedra due to its complicated geometric structures. Moreover, even with
an upper bound on $\mathsf{GSA}$, we still do not know how to obtain Gaussian
anti-concentration. Here, to move from mollifier-closeness to CDF closeness,
we prove a _Boolean_ anti-concentration for positive spectrahedra, which is in
fact _stronger_ than Gaussian anti-concentration, inspired by the Boolean
anti-concentration for polytopes in [OST19].
##### Regularity condition.
Before explaining the Boolean anti-concentration, we need to revisit the
regularity condition, which is also used for polytopes. In [HKM13, ST17], it
is assumed that every halfspace (or row in the matrix $W$) satisfies
$\|W^{i}\|_{2}=1$ and $\|W^{i}\|_{4}\leq\tau$. One important question is: what
is a regularity assumption for spectrahedra and for which assumptions can we
_show_ anti-concentration? A natural possibility is to see if Nazarov’s result
[Naz03] holds for spectrahedra (i.e., show anti-concentration in the weaker
_Gaussian_ setting). To the best of our knowledge, this has firstly not been
studied in literature. Moreover, it is not hard to see that, in order for the
proof of Nazarov to work for spectrahedra, one can make a very _strong_
assumption that _every_ $A^{i}$ satisfies $\lambda_{\min}(A^{i})\geq 1$.
However, this seems to significantly restrict the class of spectrahedra.
In order to resolve this, we propose $\left(\tau,M\right)$-regularity as
defined in Eq. (1) and prove a stronger statement, i.e., Boolean anti-
concentration for $(\tau,M)$-regular positive spectrahedra. We use this
statement to go from closeness between the mollifier
$\widetilde{\mathcal{O}}\left(\sum_{i}\bm{x}_{i}A^{i}-B\right)$ and
$\widetilde{\mathcal{O}}\left(\sum_{i}\bm{g}_{i}A^{i}-B\right)$ (which we
already established in Eq. (3)) to closeness between
$\left[\sum_{i}\bm{x}_{i}A^{i}\preceq B\right]$ and
$\left[\sum_{i}\bm{g}_{i}A^{i}\preceq B\right]$. In this direction, we prove a
Littlewood-Offord type theorem for positive spectrahedra.
###### Result 3 (Littlewood-Offord for positive spectrahedra).
If $(A^{1},\ldots,A^{n})$ are $(\tau,M)$-regular. Then every $\Lambda$, we
have
$\Pr_{\bm{x}\sim\mathcal{U}_{n}}\left[\lambda_{\max}\left(\sum_{i}\bm{x}_{i}A^{i}-B\right)\in[-\Lambda,\Lambda]\right]\leq
O(\Lambda).$
The classic Littlewood-Offord theorem [LO39, Erd45] anti-concentration
inequality for a halfspace $w\in\mathbb{R}^{n}$ (satisfying $|w_{i}|\geq 1$)
and $\alpha\in\mathbb{R}$ proves a bound on the probability that
$\sum_{i}w_{i}\bm{x}_{i}\in[\alpha,\alpha+2]$ (where
$\bm{x}\sim\mathcal{U}_{n}$). In [OST19] they generalized this for
_intersections_ of halfspaces and in the result above we show a matrix-version
of Littlewood-Offord theorem. Intuitively, our statement shows the largest
eigenvalue of a positive spectrahedron cannot all be very-concentrated in a
small region (i.e., small eigenvalue regions have small measure over the
Boolean cube).
The proof of our result is similar to the proofs in [Kan14a, OST19] which show
anti-concentration for intersections of unate functions. There are a couple of
subtleties for us: in [OST19], they perform random “bucketing” of the
coordinates in a polytope and show that with high probability, each bucket has
“significant” weight, which follows immediately from the Paley-Zygmund
inequality. However, for us, random bucketing does not produce a positive
spectrahedron (the same issue which we faced in Theorem 2), so instead we need
to bucket in a non-standard way to go from a positive spectrahedron to a
bucket which corresponds to a unate function. Next, to show that each bucket
has significant weight (which in our case corresponds to large smallest
eigenvalue), we invoke the matrix Chernoff bound for negatively correlated
variables, proving our result. We remark that higher-dimensional extensions of
the Littlewood-Offord theorem [FF88, TV12] do not talk of eigenspectra of
matrices and differs from our result.
Using the standard bits-to-Gaussians trick, this also gives us Gaussian anti-
concentration (i.e., the positive spectrahedra analogue of Nazarov’s result
[Naz03] which is unknown as far as we are aware). Putting this together with
our invariance principle statement we obtain our main result.
###### Result 4 (Fooling positive spectrahedra).
For every $(\tau,M)$-regular positive spectrahedron $S$,
$\displaystyle\big{|}\mathop{\mathbb{E}}_{\bm{x}\sim\mathcal{U}_{n}}[\bm{x}\in
S]-\mathop{\mathbb{E}}_{\bm{g}\sim\mathcal{G}^{n}}[\bm{g}\in
S]\big{|}\leq\operatorname{poly}(M,\log k,\tau).$ (4)
Apart from the applications of constructing pseudorandom generators (which we
discuss in the next section) we believe that our invariance principle for the
Bentkus mollifier of _arbitrary_ spectrahedra, opening up the Bentkus
mollifier (i.e., understanding the Bentkus functions which were almost used as
a black-box in [HKM13, ST17, OST19]), the Littlewood-Offord theorem and noise
sensitivity for positive spectrahedra could be of independent interest.
### 1.6 Applications
#### 1.6.1 Pseudorandom generators
We now briefly discuss how to use the invariance principle to obtain our
pseudorandom generator. Our construction is based on the Meka-Zuckerman [MZ13]
$\mathsf{PRG}$ construction for fooling halfspaces. We note in the passing
that this same $\mathsf{PRG}$ (with different parameters) was also used by
[HKM13, ST17] and slight modification of it by [OST19]. We omit the details of
the $\mathsf{PRG}$ construction here referring the interested reader to
Section 6.3 for an explicit construction.
One subtlety in order to go from invariance principle to fooling the MZ-
generator is the following: recall that our invariance principles showed that
expected value under the uniform distribution was close to the expected value
under the Gaussian distribution. However, in order to fool the MZ-generator
one needs to show that the invariance principle proofs holds also for $k$-wise
independent distributions. In this direction, we use a neat trick from [OST19]
that shows that in order to show invariance principles for $k$-wise
independent distributions, it suffices to show just Boolean anti-
concentration. Second we crucially use the fact that the matrix Rosenthal
inequality can be derandomized by analyzing its the original proof. Put
together, this shows that our invariance principle proof holds for $k$-wise
independent distributions and gives us our main $\mathsf{PRG}$ result.
###### Result 5 (PRG for positive spectrahedra).
Let $S$ be a $(\tau,M)$-regular positive spectrahedron. There exists a
$\mathsf{PRG}$ $G:\\{0,1\\}^{r}\rightarrow\\{-1,1\\}^{n}$ with $r=(\log
n)\cdot\operatorname{poly}(\log k,M,1/\delta)$ that $\delta$-fools $S$ with
respect to the uniform distribution for every
$\tau\leq\operatorname{poly}(\delta/(\log k\cdot M))$.
#### 1.6.2 Learning theory
Learning geometric objects is a fundamental problem in computational learning
theory. An application of upper bounding noise sensitivity (in Theorem 2) is
in agnostic learning. The agnostic learning framework introduced by [KSS94,
Hau92] is the following: let
$\mathcal{C}\subseteq\\{c:\\{-1,1\\}^{n}\rightarrow\\{0,1\\}\\}$ be a concept
class and $\mathcal{D}:\\{-1,1\\}^{n}\times\\{0,1\\}\rightarrow[0,1]$ be a
distribution. Define
$\textsf{opt}(\mathcal{C})=\min_{c\in\mathcal{C}}\Pr_{(x,b)\sim\mathcal{D}}[c(x)\neq
b],$ i.e., what is the _best_ approximation to $\mathcal{D}$ from within the
concept class. The goal of an agnostic learner is the following: given many
samples $(x,b)\sim\mathcal{D}$, the goal of a learner is to produce a
hypothesis $h:\\{-1,1\\}^{n}\rightarrow\\{0,1\\}$ which satisfies
$\Pr_{(x,b)\sim\mathcal{D}}[h(x)\neq
b]\leq\textsf{opt}(\mathcal{C})+\varepsilon.$
Note that if $\textsf{opt}(\mathcal{C})=0$, this is the standard PAC learning
framework and agnostic learning models learnability under adversarial noise. A
natural restriction of this model is when the marginal of $\mathcal{D}$ on the
first $n$ bits is the uniform distribution on $\\{0,1\\}^{n}$. It is a
folklore result [KOS04] that a function $f$ having low noise sensitivity can
be approximated by low-degree polynomials (see [HKM13, Lemma 2.7] for an
explicit statement). Furthermore, the well-known L1-polynomial regression
algorithm [KKMS08] shows how to learn low-degree polynomials in the agnostic
framework. Putting these two connections together gives us the following
theorem.
###### Result 6 (Learning positive spectrahedra).
The concept class of positive spectrahedra (in $n$ variables with $k\times k$
symmetric matrices) can be agnostically learned under the uniform distribution
in time $n^{O(\log k)}$ for every constant error parameter.
The previous best known result [KOS08] for learning positive spectrahedra even
in the PAC model was $2^{O(n^{1/4})}$ (as far as we are aware); our result
provides a substantially better complexity.
#### 1.6.3 Discrepancy sets for spectrahedra
Understanding discrepancy sets for convex objects is a fundamentally important
problem in the fields of convex geometry, optimization, and a range of other
areas. Prior works of [HKM13, ST17, OST19] constructed such discrepancy sets
for polytopes, but a natural question is to extend their construction to
spectrahedra. In our context, one application of our main result can be viewed
as the following: consider the set of all possible positive spectrahedra (over
the Boolean cube)
$S=\\{x\in\left\\{-1,1\right\\}^{n}:\sum_{i}x_{i}A^{i}\preceq B\\}$, then can
we construct a _small_ subset of the Boolean cube $\\{-1,1\\}^{n}$ such that
this set $\delta$-approximates the $\left\\{-1,1\right\\}^{n}$-volume of every
positive spectrahedron? One way to construct such a set is to construct a
$\mathsf{PRG}$ for the class of functions. So an immediate corollary of our
$\mathsf{PRG}$ for positive spectrahedra is the following theorem.888We remark
that counting integer solutions to positive spectrahedra is not as _naturally
motivated_ as that for polytopes, but nevertheless understanding discrepancy
sets for geometric objects is a fundamental question.
###### Result 7 (Discrepancy set for positive spectrahedra).
There is a deterministic algorithm which, given a $(\tau,M)$-regular positive
spectrahedron $S$, runs in time $\exp(\log n,\log k,M,1/\delta)$ and outputs a
$\delta$-approximation of the number of points in $\left\\{-1,1\right\\}^{n}$
contained in $S$ as long as $\tau\leq\operatorname{poly}(\delta/(M\log k))$.
#### 1.6.4 Intersection of (structured) polynomial threshold functions
Constructing $\mathsf{PRG}$s for $\mathsf{PTF}$s has received a lot of
attention. However, the best known seed length for fooling a degree-$k$
$\mathsf{PTF}$ on $n$ bits scales as $O(\log n\cdot 2^{k})$ (over the Boolean
space). A simple observation we make is that fooling spectrahedra (on $n$ bits
with $k\times k$ matrices) can be in fact be viewed as the more challenging
task of fooling an _intersection_ of $k$ many degree-$k$ $\mathsf{PTF}$s.
Recall that a spectrahedron is given by
$S=\\{x\in\mathbb{R}^{n}:B-\sum_{i}x_{i}A^{i}\succeq 0\\}.$ Without loss of
generality, we may assume that the measure of $x$ satisfying
$\det\left(\sum_{i}B-x_{i}A^{i}\right)=0$ is zero. Sylvester’s criterion
implies that a matrix $M$ (which in our case is $B-\sum_{i}x_{i}A^{i}$) is
positive definite _if and only if_ the determinant of the $k$ principle minors
of $M$ are positive. Hence, an alternate characterization of $S$ is the set of
$x\in\mathbb{R}^{n}$ for which
$S=\bigwedge_{r=1}^{k}\left[\textsf{det}\left(B-\sum_{i}x_{i}A^{i}\right)_{r\times
r}>0\right]=\bigwedge_{r=1}^{k}\operatorname{sign}[p_{r}(x)]$
modulo a zero-measure set, where $M_{r\times r}$ means the top left $r\times
r$ principle minor of $M$. Clearly each determinantal expression produces a
polynomial $p_{r}$ of degree at most $r$. So, our main result about fooling
$S$, shows that there is a _structured_ class of _intersections_ of degree-$k$
$\mathsf{PTF}$s (i.e., the class of polynomials which can be written as in
terms of the above) which can be fooled by a $\mathsf{PRG}$ with seed length
$O(\log n\cdot\log k\cdot M/\delta)$, which is exponentially better than using
existing $\mathsf{PRG}$s for $\mathsf{PTF}$s.
We remark that apriori, it is not even clear why should an _arbitrary_
polynomial even correspond to a spectrahedron as above? However, a well-known
result of [HMV06, GM12] states that an arbitrary degree-$d$ polynomial
$p\in\mathbb{R}[x_{1},\ldots,x_{n}]$ with real coefficients has a symmetric
_determinantal representation_ ,999See [Qua12] for a simple linear algebraic
proof of this statement. i.e., there exists symmetric
$A^{0},A^{1},\ldots,A^{n}$ such that
$p(x_{1},\ldots,x_{n})=\textsf{det}\left(A^{0}+\sum_{i}x_{i}A^{i}\right).$
where $A^{i}\in\textsf{Sym}\binom{n+d}{d}$. So, if we could fool arbitrary
spectrahedra that might be a promising avenue to fool $\mathsf{PTF}$s and
intersections of $\mathsf{PTF}$s.
### 1.7 Future work
Our work opens this new line of research into understanding $\mathsf{PRG}$s
for spectrahedra with several novel techniques. This raises several questions
for future work.
_1. Can we remove regularity for positive spectrahedra?_ One of the crucial
techniques that Servedio and Tan [ST17] introduced (inspired by a prior work
of Servedio [Ser06]) was decomposing a polytope into head and tail variables
(i.e., tail coordinates in a halfspace which satisfy regularity and head
coordinates are the dominant variables). They express the head variables as
CNF, use the result of Bazzi [Baz09] to fool the head variables and invariance
principles for tail variables. However, in our setting breaking up a single
spectrahedron into head and tail variables is unclear and even if possible,
what is the analogue of the CNF for our setting?
_2. Can we fool arbitrary spectrahedra?_ Besides the difficulty in removing
the regularity condition, another fundamental barrier we face here is, anti-
concentration. What is the Gaussian surface area of a spectrahedron, even this
is unknown (as far as we are aware). Our techniques such as bucketing, using
Kane’s result [Kan14a], and Boolean anti-concentration [OST19] crucially use
the assumption of positivity. Going beyond this, might require new
understanding on the geometric structures (like average sensitivity, noise
sensitivity) about arbitrary spectrahedra.
_3. A general invariance principle for spectral functions?_ Here, we showed
our invariance principle specifically for the Bentkus mollifier. However, like
the result of [HKM13] can we prove a general invariance principle for
arbitrary smooth spectral functions? Given the applications of invariance
principles, they are now considered to be powerful techniques in computational
complexity theory. Having an invariance principle for spectral functions could
find more applications such as deciding noisy entangled quantum games [Yao19].
_4. Can we fool spectrahedral caps?_ Let
$S_{n-1}=\\{x\in\mathbb{R}^{n}:\|x\|_{2}=1\\}$ denote the $n$-dimensional
sphere, then a _spectrahedral cap_ is the set of $S_{n-1}$ that is “cut” by a
spectrahedron, i.e., for a spectrahedron $S$, we define the spectrahedral cap
$C_{S}$ as $C_{S}=S_{n-1}\cap S$. In the polytope-setting, fooling spherical
caps has received a lot of attention classically [HKM13, KM15] (with almost
optimal seed length $\mathsf{PRG}$s). Can we similarly fool spectrahedral
caps?
_5. Fooling polynomial threshold functions?_ Can we make progress in finding
better $\mathsf{PRG}$s for $\mathsf{PTF}$s using techniques we developed here
for fooling arbitrary spectrahedra?
##### Acknowledgements.
We thank Oded Regev for pointing out a minor inconsequential error of the
previous version. We also thank Jop Briët and Minglong Qin for several helpful
comments. This collaboration earlier faced some bureaucratic issues. We are
deeply grateful for the support from Jelani Nelson, Kewen Wu, Yitong Yin and
others in the TCS community. P.Y. was supported by the National Key R&D
Program of China 2018YFB1003202, National Natural Science Foundation of China
(Grant No. 61972191), the Program for Innovative Talents and Entrepreneur in
Jiangsu, the Fundamental Research Funds for the Central Universities
0202/14380068 and Anhui Initiative in Quantum Information Technologies Grant
No. AHY150100. Part of the work was done when P.Y. and S.A. were participating
in the program ”Quantum Wave in Computing” held at Simons Institute for the
Theory for Computing.
##### Organization.
In Section 2 we introduce the mathematical aspects which we use in this paper,
and state various lemmas in random matrix theory and multidimensional
calculus. In Section 3, we introduce the Bentkus mollifier and discuss various
properties. In Section 4 we state our main theorem regarding spectral
derivatives of smooth functions and go on to bound the spectral derivatives
for the Bentkus function (proving a technical lemma in Appendix A). In Section
5 we prove an upper bound on the noise sensitivity of positive spectrahedra as
well as our Littlewood-Offord theorem for this class. In Section 6 we prove
our invariance principle theorem and go on to construct a pseudorandom
generator for the class of positive spectrahedra.
## 2 Preliminaries
For an integer $n\geq 1$, let $[n]$ represent the set
$\left\\{1,\ldots,n\right\\}$. Given a finite set $\mathcal{X}$ and a natural
number $k$, let $\mathcal{X}^{k}$ be the set
$\mathcal{X}\times\cdots\times\mathcal{X}$, the Cartesian product of
$\mathcal{X}$, $k$ times. Given $a=(a_{1},\ldots,a_{k})$ and a set
$S\subseteq[k]$, we write $a_{S}$ and $a_{-S}$ to represent the projections of
$a$ to the coordinates specified by $S$ and the coordinates outside $S$,
respectively. For any $i\in[k]$, $a_{-i}$ represents
$a_{1},\ldots,a_{i-1},a_{i+1},\ldots,a_{n}$ and $a_{<i}$ represents
$a_{1},\ldots,a_{i-1}$. $a_{\leq i},a_{>i},a_{\geq i}$ are defined similarly.
For a distribution $\mu$ on $\mathcal{X}$, let $\mu\left(x\right)$ represent
the probability of $x\in\mathcal{X}$ according to $\mu$. Let $X$ be a random
variable distributed according to $\mu$. We use the same symbol to represent a
random variable and its distribution whenever it is clear from the context.
The expectation of a function $f$ on $\mathcal{X}$ is defined as
$\mathbb{E}\left[f(X)\right]=\mathbb{E}_{\bm{x}\sim
X}\left[f(\bm{x})\right]=\sum_{x\in\mathcal{X}}\mathrm{Pr}\>\\!\\!\left[X=x\right]\cdot
f\left(x\right)=\sum_{x}\mu\left(x\right)\cdot f\left(x\right)$, where
$\bm{x}\sim X$ represents that $\bm{x}$ is drawn according to $X$. For any
event $\mathcal{E}_{x}$ on $x$, $\left[\mathcal{E}\left(x\right)\right]$
represents the indicator function of $\mathcal{E}$. In this paper, the lower-
cased letters in bold $\bm{x},\bm{y},\bm{z}\cdots$ are reserved for random
variables.
##### Distributions.
Throughout the paper, we denote $\mathcal{G}$ (where
$\mathcal{G}=\mathcal{N}(0,1)$) to be a standard normal distribution over
$\mathbb{R}$ with mean $0$ and variance $1$. We denote $\mathcal{U}_{n}$ to be
the uniform distribution on $\\{-1,1\\}^{n}$. We say a joint distribution
$X=(\bm{x}_{1},\ldots,\bm{x}_{n})$ is $t$-wise uniform if the marginal
distribution $X_{S}$ for any subset $S\subseteq[n]$ of size $\left|S\right|=t$
is uniformly distributed (observe that the uniform distribution is clearly
$t$-wise independent for every $t\geq 1$). A distribution $\mathcal{H}$ on
functions $[n]\rightarrow[m]$ is said to be an $r$-wise uniform hash family if
for $\bm{h}\sim\mathcal{H}$,
$\left(\bm{h}\left(1\right),\ldots,\bm{h}\left(n\right)\right)$ is $r$-wise
uniform.
### 2.1 Derivatives and multidimensional Taylor expansion
We denote $\mathcal{C}^{d}$ as the set of all real functions that are $d$-time
differentiable. For $f:{\mathbb{R}}\rightarrow{\mathbb{R}}$ in
$\mathcal{C}^{d}$, we use $f^{\left(d\right)}$ to denote the $d$-th derivative
of $f$. Given a function $F:{\mathbb{R}}^{k}\rightarrow{\mathbb{R}}$ and a
$k$-dimensional multi-index
$\alpha=\left(\alpha_{1},\ldots,\alpha_{m}\right)\in\mathbb{N}^{k}$,
$\partial_{\alpha}F$ denotes the mixed partial derivative taken $\alpha_{i}$
times in the $i$-th coordinate.
###### Fact 1 ([Rud86]).
Let $k\in\mathbb{N}$ and $f:{\mathbb{R}}^{k}\rightarrow{\mathbb{R}}$ be a
$\mathcal{C}^{d}$ function. Then for all $x,y\in{\mathbb{R}}^{k}$,
$f\left(x+y\right)=\sum_{\alpha\in\mathbb{N}^{k}:\left|\alpha\right|\leq
d-1}\frac{\partial_{\alpha}f\left(x\right)}{\alpha!}\prod_{i=1}^{m}y_{i}^{\alpha_{i}}+\textsf{err}\left(x,y\right),$
where $\alpha!=\alpha_{1}!\cdots\alpha_{m}!$,
$\left|\alpha\right|=\sum_{i}\alpha_{i}$ and
$\left|\textsf{err}\left(x,y\right)\right|\leq\sup_{v\in{\mathbb{R}}^{k}}\sum_{\alpha\in\mathbb{N}^{k}:\left|\alpha\right|=d}\left|\partial_{\alpha}f\left(v\right)\right|\max_{i}\left|y_{i}\right|^{d}.$
For a $t$-time differentiable function $f:\mathbb{R}^{k}\rightarrow\mathbb{R}$
and $s\leq t$, define
$\|f^{(s)}\|_{1}=\max\Big{\\{}\sum_{p_{1},p_{2},\ldots,p_{s}\in[k]}|\partial_{p_{1}}\cdots\partial_{p_{s}}f(x)|:x\in\mathbb{R}^{k}\Big{\\}}$
###### Definition 2.
Let $f:{\mathbb{R}}\rightarrow{\mathbb{R}}$. For any distinct inputs
$x_{1},\ldots,x_{n}\in{\mathbb{R}}$, the divided difference is defined
recursively as follows.
$\displaystyle f^{[0]}=f,$ $\displaystyle
f^{[i]}\left(x_{1},\ldots,x_{i+1}\right)=\frac{f^{[i]}\left(x_{1},\ldots,x_{i-1},x_{i}\right)-f^{[i]}\left(x_{1},\ldots,x_{i-1},x_{i+1}\right)}{x_{i}-x_{i+1}}.$
For other values of $x_{1},\ldots,x_{i+1}$, $f^{[i]}$ is defined by continuous
extension.
###### Fact 3 (Mean value theorem for divided difference [Boo05]).
For every $f\in\mathcal{C}^{n}$ and $x_{1},\ldots,x_{n+1}$, there exists
$\xi\in(\min\left\\{x_{1},\ldots,x_{n+1}\right\\},\max\left\\{x_{1},\ldots,x_{n+1}\right\\})$
such that
$f^{[n]}\left(x_{1},\ldots,x_{n+1}\right)=\frac{f^{(n)}\left(\xi\right)}{n!}.$
### 2.2 Combinatorial properties of Boolean functions
Let $f:\\{0,1\\}^{n}\rightarrow\\{0,1\\}$,
$g:\mathbb{R}^{n}\rightarrow\\{0,1\\}$ and $S$ be a Borel set in
${\mathbb{R}}^{n}$. We define the following combinatorial properties of
Boolean-valued functions $f,g$.
1. 1.
Average sensitivity: $\mathsf{AS}(f)=\sum_{i=1}^{n}\Pr_{\bm{x}}[f(\bm{x})\neq
f(\bm{x}\oplus e_{i})]$, where the probability is taken uniformly in
$\\{0,1\\}^{n}$.
2. 2.
$\varepsilon$-Noise sensitivity:
$\mathsf{NS}_{\varepsilon}(f)=\Pr_{\bm{x},\bm{y}}[f(\bm{x})\neq f(\bm{y})]$
where the probability is taken according to the distribution: $\bm{x}$ is
uniformly random in $\\{0,1\\}^{n}$ and $\bm{y}$ is obtained from $\bm{x}$ by
independently flipping each $\bm{x}_{i}$ with probability $\varepsilon$.
We refer interested readers to [O’D14] for more on these parameters and their
applications to analysis of Boolean functions.
### 2.3 Matrix analysis and Random matrices
For any integer $k>0$, we use $\mathsf{Mat}_{k}$ and $\mathsf{Sym}_{k}$ to
represent the set of $k\times k$ real matrices and symmetric matrices,
respectively. For any matrix $X$, $\mbox{$\|{X}\|$}_{p}$ represents the
Schattern $p$-norm of $X$ and $\|{X}\|$ represents the spectral norm of $X$.
$\mathbb{I}_{k}$ represents a $k\times k$ identity matrix. The subscript $k$
may be omitted whenever the dimension is clear from the context. We need the
following results in matrix analysis.
###### Fact 4.
[Bha00] For any $k\times k$ real symmetric matrix $A$, let $B$ be its upper
triangle part of $A$. Namely $B_{i,j}=A_{i,j}$ if $i\leq j$ and is $0$
otherwise. Then $\mbox{$\|{B}\|$}\leq\frac{\ln k}{\pi}\mbox{$\|{A}\|$}$.
###### Fact 5.
[Tro12, Theorem 1.1] Let $n,k\geq 1$ be integers and $X_{1},\ldots,X_{n}$ be
independent random $k\times k$ real symmetric matrices satisfy $0\preceq
X_{i}\preceq R$ for $i\in[n]$. Set
$\mu=\lambda_{\min}\left(\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left[X_{i}\right]\right).$
Then
$\mathrm{Pr}\>\\!\\!\left[\lambda_{\min}\left(\sum_{i=1}^{n}X_{i}\right)\leq\left(1-\delta\right)\mu\right]\leq
k\cdot\left(\frac{e^{-\delta}}{\left(1-\delta\right)^{1-\delta}}\right)^{\mu/R}$
for every $\delta\in[0,1)$.
###### Fact 6.
For every integer $m\geq 1$ and $A_{1},\ldots,A_{n}\in\mathsf{Sym}_{k}$ it
holds that
$\mathbb{E}\left[\mbox{$\|{\sum_{i}\bm{g}_{i}A^{i}}\|$}^{m}\right]\leq\left(1+2m\lceil\log
k\rceil\right)^{m/2}\cdot\mbox{$\|{\sum_{i}(A^{i})^{2}}\|$}^{m/2}$
and
$\mathbb{E}\left[\mbox{$\|{\sum_{i}\bm{x}_{i}A^{i}}\|$}^{m}\right]\leq\left(1+2m\lceil\log
k\rceil\right)^{m/2}\cdot\mbox{$\|{\sum_{i}(A^{i})^{2}}\|$}^{m/2},$
where the expectations are taken over $\bm{x}\sim\mathcal{U}_{n}$ and
$\bm{g}\sim\mathcal{G}^{n}$. Additionally, the second inequality still holds
if $\bm{x}$ is $2m\lceil\log k\rceil$-wise uniform.
###### Proof.
It suffices to prove the second inequality as the first one follows by the
standard bits-to-Gaussians tricks [O’D14, Chapter 11]. Let
$B=\sum_{i}\bm{x}_{i}A^{i}$ where $\bm{x}\sim\mathcal{U}_{n}$. The proof
closely follows the argument in [Tro16], where Tropp proved the $m=1$ case.
For any integer $p\geq 1$, it is proved in [Tro16, Eqs. (4.9, 4.11)] that
$\mathbb{E}\left[\mbox{\rm Tr}~{}B^{2p}\right]\leq
k\cdot\left(\frac{2p+1}{e}\right)^{p}\cdot\mbox{$\|{\sum_{i}(A^{i})^{2}}\|$}^{p}.$
Thus
$\mathbb{E}\left[\mbox{$\|{B}\|$}^{m}\right]\leq\left(\mathbb{E}\left[\mbox{\rm
Tr}~{}B^{2pm}\right]\right)^{1/2p}\leq
k^{1/2p}\cdot\left(\frac{2pm+1}{e}\right)^{m/2}\cdot\mbox{$\|{\sum_{i}(A^{i})^{2}}\|$}^{m/2}.$
Setting $p=\lceil\log k\rceil$, we conclude the result. Since the proof
involves only $2m\cdot\lceil\log k\rceil$ powers of $B$, it also holds true
for $\bm{x}$ being drawn from a $2m\cdot\lceil\log k\rceil$-wise uniform
distribution. ∎
###### Fact 7 (Matrix Rosenthal inequality [MJC+14, Corollary 7.4]).
Let $X_{1},\ldots,X_{n}$ be centered, independent random real symmetric
matrices. Then
$\displaystyle\left(\mathbb{E}\left[\mbox{$\|{\sum_{i}X_{i}}\|$}_{4p}^{4p}\right]\right)^{\frac{1}{4p}}\leq\sqrt{4p-1}\mbox{$\|{\left(\sum_{i}\mathbb{E}\left[X_{i}^{2}\right]\right)^{\frac{1}{2}}}\|$}_{4p}+\left(4p-1\right)\left(\sum_{i}\mathbb{E}\left[\mbox{$\|{X_{i}}\|$}_{4p}^{4p}\right]\right)^{\frac{1}{4p}}.$
This inequality still holds if $X_{1},\ldots,X_{n}$ are $4p$-wise independent.
### 2.4 Matrix functions, spectral functions and Fréchet derivatives
Let $f:\mathbb{R}^{k}\rightarrow\mathbb{R}$ and
$\lambda:\mathsf{Sym}_{k}\rightarrow\mathbb{R}^{k}$ where
$\lambda(X)=\left(\lambda_{1}(X),\ldots,\lambda_{k}(X)\right)$ are the
eigenvalues of $M$ sorted in a non-increasing order. We refer to
$\lambda_{\max}=\lambda_{1}$ interchangeably. Let
$F=f\circ\lambda:\mathsf{Sym}_{k}\rightarrow\mathbb{R}$.
If $f:{\mathbb{R}}\rightarrow{\mathbb{R}}$ is an analytic function in
${\mathbb{R}}$, namely its Taylor series converges in ${\mathbb{R}}$, we
define $f\left(X\right)$ for general matrices using its Taylor expansion. It
is not hard to see that the Taylor series still converges with matrix inputs.
If $X$ is symmetric with a spectral decomposition $X=UDU^{T}$, where
$D=\mbox{\rm
diag}\left(\lambda_{1}\left(X\right),\ldots,\lambda_{k}\left(X\right)\right)$,
then $f\left(X\right)=U\mbox{\rm
diag}\left(f\left(\lambda_{1}\left(X\right)\right),\ldots,\lambda_{k}\left(X\right)\right)U^{T}$.
The Fréchet derivatives are a notion of derivatives defined in Banach space.
In this paper, we only concern about the Fréchet derivatives on matrix spaces.
Readers may refer to [Col12] for a more thorough treatment. The Fréchet
derivatives are the maps that are defined as follows.
###### Definition 8.
Given integers $m,n\geq 1$, a map
$F:\mathsf{Mat}_{m}\rightarrow\mathsf{Mat}_{n}$ and $P,Q\in\mathsf{Mat}_{m}$,
the Fréchet derivative of $F$ at $P$ with respect to $Q$ is defined to be
$DF\left(P\right)\left[Q\right]=\frac{d}{dt}F\left(P+tQ\right)|_{t=0}.$
The $k$-th order Fréchet derivative of $F$ at $P$ with respect to
$\left(Q_{1},\ldots,Q_{k}\right)$ is defined recursively as
$D^{k}F\left(P\right)\left[Q_{1},\ldots,Q_{k}\right]=\frac{d}{dt}D^{k-1}F\left(P+tQ_{k}\right)\left[Q_{1},\ldots,Q_{k-1}\right]|_{t=0}.$
Fréchet derivatives share many common properties with the derivatives in
Euclidean spaces, such as linearity, composition rules, Taylor expansions,
etc. We refer the interested reader to [Col12, Bha13] for more. Some basic
properties of Fréchet derivatives are summarized in the following fact.
###### Fact 9.
[Bha13, Chapter X.4] Given $F,G:\mathsf{Mat}_{n}\rightarrow\mathsf{Mat}_{m}$
and $P,Q_{1},\ldots,Q_{k}\in\mathsf{Mat}_{n}$, it holds that
1. 1.
$D\left(F+G\right)\left(P\right)\left[Q\right]=DF\left(P\right)\left[Q\right]+DG\left(P\right)\left[Q\right]$.
2. 2.
$D\left(F\cdot
G\right)\left(P\right)\left[Q\right]=DF\left(P\right)\left[Q\right]\cdot
G\left(P\right)+F\left(P\right)\cdot DG\left(P\right)\left[Q\right]$.
3. 3.
If $m=n$, $D\left(F\circ
G\right)\left(P\right)\left[Q\right]=\left(D\left(G\circ
F\right)\left(P\right)\circ DF\left(P\right)\right)\left[Q\right]$.
4. 4.
$D^{k}F\left(P\right)\left[Q_{1},\ldots,Q_{k}\right]=D^{k}F\left(P\right)\left[Q_{\sigma\left(1\right)},\ldots,Q_{\sigma\left(k\right)}\right]$
for every $k>0$ and permutation $\sigma\in S_{k}$.
The following fact states that Fréchet derivatives can be expressed as divided
differences.
###### Fact 10.
[BLZ05] Let $f:{\mathbb{R}}\rightarrow{\mathbb{R}}$ be twice differentiable
and $X=\mbox{\rm diag}\left(x_{1},\ldots,x_{k}\right)$ be a diagonal matrix
whose spectrum is in ${\mathbb{R}}$. For any matrix $A,B$, the following
holds101010In [BLZ05, Lemma 3.8] this fact is proven when $A=B$ is a symmetric
matrix and one can easily generalize their proof to obtain Eqs. (5), (6) for
general matrices $A,B$.
1. 1.
$Df\left(X\right)\left[A\right]=\left(f^{[1]}\left(x_{i_{1}},x_{i_{2}}\right)A_{i_{1},i_{2}}\right)_{1\leq
i_{1},i_{2}\leq k}.$ (5)
2. 2.
$D^{2}f\left(X\right)\left[A,B\right]=\left(\sum_{j=1}^{k}f^{[2]}\left(x_{i_{1}},x_{j},x_{i_{2}}\right)A_{i_{1},j}B_{j,i_{2}}\right)_{1\leq
i_{1},i_{2}\leq k}.$ (6)
###### Fact 11 (Dyson’s expansion [Bha13, Chapter X.4]).
Let $f\left(x\right)=e^{x}$. For any $X\in\mathsf{Sym}_{k}$ and
$A\in\mathsf{Mat}_{k}$, it holds
$Df\left(X\right)\left[A\right]=\int_{0}^{1}du~{}e^{\left(1-u\right)X}Ae^{uX}.$
###### Lemma 12.
Let $f\left(x\right)=e^{-x^{2}/2}$. For any $X\in\mathsf{Sym}_{k}$ and
$A,B\in\mathsf{Mat}_{k}$, it holds that
$\displaystyle D^{2}f\left(X\right)\left[A,B\right]$ $\displaystyle=$
$\displaystyle\frac{1}{4}\int_{0}^{1}du\int_{0}^{1}dv~{}\left(1-u\right)e^{-\left(1-u\right)\left(1-v\right)X^{2}/2}\left(XB+BX\right)e^{-\left(1-u\right)vX^{2}/2}\left(XA+AX\right)e^{-uX^{2}/2}$
$\displaystyle+\frac{1}{4}\int_{0}^{1}du\int_{0}^{1}dv~{}ue^{-\left(1-u\right)X^{2}/2}\left(XA+AX\right)e^{-u\left(1-v\right)X^{2}/2}\left(XB+BX\right)e^{-uvX^{2}/2}$
$\displaystyle-\frac{1}{2}\int_{0}^{1}du~{}e^{-\left(1-u\right)X^{2}/2}\left(AB+BA\right)e^{-uX^{2}/2}.$
In particular, if $A=B=H$ is a symmetric matrix ,then
$\displaystyle D^{2}f\left(X\right)\left[H,H\right]$ $\displaystyle=$
$\displaystyle\frac{1}{4}\int_{0}^{1}du\int_{0}^{1}dv~{}\left(1-u\right)e^{-\left(1-u\right)\left(1-v\right)X^{2}/2}\left(XH+HX\right)e^{-\left(1-u\right)vX^{2}/2}\left(XH+HX\right)e^{-uX^{2}/2}$
$\displaystyle+\frac{1}{4}\int_{0}^{1}du\int_{0}^{1}dv~{}\left(u\right)e^{-\left(1-u\right)X^{2}/2}\left(XH+HX\right)e^{-u\left(1-v\right)X^{2}/2}\left(XH+HX\right)e^{-uvX^{2}/2}$
$\displaystyle-\int_{0}^{1}du~{}e^{-\left(1-u\right)X^{2}/2}H^{2}e^{-uX^{2}/2}.$
Note that $f\left(x\right)=e^{-x^{2}/2}$ is analytical in ${\mathbb{R}}$. Thus
it is valid to define $f$ on arbitrary matrices.
###### Proof.
For any $t\in(0,1)$, we define $g\left(x\right)=e^{-tx^{2}}$. By the
definition of Fréchet derivatives
$\displaystyle Dg\left(X\right)\left[A\right]=\lim_{\varepsilon\rightarrow
0}\frac{1}{\varepsilon}\left(e^{-t\left(X+\varepsilon
A\right)^{2}}-e^{-tX^{2}}\right)$ $\displaystyle=$
$\displaystyle\lim_{\varepsilon\rightarrow
0}\frac{1}{\varepsilon}\left(e^{-t\left(X^{2}+\varepsilon\left(XA+AX\right)+\varepsilon^{2}A^{2}\right)}-e^{-tX^{2}}\right)$
$\displaystyle=$ $\displaystyle\lim_{\varepsilon\rightarrow
0}\frac{1}{\varepsilon}\left(e^{-t\left(X^{2}+\varepsilon\left(XA+AX\right)\right)}+O\left(\varepsilon^{2}\right)-e^{-tX^{2}}\right)$
$\displaystyle=$ $\displaystyle Dh(-tX^{2})[-t(XA+AX)]$ $\displaystyle=$
$\displaystyle-t\int_{0}^{1}du~{}e^{-\left(1-u\right)tX^{2}}\left(XA+AX\right)e^{-utX^{2}},$
where the second equality is from the fact that $\mbox{$\|{e^{X+\varepsilon
Y}-e^{X}}\|$}=O\left(\varepsilon\right)$, third equality holds for
$h(x)=e^{x}$ and the last equality is from Fact 11. Setting $t=\frac{1}{2}$,
we have
$Df\left(X\right)\left[A\right]=-\frac{1}{2}\int_{0}^{1}du~{}e^{-\left(1-u\right)X^{2}/2}\left(XA+AX\right)e^{-uX^{2}/2}.$
Taking one more derivative on $X$ with respect to $B$, we conclude the result
(using properties of Fréchet derivatives in items 2,3 of Fact 9). ∎
### 2.5 spectrahedra and Positive spectrahedra
###### Definition 13.
Given $\tau,M>0$, we say a sequence of $k\times k$ positive semidefinite
matrices $\left(A_{1},\ldots,A_{n}\right)$ is $\left(\tau,M\right)$-regular if
$\mathbb{I}\preceq\sum_{i=1}^{n}\left(A^{i}\right)^{2}\preceq
M\cdot\mathbb{I}~{}\mbox{and}~{}A^{i}\preceq\tau\cdot\mathbb{I}\text{ for
every }i\in[m]$ (7)
A _spectrahedron_ $S\subseteq{\mathbb{R}}^{k}$ is a feasible region of a
semidefinite program. Namely, the set
$S=\left\\{x\in{\mathbb{R}}^{n}:\sum_{i}x_{i}A^{i}\preceq B\right\\}$ for some
symmetric matrices $A_{1},\ldots,A_{n},B$. We say $S$ is a _positive
spectrahedron_ if either all $A^{i}$s are positive semidefinite or all
$A^{i}$s are negative semidefinite $(\mathsf{NSD})$. Moreover, it is
$\left(\tau,M\right)$-regular if either $\left(A_{1},\ldots,A_{n}\right)$ or
$\left(-A_{1},\ldots,-A_{n}\right)$ is $\left(\tau,M\right)$-regular.
We say $S$ is an intersection of positive spetrahedrons if $S=S_{1}\cap S_{2}$
where $S_{1}$ and $S_{2}$ are positive spectrahedra whose matrices are all
positive semidefinite and negative semidefinite, respectively. Note that it
suffices to consider the intersections of two spetrahedrons as one can pack
all $\mathsf{PSD}$ matrices into one large block-diagonal matrix (looking
ahead this will only affect the parameters in our main results by a
logarithmic factor). Packing the corresponding $B_{i}$s, one get a positive
spectrahedron. Same for all negative semidefinite matrices.
### 2.6 Pseudorandomness
###### Definition 14.
A function $g:\left\\{-1,1\right\\}^{r}\rightarrow\left\\{-1,1\right\\}^{n}$
with seed length $r$, is said to $\delta$-fool a function
$f:\left\\{-1,1\right\\}^{n}\rightarrow{\mathbb{R}}$ if
$\left|\operatorname*{\mathbb{E}}_{\bm{s}\sim\mathcal{U}_{r}}\left[f\left(g\left(\bm{s}\right)\right)\right]-\operatorname*{\mathbb{E}}_{\bm{u}\sim\mathcal{U}_{n}}[f\left(\bm{u}\right)]\right|\leq\delta.$
The function $g$ is said to be an efficient pseudorandom generator
$(\mathsf{PRG})$ that $\delta$-fools a class $\mathcal{F}$ of $n$-variable
functions if $g$ is computable by a deterministic uniform poly$(n)$-time
algorithm and $g$ fools all function $f\in\mathcal{F}$.
### 2.7 Tensors
For $\ell\geq 1$, let $T^{\ell}$ be an $\ell$-tensor, i.e.,
$T^{\ell}:(\mathbb{R}^{k})^{\times\ell}\rightarrow\mathbb{R}$. Note that an
$\ell$-tensor is defined uniquely by the coefficients
$\\{T_{i_{1},\ldots,i_{\ell}}:i_{1},\ldots,i_{\ell}\in[k]\\}$. Below we abuse
notation by letting $T(i_{1},\ldots,i_{\ell})=T_{i_{1},\ldots,i_{\ell}}$.
Often we will use the natural bijection between $2\ell$-tensors acting on
$\mathbb{R}^{k}$ and $\ell$-tensors acting on $\mathsf{Mat}_{k}$, i.e., for a
$2\ell$-tensor $T:(\mathbb{R}^{k})^{\times 2\ell}\rightarrow\mathbb{R}$
defined as
$T(x^{1},\ldots,x^{2\ell})=\sum_{i_{1},\ldots,i_{2\ell}\in[k]}T(i_{1},\ldots,i_{\ell},i_{\ell+1},\ldots,i_{2\ell})x^{1}_{i_{1}}\cdots
x^{2\ell}_{i_{2\ell}},$
we can also view $T$ as
$T^{\prime}:(\mathsf{Mat}_{k})^{\times\ell}\rightarrow\mathbb{R}$ defined by
rearranging the terms above to obtain:
$T^{\prime}(X^{1},\ldots,X^{\ell})=\sum_{i_{1},j_{1}\in[n]}\sum_{i_{2},j_{2}\in[k]}\cdots\sum_{i_{\ell},j_{\ell}\in[k]}T(i_{1},\ldots,i_{\ell},j_{1},\ldots,j_{\ell})X^{1}_{i_{1},j_{1}}\cdots
X^{\ell}_{i_{\ell},j_{\ell}}$
Finally, we define a “permutation folding” operator which takes a
$(2\ell)$-tensor on $\mathbb{R}^{k}$ as defined above and produces a
permutation to produce an $\ell$-tensor on $\mathsf{Mat}_{k}$.
###### Definition 15.
[Sen07][Definition of $\mbox{\rm diag}^{\sigma}T$] Let
$T:(\mathbb{R}^{k})^{\times t}\rightarrow\mathbb{R}$ be a $k$-tensor and
$\sigma\in S_{k}$. Then we define $\mbox{\rm
diag}^{\sigma}T:(\mathsf{Mat}_{k})^{\times t}\rightarrow\mathbb{R}$ as the
following map
$\displaystyle\left(\mbox{\rm
diag}^{\sigma}T\right)\left((i_{1},j_{1})\ldots,(i_{k},j_{k})\right)=T(i_{1},\ldots,i_{k})\quad\text{
iff }\vec{i}=\sigma\vec{j},$ (8)
and $0$ otherwise.
## 3 Bentkus mollifier
In this paper, we are interested in smooth approximators of the function
$\psi:{\mathbb{R}}^{k}\rightarrow{\mathbb{R}}$ defined as
$\psi\left(x\right)=\left[\max_{i}x_{i}\leq 0\right].$ (9)
To this end, we introduce the Bentkus mollifier defined by Bentkus in [Ben90]
and establish several new properties. Readers may refer to [Ben90, FK20] for a
more thorough treatment.
###### Definition 16.
[Ben90] Let $g:{\mathbb{R}}\rightarrow{\mathbb{R}}$ be a function defined as
$g\left(x\right)=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{-t^{2}/2}dt$ (10)
For every integer $k\geq 1$, define
$G:{\mathbb{R}}^{k}\rightarrow~{}{\mathbb{R}}$ as
$G\left(x_{1},\ldots,x_{k}\right)=\prod_{i=1}^{k}g\left(x_{i}\right).$ (11)
The subscript $k$ may be omitted whenever it is clear from the context.
### 3.1 Properties of the mollifier and its derivatives
From the definition of $g$ in Eq. (10), it is easy to calculate that
$\displaystyle g^{\prime}\left(x\right)=\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}$
(12) $\displaystyle
g^{\prime\prime}\left(x\right)=-\frac{x}{\sqrt{2\pi}}\exp\left(-x^{2}/2\right)$
(13) $\displaystyle
g^{\prime\prime\prime}\left(x\right)=\frac{1}{\sqrt{2\pi}}\left(x^{2}-1\right)\exp\left(-x^{2}/2\right).$
(14)
In order to simplify calculations, we introduce the function
$\bar{g}\left(x\right)=\frac{g^{\prime}\left(x\right)}{g\left(x\right)}.$ (15)
###### Fact 17.
[FK20, Page 10] It holds that
$\displaystyle\overline{g}^{\prime}(u)=-(u+\overline{g}(u))\cdot\overline{g}(u);$
(16)
$\displaystyle\overline{g}^{\prime\prime}\left(u\right)=\left(u^{2}-1\right)\overline{g}\left(u\right)+3u\overline{g}\left(u\right)^{2}+2\overline{g}\left(u\right)^{3}.$
(17)
Also $\overline{g}$ is positive and monotone decreasing in ${\mathbb{R}}$.
$\overline{g}^{\prime}$ is negative in ${\mathbb{R}}$.
###### Fact 18.
[Fel68, Section 7.1] For any $x\geq 0$, it holds that
$\frac{e^{-x^{2}/2}}{\sqrt{2\pi}}\left(\frac{1}{x}-\frac{1}{x^{3}}\right)\leq
1-g\left(x\right)\leq\frac{e^{-x^{2}/2}}{x\sqrt{2\pi}}.$
The following lemma immediately follows from Fact 17 and Fact 18.
###### Lemma 19.
For any $\Delta\geq 1$ and $x\in{\mathbb{R}}$ with $\left|x\right|\leq\Delta$,
it holds that
$\left|\overline{g}\left(x\right)\right|\leq
2\Delta,\left|\overline{g}^{\prime}\left(x\right)\right|\leq
3\Delta\left|\overline{g}\left(x\right)\right|,\left|\overline{g}^{\prime\prime}\left(x\right)\right|\leq
15\Delta^{2}\left|\overline{g}\left(x\right)\right|.$
### 3.2 Properties of the spectral norm of the mollifier
In this section, we establish several properties of Bentkus mollifier, which
hasn’t been studied to the best of our knowledge. We first state a crucial
fact that Bentkus proved about the derivatives of the mollifier, which is the
only fact needed and used by prior works [HKM13, ST17, CDS19, OST19].
###### Fact 20.
[Ben90] It holds that for any integer $t,k\geq 1$
$\sup_{x\in{\mathbb{R}}^{k}}\mbox{$\|{G^{(t)}\left(x\right)}\|$}_{1}\leq
C_{t}\log^{t/2}(k+1)$ (18)
for some constant $C_{t}$ only depending on $t$.
###### Lemma 21.
For any $x\in{\mathbb{R}}^{k}$, if there exist more than $3\log k$ indices
satisfying $x_{i}\leq 0$, then $\mbox{$\|{G^{(1)}\left(x\right)}\|$}_{1}\leq
O\left(\frac{1}{k^{2}}\right)$.
###### Proof.
Note that $g\left(z\right)\leq\frac{1}{2}$ if $z\leq 0$. Let
$T=\left\\{i:x_{i}\leq 0\right\\}$. Then
$\displaystyle\mbox{$\|{G^{(1)}\left(x\right)}\|$}_{1}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{k}\left|g^{\prime}\left(x_{i}\right)\prod_{j\neq
i}g\left(x_{j}\right)\right|$ $\displaystyle=$ $\displaystyle\sum_{i\in
T}\left|g^{\prime}\left(x_{i}\right)\prod_{j\neq
i}g\left(x_{j}\right)\right|+\left|\sum_{i\notin
T}g^{\prime}\left(x_{i}\right)\prod_{j\neq i}g\left(x_{j}\right)\right|$
$\displaystyle\leq$
$\displaystyle\frac{\left|T\right|}{2^{\left|T\right|-1}}+\frac{1}{2^{\left|T\right|}}\left|\sum_{i\notin
T}g^{\prime}\left(x_{i}\right)\prod_{\begin{subarray}{c}j\neq i:\\\ j\notin
T\end{subarray}}g\left(x_{j}\right)\right|\leq\frac{\left|T\right|}{2^{\left|T\right|-1}}+\frac{2\sqrt{2\log
k}}{2^{\left|T\right|}},$
where the equality used that the terms are all positive and the second
inequality is from Fact 20 and that
$\left|\sum_{i\notin
T}g^{\prime}\left(x_{i}\right)\prod_{\begin{subarray}{c}j\neq i:\\\ j\notin
T\end{subarray}}g\left(x_{j}\right)\right|=\mbox{$\|{G^{(1)}\left(x_{T^{c}}\right)}\|$}_{1}.$
The upper bound is $O\left(\frac{1}{k^{2}}\right)$ if $\left|T\right|\geq
3\log k$. ∎
###### Claim 22.
For any $x>y$, it holds that
$\left|\frac{g\left(x\right)g^{\prime}\left(y\right)-g^{\prime}\left(x\right)g\left(y\right)}{x-y}\right|\leq\left(1+\left|x\right|\right)\exp\left(-\frac{y^{2}}{2}\right)=\left(1+\left|x\right|\right)g^{\prime}(y)\cdot\sqrt{2\pi}.$
(19)
###### Proof.
$\displaystyle\left|\frac{g\left(x\right)g^{\prime}\left(y\right)-g^{\prime}\left(x\right)g\left(y\right)}{x-y}\right|$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\left|\int_{-\infty}^{0}\frac{\exp\left(-\frac{1}{2}\left(y^{2}+\left(t+x\right)^{2}\right)\right)-\exp\left(-\frac{1}{2}\left(x^{2}+\left(t+y\right)^{2}\right)\right)}{x-y}dt\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{2\pi}\exp\left(-\frac{x^{2}+y^{2}}{2}\right)\int_{-\infty}^{0}\left|\exp\left(-\frac{t^{2}}{2}\right)\frac{\exp\left(-ty\right)-\exp\left(-tx\right)}{x-y}\right|dt$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\exp\left(-\frac{x^{2}+y^{2}}{2}\right)\int_{-\infty}^{0}\left|\exp\left(-\frac{t^{2}}{2}-tx\right)\frac{1-\exp\left(-t(y-x)\right)}{y-x}\right|dt$
$\displaystyle\leq$
$\displaystyle\frac{1}{2\pi}\exp\left(-\frac{x^{2}+y^{2}}{2}\right)\int_{-\infty}^{0}\left|\exp\left(-\frac{t^{2}}{2}-tx\right)t\right|dt$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\exp\left(-\frac{y^{2}}{2}\right)\int_{-\infty}^{0}\left|\exp\left(-\frac{1}{2}\left(t+x\right)^{2}\right)t\right|dt$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\exp\left(-\frac{y^{2}}{2}\right)\left(\exp\left(-\frac{x^{2}}{2}\right)+\sqrt{2\pi}x-x\int_{x}^{\infty}e^{-t^{2}/2}dt\right)$
$\displaystyle\leq$
$\displaystyle\left(1+\left|x\right|\right)\exp\left(-\frac{y^{2}}{2}\right),$
where the second inequality used $|1-e^{-z}|\leq|z|$ for $z\geq 0$. ∎
For every $\theta>0$, we define the Bentkus mollifier as follows.
$G_{\theta}\left(x\right)=\Pr_{\bm{g}\sim\mathcal{G}^{k}}\big{[}x+\theta\bm{g}\leq
0\big{]}$ (20)
It is not hard to verify that
$G_{\theta}\left(x\right)=\prod_{i=1}^{n}\int_{-\infty}^{-\frac{x_{i}}{\theta}}\frac{1}{\sqrt{2\pi}}e^{-x_{i}^{2}/2}=G\left(-\frac{x_{1}}{\theta},\cdots,-\frac{x_{k}}{\theta}\right).$
(21)
The following fact states that
$G_{\theta}\left(\cdot+\alpha\right)/G_{\theta}\left(\cdot-\alpha\right)$ is a
good approximator of $\psi$ defined in Eq. (9) except a small inner/outer
region near the “boundary” which is made precise below.
###### Fact 23 (Lemma 6.7 and Fact 6.8 in [OST19]).
For any $\delta,\theta\in(0,1)$, $x\in{\mathbb{R}}^{k}$ there exists
$\Lambda=\Theta\left(\theta\cdot\sqrt{\log(k/\delta)}\right)$ and
$\alpha=\Theta\left(\theta\cdot\sqrt{\log(k/\delta)}\right)$ such that the
following holds.
1. 1.
$\left|G_{\theta}\left(x+\alpha\right)-\psi\left(x\right)\right|\leq\delta$ if
$\max_{i}x_{i}\leq-\Lambda$.
2. 2.
$\left|G_{\theta}\left(x-\alpha\right)-\psi\left(x\right)\right|\leq\delta$ if
$\max_{i}x_{i}\geq\Lambda$.
3. 3.
$G_{\theta}\left(x+\alpha\right)-\delta\leq\psi\left(x\right)\leq
G_{\theta}\left(x-\alpha\right)+\delta$ for all $x\in{\mathbb{R}}^{k}$.
where $x+\alpha=\left(x_{1}+\alpha,\ldots,x_{k}+\alpha\right)$
Let $A^{i}=\mbox{\rm diag}\left(A^{i}_{1},A^{i}_{2}\right)$ and $D=\mbox{\rm
diag}\left(D_{1},D_{2}\right)$ be block diagonal matrices. To keep the
notations succinct, we set $A\left(x\right)=\sum_{i}x_{i}A^{i}-D$.
###### Fact 24.
[OST19, Lemma 6.9] Let $k,\delta,\theta,\Lambda,\alpha$ be the parameters
satisfying Fact 23. Let
$\Psi,\Psi_{\theta}:\mathsf{Sym}_{k}\rightarrow{\mathbb{R}}$ be the functions
defined as $\Psi\left(M\right)=\psi\left(\lambda\left(M\right)\right)$,
$\Psi_{\theta}\left(M\right)=G_{\theta}\left(\lambda\left(M\right)\right)$,
where $\psi$ is defined in Eq. (9) and $G_{\theta}$ is defined in Eq. (20),
$\bm{x}$ and $\bm{x}^{\prime}$ be two random variables in ${\mathbb{R}}^{k}$
satisfying that
$\left|\operatorname*{\mathbb{E}}\left[\Psi_{\theta}\left(A\left(\bm{x}\right)+\beta\mathbb{I}\right)\right]-\operatorname*{\mathbb{E}}\left[\Psi_{\theta}\left(A\left(\bm{x}^{\prime}\right)+\beta\mathbb{I}\right)\right]\right|\leq\eta,$
for both $\beta=\alpha$ and $\beta=-\alpha$. Then, it holds that
$\left|\operatorname*{\mathbb{E}}\left[\Psi\left(A\left(\bm{x}\right)\right)\right]-\operatorname*{\mathbb{E}}\left[\Psi\left(A\left(\bm{x}^{\prime}\right)\right)\right]\right|\leq\eta+3\delta+\Pr\left[\lambda_{\max}\left(A\left(\bm{x}\right)\right)\in(-\Lambda,\Lambda]\right].$
## 4 Computing spectral derivatives
In this section use the result by Sendov [Sen07] to bound the spectral
derivatives of functions.
### 4.1 Formulas for spectral derivatives
Before we describe the main theorem of this section, we need the following
notation introduced by Sendov in [Sen07] to calculate the high-order Fréchet
derivatives of spectral functions.
###### Definition 25.
[Sen07] Let $t\geq 1$ and $x\in\mathbb{R}^{t}$. Let
$T:(\mathbb{R}^{k})^{\times t}\rightarrow\mathbb{R}$ be a $t$-tensor. For
every, $\ell\in[t]$, define a $(t+1)$-tensor
$T^{\ell}_{\operatorname{out}}:(\mathbb{R}^{k})^{\times(t+1)}\rightarrow\mathbb{R}$
as follows
$(T^{\ell}_{\operatorname{out}})(i_{1},\ldots,i_{t+1})=\begin{cases}0&i_{\ell}=i_{t+1}\\\
\frac{T(i_{1},\ldots,i_{\ell-1},i_{t+1},i_{\ell+1},\ldots,i_{t})-T(i_{1},\ldots,i_{\ell-1},i_{\ell},i_{\ell+1},\ldots,i_{t})}{x_{i_{t+1}}-x_{i_{\ell}}}&i_{\ell}\neq
i_{t+1}.\end{cases}$
Finally, for every $\ell\in[t]$, define
$T_{\sigma}(x)=\begin{cases}\nabla f(x)&\ell=1,\sigma=(1)\\\
\left(T(x)\right)^{\ell}_{\operatorname{out}}&\ell\leq t-1\\\ \nabla
T_{\sigma}(x)&\ell=t,\end{cases}$
where $\sigma(\ell)$ is defined as follows: let $\sigma$ be a permutation of
$[k]$ given in the cycle decomposition, then $\sigma(\ell)$ is a permutation
of $[k+1]$ elements whose cycle representation is the same as $\sigma$ except
that the element $k+1$ is inserted after the $\ell$th element and before the
$(\ell+1)$th element in the cycle representation of $\sigma$.111111For better
intuition, consider a simple example: let $\sigma=(12)(3)$ be a permutation on
$[3]$, then $\sigma(\cdot)$ is a permutation on $[4]$ defined as follows:
$\sigma(1)$ is $(142)(3)$, similarly $\sigma(2)=(124)(3)$,
$\sigma(3)=(12)(34)$, $\sigma(4)=(12)(3)(4)$.
We are now ready to state the Sendov’s formula for high-order Fréchet
derivatives of spectral functions.
###### Theorem 26.
[Sen07] Let $F:\mathsf{Sym}_{k}\rightarrow\mathbb{R}$ be a spectral function
(i.e., $F=f\circ\lambda$ for $f:\mathbb{R}^{k}\rightarrow\mathbb{R}$). Then
for any $X\in\mathsf{Sym}_{k}$ satisfying that all the eigenvalues are
distinct, $F$ is $t$-times differentiable at $X$ _if and only if_ $f$ is
$t$-times differentiable at $\lambda(X)$. If
$f(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}g\left(x_{i}\right)$ for
$g:{\mathbb{R}}\rightarrow{\mathbb{R}}$, then for any $X\in\mathsf{Sym}_{k}$,
$F$ is $t$-times Fréchet differentiable at $X$ _if and only if_ $f$ is
$t$-times differentiable at $\lambda(X)$, i.e., the distinctness of the
eigenvalues is not necessary anymore.
Moreover, for every $\sigma\in S_{t},x\in\mathbb{R}^{k}$, let
$T_{\sigma}(x):(\mathbb{R}^{k})^{\times t}\rightarrow\mathbb{R}$ be a
$t$-tensor as defined in Definition 25 (which depends on the function
$f$).121212Think of $x\in\mathbb{R}^{k}$ as the eigenvalues of
$X\in\mathsf{Sym}_{k}$, i.e., $x=\lambda(X)$. Then, for every
$U_{1},\ldots,U_{t}\in\mathsf{Sym}_{k}$, we have
$D^{t}F\left(X\right)\left[U_{1},\ldots,U_{t}\right]=\left(\sum_{\sigma\in
S_{t}}\mbox{\rm
diag}^{\sigma}T_{\sigma}(\lambda(X))\right)(V^{T}U_{1}V,\ldots,V^{T}U_{t}V),$
where $V$ satisfies $X=V\left(\mbox{\rm diag}(\lambda(X)\right)V^{T}$ and
$\mbox{\rm diag}^{\sigma}T:(\mathsf{Mat}_{k})^{t}\rightarrow\mathbb{R}$ is a
$t$-tensor on the set $\mathsf{Sym}_{k}$ (as defined in Definition 15).
### 4.2 Third order Fréchet derivatives of smooth functions
In this section, we explicitly compute the third order Fréchet derivatives of
spectral functions.
###### Theorem 27.
Let $k,n\geq 1$. Let $f:\mathbb{R}^{k}\rightarrow\mathbb{R}$ be a $3$-times
differentiable symmetric function and
$\lambda:\mathsf{Sym}_{k}\rightarrow\mathbb{R}^{k}$ be the map
$\lambda(M)=\left(\lambda_{1}(M),\ldots,\lambda_{k}(M)\right)$ for every
$M\in\mathsf{Sym}_{k}$. Let $F:\mathsf{Sym}_{k}\rightarrow\mathbb{R}$ be
defined as $F(M)=(f\circ\lambda)(M)$ for all $M\in\mathsf{Sym}_{k}$. Then, for
every $P\in\mathsf{Sym}_{k}$ with _distinct_ eigenvalues and
$H\in\mathsf{Sym}_{k}$, let $P=V\left(\mbox{\rm
diag}\left(\lambda\left(P\right)\right)\right)V^{T}$ be a spectral
decomposition of $P$ and $H=VQV^{T}$. Then
$D^{3}F\left(P\right)\left[Q,Q,Q\right]$ is the summation of the following
terms.
1. 1.
$\sum_{i_{1}}\nabla^{3}_{i_{1},i_{1},i_{1}}f\left(x\right)H_{i_{1},i_{1}}^{3}$
2. 2.
$\sum_{i_{1}\neq
i_{2}}\nabla^{3}_{i_{1},i_{2},i_{1}}f\left(x\right)H_{i_{1},i_{1}}^{2}H_{i_{2},i_{2}}$
3. 3.
$\sum_{i_{1}\neq i_{2}\neq
i_{3}}(\nabla^{3}_{i_{1},i_{2},i_{3}}f\left(x\right))\cdot
H_{i_{1},i_{1}}H_{i_{2},i_{2}}H_{i_{3},i_{3}}$
4. 4.
$\sum_{i_{1}\neq
i_{2}}\left(\frac{\nabla^{2}_{i_{2},i_{2}}-\nabla^{2}_{i_{1},i_{2}}}{x_{i_{2}}-x_{i_{1}}}-\frac{\nabla_{i_{2}}-\nabla_{i_{1}}}{(x_{i_{2}}-x_{i_{1}})^{2}}\right)f\left(x\right)H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}$
5. 5.
$\sum_{i_{1}\neq i_{2}\neq
i_{3}}\frac{\nabla^{2}_{i_{2},i_{3}}-\nabla^{2}_{i_{1},i_{3}}}{x_{i_{2}}-x_{i_{1}}}f\left(x\right)H_{i_{1},i_{2}}^{2}H_{i_{3},i_{3}}$
6. 6.
$\sum_{i_{1}\neq i_{2}\neq
i_{3}}\left(\frac{\nabla_{i_{3}}-\nabla_{i_{1}}}{(x_{i_{3}}-x_{i_{2}})(x_{i_{3}}-x_{i_{1}})}-\frac{\nabla_{i_{2}}-\nabla_{i_{1}}}{(x_{i_{3}}-x_{i_{2}})(x_{i_{2}}-x_{i_{1}})}\right)f\left(x\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}$
7. 7.
$\sum_{i_{1}\neq i_{2}\neq
i_{3}}\left(\frac{\nabla_{i_{2}}-\nabla_{i_{3}}}{(x_{i_{3}}-x_{i_{1}})(x_{i_{2}}-x_{i_{3}})}-\frac{\nabla_{i_{2}}-\nabla_{i_{1}}}{(x_{i_{3}}-x_{i_{1}})(x_{i_{2}}-x_{i_{1}})}\right)f\left(x\right)H_{i_{1},i_{3}}H_{i_{2},i_{1}}H_{i_{3},i_{2}},$
where
$x=\left(\lambda_{1}\left(P\right),\ldots,\lambda_{k}\left(P\right)\right)$.
###### Proof.
To prove this theorem, we first apply Theorem 26 for $t=3$ to obtain
$\displaystyle D^{3}F\left(P\right)\left[Q,Q,Q\right]=\left(\sum_{\sigma\in
S_{3}}\mbox{\rm diag}^{\sigma}T_{\sigma}(\lambda(P))\right)(H,H,H).$ (22)
We next carefully express each quantity in the summation using the definition
of these tensors and upper bound each term. To this end, we break down all the
six elements of $S_{3}$ and analyze them separately as follows.
Case 1: $\sigma=(1)(2)(3)$. Then $T_{\sigma}(x)=\nabla^{3}f(x)$.
Case 2: $\sigma=(12)(3)$. First, observe that considering $\sigma=(12)$ we get
$\left(T_{(12)}(x)\right)_{i_{1},i_{2}}=\begin{cases}0&i_{1}=i_{2}\\\
\frac{1}{x_{i_{2}}-x_{i_{1}}}\cdot\left(\nabla_{i_{2}}-\nabla_{i_{1}}\right)f\left(x\right)&i_{1}\neq
i_{2}\\\ \end{cases}$
Now, in order to compute $T_{(12)(3)}$, we need to compute $\nabla
T_{(12)}(x)$ which can be written as follows
$\displaystyle\left(T_{(12)(3)}(x)\right)_{i_{1},i_{2},i_{3}}$
$\displaystyle=\begin{cases}0&i_{1}=i_{2}\\\
\frac{1}{x_{i_{3}}-x_{i_{1}}}\cdot\left(\nabla^{2}_{i_{3},i_{3}}-\nabla^{2}_{i_{1},i_{3}}\right)f\left(x\right)-\frac{1}{(x_{i_{3}}-x_{i_{1}})^{2}}\cdot\left(\nabla_{i_{3}}-\nabla_{i_{1}}\right)f\left(x\right)&i_{2}=i_{3}\neq
i_{1}\\\
\frac{1}{x_{i_{2}}-x_{i_{3}}}\cdot\left(\nabla^{2}_{i_{2},i_{3}}-\nabla^{2}_{i_{3},i_{3}}\right)f\left(x\right)+\frac{1}{(x_{i_{2}}-x_{i_{3}})^{2}}\cdot\left(\nabla_{i_{2}}-\nabla_{i_{3}}\right)f\left(x\right)&i_{1}=i_{3}\neq
i_{2}\\\
\frac{1}{x_{i_{2}}-x_{i_{1}}}\cdot\left(\nabla^{2}_{i_{2},i_{3}}-\nabla^{2}_{i_{1},i_{3}}\right)f\left(x\right)&i_{1}\neq
i_{2}\neq i_{3}\\\ \end{cases}$
Case 3: $\sigma=(13)(2)$. First note that for $\sigma=(1)(2)$, we have
$T_{(1)(2)}=\nabla^{2}f$ and $\sigma(1)=(13)(2)$. So, we need to compute
$\left(\nabla^{2}f\right)f\left(x\right)^{1}_{\operatorname{out}}$ and we get
$\left(T_{(13)(2)}(x)\right)_{i_{1},i_{2},i_{3}}=\begin{cases}0&i_{1}=i_{3}\\\
\frac{1}{x_{i_{3}}-x_{i_{1}}}\cdot\left(\nabla^{2}_{i_{3},i_{2}}-\nabla^{2}_{i_{1},i_{2}}\right)f\left(x\right)&i_{1}\neq
i_{3}\end{cases}$
Case 4: $\sigma=(1)(23)$. First note that for $\sigma=(1)(2)$, we have
$T_{(1)(2)}=\nabla^{2}f$ and $\sigma(2)=(1)(23)$. So, we need to compute
$\left(\nabla^{2}f\right)f\left(x\right)^{2}_{\operatorname{out}}$ and we get
$\left(T_{(1)(23)}(x)\right)_{i_{1},i_{2},i_{3}}=\begin{cases}0&i_{2}=i_{3}\\\
\frac{1}{x_{i_{3}}-x_{i_{2}}}\cdot\left(\nabla^{2}_{i_{3},i_{1}}-\nabla^{2}_{i_{2},i_{1}}\right)f\left(x\right)&i_{2}\neq
i_{3}\end{cases}$
Case 5: $\sigma=(123)$. Let $\sigma=(12)$, then $\sigma(2)=(123)$. So we need
to compute $\left(T_{(12)}\right)f\left(x\right)^{2}_{\operatorname{out}}$ and
we obtain
$\displaystyle\left(T_{(123)}(x)\right)_{i_{1},i_{2},i_{3}}$
$\displaystyle=\begin{cases}\frac{1}{(x_{i_{2}}-x_{i_{1}})^{2}}\cdot\left(\nabla_{i_{2}}-\nabla_{i_{1}}\right)f\left(x\right)&i_{2}\neq
i_{3}=i_{1}\\\
\frac{1}{(x_{i_{3}}-x_{i_{1}})^{2}}\cdot\left(\nabla_{i_{3}}-\nabla_{i_{1}}\right)f\left(x\right)&i_{1}=i_{2}\neq
i_{3}\\\
\frac{1}{(x_{i_{3}}-x_{i_{2}})(x_{i_{3}}-x_{i_{1}})}\cdot\left(\nabla_{i_{3}}-\nabla_{i_{1}}\right)f\left(x\right)-\frac{1}{(x_{i_{3}}-x_{i_{2}})(x_{i_{2}}-x_{i_{1}})}\cdot\left(\nabla_{i_{2}}-\nabla_{i_{1}}\right)f\left(x\right)&i_{1}\neq
i_{3}\neq i_{2}\\\ 0&\text{ otherwise}\end{cases}$
Case 6: $\sigma=(132)$. Let $\sigma=(12)$, then $\sigma\tau(1)=(132)$. So we
need to compute
$\left(T_{(12)}\right)f\left(x\right)^{1}_{\operatorname{out}}$ and we obtain.
$\displaystyle\left(T_{(132)}(x)\right)_{i_{1},i_{2},i_{3}}$
$\displaystyle=\begin{cases}-\frac{1}{(x_{i_{2}}-x_{i_{1}})^{2}}\cdot\left(\nabla_{i_{2}}-\nabla_{i_{1}}\right)f\left(x\right)&i_{1}\neq
i_{3}=i_{2}\\\
\frac{1}{(x_{i_{3}}-x_{i_{2}})^{2}}\cdot\left(\nabla_{i_{3}}-\nabla_{i_{2}}\right)f\left(x\right)&i_{2}=i_{1}\neq
i_{3}\\\
\frac{1}{(x_{i_{3}}-x_{i_{1}})(x_{i_{2}}-x_{i_{3}})}\cdot\left(\nabla_{i_{2}}-\nabla_{i_{3}}\right)f\left(x\right)-\frac{1}{(x_{i_{3}}-x_{i_{1}})(x_{i_{2}}-x_{i_{1}})}\cdot\left(\nabla_{i_{2}}-\nabla_{i_{1}}\right)f\left(x\right)&i_{1}\neq
i_{3}\neq i_{2}\\\ 0&\text{ otherwise}\end{cases}$
Using the above cases we can now rewrite Eq. (22) as
$\sum_{\sigma\in
S_{3}}T_{\sigma}(x)(H,H,H)=\sum_{\sigma}\sum_{\begin{subarray}{c}i_{1},i_{2},i_{3}\end{subarray}}\left(T_{\sigma}(x)\right)_{i_{1},i_{2},i_{3}}H_{i_{1},i_{\sigma(1)}}H_{i_{2},i_{\sigma(2)}}H_{i_{3},i_{\sigma(3)}}$
Let’s write this out as follows: by $T_{i}$, we mean $T_{case(i)}$ above
$\displaystyle\sum_{i_{1},i_{2},i_{3}}$
$\displaystyle(T_{1})_{i_{1},i_{2},i_{3}}H_{i_{1},i_{1}}H_{i_{2},i_{2}}H_{i_{3},i_{3}}+(T_{2})_{i_{1},i_{2},i_{3}}H_{i_{1},i_{2}}H_{i_{2},i_{1}}H_{i_{3},i_{3}}+(T_{3})_{i_{1},i_{2},i_{3}}H_{i_{1},i_{3}}H_{i_{2},i_{2}}H_{i_{3},i_{1}}$
$\displaystyle+(T_{4})_{i_{1},i_{2},i_{3}}H_{i_{1},i_{1}}H_{i_{2},i_{3}}H_{i_{3},i_{2}}+(T_{5})_{i_{1},i_{2},i_{3}}H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}+(T_{6})_{i_{1},i_{2},i_{3}}H_{i_{1},i_{3}}H_{i_{2},i_{1}}H_{i_{3},i_{2}}$
and in particular, since $H$ is symmetric the above simplifies to
$\displaystyle\begin{aligned}
\sum_{i_{1},i_{2},i_{3}}&(T_{1})_{i_{1},i_{2},i_{3}}H_{i_{1},i_{1}}H_{i_{2},i_{2}}H_{i_{3},i_{3}}+(T_{2})_{i_{1},i_{2},i_{3}}H_{i_{1},i_{2}}^{2}H_{i_{3},i_{3}}+(T_{3})_{i_{1},i_{2},i_{3}}H_{i_{1},i_{3}}^{2}H_{i_{2},i_{2}}\\\
&+(T_{4})_{i_{1},i_{2},i_{3}}H_{i_{1},i_{1}}H_{i_{2},i_{3}}^{2}+(T_{5})_{i_{1},i_{2},i_{3}}H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}+(T_{6})_{i_{1},i_{2},i_{3}}H_{i_{1},i_{3}}H_{i_{2},i_{1}}H_{i_{3},i_{2}}\end{aligned}$
(23)
Now, we will break up this sum into $5$ cases as follows which will give us
our theorem statement.
##### Case (i): $i_{1}=i_{3}\neq i_{2}$.
Then Eq. (23) reduces to the following
$\displaystyle\sum_{i_{1},i_{2}}H_{i_{1},i_{1}}^{2}H_{i_{2},i_{2}}\left(T_{1}+T_{3}\right)+H_{i_{1},i_{1}}H_{i_{2},i_{1}}^{2}\left(T_{2}+T_{4}+T_{5}+T_{6}\right)$
(24)
Note that when we say $T_{q}$ above, we mean
$(T_{q})_{i_{1},i_{2},i_{3}}=(T_{q})_{i_{1},i_{2},i_{1}}$ (since
$i_{3}=i_{1}$). Let us now plug in the values of the corresponding $T_{q}$s
into the formula and rewrite the above as follows
$\displaystyle\begin{aligned} &\sum_{i_{1}\neq
i_{2}}H_{i_{1},i_{1}}^{2}H_{i_{2},i_{2}}\left(\nabla^{3}_{i_{1},i_{2},i_{1}}f\left(x\right)+0\right)+\\\
&\quad+H_{i_{1},i_{1}}H_{i_{2},i_{1}}^{2}\left(\frac{\nabla^{2}_{i_{2},i_{1}}-\nabla^{2}_{i_{1},i_{1}}}{x_{i_{2}}-x_{i_{1}}}+\frac{\nabla_{i_{2}}-\nabla_{i_{1}}}{(x_{i_{2}}-x_{i_{1}})^{2}}+\frac{\nabla^{2}_{i_{1},i_{1}}-\nabla^{2}_{i_{2},i_{1}}}{x_{i_{1}}-x_{i_{2}}}+\frac{\nabla_{i_{2}}-\nabla_{i_{1}}}{(x_{i_{2}}-x_{i_{1}})^{2}}\right)f\left(x\right)\\\
&=\sum_{i_{1}\neq
i_{2}}H_{i_{1},i_{1}}^{2}H_{i_{2},i_{2}}\left(\nabla^{3}_{i_{1},i_{2},i_{1}}f\left(x\right)\right)+2H_{i_{1},i_{1}}H_{i_{2},i_{1}}^{2}\left(\frac{\nabla^{2}_{i_{2},i_{1}}-\nabla^{2}_{i_{1},i_{1}}}{x_{i_{2}}-x_{i_{1}}}+\frac{\nabla_{i_{2}}-\nabla_{i_{1}}}{(x_{i_{2}}-x_{i_{1}})^{2}}\right)f\left(x\right)\end{aligned}$
(25)
Case (ii): $i_{1}=i_{2}\neq i_{3}$. Then Eq. (23) reduces to
$\displaystyle\sum_{i_{1},i_{3}}H_{i_{1},i_{1}}^{2}H_{i_{3},i_{3}}\left(T_{1}+T_{2}\right)+H_{i_{1},i_{1}}H_{i_{3},i_{1}}^{2}\left(T_{3}+T_{4}+T_{5}+T_{6}\right)$
(26)
The above simplies to the following
$\displaystyle\begin{aligned} &\sum_{i_{1}\neq
i_{3}}H_{i_{1},i_{1}}^{2}H_{i_{3},i_{3}}\left(\nabla^{3}_{i_{1},i_{1},i_{3}}f\left(x\right)+0\right)+\\\
&\quad+H_{i_{1},i_{1}}H_{i_{3},i_{1}}^{2}\left(\frac{\nabla^{2}_{i_{3},i_{1}}-\nabla^{2}_{i_{1},i_{1}}}{x_{i_{3}}-x_{i_{1}}}+\frac{\nabla^{2}_{i_{3},i_{1}}-\nabla^{2}_{i_{1},i_{1}}}{x_{i_{3}}-x_{i_{1}}}+\frac{\nabla_{i_{3}}-\nabla_{i_{1}}}{(x_{i_{3}}-x_{i_{1}})^{2}}+\frac{\nabla_{i_{3}}-\nabla^{2}_{i_{1}}}{(x_{i_{3}}-x_{i_{1}})^{2}}\right)f\left(x\right)\\\
&=\sum_{i_{1}\neq
i_{3}}H_{i_{1},i_{1}}^{2}H_{i_{3},i_{3}}\left(\nabla^{3}_{i_{1},i_{1},i_{3}}f\left(x\right)\right)+2H_{i_{1},i_{1}}H_{i_{3},i_{1}}^{2}\left(\frac{\nabla^{2}_{i_{3},i_{1}}-\nabla^{2}_{i_{1},i_{1}}}{x_{i_{3}}-x_{i_{1}}}+\frac{\nabla_{i_{3}}-\nabla_{i_{1}}}{(x_{i_{3}}-x_{i_{1}})^{2}}\right)f\left(x\right)\end{aligned}$
(27)
Case (iii): $i_{2}=i_{3}\neq i_{1}$. Then Eq. (23) reduces to
$\displaystyle\sum_{i_{1},i_{2}}H_{i_{2},i_{2}}^{2}H_{i_{1},i_{1}}\left(T_{1}+T_{4}\right)+H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}\left(T_{2}+T_{3}+T_{5}+T_{6}\right)$
(28)
The above simplifies to the following
$\displaystyle\begin{aligned} &\sum_{i_{1}\neq
i_{2}}H_{i_{2},i_{2}}^{2}H_{i_{1},i_{1}}\left(\nabla^{3}_{i_{1},i_{2},i_{2}}f\left(x\right)+0\right)+\\\
&\quad+H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}\left(\frac{\nabla^{2}_{i_{2},i_{2}}-\nabla^{2}_{i_{1},i_{2}}}{x_{i_{2}}-x_{i_{1}}}-\frac{\nabla_{i_{2}}-\nabla_{i_{1}}}{(x_{i_{2}}-x_{i_{1}})^{2}}+\frac{\nabla^{2}_{i_{2},i_{2}}-\nabla^{2}_{i_{1},i_{2}}}{x_{i_{2}}-x_{i_{1}}}-\frac{\nabla_{i_{2}}-\nabla_{i_{1}}}{(x_{i_{2}}-x_{i_{1}})^{2}}\right)f\left(x\right)\\\
&=\sum_{i_{1}\neq
i_{2}}H_{i_{2},i_{2}}^{2}H_{i_{1},i_{1}}\left(\nabla^{3}_{i_{1},i_{2},i_{2}}f\left(x\right)\right)+2H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}\left(\frac{\nabla^{2}_{i_{2},i_{2}}-\nabla^{2}_{i_{1},i_{2}}}{x_{i_{2}}-x_{i_{1}}}-\frac{\nabla_{i_{2}}-\nabla_{i_{1}}}{(x_{i_{2}}-x_{i_{1}})^{2}}\right)f\left(x\right)\end{aligned}$
(29)
Case (i)+ Case (ii)+ Case (iii). We first upper bound these three cases to get
the desired upper bound in the theorem statement. First summing the three
cases, we have
$\displaystyle\begin{aligned} \sum_{i_{1}\neq
i_{2}}&H_{i_{1},i_{1}}^{2}H_{i_{2},i_{2}}\left(\nabla^{3}_{i_{1},i_{2},i_{1}}+\nabla^{3}_{i_{1},i_{2},i_{2}}+\nabla^{3}_{i_{2},i_{1},i_{1}}\right)f\left(x\right)\\\
&+6\sum_{i_{1}\neq
i_{2}}H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}\underbrace{\left(\frac{\nabla^{2}_{i_{2},i_{2}}-\nabla^{2}_{i_{1},i_{2}}}{x_{i_{2}}-x_{i_{1}}}-\frac{\nabla_{i_{2}}-\nabla_{i_{1}}}{(x_{i_{2}}-x_{i_{1}})^{2}}\right)f\left(x\right)}_{(\star)}\end{aligned}$
(30)
##### Case (iv): $i_{2}=i_{3}=i_{1}$.
Then Eq. (23) reduces to
$\displaystyle\sum_{i_{1}}H_{i_{1},i_{1}}^{3}\left(T_{1}+T_{2}+T_{3}+T_{4}+T_{5}+T_{6}\right)=\sum_{i_{1}}H_{i_{1},i_{1}}^{3}\nabla^{3}_{i_{1},i_{1},i_{1}}f$
(31)
##### Case (v): $i_{2}\neq i_{3}\neq i_{1}$.
Then Eq. (23) stays the same and we get
$\displaystyle\begin{aligned}
\sum_{i_{1},i_{2},i_{3}}&(\nabla^{3}_{i_{1},i_{2},i_{3}}f)\cdot
H_{i_{1},i_{1}}H_{i_{2},i_{2}}H_{i_{3},i_{3}}\\\
&+\frac{\nabla^{2}_{i_{2},i_{3}}-\nabla^{2}_{i_{1},i_{3}}}{x_{i_{2}}-x_{i_{1}}}f\left(x\right)H_{i_{1},i_{2}}^{2}H_{i_{3},i_{3}}+\frac{\nabla^{2}_{i_{3},i_{2}}-\nabla^{2}_{i_{1},i_{2}}}{x_{i_{3}}-x_{i_{1}}}f\left(x\right)H_{i_{1},i_{3}}^{2}H_{i_{2},i_{2}}+\frac{\nabla^{2}_{i_{3},i_{1}}-\nabla^{2}_{i_{2},i_{1}}}{x_{i_{3}}-x_{i_{2}}}f\left(x\right)H_{i_{1},i_{1}}H_{i_{2},i_{3}}^{2}\\\
&+\left(\frac{\nabla_{i_{3}}-\nabla_{i_{1}}}{(x_{i_{3}}-x_{i_{2}})(x_{i_{3}}-x_{i_{1}})}-\frac{\nabla_{i_{2}}-\nabla_{i_{1}}}{(x_{i_{3}}-x_{i_{2}})(x_{i_{2}}-x_{i_{1}})}\right)f\left(x\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\\\
&+\left(\frac{\nabla_{i_{2}}-\nabla_{i_{3}}}{(x_{i_{3}}-x_{i_{1}})(x_{i_{2}}-x_{i_{3}})}-\frac{\nabla_{i_{2}}-\nabla_{i_{1}}}{(x_{i_{3}}-x_{i_{1}})(x_{i_{2}}-x_{i_{1}})}\right)f\left(x\right)H_{i_{1},i_{3}}H_{i_{2},i_{1}}H_{i_{3},i_{2}}\end{aligned}$
(32)
This concludes the proof of the theorem statement. ∎
### 4.3 Main theorem: Fréchet derivatives of Bentkus function
We now state the main theorem which bounds all the terms that appear in the
theorem in the previous section. Let
$G:{\mathbb{R}}^{k}\rightarrow{\mathbb{R}}$ be the Bentkus function given in
Definition 16.
###### Theorem 28.
Let $k\geq 1$ be an integer and $\Psi:\mathsf{Sym}_{k}\rightarrow{\mathbb{R}}$
be a function defined as
$\psi\left(M\right)=\left(G\circ\lambda\right)\left(M\right)$ where $G$ is
given in Definition 16. Given $\Delta\geq 1$ $X\in\mathsf{Sym}_{k}$ with
eigenvalues $\lambda\left(X\right)=\left(x_{1},\ldots,x_{k}\right)$ satisfying
that $\mbox{$\|{X}\|$}\leq\Delta$, it holds that
$\left|D^{3}\Psi\left(X\right)\left[H,H,H\right]\right|\leq
O\left(\Delta^{2}\cdot\log^{3}k\cdot\|H\|^{3}\right).$
The following corollary simply follows from the definition of $G_{\theta}$ in
Eq. 20 and the chain rule of Fréchet derivatives in Fact 9.
###### Corollary 29.
Let $k\geq 1$ be an integer and $\theta>0,\alpha\in{\mathbb{R}}$ and
$\Psi_{\theta}:\mathsf{Sym}_{k}\rightarrow{\mathbb{R}}$ be a function defined
as
$\Psi_{\theta}\left(M\right)=\left(G_{\theta}\circ\lambda\right)\left(M+\alpha\mathbb{I}\right)$,
where $G_{\theta}$ is given in Eq. (20). Given $\Delta\geq 1$,
$X\in\mathsf{Sym}_{k}$ with eigenvalues
$\lambda\left(X\right)=\left(x_{1},\ldots,x_{k}\right)$ satisfying that
$\mbox{$\|{X}\|$}\leq\Delta$, it holds that
$\left|D^{3}\Psi_{\theta}\left(X+\alpha\mathbb{I}\right)\left[H,H,H\right]\right|\leq
O\left(\frac{\Delta^{2}+\alpha^{2}}{\theta^{3}}\cdot\log^{3}k\cdot\|H\|^{3}\right).$
In order to prove the theorem above, We upper bound all the terms listed in
Theorem 27 individually in the following sections (in increasing order of
difficulty). Given the calculations are fairly technical we break down the
analysis in the following sections for modularity and reader convenience. In
Section 4.4.1 we bound the first three terms in Theorem 27 (this is the easy
case since the analysis is very similar to what happens in [HKM13] by directly
using known properties of the Bentkus function), in Section 4.4.2 and 4.4.3 we
bound the fourth and fifth term (this already deviates from the analysis of
[HKM13]) and finally in Section 4.5 we bound the sixth and seventh term (this
calculation is fairly involved and deviates significantly from prior works,
since we need to deal with various aspects of Fréchet derivatives, new
properties of Bentkus function and the _non-diagonal_ entries of the matrices
$H$ which is unique to the matrix-spectrahedron case and is not faced in
[HKM13, ST17, OST19]).
As spectral functions and spectral norms are unitarily invariant, Assume that
$X=\mbox{\rm diag}\left(x_{1},\ldots,x_{k}\right)$ is diagonal without loss of
generality. To adopt Theorem 27, we assume that all the $x_{1},\ldots,x_{n}$
are distinct. We claim that the general case follows by the continuity
argument: notice that
$\ln(G_{\theta}\left(x\right))=\sum_{i=1}^{n}\ln(g\left(-\frac{x_{i}}{\theta}\right))$
from Eq. (21). Thus by Theorem 26, $\ln(G_{\theta}\circ\lambda)$ is infinitely
Fréchet differentiable at any $X\in\mathsf{Sym}_{k}$ as $\ln(G_{\theta})$ is
infinitely times differentiable. This further implies by definition that
$\Psi_{\theta}$ is infinitely Fréchet differentiable.
### 4.4 Bounding terms ($1$)-($5$) in Theorem 27 for Bentkus function
Let $G:{\mathbb{R}}^{k}\rightarrow{\mathbb{R}}$ be the Bentkus function given
in Definition 16. Recall that $G(x)=\prod_{i}g(x_{i})$, where
$g\left(x\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{-x}e^{-t^{2}/2}dt$.
Recall the notation
$g^{\prime}\left(x\right)=\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}$ and
$\overline{g}\left(x\right)=g^{\prime}(x)/g(x)$.
#### 4.4.1 Bounding terms $(1,2,3)$ in Theorem 27
###### Lemma 30 (Bounding terms $(1,2,3)$).
The following three terms
$\left|\sum_{i_{1}}\nabla^{3}_{i_{1},i_{1},i_{1}}G\left(x\right)H_{i_{1},i_{1}}^{3}\right|,\quad\left|\sum_{i_{1}\neq
i_{2}}\nabla^{3}_{i_{1},i_{2},i_{1}}G\left(x\right)H_{i_{1},i_{1}}^{2}H_{i_{2},i_{2}}\right|,\quad\left|\sum_{i_{1}\neq
i_{2}\neq i_{3}}\nabla^{3}_{i_{1},i_{2},i_{3}}G\left(x\right)\cdot
H_{i_{1},i_{1}}H_{i_{2},i_{2}}H_{i_{3},i_{3}}\right|$
can be upper bound by $O(\log^{1.5}k\cdot\|H\|^{3})$.
###### Proof.
The first upper bound is straightforward. Observe that
$\left|\sum_{i_{1}}\nabla^{3}_{i_{1},i_{1},i_{1}}G\left(x\right)H_{i_{1},i_{1}}^{3}\right|\leq\max_{i}|H_{i,i}|^{3}\cdot\sum_{i_{1}}\left|\nabla^{3}_{i_{1},i_{1},i_{1}}G\left(x\right)\right|\leq\max_{i}|H_{i,i}|^{3}\cdot\|G^{(3)}\left(x\right)\|_{1}\leq\|H\|^{3}\cdot\log^{1.5}k,$
where the second inequality follows by definition of $\|G^{(3)}\|_{1}$ and the
last inequality used $\max_{i,j}|H_{i,j}|\leq\|H\|$ (the latter being the
spectral norm of $H$) and Fact 20 to conclude $\|G^{(3)}\|_{1}\leq
O\left(\log^{1.5}k\right)$. Similarly, the remaining two terms can also be
bounded exactly as above (by observing that $\sum_{i_{1}\neq
i_{2}}\nabla^{3}_{i_{1},i_{2},i_{1}}G$ and $\sum_{i_{1}\neq i_{2}\neq
i_{3}}(\nabla^{3}_{i_{1},i_{2},i_{3}}G)$ appear in the expression of
$\|G^{(3)}\|_{1}$). ∎
#### 4.4.2 Bounding term ($4$) in Theorem 27
In order to bound the remaining terms in Theorem 27, we need the following
claim.
###### Claim 31.
It holds that
1. 1.
$\sum_{i_{1}\neq
i_{2}}\overline{g}\left(x_{i_{1}}\right)\left|G\left(x\right)H_{i_{2},i_{2}}H_{i_{1},i_{2}}^{2}\right|\leq
O\left(\sqrt{\log k}\cdot\mbox{$\|{H}\|$}^{3}\right).$
2. 2.
$\sum_{i_{1}\neq
i_{2}}\overline{g}\left(x_{i_{2}}\right)\left|G\left(x\right)H_{i_{2},i_{2}}H_{i_{1},i_{2}}^{2}\right|\leq
O\left(\sqrt{\log k}\cdot\mbox{$\|{H}\|$}^{3}\right).$
3. 3.
$\sum_{i_{1}\neq i_{2}\neq
i_{3}}\left|\overline{g}\left(x_{i_{2}}\right)\overline{g}\left(x_{i_{3}}\right)G\left(x\right)H_{i_{1},i_{2}}^{2}H_{i_{3},i_{3}}\right|\leq
O\left(\log k\cdot\mbox{$\|{H}\|$}^{3}\right)$.
###### Proof.
For Item 1, we have
$\sum_{i_{1}\neq
i_{2}}\overline{g}\left(x_{i_{1}}\right)\left|G\left(x\right)H_{i_{2},i_{2}}H_{i_{1},i_{2}}^{2}\right|\leq\sum_{i_{1}}\overline{g}\left(x_{i_{1}}\right)G\left(x\right)\cdot\max_{i_{1}}\sum_{i_{2}}\left|H_{i_{2},i_{2}}H_{i_{1},i_{2}}^{2}\right|\leq\mbox{$\|{G^{(1)}\left(x\right)}\|$}_{1}\cdot\mbox{$\|{H}\|$}^{3}$
where the last inequality is because
$\max_{i_{1}}\sum_{i_{2}}\left|H_{i_{2},i_{2}}H_{i_{1},i_{2}}^{2}\right|\leq\mbox{$\|{H}\|$}\max_{i_{1}}\left(H^{2}\right)_{i_{1},i_{1}}\leq\mbox{$\|{H}\|$}^{3},$
(33)
using the fact that $\max_{ij}|H_{ij}|\leq\|H\|$. Using Fact 20 shows the
first inequality. Item 2 follows by the same reason.
For Item 3, we have
$\displaystyle\sum_{i_{1}\neq i_{2}\neq
i_{3}}\left|\overline{g}\left(x_{i_{2}}\right)\overline{g}\left(x_{i_{3}}\right)G\left(x\right)H_{i_{1},i_{2}}^{2}H_{i_{3},i_{3}}\right|$
$\displaystyle=$ $\displaystyle\sum_{i_{2}\neq
i_{3}}\left|\overline{g}\left(x_{i_{2}}\right)\overline{g}\left(x_{i_{3}}\right)G\left(x\right)\right|\max_{i_{2},i_{3}}\sum_{i_{1}}\left|H_{i_{1},i_{2}}^{2}H_{i_{3},i_{3}}\right|$
$\displaystyle\leq$ $\displaystyle O\left(\log
k\cdot\mbox{$\|{H}\|$}^{3}\right)$
where the inequality is because $\sum_{i_{2}\neq
i_{3}}\left|\overline{g}\left(x_{i_{2}}\right)\overline{g}\left(x_{i_{3}}\right)G\left(x\right)\right|$
appears in $G^{(2)}(x)$ and then we use Fact 20 to upper bound it by $\log k$
and additionally we use that
$\max_{i_{2},i_{3}}\sum_{i_{1}}\left|H_{i_{1},i_{2}}^{2}H_{i_{3},i_{3}}\right|\leq\mbox{$\|{H}\|$}\cdot\max_{i_{2}}\left(H^{2}\right)_{i_{2},i_{2}}\leq\mbox{$\|{H}\|$}^{3}.$
(34)
∎
###### Lemma 32 (Bounding terms $(4)$ in Theorem 27).
We have
$\sum_{i_{1}\neq
i_{2}}H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}\left(\frac{\nabla^{2}_{i_{2},i_{2}}G-\nabla^{2}_{i_{1},i_{2}}G}{x_{i_{2}}-x_{i_{1}}}-\frac{\nabla_{i_{2}}G-\nabla_{i_{1}}G}{(x_{i_{2}}-x_{i_{1}})^{2}}\right)\leq
O\left(\Delta^{2}\cdot\sqrt{\log k}\mbox{$\|{H}\|$}^{3}\right),$
where $\Delta=\max_{i}|x_{i}|$.
###### Proof.
First observe that
$\nabla_{i_{2}}G(x)=g^{\prime}(x_{i_{2}})\prod_{j\neq
i_{2}}G(x_{j})=\overline{g}(x_{i_{2}})\cdot G(x),$
and similarly we have
$\nabla_{i_{2},i_{2}}G(x)=\overline{g}(x_{i_{2}})\nabla_{i_{2}}G(x)+G(x)\nabla_{i_{2}}\overline{g}(x_{i_{2}})=\left(\overline{g}(x_{i_{2}})^{2}-(x_{i_{2}}+\overline{g}(x_{i_{2}}))\overline{g}(x_{i_{2}})\right)G(x)=-x_{i_{2}}\overline{g}(x_{i_{2}})G(x),$
where we used Fact 17. Now, let us start upper bounding the lemma statement as
follows
$\displaystyle\left|\sum_{i_{1}\neq
i_{2}}H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}\left(\frac{\nabla^{2}_{i_{2},i_{2}}G-\nabla^{2}_{i_{1},i_{2}}G}{x_{i_{2}}-x_{i_{1}}}-\frac{\nabla_{i_{2}}G-\nabla_{i_{1}}G}{(x_{i_{2}}-x_{i_{1}})^{2}}\right)\right|$
$\displaystyle\leq\sum_{i_{1}\neq
i_{2}}\Big{|}-\frac{\overline{g}(x_{i_{1}})\overline{g}(x_{i_{2}})+x_{i_{2}}\overline{g}(x_{i_{2}})}{x_{i_{2}}-x_{i_{1}}}-\frac{\overline{g}(x_{i_{2}})-\overline{g}(x_{i_{1}})}{(x_{i_{2}}-x_{i_{1}})^{2}}\Big{|}\cdot|G(x)\cdot
H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}|$ $\displaystyle=\sum_{i_{1}\neq
i_{2}}\Big{|}-\frac{\overline{g}(x_{i_{1}})\overline{g}(x_{i_{2}})+x_{i_{2}}\overline{g}(x_{i_{2}})}{x_{i_{2}}-x_{i_{1}}}-\frac{\overline{g}^{\prime}(\xi_{i_{1},i_{2}})}{x_{i_{2}}-x_{i_{1}}}\Big{|}\cdot|G(x)\cdot
H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}|$ $\displaystyle=\sum_{i_{1}\neq
i_{2}}\Big{|}\frac{\overline{g}(x_{i_{1}})\overline{g}(x_{i_{2}})+x_{i_{2}}\overline{g}(x_{i_{2}})}{x_{i_{2}}-x_{i_{1}}}-\frac{(\xi_{i_{1},i_{2}}+\overline{g}(\xi_{i_{1},i_{2}}))\overline{g}(\xi_{i_{1},i_{2}})}{x_{i_{2}}-x_{i_{1}}}\Big{|}\cdot|G(x)\cdot
H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}|$ $\displaystyle\leq\sum_{i_{1}\neq
i_{2}}\underbrace{\Big{|}\frac{x_{i_{2}}\overline{g}(x_{i_{2}})-\xi_{i_{1},i_{2}}\overline{g}(\xi_{i_{1},i_{2}})}{x_{i_{2}}-x_{i_{1}}}\Big{|}\cdot|G(x)\cdot
H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}|}_{:=(1)}+\underbrace{\Big{|}\frac{\overline{g}(x_{i_{1}})\overline{g}(x_{i_{2}})-\overline{g}(\xi_{i_{1},i_{2}})^{2}}{x_{i_{2}}-x_{i_{1}}}\Big{|}\cdot|G(x)\cdot
H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}|}_{:=(2)},$
where the first equality used the mean-value theorem to obtain a
$\xi_{i_{1},i_{2}}\in[x_{i_{1}},x_{i_{2}}]$, second equality used Eq. (16). We
now bound both these terms separately as follows.
##### Term 1 upper bound.
Note that $\xi_{i_{1},i_{2}}$ is between $x_{i_{1}}$ and $x_{i_{2}}$. The
first term is upper bounded by
$\displaystyle\sum_{i_{1}\neq
i_{2}}\Big{|}\frac{x_{i_{2}}\overline{g}(x_{i_{2}})-\xi_{i_{1},i_{2}}\overline{g}(\xi_{i_{1},i_{2}})}{x_{i_{2}}-\xi_{i_{1},i_{2}}}\Big{|}\cdot|G(x)\cdot
H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}|$ $\displaystyle=$
$\displaystyle\sum_{i_{1}\neq
i_{2}}\left|\left(1-\eta_{i_{1},i_{2}}^{2}\right)\overline{g}\left(\eta_{i_{1},i_{2}}\right)-\eta_{i_{1},i_{2}}\overline{g}\left(\eta_{i_{1},i_{2}}\right)^{2}\right|\cdot\left|G(x)\cdot
H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}\right|$ $\displaystyle\leq$ $\displaystyle
2\Delta^{4}\sum_{i_{1}\neq
i_{2}}\overline{g}\left(\eta_{i_{1},i_{2}}\right)\left|G(x)\cdot
H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}\right|$
for some $\eta_{i_{1},i_{2}}$ between $x_{i_{2}}$ and $\xi_{i_{1},i_{2}}$,
where we apply a mean value theorem for the function
$x\overline{g}\left(x\right)$ for the equality and Lemma 19 for the
inequality. Note that $\overline{g}\left(\cdot\right)$ is nonnegative and
monotone decreasing by Fact 17. Thus the first term is upper bounded by
$\displaystyle 2\Delta^{2}\sum_{i_{1}\neq
i_{2}}\max\left\\{\overline{g}\left(x_{i_{1}}\right),\overline{g}\left(x_{i_{2}}\right)\right\\}\left|G(x)\cdot
H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}\right|$
which, in turn, is upper bounded by $O\left(\Delta^{4}\cdot\sqrt{\log
k}\cdot\mbox{$\|{H}\|$}^{3}\right)$ from Fact 20 and Eqs (33), (34).
##### Term 2 upper bound.
By triangle inequality we upper bound the second term by
$\displaystyle\begin{aligned} &\sum_{i_{1}\neq
i_{2}}\Big{|}\frac{\overline{g}(x_{i_{1}})\overline{g}(x_{i_{2}})-\overline{g}(\xi_{i_{1},i_{2}})^{2}}{x_{i_{2}}-x_{i_{1}}}\Big{|}\cdot|G(x)\cdot
H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}|\\\ &\leq\sum_{i_{1}\neq
i_{2}}\Big{|}\frac{\overline{g}(x_{i_{1}})\overline{g}(x_{i_{2}})-\overline{g}(x_{i_{1}})^{2}}{x_{i_{2}}-x_{i_{1}}}\Big{|}\cdot|G(x)\cdot
H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}|+\sum_{i_{1}\neq
i_{2}}\Big{|}\frac{\overline{g}(x_{i_{1}})^{2}-\overline{g}(\xi_{i_{1},i_{2}})^{2}}{x_{i_{2}}-x_{i_{1}}}\Big{|}\cdot|G(x)\cdot
H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}|.\end{aligned}$ (35)
We first upper bound the first quantity in Eq. (35) as follows.
$\displaystyle\begin{aligned} &\sum_{i_{1}\neq
i_{2}}\Big{|}\frac{\overline{g}(x_{i_{1}})\overline{g}(x_{i_{2}})-\overline{g}(x_{i_{1}})^{2}}{x_{i_{2}}-x_{i_{1}}}\Big{|}\cdot|G(x)\cdot
H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}|\\\ &=\sum_{i_{1}\neq
i_{2}}\Big{|}\frac{\overline{g}(x_{i_{2}})-\overline{g}(x_{i_{1}})}{x_{i_{2}}-x_{i_{1}}}\Big{|}\cdot|G(x)|\cdot|\overline{g}(x_{i_{1}})|\cdot|H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}|\\\
&=\sum_{i_{1}\neq
i_{2}}|\overline{g}^{\prime}(\zeta_{i_{1},i_{2}})|\cdot|G(x)|\cdot|\overline{g}(x_{i_{1}})|\cdot|H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}|\quad\quad\quad\mbox{}\\\
&\leq 6\Delta^{2}\cdot\sum_{i_{1}\neq
i_{2}}|G(x)|\cdot|\overline{g}(x_{i_{1}})|\cdot|H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}|\quad\quad\quad\quad\quad\mbox{}\\\
&\leq
6\Delta^{2}\mbox{$\|{G^{(1)}}\|$}_{1}\mbox{$\|{H}\|$}^{3}.\quad\quad\quad\quad\quad\mbox{}\\\
&\leq O\left(\Delta^{2}\cdot\sqrt{\log
k}\cdot\mbox{$\|{H}\|$}^{3}\right),\end{aligned}$ (36)
where $\zeta_{i_{1},i_{2}}$ between $x_{i_{1}}$ and $x_{i_{2}}$, first
inequality uses Fact 19, the second inequality uses Eqs. (33), (34) and the
last inequality is from Fact 20.
We now bound the second term in Eq. (35) as follows
$\displaystyle\sum_{i_{1}\neq
i_{2}}\Big{|}\frac{\overline{g}(x_{i_{1}})^{2}-\overline{g}(\xi_{i_{1},i_{2}})^{2}}{x_{i_{2}}-x_{i_{1}}}\Big{|}\cdot|G(x)\cdot
H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}|$ $\displaystyle\leq\sum_{i_{1}\neq
i_{2}}\Big{|}\frac{\overline{g}(x_{i_{1}})^{2}-\overline{g}(\xi_{i_{1},i_{2}})^{2}}{\xi_{i_{1},i_{2}}-x_{i_{1}}}\Big{|}\cdot|G(x)\cdot
H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}|\quad\quad\quad\quad$ (for $\xi$ is between
$x_{i_{1}}$ and $x_{i_{2}}$) $\displaystyle=2\sum_{i_{1}\neq
i_{2}}\left|\overline{g}\left(\eta_{i_{1},i_{2}}\right)\overline{g}^{\prime}\left(\eta_{i_{1},i_{2}}\right)\right|\cdot\left|G\left(x\right)\right|\cdot\left|H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}\right|\quad\quad\quad\quad\mbox{(for
some $\eta_{i_{1},i_{2}}$ between $x_{i_{1}}$ and $\xi_{i_{1},i_{2}}$)}$
$\displaystyle\leq 12\Delta^{2}\sum_{i_{1}\neq
i_{2}}\left|\overline{g}\left(\eta_{i_{1},i_{2}}\right)\right|\cdot\left|G\left(x\right)\right|\cdot\left|H_{i_{2},i_{2}}H_{i_{2},i_{1}}^{2}\right|\quad\quad\quad\quad\quad$
(Lemma 19) $\displaystyle\leq 12\Delta^{2}\sum_{i_{1}\neq
i_{2}}\max\left\\{\overline{g}\left(x_{i_{1}}\right),\overline{g}\left(x_{i_{2}}\right)\right\\}\cdot\left|G\left(x\right)\right|\cdot\left|H_{i_{2},i_{2}}H_{i_{1},i_{2}}^{2}\right|.\quad\quad\quad\quad\quad$
(Fact 17)
Further applying Fact 20 and putting together Eqs. (34)(33), we conclude that
it can be upper bounded by $O\left(\Delta^{2}\sqrt{\log
k}\mbox{$\|{H}\|$}^{3}\right)$. ∎
#### 4.4.3 Bounding term ($5$) in Theorem 27
###### Lemma 33 (Bounding terms $(5)$ in Theorem 27).
We have
$\displaystyle\left|\sum_{i_{1}\neq i_{2}\neq
i_{3}}\frac{\nabla^{2}_{i_{2},i_{3}}G\left(x\right)-\nabla^{2}_{i_{1},i_{3}}G\left(x\right)}{x_{i_{2}}-x_{i_{1}}}H_{i_{1},i_{2}}^{2}H_{i_{3},i_{3}}\right|\leq
O\left(\Delta\cdot\log k\cdot\mbox{$\|{H}\|$}^{3}\right).$
###### Proof.
$\displaystyle\left|\sum_{i_{1}\neq i_{2}\neq
i_{3}}\frac{\nabla^{2}_{i_{2},i_{3}}G\left(x\right)-\nabla^{2}_{i_{1},i_{3}}G\left(x\right)}{x_{i_{2}}-x_{i_{1}}}H_{i_{1},i_{2}}^{2}H_{i_{3},i_{3}}\right|$
$\displaystyle=$ $\displaystyle\left|\sum_{i_{1}\neq i_{2}\neq
i_{3}}\frac{\overline{g}\left(x_{i_{3}}\right)\left(\overline{g}\left(x_{i_{2}}\right)-\overline{g}\left(x_{i_{1}}\right)\right)}{x_{i_{2}}-x_{i_{1}}}G\left(x\right)H_{i_{1},i_{2}}^{2}H_{i_{3},i_{3}}\right|$
$\displaystyle=$ $\displaystyle\left|\sum_{i_{1}\neq i_{2}\neq
i_{3}}\overline{g}^{\prime}\left(\xi_{i_{1},i_{2}}\right)\overline{g}\left(x_{i_{3}}\right)G\left(x\right)H_{i_{1},i_{2}}^{2}H_{i_{3},i_{3}}\right|$
(for some $\xi_{i_{1},i_{2}}$ between $x_{i_{1}}$ and $x_{i_{2}}$)
$\displaystyle\leq$ $\displaystyle 3\Delta\sum_{i_{1}\neq i_{2}\neq
i_{3}}\left|\max\left\\{\overline{g}\left(x_{i_{1}}\right),\overline{g}\left(x_{i_{2}}\right)\right\\}\overline{g}\left(x_{i_{3}}\right)G\left(x\right)H_{i_{1},i_{2}}^{2}H_{i_{3},i_{3}}\right|\quad\quad\quad\quad\quad$
(Fact 17 and Lemma 19) $\displaystyle\leq$ $\displaystyle
O\left(\Delta\cdot\log k\cdot\mbox{$\|{H}\|$}^{3}\right),$
where the last inequality is from Fact 20 and Eqs. (33)(34). ∎
### 4.5 Bounding terms $(6,7)$ in Theorem 27 for Bentkus function
Let $G:{\mathbb{R}}^{k}\rightarrow{\mathbb{R}}$ be the Bentkus function given
in Definition 16. Recall that $G(x)=\prod_{i}g(x_{i})$, where
$g\left(x\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{-x}e^{-t^{2}/2}dt$.
Recall the notation
$g^{\prime}\left(x\right)=\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}$ and
$\overline{g}\left(x\right)=g^{\prime}(x)/g(x)$. The terms are restated here
for convenience.
###### Lemma 34 (Bounding terms $(6,7)$ in Theorem 27).
$\left|\sum_{i_{1}\neq i_{2}\neq
i_{3}}\frac{\frac{\overline{g}\left(x_{i_{1}}\right)-\overline{g}\left(x_{i_{3}}\right)}{x_{i_{3}}-x_{i_{1}}}-\frac{\overline{g}\left(x_{i_{1}}\right)-\overline{g}\left(x_{i_{2}}\right)}{x_{i_{2}}-x_{i_{1}}}}{x_{i_{3}}-x_{i_{2}}}G\left(x\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\leq
O\left(\Delta\log^{2}k\mbox{$\|{H}\|$}^{3}\right)$ (37)
This is the most involved part. Note that the left hand side is unchanged if
we zero out all diagonal entries of $H$. And further note that
$\mbox{$\|{H-\mbox{\rm diag}\left(H\right)}\|$}\leq 2\mbox{$\|{H}\|$}$ where
$\mbox{\rm diag}\left(H\right)$ is a diagonal matrix obtained by diagonalizing
$H$. Thus, we may assume that the diagonal elements in $H$ are zeros without
loss of generality. We break down the analysis into two cases (the first one
being the simpler case).
#### 4.5.1 Case 1: Many negative $x_{i}$s.
The simpler case is when the number of negative $x_{i}$s is “large”.
###### Lemma 35.
If $\left|\left\\{i:x_{i}<0\right\\}\right|>3\log k$, then the quantity in Eq.
(37) is upper bounded by $O\left(\Delta^{2}\frac{\sqrt{\log
k}}{k}\cdot\mbox{$\|{H}\|$}^{3}\right)$.
###### Proof.
Applying Fact 3 a mean value theorem of divided difference and Lemma 19, the
term in Eq. (37) is upper bounded by
$\displaystyle O\left(\Delta^{2}\sum_{i_{1}\neq i_{2}\neq
i_{3}}\overline{g}(\zeta_{i_{1},i_{2},i_{3}})\right)G\left(x\right)\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|$
$\displaystyle\leq$ $\displaystyle O\left(\Delta^{2}\sum_{i_{1}\neq i_{2}\neq
i_{3}}\max\left\\{\overline{g}\left(x_{i_{1}}\right),\overline{g}\left(x_{i_{2}}\right),\overline{g}\left(x_{i_{3}}\right)\right\\}\right)G\left(x\right)\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|$
$\displaystyle\leq$ $\displaystyle
O\left(\Delta^{2}\mbox{$\|{G^{(1)}}\|$}_{1}\max_{i_{1}}\sum_{i_{2},i_{3}}\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
$\displaystyle\leq$ $\displaystyle O\left(\Delta^{2}\cdot
k\cdot\mbox{$\|{G^{(1)}\left(x\right)}\|$}_{1}\cdot\max_{i_{1},i_{2}}\sum_{i_{3}}\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
$\displaystyle\leq$ $\displaystyle O\left(\Delta^{2}\cdot
k\cdot\mbox{$\|{G^{(1)}\left(x\right)}\|$}_{1}\cdot\mbox{$\|{H}\|$}^{3}\right)$
$\displaystyle\leq$ $\displaystyle O\left(\frac{\Delta^{2}\sqrt{\log
k}}{k}\cdot\mbox{$\|{H}\|$}^{3}\right)$
where the first inequality is from the positivity and monotonicity of
$\overline{g}\left(\cdot\right)$ due to Fact 17 to conclude that
$|\overline{g}(\zeta_{i_{1},i_{2},i_{3}})|\leq\max\\{|\overline{g}(x_{i_{1}})|,|\overline{g}(x_{i_{2}})|,|\overline{g}(x_{i_{3}})|\\}$;
the second last inequality is from the following fact
$\sum_{i_{3}}\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\leq\mbox{$\|{H}\|$}\sqrt{\left(\sum_{i_{3}}H_{i_{1},i_{3}}^{2}\right)\left(\sum_{i_{3}}H_{i_{2},i_{3}}^{2}\right)}\leq\mbox{$\|{H}\|$}^{3};$
(38)
the last inequality is from Lemma 21 (which uses that the number of negative
$x_{i}$s is $\leq 3\log k$). ∎
#### 4.5.2 Case 2: A few negative $x_{i}$s
We now assume that $\left|\left\\{i:x_{i}<0\right\\}\right|\leq 3\log k$ and
this case the most complicated and upper bounding it is the most technical. We
push this proof to Appendix A.
###### Proof of Theorem 28.
Combining Theorem 27 and Lemmas 30, 32, 33, 34, we obtain our result. ∎
## 5 Properties of positive spectrahedra
### 5.1 Average sensitivity and Noise sensitivity
In this section we prove certain combinatorial properties of positive
spectrahedra. Understanding the average sensitivity and noise sensitivity is a
fundamental question in Boolean analysis and learning theory. Proving bounds
on these quantities for geometric objects has also received a lot of
attention. In this direction, [HKM13, Kan14a] proved upper bounds on average
sensitivity of halfspaces and the well-known Peres’ theorem [Per04] bounds the
noise-sensitivity of halfspaces. In this section, we show analogous bounds to
these papers also hold for positive spectrahedra.
###### Theorem 36 (Matrix version of Peres theorem).
Let $S$ be a positive spectrahedron defined as
$S=\big{\\{}x\in\mathbb{R}^{n}:\sum_{i}x_{i}A^{i}\preceq B,\hskip
5.69054ptA^{1},\ldots,A^{n},B\in\mathsf{Sym}_{k}\text{ and }A^{i}\text{ is
}\mathsf{PSD}\text{ for }i\in[n]\big{\\}}.$
Let $f(x)=[x\in S]$ for $x\in\\{-1,1\\}^{n}$. Then the $\varepsilon$-noise
sensitivity of $f$ is $\mathsf{NS}_{\varepsilon}(f)=O(\sqrt{\varepsilon})$.
###### Theorem 37.
Let $S^{1},S^{2}$ be $2$ distinct positive spectrahedra specified by
$\\{A^{1}_{j},\ldots,A^{n}_{j},B_{j}\\}_{j\in[2]}$ respectively, where
$A^{i}_{1}\succeq 0$ and $A^{i}_{2}\preceq 0$ for all $i$. Let
$F(x)=\bigwedge_{j=1}^{2}\left[\sum_{i}x_{i}A^{i}_{j}\preceq B_{j}\right]$
be an intersection of positive spectrahedra. Then $\mathsf{AS}(F)\leq
O(\sqrt{n}).$
The proof of Theorem 36 follows closely the proof of Kane [Kan14a] who showed
that $k$-facet polytopes have $\varepsilon$-noise sensitivity at most
$O(\sqrt{\varepsilon\log k})$. Before stating the Kane’s result, we need to
introduce the following notion.
###### Definition 38 (Unate function).
A function $f:\left\\{-1,1\right\\}^{n}\rightarrow\\{0,1\\}$ is _unate_ if it
satisfies the following: for every $i\in[n]$, $f$ is either increasing or
decreasing with respect to the $i$th coordinate, i.e., for every $i\in[n]$,
either $f(x_{1},\ldots,x_{i-1},-1,x_{i+1},\ldots,x_{n})\leq
f(x_{1},\ldots,x_{i-1},1,x_{i+1},\ldots,x_{n})$ for all $x$ or
$f(x_{1},\ldots,x_{i-1},-1,x_{i+1},\ldots,x_{n})\geq
f(x_{1},\ldots,x_{i-1},1,x_{i+1},\ldots,x_{n})$ for all $x$.
In particular, Kane proved the following stronger statement.
###### Theorem 39.
[Kan14a] Let $f_{1},\ldots,f_{k}:\\{-1,1\\}^{n}\rightarrow\\{0,1\\}$ be unate
functions and let $F:\\{-1,1\\}^{n}\rightarrow\\{0,1\\}$ be defined as
$F(x)=\bigwedge_{i}f_{i}(x)$. Then the average sensitivity of $F$ satisfies
$\mathsf{AS}(F)\leq O(\sqrt{n\log(k+1)})$.131313There is a $+1$ compared to
Kane’s result to ensure that the result is valid for $k=1$.
It is not hard to see that a positive spectrahedron is a unate function so
Theorem 39 holds for us as well for $k=1$. Hence we have the following
corollary.
###### Corollary 40.
Let $S$ be as defined in Theorem 36. Let
$F:\\{-1,1\\}^{n}\rightarrow\\{0,1\\}$ be defined as $F(x)=1$ if and only if
$x\in S$. Then $\mathsf{AS}(F)\leq O(\sqrt{n})$.
Recall that we are interested in noise sensitivity of $F$. In the same paper,
Kane [Kan14a] adapts the well-known techniques of [DGJ+10] to show that the
$\varepsilon$-noise sensitivity of _intersections_ of halfspaces is at most
$O(\sqrt{\varepsilon\log k})$ and remarks that such a bound _does not_ hold
for the intersections of unate functions. Below, we show that one can modify
the proof of [DGJ+10] to also show that the noise sensitivity of positive
spectrahedra can be bounded by the “average 2-sensitivity” of positive
spectrahedra which we show is $O(\sqrt{\varepsilon})$ by modifying Kane’s
proof in Theorem 39. This proves Theorem 36.
###### Proof of Theorem 36.
In order to prove the theorem, we first show that for a function
$f:\\{-1,1\\}^{n}\rightarrow\\{0,1\\}$ defined as
$f(x)=\left[\sum_{i=1}^{n}x_{i}A^{i}\preceq B\right]$
for $A^{1},\ldots,A^{n},B\in\mathsf{Sym}_{k}$ and $A^{i}$ is $\mathsf{PSD}$
for $i\in[n]$, the $\varepsilon$-noise sensitivity of $f$ satisfies
$\mathsf{NS}_{\varepsilon}(f)=\Pr_{\begin{subarray}{c}(\bm{x},\bm{y})\\\
\varepsilon-\text{correlated}\end{subarray}}[f(\bm{x})\neq f(\bm{y})]\leq
O(\sqrt{\varepsilon}).$
For simplicity let us assume that $\varepsilon=1/m$, for some integer $m$
which divides $n$ (since $\mathsf{NS}_{\varepsilon}$ is a non-decreasing
function in $\varepsilon$, we can even round $\varepsilon$ down to satisfy
this condition).
In order to analyze $\mathsf{NS}_{\varepsilon}(f)$ we first observe that one
can generate an $\varepsilon$-correlated pair of strings
$(x,y)\in\\{-1,1\\}^{n}$ as follows141414We deviate from [DGJ+10] in this
process of generating correlated strings. The reason for this modification is,
we require that within every bucket, the induced spectrahedron has to be
either $\mathsf{PSD}$ or $\mathsf{NSD}$, which isn’t guaranteed in the
original bucketting procedure of [DGJ+10, Kan14a]:
1. 1.
Pick a uniformly random string $\bm{z}\sim\mathcal{U}_{n}$.
2. 2.
Randomly partition $[n]$ into $m$ disjoint buckets
$C_{1},\ldots,C_{m}\subseteq[n]$ such that $\cup_{i}C_{i}=[n]$. Furthermore,
for $z\in\\{-1,1\\}^{n}$ (picked in step $1$), split each bucket as follows:
for every $\ell\in[m]$, split $C_{\ell}$ into $C_{\ell,1}$ and $C_{\ell,-1}$
such that $C_{\ell,1}$ corresponds to the _positive_ coordinates in
$z_{C_{\ell}}$ and $C_{\ell,-1}$ corresponds to the _negative_ coordinates in
$z_{C_{\ell}}$. So overall there are $2m$ disjoint buckets
$\\{C_{\ell,s}:\ell\in[m],s\in\\{-1,1\\}\\}$ such that
$\cup_{\ell,s}C_{\ell,s}=[n]$. Set $\tilde{C}_{\ell}=C_{\ell,1}$ if $\ell\leq
m$ and $\tilde{C}_{\ell}=C_{\ell-m,-1}$ if $\ell>m$.
3. 3.
Corresponding to each bucket $\tilde{C}_{\ell}$, pick a uniformly random bit
$\mathbf{b}_{\ell}\sim\mathcal{U}_{1}$.
4. 4.
We obtain $\bm{x}$ as follows: for every $\ell\in[2m]$, obtain $\bm{x}$ from
$\bm{z}$ by multiplying all the bits in $\bm{z}_{\tilde{C}_{\ell}}$ by
$\mathbf{b}_{\ell}$.
5. 5.
We obtain $\bm{y}$ as follows: pick a uniformly random $\ell\in[m]$ and flip
the signs of $\bm{x}_{i}$ (obtained in step $4$) for all the indices $i$ in
$C_{\ell}$, i.e., $\bm{y}_{i}=-\bm{x}_{i}$ if $i\in C_{\ell}$ and
$\bm{y}_{i}=\bm{x}_{i}$ otherwise.
Observe that the the $(\bm{x},\bm{y})$ obtained in step $(4,5)$ are uniform
and $\varepsilon$-correlated. To see this, first observe that the probability
of obtaining $x\in\\{-1,1\\}^{n}$ is given by
$\displaystyle\Pr_{\begin{subarray}{c}\bm{z}\sim\mathcal{U}_{n},\\\
\\{C_{k}\\},\bm{b}\sim\mathcal{U}_{2m}\end{subarray}}\left[\bm{x}=x\right]$
$\displaystyle=\Pr_{\bm{z},C,\bm{b}}\left[\bm{z}_{\tilde{C}_{1}}\cdot\bm{b}_{1}=x_{\tilde{C}_{1}},\ldots\bm{z}_{\tilde{C}_{2m}}\cdot\bm{b}_{2m}=x_{\tilde{C}_{2m}}\right]$
$\displaystyle=\prod_{i=1}^{2m}\Pr_{\bm{z},C,\bm{b}}\left[\bm{z}_{\tilde{C}_{i}}\cdot\bm{b}_{i}=x_{\tilde{C}_{i}}|\bm{z}_{\tilde{C}_{<i}}\cdot\bm{b}_{<i}=x_{\tilde{C}_{<i}}\right]$
$\displaystyle=\prod_{i=1}^{2m}\Pr_{\bm{z},C,\bm{b}}\left[\bm{z}_{\tilde{C}_{i}}\cdot\bm{b}_{i}=x_{\tilde{C}_{i}}\right]=\prod_{i=1}^{2m}\frac{1}{2^{|\tilde{C}_{i}|}}=\frac{1}{2^{n}},$
where the third equality is because $\bm{z},\bm{b}$ are uniformly random and
final equality is because $\cup_{k}\tilde{C}_{k}=~{}[n]$ and
$\left\\{\tilde{C}_{i}\right\\}_{1\leq i\leq 2m}$ are disjoint. In order to
see $(\bm{x},\bm{y})$ are $\varepsilon$-correlated, observe that for a fixed
$i\in[n]$ the probability $\bm{x}_{i}$ differs from $\bm{y}_{i}$ is exactly
the probability $i$ lies in the bucket $C_{\ell}=C_{\ell,1}\cup C_{\ell,-1}$
picked in Step (5) above. The probability of picking a bucket $C_{\ell}$ is
exactly $1/m=\varepsilon$. This event happens independently over all the
coordinates $i\in[m]$, hence $y$ is $\varepsilon$-correlated with $x$.
Now that we have shown $(\bm{x},\bm{y})$ are $\varepsilon$-correlated, we next
observe that for a fixed $z$ and buckets
$\tilde{C}_{1},\ldots,\tilde{C}_{2m}$, we can write
$f:\\{-1,1\\}^{n}\rightarrow\\{0,1\\}$ as a function
$g:\\{-1,1\\}^{2m}\rightarrow\\{0,1\\}$ defined as
$g(b)=\left[\sum_{q=1}^{2m}b_{q}\sum_{j\in\tilde{C}_{q}}z_{j}A^{j}\preceq
B\right].$ (39)
Similarly, one can define $f(y)$ as $g(b^{\prime})$ where $b^{\prime}$ is
obtained from $b$ by picking a uniformly random $\boldsymbol{\ell}\sim[m]$ and
flipping $b_{\boldsymbol{\ell}},b_{\boldsymbol{\ell}+m}$ where
$C_{\boldsymbol{\ell}}$ is the bucket chosen in Step $(5)$. Furthermore,
observe that
$\mathsf{NS}_{\varepsilon}(f)=\Pr_{\begin{subarray}{c}(\bm{x},\bm{y})\\\
\varepsilon-\text{correlated}\end{subarray}}[f(\bm{x})\neq
f(\bm{y})]=\Pr_{\begin{subarray}{c}\bm{b}\sim\mathcal{U}_{2m},\\\
\boldsymbol{\ell}\sim[m]\end{subarray}}[g(\bm{b})\neq
g(\bm{b}^{\boldsymbol{\ell},\boldsymbol{\ell}+m})],$
where $\bm{b}^{i,j}$ is obtained by flipping the $i,j$th coordinates in
$\bm{b}$ and $\boldsymbol{\ell}$ are chosen uniformly random in $[m]$. We can
further upper bound the quantity above by
$\displaystyle\begin{aligned}
\mathsf{NS}_{\varepsilon}(f)&=\Pr_{\begin{subarray}{c}\bm{b}\sim\mathcal{U}_{2m},\\\
\boldsymbol{\ell}\sim[m]\end{subarray}}[g(\bm{b})\neq
g(\bm{b}^{\boldsymbol{\ell},\boldsymbol{\ell}+m})]\\\
&\leq\Pr_{\begin{subarray}{c}\bm{b}\sim\mathcal{U}_{2m},\\\
\boldsymbol{\ell}\sim[m]\end{subarray}}[g(\bm{b})\neq
g(\bm{b}^{\boldsymbol{\ell}})]+\Pr_{\begin{subarray}{c}\bm{b}\sim\mathcal{U}_{2m},\\\
\boldsymbol{\ell}\sim[m]\end{subarray}}[g(\bm{b}^{\boldsymbol{\ell}})\neq
g(\bm{b}^{\boldsymbol{\ell},\boldsymbol{\ell}+m})]\\\
&=\Pr_{\begin{subarray}{c}\bm{b}\sim\mathcal{U}_{2m},\\\
\boldsymbol{\ell}\sim[m]\end{subarray}}[g(\bm{b})\neq
g(\bm{b}^{\boldsymbol{\ell}})]+\Pr_{\begin{subarray}{c}\bm{b}\sim\mathcal{U}_{2m},\\\
\boldsymbol{\ell}\sim[m]\end{subarray}}[g(\bm{b})\neq
g(\bm{b}^{\boldsymbol{\ell}+m})]=\frac{1}{m}\mathsf{AS}(g),\end{aligned}$ (40)
where the second equality used the fact that $\bm{b},\boldsymbol{\ell}$ are
uniform over their respective domains and the last equality used the
definition of $\mathsf{AS}(g)$ to obtain
$\mathsf{AS}(g)=\sum_{\ell=1}^{2m}\Pr[g(\bm{x})\neq
g(\bm{x}^{\ell})]=\sum_{\ell=1}^{m}\Pr[g(\bm{x})\neq
g(\bm{x}^{\ell})]+\sum_{\ell=m+1}^{2m}\Pr[g(\bm{x})\neq g(\bm{x}^{\ell})].$
We now finally upper bound the average sensitivity of $g$. Observe that
$\sum_{j\in\tilde{C}_{q}}\bm{z}_{j}A^{j}$ is either $\mathsf{PSD}$ or
$\mathsf{NSD}$ (since all the $\bm{z}_{j}$ in the bucket $\tilde{C}_{q}$ have
the same sign and $A^{j}$s are all $\mathsf{PSD}$ by definition). From Eq.
(39), observe that $g$ is a unate function. Hence, we have
$\displaystyle\mathsf{NS}_{\varepsilon}(f)\leq\frac{1}{m}\mathsf{AS}(g)\leq
O\left(\sqrt{\frac{1}{m}}\right)=O(\sqrt{\varepsilon}),$ (41)
where the first inequality is by Eq. (40), second inequality uses Theorem 39
and the last equality used the definition of $m=1/\varepsilon$. This concludes
the proof of the theorem. ∎
We now prove Theorem 37 which bounds the average sensitivity of intersections
of positive spectrahedra.
###### Proof of Theorem 37.
The proof is very similar to the proof of the theorem above. Let $m=\lceil
1/\varepsilon\rceil$. We follow the same bucketing steps $(1)-(5)$ in Theorem
36 to obtain a $g:\\{-1,1\\}^{2m}\rightarrow\\{0,1\\}$ given by
$g(b)=\left[\sum_{q=1}^{2m}b_{q}\sum_{j\in C_{q}}z_{1}A^{j}_{1}\preceq
B_{1}\right]\cdot\left[\sum_{q=1}^{2m}b_{q}\sum_{j\in
C_{q}}z_{1}A^{j}_{2}\preceq B_{2}\right].$
Observe that $g$ is an intersection of positive spectrahedra and by definition
each positive spectrahedron is a unate function. So, by Theorem 39, we have
$\mathsf{AS}(g)\leq\sqrt{1/\varepsilon}=O(\sqrt{m}).$
This concludes the proof of the corollary. ∎
### 5.2 Boolean Anti-concentration: Littlewood Offord for spectrahedra
We now prove the main lemma which shows that the largest eigenvalues of
positive spectrahedra cannot be very concentrated. In particular, we show that
for a uniformly random $\bm{x}\sim\mathcal{U}_{n}$, the probability that the
random matrix $D=\sum_{i}\bm{x}_{i}A^{i}-B$ has the largest eigenvalue in a
small interval is fairly small. This anti-concentration statement will be
crucial in our invariance principle proof when we move from the Bentkus
mollifier to our CDF function. In the passing we remark that, prior to this
work, we aren’t even aware if the weaker _Gaussian_ analogue of this statement
was known (in particular, the results of [HKM13, ST17] only require Gaussian
anti-concentration for which they use a result of Nazarov [Naz03] as a black-
box).
In order to prove our main theorem (stated below), we follow the result of
[OST19, Kan14a] closely since they are able to handle intersections of unate
functions which is the case for positive spectrahedra. However, there are two
subtleties.
1. ($i$)
In [OST19] they bucket the set of halfspaces (which form the polytope) and
show that each bucket has significant weight. Crucially for them, they use the
fact that intersections of halfspaces are still unate functions. But this is
not the case for positive spectrahedra. For this, we need to modify the
bucketing procedure (akin to what happens in the proof of Theorem 36) so that
this bucketing of positive spectrahedra still results in a unate function.
2. ($ii$)
In [OST19] they prove an analogue of Lemma 45 which shows that each bucket has
“significant weight”. However our proof deviates significantly from the proof
in [OST19]. For them, proving the statement in the lemma (for diagonal
matrices), follows directly from Paley-Zygmund inequality, but as far as we
are aware, we do not have a matrix-version of this inequality. Due to this
difficulty, we modify their proof and use the matrix Chernoff bound to prove
the statement above.
###### Theorem 41.
Let $k\geq 0$ be an integer and $\tau\leq\frac{1}{100\sqrt{\log k}}$. Let
$\\{B_{1},B_{2}\\}\subseteq\mathsf{Sym}_{k}$, $\\{A^{i}_{1}\\}_{i\in[n]}$ and
$\\{A^{i}_{2}\\}_{i\in[n]}$ be sequences of $\mathsf{PSD}$ and $\mathsf{NSD}$
matrices, respectively. They satisfy that for all $i\in[n],j\in[2]$,
$A^{i}_{1}\preceq\tau\cdot\mathbb{I},A^{i}_{2}\succeq-\tau\mathbb{I}$ and
$\sum_{i=1}^{n}(A^{i}_{j})^{2}\succeq\mathbb{I}$. Then for every $\Lambda\geq
20\tau\log k$, we have
$\Pr_{\bm{x}\sim\mathcal{U}_{n}}\left[\exists j\in[2]\text{ s.t.
}\lambda_{\max}\left(\sum_{i}\bm{x}_{i}A^{i}_{j}-B_{j}\right)\in(-\Lambda,\Lambda]\right]\leq
O(\Lambda).$
Using the standard bits-to-Gaussians trick [O’D14, Chapter 11], we have the
following corollary.
###### Corollary 42.
Let $k\geq 0$ be an integer and $\tau\leq\frac{1}{\log k}$. Let
$\\{B_{1},B_{2}\\}\subseteq\mathsf{Sym}_{k}$, $\\{A^{i}_{1}\\}_{i\in[n]}$ and
$\\{A^{i}_{2}\\}_{i\in[n]}$ be sequences of $\mathsf{PSD}$ and $\mathsf{NSD}$
matrices, respectively. They satisfy that for all $i\in[n],j\in[2]$,
$A^{i}_{1}\preceq\tau\cdot\mathbb{I},A^{i}_{2}\succeq-\tau\mathbb{I}$ and
$\sum_{i}(A^{i}_{j})^{2}\succeq\mathbb{I}$ . Then for every $\Lambda\geq
20\tau\log k$, we have
$\Pr_{\bm{g}\sim\mathcal{G}^{n}}\left[\exists j\in[2]\text{ s.t.
}\lambda_{\max}\left(\sum_{i}\bm{g}_{i}A^{i}_{j}-B_{j}\right)\in(-\Lambda,\Lambda]\right]\leq
O(\Lambda).$
In order to prove this theorem we will use the following two lemmas by
[OST19]. Before stating these lemmas, we introduce a few definitions from
[OST19] (adapted to our setting of positive spectrahedra). For the rest of the
section, we let $F:\\{-1,1\\}^{n}\rightarrow\\{0,1\\}$ be the indicator of an
intersection of positive spectrahedra, i.e., for every $j\in[2]$, let
$F_{j}(x)=\left[\sum_{i=1}^{n}x_{i}A_{j}^{i}\preceq B_{j}\right]$, where
$\left\\{A^{i}_{j}\right\\}_{j\in\left\\{1,2\right\\}}$ are sequences of
$\mathsf{PSD}$($\mathsf{NSD}$) matrices and
$\displaystyle
F(x)=\bigwedge_{j=1}^{2}F_{j}(x)=\bigwedge_{j=1}^{2}\left[\sum_{i=1}^{n}x_{i}A_{j}^{i}\preceq
B_{j}\right].$ (42)
1. 1.
For a set $S\subseteq\\{-1,1\\}^{n}$, let $\mathcal{E}(S)$ be the fraction of
$n\cdot 2^{n-1}$ edges which have one endpoint in $S$ and one endpoint in
$S^{c}$ (i.e., complement of $S$).
2. 2.
We let $H_{j}\subseteq\\{-1,1\\}^{n}$ be the _indicator-set_ for $F_{j}$,
i.e., $x\in H_{j}$ if and only if $F_{j}(x)=1$. Additionally, suppose we have
sets $\\{\bar{H}_{1},\bar{H}_{2}\\}$ such that $H_{j}\subseteq\bar{H}_{j}$
such that $\bar{H}_{j}$ are also the indicator-sets of unate functions. Let
$\partial H_{j}=\bar{H}_{j}\backslash H_{j}$.
3. 3.
For $\alpha\in[0,1]$, we say $\partial H_{j}$ is _$\alpha$ -semi thin_ if for
every $x\in H_{j}$, at least an $\alpha$-fraction of its hypercube-neighbours
(i.e., set of $y\in\\{-1,1\\}^{n}$ for which $d(x,y)=1$) are outside $\partial
H_{j}$.
4. 4.
We now define a few sets: let
$F=\bar{H}_{1}\cap\bar{H}_{2},\qquad
F^{\circ}={H}_{1}\cap{H}_{2},\qquad\partial F=F\backslash F^{\circ}$
With this terminology, we have the following lemma that bounds the number of
edges that cross $F$.
###### Lemma 43 ([OST19, Theorem 7.18]).
For $j\in[2]$, let $H_{j}$ be as defined above. Suppose $H_{j}$ is
$\alpha$-semi thin, then
$\mathsf{vol}(\partial F)\leq O\left(\frac{1}{\alpha\sqrt{n}}\right)$
Using this lemma, we get the following theorem (which is the analogue of
[OST19, Theorem 7.19]).
###### Theorem 44.
Let $\lambda>0,\alpha\in[0,1],\\{B_{1},B_{2}\\}\subseteq\mathsf{Sym}_{k}$. Let
$\\{A^{i}_{j}\\}_{i\in[n],j\in[2]}\subseteq\mathsf{Sym}_{k}$ satisfy that
$A^{i}_{1}\succeq 0,A^{i}_{2}\preceq 0$ for all $i\in[n]$. At least
$\alpha$-fraction of $i\in[n]$ satisfy that
$A^{i}_{1}\succeq\lambda\cdot\mathbb{I}$ and
$A^{i}_{2}\preceq-\lambda\cdot\mathbb{I}$. Then, we have
$\Pr_{\bm{x}\sim\mathcal{U}_{n}}\left[\exists j\in[2]\text{ s.t.
}\lambda_{\max}\left(\sum_{i}\bm{x}_{i}A^{i}_{j}-B_{j}\right)\in(-2\lambda,0]\right]\leq
O\left(\frac{1}{\alpha\sqrt{n}}\right).$
###### Proof.
Let $\\{A^{i}_{j}\\},\\{B_{j}\\}$ be as in the theorem statement. Let
$H_{j}=\Big{\\{}x\in\\{-1,1\\}^{n}:\lambda_{\max}\left(\sum_{i}x_{i}A^{i}_{j}-B_{j}\right)\leq-2\lambda\Big{\\}},\quad\bar{H}_{j}=\Big{\\{}x\in\\{-1,1\\}^{n}:\lambda_{\max}\left(\sum_{i}x_{i}A^{i}_{j}-B_{j}\right)\leq
0\Big{\\}}.$
Clearly we then have that
$\partial
H_{j}=\left\\{x\in\\{-1,1\\}^{n}:\lambda_{\max}\left(\sum_{i}x_{i}A^{i}_{j}-B_{j}\right)\in(-2\lambda,0]\right\\}$
and
$\partial F=\left\\{x\in\\{-1,1\\}^{n}:\exists j\in[2]\text{ s.t.
}\lambda_{\max}\left(\sum_{i}x_{i}A^{i}_{j}-B_{j}\right)\in(-2\lambda,0]\right\\}.$
Since we assumed that at least an $\alpha$-fraction of $i$s satisfied
$A^{i}_{1}\succeq\lambda\cdot\mathbb{I}$ and
$A^{i}_{2}\preceq-\lambda\cdot\mathbb{I}$, it follows that $H_{j}$ is
$\alpha$-semi thin, hence we can apply Lemma 43 to obtain the theorem
statement. ∎
Using this theorem, we are now ready to prove our main technical lemma which
says that we can always “randomly bucket” our positive spectrahedron so that
many of these buckets have “pretty large” smallest eigenvalue.
###### Lemma 45.
Let $\\{A^{i}\\}_{i\in[n]}\subseteq\mathsf{Sym}_{k}$ be a sequence of positive
semidefinite matrices which is $\left(\tau,M\right)$-regular with
$\tau\leq\frac{1}{100\sqrt{\log k}}$. Let $m\geq\frac{1}{10\tau^{2}\log k}$
and $\pi:[n]\rightarrow[m]$ be a random hash function that independently
assigns each $i\in[n]$ to a uniformly random bucket in $[m]$. For $c\in[m]$,
let
$\sigma_{c}=\sum_{j\in\pi^{-1}(c)}A^{j}$
and we say the bucket $c\in C$ is _good_ if $\sigma_{c}\succeq\frac{1}{2\tau
m}\cdot\mathbb{I}$. Then,
$\Pr\left[\text{at most }3m/4\text{ buckets }c\in[m]\text{ are good
}\right]\leq\exp\left(-m/4\right).$
###### Proof.
Let $\bm{z}_{i}\in\\{0,1\\}$ be a random variable satisfying
$\Pr[\bm{z}_{i}=1]=1/m$. Let $Z_{i}=\bm{z}_{i}\cdot A^{i}$, hence one can
write $\sigma_{c}=\sum_{i}Z_{i}$. In particular, this implies
$\mathbb{E}\left[\sigma_{c}\right]=\frac{1}{m}\sum_{i}A^{i}\succeq\frac{1}{\tau\cdot
m}\sum_{i}\left(A^{i}\right)^{2}\succeq\frac{1}{\tau m},$
where we used $A^{i}\preceq\tau\cdot\mathbb{I}$. Applying Fact 5 (for
$\delta=1/2$, $\mu=1/\tau m$, $R=\tau$) we have
$\mathrm{Pr}\>\\!\\!\left[\sum_{i}Z_{i}\succeq\frac{1}{2\tau
m}\mathbb{I}\right]\geq
1-k\cdot\left(\frac{2}{e}\right)^{\frac{1}{2\tau^{2}m}}\geq\frac{9}{10}$
For $j\in[n]$ and $c\in[m]$ define random variables
$Y_{c,j}=\begin{cases}1~{}\mbox{if $\pi(j)=c$}\\\
0~{}\mbox{otherwise},\end{cases}\quad\text{ and }\qquad
X_{j}=\left[\sum_{c=1}^{m}Y_{c,j}\sigma_{c}\succeq\frac{1}{2\tau
m}\mathbb{I}\right].$
Using the Claim 46 below, $X_{1},\ldots,X_{n}$ are negatively associated. Thus
we may apply the Chernoff bound to $\sum_{i=1}^{m}X_{i}$ which has mean at
least $3m/4$, which gives us the lemma statement.
###### Claim 46.
The random variables $X_{1},\ldots,X_{n}$ are negatively associated.
###### Proof.
From [DP09, Page 35, Example 3.1], the set of random variables
$\left\\{Y_{c,j}\right\\}_{1\leq c\leq m}$ are negatively associated for
$j\in[n]$. Note that $\left\\{Y_{1,j},\ldots,Y_{m,j}\right\\}_{j\in[n]}$ are
$n$ independent families of random variables. By [DP09, Page 35],
$\left\\{Y_{c,j}\right\\}_{c\in[m],j\in[n]}$ are negatively associated. Given
$\sigma_{1},\ldots,\sigma_{m}$,
$\left[\sum_{c=1}^{m}Y_{c,j}\sigma_{c}\succeq\frac{1}{2\tau
m}\mathbb{I}\right]$ is a monotone non-decreasing function of
$Y_{c,1},\ldots,Y_{c,n}$. Thus from [DP09, Page 35], $X_{1},\ldots,X_{m}$ are
negatively associated. ∎
The proof of this claim concludes the proof of the lemma. ∎
We are now ready to proof our main theorem.
###### Proof of Theorem 41.
For $j\in[2]$, let $f_{j}(x)=\sum_{i=1}^{n}x_{i}A^{i}_{j}$. Let
$\pi:[n]\rightarrow[2m]$ be a random hash function that independently assigns
each $i\in[n]$ to uniformly random bucket in $[2m]$. Let
$C_{1},\ldots,C_{2m}\subseteq[n]$ be the buckets and $z\in\\{-1,1\\}^{2m}$ be
uniformly random. Consider the function
$g_{j}:\\{-1,1\\}^{2k}\rightarrow\mathsf{Sym}_{k}$ defined as
$g_{j}(z)=\sum_{q=1}^{2m}z_{q}\cdot\sum_{i\in C_{q}}A^{i}_{j}.$
For $q\in[2m]$, define $\bar{A}^{q}_{j}=\sum_{i\in C_{q}}A^{i}_{j}$, so
$g_{j}(z)=\sum_{q}z_{q}\bar{A}^{q}_{j}$. Observe that distribution of $f_{j}$
and $g_{j}$ are the same, i.e., for every $D\in\mathsf{Sym}_{k}$ we have
$\displaystyle\Pr_{\bm{z}\sim\mathcal{U}_{2m},\\{C_{i}\\}}[g_{j}(\bm{z})=D]=\Pr_{\bm{x}\sim\mathcal{U}_{n}}[f_{j}(\bm{x})=D].$
(43)
In order to see this we argue that the $n$-bit string $w\in\\{-1,1\\}^{n}$
defined as $w_{i}=z_{q}$ iff $i\in C_{q}$, is uniformly random. To show this,
we first prove the following: for $z\in\\{-1,1\\}^{2m}$, let
$S=\\{q\in[2m]:z_{q}=1\\}$ and $T=\cup_{q\in S}C_{q}$. Then, observe that for
every $T\subseteq[n]$, we have $\Pr_{\bm{z},\\{C_{q}\\}}[\textbf{T}=T]=2^{-n}$
(for every $i\in[n]$, the probability of $i\in C_{q}$ is $1/(2m)$ and the
probability $C_{q}$ is included in $T$ is $1/2$ since $z_{q}$ is a uniformly
random bit, hence for every $i\in[n]$, we have $\Pr_{\bm{z},\\{C_{q}\\}}[i\in
T]=\sum_{i=1}^{2m}(1/2m)\cdot(1/2)=1/2$ and this is independent for every
$i\in[n]$ by construction). It is now easy to see that $w$ is uniformly random
because
$\Pr_{\bm{z},\\{C_{j}\\}}[W=w]=\sum_{T}\Pr[\textbf{T}=T]\cdot\Pr[W=w|\textbf{T}=T]=\frac{1}{2^{n}}\sum_{T}\Pr[W=w|\textbf{T}=T]=2^{-n},$
where the last equality used the fact that once we fix $T$, then all the bits
of $w$ which are $1$ are fixed.
For $m=\frac{1}{20\tau^{2}\log k}$, let $\pi:[n]\rightarrow[2m]$ be a random
hash that buckets these $n$ variables (jointly for $j\in[2]$). By Lemma 45, we
argued that, with probability at least $1-e^{-m/2}$, at least $9m/5$ of the
$2m$ buckets are good for $j=1$, i.e., a good bucket $q\in[2m]$ for $j=1$
satisfies $\sum_{i\in\pi^{-1}(q)}A^{i}_{1}\succeq\frac{1}{4\tau
m}\cdot\mathbb{I}$. For the same reason, with probability at least
$1-e^{-m/2}$, at least $9m/5$ of the $2m$ buckets are good for $j=2$, i.e., a
good bucket $q\in[2m]$ for $j=2$ satisfies
$\sum_{i\in\pi^{-1}(q)}A^{i}_{2}\preceq-\frac{1}{4\tau m}\cdot\mathbb{I}$.
Applying a union bound, at least $8m/5$ of $2m$ buckets are good for every
$j\in[2]$ with probability at least $1-2\cdot e^{-m/2}$.
By the argument in the start of the proof, we know that after bucketing, we
can convert each $f_{j}$ into a function
$g_{j}:\\{-1,1\\}^{2m}\rightarrow\mathsf{Sym}_{k}$ such that $f_{j}$ and
$g_{j}$ have the same distribution. Now we can invoke Theorem 44 as follows:
we know that a $4/5$-fraction of $q\in[2m]$ satisfy
$\bar{A}^{q}_{1}\succeq\frac{1}{4\tau m}\cdot\mathbb{I}$ and
$\bar{A}^{q}_{2}\preceq-\frac{1}{4\tau m}\cdot\mathbb{I}$, so we have
$\displaystyle\Pr_{\bm{z}\sim\mathcal{U}_{m}}\left[\exists
j\in[2]~{}\text{s.t.}~{}\lambda_{\max}\left(\sum_{q=1}^{m}\bm{z}_{q}\bar{A}^{q}_{j}-B_{j}\right)\in(-1/2\tau
m,0]\right]\leq O\left(\sqrt{\frac{1}{m}}\right)+2e^{-m/2}.$
We now prove the main theorem statement. In order to do so, first observe
that, we can partition the bound on the LHS into $\lceil 2\Lambda\tau m\rceil$
intervals as $\Lambda\geq 1/2\tau m$ from our choice of parameters.151515To be
precise, for a vector $v\in\mathbb{R}^{k}$, observe that the event
$\left[\forall i\in[k]:v_{i}\leq b_{i}+\Lambda,\text{ and }\exists
j\in[k]:v_{j}\geq b_{j}-\Lambda\right]$ can be broken down into the
intersections of $\Lambda/2\tau m$ events given by
$\bigwedge_{\ell=0}^{2\Lambda\tau m-1}[\forall i\in[k]:v_{i}\leq
b_{i}+\Lambda-\ell/2\tau m,\text{ and }\exists
j\in[\ell]:v_{j}>b_{j}-\Lambda-(\ell+1)/2\tau m]$. and by a union bound we
have
$\displaystyle\Pr_{\bm{x}\sim\mathcal{U}_{n}}\left[\exists j\in[2]\text{ s.t.
}\lambda_{\max}\left(\sum_{i}\bm{x}_{i}A^{i}_{j}-B_{j}\right)\in(-\Lambda,\Lambda]\right]$
$\displaystyle\leq$ $\displaystyle O\left(\Lambda\cdot\tau\cdot
m\left(\sqrt{\frac{1}{m}}+\exp(-\Omega(m/2))\right)\right)$
From the choice of the parameters, the first term above dominates. And thus
$\Pr_{\bm{x}\sim\mathcal{U}_{n}}\left[\exists j\in[2]\text{ s.t.
}\lambda_{\max}\left(\sum_{i}\bm{x}_{i}A^{i}_{j}-B_{j}\right)\in[-\Lambda,0]\right]\leq
O\left(\Lambda\right).$
Similarly one can also show when the LHS of the equation above is replaced
with $(0,\Lambda]$. Hence we get our theorem statement. ∎
## 6 Invariance principle for positive spectrahedra
In this section, we establish our main invariance principle.
### 6.1 Invariance principle for the spectral Bentkus mollifier
We now prove our main lemma which is an invariance principle for the Bentkus
mollifier. We remark that our analysis is the standard Lindeberg-style
argument for proving invariance principles, but when applied to the spectral
Bentkus mollifier. We first write out the Fréchet series for the Bentkus
mollifier, which we then upper bound using our main Theorem 28. In order to
upper bound the error terms in the Fréchet series, we use the matrix Rosenthal
inequality (in Fact 7) to bound the moments of random matrices (we remark that
this inequality will also be useful in our $\mathsf{PRG}$ construction).
Superficially, our proof techniques resemble the previous invariance principle
proofs used in [HKM13, ST17, OST19], but the quantities we need to bound are
very different from their analysis since we are dealing with matrices.
###### Lemma 47.
Let $k\geq 1,\theta,\tau\in(0,1)$ and
$\Psi_{\theta}:\mathsf{Sym}_{k}\rightarrow{\mathbb{R}}$ be defined as
$\Psi_{\theta}\left(Q\right)=\left(G_{\theta}\circ\lambda\right)\left(Q\right)$
where $G_{\theta}$ is the Bentkus mollifier defined in Eq. (20). Let
$S_{1},S_{2}$ be $(\tau,M)$-regular positive spectrahedra specified by
matrices $\\{A^{1}_{1},\ldots,A^{n}_{1},B_{1}\\}$ and
$\\{A^{1}_{2},\ldots,A^{n}_{2},B_{2}\\}$ respectively. Let $A^{i}=\mbox{\rm
diag}\left(A^{i}_{1},A^{i}_{2}\right)$ and $B=\mbox{\rm
diag}\left(B_{1},B_{2}\right)$ be block diagonal matrices. Then
$\displaystyle\left|\operatorname*{\mathbb{E}}_{\bm{x}\sim\mathcal{U}_{n}}\left[\Psi_{\theta}\left(\sum_{i=1}^{n}\bm{x}_{i}A^{i}-B\right)\right]-\operatorname*{\mathbb{E}}_{\bm{g}\sim\mathcal{G}^{n}}\left[\Psi_{\theta}\left(\sum_{i=1}^{n}\bm{g}_{i}A^{i}-B\right)\right]\right|\leq
O\left(\frac{\log^{7}k}{\theta^{3}}\cdot(M+\|B\|^{2})\cdot(M\cdot\tau)^{1.5}\right).$
This inequality still holds if $\bm{x}$ is $\left(80\log k\right)$-wise
uniform.
###### Proof.
Let $t=\lceil 1/\tau\rceil$. Let $\mathcal{H}=\\{h:[n]\rightarrow[t]\\}$ be a
family of $(80\log k)$-wise uniform hashing functions, i.e., for every subset
$I\subseteq[n]$ of size at most $80\log k$, and $b\in[t]^{I}$, we have
$\Pr_{\bm{h}\in\mathcal{H}}\left[\bm{h}(i)=b_{i}\right]=\frac{1}{t^{|I|}},$
where the probability is taken over a uniformly random function
$h\in\mathcal{H}$. Fix an $h\in\mathcal{H}$ (think of $h$ as a partition of
$[n]$ into $t$ blocks $S_{1},\ldots,S_{t}\subseteq[n]$, where
$S_{i}=h^{-1}(i)$ for all $i\in[t]$). For $\bm{x}\sim\mathcal{U}_{n}$ and
$\bm{y}\sim\mathcal{G}^{n}$ let us divide $\bm{x},\bm{y}$ into blocks
$\bm{x}^{1},\ldots,\bm{x}^{t}$ and $\bm{y}^{1},\ldots,\bm{y}^{t}$ according to
$h$. It is not hard to see that $\bm{x}^{i}\sim\mathcal{U}^{|h^{-1}(i)|}$ and
$\bm{y}^{i}\sim\mathcal{G}^{|h^{-1}(i)|}$. We now upper bound the quantity
$\displaystyle\left|\mathop{\mathbb{E}}_{\bm{x}\sim\mathcal{U}_{n}}\left[\Psi_{\theta}\left(\sum_{i=1}^{n}\bm{x}_{i}A^{i}-B\right)\right]-\mathop{\mathbb{E}}_{\bm{y}\in\mathcal{G}^{n}}\left[\Psi_{\theta}\left(\sum_{i=1}^{n}\bm{y}_{i}A^{i}-B\right)\right]\right|$
(44)
by the standard hybrid argument. Let $\\{Z^{0},\ldots,Z^{t}\\}$ be a set of
random variable on $n$ coordinates such that $Z^{0}$ is the uniform
distribution on $\\{-1,1\\}^{n}$ and $Z^{t}$ is uniform in $\mathcal{G}^{n}$.
To this end, define $Z^{\ell}$ as follows: for $j\in[\ell]$, let
$Z^{\ell}_{|h^{-1}(j)}=\bm{y}^{j}$ and for $\ell<j\leq t$ let
$Z^{\ell}_{|h^{-1}(j)}=\bm{x}^{j}$. It is easy to see that
$Z^{0}\sim\mathcal{U}_{n}$ and $Z^{t}\sim\mathcal{G}^{n}$. We now can upper
bound Eq. (44) as
$\displaystyle\begin{aligned}
&\left|\mathop{\mathbb{E}}_{\bm{x}\sim\mathcal{U}_{n}}\left[\Psi_{\theta}\left(\sum_{i=1}^{n}\bm{x}_{i}A^{i}-B\right)\right]-\mathop{\mathbb{E}}_{\bm{y}\sim\mathcal{G}^{n}}\left[\Psi_{\theta}\left(\sum_{i=1}^{n}\bm{y}_{i}A^{i}-B\right)\right]\right|\\\
&=\left|\sum_{\ell=1}^{t}\mathop{\mathbb{E}}_{\begin{subarray}{c}\bm{x}\sim\mathcal{U}_{n}\\\
\bm{y}\sim\mathcal{G}^{n}\end{subarray}}\left[\Psi_{\theta}\left(\sum_{i=1}^{n}Z^{\ell}_{i}A^{i}-B\right)\right]-\mathop{\mathbb{E}}_{\begin{subarray}{c}\bm{x}\sim\mathcal{U}_{n}\\\
\bm{y}\sim\mathcal{G}^{n}\end{subarray}}\left[\Psi_{\theta}\left(\sum_{i=1}^{n}Z^{\ell-1}_{i}A^{i}-B\right)\right]\right|\\\
&\leq\sum_{\ell=1}^{t}\left|\mathop{\mathbb{E}}_{\begin{subarray}{c}\bm{x}\sim\mathcal{U}_{n}\\\
\bm{y}\sim\mathcal{G}^{n}\end{subarray}}\left[\Psi_{\theta}\left(\sum_{i=1}^{n}Z^{\ell}_{i}A^{i}-B\right)\right]-\mathop{\mathbb{E}}_{\begin{subarray}{c}\bm{x}\sim\mathcal{U}_{n}\\\
\bm{y}\sim\mathcal{G}^{n}\end{subarray}}\left[\Psi_{\theta}\left(\sum_{i=1}^{n}Z^{\ell-1}_{i}A^{i}-B\right)\right]\right|\end{aligned}$
(45)
We now upper bound each of the $t$ quantities on the RHS of Eq. (45). Fix
$\ell\in[t]$ and let us assume for simplicity that $h^{-1}(\ell)=[m]$. By
definition of $Z^{\ell}$ we observe that $Z^{\ell}_{j}=Z^{\ell+1}_{j}$ for all
$j\in\\{m+1,\ldots,n\\}$ and in fact we have
$Z^{\ell}=(\bm{x}_{1},\ldots,\bm{x}_{m},Z_{m+1},\ldots,Z_{n}),\quad
Z^{\ell+1}=(\bm{y}_{1},\ldots,\bm{y}_{m},Z_{m+1},\ldots,Z_{n}),$
where $\bm{x}_{i}\sim\mathcal{U}_{1}$ and $y_{i}\in\mathcal{G}$ is uniform in
their respective domains. Crucially note that $Z_{m+1},\ldots,Z_{n}$ is
independent of the $\bm{x}_{i}$s or $\bm{y}_{i}$s by definition of
$Z^{\ell},Z^{\ell+1}$. Rewriting the $\ell$-th term in Eq. (45), we get
$\displaystyle\left|\mathop{\mathbb{E}}_{\begin{subarray}{c}\bm{x}\sim\mathcal{U}_{n}\\\
\bm{y}\sim\mathcal{G}^{n}\end{subarray}}\left[\Psi_{\theta}\left(\underbrace{\sum_{i=1}^{m}\bm{x}_{i}A^{i}}_{Q}+\underbrace{\sum_{i=m+1}^{n}Z_{i}A^{i}-B}_{P}\right)\right]-\mathop{\mathbb{E}}_{\begin{subarray}{c}\bm{x}\sim\mathcal{U}_{n}\\\
\bm{y}\sim\mathcal{G}^{n}\end{subarray}}\left[\Psi_{\theta}\left(\underbrace{\sum_{i=1}^{m}\bm{y}_{i}A^{i}}_{R}+\underbrace{\sum_{i=m+1}^{n}Z_{i}A^{i}-B}_{P}\right)\right]\right|$
(46)
Let us analyze both these quantities separately. We can first write the
Fréchet series for both these expressions as
$\displaystyle\Psi_{\theta}(Q+P)=\Psi_{\theta}(P)+D\Psi_{\theta}\left(P\right)\left[Q\right]+\frac{1}{2}D^{2}\Psi_{\theta}\left(P\right)\left[Q,Q\right]+\frac{1}{6}D^{3}\Psi_{\theta}\left(P^{\prime}\right)\left[Q,Q,Q\right]$
(47)
where $P^{\prime}=P+\xi Q$ for some $\xi\in[0,1]$.161616This follows directly
from the mean value theorem for Fréchet derivatives [AP95].
$\displaystyle\Psi_{\theta}(R+P)=\Psi_{\theta}(P)+D\Psi_{\theta}\left(P\right)\left[R\right]+\frac{1}{2}D^{2}\Psi_{\theta}\left(P\right)\left[R,R\right]+\frac{1}{6}D^{3}\Psi_{\theta}\left(P^{\prime\prime}\right)\left[R,R,R\right],$
(48)
where $P^{\prime\prime}=P+\xi^{\prime}R$ for some $\xi\in[0,1]$.
Now, observe that since the first moment and the second moment of $\bm{x}$
match with the standard normal distributions. Thus we have that
$\displaystyle\begin{aligned}
\operatorname*{\mathbb{E}}_{\begin{subarray}{c}\bm{x}\sim\mathcal{U}_{n}\\\
\bm{y}\sim\mathcal{G}^{n}\end{subarray}}\left[D\Psi_{\theta}\left(P\right)\left[R\right]\right]&=\operatorname*{\mathbb{E}}_{\begin{subarray}{c}\bm{x}\sim\mathcal{U}_{n}\\\
\bm{y}\sim\mathcal{G}^{n}\end{subarray}}\left[D\Psi_{\theta}\left(P\right)\left[Q\right]\right]\\\
\operatorname*{\mathbb{E}}_{\begin{subarray}{c}\bm{x}\sim\mathcal{U}_{n}\\\
\bm{y}\sim\mathcal{G}^{n}\end{subarray}}\left[D^{2}\Psi_{\theta}\left(P\right)\left[R,R\right]\right]&=\operatorname*{\mathbb{E}}_{\begin{subarray}{c}\bm{x}\sim\mathcal{U}_{n}\\\
\bm{y}\sim\mathcal{G}^{n}\end{subarray}}\left[D^{2}\Psi_{\theta}\left(P\right)\left[Q,Q\right]\right].\end{aligned}$
(49)
So by taking the difference of Eq. (48) and Eq. (47), only the third order
spectral derivatives remain to be bounded. For this, we now use the Corollary
29 and obtain
$\displaystyle\big{|}D^{3}\Psi_{\theta}\left(P^{\prime}\right)\left[Q,Q,Q\right]\big{|}\leq
O\left(\frac{\Delta_{1}^{2}}{\theta^{3}}\log^{3}k\cdot\mbox{$\|{Q}\|$}^{3}\right)$
(50)
$\displaystyle\big{|}D^{3}\Psi_{\theta}\left(P^{\prime\prime}\right)\left[R,R,R\right]\big{|}\leq
O\left(\frac{\Delta_{2}^{2}}{\theta^{3}}\log^{3}k\cdot\mbox{$\|{R}\|$}^{3}\right).$
(51)
where $\Delta_{1}=\mbox{$\|{P^{\prime}}\|$}$ and
$\Delta_{2}=\mbox{$\|{P^{\prime\prime}}\|$}$.
Thus, the absolute value of Eq. (46) is upper bounded by
$\frac{\log^{3}k}{\theta^{3}}\mathbb{E}\left[\Delta_{1}^{2}\mbox{$\|{Q}\|$}^{3}+\Delta_{2}^{2}\mbox{$\|{R}\|$}^{3}\right]\leq\frac{\log^{3}k}{\theta^{3}}\left(\mathbb{E}\left[\mbox{$\|{P^{\prime}}\|$}^{4}\right]^{1/2}\mathbb{E}\left[\mbox{$\|{Q}\|$}^{6}\right]^{1/2}+\mathbb{E}\left[\mbox{$\|{P^{\prime\prime}}\|$}^{4}\right]^{1/2}\mathbb{E}\left[\mbox{$\|{R}\|$}^{6}\right]^{1/2}\right),$
(52)
where the inequality is by Cauchy-Schwarz inequality.
Using 6 and the fact that $\sum_{i}(A^{i})^{2}\preceq M\cdot\mathbb{I}$, we
have
$\displaystyle\mathbb{E}\left[\mbox{$\|{P^{\prime}}\|$}^{4}\right]\leq
O\left(\log^{2}k\cdot
M^{2}+\mbox{$\|{B}\|$}^{4}\right),\quad\mathbb{E}\left[\mbox{$\|{P^{\prime\prime}}\|$}^{4}\right]\leq
O\left(\log^{2}k\cdot M^{2}+\mbox{$\|{B}\|$}^{4}\right)$ (53)
We now upper bound the last term in Eq. (52) using the following claim.
###### Claim 48.
It holds that $\mathbb{E}\left[\mbox{$\|{Q}\|$}^{6}\right]\leq
O\left(\log^{6}k\cdot\tau^{3}\cdot M^{3}\right)$,
$\mathbb{E}\left[\mbox{$\|{R}\|$}^{6}\right]\leq
O\left(\log^{6}k\cdot\tau^{3}\cdot M^{3}\right)$.
Before proving this claim, observe that combining Claim 48 with Eq. (53),
(52), we can upper bound Eq. (52) (and in turn Eq. (46)) by
$\displaystyle O\left(\frac{\log^{3}k}{\theta^{3}}\cdot\left(M\log
k+\|B\|^{2}\right)\cdot\left(\log^{3}k\cdot\tau^{1.5}\cdot
M^{1.5}\right)\right)\leq
O\left(\frac{\log^{7}k}{\theta^{3}}\cdot(M+\|B\|^{2})\cdot(M\cdot\tau)^{1.5}\right)$
Putting together this inequality with Eq. (45), we finally get
$\displaystyle\Big{|}\mathop{\mathbb{E}}_{x\sim\mathcal{U}_{n}}\left[\Psi_{\theta}\left(\sum_{i=1}^{n}x_{i}A^{i}\right)\right]-\mathop{\mathbb{E}}_{y\sim\mathcal{G}^{n}}\left[\Psi_{\theta}\left(\sum_{i=1}^{n}y_{i}A^{i}\right)\right]\Big{|}\leq
O\left(\frac{\log^{7}k}{\theta^{3}}\cdot(M+\|B\|^{2})\cdot(M\cdot\tau)^{1.5}\right),$
concluding the theorem proof. We now prove the claim above.
###### Proof of Claim 48.
Note that $Q=\sum_{i=1}^{n}\bm{x}_{i}A^{i}$, where
$\left(\bm{x}_{1},\ldots,\bm{x}_{n}\right)$ is i.i.d. with
$\mathrm{Pr}\>\\!\\!\left[\bm{x}_{i}=1\right]=\mathrm{Pr}\>\\!\\!\left[\bm{x}_{i}=-1\right]=\frac{1}{2t}$
and $\mathrm{Pr}\>\\!\\!\left[\bm{x}_{i}=0\right]=1-1/t$. Then using Fact 7,
we have
$\displaystyle\mathbb{E}\left[\mbox{$\|{Q}\|$}_{8p}^{8p}\right]^{1/8p}$
$\displaystyle\leq$
$\displaystyle\sqrt{8p-1}\Big{\|}\left(\frac{1}{t}\sum_{i}\left(A^{i}\right)^{2}\right)^{1/2}\Big{\|}_{8p}+\left(8p-1\right)\left(\frac{1}{t}\sum_{i}\mbox{$\|{A^{i}}\|$}_{8p}^{8p}\right)^{1/8p}$
$\displaystyle\leq$ $\displaystyle\sqrt{8p-1}\cdot\sqrt{\frac{M}{t}}\cdot
k^{\frac{1}{8p}}+\left(8p-1\right)\left(\frac{\tau^{8p-2}\cdot k\cdot
M}{t}\right)^{1/8p}$
where the second inequality used $\sum_{i}\left(A^{i}\right)^{2}\preceq
M\cdot\mathbb{I}$ for both terms and $0\preceq A^{i}\preceq\tau\mathbb{I}$ for
upper bounding the second term. Setting $p=10\log k$, $t=1/\tau$ we have
$\left(\mathbb{E}\left[\mbox{$\|{Q}\|$}_{8p}^{8p}\right]\right)^{1/8p}\leq
O\left(\sqrt{\log k}\cdot\sqrt{\tau}\cdot\sqrt{M}+\log
k\cdot\tau\cdot(M/\tau)^{1/(80\log k)}\right)=O\left(\log
k\cdot\sqrt{\tau}\cdot\sqrt{M}\right).$
Thus, we have
$\mathbb{E}\left[\mbox{$\|{Q}\|$}^{6}\right]\leq\left(\mathbb{E}\left[\mbox{$\|{Q}\|$}_{8p}^{8p}\right]\right)^{\frac{3}{4p}}\leq
O\left(\log^{6}k\cdot\tau^{3}\cdot M^{3}\right),$
where in the first inequality note that the LHS is the spectral norm and the
RHS is the $(8p)$-Schatten norm. This proves the first inequality in the claim
statement. The second inequality in the claim follows by the exact same
argument (since Fact 7 applies to even $\sum_{i}{\bm{g}}_{i}A^{i}$). ∎
The proof of this claim concludes the proof of the theorem. Additionally,
observe that since the largest Schatten power of $Q$ that we use is $8p=80\log
k$, the proof of this theorem also works for $\bm{x}$ that is $(80\log
k)$-wise uniform. ∎
### 6.2 Invariance principle for positive spectrahedra
We are now ready to prove our main theorem, which involves combining our anti-
concentration Theorem 41 and our invariance principle for Bentkus mollifier in
Lemma 47.171717We remark that our theorem statements should also hold true for
a larger class of _proper distributions_ as considered in [HKM13], which
requires one to extend our main Theorem 22 to show that even the $4$th order
spectral derivatives can be bounded by $\|f^{(4)}\|_{1}$. We believe this
should be possible and leave this to be made rigorous for future work.
###### Theorem 49.
Let $k\geq 1$, $M\geq 1,\gamma\geq 1,\tau\in[0,1],\delta\in[0,1]$. Let
$S_{1},S_{2}$ be $(\tau,M)$-regular positive spectrahedra specified by
matrices $\\{A^{1}_{1},\ldots,A^{n}_{1},B_{1}\\}\in\mathsf{Sym}_{k}$ and
$\\{A^{1}_{2},\ldots,A^{n}_{2},B_{2}\\}\in\mathsf{Sym}_{k}$ respectively
satisfying $\mbox{$\|{B_{1}}\|$},\mbox{$\|{B_{2}}\|$}\leq~{}\gamma$. Let
$S=S_{1}\cap S_{2}$. If $\mu$ is a $\left(80\log k\right)$-wise uniform
distribution over $\\{-1,1\\}^{n}$, then
$\left|\operatorname*{\mathbb{E}}_{\bm{x}\sim\mu}[\bm{x}\in
S]-\operatorname*{\mathbb{E}}_{\bm{g}\sim\mathcal{G}^{n}}[\bm{g}\in
S]\right|\leq C\cdot\left(M+\gamma^{2}\right)^{1/5}\cdot\log^{7/5}k\cdot
M^{3/10}\cdot\tau^{3/10},$
for some universal constant $C>0$.
###### Proof.
Again for notational simplicity, let $A^{i}=\mbox{\rm
diag}\left(A^{i}_{1},A^{i}_{2}\right)$ and $B=\mbox{\rm
diag}\left(B_{1},B_{2}\right)$ be block diagonal matrices. We conclude the
result by combining Fact 24, Lemma 47 and Corollary 41 as follows: first Lemma
47 implies
$\displaystyle\begin{aligned}
&\left|\operatorname*{\mathbb{E}}_{\bm{x}\sim\mu}\left[\Psi_{\theta}\left(\sum_{i=1}^{n}\bm{x}_{i}A^{i}-B\right)\right]-\operatorname*{\mathbb{E}}_{\bm{g}\sim\mathcal{G}^{n}}\left[\Psi_{\theta}\left(\sum_{i=1}^{n}\bm{g}_{i}A^{i}-B\right)\right]\right|\leq
O\left(\frac{\log^{7}k}{\theta^{3}}\cdot(M+\|B\|^{2})\cdot(M\cdot\tau)^{1.5}\right),\end{aligned}$
In particular, using Fact 24 (for $D=B-\beta\cdot\mathbb{I}$ and
$D=B+\beta\cdot\mathbb{I}$), the “if” condition of Fact 24 is satisfied with
$\eta=O\left(\frac{\log^{7}k}{\theta^{3}}\cdot\left(M+(\gamma+\beta)^{2}\right)\cdot(M\cdot\tau)^{1.5}\right)$
where $\beta=O(\theta\cdot\sqrt{\log k/\delta})$. In particular, Fact 24 and
Corollary 42 now together imply that
$\displaystyle\left|\operatorname*{\mathbb{E}}_{\bm{x}\sim\mu}\left[\Psi\left(\sum_{i=1}^{n}\bm{x}_{i}A^{i}-B\right)\right]-\operatorname*{\mathbb{E}}_{\bm{g}\sim\mathcal{G}^{n}}\left[\Psi\left(\sum_{i=1}^{n}\bm{g}_{i}A^{i}-B\right)\right]\right|$
$\displaystyle\leq\gamma+3\delta+\Pr_{\bm{g}\sim\mathcal{G}^{n}}\left[\lambda_{\max}\left(\sum_{i=1}^{n}\bm{g}_{i}A^{i}-B\right)\in[-\Lambda,\Lambda]\right]$
$\displaystyle=O\left(\frac{\log^{7}k}{\theta^{3}}\cdot\left(M+\left(\gamma+\theta\cdot\sqrt{\log(k/\delta)}\right)^{2}\right)\cdot(M\cdot\tau)^{1.5}+\delta+\Lambda\right)$
$\displaystyle\leq
O\left(\frac{\log^{7}k}{\theta^{3}}\cdot\left(M+\left(\gamma+\sqrt{\log(k/\delta)}\right)^{2}\right)\cdot(M\cdot\tau)^{1.5}+\delta+\Lambda\right)$
Let us fix
$\displaystyle\theta\leftarrow\delta,\quad\theta\leftarrow\Lambda,\quad\left((M\cdot\tau)^{1.5}\cdot\log^{7}k\cdot\left(M+\left(\gamma+\sqrt{\log
k}\right)^{2}\right)\right)^{1/5}\leftarrow\theta.$
This gives us
$\displaystyle\left|\operatorname*{\mathbb{E}}_{\bm{x}\sim\mu}\left[\Psi\left(\sum_{i=1}^{n}\bm{x}_{i}A^{i}-B\right)\right]-\operatorname*{\mathbb{E}}_{\bm{g}\sim\mathcal{G}^{n}}\left[\Psi\left(\sum_{i=1}^{n}\bm{g}_{i}A^{i}-B\right)\right]\right|\leq\left((M\cdot\tau)^{1.5}\cdot\log^{7}k\cdot(M+\gamma^{2})\right)^{1/5}.$
∎
### 6.3 Application: Pseudorandom generators for positive spectrahedra.
We are now ready to describe our pseudorandom generator for fooling positive
spectrahedra. Our $\mathsf{PRG}$ is based on the well-known construction of
Meka and Zuckerman [MZ13] which we describe now. We remark that the same
$\mathsf{PRG}$ (with minor modifications and different parameter settings) was
used in [MZ13, HKM13, ST17] in order to obtain $\mathsf{PRG}$s for polytopes.
##### Meka-Zuckerman PRG.
We begin by describing the Meka-Zuckerman $\mathsf{PRG}$. Let us fix the
parameters $\delta\in(0,1)$, $\tau=\Omega(\delta^{10/3}/(\log^{5}k\cdot
M\cdot(M+\gamma^{2})))$ so that we have
$\left(M+\gamma^{2}\right)^{1/5}\cdot\log^{7/5}k\cdot
M^{3/10}\cdot\tau^{3/10}=\delta$ (where the LHS of this equality is the upper
bound obtained in our invariable principle proof). Let $t=\lceil 1/\tau\rceil$
and consider the family of $(80\log k)$-wise uniform functions
$\mathcal{H}=\\{h:[n]\rightarrow[t]\\}$, i.e., for every for every subset
$I\subseteq[n]$ of size at most $80\log k$, and $b\in[t]^{I}$, we have
$\Pr_{\bm{h}\in\mathcal{H}}\left[\bm{h}(i)=b_{i}\right]=\frac{1}{t^{|I|}},$
where the probability is taken over a uniformly random function
$h\in\mathcal{H}$. Efficient constructions of such hash function families are
known with $|\mathcal{H}|=O(n^{80\log k})$. For simplicity (as in the proof of
[MZ13, HKM13]), we also assume that for every $j\in[t]$, we have
$|h^{-1}(j)|=n/t$. Let $m=n/t$ and
$G_{0}:\\{0,1\\}^{s}\rightarrow\\{-1,1\\}^{m}$ generate a $(80\log k)$-wise
uniform distribution over $\\{-1,1\\}^{m}$, i.e., for every $I\subseteq[n]$ of
size at most $80\log k$ and $b\in\\{-1,1\\}^{I}$, we have
$\Pr_{\begin{subarray}{c}\bm{z}\in\\{0,1\\}^{s}\\\
\bm{x}=G_{0}(\bm{z})\end{subarray}}[\bm{x}_{i}=\bm{b}_{i}\text{ for all }i\in
I]=\frac{1}{2^{|I|}},$
where the probability is taken over uniformly random $z\in\\{0,1\\}^{s}$. It
is well-known by [NN93] that efficient constructions of generators $G_{0}$ are
known for $s=O(\log k\log n)$. Finally, we are ready to describe the Meka-
Zuckerman generator: for a given hash function family $\mathcal{H}$ and
generator $G_{0}$, define
$G:\mathcal{H}\times(\\{0,1\\}^{s})^{t}\rightarrow\\{-1,1\\}^{n}$ by
$G(h,z^{1},\ldots,z^{t})=x,\qquad\text{ where
}x_{|h^{-1}(i)}=G_{0}(z^{i})\text{ for }i\in[t].$
Clearly the seed length of this generator is
$O\left((\log n)(\log k)+(\log n)(\log k)\frac{1}{\tau}\right)=O((\log n)(\log
k)/\tau)=(\log n)\cdot\operatorname{poly}(\log k,M,1/\delta,\gamma),$
where the first term is the logarithm of the number of elements of the hash
function family $|\mathcal{H}|$, the second term because we have $s=O((\log
n)(\log k))$ and recall that we picked $t=O(1/\tau)$ and the final equality
used the bound on $\tau$ we fixed at the start of the proof.
We now restate our main theorem and prove it.
###### Theorem 50.
Let $\delta\in(0,1)$, $k,n,M\geq 1$ and $\tau\leq\delta^{10/3}/(\log^{5}k\cdot
M\cdot(M+\gamma^{2}))$. Let $S_{1},S_{2}$ be $(\tau,M)$-regular positive
spectrahedra specified by matrices
$\\{A^{1}_{1},\ldots,A^{n}_{1},B_{1}\\}\in\mathsf{Sym}_{k}$ and
$\\{A^{1}_{2},\ldots,A^{n}_{2},B_{2}\\}\in\mathsf{Sym}_{k}$ with
$\|B_{1}\|,\|B_{2}\|\leq\gamma$. Let $S=S_{1}\cap S_{2}$. There exists a
$\mathsf{PRG}$ $G:\\{0,1\\}^{r}\rightarrow\\{-1,1\\}^{n}$ with
$r=(\log n)\cdot\operatorname{poly}(\log k,M,1/\delta,\gamma)$
that $\delta$-fools $S$ with respect to the uniform distribution.
The proof of this theorem is a generic statement that allows one to go from
invariance principles proven using the proof techniques to construct
$\mathsf{PRG}$s. The proof uses the same proof ideas of Harsha, Klivans and
Meka [HKM13, Section 7.2] (except that now we directly proved _Boolean_ anti-
concentration instead of the weaker _Gaussian_ anti-concentration as proven by
[HKM13]). We provide the proof below for completeness.
###### Proof.
Again for notational simplicity, let $A^{i}=\mbox{\rm
diag}\left(A^{i}_{1},A^{i}_{2}\right)$ and $B=\mbox{\rm
diag}\left(B_{1},B_{2}\right)$ be block diagonal matrices. The $\mathsf{PRG}$
$G$ will be the Meka-Zuckerman $\mathsf{PRG}$ defined above, so the seed
length $r=(\log n)\cdot\operatorname{poly}(\log k,M,1/\delta,\gamma)$
immediately follows.
$\displaystyle\begin{aligned}
\left|\operatorname*{\mathbb{E}}_{\bm{x}\sim\mathcal{U}_{r}}\left[\Psi_{\theta}\left(\sum_{i=1}^{n}\left(G(\bm{x})\right)_{i}A^{i}-B\right)\right]-\operatorname*{\mathbb{E}}_{\bm{g}\sim\mathcal{G}^{n}}\left[\Psi_{\theta}\left(\sum_{i=1}^{n}\bm{g}_{i}A^{i}-B\right)\right]\right|\leq
O\left(\frac{\log^{7}k}{\theta^{3}}\cdot(M+\|B\|^{2})\cdot(M\cdot\tau)^{1.5}\right),\end{aligned}$
(54)
where we used the fact that $G(x)$ for uniformly random $x\in\\{0,1\\}^{r}$
generates a $(80\log k)$-wise uniform distribution and Lemma 47 holds for
every $(80\log k)$-wise uniform distribution $\mu$. Repeating the same
calculation that we did in the proof of Theorem 49, we get
$\displaystyle\left|\operatorname*{\mathbb{E}}_{\bm{x}\sim\mathcal{U}_{r}}\left[\Psi\left(\sum_{i=1}^{n}\left(G(\bm{x})\right)_{i}A^{i}-B\right)\right]-\operatorname*{\mathbb{E}}_{\bm{g}\sim\mathcal{G}^{n}}\left[\Psi\left(\sum_{i=1}^{n}\bm{g}_{i}A^{i}-B\right)\right]\right|$
$\displaystyle\leq\gamma+3\delta+\Pr_{\bm{g}\sim\mathcal{G}^{n}}\left[\lambda_{\max}\left(A\left(\bm{g}\right)\right)\in(-\Lambda,\Lambda]\right]$
$\displaystyle=O\left(\frac{\log^{7}k}{\theta^{3}}\cdot(M+\|B\|^{2})\cdot(M\cdot\tau)^{1.5}+\delta+\Lambda\right),$
and using our assumption on $\tau$ (and the same parameters as in Theorem 49),
this implies that
$\left|\operatorname*{\mathbb{E}}_{\bm{x}\sim\mathcal{U}_{r}}[G(\bm{x})\in
S]-\operatorname*{\mathbb{E}}_{\bm{g}\sim\mathcal{G}^{n}}[\bm{g}\in
S]\right|\leq\delta,$
hence proving our theorem statement. ∎
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## Appendix A Proof of Lemma 34: Case 2
Recall that the goal is to prove the following inequality
$\left|\sum_{i_{1}\neq i_{2}\neq
i_{3}}\frac{\frac{\overline{g}\left(x_{i_{1}}\right)-\overline{g}\left(x_{i_{3}}\right)}{x_{i_{3}}-x_{i_{1}}}-\frac{\overline{g}\left(x_{i_{1}}\right)-\overline{g}\left(x_{i_{2}}\right)}{x_{i_{2}}-x_{i_{1}}}}{x_{i_{3}}-x_{i_{2}}}G\left(x\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\leq
O\left(\Delta\log^{2}k\mbox{$\|{H}\|$}^{3}\right)$ (55)
First observe that the LHS of the inequality above can be rephrased as
follows.
$\displaystyle\left|\sum_{i_{1}\neq i_{2}\neq
i_{3}}\frac{\frac{\overline{g}\left(x_{i_{1}}\right)-\overline{g}\left(x_{i_{3}}\right)}{x_{i_{3}}-x_{i_{1}}}-\frac{\overline{g}\left(x_{i_{1}}\right)-\overline{g}\left(x_{i_{2}}\right)}{x_{i_{2}}-x_{i_{1}}}}{x_{i_{3}}-x_{i_{2}}}G\left(x\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|$
$\displaystyle=\left|2\sum_{i_{1}\neq i_{2}\neq i_{3}\atop
x_{i_{3}}>x_{i_{2}}}\frac{\frac{g^{\prime}\left(x_{i_{1}}\right)g\left(x_{i_{3}}\right)-g\left(x_{i_{1}}\right)g^{\prime}\left(x_{i_{3}}\right)}{x_{i_{3}}-x_{i_{1}}}g\left(x_{i_{2}}\right)-\frac{g^{\prime}\left(x_{i_{1}}\right)g\left(x_{i_{2}}\right)-g\left(x_{i_{1}}\right)g^{\prime}\left(x_{i_{2}}\right)}{x_{i_{2}}-x_{i_{1}}}g\left(x_{i_{3}}\right)}{x_{i_{3}}-x_{i_{2}}}G\left(x_{-\left\\{i_{1},i_{2},i_{3}\right\\}}\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|$
(56)
Providing an upper bound on this consists of several lemmas and the result is
concluded by combing all of them via triangle inequalities. To keep the
expressions short, we use the following notations to represent Eq. (55), which
are clear in the context.
$\displaystyle\left|2\sum_{\begin{subarray}{c}i_{1}\neq i_{2}\neq i_{3}\\\
x_{i_{3}}>x_{i_{2}}\end{subarray}}\frac{\frac{\langle
i_{1}\rangle^{\prime}\langle i_{3}\rangle-\langle i_{1}\rangle\langle
i_{3}\rangle^{\prime}}{[i_{3}-i_{1}]}\langle i_{2}\rangle-\frac{\langle
i_{1}\rangle^{\prime}\langle i_{2}\rangle-\langle i_{1}\rangle\langle
i_{2}\rangle^{\prime}}{[i_{2}-i_{1}]}\langle
i_{3}\rangle}{[i_{3}-i_{2}]}\right|,$ (57)
where we implicitly hide the
$G\left(x_{-\left\\{i_{1},i_{2},i_{3}\right\\}}\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}$
term. We first give a sketch of how we are going to upper bound this
inequality and break it into subsections.
$\eqref{eq:mainequationwecare}=\underbrace{\frac{\langle i_{1}\rangle\langle
i_{3}\rangle^{\prime}-\langle i_{1}\rangle^{\prime}\langle
i_{3}\rangle}{[i_{3}-i_{1}]}\cdot\frac{\langle i_{3}\rangle-\langle
i_{2}\rangle}{[i_{3}-i_{2}]}}_{Section~{}\ref{sec:2.1},\hskip
2.84526ptLemma~{}\ref{lem:3}}-\underbrace{\frac{\frac{\langle
i_{1}\rangle\langle i_{3}\rangle^{\prime}-\langle i_{1}\rangle^{\prime}\langle
i_{3}\rangle}{[i_{3}-i_{1}]}-\frac{\langle i_{1}\rangle\langle
i_{2}\rangle^{\prime}-\langle i_{1}\rangle^{\prime}\langle
i_{2}\rangle}{[i_{2}-i_{1}]}}{[i_{3}-i_{2}]}\langle i_{3}\rangle}_{(\star)}.$
(58)
We now break up $(\star)$ into two cases
$\displaystyle(\star)=\underbrace{(\star)\cdot\mathbb{I}[\min\\{x_{i_{1}},x_{i_{3}}\\}>x_{i_{2}}]}_{(\dagger)}+\underbrace{(\star)\cdot\mathbb{I}[x_{i_{1}}<x_{i_{2}}<x_{i_{3}}]}_{(\dagger\dagger)}.$
(59)
Note that there are the only two cases we need to handle since by symmetry
between $i_{2}$ and $i_{3}$, we can assume $x_{i_{3}}>x_{i_{2}}$, without loss
of generality. Now we bound these two terms, separately.
$(\dagger)=\underbrace{\frac{\frac{\langle i_{3}\rangle^{\prime}-\langle
i_{1}\rangle^{\prime}}{[i_{3}-i_{1}]}-\frac{\langle
i_{2}\rangle^{\prime}-\langle
i_{1}\rangle^{\prime}}{[i_{2}-i_{1}]}}{[i_{3}-i_{2}]}\langle
i_{1}\rangle\langle i_{3}\rangle}_{Section~{}\ref{sec:2.2},\hskip
2.84526ptLemma~{}\ref{lem:5}}-\underbrace{\frac{\frac{\langle
i_{3}\rangle-\langle i_{1}\rangle}{[i_{3}-i_{1}]}-\frac{\langle
i_{2}\rangle-\langle i_{1}\rangle}{[i_{2}-i_{1}]}}{[i_{3}-i_{2}]}\langle
i_{1}\rangle^{\prime}\langle i_{3}\rangle}_{Section~{}\ref{sec:2.2},\hskip
2.84526ptLemma~{}\ref{lem:333}}.$
and
$(\dagger\dagger)=\underbrace{\frac{\frac{\langle i_{1}\rangle\langle
i_{3}\rangle^{\prime}-\langle i_{3}\rangle\langle
i_{3}\rangle^{\prime}}{[i_{3}-i_{1}]}-\frac{\langle i_{1}\rangle\langle
i_{2}\rangle^{\prime}-\langle i_{2}\rangle\langle
i_{2}\rangle^{\prime}}{[i_{2}-i_{1}]}}{[i_{3}-i_{2}]}\langle
i_{3}\rangle}_{Section~{}\ref{sec:2.3},\hskip
2.84526ptLemma~{}\ref{lem:casev111}}+\underbrace{\frac{\frac{\langle
i_{3}\rangle^{\prime}-\langle i_{1}\rangle^{\prime}}{[i_{3}-i_{1}]}\langle
i_{3}\rangle-\frac{\langle i_{2}\rangle^{\prime}-\langle
i_{1}\rangle^{\prime}}{[i_{2}-i_{1}]}\langle
i_{2}\rangle}{[i_{3}-i_{2}]}\langle i_{3}\rangle}_{(\P)\hskip
2.84526ptSection~{}\ref{sec:2.3}},$ (60)
and
$\displaystyle(\P)=\underbrace{\frac{\langle i_{3}\rangle^{\prime}-\langle
i_{1}\rangle^{\prime}}{[i_{3}-i_{1}]}\cdot\frac{\langle i_{3}\rangle-\langle
i_{2}\rangle}{[i_{3}-i_{2}]}\cdot\langle
i_{3}\rangle}_{Section~{}\ref{sec:2.4},\hskip
2.84526ptLemma~{}\ref{lem:444}}+\underbrace{\frac{\frac{\langle
i_{3}\rangle^{\prime}-\langle
i_{1}\rangle^{\prime}}{[i_{3}-i_{1}]}-\frac{\langle
i_{2}\rangle^{\prime}-\langle
i_{1}\rangle^{\prime}}{[i_{2}-i_{1}]}}{[i_{3}-i_{2}]}\langle
i_{2}\rangle\langle i_{3}\rangle}_{Section~{}\ref{sec:2.4},\hskip
2.84526ptLemma~{}\ref{lem:555}}$ (61)
Finally in order to upper bound Eq. (57), we simply bound each of these terms
by $O\left(\Delta\log^{3}k\mbox{$\|{H}\|$}^{3}\right)$ in the respective
sections (as underbraced by the terms).
### A.1 Upper bounding first term in Eq. (58)
###### Lemma 51.
$\left|\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop x_{i_{3}}>x_{i_{2}}}\frac{\langle
i_{1}\rangle\langle i_{3}\rangle^{\prime}-\langle i_{1}\rangle^{\prime}\langle
i_{3}\rangle}{[i_{3}-i_{1}]}\cdot\frac{\langle i_{3}\rangle-\langle
i_{2}\rangle}{[i_{3}-i_{2}]}\right|\leq
O\left(\Delta^{2}\cdot\log^{2}k\cdot\mbox{$\|{H}\|$}^{3}\right).$
###### Proof of Lemma 51.
We apply Claim 22 to the first sum and obtain
$O\left(\Delta\max\left\\{g^{\prime}\left(x_{i_{1}}\right),g^{\prime}\left(x_{i_{3}}\right)\right\\}\right)$
(note that we have $\max\\{\cdot,\cdot\\}$ to compensate for the fact that
$x_{i_{1}}\geq x_{i_{3}}$ or $x_{i_{3}}\geq x_{i_{1}}$). Therefore, the left
hand side in Lemma 51 can be upper bounded by
$\displaystyle O\left(\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}}}\Delta\left|\max\left\\{g^{\prime}\left(x_{i_{1}}\right),g^{\prime}\left(x_{i_{3}}\right)\right\\}\cdot\frac{g\left(x_{i_{3}}\right)-g\left(x_{i_{2}}\right)}{x_{i_{3}}-x_{i_{2}}}\cdot
G\left(x_{-\left\\{i_{1},i_{2},i_{3}\right\\}}\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
$\displaystyle\leq$ $\displaystyle O\left(\sum_{i_{1}\neq i_{2}\neq
i_{3}:\atop x_{i_{3}}>x_{i_{2}}\geq 0,x_{i_{1}}\geq
0}\left(\cdots\right)+\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}}\geq 0,x_{i_{1}}<0}\left(\cdots\right)+\sum_{i_{1}\neq
i_{2}\neq i_{3}:\atop x_{i_{3}}>x_{i_{2}},x_{i_{2}}<0,x_{i_{1}}\geq
0}\left(\cdots\right)+\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}},x_{i_{1}}<0,x_{i_{2}}<0}\left(\cdots\right)\right)$
First term in Eq. (LABEL:eqn:casev1). Note that
$g\left(x\right)\geq\frac{1}{2}$ if $x\geq 0$. Since $g^{\prime}$ is monotone
decreasing in the interval $[0,\infty)$, the first summation is upper bounded
by
$\displaystyle O\left(\left|\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}}\geq 0,x_{i_{1}}\geq
0}\Delta\left|\max\left\\{g^{\prime}\left(x_{i_{1}}\right)g^{\prime}\left(x_{i_{2}}\right)G\left(x_{-i_{3}}\right),g^{\prime}\left(x_{i_{3}}\right)g^{\prime}\left(x_{i_{2}}\right)G\left(x_{-i_{1}}\right)\right\\}H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right|\right)$
(63) $\displaystyle\leq$ $\displaystyle
O\left(\Delta\cdot\|G^{(2)}\|_{1}\cdot\max_{i_{1},i_{2}}\sum_{i_{3}}\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
$\displaystyle\leq$ $\displaystyle O\left(\Delta\cdot\log
k\cdot\max_{i_{1},i_{2}}\sum_{i_{3}}\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)\leq
O\left(\Delta\cdot\log k\cdot\mbox{$\|{H^{3}}\|$}\right)$
where the second inequality is from Fact 20 and the last inequality follows by
Eq. (38).
Second term in Eq. (LABEL:eqn:casev1). The second summation is upper bounded
as follows. Again by the mean value theorem, we observe that
$\displaystyle O\left(\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}}\geq
0,x_{i_{1}}<0}\Delta\left|\max\left\\{g^{\prime}\left(x_{i_{1}}\right)g^{\prime}\left(x_{i_{2}}\right),g^{\prime}\left(x_{i_{3}}\right)g^{\prime}\left(x_{i_{2}}\right)\right\\}G\left(x_{-i_{1}}\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
$\displaystyle\leq$ $\displaystyle
O\left(\Delta\cdot\sum_{i_{1}:x_{i_{1}}<0}\mbox{$\|{G^{(1)}\left(x_{-i_{1}}\right)}\|$}_{1}\max_{i_{2}}\sum_{i_{3}}\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|+\mbox{$\|{G^{(2)}\left(x_{-i_{1}}\right)}\|$}_{1}\max_{i_{2},i_{3}}\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
$\displaystyle\leq$ $\displaystyle
O\left(\Delta\cdot\log^{1.5}k\cdot\mbox{$\|{H}\|$}^{3}\right),$
where the last inequality is from Fact 20, Eq. (38) and the assumption that
$\left|\left\\{i:x_{i}\leq 0\right\\}\right|\leq 3\log k$.
Third term in Eq. (LABEL:eqn:casev1). Using the fact that $g^{\prime}(\cdot)$
is bounded by a constant, the third summation is upper bounded by
$\displaystyle O\left(\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}},x_{i_{2}}<0,x_{i_{1}}\geq
0}\Delta\left|\max\left\\{g^{\prime}\left(x_{i_{1}}\right),g^{\prime}\left(x_{i_{3}}\right)\right\\}\cdot
G\left(x_{-\left\\{i_{2},i_{3}\right\\}}\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
(64) $\displaystyle=$ $\displaystyle O\left(\sum_{i_{1}\neq i_{2}\neq
i_{3}:\atop x_{i_{3}}>x_{i_{2}},x_{i_{1}}\geq 0,x_{i_{2}}<0,x_{i_{3}}\geq
0}\left(\cdots\right)+\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}},x_{i_{1}}\geq
0,x_{i_{2}}<0,x_{i_{3}}<0}\left(\cdots\right)\right).$
For the first summation in Eq. (64), using the fact that
$g\left(x\right)\geq\frac{1}{2}$ when $x\geq 0$, it is upper bounded by
$\displaystyle O\left(\Delta\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}},x_{i_{1}}\geq 0,x_{i_{2}}<0,x_{i_{3}}\geq
0}\left|\max\left\\{g^{\prime}\left(x_{i_{1}}\right),g^{\prime}\left(x_{i_{3}}\right)\right\\}\cdot
G\left(x_{-\left\\{i_{2}\right\\}}\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
$\displaystyle\leq$ $\displaystyle
O\left(\Delta\sum_{i_{2}:x_{i_{2}}<0}\|G^{(1)}\|_{1}\max_{i_{1}}\sum_{i_{3}}\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
$\displaystyle\leq$ $\displaystyle
O\left(\Delta\sum_{i_{2}:x_{i_{2}}<0}\sqrt{\log
k}\max_{i_{1}}\sum_{i_{3}}\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
$\displaystyle\leq$ $\displaystyle
O\left(\Delta\cdot\log^{1.5}k\cdot\mbox{$\|{H}\|$}^{3}\right),$
where the second inequality is from Fact 20, and the last inequality used Eq.
(38) and the assumption that $\left|\left\\{i:x_{i}\leq 0\right\\}\right|\leq
3\log k$.
In order to upper bound the second summation in Eq. (64), first observe that
both $g\left(\cdot\right)$ and $G\left(\cdot\right)$ are positive and upper
bounded by $1$. Thus, Eq. (64) can be bounded as
$\displaystyle O\left(\Delta\sum_{i_{2}\neq i_{3}:\atop
x_{i_{2}}<0,x_{i_{3}}<0}\sum_{i_{1}}\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)\leq
O\left(\Delta\cdot\log^{2}k\cdot\mbox{$\|{H}\|$}^{3}\right).$
where we again use Eq. (38) and the assumption that $\left|\left\\{i:x_{i}\leq
0\right\\}\right|\leq 3\log k$.
Fourth term in Eq. (LABEL:eqn:casev1). The last summation is upper bounded by
$O\left(\Delta\cdot\log^{2}k\cdot\mbox{$\|{H}\|$}^{3}\right)$ using the same
arguments to upper bound the second summation in Eq. (64). ∎
### A.2 Upper bounding $(\dagger)$ first term in $(\star)$ in Eq. (59)
We upper bound the quantity in $(\star)$ in two cases that
$x_{i_{1}}>x_{i_{2}}$ and $x_{i_{2}}>x_{i_{1}}$. In order to prove this lemma
we need the following lemmas and claims.
###### Claim 52.
For integer $k\geq 1$, $X\in\mathsf{Sym}_{k}$ and $H\in\mathsf{Mat}_{k}$ it
holds that
$\big{\|}\left(XH+HX\right)e^{-\frac{X^{2}}{2}}\big{\|}_{2}\leq
2\mbox{$\|{X}\|$}\cdot\mbox{$\|{He^{-X^{2}/2}}\|$}_{2}$
and
$\big{\|}e^{-\frac{X^{2}}{2}}\left(XH+HX\right)\big{\|}_{2}\leq
2\mbox{$\|{X}\|$}\cdot\mbox{$\|{e^{-X^{2}/2}H}\|$}_{2}$
###### Proof.
As the Schattern norm is unitarily invariant, we assume that $X=\mbox{\rm
diag}\left(x_{1},\ldots,x_{n}\right)$ is diagonal without loss of generality.
Then
$\mbox{$\|{\left(XH+HX\right)e^{-X^{2}/2}}\|$}_{2}^{2}=\sum_{i,j}H_{i,j}^{2}\left(x_{i}+x_{j}\right)^{2}e^{-x_{j}^{2}}\leq
4\mbox{$\|{X}\|$}^{2}\cdot\sum_{i,j}H_{i,j}^{2}e^{-x_{j}^{2}}=4\Delta^{2}\mbox{$\|{He^{-X^{2}/2}}\|$}_{2}^{2}.$
The second inequality follows by the same argument. ∎
###### Lemma 53.
Given an integer $k\geq 1$, $u_{1},u_{2},u_{3}\geq 0$ satisfying
$u_{1}+u_{2}+u_{3}=1$ and
$X\in\mathsf{Sym}_{k},H_{1},H_{2},H_{3}\in\mathsf{Mat}_{k}$, if
$u_{1},u_{3}\leq\frac{1}{2}$, then it holds that
$\displaystyle\left|\mbox{\rm
Tr}\left[e^{-u_{1}X^{2}}H_{1}e^{-u_{2}X^{2}}H_{2}e^{-u_{3}X^{2}}H_{3}\right]\right|$
$\displaystyle\leq$
$\displaystyle\left(\mbox{$\|{H_{1}e^{-\frac{1}{2}X^{2}}}\|$}_{2}+\mbox{$\|{e^{-\frac{1}{2}X^{2}}H_{1}}\|$}_{2}\right)\cdot\left(\mbox{$\|{H_{2}e^{-\frac{1}{2}X^{2}}}\|$}_{2}+\mbox{$\|{e^{-\frac{1}{2}X^{2}}H_{2}}\|$}_{2}\right)\cdot\mbox{$\|{H_{3}}\|$}.$
###### Proof.
Using the inequality $\left|\mbox{\rm
Tr}ABC\right|\leq\mbox{$\|{A}\|$}_{2}\cdot\mbox{$\|{B}\|$}_{2}\cdot\mbox{$\|{C}\|$}$
(where $\|\cdot\|_{2}$ is the standard Frobenius norm and $\|\cdot\|$ is the
spectral norm), we have
$\displaystyle\left|\mbox{\rm
Tr}\left[e^{-u_{1}X^{2}}H_{1}e^{-u_{2}X^{2}}H_{2}e^{-u_{3}X^{2}}H_{3}\right]\right|\
$ $\displaystyle\leq$
$\displaystyle\mbox{$\|{e^{-u_{1}X^{2}}H_{1}e^{\left(u_{1}-\frac{1}{2}\right)X^{2}}}\|$}_{2}\cdot\mbox{$\|{e^{-u_{3}X^{2}}H_{2}e^{\left(u_{3}-\frac{1}{2}\right)X^{2}}}\|$}_{2}\cdot\mbox{$\|{H_{3}}\|$}.$
We conclude the result by Lemma 54. ∎
###### Lemma 54.
Given diagonal matrices $A=\mbox{\rm
diag}\left(a_{1},\ldots,a_{k}\right),B=\mbox{\rm
diag}\left(b_{1},\ldots,b_{k}\right)$ with $a_{1}\geq\cdots\geq a_{k}\geq 0$
and $b_{1}\geq\cdots\geq b_{k}\geq 0$ and an arbitrary matrix $H$, it holds
that
$\mbox{$\|{AHB}\|$}_{2}^{2}+\mbox{$\|{BHA}\|$}_{2}^{2}\leq\mbox{$\|{HAB}\|$}_{2}^{2}+\mbox{$\|{ABH}\|$}_{2}^{2}.$
In particular,
$\mbox{$\|{AHB}\|$}_{2}\leq\mbox{$\|{HAB}\|$}_{2}+\mbox{$\|{ABH}\|$}_{2}.$
###### Proof.
Note that
$\displaystyle\left(\mbox{$\|{HAB}\|$}_{2}^{2}+\mbox{$\|{ABH}\|$}_{2}^{2}\right)-\left(\mbox{$\|{AHB}\|$}_{2}^{2}+\mbox{$\|{BHA}\|$}_{2}^{2}\right)$
$\displaystyle=$
$\displaystyle\sum_{i,j}H_{i,j}^{2}\left(a_{i}^{2}b_{i}^{2}+a_{j}^{2}b_{j}^{2}-a_{i}^{2}b_{j}^{2}-a_{j}^{2}b_{i}^{2}\right)$
$\displaystyle=$
$\displaystyle\sum_{i,j}\left(a_{i}^{2}-a_{j}^{2}\right)\left(b_{i}^{2}-b_{j}^{2}\right)\geq
0,$
where the first equality is from the symmetry. ∎
###### Lemma 55.
Given an integer $k\geq 1$, matrices $A,B,C\in\mathsf{Mat}_{k}$ and
$X\in\mathsf{Sym}_{k}$ with $\mbox{$\|{X}\|$}\leq\Delta$, it holds that
$\displaystyle\left|\mbox{\rm
Tr}\left[D^{2}\left(e^{-X^{2}/2}\right)\left[A,B\right]C\right]\right|$
$\displaystyle\leq$ $\displaystyle
4\Delta^{2}\cdot\max\begin{Bmatrix}\left(\mbox{$\|{Ae^{-X^{2}/2}}\|$}_{2}+\mbox{$\|{e^{-X^{2}/2}A}\|$}_{2}\right)\cdot\left(\mbox{$\|{Be^{-X^{2}/2}}\|$}_{2}+\mbox{$\|{e^{-X^{2}/2}B}\|$}_{2}\right)\cdot\mbox{$\|{C}\|$},\\\
\left(\mbox{$\|{Ae^{-X^{2}/2}}\|$}_{2}+\mbox{$\|{e^{-X^{2}/2}A}\|$}_{2}\right)\cdot\left(\mbox{$\|{Ce^{-X^{2}/2}}\|$}_{2}+\mbox{$\|{e^{-X^{2}/2}C}\|$}_{2}\right)\cdot\mbox{$\|{B}\|$},\\\
\left(\mbox{$\|{Be^{-X^{2}/2}}\|$}_{2}+\mbox{$\|{e^{-X^{2}/2}B}\|$}_{2}\right)\cdot\left(\mbox{$\|{Ce^{-X^{2}/2}}\|$}_{2}+\mbox{$\|{e^{-X^{2}/2}C}\|$}_{2}\right)\cdot\mbox{$\|{A}\|$}\end{Bmatrix}.$
###### Proof.
Combining Lemma 12, Lemma 53 and the inequality that
$\mbox{$\|{\left(XA+AX\right)e^{-X^{2}/2}}\|$}_{2}+\mbox{$\|{e^{-X^{2}/2}\left(XA+AX\right)}\|$}_{2}\leq
2\Delta\left(\mbox{$\|{Ae^{-X^{2}/2}}\|$}_{2}+\mbox{$\|{e^{-X^{2}/2}A}\|$}_{2}\right)$
and
$\mbox{$\|{\left(XB+BX\right)e^{-X^{2}/2}}\|$}_{2}+\mbox{$\|{e^{-X^{2}/2}\left(XB+BX\right)}\|$}_{2}\leq
2\Delta\left(\mbox{$\|{Be^{-X^{2}/2}}\|$}_{2}+\mbox{$\|{e^{-X^{2}/2}B}\|$}_{2}\right)$
and
$\mbox{$\|{XA+AX}\|$}\leq 2\Delta\mbox{$\|{A}\|$},\hskip
11.38109pt\mbox{$\|{XB+BX}\|$}\leq 2\Delta\mbox{$\|{B}\|$},$
we conclude the result. ∎
###### Lemma 56.
$\left|\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{1}}>x_{i_{2}},x_{i_{3}}>x_{i_{2}}}\frac{\frac{\langle
i_{3}\rangle^{\prime}-\langle
i_{1}\rangle^{\prime}}{[i_{3}-i_{1}]}-\frac{\langle
i_{2}\rangle^{\prime}-\langle
i_{1}\rangle^{\prime}}{[i_{2}-i_{1}]}}{[i_{3}-i_{2}]}\langle
i_{1}\rangle\langle i_{3}\rangle\right|\leq
O\left(\Delta^{2}\cdot\log^{2.5}k\mbox{$\|{H}\|$}^{3}\right).$
###### Proof of Lemma 56.
We break the summation into two summations
$\left|\sum_{i_{1}\neq i_{2}\neq i_{3}\atop
x_{i_{1}}>x_{i_{3}}>x_{i_{2}}}\left(\cdots\right)\right|+\left|\sum_{i_{1}\neq
i_{2}\neq i_{3}\atop x_{i_{3}}>x_{i_{1}}>x_{i_{2}}}\left(\cdots\right)\right|$
(65)
For the first summation, we define
$A_{i,j}=\begin{cases}H_{i,j},&\mbox{if $x_{i}<x_{j}$}\\\
0,&\mbox{otherwise}.\end{cases}$
and Then $\mbox{$\|{A}\|$}\leq\log k\cdot\mbox{$\|{H}\|$}$ by Fact 4 (without
loss of generality, we may assume that $x_{i}$s are sorted in increasing
order. Further notice that all the diagonal entries of $H$ are zeros. Thus $A$
is the upper triangle part of $H$). We first bound the first term in Eq. (65).
In this direction, we first rewrite it as
$\displaystyle\frac{1}{\sqrt{2\pi}}\sum_{i_{2}}G\left(x_{-i_{2}}\right)\left(\left(D^{2}\left(e^{-X^{2}/2}\right)[A,A^{T}]H\right)_{i_{2},i_{2}}\right)=\frac{1}{\sqrt{2\pi}}\sum_{i_{2}{x_{i_{2}}<0}}\left(\cdots\right)+\frac{1}{\sqrt{2\pi}}\sum_{i_{2}:x_{i_{2}}\geq
0}\left(\cdots\right)$ (66)
where $X=\mbox{\rm diag}\left(x_{1},\ldots,x_{k}\right)$ and we implicitly
used that we are summing over terms with $x_{i_{2}}<x_{i_{3}}$. Note that $A$
is obtained from $H$ by zeroing out part of entries. Thus
$\max\left\\{\mbox{$\|{Ae^{-X^{2}/2}}\|$}_{2},\mbox{$\|{A^{T}e^{-X^{2}/2}}\|$}_{2},\mbox{$\|{e^{-X^{2}/2}A}\|$}_{2},\mbox{$\|{e^{-X^{2}/2}A^{T}}\|$}_{2}\right\\}\leq\mbox{$\|{He^{-X^{2}/2}}\|$}_{2},$
(67)
To upper bound first summation in Eq. (66), we apply Lemma 55, Eq. (67) and
inequalities $\mbox{$\|{A}\|$}\leq\log k\cdot\mbox{$\|{H}\|$}$ and
$\mbox{$\|{HE_{i_{2},i_{2}}}\|$}\leq\mbox{$\|{H}\|$}$ and obtain
$\left|\mbox{\rm
Tr}\left(D^{2}\left(e^{-X^{2}/2}\right)[A,A^{T}]HE_{i_{2},i_{2}}\right)\right|\leq
16\Delta^{2}\log k\mbox{$\|{He^{-X^{2}/2}}\|$}_{2}^{2}\cdot\mbox{$\|{H}\|$}.$
(68)
Thus, the first summation in Eq. (66) is upper bounded by
$\displaystyle\frac{\Delta^{2}\cdot\log
k}{\sqrt{2\pi}}\sum_{i_{2}:x_{i_{2}}<0}G\left(x_{-i_{2}}\right)\mbox{$\|{He^{-X^{2}/2}}\|$}_{2}^{2}\cdot\mbox{$\|{H}\|$}$
(69) $\displaystyle=$ $\displaystyle\left(\Delta^{2}\log
k\cdot\sum_{i_{2}:x_{i_{2}}<0}G\left(x_{-i_{2}}\right)\sum_{i_{1},i_{3}}e^{-x_{i_{3}}^{2}}H_{i_{1},i_{3}}^{2}\mbox{$\|{H}\|$}\right)$
$\displaystyle\leq$
$\displaystyle\left(\Delta^{2}\log^{2}k\cdot\max_{i_{2}}\sum_{i_{1}\neq
i_{3}}g^{\prime}\left(x_{i_{3}}\right)\cdot G\left(x_{-i_{2}}\right)\cdot
H_{i_{1},i_{3}}^{2}\mbox{$\|{H}\|$}\right)$ $\displaystyle\leq$ $\displaystyle
O\left(\Delta^{2}\log^{2.5}k\mbox{$\|{H}\|$}^{3}\right),$
where the first inequality is from the assumption that
$\left|\left\\{i:x_{i}<0\right\\}\right|\leq 3\log k$ and the second
inequality is from Fact 20.
For the second summation in Eq. (66), we define
$\tilde{H}_{i,j}=\begin{cases}\frac{H_{i,j}}{g\left(x_{j}\right)},&\mbox{if
$x_{j}\geq 0$}\\\ 0,&\mbox{otherwise}.\end{cases}$
Then $\mbox{$\|{\tilde{H}}\|$}\leq 2\mbox{$\|{H}\|$}$ as
$g\left(x_{i}\right)\geq\frac{1}{2}$ if $x_{i}\geq 0$. Again applying Eq.
(67), we can verify that the second summation in Eq. (66) is equal to
$\displaystyle\left|\frac{1}{\sqrt{2\pi}}G\left(x\right)\mbox{\rm
Tr}~{}D^{2}\left(e^{-X^{2}/2}\right)[A,A^{T}]\tilde{H}\right|\leq\frac{16\Delta^{2}\log
k}{\sqrt{2\pi}}G\left(x\right)\mbox{$\|{He^{-X^{2}/2}}\|$}_{2}^{2}\mbox{$\|{H}\|$}\leq
O\left(\Delta^{2}\cdot\log^{1.5}k\mbox{$\|{H}\|$}^{3}\right).$
where the first inequality is from Lemma 55 and the second inequality is from
Fact 20.
Finally, the second summation in Eq. (65) can be upper bounded using the
verbatim same arguments by
$O\left(\Delta^{2}\cdot\log^{2.5}k\cdot\mbox{$\|{H}\|$}^{3}\right)$. This
proves the lemma statement. ∎
###### Lemma 57.
$\left|\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}},x_{i_{1}}>x_{i_{2}}}\frac{\frac{\langle
i_{3}\rangle-\langle i_{1}\rangle}{[i_{3}-i_{1}]}-\frac{\langle
i_{2}\rangle-\langle i_{1}\rangle}{[i_{2}-i_{1}]}}{[i_{3}-i_{2}]}\langle
i_{1}\rangle^{\prime}\langle i_{3}\rangle\right|\leq
O\left(\Delta\cdot\log^{1.5}k\cdot\mbox{$\|{H}\|$}^{3}\right)$
###### Proof.
$\displaystyle\left|\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}},x_{i_{1}}>x_{i_{2}}}\frac{\frac{\langle
i_{3}\rangle-\langle i_{1}\rangle}{[i_{3}-i_{1}]}-\frac{\langle
i_{2}\rangle-\langle i_{1}\rangle}{[i_{2}-i_{1}]}}{[i_{3}-i_{2}]}\langle
i_{1}\rangle^{\prime}\langle i_{3}\rangle\right|=\left|\sum_{i_{1}\neq
i_{2}\neq i_{3}:\atop x_{i_{3}}>x_{i_{2}},x_{i_{1}}>x_{i_{2}}\geq
0}\left(\cdots\right)+\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}},x_{i_{1}}>0,x_{i_{2}}<0}\left(\cdots\right)\right|$ (70)
To upper bound the first summation in Eq. (70), we apply Fact 3 and upper
bound the first summation by
$\displaystyle O\left(\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}},x_{i_{1}}>x_{i_{2}}\geq
0}\left|g^{\prime\prime}\left(\xi_{i_{1},i_{2},i_{3}}\right)g^{\prime}\left(x_{i_{1}}\right)G\left(x_{-\left\\{i_{1},i_{2}\right\\}}\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
$\displaystyle\leq$ $\displaystyle O\left(\left|\Delta\cdot\sum_{i_{1}\neq
i_{2}\neq i_{3}:\atop x_{i_{3}}>x_{i_{2}},x_{i_{1}}>x_{i_{2}}\geq
0}g^{\prime}\left(x_{i_{2}}\right)g^{\prime}\left(x_{i_{1}}\right)G\left(x_{-\left\\{i_{1},i_{2}\right\\}}\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
$\displaystyle\leq$ $\displaystyle
O\left(\mbox{$\|{G^{(2)}\left(x\right)}\|$}_{2}\max_{i_{1},i_{2}}\sum_{i_{3}}\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
$\displaystyle\leq$ $\displaystyle O\left(\Delta\cdot\log
k\cdot\mbox{$\|{H}\|$}^{3}\right)$
where the last inequality is from Fact 20 and Eq. (38). Note that
$\left|g^{\prime\prime}\left(\xi\right)\right|\leq\Delta$ for any
$\xi\in[x_{i_{2}},\max\left\\{x_{i_{1}},x_{i_{3}}\right\\}]$ by Eq. (13).
Applying Fact 3, the second summation in Eq. (70) is upper bounded by
$\displaystyle
O\left(\Delta\sum_{i_{2}:x_{i_{2}}<0}\sum_{i_{1},i_{3}}g^{\prime}\left(x_{i_{1}}\right)G\left(x_{-\left\\{i_{1},i_{2}\right\\}}\right)\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
$\displaystyle\leq$ $\displaystyle O\left(\Delta\cdot\log
k\cdot\max_{i_{2}}\mbox{$\|{G^{(1)}\left(x_{-i_{2}}\right)}\|$}_{1}\cdot\max_{i_{1}}\sum_{i_{3}}\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
$\displaystyle\leq$ $\displaystyle
O\left(\Delta\cdot\log^{1.5}k\cdot\mbox{$\|{H}\|$}^{3}\right)$
where the first inequality is from the assumption that
$\left|\left\\{i:x_{i}<0\right\\}\right|\leq 3\log k$ and the second
inequality is from Fact 20 and Eq. (38). ∎
### A.3 Upper bounding first term in $(\dagger\dagger)$ in Eq. (60)
We now bound the first term in Eq. (60) when $x_{i_{3}}>x_{i_{2}}>x_{i_{1}}$.
Recall that the goal is to upper bound the following lemma.
###### Lemma 58.
$\left|\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}}>x_{i_{1}}}\frac{\frac{\langle i_{1}\rangle\langle
i_{3}\rangle^{\prime}-\langle i_{3}\rangle\langle
i_{3}\rangle^{\prime}}{[i_{3}-i_{1}]}-\frac{\langle i_{1}\rangle\langle
i_{2}\rangle^{\prime}-\langle i_{2}\rangle\langle
i_{2}\rangle^{\prime}}{[i_{2}-i_{1}]}}{[i_{3}-i_{2}]}\langle
i_{3}\rangle\right|\leq
O\left(\Delta\cdot\log^{1.5}k\cdot\mbox{$\|{H}\|$}^{3}\right)$
###### Proof of Lemma 58.
$\displaystyle\left|\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}}>x_{i_{1}}}\frac{\frac{\langle i_{1}\rangle\langle
i_{3}\rangle^{\prime}-\langle i_{3}\rangle\langle
i_{3}\rangle^{\prime}}{[i_{3}-i_{1}]}-\frac{\langle i_{1}\rangle\langle
i_{2}\rangle^{\prime}-\langle i_{2}\rangle\langle
i_{2}\rangle^{\prime}}{[i_{2}-i_{1}]}}{[i_{3}-i_{2}]}\langle
i_{3}\rangle\right|$ (71) $\displaystyle\leq$ $\displaystyle\sum_{i_{1}\neq
i_{2}\neq i_{3}:\atop x_{i_{3}}>x_{i_{2}}>x_{i_{1}}}\left|\frac{\frac{\langle
i_{1}\rangle\langle i_{2}\rangle^{\prime}-\langle i_{3}\rangle\langle
i_{2}\rangle^{\prime}}{[i_{3}-i_{1}]}-\frac{\langle i_{1}\rangle\langle
i_{2}\rangle^{\prime}-\langle i_{2}\rangle\langle
i_{2}\rangle^{\prime}}{[i_{2}-i_{1}]}}{[i_{3}-i_{2}]}\langle
i_{3}\rangle\right|+\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}}>x_{i_{1}}}\left|\frac{\frac{\langle i_{1}\rangle\langle
i_{3}\rangle^{\prime}-\langle i_{3}\rangle\langle
i_{3}\rangle^{\prime}}{[i_{3}-i_{1}]}-\frac{\langle i_{1}\rangle\langle
i_{2}\rangle^{\prime}-\langle i_{3}\rangle\langle
i_{2}\rangle^{\prime}}{[i_{3}-i_{1}]}}{[i_{3}-i_{2}]}\langle
i_{3}\rangle\right|$ $\displaystyle=$ $\displaystyle\sum_{i_{1}\neq i_{2}\neq
i_{3}:\atop x_{i_{3}}>x_{i_{2}}>x_{i_{1}}}\left|\frac{\frac{\langle
i_{1}\rangle-\langle i_{3}\rangle}{[i_{3}-i_{1}]}-\frac{\langle
i_{1}\rangle-\langle i_{2}\rangle}{[i_{2}-i_{1}]}}{[i_{3}-i_{2}]}\langle
i_{2}\rangle^{\prime}\langle i_{3}\rangle\right|+\sum_{i_{1}\neq i_{2}\neq
i_{3}:\atop x_{i_{3}}>x_{i_{2}}>x_{i_{1}}}\left|\frac{\langle
i_{1}\rangle-\langle i_{3}\rangle}{[i_{3}-i_{1}]}\cdot\frac{\langle
i_{3}\rangle^{\prime}-\langle
i_{2}\rangle^{\prime}}{[i_{3}-i_{2}]}\cdot\langle i_{3}\rangle\right|$
The first term is upper bounded by
$O\left(\Delta\cdot\log^{1.5}k\cdot\mbox{$\|{H}\|$}^{3}\right)$ using the same
argument in Lemma 57. The second term can be rephrased as
$\displaystyle\left|\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}}>x_{i_{1}}}\frac{g\left(x_{i_{1}}\right)-g\left(x_{i_{3}}\right)}{x_{i_{3}}-x_{i_{1}}}\cdot\frac{g^{\prime}\left(x_{i_{3}}\right)-g^{\prime}\left(x_{i_{2}}\right)}{x_{i_{3}}-x_{i_{2}}}g\left(x_{i_{3}}\right)G\left(x_{-i_{1}}\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|$
(72) $\displaystyle\leq$ $\displaystyle\left|\sum_{i_{1}\neq i_{2}\neq
i_{3}:\atop x_{i_{3}}>x_{i_{2}}>x_{i_{1}},x_{i_{1}}\geq
0}\left(\cdots\right)\right|+\left|\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}}>x_{i_{1}},x_{i_{1}}<0}\left(\cdots\right)\right|$
For the first summation in Eq. (72), we apply the mean value theorem for both
$g$ and $g^{\prime}$. From Eq. (13) it is upper bounded by
$\displaystyle\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}}>x_{i_{1}},x_{i_{1}}\geq
0}\Delta\left|g^{\prime}\left(x_{i_{1}}\right)g^{\prime}\left(x_{i_{2}}\right)g\left(x_{i_{3}}\right)G\left(x_{-i_{1}}\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|$
$\displaystyle\leq$ $\displaystyle
O\left(\Delta\cdot\mbox{$\|{G^{(2)}\left(x\right)}\|$}_{1}\mbox{$\|{H}\|$}^{3}\right)$
$\displaystyle\leq$ $\displaystyle O\left(\Delta\cdot\log
k\cdot\mbox{$\|{H}\|$}^{3}\right).$
For the second term in Eq. (72), it is not hard to verify that
$\left|\frac{g^{\prime}\left(x_{i_{3}}\right)-g^{\prime}\left(x_{i_{2}}\right)}{x_{i_{3}}-x_{i_{2}}}\right|\leq\Delta\max\left\\{g^{\prime}\left(x_{i_{3}}\right),g\left(x_{i_{2}}\right)\right\\}$
(73)
Further notice that $\left|g^{\prime}\left(\cdot\right)\right|\leq 1$.
Applying the mean value theorem to $g$, we upper bound the second summation in
72 by
$\displaystyle O\left(\Delta\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}}>x_{i_{1}},x_{i_{1}}<0}\max\left\\{g^{\prime}\left(x_{i_{3}}\right),g^{\prime}\left(x_{i_{2}}\right)\right\\}g\left(x_{i_{3}}\right)G\left(x_{-i_{1}}\right)\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
$\displaystyle\leq$ $\displaystyle O\left(\Delta\cdot\log
k\cdot\max_{i_{1}}\cdot\mbox{$\|{G^{(1)}\left(x_{-i_{1}}\right)}\|$}_{1}\cdot\max_{i_{2}}\sum_{i_{3}}\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)$
$\displaystyle\leq$ $\displaystyle
O\left(\Delta\cdot\log^{1.5}k\cdot\mbox{$\|{H}\|$}^{3}\right)$
where the first inequality is from the assumption that
$\left|\left\\{i:x_{i}<0\right\\}\right|\leq 3\log k$ and the second
inequality is from Fact 20 Eq. (38). ∎
### A.4 Upper bounding the second term ($\P$) in Eq. (60)
Let us rewrite ($\P$) as a sum of two term as in Eq. (61). We first upper
bound the first easy term.
###### Lemma 59.
$\left|\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}}>x_{i_{1}}}\frac{\langle i_{3}\rangle^{\prime}-\langle
i_{1}\rangle^{\prime}}{[i_{3}-i_{1}]}\cdot\frac{\langle i_{3}\rangle-\langle
i_{2}\rangle}{[i_{3}-i_{2}]}\cdot\langle i_{3}\rangle\right|\leq
O\left(\Delta\cdot\log^{2}k\cdot\mbox{$\|{H}\|$}^{3}\right)$
###### Proof of Lemma 59.
We split the summation into two cases that $x_{i_{2}}\geq 0$ and
$x_{i_{2}}<0$. For the case that $x_{i_{2}}\geq 0$, we apply the mean value
theorem to $g\left(\cdot\right)$ and Eq. (73), it is upper bounded by
$O\left(\Delta\cdot\log k\cdot\mbox{$\|{H}\|$}^{3}\right)$. For the case that
$x_{i_{2}}<0$, we have $x_{i_{1}}<0$. Note that
$\left|g^{\prime}\left(\cdot\right)\right|\leq 1$. Thus it is upper bounded by
$O\left(\sum_{i_{1}\neq i_{2}:\atop
x_{i_{1}}<x_{i_{2}}<0}\sum_{i_{3}}\left|H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\right)\leq
O\left(\log^{2}k\cdot\mbox{$\|{H}\|$}^{3}\right).$
∎
Next, our goal is to prove an upper bound on the second term in Eq. (61).
###### Lemma 60.
$\left|\sum_{i_{1}\neq i_{2}\neq i_{3}:\atop
x_{i_{3}}>x_{i_{2}}>x_{i_{1}}}\frac{\frac{\langle
i_{3}\rangle^{\prime}-\langle
i_{1}\rangle^{\prime}}{[i_{3}-i_{1}]}-\frac{\langle
i_{2}\rangle^{\prime}-\langle
i_{1}\rangle^{\prime}}{[i_{2}-i_{1}]}}{[i_{3}-i_{2}]}\langle
i_{2}\rangle\langle i_{3}\rangle\right|\leq O\left(\sqrt{\log
k}\cdot\mbox{$\|{H}\|$}^{3}\right).$
Before we prove this lemma, we first prove a “simpler” proposition which will
be crucial in upper bound the above.
###### Proposition 61.
$\left|\sum_{i_{1}\neq i_{2}\neq i_{3}}\frac{\frac{\langle
i_{1}\rangle^{\prime}-\langle
i_{3}\rangle^{\prime}}{[i_{3}-i_{1}]}-\frac{\langle
i_{1}\rangle^{\prime}-\langle
i_{2}\rangle^{\prime}}{[i_{2}-i_{1}]}}{[i_{3}-i_{2}]}\right|\leq
O\left(\Delta^{2}\cdot\sqrt{\log k}\cdot\mbox{$\|{H}\|$}^{3}\right)$
###### Proof.
Using Fact 10,
$\displaystyle\left|\sum_{i_{1}\neq i_{2}\neq
i_{3}}\frac{\frac{g^{\prime}\left(x_{i_{3}}\right)-g^{\prime}\left(x_{i_{1}}\right)}{x_{i_{3}}-x_{i_{1}}}-\frac{g^{\prime}\left(x_{i_{2}}\right)-g^{\prime}\left(x_{i_{1}}\right)}{x_{i_{2}}-x_{i_{1}}}}{x_{i_{3}}-x_{i_{2}}}G\left(x\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|=\frac{1}{\sqrt{2\pi}}\left|\mbox{\rm
Tr}\left[D^{2}\left(e^{-\frac{X^{2}}{2}}\right)[H,H]\cdot
H\right]\right|G\left(x\right),$
where $X=\mbox{\rm diag}\left(x_{1},\ldots,x_{n}\right)$. Using Lemma 12, it
suffices to upper bound
$G\left(x\right)\left|\mbox{\rm
Tr}\left[e^{-\frac{uX^{2}}{2}}\left(XH+HX\right)e^{-\frac{v\left(1-u\right)X^{2}}{2}}\left(XH+HX\right)e^{-\frac{\left(1-v\right)\left(1-u\right)X^{2}}{2}}H\right]\right|$
(74)
and
$G\left(x\right)\left|\mbox{\rm
Tr}\left[e^{\frac{\left(u-1\right)X^{2}}{2}}H^{2}e^{-\frac{uX^{2}}{2}}H\right]\right|$
(75)
Note that $u+v\left(1-u\right)+\left(1-v\right)\left(1-u\right)$=1. At least
two of these three quantities are at most $\frac{1}{2}$. We upper bound Eq.
(74) in the following three cases.
If $u\leq\frac{1}{2}$ and $\left(1-u\right)\left(1-v\right)\leq\frac{1}{2}$,
using Claim 52, Lemma 53 and the fact that
$\mbox{$\|{He^{-X^{2}/2}}\|$}_{2}=\mbox{$\|{e^{-X^{2}/2}H}\|$}_{2}$ as $H$ is
symmetric Eq. (74) is upper bounded by
$\displaystyle\big{\|}\left(XH+HX\right)e^{-\frac{X^{2}}{2}}\big{\|}_{2}^{2}\mbox{$\|{H}\|$}\leq
16\Delta^{2}\mbox{$\|{He^{-X^{2}/2}}\|$}_{2}^{2}\cdot\mbox{$\|{H}\|$}.$
If $u\leq\frac{1}{2}$ and $v\left(1-u\right)\leq\frac{1}{2}$, then the Eq.
(74) is upper bounded by
$\displaystyle\mbox{$\|{\left(XH+HX\right)e^{-\frac{X^{2}}{2}}}\|$}_{2}\cdot\mbox{$\|{He^{-\frac{X^{2}}{2}}}\|$}_{2}\mbox{$\|{XH+HX}\|$}$
$\displaystyle\leq$ $\displaystyle
2\Delta\mbox{$\|{He^{-X^{2}/2}}\|$}_{2}^{2}\cdot\mbox{$\|{XH+HX}\|$}$
$\displaystyle\leq$ $\displaystyle
4\Delta^{2}\mbox{$\|{He^{-X^{2}/2}}\|$}_{2}^{2}\cdot\mbox{$\|{H}\|$}.$
where the second last inequality is by Claim 52. The case that
$u\left(1-v\right)\leq\frac{1}{2}$ and $v\left(1-u\right)\leq\frac{1}{2}$
follows similarly. Also Eq. (75) can be upper bounded with similar arguments.
Thus
$G\left(x\right)\cdot\left|\mbox{\rm
Tr}~{}e^{\left(u-1\right)X^{2}/2}H^{2}e^{-uX^{2}/2}H\right|\leq
16\Delta^{2}G\left(x\right)\mbox{$\|{H}\|$}\cdot\mbox{$\|{He^{-X^{2}/2}}\|$}_{2}^{2}.$
(76)
Therefore,
$\displaystyle\left|\sum_{i_{1}\neq i_{2}\neq
i_{3}}\frac{\frac{g^{\prime}\left(x_{i_{3}}\right)-g^{\prime}\left(x_{i_{1}}\right)}{x_{i_{3}}-x_{i_{1}}}-\frac{g^{\prime}\left(x_{i_{2}}\right)-g^{\prime}\left(x_{i_{1}}\right)}{x_{i_{2}}-x_{i_{1}}}}{x_{i_{3}}-x_{i_{2}}}G\left(x\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|$
$\displaystyle\leq$ $\displaystyle\left(G(x)\cdot\mbox{\rm
Tr}\left[D^{2}\left(e^{-\frac{X^{2}}{2}}\right)[H,H]\cdot H\right]\right)$
$\displaystyle\leq$ $\displaystyle
O\left(\Delta^{2}G\left(x\right)\mbox{$\|{H}\|$}\mbox{$\|{He^{-X^{2}/2}}\|$}_{2}^{2}\right)$
$\displaystyle=$ $\displaystyle
O\left(\Delta^{2}\sum_{i_{1},i_{2}}e^{-x_{i_{2}}^{2}}H_{i_{1},i_{2}}^{2}G\left(x\right)\cdot\mbox{$\|{H}\|$}\right)$
$\displaystyle\leq$ $\displaystyle
O\left(\Delta^{2}\sum_{i_{1},i_{2}}g^{\prime}\left(x_{i_{2}}\right)G\left(x\right)H_{i_{1},i_{2}}^{2}\cdot\mbox{$\|{H}\|$}\right)\quad$
$\displaystyle\leq$ $\displaystyle
O\left(\Delta^{2}\sum_{i_{1},i_{2}}g^{\prime}\left(x_{i_{2}}\right)G\left(x_{-i_{2}}\right)H_{i_{1},i_{2}}^{2}\cdot\mbox{$\|{H}\|$}\right)$
$\displaystyle\leq$ $\displaystyle
O\left(\Delta^{2}\mbox{$\|{G^{(1)}\left(x\right)}\|$}_{1}\cdot\left(\max_{i_{2}}\sum_{i_{1}}H_{i_{1},i_{2}}^{2}\right)\cdot\mbox{$\|{H}\|$}\right)$
$\displaystyle\leq$ $\displaystyle
O\left(\Delta^{2}\mbox{$\|{G^{(1)}\left(x\right)}\|$}_{1}\cdot\max_{i_{2}}\left(H^{2}\right)_{i_{2},i_{2}}\cdot\mbox{$\|{H}\|$}\right)$
$\displaystyle\leq$ $\displaystyle O\left(\Delta^{2}\cdot\sqrt{\log
k}\cdot\mbox{$\|{H}\|$}^{3}\right),$ (78)
where the second inequality used $e^{-x_{i}^{2}/2}\leq 1$, third inequality
used $g(x)\in[0,1]$ and the last inequality is from Fact 20. ∎
We are now ready to prove the main lemma. Note that end of the day we need to
bound the inequality in Lemma 60 which can be written as
$\displaystyle\left|\sum_{i_{1}\neq i_{2}\neq i_{3}\atop
i_{1}<i_{2},i_{1}<i_{3}}\frac{\frac{g^{\prime}\left(x_{i_{1}}\right)-g^{\prime}\left(x_{i_{3}}\right)}{x_{i_{3}}-x_{i_{1}}}-\frac{g^{\prime}\left(x_{i_{1}}\right)-g^{\prime}\left(x_{i_{2}}\right)}{x_{i_{2}}-x_{i_{1}}}}{x_{i_{3}}-x_{i_{2}}}G\left(x_{-\left\\{i_{1}\right\\}}\right)H_{i_{1},i_{2}}H_{i_{2},i_{3}}H_{i_{3},i_{1}}\right|\leq
O\left(\Delta^{2}\cdot\log^{2.5}k\cdot\mbox{$\|{H}\|$}^{3}\right)$ (79)
Observe that in this section we are concerned with
$x_{i_{1}}<x_{i_{2}}<x_{i_{3}}$ so the summation in this lemma and the
equation above are over the same indices.
###### Proof of Lemma 60.
By the paragraph above, proving this lemma is equivalent to proving Eq. (79).
The left hand side of Eq. (79) can be expressed as
$\displaystyle\left|\frac{1}{\sqrt{2\pi}}\sum_{i_{1}}G\left(x_{-i_{1}}\right)\left(\mbox{\rm
Tr}~{}D^{2}\left(e^{-X^{2}/2}\right)\left[A^{i_{1}},\left(A^{i_{1}}\right)^{T}\right]H\right)\right|=\left|\frac{1}{\sqrt{2\pi}}\sum_{i_{1}:x_{i_{1}}<0}\left(\cdots\right)+\frac{1}{\sqrt{2\pi}}\sum_{i_{1}:x_{i_{1}}\geq
0}\left(\cdots\right)\right|$ (80)
For the first summation above, let
$\left(A^{i_{1}}\right)_{i,j}=\begin{cases}H_{i,i_{1}},&\mbox{if $j=i_{1}$ and
$i>i_{1}$}\\\ 0,&\mbox{otherwise}.\end{cases}$
Note that $\mbox{$\|{He^{-X^{2}/2}}\|$}_{2}=\mbox{$\|{e^{-X^{2}/2}H}\|$}_{2}$
as $H$ is symmetric. Using the same argument as Eq. (67), we have
$\max\left\\{\mbox{$\|{A^{i_{1}}e^{-X^{2}/2}}\|$}_{2},\mbox{$\|{\left(A^{i_{1}}\right)^{T}e^{-X^{2}/2}}\|$}_{2},\mbox{$\|{e^{-X^{2}/2}A^{i_{1}}}\|$}_{2},\mbox{$\|{e^{-X^{2}/2}\left(A^{i_{1}}\right)^{T}}\|$}_{2}\right\\}\leq\mbox{$\|{He^{-X^{2}/2}}\|$}_{2}.$
Further notice that
$\mbox{$\|{A^{i_{1}}e^{-X^{2}/2}}\|$}_{2}\leq\mbox{$\|{He^{-X^{2}/2}}\|$}_{2},\mbox{$\|{A^{i_{1}}}\|$}\leq\mbox{$\|{H}\|$},$.
Following the same proof of Proposition 61, we can upper bound the first
summation in Eq. (80) by
$\displaystyle\frac{1}{\sqrt{2\pi}}\sum_{i_{1}:x_{i_{1}}<0}G\left(x_{-i_{1}}\right)\mbox{$\|{He^{-X^{2}/2}}\|$}_{2}^{2}\cdot\mbox{$\|{H}\|$}\leq
O\left(\Delta^{2}\cdot\log^{2.5}k\mbox{$\|{H}\|$}^{3}\right),$
where the inequality follows from the argument in Eq. (69). For the second
summation, define
$B_{i_{1},i_{2}}=\begin{cases}\frac{H_{i_{1},i_{2}}}{\sqrt{g\left(x_{i_{1}}\right)}},&\mbox{if
$x_{i_{1}}\geq 0$}\\\ 0,&\mbox{otherwise}.\end{cases}$
Note that
$\max\left\\{\mbox{$\|{Be^{-X^{2}/2}}\|$}_{2},\mbox{$\|{B^{T}e^{-X^{2}/2}}\|$}_{2},\mbox{$\|{e^{-X^{2}/2}B}\|$}_{2},\mbox{$\|{e^{-X^{2}/2}B^{T}}\|$}_{2}\right\\}\leq\sqrt{2}\mbox{$\|{He^{-X^{2}/2}}\|$}_{2}$
and $\mbox{$\|{B}\|$}\leq\sqrt{2}\mbox{$\|{H}\|$}$ (since $g(x)\geq 1/2$ for
$x\geq 0$). Then the second summation in Eq. (80) is equal to
$\displaystyle\left|G\left(x\right)\frac{1}{\sqrt{2\pi}}\mbox{\rm
Tr}~{}D^{2}\left(e^{-X^{2}/2}\right)\left[B,B^{T}\right]A\right|\leq\frac{1}{\sqrt{\pi}}G\left(x\right)\mbox{$\|{He^{-X^{2}/2}}\|$}_{2}\mbox{$\|{H}\|$}\leq
O\left(\Delta^{2}\cdot\log^{1.5}k\cdot\mbox{$\|{H}\|$}^{3}\right)$
where the inequality follows from the argument in Eq. (69). ∎
|
# Vibrational Quenching of CN- in Collisions with He and Ar
Barry Mant Institute for Ion Physics and Applied Physics, University of
Innsbruck, Technikerstr. 25/3, 6020 Innsbruck, Austria Ersin Yurtsever
Department of Chemistry, Koç University, Rumelifeneri yolu, Sariyer, TR-34450,
Istanbul, Turkey Lola González-Sánchez Departamento de Química Física,
University of Salamanca, Plaza de los Caídos sn, 37008 Salamanca, Spain
Roland Wester Institute for Ion Physics and Applied Physics, University of
Innsbruck, Technikerstr. 25/3, 6020 Innsbruck, Austria Franco A. Gianturco
<EMAIL_ADDRESS>Institute for Ion Physics and Applied Physics,
University of Innsbruck, Technikerstr. 25/3, 6020 Innsbruck, Austria
###### Abstract
The vibrational quenching cross sections and corresponding low-temperature
rate constants for the $\nu=1$ and $\nu=2$ states of CN-(${}^{1}\Sigma^{+}$)
colliding with He and Ar atoms have been computed ab initio using new three
dimensional potential energy surfaces. Little work has so far been carried out
on low-energy vibrationally inelastic collisions for anions with neutral
atoms. The cross sections and rates calculated at energies and temperatures
relevant for both ion traps and astrochemical modelling, are found by the
present calculations to be even smaller than those of the similar
C${}_{2}^{-}$/He and C${}_{2}^{-}$/Ar systems which are in turn of the order
of those existing for the collisions involving neutral diatom-atom systems.
The implications of our finding in the present case rather small computed rate
constants are discussed for their possible role in the dynamics of molecular
cooling and in the evolution of astrochemical modelling networks.
††preprint: AIP/123-QED
## I Introduction
Vibrationally inelastic collisions are fundamental processes in chemical
physics and molecular dynamics. Gas phase collisions which can excite or
quench a vibrational mode in a molecule have been studied both experimentally
and theoretically for decades Takayanagi (1963); Secrest (1973); Krajnovich,
Parmenter, and Catlett (1987); Secrest and Robert Johnson (1966); Eastes and
Secrest (1972) and are generally well understood. Typically the scattering
cross sections and corresponding rates are relatively small Campbell _et al._
(2008) due to the generally large energy spacing between vibrational levels
which require strong interaction forces between the colliding species to
induce transitions. On the other hand, these processes still attract a great
deal of attention and study as they have important applications in fields such
as cold molecules, where collisions are used to quench internal molecular
motion,Kozyryev _et al._ (2015); Caruso _et al._ (2012); Rellergent _et
al._ (2013) or astrochemistry, where accurate rate constants are necessary to
model the evolution of gas clouds and atmospheres.van der Tak _et al._
(2020); Balança and Dayou (2017); Toboła _et al._ (2008); Lique and
Spielfiedel (2007); Lique _et al._ (2006) There are also exceptional systems
such as the dramatic case of BaCl+ \+ Ca where laser cooled calcium atoms can
efficiently quench vibrational motion with rates similar to rotational
transitions. Rellergent _et al._ (2013); Stoecklin _et al._ (2016)
There continues to be many studies of diatom-atom vibrationally inelastic
collisions for both neutral Balança and Dayou (2017); Yang _et al._ (2016);
Kozyryev _et al._ (2015); Kalugina, Lique, and Marinakis (2014) and cationic
species.Iskandarov _et al._ (2017); Stoecklin and Voronin (2011, 2008);
Stoecklin _et al._ (2016) This is to be contrasted by with the case for
anions, where very little work has been carried out on vibrationally inelastic
collision processes. Recently we have tried to change this trend and have
investigated vibrational quenching of the C${}_{2}^{-}$ anion in collisions
with noble gas atoms.Mant _et al._ (2020a) This molecule is of direct
interest as a possible candidate for laser cooling mechanisms Yzombard _et
al._ (2015) but a first step will require the cooling of internal motion via
collisions since spontaneous dipole emission is forbidden for the
rovibrational excited states of this homonuclear species. The cross sections
and rate constants for vibrational transitions were found by our calculations
to be small, i.e. of the order of those for neutral species.
In this article we report the vibrational quenching of yet another important
anion, CN- in collisions with He and Ar atoms. The cyanide anion is a well
studied molecule, particularly its spectroscopic properties have attracted a
great deal of attention and investigations Bradforth _et al._ (1993); Forney,
Thomson, and Jacox (1992); Gottlieb _et al._ (2007); Botschwina (1985);
Peterson and Claude Woods (1987); J and Dateo (1999) as well as the
determination of its photodetachment energyBerkowitz, Chupka, and Walter
(1969); Klein, McGinnis, and Leone (1983); Bradforth _et al._ (1993). Recent
work in our group has further clarified important aspects of its
photodetachment behaviour at threshold from cold trap experiments. Simpson
_et al._ (2020) This molecule has also been detected in the envelope of a
carbon starAgúndez, M. _et al._ (2010) after its rotational constants were
carefully measured.Gottlieb _et al._ (2007) Collisional processes of the
anion with the astrochemically relevant He and H2 speciesGonzález-Sánchez _et
al._ (2020); Kłos and Lique (2011) for rotational transitions have recently
been studied and we have also investigated the rotational cooling of this
molecular anion with He, Ar and H2 as buffer gasses.González-Sánchez _et al._
(2020) The CN- anion is also thought to be an important participant as well in
reactions in the interstellar mediumPetrie (1996); Romanzin _et al._ (2016);
Jerosimić, Gianturco, and Wester (2018); Satta _et al._ (2015) and in the
atmosphere of TitanBiennier _et al._ (2014) where it has been detected.Coates
_et al._ (2007); Vuitton _et al._ (2009)
We note in passing that the corresponding neutral species CN was one of the
first molecules to be detected in spaceMcKellar (1940) and cross sections and
rates for this species have been investigated and obtained for various ro-
vibrational processes in collisions with He and H2.Yang _et al._ (2016);
Burton _et al._ (2018); Lique _et al._ (2010); Lique and Kłos (2011);
Kalugina, Lique, and Kłos (2012); Kalugina, Kłos, and Lique (2013); Kalugina
and Lique (2015) The cyanide cation is also suspected to be important to
astrochemical processes but has yet to be detected. The cation’s vibrational
energies have recently been measuredDoménech _et al._ (2020) as well as a
study has been carried out on its rotational transitions induced by He
collisions.Anusuri (2020)
Vibrationally inelastic collisions involving the CN- molecular anion with
neutral atoms are a type of process rarely studied for such systems. Although
CN- can of course lose energy through spontaneous emission, its wide relevance
justifies providing an accurate assessment of the vibrational quenching
processes involving He and Ar, typical buffer gases in ion traps.
The paper is organised as follows: Section II presents the CN- potential
energy and dipole moment curves along with the anion’s vibrational energy
levels and Einstein A coefficients. The potential energy surfaces for the
CN-/He and CN-/Ar systems are then discussed in Section III. The quantum
scattering methodology is described in Section IV and scattering cross
sections and rates are discussed in Section V. Conclusions are given in
Section VI.
## II CN- Potential Energy Curve and Dipole Moment
Electronic energies for the ground ${}^{1}\Sigma^{+}$ state of the CN- anion
were calculated at 19 internuclear distances $r$ to obtain the anion’s
potential energy curve (PEC). Calculations were carried out using the MOLPRO
suite of quantum chemistry codes Werner _et al._ (2012, 2019) at the CCSD(T)
level of theory Hampel, Peterson, and Werner (1992); Deega and Knowles (1994)
employing an aug-cc-pV5Z basis set. Woon and Dunning Jr (1993, 1994) The
expectation value of the non-relaxed CCSD dipole moment at each $r$ distance
was also obtained. The ab initio energies and dipole moment curve (DMC) for
CN- are shown in Fig. 1.
Figure 1: Ab initio energies, PEC fit and DMC for CN- (${}^{1}\Sigma^{+}$).
The horizontal lines show the first six vibrational energies.
The LEVEL program Le Roy (2017) was used to obtain the vibrational energies
and wavefunctions, for the CN- molecule. The ab initio energies were used as
input, interpolated using a cubic spline and extrapolated to $r$ values below
and above the range of calculated energies using functions implemented in
LEVEL. The relative energies of the first three vibrational levels along with
the rotational constants for each state are shown in Table 1 and compared with
previously published calculated theoretical and experiment values. The
agreement with previous calculations and experimental values is quite good and
certainly sufficient to evaluate the cross sections and rates constants of
inelastic collisions considered below.
Table 1: Comparison of vibrational energies and rotational constants with previous theoretical and experimental values. Literature values calculated from Dunham parameters provided. Units of cm-1. | | Relative energy | $B_{\nu}$
---|---|---|---
$\nu_{0}$ | This work | 0 | 1.864
| Calc. J and Dateo (1999) | 0 | 1.868
| Exp. Gottlieb _et al._ (2007) | 0 | 1.872
$\nu_{1}$ | This work | 2040 | 1.845
| Calc. J and Dateo (1999) | 2045 | 1.851
| Exp. Bradforth _et al._ (1993) | 2035 ($\pm$ 40) |
| Exp. Forney, Thomson, and Jacox (1992) | 2053 (Neon) |
$\nu_{2}$ | This work | 4055 | 1.831
| Calc. J and Dateo (1999) | 4065 | 1.834
We have recently evaluated the dipole moment of CN- at its equilibrium bond
length $r_{eq}$ using a variety of ab initio methods and basis sets González-
Sánchez _et al._ (2020) and used it to evaluate the Einstein A coefficients
for pure rotational transitions. The best estimate of that work of 0.71 D is
in quite good agreement with the value of the DMC at $r_{e}$ of 0.65 D
computed here. The LEVEL program was also used to calculate the Einstein A
coefficients for ro-vibrational transitions of CN- using the ab initio
calculated DMC. The values of
$A_{\nu^{\prime}j^{\prime},\nu^{\prime\prime}j^{\prime\prime}}$ for the first
two vibrational states of the anion are shown in Table 2 and compared to those
of neutral CN. Brooke _et al._ (2014) The values for the anion and neutral
molecule are broadly similar which is reasonable considering they have very
similar bond lengths and vibrational energies. Brooke _et al._ (2014) The
slightly larger values for neutral CN are a result of the larger dipole moment
for the neutral molecule. Brooke _et al._ (2014)
Table 2: Einstein A coefficients $A_{\nu^{\prime},\nu^{\prime\prime}}$ for selected CN- (${}^{1}\Sigma^{+}$) vibrational transitions compared to those for neutral CN (${}^{2}\Sigma^{+}$) calculated by Brooke et al. Brooke _et al._ (2014) For CN- the P(1) branch values were used to compare to the Q-branch values for CN. Units of s-1. Transition | CN- | CN
---|---|---
$\nu_{1}\rightarrow\nu_{0}$ | 6.60 | 8.85
$\nu_{2}\rightarrow\nu_{1}$ | 12.50 | 16.50
$\nu_{2}\rightarrow\nu_{0}$ | 0.36 | 0.66
## III CN-/He and CN-/Ar Potential Energy Surfaces and Vibrationally
Averaged Matrix Elements
Figure 2: Contour plots for CN-(${}^{1}\Sigma^{+}$)/He (left) and
CN-(${}^{1}\Sigma^{+}$)/Ar (right) of vibrationally averaged matrix elements
$V_{0,0}(R,\theta)$ (top) and $V_{0,1}(R,\theta)$ (bottom) projected onto
Cartesian coordinates. Energies in cm-1. See main text for further details
The interaction energies between CN- in its ground ${}^{1}\Sigma^{+}$
electronic state with He and Ar atoms were calculated using ab initio methods
implemented in the MOLPRO suite of codes.Werner _et al._ (2012, 2019)
Geometries were defined on a Jacobi grid with $R$ (the distance from the
centre of mass of CN- to the atom) ranging from 2.5 to 20 Å and $\theta$ (the
angle between $R$ and the CN- internuclear axis $r$) from 0 (C side) to 180∘
in 15∘ and 10∘ intervals for He and Ar respectively. Seven values of the CN-
bond length for each system between $r=1.00$-1.42 Å were used including the
equilibrium value of $r_{eq}=1.181431$ Å. This is sufficient to cover the
vibrational levels of interest in the present study. Interaction potential
energies between CN- and the noble gas atoms were determined by subtracting
the asymptotic energies for each bond length.
For CN-/He, energies were calculated using the Multi-configurational self-
consistent field (MCSCF) method Werner and Knowles (1985); Knowles and Werner
(1985) with 8 occupied orbitals and 2 closed orbitals followed by a 1-state
multi-reference configuration interaction (MRCI) Shamasundar, Knizia, and
Werner (2011) calculation. An aug-cc-pV5Z basis Kendall, Dunning Jr, and
Harrison (1992) was employed. In our earlier discussion of the CN-/He PES
González-Sánchez _et al._ (2020) we discuss in detail the reasons why we
followed both methods for this system and compared the CASSCF+MRCI results
with the CCSD(T) with similar basis set expansions, finding them to be
coincident in values. In particular, we corrected for the size-consistency
possible shortcomings of the CASSCF+MRCI vs the CCSD(T) methods by correcting
the latter results using the Davidson´s correction as implemented in MOLPRO.
In our earlier workGonzález-Sánchez _et al._ (2020) we showed that this
correction brought the two sets of potential calculations to yield the same
potential values over a broad range of the employed grid. As an example, we
note here that from our CBS (Complete Basis Set) extrapolated CCSD(T)
calculations on CN-/He system we find the minimum energy configuration as
theta=40 deg., R=3.95 Å with BSSE corrected energy at 49.522 cm-1. CBS is
calculated by the default procedure in MOLPRO: it is the so-called L3
extrapolation discussed in there. The results within the CAS(8,4) within the
CASSCF+MRCI gave a theta=40 deg., R=4.00 and an energy of 50.39 cm-1 for its
minimum configuration, showing the two methods to provide essentially the same
results.
For the CN-/Ar system, energies were calculated using the CCSD(T) method Deega
and Knowles (1994) with complete basis set (CBS) extrapolation using the aug-
cc-pVTZ, aug-cc-pVQZ and aug-cc-pV5Z basis sets. Wilson and van Mourik (1996);
Woon and Dunning Jr (1993) The basis-set-superposition-error (BSSE) was also
accounted for all calculated points using the counterpoise procedure. Boys and
Bernardi (1970)
The three-dimensional PESs were fit to an analytical form using the method of
Werner, Follmeg and Alexander Werner, Follmeg, and Alexander (1988); Balança
and Dayou (2017) where the interaction energy is given as
$V_{\mathrm{int}}(R,r,\theta)=\sum_{n=0}^{N_{r}-1}\sum_{l=0}^{N_{\theta}-1}P_{l}(\cos\theta)A_{ln}(R)(r-r_{eq})^{n},$
(1)
where $N_{r}$ = 7 and $N_{\theta}$ = 13 or 19 respectively are the number of
bond lengths $r$ and angles $\theta$ in the ab initio grid,
$P_{l}(\cos\theta)$ are the Legendre polynomials and $r_{eq}=1.181431$ Å is
the equilibrium bond length of CN-. For each bond length $r_{m}$ and angle
$\theta_{k}$, one-dimensional cuts of the PESs
$V_{\mathrm{int}}(R,r_{m},\theta_{k})$ were fit to
$\displaystyle
B_{km}(R)=\exp(-a_{km}R)\left[\sum_{i=0}^{i_{\mathrm{max}}}b_{km}^{(i)}R^{i}\right]$
$\displaystyle-\frac{1}{2}\left[1+\tanh(R)\right]\left[\sum_{j=j_{\mathrm{min}}}^{j=j_{\mathrm{max}}}c_{km}^{j}R^{-j}\right],$
(2)
where the first terms account for the short range part of the potential and
the second part for the long range terms combined using the
$\frac{1}{2}\left[1+\tanh(R)\right]$ switching function. For each $r_{m}$ and
$\theta_{k}$ Eq. 2 was least squares fit to the ab initio data (around 40 $R$
points) using $i_{\mathrm{max}}=2$, $j_{\mathrm{min}}=4$ and
$j_{\mathrm{max}}=10$ for eight variable parameters. The average root-mean-
square error (RMSE) for each fit was 0.21 cm-1 for CN-/He and 0.27 cm-1 for
CN-/Ar. From the 1D potential fits $B_{km}(R)$, the radial coefficients
$A_{ln}(R)$ can be determined from the matrix product
$\mathbf{A}(R)=\mathbf{P}^{-1}\mathbf{B}(R)\mathbf{S}^{-1}$ where the matrix
elements of $\mathbf{P}$ and $\mathbf{S}$ are given as
$P_{kl}=P_{l}(\cos\theta_{k})$ and $S_{nm}=(r_{m}-r_{eq})^{n}$ respectively.
The analytical representation of the PES, Eq. 1, gives a reasonable
representation of the ab initio interaction energies. An overall RMSE of 82
cm-1 for all points used in the fit was obtained for CN-/He but this drops to
0.26 cm-1 for $V<500$ cm-1. For CN-/Ar an overall RMSE of 21 cm-1 was
obtained, a value which went down to 1.5 cm-1 for $V<500$ cm-1.
The scattering calculations described in the next section require the
interaction potential to be averaged over the vibrational states of CN-
$\chi_{\nu}(r)$, which were obtained from LEVEL as described in Section II, as
$V_{\nu,\nu^{\prime}}(R,\theta)=\langle\chi_{\nu}(r)|V_{\mathrm{int}}(R,r,\theta)|\chi_{\nu^{\prime}}(r)\rangle.$
(3)
Fig. 2 shows the diagonal terms $V_{0,0}(R,\theta)$ for both systems. As
expected for a molecule with a strong bond, so that the ground state
vibrational wavefunction is strongly peaked around $r_{eq}$, the contour plots
of the $V_{0,0}(R,\theta)$ for each system are very similar to our earlier
rigid-rotor (RR) PESs which were obtained without the vibrational averaging.
González-Sánchez _et al._ (2020, 2020) Both system’s PES have a fairly
similar appearance with the most attractive part of the potential located on
the nitrogen end of CN-. The well depth is the main difference which increases
as expected from He to Ar due to the increasing number of electrons on the
atoms and on the much larger dipole polarizabiliy that dominates the long-
range attractive terms with a value of 1.383 $a_{0}^{3}$ for He and 11.070
$a_{0}^{3}$ for Ar. Gaiser and Fellmuth (2018)
The off diagonal $V_{0,1}(R,\theta)$ terms which directly drive vibrationally
inelastic $\nu=1$ to $\nu=0$ transitions are also shown in Fig. 2. At short
distances the coupling terms are repulsive, becoming negligible rather quickly
at longer distances, as is the case for many other atom-diatom systems where
the vibrational coupling features are largely short-range coupling regions.
The interaction of CN- with Ar is more repulsive at close range and for a
broader range of geometries than is the case for He. These findings suggest
already that low-energy collisions with Ar will be likely to induce larger
vibrational cross sections than for the same collisions involving He atoms.
Such expected behaviour will be in fact confirmed below by our actual
calculations.
The PESs for CN-/He and CN-/Ar can be compared to similar systems such as
C${}_{2}^{-}$/He and C${}_{2}^{-}$/Ar which we have recently investigated.Mant
_et al._ (2020a) The location of the minimum interaction energy for both
anions interacting with He and Ar respectively are very similar with the main
different being the perpendicular angle of the well for C${}_{2}^{-}$. The
off-diagonal matrix elements for these systems are also similar in magnitude
and range but being slightly larger for the interaction of He and Ar with
C${}_{2}^{-}$, explaining the larger quenching rates for this anion (see
below). The PES for the corresponding neutral systems CN/He and CN/Ar which
were reported by Saidani et al. can also be compared.Saidani _et al._ (2013)
In this case the well depth for He interacting with both CN and CN- is similar
but for Ar the interaction with the anion is somewhat weaker. As expected the
interaction potential for He and Ar interacting with the anion extends further
than the corresponding neutral systems. The off-diagonal elements for the
neutral and anionic systems are broadly similar.
The close-coupling (CC) scattering calculations to be discussed in the next
section require to have the vibrationally averaged matrix elements in the form
of the familiar multipole expansion given as
$V_{\nu,\nu^{\prime}}(R,\theta)=\sum_{\lambda}^{\lambda_{\rm{max}}}V_{\nu,\nu^{\prime}}^{\lambda}(R)P_{\lambda}(\cos\theta).$
(4)
Fig. 3 shows the multipole expansion coefficients for the first three
$V_{0,0}(R,\theta)$ terms for both systems. As anticipated from the broad
spatial similarity of the contour plots, the multipole expansion for the
vibrationally averaged matrix elements are very close to those obtained from
considering the anion as a rigid rotor. This justifies our previous treatment
of purely rotationally inelastic transitions where we considered the anion to
behave as a rigid rotor (RR)González-Sánchez _et al._ (2020, 2020) and we
refer the reader to these works for a discussion of pure rotational
transitions.
It is also worthy of note about the diagonal coupling matrix elements reported
in that Fig. 3 how the much more polarizable Ar projectile gives the three
lowest multipolar terms as attractive contributions to the interaction,
thereby indicating that their collective effects during the interaction would
be to draw the heavier partner closer to the anion. On the other hand, the
same three coefficients for the lighter He partner (left-hand panel in Fig. 3)
exhibit much shallower attractive wells and only for two of the coefficients,
with the $\lambda$ = 1 coefficient showing instead a slightly repulsive
behaviour at intermediate distances.
Figure 3: $V_{0,0}^{\lambda}(R)$ expansion coefficients for $\lambda$ = 0, 1
and 2 terms for CN-/He (left) and CN-/Ar (right). The rigid rotor (RR) values
are also plotted as dashed lines but essentially overlap the vibrationally
averaged coefficients discussed in the present work.
The off-diagonal expansion coefficients $V_{0,1}^{\lambda}$ are shown in Fig.
4. All terms quickly approach zero as $R$ is increased. For both systems the
$V_{0,1}^{\lambda}(R)$ coefficients are mostly steeply repulsive as $R$
decreases. As expected from the contour plots, the $V_{0,1}^{\lambda}(R)$
terms are seen to be much more repulsive for the CN-/Ar interaction, with
their turning points located at larger distances than happens for the He
partner. Such features of the interactions again suggest a larger dynamical
vibrational inelasticity for the case of Ar atoms than for the He collision
partners.
Figure 4: $V_{1,0}^{\lambda}(R)$ expansion coefficients for $\lambda$ = 0, 1
and 2 terms for CN-/He (left) and CN-/Ar (right).
## IV Quantum Scattering Calculations
Quantum scattering calculations were carried out using the coupled channel
(CC) method to solve the Schrödinger equation for scattering of an atom with a
diatomic molecule as implemented in our in-house code, ASPIN. López-Duránn,
Bodo, and Gianturco (2008) The method has been described in detail many times
before, from one of its earliest, now classic formulations Arthurs and
Dalgarno (1960) to one of its more recent, computation-oriented visitation
from our own work López-Duránn, Bodo, and Gianturco (2008). Therefore, only a
brief summary of the method will be given here with all equations given in
atomic units. By starting with the form employed for any given total angular
momentum $\mathbf{J=l+j}$ the scattering wavefunction is expanded as
$\Psi^{JM}(R,r,\Theta)=\frac{1}{R}\sum_{\nu,j,l}f_{\nu
lj}^{J}(R)\chi_{\nu,j}(r)\mathcal{Y}_{jl}^{JM}(\hat{\mathbf{R}},\hat{\mathbf{r}}),$
(5)
where $l$ and $j$ are the orbital and rotational angular momentum
respectively, $\mathcal{Y}_{jl}^{JM}(\hat{\mathbf{R}},\hat{\mathbf{r}})$ are
coupled-spherical harmonics for $l$ and $j$ which are eigenfunctions of $J$.
$\chi_{\nu,j}(r)$ are the radial part of the ro-vibrational eigenfunctions of
the molecule. The values of $l$ and $j$ are constrained, via Clebsch-Gordan
coefficients, such that their final summation is compatible with the specific
total angular momentum $J$ one is considering. Arthurs and Dalgarno (1960);
López-Duránn, Bodo, and Gianturco (2008) $f_{\nu lj}^{J}(R)$ are the radial
expansion functions which need to be determined from the propagation of the
radial coupled equations.
Substituting the expansion into the Schrödinger equation with the Hamiltonian
for atom-diatom scattering as defined in detail in Arthurs and Dalgarno
(1960); López-Duránn, Bodo, and Gianturco (2008), leads to the CC equations
for each contributing $J$
$\left(\frac{d^{2}}{dR^{2}}+\mathbf{K}^{2}-\mathbf{V}-\frac{\mathbf{l}^{2}}{R^{2}}\right)\mathbf{f}^{J}=0.$
(6)
Here each element of $\mathbf{K}=\delta_{i,j}2\mu(E-\epsilon_{i})$ (where
$\epsilon_{i}$ is the channel asymptotic energy), $\mu$ is the reduced mass of
the system, $\mathbf{V}=2\mu\mathbf{U}$ is the interaction potential matrix
between channels and $\mathbf{l}^{2}$ is the matrix of orbital angular
momentum. For the ro-vibrational scattering calculations of interest in the
present study,the matrix elements $\mathbf{U}$ are given explicitly as
$\displaystyle\langle\nu
jlJ|V|\nu^{\prime}j^{\prime}l^{\prime}J\rangle=\int_{0}^{\infty}\mathrm{d}r\int\mathrm{d}\hat{\mathbf{r}}\int\mathrm{d}\hat{\mathbf{R}}$
$\displaystyle\chi_{\nu,j}(r)\mathcal{Y}_{jl}^{JM}(\hat{\mathbf{R}},\hat{\mathbf{r}})^{*}|V(R,r,\theta)|\chi_{\nu^{\prime},j^{\prime}}(r)\mathcal{Y}_{j^{\prime}l^{\prime}}^{JM}(\hat{\mathbf{R}},\hat{\mathbf{r}}).$
(7)
Since the intermolecular potential $V(R,r,\theta)$ is expressed as in Eq. 4,
then Eq. 7 can be written as
$\langle\nu
jlJ|V|\nu^{\prime}j^{\prime}l^{\prime}J\rangle=\sum_{\lambda=0}^{\infty}V_{\nu,\nu^{\prime}}^{\lambda}(R)f^{J}_{\lambda
jlj^{\prime}l^{\prime}},$ (8)
where the $f^{J}_{\lambda jlj^{\prime}l^{\prime}}$ terms are the Percival-
Seaton coefficients
$f^{J}_{\lambda
jlj^{\prime}l^{\prime}}=\int\mathrm{d}\hat{\mathbf{r}}\int\mathrm{d}\hat{\mathbf{R}}\quad\mathcal{Y}_{jl}^{JM}(\hat{\mathbf{R}},\hat{\mathbf{r}})^{*}P_{\lambda}(\cos\theta)\mathcal{Y}_{j^{\prime}l^{\prime}}^{JM}(\hat{\mathbf{R}},\hat{\mathbf{r}}),$
(9)
for which analytical forms are known. López-Duránn, Bodo, and Gianturco (2008)
Eq. 8 also makes use of the widely known approximation
$V_{\nu,\nu^{\prime}}^{\lambda}(R)\approx V_{\nu
j\nu^{\prime}j^{\prime}}^{\lambda}(R),$ (10)
for all $j$ such that the effect of rotation on the vibrational matrix
elements is ignored for reasons that shall be further discussed below.
The CC equations are propagated outwards from the classically forbidden region
to a sufficient distance where the scattering matrix $\mathbf{S}$ can be
obtained. The inelastic ro-vibrational state-changing cross sections are
obtained as
$\sigma_{\nu j\rightarrow\nu^{\prime}j^{\prime}}=\frac{\pi}{(2j+1)k_{\nu
j}^{2}}\sum_{J}(2J+1)\sum_{l,l^{\prime}}|\delta_{\nu
lj,\nu^{\prime}l^{\prime}j^{\prime}}-S^{J}_{\nu
lj,\nu^{\prime}l^{\prime}j^{\prime}}|^{2}.$ (11)
To converge the CC equations, a rotational basis set was also used: for both
systems it included up to $j=20$ rotational functions for each vibrational
state. The CC equations were propagated between 1.7 and 100.0 Å using the log-
derivative propagator Manolopoulos (1986) up to 60 Å and the variable-phase
method at larger distances. Martinazzo, Bodo, and Gianturco (2003) The
potential energy was interpolated between calculated
$V_{\nu,\nu^{\prime}}^{\lambda}(R)$ values using a cubic spline. For $R<2.5$ Å
the $V_{\nu,\nu^{\prime}}^{\lambda}(R)$ were extrapolated as
$\frac{a_{\lambda}}{R}+b_{\lambda}R$ while for $R>20$ Å the $\lambda=0$ terms
were extrapolated as $\frac{c}{R^{4}}+\frac{d}{R^{6}}$. As our ab initio
calculated interaction energies were computed to $R=25$ Å where the
interaction energy is negligible for the temperatures of interest here, the
extrapolated form has also a negligible effect on cross sections. Mant _et
al._ (2020b)
A number of parameters of the calculation were checked for convergence. The
scattering cross sections differed by around 10-15% on going from 10 to 19
$\lambda$ terms. This is less precise than for rotationally inelastic cross
sections where convergence to around 1% is typical and is due to the very
small cross sections for these processes which makes obtaining precise and
stable values more difficult to achieve. For production calculations, 10
$\lambda$ terms were included for each $V_{\nu,\nu^{\prime}}(R)$ as a
compromise between accuracy and computational time. The effect of the
vibrational basis set was also considered. It was found that for the $\nu=1$
and $\nu=2$ levels, which are the states of interest here (see next section),
it was sufficient to only include these states. Including the $\nu=3$ state
had a negligible effect on the $\nu=1$ and $\nu=2$ quenching cross sections.
The rotational $j=20$ basis gave convergence to better than 1% for the CN-/He
while for CN-/Ar convergence to about 10% was achieved.
Scattering calculations were carried out for collision energies between 1 and
1000 cm-1 using steps of 0.1 cm-1 for energies up to 100 cm-1, 0.2 cm-1 for
100-300 cm-1, 1.0 cm-1 for 300-500 cm-1 and 10.0 cm-1 for 500-1000 cm-1. This
energy grid was used to ensure that important features such as resonances
appearing in the cross sections were accounted for and their contributions
included when the corresponding rates were calculated. At low collision
energies, the positions and widths of such resonances will be very sensitive
to the details of the PES.
For CN-/Ar the number of partial waves was increased with increasing energy as
usual, requiring $J=120$ for the highest energies considered. For the CN-/He
system however, inverse behaviour was encountered: at low scattering energies
below 100 cm-1, more partial was were required (up to $J=80$) to converge the
vibrationally inelastic partial cross sections than at higher energies where
only up to $J=35$ was required. We suspect this is due to the very small cross
sections so that at low energies it becomes difficult to converge the
calculations as all partial waves contribute uniformly very small values so
that many more of them need inclusion for an acceptable convergent behaviour
to occur.
Vibrationally inelastic cross sections were computed for the $\nu=1$ and
$\nu=2$ states of CN- for collisions with He. Due to time and memory
constraints, only $\nu=1$ states of CN- were considered for Ar collisions. We
think, however, that such calculations are already sufficient for our results
to make convincingly our main points, as discussed further below.
## V Vibrationally Inelastic Cross Sections & Rate Coefficients
Figure 5: Scattering cross sections for vibrationally inelastic collisions of
CN- with He and Ar. Figure 6: Rate constants $k_{\nu\to\nu^{\prime}}(T)$ for
vibrationally inelastic transitions in CN-/He and Ar collisions. Also shown
are the corresponding values for C${}_{2}^{-}$/He and Ar.Mant _et al._
(2020a)
Fig. 5 compares vibrationally inelastic rotationally elastic (for
$j=j^{\prime}=0$) cross sections for the de-excitation
$\nu=1\rightarrow\nu=0$, $\nu=2\rightarrow\nu=1$ and $\nu=2\rightarrow\nu=0$
transitions for CN- colliding with He and $\nu=1\rightarrow\nu=0$ for CN-/Ar.
At low collisions energies below 100 cm-1 the cross sections for He are very
small, orders of magnitude less than rotationally inelastic collisions for
this system. González-Sánchez _et al._ (2020) The cross sections show
resonances at lower collision energies due to shape and/or Feshbach
resonances. As expected due to the larger energy difference, the
$\nu=2\rightarrow\nu=0$ process is smaller than the $\nu=2\rightarrow\nu=1$
and $\nu=1\rightarrow\nu=0$ cross sections. At collision energies above 100
cm-1, the cross sections rapidly increase in value, a behaviour typically
observed also in other systems for vibrationally inelastic cross sections.
Lique _et al._ (2006); Lique and Spielfiedel (2007); Toboła _et al._ (2008);
Balança and Dayou (2017)
The CN-/Ar cross sections are found to be about four orders of magnitude
larger than those we have obtained for He at lower energies, also showing many
distinct resonance features which are brought about by the presence of a
stronger interaction with the molecular anion. The detailed analysis of such a
forest of resonances would also be interesting and perhaps would be warranted
in the case of existing experimental data on such processes, of which we are
not aware till now, but would require a substantial extension of the present
work. Thus, we do not intend to carry it out now, being somewhat outside the
main scope of the present study, and are leaving it for future extension of
this study in our own laboratory.
The far larger cross sections we found for the Ar projectile are obviously a
consequence of the deeper attractive well for the $V_{\nu,\nu}(R,\theta)$
diagonal matrix elements and the larger off-diagonal
$V_{\nu,\nu^{\prime}}(R,\theta)$ matrix elements (see Fig. 2), i.e. they stem
from distinct differences in the strengths of the coupling potential terms
that drive the inelastic dynamics for the Ar collision partner.
The general features of the vibrationally inelastic cross sections shown in
Fig. 5 are indeed similar to those which we have obtained earlier for the
C${}_{2}^{-}$ anion colliding with He, Ne and Ar set of systems that we have
recently studied. Mant _et al._ (2020a) For both of the anions, we have found
that the vibrational quenching cross sections with He are uniformly very
small, while we also found that they increase by orders of magnitude when the
larger and more polarizable Ar atom becomes the collisional partner for either
of these anionic molecules. Although such general behaviour could be
reasonably expected from what we know in these systems about their interaction
forces, it is nevertheless reassuring to obtain quantitative confirmation on
the extent of the size differences from detailed, and in principle exact,
scattering calculations.
The computed inelastic cross sections of the previous section can in turn be
used to obtain the corresponding thermal rate constants over ranges of
temperature of interest for placing the present anion in cold environments.
The corresponding $k_{\nu\to\nu^{\prime}}(T)$ can be evaluated, in fact, as
the convolution of the computed inelastic cross sections over a Boltzmann
distribution of the relative collision energies of the interacting partners as
$k_{\nu\to\nu^{\prime}}(T)=\left(\displaystyle\frac{8}{\pi\mu
k_{B}^{3}T^{3}}\right)^{1/2}\int_{0}^{\infty}E_{c}\sigma_{\nu\to\nu^{\prime}}(E_{c})e^{-E_{c}/k_{B}T}dE_{c}$
(12)
where $E_{c}=\mu v^{2}/2$ is the kinetic energy in the collision calculations.
The rate constants were computed between 5 and 100 K in 1 K intervals. Fig. 6
shows the rates for vibrationally inelastic rotationally elastic
($j=j^{\prime}=0$) transitions corresponding to the cross sections in Fig. 5.
The figure also shows rates for the corresponding transitions of the similar
C${}_{2}^{-}$/He and Ar systems. For CN-/He the rate constants for vibrational
quenching are very small, even lower than those for C${}_{2}^{-}$/He and
around nine orders of magnitude lower than those for CN-/He rotationally
inelastic collisions. González-Sánchez _et al._ (2020) For CN-/Ar the
$\nu=1\rightarrow\nu=0$ rate constants are about four orders of magnitude
larger than those for He but about three orders of magnitude less than for the
corresponding transition for C${}_{2}^{-}$/Ar. The $\nu=2\rightarrow\nu=1$
rate constants for CN-/He is broadly similar to those for
$\nu=1\rightarrow\nu=0$ while as expected the $\nu=2\rightarrow\nu=0$ rate
constants are slightly smaller.
The increase in rate constants on going from He to Ar in collisions are
similar to what was found for rotationally inelastic collisions for CN-
González-Sánchez _et al._ (2020) and C${}_{2}^{-}$ Mant _et al._ (2020c) and
as shown in Fig. 6, vibrationally inelastic collisions for C${}_{2}^{-}$. Mant
_et al._ (2020a) This trend, while seemingly in expectation with the stronger
interaction potential for the larger atom is not easy to predict a priori.
Kato et al. and Ferguson measured vibrational quenching rates for
N${}_{2}^{+}$ in collisions with He, Ne, Ar, Xe and Kr Kato, Bierbaum, and
Leone (1995) and O${}_{2}^{+}$ with He, Ne and Ar atoms Ferguson (1986)
respectively at 300 K. For both cations, quenching rates increased with the
size of atom, suggesting that the polarizability of the colliding atom plays
an important role. In contrast Saidani et al calculated quenching rates for CN
with He and Ar over a wide range of temperatures and found that cross sections
and rates for Ar were orders of magnitude lower than those for He. Saidani
_et al._ (2013) However, ionic interactions are driven by different forces
than those acting between neutrals, so it is not obvious how such a result
relates to the present findings for an anion. Analytical models can also be
used to gain insight into vibrational quenching such as the work of
Dashevskaya et al. where the quenching rate for $\nu=1\rightarrow\nu=0$ for
N2/He was calculated over a large range of temperatures from 70-3000 K.
Dashevskaya _et al._ (2006) The rates obtained were in good agreement with
experiment and similar to those found here for CN-/He at 100 K. It would be
interesting to apply these models to the anion-neutral collisions of interest
here.
The work we have presented here, and the similar findings from our previous
study Mant _et al._ (2020a) on a different diatomic anion like C${}_{2}^{-}$,
strongly suggests that the process of vibrational inelasticity in multiply
bonded anionic molecules by low-T collisions with neutral noble gases is
rather inefficient. The rates for quenching found here are even smaller than
those we had found earlier for C${}_{2}^{-}$, also uniformly smaller than
those known for many neutral diatomic molecules and cationsMant _et al._
(2020a).
The quenching rates and Einstein A coefficients which we have mentioned and
shown earlier in this work, can be used to consider the properties of the
critical density $n_{\text{crit}}^{i}(T)$ for CN- vibrations which is given as
$n_{\text{crit}}^{i}(T)=\frac{A_{ij}}{\sum_{j\neq i}k_{ij}(T)}.$ (13)
This quantity gives the gas density values which would be required so that
collisional state-changing processes match in size those which lead to
collision-less emission via spontaneous decay. It is used in astronomical
contexts to asses the possible densities required for local thermal
equilibrium (LTE) to be reached and are usually applied for rotational
transitions in molecules that can occur in the interstellar medium (ISM).
In the present case of CN-/He, when we apply Eq. 13 to the $\nu=1$ and $\nu=2$
vibrational state-changing process gives $n_{\text{crit}}^{i}(T)\approx
10^{19}-10^{20}$ cm-3 at 100 K. Current kinetics models which describe the
density conditions in molecular clouds indicate a wide variety of densities
being present: from diffuse molecular clouds estimated at around 102 cm-3 to
dense molecular clouds which are considered to be between 103 \- 106 cm-3.
Snow and B.J. (2006); Agúndez and Cernicharo (2006) The critical density
obtained here for the vibrational decay of CN- interacting with environmental
He atoms was found to be orders of magnitude larger than those expected in the
ISM regions where CN- has been detected, clearly suggesting that thermal
equilibrium for these process will likely never be attained and that the
presently computed radiative transitions determine that CN- populates
essentially only the ground vibrational level in the ISM.
## VI Conclusions
The cross sections and corresponding rate constants for vibrationally
inelastic transitions of CN- colliding with He and Ar atoms have been
calculated using new ab initio potential energy surfaces. As for atom-diatom
vibrationally inelastic collisions, the rate constants for both CN-/He and
CN-/Ar are very small, even smaller than those for corresponding values of the
similar C${}_{2}^{-}$/He and C${}_{2}^{-}$/Ar systems. Although more work is
required before definitive conclusions can be drawn, it appears from the
present calculations that vibrationally inelastic collisions of molecular
anions with neutral atoms (or at least noble gas atoms) are similar to neutral
molecule-atom collisions in that they generate similarly small transition
probabilities and their collision mechanisms for transferring relative energy,
at sub-thermal and thermal conditions, to the vibrational internal motion of
the anion is rather inefficient. This is in contrast with the generally more
efficient collisional energy transfer probabilities which are found for
molecular cation-atom systems in the current literature Mant _et al._
(2020a).
For the anion of interest here this is not a crucial concern when wanting to
find alternative paths which are more efficient in cooling its internal
vibrational motion, since CN- can dissipate energy through spontaneous dipole
emission (Section II). On the other hand, in the case of homonuclear anions
such as C${}_{2}^{-}$ (of current interest for laser cooling cycles in cold
traps Yzombard _et al._ (2015)) where this process is forbidden, collisions
are likely to be the primary means for quenching its vibrational motion. In
such cases high gas pressures and the use of larger noble buffer gases seem to
be required.
The present calculations confirm that collisional energy transfer paths which
involve vibrational degrees of freedom for a molecular anion under cold trap
conditions are invariably very inefficient and are several orders of magnitude
smaller that the collisional energy-changing paths which involve their
rotational degrees of freedom. One can therefore safely estimate that these
two paths to energy losses are markedly decoupled with one another and can be
treated on a separate footing within any kinetics modelling of their
behaviour.
###### Acknowledgements.
We acknowledge the financial support of the Austrian FWF agency through
research grant n. P29558-N36. One of us (L.G-S) further thanks MINECO (Spain)
for grant PGC2018-09644-B-100.
## VII Data Availability Statement
Fortran programs and subroutines for the CN-/He and CN-/Ar PESs used are
available in the Supplementary Material along with the vibrational coupling
coefficients and vibrational quenching rate constants.
## VIII Supplementary material
The multipolar coefficients for the Legendre expansion of the new vibrational
PESs for CN-/He and CN-/Ar are provided via Fortran program routines, as well
as the coupling coefficients for the vibrational dynamics. They are all are
available as Supplementary Material to the present publication. That
Supplementary Material also contains subroutines for the inelastic and elastic
rate coefficients for the two systems studied in the present paper.
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|
∎
e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>e3e-mail:
<EMAIL_ADDRESS>
11institutetext: INFN TIFPA, Via Sommarive, 14 I-38123 Trento, Italy
22institutetext: CERN, 1211 Genève 23, Switzerland 33institutetext: University
of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand
# Optimising longitudinal and lateral calorimeter granularity for software
compensation in hadronic showers using deep neural networks
Coralie Neubüsere1,addr1 Jan Kieselere2,addr2 Paul Lujane3,addr3
(Received: date / Accepted: date)
###### Abstract
We investigate the effect of longitudinal and transverse calorimeter
segmentation on event-by-event software compensation for hadronic showers. To
factorize out sampling and electronics effects, events are simulated in which
a single charged pion is shot at a homogenous lead glass calorimeter, split
into longitudinal and transverse segments of varying size. As an approximation
of an optimal reconstruction, a neural network-based energy regression is
trained. The architecture is based on blocks of convolutional kernels
customized for shower energy regression using local energy densities; biases
at the edges of the training dataset are mitigated using a histogram
technique. With this approximation, we find that a longitudinal and transverse
segment size less than or equal to 0.5 and 1.3 nuclear interaction lengths,
respectively, is necessary to achieve an optimal energy measurement. In
addition, an intrinsic energy resolution of $8\%/\sqrt{E}$ for pion showers is
observed.
###### Keywords:
Machine learning calorimeters hadronic showers
###### pacs:
07.05.Mh 29.40.Vj
††journal: Eur. Phys. J. C
## 1 Introduction
Both existing high-energy physics experiments, such as those at the CERN LHC,
and future experiments at future colliders, like the Future Circular Collider
(FCC), rely heavily on the performance of hadron calorimeters and their
particle flow capabilities for measuring jet and missing transverse momentum
($p_{T}$) Sirunyan:2017ulk ; Aaboud:2017aca ; Ruan:2014paa ; Thomson:2009rp ;
Marshall:2012ry ; Marshall:2013bda ; Marshall:2015rfa ; Sefkow:2015hna ;
Tran:2017tgr . Hadron calorimeters are currently characterized not only in
terms of their intrinsic energy resolution, but by their imaging capabilities,
which allow for offline corrections using smart algorithms. Due to the diverse
composition of hadronic showers and the differences in the calorimeter
response, a correct energy measurement becomes challenging. In general, the
components of hadronic showers can be divided into electromagnetic (EM) and
hadronic parts. The hadronic part of the shower consists of particles such as
neutrinos and neutrons which are partially invisible to the detector. This can
be affected by the chosen active detector material, where, e.g., plastic
scintillators allow for neutron detection via strong interaction with the
atomic nucleus. The undetectable particles in the hadronic shower result in an
unequal detector response; that is, $e/h\neq 1$, where $e$ and $h$ are the
calorimeter response to electromagnetic and hadronic shower fractions,
respectively.
Many hadronic calorimeters currently in use and planned for future experiments
are sampling calorimeters, which consist of alternating active and passive
absorber layers CERN-LHCC-97-031 ; CERN-LHCC-96-041 ; HGCAL-TDR ;
Neubuser:2705432 . The sampling of the hadronic shower allows for tuning of
the hadronic and electromagnetic shower responses. In the past, the $e/h$
ratio has been adjusted closer to 1 by either suppressing the electromagnetic
response, e.g., by using high-$Z$ absorbers, or by enhancing the hadronic
response, using neutron-sensitive active materials. Calorimeters that have a
ratio $e/h\sim 1$ are called “compensating” calorimeters. These optimizations
in the active and passive materials often require a decreased sampling
fraction (ratio of active/passive material), which itself degrades the
calorimeter energy resolution by increasing the stochastic term $\alpha$ of
$\frac{\sigma_{E}}{\left<E\right>}=\frac{\alpha}{E}\oplus c.$ (1)
The stochastic term is dominated by the sampling fraction and frequency for
sampling calorimeters, and expresses the dependence of the calorimeter
resolution on the fluctuations of the number of particles within the hadronic
shower (following a Poisson distribution). The constant term $c$ expresses
energy-dependent uncertainties, like the fluctuations on the EM-to-hadronic
shower fraction, which is logarithmically increasing with energy, or energy
losses due to particles escaping the detector, caused by limited calorimeter
sizes. The first can be removed either by intrinsic compensation, or by an
event-by-event measurement of the EM fraction, which is called software
compensation.
Due to the cost and mechanical stability benefits, absorbers made of steel or
lead are widely in use. These materials have been found to require very small
sampling fractions in, e.g., scintillator-steel calorimeters in order to
achieve compensating behavior. Since such low sampling fractions would degrade
the performance, especially for particles at low energies ($<50$ GeV), the
solution to correct for fluctuations in the electromagnetic shower fraction is
to use software compensation techniques.
In order to allow algorithms to distinguish between the dense electromagnetic
shower core and other shower parts, such as e.g. disappearing tracks, the
granularity of the calorimeter plays a key role. The first attempt in so-
called imaging calorimetry has been made by the CALICE collaboration, which
started a R&D program of calorimeters for a future e-e+ linear collider
Israeli:2018byq ; Chefdeville:2019zzq , where the calorimeter designs have
been optimised for particle flow algorithms Marshall:2012ry . These algorithms
allow for jet energy measurements using the best suited sub-detector to
reconstruct each jet sub-particle. The prototypes of these calorimeters have
been realised with active layers made of silicon for the EM shower part and
scintillator or resistive plate chambers for the measurement of hadronic
showers. The active layers were tested and interleaved within both steel and
tungsten absorber stacks Chefdeville:2015 ; Adloff:2012gv and achieved such
good results in testbeams Eigen:2019ccp that the CMS collaboration decided to
adopt this concept in a full silicon-tungsten/scintillator-steel endcap
calorimeter HGCAL-TDR ; Quast:2017gnq . The developments in, e.g., silicon
photomultiplier (SiPM) technologies have been the key to measure the
scintillation light produced in calorimeter cell sizes of $3\times 3\times
0.5$ cm3 Sefkow:2018rhp . The impact of software compensation techniques on
the performance of particle flow algorithms has been studied in a specific
detector design Tran:2017tgr , and proven to provide a significant improvement
to the jet energy measurement by using a corrected calorimeter cluster which
is matched to tracks in the tracking system.
The next step towards a calorimeter design optimized for the use of software
compensation techniques is to study the necessary granularity that allows an
algorithm to determine most accurately the hadronic shower energy.
In this paper, we will discuss the performance of a software compensation
technique using a deep neural network (DNN), with a specific focus on the
dependence on the transverse and longitudinal granularity. Therefore, a
homogenous model calorimeter has been studied in full Geant4 simulations. The
performance is evaluated in terms of energy resolution and linearity for
single charged pions. The goal is to determine the minimal granularity of a
calorimeter needed to achieve the best energy measurement using DNNs. As the
choice of granularity can influence the detector design and cost, a
measurement of the impact of this choice is necessary in order to optimize the
design. Here, the DNN is utilised as a generic close-to-optimal reconstruction
algorithm that can be optimised to the granularity in an automatised fashion.
## 2 Calorimeter and dataset
The studied calorimeter is a homogeneous lead tungstate calorimeter, which
follows the EM calorimeter concept of the CMS experiment CERN-LHCC-97-033 .
However, we do not consider any passive absorber material, assuming that the
impact on the calorimeter performance of the sampling fraction and the
longitudinal and transverse segmentation are uncorrelated. The dimensions are
$1\times 1\times 2.5$ m3, which ensures complete shower containment within the
calorimeter volume and corresponds to $10\,\lambda$ and $200\,X_{0}$ of total
depth. The longitudinal and transverse segmentation is increased from no
segmentation up to $30\times 30$ segments in $x$ and $y$, and from 1 to 60
segments in the lateral direction. A list of the configurations can be found
in Table 1.
The data set consists of approximately $5\times 10^{6}$ charged pion events,
generated using the FTFP_BERT physics list of Geant4 10.04 patch 0. The
training data set comprises pions with energies sampled from a flat
distribution between 1 and 110 GeV. The test data set covers 11 discrete
energies of 5 to 105 GeV in 10 GeV steps. The Geant4 simulation has been
performed in the highest granularity, while for the tests and training of
different segmentation configurations, the same dataset has been used. For
this purpose, the energy deposits in the cells have been merged corresponding
to the tested cell sizes. This method avoids inconsistencies that are
otherwise to be expected due to the different number of surfaces and material
borders through which Geant4 propagates the particles.
Stage | Longitudinal | Depth of layers
---|---|---
segments | in cm | in ${\lambda}_{\pi}$ | in $X_{0}$
0 | 1 | 250 | 9.8 | 198
1 | 6 | 41.7 | 1.6 | 33
2 | 10 | 25 | 1.0 | 20
3 | 12 | 20.8 | 0.8 | 16
4 | 15 | 16.7 | 0.7 | 13
5 | 20 | 12.5 | 0.5 | 10
6 | 30 | 8.3 | 0.3 | 7
7 | 60 | 4.2 | 0.2 | 3
Stage | Transverse | Size of cells
segments | in cm2 | in $\lambda_{n}$ | in $X_{0}$
default | $1\times 1$ | $100\times 100$ | $3.9\times 3.9$ | $79\times 79$
A | $3\times 3$ | $33\times 33$ | $1.3\times 1.3$ | $26\times 26$
B | $5\times 5$ | $20\times 20$ | $0.8\times 0.8$ | $16\times 16$
C | $10\times 10$ | $10\times 10$ | $0.4\times 0.4$ | $8\times 8$
D | $15\times 15$ | $6.7\times 6.7$ | $0.3\times 0.3$ | $5\times 5$
E | $30\times 30$ | $3.3\times 3.3$ | $0.1\times 0.1$ | $3\times 3$
Table 1: Granularity configurations considered in this analysis.
## 3 Neural network architecture and training
At the core of the neural network architecture used here is a software
compensation block that uses convolutional neural network (CNN) layers
lecun1998gradient to achieve local identification of the subshowers, similar
to the one introduced in Ref. Neubuser:2705432 , which is used as a subblock
in the overall model. This subblock consists of 3 parallel paths: in the first
path, the energy of all cells within the kernel range $K$ is summed up and
forwarded to the next block, while this kernel is moved with a stride of size
$K$; the second path consists of a CNN layer with the same kernel size and
$F=16$ filters; the third path contains in total three subsequent CNN layers,
out of which the first two have kernel sizes (in x, y, and depth) of
$K_{a}=(1,\ k,\ 3)$ and $K_{b}=(k,\ 1,\ 3)$, with no stride and 32 filters,
each. Here, $k$ is an adjustable parameter depending on granularity, as
described later. The final layer of this path is a CNN layer with a kernel
size of $K$ with a stride of $K$ and $F$ filters, such that the output of all
paths can be combined. This combination is done by adding the output of the
CNN layers of all paths feature by feature. All layers use a tanh activation
function. The weights of the layers in the third path are initialised with a
Gaussian distribution centred at 0 with a width of $10^{-3}$, and receive a
small $L2$ regularisation of $10^{-5}$. This structure is optimised to derive
small corrections to the simple energy sum by detecting the different shower
shape of electromagnetic subshowers.
In the final model, the input is passed through a batch normalisation layer
ioffe2015batch , normalising all inputs except for the per-cell energy. If
less than 6 calorimeter layers are present or the transverse granularity in
either direction is less than 6, the input is directly flattened and passed to
3 dense layers, the first two of which contain 128 and 64 nodes using ELU
activation elu_activation , before being finally passed to the energy
prediction layer with 1 node. In all other cases, the input is first passed
through a set of the subblocks described above before being fed through the
same structure with dense layers. These subblocks adapt to the input: if the
corresponding granularity is less than $6\times 6$ cells in the transverse
directions, a stride of $1\times 1$ is used, and the input $k$ for the kernel
size determination is set to $k=1$. Otherwise, a stride of $2\times 2$ and
$k=3$ are used in these directions. The subblock is repeated until the
dimensionality in $x$, $y$, or depth is less than or equal to 6. At this
point, the output is fed to the three final dense layers.
The model is trained using the Adam optimiser kingma2014adam using TensorFlow
tensorflow and Keras keras within the DeepJetCore framework DJC . The
training consists of five steps: the first four steps use a loss function
$L_{\mathrm{calo}}$ that follows the expected calorimeter resolution:
$L_{\mathrm{calo}}=\frac{(E_{\mathrm{true}}-E_{\mathrm{pred}})^{2}}{E_{\mathrm{true}}}\mathrm{.}$
(2)
These steps are trained for 1, 19, 60, and 20 epochs with learning rates of
$10^{-4}$, $10^{-4}$, $10^{-5}$, and $10^{-5}$, and batch sizes of 256, 512,
1280, and 1280. Between the third and fourth step, the batch normalisation is
frozen.
The mean and expectation value for $E_{\mathrm{true}}$ differ at the edges of
the training sample. This typically leads to edge effects, which introduce a
bias towards higher predicted values at the low edge, and towards lower
predicted values at the high edge.
To mitigate the effect, we freeze all layers except for the last dense layers,
and introduce a loss that follows a $\chi^{2}$ distribution taking the
difference of the average predicted and truth energy in bins of
$E_{\mathrm{true}}$, and accounting for the number of samples in that bin. The
bin boundaries are randomly chosen for each batch to avoid a global bias.
Using this loss, the model is trained for another 50 epochs with a learning
rate of $10^{-5}$ and a batch size of 1280.
(a) (b)
Figure 1: Results for a scenario with 15 longitudinal layers (stage 4) and no
transverse segmentation. (a) Energy distribution for 45 GeV pions. The width
as computed from the Gaussian fit (black line) and from the RMS are shown. (b)
Energy resolution as a function of the particle energy. The resolution is
computed two ways, using the Gaussian fit (open circles) and using the RMS
(filled squares).
## 4 Results
The energy resolution is evaluated as the ratio of the width to the most
probable value of the distribution of the reconstructed energy. These
distributions, as shown for example in Figure 1a, follow a Gaussian function.
The standard deviation can thus be extracted from a fit. This fit is limited
within 2$\sigma$ around the most probable value $\mu$, following the procedure
widely used in calorimeter performance studies. As a comparison and
validation, the energy resolution has also been evaluated from the root mean
square (RMS) and mean, which is sensitive to the tails of the distribution.
The energy resolution over the full available energy range is shown for stage
4, which corresponds to a granularity of 15 longitudinal layers, in Figure 1b.
The points are fitted following Equation 1, and the values of the stochastic
and constant term are shown in the legend. An overall 10–20% degradation in
energy resolution from the Gaussian fit to the RMS method is observed.
In the following, the energy resolutions obtained for different granularities
will refer to the results obtained from the Gaussian fit.
The results, in terms of the stochastic term $\alpha$ and constant term $c$
for all studied longitudinal and transverse granularities, are summarized in
Table 2. The theory of the different contributions to the energy resolution of
hadronic showers Fabjan:1989ti considers that the stochastic term is in fact
a sum of two major effects,
$\alpha=\alpha_{\mathrm{int}}\oplus\alpha_{\mathrm{sampl}}$, where the first
intrinsic term is irreducible and determined by the fluctuations of the
initial energy that is transformed into ionising shower particles, and the
second is the term due to the sampling fraction. These losses are material
dependent, due to material-dependent nuclear binding energy losses, and have
been found to be on the order of 19%/11% in the ZEUS uranium/lead-scintillator
calorimeter prototypes Tiecke:1989nz .
We assume that the DNN is able to identify and re-weight the electromagnetic
and hadronic shower fractions, due to the topological differences of EM and
hadronic subshowers ($\lambda_{\pi}/X_{0}\sim 27$). Thus, we expect the
constant term to decrease. Table 2 shows the resulting measured stochastic and
constant terms (using both the Gaussian fit and the RMS to obtain the
resolution) for three different sets of scenarios: first, the different
longitudinal granularities with no transverse segmentation, the results for
which are plotted in Figure 2; second, longitudinal stage 0 with different
transverse granularities (Figure 3); and third, longitudinal stage 5 with
different transverse granularities (Figure 4). Overall, at the finest
granularities, we observe that the constant term goes to zero, while the
stochastic term decreases by approximately 50% with respect to the scenario
with no segmentation, reaching a minimum of 8%, which can be considered the
intrinsic stochastic term $\alpha_{\mathrm{int}}$.
The constant term is consistently removed as soon as the first segmentation in
transverse granularity into $3\times 3$ cells is implemented. Figure 5 shows
an event display of a 35 GeV pion shower; the bottom shows the impact of a
$3\times 3$ transverse segmentation. We can see that already at this stage, a
significant enough energy fraction of 9% (shown as
$E_{\mathrm{out}}/E_{\mathrm{tot}}$ in the legend) is found in the outer
quadrants. In comparison, the same shower is represented in 3D on the top, and
visualises the imaging power of the finest chosen granularity of the
homogeneous PbW calorimeter.
stage | stochastic term [%] | constant term [%]
---|---|---
| RMS | Gauss | RMS | Gauss
0 | 20.5 | 17.3 | 3.0 | 2.6
1 | 19.7 | 15.9 | 2.2 | 2.0
2 | 17.8 | 14.0 | 1.8 | 1.5
3 | 17.2 | 13.6 | 1.7 | 1.3
4 | 16.0 | 12.9 | 1.5 | 1.1
5 | 15.4 | 12.1 | 1.3 | 0.8
6 | 14.6 | 11.6 | 1.1 | 0.6
7 | 13.0 | 10.9 | 1.0 | 0.5
0A | 20.3 | 15.1 | 1.3 | 0
0B | 20.0 | 14.6 | 1.2 | 0
0C | 18.2 | 13.6 | 1.3 | 0
0D | 18.6 | 13.6 | 1.1 | 0
0E | 17.9 | 13.4 | 1.3 | 0
5A | 11.4 | 8.6 | 0.6 | 0
5B | 10.6 | 8.1 | 0.6 | 0
5C | 11.0 | 8.1 | 0 | 0
5D | 10.9 | 7.9 | 0 | 0
5E | 10.9 | 7.9 | 0 | 0
Table 2: Summary of energy resolution fit results. The top set shows the
different longitudinal segmentation scenarios with no transverse segmentation,
while the other two sets show two specific longitudinal stages with different
transverse segmentation scenarios, as described in Table 1.
(a) (b)
Figure 2: Energy resolution (a) and linearity (b) for different longitudinal
granularities and no transverse segmentation. The curves correspond to the fit
with Equation 1.
(a) (b)
Figure 3: Energy resolution (a) and linearity (b) for different transverse
granularities with 1 longitudinal layer (stage 0). The curves show the fit to
the form given in Equation 1.
(a) (b)
Figure 4: Energy resolution (a) and linearity (b) for different transverse
granularities with 20 longitudinal layers (stage 5). The curves show the fit
to the form given in Equation 1.
(a) (b)
Figure 5: Event display of a 35 GeV pion shower in the homogeneous PbW
calorimeter. (a) A 3D view at the finest granularity. (b) A front view of the
same shower, with a grid overlaid corresponding to the coarsest applied
transverse segmentation.
Figure 6 summarizes the energy resolution as a function of longitudinal and
transverse granularity. We observe that the behavior of the resolution as a
function of granularity exhibits the same pattern regardless of the incident
particle energy. For the transverse granularity, the resolution reaches an
optimal value at a cell size of $\approx 1\lambda_{\pi}$, and finer
segmentation does not yield any appreciable further benefit. In the
longitudinal direction, the energy resolution continues to improve as the
layer size is decreased, reaching the minimum at the finest granularity
considered ($\approx 0.2\lambda_{\pi}$ or $\approx 3X_{0}$).
Figure 7 summarizes the fitted parameters $\alpha$ and $c$ in the energy
resolution function in Equation 1, as a function of longitudinal and
transverse granularity. In the transverse direction, we observe that the
constant term goes to zero at a cell size of $\approx 1\lambda_{\pi}$
($25X_{0}$), and further decrease in the cell size does not further improve
the stochastic term $\alpha$. In the longitudinal case, a layer width in the
region 7–10 $X_{0}$ appears to offer the best balance between the obtained
resolution and the detector complexity.
(a) (b)
Figure 6: Energy resolution as a function of the longitudinal (a) and
transverse (b) granularity. Three different particle energies are considered:
15 GeV (red circles), 55 GeV (purple squares), and 85 GeV (dark blue
triangles). In the upper plot, no transverse segmentation is used, while on
the bottom, two different longitudinal segmentations are shown: 1 layer
(dashed lines) and 20 layers (solid lines).
(a) (b)
Figure 7: Values of the parameters $\alpha$ (black line) and $c$ (blue line)
in the energy resolution function
$\frac{\sigma_{E}}{\left<E\right>}=\frac{\alpha}{E}\oplus c$ as a function of
the longitudinal (a) and transverse (b) granularity. In the upper plot, no
transverse segmentation is used, while on the bottom, two different
longitudinal segmentations are shown: 1 layer (dashed lines) and 20 layers
(solid lines).
## 5 Conclusions
When calorimeters are designed for new high-energy physics experiments, often
the approach has been to pick a technology before optimising the
reconstruction of jet particles. From the perspective of testing various
options, this not only requires significant computing power due to the
introduced details of signal processing (digitisation) in the simulations, but
also means that the simulations are unable to answer basic questions due to
the high complexity. For example, a smaller cell size improves the spatial and
pointing resolution, which should help the particle-flow algorithm to
reconstruct the jet. However, the signal height per cell decreases, which
introduces an energy loss due to a lower signal-to-noise ratio. Thus, a high-
level optimisation becomes blind to the individual impact for each effect.
Instead, a different approach could be to first identify the necessary input
for reconstruction algorithms which allows for optimal performance, before
selecting the detector technology.
Moving towards that approach, we have defined a model calorimeter to identify
the necessary cell granularity for a DNN to perform an optimal energy
reconstruction. In this model, the impact of the sampling fraction has been
intentionally excluded. Even though we are aware that the type of chosen
active and passive material will impact the shower development, we believe
that this study can be used in order to design a future hadronic calorimeter
which allows for optimal energy measurements using DNNs.
These studies suggest that a hadronic calorimeter (with
$\lambda_{\pi}/X_{0}\sim 27$) should feature cell sizes of at most 1 nuclear
interaction length, and longitudinal layers of 7–10 $X_{0}$ thickness, in
order to allow for an optimal software compensation and thus to reach the
intrinsic stochastic term of 8%. Following this approach, one could imagine
further study to determine the optimal cell and layer sizes as a function of
the $\lambda_{\pi}/X_{0}$ ratio. However, this exceeds the scope of this
paper.
## Acknowledgments
The training of the models was performed on the GPU clusters of the CERN CMG
group.
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|
# On the Readability of Abstract Set Visualizations
Markus Wallinger, Ben Jacobsen, Stephen Kobourov, and Martin Nöllenburg M.
Wallinger and M. Nöllenburg are with the Algorithms and Complexity Group, TU
Wien, Vienna, Austria B. Jacobsen and S. Kobourov are with the Dept. of
Computer Science, University of Arizona, Tucson, AZ
###### Abstract
Set systems are used to model data that naturally arises in many contexts:
social networks have communities, musicians have genres, and patients have
symptoms. Visualizations that accurately reflect the information in the
underlying set system make it possible to identify the set elements, the sets
themselves, and the relationships between the sets. In static contexts, such
as print media or infographics, it is necessary to capture this information
without the help of interactions. With this in mind, we consider three
different systems for medium-sized set data, LineSets, EulerView, and
MetroSets, and report the results of a controlled human-subjects experiment
comparing their effectiveness. Specifically, we evaluate the performance, in
terms of time and error, on tasks that cover the spectrum of static set-based
tasks. We also collect and analyze qualitative data about the three different
visualization systems. Our results include statistically significant
differences, suggesting that MetroSets performs and scales better.
###### Index Terms:
set visualization, usability study, quantitative evaluation.
## 1 Introduction
Set systems naturally model data with categorical attributes that occur
frequently in data science and analytics in various application domains.
Examples are actors in social networks with different (overlapping) community
structures they are members of, patients in health care data and the different
types of symptoms they show, or artists and bands that belong to various
musical genres. More generally, a set system $(\mathcal{U},\mathcal{E})$ is
comprised of a universe $\mathcal{U}$ of _elements_ (actors, patients, or
artists) that are grouped into different _subsets_ $S\subseteq\mathcal{U}$,
which together form a subset family $\mathcal{E}\subseteq 2^{\mathcal{U}}$
(communities, symptoms, or genres). Set systems can also be modeled as
_hypergraphs_ , where $\mathcal{U}$ is the vertex set and $\mathcal{E}$ is the
set of _hyperedges_. Each hyperedge $S\in\mathcal{E}$ is a subset
$S\subseteq\mathcal{U}$ of vertices. Hypergraphs generalize graphs, which have
the additional restriction that (hyper-)edges must consist of exactly two
vertices.
The most popular visualization style for small set systems are Euler (and
Venn) diagrams, showing sets as closed shapes with possible overlaps
indicating set relations such as intersection or containment. However, Euler
diagrams do not necessarily show individual elements and do not scale well.
Therefore many different visualization styles have been proposed in the
literature, ranging from overlay techniques for pre-embedded elements in the
plane to more scalable node-link and matrix-based techniques. A state-of-the-
art report by Alsallakh et al. [1] includes a taxonomy for classifying the
different visualization techniques as well as a collection of 26 general tasks
in three categories to be solved using set visualizations. One of the
observations from their survey is a “clear lack of empirical user studies that
assess the effectiveness of different techniques in performing different
tasks” [1].
There are a few previous empirical evaluations. Some considered spatial set
systems with embedded elements [2], or focused on different visual parameters
in a single type of visualization style such as Euler diagrams [3] or LineSets
[4]. Others studied specific tasks in combined visualizations of set systems
and underlying networks [5, 6] or compared visualization techniques that do
not show individual elements such as linear and mosaic diagrams [7].
The focus of our study is on evaluating the readability of set visualizations
for abstract, non-spatial data generated by publicly available visualization
systems, by measuring performance of representative element-based and set-
based tasks. In particular, this means that both individual elements in
$\mathcal{U}$ and all the sets in $\mathcal{E}$ must be shown and labeled.
Further, we aimed at visualization techniques that can be intuitively
understood by non-experts and thus serve as candidates for set visualizations
to be used by data journalists, e.g., in static infographics for print media,
or for sharing images about interesting datasets in social media. This focus
of the study limits the size of datasets to less than ten sets and fewer than
a hundred elements. Datasets larger than that require interaction and
navigation (e.g., filtering, zooming, panning).
With this in mind, our search narrowed down to three set visualization
systems: EulerView [8], a technique representing the popular class of Euler
diagrams, LineSets [4], a technique representing overlay-based set
visualizations, and MetroSets [9], a graph-based technique using the metro map
metaphor; see Figure 1.
(a)
(b)
(c)
Figure 1: Our study evaluates the readability of set visualizations for
abstract, non-spatial data generated by publicly available visualization
systems (EulerView (left), LineSets (middle), MetroSets (right)) by measuring
their performance on representative element-based and set-based tasks.
For our evaluation, we performed an online experiment with a total of 120
participants who solved six different tasks on stimuli of two sizes generated
by the three selected systems. We used small ($|\mathcal{U}|=30$ and
$|\mathcal{E}|=6$) and large ($|\mathcal{U}|=60$ and $|\mathcal{E}|=8$) real-
world set system data derived from Spotify’s database of artists and genres.
Note that the terms “small” and “large” are used here simply to distinguish
between the two sizes, rather than as descriptive of the actual sizes of the
underlying data. Nevertheless, such small datasets do occur in research
papers, popular science, and social media, whereas truly large datasets
require visual analytic systems.
The results show statistically significant quantitive and qualitative
differences, suggesting that MetroSets scales better and performs more
consistently than EulerView or LineSets, and is also better-liked by users.
## 2 Related Work
### 2.1 Set Visualization Techniques
The survey of Alsallakh et al. [1] provides a classification of set
visualization techniques into six different groups. The most commonly used
techniques belong to the family of Euler and Venn diagrams, which represent
each set as a region enclosed by a simple, closed curve and set relations by
corresponding intersections and inclusions of these regions. Most of these
diagrams focus on representing the sets, but not necessarily the individual
elements. Only two Euler-based techniques show individual elements [8, 10].
Area-proportional Euler diagrams [11, 12, 13] do not show the elements, but
indicate set cardinalities by scaling the region sizes proportionally. Some
methods for generating Euler diagrams use simple shapes such as circles and
ellipses to represent the sets [14, 12, 11, 15], others are more widely
applicable but also produce more complex shapes [10, 8, 16, 17, 18]. Well-
formed Euler diagrams do not always exist and some techniques may produce
inconsistent diagrams showing non-existing set intersections.
The second class of techniques are overlay-based set visualizations. Here, all
the elements are explicitly shown at pre-defined positions in the plane, which
could represent geolocations in spatial datasets or positions of nodes in a
node-link diagram of underlying network data. The techniques create visual
overlays that indicate the set memberships of the input point set. Bubble Sets
[19] create isocontours for each set, which enclose their corresponding points
similarly to regions in Euler diagrams. Pairs of isocontours, however, may
intersect even if their two sets have an empty intersection in the data.
LineSets [4] is based on the idea of connecting the elements of each set by a
smooth and short curve reminiscent of lines in metro maps. Kelp Diagrams [20]
and KelpFusion [2] extend and combine ideas from LineSets and Bubble Sets by
parameterizing set representations from sparse spanning graphs to convex
hulls, with the middle range resulting in bubble shapes for local point
clusters in a set and edge-like links to connect elements further apart. All
overlay techniques inherently require a given embedding of the set elements
and so these methods are not directly applicable for abstract set systems.
Visualizations based on bipartite graphs represent elements and sets as two
types of vertices. Each element is connected by an edge to all sets containing
that element, which yields a bipartite graph. General visualization techniques
for (bipartite) graphs can then be employed and have been integrated in
several systems [21, 22, 23, 24]. However, for complex set systems, the
resulting layouts are dense with many edge crossings and there is little
support for set-related tasks [1].
Due to their equivalence to set systems, hypergraph layout methods are also
relevant. Johnson and Pollak [25] introduced different notions of hypergraph
planarity. Many papers study, from a theoretical perspective, support graphs
(or supports in short), which are graphs defined on the vertex set
$\mathcal{U}$, such that the elements of each hyperedge $S\in\mathcal{E}$
induce a connected subgraph. A drawing of the support graph could then be used
as the basis for visualizing the set system. Of particular interest are planar
supports [26, 27], tree supports [28, 29], and path supports [30]. Not all
hypergraphs admit all types of (planar) supports, which may limit their
applicability in practice. The recent MetroSets system [9] computes a (not
necessarily planar) path support as the basis for rendering abstract set
systems in the style of a metro map.
Another class of techniques are matrix-based approaches, which represent sets
and elements as the rows and columns of a matrix. An element contained in a
set is indicated by coloring or marking the respective matrix cell with a
glyph. More complex set-related tasks typically require interaction, and most
matrix-based systems are designed for interactive analysis and exploration.
Examples are ConSet [31], OnSet [32], RainBio [33], or UpSet [34]. Matrix-
based methods scale well, but have a strong dependency on row and column
ordering, are less intuitive for non-experts, and require interaction with a
non-trivial interface for the more complex set visualization tasks. Closely
related are linear diagrams [35, 7, 36], where again each set is a row in a
table. The columns, however, do not represent individual elements, but rather
non-empty set intersections. Any marked or colored cell in the table indicates
that a set is part of a non-empty set intersection with all the other sets
being marked in the same column.
Finally, there are many aggregation-based techniques implemented as components
in interactive visual analytics applications, e.g. [24, 37, 38]. These
techniques are designed for analyzing large-scale data and aggregate elements
into groups, while maintaining all set relations. Since they do not support
element-based tasks, we did not consider them for our study.
### 2.2 Evaluations of Set Visualization Techniques
Several aspects of set visualizations have been empirically studied in the
literature before, but are limited to evaluating parameters of a single
visualization style, comparing overlay-based techniques with pre-embedded
elements only, or excluding element-based tasks completely. All studies
mentioned below build their experimental design on static images generated
with their respective target systems and approaches (like ours). Alper et al.
[4] performed a readability evaluation of their overlay-based system LineSets
in comparison to Bubble Sets [19]. In their study 12 participants performed
four element-based and set-based tasks on spatial and social network data with
3–5 sets and 50–200 elements; accuracy and time were recorded. For detailed
tasks and preference ratings, LineSets outperformed Bubble Sets; otherwise
there were no significant differences. Meulemans et al. [2] evaluated
KelpFusion, LineSets, and Bubble Sets with 13 participants. Their geographic
datasets had 4 or 5 sets and 12–49 elements; the four tasks were similar to
those by Alper et al. [4]. The results showed no significant difference in
performance or preference between KelpFusion and LineSets, but both
outperformed Bubble Sets. Both studies considered only pre-embedded set
systems.
Two studies evaluated techniques that show combined set and network
visualizations. Rodgers et al. [6] included five different systems in their
crowdsourced study: Bubble Sets, LineSets, KelpFusion, EulerView, and their
system, SetNet. They used datasets with 11–64 elements, 3–7 sets, and 42–162
edges in the network. All tasks involved both the network structure and the
set information. Their results indicated that SetNet and EulerView
significantly outperform the other three systems on the combined set+network
tasks. Baimagambetov et al. [5] compared SetNet, Bubble Sets, and
WebCola111see https://ialab.it.monash.edu/webcola/ on instances with 2–8 sets,
10–100 elements, and 40–170 network edges. Their study did not assess
readability, but rather quantitatively evaluated the frequency of inaccuracies
and properties that have previously been empirically confirmed to be visually
ineffective. Bubble Sets turned out to be least inaccurate system and in terms
of ineffectiveness there was no clear winner.
Further evaluations considered techniques such as Euler diagrams, linear
diagrams, and mosaic diagrams, all of which do not represent individual
elements and thus used only set-based tasks [3, 7]. They showed that linear
diagrams outperformed Euler diagrams and were on par with mosaic diagrams.
Thus, despite several previous specialized evaluations, a broader study of
abstract set visualization across different classes of techniques as well as
including both set- and element-based tasks is missing. Hence the observation
of Alsallakh et al. [1] that there is a lack of empirical studies assessing
the effectiveness of set visualization techniques, remains valid, especially
regarding fundamental element-based and set-based tasks on abstract set data
using visualizations that show the individual set elements.
## 3 Principles of Abstract Set Visualization
The design of set system visualizations is guided by the types of tasks they
should support, as well as by the type of information that needs to be
represented. Alsallakh et al. [1] present a taxonomy of tasks classified into
the following three categories.
1. 1.
Element-based tasks are concerned with specific elements and their respective
relationship to the sets. For example: What music genre(s) does ’Van Halen’
belong to?
2. 2.
Set-based tasks are concerned with the relationship between different sets
without taking individual elements into account. For example: Which music
genres overlap with ’Rock’?
3. 3.
Attribute-based tasks are concerned with attributes of set elements and their
relationship of distribution in regards to set membership. For example: Do
artists in the ’Rock’ genre sell more records than artists in the ’Hip Hop’
genre?
We focus on element-based tasks and set-based tasks in this study as they
represent elementary tasks on abstract set systems that can be performed on
static (rather than interactive) visualizations that appear in research
papers, newspaper articles and social media. According to [1] the support of
the different types of tasks is closely tied to the type of information
represented in the visualization.
* •
Representing set information only: the focus is on the relationships between
sets and individual elements might not be explicitly represented.
* •
Representing individual elements: Individual elements are represented
explicitly, making it possible to also encode additional attribute
information.
One important aspect of set visualization techniques is their visual
scalability, i.e., whether a visualization remains comprehensible when the
number of elements and/or the number of sets increases. Naturally, scalability
is closely related to the size of the set system. However, scalability also
depends on the information that is represented. For example, visualizations
that only represent set information are not adversely affected by increasing
the number of elements. Even when only set information is represented,
scalability depends not only on the visual metaphor, but in some cases (e.g.,
Euler and Venn diagrams) also on the structure and number of the intersection
relationships of the underlying set system.
Explicitly representing elements is associated with increased visual
complexity as individual elements need to be depicted. Furthermore, it is non-
trivial to assign primacy to either the depiction of the sets or of the
elements, as one can influence the comprehension of the other negatively.
## 4 Systems
In our study we focus on systems that produce static visualizations that can
be used to argue about the set system and to perform elementary tasks, rather
than interactive visual analytics systems. The general requirements
determining whether a given system could be used in our study include the
following criteria:
1. 1.
All elements are explicitly depicted.
2. 2.
All elements are identifiable by an associated label.
3. 3.
All sets are identifiable by a given label or legend.
4. 4.
No interactivity is required to reason about relationships between sets and
between elements and sets.
5. 5.
The visualization system is implemented and its source code available.
Most systems that were taken into consideration do not fulfil all of the above
criteria: some do not depict all elements (e.g., Eulerr [11], VennEuler [12],
Linear Diagrams [35]), others do not label the elements (e.g., eulerGlyphs
[39]), still others focus on different tasks (e.g., network based tasks such
as SetNet [6]). Many do not provide implementation (e.g., untangling Euler
diagrams [10]). Eventually we narrowed our selection down to three systems:
EulerView [8], LineSets [4], and MetroSets [9].
EulerView is an Euler-based technique. Notably, while many Euler diagram
systems do not represent individual elements, EulerView explicitly visualizes
elements and their respective labels.
LineSets is a representative of the overlay-based visualization approaches. As
such, it requires initial positions for the elements. To resolve this issue,
we contacted the authors and followed their suggestion to pre-process the data
with a force-based layout [40].
MetroSets is visually similar to LineSets but is designed with abstract data
in mind. It is the most recent of the three techniques and combines ideas from
graph-based and overlay-based approaches.
While other overlay-based systems also meet our criteria (e.g. BubbleSets
[19]), we chose to use LineSets because of its strong conceptual and visual
similarity to MetroSets.
For more detailed descriptions and implementation details of the three
selected systems we refer the reader to the supplemental material.
## 5 Controlled Study
As mentioned in Section 3 representing information is closely coupled with
tasks that can be performed on a set visualization and therefore task
performance can give a general idea about the effectiveness of the
visualization. Traditionally, task performance is measured by accuracy (number
of tasks solved correctly) and task completion time, with better performance
associated with high accuracy and/or low completion time. The aim of our study
is to evaluate the effectiveness of the three visualization systems on a set
of carefully selected tasks that span the spectrum of elementary element-based
and set-based tasks. We assessed the different systems by performing a
controlled human subjects study with two experiments (that differ in the size
of the underlying data). We collected quantitative data (time and accuracy) as
well as qualitative data (Likert-scale subjective ratings).
While lab-experiments can be better controlled, we opted for a fully online
setting due to difficulties associated with in-person gatherings in the year
2020. With this in mind, we used social media posts (e.g., reddit, facebook,
instagram) to recruit participants. We stopped collecting data after reaching
120 fully completed instances. A repository with all our datasets, stimuli,
responses of participants and evaluation code can be found on OSF222see
https://osf.io/nvd8e/.
### 5.1 Participants and Setting
Participants in the study were at least 18 years of age, without known color
vision deficiency. 53% of participants reported being younger than 30,
although we had non-trivial numbers of participants aged 55 or older. 57% of
the participants were male, 36% were female, and the rest identified as other
or preferred not to answer. The majority of the participants were well-
educated (73% with a post-secondary degree or higher) but did not report high
familarity with data visualization systems (average self-rating of 2.7 out of
5.0). Nevertheless, many were familiar with Euler or Venn diagrams, and some
with overlay-based set visualization. A summary of the demographic data is in
Figure 2.
The study itself was realized using the LimeSurvey open-source online survey
tool;333see https://www.limesurvey.org/en and hosted online. We used the
standard functionality of question types provided by LimeSurvey but modified
the layout to place images and questions side-by-side to reduce the need for
scrolling; see Figure 3.
In total 44 question, excluding transition screens, had to be answered to
complete the study: 7 preliminary screening questions, 8 demographic
questions, 6 task training questions, 18 testing questions, and 5 qualitative
questions.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 2: Demography of the 120 participants.
The participants were randomly assigned to either the small or large
experiment (details about the two are provided in Section 5.3).
### 5.2 Datasets
We used a different real-world dataset for each task to minimize the variance
that might result from one dataset being well-suited for a certain system. We
created a hypergraph (4549 hyperedges, 347686 vertices) of artists and music
genres of Spotify data and automatically extracted subgraphs which where
subsequently used to generate the stimuli in all three systems. In the context
of the Spotify hypergraph, artists are set elements represented by vertices,
and genres are sets represented by hyperedges.
We used the following approach to extract the subgraphs. We set a target
number of elements and sets; 30 elements and 6 sets for the small datasets and
60 elements and 8 sets for the large dataset. We also specified the number of
sets that elements should belong to. For subgraphs of the small dataset we
required the graph to have ten elements that are members in exactly one set,
ten elements that are members in two sets, six elements that are members in
three sets and four elements that are members in four sets. For the large
dataset we doubled the number of elements in each membership degree category,
but did not add another category.
To extract a single subgraph we manually declared a seed set in the hypergraph
and iteratively added more sets that overlap with the current sets until the
target number of sets was reached, allowing for enough elements to be added in
each membership category. We proceeded by iteratively picking random elements
from the set of six or eight sets that fulfilled the membership requirement
until the target number of vertices was reached. If no valid subgraph was
extracted after a fixed number of iterations, we concluded that none existed
and started the process again with another seed node.
The previous step gave us a potential candidate that had to fulfill further
requirements. First, we excluded hypergraphs with multiple disjoint
components. Second, to avoid trivial instances, we excluded hypergraphs with
fewer than three elements per set. Third, as the study targeted an English-
speaking audience, we also excluded hypergraphs with vertices containing non-
Latin characters in their labels.
After generating the first set of stimuli we decided to additionally exclude
elements with long labels (greater than eight for the large dataset and
greater than twelve for the small dataset). The reasons behind this decision
are that all three systems struggle handling long labels, and long individual
labels distract from the overall visualization.
With our extraction approach we were able to create pseudo-realistic datasets
from real-world data with the advantage of having control over relevant
properties. Note that it would have been much easier to use a synthetic
dataset. We considered that possibility, but decided against it for several
reasons. Primarily, we believed that using real-world data would make the
study more enjoyable, thereby increasing participation. We discuss potential
problems with this approach in Section 7.
### 5.3 Size
We determined the two dataset sizes (small and large) based on observations we
made in the three systems during the pilot experiment. Our goal was to choose
sizes which reflected the capabilities of all three systems, while avoiding
situations where the extreme difficulty (or ease) of a task might obscure the
differences between systems.
Recall that the terms “small” and “large” are used here simply to distinguish
between the two sizes, rather than as descriptive of the actual sizes of the
underlying data. The size of the small dataset (30 elements and 6 sets) can be
considered a typical size for set systems where naive approaches, e.g. classic
Euler diagrams, start to struggle. We determined the size of the large dataset
(60 elements and 8 sets) to be the largest such that the resulting
visualizations were consistently readable on an ordinary monitor without user
interaction.
### 5.4 Tasks
We used the taxonomy of Alsallakh et al. [1] to select elementary tasks of set
systems that participants would perform. Specifically, we selected three
element-based (T1–T3) and three set-based (T4–T6) tasks that can be performed
on all three systems without the need for interaction. We re-worded technical
terms (e.g., element, set, hypergraph) using a more natural language for
better accessibility.
* T1
Find/Select elements that belong to a specific set: Check all of the ’Genre’
artists below; Three artists were given as possible answers.
* T2
Find sets containing a specific element: What genre(s) does ’Artist’ belong
to; All sets were given as possible answers.
* T3
Find/Select elements based on their set memberships: Check all of the artists
below that belong to both ’Genre 1’ and ’Genre 2’; Three artists were given as
possible answers.
* T4
Analyze intersection relation: Please check below if any artist(s) belong to
both of the following pairs of genres; Three pairs of genres were given as
possible answer.
* T5
Identify set intersections belonging to a specific set: Which genres overlap
with ’Genre’; All sets except the specified one were given as possible
answers.
* T6
Analyze and compare set- and intersection cardinalities: How many artists are
both ’Genre 1’ and ’Genre 2’; Numbers from 0–10 were given as possible
answers.
We assigned one small and one large dataset to each task and used different
approaches to select a balanced set of possible answers. We excluded tasks
which were ambiguous or impossible to solve (e.g., nodes overlapping lines in
MetroSets, overlapping labels in EulerView, covered up lines between two
elements in LineSets).
As our question types were limited to those provided by LimeSurvey, we elected
to use multiple choice for every task except for T6. Each option in a multiple
choice question can be thought of as a subtask and we counted tasks T1-T5 as
correct only when a participant answered all subtasks correctly.
Task T2 required a participant to select all sets an element is member of. For
the sake of fairness we picked elements with the exact same set membership
degree. The same reasoning was applied to task T5, where participants had to
select all sets intersecting with a given set, and we selected sets that had
the same cardinality of intersections.
Lastly, participants had to correctly count the number of elements in the
intersection of two sets in task T6. We selected pairs of sets with a similar
number of elements (6-8) in their intersections.
### 5.5 Stimuli
We generated static images, or stimuli, for all combinations of dataset and
system. The images for MetroSets and EulerView were rendered in their native
implementation. The ‘balanced’ pipeline preset was used for MetroSets. For
LineSets, our implementation provided output in the form of a dot file. The
final visualization was then rendered to a PNG file using graphviz[41], with
parameters chosen to match those used in the original paper introducing
LineSets[4]. To minimize possible confounding factors, we used the color
scheme from MetroSets for our LineSets stimuli.
We modified all three systems to use the same font: PT Narrow Sans. The
EulerView module automatically removes label overlaps by only showing a subset
of non-overlapping labels. As this approach is not suitable for our study, we
fixed the font size to the largest possible so that we could remove overlaps
by setting the label anchor to the left, right, or top of the glyph (instead
of only the bottom by default).
As all systems generate slightly different output, we used image manipulation
software to normalize the difference. We created a blank image with 2000px
width and 1385px height. We then scaled the images generated by the systems to
fit into the canvas with space to fit a legend into the bottom right corner of
the image. In the case of LineSets, we found that it was often impossible to
fit the stimuli alongside the legend without making labels unreadably small.
To resolve this, we automatically rotated the initial layout provided to
LineSets by 90 degrees whenever doing so reduced its height. We did not
encounter this issue with the other systems, and therefore did not rotate any
other stimuli.
MetroSets is the only system that automatically creates an image that has a
legend. While sets in [8] are by default annotated with a label, we decided to
remove any potential confounding factors and instead created a legend with two
columns of circular (50px diameter) texture samples and the associated label.
We created a similar legend for LineSets using the same design as MetroSets
legend. All legends use 22pt Roboto Mono as header and 18pt Roboto for set
labels.
To reduce the time required to find elements in the visualization for tasks
T1-T3, we highlight the relevant elements by adding a colored 2px rectangular
border around them; see Figure 3. The main reason for adding highlighting was
to exclude the time required to search for elements, as this could be
considered its own task and would add noise to our overall time measurements.
We applied an additional sharpening effect to all EulerView stimuli, as the
images tended to be blurry after normalization.
We communicated with the authors of all three systems to ensure that we
generate visual stimuli that represent each system fairly. We also discussed
the slight modifications we made (e.g., using the same fonts, using the same
colors, reducing label overlaps). All visual stimuli used in the study can be
found in the supplemental material.
### 5.6 Experimental Procedure
For each condition (EulerView, LineSets, MetroSets) we conducted two
experiments, one on the small and one on the large dataset. The experimental
section of the study had three set-based tasks and three element-based tasks.
This resulted in
$3\>\mathrm{conditions}\times(3+3)\>\mathrm{tasks}=18\>\mathrm{trials}$.
Participants were randomly assigned to work with either small or large
datasets.
Both experiments follow the five-phase template; (1) information consent and
screening, (2) demographic questions, (3) tutorial, (4) formal study and (5)
post-task questionnaire.
In phase one participants were given the background information and procedure
of the study before they had to give consent to the study policy. The policy
stated that only participants of age 18 and above with no known color vision
deficiency are allowed to participate. After giving consent the participants
were first shown an image with excerpts of all three visualization styles. The
excerpts show the worst case label size and we ask a yes/no question if the
labels are readable in all three systems. Additionally, we state that the
window size should not be altered during the study and automatically logged
the window information. We assume that participants did not change their
screen size during the experiment. This information was used to screen and
exclude participants who were potentially not able to accurately identify set
elements. Afterwards the participant completed a subset of six plates of the
Ishihara Test [42] to screen for potential issues with color perception.
Failing to give the correct answer could be due to different potential causes
(e.g. viewing angle or uncalibrated monitor).
In phase two the participants provided (optional) demographic information,
summarized in Figure 2. In phase three the participants were introduced to the
different tasks and visual stimuli. We had a total of six tasks: three set-
based tasks and three element-based tasks. We randomly assigned a set-based
task and a element-based task to each system. Participants were not able to
proceed to the next task until a correct answer was given. The tutorial
portion of the study does not contribute to the results.
Phase four contains the quantitative portion of the study. Each participant
was presented with 18 tasks split into 6 blocks of three tasks. Each block
consisted of exactly one question per system of different types of our
potential set of 6 tasks; for an example task see Figure 3. We randomized the
order in which the blocks were shown to the participant, as well as the order
of questions inside each block. Before each block the participants were shown
a white screen with text letting them know that they can take a break.
In phase five the participants provided qualitative feedback with four Likert
scale questions and one fillable text field. Specifically, we asked the
following questions:
* •
How clearly could you identify to which genre(s) an artist belonged to?
* •
How clearly could you identify the overlap between different genres?
* •
How often did you use pre-existing knowledge about music to answer an
question?
* •
How interested are you in using each style again?
* •
Please share any thoughts you have about the different styles!
Figure 3: Example of a element-based task (T2) in combination with the
MetroSets system.
### 5.7 Pilot Study
Before recruiting participants we asked a group of four experts and six lay
people to give their opinion on the design of the study. We asked the lay
people for their opinions on duration, wording, and experience. The experts
where given a questionnaire and asked for further comments/suggestions. From
the information gathered we made several modifications to the study design.
The first major change was to not include all tasks of the small and large
dataset in a single experiment, as the study duration was too long (about 45
minutes on average) and caused participants to tire. As a result, we split the
experiment into two experiments: one with the small dataset and another with
the large dataset. We also reduced the number of questions we asked for each
task.
As suggested by several of the experts, we added highlighting for all element-
based tasks (T1, T2, and T3). Otherwise the time required to search for the
relevant elements dominates the time measurements.
Initially, T4 asked a participant to identify if a pair of given sets
intersect with a simple yes/no question. We modified this question, as it had
a $50\%$ chance of being guessed correctly, and instead asked if three
presented pairs of genres intersect.
The qualitative evaluation was streamlined and shortened from eight Likert
type questions to four. We also added a free form text field, asking the
participants to leave comments.
### 5.8 Hypotheses
The goal of our study is to evaluate the differences in readability in terms
of the performance of both element-based and set-based tasks among abstract
set visualization techniques. We selected one representative each from the
three classes of techniques [1] that show individual elements and are suitable
as visualizations for non-experts, namely Euler diagrams, overlay-based
techniques, and graph-based techniques.
We formulate three different hypotheses regarding element-based task
performance, set-based task performance, and visual scalability. For the
purpose of this study, we say that a technique $A$ outperforms a technique $B$
for some task(s) of interest if the results of $A$ are significantly more
accurate than $B$, using a significance threshold of $0.05$. Further, we say
that technique $A$ scales better than $B$ if the accuracy of $B$ decreases
significantly more than the accuracy of $A$ when used with a larger dataset.
MetroSets and LineSets use a similar visualization style, but MetroSets can
achieve lower visual complexity by freely placing the elements and
schematizing the line trajectories. In addition, previous studies have shown
advantages of LineSets over Euler-like visualizations [2, 4]. Finally, as we
observed in Section 4, both LineSets and EulerView can sometimes produce
ambiguous results, and we expect this to occur more frequently with more
complicated datasets. Based on these facts, our hypotheses are as follows:
* •
H1: For element-based tasks MetroSets will outperform LineSets and LineSets
will outperform EulerView.
* •
H2: For set-based tasks MetroSets will outperform LineSets and LineSets will
outperform EulerView.
* •
H3: MetroSets will scale better than LineSets and EulerView.
### 5.9 Summary of Supplementary Material
We provide all related material of the study in the supplementary materials:
the code for dataset extraction and input format parsing, stimuli for pilot
and study, images of a full walkthrough of the study, the raw results, the
statistical analysis code, and the results of our ANOVA tests. The code for
MetroSets can be found on OSF, the LineSets implementation in GMAP is
available on GitHub. We used Tulip 3.5 to build and run the EulerView module.
## 6 Results and Analysis
### 6.1 Data
After filtering out participants who failed to complete the entire study or
gave incorrect answers on the Ishihara Test, we were left with a total of 116
respondents. Exactly half of these respondents worked with small datasets,
while the other half performed tasks with large datasets.
### 6.2 Methods
To analyze the relationship between style of visualization, dataset size, and
accuracy, we performed a 2-way ANOVA test [43], including an interaction term
meant to capture the differences in scalability between the systems. For each
statistically significant result ($\alpha=.05$), we performed post-hoc
analysis using Tukey’s test [44]. This experiment was repeated separately for
each of the 6 tasks we asked participants to perform.
### 6.3 Accuracy
Our ANOVA showed that all factors were statistically significant for all
tasks, with the exception of size, which did not have a significant impact for
task T4 or T6. Post-hoc testing confirmed significant differences on all six
tasks, which are presented in Figure 4 and described explicitly below.
On task 1, there was no significant difference between the three systems on
the small dataset. However, on the large dataset, EulerView performed
significantly worse than either MetroSets or LineSets ($p<.001$, mean
difference $\approx 22\%$).
Conversely, on task 2, there was no significant difference between the three
systems on the large dataset. However, on the small dataset, MetroSets was
significantly more accurate than LineSets ($p=.002$, mean difference $\approx
25\%$), which in turn was significantly more accurate than EulerView
($p<.001$, mean difference $\approx 36\%$).
On task 2, the participants performed better with the larger dataset. Looking
more carefully at the stimuli provides a plausible explanation. The smaller
dataset here contained four overlapping sets to be identified, while the
larger dataset contained only three. This is likely the cause of the
difference observed: a priori, increasing the size of the visualization should
have no impact on performance for this task, since it only requires looking at
the area around a single, highlighted element.
On task 3, there was again no significant difference between the three systems
on the small dataset. However, with the large dataset, participants accuracy
with EulerView was barely better than guessing (mean accuracy of approximately
15%). Otherwise, MetroSets performed slightly better on the larger dataset
than the smaller ($p=.046$, mean difference $\approx 15\%$).
All told, with respect to element based tasks (T1-T3), both LineSets and
MetroSets maintained average accuracy of approximately 92%. EulerView
performed considerably worse, with an average accuracy of roughly 65%.
On task 4, there was no significant difference between LineSets and MetroSets,
regardless of the size of the dataset. However, EulerView performed
significantly worse than either on both the large dataset ($p<.001$, mean
difference $\approx 39\%$) and the small one ($p<.001,$ mean difference
$\approx 61\%)$. Its performance on the small dataset was significantly worse
than on the large ($p=.006$, mean difference $\approx 22\%$).
On task 5, there was no significant difference between the three systems on
the small dataset. However, on the large dataset, both EulerView and LineSets
performed very poorly, with average accuracy of roughly 14%. MetroSets
performed significantly better on the large dataset ($p<.001$, mean difference
$\approx 56\%$). MetroSets’ performance on the larger dataset was
significantly worse than its performance on the smaller dataset, however
($p=.003$, mean difference $\approx 24\%$).
On task 6, MetroSets performed significantly better than EulerView on the
small dataset ($p<.001,$ mean difference $\approx 36\%$). However, both
EulerView and MetroSets performed significantly better than LineSets on the
large dataset $(p<.001$, mean difference $\approx 53\%$).
Considering all set-based tasks (T4-T6), MetroSets maintained an average
accuracy of approximately $85\%$. Meanwhile, LineSets had an average accuracy
of roughly $64\%$, while EulerView’s accuracy was only $50\%$.
The supplementary materials contains the mean and standard deviation for the
accuracy of participants on each task, system, and dataset size.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Figure 4: Task accuracy results: each plot corresponds to a single task, and
contains means and adjusted 95% confidence intervals for the accuracy
associated with each pair (size, system). A horizontal guideline has been
added to the top of each confidence interval to make it easier to determine if
two intervals overlap; if the interval for one system lies completely above
the guideline for another system, then it means the first system performed
significantly better ($\alpha=0.05$). Likewise, if the guideline for one
system passes through the interval for another system, then the two are not
significantly different.
### 6.4 Time
While we recorded the time taken by participants on each task, this data is
difficult to analyze rigorously. As the study was online, we cannot confirm
that our participants remained focused and attentive throughout the study.
This fact, in combination with the general challenges associated with
analyzing response time data (such as high skew and the tendency for high
variance even when a single subject performs the same task multiple times)
[45], led us to decide against detailed time analysis. Instead, we visualize
the distribution of response times for each question, size, and system in
Figure 5. In addition, descriptive statistics are given in the supplementary
materials.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5: Violin plots summarizing the distribution of times taken by
participants on each system, size, and task. Each violin plot represents the
distribution of times taken on that task with the corresponding system and
size. The red region on the left represents the distribution of times taken on
incorrect answers, while the blue region on the right represents the
distribution of times for correct answers. The fatter a region is at a point,
the more participants completed the task in roughly that much time. Note that
a handful of extreme outliers are not visualized due to space constraints.
Comparing this data to our accuracy results in Figure 4 suggests that the
variance in response time was greatest when participants struggled to find the
right answer: see, e.g., the performance of EulerView on the small dataset for
tasks T2 and T4. In general, however, there is no obvious pattern to the time
taken as a function of the system. Each of the three techniques was the
fastest for at least one task.
### 6.5 Qualitative Feedback
After completing the study, participants were invited to provide qualitative
feedback on the systems and the study. This feedback took the form of four
Likert-scale questions and a free-form text field. The distribution of answers
for the Likert scale questions is presented in Figure 6.
We analyzed the results of the Likert scale questions using the Kruskal-Wallis
test [46]. This test showed that there was no significant difference in the
frequency with which participants used pre-existing knowledge to answer
questions for each system ($p=0.97$). We do find significant differences for
the other questions asking about interest in the different visualization
styles, ability to perform element-based tasks, and ability to perform set-
based tasks ($p<0.01$ in all three cases).
For each of the three questions with significant differences, we performed
pairwise Mann-Whitney U tests[47] between the systems, using Bonferroni
correction[48]. This post-hoc analysis again revealed significant results
($p<0.01$ for all comparisons). We use the AUC measure for effect size[49],
which has the intuitive interpretation as the probability that a randomly
chosen rating for one system will be larger than a randomly chosen rating for
the second. We summarize the results below:
* •
For element-based tasks, the participants found it easier to identify the
genres to which an artist belonged with MetroSets than with LineSets (AUC =
0.84) and with LineSets over EulerView (AUC = 0.82). The difference between
MetroSets and EulerView was even greater (AUC = 0.97).
* •
For set-based tasks, particpants found it easier to identify the overlap
between genres in MetroSets than with LineSets (AUC = 0.8) and with LineSets
over EulerView (AUC = 0.75). They also found MetroSets easier than EulerView
(AUC = 0.92).
* •
The participants reported greater interest in using MetroSets than LineSets
(AUC = 0.8), and greater interest in using LineSets than EulerView (AUC =
0.7). They also preferred MetroSets to EulerView (AUC = 0.9).
For the free-form response, we collected a total of 35 ($\sim 60\%$) responses
for the small dataset and 41 ($\sim 71\%$) responses for the large dataset.
The general opinion is that regions in EulerView are hard to distinguish,
especially if the region depicts a intersection of more than two sets. The
following response is a typical example: “EulerView works great until you have
more than 3 overlapping genres. For smaller sets of genres, I think EulerView
would be preferable, but when artists belong to too many genres, it becomes
difficult to identify all the patterns in the space.”
Participants had positive opinions about MetroSets and LineSets, with a slight
preference for MetroSets when listing all three systems. Overlapping lines in
LineSets was a recurring issue: “Some instances of LineSets were very clear.
Some other instances were unclear due to a lot of things going on in a very
small region. I had the impression that such situations were avoided with
MetroSets.” This is in line with the observation we made in the statistical
analysis that LineSets does not scale as well as MetroSets. The color choice
in all three systems was frequently critiqued. In particular, participants had
difficulty distinguishing parallel lines with similar colors in MetroSets. A
list of all responses can be found in the supplementary material.
(a)
(b)
(c)
Figure 6: Responses to post-questionnaire Likert scale questions. In question
(a) and (b) the answer options ranged from 1 (poorly) to 5 (clearly). For
question (c) the range was 1 (not) to 5 (very).
### 6.6 Findings
We hypothesized that, for both element and set based tasks, MetroSets would
outperform LineSets and LineSets would outperform EulerView (H1 and H2). Our
qualitative analysis supports these hypotheses: participants significantly
preferred MetroSets over LineSets and LineSets over EulerView for both
element-based and set-based tasks. Our quantitative analysis partially
supports H1: MetroSets and LineSets both performed significantly better than
EulerView on all element-based tasks, but there was no conclusive difference
between MetroSets and LineSets. Likewise, our data partially supports H2: on
all of the set-based tasks, MetroSets outperformed either LineSets, EulerView,
or both. However, there is no clear order between LineSets and EulerView:
LineSets did much better on task T4, while EulerView did much better on task
T6. The two performed comparably on task T5.
Finally, we hypothesized that MetroSets would scale better than LineSets or
EulerView (H3). Our evidence supports this hypothesis: For all of the tasks
except T5, MetroSets’ performance did not significantly degrade on the larger
dataset, and the difference on task T5 was considerably larger for the other
systems. Meanwhile, LineSets’ performance degraded significantly on tasks T5
and T6, while EulerView’s performance degraded significantly on tasks T1, T3,
and T5.
We additionally performed exploratory analysis to determine if there were any
meaningful correlations between demographic data, accuracy, and responses to
the post-test questionnaire. In general, there is no obvious pattern between
responses in one category and those in another. This is somewhat surprising:
our demographic questions include factors such as familiarity with the
visualization types, which intuitively could be relevant.
## 7 Discussion
We next discuss some of the limitations of our study, as well as implications
for improving existing set visualization approaches.
### 7.1 Limitations
While we aimed for a study design that allows to generalize the findings to
visualizing set systems of similar characteristics, i.e., small to medium-
sized data with less than 10–12 sets, less than 80–100 elements, and elements
being members of 1–4 sets, we are aware of many limitations of our experiment.
We list just a few examples below. We chose to use real-world music data to
arouse the interest in voluntary participation in our study. This prevented us
from having fine control over the visual stimuli; nonetheless, all three
techniques used the same datasets and so the effect of the real-world data
should apply to all. The use of real-world data also is associated with the
risk that some participants might use prior knowledge when performing the
tasks. However, the majority (84%) of participants (self) reported never or
almost never using pre-existing knowledge of music to help them answer
questions.
In the interest of keeping the study relatively short, we split the
participants into two groups: one working with the small datasets and the
other with the large datasets. Traditional confounding factors for between
subjects studies likely apply here, hopefully ameliorated by the reasonably
large number of participants in both settings. As we decided to run the
experiment online in a web browser rather than in a controlled lab setting,
participants were using a variety of different screen sizes, input devices,
and environmental conditions, which may have affected their task performance.
We attempted to mitigate this issue by excluding experimental trials from
smartphones or with screen resolutions below a minimum threshold.
Some pairs of line colors could be difficult to discern and some of the
textures used in EulerView were difficult to perceive than others. We
attempted to mitigate this issue by using the same color scheme for both
MetroSets and LineSets and by selecting tasks of similar difficulty levels
across the different visualizations. However, it is possible that confounding
factors associated with colors and textures do exists.
We also acknowledge that our experiment only compares combinations of designs
and implementations, and that the differences observed could be due to either.
For example, in generating the LineSets stimuli, we contacted the original
authors and followed their recommendation to use a standard force-based graph
layout algorithm to determine the positions of elements. It is entirely
possible that an implementation of LineSets using a different layout method
would perform better. More generally, each of the designs we considered might
admit better implementations than are currently available.
### 7.2 Implications
A recurring theme in the qualitative feedback was that participants had
difficulty distinguishing colors in all three systems. The problem was most
acute with EulerView, where regions often became muddled when many sets
overlapped simultaneously. This difficulty may partly account for EulerView’s
comparatively poor performance in our experiment. However, given the strengths
of EulerView relative to other Euler-based visualization systems, it may be
worthwhile to revisit the technique, for example by using the smoother
boundary contours proposed by Simonetto et al. [16].
MetroSets and LineSets used the same color scheme in our experiment, and in
both cases participants complained that some colors (particularly red and
pink) were too difficult to distinguish. Usability of both systems would be
improved by better use of colors. Borrowing from EulerView, it might be
beneficial to explore using a smaller categorical color scheme supplemented by
textures, such as dots or dashes along different lines.
For the most part, LineSets performed well in our study. However, it did
occasionally produce ambiguous results, resulting in very low accuracy on some
tasks. These problems are likely fixable. For example, representing elements
with a glyph and then labeling, rather than representing elements directly
with a label will likely help. From an algorithmic perspective, it may also be
useful to compute curves collectively, rather than independently. In this way,
it would be easy to detect overlapping lines and to split them apart.
## 8 Conclusions
We evaluated three different systems for set visualization via a human-
subjects experiment comparing their effectiveness both quantitatively and
qualitatively. Our results include statistically significant differences,
suggesting that MetroSets scales better and performs more consistently than
EulerView or LineSets, and is also better-liked. We additionally considered
implications for the design of all three systems.
## Acknowledgments
The authors would like to thank all of the participants in our experiment, and
especially the experts of our pilot study (Daniel Archambault, Helen Purchase,
Silvia Miksch). This work is supported by NSF grants CCF-1740858, CCF-1712119,
and DMS-1839274 and by the Vienna Science and Technology Fund (WWTF) through
project ICT19-035.
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| Markus Wallinger is a PhD student in the Algorithms and Complexity group of
TU Wien. He received his BS degree in Computer Science from the University of
Innsbruck and his MS degree in Visual Computing from TU Wien. His research
focuses on algorithms for information visualization, graph theory and linear
orders.
---|---
| Ben Jacobsen is an undergraduate student at the University of Arizona
majoring in Mathematics and Computer Science. His interests include graph
theory, computer security, and the intersection of computer science,
mathematics, and philosophy.
---|---
| Stephen Kobourov is a Professor at the Department of Computer Science at
the University of Arizona. He received a BS degree in Mathematics and Computer
Science from Dartmouth College and MS and PhD degrees from Johns Hopkins
University. His research interests include information visualisation, graph
theory, and geometric algorithms.
---|---
| Martin Nöllenburg is a Professor in the Algorithms and Complexity Group of
TU Wien, Vienna, Austria. He received his PhD and habilitation degrees from
Karlsruhe Institute of Technology (KIT), Germany, in 2009 and 2015,
respectively. His research interests include graph drawing algorithms,
computational geometry, and information visualization with a focus on
applications in network visualization and computational cartography.
---|---
## Appendix A Set Visualization Systems
### A.1 EulerView
EulerView by Simonetto et al. [8] generates Euler-like diagrams of set
systems. Euler diagrams consists of simple closed shapes in the two-
dimensional plane that depict sets. Each curve of the boundary of the shape
divides the plane into two regions. Elements inside the enclosed region are
the members of the respective set, whereas elements outside the curve are not.
The EulerView algorithm extends Euler diagrams by allowing holes in regions or
split regions, thus avoiding undrawable instances.
The implementation of EulerView is realized as a module in the Tulip [50] data
visualization framework. Set elements are represented by white square glyphs
with a black border and an attached label by default. Even though most
attributes, such as glyph size, glyph shape, or label position, could be
manipulated, we opted to keep the diagrams similar in style to the examples
given in the original paper [8], as any manipulation changed the rendering of
textures drastically. The module does not incorporate the later improvements
of smoother boundary contours proposed by Simonetto et al. [16]. We were not
able to use the EulerView module with a recent version of Tulip, even after
contacting the original author. Eventually, we were able to use Tulip version
3.5 to create the output.
A major advantage of EulerView (and Euler-based systems generally) is that all
elements with identical set membership are grouped together in the final
visualization. It additionally solves several major issues plaguing Euler-
based visualizations, such as the absence of element labels and the
possibility of implying non-existent set overlaps. Despite these strengths,
there are also certain weaknesses inherent in this type of visualizations. In
particular, the textures used to differentiate different regions often become
hard to distinguish when more than two or three sets overlap. This problem is
exacerbated by the fact that the boundaries of different regions often
overlap, making it difficult to identify them through their outlines.
### A.2 LineSets
LineSets by Alper et al. [4] is an overlay-based system for set visualization.
It represents elements as points in the 2D plane and sets as continuous
curves. If a set contains an element, then the corresponding curve passes
through the corresponding point. Like other overlay-based systems, LineSets
expects positions for each element as part of the input, and so the main
algorithmic work consists of choosing the order in which each curve visits all
elements that correspond to that set. Once such an order is determined, the
curve is overlaid on top of the elements. Within these broad constraints,
there are different ways to implement LineSets. We use the implementation
provided in GMap[51], which is available on github.
To apply LineSets to abstract data with no intrinsic spatial dimension, we
follow the original designers of the system and create an initial embedding
for the data using a force-based algorithm. Specifically, we treat the input
as a graph, where two nodes are connected by an edge if they share at least
one set, and generate a layout for it using the force-directed algorithm by
Fruchterman-Reingold algorithm [40], as implemented in NetworkX [52]. The
positions generated for each node are then included in the input to LineSets.
It should be noted that, while this approach follows that taken by the authors
of LineSets, it is not a fundamental component of the design. Alternative
methods for determining layout, such as multi-dimensional scaling, would be
equally viable, and the performance of LineSets in our study may to some
extent reflect our particular choice of layout algorithm.
While LineSets visualizations are generally easy to interpret, two types of
ambiguities can arise in some situations. First, because labels are opaque and
placed directly on top of the curves representing sets, it is sometimes
difficult to determine whether a line actually goes through an element, or
simply passes nearby. Second, lines travelling along the same sub-curve can
overlap one another, making it hard to tell at a glance exactly how many lines
connect a given pair of elements.
### A.3 MetroSets
The core concept behind MetroSets [9] is to apply the metro map metaphor to
set visualization: set elements are metro stations and different sets are
different metro lines. Whenever a line passes through a station the associated
element is considered a member of the set depicted by the line. If an element
is in multiple sets it will be depicted as an interchange station. MetroSets
creates a schematic drawing that adheres to typical metro map design rules
[53, 54, 55]: colored octolinear (horizontal, vertical, or 45∘-diagonals)
lines, labeled stations, uniform distance between stations, straight lines.
MetroSets uses a 4-step pipeline approach with two or more algorithms for each
step that can be specified when creating a visualization. In our study, we use
the default pipeline. Elements are depicted by circles whose diameter is
proportional to their set membership degree and lines are drawn as sequences
of octolinear poly-lines between stations. Each line is given a distinct color
(if less than 21 sets are present) from the Tableau 20 color palette. Finally,
each set element has a label assigned that is placed next to the corresponding
station in $45^{\circ}$ steps and adheres to the design criteria for metro map
labeling [56]. The resulting drawing is rendered using the D3 javascript
library444see https://d3js.org/ and a legend is added at the bottom right
corner.
While the visual style of MetroSets is similar to LineSets, there are several
important differences. In particular, MetroSets avoids issues with lines
overlapping each other by adding regular white space between parallel lines.
In addition, by placing labels near the elements, rather then on top of them,
MetroSets reduces potential ambiguity when determining whether a line passes
through a given element.
## Appendix B Additional statistical results
This section contains further descriptive statistics and visualizations of the
time and accuracy data we gathered through our experiments. Table I presents
the mean and standard deviation of accuracy for each task, size and system,
while Table II does the same for time. Note that the mean and standard
deviation are not robust against outliers, which are present in large number
here because of the inherently skewed nature of response time data: Figure 5
visualizes the full distribution of response times for each task, size and
system.
TABLE I: Accuracy of Participants
Task | | E30 | L30 | M30 | E60 | L60 | M60
---|---|---|---|---|---|---|---
T1 | mean
std | 0.97
0.18 | 0.97
0.18 | 1.00
0.00 | 0.74
0.44 | 0.98
0.13 | 0.95
0.22
T2 | mean
std | 0.33
0.47 | 0.69
0.47 | 0.95
0.22 | 0.88
0.33 | 0.93
0.26 | 0.83
0.38
T3 | mean
std | 0.86
0.35 | 0.97
0.18 | 0.83
0.38 | 0.16
0.37 | 0.95
0.22 | 0.98
0.13
T4 | mean
std | 0.33
0.47 | 0.97
0.18 | 0.93
0.26 | 0.55
0.50 | 0.95
0.22 | 0.91
0.28
T5 | mean
std | 0.78
0.42 | 0.95
0.22 | 0.95
0.22 | 0.16
0.37 | 0.14
0.35 | 0.71
0.46
T6 | mean
std | 0.48
0.50 | 0.62
0.49 | 0.84
0.37 | 0.72
0.45 | 0.21
0.41 | 0.76
0.43
This table summarizes statistics describing the accuracy of participants. Each
row represents a single task, while each column represents a particular system
and size.
TABLE II: Timing of Participants
Task | | E30 | L30 | M30 | E60 | L60 | M60
---|---|---|---|---|---|---|---
T1Time | mean
std | 19.96
9.08 | 19.91
10.34 | 21.38
9.44 | 34.67
21.30 | 29.59
37.25 | 24.77
11.72
T2Time | mean
std | 47.01
51.85 | 28.76
9.65 | 27.72
9.04 | 34.07
16.16 | 24.67
10.74 | 24.43
10.05
T3Time | mean
std | 29.10
12.56 | 27.94
17.00 | 37.42
20.54 | 42.36
24.52 | 32.57
15.73 | 33.78
16.72
T4Time | mean
std | 51.80
24.90 | 37.45
11.04 | 41.41
19.96 | 61.10
28.72 | 53.66
19.42 | 60.07
22.58
T5Time | mean
std | 35.96
13.52 | 43.84
16.14 | 36.29
11.69 | 55.35
26.80 | 54.49
25.97 | 50.70
33.13
T6Time | mean
std | 32.11
13.10 | 39.04
17.64 | 31.39
9.71 | 52.42
33.86 | 53.44
29.71 | 43.01
21.93
This table summarizes statistics describing the time taken on each task. Each
row represents a single task, while each column represents a particular system
and size.
|
# Lessons from the German Tank Problem
George Clark<EMAIL_ADDRESS>Department of Mathematics and Statistics,
Williams College, Williamstown, MA 01267 , Alex Gonye<EMAIL_ADDRESS>Department of Mathematics and Statistics, Williams College, Williamstown, MA
01267 and Steven J. Miller
sjm1<EMAIL_ADDRESS>Department of Mathematics
and Statistics, Williams College, Williamstown, MA 01267 Department of
Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213
###### Abstract.
During World War II the German army used tanks to devastating advantage. The
Allies needed accurate estimates of their tank production and deployment. They
used two approaches to find these values: spies, and statistics. This note
describes the statistical approach. Assuming the tanks are labeled
consecutively starting at 1, if we observe $k$ serial numbers from an unknown
number $N$ of tanks, with the maximum observed value $m$, then the best
estimate for $N$ is $m(1+1/k)-1$. This is now known as the German Tank
Problem, and is a terrific example of the applicability of mathematics and
statistics in the real world. The first part of the paper reproduces known
results, specifically deriving this estimate and comparing its effectiveness
to that of the spies. The second part presents a result we have not found in
print elsewhere, the generalization to the case where the smallest value is
not necessarily 1. We emphasize in detail why we are able to obtain such
clean, closed-form expressions for the estimates, and conclude with an
appendix highlighting how to use this problem to teach regression and how
statistics can help us find functional relationships.
###### Key words and phrases:
German Tank Problem, Binomial Identities, Regression
###### 2010 Mathematics Subject Classification:
62J05, 60C05 (primary), 05A10 (secondary)
The third named author was partially supported by NSF Grant DMS1561945. Parts
of this paper were given by the first two named authors as their senior
colloquium; we thank our colleagues in the department for comments. The third
author has presented this topic at several conferences and programs, and
thanks the participants for many suggestions; for video see
https://www.youtube.com/watch?v=quV-MCB8Ozs or https://youtu.be/I3ngtIYjw3w.
We thank our colleagues at Williams and audiences at several talks over the
years, and the referee, for valuable feedback that improved the exposition,
and Jason Zhou for pointing out some typos in an earlier draft.
###### Contents
1. 1 Introduction
2. 2 Derivation with a known minimum
1. 2.1 The probability that the sample maximum is $m$
2. 2.2 The best guess for $\mathaccent 866{N}$
3. 3 Derivation with an unknown minimum
1. 3.1 The probability that the spread is $s$
2. 3.2 The best guess for $\mathaccent 866{N}$
4. 4 Comparison of Approaches
5. A The German Tank Problem and Linear Regression
1. A.1 Theory of Regression
2. A.2 Issues in Applying to the German Tank Problem
3. A.3 Resolving Implementation Issues
4. A.4 Determining the Functional Form
## 1\. Introduction
In this paper we revisit a famous and historically important problem which has
since become a staple in many probability and statistics classes: the German
Tank Problem. This case study illustrates that one does not need to use the
most advanced mathematics to have a tremendous impact on real world problems;
the challenge is frequently in creatively using what one knows.111Another
great example is the famous Battle of Midway and the role the cryptographers
played in figuring out the Japanese target; see for example [1].
Figure 1. Left: Europe before the start of major hostilities. Right: Europe
in 1942. Images from Wikimedia Commons from author San Jose.
By the end of 1941 most of continental Europe had fallen to Nazi Germany and
the other Axis powers, and by 1942 their forces had begun their significant
advance in the Eastern Front deep into the Soviet Union; see Figure 1 for an
illustration of their rapid progress, or
https://www.youtube.com/watch?v=WOVEy1tC7nk for an animation of territorial
gains day by day. A key component to their rapid conquests lay with their
revolutionary use of tanks in modern warfare. While other militaries, most
notably France, used tanks as a modern, armored form of cavalry, the Germans
were the first to fully utilize tanks’ speed and strength to their advantage.
Tanks would move rapidly and punch through enemy lines creating gaps which
German infantry would stream through. Once through the holes in the line, the
Germans would wreak havoc on lines of communication, creating logistical
nightmares for the combatants left on the front lines. This lightning fast
warfare has become dubbed Blitzkrieg (or lighting war) by many historians.
With the Nazis utilizing tanks with such devastating results, it was essential
for the Allies to stop them. A key component to the solution was figuring out
how many tanks the Germans were building, or had deployed in various theaters,
in order to properly allocate resources. As expected, they tried spying (both
with agents and through decrypting intercepted messages) to accurately
estimate these numbers. It was essential that the Allies obtain accurate
values, as there is a tremendous danger in both under- and over-estimating the
enemy’s strength. The consequence of underestimating is clear, as one could
suddenly be outnumbered in battle. Overestimating is also bad, as it can lead
to undo caution and failure to exploit advantages, or to committing too many
resources in one theater and thus not having enough elsewhere. The U.S. Civil
War provides a terrific example of these consequences, where General George
McClellan would not have his Union army take the field against the
Confederates because, in his estimation, his forces were greatly outnumbered
though in fact they were not. This situation led to one of President Abraham
Lincoln’s many great quips: _If General McClellan isn’t going to use his army,
I’d like to borrow it for a time._ 222There are many different phrasings of
this remark; this one is taken from
https://thehistoriansmanifesto.wordpress.com/2013/05/13/best-abraham-lincoln-
quotes/. Considering how close Pickett’s charge came to succeeding at
Gettysburg, or what would have happened if Sherman hadn’t taken Atlanta before
the 1864 elections (where McClellan, now as the Democrat nominee for
president, was running against Lincoln on a platform of a negotiated peace),
the paralysis from incorrect analysis could have changed the outcome of the
war.
Returning to World War II and the problem of determining the number of tanks
produced and facing them in the field, the Allies understandably wanted to
have a way to evaluate the effectiveness of their estimates. During a battle,
they realized that the destroyed and captured tanks had serial numbers on
their gearboxes which could help with this problem. Assuming that the serial
numbers are sequential and start with 1, given a string of observed numbers
one can try to estimate the largest value. This discovery contributed to the
birth of the statistical method, the use of a population maximum formula, and
changed both the war and science.333There are advantages in consecutive
labeling; for example, it simplifies some maintenance issues as it is clear
which of two tanks is older.
The original variant of the problem assumes the first tank is numbered 1,
there are an unknown number $N$ produced (or in theater), that the numbers are
consecutive, and that $k$ values are observed with the largest being $m$.
Given this information, the goal is to find $\mathaccent 866{N}$, the best
estimate of $N$. The formula they derived is
$\mathaccent 866{N}\ =\ m\left(1+\frac{1}{k}\right)-1.$
For the reader unfamiliar with this subject, we deliberately do not state here
in the introduction how well this formula does versus what was done by spies
and espionage.444That said, as this paper is appearing in a mathematics
journal and not a bulletin of a spy agency, we invite the reader to conjecture
which method did better. While this story has been well told before (see for
example [3, 4, 5, 6]), our contribution is to extend the analysis to consider
the more general case, namely what happens when we do not know the smallest
value. To our knowledge this result has not been isolated in the literature;
we derive in §3 that if $s$ is the spread between the smallest and the largest
of the observed $k$ serial numbers, then
$\mathaccent 866{N}\ =\ s\left(1+\frac{2}{k-1}\right)-1.$
In Appendix A we show how to use regression to show that these are reasonable
formulas, and thus the German Tank Problem can also be used to introduce some
problems and subtleties in regression analysis, as well as serve as an
introduction to mathematical modeling.
As it is rare to have a clean, closed-form expression such as the ones above,
we briefly remark on our fortune. The key observation is that we have a
combinatorial problem where certain binomial identities are available, and
these lead to tremendous simplifications.
## 2\. Derivation with a known minimum
In this section we prove
$\mathaccent 866{N}\ =\ m\left(1+\frac{1}{k}\right)-1$
when we observe $k$ tanks, the largest labeled $m$, and knowing that the
smallest number is 1 and the tanks are consecutively numbered. Before proving
it, as a smell test we look at some extreme cases. First, we never obtain an
estimate that is less than the largest observed number. Second, if there are
many tanks and we observe just one (so $k=1$), then $\mathaccent 866{N}$ is
approximately $2m$. This is very reasonable, and essentially just means that
if we only have one data point, it’s a good guess that it was in the middle.
Further, as $k$ increases the amount we must inflate our observed maximum
value decreases. For example, when $k=2$ we inflate $m$ by approximately a
factor of $3/2$, or in other words this is saying our observed maximum value
is probably about two-thirds of the true value. Finally, if $k$ equals the
number of tanks $N$, then $m$ must also equal $N$, and the formula simplifies
to $\mathaccent 866{N}=N$.
We break the proof into two parts. While we are fortunate in that we are able
to obtain a closed-form expression, if we have a good guess as to the
relationship we can use statistics to test its reasonableness; we do that in
Appendix A. For the proof we first determine the probability that the observed
largest value is $m$. Next we compute the expected value, and show how to pass
from that to an estimate for $N$. We need two combinatorial results.
The first is Pascal’s identity:
$\displaystyle{n+1\choose r}\ =\ {n\choose r}+{n\choose r-1}.$ (2.1)
There are many approaches to proving this; the easiest is to interpret both
sides as two different ways of counting how many ways we can choose $r$ people
from a group of $n+1$ people, where exactly $n$ of these people are in one set
and exactly one person is in another set. It is easier to see if we rewrite it
as
${n+1\choose r}\ =\ {1\choose 0}{n\choose r}\ +\ {1\choose 1}{n\choose r-1};$
this is permissible because ${1\choose 0}={1\choose 1}=1$. Note the left side
is choosing $r$ people from the combined group of $n+1$ people, while the
right is choosing $r$ people with the first summand corresponds to not
choosing the person from the group with just one person, and the second
summand to requiring we chose that person. $\Box$
The second identity involves sums of binomial coefficients:
$\displaystyle\sum_{m=k}^{N}{m\choose k}\ =\ {N+1\choose k+1}.$ (2.2)
We can prove this by induction on $N$, noting that $k$ is fixed. The base case
is readily established. Letting $N=k$, we find
$\sum_{m=k}^{N}{m\choose k}\ =\ \sum_{m=k}^{k}{m\choose k}\ =\ {k\choose k}\
=\ 1\ =\ {k+1\choose k+1}.$
For the inductive step, we assume
$\sum_{m=k}^{N}{m\choose k}\ =\ {N+1\choose k+1}.$
Then
$\displaystyle\sum_{m=k}^{N+1}{m\choose k}$ $\displaystyle\ =\ $
$\displaystyle\left(\sum_{m=k}^{N}{m\choose k}\right)+{N+1\choose k}$
$\displaystyle\ =\ $ $\displaystyle{N+1\choose k+1}+{N+1\choose k}$
$\displaystyle\ =\ $ $\displaystyle{N+2\choose k+1},$
where the last equality follows from Pascal’s identity, (2.1). This completes
the proof. $\Box$
While this identity suffices for the original formulation of the German Tank
Problem, when we do not know the starting serial number the combinatorics
become slightly more involved, and we need a straightforward generalization:
$\sum_{\ell=a}^{b}{\ell\choose a}\ =\ {b+1\choose a+1};$ (2.3)
the proof follows similarly.
### 2.1. The probability that the sample maximum is $m$
Let $M$ be the random variable for the maximum number observed, and let $m$ be
the value we see. Note that there is zero probability of observing a value
smaller than $k$ or larger than $N$. We claim for $k\leq m\leq N$ that
${\rm Prob}(M=m)\ =\ \frac{{m\choose k}-{m-1\choose k}}{{N\choose k}}\ =\
\frac{{m-1\choose k-1}}{{N\choose k}}.$
We give two proofs. The first is to note there are ${N\choose k}$ ways to
choose $k$ numbers from $N$ when order does not matter. The probability that
the largest observed is exactly $m$ equals the probability the largest is at
most $m$ minus the probability the largest is at most $m-1$. The first
probability is just ${m\choose k}/{N\choose k}$, as if the largest value is at
most $m$ then all $k$ observed numbers must be taken from $\\{1,2,\dots,m\\}$.
A similar argument gives the second probability is ${m-1\choose k}/{N\choose
k}$, and the claim now follows by using Pascal’s identity to simplify the
difference of the binomial coefficients.
We could also argue as follows. If the largest is $m$ then we have to choose
that serial number, and now we must choose $k-1$ tanks from the $m-1$ smaller
values; thus we find the probability is just ${m-1\choose k-1}/{N\choose k}$.
$\Box$
###### Remark 2.1.
Interestingly, we can use the two equivalent arguments above as yet another
way to prove the Pascal identity.
### 2.2. The best guess for $\mathaccent 866{N}$
We now compute the best guess for $N$ by first finding the expected value of
$M$. Recall the expected value of a random variable $M$ is the sum of all the
possible values of $M$ times the probability of observing that value. We write
$\mathbb{E}[M]$ for this quantity, and thus we must compute
$\mathbb{E}[M]\ :=\ \sum_{m=k}^{N}m{\rm Prob}(M=m)$
(note we only need to worry about $m$ in this range, as for all other $m$ the
probability is zero and thus does not contribute). Once we find a formula for
$\mathbb{E}[M]$ we will convert that to one for the expected number of tanks.
Our first step is to substitute in the probability that $M$ equals $m$,
obtaining
$\mathbb{E}[M]\ =\ \sum_{m=k}^{N}m\frac{{m-1\choose k-1}}{{N\choose k}}.$
Fortunately this sum can be simplified into a nice closed-form expression; it
is this simplification that allows us to obtain a simple formula for
$\mathaccent 866{N}$. We expand the binomial coefficients in the expression
for $\mathbb{E}[M]$ and then use our second combinatorial identity, (2.2), to
simplify the sum of ${m\choose k}$ which emerges as we manipulate the
quantities below. We find
$\displaystyle\mathbb{E}[M]$ $\displaystyle\ =\ $
$\displaystyle\sum_{m=k}^{N}m\frac{{m-1\choose k-1}}{{N\choose k}}$
$\displaystyle\ =\ $
$\displaystyle\sum_{m=k}^{N}m\frac{(m-1)!}{(k-1)!(m-k)!}\frac{k!(N-k)!}{N!}$
$\displaystyle=$
$\displaystyle\sum_{m=k}^{N}\frac{m!k}{k!(m-k)!}\frac{k!(N-k)!}{N!}$
$\displaystyle=$ $\displaystyle\frac{k\cdot
k!(N-k)!}{N!}\sum_{m=k}^{N}{m\choose k}$ $\displaystyle=$
$\displaystyle\frac{k\cdot k!(N-k)!}{N!}{N+1\choose k+1}$ $\displaystyle=$
$\displaystyle\frac{k\cdot k!(N-k)!}{N!}\frac{(N+1)!}{(k+1)!(N-k)!}$
$\displaystyle=$ $\displaystyle\frac{k(N+1)}{k+1}.$
As we have such a clean expression, it’s trivial to solve for $N$ in terms of
$k$ and $\mathbb{E}[M]$:
$N\ =\ \mathbb{E}[M]\left(1+\frac{1}{k}\right)-1.$
Thus if we substitute in $m$ (our observed value for $M$) as our best guess
for $\mathbb{E}[M]$, we obtain our estimate for the number of tanks produced:
$\mathaccent 866{N}\ =\ m\left(1+\frac{1}{k}\right)-1,$
completing the proof. $\Box$
###### Remark 2.2.
A more advanced analysis can prove additional results about our estimator, for
example, whether or not it is unbiased.
###### Remark 2.3.
There are many ways to see this formula is reasonable. The first is to try
extreme cases, such as $k=N$ (which forces $m$ to be $N$ and gives $N$ as the
answer), or to try $k=1$. In that case we expect our one observation to be
around $N/2$, and thus a formula that has the best guess being doubling the
observation is logical. We can also get close to this formula from by trying
to guess the functional form (for more details see Appendix A). We know our
best guess must be at least $m$, so let’s write it as $m+f(m,k)$. For a fixed
$k$ as $m$ increases we might expect our guess to increase, while for fixed
$m$ as $k$ increases we would expect a smaller boost. These heuristics suggest
$f(m,k)$ increases with $m$ and decreases with $k$; the simplest such function
is $bm/k$ for some constant $b$. This leads to a guess of $m+bm/k$, and again
looking at extreme cases we get very close to the correct formula.
## 3\. Derivation with an unknown minimum
Not surprisingly, when we do not know the lowest serial number the resulting
algebra becomes more involved; fortunately, though, with a bit of work we are
still able to get nice closed-form expressions for the needed sums and obtain
again a clean answer for the estimated number of tanks. We still assume the
tanks are numbered sequentially, and focus on the spread (the difference
between the largest and smallest observed values). Similar to the previous
section, we derive a formula to inflate the observed spread to be a good
estimate of the number of total tanks.
We first set some notation:
* •
the minimum tank serial value, $N_{1}$,
* •
the maximum tank serial value, $N_{2}$,
* •
the total number of tanks, $N$ ($N=N_{2}-N_{1}+1$),
* •
the observed minimum value, $m_{1}$ (with corresponding random variable
$M_{1}$),
* •
the observed maximum value, $m_{2}$ (with corresponding random variable
$M_{2}$),
* •
the observed spread $s$ (with corresponding random variable $S$).
As $s=m_{2}-m_{1}$, in the arguments below we can focus on just $s$ and $S$.
We will prove the best guess is $s\left(1+\frac{2}{k-1}\right)-1$.
###### Remark 3.1.
There are two differences between this formula and the case when the smallest
serial number is known. The first is we divide by $k-1$ and not $k$; however,
as we cannot estimate a spread with one observation this is reasonable. Note
the similarity here with the sample standard deviation, where we divide by one
less than the number of observations; while one point suffices to estimate a
mean, we need at least two for the variance. The second difference is that we
have a factor of 2, which can be interpreted as movement in both directions.
### 3.1. The probability that the spread is $s$
We claim that if we observe $k$ tanks then for $k-1\leq s\leq N_{2}-N_{1}$ we
have
${\rm Prob}(S=s)\ =\
\frac{\sum_{m=N_{1}}^{N_{2}-s}{{s-1}\choose{k-2}}}{{{N_{2}-N_{1}+1}\choose{k}}}\
=\
\frac{\left(N_{2}-N_{1}+1-s\right){{s-1}\choose{k-2}}}{{{N_{2}-N_{1}+1}\choose{k}}}\
=\ \frac{\left(N-s\right){{s-1}\choose{k-2}}}{{N\choose k}},$
and for all other $s$ the probability is zero.
To see this, note that the spread $s$ must be at least $k-1$ (as we have $k$
observations), and cannot be larger than $N_{2}-N_{1}$. If we want a spread of
$s$, if the smallest observed value is $m$ then the largest is $m+s$. We must
choose exactly $k-2$ of the $s-1$ numbers in $\\{m+1,m+2,\dots,m+s-1\\}$;
there are ${s-1\choose k-2}$ ways to do so. This proves the first equality,
the sum over $m$. As all the summands are the same we get the second equality,
and the third follows from our definition of $N$. $\Box$
### 3.2. The best guess for $\mathaccent 866{N}$
We argue similarly as in the previous section. In the algebra below we will
use our second binomial identity, (2.2); relabeling the parameters it is
$\displaystyle\sum_{\ell=a}^{b}{\ell\choose a}\ =\ {b+1\choose a+1}.$ (3.1)
We begin by computing the expected value of the spread. We include all the
details of the algebra; the idea is to manipulate the expressions and pull out
terms that are independent of the summation variable, and rewrite expressions
so that we can identify binomial coefficients and then apply our combinatorial
results. We have
$\displaystyle\mathbb{E}[S]$ $\displaystyle\ =\ $
$\displaystyle\sum_{s=k-1}^{N-1}s{\rm Prob}(S=s)$ $\displaystyle=$
$\displaystyle\sum_{s=k-1}^{N-1}s\frac{\left(N-s\right){{s-1}\choose{k-2}}}{{N\choose
k}}$ $\displaystyle=$ $\displaystyle{N\choose
k}^{-1}\sum_{s=k-1}^{N-1}s\left(N-s\right){{s-1}\choose{k-2}}$ $\displaystyle\
=\ $ $\displaystyle{N\choose
k}^{-1}N\sum_{s=k-1}^{N-1}\frac{s(s-1)!}{(s-k+1)!(k-2)!}-{N\choose
k}^{-1}\sum_{s=k-1}^{N-1}\frac{s^{2}(s-1)!}{(s-k+1)!(k-2)!}$ $\displaystyle\
=\ $ $\displaystyle{N\choose
k}^{-1}N\sum_{s=k-1}^{N-1}\frac{s!(k-1)}{(s-k+1)!(k-1)!}-{N\choose
k}^{-1}\sum_{s=k-1}^{N-1}\frac{ss!(k-1)}{(s-k+1)!(k-1)!}\ =\ T_{1}-T_{2}.$
We first simplify $T_{1}$; below we always try to multiply by 1 in such a way
that we can combine ratios of factorials into binomial coefficients:
$\displaystyle T_{1}$ $\displaystyle\ =\ $ $\displaystyle{N\choose
k}^{-1}N\sum_{s=k-1}^{N-1}\frac{s!(k-1)}{(s-k+1)!(k-1)!}$ $\displaystyle=$
$\displaystyle{N\choose k}^{-1}N(k-1)\sum_{s=k-1}^{N-1}{s\choose k-1}$
$\displaystyle=$ $\displaystyle{N\choose k}^{-1}N(k-1){N\choose k}\ =\
N(k-1),$
where we used (3.1) with $a=k-1$ and $b=N-1$.
Turning to $T_{2}$ we argue similarly, at one point replacing $s$ with
$(s-1)+1$ to assist in collecting factors into a binomial coefficient:
$\displaystyle T_{2}$ $\displaystyle\ =\ $ $\displaystyle{N\choose
k}^{-1}\sum_{s=k-1}^{N-1}\frac{ss!(k-1)}{(s-k+1)!(k-1)!}$ $\displaystyle\ =\ $
$\displaystyle{N\choose
k}^{-1}\sum_{s=k-1}^{N-1}\frac{(s+1-1)s!(k-1)}{(s-(k-1))!(k-1)!}$
$\displaystyle\ =\ $ $\displaystyle{N\choose
k}^{-1}\sum_{s=k-1}^{N-1}\frac{(s+1)!(k-1)k}{(s+1-k)!(k-1)!k}-{N\choose
k}^{-1}\sum_{s=k-1}^{N-1}\frac{s!(k-1)}{(s-(k-1))!(k-1)!}$ $\displaystyle\ =\
$ $\displaystyle{N\choose
k}^{-1}\sum_{s=k-1}^{N-1}\frac{(s+1)!k(k-1)}{(s+1-k)!k!}-{N\choose
k}^{-1}\sum_{s=k-1}^{N-1}(k-1){s\choose k-1}$ $\displaystyle\ =\ $
$\displaystyle{N\choose k}^{-1}\sum_{s=k-1}^{N-1}k(k-1){s+1\choose
k}-{N\choose k}^{-1}\sum_{s=k-1}^{N-1}(k-1){s\choose k-1}\ =\ T_{21}+T_{22}.$
We can immediately evaluate $T_{22}$ by using (2.3) with $a=k-1$ and $b=N-1$,
and find
$T_{22}\ =\ {N\choose k}^{-1}(k-1){N\choose k}\ =\ k-1.$
Thus all that remains is analyzing $T_{21}$:
$T_{21}\ =\ {N\choose k}^{-1}\sum_{s=k-1}^{N-1}{s+1\choose k}k(k-1).$
We pull $k(k-1)$ outside the sum, and letting $w=s+1$ we see that
$T_{21}\ =\ {N\choose k}^{-1}k(k-1)\sum_{w=k}^{N}{w\choose k},$
and then from (2.3) with $a=k$ and $b=N$ we obtain
$T_{21}\ =\ {N\choose k}^{-1}k(k-1)\sum_{w=k}^{N}{w\choose k}\ =\ {N\choose
k}^{-1}k(k-1){N+1\choose k+1}.$
Thus substituting everything back yields
$\mathbb{E}[S]\ =\ N(k-1)+(k-1)-{N\choose k}^{-1}k(k-1){N+1\choose k+1}.$
We can simplify the right hand side:
$\displaystyle(N+1)(k-1)-k(k-1)\frac{\frac{(N+1)!}{(N-k)!(k+1)!}}{\frac{N!}{(N-k)!k!}}$
$\displaystyle\ =\ $
$\displaystyle(N+1)(k-1)-k(k-1)\frac{(N+1)!(N-k)!k!}{N!(N-k)!(k+1)!}$
$\displaystyle\ =\ $ $\displaystyle(N+1)(k-1)-k(k-1)\frac{N+1}{k+1}$
$\displaystyle\ =\ $ $\displaystyle(N+1)(k-1)-\frac{k(k-1)(N+1)}{k+1}$
$\displaystyle\ =\ $ $\displaystyle(N+1)(k-1)\left(1-\frac{k}{k+1}\right)$
$\displaystyle\ =\ $ $\displaystyle(N+1)\frac{k-1}{k+1},$
and thus obtain
$\mathbb{E}[S]\ =\ (N+1)\frac{k-1}{k+1}.$
The analysis is completed as before, where we pass from our observation of $s$
for $S$ to a prediction $\mathaccent 866{N}$ for $N$:
$\mathaccent 866{N}\ =\ \frac{k+1}{k-1}s-1\ =\
s\left(1+\frac{2}{k-1}\right)-1,$
where the final equality is due to rewriting the algebra to mirror more
closely the formula from the case where the first tank is numbered 1. Note
that this formula passes the same smell checks the other did; for example
$s\frac{2}{k-1}-1$ is always at least 1 (remember $k$ is at least 2), and thus
the lowest estimate we can get for the number of tanks is $s+1$.
## 4\. Comparison of Approaches
So, which did better: statistics or spies? Once the Allies won the war, they
could look into Albert Speer’s, the Nazi Minister of Armaments, records to see
the exact number of tanks produced each month; see Figure 2.
Figure 2. Comparison of estimates from statistics and spies to the true
values. Table from [6].
The meticulous German record keeping comes in handy for the vindication of the
statisticians; these estimates were astoundingly more accurate. While
certainly not perfect (an underestimation of 30 tanks could have pretty dire
consequences when high command is allocating resources), the statistical
analysis was tremendously superior to the intelligence estimates, which were
off by factors of 5 or more. We mentioned earlier the lessons to be learned
from McClellan’s caution. He was the first General of the Army of the Potomac
(which was the Union army headquartered near Washington), and he repeatedly
missed opportunities to deliver a debilitating blow to General Robert E. Lee’s
army of Northern Virginia, most famously during Lee’s retreat from Antietam.
Despite vastly outnumbering Lee in men and supplies, McClellan chronically
overestimated Lee’s forces, causing him to be overly cautious and far too
timid a commander. Ultimately the Civil War would drag on for four years and
costing over 650,000 American lives, and one wonders how the outcome would
have been different if McClellan had been more willing to take the field.
We encourage the reader to write some simple code to simulate both problems
discussed here (or see [5]), namely when we know and when we don’t know the
number of the lowest tank. These problems provide a valuable warning on how
easy it is to accidentally convey information. In many situations today
numbers are randomly generated to prevent such an analysis. Alternatively,
sometimes numbers are deliberately started higher to fool an observer into
thinking that more is present than actually is (examples frequently seen are
the counting done during a workout, putting money in the tip jar at the start
of the shift to encourage future patrons to be generous, or checkbooks
starting with the first check as 100 or higher so the recipient does not
believe it is from a new account).
## Appendix A The German Tank Problem and Linear Regression
The German Tank Problem is frequently used in probability or discrete math
classes, as it illustrates the power of those two disciplines to use binomial
identities to great advantage. It’s also seen in statistics classes in
discussing how to find good estimators of population values. Focusing on these
examples, however, neglects another great setting where it may be used
effectively: as an application of the power of Linear Regression (or the
Method of Least Squares). We quickly review how these methods yield the best-
fit line or hyperplane, and then generalize to certain non-linear
relationships. We show how simulations can be used to provide support for
formulas. This is extremely important, as often we are unable to prove
conjectured relationships. Returning to World War II, the Allies could run
trials (say drawing numbered pieces of paper from a bag) to model the real
world problem, and use the gathered data to sniff out the relationship $m$,
$k$ and $N$.
Additionally, we use this appendix as an opportunity to discuss some of the
issues that can arise when implementing the Method of Least Squares to find
the best fit line. While these do not occur in most applications, it is worth
knowing that they can happen and seeing solutions.
### A.1. Theory of Regression
Suppose we believe there are choices of $a$ and $b$ such that given an input
$x$ we should observe $y=ax+b$, but we don’t know what these values are. We
could observe a large number of pairs of data $\\{x_{i},y_{i}\\}_{i=1}^{I}$,
and use these to find the values of $a$ and $b$ that minimize the sum of the
squares of the errors555We cannot just add the errors, as then a positive
error could cancel with a negative error. We could take the sum of the
absolute values, but the absolute value function is not differentiable; it is
to have calculus available that we measure errors by sums of squares. between
the observed and predicted values:
$E(a,b)\ =\ \sum_{i=1}^{I}\left(y_{i}-(ax_{i}+b)\right)^{2}.$
By setting
$\frac{\partial E}{\partial a}\ =\ \frac{\partial E}{\partial b}\ =\ 0,$
after some algebra666The resulting matrix is invertible, and hence there is a
unique solution, so long as at least two of the $x_{i}$’s differ. One can see
this through some algebra, where the determinant of the matrix is essentially
the variance of the $x_{i}$’s; if they are not all equal then the variance is
positive. If the $x_{i}$’s are all equal the inverse does not exist, but in
such a case we should not be able to predict how $y$ varies with $x$ as we are
not varying $x$! we find the best fit values are
$\left(\begin{array}[]{c}\mathaccent 866{a}\\\ \mathaccent
866{b}\end{array}\right)\ =\
\left(\begin{array}[]{cc}\sum_{i=1}^{I}x_{i}^{2}&\sum_{i=1}^{I}x_{i}\\\
\sum_{i=1}^{I}x_{i}&\sum_{i=1}^{I}1\end{array}\right)^{-1}\left(\begin{array}[]{c}\sum_{i=1}^{I}x_{i}y_{i}\\\
\sum_{i=1}^{I}y_{i}\end{array}\right);$
see for example the supplemental material online for [2]. What matters is that
the relation is linear in the unknown parameters $a$ and $b$ (or more
generally $a_{1},\dots,a_{\ell}$); similar formulas hold for
$y\ =\ a_{1}f_{1}(x)+\cdots+a_{\ell}f(x_{\ell}).$
For a linearly algebraic approach to regression see for example [3].
Regression is a rich subject; we wish to try to find the best fit parameters
to relate $N$ to $m$ and $k$; however, we’ll shortly see that our initial
guess at a relationship is non-linear. Fortunately, by taking logarithms, we
can convert many non-linear relations to linear ones, and thus the formulas
above are available again. The idea is that by doing extensive simulations we
can gather enough data to make a good conjecture on the relationship.
Sometimes, as will be the case with the German Tank Problem, we are able to do
a phenomenal job in predicting the functional form and coefficients, while
other times we can only get some values with confidence.
To highlight these features we first quickly review a well-known problem: The
Birthday Paradox (see for example [2]). The standard formulation assumes we
have a year with $D$ days, and asks how many people do we need in a room to
have a 50% chance that at least two share a birthday, under the assumption
that the birthdays are independent and uniformly distributed from 1 to $D$. A
straightforward analysis shows the answer is approximately $D^{1/2}\sqrt{\log
4}$. We now consider the closely related but less well-known problem of what
is the expected number of people $P$ we need in a room before there is a
match.777As a nice exercise, use linearity of expectation to show that we
expect at least two people to share a birthday when $P=D^{1/2}\sqrt{2}+1$.
Based on the first problem it is reasonable to expect the answer to also be on
the order of $D^{1/2}$, but what is the constant factor? We can try a relation
of the form $P=BD^{a}$, and then taking logs (and setting $b=\log B$) we would
get $\log P=a\log D+b$. See Figure 3.
Figure 3. Plot of best fit line for $P$ as a function of $D$. We twice ran
10,000 simulations with $D$ chosen from $10,000$ to $100,000$. Best fit values
were $a\approx 0.506167,b\approx-0.0110081$ (left) and $a\approx 0.48141$,
$b\approx 0.230735$ (right).
The two simulations both have similar values for $a$, with both of them
consistent with an exponent of 1/2. Unfortunately the values for $b$ wildly
differ, though of the two parameters we care more about $a$ as it tells us how
our answer changes with the number of days. There is an important lesson here:
data analysis can often suggest much of the answer, but it is not always the
full story and there is a role for theory in supplementing such analysis.
### A.2. Issues in Applying to the German Tank Problem
Building on this lesson, we return to the German Tank Problem. What is a
reasonable choice for $N$ as a function of $m$ and $k$? Clearly $N$ is at
least $m$, so we try $N=m+f(m,k)$, which transfers the problem to estimating
$f(m,k)$. We expect that as $m$ increases this should increase, and as $k$
increases it should decrease. Looking at extreme cases is useful; if $k=N$
then $f(N,N)$ should vanish, as then $m$ must equal $N$. The simplest function
that fits this is $f(m,k)=b\cdot m/k$ with $b$ as our free parameter, and we
are led to conjecturing a relationship of the form
$N\ =\ m+b\frac{m}{k}\ =\ m\left(1+\frac{b}{k}\right).$
Note that this guess is quite close to the true answer, but because the
observed quantities $m$ and $k$ appear as they do, it is not a standard
regression problem. We could try to fix this by looking at $N-m$, the number
of tanks we need to add to our observed largest value to get the true number.
We could then try to write this as a linear function of the ratio $m/k$:
$N-m\ =\ a\frac{m}{k}+b,$
where we allowed ourselves a constant term to increase our flexibility of what
we can model. While for $a=-b=1$ this reproduces the correct formula, finding
the best fit values leads to a terrible fit, as evidenced in Figure 4.
Figure 4. Plot of best fit line for $N-m$ as a function of $m/k$. We ran
10,000 simulations with $N$ chosen from $[100,2000]$ and $k$ from $[10,50]$.
Best fit values for $N-m=a(m/k)+b$ for this simulation were $a\approx
0.793716$, $b\approx 7.10602$.
Why is the agreement so poor, given that proper choices exist? The problem is
the way $m$ and $k$ interact, and in the set-up above we have the observed
quantity $m$ both as an input variable and as an output in the relation. We
thus need a way to separate $m$ and $k$, keeping both on the input side. As
remarked, we can do this through logarithms; we discuss another approach in
the next subsection.
### A.3. Resolving Implementation Issues
We look at our best fit line for two choices of $k$; The left side of Figure 5
does $k=1$ while Figure 6 is $k=5$. Both of these show a terrible fit of $N$
as a linear function of $m$ (for a fixed $k$). In particular, when $k=1$ we
expect $N$ to be $2m-1$ but our best fit line is about $.784m+2875$; this is
absurd as for large $m$ we predict $N$ to be less than $m$! Note, however, the
situation is completely different if instead we plot $m$ against $N$ (the
right hand side of those figures). Clearly if $N$ linearly depends on $m$ then
$m$ linearly depends on $N$. When we do the fits this way, the results are
excellent.
Figure 5. Left: Plot of $N$ vs maximum observed tank $m$ for fixed $k=1$.
Theory: $N=2m-1$, best fit $N=.784m+2875$. Right: Plot of maximum observed
tank $m$ vs $N$ for fixed $k=1$. Theory: $m=.5N+.5$, best fit $m=.496N+10.5$.
Figure 6. Left: Plot of $N$ vs maximum observed tank $m$ for fixed $k=5$.
Theory: $N=1.2m-1$, best fit $N=1.037m+749$. Right: Plot of maximum observed
tank $m$ vs $N$ for fixed $k=5$. Theory: $m=.883N+.883$, best fit
$m=.828N+25.8$.
Note that from the point of view of an experiment, it makes more sense to plot
$m$ as the dependent variable and $N$ as the independent, input variable. The
reason is the way we simulate; we fix a $k$ and an $N$ and then choose $k$
distinct numbers uniformly from $\\{1,\dots,N\\}$.
We end with another approach which works well, and allows us to view $N$ as a
function of $m$. Instead of plotting each pair $(m,N)$ for a fixed $k$, we
instead fix $k$, choose an $N$, and then do 100 trials. For each trial we
record the largest serial number $m$, and then we average these, and plot
$(\overline{m},N)$ where $\overline{m}$ is the average. This greatly decreases
the variability, and we now obtain a nearly perfect straight line and fit; see
Figure 7.
Figure 7. Plot of $N$ vs maximum observed tank $m$ for fixed $k=1$. Theory:
$N=1.5m-1$, best fit $N=1.496m+171.2$.
### A.4. Determining the Functional Form
We consider the more general relation
$N\ =\ Cm^{a}\left(1+\frac{b}{k}\right),$
where we expect $C=a=b=1$; note this won’t have the $-1$ summand we know
should be there, but for large $m$ that should have negligible impact. Letting
$C=e^{c}$ for notational convenience, we find
$\log N\ =\ c+a\log(m)+\log\left(1+\frac{b}{k}\right).$
If $x$ is large then $\log(1+1/x)\approx 1/x$, so we try the approximation
$\log N\ \approx\ c+a\log(m)+b\frac{1}{k}.$
Figure 8 shows the analysis when $C=1$ (so $c=0$), as the analysis then
reduces to the usual case with two unknown parameters. We chose to take $C=1$
from the lesson we learned in the analysis of the Birthday Problem.
Figure 8. Plot of $\log N$ against $\log m$ and $1/k$. We ran 10,000
simulations with $N$ chosen from $[100,2000]$ and $k$ from $[10,50]$. The data
is well-approximated by a plane (we do not draw it in order to prevent our
image from being too cluttered).
The best fit values of the parameters are $a=0.999911$ and $b=0.961167$, which
are reasonably close to $a=b=1$. Thus these numerics strongly support our
conjectured relation $N=m(1+1/k)$, and shows the power of statistics. While we
were able to see the arguments needed to prove this relation essentially
holds, imagine we could not prove it but still have our heuristic arguments
and analysis of extreme cases which suggest it is true. By simulating data and
running the regression, we see that our formula does a stupendous job
explaining our observations, and thus gain confidence to use it in the field.
We end with one last approach. Let us guess a relationship of the form
$N=a(k)m+b(k)$, where $a(k)=1+f(k)$ (we write $a(k)$ as $1+f(k)$ as we know
there have to be at least $m$ tanks). We can fix $k$, and find the best fit
values of $a(k)$ and $b(k)$. In Figure 9 we plot the best fit slope $a(k)$
versus $k$, as well as a log-log plot. For the log-log plot we look at
$a(k)-1$, subtracting the known component. We see a beautiful linear relation,
and thus even if we did not know it should be $m$ plus a constant times $m/k$,
the data suggests that beautifully! Specifically, we found the best fit line
was $\log(a(k)-1)=-.999\log(k)-.007$, suggesting that $a(k)=1+1/k$; we obtain
the correction functional form just by running simulations!
Figure 9. Left: Plot of $a(k)$, the slope in $N=a(k)m+b$, versus $k$. Right:
Log-Log Plot of $a(k)-1$ versus $k$. In $\log(a(k)-1)$ versus $\log k$, theory
is $\log(a(k)-1)=-1\log(k)$, best fit line is
$\log(a(k)-1)=-.999\log(k)-.007$.
## References
* [1] M. Cozzens and S. J. Miller, _The Mathematics of Encryption: An Elementary Introduction_ , AMS Mathematical World series 29, Providence, RI, 2013.
* [2] S. J. Miller, _The Probability Lifesaver_ , Princeton University Press, Princeton, NJ, 2018. https://web.williams.edu/Mathematics/sjmiller/public_html/probabilitylifesaver/index.htm.
* [3] G. Strang, _Introduction to Linear Algebra_ , Fifth Edition, Wellesley-Cambridge Press, 2016.
* [4] Probability and Statistics Blog, _How many tanks? MC testing the GTP_ , https://statisticsblog.com/2010/05/25/how-many-tanks-gtp-gets-put-to-the-test/.
* [5] Statistical Consultants Ltd, _The German Tank Problem_ , https://www.statisticalconsultants.co.nz/blog/the-german-tank-problem.html.
* [6] WikiEducator, _Point Estimation - German Tank Problem_ , https://wikieducator.org/Point_estimation_-_German_tank_problem.
* [7] Wikipedia, _German Tank Problem_ , Wikimedia Foundation, https://en.wikipedia.org/wiki/German_tank_problem.
|
# Successive occurrences of quasi-circular ribbon flares in a fan-spine-like
configuration involving hyperbolic flux tube
Prabir K. Mitra1,2and Bhuwan Joshi1
1Udaipur Solar Observatory, Physical Research Laboratory, Udaipur 313 001,
India
2Department of Physics, Gujarat University, Ahmedabad 380 009, India Contact
e-mail<EMAIL_ADDRESS>
###### Abstract
We present a comprehensive analysis of the formation and evolution of a fan-
spine-like configuration that developed over a complex photospheric
configuration where dispersed negative polarity regions were surrounded by
positive polarity regions. This unique photospheric configuration, analogous
to the geological “atoll" shape, hosted four homologous flares within its
boundary. Computation of the degree of squashing factor ($Q$) maps clearly
revealed an elongated region of high $Q$-values between the inner and outer
spine-like lines, implying the presence of an hyperbolic flux tube (HFT). The
coronal region associated with the photospheric atoll configuration was
distinctly identified in the form of a diffused dome-shaped bright structure
directly observed in EUV images. A filament channel resided near the boundary
of the atoll region. The activation and eruption of flux ropes from the
filament channel led to the onset of four eruptive homologous quasi-circular
ribbon flares within an interval of $\approx$11 hours. During the interval of
the four flares, we observed continuous decay and cancellation of negative
polarity flux within the atoll region. Accordingly, the apparent length of the
HFT gradually reduced to a null-point-like configuration before the fourth
flare. Prior to each flare, we observed localised brightening beneath the
filaments which, together with flux cancellation, provided support for the
tether-cutting model of solar eruption. The analysis of magnetic decay index
revealed favourable conditions for the eruption, once the pre-activated flux
ropes attained the critical heights for torus instability.
###### keywords:
Sun: activity – Sun: filaments, prominence – Sun: flares – Sun: magnetic
fields – sunspots
††pubyear: 2020††pagerange: Successive occurrences of quasi-circular ribbon
flares in a fan-spine-like configuration involving hyperbolic flux
tube–LABEL:lastpage
## 1 Introduction
Solar flares are sudden, localized enhancement of brightness in the solar
atmosphere during which energy up to $\sim$1032 erg can be released in the
entire electromagnetic spectrum (see review articles by Fletcher et al., 2011;
Benz, 2017). It is well understood that magnetic field remains at the helm of
all the catastrophic processes occurring in the solar atmosphere including
flares, as the energy released during flares is supplied from the magnetic
energy that is stored in the flaring region prior to the flare (see e.g.,
Shibata & Magara, 2011). Therefore, the pre-flare magnetic configuration plays
a crucial role in determining the trigger and subsequent evolution of solar
flares and associated eruptive phenomena (e.g., Joshi et al., 2015, 2017c;
Hernandez-Perez et al., 2019; Mitra et al., 2020a; Qiu et al., 2020).
Traditionally, solar flares were observed to be associated with a pair of
ribbon like brightening identified in chromospheric H$\alpha$ images, which
were situated on the opposite sides of a polarity inversion line (PIL). To
explain such parallel ribbon flares, a ‘standard flare model’ was proposed
combining the works of Carmichael (1964); Sturrock (1966); Hirayama (1974);
Kopp & Pneuman (1976), which is also known as the ‘CSHKP’ model (Shibata &
Magara, 2011). According to this model, magnetic reconnection takes place
along a vertical current sheet, formed between the inflowing magnetic fields
beneath an erupting prominence. During reconnection, magnetic energy gets
transformed into heat and particle accelerations resulting in localised sudden
flash in the solar atmosphere and highly accelerated electrons that are
projected with almost relativistic speeds toward the lower, denser
chromospheric layer of the Sun along the reconnected field lines. The
accelerated electrons collide with the dense chromospheric plasma giving rise
to hard X-ray (HXR) footpoint sources in association with EUV and optically
observable conjugate ribbons termed as chromospheric flare ribbons (Fletcher
et al., 2013; Musset et al., 2015; Joshi et al., 2017a; Kazachenko et al.,
2017). Despite the general success of the CSHKP model toward explaining the
commonly observed features of eruptive parallel ribbon flares i.e., footpoint
and looptop HXR sources, post-flare arcade, hot cusp etc., several studies
have reported flaring activities to involve complex structures of flare
ribbons and dynamics of overlying coronal loops implying that CSHKP model
alone cannot explain all the flares (e.g., Veronig et al., 2006; Joshi et al.,
2009; Kushwaha et al., 2015; Mitra et al., 2018).
It is also important to note that the CSHKP model, being a 2D model, can not
explain the 3D aspects of typical solar flares such as, evolution of shear
from the pre-flare loops to post-flare arcades; relative positions, shapes,
and motions of the flare ribbons etc. To incorporate these features, the CSHKP
model has been extended in 3D with numerical simulations (Aulanier et al.,
2012, 2013). The 3D standard flare model suggests that small-scale current
sheets are generated between the highly sheared pre-flare magnetic field
configuration. Reconnection on these current sheets drives the transfer of
differential magnetic shear, from the pre- to the post-eruptive configuration.
With the evolution of the flare, as the eruption of the flux rope initiates,
magnetic loops enveloping it straighten vertically and the current sheet
extends along with them. Thus, magnetic reconnection continues beneath the
erupting flux rope which is in line with the 2D standard flare model.
Further, the formation of the flux rope and the triggering of its eruption
goes beyond the scope of the CSHKP model. Flux ropes are defined by a set of
magnetic field lines that are wrapped around each other in a braided fashion
or wrapped around a central axis (braided and twisted flux ropes,
respectively, see; Prior & Yeates, 2016). Observationally, a flux rope can be
identified in different forms: filament (Zirin, 1988; Martin, 1998),
prominence (Tandberg-Hanssen, 1995; Parenti, 2014), coronal cavity (Forland et
al., 2013; Gibson, 2015), hot channel (Zhang et al., 2012; Cheng et al., 2013;
Mitra & Joshi, 2019; Sahu et al., 2020), coronal sigmoid (Rust & Kumar, 1996;
Manoharan et al., 1996; Joshi et al., 2017a; Mitra et al., 2018) etc. The
processes involved in the triggering of a flux rope from its stable condition
and its successive evolution within the source region are rather complex and
debatable (see e.g., Chatterjee & Fan, 2013; Kumar et al., 2016; Prasad et
al., 2020). Different mechanisms have been proposed in this regard, which can
be classified in two general groups: ideal instability and resistive
instability. Ideal instability models put forward the idea that a flux rope
can attain eruptive instability if the values of some parameters go beyond a
critical value e.g., decay index ($n$; $n$=$-\frac{dlog(B_{h})}{dlog(h)}$;
$B_{h}$ and $h$ being horizontal magnetic field and height, respectively) more
than 1.5 for torus instability (Kliem & Török, 2006) or the twist of the flux
rope more than $\approx$3.5$\pi$ for kink instability (Török et al., 2004). On
the other hand, resistive instability models recognize the role of initial
small-scale reconnection in the active region as the triggering mechanism of a
flux rope (or core field) e.g., initial reconnection beneath a flux rope or
sheared arcade for the case of tether-cutting model (Moore & Roumeliotis,
1992; Moore et al., 2001) or reconnection at a coronal null well above the
core field for the case of breakout model (Antiochos et al., 1999). Once the
MFR attains eruptive motion, magnetic reconnection initiates beneath the flux
rope and two parallel ribbons are observed along with other observable flare
signatures explained in the CSHKP model.
Morphologically, a completely different category of flares is circular (or
quasi-circular) ribbon flares (Masson et al., 2009) which are usually
associated with a fan-spine configuration in a 3D null-point topology (Lau &
Finn, 1990; Sun et al., 2013). Coronal null-points are locations in the solar
corona, where the strengths of all the three components of magnetic field
become locally zero (see review by Longcope, 2005). Magnetic field beyond the
immediate neighbourhood of the null-point is characterised by a spine line and
a fan surface. Depending on the sign of the null-point (positive or negative),
magnetic field lines approach the null-point along the spine line and move
away from it along the fan surface (for positive null; see Figure 4 in
Longcope, 2005); or, approach the null along the fan surface and recede from
it along the spine line (negative null). In the context of such 3D null-point
configurations generating fan and spine lines, the anemone-type active regions
where a compact magnetic region is surrounded by magnetic regions of opposite
polarity (see, Shibata et al., 1994) are of special significance. The inner
compact region is connected with the surrounding opposite polarity region by
small-scale closed magnetic loops while a set of relatively large field lines
connect the surrounding polarity to a remote region of polarity similar to
that of the inner compact region. In this way, the two sets of field lines
constitute two sets of fan lines (inner and outer fan lines) and two sets of
spine lines (inner and outer spine lines). The two sets of fan lines are
separated by a dome-shaped surface (i.e., fan separatrix) characterised by
high degree of squashing factor ($Q$; see, Priest & Démoulin, 1995; Titov et
al., 2002), which intersects the spine lines at the null-point (Sun et al.,
2013). In general, domains corresponding to drastic changes in the magnetic
field connectivity gradient are identified as quasi-separatrix layers (QSLs;
see e.g., Janvier et al., 2013). While the values of $Q$ corresponding to QSLs
are high ($\gg$2; see, Aulanier, G. et al., 2005), null-points can be
characterised by $Q\rightarrow\infty$. The finite values of $Q$ at the QSLs
imply that although magnetic fields show drastic change in the connectivity
they are still continuous, which is contrary to the cases of null-points where
magnetic field becomes discontinuous. In complex photospheric configurations,
e.g., those formed by two bipolar sunspots, a pair of photospheric null-points
of opposite signs may exist which are connected by ‘separators’ (Titov et al.,
2002). A separator can be identified by narrow elongated strips of high
$Q$-values, i.e. a QSL, with a pair of null-points at both of its ends.
Further, a generalisation of the concept of separator lines reveals a special
geometrical feature called ‘Hyperbolic Flux Tubes’ (HFTs; Titov et al., 2002)
which can be understood as the intersection of two QSLs. The middle of an HFT
is characterised by ‘X’-type cross section comprised of high $Q$-values. Such
structures of high $Q$-values are preferred sites for the formation of current
sheets and initiation of magnetic reconnection (Titov et al., 2003). Moreover,
magnetic field lines can constantly change their connectivities along the QSLs
as a consequence of local diffusion in the region, allowing neighboring field
lines to exchange connectivities (Aulanier et al., 2006). This can be observed
as an apparent slipping or flipping motion of loop connectivities and are
termed as ‘slipping reconnction’ (see e.g., Priest & Démoulin, 1995; Aulanier
et al., 2006; Janvier et al., 2013).
With the advancements of observational facilities and numerical techniques, in
the recent years a number of studies have reported flaring activities that
involved both the circular and parallel ribbons where the parallel ribbons
usually reside at the inside edge of the circular ribbon (e.g., Joshi et al.,
2015, 2017b; Hernandez-Perez et al., 2017; Li et al., 2017; Xu et al., 2017;
Li et al., 2018a; Li et al., 2018b; Hou et al., 2019; Shen et al., 2019; Devi
et al., 2020). Such events usually develop as a small flux rope erupts within
a fan-dome which then triggers reconnection at the null-point. The null-point
reconnection itself is a complex, multi-stage mechanism which initially
includes slipping reconnection at the quasi-separatrix surface at the fan-dome
giving rise to the circular or quasi-circular ribbon and during the subsequent
stage, interchange reconnection takes place between the inner close fan lines
and the outer open spine lines causing the remote brightening (see, Masson et
al., 2009). In response to the interchange reconnection at the coronal null-
point, collimated ejection of plasma i.e., coronal jets or H$\alpha$-surges
have been identified in several studies (Pariat et al., 2009a, 2010). These
findings point toward the fact that magnetic configurations on the Sun could
be very complex and more studies are extremely essential toward reaching at a
general understanding of the complex sunspot configurations and associated
flaring activities.
In this article, we report four homologous quasi-circular ribbon flares from
the active region NOAA 11977, which were triggered by erupting filaments from
the circular ribbon region. With the help of high resolution images of
Atmospheric Imaging Assembly (AIA; Lemen et al., 2012) and Helioseismic and
Magnetic Imager (HMI; Schou et al., 2012) on board the Solar Dynamics
Observatory (SDO; Pesnell et al., 2012), we study the evolution of the active
region and the complex flares in detail. Magnetic field modelling based on a
‘Non-linear Force Free Field’ (NLFFF) method has revealed a fan-spine-like
configuration associated with the flaring region. The most important finding
of this study is the absence of coronal null-point in the fan-spine-like
configuration. Instead, the calculation of $Q$ revealed the presence of an HFT
between the inner and outer spine-like lines. In Section 2, we provide a brief
description of the observational data sources and the image analysis
techniques along with the numerical methods used in this article. We discuss
the morphology and evolution of the active region as well as give a brief
account of all the flares produced by it in Section 3. Results obtained from
imaging analysis of the two circular ribbon flares and NLFFF extrapolation are
presented in Sections 4 and 5. We discuss and interpret the results in Section
6.
## 2 Observational Data and Analysis Techniques
For EUV imaging, we have utilised the 12 s cadence, 4096$\times$4096 pixel
full disk observations from the AIA on board the SDO with pixel resolution of
0$\aas@@fstack{\prime\prime}$6\. For the chromospheric imaging of the Sun, we
have used the 2048$\times$2048 pixel full disk images in the H$\alpha$
passband with a pixel resolution of $\approx$1$\aas@@fstack{\prime\prime}$0,
obtained from the archive Global Oscillation Network Group (GONG; Harvey et
al., 1996, 2011). We have studied the photospheric structures associated with
the active region NOAA 11977 by using the 45 s cadence, 4096$\times$4096 pixel
full disk continuum and line-of-sight (LOS) magnetogram observations with
spatial sampling resolution of 0$\aas@@fstack{\prime\prime}$5 pixel-1 by HMI
on board the SDO. The HMI LOS intensity and magnetogram images were further
processed with the IDL-based algorithm ‘hmi_prep’ to co-align them with AIA
pixel resolution. Coronal magnetic fields were extrapolated by employing the
optimisation based Non-Linear Force Free Field (NLFFF) extrapolation method
developed by Wiegelmann & Inhester (2010); Wiegelmann et al. (2012). For the
purpose, we have used the vector magnetogram data from the
‘hmi.sharp$\\_$cea$\\_$720s’ series of HMI at a reduced spatial resolution of
1$\aas@@fstack{\prime\prime}0$ pixel-1 as the input boundary condition.
Extrapolations were done within a volume of dimensions
453$\times$270$\times$240 pixels which corresponds to the physical dimension
of $\approx$328$\times$196$\times$174 Mm3. Based on the NLFFF extrapolation
results, we calculated the degree of squashing factor ($Q$) and twist number
($T_{w}$) in the extrapolation volume by using the IDL-based code developed by
Liu et al. (2016). In order to locate 3D null-points within the extrapolation
volume, we used the trilinear method as suggested by Haynes & Parnell (2007).
For the purpose, the whole active region volume was divided into grid cells of
dimension 2$\times$2$\times$2 pixels. The first step of the trilinear method
is to quickly scan through every grid cell by examining the signs of each
component of magnetic field at all the eight corners of the grid cells. If any
of the three components have same sign at all the eight corners, a null-point
can not reside within the grid cell and therefore, the corresponding cell is
excluded from further analysis. Each of the remaining other cells is then
further divided into 100$\times$100$\times$100 sub-grid cells and the
threshold $\bigtriangleup x\leqslant 2$ sub-grid cell-width was used for
locating null-points. For visualizing the modelled field lines and the
distribution of $Q$ in the active region volume, we have used the
Visualization and Analysis Platform for Ocean, Atmosphere, and Solar
Researchers (VAPOR111https://www.vapor.ucar.edu/; Clyne et al., 2007)
software.
## 3 Structure and Evolution of the active region NOAA 11977
The active region NOAA 11977 appeared on the eastern limb of the Sun on 2014
February 11 as a simple $\alpha$-type active region. It quickly transformed
into a relatively more complex $\beta$-type on the very next day. The active
region gradually developed into $\beta\gamma$-type on 2014 February 14 and
remained so for the next four days. Notably, the active region started to
decay in its area since 2014 February 15. On 2014 February 16, an intriguing
configuration of magnetic fields, involving complex distribution and topology,
developed in the westernmost part of the active region which we study
comprehensively in Section 3.2. Notably, one M and three C-class flares
originated from this region within an interval of $\approx$11 hours on
February 16 (see Table 1). The magnetic complexity of the active region
reduced to $\beta$-type on 2014 February 19. The active region disappeared
from the western limb of the Sun on 2014 February 23 as an $\alpha$-type
sunspot.
Figure 1: The morphology of the active region NOAA 11977 on the photosphere
(panels (a) and (b)) and different coronal temperatures (panels (c) and (d)).
The flaring activity occurred from the region shown within the white box in
panels (b)–(d).
### 3.1 Morphology of the active region NOAA 11977
In Figure 1, we show a comparison of the photospheric structure of the active
region with its coronal configuration, prior to the M-class flare on 2014
February 16. We find that, the active region was comprised of a few prominent
sunspots and many pores. We consider two subregions of the active region: the
leading sunspot group and the trailing sunspot group (shown by the dashed and
dotted boxes in Figure 1(a), respectively). Comparison of a co-temporal LOS
magnetogram (Figure 1(b)) of the active region with the intensity image
reveals that the leading sunspot group was consisted of mostly positive
polarity while the trailing part of the active region was dominated by
negative polarity magnetic field. However, the most interesting aspect of the
active region, in the context of our analysis, is the magnetic configuration
that developed in the extreme western part of it where magnetic patches of
positive and negative polarities formed a configuration similar to an ‘atoll’
(within the white box in Figure 1(b)). AIA images suggest that the active
region, on the whole, consisted of different sets of coronal loops of varying
spatial extents and projected heights. In Figures 1(c) and (d), we recognize
some of these loops by arrows of different colors (white, green and red).
We note striking coronal structures over the photospheric atoll region which
can be easily distinguished by its peculiar morphology. The region is outlined
by the white boxes in Figures 1(c) and (d). The EUV images of the coronal
region, over the atoll-shaped magnetic structure in the photosphere, clearly
reveal enhanced structured brightening within a sharp boundary that forms an
oval-shaped feature. All the four events of 2014 February 16 occurred within
this region.
### 3.2 Formation of the ‘magnetic atoll’ region
In order to understand the formation and development of the magnetic atoll
region, in Figures 2(a)–(f), we present a series of LOS magnetograms
corresponding to the photospheric region (shown within the white box in Figure
1(b)). Since, the diffused brightening region was intrinsically related with
the magnetic atoll region, we have also plotted co-temporal AIA 304 Å images
of the same region in Figures 2(g)–(l). Our observations reveal that the atoll
region started to develop from 2014 February 15 with the emergence a of few
small negative polarity flux regions. In Figure 2(a), we indicate these
emerging negative polarity regions by the yellow arrows. With time, negative
polarity magnetic flux kept emerging in this region and by the first hour of
2014 February 16, a relatively large magnetic patch of negative polarity was
developed (indicated by the blue arrow in Figure 2(c)). Co-temporal AIA 304 Å
images suggest that the distinct oval-shaped region displaying enhanced-
diffused brightening started to build-up from around this time onward (cf.
Figures 2(i) and (c)). Gradually, this diffused brightening region became more
prominent and extended spatially (cf. Figures 2(i)–(l)). In the same duration,
positive flux also emerged in this region which surrounded the negative
polarity patches in the south-west direction (cf. the regions indicated by the
pink arrows in Figures 2(c) and (f)). In general, the atoll region can be
characterised by newly emerged and spatially dispersed prominent patches of
negative magnetic flux surrounded by positive polarity regions from two sides
i.e., northeast (by the positive polarity sunspot) and southwest (by the
dispersed positive polarity patches). Further, from HMI LOS magnetograms
during the early hours of 2014 February 16, we could identify several
instances of flux emergence and cancellation in the magnetic atoll region. We
have identified a few events of flux changes by blue, red and green arrows in
Figures 2(c)–(f). During the same interval, we also observed formation of a
filament channel in the diffused brightening region along the PIL between the
positive polarity sunspot and adjacent negative polarity patches. We have
indicated it by a green arrow in Figure 2(j).
Figure 2: Series of HMI LOS magnetograms showing the development of the
magnetic ‘atoll’ region (panels (a)–(f)). Arrows of different color indicate
few prominent locations of negative magnetic flux emergence. Panels (g)–(l)
are AIA 304 Å images associated with the magnetic atoll region displaying the
formation of the diffused circular brightening region. All the images are
derotated to 2014 February 16 09:20 UT.
## 4 Four successive flares from the magnetic atoll region
The magnetic atoll region produced four successive flares on 2014 February 16.
In Figure 3(a), we plot SXR flux in the GOES 1–8 Å channel along with EUV
intensities of AIA 94 and 304 Å channels from 2014 February 16 00:00 UT–20:00
UT which displays the onset and temporal evolutions of the four flares (see
summary of events in Table 1). These multi-wavelength lightcurves suggest that
the first flare of GOES class M1.1 initiated at $\approx$09:20 UT. This short
lived flare reached to its peak at $\approx$09:26 UT. The second flare (GOES
class C3.4) initiated at $\approx$13:48 UT and after a relatively extended
rise phase of $\approx$12 minutes, reached its peak intensity. Interestingly,
the subsequent two flares originating from this region were rather impulsive
and short lived. According to the GOES 1–8 Å flux profile, these C class
flares originated at $\approx$17:35 UT and 19:20 UT while their impulsive
phases lasted for only $\approx$3 minutes each. In Figure 3(a), we indicate
the durations of the four flares by gray-shaded intervals and assign the
notations ‘F1’, ‘F2’, ‘F3’ and ‘F4’ to the flares in the chronological order.
Table 1: Summary of the flares occurred on 2014 February 16 from the magnetic atoll region of NOAA 11977 Flare | Location | Flare timing (UT) | GOES class | Remarks
---|---|---|---|---
Id. | | Start/ Peak/ End | |
F1 | S10E00 | 09:20/ 09:26/ 09:29 | M1.1 | Observation of quasi-circular ribbon
F2 | S10W01 | 13:48/ 14:00/ 14:05 | C3.4 | Observation of shortened quasi-circular ribbon
F3 | S10W03 | 17:35/ 17:38/ 17:40 | C1.2 | Observation of arc-shaped ribbon
F4 | S10W04 | 19:20/ 19:23/ 19:28 | C1.7 | Observation of arc-shaped ribbon
Figure 3: Panel (a): GOES 1–8 Å flux (blue curve), AIA 304 Å (red curve) and
AIA 94 Å (green curve) intensities showing the onset and evolution of four
flares from the diffused brightening region. The duration of the flares are
indicated by four shaded regions. The individual fares are named as ‘F1’,
‘F2’, ‘F3’ and ‘F4’. Panels (b)–(e): AIA 94 Å images of the diffused
brightening region during the peak phases of the four flares. The red dotted
curves in panels (b) and (c) indicate the quasi-circular ribbon observed
during F1 and F2 while the yellow arrows in panels (d)–(e) indicate arc-shaped
ribbons during F3 and F4 flares, respectively. Panels (j)–(i): AIA 304 Å
images showing the peak phases of the four flares. The green arrows in
different panels indicate the quasi-circular (panel (f)), semi-circular (panel
(g)) and arc-shaped (panels (h) and (i)) ribbons observed during the flares.
The black arrows indicate the erupting filament. Panels (j)–(m): HMI LOS
magnetograms of the ‘magnetic atoll’ region prior to the onset of the F1–F3
flares (panels (j)–(l)) and after the F4 flare (panel (m)). The blue arrows in
panel (j) indicate few negative flux region which disappeared with time.
Contours of co-temporal EUV intensities in AIA 304 Å are overplotted on the
bottom row. Contour levels are [4, 80]%, [12, 60]%, [11, 60]% and [9, 40]% of
the corresponding peak intensities in panels (j), (k), (l) and (m),
respectively. An animation associated with this figure is attached in the
online supplementary materials.
In Figures 3(b)–(e) and (f)–(i), we show AIA 94 Å and AIA 304 Å images,
respectively, of the diffused brightening region during the peak phases of the
circular ribbon flares. From these images it becomes evident that, all the
four flares were associated with some degree of ribbon like brightening along
the circumference of the diffused brightening region. We observed quite
prominent signatures of an extended quasi-circular ribbon brightening during
F1, which is delineated by the red dotted curve in Figure 3(b) and the green
arrows in Figure 3(f). The quasi-circular ribbon shortened significantly
during F2 which we have indicated by the red dotted curve in Figure 3(c) and
the green arrows in Figure 3(g). During F3 and F4 the flare ribbons along with
only a small portion of the boundary of the diffused brightening region
displayed enhanced emission. This arc-shaped flare brightening is indicated by
the yellow arrows in Figures 3(d)–(e) and the green arrows in Figures
3(h)–(i).
In Figures 3(j)–(m), we plot preflare HMI LOS magnetograms corresponding to
all the four flares, respectively. We find that the negative polarity flux
regions from the magnetic atoll region continuously decayed with the evolution
of each flares. This cancellation of negative flux was much more prominent in
the southern part of the atoll region than the northern part (cf. the negative
flux regions indicated by blue arrows in Figures 3(j) and (m)). For a better
understanding of the relation between the photospheric flux regions and the
flare emission, we plotted contours of EUV intensities in AIA 304 Å channel
over the LOS magnetograms in Figures 3(j)–(m). From the overplotted contours,
it becomes clear that the quasi-circular ribbon brightening during the peak
phases of the flares were co-spatial with the positive polarity flux regions
situated at the boundary of the photospheric atoll region.
In Figure 4, we plot chromospheric images of the diffused brightening region
in the H$\alpha$ passband during the pre-flare phases of each flares. From
these images, we clearly identified impressions of the filaments, implying
recursive development of filaments at the filament channel situated along the
PIL between the positive polarity sunspot and the parasitic negative polarity
patches (Figure 2 and Section 3.2). It is worth mentioning that the evolution
of the quasi-circular ribbon flares in the H$\alpha$-passband (not shown here)
was broadly similar to that observed in EUV channels.
In order to have a comprehensive understanding towards the influence of
photospheric flux on the evolution of the diffused brightening region as well
as on the homologous flares originated from it, in Figure 5, we examine the
evolution of photospheric flux over a prolonged period that also covers the
time-span of the four flares. For the purpose, we emphasise on the small-scale
flux changes within the atoll region (in Figures 5(b1)–(b8)) and the temporal
evolution of flux from it (Figure 5(c)) by analysing HMI LOS magnetograms. The
atoll region is enclosed in Figure 5(a) by the black box. We clearly
identified several instances of small-scale flux cancellation; a few of them
are indicated by different colours of ovals and arrows in Figures 5(b1)–(b8).
During the pre-flare phase of F1 (Figures 5(b1)–(b2)) we observed cancellation
of positive flux at the region of the PIL of the filament, which is indicated
by the yellow ovals. During the interval between F1 and F2, we could identify
multiple instances of flux cancellation of both polarities at the PIL region
situated at the northern part of the atoll region (cf. the arrow heads of the
red, blue and green arrows in Figures 5(b3)–(b4)). Between F3 and F4, a major
part of a relatively prominent negative flux region got decayed (cf. the
region enclosed by the green ovals in Figures 5(b5)–(b6)). We further observed
cancellation of positive flux in the atoll region which we have highlighted
within the orange ovals in Figures 5(b5)–(b6). This region was observed with
flux cancellation between F3 and F4 also which is indicated by the red arrows
in Figures 5(b7)–(b8). We also observed cancellation of small negative fluxes
from the atoll region in this duration which are indicated by the blue arrows
in Figures 5(b7)–(b8).
In general, the atoll region experienced a significant decrease of negative
flux (Figures 3(j)–(m) and 5(b1)–(b8)) which is readily recognised by the
temporal evolution of negative flux (blue curve in Figure 5(c)). The flux
profiles suggest that in the first part of 2014 February 16, the flaring
region (enclosed by the black box in Figure 5(a)), underwent a rapid increase
of negative flux beside relatively moderate positive flux enhancement (see
00:00 UT – 08:00 UT in Figure 5(c)). Notably, this period can be characterised
by the formation of the atoll and associated diffused brightening region
(Section 3.2 and Figure 2). The first flare (F1) initiated when negative flux
reached to its peak; afterwards negative flux from the region remained
relatively unchanged for $\approx$3 hours and then continuously decayed till
$\approx$19:30 UT. After this time, negative flux maintained an approximately
constant level for $\approx$1.5 hours when the last flare from this region
(F4) took place. From $\approx$20:00 UT, negative flux primarily decayed
characterising the decay of the overall diffused brightening region. On the
other hand, positive flux in this region displayed stepwise enhancement till
$\approx$20:00 UT after which decayed slowly till end end of the period of our
calculation.
### 4.1 Morphological evolution of quasi-circular ribbon flares
All the four flares originating from the oval-shaped diffused brightening
region initiated with the activation and eruption of the filaments situated
along the PIL between the positive polarity sunspot the negative flux regions.
We recall that the coronal region showing oval-shaped diffused brightening was
lying over the peculiar magnetic atoll region at the photosphere (see Section
3.2 and Figure 2). Further, during all the four flares, we observed flare
ribbons along the boundary of the diffused brightening region which forms
quasi-circular ribbons (Figure 3). The quasi-circular ribbon was the most
prominent and extended during F1 of GOES class M1.1. The subsequent three
flares were very similar in their onset and evolution. However, as explained
in Section 4 (Figure 3), the length of quasi-circular ribbon decreased
progressively during the subsequent flares (i.e., F2–F4). More importantly,
during F2–F4, the filament evolved with a complete blow-out type eruption from
the diffused brightening region while such catastrophic eruption of the
filament was not observed during F1. In this section, we focus on the detailed
morphological evolution of F1 and F2 flares.
Figure 4: Selective of H$\alpha$ images of the diffused brightening region
during the pre-flare phases of the quasi-circular ribbon flares, recorded by
different stations of the GONG network. The corresponding GONG stations are
indicated in the titles in each panel as Uh (Udaipur Solar Observatory,
India), Ch (Cerro Tololo Inter-American Observatory, Chile) and Bh (Big Bear
Solar Observatory, USA). The blue arrows indicate the presence of small
filaments prior to the flares. Figure 5: Panel (a): LOS magnetogram of the
active region NOAA 11977 where the magnetic atoll region is enclosed within
the black box. Panels (b${}_{\textrm{1}}$)–(b${}_{\textrm{8}}$): Selective LOS
magnetograms of the atoll region displaying several instances of flux
cancellation. We have indicated few of such flux cancellation by arrows and
ovals of different colours in different panels. Panel (c): Evolution of
magnetic flux in the magnetic atoll region indicated in panel (a). The shaded
intervals in panel (c) indicate the durations of flares occurred from the
magnetic atoll region.
#### 4.1.1 Filament eruption and quasi-circular ribbon during F1
In Figure 6, we plot a series of AIA 304 Å images of the diffused brightening
region to investigate the evolution and eruption of the filament in
association with F1 i.e., the GOES M1.1 flare. Our observations suggest that
the impression of the filament at the northern boundary of the diffused
brightening region was first identified at around $\approx$02:30 UT (Section
3.2 and Figure 2). In Figure 6(b), we indicate the initial filament by the
blue arrows and we will refer this filament as FL1 henceforth. A comparison of
the location of FL1 with the LOS magnetograms in Figure 6(a) immediately
reveals that the filament was situated along the PIL between the positive
polarity sunspot and the dispersed negative magnetic field regions. The
filament was associated with a series of localised small-scale brightenings
(indicated by the green arrows in Figures 6(c) and (d)) during an extended
period prior to the onset of the GOES M-class flare (i.e, F1). The filament
was observed most prominently at $\approx$09:20 UT which is indicated by the
blue dotted curve in Figure 6(e). After $\approx$09:21 UT, a subtle
enhancement in the brightness at the northern leg of the filament was observed
which was immediately followed by eruption of the filament from its northern
end. In Figure 6(f), we indicate the erupting part of the filament by a blue
arrow and the localized brightening by a white arrow. The enhanced brightening
advanced southward (cf. the white arrowheads in Figures 6(d) and (c)) as the
erupting front of the filament induced upward erupting motion in the southern
part of the filament also. The southward induced eruption of the filament is
further outlined by the blue dotted curve in Figure 6(g) and indicated by the
blue arrows in Figures 6(h) and (i). Two localised ribbon-like bright
structures formed at both sides of the filament (indicated by the black arrows
in Figure 6(i)) at around 09:24 UT, which resembles with ‘standard flare
ribbons’. At the same time, a circular ribbon brightening was prominently
observed at the boundary of the diffused brightening region. After
$\approx$09:25 UT, the filamentary structure (FL1) appeared as a straight,
long structure with one end still attached to its initial location (outlined
by the blue dotted curve in Figure 6(j)). Interestingly, during this time, it
was observed as a bright structure suggesting strong heating of the
filamentary materials during the flaring process. The open end of the
filamentary structure was associated with plasma eruption after $\approx$09:25
UT. The direction of the erupting plasma is indicated by the arrows in Figure
6(j). Notably, both the parallel and circular ribbon brightening increased up
to $\approx$09:27 UT (the circular ribbon brightening is outlined by the black
dotted curve in Figure 6(g)) after which, flare brightenings from the active
region slowly decreased while eruption continued. In Figures 6(k), we indicate
the erupting plasma by white arrows. Formation of post-flare arcade was
observed after $\approx$09:32 UT which sustained till $\approx$09:42 UT. We
have indicated the post-flare arcade by black arrows in Figure 6(l).
Figure 6: Representative AIA 304 Å images depicting filament eruption and
associated M1.1 flare. The two blue arrows indicate a filament which is
referred to as ‘FL1’. The green arrows in panels (c) and (d) indicate small-
scale brightenings observed at the location of the filament. In panel (e), we
outline the ‘S’ shaped filament by dotted curve. The filament during its
initial eruption phase is shown by blue arrows in panels (f) and (h) and blue
dotted line in panel (g). The white arrows in panels (c)–(h) indicate initial
flare brightening beneath the northern end of the erupting filament. The final
stage of filament eruption from the active region is outlined by the blue
arrow in panel 9i) and blue dotted curve in panel (j). The black arrows in
panel (e) indicate parallel flare brightening during the impulsive phase of
the flare. The black arrows in panel (j) and white arrows in panel (k)
indicate erupting plasma. The black dotted curve in panel (k) indicate the
quasi-circular ribbon. Co-temporal LOS HMI magnetogram contours at levels
$\pm$[300, 500, 1000] G are plotted in panel (a). Green and blue contours
refer to positive and negative polarity, respectively. An animation associated
with this figure is attached in the online supplementary materials.
#### 4.1.2 Filament eruption and shortened quasi-circular ribbon during F2
Although, the filament situated along the northern edge of the diffused
brightening region (FL1), partially erupted during F1, the corresponding PIL
showed the presence of a filament channel throughout the lifetime of the
diffused brightening region. Before the onset of F2, another filament was
observed quite prominently at the same location (i.e., FL1) which is
highlighted by a blue dotted line in Figure 7(a). Importantly, we noticed
impressions of a second filament near FL1 which we mark as ‘FL2’ (indicated by
the green arrow in Figures 7(a)). Similar to the pre-flare phase prior to F1,
the localised region associated with the filaments, underwent small-scale
brightenings during the pre-flare phase of F2 also. We have indicated few such
episodes of brightenings by the yellow arrows in Figures 7(a) and (b). With
time, FL1 became more prominent, which is indicated by the blue arrow in
Figure 7(b). Few minutes prior to the onset of F2, the filament FL2 also
became very prominent, making a double decker flux rope system in association
with FL1. In Figure 7(c), we indicate the two filaments of the double-decker
system by blue and green dotted lines. Notably, this double-decker system is
confirmed by the presence of two intertwined flux ropes (Section 5.2).
Interestingly, the north-western end of the FL2 was associated with narrow
collimated plasma eruption even before the flare was initiated (indicated by
the yellow arrow in Figure 7(c)). Our observations suggest that the flare
onset took place as the filament FL1 went through a complete eruption at
$\approx$13:51 UT. In Figure 7(e), we have outlined the erupting FL1 by a blue
dotted line. At the same time, a part of the western boundary of the diffused
brightening region became very bright implying the formation of a circular
ribbon which we have highlighted by the black dashed curve in Figure 7(e).
Soon after the onset of the eruption, the active region was partially masked
by the cool erupting plasma (the direction of erupting plasma is indicated by
yellow arrows in Figure 7(f)). During the gradual phase, we observed a dense
post flare arcade (indicated by black arrows in Figure 7(i)) which is
expected.
Figure 7: Representative AIA 304 Å images depicting the eruption of the
filament. The filament FL1 is indicated by the blue dotted curve in panel (a)
and blue arrow in panel (b). A second filament (‘FL2) is indicated by the
green arrow in panel (a). The yellow arrows in panels (a) and (b) indicate
small-scale brightenings at the location of the filaments during the pre-flare
phase. The blue and green dotted lines in panel (c) indicate the double-decker
flux rope configuration formed by FL1 and FL2. The yellow arrow in panel (c)
indicate jet-like plasma ejection prior to the onset of the flare. The blue
dotted curve in panel (e) indicate erupting filament while the black dashed
curve in the same panel indicate a quasi-circular ribbon during the F2 flare.
The yellow arrows in panel (f) indicate the direction of the erupting plasma
during the flare. The black arrows in panel (i) indicate post-flare arcade. An
animation associated with this figure is attached in the online supplementary
materials.
## 5 Coronal Magnetic Field Modelling
### 5.1 Extrapolation set up
In order to investigate the coronal magnetic configurations prior to the onset
of the quasi-circular ribbon flares, we employed a non-linear force free field
(NLFFF) extrapolation method with a particular focus to the magnetic atoll
(i.e., flaring) region, using the vector magnetograms from the
‘hmi.sharp_cea_720s’ series at four times: 09:22 UT (prior to F1; Figure 8),
13:46 UT (prior to F2; Figure 9), 17:34 UT (prior to F3; Figure 10(a)–(d1))
and 18:58 UT (prior to F4; Figure 10(e)–(h1)) on 2014 February 16 as the input
boundary conditions. Since the photosphere is not force-free, the photospheric
magnetograms used as the input boundary conditions were preprocessed, as
explained in Wiegelmann et al. (2006). The optimization based NLFFF code,
including the part of preprocessing of input magnetic field, allows a number
of ‘free parameters’ for the user’s consideration, which are $\nu$, $w_{los}$,
$w_{trans}$, $\mu_{1}$, $\mu_{2}$, $\mu_{3}$ and $\mu_{4}$ (for a quick
summary, see Mitra et al., 2020b). In all the four instances of coronal
magnetic field extrapolation conducted in this work, we used the following
values of the free parameters:
$\nu=0.01;~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}w_{los}=1;~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}w_{trans}=\frac{B_{trans}}{max(B_{trans})};\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}\mu_{1}=\mu_{2}=1;~{}~{}~{}~{}~{}\mu_{3}=0.001;~{}~{}~{}~{}~{}~{}~{}\mu_{4}=0.01$.
Theoretically, in the NLFFF model, the angle between current ($\vec{J}$) and
magnetic field ($\vec{B}$) should be 0. However, since the NLFFF-code uses
real measurements of magnetic field, small but non-zero values of
|$\vec{J}\times\vec{B}$| is expected from the reconstructed magnetic field.
Therefore, to assess the quality of the coronal magnetic field reconstruction,
the average value of fractional flux ratio
($<|f_{i}|>=<|(\vec{\nabla}\cdot\vec{B})_{i}|/(6|\vec{B}|_{i}/\bigtriangleup
x)>$), weighted angle ($\theta_{J}$) between $\vec{J}$ and $\vec{B}$ can be
considered (see, DeRosa et al., 2015). In general, NLFFF solutions returning
the values $|\vec{J}\times\vec{B}|\lesssim 10^{-2}$, $<|f_{i}|>\lesssim
2\times 10^{-3}$, $\theta_{J}\lesssim 10^{\circ}$ are considered as good
solutions (see e.g., DeRosa et al., 2015; Thalmann et al., 2019). In Table 2,
we have listed the values of these parameters corresponding to all the four
extrapolations used in this paper. Here, it should be noted that, the
extrapolations were conducted over the entire active region (Figure 5(a)),
however, in this section and in Figures 8–10, we have only shown and discussed
the modelled coronal configuration associated with the atoll region.
Table 2: Summary of the parameters for assessing the quality of NLFFF extrapolation Time of extrapolation | <|$\vec{J}\times\vec{B}$|> | <|$f_{i}$|> | $\theta_{J}$
---|---|---|---
09:22 UT | 4.16$\times$10-3 | 6.79$\times$10-4 | 8.35∘
13:46 UT | 4.82$\times$10-3 | 7.26$\times$10-4 | 8.24∘
17:34 UT | 3.89$\times$10-3 | 6.93$\times$10-4 | 7.88∘
18:58 UT | 5.67$\times$10-3 | 6.58$\times$10-4 | 7.59∘
### 5.2 Preflare coronal magnetic connectivities
NLFFF extrapolation results at 09:22 UT readily indicates the presence of a
flux rope in the apparent location of FL1 which are drawn by blue lines in
Figures 8(a) and (d). For a better understanding of the structure of the flux
rope, we have shown only the field lines constituting the flux rope from top
and side views in Figure 8(b) and (c), respectively, where different field
lines are plotted in different colours. The flux rope is enveloped by a set of
highly sheared overlying closed loops (shown by the green lines in Figure 8).
Notably, these field lines connect the outer positive polarity regions to the
central dispersed negative polarity regions. The green lines are surrounded by
a set of open field lines (shown by yellow color in Figures 8(d)–(f)) which
originate from the outer positive polarity regions. Interestingly, the
boundary of the diffused, nearly-circular flare brightening apparently
delineates the footpoints of the open field lines (cf. the modelled yellow
lines and the background AIA 304 Å image in Figure 8(d)). In this way, the
entire structure made of the green and yellow lines resembles a spine-fan-like
configuration.However, although few low-lying null-points were located close
to the bottom boundary of the extrapolation volume, we could not find presence
of any null-point near or within our region of interest, suggesting that the
coronal configuration associated with the flaring activities reported in this
article, differed from the 3D spine-fan configuration. Further, the laterally
extended nature of the spine-like lines (indicated by the black arrows in
Figures 8(a) and (d)) is also uncharacteristic of the spine-fan configuration.
In Figure 8(e), we show the whole configuration from a different angle for an
overall visualisation. This atypical, spine-fan-like configuration is also
recognised prior to the onset of F2 which is indicated by the black arrows in
Figures 9(d)–(e)). Our extrapolation results reveal that this spine-fan-like
configuration decays significantly in spatial extent afterwards (Figures 10(a)
and (e)) vis-à-vis changes in the corresponding photospheric magnetic field
structure of the magnetic atoll region (Figures 3(j)–(m)).
Figure 8: Non-linear force free field extrapolation results of the magnetic
atoll region prior to the flare F1 (at 09:22 UT) showing the presence of a
flux rope and closed magnetic loops within the region (green lines) besides
large open field lines originating from the outer magnetic patches of the
atoll region (yellow lines). To understand the structures of the flux ropes
clearly, we provide zoomed views of the flux rope with multiple colours in
panels (b) (top view) and (c) (side view). The red patches over the background
HMI magnetogram in panels (a), (c), and (e) represent regions with high
$Q$-value ($log(Q)>2$). The arrows in panel (c) highlight the close
association between the legs of the flux rope and photospheric regions with
high $Q$-value. The Y-Z tilted vertical surface in panel (f) drawn along the
yellow lines represent the distribution of $Q$-values. In panel (g), the
distribution of $Q$ is shown along a plane passing across the magnetic atoll
region i.e., along X-Z plane. The arrow in panel (g) indicate an ‘X’-shape
formed by the regions of high $Q$. For reference, we have included the colour-
table showing the distribution of $log(Q)$ values in the box in panels (f) and
(g). Top boundary in panels (a), (b), (d) as well as the sky-coloured boundary
in panel (e) represent north. We have plotted a compass for representing the
direction in panels (c) and (f). An AIA 304 Å image of 09:26 UT has been
plotted in the background in panel (d). Figure 9: Non-linear force free field
extrapolation results on the magnetic atoll region prior to the F2 flare (at
13:46 UT) showing the presence of a flux rope (blue lines) and closed magnetic
loops within the region (green lines) as well as large open field lines
originating from magnetic patches of the atoll region (yellow lines). A set of
twisted field lines having origin in the south-western end of the polarity
inversion line and extending north-westward as a part of open field lines
i.e., ‘open flux rope’ are shown in pink colour. In panels (b) and (c), only
the two flux ropes are shown from top and side views, respectively. The red
patches over the background HMI magnetogram in panels (a) and (c) represent
regions with high Q-value ($log(Q)>2$). The arrows in panel (c) highlight the
close association between the legs of the flux ropes and photospheric regions
with high $Q$. Panels (e) and (f) are same as Figures 8(f) and (g),
respectively. Top boundary in panels (a), (b) and (d) represent north. Top
boundary in panel (c) represents east.
The model field structure during the preflare phase of F2 reveals two sets of
twisted field lines intertwined with each other in a double-decker flux rope
configuration (shown by blue and pink colours in Figures 9(a)–(c)). While the
twisted field lines shown in blue are similar to the flux rope identified
during the pre-flare phase of F1 (i.e., FL1), the apparent twist associated
with it is found to be more than the previous flux rope (see Table 3). The
other set of twisted field lines shown in pink colour is rather interesting.
One leg of these lines is situated in the PIL region where one leg of the blue
flux rope is also located. Further, we note that, the location of the pink
lines is same as the apparent location of FL2 (cf. Figures 9(b) and 7(c)).
However, while the blue flux rope is anchored to the photosphere at both the
ends, the field lines constituting the pink flux rope is anchored only at its
southern end. At the northern end, the field lines of the flux rope become a
part of the open lines instead of terminating on the photosphere (cf. the open
end of the pink lines in Figures 9(a) and the open yellow lines in Figure
9(d)). In Figures 9(b) and (c), we show only the double-decker flux rope
configuration from top and side views for a better understanding of their
structures. In Figure 10, we show the modelled magnetic configuration above
the magnetic atoll region prior to the onset of the F3 (Figures 10(a)–(d1))
and F4 (Figures 10(e)–(h1)) flares. We find that, similar to F1, prior to the
F3 and F4 flares also, single flux rope structures are identified from the
modelled coronal magnetic field which are shown from top views in Figures
10(a), (b) and 10(e), (f), respectively, and side views in Figures 10(c) and
(g), respectively.
### 5.3 Distribution of Squashing factor ($Q$)
NLFFF modelling reveals a complex coronal magnetic configuration prior to the
successive four flares. While the overall coronal structures resemble a fan-
spine-like configuration, they lack a coronal null-point. To further
investigate the fan-spine-like configuration, we calculated the squashing
factor ($Q$) in the active region using the NLFFF extrapolated magnetic fields
in the extrapolation volume. The relevant panels in Figures 8–10 display the
photospheric regions associated with high $Q$ ($log(Q)>2$) values by the red
coloured patches. From Figures 8(a), 9(a), 10(a) and (e), we readily observe
that along the elongated footpoint locations of the green lines over the
negative polarity magnetic field region (indicated by the black arrows in
Figures 8(a) and 9(a)), the $Q$-values are higher than 102 which provides
substantial evidence for the laterally extended nature of the spine-like
lines. Notably, the footpoint regions of the flux ropes are also found to be
associated with high $Q$-values which can be inferred from the red coloured
patches indicated by the arrows in Figures 8(c), 9(c), 10(c) and (g).
To understand the variation of $Q$-values along the laterally extended spine-
like lines, we draw tilted vertical planes passing through the outer spine-
like lines (i.e., the yellow lines shown in Figures 8(d)–(e), 9(d));
distribution of $Q$ along which are shown in Figure 8(f), 9(e) and 10(d) and
(h). We find that immediately over the inner fan-like lines (shown by green
color), the $Q$-values approached maximum values ($log(Q)\gtrsim$5) signifying
drastic change in the magnetic connectivity between the green and the yellow
lines. The extended arc-shaped high $Q$ region (shown in purple color and the
arrows in Figures 8(f), 9(e) 10(d) and (h)) in the absence of coronal nulls,
suggests the presence of an HFT between the green and yellow lines. In Figure
8(g), we plot $Q$-values along a plane that crosses perpendicularly the tilted
plane of Figure 8(f), i.e., it shows the variation of $Q$ across the spine-
fan-like configuration. From this panel, we readily observe the ‘X’-shape
formed by high $Q$-values (indicated by the black arrow) which further
confirms the presence of the HFT in the coronal magnetic configuration above
the atoll region. Notably, similar configurations are also found prior to the
onset of the subsequent flares (indicated by the arrow in Figures 9(f), 10(d1)
and (h1)); however, with the decay of the magnetic atoll region, the extended
coronal region of high $Q$-values over the inner fan-like lines (indicated by
arrows in Figures 8(f), 9(e) and 10(d)), gradually reduced and concentrated to
a point-like structure prior to the onset of the F4 flare (indicated by the
black arrow in Figure 10(h)).
Figure 10: NLFFF extrapolation results showing the coronal configurations
prior to the F3 (at 17:34 UT; panels (a)–(d1)) and F4 flare (at 18:58 UT;
panels (e)–(h1). Panels (a) and (e) show the flux ropes (in blue colour) and
the inner fan-like lines (in green colour) from top view. The flux ropes prior
to the F3 and F4 flares are exclusively shown in multiple colours from the top
view in panels (b) and (f), respectively, and from side views in panels (c)
and (g), respectively. Panels (d)–(d1) and (h)–(h1) are same as Figures
8(f)–(g), respectively. The arrows in panels (c) and (g) highlight the close
association between the legs of the flux ropes and photospheric regions with
high $Q$.
### 5.4 Calculation of twist number and magnetic decay index
Both GONG H$\alpha$ and AIA 304 Å images suggest the filaments going through
eruptive evolution during the flares (Figures 4, 6, 7). The presence of the
filaments are confirmed by the presence of the flux ropes identified in the
NLFFF-modelled coronal configuration (Figures 8, 9, 10). For a quantitative
assessment of the twists of the flux ropes, we compare distribution of twist
number ($T_{w}$; see, Berger & Prior, 2006) associated with the location of
the flux ropes within the extrapolation volume, defined as
$T_{w}=\frac{1}{4\pi}\int_{L}\frac{(\nabla\times\vec{B})\cdot\vec{B}}{B^{2}}dl$
(1)
where $L$ is the length of the flux rope. In Table 3, we present average twist
number ($T_{w}$) associated with the flux ropes corresponding to consecutive
flares prior to their onset (F1–F4). We find that $T_{w}$ increased
successively from F1 to F4. While $T_{w}$ prior to F1 was only $\lesssim$0.93,
it increased to $\approx$1.22 prior to the onset of F4.
To understand how the horizontal magnetic field changed with height over the
flux ropes, we calculate magnetic decay index ($n$). For this purpose, we have
considered the PILs over which the flux ropes were situated and computed
average decay index along vertical surfaces above it. In Figure 11, we plot
the variation of magnetic decay index averaged over the PILs, with height;
where we have indicated the critical heights at which the value of decay index
reached $n=1.0$ ($h_{crit}(n=1.0)$) and $n=1.5$ ($h_{crit}(n=1.5)$) prior to
all the four flares, by the dotted and dashed vertical lines. In Table 3, we
have summarised the values of $h_{crit}$ and approximate maximum heights of
the flux ropes before the onset of all the four flares. Our calculations
suggest that prior to the F1 flare, $h_{crit}$ was $\approx$25 Mm while it
slightly increased to $\approx$27–30 Mm prior to the subsequent flares.
However, the maximum heights of the flux ropes prior to the onset of the
flares were found to be only $\approx$4–5 Mm (Table 3) which are much less
compared to the critical heights.
Table 3: Summary of the twist number $|T_{w}|$ and critical height ($h_{crit}$) for magnetic decay index $n=1.0$ and 1.5, prior to the four flares from the magnetic atoll region Flare | $|T_{w}|$ | $h_{crit}(n=1.0)$ (Mm) | $h_{crit}(n=1.5)$ (Mm) | Maximum height of
---|---|---|---|---
Id. | | | | the flux rope (Mm)
F1 | $\approx$0.93$\pm$0.11 | $\approx$15 | $\approx$25 | $\approx$ 4
F2 | $\approx$1.12$\pm$0.18 | $\approx$19 | $\approx$30 | $\approx$ 4
F3 | $\approx$1.20$\pm$0.20 | $\approx$17 | $\approx$28 | $\approx$ 5
F4 | $\approx$1.22$\pm$0.20 | $\approx$17 | $\approx$27 | $\approx$ 4
Figure 11: Plot of decay index ($n$) above the PIL as a function of height,
prior to the four events. The values of decay index $n=1.0$ and 1.5 are
indicated by the pink dotted and purple dashed horizontal lines. The critical
heights corresponding to the values $n=1.0$ and 1.5 prior to the flares are
represented by the dotted and dashed vertical lines, respectively, and noted
at the bottom of the plot. Colours of these vertical lines and the values of
critical heights corresponding to the events are synchronised with the colours
of the decay index plots.
## 6 Discussion
In this article, we present a detailed analysis of four successive flares (F1,
F2, F3, F4; in the order of their occurrence) associated with quasi-circular
ribbons. These flares were originated from the active region NOAA 11977 on a
single day within a period of $\approx$11 hour (see Table 1). The flaring
region possessed a fan-spine-like configuration that involved a hyperbolic
flux tube (HFT). Observationally, the fan-spine-like configuration was
identified in the form of a region with diffused EUV brightening that spread
within a circular base prior to F1. The fan-spine-like structure decayed
following each flare and, as a consequence, the prominent quasi-circular
brightening identified prior to F1 also decomposed and simplified during the
succeeding flares.
Our observations revealed that the development of the interesting coronal
configuration prior to F1, facilitating the origin of the subsequent
morphologically similar flaring events, was eventually connected with the
emergence of negative polarity patches on the photosphere that formed a
magnetic atoll region (Figure 2). Further, the location over which negative
polarity patches emerged can be characterised by a geometric stadium
shape222http://mathworld.wolfram.com/Stadium.html with the longitudinal
dimension lying along northwest-southeast (NW-SE) direction (Figure 2(f)).
These negative polarity regions were surrounded by positive polarity regions
on northeast (NE) and southwest (SW) sides. Thus, the photospheric structure
of the flaring region consisted of a longitudinally stretched magnetic patch
bounded by regions of opposite polarity on both sides. NLFFF extrapolation
results suggested a complex configuration in which open field lines
originating from the positive polarity regions enclosed an extended fan-like
structure (Figures 8–10) such that the overall coronal magnetic configuration
resembled the topology of pseudo-streamers (Wang et al., 2007; Titov et al.,
2011, 2012; Masson et al., 2014) albeit in a much smaller spatial scale.
Notably, since pseudo-streamers connect coronal holes of the same polarity,
involving even number of PILs, two-dimensional depiction of the cross-section
of pseudo-streamers indicate the presence of X-shaped high Q-structures (Titov
et al., 2012), which may represent true 3D null-points or topological
structures such as separators connecting multiple null-points, HFTs etc.
(Gibson et al., 2017). In the absence of any null-point, the coronal magnetic
configuration, derived from our analysis, can be physically well interpreted
by considering a small-scale pseudo-streamer involving an HFT.
The presence of the HFT in the flaring region was further confirmed by their
cross-sectional X-shaped high $Q$-regions (Figures 8(g), 9(f), 10(d1) and
(h1)). The high $Q$ values of these regions imply high gradient of magnetic
field (i.e., QSLs) within the diffused brightening region. Intense current
sheets are formed naturally around QSLs as gradient in magnetic field
contributes toward the generation of current (Lau & Finn, 1990; Priest &
Titov, 1996; Aulanier, G. et al., 2005; Démoulin, 2007). Using MHD
simulations, formation of electric current has also been demonstrated in
magnetic flux ropes, coronal sigmoids, beneath an erupting flux rope etc.
(see, e.g., Wilmot-Smith et al., 2009; Pariat et al., 2009b; Aulanier et al.,
2009). Joule heating due to dissipation of these currents associated with the
QSL formed by the inner fan-like lines was most likely responsible for the
diffused EUV brightness confined within a quasi-circular border and was most
prominent during the pre-flare phase of F1 (see Figure 2(i)). Evidently, the
dome-shaped active pre-flare coronal structure, observed in EUV, was co-
spatial with the photospheric magnetic atoll region.
EUV images clearly revealed the formation of distinct quasi-circular flare
ribbons along the boundary of dome-shaped pre-flare structure as the filament
eruption proceeded (Figures 4, 6 and 7). Notably, prior to the flares, the
filament resided within the EUV-dome, i.e., modelled fan-spine-like
configuration (Figures 8–10). This is definitive signature that the quasi-
circular ribbon flares were triggered as the erupting flux rope interacted
with the fan-like separatrix surface. Using MHD simulations, it has been
established that stressed QSL regions can give rise to slipping reconnections
even without the presence of a coronal null-point and for sufficiently thin
QSLs and high resistivities, the field line footpoints can slip-run at super-
Alfvénic speeds along the intersection of the QSLs (slip running reconnection;
Aulanier et al., 2006). While studying a circular ribbon in association with a
coronal null-point topology, Masson et al. (2009) observed that slip-running
reconnection and null-point reconnection can occur sequentially. They also
found that $Q$ is a highly effective parameter which determines which mode of
reconnection will occur in a null-point topology: cut-paste type null-point
reconnetion occurs when the value of $Q$ reaches infinity and slipping (or
slip-running) reconnection occurs for lesser values of $Q$. As observational
signature of slipping/slip-running reconnection, circular ribbon and remote
brightening can be highlighted (Masson et al., 2009); while the null-point
reconnection usually gives rise to collimated eruptions i.e., coronal jets or
H$\alpha$ surges (Pariat et al., 2009a, 2010).
Notably, we observed collimated eruption of plasma prior to the onset of the
C3.4 flare (Figure 7(c)). AIA 304 Å images clearly revealed two adjacent
filaments at the flaring location that apparently crossed each other (Figure
7(c)). NLFFF extrapolation clearly identified two braided flux ropes within
the atoll region (Figure 9). Such arrangement of intertwining flux ropes is
called ‘double-decker flux rope systems’ (see e.g., Liu et al., 2012; Cheng et
al., 2014; Mitra et al., 2020b). Jet-like plasma ejections resulting from the
interaction between the flux ropes within a double-decker system has been
reported in Mitra et al. (2020b). Further, the double-decker region reported
in Mitra et al. (2020b) was associated with a set of open field lines which
guided the eruption of collimated jets. NLFFF model magnetic field structure
prior to the onset of F2 revealed that one end of one flux rope within the
double-decker system was directly connected to the open spine-like-lines
(i.e., open flux rope; shown in the pink colour in Figure 9). These findings
led us to conclude that the jet-like eruption was triggered as a result of the
interaction between the two flux ropes of the double-decker flux ropes system;
while, the open field lines was responsible for guiding eruption in a
collimated manner. We also clarify that, in our case, the collimated eruption
observed prior to the C3.4 flare should not be related with magnetic
reconnection at the HFT. Here, it is worth mentioning that magnetic structures
similar to the open flux rope shown in Figure 9, where magnetic field lines
constituting the flux rope, becomes open at one end, has been previously noted
by Lugaz et al. (2011); Janvier et al. (2016). While numerically investigating
evolution of eruptive flares from complex photospheric configurations of the
active regions NOAA 10798 and 11283, respectively, both the studies found that
opening of the field lines of flux rope structures can lead to and guide solar
eruptions resulting in the formation of CMEs.
We would like to highlight the association of flux ropes with QSL which we
noted in all the four cases (Figures 8(c), 9(c), 10(c) and (g)). Our analysis
readily revealed that the photospheric regions associated with the legs of the
flux ropes were characterised by high $Q$-values. High $Q$-values at the
boundaries of flux ropes essentially associated with two different sets of
magnetic field lines: one set forming the flux rope and the other set forming
the relatively potential, ambient magnetic field (see e.g., Savcheva et al.,
2012; Zhao et al., 2016; Janvier et al., 2016; Guo et al., 2019).
The atoll region produced four homologous flares on 2014 February 16 (Table
1). All the flares triggered as a filament lying over the PIL between the
positive polarity sunspot and negative polarity regions, got destabilised. The
erupting filament interacted with the fan-spine configuration which led to
some degree of circular ribbon brightening during all the flares (Figure 3).
Prior to the onset of each flare, we could identify instances of flux
cancellation from the PIL region (Figure 5) as well as localised brightenings
beneath the filaments (Figure 6 and 7). These observational findings support
the tether-cutting model of solar eruptions (Moore & Roumeliotis, 1992; Moore
et al., 2001). We also explored the possibility of torus and kink
instabilities as the triggering mechanism, by computing the decay index and
twist numbers. Our analysis suggests that the critical decay height for
$n=$1.5 above the PIL of the flux rope prior to all the four flares remained
consistent within the heights of $\approx$25–30 Mm (Table 3). Statistical
surveys concerning the torus instability as the triggering mechanism for
eruptive flares by Wang et al. (2017); Baumgartner et al. (2018) revealed the
ciritical decay height to lie within the ranges of $\approx$36$\pm$17 Mm and
21$\pm$10 Mm. Our results of decay height calculations are in well agreement
with these statistically established values, suggesting a favourable role
played by torus instability. However, considering the average decay index over
the PIL, as followed in the present analysis, is rather a simplistic approach.
Detailed studies devoted to the analysis of decay index (see e.g., Zuccarello
et al., 2014; Zuccarello et al., 2017; Myshyakov & Tsvetkov, 2020) have shown
that a flux rope eruption can take place even when the critical height (hcrit)
becomes sufficiently low in a few discrete locations over the flux rope.
Further, theoretical studies have shown that the critical height strongly
depend on the magnetic topology (Kliem et al., 2014). In this context we note
that, the coronal magnetic configuration associated with all the four flares
reported in this paper was much complex compared to the general cases without
spine-fan configurations. Rapid decay of magnetic field within the fan-surface
is expected which in turns results in high values of magnetic decay index.
Average twist number associated with the flux ropes increased successively
from $\approx$0.93 prior to F1 to $\approx$1.22 prior to $F_{4}$ (Table 3).
Although, the increase in twist suggest higher storage of magnetic free energy
in the flux ropes, the critical value of twist number for resulting in the
destabilisation of the flux ropes was statistically established to be
$|T_{w}|\approx$2 (Duan et al., 2019). Therefore, we could not find any
conclusive evidence for kink instability as the triggering mechanism for the
homologous flares reported in this paper.
It is also noteworthy that, although a clear circular ribbon appeared during
the M-class flare (F1), no observable signature of null-point-like
reconnection (i.e., jet/surge, breakout type eruption etc.) was observed
during it. AIA 304 Å images during the flare clearly suggested that the
erupting filament experienced an apparent sliding motion from the northern end
to the southern end within the diffused brightening region. Plasma eruption
from one end of the filament was observed only when the sliding filament
eventually reached the southern boundary of the diffused brightening region.
These observational features suggest the occurrence of predominantly QSL-
reconnection during the M1.1 flare, rather than the reconnection between the
inner and outer fan-like lines. On the other hand, all the subsequent C-class
flares (F2–F4) evolved with complete eruption of the filaments from the
diffused brightening region, clearly implying reconnection at the HFT. Here we
remember that twist number associated with the flux ropes successively
increased from F1 to F4. Additionally, negative flux within the magnetic atoll
region decreased almost monotonically following the onset of F1 flare
signifying less constraining energy stored in the fan-spine configuration
during the subsequent flares compared to F1. Therefore, we speculate that
excess energy stored in the flux rope (in the form of higher twist number) and
less energy stored in the constraining magnetic field resulted in the complete
destruction of the fan-spine-like configuration during the subsequent C-class
flares; while, during the M1.1 flare, less twisted flux rope did not have
enough energy to trigger exchange type reconnection at the HFT.
In summary, all the eruptive flares initiated from a diffused brightening
region which formed over a complex magnetic configuration where dispersed
negative polarity regions were surrounded by positive polarity regions
(magnetic atoll region). The coronal configuration associated with the
magnetic atoll region was a fan-spine-like configuration that involved an HFT
situated in the corona above the elongated parasitic negative polarity
regions; a configuration similar to those of pseudo-streamers, in a much
smaller spatial scale. All the four flares were initiated as a flux rope was
activated and erupted within the fan dome. Prior to all the flares, we
observed localised brightenings associated with the filaments as well as
small-scale flux cancellation from the PIL region which supports the tether-
cutting model of solar eruption. The magnetic decay index reached to the value
of 1.5 within low coronal heights prior to all the four flares signifying
favourable coronal conditions for driving successful eruption of the flux
ropes. Interaction between the erupting flux rope and the fan-like-separatrix
surface gave rise to circular ribbon brightening during the flares. During the
first flare, the erupting flux rope with relatively less twist, could not
trigger reconnection in the HFT; while, during the subsequent flares, the flux
ropes having relatively higher twist, could blow out the already decaying fan-
spine-like configuration leading to the complete eruption of the core fields.
We further emphasize that, occurrence of successive quasi-circular ribbon
flares from complex fan-spine-like configurations including HFTs, have been
rarely reported in the literature and subsequent studies involving theoretical
and observational analyses of similar events are essential to reach to a
general understanding of the complex coronal configurations in the solar
atmosphere.
## Acknowledgements
We would like to thank the SDO team for their open data policy. SDO is NASA’s
mission under the Living With a Star (LWS) program. This work utilises GONG
data from NSO, which is operated by AURA under a cooperative agreement with
NSF and with additional financial support from NOAA, NASA, and USAF. We are
thankful to Dr. Thomas Wiegelmann for providing the NLFFF code. We are also
thankful to the anonymous referee for his/her important comments and
suggestions which enhanced the scientific content and overall presentation of
the article.
## Data Availability
Observational data from AIA and HMI on board SDO utilised in this article are
available at http://jsoc.stanford.edu/ajax/lookdata.html. GONG H$\alpha$ data
used in this article are available at GONG data archive
(https://gong2.nso.edu/archive/patch.pl?menutype=a). The NLFFF code employed
in this article for coronal magnetic field modelling is provided by Dr. Thomas
Wiegelmann. Different aspects of the code are explained and discussed in
https://doi.org/10.1023/B:SOLA.0000021799.39465.36,
https://doi.org/10.1007/s11207-006-2092-z,
https://doi.org/10.1051/0004-6361/201014391,
https://doi.org/10.1007/s11207-012-9966-z. The IDL-based code used for the
computation of $Q$ and $T_{w}$ is available at
http://staff.ustc.edu.cn/~rliu/qfactor.html.
## Supplementary Material
Videos are attached with Figures 3, 6 and 7, which are available in the online
article.
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|
# Optimal-order convergence of Nesterov acceleration for linear ill-posed
problems
Stefan Kindermann Industrial Mathematics Institute, Johannes Kepler University
Linz, Linz, Austria (kindermann@indmath.uni-linz.ac.at).
###### Abstract
We show that Nesterov acceleration is an optimal-order iterative
regularization method for linear ill-posed problems provided that a parameter
is chosen accordingly to the smoothness of the solution. This result is proven
both for an a priori stopping rule and for the discrepancy principle. The
essential tool to obtain this result is a representation of the residual
polynomials via Gegenbauer polynomials.
> This article is dedicated to A. Neubauer on the occasion of his 60th
> birthday. His analytical insight shall be our unmatched benchmark.
## 1 Introduction
One option to calculate a regularized solution to an ill-posed operator
equation $Ax=y^{\delta}$, with $A:X\to Y$ and $X,Y$ being Hilbert spaces, is
to employ iterative regularization schemes, where approximate solutions
$x_{k}^{\delta}$ are calculated iteratively combined with a stopping rule as
regularization parameter choice. The simplest one being Landweber iteration
(cf., e.g., [4]), which has the downside of being rather slow. To speed up
convergence, acceleration schemes may be used such as the following Nesterov
acceleration:
$\displaystyle\begin{split}x_{k+1}^{\delta}&=z_{k}^{\delta}+A^{*}(y^{\delta}-Az_{k}^{\delta}),\qquad
k\geq 1\\\
z_{k}^{\delta}&=x_{k}^{\delta}+\alpha_{k}(x_{k}^{\delta}-x_{k-1}^{\delta}),\qquad
x_{0}=0,x_{1}=A^{*}y^{\delta},\end{split}$ (1)
where $\|A^{*}A\|\leq 1$ is assumed and where the sequence $\alpha_{k}$ is
chosen, for instance, as
$\displaystyle\alpha_{k}=\frac{k-1}{k+\beta},\qquad k\geq 1,\quad\beta>-1.$
(2)
Here $\beta$ is a parameter; common choices are, for example, $\beta=1$ or
$\beta=2$. We remark that other alternatives for the sequence $\alpha_{k}$ are
possible as well, but for the main analysis of this paper we only consider
(2).
This iteration (in a general nonlinear context) was suggested by Yurii
Nesterov for general convex optimization problems [9]. It is an instance of a
method that achieves the best rate of convergence (in the sense of objective
function decrease) that is generally possible for a first-order method. The
Nesterov acceleration can be employed to speed up convergence of gradient
methods in nonlinear or convex optimization. A particular successful instance
is the FISTA algorithm of Beck and Teboulle [2] for nondifferentiable convex
optimization.
In the realm of ill-posed problems, Hubmer and Ramlau [8] performed a
convergence analysis for nonlinear problems, and shows the efficiency of the
method.
The background and main motivation of the present article is the recent
interesting analysis of Neubauer [10] for ill-posed problems in the linear
case. He showed that (1) is an iterative regularization scheme and, more
important proved convergence rates, which are of optimal order only for a
priori parameter choices and in case of low smoothness of the solution and
suboptimal else. What is puzzling is that the method shows a quite unusual
”semi-saturation” phenomenon (we explain this term below in Section 3.1).
Our contribution in this article is twofold: At first, we prove a formula for
the residuals of the iteration (1) involving Gegenbauer polynomials. On this
basis, we can build a convergence rate analysis, which improves and extends
the results of Neubauer. In particular, we show that the method can always be
made an optimal-order method if the parameter $\beta$ is chosen accordingly to
the (Hölder-)smoothness index of the solution. This result holds for both an a
priori stopping rule and for the discrepancy principle.
Our analysis also explains the quite nebulous role that this parameter plays
in the iteration, as it is related to the index of the orthogonal polynomials
appearing in the residual formula.
Moreover, the above mentioned residual representation also clearly elucidates
the semi-saturation phenomenon because the iteration can be interpreted as a
mixture of a saturating iteration (Brakhage’s $\nu$-method) and a non-
saturating one (Landweber method).
In the following we employ some standard notation of regularization theory as
in [4]: $\delta=\|Ax^{\dagger}-y^{\delta}\|$ is the noise level and
$x^{\dagger}$ denotes the minimum-norm solution to the operator equation
$Ax=y$ with exact data $y=Ax^{\dagger}$. The index $\delta$ of $y^{\delta}$
indicates noisy data, and analogous, $x_{k}^{\delta}$ denotes the iterates of
(1) with noisy data $y^{\delta}$, while the lack of $\delta$ indicates exact
data $y$ and correspondingly the iteration $x_{k}$ with exact data $y$ in
place of $y^{\delta}$ in (1).
## 2 Residual polynomials for Nesterov acceleration
Our work follows the general theory of spectral filter-based regularization
methods as in [4], where the convergence analysis results from estimates of
the corresponding filter function. The first main result, Theorem 1 is quite
useful for this purpose as it represents the residual function in terms of
known polynomials.
The iteration (1) is a Krylov-space method, and the residual can be expressed
as
$y^{\delta}-Ax_{k}^{\delta}=:r_{k}(AA^{*})y^{\delta}$
with the residual polynomials satisfying the recurrence relation (cf. [10])
$\displaystyle\begin{split}r_{k+1}(\lambda)&=(1-\lambda)\left[r_{k}(\lambda)+\alpha_{k}(r_{k}(\lambda)-r_{k-1}(\lambda))\right],\quad
k\geq 1,\\\ \qquad r_{0}(\lambda)&=1,\qquad
r_{1}(\lambda)=(1-\lambda).\end{split}$ (3)
This is a simple consequence of the definition in (1). The $k$-th iterate can
be expressed via spectral filter functions
$x_{k}^{\delta}=g_{k}(A^{*}A)A^{*}y^{\delta},\qquad
g_{k}(\lambda):=\frac{1-r_{k}(\lambda)}{\lambda}.$
Observe that the three-term recursion (3) is not of the form to apply Favard’s
theorem [5], hence $r_{k}$ is not any orthogonal polynomial with respect to
some weight functions. (Note that Favard’s theorem fully characterizes three-
term recurrence relations that lead to orthogonal polynomials).
Before we proceed, we may compare the residual polynomials with other well-
known cases. For classical Landweber iteration [4], which is obtained by
setting $\alpha_{k}=0$ and thus $z_{k}^{\delta}=x_{k}^{\delta}$, the
corresponding residual functions $r_{k}=:r_{k}^{(LW)}$ is
$r_{k}^{(LW)}(\lambda)=(1-\lambda)^{k}.$
On the other hand, another class of well-known iteration methods for ill-posed
problems that are based on orthogonal polynomials are two-step semiiterative
methods [6]. They have the form
$\displaystyle x_{k+1}^{\delta}$
$\displaystyle=x_{k}^{\delta}+\mu_{k+1}(x_{k}-x_{k-1})+\omega_{k+1}A^{*}(y^{\delta}-Ax_{k}),\qquad
k>1,$
where $\mu_{k}$ and $\omega_{k}$ are appropriately chosen sequences. The
corresponding residual functions satisfy the recurrence relation
$\displaystyle r_{k+1}(\lambda)$
$\displaystyle=(1-\omega_{k+1}\lambda)r_{k}(\lambda)+\mu_{k+1}(r_{k}-r_{k-1}),\qquad
k>1,$ (4)
and thus, $r_{k}(\lambda)$ form a sequence of orthogonal polynomials. Of
special interest in ill-posed problems are the $\nu$-methods of Brakhage [3,
6], defined by the sequences, for $k>1$,
$\displaystyle\mu_{k+1}$
$\displaystyle=\frac{(k-1)(2k-2)(2k+2\nu-1)}{(k+2\nu-1)(2k+4\nu-1)(2k+2\nu-3)},$
$\displaystyle\qquad\omega_{k+1}$
$\displaystyle=4\frac{(2k+2\nu-1)(k+\nu-1)}{(k+2\nu-1)(2k+4\nu-1)},$
the initial values $x_{0}=0,$ $x_{1}=\frac{4\nu+2}{4\nu+1}T^{*}y^{\delta}$,
and with $\nu>0$ a user-selected parameter. The associated residual
polynomials $r_{k}=:r_{k}^{(\nu)}$ related to (4) with $r_{0}=1$,
$r_{1}=1-\lambda\frac{4\nu+2}{4\nu+1},$ have the representation [3]
$r_{k}^{(\nu)}(\lambda)=\frac{C_{2k}^{(2\nu)}(\sqrt{1-\lambda})}{C_{2k}^{(2\nu)}(1)},$
where $C_{n}^{(\alpha)}$ denotes the Gegenbauer polynomials (aka.
ultraspherical polynomials); cf. [1].
We now obtain the corresponding representation for the Nesterov residual
polynomials, which is the basis of this article.
###### Theorem 1.
Let $\beta>-1$. The residual polynomials for the Nesterov acceleration (1)
with (2) are
$r_{k}(\lambda)=(1-\lambda)^{\frac{k+1}{2}}\frac{C_{k-1}^{(\frac{\beta+1}{2})}(\sqrt{1-\lambda})}{C_{k-1}^{(\frac{\beta+1}{2})}(1)},\qquad
k\geq 1,$ (5)
with the Gegenbauer polynomials $C_{n}^{(\alpha)}$.
###### Proof.
Defining $h_{k}(\lambda)=r_{k}(\lambda)(1-\lambda)^{-\frac{k+1}{2}}$ and
multiplying (3) by $(1-\lambda)^{-\frac{k+2}{2}}$ leads to the relation
$\displaystyle h_{k+1}(\lambda)$
$\displaystyle=(1+\alpha_{k})\sqrt{1-\lambda}h_{k}(\lambda)-\alpha_{k}h_{k-1}(\lambda),\qquad
k\geq 2,$ (6) $\displaystyle h_{1}(\lambda)$ $\displaystyle=1,\qquad
h_{2}(\lambda)=\sqrt{1-\lambda}.$
We note that $C_{n}^{(\frac{\beta+1}{2})}(x)$ satisfy the recursion relation
(cf. [1, p. 782])
$\begin{split}C_{k}^{(\frac{\beta+1}{2})}(x)&=xc_{k}C_{k-1}^{(\frac{\beta+1}{2})}(x)-d_{k}C_{k-2}^{(\frac{\beta+1}{2})}(x),\qquad
k\geq 2\\\ C_{0}^{(\frac{\beta+1}{2})}(x)&=1,\qquad
C_{1}^{(\frac{\beta+1}{2})}(x)=(\beta+1)x\end{split}$ (7)
with
$c_{k}=\frac{2k+\beta-1}{k},\qquad d_{k}=\frac{k+\beta-1}{k}.$
Using the recurrence relation with $x=1$, leads to
$\displaystyle C_{k}^{(\frac{\beta+1}{2})}(1)$
$\displaystyle=c_{k}C_{k-1}^{(\frac{\beta+1}{2})}(1)\left(1-\theta_{k}^{-1}\right)=d_{k}C_{k-2}^{(\frac{\beta+1}{2})}(1)\left(\theta_{k}-1\right)$
with
$\theta_{k}:=\frac{c_{k}C_{k-1}^{(\frac{\beta+1}{2})}(1)}{d_{k}C_{k-2}^{(\frac{\beta+1}{2})}(1)}.$
Dividing (7) by $C_{k}^{(\frac{\beta+1}{2})}(1)$ and using this relation
yields
$\displaystyle\frac{C_{k}^{(\frac{\beta+1}{2})}(x)}{C_{k}^{(\frac{\beta+1}{2})}(1)}$
$\displaystyle=\frac{C_{k-1}^{(\frac{\beta+1}{2})}(x)}{C_{k-1}^{(\frac{\beta+1}{2})}(1)}\frac{1}{1-\theta_{k}^{-1}}-\frac{C_{k-2}^{(\frac{\beta+1}{2})}(x)}{C_{k-2}^{(\frac{\beta+1}{2})}(1)}\frac{1}{\theta_{k}-1}.$
(8)
By induction (or by well-known formulae [1, 11]), it can easily be verified
that
$\frac{C_{k-1}^{(\frac{\beta+1}{2})}(1)}{C_{k-2}^{(\frac{\beta+1}{2})}(1)}=\frac{k+\beta-1}{k-1},$
from which it follows that $\theta_{k}-1=\alpha_{k}^{-1}$ as well as
$1-\theta_{k}^{-1}=(1+\alpha_{k})^{-1}$. Thus,
$\frac{C_{k}^{(\frac{\beta+1}{2})}(x)}{C_{k}^{(\frac{\beta+1}{2})}(1)}$
satisfies the same recursion as $h_{k+1}(\lambda)$, and the corresponding
initial values for $k=0,1$ agree when setting $x=\sqrt{1-\lambda}$. This
allows us to conclude that
$h_{k}(\lambda)=\frac{C_{k-1}^{(\frac{\beta+1}{2})}(\sqrt{1-\lambda})}{C_{k-1}^{(\frac{\beta+1}{2})}(1)},$
which proves the theorem. ∎
This theorem relates the residual function of Nesterov acceleration to other
known iterations. In particular, the residual $r_{k}$ is roughly the product
of that of $\frac{k}{2}$ Landweber iterations and that of $\frac{k}{2}$
iterations of a $\nu$-method with $\nu=\frac{\beta+1}{4}$.
###### Remark 1.
Gegenbauer polynomials are special cases of Jacobi polynomials and themselves
embrace several other orthogonal polynomials as special cases. Certain values
of $\beta$ in (1) lead to various specializations in (5): The choice $\beta=0$
leads to Legendre polynomials, the often encountered choice $\beta=1$ leads to
Chebyshev polynomials of the second kind [1].
We note that the result of Theorem 1 even holds for $\beta=-1$. In this case,
only $\alpha_{1}$ is not well-defined, but it is always $0$ for $\beta>-1$.
Thus, we may extend the definition of the iteration to $\beta=-1$ by setting
$x_{k}^{\delta}:=\lim_{\beta\to-1}x_{k}^{\delta}$. (This just amounts to
slightly modifying (2) by setting $\alpha_{1}=0$ for $k=1$; the remaining
iteration is well-defined by (1) and (2).) In this case, we may use [1, Eq.
22.5.28] to conclude that the resulting polynomials are Chebyshev polynomials
of the first kind.
Before we proceed with the convergence analysis, we state for generality the
corresponding theorem for Nesterov iteration with a general sequence
$\alpha_{k}$.
###### Theorem 2.
Consider the iteration (1) with a positive sequence $\alpha_{k}$. Then the
corresponding residual function can be expressed as
$r_{k}(\lambda)=(1-\lambda)^{\frac{k}{2}}\frac{P_{k}(\sqrt{1-\lambda})}{P_{k}(1)},\qquad
k\geq 1,$ (9)
where $P_{k}$ is a sequence of orthogonal polynomials obeying the recurrence
relation
$\displaystyle\begin{split}P_{k+1}(x)&=c_{k}xP_{k}(x)-d_{k}P_{k-1}(x),\qquad
k\geq 1\\\ P_{0}(x)&=1,\qquad P_{1}(x)=c_{0}x\end{split}$ (10)
with $c_{n}$ and $d_{n}$ recursively defined to satisfy
$\displaystyle\begin{split}\frac{c_{1}c_{0}}{d_{1}}&=1+\frac{1}{\alpha_{1}}\\\
\frac{c_{k}c_{k-1}}{d_{k}}&=(1+\frac{1}{\alpha_{k}})(\alpha_{k-1}+1)\qquad
k\geq 2.\end{split}$ (11)
Conversely, given a sequence of orthogonal polynomials defined by the
recurrence relation (10) with given sequences $c_{n},d_{n}$. Then there exists
a sequence $\alpha_{k}$ (defined via (11)) such that the corresponding
Nesterov iteration (1) has a residual function as in (9).
###### Proof.
The function $h_{k}(\lambda):=r_{k}(\lambda)(1-\lambda)^{\frac{k}{2}}$
satisfies the recursion (6) with $h_{0}(\lambda)=1$ and
$h_{1}(\lambda)=\sqrt{1-\lambda}$ and for $k\geq 1$. As in the proof of
Theorem 1, we may conclude that (10) leads to a similar recursion as (8):
$\frac{P_{k+1}(x)}{P_{k+1}(1)}=\frac{P_{k}(x)}{P_{k}(1)}\frac{1}{1-\theta_{k}^{-1}}-\frac{P_{k-1}(x)}{P_{k-1}(1)}\frac{1}{\theta_{k}-1},\qquad
k\geq 1.$
with
$\theta_{k}=\frac{c_{k}P_{k}(1)}{d_{k}P_{k-1}(1)},\qquad k\geq 1.$
From (10) we can conclude by some algebraic manipulations that
$\theta_{k}=\frac{c_{k}c_{k-1}}{d_{k}}\left(1-\theta_{k-1}^{-1}\right),\qquad
k\geq 2.$
If (11) holds, then from the recursion for $\theta_{k}$, it follows that we
can perform an induction step following that
$\frac{1}{\theta_{k-1}-1}=\alpha_{k-1}$ implies
$\frac{1}{\theta_{k}-1}=\alpha_{k}$. Since $\frac{1}{\theta_{1}-1}=\alpha_{1}$
by definition, we obtain that $h_{k}(\lambda)$ and $\frac{P_{k}(x)}{P_{k}(1)}$
satisfy identical recursions and have identical initial conditions with the
setting $x=\sqrt{1-\lambda}$.
Conversely, if (10) is given and the sequence $\alpha_{k}$ is recursively
defined by (11), then it follows in a similar manner that
$\frac{P_{k}(x)}{P_{k}(1)}$ has the same recursion and initial conditions as
$h_{k}(\lambda)$ and thus both functions agree. ∎
The polynomials $P_{k}(x)$ in this theorem correspond to
$xC_{k-1}^{\frac{\beta+1}{2}}(x)$ in Theorem 1.
As an illustration, we may consider the peculiar choice of $\alpha_{k}$ in
Nesterov’s original paper [9], which is also used in the well-known FISTA
iteration [2]: First, a sequence is defined recursively,
$t_{k+1}=\frac{1}{2}\left(1+\sqrt{1+4t_{k}^{2}}\right),\qquad t_{1}=1,$
and then the sequence $\alpha_{k}$ is given by
$\alpha_{k}=\frac{t_{k}-1}{t_{k+1}}.$
Note that $t_{k+1}$ is the positive root of the equation
$t_{k+1}(t_{k+1}-1)=t_{k}^{2}.$ Using this identity, we may calculate that
$(1+\frac{1}{\alpha_{k}})(\alpha_{k-1}+1)=\frac{t_{k}}{t_{k-1}}(1+\frac{t_{k}}{t_{k+1}})(1+\frac{t_{k-1}}{t_{k}}).$
Thus, coefficients for a recurrence formula for orthogonal polynomials that
correspond to such an iteration are
$c_{k}=1+\frac{t_{k}}{t_{k+1}},\qquad d_{k}=c_{k-1}-1.$
However, this does not seem to be related to any common polynomial family, to
the knowledge of the author.
On the other hand, we may design Nesterov iterations from the recurrence
relation of classical polynomials. For instance, the Hermite polynomials obey
a relation (10) with $c_{k}=2,$ $d_{k}=2k$. Thus, the sequence $\alpha_{k}$
has to satisfy the recursion
$a_{k}:=\frac{1+a_{k-1}}{\frac{2}{k}-a_{k-1}-1}.$
We do not know if this is of any use, though.
## 3 Convergence analysis
We consider the iteration (1) with the usual $\alpha_{k}$-sequence (2) and
show that it is an optimal-order regularization methods (of course, when
combined with a stopping rule).
### 3.1 Convergence rates and semi-saturation
In the classical analysis of regularization schemes [4], one tries to bound
the error in terms of the noiselevel $\delta$:
$\|x_{k(\delta)}^{\delta}-x^{\dagger}\|\leq f(\delta)$, where $f$ is some
function decreasing to $0$ with $\delta\to 0$. Often, Hölder-type rates are
considered with $f(\delta)=C\delta^{\xi}$. For such estimates, one has to
impose smoothness conditions in form of a source condition
$\displaystyle
x^{\dagger}=(A^{*}A)^{\mu}\omega,\qquad\|\omega\|<\infty,\quad\mu>0.$ (12)
It is also well-known [4] that the optimal rate of convergence under (12) is
of the form
$\|x_{k(\delta)}^{\delta}-x^{\dagger}\|\leq O(\delta^{\frac{2\mu}{2\mu+1}}),$
and a regularization scheme that achieves this bound is called of optimal
order.
The phenomenon of saturation is the effect that for certain regularization
method, the convergence rate $f(\delta)$ does not improve even when the
smoothness is higher, i.e., $\mu$ is larger. This happens, for instance for
Tikhonov regularization at $\mu=1$ or for the $\nu$-methods at $\mu=\nu$; see
[4].
For the Nesterov iteration (1), a detailed analysis has been performed by
Neubauer [10] with the result that, assuming a usual source condition (12) and
an appropriate a priori stopping rule, the resulting iterative regularization
scheme is of optimal order for $\mu\leq\frac{1}{2}$, and, for
$\mu>\frac{1}{2}$, the convergence rates improve with $\mu$ but in a
suboptimal way. More precisely, the convergence rates proven in [10] are
$\|x_{k(\delta)}^{\delta}-x^{\dagger}\|=\begin{cases}O(\delta^{\frac{2\mu}{2\mu+1}})&\mu\leq\frac{1}{2},\\\
O(\delta^{\frac{2\mu+1}{2\mu+3}})&\mu>\frac{1}{2}.\end{cases}$
Thus, contrary to saturating methods, the order still improves beyond the
”saturation index” $\mu=\frac{1}{2}$ but in a suboptimal way. This is what we
call ”semi-saturation”, and, to the knowledge of the author, this has not been
observed yet for a classical regularization method. A further result of [10]
is that using the discrepancy principle as stopping rule, convergence rates
are proven, which are, however, always suboptimal.
Our second main contribution is an improvement of Neubauer’s result in the
sense that we show that the Nesterov iteration is of optimal order for a
smoothness index $\mu\leq\frac{\beta+1}{4}$ with an a priori stopping rule.
Moreover, contrary to [10], we also obtain optimal-order rates with the
discrepancy principle provided that $\mu\leq\frac{\beta-1}{4}$. These findings
allows one to achieve always optimal-order convergence provided $\beta$ is
chosen sufficiently large.
Moreover, the phenomenon of semi-saturation is made transparent by referring
to the representation in Theorem 1: The residual is a product of Landweber-
type and $\nu$-type residuals, and keeping in mind that Landweber iteration
does not show saturation for Hölder indices while the $\nu$-method do, it is
clear that a product as in (5) leads to the above described semi-saturation.
### 3.2 Convergence analysis
In this section we perform a convergence analysis for the iteration (1). By
Theorem 1, we may base our investigation on the known results for Landweber
iteration and the $\nu$-methods.
We collect some useful known estimates:
$\left|\frac{C_{k-1}^{(\frac{\beta+1}{2})}(\sqrt{1-\lambda})}{C_{k-1}^{(\frac{\beta+1}{2})}(1)}\right|\leq
1,\qquad 0\leq\lambda\leq 1,\beta>-1.$ (13)
This is well-known and follows from [11, Eq. (7.33.1), (4.73)]. From this we
immediately obtain that
$|r_{k}(\lambda)|\leq 1,\qquad 0\leq\lambda<1,\quad\beta>-1,$ (14)
which has already been shown in [10]. Moreover, we may conclude from (13) and
(5) as well that
$\lim_{k\to\infty}r_{k}(\lambda)\to 0,\qquad 0<\lambda<1.$ (15)
Recall that we denote by $x_{k}$ the iteration with $y^{\delta}$ replaced by
the exact data. As usual, this allows one to split the total error into an
approximation and stability term. We estimate the stability term:
###### Proposition 1.
Let $\|A^{*}A\|\leq 1$ and define $x_{k}^{\delta}$ by (1) (2) with $\beta>-1$.
Let $x_{k}$ be the corresponding noise-free iteration with $y^{\delta}$
replaced by $y=Ax^{\dagger}$. Then we have the estimate
$\|x_{k}^{\delta}-x_{k}\|\leq\sqrt{2}\sqrt{(k-1)^{2}+\frac{k+1}{2}}\delta\leq\sqrt{2}k\delta.$
(16)
###### Proof.
Following [4], it is enough to estimate
$g_{k}(\lambda)=\frac{1-r_{k}(\lambda)}{\lambda}=r_{k}^{\prime}(\tilde{\lambda}),$
where we used the mean value theorem with $\tilde{\lambda}\in(0,\lambda)$. The
derivative may be calculated from (5) as
$\displaystyle r_{k}^{\prime}(\lambda)$
$\displaystyle=\frac{k+1}{2}(1-\lambda)^{\frac{k-1}{2}}\frac{C_{k-1}^{(\frac{\beta+1}{2})}(\sqrt{1-\lambda})}{C_{k-1}^{(\frac{\beta+1}{2})}(1)}-\frac{1}{2}(1-\lambda)^{\frac{k}{2}}\left(\frac{[C_{k-1}^{(\frac{\beta+1}{2})}](\sqrt{1-\lambda})}{C_{k-1}^{(\frac{\beta+1}{2})}(1)}\right)^{\prime}.$
We use Markov’s inequality (cf. [4, Eq. (6.16)]) and (13) to conclude that
$\left|\left(\frac{[C_{k-1}^{(\frac{\beta+1}{2})}](\sqrt{1-\lambda})}{C_{k-1}^{(\frac{\beta+1}{2})}(1)}\right)^{\prime}|\right|\leq
2(k-1)^{2}\max_{0\leq\lambda\leq
1}\left|\frac{C_{k-1}^{(\frac{\beta+1}{2})}(\sqrt{1-\lambda})}{C_{k-1}^{(\frac{\beta+1}{2})}(1)}\right|\leq
2(k-1)^{2}.$
Thus,
$|g_{k}(\lambda)|\leq\frac{k+1}{2}+(k-1)^{2}.$
The result now follows with [4, Theorem 4.2] and (14). ∎
Note that this estimate is a slight improvement compared to the corresponding
estimate in [10, Equation (3.2)], which has $2k\delta$ on the right-hand side,
similar as for the $\nu$-methods.
From this we may conclude convergence:
###### Theorem 3.
Let $\|A^{*}A\|\leq 1$ and $\beta>-1$. If the iteration is stopped at a
stopping index $k(\delta)$ that satisfies $k(\delta)\delta\to 0$ and
$k(\delta)\to\infty$ as $\delta\to 0$, then we obtain convergence
$x_{k(\delta)}^{\delta}\to x^{\dagger}.$
###### Proof.
We estimate
$\displaystyle\|x_{k(\delta)}^{\delta}-x^{\dagger}\|$
$\displaystyle\leq\|x_{k(\delta)}^{\delta}-x_{k(\delta)}\|+\|x_{k(\delta)}-x^{\dagger}\|\leq\sqrt{2}k(\delta)\delta+r_{k(\delta)}(A^{*}A)x^{\dagger}.$
The first term converges to $0$ by assumption on $k(\delta)$ and the second
term does so because $k(\delta)\to\infty$ and by the dominated convergence
theorem using (14), (15) as in [4]. ∎
We now consider convergence rates, and for this, the following rather deep
estimate for orthogonal polynomials is needed. It was derived by Brakhage [3]
as well as by Hanke [4, Appendix A.2], [6] on basis of Hilb-type estimates for
Jacobi polynomials.
###### Proposition 2.
Let $\beta>-1$. Then there is a constant $c_{\beta}$ with
$\left|\lambda^{\frac{\beta+1}{4}}\frac{C_{k}^{(\frac{\beta+1}{2})}(\sqrt{1-\lambda})}{C_{k}^{(\frac{\beta+1}{2})}(1)}\right|\leq
c_{\beta}k^{-2\frac{\beta+1}{4}},\qquad 0\leq\lambda\leq 1.$ (17)
###### Proof.
For $k$ even, this is [4, Eq. (6.22)] (with $k$ there meaning $2k$ here), or
[6, Th. 4.1]. However, the result there is based on the Hilb-type formula
([11, Theorems 8.21.12, 8.21.13] which holds for all $k$ as in [3, p. 170].
Thus, by following the steps in [4, Appendix A.2], the result is obtained. ∎
Note that in case $-1<\beta<1$, the constant $c_{\beta}$ may be explicitly
calculated from [11, Eq. (7.33.5)].
The corresponding estimates for the residuals of Landweber iteration are
standard; cf. [4, Eq. (6.8)]:
$|\lambda^{\mu}(1-\lambda)^{k}|\leq c_{\mu}(k+1)^{-\mu}.$ (18)
As a consequence, we may state our main convergence rate result for an a
priori stopping rule:
###### Theorem 4.
Let $\|A^{*}A\|\leq 1$ and $\beta>-1$, and suppose that a source condition
(12) is satisfied with some $\mu>0$.
1. 1.
If $\mu\leq\frac{\beta+1}{4}$ and the stopping index is chosen as
$k(\delta)=O(\delta^{-\frac{1}{2\mu+1}}),$
then we obtain optimal order convergence
$\displaystyle\|x_{k(\delta)}^{\delta}-x^{\dagger}\|\leq
O(\delta^{\frac{2\mu}{2\mu+1}}).$
2. 2.
If $\mu>\frac{\beta+1}{4}$ and the stopping index is chosen as
$k(\delta)=O(\delta^{-\frac{1}{\mu+\frac{\beta+1}{4}+1}}),$ (19)
then we obtain suboptimal order convergence
$\displaystyle\|x_{k(\delta)}^{\delta}-x^{\dagger}\|\leq
O(\delta^{\frac{\mu+\frac{\beta+1}{4}}{\mu+\frac{\beta+1}{4}+1}}).$
###### Proof.
For $\lambda\leq 1$ the estimate (17) yields (by interpolation) and
$(1-\lambda)^{\frac{k+1}{2}}\leq 1$ that
$|r_{k}(\lambda)\lambda^{\mu}|\leq Ck^{-2\mu},\qquad\mu\leq\frac{\beta+1}{4}.$
(20)
In case of $\mu>\frac{\beta+1}{4}$, we have with additionally using (18)
$\displaystyle|r_{k}(\lambda)\lambda^{\mu}|$
$\displaystyle\leq|(1-\lambda)^{\frac{k+1}{2}}\lambda^{\mu-\frac{\beta+1}{4}}|c_{\beta}k^{-2\frac{\beta+1}{4}}$
$\displaystyle\leq
c_{\mu,\beta}(\frac{k+1}{2})^{-(\mu-\frac{\beta+1}{4})}|c_{\beta}k^{-2\frac{\beta+1}{4}}\leq
Ck^{-\left(\mu+\frac{\beta+1}{4}\right)}.$
The result now follows by standard means:
$\displaystyle\|x_{k}^{\delta}-x^{\dagger}\|$
$\displaystyle\leq\|x_{k}^{\delta}-x_{k}^{\delta}\|+\|x_{k}^{\delta}-x^{\dagger}\|\leq\sqrt{2}k\delta+\|r_{k}(A*A)(A^{*}A)^{\mu}\omega\|$
$\displaystyle\leq\begin{cases}\sqrt{2}k\delta+Ck^{-2\mu}&\mu\leq\frac{\beta+1}{4},\\\
k\delta+Ck^{-(\mu+\frac{\beta+1}{4})}&\mu>\frac{\beta+1}{4}.\end{cases}$
Solving for $k$ by equating the two terms in the last bounds yields the a
priori parameter choice and the corresponding rates. ∎
These results correspond to those of Neubauer when $\beta=1$. However, for
$\beta>1$ this is an improvement as we obtain optimal-order convergence if
$\beta$ is chosen larger than $4\mu-1$. We note that in the optimal order
case, the number of iteration needed is of order
$O(\delta^{-\frac{1}{2\mu+1}})$, which is the same order as for semiiterative
methods and for the conjugate gradient method. Thus, the Nesterov acceleration
certainly qualifies being called a fast method.
### 3.3 Discrepancy principle
With the improved estimates, we can as well strengthen the result of [10] when
the iteration is combined with the well-known discrepancy principle. Recall
that it defines a stopping index $k(\delta)$ a posteriori by the first
(smallest) $k$ that fulfils the inequality
$\|Ax_{k}^{\delta}-y^{\delta}\|\leq\tau\delta,$ (21)
where $\tau>1$ is fixed. The corresponding convergence rates can be obtained
by a slight modification of the proof in [10] and the general theory in [4].
###### Theorem 5.
Let $\|A^{*}A\|<1$, $\beta>-1$, and assume a source condition (12) satisfied.
If the iteration (1) is stopped by the discrepancy principle (21), then we
obtain the following convergence rates:
1. 1.
If $\mu+\frac{1}{2}\leq\frac{\beta+1}{4}$, then we achieve optimal order
convergence rates
$\displaystyle\|x_{k(\delta)}^{\delta}-x^{\dagger}\|\leq
O(\delta^{\frac{2\mu}{2\mu+1}})$
with stopping index being of the same order as in (19).
2. 2.
$\mu+\frac{1}{2}\geq\frac{\beta+1}{4}$, then we obtain that
$k(\delta)=O(\delta^{-\frac{1}{\frac{1}{2}+\mu+\frac{\beta+1}{4}}})$
and a rate of
$\displaystyle\|x_{k(\delta)}^{\delta}-x^{\dagger}\|\leq
O(\delta^{\frac{\mu+\frac{\beta+1}{4}-\frac{1}{2}}{\mu+\frac{\beta+1}{4}+\frac{1}{2}}}).$
###### Proof.
The proof [10, Theorem 4.1] only needs minor modifications. The estimate [10,
Eq. (4.3)]
$\|x_{k(\delta)}-x^{\dagger}\|\leq\|r_{k(\delta)}(T^{*}T)w\|^{\frac{1}{2\mu+1}}\left((\tau+1)\delta\right)^{\frac{2\mu}{2\mu+1}}$
is valid independent of our new rate results, hence it follows as in [10, Eq.
(4.4)] that $\|x_{k(\delta)}-x^{\dagger}\|\leq
o(\delta^{\frac{2\mu}{2\mu+1}})$. It remains to estimate
$\|x_{k(\delta)}^{\delta}-x^{\dagger}\|$ by (16) combined with an upper bound
for $k(\delta)$. Estimate [10, Eq. (4.2)] and the discrepancy principle yields
$\tau\delta\leq\delta+\|(T^{*}T)^{\frac{1}{2}+\mu}r_{k}(T^{*}T)w\|$
for $k=k(\delta)$. Using (20) in case that
$\mu+\frac{1}{2}\leq\frac{\beta+1}{4}$, we obtain
$(\tau-1)\delta\leq Ck(\delta)^{-2(\frac{1}{2}+\mu)},$
which yields (19), and with (16) we obtain
$\|x_{k(\delta)}^{\delta}-x^{\dagger}\|=O(\delta^{\frac{2\mu}{2\mu+1}})$,
which proves the result in the optimal case.
In case that $\mu+\frac{1}{2}>\frac{\beta+1}{4}$, the corresponding estimate
is
$(\tau-1)\delta\leq Ck(\delta)^{-(\frac{1}{2}+\mu+\frac{\beta+1}{4})},$
from which the result in the second case follow. ∎
These rates agree with those of [10] when setting $\beta=1$. There, however,
only the suboptimal case 2. was possible. Our improvement is to show that we
may achieve optimal order results even with the discrepancy principle provided
$\beta$ is sufficiently large.
###### Remark 2.
It is clear that in practice $\beta$ should be selected in the regime of
optimal rates, i.e. $\beta>4\mu-1$ for a prior choices and $\beta>4\mu+1$ for
the discrepancy principle. However, it is a rule of thumb to choose such
parameter also as small as possible, or more precisely, in such a way to come
close to the saturation point, i.e. $\beta\sim 4\mu-1$, respectively
$\beta\sim 4\mu+1$.
###### Remark 3.
For semiiterative methods, a modified discrepancy principle [6, 4] has been
defined, where the residual in (21) is replaced by an expression of the form
$(y^{\delta},s_{k}(AA^{*})y^{\delta})$ with a constructed function $s_{k}$.
This yields an order-optimal method as for the a priori stopping rule. An
adaption of this strategy for Nesterov iteration is certainly possible and
this should yield order-optimal rates for all $\mu\leq\frac{\beta+1}{4}$.
However, the strategy is quite involved and it is not completely clear to us
how to include this into the iteration efficiently. We thus do not intend to
investigate such modifications in this article.
## 4 Numerical results
In this section we present some small numerical experiments to illustrate the
semi-saturation phenomenon and to investigate the performance of Nesterov’s
iteration, in particular, with respect to the optimal-order results.
In a first example we consider a simple diagonal operator $A={\rm
diag}(\frac{1}{n^{2}})$, for $n=1,\ldots 1000$, as well as an exact solution
$x^{\dagger}=(\frac{1}{n^{4}}(-1)^{n})_{n=1}^{1000}$, which amounts to a
source condition being satisfied with index $\mu=0.75$. Thus, we are in a case
of higher smoothness, where the results of the present article really improve
those of [10]. We add standard normally distributed Gaussian noise to the
exact data and performed various iterative regularization schemes: Landweber
iteration, the $\nu$-method, and the Nesterov iteration, the latter two with
various settings of the parameters $\nu$ and $\beta$, respectively.
We calculated the stopping index either by the discrepancy principle (21) with
$\tau=1.01$ or, since we have the luxury of an available exact solution in
this synthetic example, we also calculate the oracle stopping index, which is
defined as
$k_{opt}={\rm argmin}_{k}\|x_{k}^{\delta}-x^{\dagger}\|.$
In other words, $k_{opt}$ is the theoretically optimal possible stopping
index.
Figure 1: Log-log plot of the error $\|x_{k(\delta)}^{\delta}-x^{\dagger}\|$
versus the noiselevel $\delta$ for Nesterov iteration (full line, blue),
Landweber iteration (dotted line, black), and the $\nu$-method (dashed dotted
line, red). Left: optimal stopping rule. Right: stopping by discrepancy
principle. The parameters $\beta,\nu$ are in an optimal-order regime.
Figure 2: The similar plot as in Figure 1(left) for various iteration
parameter in a suboptimal-order regime. Left: $\nu=0.4$ and $\beta=0$. Right:
$\nu=0.3$ and $\beta=-0.5$. Stopping by optimal stopping rule $k_{opt}$.
In Figure 1, we display the error $\|x_{k(\delta)}^{\delta}-x^{\dagger}\|$
against various noiselevels on a log-log scale. The curves correspond to
convergence rates for Nesterov iteration (full line, blue), Landweber
iteration (dotted line, black), and the $\nu$-method (dashed dotted line,
red). The parameter were chosen as $\beta=4$ and $\nu=1$, i.e., we are in the
optimal-order case covered by item 1 in Theorem 4 and Theorem 5. On the left-
hand side we employ the oracle stopping rule using $k_{opt}$ and on the right-
hand side we use the discrepancy principle.
As can be observed, all three methods show a similar (optimal-order) rate, as
stated in Theorems 4 and 5. In particular, this verifies one of our findings
that the discrepancy principle for Nesterov’s iteration leads to an optimal-
order method provided $\beta$ is chosen appropriately.
In Figure 2, we illustrate the semi-saturation phenomenon: Here $\beta$ and
$\nu$ are deliberately chosen as too small ($\beta=0$, $\nu=0.4$ on the left-
hand side and $\beta=-0.5$, $\nu=0.3$ on the right-hand side). We observe that
for small $\nu$, the convergence rate of the $\nu$-method is slow as a result
of its saturation. On the other hand, the Nesterov iteration also has a slower
rate than the non-saturating Landweber iteration, but, as can be expected from
our residual polynomial representation, it is in between the other two.
We remark that the $\nu$-methods show some unpleasant behaviour when $\nu$ is
chosen small. The residual is highly oscillating and for small noiselevel we
could not even reach the prescribed discrepancy, and if we did, then the
number of iteration was quite high, even higher than for Landweber iteration.
This might be attributed to our quite aggressive setting of the discrepancy
principle with $\tau=1.01$. In that respect, the Nesterov iteration was very
well-behaved, and we had no problem with a small $\beta$, which is probably
due to the robust Landweber-component in the representation (5).
The optimal-order convergence only partly illustrates the effective
performance of the methods. In Tables 1 we therefore provide the ratio of
errors values, i.e., the numbers in the table are
$\frac{\|x_{method,k}^{\delta}-x^{\dagger}\|}{\|x_{Nesterov,k_{opt}}^{\delta}-x^{\dagger}\|}$,
where $x_{Nesterov,k_{opt}}^{\delta}$ denotes Nesterov iteration with the
optimal stopping rule and $x_{method,k}^{\delta}$ the iteration of the
respective method with the respective stopping rule. All results correspond to
an optimal-order regime of parameters (those of Figure 1). The number of
iterations (both for the oracle stopping rule and the discrepancy principle)
are given in Table 2. In these tables, we also include the corresponding
results for the conjugate gradient iteration CGNE [7].
Table 1: Errors compared to Nesterov iteration: $\frac{\|x_{method,k}^{\delta}-x^{\dagger}\|}{\|x_{Nesterov,k_{opt}}^{\delta}-x^{\dagger}\|}$. | | $\delta$
---|---|---
Method | Stopping | $10^{-5}$ | $10^{-4}$ | $10^{-3}$ | $10^{-2}$ | $10^{-1}$
Nesterov | $k_{opt}$ | 1 | 1 | 1 | 1 | 1
Landweber | $k_{opt}$ | 1.15 | 0.83 | 0.96 | 1.05 | 1.06
$\nu$-Method | $k_{opt}$ | 1.02 | 1.06 | 1.01 | 1.26 | 0.97
CGNE | $k_{opt}$ | 1.02 | 0.82 | 1.05 | 1.02 | 0.84
Nesterov | Discrepancy | 1.58 | 1.10 | 1.41 | 2.84 | 1.90
Landweber | Discrepancy | 2.23 | 1.17 | 1.41 | 2.80 | 1.98
$\nu$-Method | Discrepancy | 1.02 | 1.13 | 1.00 | 1.56 | 1.88
CGNE | Discrepancy | 1.81 | 1.19 | 1.05 | 2.51 | 1.97
Table 2: Number of iterations for various methods; setting as in Table 1. | | $\delta$
---|---|---
Method | Stopping | $10^{-5}$ | $10^{-4}$ | $10^{-3}$ | $10^{-2}$ | $10^{-1}$
Nesterov | $k_{opt}$ | 371 | 163 | 65 | 26 | 15
Landweber | $k_{opt}$ | 11000 | 2193 | 512 | 145 | 36
$\nu$-Method | $k_{opt}$ | 190 | 82 | 33 | 22 | 9
CGNE | $k_{opt}$ | 10 | 6 | 4 | 3 | 2
Nesterov | Discrepancy | 260 | 111 | 39 | 13 | 1
Landweber | Discrepancy | 5106 | 1080 | 220 | 37 | 1
$\nu$-Method | Discrepancy | 190 | 96 | 33 | 10 | 1
CGNE | Discrepancy | 8 | 5 | 4 | 2 | 1
In terms of the number of iteration, the Nesterov iteration is slightly slower
than the $\nu$-methods (approximately by a constant factor of 1.5) but both
have a similar modest increase of iterations when $\delta$ is decreased. Both
need more iteration than the CGNE-method, which, of course, is the fastest one
by design. The slightly higher number of iterations might be attributed to the
better error estimate in (16). (Note that the $\nu$-methods have a $2$ in
place of $\sqrt{2}$ there). It might appear a little bit paradoxical that a
better estimate leads to slower convergence, but this is clear from the theory
as the number of iteration is a decreasing function of $\delta$ and thus also
of any factor in front of $\delta$. This factor, however, pays off when
considering the total error of the method, and we observe that Nesterov
iteration with the optimal choice $k_{opt}$ indeed has almost always a
slightly smaller error than the $\nu$-method. Surprisingly, it is in several
instances also better than the CGNE-method. However, the Nesterov method
sometimes loses some of its advantages against the $\nu$-method, when using
the discrepancy principle, but the performance is still acceptable.
Some further experiments indicate that the results are rather insensitive to
overestimating $\beta$. As stated in Remark 2, the best choice is usually
related to the smoothness index, but there was arose no serious problems when
$\beta$ was larger.
Further numerical experiments have been performed in [10]: Even though the
value of $\beta$ was not reported there, the results are consistent with our
theory with the choice $\beta=1$. The forward operator there was the Green’s
function for the solution of the 1D boundary value problem
$-u^{\prime\prime}=f$ with homogeneous boundary conditions. Exact solutions
with various smoothness are stated there: Example 5.1 with $\mu=\frac{1}{8}$,
Example 5.2 with $\mu=\frac{5}{8}$, and Example 5.3 with $\mu=\frac{17}{8}$.
We used the same problem and the same examples, but we calculated $A$ by using
a FEM-discretization of the boundary value problem and $A$ as the
corresponding solution operator. For simplicity we ignored discretization
errors and took the discretized (projected) solution as $x^{\dagger}$.
Figure 3: Convergence rates for the examples in [10]. Left: Ex. 1, smoothness
index $\mu=\frac{1}{8}$. Center: Ex. 2, smoothness index $\mu=\frac{5}{8}$.
Right: Ex. 3, smoothness index $\mu=\frac{17}{8}$. Displayed are the errors
versus the noiselevel on a logarithmic scale. A marker ’x’ indicates optimal
choice of $\beta$, and ’+’ indicates suboptimal choice $\beta=1$. The full
line indicates the optimal order rate.
The main purpose of this experiment is to verify that the discrepancy
principle ($\tau=1.1$) can be made an optimal-order method. We choose
$\beta=3.5$ for the first two examples and $\beta=9.5$ for the third, which
should in any case lead to an optimal-order situation. In Figure 3, we plotted
the error versus the relative noiselevel on a logarithmic scale for the three
examples with this choice of $\beta$, indicated by the marker ’x’. As a
comparison, we also indicated the predicted optimal rate by a solid line.
Furthermore, also shown and marked with ’+’ are the corresponding results for
$\beta=1$, i.e., in the suboptimal case.
These results clearly illustrate that for the discrepancy principle we may
achieve the optimal order rates with the correct choice of $\beta$ and for a
wrong choice of $\beta$ the rate deteriorates. For low-smoothness as in
Example 1 (left picture in Figure 3, however, there seems to occur almost no
deterioration contrary to expectation.
## 5 Conclusion
We have provided a representation of the residual polynomials for Nesterov’s
acceleration method for linear ill-posed problems as a product of Gegenbauer
polynomials and Landweber-type residuals. This allowed us to prove optimal-
order rates for an a priori stopping rule and the discrepancy principle as
long as $\beta$ in (2) is sufficiently large. The number of iteration is shown
to be of the same order as for other fast methods such as the $\nu$-method or
the conjugate gradients methods. Moreover, our representation clearly explains
the observed semi-saturation phenomenon.
Within the class of linear iterative methods, the Nesterov acceleration is an
excellent choice, as it is a fast method as well as a quite robust one.
Although, it must be conceded, that it cannot compete with the conjugate
gradient method in terms of number of iterations. However, this is compensated
by its flexibility and simplicity of use, which also allows one to easily
integrate it into existing gradient methods and also to apply it in nonlinear
cases.
## References
* [1] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, vol. 55 of National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C., 1964.
* [2] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), pp. 183–202.
* [3] H. Brakhage, On ill-posed problems and the method of conjugate gradients, in Inverse and ill-posed problems (Sankt Wolfgang, 1986), vol. 4 of Notes Rep. Math. Sci. Engrg., Academic Press, Boston, MA, 1987, pp. 165–175.
* [4] H. W. Engl, M. Hanke, and A. Neubauer, Regularization of inverse problems, vol. 375 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1996.
* [5] J. Favard, Sur les polynomes de tchebicheff, C.R. Acad. Sci. Paris, 200 (1935), pp. 2052–2053.
* [6] M. Hanke, Accelerated Landweber iterations for the solution of ill-posed equations, Numer. Math., 60 (1991), pp. 341–373.
* [7] , Conjugate gradient type methods for ill-posed problems, vol. 327 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1995.
* [8] S. Hubmer and R. Ramlau, Nesterov’s accelerated gradient method for nonlinear ill-posed problems with a locally convex residual functional, Inverse Problems, 34 (2018), pp. 095003, 30 pages.
* [9] Y. E. Nesterov, A method for solving the convex programming problem with convergence rate $O(1/k^{2})$, Dokl. Akad. Nauk SSSR, 269 (1983), pp. 543–547.
* [10] A. Neubauer, On Nesterov acceleration for Landweber iteration of linear ill-posed problems, J. Inverse Ill-Posed Probl., 25 (2017), pp. 381–390.
* [11] G. Szegő, Orthogonal polynomials, American Mathematical Society, Providence, R.I., fourth ed., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII.
|
# tRecX — an environment for solving time-dependent Schrödinger-like problems
Armin Scrinzi Ludwig Maximilian University, Theresienstrasse 37, 80333
Munich, Germany
###### Abstract
tRecX is a C++ code for solving generalized inhomogeneous time-dependent
Schrödinger-type equations $id\Psi/dt=H[t,\Psi]+\Phi$ in arbitrary dimensions
and in a variety of coordinate systems. The operator $H[t,\Psi]$ may have
simple non-linearities, as in Gross-Pitaevskii and Hartree(-Fock) problems.
Primary application of tRecX has been non-perturbative strong-field single and
double photo-electron emission in atomic and molecular physics. The code is
designed for large-scale ab initio calculations, for exploring models, and for
advanced teaching in computational physics. Distinctive numerical methods are
the time-dependent surface flux method for the computation of single and
double emission spectra and exterior complex scaling for absorption. Wave
functions and operators are handled by tree-structures with the systematic use
of recursion on the coarse-grain level. Numerical, analytic, and grid-based
discretizations can be combined and are treated on the same abstract level.
Operators are specified in the input using a script language including
symbolic algebra. User-friendly in- and output, error safety, and
documentation are integrated by design.
###### keywords:
Schrödinger solver , strong field physics , attosecond physics , recursive
structure
††journal: Journal of LaTeX Templates
PROGRAM SUMMARY
Program title: tRecX — time-dependent Recursive indeXing (tRecX=tSurff+irECS)
CPC Library link to program files: (to be added by Technical Editor)
Developer’s repository link: https://gitlab.physik.uni-muenchen.de/AG-
Scrinzi/tRecX
Code Ocean capsule: (to be added by Technical Editor)
Licensing provisions: GNU General Public License 2
Programming language: C++
Nature of problem: tRecX is a general solver for time-dependent Schrödinger-
like problems, with applications mostly in strong field and attosecond
physics. There are no technical restrictions on the spatial dimension of the
problem with up to 6 spatial dimensions realized in the strong-field double
ionization of Helium. A selection of coordinate systems is available and any
Hamiltonian involving up to second derivatives and arbitrary up to three
dimensional potentials can be defined on input by simple scripts.
Solution method: The method of lines is used with spatial discretization by a
flexible combination of one dimensional basis sets, DVR representations,
discrete vectors, expansions into higher-dimensional eigenfunctions of user-
defined operators and multi-center basis sets. Photo-emission spectra are
calculated using the time-dependent surface flux method (tSurff) in
combination with infinite range exterior complex scaling (irECS) for
absorption. The code is object oriented and makes extensive use of tree-
structures and recursive algorithms. Parallelization is by MPI. Code design
and performance allow use in production as well as for graduate level
training.
###### Contents
1. 1 Introduction
1. 1.1 Purpose and scope of this paper
2. 2 Application examples
1. 2.1 A single-electron atom in a strong laser field
2. 2.2 The Helium atom in a strong laser field
1. 2.2.1 Basis constraints and index hierarchy
2. 2.2.2 Coulomb repulsion
3. 2.3 Floquet calculation
4. 2.4 Model in two spatial dimensions
5. 2.5 Molecular model
1. 2.5.1 Orientation dependence of time-delays in photo-electron emission
6. 2.6 Multi-center bases
7. 2.7 Further tutorials
3. 3 Methods and general framework
1. 3.1 irECS and tSurff
1. 3.1.1 Discretization of complex scaled operators
2. 3.2 Recursive discretization
1. 3.2.1 Wave function expansion
2. 3.2.2 Coefficients and operator matrices
3. 3.3 Quadratures
4. 3.4 Adaptive features
5. 3.5 Control of stiffness
6. 3.6 Parallelization
1. 3.6.1 Scaling
4. 4 Main classes
1. 4.1 Index class
1. 4.1.1 Special Index constructors
2. 4.1.2 Discretization classes
2. 4.2 The template class Tree
3. 4.3 Operators classes
1. 4.3.1 Construction and optimization of an OperatorTree
2. 4.3.2 Further important operator classes
4. 4.4 Recursive algorithms
5. 4.5 Basis sets
1. 4.5.1 BasisIntegrable
2. 4.5.2 BasisDVR
3. 4.5.3 BasisGrid
4. 4.5.4 BasisVector
5. 4.5.5 BasisSub
6. 4.5.6 Multi-dimensional basis functions — BasisNdim
6. 4.6 Input, units conversion, and algebraic expressions
1. 4.6.1 General input format — class ReadInput
2. 4.6.2 Class Units
3. 4.6.3 Class Algebra
7. 4.7 TimePropagator and TimePropagatorOutput classes
1. 4.7.1 Plot
8. 4.8 Python scripts
5. 5 Conclusions
## 1 Introduction
The tRecX code package is designed to be a high-performance, yet flexible and
robust code with good maintainability and usability for Schrödinger-like time-
dependent problems. It is in use for computing the interaction of atomic and
molecular systems in non-perturbatively strong laser fields. It implements a
range of techniques such as irECS (infinite-range exterior complex scaling
[1]), tSurff (the time-dependent surface flux method [2, 3]), general and
mixed gauges [4], and the FE-DVR method for complex scaling [5, 6]. The hybrid
anti-symmetrized Coupled Channels method (haCC [7]) is going to be made
publicly available with the next release. The code has been developed for and
applied to solving several problems in strong field physics. The most
outstanding applications of tRecX are the computation of fully-differential
double electron emission spectra of the Helium atom [8, 9] at laser wave
length from 10 to 800 nm, including also elliptically polarized fields [10],
as well as strong field ionization rates and photo-emission spectra for di-
and tri-atomic linear molecules [11, 12, 13] with arbitrary alignment between
the direction of laser polarization and the molecular axis.
During the development of the code a conscious effort has been and still is
being made to adhere to good programming practice for ensuring re-usability
and maintainability. The object-oriented C++ code systematically uses abstract
and template classes for ensuring uniform and transparent code structure. For
easier accessibility by physicists these classes reflect concepts that are
familiar in physics such as the linear and more specifically Hilbert space,
operators that are usually but not necessarily linear maps, and wave
functions. Discretization of the wave function is in terms of an abstract
basis set class, whose specific implementation covers the whole range from
discrete sets of vectors, over grids, finite-elements, standard basis sets
such as spherical harmonics, all the way to expansions in terms of
eigenfunctions of a user-defined operator. These can be combined in a tree-
structured hierarchy that admits building correlated (non-product) bases from
one-dimensional factors. For performance, numerical libraries such as Lapack
[14], Eigen [15], or FFTW [16] are used on the low level. Parallelization is
through MPI with some degree of automatic load-balancing based on self-
measurement of the code.
The development of tRecX was initially motivated by several simultaneous PhD
projects all related to the time-dependent Schrödinger equation (TDSE), but
varying in dimension from 1 to 6 with different coordinate systems and
discretization strategies. Using math-type strings for input of
discretizations and operators allowed covering all these projects within the
same framework, reducing supervision overhead, code redundancy, and
programming errors. Also, non-trivial model Hamiltonians can be implemented
quickly with little compromise in computational performance. This includes,
for example, Floquet calculations or simple many-body systems.
For the use with research students and for graduate level teaching, but also
for productivity in research, error-safety and usability are important design
goals, as is adhering to good programming practice. Due to its origin in
multiple research projects the code does contain important sections that do
not conform to such best practice, but there is an ongoing effort to re-
implement those sections with modern standards. Documentation relies on code-
readability and Doxygen [17] inline documentation. Input is exclusively
through a dedicated class, which only allows documented input and machine-
generates up-to-date help. Input can be as numbers or algebraic expressions
with standard mathematical functions and combining SI, cgs(ESU), or atomic
units (a.u.).
### 1.1 Purpose and scope of this paper
We give an overview of typical uses of tRecX that do not require any
extensions to the code. In addition, the code’s potential is made clear and
possible advanced use with or without code extensions is indicated. Far from
attempting complete documentation in this place, we expose the mathematical
background, logical structure, and the principles for mapping equations into
the code. Some room is given to describing code structure and selected
classes. This information, apart from being useful in its own right, is meant
to illustrate design philosophy and principles, which we consider as a
defining constituent of the tRecX project.
The aim is to provide answers and/or useful information regarding the
following questions:
* 1.
What has been done and what is typically done using tRecX?
* 2.
Is there a possibility of using or adapting tRecX for my problem?
* 3.
What are the most important methods in tRecX? Which of them are specific for
tRecX?
* 4.
What is the code structure? How could I extend this for a new use?
We do not discuss here specific algorithms or numerical methods in greater
detail, giving the relevant references instead.
In the following, we first list examples of applications and discuss the
corresponding inputs, before introducing the main methods used in the code.
Finally, code concept and structure are illustrated at the example of its main
classes. Independent reading of the sections is aided by ample cross-
referencing and minor redundancies between the sections. No effort is made to
provide a complete manual for the code here or elsewhere. Rather, all examples
shown and further introductory and advanced examples are provided as tutorials
with the code. This together with code readability and generous Doxygen
annotation is intended to serve as a source of full documentation.
## 2 Application examples
The code source resides on a git repository [18] from where up-to-date
information on file structure and compilation should be drawn. We only single
out the subdirectory tutorial that contains input files for a range of
applications, named 00HarmonicOsc1.inp, 01HarmonicOsc2.inp, etc., where
tutorial/00 through 11 systematically introduce the most important input
features and code functionalities.
### 2.1 A single-electron atom in a strong laser field
We choose the single-electron system for introducing the general
characteristics of strong field physics problems, the discretization strategy,
and the form of operators in tRecX. Complete input at slightly different
parameters is given in tutorial/11shortPulseIR.
The single-electron time-dependent Schrödinger equation (TDSE) in strong
fields is, in atomic units (a.u. )
$i\frac{d}{dt}\Psi({\vec{r}},t)=[-\frac{1}{2}\Delta+iA_{z}(t)\partial_{z}+V(|{\vec{r}}|)]\Psi({\vec{r}},t).$
(1)
This describes an electron bound by a rotationally symmetric potential, where
the laser field is linearly polarized in $z$-direction
$\vec{\mathcal{E}}(t)=(0,0,E_{z}(t)$ and the interaction is written in dipole
approximation and velocity gauge with the vector potential
${\vec{A}}(t)=\int_{-\infty}^{t}d\tau\vec{\mathcal{E}}(\tau).$ (2)
At optical or near-infrared wave length the duration of one field oscillation
is on the scale of 100 a.u. and pulse durations reach 1000’s of a.u. . In
ionization, a wide range of momenta appears and the wave function expands to
very large size during the pulse. This requires reliable absorption at the
simulation box boundaries, if exceeding simulation sizes are to be avoided. In
tRecX, the standard method for absorption is irECS (Sec. 3.1), which allows to
work with box sizes of only a few 10’s of a.u. , although the underlying
problem expands to 1000’s of a.u. .
Technically, ”strong field” also means that rotational symmetry is strongly
broken. Still, the use of polar coordinates and an expansion into spherical
harmonics $Y^{m}_{l}$ often remains convenient and efficient. The ansatz is
$\Psi({\vec{r}},t)=\sum_{m=-M}^{M}\sum_{l=|m|}^{L}Y^{m}_{l}(\varphi,\theta)\frac{1}{r}\chi^{ml}(r,t).$
(3)
In linear polarization the $m$-quantum number is conserved and the problem is
effectively two-dimensional. The radial functions $\chi^{ml}$ need to support
a broad range of momenta, which suggests the use of higher order grid methods
with sufficient density of points. The standard choice in tRecX is a finite-
element discrete-variable method (FE-DVR) with K=10-20 collocation points per
element. The density of points is problem-dependent, typical average densities
are 2 points per atomic unit. Such an expansion is written as
$\chi^{ml}(r,t)=\sum_{n=0}^{N-1}\sum_{k=0}^{K-1}b^{n}_{k}(r)C^{mlnk}(t).$ (4)
We remark here that indices of coefficients and partial wave functions are
generally written as superscripts, while basis functions are labeled by a
subscript that counts the basis, and a superscript, that designates the set of
basis functions to which the individual function belongs. That principle is
loosely adhered to throughout the paper and broken occasionally for aesthetic
reasons.
FE-DVR can be considered as a local basis set discretization with Lagrange
polynomials as the basis functions on intervals $[r^{n},r^{n+1}]$
$b^{n}_{k}(r)=L_{k}\left(\frac{r-r^{n}}{r^{n+1}-r^{n}}\right)\text{ for
}r\in[r^{n},r^{n+1}],\qquad L_{k}(y)=\prod_{{j=0,j\neq
k}}^{K-1}\frac{y-y_{j}}{y_{k}-y_{j}}.$ (5)
The $y_{j}$ are the quadrature points for a Lobatto quadrature rule on the
interval $[0,1]$. It is sufficient to ensure continuity at the $r_{n}$, which
amounts to a linear constraint on the expansion coefficients of the form
$C^{ml,n-1,K-1}=C^{ml,n,0}$.
Using polar coordinates for ${\vec{r}}$, the full expansion can be written as
a hierarchy of sums
$\Psi(\varphi,\cos\theta,r;t)=\sum_{m=-M}^{M}e^{im\varphi}\sum_{l=|m|}^{L}P^{|m|}_{l}(\cos\theta)\sum_{n=0}^{N-1}\sum_{k=0}^{K-1}\frac{b^{n}_{k}(r)}{r}C^{mlnk}(t),$
(6)
where $P^{|m|}_{l}$ are properly normalized associated Legendre functions. For
the computation of matrix elements all operators involved can be written as
(short sums of) tensor products, for example
$-\Delta=-\mathbf{1}\otimes\mathbf{1}\otimes\frac{1}{r}\partial_{r}^{2}r-\left(\mathbf{1}\otimes\frac{\partial}{\partial\cos\theta}{\sin^{2}\theta}\frac{\partial}{\cos\theta}+\frac{1}{\sin^{2}\theta}\otimes\partial^{2}_{\varphi}\right)\otimes\frac{1}{r^{2}}.$
(7)
In this form matrix elements only involve one-dimensional integrations, which,
in FE-DVR, are performed using the underlying Lobatto quadrature scheme. We
denote quadrature schemes by pairs of nodes and weights, in present example as
$(r_{j},w_{j})$. For correct results in FE-DVR one must use the explicitly
symmetric form of any operator involving derivatives. For example, one writes
$\displaystyle\int_{r_{n}}^{r_{n_{1}}}r^{2}dr\frac{1}{r}b^{n}_{k}(r)[-\frac{1}{r}\partial_{r}^{2}r\frac{1}{r}b^{n}_{l}(r)]\to$
(8)
$\displaystyle\int_{r_{n}}^{r_{n_{1}}}dr[\partial_{r}b^{n}_{k}(r)][\partial_{r}b^{n}_{l}(r)]=\sum_{j=0}^{K-1}w_{j}[\partial_{r}b^{n}_{k}(r_{j})][\partial_{r}b^{n}_{l}(r_{j})],$
and similarly for the other coordinates. Note that in this example the Lobatto
quadrature rule gives the exact integral.
For product bases, matrices corresponding to tensor products are tensor
products of matrices. Typical bases in tRecX are not tensor products, but
rather show tree-like interdependence (Sec. 3.2). Still, matrix-vector
multiplications can be performed with essentially the same operations count as
for strict tensor products (cf. Sec. 3.3). In the given case, rotational
symmetry of the potential and dipole selection rules reduces operator matrices
to simple block-tridiagonal matrices and there is no computational advantage
in exploiting the tensor-product form.
The negative Laplacian Eq. (7) can be specified on input by the string
⬇
<1><1><d_1_d>+<1><d_(1-Q*Q)_d><1/(Q*Q)>...
...+<d_1_d><1/(1-Q*Q)><1/(Q*Q)>.
The pairs of “$\ldots$” in subsequent lines are for typesetting only and
indicate that the lines in actual input should be joined into a single line.
The symbols <d_ and _d> indicate the first derivatives of the bra and ket
basis functions, respectively, as in Eq. (8) and Q is the placeholder for the
coordinates $\varphi$, $\eta=\cos\theta$, and $r$ at the respective positions
in the tensor product. In practice, for standard operators such as the
Laplacian or partial derivatives $\partial_{x},\partial_{y}$ and
$\partial_{z}$ short hand notation such as <<Laplacian>>, <<D/DX>> etc. can be
used instead of the full definition.
Apart from possible right (_d) and left (d_) derivatives the string within the
<...> is an algebraic expressions where Q is a placeholder for the coordinate
in the respective tensor product. For the construction of admissible algebraic
expressions see Sec. 4.6.3.
The code automatically infers from the input the Dirichlet boundary condition
$\chi(r\\!=\\!0)\\!=\\!0$ and implements it by omitting the Lagrange
polynomial $b^{0}_{0}(0)=1$ from the basis. For absorption, one adds a special
“infinite” element $[r^{N-1},\infty)$ with basis functions $b^{N-1}_{k}(r)$
based on the Gauss-Radau quadrature for Laguerre-type polynomials. This leaves
the general structure of Eq. (6) unchanged and provides for highly accurate
and numerically efficient absorption, see discussion of irECS in Sec. 3.1.
As an example we consider the Hydrogen atom, $V(|{\vec{r}}|)=-\frac{1}{r}$,
and the computation of photoelectron spectra for a laser pulse with peak
intensity of $2\times 10^{14}W/cm^{2}$ at central wave length of 800 nm and a
pulse duration of $5$ optical cycles at FWHM. (One optical cycle at circular
frequency $\omega$ is $2\pi/\omega$.) In order do ensure the absence of any
unphysical dc-component from the laser pulse, pulses are defined in terms of
${\vec{A}}$ rather than $\vec{\mathcal{E}}$ through pulse shape and
polarization direction ${\vec{\alpha}}(t)$ and the peak intensity $I_{0}$
${\vec{A}}(t)={\vec{\alpha}}(t)\sqrt{\frac{I_{0}}{2\omega^{2}}}\sin(\omega
t-\phi).$ (9)
The tRecX input for the pulse above is
⬇
Laser: shape, I(W/cm2), FWHM, lambda(nm), phiCEO
cos8, 2.e14, 5 OptCyc, 800., 0
The Laser:shape and FWHM parameters determine ${\vec{\alpha}}(t)$, which by
default points into $z$-direction. Any desired polarization angle can be input
with additional parameters. Shape cos8 indicates a pulse envelope function
$\cos^{8}$, which approximates a Gaussian pulse but maintains strictly finite
pulse duration, in this case about 3000 a.u. . At the carrier envelope offset
phase $\phi=0$ the vector potential $|{\vec{A}}|$ has a node at $t=0$. The
field $\vec{\mathcal{E}}(t)$ then has its peak approximately at $t=0$ except
for very short pulses, where the factorization into carrier and envelope
becomes ill-defined and extra contributions from the time-derivative of
${\vec{\alpha}}(t)$, see (2), become non-negligible.
The discretization is specified in the form
⬇
Axis:name,nCoefficients,...
...lower end,upper end,functions,order
Phi,1
Eta,30,-1,1, assocLegendre{Phi}
Rn,80, 0, 40,polynomial,20
Rn,20, 40,Infty,polExp[0.5]
This means that we use 30 angular momenta ($L_{\text{max}}=29$) and FE-DVR
functions $b^{n}_{k}(r),n=0,1,2,3$ on equal size sub-intervals of $[0,40]$,
each of order 20 with a total of 80=20$\times$4 coefficients. The FE-DVR basis
$b^{4}_{k}$ starting at $40$, consists 20 polynomials with exponential damping
$\exp(-0.5r)$. The single function on the $\varphi$-coordinate is trivially
constant and the associated Legendre functions here effectively reduce to the
ordinary Legendre polynomials. Specifying the radial coordinate as Rn
instructs the code to use the Dirichlet boundary conditions at $0$ and a
warning will be issued, if the basis does not start from $r=0$. The remaining
inputs for time-propagation and complex scaling, will be discussed in later
examples.
At the given laser parameters tSurff was first demonstrated for a realistic
scale problem in a prototype implementation [2]. With tRecX results are
obtained within $\lesssim 3$ minutes on a modern CPU with the input listed
above which delivers relative accuracies of the photo-electron spectra of
about 10$\sim$20% in the main part of the spectrum, see Fig. 1 and also
discussion in [2]. Computation times can be further reduced by
parallelization, but gains of a factor $\lesssim 4$ on up to 8 cores remain
moderate due to the small overall size of the problem, see Sec. 3.6.1. A
complete functional input with comments on the specific choices and on
convergence is can be found in tutorial/11.
Figure 1: Dependence of calculated photo-emission spectra on the radius
$R_{c}$ (c.f. Sec. 3.1). Calculation for the Hydrogen atom with a laser pulse
duration of 3 optical cycles at wave length 800 nm and peak intensity $3\times
10^{14}W/cm^{2}$. The relative differences of $\lesssim 10\%$ between the
calculations arise, as the Coulomb tail is effectively cut off at $R_{c}$.
Numerical convergence is well below these differences. A script for producing
this and similar plots directly from multiple tRecX outputs is provided with
the code, see Sec. 4.8.
### 2.2 The Helium atom in a strong laser field
This much larger problem is used to illustrate the input of higher-dimensional
and more complex discretizations that contain basis constraints in the form of
inter-dependencies between the coordinates. Also, with electron repulsion an
operator appears that does not have tensor-product form. The
tutorial/23Helium3DSpectrum elaborates further on the following by computing
double-emission spectra, although for less demanding parameters.
The Hamiltonian of the Helium atom is
$H_{2}(t)=H(t)\otimes\mathbf{1}+\mathbf{1}\otimes
H(t)+\frac{1}{|{\vec{r}}_{1}-{\vec{r}}_{2}|},$ (10)
where $H(t)$ is the single-electron Hamiltonian from Eq. (1) with
$V(r)=-\frac{2}{r}$. We generalize the expansion (6) to two electrons with the
ansatz
$\Psi({\vec{r}}_{1},{\vec{r}}_{2})=\sum_{m_{1}=-M}^{M}\sum_{m_{2}=-M}^{M}\sum_{l_{1}=|m_{1}|}^{L}\sum_{l_{2}=|m_{2}|}^{L}Y^{m_{1}}_{l_{1}}(\varphi_{1},\theta_{1})Y^{m_{2}}_{l_{2}}(\varphi_{2},\theta_{2})\chi^{m_{1}m_{2}}_{l_{1}l_{2}}(r_{1},r_{2})$
(11)
and the radial functions
$\chi^{m_{1}m_{2}}_{l_{1}l_{2}}(r_{1},r_{2})=\sum_{n_{1},n_{2}=0}^{N-1}\sum_{k_{1},k_{2}=0}^{K-1}b^{n_{1}}_{k_{1}}(r_{1})b^{n_{2}}_{k_{2}}(r_{2})C^{m_{1}m_{2}n_{1}n_{2}}_{l_{1}l_{2}k_{1}k_{2}}(t).$
(12)
In a complete expansion for $\Psi({\vec{r}}_{1},{\vec{r}}_{2})$ analogous to
Eq. (6), the 8-index coefficients appear within a hierarchy of 8 sums, with
the number of indices related to the dimension of the problem. In tRecX, such
a discretization can be specified by the lines
⬇
#define BOX 40
#define ANG 20
#define NABS 15
Axis:name,nCoefficients,lower end,upper end,functions,order
Phi1, 3, 0.,2*pi,expIm
Eta1, ANG, -1,1, assocLegendre{Phi1}
Phi2, 3,0.,2*pi,expIm
Eta2, ANG,-1,1, assocLegendre{Phi2}
Rn1, 20, 0.,10.,polynomial,20
Rn1, 40, 10,BOX,polynomial,20
Rn1, NABS, BOX,Infty,polExp[1.]
Rn2, 20, 0.,10.,polynomial,20
Rn2, 40, 10,BOX,polynomial,20
Rn2, NABS, BOX,Infty,polExp[1.]
For convenience, the input files allow local macros, here used to define ANG
as 20 for the number of angular momenta, BOX for the simulation box size, and
NABS for number of functions for absorption. The radial axes Rn1 and Rn2 are
here cut into three different regions, the section $[0,10]$ with 20 points,
the region with lower density of 40 points on $[10,40]$, and the absorption
region beyond 40. This choice accounts for the fact that higher momenta occur
mostly near the nucleus, but, of course, this intuition needs to be verified
by convergence studies. On the $\varphi_{1}$ and $\varphi_{2}$ coordinates we
have the first three functions from the basis expIm which is defined as
$\\{1,e^{-i\varphi},e^{i\varphi},e^{-2i\varphi},e^{2i\varphi},\ldots\\}$.
#### 2.2.1 Basis constraints and index hierarchy
Nominally, the above basis has a daunting size, given by the product of the
size of each of the axes, which would be impractical for calculations. tRecX
allows to impose constraints on the bases by letting the basis one hierarchy
level depend on the preceding levels. In fact, a first such constraint has
tacitly been introduced by using the spherical harmonics $l_{i}\geq|m_{i}|$,
where the $P^{|m_{i}|}_{l_{i}}(\cos\theta_{i})$ depend on the value of
$m_{i}$. For the given problem further constraints were added by the input
⬇
BasisConstraint: axes,kind
Phi1.Phi2,M=0
Eta1.Eta2,Lshape[3;24]
The first line simply constrains the $z$-component of total angular momentum
to $0=m_{1}+m_{2}$, which reduces the 6-dimensional problem to 5 dimensions,
and, in our example reduces the basis size by a factor 3. The second
constraint accounts for the fact that because of the particular dynamics of
photo-ionization pairs of angular momenta $(l_{1},l_{2})$ with both values
large do not occur and the basis can be constrained to an L-shaped region near
the axes in the $l_{1}l_{2}$-plane, Fig. 2. Examples and numerical
demonstration of such constraints can be found in [8, 10]. This reduces the
effective dimension to near 4. The possibility to flexibly impose constraints
of this kind is one of the important features of the tree-structures in tRecX
and has been used extensively in applications.
As a result of the constraints the expansion coefficients $C$ no longer are
the components of a tensor. Rather, the indices become inter-dependent, where
we use the convention that any index can only depend on the indices to the
left of it. While operator matrices cease to be tensor products of matrices,
the hierarchy of indices still allows efficient operator application, see Sec.
3.3.
Presently only the BasisConstraint’s shown in the input documentation are
available. Extension is easy for simple cases. That includes basic cases of
spin, where spin can be added as a two-component Vec axis. A class must be
derived from IndexConstraint to handle the case. For implementation of non-
local symmetries, such as multi-particle angular momentum or exchange
symmetry, the use of constraints can become very complicated and direct
implementation through explicitly symmetrized bases (to derive from
BasisAbstract) may be more efficient both, in programming and computation.
Note, however, that time-propagation dominantly depends on the sparsity and
tensor-product structure of the operator matrix and only to a lesser degree on
the length of the coefficient vectors. Also, non-locality of a symmetrized
basis may deteriorate parallelization. These various aspects need to be
considered when deciding for explicit implementation of symmetries. At
present, tRecX mostly uses unsymmetrized, but in return sparse and factorizing
representations.
#### 2.2.2 Coulomb repulsion
Coulomb repulsion cannot be written as a finite tensor product and requires
special treatment. We use a multipole expansion and apply the radial part by
multiplication on a quadrature grid. Although this can be made exact within
the given polynomial basis, it turns out that the approximate DVR quadrature
does not compromise computation accuracy. Details of the scheme are given in
Ref. [8] for finite elements, which can be readily transferred to FE-DVR now
used by default in tRecX.
While tensor product operators can be defined through simple scripting,
Coulomb repulsion is custom-implemented. The Hamiltonian (10) can be specified
as
⬇
Operator: hamiltonian=1/2<<Laplacian>>...
...-2.<<Coulomb>>+[[eeInt6DHelium]]
Operator: interaction=iLaserAz[t]<<D/DZ>>,
where the <<...>> are automatically converted to strings of the operator
scripting discussed above, but [[eeInt6DHelium]] directs the code to a
specialized operator class for electron repulsion. The separation into
hamiltonian and interaction is for convenience only, internally the two
strings are merged into a single operator. Also note that the axes need not be
given in exactly the sequence as shown in the example, if only one ensures
that pieces belonging the same axis are in consecutive lines and that the
functions on a given coordinate axis can only depend on coordinates specified
above it: for example, Phi2 must appear above the Eta2 which carries the
associated Legendre functions assocLegendre{Phi2}. The sequence determines the
layout of the indices of the $C$’s, where storage is such that lowest axis
corresponds to the rightmost index, which runs fastest. Storage arrangement
can be modified when defining the parallel layout, See. 3.6.
Figure 2: Preponderance rules and the use of constraints. Panel (a) shows the
maximal amplitudes of angular momentum pairs $(l_{1},l_{2})$ during laser
ionization of Helium atom by a linearly polarized field. An Lshape constraint
discards unneeded pairs. Reproduced from Ref. [8]. Panel (b): maximal
amplitudes in the spherical waves $Y^{m}_{l}$ during ionization of a single-
electron atom by a near-circularly polarized field. A pronounced preponderance
for small $l+m$ is seen. Reproduced from Ref. [10].
### 2.3 Floquet calculation
The Floquet method converts a time-periodic problem into a stationary problem
by discrete Fourier expansion in time. The resulting operator has continuous
spectrum on the whole real axis, but underlying resonances can be accessed by
complex scaling. The tutorial/90Floquet shows how the method can be used
within tRecX.
The TDSE for a single-electron system in a cw field polarized in $z$-direction
is, in velocity gauge,
$i\frac{d}{dt}\Psi({\vec{r}},t)=\left[H_{0}+i\sin(\omega
t)A_{z}\partial_{z}\right]\Psi({\vec{r}},t).$ (13)
The $\Psi({\vec{r}},t)$ can be expanded into
$\Psi^{\alpha}({\vec{r}},t)=e^{-i\epsilon^{\alpha}t}\Phi^{\alpha}({\vec{r}},t),$
(14)
where the $\Phi^{\alpha}({\vec{r}},t)=\Phi^{\alpha}({\vec{r}},t+T)$ are
strictly time-periodic and in turn can be expanded into a discrete Fourier
series
$\Phi^{\alpha}({\vec{r}},t)=\sum_{n=-\infty}^{\infty}e^{in\omega
t}\Phi^{\alpha}_{n}({\vec{r}}),\quad\omega=\frac{2\pi}{T}.$ (15)
Inserting into the TDSE and arranging the $\Phi^{\alpha}_{n}$ into a vector
${\vec{\Phi}}^{\alpha}$ one finds the eigenvalue equation
${\widehat{H}}_{F}{\vec{\Phi}}^{\alpha}=\epsilon^{\alpha}{\vec{\Phi}}^{\alpha}$
(16)
with
$({\widehat{H}}_{F})_{mn}=\delta_{nm}(H_{0}+n\omega)+\frac{1}{2}(\delta_{n,m+1}-\delta_{n,m-1})A_{z}\partial_{z}$
(17)
The Floquet Hamiltonian $H_{F}$ has the complete real axis as its continuous
spectrum, into which the bound states of $H_{0}$ are embedded. For non-zero
$A_{z}$ all bound states experience an ac-Stark shift to a resonance energy
$E_{r}$ with a decay width $\Gamma_{r}$. Upon complex scaling these two
quantities appear as complex eigenvalue $E_{r}-i\Gamma/2$ of the complex
scaled $H_{F}$.
We define a discretization for the expansion (15) as
⬇
Axis: name,nCoefficients,lower end,upper end,functions,order
Vec,18
Phi, 1
Eta, 7,-1,1, assocLegendre{Phi}
Rn, 60, 0.,BOX,polynomial,30
Rn, 30, BOX,Infty,polExp[0.5]
where the first axis Vec labels a total of 18 Floquet blocks, i.e. the Fourier
components $\Phi_{n},n<18$. The Floquet Hamiltonian (17) is input as
⬇
#define KIN 1/2<<Laplacian>>-<1><1><1/Q+exp(-2.135*Q)/Q>
#define OM 0.1155<1><1><1>
#define INT A[I]/2(<delta[1]>-<delta[-1]>)<<D/DZ>>
Operator:hamiltonian=<Id>H0+<diagonal[Q-14]>OM+INT
where the define macros are used for better readability. The factor
<diagonal[Q-14]> indicates a diagonal matrix for the first axis Vec with
entries $(i-14)\delta_{ij}$.
The potential $-(1+e^{-2.135r})/r$ models the screened potential seen by one
electron in a Helium atom. That model gives qualitatively meaningful results
for single-ionization processes and approximately reproduces the first few
ground and excited state energies of the Helium atom. We use it to illustrate
non-perturbative ac-Stark shifts and the resulting intensity-dependent
$n$-photon Freeman resonances [19]. We trace the resonance positions
$E_{r}-i\Gamma_{2}$ as a functions of $A_{z}$ from field intensity $I=0$ to
$2\times 10^{14}W/cm^{2}$. The function A[I] that is used in the Hamiltonian
string together with tracing range and step size are defined in the input as
⬇
Trace: eigenvalues,from, to, steps, function
-0.903, 0, 2e14 W/cm2, 71, A[I]=sqrt(I)/0.1155
where $-0.903$ is the initial guess eigenvalue and the function $A[I]$ defines
the conversion from intensity to $A_{z}$ in a.u. for the given photon energy
of $0.1155\,au\sim 3\,eV$. The eigenproblem is solved by inverse iteration and
roots are selected for largest overlap with the preceding solution. Fig. 3
shows traces for ground and excited states, where crossings near intensities
$1.5\times 10^{14}$ indicate an 8-photon resonance. These lead to
characteristic structural changes in differential double emission spectra, as
discussed in Ref. [9].
Figure 3: Floquet energies and widths as functions of laser intensity at
photon energy 0.1115 a.u. ($\approx 400nm$). States can be identified by their
initial field-free energies. The crossings near $1.5\times 10^{14}W/cm^{2}$
occur at 8-photon transitions from the ground to the 3s and 3d-states,
respectively. Energies are w.r.t. the continuum threshold.
### 2.4 Model in two spatial dimensions
A popular model for inspirational studies in strong field physics is the “two-
dimensional Helium atom” defined by the Hamiltonian
$H(x_{1},x_{2})=\sum_{i=1,2}\left[-\frac{1}{2}\frac{\partial^{2}}{\partial
x_{i}^{2}}+iA(t)\frac{\partial}{\partial
x_{i}}-\frac{2}{\sqrt{x_{i}^{2}+a}}\right]+\frac{1}{\sqrt{(x_{1}-x_{2})^{2}+b}},$
(18)
which with values $a=0.5$ and $b=0.3$ has a ground state energy of -2.88 a.u.
and, remarkably, the exact single ionization threshold of -2 a.u. The model
owes its popularity to the fact that Fast Fourier Transform can be used for an
efficient representation of the derivatives and comparatively large spatial
domains can be used to extract spectra by standard procedures. In tRecX we use
the model mostly for exploring numerical procedures and for testing new code,
such as the first demonstration of double-emission spectra in Ref. [3]. For a
complete input example, see tutorial/20Helium2d
A Cartesian grid extending symmetrically around the origin is input as
⬇
#define BOX 20
Axis:name,nCoefficients,lower end,upper end,functions,order
X1, 10,-Infty,-BOX.,polExp[0.5]
X1, 40,-BOX,BOX,polynomial,20
X1, 10, BOX,Infty,polExp[0.5]
X2, 10,-Infty,-BOX,polExp[0.5]
X2, 40,-BOX,BOX,polynomial,20
X2, 10, BOX,Infty,polExp[0.5]
The coordinates X1,X2 illustrate the general tRecX feature that coordinates
can be numbered. Equivalently one can use, e.g., the axis names X,Y. Complex
scaling is input as
⬇
Absorption: kind, axis, theta, upper
ECS,X1,0.3,BOX
ECS,X2,0.3,BOX
with a complex scaling radius of $R_{0}=20$ at positive coordinates. The
complex scaling radius at negative coordinates defaults to $-R_{0}$, but can
also be set explicitly by specifying a value for Absorption:lower.
Using input macros for brevity, the Hamiltonian is
⬇
#define H1 (0.5<d_1_d>-<2/sqrt(Q*Q+0.5)>)
#define H2 H1<1>+<1>H1
Operator:hamiltonian=H2+<{}><1/sqrt(pow[2](X1-X2)+0.3)>
This illustrates how to define electron repulsion, which is a multiplicative
operator that is not a tensor product w.r.t. $x_{1}$ and $x_{2}$: one defers
the definition of the potential by putting a placeholder factor <{}> until one
reaches the hierarchy level of the lowest coordinate axis, here X2. On that
last level one defines the function using the axis names as the variables.
Simple multi-dimensional potentials can be input easily in this way. For more
complicated dependencies one may consider writing a specialized class instead.
A larger class of general three-dimensional potentials is covered by the Pot3d
discussed in section 2.6 below.
Spectra for emission into the first quadrant
$\mathbb{R}^{+}\times\mathbb{R}^{+}$ can be computed by inputs analogous to
the full 6-dimensional case. Other quadrants are not supported at present, but
spectra can be obtained by computations with reflected coordinate axes
$(x_{1},x_{2})\to(\pm x_{1},\pm x_{2})$. Fig. 4 shows the dependence of
spectra on the carrier-envelope phase $\phi$, Eq. (9), for a single-cycle
pulse.
Figure 4: Photo-electron spectra for the 2$\times$1-dimensional Helium model
(18), dependence on the carrier-envelope phase $\phi$, see Eq. (9). All graphs
share the same color code with normalization to overall maximum 1. Only the
quadrant where both electrons are emitted into the positive axis direction is
shown. The pulse is single-cycle at wave length 800 nm and intensity $3\times
10^{14}W/cm^{2}$. Spectra strongly depend on $\phi$ because of the pronounced
asymmetry of the single-cycle pulse.
### 2.5 Molecular model
In tRecX one can use hybrid bases where different types of basis functions are
combined to discretize the same space. A typical example is the haCC method
[7] for molecules in strong fields, which combines a Gaussian-based CI with
the numerical basis described above. Another example is a multi-center basis,
where spherical bases with different centers are combined.
For the introduction of the concept of hybrid bases we use a model that is
popular in strong field physics, realized in tutorial/221CO2Free. In that type
of model one assumes that a single or a few bound states
$\\{|\alpha\rangle,\alpha=0,1,\ldots,A-1\\}$ of some complicated Hamiltonian
$H_{a}$ are essential, but the strong field dynamics on the rest of the space
can be described by a simplified Hamiltonian $H_{b}$ with the total
Hamiltonian
$H(t)=PH_{a}P+QH_{b}Q+i{\vec{A}}(t)\cdot\vec{\nabla},\quad\text{and}\quad
Q=(1-P),P=\sum_{\alpha=0}^{A-1}|\alpha\rangle\langle\alpha|.$ (19)
For $H_{b}$ one typically uses free motion or motion in a Coulomb field. Note
that the interaction among the $|\alpha\rangle$ states and between
$|\alpha\rangle$ and the rest of the space is taken fully into account in
$H(t)$. With a single bound state $A=1$ and $H_{b}=-\frac{1}{2}\Delta$ this is
very nearly the so-called “strong field approximation” [20], which is behind
much of the theoretical understanding of strong field physics. Hamiltonian
(19) was used to investigate attosecond ($1\,as=10^{-18}s$) delays in photo-
emission from $CO_{2}$. We choose a highly simplified $CO_{2}$ model
Hamiltonian
$H_{a}=-\frac{1}{2}\Delta-\frac{\gamma(1+5e^{-r/c})}{|{\vec{r}}|}-\frac{(1-\gamma)(1+7e^{-|{\vec{r}}+{\vec{b}}|/a})}{2|{\vec{r}}+{\vec{b}}|}-\frac{(1-\gamma)(1+7e^{-|{\vec{r}}-{\vec{b}}|/a})}{2|{\vec{r}}-{\vec{b}}|},$
(20)
where $\gamma=0.5$ parameterizes the distribution of charge between the $C$
and $O$ atoms and screening was chosen as $c=3,a=1.73$. The O-atoms are
located along the $z$-axis at the equilibrium $C\\!-\\!O$ bond length
${\vec{b}}=(0,0,2.197\,au)$. With that one finds a $\Pi$-gerade state at the
$CO_{2}$ HOMO energy of $\approx-0.51a.u.\ $. For the purpose of this study it
suffices to compute the eigenstates $|\alpha\rangle$ of $H_{a}$ in a single-
center expansion. We pick the HOMO and the next higher $\Sigma$-state as
follows:
⬇
#define CO2 <1><{}><CO2Pot[BOX,GAM,CSCR,ASCR](Eta,Rn)>
#define HAM (1/2<<Laplacian>>+CO2)
Axis: subset,name,nCoefficients,lower end, upper end,...
...functions,order
Subspace,Orbital,2,7,,Eigenbasis[HAM:Complement]
Complement, Phi, 7
, Eta, 10,-1, 1,assocLegendre{Phi}
, Rn, 80, 0, 40,polynomial,20
, Rn, 20,40,Infty,polExp[0.5]
The potential parameters BOX,GAM,CSCR,ASCR are set by define’s. The function
itself was hard-coded into tRecX for efficiency, although it can be, in
principle, written as in the example of Sec. 2.4. The additional input subset
separately specifies the discretization on a Subspace and its Complement. The
basis on the subspace are two Orbitals
$\\{\Phi_{0}({\vec{r}}),\Phi_{1}({\vec{r}})\\}$, which are three-dimensional
Eigenbasis functions of the Hamiltonian HAM, which are computed in the
discretization defined in the subset named Complement. Complement is a
standard spherical expansion. The axial symmetry around $z$ is broken by the
field, which is why a total of 7 $\varphi$-functions $m=1,\pm 1,\pm 2$ are
used. This suffices as we only study two-photon transitions in the
perturbative limit.
The dipole field of the laser is specified by the fundamental $\omega$ and its
13th and 15th harmonic as
${\vec{A}}(t)=\hat{\epsilon}A_{f}\cos^{2}(\frac{t}{\tau_{0}})\sin(\omega
t+\phi)+\hat{\epsilon}A_{h}\cos^{4}(\frac{t}{\tau_{H}})[\sin(13\omega
t)+\sin(15\omega t)]$ (21)
with the polarization vector $\hat{\epsilon}$ in the $xz$-plane. The field is
input in terms of peak intensities and FWHM as
⬇
Laser:shape,I(W/cm2),FWHM, lambda(nm),phiCEO, polarAngle
cos2, 1e10, 4 OptCyc, 800, pi/2, 45
cos4, 1e11, 3 OptCyc, 800/13, 0, 45
cos4, 1e11, 3 OptCyc, 800/15, 0, 45
Here pi/2 for phiCEO at the fundamental means that node of the fundamental
field falls onto the peak intensity of the harmonics. Note that OptCyc is
w.r.t. to the first wave length in the list. A warning issued by the code will
remind the user of this fact.
The Hamiltonian (19) is specified as
⬇
Operator: hamiltonian=<0,0>HAM+<1,1>(1/2<<Laplacian>>)
Operator: interaction=<allOnes>...
...(iLaserAx[t]<<D/DX>>+iLaserAz[t]<<D/DZ>>)
The factor <allOnes> is a matrix filled with $1$’s. It refers to the hybrid
“coordinate” axis Subspace&Complement and indicates that all sub-blocks of the
interaction on the subspace and its complement are to be computed:
$H_{I}=i{\vec{A}}\cdot\vec{\nabla}=PH_{I}P+PH_{I}(1-P)+(1-P)H_{I}P+(1-P)H_{I}(1-P).$
(22)
Also note that polarization is no longer along the $z$-axis but rather in the
$xz$-plane, which is why $x$ and $z$-components of the dipole interaction are
both present.
Further possibilities to set up the Hamiltonian are to select more and
different orbitals in the subset space. Also, numerical values of small
matrices for the construction of Hamiltonian and interaction can be specified
in the input, see tutorial/221 for an illustration.
#### 2.5.1 Orientation dependence of time-delays in photo-electron emission
Delays in the laser-emission of electrons have drawn some attention as
possible indicators of a delay in tunneling emission (see, e.g., [21, 22]),
which may occur on the time scale of attoseconds. In order to correctly pose
the question, one must disentangle any possible such delay from delays not
related to tunneling that are well known to appear in scattering after
emission. The model above allows to give meaning to the notion of “scattering
after emission” by restricting the action of the binding potential to the
initial state and use the free particle Hamiltonian everywhere outside the
bound initial state. This can be compared to the full problem, or a partially
restricted problem, e.g. using only the short range part of the molecular
potential or motion in the Coulomb field instead of than free motion. The
experimental definition of delay is related to a beat in a so-called RABITT
spectrogram, see, e.g. [23] for a general discussion of attosecond techniques.
Here we only illustrate the use of tRecX for comparing alternative models
within the same computational framework without any deeper discussion of the
underlying physics. Fig. 5 shows RABITT delays computed with three different
models, the full single-electron Hamiltonian (20), the strong field-like
approximation Eq. (19) with free motion $H_{b}=-\Delta/2$ outside the ground
state, and Coulomb scattering $H_{b}=-\Delta/2-1/r$. If there were any
dependence of the delays on the alignment of the laser field with the
molecular axis, this would be considered as an effect of tunneling through the
orientation-dependent barrier. While the full model shows strong orientation
dependence, no such effect is seen with free motion or motion in the Coulomb
field. The conclusion from this simple study is that any possible effects of
tunneling delays would be completely dominated by delays incurring after
emission.
Figure 5: Dependence of RABITT emission delays on alignment between of the
molecular axis and laser polarization direction. The three calculations are
for the $CO_{2}$ single-electron model (20, dots) and the Hamiltonian (19)
with $H_{b}$ the free motion (diamonds) and motion in the Coulomb field
(squares), respectively. Size of the dots indicates emission yield. Alignment-
dependent delays are largely are due to scattering in the molecular potential.
Delays by the Coulomb potential are nearly independent of alignment.
### 2.6 Multi-center bases
When a system has singularities at several points in space the use of a multi-
center basis is advisable. The tutorial/510OffCenterScatter was used as the
starting point for the calculations published in [24].
We consider one scatterer at some larger distance from the origin. Such a
potential cannot be written as tensor product w.r.t. the original polar
coordinates, but rather is treated as a general three-dimensional potential
which is input in a category Pot3d. It is referenced in the operator
definition as the special operator [[Pot3d]]. An off-center radial potential
can be specified by the Cartesian coordinates of its origin and an algebra
string for the radial function, as for $-1/\sqrt{|{\vec{r}}-{\vec{r}}_{0}|}$
with ${\vec{r}}_{0}=(0,0,65)$ in
⬇
Pot3d: potential=radial[0,0,65,-1/Q]
A matching off-center basis with the spherical harmonics $l\leq 2$ is
specified as
⬇
PolarOffCenter:radius=5,origin=[0,0,65],Lmax=2,Mmax=0,Nmax=5
which uses polynomials of degree 4 on a sphere of radius=5 around
${\vec{r}}_{0}$. With the center placed on the $z$-axis, we have axial
symmetry around the $z$-axis and $m$-quantum numbers remain conserved. In that
case one may constrain the off-center basis to $m=0$, as in the example above.
That basis is to be combined with a standard spherical basis centered at the
origin into a hybrid basis as in
⬇
Axis: subset,name,functions,nCoefficients,...
...lower end,upper end,order
Off,Ndim,PolarOffCenter
Center,Phi,,1
,Eta,assocLegendre{Phi},4, -1, 1
, Rn, polynomial, 160, 0,80,10
, Rn, polExp[1.], 20, 80,Infty
The off-center potential and the off-center basis both break rotational
symmetry and cause partial fill-in of overlap and operator matrices. Note that
here, different from Sec. 2.5, the bases of the two subsets are not
orthogonal. The inverse of the overlap is applied through a specialized class
that implements the Woodbury formula for low-dimensional updates of an inverse
(cf. Sec. 4.3.2). Possible linear dependency and ill-conditioning of the
overlap is monitored, but does not usually pose a problem for a rather well-
localized off-center basis as in this example.
The fill-in of operator matrices occurs where the off-center functions overlap
with the radial sections of the origin-centered basis. For that reason it is
recommended to minimize the number of radial elements $[r^{n},r^{n+1}]$ where
the Center-basis overlaps with the Off-basis. In the given example, the off-
center basis has overlap with two radial sections $r\in[60,70]$.
Operators must be defined with respect to the Center discretization, as in
⬇
Operator: hamiltonian=0.5<allOnes><<Laplacian>>...
...-<allOnes><<Coulomb>>+[[Pot3d]]
The factor <allOnes> translates into a 2$\times$2-matrix filled with 1’s for
the hybrid Off&Center axis. This indicates that matrix elements between the
basis functions of the two subsets are non-zero, when the functions overlap
spatially. Matrix elements between all parts of the basis are computed using
quadratures. When any of the functions is off-center, three-dimensional
quadrature for the off-center basis employed, as typically the center-basis is
smooth across the support of the off-center basis, e.g. a small solid angle
from the sphere times a polynomial in $r$.
### 2.7 Further tutorials
Except for the representative examples above, all standard features of the
code are demonstrated with inputs in the tutorial subdirectories. The inputs
included there at the time of writing are presented with brief descriptions in
Table. 1.
00HarmonicOsc1 | 1d-HO — discretization and eigenvalues
---|---
01HarmonicOsc2 | 2d-HO — combine two discretization axes
02HarmonicOscPolar | 3d-HO — polar coordinate, input of operators
03HydrogenPolar | 3d-hydrogen atom — plot densities
04Hyd1d | “1d-hydrogen atom” — model and numerics
05irECS | irECS for the 1d hydrogen atom
06TimeProp | Basics of time propagation
07HighHarmonicGeneration | 3.5 High harmonic spectra
08Hyd1dSpectrum | Photoelectron spectrum (1d)
09HydrogenSpectrum | 3d hydrogen: photo-electron spectrum at 20 nm
10IRSpectrum | 3d hydrogen: photo-electron spectrum at 800 nm
11shortPulseIR | 2.1 3d hydrogen: strong IR pulse
12IRlongPulse | 3d hydrogen: strong and long IR pulse
13Circular400nm | 3.4 Circular polarization, 400 nm wave length
14Circular400nmLong | Circular polarization, longer pulse
15Elliptic400nm | Elliptic polarization
15TayloredField | Two-color field at general polarization
16RotatingFrame | Rotating frame: photoemission at 400 nm
a16Circular800nm | Rotating frame: photoemission at 800 nm
17MixedGauge | Mixed gauge, field as tutorial 10
19TwoColorHarmonics | Calculation of harmonics, 2-color driver
20Helium2d | 2.4 Double-emission: 1+1-dimensional He
21Helium2dIR | 3.5 1+1-dimensional He, IR pulse
22Helium6d | Ground state of the He atom
23Helium3DSpectrum | 2.2 Double emission from He
51ParabolicHarmonic | Harmonic oscillator in parabolic coordinates
70RabittDelays | Attosecond RABITT delay calculation
90Floquet | 2.3 Floquet calculation.
110Pot2d | Variants of inputting 2d potentials
111Pot2dCO2 | A simple 2d CO2 model
220HybridSubspace | Hybrid of orbital and numerical basis
221CO2Free | 2.5 Strong-field-approximation type model
510offCenterScatter | 2.6 Combine spherical with off-center basis
Table 1: List of tutorials supplied with the code. Tutorials 00-10 provide a
basic introduction of inputs. Tutorials referenced in the present paper have
links to the respective sections.
## 3 Methods and general framework
### 3.1 irECS and tSurff
Strong field problems involve photo-emission all the way to total ionization
of the initial system. Pulse durations are long on the atomic time scale and
the momentum spectrum can be very broad. In this situation efficient
absorption of outgoing flux is provided by “infinite range exterior complex
scaling” (irECS) [1]. Complex scaling is an analytic continuation technique
for Schrödinger operators by which the continuous energy spectrum is rotated
around the single or multiple continuum thresholds into the lower complex
plane leading to damping of the continuous energies in forward time-evolution.
Bound state energies remain unaffected by the transformation and a new class
of discrete eigenvalues $W_{r}=E_{r}-i\Gamma_{r}/2$ appears that belong to
square-integrable resonance states at energies $E_{r}$ with decay widths
$\Gamma_{r}$.
The transformation is achieved by scaling the coordinates ${\vec{r}}\to
e^{i\theta}{\vec{r}}$. If the scaling is only applied outside a finite radius
$R_{0}$ one speaks of exterior complex scaling (ECS). As a consequence of
analyticity, exterior complex scaling leaves the solution in the region $r\leq
R_{0}$ strictly unchanged and allows direct physics interpretation — it is a
perfect absorber. The usual discretization errors arise but any dependence on
the complex scaling angle $\theta$ can be reduced to machine precision and in
that sense there are no adjustable parameters. The choice of $\theta$ does
matter for efficiency with $\theta\sim\pi/10-\pi/6$ usually giving best
results. A particularly efficient discretization is used in irECS with
exponentially damped polynomials in the scaled region $r>R_{0}$
$b_{k}(r)=L_{k}(r)e^{-\alpha r}.$ (23)
The $L_{k}$ is any set of orthogonal or sufficiently well-conditioned
polynomials such as Laguerre or Lagrange polynomials. In tRecX, we use for
$L_{k}$ the Lagrange polynomials at the Radau quadrature points for the weight
$e^{-2\alpha r}$, which is a DVR basis (Sec. 4.5.2). The rationale of this
discretization is to simultaneously accommodate short and long wave lengths:
short wave-lengths require finer sampling but get damped by complex scaling
over a short range. Long wave lengths penetrate deeper into the absorbing
region, but need fewer discretization functions over the range. This
discretization reduces the number of functions needed for absorption per
coordinate by factors $\lesssim 4$ from the already efficient absorption by
ECS, an advantage that plays out especially in higher dimensions.
The input of the irECS parameters for the example of the two radial
coordinates in the Helium problem of Sec. 2.2 is
⬇
Absorption: kind, axis, theta, upper
ECS,Rn1,0.3,20
ECS,Rn2,0.3,20
The name upper indicates the complex scaling radius $R_{0}$ for interval
$[R_{0},\infty)$. For Cartesian coordinates (Sec. 2.4) one also needs
absorption towards negative infinity $(-\infty,X_{-}]$ which defaults to
$X_{-}=-X_{+}$, but can be set independently by lower if so desired. The
exponentially damped functions are chosen with the axes, as shown in the
applications of Sec. 2.
The time-dependent surface flux (tSurff) method constructs spectra from the
flux through a surface at some sufficiently large radius $R_{c}$. It is
specific for the dipole approximation used in laser-ionization that momenta
will get modified also after they pass any remote surface. This can be taken
into account if one has an analytic solution for the time-evolution outside
$R_{c}$. With $R_{c}$ large enough for neglecting the potentials, these are
the Volkov solutions for electronic motion in a dipole field, here given in
velocity gauge and $\delta$-normalized (w.r.t. ${\vec{k}}$)
$\chi^{V}_{\vec{k}}({\vec{r}},t)=(2\pi)^{-3/2}e^{-i\Phi({\vec{k}},t)}e^{i{\vec{k}}{\vec{r}}},$
(24)
with the ${\vec{k}}$-dependent Volkov phases
$\Phi({\vec{k}},t)=\int_{0}^{t}d\tau[{\vec{k}}-{\vec{A}}(\tau)]^{2}/2$. With
these the complete spectral amplitude at a given ${\vec{k}}$ can be written as
an integral over the surface and time
$b({\vec{k}},T)=\int_{T_{0}}^{T}\langle\chi^{V}_{\vec{k}}(t)|[-\frac{1}{2}(-i\vec{\nabla}-{\vec{A}}(t))^{2},h(r-R_{c})]|\Psi(t)\rangle
dt$ (25)
As $h$ is the Heaviside function, the commutator leads to $\delta$-functions
at $r=R_{c}$ and the integral is only over the surface. $T_{0}$ is the begin
time of the pulse, and $T$ is some time large enough such that all relevant
flux has passed $R_{c}$. The scheme written here for the single-particle
emission can be generalized to the emission of two or more particles. In
tRecX, the general form is implemented, but in practice three-particle
emission has not been studied for reasons of problem size. Further details on
the tSurff method can be found in Refs. [2, 3].
tRecX computes values and derivatives of $\Psi(t)$ on the surface and saves
them to disk. In a second sweep, the integral (25) for the spectral amplitudes
$b({\vec{k}},T)$ is computed. For multi-particle spectra the process is
recursively iterated. One can specify the desired grid for ${\vec{k}}$ using
the input category Spectrum with a choice of points and optionally a momentum
range. If Spectrum is found, tRecX automatically initiates the amplitude
computation. Alternatively, one can restart tRecX with the output directory as
input and the momentum grid specified by command line parameters.
tSurff and irECS are the two defining techniques of tRecX which also have
phonetically inspired the name as tRecX=tSurff+irECS. An alternative
interpretation of the acronym is related to the recursive discretization
discussed below in Sec. 3.2.
#### 3.1.1 Discretization of complex scaled operators
The matrix representing a complex scaled Hamiltonian is non-hermitian and has
the desired complex eigenvalues. If one uses strictly real basis functions,
the matrix for the unscaled Hamiltonian will usually be real. In that case,
the Hamiltonian matrix will become complex symmetric upon scaling, i.e.
${\widehat{H}}_{ij}={\widehat{H}}_{ji}$ without complex conjugation. This is a
computationally useful property: the right eigenvectors of ${\widehat{H}}$ are
identical to the left-eigenvectors
${\widehat{H}}{\vec{C}}_{n}={\vec{C}}_{n}W_{n}\Leftrightarrow{\vec{C}}^{T}_{n}{\widehat{H}}=W_{n}{\vec{C}}^{T}_{n}$
(26)
and the eigenvectors ${\vec{C}}_{n}$ can be selected to be pseudo-orthonormal
${\vec{C}}_{m}^{T}{\vec{C}}_{n}=\delta_{mn}.$ (27)
A modification of that general approach is used for irECS: one starts from
strictly real basis functions on the rhs. , but in the scaled region $r\geq
R_{0}$ these are multiplied by a complex factor. This creates a complex-valued
discontinuity of the logarithmic derivative in rhs. basis at $R_{0}$, that is
required by ECS. The analogous, but complex conjugated discontinuity is
required for the lhs. basis. Mathematical and implementation details of this
realization of ECS, and its numerical advantages compared to commonly used
alternatives are discussed in Refs. [1, 6]. With the lhs. differing from the
rhs. basis, also the overlap matrix becomes complex symmetric rather than
hermitian. However, algorithms remain unchanged from the hermitian case, if
some care is taken to properly use transposed instead of adjoint matrices and
vectors. For example, a pseudo-Schmidt-orthonormalization can be performed if
only one replaces the standard scalar product with its complex-symmetric
counterpart ${\vec{C}}^{\dagger}{\vec{C}}\to{\vec{C}}^{T}{\vec{C}}$, and even
a pseudo-Cholesky decomposition exists and is used. In tRecX, a keyword pseudo
indicates that the unconjugated rather than standard operation is performed.
The above approach keeps its simplicity only, when the original Hamiltonian
matrix w.r.t. the chosen basis is real. As the resulting complex symmetry of
the complex scaled problem simplifies and accelerates algorithms, an effort
should be made to find such a representation. In fact, at present tRecX does
not reliably support cases, where the original unscaled matrix would be non-
hermitian. While non-real hermitian matrices cannot ruled out in general, in
all applications shown here matrices are indeed real. For example, in the
Floquet problem a real matrix is obtained by defining a factor $i^{n}$ into
the basis of the $n$’th block, which results in the overall hermitian
definition for the interaction as (<delta[1]>-<delta[-1]>)<<D/DZ>> in Sec.
2.3.
### 3.2 Recursive discretization
The organization of operators, wave-functions, expansion coefficients, basis
sets, and multi-indices in trees is central to the design of tRecX. Trees are
used to recursively generate the objects and in virtually all other
algorithms. This makes the code largely independent of specific coordinate
systems and dimensions and allows to handle all multi-dimensional expansions
of the examples above within the same scheme. Program uniformity is ensured by
deriving all trees from a template abstract base class Tree, Sec. 4.2.
#### 3.2.1 Wave function expansion
We denote the $L$-tuple of all coordinates by
$Q=Q^{0}=(q^{0},q^{1},\ldots,q^{L-1})$ and the sub-tuple starting at $l$ by
$Q^{l}=(q^{l},\ldots,q^{L-1})$. There is some flexibility as to what is
considered as a “coordinate”: on the one hand, the finite-element index $n$ in
Eq. (6) can assume the role of a coordinate, but also all three spatial
coordinates ${\vec{r}}$ of the orbitals $\Phi_{\alpha}({\vec{r}})$ in the
hybrid discretization of Sec. 2.5 can be subsumed in a single
$q^{l}:={\vec{r}}$.
One or several sets of basis functions
$\underline{b}^{J_{l}}=\left(b^{J_{l}}_{0}(q^{l}),b^{J_{l}}_{1}(q^{l}),\ldots\right)$
are defined for a coordinate $q^{l}$, where we arranged the set as a row
vector, indicated by the underscore. The multi-index
$J_{l}=(j_{0},\ldots,j_{l-1})$ unites the labels $j_{k}$ of all functions
$b^{J_{k}}_{j_{k}}(q^{k}),k<l$ preceding the basis set
$\underline{b}^{J_{l}}(q^{l})$. In that way $\underline{b}^{J_{l}}$ can be
made to depend on the sequence of basis functions preceding it in the
coordinate hierarchy. The tuple $J=J_{L}$ is the complete set of indices for a
single expansion coefficient $C^{J}=C^{j_{0},j_{1},\ldots,j_{L-1}}$. The basis
function matching $C^{J}$ is
$B_{J}(Q)=\prod_{l=0}^{L-1}b^{J_{l}}_{j_{l}}(q^{l})$. As with coordinates, we
use the word “basis function” in a rather wide sense: basis functions in the
proper sense are trigonometric functions, associated Legendre functions, or
the Lagrange polynomials for FE-DVR discretization, etc. but we also consider
Kronecker $\delta$: $b_{j}(n)=\delta_{jn}$ as a “basis function” for a
discrete index $n$, e.g. the photon index in the Floquet model of Sec. 2.3. By
that principle all discretization methods are treated uniformly in tRecX.
All basis expansions discussed in Sec. 2 fit into the scheme, e.g. Eqs. (6)
and (11). It is important to note that, while individual functions $B_{J}$ are
products of functions of the coordinates, the total basis $\underline{B}$
(again considered as a row vector of $B_{J}$’s) is not a product basis
because of the dependence of factor functions $b^{J_{l}}_{k}$ on the complete
preceding hierarchy. A well known set of two-dimensional functions with this
structure are the spherical harmonics $Y^{m}_{l}(\phi,\theta)\propto
e^{im\phi}P^{|m|}_{l}(\cos\theta)$. Another example for the hierarchical
dependence in the products is the implementation of angular constraints as
discussed in Sec. 2.2.
With the above definitions, the wave function for a given tuple of coordinates
$Q^{l}$ is expanded recursively as
$\Psi^{J_{l}}(Q^{l})=\sum_{j=0}^{K_{J_{l}}-1}b^{J_{l}}_{j}(q^{l})\Psi^{J_{l}j}(Q^{l+1}),$
(28)
where we use the notation $J_{l}j:=(j_{0},\ldots,j_{l-1},j)$.
$\Psi(Q):=\Psi^{J_{0}}(Q^{0})$ is the complete wave function at the point $Q$
and $\Psi^{J_{L}}=:C^{J_{L}}=C^{j_{0},j_{1},\ldots,j_{L-1}}$ is a vector of
length one, i.e. the complex valued expansion coefficient at the multi-index
$J_{l}=(j_{0},j_{1},\ldots,j_{L-1})$. The recursion (28) defines the
discretization as a tree whose nodes are labeled by an index $J_{l}$, as in
Fig. 6. The subtree starting at $J_{l}$ defines a multi-coordinate wave
function component $\Psi^{J_{l}}(Q^{l})$. Every node hosts a basis
$\underline{b}^{J_{l}}=\left(b^{J_{l}}_{j}(q^{l}),\,j=0,\ldots,K_{J_{l}}-1\right)$
and each function of the basis connects to one branch of the node. Both, the
number $K_{J_{l}}$ of basis functions $b^{J_{l}}_{j}$ and their kind can be
different on every node, as, e.g., for the associated Legendre functions
$b^{J_{2}}_{l-|m|}=P^{|m|}_{l}$ in Fig. 6. Usually basis sets at given level
$l$ have equal coordinate $q^{l}$. An exception are hybrid discretizations as
in Secs. 2.5 and 2.6. In Fig. 6, on level $l=1$ the node at $J_{1}=(0)$ hosts
three-dimensional eigenfunctions
$\underline{b}^{0}=\left(\Phi_{0}({\vec{r}}).\Phi_{1}({\vec{r}})\right)$,
while its neighbor at $J_{1}=(1)$ has the node-basis
$\underline{b}^{1}=\left(1,e^{-i\phi},e^{i\phi}\right)$.
Figure 6: Index tree for the hybrid discretization of Sec. 2.5 (abbreviated).
Axis names are indicated in yellow. Nodes on the respective levels are labeled
by $J_{l}$, factor basis functions connect a node to the next-lower level.
#### 3.2.2 Coefficients and operator matrices
The recursive hierarchy is also reflected in the expansion coefficients. Every
subtree wave function $\Psi^{J_{l}}$ is associated with a vector of expansion
coefficients $\overline{C}^{J_{l}}$. The overline indicates a column vector
and emphasizes its duality to the basis $\underline{B}^{J_{l}}$.
$\overline{C}^{J_{l}}$ is the direct sum of the coefficient vectors at $l+1$:
$\overline{C}^{J_{l}}=\begin{pmatrix}\overline{C}^{J_{l}0}\\\
\overline{C}^{J_{l}1}\\\ \vdots\\\
\overline{C}^{J_{l}K_{J_{l}}}\end{pmatrix}.$ (29)
The recursion can be phrased as “a coefficient vector is a vector of
coefficient vectors”.
Finally, the recursive hierarchy of the overall multi-dimensional basis set
belonging to $\overline{C}$ can be exploited for the computation of the
operator matrices and in matrix-vector multiplication. The row-vector of
multi-dimensional basis functions $\underline{B}^{J_{l}}(Q^{l})$ for the
subtree $J_{l}$ is defined recursively as
$\underline{B}^{J_{l}}(Q^{l})=\left(b^{J_{l}}_{0}\underline{B}^{J_{l}0},\,b^{J_{l}}_{1}\underline{B}^{J_{l}1},\ldots,\,b^{J_{l}}_{K_{J_{l}}-1}\underline{B}^{J_{l}K_{J_{l}}-1}\right).$
(30)
The recursion starts from $B^{J_{L}}\equiv 1\forall
J_{L}=(j_{0},j_{1},\ldots,j_{L-1})$ and $\underline{B}^{J_{0}}=\underline{B}$
is the complete multi-dimensional basis. The full Hamiltonian matrix can be
denoted as ${\widehat{H}}=\langle\underline{B}|H|\underline{B}\rangle$ if we
interpret $\langle\underline{B}|$ as a column vector of bra-functions. The
full wave function is $\Psi=\underline{B}\overline{C}$.
Here one can clearly see that the basis Eq. (30) reduces to a tensor product,
only if the bases at all subnodes of $J_{l}$ are equal,
$\underline{B}^{J_{l}j}\equiv\underline{B}^{J_{l}0},\forall j$:
$\underline{B}^{J_{l}}=\left(b^{J_{l}}_{0}\underline{B}^{J_{l}0},\,b^{J_{l}}_{1}\underline{B}^{J_{l}0},\ldots,\,b^{J_{l}}_{K_{J_{l}}-1}\underline{B}^{J_{l}0}\right)=\underline{b}^{J_{l}}\otimes\underline{B}^{J_{l}0}.$
(31)
For each pair $I_{l},J_{l}$ of index subtrees we define a sub-block
${\widehat{H}}^{I_{j}J_{j}}$ of ${\widehat{H}}={\widehat{H}}^{I_{0},J_{0}}$,
where the $1\times 1$ blocks ${\widehat{H}}^{I_{L}J_{L}}$ are the matrix
elements. The recursive structure of the coefficients induces a recursive
block structure of the matrix as
${\widehat{H}}^{I_{l},J_{l}}=\begin{pmatrix}{\widehat{H}}^{I_{l}0,J_{l}0}&{\widehat{H}}^{I_{l}0,J_{l}1}&\cdots&{\widehat{H}}^{I_{l}0,J_{l}K_{J_{l}}-1}\\\
{\widehat{H}}^{I_{l}1,J_{l}0}&{\widehat{H}}^{I_{l}1,J_{l}1}&\cdots&{\widehat{H}}^{I_{l}1,J_{l}K_{J_{l}}-1}\\\
&\vdots&&\vdots&\\\
{\widehat{H}}^{I_{l}K_{I_{l}}-1,J_{l}0}&{\widehat{H}}^{I_{l}K_{I_{l}}-1,J_{l}1}&\cdots&\\\
\end{pmatrix}.$ (32)
This structure can be exploited for construction of the operator matrices and
also for simple representation of block-sparsity, e.g. in presence of
selection rules. One can phrase this recursively as “an operator matrix is a
matrix of operator matrices”.
If the operator $H$ is a tensor product
$\displaystyle H$ $\displaystyle=$ $\displaystyle h_{0}\otimes
h_{1}\otimes\ldots\otimes h_{L-1}=:h_{0}\otimes\ldots\otimes h_{l}\otimes
H_{l+1}$ (33) $\displaystyle H_{l}:=h_{l}\otimes H_{l+1},\quad
l=0,1,\ldots,L-1$
the operator matrix for $I_{l},J_{l}$ on level $l$ can be assembled from all
blocks at the next-lower level $(I_{l+1},J_{l+1})=(I_{l}i,J_{l}j)$ as
$\left[\langle\underline{B}^{I_{l}}|H_{l}|\underline{B}^{J_{l}}\rangle\right]_{ij}=\left[{\widehat{H}}^{I_{l},J_{l}}\right]_{ij}=\langle
b^{I_{l}}_{i}|h_{l}|b^{J_{l}}_{j}\rangle{\widehat{H}}^{I_{l}i,J_{l}j},$ (34)
where $[\ldots]_{ij}$ designates the $ij$-block of
${\widehat{H}}^{I_{l},J_{l}}$ and the indices $i$ and $j$ range from 0 to
$K_{I_{l}}-1$ and $K_{J_{l}}-1$, respectively. In practice, the matrix is not
usually constructed explicitly.
The tensor-product form of $H_{l}$ implies a tensor-product form of
${\widehat{H}}^{I_{l}J_{l}}=\langle\underline{B}^{I_{l}}|H_{l}|\underline{B}^{J_{l}}\rangle$,
only if also $\underline{B}^{I_{l}}$ and $\underline{B}^{J_{l}}$ are strict
tensor products as in Eq. (31), in which case Eq. (34) reduces to
${\widehat{H}}^{I_{l}J_{l}}=\langle\underline{b}^{I_{l}}|h_{l}|\underline{b}^{J_{l}}\rangle\otimes{\widehat{H}}^{I_{l}0,J_{l}0}.$
(35)
Yet, also when the matrix is not a tensor product, the recursive structure Eq.
(34) largely preserves the computational advantages of tensor products in
terms of the floating point count and, to a lesser degree, data compression. A
typical algorithm for matrix-vector multiplication is discussed in Sec. 3.3.
Many operators in physics can be written as short sums of tensor products and
allow efficient and transparent computation of the matrices using this scheme.
In some cases it is advantageous to exploit the recursive structure for
applying the operator matrices to coefficient vectors, as for the radial
kinetic energy in two-particle problems. If the matrix is very block-sparse,
as e.g. in case of dipole selection rules, direct block-wise application
performs better. The choice between these options is made automatically in
tRecX based on the actual operator, using non-rigorous heuristics. When
operators do not have tensor-product structure, such as electron repulsion
$|{\vec{r}}_{1}-{\vec{r}}_{2}|^{-1}$ in the Helium atom, the recursive scheme
is still used in tRecX for bookkeeping and for ensuring a uniform construction
of operator matrices.
For numerical efficiency, operators are not usually expanded to the lowest
level, but rather recursion is terminated at a “floor”level $F\leq L$ such
that the smallest operator block has typical sizes of $10\times 10\sim
400\times 400$, depending on the actual problem and choice of the
discretization. An example for large floor blocks is for the Helium atom, Sec.
2.2. There the floor level is put to the two-dimensional finite element
patches $[r_{n1},r_{n+1,1}]\times[r_{m1},r_{m+1,1}]$, with a typical number of
$K=20$ functions for each radial coordinate. Operators on the floor level are
usually not represented by full matrices. In the Helium Hamiltonian Eq. (10)
the first two terms are trivial tensor products. With basis size $K$ on both
coordinates $r_{1},r_{2}$, the operations count of matrix-vector multiplies is
$\mathcal{O}(K^{3})$ when one exploits the tensor-product form, rather than
$\mathcal{O}(K^{4})$ for general full matrix. Such structures are
automatically recognized by tRecX and implemented using derived classes of an
abstract base class OperatorFloor. Electron repulsion on this lowest level
requires application of matrices that are diagonal for each multipole term
with matrix-vector operations count $\mathcal{O}(K^{2})$. As mentioned above
and discussed in Ref. [8], this is not exact, but turns out to be an excellent
approximation. The high computational cost of electron repulsion arises not
from the radial part, but from the significant fill-in of the block sparse
matrix by widely coupling the angular momenta of the two individual electrons.
This can be controlled to some extent by truncating the multipole expansion at
less than maximal order (input OperatorFloorEE:lambdaMax).
The recursive scheme for operators and coefficients translates into simple and
transparent algorithms for matrix setup and matrix-vector operations, which
are implemented in the C++ class OperatorTree discussed in Sec. 4.3.
### 3.3 Quadratures
The code makes extensive use of numerical quadrature. This is so, by
definition, for FE-DVR basis functions, but we apply it throughout: integrals
involving trigonometric functions, the associated Legendre functions
$P^{|m|}_{l}$ or the general multi-dimensional basis functions of Sec. 2.6 are
usually all computed by quadratures. Wherever possible, exact quadrature is
used. Apart from providing a uniform and comparatively error-safe
computational scheme in the code, exact quadratures are often numerically more
stable the evaluation of complicated algebraic expressions for analytic
integrals.
The tree-structure of the expansion provides for efficient conversion to and
from product grids that tRecX uses in multi-dimensional quadratures. The wave-
function value at one point
$Q_{A_{0}}=(q^{0}_{\alpha_{0}},\ldots,q^{L-1}_{\alpha_{L-1}})$ of an
$L$-dimensional product grid is
$\displaystyle\Psi$
$\displaystyle(q^{0}_{\alpha_{0}},\ldots,q^{L-1}_{\alpha_{L-1}})=$
$\displaystyle\sum_{j_{0}=0}^{K_{J_{0}}}b^{J_{0}}_{j_{0}}(q^{0}_{\alpha_{0}})\sum_{j_{1}=0}^{K_{J_{1}}}b^{J_{1}}_{j_{1}}(q^{1}_{\alpha_{1}})\cdots\sum_{j_{L-1}=0}^{K_{J_{L-1}}}b^{J_{L-1}}_{j_{L-1}}(q^{L-1}_{\alpha_{L-1}})C^{j_{0},j_{1},\ldots,j_{L-1}}$
(36)
We abbreviate the matrix of basis function values at the grid points as
$b^{J_{l}}_{j_{l}}(q^{l}_{\alpha_{l}})=:b^{J_{l}}_{\alpha_{l}j_{l}}$ and
introduce the intermediate vectors
$\displaystyle\overline{G}_{A_{l}}^{J_{l}}=\overline{G}^{J_{l}}_{\alpha_{l}A_{l+1}}$
$\displaystyle=$
$\displaystyle\sum_{j_{l}=0}^{K_{J_{l}}}b^{J_{l}}_{\alpha_{l}j_{l}}\cdots\sum_{j_{L-1}=0}^{K_{J_{L-1}}}b^{J_{L-1}}_{\alpha_{L-1}j_{L-1}}C^{j_{0},j_{1},\ldots,j_{L-1}}$
(37)
$\displaystyle=\sum_{j_{l}=0}^{K_{J_{l}}}b^{J_{l}}_{\alpha_{l}j_{l}}\overline{G}_{A_{l+1}}^{J_{l}j_{l}}\quad\forall\alpha_{l}$
with $A_{l}=(\alpha_{l},\ldots,\alpha_{L-1})$ and the previously defined
$J_{l}=(j_{0},\ldots,j_{l-1})$. The last equality defines a recursion starting
from coefficients $C^{J_{L}}=\overline{G}_{A_{L}}^{J_{L}}$ and ending at the
vector $\overline{G}_{A_{0}}^{J_{0}}$ of the values of $\Psi$ at all grid
points.
The analogous recursion can be set up for the back-transformation from grid to
basis functions. With quadrature weights
$w^{l}_{\alpha_{l}},\alpha_{l}=0,\ldots,S_{l}-1$ at the grid points
$q^{l}_{\alpha_{l}}$ one computes the overlap matrices for coordinate $q^{l}$
at the nodes $J_{l}$
$s^{J_{l}}_{ij}=\sum_{\alpha_{l}}(b^{J_{l}})^{\dagger}_{i\alpha_{l}}w^{l}_{\alpha_{l}}b^{J_{l}}_{\alpha_{l}j}$
and from that the factor matrices for back-transformation
$d^{J_{l}}_{j_{l}\alpha_{l}}=\left[(s^{J_{l}})^{-1}b^{J_{l}\dagger}\right]_{j_{l}\alpha_{l}}w^{l}_{\alpha_{l}}.$
On complex-scaled coordinates, the adjoint $b^{J_{l}\dagger}$ must be replaced
by the transpose $b^{J_{l}T}$, see Sec. 3.1.1. The recursion for back-
transformation from $\overline{G}=\overline{G}_{A_{0}}^{J_{0}}$ to
$C^{J_{L}}=\overline{G}_{A_{L}}^{J_{L}}$ proceeds by
$\overline{G}_{A_{l+1}}^{J_{l+1}}=\overline{G}^{J_{l}j_{l}}_{A_{l+1}}=\sum_{\alpha_{l}=0}^{K^{J_{l}}_{l}-1}d^{J_{l}}_{j_{l}\alpha_{l}}\overline{G}_{\alpha_{l}A_{l+1}}^{J_{l}}\quad\forall
j_{l}$ (38)
Both recursions (37) and (38) share the same structure and are implemented in
a class OperatorMap, Sec. 4.3.2.
The recursive algorithm for the transformation to a product grid is very
similar to the algorithm for applying a tensor product of operators to a
vector and it has the same favorable operations count. For an ideal quadrature
grid with $S_{l}=K_{J_{l}}$, the transformation maintains size
$\text{len}(\overline{C})=\text{len}(\overline{G})$ and the operations count
for the transformation $\overline{C}\to\overline{G}$ is
$\left(\sum_{l=0}^{L-1}K_{J_{l}}\right)\times\text{len}(\overline{C}).$ (39)
The computational gain increases exponentially with dimension $L$ comparing to
direct application of the full transformation matrix
$(\prod_{l=0}^{L-1}K_{J_{l}})\times\text{len}(\overline{C})$. In practice the
number of quadrature points often exceeds the number of basis functions,
$S_{l}>K_{j_{l}}$, with a corresponding increase of operations count. One
prominent example are the associated Legendre functions
$P^{|m|}_{l}(\eta),l\leq L$ where we use a Legendre quadrature grid
$\eta_{k},k=0,\ldots,L$ which is shared among all $m$ and is exact for all
overlaps, but inflates the vector length from $L-|m|+1$ to $L+1$. These are
more points than, e.g., in a Lebedev quadrature grid [25], but the product
structure is maintained and with it the efficient algorithm for transformation
to the grid.
For simplicity we have treated the case where all coordinates are transformed
to a grid, but obviously transformations can be limited to a given subset of
the coordinates, as needed. In tRecX, the creation of product grids and
transformations to and from them are handled by a specialized class
DiscretizationGrid, see Sec. 4.
### 3.4 Adaptive features
In problems that are strongly driven by the external field, time step size and
required basis size can change significantly as the system evolves. Step sizes
decrease near field peaks and increase near field nodes. By default, the code
automatically controls the size of the time steps based on a standard single-
to-double step estimate, which has an overhead slightly above 50%. We have
decided to use this simple but universal control algorithm, which only
requires a well-defined consistency order of the underlying time-stepper, in
order to maintain flexibility in choosing the time-stepper. In strongly driven
systems, gain by adaptive step size can be up to a factor of 2 compared to a
step fixed at the maximal stable size. The maybe more important advantage of
step size control in tRecX is that well-defined accuracies are achieved
without the need of careful time-step adjustment. At the end of time
propagation average step size and its variance are printed, based on which one
can fix the step size once a system’s behavior in a given parameter range and
discretization is known.
A typical phenomenon of strong-field physics is a large increase in angular
momenta as the field ramps up. After the end of the pulse, those angular
momentum components gradually decay and the operator does not need to be
applied to them. Also, in absence of the pulse the interaction part of the
operator is zero. These developments are monitored in the code and operators
are only applied in regions where there is non-negligible contribution to the
time-evolution. Control is achieved by estimating the contribution to the
derivative vector based on the norm of the floor operator block
$||{\widehat{H}}^{I_{F}J_{F}}||$, which is precomputed at setup, and a norm of
the rhs. vector $\overline{C}_{J_{F}}$. As the vector norm needs to be
evaluated at every time-step, we use the simple estimate
$||\overline{C}_{J_{F}}||_{a}:=\max_{i}[|\Re(C_{i})|+|\Im(C_{i})|]$. If the
contribution to the total vector norm is below a threshold
$||{\widehat{H}}^{I_{F}J_{F}}||||\overline{C}_{J_{F}}||_{a}\leq\epsilon_{th}$
application of the block is skipped. The procedure requires some care with
choosing $\epsilon_{th}$, but can speed up computations by factors $\lesssim
2$ without loss of accuracy. Application is demonstrated in tutorial/13. The
code will print some advice when $\epsilon_{th}$ may have been chosen too
large or too small, but at present heuristics for the choice of
$\epsilon_{th}$ is incomplete. By default $\epsilon_{th}=0$, i.e. blocks are
only skipped when the operator block or the vector become exactly zero, which
happens, for example, after the end of a laser pulse with strictly finite
duration.
### 3.5 Control of stiffness
For time-propagation at present only explicit methods are used, whose
efficiency notoriously deteriorates as the norm of the operator matrix
increases. The main origin of large norm in Schrödinger-like problems is the
Laplacian, whose matrix norm grows as $p_{\max{}}^{2}\sim\delta x^{-2}$, where
$p_{\max{}}$ and $\delta x$ are the characteristic scales of maximal momentum
and spatial resolution, respectively. Usually one does not manage to restrict
the momenta in the discretization to the physically relevant level and
spurious, very high eigenvalues appear that can dramatically slow down
explicit time-steps to the level of numerical breakdown of the propagation.
This stiffness problem can be fixed, if one manages to remove spurious
eigenvalues from the operators. In tRecX, high-lying eigenvalues of the field-
free Hamiltonian are suppressed by spectral projections. In the simplest form
one replaces the full Hamiltonian matrix with a projected one
${\widehat{H}}(t)\to(\mathbf{1}-{\widehat{P}}){\widehat{H}}(t)(\mathbf{1}-{\widehat{P}}),\qquad{\widehat{P}}=\sum_{i}|i\rangle\langle
i|,$ (40)
where the $|i\rangle$ are orthonormal eigenvectors for large eigenvalues of
the field-free Hamiltonian.
With more challenging Hamiltonians like for the Helium atom or molecular
systems, the full field-free Hamiltonian has many high-lying spurious states,
they are expensive to compute, and application of the projection becomes
costly. In such cases one can use for the $|i\rangle$ eigenvectors of a
different operator, for example the Laplacian. Eigenvectors of the Laplacian
are sparse due to rotational symmetry and in case of multi-particle systems
they can be given as tensor products of single-electron vectors. This renders
calculation of the eigenvectors as well as application of the projection
computationally cheap.
The cutoff energy for removal of high-lying states is characteristically set
around 100 a.u. . This is far larger than the actual energy scale of typically
$\lesssim 10\,a.u.\ $ However, choosing the threshold that low would
compromise the results and raises the cost of applying the projection to the
point where no compute time is gained, in spite of the fact that step-size
increases inversely proportional to the cutoff energy. The energy cutoff is
first introduced in tutorial/07. Examples for using the Laplacian instead of
the full Hamiltonian for projecting are in tutorial/21 and 23.
Comparing to an outright spectral representation of the operators, removing a
small number of outlier eigenvalues from a local representation maintains all
sparsity deriving from locality of operators represented in a local basis. The
cost of removal remains low because the number of removed eigenvalues is small
compared to the basis size and the vectors may have tensor product form, as
for the Laplacian of the He atom.
### 3.6 Parallelization
tRecX is parallelized using MPI, but it will also compile without MPI, if no
MPI library is detected by Cmake. The code is aware of hardware hierarchy in
that it can distinguish between “compute nodes” assumed connected through
switches, “boards” connected by a bus, and “CPUs” assumed to have fast shared
memory access. This hierarchy, although present in the code, is not at present
exploited by the distribution algorithm. For local operators, communication
between non-overlapping elements of the FE-DVR is low. Operator locality
between elements is detected during setup and taken into account by the
default distribution algorithms for the respective coordinate systems.
The finest MPI grains are the OperatorFloor blocks
${\widehat{H}}^{I_{F}J_{F}}$. These operate between subsections of the
coefficient vectors $\overline{C}_{I_{F}}$ and $\overline{C}_{J_{F}}$ with
typical dimensions 10 to 400. The operator blocks can be distributed
arbitrarily across all MPI nodes, but communication overhead must be taken
into consideration. The corresponding class OperatorFloor has a member cost()
that determines the CPU load for its application by self-measurement during
setup. A heuristic algorithm uses these numbers to create a load-balanced
distribution of the operator. For containing communication cost, care is taken
to arrange blocks into groups that share either $I_{F}$ or $J_{F}$. At least
one of the respective sections of coefficient vectors $\overline{C}_{J_{F}}$
or $\overline{C}_{J_{F}}$ reside on the same parallel process, which then
“owns” the corresponding $I_{F}$ or $J_{F}$.
Actual communication cost is not measured by the code. Rather, it assumes
there is a sorting of the $\overline{C}_{I_{F}}$ such that compunction is
dominantly short range, as e.g. sorting by increasing angular momenta in case
of dipole interactions. Then neighboring $\overline{C}_{I_{F}}$’s are
preferably assigned to the same thread. The default for this sorting is the
sequence how the Axis:name are input. For some coordinate systems this is
overridden by internal defaults and the user can in turn can override by the
input Parallel:sort. The sorting actually used is shown in the output.
#### 3.6.1 Scaling
Problems that can be solved with tRecX vary widely in structure and also in
the methods employed. Scaling behavior strongly depends on these choices.
Memory is, in general, not a limiting factor for tRecX calculations.
Parallelization strategy focuses on large problems where run times in
sequential mode would be days or weeks, while little effort has been made to
boost parallelization for small problems with runtimes on the scale of
minutes. Into the latter category fall many problems in tRecX that would be on
time scales of hours with more traditional approaches. These gains in program
efficiency are through complex features such as exploiting tensor products and
block-sparsity, by stiffness control, the use of high order methods, or the
tSurff box-size reduction. All these features, while at times dramatically
reducing compute times and problem sizes, tend to lead to coarser graining and
enhanced communication, which necessarily deteriorates scalability.
Specifically the haCC method is inherently non-local with large communication
and therefore mostly restricted to shared memory use.
Most problems treated with tRecX are best solved on small parallel machines in
the range from 4 to 64 cores. Only large problems such as the double-
ionization of the Helium atom can profit from more extensive parallelization.
Fig. 7 shows the scaling behavior for fixed-size problems (“strong scaling”).
The two examples are hydrogen in an IR field discussed in Sec. 2.1 and a
Helium atom computation with 20 $l$-functions, $m=-1,0,1$ and 91 radial
functions for each electron, resulting in total basis size of $3\times
10^{5}$. Computations were performed at the LMU Theory machine KCS hosted at
the Leibnitz Rechenzentrum (LRZ), which consists of compute nodes connected by
infiniband and dual boards with 2$\times$16 cores on each node.
Parallelization gains can be seen up to 256 cores. Scaling remains away from
linear and as always in this situation one has to weigh time gains for
individual computations against overall throughput for multiple runs.
Figure 7: Scaling of tRecX for a double-ionization calculation of the Helium
atom at total basis size of $3\times 10^{5}$ (dots) and single-electron
problem discussed in Sec. 2.1 at basis size $1460$ (squares).
## 4 Main classes
Here we discuss the classes that form the functional and conceptional backbone
of tRecX. A complete listing of all classes is provided by the code’s Doxygen
documentation. In general, many classes in the code have a .write() member for
dumping to file and a matching .read() or constructor for recovery from file.
Mostly for debugging purposes, there is usually a .str() member that returns a
human-readable string. Also, for critical classes, there are test() functions
that provide cross-checks and usage examples. A key role is played by the
abstract template class Tree.
### 4.1 Index class
The C++ class Index represents the complete recursive basis tree defined
through (30). The class and its member data are declared as
⬇
class Index: public Tree<Index> {
mutable uint32_t _size;
uint16_t _indexBas;
uint8_t _indexAx;
char _indexKind;
....
}
These member data refer to the given node and are the only index-specific data
of the tree. The complete tree-structure, such as tree iterators, pruning,
transposition, and other tree transformations are implemented in the template
class Tree, Sec. 4.2, which is used for all tree classes of tRecX. The Index
data is squeezed into 8 bytes in an attempt to minimize storage, as Index
trees can become very large. This limits the number of different single-level
basis sets $\underline{b}$ that are pointed to by _indexBas to $2^{15}$. In
practice, also in very large computations only a few tens of different bases
appear. Whenever a tensor product basis is used, the same basis re-appears at
many nodes and has the same _indexBas, as for example the product bases for
the $r_{1},r_{2}$-discretization. The range of the axis pointer _indexAx is
$2^{7}$, which is sufficient as the number of axes is intimately related to
the dimension of the problem and hardly ever exceeds 10. _size gives the
length of $\overline{C}_{J}$ at the node. This information is redundant, but
is cached here for fast access, and similarly _indexKind is cached information
about the node’s function and position within the tree.
The Index class, as one of the code’s oldest classes, is burdened by legacy
code. In order to disentangle the current from legacy code, primary
construction is through an auxiliary derived class IndexNew which takes an
AxisTree as its input. AxisTree, in turn, reflects the definitions read from
input. The standard AxisTree is trivial with a single branch per node,
equivalent to a vector. Only when hybrid discretizations are used, as for the
molecular problem (Sec. 2.5) and for off-centers bases (Sec. 2.6), the tree
becomes non-trivial.
An Index contains the complete information about the basis $\underline{B}$,
Eq. (30). It also has a member function overlap() that returns a pointer to
$\langle\underline{B}^{J_{l}}|\underline{B}^{J_{l}}\rangle$ as long as this is
a meaningful entity for a single $\underline{B}^{J_{l}}$, i.e. when the
subspace on level $l$ does not have overlap with any other subspace on the
same level $\langle\underline{B}^{I_{l}}|\underline{B}^{J_{l}}\rangle=0$ for
$I_{l}\neq J_{l}$.
#### 4.1.1 Special Index constructors
Figure 8: Hierarchy of Index classes, sec 4.1.1.
There is a number of specialized constructors for indices, see Fig. 8. For
disentangling from the legacy code, these are usually given as the constructor
of a derived class. One useful constructor is IndexG for transforming from
basis functions to representation by grid values. The grid can be equidistant,
useful for plotting, or a quadrature grid, which is convenient for various
forms of basis transformations and quadratures. The tree-structure of the
basis allows to perform these transformations computationally efficiently, cf.
Sec. 3.3. The IndexS represents value and radial derivative at the tSurff
surface. It is constructed from a standard Index by specifying the coordinate
axis name and the surface radius $R_{s}$. IndexProd constructs a new index
tree as the tensor product of two Index trees. IndexQuot forms the “quotient”
of an Index full by a “denominator” Index den by eliminating from full all
basis levels that appear in den, ensuring consistency of the result. This
allows to extract, e.g., a single-electron factor from a two-electron basis.
Maps to these derived indices are generated automatically by a class
OperatorMap (see below) and are not invertible in general
#### 4.1.2 Discretization classes
Various classes derived form class Discretization are wrappers around specific
indices and serve as an interface for Index construction. Examples are
DiscretizationGrid (Sec. 3.3) and DiscretizationTsurffSpectra (Sec. 4.3.2).
An important derived class is DiscretizationSpectral, which constructs all or
a selected part of the eigenvalues and eigenvectors of any diagonalizable
operator matrix ${\widehat{A}}$ and presents them in the form of a diagonal
operator ${\widehat{d}}_{A}$ (class OperatorDiagonal). Further it contains
transformations ${\widehat{U}}$ and ${\widehat{V}}$ (class OperatorMap) from
and to the original basis, respectively. Note that in general
${\widehat{U}}\neq{\widehat{V}}^{\dagger}$ as the original basis as a rule is
not orthonormal and also ${\widehat{A}}$ may not be hermitian. Block-diagonal
structure of the original operator is recognized and translated into block-
diagonal transformation. Arbitrary functions of the eigenvalues can be formed
using OperatorDiagonal. E.g. one can form
$\exp(-it{\widehat{A}})={\widehat{U}}\exp(-i{\widehat{d}}_{A}){\widehat{V}}$
for time-integration of small problems, or, similarly, to implement rotations
in a spherical harmonic basis. The principal use in tRecX is in stiffness
control, Sec. 3.5.
For operators of the special form
${H}={H}_{I}\otimes\mathbf{1}+\mathbf{1}\otimes{H}_{J}$ the class
DiscretizationSpectralProduct constructs a spectral representation taking full
advantage of the fact that there is an eigenbasis of ${H}$ in the form of a
tensor product of eigenbases of ${H}_{I}$ and ${H}_{J}$. Transformations to
and from that spectral representation have tensor product form. The class can
also be used when the basis $\underline{B}$ is not tensor product, but is
related to a tensor product $\underline{B}^{I}\otimes\underline{B}^{J}$ by a
constraint as in Sec. 2.2.1.
### 4.2 The template class Tree
All trees in the code are derived from class Tree by the ”curiously recursive
template pattern” exemplified in class Index:public Tree<Index>. We list a few
key features of this class, but refer to the documented code for the complete
definition and functionality.
Tree has the private data
⬇
template <typename T> class Tree {
const T* _parent;
vector<T*> * _child;
...
that point to a node’s parent and all its children and are accessed through
member functions parent() and child(int j), respectively. Iterators along
various paths through the tree are provided. For legacy reasons these are not
realized in the standard C++ iterator syntax, but rather by member functions
returning pointers to the incremented node. The most important iterators are
descend() for descending from a node to its left-most branch and
nodeRight(Origin) the next node to the right within the subtree originating at
node Origin.
Nodes without branches are called leafs. A standard sorting of leafs is by
their position along the lower edge of the tree. The functions firstLeaf() and
nextLeaf() return leftmost leaf descending from a given node and the iterator
through the leafs. Note that in general nextLeaf() is not equivalent to
nodeRight() as a tree’s lower edge does not need to remain at the same level
depth, as in the example of Fig. 6 The index of a node is returned by
vector<int> index(). For class Index, this is exactly the tuple $J_{l}$
defined in Sec. 3.2.
Functions to add and remove branches include childAdd(T* C) and childPop().
For re-sorting trees there is a permute(...), which takes a permutation of the
tree levels as its argument and returns at tree with the levels permuted. A
typical case would be the transposition of tensor indices. With non-tensor
objects, as e.g. in Sec. 2.2.1, it may not be possible to interchange certain
indices in an unambiguous way and an exception will be raised upon the
attempt.
Finally, trees can also be realized as “views”, which do not actually own
copies of their data, but rather point do data of another tree. This is
particularly useful for re-arranging tree data into a new tree by permuting
indices without actually moving the data.
### 4.3 Operators classes
All operator classes are derived from an abstract base class with the
following key data and member function:
⬇
class OperatorAbstract {
const Index *iIndex, *jIndex;
void apply(complex<double> A,const Coefficients&X
complex<double>B,Coefficients&Y)const=0;
It symbolizes a map ${\vec{X}}\to{\vec{Y}}=Op({\vec{X}})$. The class
containing the coefficients is a tree class Coefficients: public
Tree<Coefficients> which is usually constructed from an Index* idx as
Coefficients X(idx). It mirrors the tree structure of idx and, at each of its
nodes $J_{l}$, it points to the data of ${\vec{X}}_{J_{l}}$. Derived classes
must implement the virtual abstract function apply(...) for the map
${\vec{Y}}\leftarrow A{\vec{X}}+B{\vec{Y}}$. On this abstract level, there are
no particular assumptions other than that the operator maps from a linear
space into a linear space. The two spaces do not need be equal or subspaces of
the same space, the map itself does not need to be linear.
Figure 9: Hierarchy of operators in tRecX (incomplete). An OperatorAbstract
is an instance of a map between linear spaces LinSpaceMap, specifically
between Coefficients. Implementations OperatorTree, OperatorMap, and Resolvent
are briefly described in the text.
A large number of diverse operators are derived from OperatorAbstract, part of
who are shown in the Doxygen-generated class hierarchy in Fig. 9. Particularly
important is
⬇
class OperatorTree: public Tree<OperatorTree>,
public OperatorAbstract {
protected:
OperatorFloor * oFloor;
...
which implements the hierarchy of block-matrices (32). The oFloor pointer is
only non-null at the leafs of the operator tree. The class OperatorFloor
implements all forms of maps in a numerically efficient way, for example,
multiplication of a vector by a full or diagonal matrix, multiplication by a
tensor product of small matrices, but also more complicated maps as, for
example in the electron-electron interaction for a given multipole-
contribution. Again, the map may be also non-linear, as in a Gross-Pitaevskii
operator. These various forms are realized as derived classes of the abstract
base class OperatorFloor.
#### 4.3.1 Construction and optimization of an OperatorTree
The primary constructor of OperatorTree takes an operator definition string as
in the examples of Sec. 2 that matches Index and recursively sets up the full
operator. As a rule, no complete matrix is constructed. Mostly, the tree
contains only the non-zero OperatorFloor’s. If tensor product structure is
detected in the operator, it is exploited if found to be numerically
advantageous by some (approximate) internal algorithm. When multiple terms
contribute to the same OperatorFloor these are summed into a single
OperatorFloor where this is possible and numerically profitable.
#### 4.3.2 Further important operator classes
From the whole list of operators we further single out the following classes
for their more general relevance:
##### OperatorInverse
Calculates the inverses of overlap matrices using Woodbury-like methods
consisting of a cheap direct inverse with some low rank update for completing
the exact inverse.
##### MapGauge
Implements general Gauge transformations.
##### OperatorMap
Given two Index objects for discretizations $\underline{B}$ and
$\underline{B}^{\prime}$ this is the map
$\underline{B}\to\underline{B}^{\prime}$, where this is logically possible and
meaningful. Typical examples are maps to and from grids, Sec. 3.3. Another
application is in class DiscretizationTsurffSpectra for the transformation
from surface values to a grid of momentum points, where the momentum spectra
are accumulated. Such transformations are not necessarily lossless.
##### Resolvent
Given ${\widehat{H}}$ and an overlap ${\widehat{S}}$ as OperatorAbstract’s,
this class constructs the resolvent operator
$({\widehat{H}}-z{\widehat{S}})^{-1}$ with a complex $z$. At present the
implementation is through Eigen’s sparse LU-decomposition and is limited by
basis sizes. For banded matrices, like in the Floquet example discussed above,
Resolvent can be constructed for very large dimensions.
### 4.4 Recursive algorithms
Recursive structures provide for tRecX’s flexibility, but in addition they
generate compact and comparatively transparent code. The basic pattern is:
⬇
function(tree):
if isLeaf: specific action on leaf
else: for c in children: function(c)
As examples, we discuss the apply member function of the class OperatorTree
and an OperatorTree constructor. For apply, which implements
${\vec{Y}}\leftarrow b{\vec{Y}}+a{\widehat{O}}[{\vec{X}}]$, the simplified
pseudo-code is
⬇
1: O.apply(a,X,b,Y):
2: Y<-b*Y
3: if isLeaf:
4: OperatorFloor.apply(a,X,1,Y)
5: else:
6: for cO in O.children:
0: cO.apply(a,X.child(cO.jdx),1,Y.child(cO.idx))
Here an OperatorFloor class implements the specialized action in a efficient
way, typically through LAPACK or Eigen. The code plays out its efficiency for
block-sparse matrices, where zero-blocks never appear in the loop, line 6. The
price to pay is that one needs to locate the block operator’s left- and right
hand indices in the coefficient vectors ${\vec{X}}$ and ${\vec{Y}}$, here
symbolical written as X.child(cO.jdx) and Y.child(cO.idx), respectively. If
one ensures that OperatorFloor.apply is a sufficiently coarse-grain operation,
say multiplication by a 20$\times$20 matrix, the overhead from the recursion
remains small. Clearly, for full matrices or matrices with very regular
structure such as band-matrices, the algorithm is at a disadvantage. Where
such performance losses are identified, OperatorTree should be replaced by a
more specialized class derived from OperatorAbstract.
After all setup is done the OperatorTree a “flattened” view of the tree is
created for use in propagation. This is a vector of pointers to the leafs of
the OperatorTree, which are automatically distributed for parallelization (see
sec. 3.6). In the process, direct pointers from the operator indices to the
respective sections of the ${\vec{X}}$ and ${\vec{Y}}$ vectors are set up,
eliminating all overhead from that place.
A second example of recursion in tRecX is the pseudo-code of a basic
OperatorTree constructor, for the case of a strict tensor product
opDef="0.5<def0><def1>...<defL>"
⬇
OperatorTree(opDef,iIndex,jIndex,mult):
# e.g., a,f,r=0.5,"<def0>","<def1>...<defL>"
a,f,r=getFactors(opDef)
# mat(i,j)=<b[i]|op[f]|b[j]>
Matrix mat=getMatrix(f,iIndex.basis,jIndex.basis)
if r=="":
mat*=a*mult
oFloor=OperatorFloor(mat)
else where mat(i,j)!=0:
mult*=a*mat(i,j)
childAdd(OperatorTree(r,
iIndex.child(i),jIndex.child(j),mult)
The opDef strings are split into the scalar prefactor a, the first tensor
factor f, and the remainder r by getFactors. This is mainly located in class
OperatorDefinition, with a few additional classes due to legacy code. Then
getMatrix interprets the tensor factor string s=<defN> and constructs the
corresponding factor matrix. If one has arrived at the last factor, the
remainder r becomes empty. The matrix mat is multiplied by scalar factors mult
and a and its matrix-vector application is implemented depending on its
structure, e.g., for full, diagonal or banded matrices. If the remainder is
not empty one advances to the next tensor factor. In this simple example,
tensor structure is multiplied out rather than preserved.
The actual tRecX code is more complex, admitting for tensor products, the sum
of terms, and handling of special operators such as [[eeInt6DHelium]] in Sec.
2.2.2. Also syntax and consistency of the defining string opDef with the
actual left and right indices iIndex,jIndex are checked throughout and errors
throw exceptions. At the end of construction the OperatorTree is post-
processed where multiple operators for the same index pair are fused into
single blocks and zero blocks that may have appeared after summation are
eliminated.
### 4.5 Basis sets
All bases are derived from a class BasisAbstract with the pure virtual
function size() giving the number of functions in the basis. The word “basis”
is used in a general way for any set of defining properties for the discrete
representation on a given $q^{l}$. This includes a discrete set of functions,
but also grids, or an orthonormal set of unit vectors in a discrete space.
Bases need not be orthonormal, although this is ensured wherever it is
possible and meaningful.
#### 4.5.1 BasisIntegrable
BasisIntegrable is an abstract class is for single-variable basis functions
that can be integrated over:
⬇
class BasisIntegrable: public BasisAbstract {
protected:
double _lowBound,_upBound;
public:
virtual void valDer(
const vector<complex<double> > &X,
vector<complex<double>> &V,
vector<complex<double>> &D,...) const=0;
virtual void quadRule(
int N, vector<double> &QuadX,
vector<double> &QuadW) const=0;
virtual unsigned int order() const=0;
...
The functions are supported on the interval [_lowBound,_upBound], which may
also be infinite. The pure virtual function valDer(...) must be implemented to
return the value and first derivative matrices $V_{ij}=b_{j}(x_{i})$ and
$D_{ij}=b^{\prime}_{j}(x_{i})$. Any BasisIntegrable must provide $N$-point
quadrature rules $q_{k},w_{k}$ in QuadX,QuadW through quadRule(...). Finally,
there is the concept “order” of a BasisIntegrable. This can be understood as
the minimal number of quadrature points needed for the correct evaluation of
overlap matrix elements. For example, in a DVR basis with Dirichlet boundary
conditions an the lower boundary, the first Lagrange polynomial is omitted,
leading to size()=order()-1.
A simple example of a BasisIntegrable that is only used for debugging purposes
are the monomials $\\{1,x,x^{2},\ldots\\}$:
⬇
class BasisMonomial:public BasisIntegrable{
int _order;
public:
BasisMonomial(int Order, double Low, double Up)
:BasisIntegrable(Low,Up),_order(Order){}
unsigned int size() const{return _order;}
void quadRule(...) const{
...shift-and-scale Legendre quadrature...
}
void valDer(const vector<complex<double>> &X,
vector<complex<double>> &V,
vector<complex<double>> &D,...
) const {
V.assign(X.size(),1.);
D.assign(X.size(),0.);
for(int k=1;k<size();k++){
for(int i=0;i<X.size();i++){
V.push_back(V[X.size()*(k-1)+i]*X[i]);
D.push_back(D[X.size()*(k-1)+i]*X[i]
+V[size()*(k-1)+i] );
}}}
unsigned int order() const{return _order;}
};
Note the use of “automatic differentiation” for the evaluation of the
derivatives. This transparent and efficient approach to determining
derivatives is used throughout tRecX.
#### 4.5.2 BasisDVR
An important BasisIntegrable implementation is BasisDVR, where the most
important data members are
⬇
class BasisDVR: public BasisIntegrable{
vector<double> _dvrX,_dvrW;
int _nBeg;
int _size;
...
_dvrX and _dvrW are the nodes and weights for quadrature rule. The rule is
Lobatto for finite intervals and Radau for semi-infinite intervals. There are
at most _dvrX.size() different Lagrange polynomials, for which values and
derivatives can be evaluated anywhere within the basis’ interval. Dirichlet
boundary conditions are determined through _nBeg and _size. _nBeg=0 means the
Lagrange polynomial for _dvrX[0]=_lowBound is included, and =1, where that
polynomial is omitted. Similarly, _nBeg+_size=_dvrX.size()-1 means the
Lagrange polynomial at _dvrX.back()=_upBound is omitted for Dirichlet
condition at that point. The values are set upon construction.
#### 4.5.3 BasisGrid
BasisGrid is not a BasisIntegrable, rather it derives directly from
BasisAbstract:
⬇
class BasisGrid: public BasisAbstract {
vector<double> _mesh;
unsigned int size() const {return _mesh.size();}
...
The only class-specific member data is _mesh, which holds the grid points.
Values of a BasisGrid are only defined at the grid points, but a member
function for Newton-interpolation between these points is provided.
The class is mostly for transforming BasisIntegrable’s to grids. Assume a wave
function $|\Psi\rangle=|\underline{B}\rangle\overline{C}$ is given in terms of
an Index cIdx containing BasisIntegrable’s. A new IndexG gridIdx(cIdx) is
created where the BasisIntegrable’s are replaced by the desired BasisGrid’s.
In the process an OperatorMap mapFrom is automatically created which
transforms $\Psi$ to its representation on the multi-dimensional grid.
Assuming Coefficients X(cIdx) contains the $\overline{C}$, then
⬇
mapFromParent()->apply(1,X,0,Y)
fills Coefficients Y(gridIdx) with $\Psi({\vec{x}}_{i})$ at the multi-
dimensional grid points ${\vec{x}}_{i}$. The assignment between values
$\Psi({\vec{x}}_{i})$ and ${\vec{x}}_{i}$ is given through the structure
information contained in gridIdx.
A class BasisGridQuad is derived from BasisGrid, with an additional member
vector<double> _weights for integration weights at the _mesh. This allows a
lossless transformation between of BasisIntegrable to a Gauss quadrature grid
that is exact for the basis. This procedure is used on several occasions,
e.g., for the efficient multiplication by the Volkov phases on a grid of
${\vec{k}}$-values (see Sec. 3.1), when the spectral amplitudes are given in
terms of spherical harmonics.
#### 4.5.4 BasisVector
BasisVector is a simple and useful class for discrete coordinate indices,
where only $l$ matters and the value of $q^{l}$ has no significance. It is
fully defined by its size
⬇
class BasisVector : public BasisAbstract{
unsigned int _size;
unsigned int size() const{return _size;}
...
This is used, for example to label the Floquet blocks in sec. 2.3.
#### 4.5.5 BasisSub
A subset of a given BasisAbstract is selected by BasisSub
⬇
class BasisSub: public BasisAbstract{
vector<int> _subset;
const BasisAbstract* _bas;
...
where the vector<int>_subset lists function numbers from _bas to be included
with BasisSub. This is used when imposing basis constraints or in general when
pruning branches from an Index.
#### 4.5.6 Multi-dimensional basis functions — BasisNdim
As illustrated in Sec. 2.5 and further discussed in Sec. 3.2, formal
coordinates $q^{l}$ may also be multi-dimensional. Functions with higher-
dimensional arguments appear as orbitals but also as intermediate objects when
mixing coordinates systems, for example for a multi-center expansion. The
class is more complex than the examples given so far. The general strategy is
to store the values and partial derivatives of all functions at a suitable
quadrature grid. This may require substantial memory, but we have not
exhausted standard size storage of a few GB in applications so far. The
quadrature grid may refer to a different coordinate system _quadCoor than the
basis’s coordinate system _ndimCoor. From this follows the class signature:
⬇
class BasisNdim : public BasisAbstract{
string _ndimCoor;
string _quadCoor;
vector<vector<double>> _quadGrid;
vector<double> _quadWeig;
vector<vector<vector<complex<double>>>> _valDer;
...
Matrix elements can be computed for operators given in terms of standard
strings, where the transformation between different coordinate systems is done
automatically adhering to the philosophy of “automatic differentiation”. For
further details we refer to the in-line documentation of the code.
### 4.6 Input, units conversion, and algebraic expressions
An attempt is made to make input human-readable, error-safe, and self-
explanatory. Rather than listing available inputs in some separate manual, the
code itself enforces input documentation and input sanity checks. Erroneous
input triggers error messages showing line number in the input file and valid
options, emits a warning about suspicious input or throws a run-time errors
when inconsistent input is detected. A dynamically generated list of possible
input is displayed, when the code is run without parameters. In general,
plausible guesses for the input will be accepted or trigger information on
valid alternatives. More details are shown in the following.
#### 4.6.1 General input format — class ReadInput
All user input is controlled by a class ReadInput with a prescribed
designation of any input item in the format Category: name as illustrated in
the examples of Sec. 2. An overloaded read(...) method requires to supply a
default value or to state explicitly that there cannot be a default and brief
documentation for every input item. Inputs can be of all standard types, which
also includes vector’s. Input is usually read from file but can be overruled
by command line flags of the format -Category:name=value by default or
abbreviated flags, that can be specified in read(...). In the input file,
several name’s can follow the Category specifier, the sequence of the names is
arbitrary. Also, the same category can appear in repeated lines, such as in
Sec. 2.5 where we have Operator:hamiltonian and Operator:interaction. We
follow the convention of having Category’s start with upper case and name’s
with lower case letters.
There is a simple syntax to restrict admissible input values. A member
function ReadInput::finish() checks all inputs from file and from the command
line for correct Category and name and will stop if a given pair Category:name
in the file does not actually appear in the code, reducing the likelihood of
misprint errors. In addition, finish() a list of all admissible inputs in a
file tRecX.doc, which also explains the input as documented in read(...) and
the default input values. The contents of this file is shown as a help when
running tRecX without any input.
The above is meant to illustrate the general strategy for enforcing
documentation and enhancing usability and error safety. Full features can be
found in the inline-documentation and are illustrated by a usage example in
the test() member function.
Another feature for productivity and error safety is the possibility to freely
choose input units and to use algebraic expressions as inputs. Default are
a.u. unless the input name specifies a different unit. In the example of
section 2.5
⬇
Laser:shape,I(W/cm2),FWHM,lambda(nm), phiCEO,polarAngle
cos2, 1e10, 4 OptCyc, 800, pi/2, 45
cos4, 1e11, 3 OptCyc, 800/13, 0, 45
cos4, 1e11, 3 OptCyc, 800/15, 0, 45
the intensity is expected with the strong-field convention as $W/cm^{2}$. The
units in brackets at I(W/cm2) form a functional part of the Category:name. The
name’s input units can be overruled by specifying e.g. 1e-2 au instead. That
value will be converted by read(...) to $W/cm^{2}$, $10^{-2}a.u.\
=3.52\ldots\times 10^{-14}W/cm^{2}$ with full available precision. Another
example is with lambda(nm), where, e.g. one could equivalently use the input
string 800e-9 m.
#### 4.6.2 Class Units
Unit conversions are performed by a class Units which at present recognizes
atomic units au, cgs ESU, and SI units plus a few units that customarily used
in strong field physics such as $W/cm^{2}$, $nm$ and Rydberg energy Ry. The
duration of an optical cycle OptCyc is computed from the wave-length of the
field component in the first line after Laser: the listing above produces 1
OptCyc $=2\pi(800\,nm)/c$ (converted to a.u. ).
#### 4.6.3 Class Algebra
Input values can be specified as algebraic expressions of constants, as in
800/13 or pi/2. The strings are interpreted by the same class Algebra that is
used for the definition of operators. It can do standard complex algebra,
where complex numbers are specified as in 2+i*3.1415. It recognizes a few
constants such as pi and hbar ($\hbar$ in SI units). Further constants can be
added from the input, as documented in the command line help.
When used for constructing functions of a single coordinate, the character Q
represents the coordinate in expressions such as pow[2](cos(Q/2)), which on a
Phi-axis would evaluate to $\cos^{2}(\phi/2)$. The most frequent mathematical
functions are available, see the tutorials for examples. When attempting to
input a malformed algebra, a diagnostic of the error will be displayed which
also includes the full list of presently implemented functions.
### 4.7 TimePropagator and TimePropagatorOutput classes
Time propagation is controlled through a wrapper class TimePropagator. It
takes start and end times, accuracy or step size and output intervals as its
main control parameters. For solving the ordinary differential equation in
time it needs a class of abstract type ODEstep. A range of those steppers have
been implemented including a general (explicit) Runge-Kutta, a specialized
classical 4-stage Runge-Kutta, and Arnoldi solver, and several experimental
solvers. At present, only the classical Runge-Kutta is used, as it was found
to be the overall most efficient across the large variety of problems treated
with tRecX. The notorious stiffness problem of explicit methods is controlled
by removing few extremely high-lying spectral values from the problem, see
Sec. 3.5. Although this does deliver a workable and rather efficient solution,
we do not consider the development of time-steppers as concluded.
The class TimePropagatorOutput controls which information is output during
time-propagation. One category of outputs are expectation values of operators,
by default the overlap $\langle\Psi(t)|\Psi(t)\rangle$ and field-free
Hamiltonian $\langle\Psi(t)|H_{0}|\Psi(t)\rangle$, where $H_{0}$ is the
operator specified as Operator:hamiltonian. Further expectation values can be
defined at the input, for example the dipole values in various gauges. Another
category are Coefficients for ${A}|\Psi(t)\rangle$. In this way the values and
derivatives at the tSurff-radius $R_{c}$ are written to disc, but ${A}$ can
also be user-defined. More transformations can be easily added by editing
main_trecx.cpp.
#### 4.7.1 Plot
One can plot densities of the kind $|\Psi({\vec{q}},t)|^{2}$ or more generally
$\overline{\Psi}({\vec{q}},t)[{A}\Psi]({\vec{q}},t)$ with a user-defined
operator ${A}$. This is handled by class Plot, which is constructed from input
as, for example,
⬇
Plot: axis,points,lowerBound,upperBound
Rn,101,0.,20.
Eta,31,-1.,1.
where the density is plotted w.r.t. the discretization’s axes Rn and Eta the
two-dimensional region $[0,20]\times[-1,1]$ with $101\times 31$ equidistant
grid points. Coordinates not listed are assumed to be integrated or summed
over. Output will be in ASCII format and readable, e.g., by Gnuplot, but also
by tRecX’s plot.py script. The order of inputs lines in Plot determines the
sorting of the density values, such that the first axis, Rn in the example,
runs fastest. For higher-dimensional plots the further dimensions will appear
as additional columns in the two-dimensional output file. Explanation of the
input for plots can be found in tRecX.doc, for the full features of the class
we refer to the Doxygen and inline documentation of the code.
### 4.8 Python scripts
There is a limited number of convenience python scripts in the SCRIPTS
subdirectory. These have mostly grown out of practice and certainly do not
comply with good programming requirements. Yet, given their proven usefulness
in practice, we include them with the distribution.
For submission to compute queues one can adjust submit_tRecX.py, which is
currently set up for SLURM and should be adaptable to similar queuing systems
with little effort. It ensures generation of properly named run-directories
before starting the actual tRecX code. By this one can submit multiple jobs
without the need to manually ensure proper run-directory numbering. It also
creates a short submit name for the job for display by the SLURM queue
overview.
Virtually all ASCII files that appear in the run directory can be plotted
using plot.py. It produces one- and two-dimensional graphs from selected
columns of a file, compares multiple runs, can annotate curves with the actual
parameters used in the run etc. Brief instructions and a full list of command
line flags are displayed by running plot.py without any arguments.
Running multiple calculations with varying parameters, either for ensuring
convergence or for analyzing a physical phenomenon is a frequent mode of using
tRecX. The script lRuns.py lists all or a selected subset of runs showing
basic information such as status of the computation, run time, wave function
norm, and energy. In addition, the user can select any set of input parameters
for display. Usage instructions are shown when running lRuns.py on the command
line without any parameters.
## 5 Conclusions
The purpose of tRecX is three-fold: applications, training and education, and
community development.
The code produces accurate solutions for TDSEs that appear in ultrafast and
strong field physics. In the present public version a wide range of standard
problems such as high harmonic generation, fully differential spectra for
single ionization, Floquet and various model systems can be solved by adapting
the given tutorial inputs. Also, with the use of significant computer
resources, fully differential double emission spectra can be computed. With
tSurff as one of its key methods, computer resource consumption remains low,
on the scale of a few minutes for single-electron calculation of standard
tasks, and within the range of the feasible for long-wavelength double
emission. Forthcoming releases will include haCC, which integrates Gaussian-
based quantum chemical wave functions with the discretizations discussed here.
This allows to compute emission from multi-electron systems.
A designated part of the tRecX development is to ensure user experience that
is acceptable to a somewhat wider range of specialist users, including
experimentalists who want to generate standard results or study simple models
as well as theorists with more complex demands. We consider error safe and
intuitive input, extensive consistency checks, and structurally enforced
documentation as essential for achieving that goal.
Finally, on the developer level, the systematic C++ object orientation has
allowed development and maintenance of the code by a very small group. The
full research code is also used in training and education on the undergraduate
and graduate level. In course of such projects, attention to understandable
and consistent code structure it taught and enforced. Student projects have
non-trivially contributed to the code in specialized applications, such as the
use of parabolic coordinates, Coulomb scattering, and double- and triple
breakup (not included in the public release yet).
For standard use, tRecX in its present form will be made available at the “AMP
gateway”, a collaborative effort for low-threshold use of atomic physics codes
[26]. At present, a preliminary installation is available a that site.
The experience with student projects shows that substantial structural
contributions from a community are possible without endangering code integrity
or maintainability. Possible first such projects would likely be
collaborative, but also unsupervised extensions may well be feasible. A formal
invitation for contributions is extended here.
## Acknowledgment
Key initial contributions to the code were made by Vinay Pramod Majety and
Alejandro Zielinski, with further contributions by, in alphabetic order,
Christoph Berger, Jonas Bucher, Florian Egli, Jacob Liss, Mattia Lupetti, Jörn
Stöhler, Jonathan Rohland, Andreas Swoboda, Hakon Volkmann, Markus and Michael
Weinmueller, and Jinzhen Zhu. Funding was provided by the DFG excellence
cluster EXC 158 “Munich Center for Advanced Photonics” (MAP), the Austrian
Science Foundation project ViCoM (F41) and the DFG priority program 1840
(QUTIF).
## References
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* [2] Liang Tao and Armin Scrinzi. Photo-electron momentum spectra from minimal volumes: the time-dependent surface flux method. New Journal of Physics, 14(1):013021, Jan 2012.
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|
11institutetext: Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France
11email<EMAIL_ADDRESS>22institutetext: LATMOS/IPSL, UVSQ Université
Paris-Saclay, Sorbonne Université, CNRS, Guyancourt, France
# Characterisation of the hydrospheres of TRAPPIST-1 planets
Lorena Acuña 11 Magali Deleuil 11 Olivier Mousis 11 Emmanuel Marcq 22 Maëva
Levesque 11 Artyom Aguichine 11
(Received 12 November 2020; accepted 19 January 2021)
###### Abstract
Context. Planetary mass and radius data are showing a wide variety in
densities of low-mass exoplanets. This includes sub-Neptunes, whose low
densities can be explained with the presence of a volatile-rich layer. Water
is one of the most abundant volatiles, which can be in the form of different
phases depending on the planetary surface conditions. To constrain their
composition and interior structure, it is required to develop models that
calculate accurately the properties of water at its different phases.
Aims. We present an interior structure model that includes a multiphase water
layer with steam, supercritical and condensed phases. We derive the
constraints for planetary compositional parameters and their uncertainties,
focusing on the multiplanetary system TRAPPIST-1, which presents both warm and
temperate planets.
Methods. We use a 1D steam atmosphere in radiative-convective equilibrium with
an interior whose water layer is in supercritical phase self-consistently. For
temperate surface conditions, we implement liquid and ice Ih to ice VII phases
in the hydrosphere. We adopt a MCMC inversion scheme to derive the probability
distributions of core and water compositional parameters
Results. We refine the composition of all planets and derive atmospheric
parameters for planets b and c. The latter would be in a post-runaway
greenhouse state and could be extended enough to be probed by space mission
such as JWST. Planets d to h present condensed ice phases, with maximum water
mass fractions below 20%.
Conclusions. The derived amounts of water for TRAPPIST-1 planets show a
general increase with semi-major axis, with the exception of planet d. This
deviation from the trend could be due to formation mechanisms, such as
migration and an enrichment of water in the region where planet d formed, or
an extended CO2-rich atmosphere.
###### Key Words.:
planets and satellites: interiors – planets and satellites: composition –
planets and satellites: atmospheres – planets and satellites: individual:
TRAPPIST-1 – methods: statistical – methods: numerical
## 1 Introduction
Ongoing space missions such as CHEOPS (Benz, 2017) and TESS (Ricker et al.,
2015), and their follow-up with ground-based radial velocity telescopes, are
confirming the existence of low-mass exoplanets with a wide range of
densities. These densities range from the values typically inferred for the
Earth or Mercury to those measured in Uranus and Neptune. The exoplanets in
the former class are mainly composed of a Fe-rich core and a silicate mantle,
while the latter class has a layer that is rich in volatiles. Water is the
most abundant and least dense volatile after H and He (Forget & Leconte,
2014), which makes it a likely species to constitute the volatile reservoir in
these planets. Several studies have investigated the interior structure and
composition of water-rich planets (Sotin et al., 2007; Seager et al., 2007;
Dorn et al., 2015; Zeng et al., 2019), but focused mainly on its condensed
phases. Nonetheless, many sub-Neptunes are close to their host star and
receive enough irradiation to trigger a runaway greenhouse state in which
water is present as steam. In some cases, the high pressure and temperature
conditions can render the hydrosphere in supercritical and plasma, or even
superionic phases (Mazevet et al., 2019; French et al., 2016). Therefore, it
is crucial to include the modeling of all possible phases of water to provide
an accurate description of its presence on the planetary surface. Moreover,
the surface conditions are determined by the greenhouse effect caused by
atmospheric gases, making the modelling of radiative-convective equilibrium in
atmospheres a key parameter to determine in which phase water could be present
on the surface. Most of interior structure models represent the planetary
atmosphere as a gas layer with a simplified isothermal temperature profile
(Dorn et al., 2018, 2017b), which is very different from the temperature
profile in the convective deep layers of thick atmospheres (Marcq, 2012).
Multiplanetary systems are unique environments that present both planets that
can hold condensed phases as well as highly-irradiated planets with steam
atmospheres. In this study, we develop a planet interior model suitable for
the different conditions at which water can be found in low-mass planets. Our
implementation includes a supercritical water layer, introduced in Mousis et
al. (2020), coupled with a 1D radiative-convective atmosphere model (Marcq,
2012; Marcq et al., 2017; Pluriel et al., 2019) to calculate the total radius
of the highly-irradiated planets with water self-consistently. Furthermore,
for temperate planets, we have updated the interior model presented in Brugger
et al. (2016, 2017) to include ice phases Ih, II, III, V and VI. We introduce
these models in a MCMC Bayesian analysis scheme adapted from Dorn et al.
(2015). This allows us to derive the water mass fraction (WMF) and core mass
fraction (CMF) with their associated confidence intervals that reproduce the
observed radius, mass and stellar composition measurements.
We use this model to explore the possible water content of the TRAPPIST-1
system, an ultra-cool M dwarf that hosts seven low-mass planets in close-in
orbits. Three of these planets are located in the habitable zone (Grimm et
al., 2018), meaning that they can hold liquid water or ice Ih on their
surfaces. Although all planets in TRAPPIST-1 system have masses and radii that
are characteristic of rocky planets, their differences in density indicate
that each planet has a different volatile content. This makes this planetary
system ideal for testing planet interior, atmospheric structure and formation
scenarios.
In Sect. 2 we describe the complete interior structure model, including the
new updates for the supercritical and ice phases, the coupling between the
interior and the atmosphere for steam and supercritical planets, and the MCMC
Bayesian algorithm. The parameters for the TRAPPIST-1 planets used in this
study are summarized in Sect. 3, including mass, radius and Fe/Si molar ratio.
The results of our analysis of the hydrospheres of TRAPPIST-1 planets are
described in Sect. 4. We compare our results with previous works and discuss
the implications of our water estimates for planet formation in Sect. 5. We
finally expose our conclusions in Sect. 6.
## 2 Planetary structure model
For consistency, we recall the main principles of the interior structure
model. The basis of our model is explained in Brugger et al. (2016, 2017). The
1D interior structure model takes as input the mass and the composition of the
planet, which is parameterized by the CMF and WMF. The structure of the planet
is stratified in 3 layers: a core, a mantle and a hydrosphere. The pressure,
temperature, gravity acceleration and density are computed at each point of
the one-dimensional spatial grid along the radius of the planet. The pressure,
$P(r)$, is obtained by integrating the hydrostatic equilibrium (Eq. 1); the
gravitational acceleration, $g(r)$, by solving Gauss’s theorem (Eq. 2); the
temperature, $T(r)$, with the adiabatic gradient (Eq. 3); and the density,
$\rho(r)$, with the Equation of State (EOS). $m$ is the mass at radius $r$,
$G$ is the gravitational constant, and $\gamma$ and $\phi$ are the Gruneisen
and the seismic parameters, respectively. Their formal macroscopic definitions
are shown in equation 4, where $E$ is the internal energy and $V$ is the
volume. The Gruneisen parameter is a thermodynamic parameter that describes
the dependence of the vibrational properties of a crystal with the size of its
lattice. It relates the temperature in a crystalline structure to the density,
which is calculated by the EOS. The seismic parameter defines how seismic
waves propagate inside a material. It is related to the slope of the EOS at
constant temperature (Brugger et al., 2017; Sotin et al., 2007).
$\dfrac{dP}{dr}=-\rho g$ (1)
$\dfrac{dg}{dr}=4\pi G\rho-\dfrac{2Gm}{r^{3}}$ (2)
$\dfrac{dT}{dr}=-g\dfrac{\gamma T}{\phi}$ (3)
$\cases{\phi}=\dfrac{dP}{d\rho}\\\ \gamma=V\
\left(\dfrac{dP}{dE}\right)_{V}{}$ (4)
The boundary conditions are the temperature and pressure at the surface, and
the gravitational acceleration at the center of the planet. The value of the
latter is zero. The total mass of the planet is calculated with Eq. 5, which
is derived from the conservation of mass (Brugger et al., 2017; Sotin et al.,
2007). Once the total input mass of the planet is reached and the boundary
conditions are fulfilled, the model has converged.
$\dfrac{dm}{dr}=4\pi r^{2}\rho$ (5)
Depending on the surface conditions, the hydrosphere can be present in
supercritical, liquid or ice states. For each of these phases of water, we use
a different EOS and Gruneisen parameter to compute their P-T profiles and
density accurately. In Sect. 2.1 we describe the updates to the supercritical
water layer with respect to the model depicted in Mousis et al. (2020), while
in Sect. 2.2 we present the implementation of the hydrosphere in ice phases.
Finally, the coupling between the atmosphere and the interior model with
planets whose hydrosphere is in steam or supercritical phases is explained in
Sect. 2.3, followed by the description of the MCMC algorithm in Sect. 2.4.
### 2.1 Supercritical water
If the planet is close enough to its host star, the upper layer of the
hydrosphere corresponds to a hot steam atmosphere, whose temperature at the
base is determined by the radiative-convective balance calculated by the
atmosphere model (Marcq, 2012; Marcq et al., 2017). When the pressure and
temperature at the surface, which is defined as the base of the hydrosphere
layer, are above the critical point of water, we include a supercritical water
layer extending from the base of the hydrosphere to a height corresponding to
the phase change to steam (Mousis et al., 2020). We updated the EOS for this
layer to the EOS introduced by Mazevet et al. (2019), which is a fit to the
experimental data provided by the International Association for the Properties
of Water and Steam (IAPWS) (Wagner & Pruß, 2002) for the supercritical regime,
and quantum molecular dynamics (QMD) simulations data for plasma and
superionic water (French et al., 2009). The IAPWS experimental data span a
temperature range of 251.2 to 1273 K and of 611.7 to 109 Pa in pressure, while
their EOS can be extrapolated up to 5000 K in temperature and 1011 Pa in
pressure (Wagner & Pruß, 2002). The validity range of the EOS presented in
Mazevet et al. (2019) includes that of the IAPWS plus the region in which the
QMD simulations are applicable, which corresponds to a temperature from 1000 K
to 105 K and densities in the 1–$10^{2}$ g/cm3 range. These densities are
reached at high pressures, i.e., in the 109–1012 Pa range. Following Eq. 3,
the adiabatic gradient of the temperature is specified by the Gruneisen and
the seismic parameters. These are dependent on the derivatives of the pressure
with respect to the density and the internal energy (Eq. 4). We make use of
the specific internal energy and density provided by Mazevet et al. (2019) to
calculate them.
### 2.2 Ice phases
Table 1: EOS and reference thermal parameters for ices Ih, II, III, V and VI.
This includes the reference values for the density $\rho_{0}$, the temperature
$T_{0}$, the bulk modulus $K_{T_{0}}$ and its derivative $K^{\prime}_{T_{0}}$,
the heat capacity $C_{p}(T_{0})$, and the thermal expansion coefficient
$\alpha_{0}$.
Phase | $\rho_{0}$ [kg m-3] | $T_{0}$ [K] | $K_{T_{0}}$ [GPa] | $K^{\prime}_{T_{0}}$ | $C_{p}(T_{0})$ [J kg-1 K-1] | $\alpha_{0}$ [$10^{-6}$ K-1] | References
---|---|---|---|---|---|---|---
Ih | 921.0 | 248.15 | 9.50 | 5.3 | 1913.00 | 147 | 1, 8
II | 1169.8 | 237.65 | 14.39 | 6.0 | 2200.00 | 350 | 1, 2, 7
III | 1139.0 | 237.65 | 8.50 | 5.7 | 2485.55 | 405 | 3, 4, 5, 7
V | 1235.0 | 237.65 | 13.30 | 5.2 | 2496.63 | 233 | 1, 4, 5, 7
VI | 1270.0 | 300.00 | 14.05 | 4.0 | 2590.00 | 146 | 4, 6, 7
(1) Gagnon et al. (1990); (2) Báez & Clancy (1995); (3) Tulk et al. (1997);
(4) Tchijov et al. (2004); (5) Shaw (1986); (6) Bezacier et al. (2014); (7)
Choukroun & Grasset (2010); (8) Feistel & Wagner (2006)
We extended the hydrosphere in Brugger et al. (2016, 2017) with liquid and
high pressure ice VII by adding 5 more condensed phases: ice Ih, II, III, V
and VI. EOS for ice Ih has been developed by Feistel & Wagner (2006) with
minimization of the Gibbs potential function from the fit of experimental
data. It covers all the pressure and temperature range in which water forms
ice Ih.
Fei et al. (1993) proposed a formalism to derive the EOS of ices II, III and
V. These EOS have the form of $V=V(P,T)$, which can be found by integrating
the following differential equation (Tchijov et al., 2004):
$\dfrac{dV}{V}=\alpha dT-\beta dP$ (6)
where $\alpha$ is the thermal expansion coefficient and $\beta$ the isothermal
compressibility coefficient. If the relationship between the specific volume,
$V$, and the pressure, $P$, at a constant temperature $T=T_{0}$ is determined,
Eq. 6 can be integrated as:
$V(P,T)=V(P,T_{0})\
exp\left(\int_{T_{0}}^{T}\alpha(P,T^{\prime})dT^{\prime}\right)$ (7)
Fei et al. (1993) proposed the following expression for the thermal expansion
coefficient $\alpha$:
$\alpha(P,T)=\alpha(P_{0},T)\dfrac{\left(1+\dfrac{K^{\prime}_{T_{0}}}{K_{T_{0}}}P\right)^{-\eta}}{\left(1+\dfrac{K^{\prime}_{T_{0}}}{K_{T_{0}}}P_{0}\right)^{-\eta}}=-\dfrac{1}{\rho}\dfrac{d\rho(T)}{dT}\dfrac{\left(1+\dfrac{K^{\prime}_{T_{0}}}{K_{T_{0}}}P\right)^{-\eta}}{\left(1+\dfrac{K^{\prime}_{T_{0}}}{K_{T_{0}}}P_{0}\right)^{-\eta}}$
(8)
where $\eta$ is an adjustable parameter estimated from the fitting of
experimental data. Its value is 1.0 for ice II and ice III (Leon et al., 2002)
and 7.86 for ice V (Shaw, 1986). $\rho$ is the density, $\alpha(P_{0},T)$ is
the coefficient of thermal expansion at a reference pressure $P_{0}$,
$K_{T_{0}}$ is the isothermal bulk modulus at the reference temperature
$T_{0}$, and $K^{\prime}_{T_{0}}$ is the first derivative of the isothermal
bulk modulus at the reference temperature. Hence, by substituting Eq. 8 in Eq.
7 and integrating, we obtain the following EOS for high-pressure ice:
$V(P,T)=V(P,T_{0})\
exp\left[ln\left(\dfrac{\rho(T_{0})}{\rho(T)}\right)\dfrac{\left(1+\dfrac{K^{\prime}_{T_{0}}}{K_{T_{0}}}P\right)^{-\eta}}{\left(1+\dfrac{K^{\prime}_{T_{0}}}{K_{T_{0}}}P_{0}\right)^{-\eta}}\right]$
(9)
The final expression (Eq. 9) requires the knowledge of the variation of the
specific volume, $V(P,T_{0})$, with pressure at the reference temperature
$T_{0}$. Moreover, the variation of the density with temperature, $\rho(T)$,
and the bulk modulus with its derivative at the reference temperature,
$K_{T_{0}}$ and $K^{\prime}_{T_{0}}$, must also be provided. In Table 1 we
specify the data and references to obtain these parameters for each ice phase.
In the case of ice VI, we adopt the second-order Birch-Murnaghan (BM2)
formulation, which is:
$P=\dfrac{3}{2}K_{T_{0}}\left[\left(\dfrac{\rho}{\rho_{0}}\right)^{\dfrac{7}{3}}-\left(\dfrac{\rho}{\rho_{0}}\right)^{\dfrac{5}{3}}\right],$
(10)
where $\rho_{0}$ is the reference density for ice VI. We also introduce a
thermal correction to the density since the pressure also depends on the
temperature:
$\rho(T)=\rho_{0}\exp\left(\alpha_{0}\left(T-T_{0}\right)\right)$ (11)
where $\alpha_{0}$ is the reference coefficient of thermal expansion.
Interfaces between liquid and ice layers are established by phase transition
functions from Dunaeva et al. (2010).
### 2.3 Interior-atmosphere coupling
Figure 1: Structural diagram of the coupling between the interior structure
model and the atmosphere model. $T_{base}$ is the temperature at the bottom of
the steam atmosphere in radiative-convective equilibrium. $z$ and $M_{atm}$
denote the atmospheric thickness and mass, respectively. $R_{bulk}$ and
$M_{bulk}$ correspond to the planet bulk radius and mass, respectively.
$R_{guess}$ refers to the initial guess of the bulk radius, while
$R_{interior}$ is the output bulk radius of the interior structure model in
each iteration.
We use a one-dimensional atmosphere model designed to compute radiative
transfer and pressure-temperature ($P$, $T$) profiles for water and CO2
atmospheres (Marcq, 2012; Marcq et al., 2017). The formation of water clouds
is considered in the computation of the albedo. The atmosphere is in radiative
equilibrium, and presents a composition of 99% water and 1% CO2. The density
of steam is obtained using a non-ideal EOS (Haar et al., 1984).
If the surface pressure is below 300 bar, the atmosphere and the interior are
coupled at the atmosphere-mantle boundary and water does not reach the
supercritical regime. However, if the surface pressure is greater than 300
bar, the atmosphere and the interior are coupled at this pressure level and a
layer of water in supercritical phase forms between the atmosphere and the
mantle. The pressure level at 300 bar is close enough to the critical point of
water at 220 bar to avoid the atmosphere model take over pressures and
temperatures where the temperature profile is adiabatic.
The top-of-atmosphere pressure is set to 20 mbar, which corresponds to the
observable transiting radius (Mousis et al., 2020; Grimm et al., 2018). We
denote the radius and mass from the center of the planet to this pressure
level the total radius and mass, $R_{total}$ and $M_{total}$, respectively. We
also define the radius and the mass that comprise the core, mantle and
supercritical layers as the bulk radius and mass, $R_{bulk}$ and $M_{bulk}$,
respectively. The atmosphere model provides the Outgoing Longwave Radiation
(OLR), albedo, thickness and mass of the atmosphere as a function of the bulk
mass and radius, and the surface temperature. If the atmosphere of the planet
is in radiative equilibrium, the OLR is equal to the radiation the planet
absorbs from its host star, $F_{abs}$. The OLR depends on the effective
temperature since OLR $=\sigma T_{\mathrm{eff}}^{4}$, where $\sigma$
corresponds to the Stefan-Boltzmann constant. To calculate the absorbed
radiation $F_{abs}$, we first compute the equilibrium temperature, which is
$T_{\mathrm{eq}}=(1-A_{B})^{0.25}\left(\frac{R_{\star}}{2a_{d}}\right)^{0.5}T_{\star},$
(12)
where $A_{B}$ is the planetary albedo, $R_{\star}$ and $T_{\star}$ are the
radius and effective temperature of the host star, respectively. $a_{d}$ is
the semi-major axis of the planet. The absorbed radiation is then calculated
as
$F_{abs}=\sigma\ T_{\mathrm{eq}}^{4}.$ (13)
Figure 1 shows the algorithm we implemented to couple the planetary interior
and the atmosphere. The interior structure model calculates the radius from
the center of the planet to the base of the steam atmosphere. For a fixed set
of bulk mass and radius, the OLR depends on the surface temperature.
Consequently, the surface temperature at which the OLR equals the absorbed
radiation corresponds to the surface temperature that yields radiative
equilibrium in the atmosphere. This is estimated with a root-finding method.
Since the bulk radius is an output of the interior model ($R_{interior}$) and
an input of the atmosphere model, we first need to calculate the surface
temperature for a certain mass and composition with an initial guess bulk
radius. Then this surface temperature is the input for the interior model,
which provides the bulk radius. With this bulk radius, we can generate a new
value of the surface temperature. This scheme is repeated until the bulk
radius converges to a constant value, to which we add the thickness of the
atmosphere, $z$, to get the total radius of the planet $R_{total}$. The total
mass $M_{total}$ is obtained as the sum of the bulk mass $M_{bulk}$ plus the
atmospheric mass $M_{atm}$. The tolerance used to determine if the bulk radius
has achieved convergence is 2% of the bulk radius in the previous iteration.
This is approximately 0.02 $R_{\oplus}$ for an Earth-sized planet.
### 2.4 MCMC Bayesian analysis
We adapted the MCMC Bayesian analysis algorithm described in Dorn et al.
(2015) to our coupled interior and atmosphere model. The input model
parameters are the bulk planetary mass $M_{bulk}$, the CMF and the WMF.
Therefore, m = $\left\\{M_{bulk},CMF,WMF\right\\}$, following the notation in
Dorn et al. (2015). Depending on the planetary system and their available
data, we can have observational measurements of the total planetary mass and
radius and the stellar composition, or only the total planetary mass and
radius. The available data in the former case is denoted as d =
$\left\\{M_{obs},R_{obs},Fe/Si_{obs}\right\\}$, while the data in the latter
case is represented as d = $\left\\{M_{obs},R_{obs}\right\\}$. The
uncertainties on the measurements are $\sigma(M_{obs}),\sigma(R_{obs})$, and
$\sigma(Fe/Si_{obs})$.
The CMF and WMF prior distributions are uniform distributions between 0 and a
maximum limit. This maximum limit is 75% for the CMF, which is derived from
the maximum estimated Fe/Si ratio of the proto-Sun (Lodders et al., 2009).
With this limit on the CMF, we are assuming that the exoplanets have not been
exposed to events during or after their formation that could have stripped
away all of their mantle, such as mantle evaporation or giant impacts. In
addition, the maximum WMF is set to 80%, which is the average water proportion
found in comets in the solar system (McKay et al., 2019). The prior
distribution for the mass is a Gaussian distribution whose mean and standard
deviations correspond to the central value and uncertainties of the
observations.
The MCMC scheme first starts by randomly drawing a value for each of the model
parameters from its respective prior distributions. This set of values is
designated as m${}_{1}=\left\\{M_{bulk,1},CMF_{1},WMF_{1}\right\\}$. The index
$i=1$ corresponds to the first proposed set of input values within the first
chain, $n=1$. The model calculates the total mass and radius and the
theoretical Fe/Si mole ratio, which are the set of output parameters
g(m${}_{1})=\left\\{R_{1},M_{1},Fe/Si_{1}\right\\}$. The likelihood of a set
of model parameters is then calculated via the following relationship (Dorn et
al., 2015):
$L(\textbf{m}_{i}\mid\textbf{d})=C\ exp\
\Biggl{(}-\frac{1}{2}\Biggl{[}\left(\frac{(R_{i}-R_{obs})}{\sigma(R_{obs})}\right)^{2}+\left(\frac{(M_{i}-M_{obs})}{\sigma(M_{obs})}\right)^{2}\\\
+\left(\frac{(Fe/Si_{i}-Fe/Si_{obs})}{\sigma(Fe/Si_{obs})}\right)^{2}\Biggr{]}\Biggr{)}$
(14)
where the normalization constant of the likelihood function $C$ is defined as:
$C=\dfrac{1}{(2\pi)^{3/2}\left[\sigma^{2}(M_{obs})\cdot\sigma^{2}(R_{obs})\cdot\sigma^{2}(Fe/Si_{obs})\right]^{1/2}}.$
(15)
When the Fe/Si mole ratio is not available as data, the square residual term
of the Fe/Si mole ratio is removed from Eq. 14, as well as its squared
uncertainty in Eq. 15.
Subsequently we draw a new set of input parameters,
m${}_{2}=\left\\{M_{bulk,2},CMF_{2},WMF_{2}\right\\}$ from the prior
distributions within the same chain, $n$. We assure that the absolute
difference between the values for $i=1$ and $i=2$ is lower than a fixed step,
which is the maximum size of the perturbation. This guarantees that the new
state m2 is uniformly bounded and centered around the old state, m1. The
maximum perturbation size is selected so that the acceptance rate of the MCMC,
which is defined as the ratio between the number of models that are accepted
and the number of proposed models, is above 20%. After m2 is chosen, the
forward model calculates its corresponding output parameters and obtains their
likelihood $L(\textbf{m}_{2}\mid\textbf{d})$, as shown in Eq. 14. The
acceptance probability is estimated with the log-likelihoods
$l(\textbf{m}\mid\textbf{d})=log(L(\textbf{m}\mid\textbf{d}))$ as:
$P_{accept}=min\left\\{1,e^{(l(\textbf{m}_{2}\mid\textbf{d})-l(\textbf{m}_{1}\mid\textbf{d}))}\right\\}$
(16)
If $P_{accept}$ is greater than a number drawn from a uniform distribution
between 0 and 1, m2 is accepted and the chain moves to the state characterised
by m2, starting the next chain $n+1$. Otherwise, the chain remains in the
state of m1 and a different set of model parameters is proposed as m3. To make
sure that the posterior distributions converge and that all parameter space is
explored, we run $10^{4}$ chains. In other words, with acceptance rates
between 0.2 and 0.6, the MCMC proposes between 1.6 and 5 $\times$ $10^{4}$
sets of model inputs.
## 3 System parameters of TRAPPIST-1
Agol et al. (2020) have performed an analysis of TTVs that includes all
transit data from Spitzer since the discovery of the system. We adopt these
data for the mass, radius and semi-major axis in our interior structure
analysis (Table 2).
TRAPPIST-1 does not have available data regarding its chemical composition.
However, the Fe/Si abundance ratio can be estimated assuming that TRAPPIST-1
presents a similar chemical composition to that of other stars of the same
metallicity, age and stellar population. As proposed by Unterborn et al.
(2018), we select a sample of stars from the Hypatia Catalog (Hinkel et al.,
2014, 2016, 2017). We choose the set of stars by constraining the C/O mole
ratio to be less than 0.8, and the stellar metallicity between -0.04 and 0.12,
as this is the metallicity range calculated for TRAPPIST-1 by Gillon et al.
(2017). We discard thick disk stars since TRAPPIST-1 is likely a thin disk
star. Our best-fit Gaussian to the distribution of the Fe/Si mole ratio shows
a mean of 0.76 and a standard deviation of 0.12. Since this Fe/Si value is an
estimate based on the chemical composition of a sample of stars that belong to
the same stellar population of TRAPPIST-1, we present two scenarios for each
planet. In scenario 1, the only available data are the planetary mass and
radius, while scenario 2 includes the estimated stellar Fe/Si mole ratio to
constrain the bulk composition.
For temperate planets that cannot have a steam atmosphere, we set the surface
temperature in our interior model to their equilibrium temperatures assuming
an albedo zero (Table 2). Although surface temperatures for thin atmospheres
are lower than that obtained with this assumption, the dependence of the bulk
radius on surface temperature for planets with condensed water is low. For
example, if we assume a pure water planet of 1 $M_{\oplus}$ with a surface
pressure of 1 bar, the increase in radius due to a change of surface
temperature from 100 K to 360 K is 0.002 $R_{\oplus}$, which is less than 0.2%
of the total radius, i.e., 10 times less than our convergence criterion.
Additionally, the atmospheres of TRAPPIST-1 planets in the habitable zone and
farther are significantly thinner than those of the highly-irradiated planets.
Lincowski et al. (2018) estimate thicknesses of approximately 80 km for
temperate planets in TRAPPIST-1, which is negligible compared to their total
radius. Therefore, we only calculate the atmospheric parameters (OLR, surface
temperature, albedo and thickness of the atmosphere) for planets that present
their hydrospheres in steam phase.
Planet | $M$ [$M_{\oplus}$] | $R$ [$R_{\oplus}$] | $a_{d}$ [$10^{-2}$ AU] | $T_{eq}[K]$
---|---|---|---|---
b | 1.374$\pm$0.069 | 1.116${}^{+0.014}_{-0.012}$ | 1.154 | 398
c | 1.308$\pm$0.056 | 1.097${}^{+0.014}_{-0.012}$ | 1.580 | 340
d | 0.388$\pm$0.012 | 0.788${}^{+0.011}_{-0.010}$ | 2.227 | 286
e | 0.692$\pm$0.022 | 0.920${}^{+0.013}_{-0.012}$ | 2.925 | 250
f | 1.039$\pm$0.031 | 1.045${}^{+0.013}_{-0.012}$ | 3.849 | 218
g | 1.321$\pm$0.038 | 1.129${}^{+0.015}_{-0.013}$ | 4.683 | 197
h | 0.326$\pm$0.020 | 0.775$\pm$0.014 | 6.189 | 172
Table 2: Masses, radii and semi-major axis for all planets in TRAPPIST-1 (Agol
et al., 2020). Equilibrium temperatures are calculated assuming a null albedo,
with the stellar effective temperature, stellar radius and semi-major axis
provided by Agol et al. (2020).
## 4 Characterisation of hydrospheres
### 4.1 CMF and WMF posterior distributions
Planet | $M_{ret}\ [M_{\oplus}]$ | $R_{ret}\ [R_{\oplus}]$ | CMF | WMF | Fe/Siret
---|---|---|---|---|---
b | 1.375$\pm$0.041 | 1.116$\pm$0.013 | 0.261$\pm$0.146 | (3.1${}_{-3.1}^{+5.0}$) $10^{-5}$ | 1.00$\pm$0.56
c | 1.300$\pm$0.036 | 1.103$\pm$0.015 | 0.239$\pm$0.084 | (0.0${}_{-0.0}^{+4.4}$) $10^{-6}$ | 0.71$\pm$0.26
d | 0.388$\pm$0.007 | 0.790$\pm$0.010 | 0.409$\pm$0.167 | 0.084$\pm$0.071 | 1.22${}^{+1.30}_{-1.22}$
e | 0.699$\pm$0.013 | 0.922$\pm$0.015 | 0.447$\pm$0.123 | 0.094$\pm$0.067 | 1.75$\pm$1.17
f | 1.043$\pm$0.019 | 1.047$\pm$0.015 | 0.409$\pm$0.140 | 0.105$\pm$0.073 | 1.44$\pm$1.14
g | 1.327$\pm$0.024 | 1.130$\pm$0.016 | 0.399$\pm$0.144 | 0.119$\pm$0.080 | 1.33$\pm$1.29
h | 0.327$\pm$0.012 | 0.758$\pm$0.013 | 0.341$\pm$0.192 | 0.081${}^{+0.089}_{-0.081}$ | 0.13${}^{+1.80}_{-0.13}$
Table 3: Output parameters retrieved by the MCMC method for all TRAPPIST-1 planets: total mass ($M_{ret}$) and radius ($R_{ret}$), CMF, WMF and Fe/Si molar ratio. In this case the mass and radius are considered as input data (scenario 1). Planet | $M_{ret}\ [M_{\oplus}]$ | $R_{ret}\ [R_{\oplus}]$ | CMF | WMF | Fe/Siret
---|---|---|---|---|---
b | 1.359$\pm$0.043 | 1.124$\pm$0.016 | 0.259$\pm$0.032 | (0.0${}_{-0.0}^{+3.4}$) $10^{-6}$ | 0.79$\pm$0.10
c | 1.299$\pm$0.034 | 1.103$\pm$0.014 | 0.257$\pm$0.031 | (0.0${}_{-0.0}^{+2.7}$) $10^{-6}$ | 0.79$\pm$0.11
d | 0.387$\pm$0.007 | 0.792$\pm$0.010 | 0.241$\pm$0.032 | 0.036$\pm$0.028 | 0.76$\pm$0.12
e | 0.695$\pm$0.012 | 0.926$\pm$0.012 | 0.249$\pm$0.031 | 0.024${}_{-0.024}^{+0.027}$ | 0.78$\pm$0.12
f | 1.041$\pm$0.019 | 1.048$\pm$0.013 | 0.240$\pm$0.031 | 0.037$\pm$0.026 | 0.76$\pm$0.12
g | 1.331$\pm$0.023 | 1.131$\pm$0.015 | 0.235$\pm$0.031 | 0.047$\pm$0.028 | 0.75$\pm$0.12
h | 0.326$\pm$0.011 | 0.758$\pm$0.013 | 0.232$\pm$0.032 | 0.055$\pm$0.037 | 0.75$\pm$0.12
Table 4: Output parameters retrieved by the MCMC method for all TRAPPIST-1
planets: total mass ($M_{ret}$) and radius ($R_{ret}$), CMF, WMF and Fe/Si
molar ratio. In this case the Fe/Si mole ratio estimated by following
Unterborn et al. (2018) is also included as data (scenario 2).
Tables 3 and 4 show the retrieved parameters, including the total planetary
mass and radius, and the Fe/Si mole ratio. In both scenarios, we retrieve the
mass and radius within the 1$\sigma$–confidence interval of the measurements
for all planets. In scenario 1, where only the mass and radius data are
considered, we retrieve Fe/Si mole ratios without any assumptions on the
chemical composition of the host star. Although the uncertainties of these
estimates are more than 50% in some cases, we can estimate a common Fe/Si mole
ratio for the planetary system. This common Fe/Si range is determined by the
overlap of the 1$\sigma$ confidence intervals of all planets, which
corresponds to Fe/Si = 0.45-0.97. This interval is compatible with the Fe/Si
mole ratio of 0.76$\pm$0.12 proposed by Unterborn et al. (2018). This overlap
can also be seen in Fig. 2, which presents the 1$\sigma$–confidence regions
derived from the 2D marginalized posterior distributions of the CMF and WMF.
The minimum value of the common CMF is determined by the lower limit of the
confidence region of planet g, which is approximately 0.23, whereas the common
maximum CMF value corresponds to the upper limit of planets b and c, which is
0.4. This is partially in agreement with the CMF obtained in scenario 2, when
we assume the Fe/Si mole ratio proposed by Unterborn et al. (2018), which are
found between 0.2 and 0.3 (Table 4). Thus, the CMF of the TRAPPIST-1 planets
could be compatible with an Earth-like CMF (CMF${}_{\oplus}=$ 0.32).
Figure 2: Top panel:1$\sigma$–confidence regions derived from the two-
dimensional posterior distributions for the first scenario, where only the
masses and radii are available as data. Bottom panel: 1$\sigma$ confidence
regions derived from the two-dimensional posterior distributions for the
second scenario, where the Fe/Si abundance ratio from Unterborn et al. (2018)
is considered together with the mass and radius for each planet. The axis of
the ternary diagram indicate the CMF, the WMF and the mantle mass fraction MMF
= 1 - CMF - WMF.
In scenario 1, the retrieved WMF for all planets in the system are below 20%
within their uncertainties. This maximum WMF limit reduces to 10% for scenario
2. This indicates that the TRAPPIST-1 system is poor in water and other
volatiles, especially the inner planets b and c. Both planets are compatible
with a dry composition in both scenarios, although the presence of an
atmosphere cannot be ruled out given the possible CMF range estimated in
scenario 1.
### 4.2 Water phases
Figure 3: OLR and absorbed radiation as a function of surface temperature for
the steam atmospheres of TRAPPIST-1 b, c and d. Vertical dotted lines indicate
the surface temperature at which the absorbed flux is equal to the OLR for
planets b and c.
Figure 3 shows the OLR calculated by the atmosphere model and the absorbed
radiation (Eq. 12 and 13) for planets b, c and d. For temperatures lower than
$\sim$ $T_{surf}=2000$ K, the OLR has little dependency on the surface
temperature. This is caused by the nearly constant temperature (between 250
and 300 K) of the radiating layers in the thermal IR range (Goldblatt et al.,
2013) and it is related to the runaway greenhouse effect (Ingersoll, 1969). We
obtain a constant OLR or an OLR limit (Nakajima et al., 1992) of 274.3, 273.7
and 254.0 $W/m^{2}$ for planets b, c and d, respectively. These are close to
the OLR limit obtained by Katyal et al. (2019) of 279.6 $W/m^{2}$ for an
Earth-like planet. The small difference is due to their different surface
gravities. As explained in Sect. 2.3, if the atmosphere can find a surface
temperature at which the OLR and the absorbed radiation are equal, their
atmospheres are in global radiative balance. This is the case for planets b
and c, whose surface temperatures are approximately 2450 K and 2250 K,
respectively. These are above the temperatures where the blanketting effect is
effective, named $T_{\varepsilon}$ in Marcq et al. (2017) implying that the
atmospheres of planets b and c are in a post-runaway state. However, planet d
is not in global radiative balance since its absorbed radiation never exceeds
its OLR. This means that planet d would be cooling down, and an internal flux
of approximately 33 $W/m^{2}$ would be required to supply the extra heat to
balance its radiative budget. TRAPPIST-1 inner planets are likely to present
an internal heat source due to tidal heating (Barr et al., 2018; Dobos et al.,
2019; Turbet et al., 2018). The tidal heat flux estimated for planet d is
$F_{tidal}=0.16\ W/m^{2}$ (Barr et al., 2018), which is one order of magnitude
lower than needed for radiative-convective balance of a steam atmosphere. Due
to the blanketting effect of radiation over the surface of planet d, the OLR
limit is larger than the absorbed radiation and hence the planet can cool
enough to present its hydrosphere in condensed phases.
Figure 4: ($P$,$T$) profiles of the hydrospheres of TRAPPIST-1 planets. The
dashed-dotted grey horizontal line indicates the 20 mbar pressure level (see
text). Thicker lines indicate the profile for the minimum WMF estimated for
each planet in scenario 1, while thinner lines mark the profile for the
maximum WMF under the same scenario. The minimum WMF of planets b, c and h is
zero.
Figure 4 shows the ($P$,$T$) profiles and the different phases of water we can
find in the hydrospheres of the TRAPPIST-1 planets. The maximum WMF of planets
b and c are $8.1\ \times\ 10^{-5}$ and $4.4\ \times\ 10^{-6}$, which
correspond to a surface pressure of 128.9 bar and 4.85 bar, respectively.
The thermal structure of their steam atmospheres are dominated by a lower,
unsaturated troposphere where water condensation does not occur. Then the
atmosphere consists of a middle, saturated troposphere where cloud formation
would be possible, extending up to 10 mbar, and finally an isothermal
mesosphere above. Since we consider a clear transit radius of 20 mbar (Grimm
et al., 2018; Mousis et al., 2020) the presence of clouds above this pressure
level would flatten the water features in the planetary spectrum (Turbet et
al., 2019; Katyal et al., 2020). On the other hand, planets d and e could
present water in liquid phase, which could be partially or completely covered
in ice Ih. While the hydrosphere of planet h is not massive enough to attain
the high pressures required for ice VII at its base, planets d, to g can reach
pressures up to a 100 GPa.
### 4.3 Retrieval of atmospheric parameters
Figure 5: 2D and 1D marginal posterior distributions for the atmospheric
parameters (surface temperature $T_{surf}$, atmospheric thickness $z_{atm}$,
albedo and atmospheric mass $M_{atm}$), and bulk mass and radius, $M_{bulk}$
and $R_{bulk}$, of TRAPPIST-1 b (left panel) and c (right panel). These have
been derived under scenario 1, where we do not consider Fe/Si data.
Figure 5 shows the output atmospheric parameters (surface temperature,
atmospheric thickness, albedo and atmospheric mass) of TRAPPIST-1 b and c for
a water-dominated atmosphere in scenario 1. The total thickness of an
atmosphere is related to its scale height, which is defined as $H=RT/\mu g$,
where R = 8.31 J/K mol is the gas constant, $T$ is the mean atmospheric
temperature, $\mu$ the mean molecular mass and $g$ the surface gravity
acceleration. For planets b and c, the mean atmospheric temperatures are 940.4
and 499.4 K, and their surface gravities are 10.8 and 10.7 $m/s^{2}$,
respectively. The mean molecular mass for a 99% water and 1% CO2 atmosphere is
18.3 g/mol. The mean temperature increases with surface temperature, while the
mean molecular mass is determined by the composition of the atmosphere.
For the same composition and surface gravity, the scale height and therefore
the thickness of the atmosphere is directly correlated to the surface
temperature. As shown in Fig. 5, the atmospheric thickness, $z_{atm}$
increases with the surface temperature $T_{surf}$. This is known as the
runaway greenhouse radius inflation effect (Goldblatt, 2015; Turbet et al.,
2019), where a highly irradiated atmosphere is more extended than a colder one
despite having similar compositions. For planet b, its atmosphere can extend
up to 450 km, while planet c presents a maximum extension of 300 km. The
minimum limit for the thicknesses is zero, which corresponds to the case of a
dry composition. Ortenzi et al. (2020) estimated that for a planet of 1-1.5
$M_{\oplus}$ the maximum atmospheric thickness due to the outgassing of an
oxidised mantle is 200 km, which is compatible with the ranges we have
obtained for the atmospheric thicknesses. Scenario 2 shows the same trends for
the atmospheric parameters but with lower atmospheric mass and surface
pressure. With their WMF posterior distributions centered in zero and low
standard deviation, the surface pressure is below 1 bar and atmospheric
thicknesses below 100 km in most of the accepted models, which means that in
scenario 2 planets b and c are most likely dry rocky planets.
## 5 Discussion
### 5.1 WMF comparison with previous works
Figure 6: Water mass fraction as a function of the distance to the star for
the TRAPPIST-1 system. Upper panel shows our estimates for scenario 1 and
those of Barr et al. (2018), where only mass and radius data were taken into
account. The lower panel corresponds to scenario 2, whose CMF is constrained
in a narrow range between 0.2 and 0.3, while for Agol et al. (2020) we show
the WMF for a CMF of 0.25.
Agol et al. (2020) use the interior and atmosphere models presented in Dorn et
al. (2018) and Turbet et al. (2020b) to obtain the WMF estimated of the
TRAPPIST-1 planets with updated and more precise radii and masses data from
Spitzer TTVs (Agol et al., 2020). We thus limited the comparison to the sole
results of Agol et al. (2020) with the same input values. By doing so, we can
be certain that the variations in WMF estimates are due to our different
modelling approach. Figure 6 shows that planets b and c are most likely dry in
scenario 2, where the resulting CMF are between 0.2 and 0.3 for the whole
system. We obtain maximum estimates of 3.4 $\times$ $10^{-6}$ and 2.7 $\times$
$10^{-6}$ for b and c, respectively. For the same density, the estimated value
of the WMF depends on the CMF that is considered. Therefore we compare WMF
estimates for similar CMF between this work and Agol et al. (2020). We show
our WMF in scenario 2, since the CMF of all planets spans a narrow range
between 0.2 and 0.3, which are the most similar values to one of the CMF
assumed by Agol et al. (2020), CMF = 0.25. Our WMF for the steam planets of
the TRAPPIST-1 system are in agreement with Agol et al. (2020), who calculated
a maximum WMF of $10^{-5}$ for a constant CMF of 0.25. We are able to reduce
the maximum limit of the water content of the highly irradiated planets
compared to previous studies and establish the most likely WMF with our
coupled atmosphere-interior model. The calculation of the total radius
requires a precise determination of the atmospheric thickness. This depends
strongly on the surface temperature and the surface gravity, which are
obtained with radiative transfer in the atmosphere, and the calculation of the
gravity profile for a bulk mass and composition in the interior self-
consistently.
In the case of planet d, we estimate a WMF of 0.036 $\pm$ 0.028, while Agol et
al. (2020) obtain an upper limit of $10^{-5}$. The latter estimate considers
that water is in vapor form, which is less dense than condensed phases, while
our model shows that the surface conditions allow liquid or ice phases,
resulting in a higher WMF. This discrepancy in the possible water phases on
the surface of planet d is due to different atmospheric compositions. We
consider a water-dominated atmosphere with 1% CO2, while Agol et al. (2020)
and Turbet et al. (2020b) assume a N2 and H2O mixture. This difference in
composition changes radiative balance since CO2 is a strong absorber in the IR
compared to N2, which is a neutral gas. Nonetheless, N2 is subject to stellar
wind-driven escape and it is unlikely to be stable for the inner planets of
TRAPPIST-1, while CO2 is more likely to survive thermal and ion escape
processes (Turbet et al., 2020a).
Our WMF for planets e to h are in agreement within uncertainties with Agol et
al. (2020), although their central values are significantly lower. The EOS
employed to compute the density of the water layers in Agol et al. (2020) is
also used in Dorn et al. (2018) and Vazan et al. (2013), which agrees well
with the widely-used SESAME and ANEOS EOSs (Baraffe et al., 2008). These EOS
are not consistent with experimental and theoretical data since they
overestimate the density at pressures higher than 70 GPa (Mazevet et al.,
2019). This yields an underestimation of the WMF for the same total planetary
density and CMF.
For the specific case of scenario 1, with no assumptions on the stellar
composition and the Fe/Si mole ratio, we compared our CMF and WMF with those
obtained in Barr et al. (2018) (Figure 6 and Table 5). These authors use
masses and radii data given by Wang et al. (2017). They obtain lower masses
compared to Agol et al. (2020) while their radii are approximately similar,
which would explain why Barr et al. (2018) tend to overestimate the water
content of the TRAPPIST-1 planets. Moreover, most of the mass uncertainties in
Wang et al. (2017) are 30-50%, while the mass uncertainties obtained by Agol
et al. (2020) are 3-5%. This causes Barr et al. (2018) to calculate wider CMF
and WMF 1$\sigma$ confidence intervals. In addition, there are differences
between our interior modelling approach and that of Barr et al. (2018). For
example, according to their results, planet b can have up to 50% of its mass
as water. This high WMF value is due to the assumption that the hydrosphere is
in liquid and ice I phases, and high-pressure ice polymorphs (HPPs), which are
more dense than the steam atmosphere we consider. In contrast, the CMF seems
to be closer to our estimates, especially for planet b, d and e, where their
maximum CMF is approximately 0.40, in agreement with our CMF 1$\sigma$
intervals.
Planet | CMF
---|---
| Barr et al. (2018) | This study (2020)
b | 0.00-0.43 | 0.12-0.41
c | 0.00-0.98 | 0.16-0.32
d | 0.00-0.39 | 0.24-0.58
e | 0.00-0.40 | 0.32-0.57
f | 0.00 | 0.24-0.58
g | 0.00 | 0.26-0.54
h | 0.00 | 0.15-0.53
Table 5: Comparison between our one-dimensional 1$\sigma$ confidence regions
for the CMF and those of Barr et al. (2018). We show only estimates for
scenario 1, since Barr et al. (2018) did not consider any constraints on the
Fe/Si ratio based on stellar composition.
We can also discuss the possible habitability of the hydrospheres of the
TRAPPIST-1 planets by comparing our WMF estimates with the layer structure as
a function of planetary mass and water content obtained by Noack et al. (2016)
. According to Noack et al. (2016), a habitable hydrosphere must be structured
in a single liquid water ocean or in several ice layers that enable the
formation of a lower ocean layer. This lower ocean would be formed by the heat
supplied by the mantle that melts the high pressure ice in the ice-mantle
boundary (Noack et al., 2016). For planet d, a surface liquid ocean would form
for all its possible WMF if the atmosphere allows for the presence of
condensed phases. For planets e, f and g, the hydrosphere could be stratified
in a surface layer of ice Ih and a liquid or an ice II-VI layer. In the case
we had low-pressure ices II-VI, their base could be melted by the heat
provided by the mantle, and form a lower ocean layer as suggested by Noack et
al. (2016). At WMF $\geq$ 0.10, less than 50% of the possible configurations
enable a habitable sub-surface ocean layer, and at a WMF $\geq$ 0.14, the
hydrosphere is uninhabitable. In scenario 1, planets e to g reach these values
within uncertainties, although their minimum values extend down to 0-0.03 in
WMF, which would be the habitable regime.
### 5.2 System formation and architecture
Figure 7: Mass-radius relationships for planets with CO2-dominated atmospheres
assuming different CMF. The surface pressure is 300 bar. The black dot and its
error bars indicate the location and uncertainties of planet d in the mass-
radius diagram.
In the case of scenario 1, where no Fe/Si data is assumed, the WMF increases
with the distance to the star with the exception of planet h, whose WMF is
similar to that of planet d. In the case of scenario 2, where a common Fe/Si
of 0.76 $\pm$ 0.12 is assumed for the whole system, the WMF increases with the
distance to the star (Fig. 6) with the exception of planet d whose WMF is
similar to that of planet f, which is more water-rich than planet e. This
slight deviation from the observed trend could be explained by migration,
where planet d could have formed beyond the snow line and then migrated
inwards (Raymond et al., 2018). In addition, pebble ablation and water
recycling back into the disk could have been less efficient in the case of
planet d, compared to planet e (Coleman et al., 2019). On the other hand, the
gas at the distance at which planet d formed could have been more enriched in
volatiles than the outer planets, accreting more water ice than planet e in a
’cold finger’ (Stevenson & Lunine, 1988; Cyr et al., 1998). Pebble formation
in the vicinity of the water iceline can induce important enhancements of the
water ice fraction in those pebbles due to the backward diffusion of vapor
through the snowline and the inward drift of ice particles. Therefore, if a
planet forms from this material, it should be more water-rich than those
formed further (Mousis et al., 2019). These formation scenarios could explain
the high WMF of planet d when we assume that its water layer is in condensed
phases. Post-formation processes could also have shaped the trend of the WMF
with axis, such as atmospheric escape due to XUV and X-ray emission from their
host star. Bolmont et al. (2017) estimated a maximum water loss of 15 Earth
Oceans (EO) for TRAPPIST-1 b and c and 1 EO for planet d. If we assume that
the current WMF are the central values of the posterior distributions we
derived in scenario 1, planets b, c and d would have had an initial WMF of
2.37 $\times\ 10^{-3}$, 2.50 $\times 10^{-3}$ and 0.085, respectively.
Therefore, atmospheric escape would have decreased the individual WMF of each
planet, but the increase of WMF with distance from the star would have been
preserved.
In addition to the WMF-axis trend, we can differentiate the very water-poor,
close-in planets, b and c, from the outer, water-rich planets, d to h. This
has been reported as a consequence of pebble accretion in the formation of
other systems, such as the Galilean moons. While Io is dry, Callisto and
Ganymede are water-rich, with Europa showing an intermediate WMF of 8% (Ronnet
et al., 2017). Pebble-driven formation can produce planets with WMF $\geq 15$%
if these are formed at the water ice line (Coleman et al., 2019; Schoonenberg
et al., 2019). In contrast, planets formed within the ice line would present
WMF less than 5% (Liu et al., 2020; Coleman et al., 2019), which is close to
the mean value we calculated for planet h, 5.5%. The maximum WMF limit in the
first scenario is approximately 20%. This maximum limit is significantly lower
than the typical WMF generated by planetesimal accretion scenario, which is
50-40% (Miguel et al., 2020). Therefore, our results are consistent with the
pebble-driven formation scenario.
### 5.3 Alternative atmospheric compositions
However, the atmosphere of planet d could be dominated by other atmospheric
gases different from H2O-based mixtures, which could produce an extended
atmosphere and increase the total planetary radius. Hydrogen-dominated
atmospheres have been deemed unlikely as one of the possible atmospheric
compositions for all planets in the TRAPPIST-1 system, both cloud-free (de Wit
et al., 2016, 2018) and with high-altitude clouds and hazes (Grimm et al.,
2018; Ducrot et al., 2020). Similarly, CH4-dominated atmospheres are not
probable given the photometry data of the Spitzer Space Telescope (Ducrot et
al., 2020). Therefore, our best candidate to explain the low density of planet
d in a water-poor scenario is CO2. We find that a CO2-dominated atmosphere
with 1% water vapor in planet d would be in radiative-convective equilibrium
by computing the OLR and absorbed radiation, as we did for water-dominated
atmospheres. The resulting surface temperature is approximately 950 K, which
is slightly higher than the surface temperature of Venus (700 K) with a higher
water vapor mixing ratio. Figure 7 introduces the mass-radius relationships
for different CMF, assuming a CO2-dominated atmosphere with a surface pressure
of 300 bar. It shows that planet d is compatible with a planet with a
CO2-dominated atmosphere and CMF between 0.2 and 0.3, which is a very likely
CMF range for TRAPPIST-1 planets based on our analysis. Surface pressures
lower than 300 bar would yield lower atmospheric thicknesses, so it would be
necessary to consider a lower CMF to explain the observed density of planet d.
CO2 in the case of planet d can be provided by volcanic outgassing (Ortenzi et
al., 2020), since its internal heat flux produced by tidal heating is in the
range 0.04-2 $W/m^{2}$, which favours plate tectonics (Papaloizou et al.,
2018). Secondary CO2-dominated atmospheres could have traces of O2, N2 and
water vapor.
## 6 Conclusions
We presented an interior structure model for low-mass planets at different
irradiations that is valid for a wide range of water phases, derived from the
approaches of Brugger et al. (2017) and Mousis et al. (2020). For highly-
irradiated planets, we couple a 1D water steam atmosphere in radiative-
convective equilibrium with a high-pressure convective layer in supercritical
phase. The density in this layer is computed by using an accurate EOS for
high-pressure and high-temperature water phases. For temperate planets whose
surface conditions allow the formation of condensed phases, we implemented a
hydrosphere with liquid water and ice phases Ih, II, III, V, VI and VII. In
addition, we adapted the MCMC Bayesian algorithm described in Dorn et al.
(2015) to our interior model to derive the posterior distributions of the
compositional parameters, WMF and CMF, given mass, radius and stellar
composition data. We then applied our interior model to the particular case of
TRAPPIST-1 planets using the latest mass and radius data from Spitzer (Agol et
al., 2020).
The hydrospheres of TRAPPIST-1 planets have been characterised by calculating
their P-T profiles and thermodynamic phases. Planets b and c are warm enough
to present steam atmospheres. They could hold post-runaway greenhouse
atmospheres with thicknesses up to 450 km and surface temperatures up to 2500
K, which means that they are extended enough to be suitable targets for
atmospheric characterisation by future space-based facilities such as JWST.
Moreover, planets d to g present their hydrospheres in condensed phases. These
hydrospheres can contain high-pressure ices that start to form at
$10^{9}-10^{10}$ Pa.
We have obtained CMF and WMF probability distributions for all planets in the
system. We found that the Fe/Si mole ratio of the system is in the 0.45-0.97
range without considering any assumption on the chemical composition of the
stellar host. This Fe/Si range corresponds to a CMF value in the 0.23-0.40
range, making the CMF of TRAPPIST-1 planets compatible with an Earth-like
value (0.32). In addition, our WMF estimates agree within uncertainties with
those derived by Agol et al. (2020), although their most likely values are
considerably lower for planets with condensed phases. In the case of planets
with steam hydrospheres, their densities are compatible with dry rocky planets
with no atmospheres. Nevertheless, we cannot rule out the presence of an
atmosphere with the Fe/Si range we have derived without any assumption on the
chemical composition of the host star. When considering a possible estimate of
the Fe/Si ratio of the host star (scenario 2), we obtained lower maximum
limits of the WMF for planets b and c compared to previously calculated limits
by Agol et al. (2020) for a similar CMF of 0.25. Our estimated WMF in steam
and condensed phases are consistent with an increase of WMF with progressing
distance from the host star. This trend, as well as the maximum WMF we
calculated, favour pebble-driven accretion as a plausible formation mechanism
for the TRAPPIST-1 system. However, planet d presents a slightly higher WMF
than planet e. This could be due to processes that took place during planet
formation, such as migration, a low-efficient ablation of pebbles and gas
recycling or an enhancement of the water ice fraction in pebbles at the
distance of the disc where planet d formed. An extended atmosphere dominated
by greenhouse gases different from a water-dominated atmosphere, such as CO2,
could also explain the low-density of planet d compared to planet e.
Future work should include more atmospheric processes and species that
determine the mass-radius relations of planets with secondary atmospheres in
the Super-Earth and sub-Neptune regime. These can vary the atmospheric
thickness and increase the total planetary radius with varying atmospheric
masses while other compositional parameters change the bulk radius. These
should be integrated in one single interior-atmosphere model, combined in a
MCMC Bayesian framework such as the one we used in this study. This
statistical approach has been employed with interior models for planets with
H/He-dominated atmospheres (Dorn et al., 2017b, a, 2018), or dry planets
(Plotnykov & Valencia, 2020), but not for planets with secondary, CO2 and
steam-dominated atmospheres. The integrated model should also include a
description of escape processes, such as hydrodynamic or Jeans escapes, which
is particularly interesting to explore the lifetime of secondary atmospheres.
Close-in, low-mass planets are likely to outgas atmospheric species, such as
CO2, and form O2 via photodissociation of outgased H2O, during their magma
ocean stage or due to plate tectonics (Chao et al., 2020). Thus, a mixture of
these gases should be considered to study the thermal structure of planets
with secondary atmospheres. Planets b and c in the TRAPPIST-1 system could
present magma oceans due to their high surface temperatures (T $\geq$ 1300 K)
(Barr et al., 2018; Chao et al., 2020), and the maximum surface pressure we
have obtained here can be used to assess the current outgassing rate in magma
ocean studies (Noack et al. (2017), Baumeister et al. submitted) and better
constrain the WMF for the interior magma ocean models (e.g Katyal et al.
(2020)) in the future.
###### Acknowledgements.
MD and OM acknowledge support from CNES. We acknowledge the anonymous referee
whose comments helped improve and clarify this manuscript.
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# Quasirandom Graphs and the Pantograph Equation
Asaf Shapira School of Mathematics, Tel Aviv University, Tel Aviv 69978,
Israel. Email<EMAIL_ADDRESS>Mykhaylo Tyomkyn Department of Applied
Mathematics, Charles University, Czech Republic. Email:
<EMAIL_ADDRESS>
###### Abstract
The pantograph differential equation and its solution, the deformed
exponential function, are remarkable objects that appear in areas as diverse
as combinatorics, number theory, statistical mechanics, and electrical
engineering. In this article we describe a new surprising application of these
objects in graph theory, by showing that the set of all cliques is not forcing
for quasirandomness. This provides a natural example of an infinite family of
graphs, which is not forcing, and answers a natural question posed by P. Horn.
## 1 The pantograph equation and its solution.
The best known differential equation
$y^{\prime}(x)=py(x),\ y(0)=1$
is solved by the exponential function $y=e^{px}$. Consider a visually similar
but very different equation
$y^{\prime}(x)=y(px),\ y(0)=1,$
where $0<p\leq 1$. This is a special case of the _pantograph_ equation, one of
the most studied examples of _delay differential equations_ , where the value
of the derivative of $y$ at time $x$ is a function of the value of $y$ at an
earlier time $px$. The pantograph equation owes its name to a component of the
electric locomotive, connecting it to the overhead wire, and was first studied
in the late 1960’s by British Railways. The equation results from the study of
the pantograph’s movement, where it is crucial that the pantograph stays in
constant contact with the wire, in order to collect current for the locomotive
without interruptions; see [4] for more historical and physics background.
The pantograph equation and its generalizations have found many applications
in physics (see [10]) and mathematics. It can be verified that its unique
solution is the so-called _deformed exponential function_
$f_{p}(x)=\sum_{j=0}^{\infty}\frac{x^{j}}{j!}p^{\binom{j}{2}}=1+x+\frac{x^{2}}{2}p+\frac{x^{3}}{6}p^{3}+\cdots;$
(1)
note that when $p=1$ we have $f_{1}(x)=e^{x}$, hence the name. In fact, the
deformed exponential was first studied by Mahler in 1940 [12], nearly 30 years
before the pantograph equation made an appearance. Mahler’s motivation was in
number theory: he used the function $f_{p}$ to derive an asymptotic formula
for the number of partitions of a large integer $n$ into powers of a fixed
integer $r$. Later the deformed exponential was found to appear naturally in
many other contexts. In combinatorics it is connected to the Tutte polynomial
of complete graphs [22], the enumeration of acyclic digraphs [16], and
inversions of trees [13]. In statistical mechanics the function $f_{p}$
appears as the partition function of one-site lattice gas [17], and in complex
analysis it is related to the Whittaker and Goncharov constants [1]. The
function $f_{p}$ continues to be the focus of current research; see for
example the recent paper [23] which studies the asymptotics of its roots.
In this article, we present an unexpected application of certain properties of
the deformed exponential function to the theory of quasirandom graphs.
## 2 Quasirandom graphs and forcing families.
_Quasirandomness_ , or _pseudorandomness_ , is a phenomenon occurring in
several areas of discrete mathematics: number theory, group theory,
combinatorics, and graph theory. It can be loosely described as the study of
properties of truly random objects in deterministic ones; see [21] for a
general survey, and [7] for a survey on pseudorandom graphs.
Let us first focus on counting copies of a fixed small graph $H$ inside a
large graph $G$. It will be more convenient to count _labeled_ copies, that
is, injective mappings from the vertex set of $H$ to that of $G$ that map
edges to edges. Let us illustrate this with some examples.
###### Example 1.
If $G$ is a complete bipartite graph111A complete bipartite graph is a graph
on the vertex set $A\cup B$, where $A$ and $B$ are disjoint, and whose edge
set is $A\times B$. More generally, a complete $k$-partite graph is a graph
whose vertex set is composed of $k$ disjoint sets $A_{1},\ldots,A_{k}$ and
whose edge set is $\bigcup_{i<j}A_{i}\times A_{j}$. In other words, every two
vertices in distinct $A_{i},A_{j}$ are connected by an edge. with both parts
of size $n/2$, where $n$ is a large even number, and $H$ is a star with $k$
edges, i.e., a complete bipartite graph with part sizes $1$ and $k$, then $G$
contains
$n\cdot\frac{n}{2}\left(\frac{n}{2}-1\right)\cdots\left(\frac{n}{2}-k+1\right)=(1+o(1))2^{-k}n^{k+1}$
labeled copies of $H$.
In the above example, as well as throughout the rest of the article, any
$o(1)$ expression should be read as a function tending to $0$ as $n$ goes to
$\infty$. The $o()$-notation is an equivalent, yet more convenient-to-use form
of the usual $\epsilon$-$\delta$-$n_{0}$ formalism. In particular, we can use
multiple $o(1)$-expressions in the same formula, saving us the need to
introduce multiple $\epsilon$’s (two $o(1)$-expressions are not assumed to be
identical). Note that here and later the $o()$-notation assumes that $n$ tends
to infinity and treats all other variables (e.g., $k$ in Example 1) as
constant parameters.
###### Example 2.
If $G$ is a clique222A clique or complete graph is a graph on a vertex set
$V$, whose edge set consists of all pairs $\\{u,v\\}\subseteq V$. on $pn$
vertices, where $n$ is large and $0<p\leq 1$, and $H$ is a cycle of length
$k$, then $G$ has
$pn(pn-1)\cdots(pn-k+1)=(1+o(1))p^{k}n^{k}$
labeled copies of $H$.
For $0<p<1$ and a large integer $n$, the _$p$ -random graph_ on $n$ vertices
[3], also known as the binomial random graph and denoted by $G(n,p)$, is
obtained by taking $n$ labeled vertices and including every edge between them
randomly and independently with probability $p$. Then, a standard
probabilistic argument using Chebyshev’s inequality (also known as the second
moment method [3]) implies the following.
###### Example 3.
For any fixed graph $H=(V,E)$, the random graph $G(n,p)$ contains with high
probability (i.e., with probability tending to $1$ as $n$ grows)
$(1+o(1))p^{|E|}n^{|V|}$
labeled copies of $H$. In particular, $G(n,p)$ with high probability contains
$(1+o(1))pn^{2}$ labeled edges and $(1+o(1))p^{4}n^{4}$ labeled copies of the
$4$-cycle $C_{4}$.
The reason we singled out the edges and $C_{4}$’s in the last example is the
following seminal result of Chung, Graham, and Wilson [2, Theorem 1]. In what
follows, when speaking about “large graphs,” we mean, formally, sequences of
graphs with the number of vertices $n$ tending to $\infty$.
###### Theorem 1 ([2]).
Suppose that $0<p<1$. The following properties of (large) $n$-vertex graphs
$G$ are equivalent.
1. (P1)
$G$ has $(1+o(1))pn^{2}$ labeled edges and
$(1+o(1))p^{4}n^{4}$
labeled copies of $C_{4}$.
2. (P2)
For every fixed graph $H=(V,E)$, the number of labeled copies of $H$ in $G$ is
$(1+o(1))p^{|E|}n^{|V|}.$
3. (P3)
For every $c>0$ and vertex set $S\subseteq V(G)$ of size $|S|\geq cn$, the
number of labeled edges between vertices in $S$ is
$(1+o(1))p|S|^{2}.$
Observe that, by Example 3, a random graph $G(n,p)$ satisfies with high
probability property (P2) and thus, a fortiori, also (P1). Similarly, a
standard application of the Chernoff bound [3] shows that $G(n,p)$ satisfies
with high probability property (P3). The remarkable aspect of Theorem 1 is
that every (deterministic) graph that satisfies the seemingly very weak
property (P1) must also satisfy the much stronger properties (P2) and (P3). In
fact, the main result of [2] exhibited a number of further equivalent
conditions (which we do not state here formally, for brevity).
A large graph satisfying either (P1), (P2), or (P3) (and therefore all three
of them), is called _$p$ -quasirandom_. A graph is _quasirandom_ if it is
$p$-quasirandom for some $0<p<1$. The notion of quasirandomness is central to
extremal combinatorics: for instance, Szemerédi’s famous regularity lemma [20]
states (vaguely speaking) that the vertices of a large graph can be
partitioned into a bounded number of parts, so that “almost all” bipartite
graphs between those parts are quasirandom, i.e., resemble a truly random
subgraph of a complete bipartite graph.
A very sensible question to ask at this point is, whether any
deterministically constructed quasirandom graphs are known to exist. The
answer is yes, and one prominent class of examples are the _Paley graphs_
(arising from a similar construction for matrices in [15]). Their
quasirandomness can be deduced from properties of quadratic residues.
###### Example 4.
Let $n=4k+1$ be a prime, so that $x$ is a quadratic residue modulo $n$ if and
only if $-x$ is one. Let $G$ be a graph on the vertex set $\\{0,\dots,n-1\\}$,
where $xy$ is an edge whenever $x-y$ is a quadratic residue modulo $n$. Then
$G$ is $1/2$-quasirandom.
On the cautious side we would like to add that some properties of the truly
random graph $G(n,p)$ are not captured by quasirandomness. For example, the
largest clique in $G(n,p)$ is with high probability of order $\log n$, whereas
in quasirandom graphs it can be of “almost linear” size.
The discussion above led the authors of [2] to define the notion of a
_forcing_ graph family.
###### Definition.
A family of graphs $\mathcal{F}$ is _forcing_ if the following holds for every
$0<p<1$. Suppose $G$ is an $n$-vertex graph so that for every
$F\in\mathcal{F}$ the graph $G$ contains $(1+o(1))n^{|V(F)|}p^{|E(F)|}$
labeled copies of $F$. Then $G$ is $p$-quasirandom.
By this definition, Theorem 1 states that the pair $\\{K_{2},C_{4}\\}$ is
forcing. To obtain a better understanding of the quasirandomness phenomenon,
one should look for a classification of forcing families. This natural inverse
question to Theorem 1 was raised by Chung, Graham, and Wilson in the same
paper [2].
In the subsequent years, this topic has seen a significant amount of research
(see [5] and the references therein), resulting in discoveries of a number of
further forcing families. It is well known that for any nonbipartite graph $H$
the pair $\\{K_{2},H\\}$ is not forcing, and the main open question in this
area is the _forcing conjecture_ by Skokan and Thoma [18], saying that for
every connected bipartite graph $H$ that is not a tree the pair
$\\{K_{2},H\\}$ is forcing. In [18] this was proved for every complete
bipartite graph $H$ (more generally, it was shown in [18] that
$\\{H_{1},H_{2}\\}$ is forcing for any pair of distinct complete bipartite
graphs), but the full conjecture is still wide open.
In the light of the above, it is even more challenging to decide whether an
_infinite_ family of graphs is forcing. Consider, for instance, four of the
most natural such families, namely the sets of all cycles, stars, trees, and
cliques. It is easy to see that the family of all cycles is not forcing for
all $0<p<1$, since, by Example 2, the graph comprising a clique on $pn$
vertices and $(1-p)n$ isolated vertices has the “correct” number
$(1+o(1))p^{\ell}n^{\ell}$ of labeled cycles of length $\ell$, but fails to
satisfy (P3) of Theorem 1. Similarly, the family of all stars is not forcing
by Example 1 — in fact, this example shows that the family of all trees is not
forcing.
The situation is less clear for the set of all finite cliques
$\mathcal{K}=\\{K_{2},K_{3},\dots\\}$, where $K_{j}$ denotes the complete
graph on $j$ vertices. Horn [24] asked whether $\mathcal{K}$, or perhaps some
finite subset of $\mathcal{K}$, is forcing (the latter would clearly imply the
former) — this would mean that any large graph having
$(1+o(1))p^{\binom{j}{2}}n^{j}$ labeled copies of $K_{j}$ for each $j\geq 2$
satisfies property (P3) of Theorem 1. Both questions have been open until now,
and our aim in this article is to answer them in the negative.
First we give an elementary proof of the fact that any set of finitely many
cliques is not forcing.
###### Theorem 2.
For any $k\geq 2$ and $0<p\leq 1/4$ there exist arbitrarily large $n$-vertex
graphs $G_{k,p}(n)$ with $(1+o(1))p^{\binom{j}{2}}n^{j}$ labeled copies of
$K_{j}$ for all $j=2,\dots,k$, and an independent set of size at least $n/2$.
Therefore, the family $\mathcal{K}_{k}=\\{K_{2},\dots,K_{k}\\}$ is not
forcing.
The theorem above deals only with finitely many cliques and only with $p\leq
1/4$.333Strictly speaking, in order to prove that a family $\mathcal{F}$ is
not forcing it is enough to exhibit a counterexample for just one $0<p<1$.
Still, it is more desirable to give examples for all $0<p<1$. By applying some
properties of the deformed exponential function defined in (1), we prove our
main result, extending Theorem 2 to all cliques and to all values of $p$.
###### Theorem 3.
The infinite family $\mathcal{K}=\\{K_{2},K_{3},\dots\\}$ is not forcing for
any $0<p<1$.
## 3 Proofs of the main results.
### 3.1 The finite case.
First we deal with the case of finitely many cliques.
###### Proof of Theorem 2..
We claim that for fixed $k\geq 2$ and $0<p\leq 1/4$ there exist $k$ real
nonnegative numbers $(c_{1},\dots,c_{k})=(c_{1}(k,p),\dots,c_{k}(k,p))$ with
the following properties.
1. (i)
For all $1\leq j\leq k$,
$\sum_{A\in\binom{[k]}{j}}\prod_{i\in A}c_{i}=\frac{p^{\binom{j}{2}}}{j!},$
where $\binom{[k]}{j}$ stands for the set of all $j$-element subsets of
$\\{1,\dots,k\\}$.
2. (ii)
$\max c_{i}\geq 3/4$.
Given these numbers, define $G=G_{k,p}(n)$ to be a graph on $n$ vertices as
follows. Partition $V(G)$ into sets $V_{1},\dots,V_{k}$, such that for all
$1\leq i\leq k$ we have $||V_{i}|-c_{i}n|<1$; this is possible since, by (i),
$\sum_{i=1}^{k}c_{i}=1$. Let $E(G)$ be the set of all edges $uv$ where $u\in
V_{i},v\in V_{j}$ for $i\neq j$. In other words, $G$ is the complete
$k$-partite graph on $(V_{1},\dots,V_{k})$.
By this construction, for every $j=2,\dots,k$ the graph $G_{k,p}(n)$ has
$\displaystyle j!\sum_{A\in\binom{[k]}{j}}\prod_{i\in A}|V_{i}|$
$\displaystyle=j!\sum_{A\in\binom{[k]}{j}}\prod_{i\in A}(c_{i}n\pm
1)=(1+o(1))j!\sum_{A\in\binom{[k]}{j}}\prod_{i\in A}c_{i}n^{j}$
$\displaystyle=(1+o(1))p^{\binom{j}{2}}n^{j}$
labeled copies of $K_{j}$. Here $c_{i}n\pm 1$ stands for an integer within $1$
of $c_{i}n$. On the other hand, by (ii), it has an independent set of size at
least $n\cdot\max c_{i}-1\geq n/2$. Hence, this graph satisfies the assertions
of Theorem 2. Note that the fact that $G_{k,p}(n)$ has an independent set of
size at least $n/2$ implies it fails property (P3) of Theorem 1 for $c=1/2$.
Therefore, $G_{k,p}$ is not $p$-quasirandom for any $0<p<1$, and we conclude
that $\mathcal{K}_{k}$ is not forcing.
It remains to construct the sequence $(c_{1},\dots,c_{k})$ with the properties
above. To this end, consider the real polynomial function
$f_{p,k}(x)=\sum_{j=0}^{k}\frac{{p^{\binom{j}{2}}}}{j!}x^{j},$ (2)
which is a truncated version of the deformed exponential function (1).
Kurtz [8] established the following useful criterion for a polynomial to have
only real roots, which can be viewed as a converse to Newton’s inequalities.
###### Proposition 4 ([8], Theorem 2).
If the coefficients of a real polynomial $P(x)=\sum_{i=0}^{n}b_{i}x^{i}$
satisfy $b_{i}>0$ for all $i$ and $b_{i}^{2}>4b_{i-1}b_{i+1}$ for all $1\leq
i\leq n-1$, then all the roots of $P$ are real and distinct.444Note that the
constant $4$ is best possible for $n=2$
In order to apply Proposition 4 to the polynomial $f_{p,k}(x)$ all we need to
check is that
$\frac{p^{2\binom{j}{2}}}{j!j!}>\frac{4p^{\binom{j+1}{2}+\binom{j-1}{2}}}{(j+1)!(j-1)!}$
holds for every $j=1,\dots,k-1$. This simplifies to
$\frac{1}{j}>\frac{4p}{j+1},$
which is evidently true for all $p\leq 1/4$. Thus, $f_{p,k}$ has $k$ distinct
real roots $a_{1},\dots,a_{k}$, and can be written as
$f_{p,k}(x)=\frac{p^{\binom{k}{2}}}{k!}\prod_{i=1}^{k}(x-a_{i}).$ (3)
Since the coefficients of $f_{p,k}$ are positive, $f_{p,k}(x)>0$ for $x\geq
0$, and thus all of the roots are negative. Writing $c_{i}:=-1/a_{i}$ (so that
$a_{i}=-1/c_{i}$), we obtain
$\displaystyle f_{p,k}(x)$
$\displaystyle=\frac{p^{\binom{k}{2}}}{k!}\prod_{i=1}^{k}\left(x+\frac{1}{c_{i}}\right)=\frac{p^{\binom{k}{2}}}{k!}\left(\prod_{i=1}^{k}c_{i}\right)^{-1}\prod_{i=1}^{k}(1+c_{i}x)$
$\displaystyle=\frac{p^{\binom{k}{2}}}{k!}(-1)^{k}\prod_{i=1}^{k}a_{i}\prod_{i=1}^{k}(1+c_{i}x).$
(4)
We will now show that the above-defined $c_{1},\dots,c_{k}$ satisfy the
properties (i) and (ii) stated at the beginning of the proof. Evaluating the
constant term in (3) gives
$\frac{p^{\binom{k}{2}}}{k!}(-1)^{k}\prod_{i=1}^{k}a_{i}=1,$
which, combined with (3.1), implies
$f_{p,k}(x)=\prod_{i=1}^{k}(1+c_{i}x).$ (5)
Next, evaluating in (5) the coefficient of $x^{j}$ for all $1\leq j\leq k$
gives
$\sum_{A\in\binom{[k]}{j}}\prod_{i\in A}c_{i}=\frac{p^{\binom{j}{2}}}{j!},$
(6)
establishing property (i). In particular, (6) implies $\sum_{i=1}^{k}c_{i}=1$
and $\sum_{1\leq i<j\leq k}c_{i}c_{j}=p/2$. Therefore,
$\displaystyle\max c_{i}$ $\displaystyle=\max
c_{i}\cdot\sum_{i=1}^{k}c_{i}\geq\sum_{i=1}^{k}c_{i}^{2}=\left(\sum_{i=1}^{k}c_{i}\right)^{2}-2\sum_{1\leq
i<j\leq k}c_{i}c_{j}=1-p\geq\frac{3}{4},$
establishing property (ii). ∎
It turns out that the polynomial $f_{p,k}$ has imaginary roots when $p>1/2$;
hence the approach we used in the proof above cannot cover all $0<p<1$.
Therefore, to extend this to all $0<p<1$ and to handle the set of all cliques,
we need a slightly different approach.
### 3.2 The general case.
We now prove that $\mathcal{K}$, the family of all finite cliques, is not
forcing. As a first step towards the proof of Theorem 3, we will construct an
“infinite graph” satisfying its assertion. We briefly mention that in the
theory of graph limits (see [11]) such an object is called a graphon.
###### Lemma 5.
For every $0<p<1$ there is a graph $W_{p}$ whose vertex set is the interval
$[0,1]$ and which satisfies:
1. (1)
For every $k\geq 2$, if we randomly and (Lebesgue-)uniformly select $k$
vertices, $v_{1},\dots,v_{k}$ from $[0,1]$, then the probability that for
every $i<j$ the vertices $v_{i},v_{j}$ are connected by an edge in $W_{p}$ is
$p^{\binom{k}{2}}$.
2. (2)
There is an interval $I\subseteq[0,1]$ of length $1-p$ so that $\\{x,y\\}$ is
not an edge of $W_{p}\ $ for every $x,y\in I$.
Observe that assertion (1) is a continuous counterpart to the property of a
finite $n$-vertex graph containing the “correct” number of cliques. Similarly,
assertion (2) states that $W_{p}$ has an independent set on a $(1-p)$-fraction
of its vertices — a finite graph with this property would fail to satisfy (P3)
of Theorem 1.
###### Proof of Lemma 5.
For a sequence of positive real numbers $\mathcal{C}=(c_{1},c_{2},\dots)$ and
an integer $k\geq 1$, we use $\sigma_{k}(\mathcal{C})$ to denote the formal
expression $\sum_{A\in\binom{\mathbb{N}}{k}}\prod_{j\in A}c_{j}$. Similarly to
the proof of Theorem 2, we claim that for every $0<p<1$ there exists a
sequence $\mathcal{C}=(c_{1},c_{2},\dots)$ of positive reals $1>c_{1}\geq
c_{2}\geq\dots>0$ with the following properties.
1. (i)
For each $k\geq 1$, $\sigma_{k}(\mathcal{C})$ is convergent, with
$\sigma_{k}(\mathcal{C})=p^{\binom{k}{2}}/k!$.
2. (ii)
$\max c_{i}=c_{1}\geq 1-p.$
With such a sequence at hand, we partition the $[0,1]$-interval into
infinitely many intervals $V_{1},V_{2},\dots$, such that $|V_{i}|=c_{i}$ for
all $i$ (note that $\sum_{i=1}^{\infty}c_{i}=\sigma_{1}(\mathcal{C})=1$), and
we let $W_{p}$ be the graph on the vertex set $[0,1]$, where $x$ and $y$ are
connected by an edge if and only if they belong to different intervals $V_{i}$
and $V_{j}$.
Then for every fixed $k\geq 2$, a random sample of $k$ vertices
$v_{1},\dots,v_{k}$ from $[0,1]$ forms a clique in $W_{p}$ with probability
$k!\sigma_{k}(\mathcal{C})=p^{\binom{k}{2}}$. Moreover, for the interval
$V_{1}$ we have $|V_{1}|=c_{1}\geq 1-p$, and no $x,y\in V_{1}$ are connected
by an edge in $W_{p}$. Hence, $W_{p}$ satisfies both assertions of the lemma.
To construct the desired sequence $\mathcal{C}$, we consider the deformed
exponential function $f_{p}(z)$ over a complex variable $z$, defined in (1).
It was shown in [6] and [14] that for every $0<p<1$ the function $f_{p}$ can
be represented as a product:
$f_{p}(z)=\prod_{i=1}^{\infty}\left(1-\frac{z}{a_{i}}\right),$ (7)
where the $a_{i}$, the roots of $f_{p}$, are real (which means they must be
negative, as the power series of $f_{p}$ has only positive signs).555To be
more detailed, $f_{p}(z)$ is an _entire function_ , that is, $f_{p}$ is
defined on all of $\mathbb{C}$, and is holomorphic everywhere (for more
background information about entire functions see [9, 19]). The _order_ of an
entire function $g(z)=\sum_{j=0}^{\infty}a_{j}z^{j}$ is given by the formula
$\rho=\limsup_{n\rightarrow\infty}\frac{n\log n}{\log(1/|a_{n}|)}$ ([9,
Section 1.3, Theorem 2]). Thereby, the deformed exponential $f_{p}$ is of
order $0$. By Hadamard’s factorization theorem ([19, Theorem 5.1]) an entire
function of order $0$, taking value $1$ at $x=0$, can be represented as a
product (7), where the $a_{i}$ are its complex roots. In [6, 14] it was shown
that for any $0<p<1$ all roots of $f_{p}$ are real. Thus, we can set
$c_{i}=-1/a_{i}$ and $\mathcal{C}:=(c_{1},c_{2},\dots)$, where the elements
are indexed in descending order, to obtain
$f_{p}(z)=\prod_{i=1}^{\infty}(1+c_{i}z).$
By comparing the coefficients of $z^{k}$ for every $k\geq 1$ we obtain that
each $\sigma_{k}(\mathcal{C})$ is convergent, with
$\sigma_{k}(\mathcal{C})=p^{\binom{k}{2}}/k!$. Furthermore, since all $c_{i}$
are positive, and $\sigma_{1}$ and $\sigma_{2}$ are (absolutely) convergent,
for $c_{1}=\max\mathcal{C}$ we get
$\displaystyle c_{1}$ $\displaystyle=c_{1}\cdot
1=c_{1}\cdot\sigma_{1}(\mathcal{C})\geq\sum_{i=1}^{\infty}c_{i}^{2}=(\sum_{i=1}^{\infty}c_{i})^{2}-2\sum_{1\leq
i<j}c_{i}c_{j}$
$\displaystyle=\sigma_{1}(\mathcal{C})^{2}-2\sigma_{2}(\mathcal{C})=1-p.$
∎
Lemma 5 gives an example of a “complete infinite-partite” graph that contains
for every $j\geq 2$ the same fraction of copies of $K_{j}$ as the random graph
$G(n,p)$. Such a construction of course cannot be achieved for finite graphs.
Instead, for every $k\geq 2$ we show how to turn $W_{p}$ into large graphs
containing the correct number of labeled copies of $K_{j}$ for every $j\leq
k$.
###### Proof of Theorem 3.
Take the infinite graph $W_{p}$ defined in Lemma 5, select $n$ vertices
$v_{1},\dots,v_{n}$ from it uniformly at random, and let $G_{p}(n)$ be the
graph induced on these vertices. That is, the edges of $G$ are the edges of
$W_{p}$ between $v_{1},\dots,v_{n}$.
Fix $k\geq 2$ and $\epsilon>0$. By assertion (1) of Lemma 5 and the law of
large numbers, for a sufficiently large $n=n(k,\epsilon)$ the graph $G_{p}(n)$
will, with probability greater than $1/2$, contain between
$(1-\epsilon)n^{j}p^{\binom{j}{2}}$ and $(1+\epsilon)n^{j}p^{\binom{j}{2}}$
labeled copies of $K_{j}$ for all $j=2,\dots,k$. Additionally, assertion (2)
of Lemma 5 together with the law of large numbers imply that, for large $n$,
with probability greater than $1/2$, $G_{p}(n)$ will have an independent set
of size at least $(1-p)n/2$. Thus, with positive probability, there exists a
graph $G_{k,p,\epsilon}(n)$ that has both properties. In other words,
$G_{k,p,\epsilon}(n)$ is an $n$-vertex graph containing between
$(1-\epsilon)n^{j}p^{\binom{j}{2}}$ and $(1+\epsilon)n^{j}p^{\binom{j}{2}}$
labeled $j$-cliques for $j=2,\dots,k$ and an independent set of size at least
$(1-p)n/2$.
Now, select a sequence $\epsilon_{1}>\epsilon_{2}>\dots>0$ such that
$\lim_{k\rightarrow\infty}\epsilon_{k}=0$, for instance, $\epsilon_{k}=1/k$.
For each $k$ take a graph $G_{k}=G_{k,p,\epsilon_{k}}(n)$ (for some $n$), and
consider the sequence $G_{2},G_{3},\dots$; for the order $n=|V(G_{k})|$ of the
graphs we have $n\geq k$ (as $G_{k}$ contains a $k$-clique), so $n$ tends to
$\infty$. Moreover, by construction, for every $j\geq 2$ the graphs in the
sequence contain $(1+o(1))p^{\binom{j}{2}}n^{j}$ labeled copies of $K_{j}$,
while also containing an independent set of size at least $(1-p)n/2$. In
particular, $G=G_{k}$ fails property (P3) of Theorem 1, implying that $G$ is
not $p$-quasirandom. Therefore, the set of all cliques
$\mathcal{K}=\\{K_{2},K_{3},\dots\\}$ is not forcing for any $0<p<1$. ∎
###### Remark.
The proof of Lemma 5 relied on the fact that the roots of $f_{p}$ are all
real. It is worth noting that a lot more is known about these numbers; for
instance, the $k$th largest root (denoted $a_{k}$ above) is known to be of
order $-kp^{1-k}$. Wang and Zhang [23] very recently established the
asymptotics of the roots of $f_{p}$ up to arbitrary lower order terms.
## Acknowledgments.
We thank two anonymous referees for their helpful remarks. The first author
was supported in part by ISF Grant 1028/16, ERC Consolidator Grant 863438 and
NSF-BSF Grant 20196. The second author was supported in part by ERC Synergy
grant DYNASNET 810115 and the H2020-MSCA-RISE project CoSP- GA No. 823748.
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|
# Bayesian GARCH Modeling of Functional Sports Data
Patric Dolmeta Department of Decision Sciences, Università Bocconi, Milano,
Italy Raffaele Argiento Department of Statistical Sciences, Università
Cattolica del Sacro Cuore, Milano, Italy & Collegio Carlo Alberto, Torino,
Italy Silvia Montagna<EMAIL_ADDRESS>ESOMAS Department, University
di Torino, Torino, Italy & Collegio Carlo Alberto, Torino, Italy
###### Abstract
The use of statistical methods in sport analytics has gained a rapidly growing
interest over the last decade, and nowadays is common practice. In particular,
the interest in understanding and predicting an athlete’s performance
throughout his/her career is motivated by the need to evaluate the efficacy of
training programs, anticipate fatigue to prevent injuries and detect
unexpected of disproportionate increases in performance that might be
indicative of doping. Moreover, fast evolving data gathering technologies
require up to date modelling techniques that adapt to the distinctive features
of sports data. In this work, we propose a hierarchical Bayesian model for
describing and predicting the evolution of performance over time for shot put
athletes. To account for seasonality and heterogeneity in recorded results, we
rely both on a smooth functional contribution and on a linear mixed effect
model with heteroskedastic errors to represent the athlete-specific
trajectories. The resulting model provides an accurate description of the
performance trajectories and helps specifying both the intra- and inter-
seasonal variability of measurements. Further, the model allows for the
prediction of athletes’ performance in future seasons. We apply our model to
an extensive real world data set on performance data of professional shot put
athletes recorded at elite competitions.
Keywords: Performance analysis Bayesian functional data analysis GARCH models
Sport analytics Latent factor modelling
## 1 Introduction
Shot put is a track and field event involving throwing (“putting”) the shot, a
metal ball (7.26kg/16lb for men, 4kg/8.8lb for women), with one hand as far as
possible from a seven-foot diameter (2.135m) circle. In order for each put to
be considered valid, the shot must not drop below the line of the athlete’s
shoulders and must land inside a designated 35-degree sector. Athletes
commonly put four to six times per competition, and their best performance is
recorded. Figure 1 displays the results of elite shot put competitions for
four athletes with careers of different lengths.
Figure 1: Each panel displays the performance results (points) of a
professional shot put athlete throughout his/her career at elite competitions.
Performance is measured in meters (length of the throw) and plotted against
the days elapsed since January 1st of each athlete’s career starting year. The
dark vertical lines represent seasons changing points, namely new year days.
Furthermore, several athlete-specific variables (e.g., age, gender, doping
history, nationality, etc.) that might be of interest in a sport analytics
perspective are naturally gathered during elite competitions or can be easily
retrieved retrospectively. Shot put competitions have been part of the modern
Olympics since their revival, hence there exist complete and long-established
data sets for shot put performances that can be exploited for performance
analysis.
Sportive competitions might display some sort of seasonality in the results.
We underline that here the term “seasonality” is not used to indicate a
cyclical behaviour of the results over time as in the literature of time
series but, rather, a time dependent gathering of observations. On one hand
competitions are traditionally concentrated in some months of the year, and on
the other hand weather and environmental conditions may affect the
performances or even the practicability of the sport itself. Straightforward
examples are sports leagues that run on a different schedule and host their
season openers, playoffs and championships at different times of the year or,
even more dramatically, winter sports as opposed to outdoor water sports. Shot
put events range over the whole year, with indoor competitions held during
Winter months and major tournaments, like the Olympics, Diamond League and
World Championship organised during Summer. Therefore, it is reasonable to say
that seasons coincides with calendar years. In Figure 1, vertical lines
represent new years’ days: the time point at which seasons change. To provide
an accurate representation of the data, seasonality effects need to be taken
into account. Note that, from a modelling perspective, the amplitude of
seasons is arbitrary, and ideally any model should be easily adapted to any
sport with any type of seasonality.
There exist a well established literature of both frequentist and Bayesian
contributions to performance data analysis in various sports. We mention,
among others, Malcata et al. (2014) and Mengersen et al. (2016) in triathlon,
Costa et al. (2010) and Costa et al. (2013) in swimming, and Koulis et al.
(2014) in cricket. Albert (2016) describe batting performance in baseball,
Wimmer et al. (2011) performance evolution in decathlon, Yousefi and Swartz
(2013) in golf, and Montagna et al. (2020) in tennis. Further, Vaci et al.
(2019) rely on a Bayesian latent variable model to investigate age-related
changes across the complete lifespan of basketball athletes on the basis of
exponential functions.
In this work, we are interested in describing the evolution of performances of
professional shot put athletes throughout their careers. To this end, we
propose a Bayesian hierarchical model where results of each athlete are
represented as error prone measurements of some underlying unknown function.
The distinctive contribution of our approach is the form of such function,
which has an additive structure with three components. First, we consider a
smooth functional contribution for capturing the overall variability in
athletes’ performances. For this component, we follow the approach in Montagna
et al. (2012), who propose a Bayesian latent factor regression model for
detecting the doping status of athletes given their shot put performance
results and other covariates. However, the authors limit the analysis to data
collected from 2012, whereas we aim at describing the trajectories in
performance over the whole time span available for our data (1996 to 2016).
For this time span, a global smoothness assumption for the trajectories could
be too restrictive. Indeed, data may exhibit jumps localised at fixed and
shared time points among all athletes. See, for example, athlete 303 in Figure
1, whose results show a jump between the second and fourth season (calendar
year) of his/her career. The presence of jumps between seasons is even more
striking when yearly average performances are considered. We highlight this
behaviour by showing yearly averages via thick horizontal lines in Figure 1.
Accordingly, the description of this yearly dependent contribution is
delegated to a mixed effect model, that quantifies the seasonal mean for each
athlete as a deviation from a grand mean. This component captures the inter-
seasonal variability of the data set, whereas the smooth functional component
describes the intra-seasonal evolution of performances. Finally, we complete
our model specification accounting for the effect of a selection of time
dependent covariates through a regressive component. We embed our model in a
Bayesian framework by proposing suitable prior distributions for all
parameters of interest. We believe the presented model represents a flexible
tool to analyse evolution of performances in measurable sports, namely, all
those disciplines for which results can be summarised by a unique measure
(e.g., distance, time or weight).
The rest of the paper is organised as follows. In Section 2, we describe the
motivating case study. In Section 3, we present the proposed model and briefly
discuss possible alternative settings. Moreover, we elicit priors for our
Bayesian approach. In Section 4, we outline the algorithm for posterior
computation. In Section 5, we discuss posterior estimates, argue on the
performance of the model and interpret the model’s parameters form a sports
analytic perspective. Conclusions are presented in Section 6. Finally, the
Appendix includes the complete description of the MCMC algorithm.
## 2 The World Athletics shot put data set
World Athletics (WA) is the world governing body for track and field athletic
sports. It provides standardized rules, competition programs, regulated
technical equipment, a list of official world records and verified
measurements. The data at our disposal was obtained with permission from an
open results database (www.tilastopaja.eu) following institutional ethical
approval (Prop_72_2017_18). The data set comprises 56,000 measurements of WA
recognized elite shot put competitions for 1,115 athletes from 1976 to 2016.
For each athlete, the data set reports the date of the event, the best result
in meters, the finishing position, an indication of any doping violation
during the athlete’s career as well as demographic information (athlete’s
name, WA ID number, date of birth, sex and country of birth).
In this work, we restrict our analysis to results for athletes performing
after 1996. Indeed, we pursue consistency of measurement accuracy, and 1996
represents a turning point in anti-doping regulation and fraud detection
procedures. The resulting data set is still sufficiently broad for our
purposes. It contains 41,033 observations for 653 athletes (309 males and 344
females). The outcome of interest is the shot distance, which ranges from a
minimum if $10.6$ up to a maximum if $22.56$ meters, with a mean of $17.30$
meters.
As shown in Figure 1 for a selection of athletes, data are collected over
time. Hereafter we will denote as $t_{ij}$ the time at which the $j$-th
observation for athlete $i$ is recorded. $t_{ij}$ corresponds to the time
elapsed from January 1st of each athlete’s career starting year to the date of
the competition. Accordingly, equal time values for different athletes are
likely to specify distinct years, but the same moment in those athletes’
careers. Moreover, different athletes will have observations ranging over a
large time span, according to the length of their careers. Having described
seasons as calendar years, athletes will also compete in a different number of
seasons. Figure 2 shows the number of athletes per season as well as boxplots
of the distribution of their mean performances across the various seasons.
Athletes with the longest careers have been playing for 19 years. A general
increasing trend in performance can be observed as a function of career length
(right panel in Figure 2) or, equivalently, the age of the athlete. In the
following, we will discuss two different modelling choices for age,
respectively, accounting for its time dependence and considering age as a
fixed quantity, namely the age of the athlete at the beginning of his/her
career.
Table 1 reports descriptive statistics suggesting how sex, environment and
doping have an effect on the average value of the result. We point out that in
our dataset we only have 18 athletes who tested positive for doping at some
point in their career. Information on the date the test was taken (or if
multiple tests were taken) is not available in the data. As expected,
performances for men are, on average, higher than for women. Similar effects,
despite less evident in magnitude, also hold true for the variable
environment, which takes values indoor and outdoor. Regarding environment, a
further remark is in order. We already pointed out that major WA events take
place outdoor (27,800 observations) during Summer months, whereas less
competitive events are held inside between November and March (13,200
observations). Figure 3 displays results in gray when recorded outdoor, and in
black otherwise. We can clearly see that field events gather in Summer months,
whereas indoor events take place during Winter (solid lines).
Figure 2: Left: total number of athletes per season. Right: each boxplot shows the distribution of the athletes’ mean performances within each season. Table 1: Performance results conditioned on covariates. | Mean | Sd | Max | Min
---|---|---|---|---
Total | 17.30 | 1.78 | 22.56 | 10.6
Women | 16.09 | 1.35 | 21.70 | 10.6
Men | 18.55 | 1.21 | 22.56 | 12.93
Not Doped | 17.30 | 1.79 | 22.56 | 10.6
Doped | 17.77 | 1.35 | 20.88 | 13.55
Indoor | 17.17 | 1.70 | 22.23 | 12.15
Outdoor | 17.38 | 1.81 | 22.56 | 10.6
Figure 3: Performance results displayed according to the variable environment,
which takes values indoor (black) and outdoor (gray). Vertical lines
corresponding to the label ticks represent season changes, whereas the two
enclosing it indicate the boundaries of winter months.
## 3 The model
Let $n$ denote the total number of athletes in the study. We assume that shot
put performances for athlete $i$ are given by noisy measurements of an
underlying function $g_{i}(t_{ij})$:
$y_{ij}=g_{i}(t_{ij})+\epsilon_{ij}$ (1)
with $\epsilon_{ij}\stackrel{{\scriptstyle iid}}{{\sim}}N(0,\psi^{2})$
independent errors. Recall $t_{ij}$ is the time at which the $j$-th
observation for athlete $i$ is collected, for $j=1,\ldots,n_{i}$, where
$n_{i}$ is the total number of measurements available on athlete $i$.
We further suggest an explicit functional form for $g_{i}(t_{ij})$:
$g_{i}(t_{ij})=f_{i}(t_{ij})+\mu_{is}+\boldsymbol{x}_{i}(t_{ij})\boldsymbol{\beta}$
(2)
where $f_{i}(t)$ is a smooth functional component for intra-seasonal
variability, $\mu_{is}$ a season-specific intercept, and
$\boldsymbol{x}_{i}(t)\boldsymbol{\beta}$ is an additional multiple regression
component. Here $s\in\\{1,2,\ldots,S_{i}\\}$ indicates the season in which the
shot was recorded. Specifically,
$\mu_{is}\equiv\mu_{i}(t_{ij})=\sum_{s=1}^{S_{i}}\mu_{is}\
\mathbb{I}_{(t_{i}^{s},t_{i}^{s+1})}(t_{ij})$ is an athlete-specific step
function taking value $\mu_{is}$ for all time points in season $s$, delimited
by $t_{i}^{s}$ and $t_{i}^{s+1}$. For a complete treatment of the notation
used insofar, please refer to Table 2.
We will now discuss each of the three terms in Eq. (2) in more detail.
Table 2: Mathematical notation Symbol | Meaning
---|---
i | Index identifying the athlete
j | Index identifying a specific observation
n | Total number of athletes
N | Total number of observations
$t_{ij}$ | Time point at which the j’th observation of athlete i is recorded
$S_{i}$ | Total number of seasons for athlete $i$
s | The currently considered season
$g_{is}$ | Number of observations in season $s$ for athlete $i$
r | The number of covariates to be considered
$y_{ij}$ | Response variable at time $j$ for athlete $i$
### 3.1 The functional component
The functional component $f_{i}(t)$ is meant to capture the subject-specific
global evolution of the response variable. It explains the global dependence
of the data from time. We require that these functions display a smooth
behaviour: the latter is assured by assuming $\\{f_{i}(t)\\}_{i=1}^{n}$ are
linear combinations of smooth basis functions, $\\{b_{m}(t)\\}_{m=1}^{p}$.
Note that, both the nature and the number $p$ of these bases are to be
determined according to some properties we wish them to satisfy. In
particular, we assume:
$f_{i}(t)=\sum_{{m}=1}^{p}\theta_{im}b_{m}(t)$ (3)
where $\\{b_{m}(t)\\}_{m=1}^{p}$ represent the B-spline basis (de Boor, 1978)
and $\\{\theta_{im}\\}_{m=1}^{p}$ are subject-specific coefficients.
We briefly recall that the B-spline basis of degree $k$ on $[L,U]$ is a
collection of $p$ polynomials defined recursively on a sequence of points,
known as knots, and indicated with $L\equiv t_{1}\leq\ldots\leq
t_{p+k+1}\equiv U$. We follow the common approach of choosing $k=3$, leading
to cubic splines (see, for instance, Marsden, 1974). Moreover, we assume the
knot sequence to be equispaced and $(k+1)$-open. That is, the first and last
$k+1$ knots are identified with the extremes of the definition interval,
whereas the remaining $p-k-1$ knots divide said interval into sets of the same
length. Under these assumptions, each basis function $b_{j}(t)$ has compact
support over $k+1$ knots, precisely $[t_{j},t_{j+k+1}]$. Moreover, together
they span the space of piecewise polynomial functions of degree $k$ on $[L,U]$
with breakpoints $\\{t_{n}\\}_{n=1}^{p+K+1}$. Finally, such functions are
twice continuously differentiable at the breakpoints, de facto eliminating any
visible type of discontinuity and providing a smooth result.
The number of basis functions is chosen to be large enough for sufficiently
many basis functions to have a completely enclosed support in a given season.
In particular, rescaling time to $[0,1]$ (for computational purposes and
easier prior definition) seasons have amplitude $0.054$ and, if we want at
least a local basis to be completely supported in one of them, we need $4$
knots to fall in it. Accordingly, we require about 75 internal knots and 80
degrees of freedom.
With the sake of tractability, a low dimensional representation of the
individual curves is of interest. Following the approach by Montagna et al.
(2012), we exploit a sparse latent factor model on the basis coefficients:
$\theta_{im}=\sum_{l=1}^{k}\lambda_{ml}\eta_{il}+\xi_{im}$ (4)
where $\lambda_{ml}$ are the entries of a $(p\times k)$ factor loading matrix
$\boldsymbol{\Lambda}$, and $\boldsymbol{\eta_{i}}$ is a vector of $k$ latent
factors for subject $i$. Finally,
$\boldsymbol{\xi_{i}}=(\xi_{i1},\ldots,\xi_{ip})$ is a residual vector,
independent of all other variables in the model. We assume:
$\boldsymbol{\eta_{i}}\stackrel{{\scriptstyle
iid}}{{\sim}}N_{k}(\boldsymbol{0},\boldsymbol{I})$ (5)
and the error terms $\boldsymbol{\xi_{i}}$ are assumed to have normal
distribution with diagonal covariance matrix,
$\boldsymbol{\xi_{i}}\stackrel{{\scriptstyle
iid}}{{\sim}}N_{p}(\boldsymbol{0},diag(\sigma_{1}^{-2},\ldots,\sigma_{p}^{-2}))$,
with $\sigma_{j}^{-2}\stackrel{{\scriptstyle
iid}}{{\sim}}Ga(a_{\sigma},b_{\sigma})$.
For the modelling of the factor loading matrix $\boldsymbol{\Lambda}$, we
follow the approach in Bhattacharya and Dunson (2011). Specifically, we adopt
a multiplicative gamma process shrinkage prior which favours an unknown but
small number of factors $k$. The prior is specified as follows:
$\displaystyle\lambda_{ml}|\phi_{ml}^{-1},\tau_{l}^{-1}\stackrel{{\scriptstyle
iid}}{{\sim}}N\left(0,\phi_{ml}^{-1}\tau_{l}^{-1}\right)\qquad\text{with}$ (6)
$\displaystyle\phi_{ml}\sim
Ga\biggl{(}\frac{\nu_{\phi}}{2},\frac{\nu_{\phi}}{2}\biggr{)}\qquad\tau_{l}=\prod_{v=1}^{h}\varpi_{v}$
$\displaystyle\varpi_{1}\sim Ga(a_{1},1)\qquad\varpi_{v}\sim Ga(a_{v},1)$
Here $\tau_{l}$ is a global stochastically increasing shrinkage parameter for
the $l$-th column which favors more shrinkage as the column index increases.
Similarly, $\phi_{ml}$ are local shrinkage parameters for the elements in the
$l$-th column. Bhattacharya and Dunson (2011) describe a procedure to select
the number of factors $k\ll p$ adaptively. We follow their lead, thus $k$ is
not set a priori but automatically tuned as the Gibbs sampler progresses.
Refer to Bhattacharya and Dunson (2011) for details. Note that, by combining
Eq. (3) and Eq. (4) one gets:
$f_{i}(t)=\sum_{l=1}^{k}\eta_{il}\Phi_{l}(t)+r_{i}(t)$
where $\Phi_{l}(t)=\sum_{m=1}^{p}\lambda_{ml}b_{m}(t)$ is a new, unknown non-
local basis function learnt from the data and $r_{i}(t)$ a functional error
term. Although the number of pre-specified basis function $p$ is potentially
large, the number of “operative” bases $k$ is $k\ll p$ and learnt from the
data.
### 3.2 The seasonal component
Early graphical displays and straightforward exploratory analysis suggest a
significant variability of the average response across seasons as displayed in
Figure 1. Namely, performances prove to be gathered over predetermined time
intervals, the seasons (calendar years). However, it is reasonable to expect
some degree of dependence for the average performance across seasons. To model
such dependence, an autoregressive model for seasonal intercepts can be
proposed. The idea behind this choice is to allow for borrowing of information
across seasons, in the sense that the seasonal intercept $\mu_{is}$ at season
$s$ is influenced by the intercept at season $s-1$ through the autoregressive
coefficient $\rho_{i}$. Namely,
$\mu_{is}\ |\ \rho_{i},\sigma_{\mu}^{2}\stackrel{{\scriptstyle
iid}}{{\sim}}N\left(\rho_{i}\mu_{i(s-1)},\sigma_{\mu}^{2}\right)$ (7)
However, when we first implemented this model, we noted how residuals
presented a pattern which we would like to intercept with a finer model.
Therefore, we consider a random intercept model with Normal Generalized
Autoregressive Conditional Heteroskedastic (GARCH) errors (Bollerslev, 1986).
Specifically,
$\displaystyle\mu_{is}\ |\ m,h_{is}=m+\zeta_{is}\stackrel{{\scriptstyle
iid}}{{\sim}}N(m,h_{is})$ (8) $\displaystyle
h_{is}=\alpha_{0}+\alpha_{1}\zeta_{is-1}^{2}+\varpi h_{is-1}$ (9)
where $\alpha_{0}>0,\alpha_{1}\geq 0$ and $\varpi\geq 0$ to ensure a positive
conditional variance and $\zeta_{is}=\mu_{is}-m$ with $h_{i0}=\zeta_{i0}:=0$
for convenience. The additional assumption of wide-sense stationarity with
$\displaystyle\mathbb{E}(\zeta_{t})=0$
$\displaystyle\mathbb{V}ar(\zeta_{t})=\alpha_{0}(1-\alpha_{1}-\varpi)^{-1}$
$\displaystyle\mathbb{C}ov(\zeta_{t},\zeta_{s})=0\text{ for }t\neq s$
is guaranteed by requiring $\alpha_{1}+\varpi<1$, as proven by Bollerslev
(1986).
Three parameters of the seasonal component require prior specification: the
overall mean $m$ and the conditional variance parameters, $\varpi$ and
$\boldsymbol{\alpha}=(\alpha_{0},\alpha_{1})^{\top}$. For the autoregressive
and heteroskedastic parameters of the GARCH model, we propose non-informative
priors satisfying the positivity constraint. For the overall mean parameter,
we rely on a more informative Normal prior centered around the mean suggested
by posterior analysis of preliminary versions of the model. In particular:
$\displaystyle m\sim N(\mu_{m_{0}},\Sigma_{m_{0}})$
$\displaystyle\boldsymbol{\alpha}\sim N_{2}(\mu_{\alpha},\Sigma_{\alpha})\
\mathbb{I}\\{\boldsymbol{\alpha}>0\\}$ (10) $\displaystyle\varpi\sim
N(\mu_{\varpi},\Sigma_{\varpi})\ \mathbb{I}\\{\varpi\geq 0\\}$
where $\boldsymbol{\alpha}=(\alpha_{0},\alpha_{1})$ is a bidimensional vector.
We complete the model specification assuming that the parameters are
statistically independent and noticing that the hypothesis needed for wide-
sense stationarity do not translate into actual prior conditions on the
parameters. Hence, one of the objects of our analysis becomes to test whether
the constraint $\alpha_{1}+\varpi<1$ holds true.
### 3.3 Covariates
We consider the effect of three covariates, gender, age and environment, and
assume conjugate prior choices for the covariates coefficients:
$\displaystyle\boldsymbol{\beta}\stackrel{{\scriptstyle
iid}}{{\sim}}N(\boldsymbol{\boldsymbol{\beta}_{0}},\sigma_{\beta}^{2}\boldsymbol{\mathbb{I}})$
$\displaystyle\sigma_{\beta}^{-2}\sim
Ga\biggl{(}\frac{\nu_{\beta}}{2},\frac{\nu_{\beta}\sigma_{\beta}^{2}}{2}\biggr{)}$
(11)
## 4 The Bayesian update
Because of the additive nature of the overall sampling model (1) - (2), we are
able to exploit a blocked Gibbs sampler grouping together the parameters of
the three modelling components described in Section 3. Note first that,
because of the high dimensionality of the problem, it is computationally
convenient to choose conditionally conjugate prior distributions for the
parameters. Indeed, conjugacy guarantees analytical tractability of posterior
distributions. In some cases, specifically for the conditional variances of
GARCH errors, no conjugate model exists and updates rely on an adaptive
version of the Metropolis Hastings algorithm for posterior sampling.
Algorithm 1 outlines our sampling scheme, while details are presented in
Appendix A. As far as the parameters of the functional component
$\boldsymbol{\theta}_{i}$ are concerned, we follow Montagna et al. (2012) by
choosing conditionally conjugate prior distributions so that the update
proceeds via simple Gibbs sampling steps. Analogously, the update of the
regression coefficients $\boldsymbol{\beta}$ and the error term $\psi$
proceeds straightforwardly by sampling from their full conditional posterior
distributions. Conjugate priors for the GARCH parameters $m,\varpi$ and
$\boldsymbol{\alpha}$ are not available, therefore we resort to adaptive
Metropolis schemes to draw values from their full conditionals. Specifically,
we build an adaptive scale Metropolis such that the covariance matrix of the
proposal density adapts at each iteration to achieve an _optimal_ acceptance
rate (see Haario et al., 2001).
Further details about the algorithm can be found in the Appendix A, whereas
code is available at https://github.com/PatricDolmeta/Bayesian-GARCH-Modeling-
of-Functional-Sports-Data.
Data: $y_{ij}=(y_{11},\ldots,y_{nn_{n}})$
Set the required MCMC sample size $G$, the burn-in period $g_{0}$ and the
thinning parameter $g_{s}$.
[0mm]
Initialise
$\boldsymbol{\theta}_{i}^{(0)},\boldsymbol{\mu}_{i}^{(0)},m^{(0)},\varpi^{(0)},\boldsymbol{\alpha}^{(0)},\boldsymbol{\beta}_{i}^{(0)},\psi_{i}^{(0)}$
For $g=0,\ldots,G$
Update functional component
Set partial residuals
$y_{ij}^{(1)(g)}=y_{ij}-\mu_{i,s}^{(g)}-\boldsymbol{x_{i}}(t_{ij})\boldsymbol{\beta}^{(g)}$
Update $\boldsymbol{\theta}_{i}^{(g+1)}$ on the base of Appendix A.1
Update seasonal component
Set partial residuals
$y_{ij}^{(2)(g)}=y_{ij}-f_{i}(t_{ij})^{(g)}-\boldsymbol{x_{i}}(t_{ij})\boldsymbol{\beta}^{(g)}$
Update
$\boldsymbol{\mu}_{i}^{(g+1)},m^{(g+1)},\varpi^{(g+1)},\boldsymbol{\alpha}^{(g+1)}$
on the base of Appendix A.2
Update regressive component
Set partial residuals
$y_{ij}^{(3)(g)}=y_{ij}-f_{i}(t_{ij})^{(g)}-\mu_{i,s}^{(g)}$
Update $\boldsymbol{\beta}_{i}^{(g+1)}$ on the base of Appendix A.3
Update error term
Set partial residuals
$\epsilon_{ij}^{(g)}=y_{ij}-f_{i}(t_{ij})^{(g)}-\mu_{i,s}^{(g)}-\boldsymbol{x_{i}}(t_{ij})\boldsymbol{\beta}^{(g)}$
Update $\psi^{(g+1)}$ on the base of Appendix A.4
Return
$\boldsymbol{\theta}_{i}^{(g)},\boldsymbol{\mu}_{i}^{(g)},m^{(g)},\varpi^{(g)},\boldsymbol{\alpha}^{(g)},\boldsymbol{\beta}_{i}^{(g)},\psi_{i}^{(g)}$
for $g=g_{0},g_{0}+g_{s},g_{0}+2g_{s},\ldots,G$
Algorithm 1 Gibbs Sampler
## 5 Posterior analysis
The idea of estimating trajectories for athletes’ performances is a natural
pursuit for the model specification we adopted. Indeed, describing
observations as error prone measurements of an unknown underlying function
suggests evaluating such function, once retrieved, on any number of points of
interest. In practice, we will generate a fine grid of $T$ equispaced time
points: $\\{t_{k}\\}_{k=1}^{T}$ between $0\equiv t_{1}$ and $1\equiv t_{T}$
and evaluate the function on this grid.
In particular, we start by evaluating the athlete-specific functional
component by exploiting the basis function representation. Being:
$\boldsymbol{\Theta}_{i}=\begin{bmatrix}\theta_{i1}^{(1)}&\theta_{i2}^{(1)}&\ldots&\theta_{ip}^{(1)}\\\
\theta_{i1}^{(2)}&\theta_{i2}^{(2)}&\ldots&\theta_{ip}^{(2)}\\\
\vdots&\vdots&\ddots&\vdots\\\
\theta_{i1}^{(G)}&\theta_{i2}^{(G)}&\ldots&\theta_{ip}^{(G)}\end{bmatrix}$
the matrix of individual-specific spline basis coefficients for all iterations
$g=1,\ldots G$ and
$\boldsymbol{b}^{\top}=\begin{bmatrix}b_{1}(t_{1})&b_{1}(t_{2})&\ldots&b_{1}(t_{k})&\ldots&b_{1}(t_{T})\\\
b_{2}(t_{1})&b_{2}(t_{2})&\ldots&b_{2}(t_{k})&\ldots&b_{2}(t_{T})\\\
\vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\\
b_{p}(t_{1})&b_{p}(t_{2})&\ldots&b_{p}(t_{k})&\ldots&b_{p}(t_{T})\end{bmatrix}$
all values of a $p$-dimensional, degree-$3$, spline basis on a set of $T+1$
equispaced knots in the unit interval, the estimated contribution of the
functional component to the overall trajectory is, at each iteration:
${f}_{i}^{(g)}(t)=\sum_{m=1}^{p}{\theta}_{im}^{(g)}b_{m}(t)={\Theta}_{i}^{(g)}\boldsymbol{b}_{t}^{\top}\quad\text{
for }t={t_{1},\ldots,t_{T}},$
where ${\Theta}_{i}^{(g)}$ corresponds to the $i$-th row of matrix
$\boldsymbol{\Theta}_{i}$.
As for the seasonal linear mixed effect, we modelled it as a piecewise
continuous function taking individual- and season-specific values. Hence, when
retrieving its estimated effect on any point in the time grid, we need to
determine which season it belongs to. As discussed in Section 3, time is
rescaled so that equal values across individuals indicate the same day of the
year, possibly in different years. Therefore, season changes, that occur at
new year’s days, can be easily computed by straightforward proportions. At
this point, the season to which $t_{k}$ belongs to is obtained by comparison
with the season thresholds. In the following Equation, the indicator variable
$\chi_{(t\in s)}$ determines to which season each time point belongs to.
Accordingly, the estimated contribution of the seasonal component to the
overall trajectory is, at each iteration:
${\mu}_{i}^{(g)}(t)=\sum_{s=1}^{S_{i}}{\mu}_{is}^{(g)}\chi_{(t\in
s)}\quad\text{ for }t={t_{1},\ldots,t_{T}}.$
Lastly, the regressive component has to be taken into account. The estimated
contribution of the regressive component to the overall trajectory is, at each
iteration:
$\sum_{l=1}^{r}x_{il}(t){\beta}_{l}^{(g)}=\boldsymbol{x}_{i}(t){\boldsymbol{\beta}}^{(g)}\quad\text{
for }t={t_{1},\ldots,t_{T}}.$
Given the three components, the overall estimate of the underlying function is
obtained by adding these three components. In particular, the estimated mean
trajectory can be written as:
$\widehat{y_{i}(t)}=\frac{1}{G}\sum_{g=1}^{G}{f}_{i}^{(g)}(t)+{\mu}_{i}^{(g)}(t)+\boldsymbol{x}_{i}(t){\boldsymbol{\beta}}^{(g)}\quad\text{
for }t={t_{1},\ldots,t_{T}}$ (12)
Similarly, $95\%$ credible intervals can be computed to quantify uncertainty
around our point estimate.
### 5.1 Model application
In this Section, we fit different specifications of our model to the data
described in Section 2.
In general, we consider the additive structure of the sampling model
illustrated in Equation 2. Table 3 reports an overview on of the six models we
compare. In Model $M_{1}$, the B-sline basis functions have 80 degrees of
freedom, the seasonal component has GARCH errors and three regressors are
taken into account: sex, age and environment. $M_{2}$ represents a slight
modification of $M_{1}$ given by the fixed-age implementation. Here we
consider the covariate age not as a time dependent variable, but as a fixed
value given by the age at the beginning of each athlete’s career. In model
$M_{3}$ a simpler dependence structure among the seasonal effects is used.
Namely, we assume an autoregressive model for $\mu_{i}s$ (see Eq. 7). For
model $M_{4}$, we simply consider a larger number of basis functions, i.e.
120, accounting for up to three splines having support in a season and hence
meant to better capture the intra-seasonal variability. Finally, models
$M_{5}$ and $M_{6}$ allow for doping as additional covariate, both in the case
of the time-dependent and time-independent specification of age.
Priors were chosen as discussed in Section 3, and with hyperparameter choices
summarized in Table 4. To argue on the choice of the informative prior for the
overall mean parameter $m$, in Table 5 we also report the results under a
slight modification of model $M_{1}$, that we denote $M_{1}^{(2)}$, yielding a
vague prior for $m$.
Table 3: Models name and description. Symbol | Meaning
---|---
$M_{1}$ | 80 df B-splines, GARCH, covariates: sex, age (time dependent), env.
$M_{2}$ | 80 df B-splines, GARCH, covariates: sex, age (time constant), env.
$M_{3}$ | 80 df B-splines, AR, covariates: sex, age (t. dep.), env.
$M_{4}$ | 120 df B-splines, GARCH, covariates: sex, age (t. dep.), env.
$M_{5}$ | 80 df B-splines, GARCH, covariates: sex, age (t. dep.), env., doping
$M_{6}$ | 80 df B-splines, GARCH, covariates: sex, age (t. const.), env., doping
Table 4: Hyperparameter choices. In the first column, we refer to the Equation where the hyperparameter first appears. Ref. | Hyp. | Value | Description
---|---|---|---
(4) | $a_{\sigma}$ | 1.0 | $1^{st}$ Gamma coeff. of error term in the factor exp.
(4) | $b_{\sigma}$ | 0.3 | $2^{nd}$ Gamma coeff. of error term in the factor exp.
(6) | $\nu_{\phi}$ | 9 | Gamma coeff.s of local shrink. param. $\phi_{ml}$
(6) | $a_{1}$ | 2.1 | $1^{st}$ Gamma coeff. of the $1^{st}$ global shrink. factor $\delta_{1}$
(6) | $b_{1}$ | 1.0 | $2^{nd}$ Gamma coeff. of the $1^{st}$ global shrink. factor $\delta_{1}$
(6) | $a_{l}$ | 2.1 | $1^{st}$ Gamma coeff. of the $l$-th global shrink. factor $\delta_{l}$
(6) | $b_{l}$ | 1.0 | $2^{nd}$ Gamma coeff. of the $l$-th global shrink. factor $\delta_{l}$
(10) | $\mu_{m_{0}}$ | -0.2 | Mean of the overall mean $m$
(10) | $\Sigma_{m_{0}}$ | 0.0001 | Variance of the overall mean $m$
(10) | $\mu_{\alpha}$ | (0.0, 0.0) | Mean vector of the $\boldsymbol{\alpha}$ GARCH coeff.
(10) | $\Sigma_{\alpha}$ | $\mathbb{I}_{2}$ | Covariance matrix of the $\boldsymbol{\alpha}$ GARCH coeff.
(10) | $\mu_{\varpi}$ | 0.0 | Mean of the $\varpi$ GARCH coeff.
(10) | $\Sigma_{\varpi}$ | 1 | Variance of the $\varpi$ GARCH coeff.
(11) | $\nu_{\beta}$ | 0.5 | $1^{st}$ Gamma coeff.s regression param.
(11) | $\sigma_{\beta}$ | 0.5 | $2^{nd}$ Gamma coeff.s of regression param.
(1) | $\mu_{\psi}$ | 1.0 | Mean of the error variance $\psi$
(1) | $\sigma_{\psi}$ | 1.0 | Variannce of the error variance $\psi$
For all experiments, inference is obtained via posterior samples drawn by the
Gibbs sampler introduced in Section 4. In particular, we ran $20,000$
iterations with a burn-in period of $60\%$ and a thinning of $5$. Performances
are compared by means of the logarithm of the pseudo marginal likelihood
(LPML) index (Geisser and Eddy, 1979). This estimator for the log marginal
likelihood is based on conditional predictive densities and provides an
overall comparison of model fit, with higher values denoting better performing
models.
Table 5: Model and hyperparameter comparison for the models in Table 3. Model | LPML
---|---
$M_{1}$ | $\boldsymbol{-45943}$
$M_{1}^{(2)}$ | -46573
$M_{2}$ | $\boldsymbol{-45472}$
$M_{3}$ | -46544
$M_{4}$ | -46314
$M_{5}$ | -48565
$M_{6}$ | -48122
Performances for the different models are fairly similar: as a matter of fact,
the model specifications do not differ in a significant way. Despite having a
slightly lower LPML than the best performing model, $M_{2}$, we prefer looking
at results for model $M_{1}$ with 80 degrees od freedom splines, GARCH errors
and three regressors with time-dependent age definition because regression
parameters prove to be significant in this setting. As far as the estimation
of trajectories describing the evolution of athletes’ performances is
concerned, we use the method discussed in Section 5. Figure 4 displays the
estimate (with $95\%$ credible bounds) for a random selection of athletes
(black) together with one-season-ahead performance prediction (grey). The
results are graphically pleasing in terms of model fit, but some comments are
of order. First, we acknowledge that the seasonal random intercept captures
the majority of the variability in the data. Second, the functional component,
which is meant to capture the overall variability in the data set, reduces to
capture the intra-seasonal variability. Interestingly, the number on non-local
bases selected by the adaptive procedure in Bhattacharya and Dunson (2011) is
exactly equal to the number of seasons in the data set. This effect seems to
be consistent with the choice of degrees of freedom, that limits the support
of each spline to a unique season. Finally, the effect of covariates is very
small in magnitude. Note that, because sex is time-constant, in the estimated
trajectory we expect to recognise the contribution of age as a linear trend
and the environmental effect as a diversification of summer and winter
performances.
Figure 4: Performance trajectory estimates for a random selection of athletes.
The $x$-axis denotes the time measured in days from January 1st of the first
season of career, whereas on the y-axis there is the length of throw in
meters. Vertical lines represent calendar years (seasons in our notation). The
final part of each trajectory (grey) for which no observations are available,
represents one-season-ahead performance prediction.
In Figure 5 we underline the effect of the three additive components of our
functional model. The first panel (top-left) displays the observed data and
the estimated trajectory for athlete 226. The top-right panel shows the
seasonal contribution (i.e, an estimation of $\mu_{is};\ i=226;\
s=1,\ldots,S_{226}$). The third panel (bottom-left) reports the functional
contribution (i.e, an estimate of $f_{226}(t)$). Finally, the bottom-right
panel shows the effect of covariates. As anticipated, within the trajectory
estimation of a unique athlete, we only recognise a performance drop predicted
during winter months (negative effect of indoor environment on performances)
in the bottom-right panel. The tight credible intervals for this component
assures estimates to be significant, despite small, as we will discuss in
further detail in the next Section.
Figure 5: Single contributions to the whole additive model as in Equation 12.
The first panel is the complete additive model, whereas the second (top-right)
displays the estimate of the seasonal random intercept. The third panel
(bottom-left) represents the functional contribution, while the bottom-right
panel displays the regressive component.
### 5.2 Parameters interpretation
The regression parameters can be easily interpreted from a sports’ analytics
perspective. It is important to stress that, to improve convergence of the
MCMC algorithm, we fitted our model centering athlete-specific data around
their average. Accordingly, in our experiments the raw data $y_{ij}$ were
substituted by the centered points:
$\tilde{y}_{ij}=y_{ij}-\frac{\sum_{j=1}^{n_{i}}y_{ij}}{n_{i}}=y_{ij}-\overline{y}_{i}\quad\text{for
}i=1,\ldots,n\text{ and }\ j=1,\ldots,n_{i}$
as data-input for the model. We have to take into account this transformation
when interpreting the regression parameters, especially when dealing with
dummy variables.
We report the posterior mean estimate of the regression coefficients, their
standard deviation, the effective sample size (ESS) and the $95\%$ posterior
credible bounds for the most interesting models. Table 6 displays the results
under model $M_{1}$, where covariates are sex ($x_{1}$), age ($x_{2}$) and
environment ($x_{3}$). Even if the covariate effect is small in size, we
observe that the $95\%$ credible intervals do not contain zero, showing a
significant effect. In particular $\beta_{2}$ is positive, suggesting that
athletes, on average, increase their performances throughout their career.
Further, $\beta_{3}$ is positive, meaning that an athlete is likely to perform
better outdoors than indoors. Finally, $\beta_{1}$ is negative. Since sex is a
time-constant dummy variable, its coefficient quantifies the difference in
variability of the athlete’s performance around his/her average
$\overline{y}_{i}$. We conclude that female’s trajectories express less
variability around their average than men’s. Parameter estimation under the
other models considered in the paper are similar in sign with respect to the
ones just discussed here. We report complete results in Appendix B.
A final comment on results obtained using doping as additional regressor is
required. We stressed that LPML performances for models $M_{5}$ and $M_{6}$
are quite low, however, this may be due to the fact that the data set is
imbalanced, i.e., there are too few doped athletes (18 out of 653). In fact,
the ESS of the parameter corresponding to doping is very low. Nevertheless, it
is interesting to observe that estimates are similar for all common parameters
and that the coefficient of the doping regressor is negative (even if the
credible intervals contain zero). We conclude that the use of performance-
enhancing drugs seems to have a negative effect on the variability of
athletes’ performances.
Table 6: Posterior mean estimate of the regression coefficients for model $M_{1}$(Table 3), together with the standard deviation of their estimate, effective sample size (ESS) with respect to 1600 retained samples, and $95\%$ posterior credible bounds. Coeff | Mean | Sd | ESS | $2.5\%$ | $97.5\%$
---|---|---|---|---|---
$\beta_{1}$ | -0.120 | 0.0270 | 190 | -0.175 | -0.0675
$\beta_{2}$ | 6.22 e-03 | 9.95 e-04 | 170 | 4.20 e-03 | 8.20 e-03
$\beta_{3}$ | 0.0453 | 9.55 e-03 | 1600 | 0.0269 | 0.0643
## 6 Discussion
We proposed an additive hierarchical Bayesian model for the analysis of
athletes’ performances in a longitudinal context. Following Montagna and
Hopker (2018), we proposed a smooth functional contribution for explaining the
overall variability in the data set. The functions are represented by means of
a high-dimensional set of pre-specified basis functions and a factor model on
the basis coefficients ensures dimensionality reduction. We enriched the model
by allowing for time-dependent covariates to affect estimates through a
regressive component. Finally, we addressed the issue of seasonal gathering of
sports data introducing a mixed effect model with GARCH errors which provides
evolving random intercepts over different time intervals in the data set.
While the motivation of our work comes from the analysis of shot put
performance data, the methodology presented in this work is applicable to the
analysis of performance data collected in all measurable sports.
The Bayesian latent factor methodology was originally developed for very
sparse longitudinal data, with the purpose of capturing a global trend in
subject-specific trajectories. We balanced the model with the requirement of
smoothness using a B-spline basis system and adding a seasonal random
intercept. However, it is evident that the latter explains the majority of
variability in the dataset. Therefore, it might be worth considering a
functional basis that batter captures the intra-seasonal variability. Further,
we observed that the contribution of the regressive component is consistent
across various modelling choices.
Finally, we looked at the effect of doping on results. Despite not being
significant, the negative effect seems to suggest that performances of doped
athletes are less variable. This is in line with previous literature
suggesting that doping is more likely used to enhance performances in periods
of decreasing fitness than to consolidate already good performances, generally
exposed to strict controls. We think this aspect deserves further
investigation, considering, for instance, more specific modeling techniques
and a less imbalanced data set.
In conclusion, the attempt to extend exiting tools of functional statistics to
the modelling of (shot put) performance data seems promising because of the
adaptability of these methodologies to all sorts of performance longitudinal
data in measurable sports.
## Acknowledgements
We would like to thank Prof. James Hopker (University of Kent, School of Sport
and Exercise Sciences) for sharing the shot put data set, and for the helpful
comments.
## Conflict of interest
The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.
## Availability of data and material
The raw data set is available at www.tilastopaja.eu, whereas the data prepared
for our analysis at https://github.com/PatricDolmeta/Bayesian-GARCH-Modeling-
of-Functional-Sports-Data.
## Code availability
The code for the Bayesian analysis and the output production (numerical and
graphical) is available at https://github.com/PatricDolmeta/Bayesian-GARCH-
Modeling-of-Functional-Sports-Data.
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## Appendix A Posterior computations
In this Appendix, we discuss in greater detail the full conditional updates
anticipated in Section 4 and referred to in Algorithm 1.
### A.1 Functional model
With reference to the hierarchical model discussed in Section 3.1, we get the
following updates:
1. 1.
Recalling that $\boldsymbol{\Lambda}$ is a $q\times k$ matrix, we update its
$j$-th row as:
$\displaystyle\pi(\boldsymbol{\lambda_{j}}|\ldots)\sim
N_{k}((\boldsymbol{D^{-1}_{j}}+\sigma_{j}^{-2}\boldsymbol{\eta^{\top}}\boldsymbol{\eta})^{-1}\boldsymbol{\eta^{\top}}\sigma_{j}^{-2}\boldsymbol{\theta}^{(j)},(\boldsymbol{D^{-1}_{j}}+\sigma_{j}^{-2}\boldsymbol{\eta^{\top}}\boldsymbol{\eta})^{-1})$
where
$\displaystyle\boldsymbol{D^{-1}_{j}}=diag(\phi_{j1}^{-1}\tau_{1}^{-1},\ldots,\phi_{jk}^{-1}\tau_{k}^{-1})$
$\displaystyle\boldsymbol{\eta^{\top}}=(\eta_{1},\ldots,\eta_{k})$
$\displaystyle\boldsymbol{\theta}^{(j)}=(\theta_{j1},\ldots,\theta_{jn})$
2. 2.
Concerning the local shrinkage parameters, we get:
$\pi(\phi_{jh}|\ldots)\sim
Ga\biggl{(}\frac{\nu_{\phi}+1}{2},\frac{\nu_{\phi}+\tau_{h}\lambda_{jh}^{2}}{2}\biggr{)}$
3. 3.
The first global shrinkage parameter is updated as follows:
$\pi(\delta_{1}|\ldots)\sim
Ga\biggl{(}a_{1}+\frac{qk}{2},1+\frac{1}{2}\sum_{l=1}^{k}\tau_{l}^{(1)}\sum_{j=1}^{q}\phi_{jl}\lambda_{jl}^{2}\biggr{)}$
4. 4.
Whereas for the remaining ones:
$\displaystyle\pi(\delta_{h}|\ldots)\sim
Ga\biggl{(}a_{h}+\frac{q(k-h-1)}{2},1+\frac{1}{2}\sum_{l=1}^{k}\tau_{l}^{(h)}\sum_{j=1}^{q}\phi_{jl}\lambda_{jl}^{2}\biggr{)}$
with $\displaystyle\tau_{l}^{(h)}=\prod_{t=1,t\neq h}^{l}\delta_{t}$
5. 5.
Variances of factor residuals can be sampled as:
$\pi(\sigma_{j}^{-2}|\ldots)\sim
Ga\biggl{(}a_{\sigma}+\frac{n}{2},b_{\sigma}+\frac{\sum_{i=1}^{n}(\boldsymbol{\theta}^{(j)}-\boldsymbol{\Lambda}\eta_{i})^{2}}{2}\biggr{)}$
6. 6.
Latent factors for a specific individual have conditional posterior
distribution:
$\displaystyle\pi(\boldsymbol{\eta}_{i}|\ldots)\sim
N(\boldsymbol{A_{i}}^{-1}\times\boldsymbol{B_{i}},\boldsymbol{A_{i}}^{-1})$
being
$\displaystyle\boldsymbol{A_{i}}=\boldsymbol{\Lambda}^{\top}\boldsymbol{B_{i}}^{\top}(\psi^{2}\boldsymbol{I}_{n_{i}}+\boldsymbol{B_{i}}\boldsymbol{\Sigma}\boldsymbol{B_{i}}^{\top})^{-1}\boldsymbol{B_{i}}\boldsymbol{\Lambda}+I_{k}$
$\displaystyle\boldsymbol{B_{i}}=\boldsymbol{\boldsymbol{\Lambda}}^{\top}\boldsymbol{B_{i}}^{\top}(\psi^{2}\boldsymbol{I}_{n_{i}}+\boldsymbol{B_{i}}\boldsymbol{\Sigma}\boldsymbol{B_{i}}^{\top})^{-1}y_{i}$
7. 7.
Finally, coefficients of the spline representation are sampled from:
$\displaystyle\pi(\boldsymbol{\theta}_{i}|\ldots)\sim
N(\boldsymbol{C_{i}}^{-1}\times\boldsymbol{D_{i}},\boldsymbol{C_{i}}^{-1})$
being
$\displaystyle\boldsymbol{C_{i}}=\psi^{2}\boldsymbol{B_{i}}^{\top}\boldsymbol{B_{i}}+\boldsymbol{\Sigma}^{-1}$
$\displaystyle\boldsymbol{D_{i}}=\psi^{2}\boldsymbol{B_{i}}^{\top}y_{i}+\boldsymbol{\Sigma}^{-1}\boldsymbol{\Lambda}\boldsymbol{\eta}_{i}$
### A.2 Seasonal GARCH model
Concerning the seasonal component, let us start noticing that the likelihood
function of the GARCH model is:
$f(\ \\{\mu_{i,s}\\}_{i=1,\ldots,n}^{s=1,\ldots,S_{i}}\ |\
\varpi,h_{i,s})=\prod_{i=1}^{n}\prod_{s=1}^{S_{i}}\frac{1}{\sqrt{2\pi
h_{i,s}}}exp\left\\{-\frac{\stackrel{{\scriptstyle\zeta_{i,s}^{2}}}{{\overbrace{(\mu_{i,s}-m)^{2}}}}}{2h_{i,s}}\right\\}$
(13)
whereas the terms in the joint sampling distribution depending on $\mu_{i,s}$
are:
$\displaystyle\prod_{i=1}^{n}\prod_{s=1}^{S_{i}}\prod_{j\in
s}\frac{1}{\sqrt{2\pi\psi^{2}}}exp\left\\{-\frac{(y_{i,j}^{(2)}-\mu_{i,s})^{2}}{2\psi^{2}}\right\\}$
where $\displaystyle
y_{i,j}^{(2)}=y_{i,j}-f_{i}(t_{i,j})-\boldsymbol{x}(t_{i,j})\boldsymbol{\beta}$
is a convenient writing for partial residuals, especially from the
implementation point of view.
Consequently, $\mu_{i,s}$ can be updated taking samples from:
$\displaystyle q_{\mu}(\mu_{i,s}\ |\ \varpi,y_{i,j}^{(2)},\psi^{2})$
$\displaystyle\propto exp\left\\{-\sum_{j\in
s}\frac{\mu_{i,s}^{2}-2\mu_{i,s}y_{i,j}^{(2)}}{2\psi^{2}}\right\\}\times
exp\left\\{-\frac{\mu_{i,s}^{2}-2\mu_{i,s}m}{2h_{i,s}}\right\\}$
$\displaystyle\sim N(\hat{\mu}_{\mu},\hat{\Sigma}_{\mu})$
$\displaystyle\text{Where }\hat{\Sigma}_{\mu}=\biggl{(}\sum_{j\in s}$
$\displaystyle\frac{1}{\psi^{2}}+\frac{1}{h_{i,s}}\biggr{)}^{-1}\text{ and }\
\hat{\mu}_{\mu}=\hat{\Sigma}_{\mu}^{-1}\biggl{(}\sum_{j\in
s}\frac{y_{i,j}^{(2)}}{\psi^{2}}+\frac{m}{h_{i,s}}\biggr{)}$
Posterior updates for other parameters are not straightforward in general
because of the recursive definition of the conditional variance. As a
consequence, we will rely on Metropolis Hastings steps within a Gibbs sampling
algorithm. We iteratively generate:
$\displaystyle m^{[j]}\sim p(m\ |\ \alpha^{[j-1]},\varpi^{[j-1]},\mu_{i,s})$
$\displaystyle\alpha^{[j]}\sim p(\alpha\ |\ m^{[j]},\varpi^{[j-1]},\mu_{i,s})$
$\displaystyle\varpi^{[j]}\sim p(\varpi\ |\ m^{[j]},\alpha^{[j]},\mu_{i,s})$
#### A.2.1 Update $m$
The posterior distribution for $m$ is based on the GARCH model under the
assumption that the conditional variances $\\{h_{i,s}\\}_{i}^{s}$ are fixed,
known and given by the writing in Eq. (9). In this case the likelihood
function is:
$f(\ \\{\mu_{i,s}\\}_{i=1,\ldots,n}^{s=1,\ldots,S_{i}}\ |\
\boldsymbol{\alpha},\varpi)=\prod_{i=1}^{n}\prod_{s=1}^{S_{i}}\frac{1}{\sqrt{2\pi
h_{i,s}}}exp\left\\{-\frac{(\mu_{i,s}-m)^{2}}{2h_{i,s}}\right\\}$
Accordingly, the posterior samples for $m$ will be drawn from:
$\displaystyle q_{m}(m\ |\ \tilde{\boldsymbol{\alpha}},\tilde{\varpi})$
$\displaystyle\propto
exp\left\\{-\sum_{i=1}^{n}\sum_{s=1}^{S_{i}}\frac{m^{2}-2m\mu_{i,s}}{2h_{i,s}}\right\\}\times
exp\left\\{-\frac{m^{2}-2m\mu_{m_{0}}}{2\Sigma_{m_{0}}}\right\\}$
$\displaystyle\sim N(\hat{\mu}_{m},\hat{\Sigma}_{m})$
$\displaystyle\text{Where }\hat{\Sigma}_{m}=\biggl{(}$
$\displaystyle\sum_{i=1}^{n}\sum_{s=1}^{S_{i}}\frac{1}{h_{i,s}}+\frac{1}{\Sigma_{m_{0}}}\biggr{)}^{-1}\text{
and }\
\hat{\mu}_{m}=\hat{\Sigma}_{m}^{-1}\biggl{(}\sum_{i=1}^{n}\sum_{s=1}^{S_{i}}\frac{\mu_{i,s}}{h_{i,s}}+\frac{\mu_{m_{0}}}{\Sigma_{m_{0}}}\biggr{)}$
It is here of capital importance that $h_{i,s}$ are obtained by means of Eq.
(9), using previous realisations the GARCH coefficients from the MH sampler.
#### A.2.2 Update $\boldsymbol{\alpha}$
To generate samples from $\boldsymbol{\alpha}$ we can not exploit conjugacy,
therefore we would like to rely on a normal proposal distribution. Indeed,
symmetry of proposal distributions significantly eases computations of
acceptance probabilities in Metropolis algorithms. Unfortunately, we are given
a non-negativity constraint for the parameter, which forces us to apply a
bidimensional transformation from the positive quadrant into the real plane:
let say $(\theta_{0},\theta_{1})=(log(\alpha_{0}),log(\alpha_{1}))$. After
transformation of the starting parameter, we propose a new sample employing a
random walk sampler.
That is, a proposal of the form
$\boldsymbol{\theta}^{\ast}=\boldsymbol{\theta}+\boldsymbol{\zeta}\boldsymbol{\epsilon}$
where $\boldsymbol{\epsilon}$ is a multivariate standard normal random vector
and $\boldsymbol{\zeta}$ a covariance matrix, defined according to an Adaptive
scaling within the Adaptive Metropolis–Hastings algorithm (ASWAM) by Haario et
al. (2001). In this approach, both the covariance matrix of the proposal is
adapted to the covariance matrix of the target density and the scale parameter
is updated to achieve an average acceptance rate of 0.234 (proven to be
optimal in different scenarios).
Provided it will be accepted, the actual parameter will be retrieved applying
the inverse transform to $\boldsymbol{\theta}$. As discussed above, the
acceptance probability will not depend on the proposal distribution.
Therefore:
$\lambda_{\alpha}=min\left\\{\frac{p(\boldsymbol{\theta}^{\ast}\ |\ \varpi,\
m,\ \mu_{i,s})}{p(\tilde{\boldsymbol{\theta}}\ |\ \varpi,\ m,\
\mu_{i,s})},1\right\\}$ (14)
With $p(\boldsymbol{\theta}\ |\ \varpi,m,\mu_{i,s})$ the full conditional
density of $\boldsymbol{\theta}$ given the data and the rest of the
parameters, which is clearly obtained from $p(\boldsymbol{\alpha}\ |\
\varpi,m,\mu_{i,s})$ by change of variables.
#### A.2.3 Update $\varpi$
Similar arguments apply for $\varpi$. Samples will be proposed according to a
normal distribution centered around $\tilde{\gamma}=log(\tilde{\varpi})$:
$\gamma^{\ast}\sim N(\gamma|\tilde{\gamma},\Sigma_{\varpi})$
Here, $\Sigma_{\varpi}$ is defined, for each iteration $g$ as:
$\sqrt{\Sigma_{\varpi}}^{(g+1)}=\zeta^{(g+1)}=\rho\bigl{(}\zeta^{(g)}*w^{(g)}\bigl{(}\lambda_{\theta}^{(g)}-\overline{\lambda}\bigr{)}\bigr{)}$
to achieve an average optimal acceptance rate of $\overline{\lambda}=0.234$
Haario et al. (2001).
Actual samples will be retrieved applying the inverse transform and accepted
according to:
$\lambda_{\varpi}=min\left\\{\frac{p(\gamma^{\ast}\ |\ \boldsymbol{\alpha},\
m,\ \mu_{i,s})}{p(\tilde{\gamma}\ |\ \boldsymbol{\alpha},\ m,\
\mu_{i,s})},1\right\\}$ (15)
where again $p(\gamma\ |\ \boldsymbol{\alpha},m,\mu_{i,s})$ is the full
conditional density of $\gamma$ given by change of variables from $p(\varpi\
|\ \boldsymbol{\alpha},m,\mu_{i,s})$
### A.3 Multiple regression model
On the base of the prior settings and the overall sampling scheme 1 we get:
1. 1.
Regression coefficient, jointly, as:
$\displaystyle\pi(\boldsymbol{\beta}|\ldots)\sim N(A^{-1}\times B,A^{-1})$
being $\displaystyle
A=\boldsymbol{x^{\top}}\boldsymbol{x}/\psi^{2}+\Sigma_{\boldsymbol{\beta}_{0}}$
$\displaystyle
B=\Sigma_{\boldsymbol{\beta}_{0}}\boldsymbol{\beta}_{0}+\boldsymbol{x^{\top}}\boldsymbol{y^{(3)}}/\psi^{2}$
2. 2.
The variance hyperparameter instead is updated as:
$\pi(\sigma_{\beta}^{-2}|\ldots)\sim
Ga\biggl{(}\frac{N+\nu_{\beta}}{2},\frac{\nu_{\beta}\sigma_{\beta}^{2}+\sum_{i=1}^{n}\sum_{j=1}^{n_{i}}(y_{ij}-f_{i}(t_{ij})-\mu_{is}-\boldsymbol{x}\boldsymbol{\beta})^{2}}{2}\biggr{)}$
### A.4 Error term
Finally, the full conditional posterior distribution for the model variance
is:
$\displaystyle\pi(\psi^{-2}|\ldots)\sim
Ga\biggl{(}\frac{N+\nu_{\psi}}{2},\frac{\nu_{\psi}\sigma_{\beta}^{2}+\sum_{i=1}^{n}\sum_{j=1}^{n_{i}}\epsilon_{i,j}^{2}}{2}\biggr{)}$
being
$\displaystyle\epsilon_{i,j}=y_{i,j}-f_{i}(t_{i,j})-\mu_{i,s}-\boldsymbol{x}\boldsymbol{\beta}$
## Appendix B Regression coefficients
Table 7: Posterior mean estimate of the regression coefficients for model $M_{2}$(Table 3), together with the standard deviation of their estimate, effective sample size (ESS) with respect to 1600 retained samples, and $95\%$ posterior credible bounds. Coeff | Mean | Sd | ESS | $2.5\%$ | $97.5\%$
---|---|---|---|---|---
$\beta_{1}$ | -5.58 e-03 | 0.0263 | 223 | -0.0585 | 2.00 e-03
$\beta_{2}$ | 3.34 e-04 | 1.16 e-03 | 107 | -2.60 e-03 | 2.00 e-03
$\beta_{3}$ | 0.0570 | 9.09 e-03 | 1600 | 0.0395 | 0.0743
Table 8: Posterior mean estimate of the regression coefficients for model $M_{5}$(Table 3), together with the standard deviation of their estimate, effective sample size (ESS) with respect to 600 retained samples, and $95\%$ posterior credible bounds. Coeff | Mean | Sd | ESS | $2.5\%$ | $97.5\%$
---|---|---|---|---|---
$\beta_{1}$ | -0.120 | 0.0285 | 77 | -0.178 | -0.0690
$\beta_{2}$ | 6.71 e-03 | 1.11 e-03 | 64 | 4.40 e-3 | 8.10 e-3
$\beta_{3}$ | 0.0373 | 0.0108 | 600 | 0.0168 | 0.0580
$\beta_{4}$ | -0.116 | 0.0761 | 70 | -0.267 | -0.0365
Table 9: Posterior mean estimate of the regression coefficients for model $M_{6}$(Table 3), together with the standard deviation of their estimate, effective sample size (ESS), and $95\%$ posterior credible bounds. Coeff | Mean | Sd | ESS | $2.5\%$ | $97.5\%$
---|---|---|---|---|---
$\beta_{1}$ | -0.0104 | 0.0250 | 74 | -0.0617 | 0.0362
$\beta_{2}$ | -4.60 e-04 | 1.14 e-03 | 67 | 2.80 e-03 | 1.90 e-03
$\beta_{3}$ | 0.0578 | 9.37 e-03 | 600 | 0.0408 | 0.0756
$\beta_{4}$ | -0.0103 | 0.0638 | 91 | -0.134 | 0.115
|
# On Provable Backdoor Defense in Collaborative Learning
Ximing Qiao, Yuhua Bai, Siping Hu, Ang Li, Yiran Chen, Hai Li
###### Abstract
As collaborative learning allows joint training of a model using multiple
sources of data, the security problem has been a central concern. Malicious
users can upload poisoned data to prevent the model’s convergence or inject
hidden backdoors. The so-called backdoor attacks are especially difficult to
detect since the model behaves normally on standard test data but gives wrong
outputs when triggered by certain backdoor keys. Although Byzantine-tolerant
training algorithms provide convergence guarantee, provable defense against
backdoor attacks remains largely unsolved. Methods based on randomized
smoothing can only correct a small number of corrupted pixels or labels;
methods based on subset aggregation cause a severe drop in classification
accuracy due to low data utilization. We propose a novel framework that
generalizes existing subset aggregation methods. The framework shows that the
subset selection process, a deciding factor for subset aggregation methods,
can be viewed as a code design problem. We derive the theoretical bound of
data utilization ratio and provide optimal code construction. Experiments on
non-IID versions of MNIST and CIFAR-10 show that our method with optimal codes
significantly outperforms baselines using non-overlapping partition and random
selection. Additionally, integration with existing coding theory results shows
that special codes can track the location of the attackers. Such capability
provides new countermeasures to backdoor attacks.
## 1 Introduction
As data acquisition increasingly becomes the bottleneck of modern machine
learning, collaborative learning on crowd-sourced data gains importance
(Vaughan 2017). Unfortunately, the distributed nature of collaborative
learning opens up an immense attack surface. The system is highly vulnerable
to data poisoning attacks (Muñoz-González et al. 2017) and model poisoning
attacks (Baruch, Baruch, and Goldberg 2019). Depending on the training
protocol, a few malicious users, usually referred to as Byzantine workers, can
prevent the model’s convergence or introduce hidden backdoor by uploading
false data or false gradients. Extensive research has been conducted on
developing Byzantine-tolerant training algorithms (Mhamdi, Guerraoui, and
Rouault 2018; Blanchard et al. 2017; Yin et al. 2018), which target on
learning the correct functionally even when a number of users are corrupted.
The central idea around these methods is that false gradients are
statistically different from normal gradients, and the false gradients can be
detected and removed by an outlier detection process. Under sufficient
assumptions such as smoothness and bounded variance, convergence of the model
can be guaranteed.
The problem that Byzantine-tolerant training algorithms cannot solve is the
backdoor problem (Li et al. 2020). The backdoor attack is a hybrid of
training-stage attack and inference-stage attack, which is very different from
pure training-stage attacks such as the convergence-preventing attacks, or
pure inference-stage attacks such as adversarial examples (Goodfellow, Shlens,
and Szegedy 2014). On one hand, backdoor attacks preserve the model’s normal
functionality on benign data so that the backdoor is hidden. On the other
hand, a set of attacker-specified triggers (a.k.a. backdoor triggers), such as
small patches with particular pixel patterns, can control the model’s behavior
when those triggers appear in the model’s input. Injecting a backdoor can be
viewed as a multitask learning, where the main task is the correct
classification on benign data, and a side task is the attacker-controlled
behavior on data with backdoor triggers. Even if Byzantine-tolerant training
algorithms can guarantee the convergence, learning of the side task cannot be
prevented. Attacks that successfully bypass Byzantine defenses are shown in
various literature (Baruch, Baruch, and Goldberg 2019; Bagdasaryan et al.
2018).
Figure 1: (a) The attacker poisons local data/model to identify the trigger as
label “1”. (b) During training, the model only aggregates data/gradients from
the solid lines, so that a few attackers cannot affect all models. (c) During
inference, a decoder identifies the correct output from $m$ potentially
erroneous outputs.
Outside the scope of collaborative learning, defense against backdoor attacks
is widely studied but mostly based on heuristics. Typical strategies include
rejecting data that produce abnormal activations (Tran, Li, and Madry 2018),
reverse engineering backdoor triggers (Wang et al. 2019), masking input
regions to remove backdoor triggers (Chou et al. 2018), and etc. These
defenses are effective against several classical forms of backdoor attacks,
but not against some newer variants (Tan and Shokri 2019; Salem et al. 2020).
Very recently, provable backdoor defenses have been introduced to provide
certified robustness against a certain degree of data poisoning, regardless of
the exact attack method. In (Weber et al. 2020) and (Wang et al. 2020), the
authors extend the randomized smoothing method, a popular algorithm that
provides certified robustness against adversarial examples. Assuming that the
attacker’s modification to the training dataset is norm limited (e.g., the
perturbation to training images is $L_{2}$ limited or the change of training
labels is $L_{0}$ limited), the defender can introduce noise to the training
process and create a “smoothed” classifier. An ensemble model is trained on
noisy versions of the training dataset and a smoothed output is produced by a
majority vote. The main drawback of this method is that it requires to train
up to $10^{3}$ models but can only correct a few pixels of modification. The
norm limit is too strict and usually not practical in the context of
collaborative learning, where each user controls a large portion of training
data and can make arbitrary modifications.
Current state-of-the-art methods are based on subset aggregation/bootstrap
aggregation, as in (Levine and Feizi 2020) and (Jia et al. 2020). An ensemble
model is trained on subsets of the training dataset, and then use majority
vote to produce the output. Selection of the data subsets can be deterministic
(Levine and Feizi 2020) or random (Jia et al. 2020). In the context of
collaborative learning, the deterministic method can be interpreted as
follows: given $n$ users in total, we can split the users into $m$ groups and
train $m$ models on the $m$ groups, then apply majority vote on the $m$
outputs. The majority vote is robust against $k$ attackers, if the count of
the highest vote is $2k+1$ larger than the count of the second-highest vote.
The intuition is that since $k$ attackers can at most change $k$ out of the
$m$ outputs, the second-highest vote can never surpass the highest vote. The
random version uses random subsets of users and provides a probabilistic
robustness guarantee. Although subset selection reduces the training cost, it
also negates the benefit of collaborative learning, which is to train models
on more data. To defend $k$ attackers, each model can be trained on at most
$1/(2k+1)$ portion of the total available data. In extreme case, users have no
collaboration at all and simply use local models to vote for their final
result.
In this work, we find that simple subset selection methods such as random
selection or non-overlapping partition cannot suffice the complicated
condition of collaborative learning, especially on non-IID datasets. Centered
around the idea that each model in the ensemble should be trained on as much
data as possible, we seek new subset selection algorithms and new aggregation
methods. We show that the optimal subset selection can be interpreted as a
code design problem. As illustrated in Figure 1, training of the ensemble
model is characterized by a binary code matrix and the majority vote
aggregation is generalized to a decoder. The new framework allows us to
provide a theoretical bound of data utilization ratio and derive the optimal
code construction. The main result is that with additional assumptions of the
independence and determinism of the backdoor attack, we can correctly decode
the final result even when the majority of the models give wrong outputs. This
leads to a data utilization ratio beyond the $1/(2k+1)$ bound of majority
vote, and results in a significant boost in classification accuracy.
Additionally, the new framework allows us to borrow rich results from coding
theory and create novel applications of the defense. One example is based on
superimposed codes (Kautz and Singleton 1964) originated from communication
and group testing. The codes allow us to track the location of the attacker
(whenever a backdoor is triggered), providing new countermeasures to the
backdoor attack.
## 2 Problem Formulation
### Subset Selection and Code Matrix
In this section, we build the basic framework and show that subset selection
can be formulated as a code design problem. Here we first introduce the matrix
representation of the subset selection. Assume that we train $m$ model using
data from $n$ users and each model is trained on a subset of users. The
selection of users can be represented by a $m\times n$ binary matrix, referred
to as a code $H$. This matrix has its element $H_{ij}=1$ when model $i$
accepts the data (or gradients) from user $j$ and $H_{ij}=0$ when it rejects
the data (or gradients).
The matrix representation allows us to describe subset aggregation-based
defenses in a uniform way. In (Levine and Feizi 2020), their defense splits
the $n$ users in to $m$ groups without overlap, and then train $m$ models on
those $m$ groups. The equivalent code is a matrix with one-hot columns (since
there is no overlap). It can also be viewed as a diagonal matrix extended by
repeating columns. In (Jia et al. 2020), their train $m$ models using $m$
independent random subsets of users (with possible overlaps). The equivalent
code is a random matrix with a constant number of 1’s per row. Examples of the
codes are given in Figure 2(a) and 2(b) for $n=6$.
### Defense Objectives and Notation
With the matrix representation, we can specify objectives of defense by
properties of code matrices. In this work, we are interested in the three
types of defenses listed below. Assuming that there are less than or equal to
$k$ attackers among the $n$ users, we can have:
* •
Backdoor Detection Codes (BDC) that detect whether an attack happens without
necessarily giving the true label;
* •
Backdoor Correction Codes (BCC) that detect the attack and recover the true
label;
* •
Backdoor Tracking Codes (BTC) that detect the attack, recover the true label,
and identify the location of the attackers.
To define the codes more specifically, we introduce $r$ as row weight, the
number of 1’s in a row. We say a code matrix $H$ has a row weight $r$ if the
number of 1’s in every row of $H$ is greater than or equal to $r$. The row
weight is an important characteristic of a code as it represents the amount of
data used to train each model. When each user contribute the same amount of
data, the data utilization ratio can be defined as $r/n$. For a given set of
$k$, $r$, and $n$, we search for codes that satisfy the above properties with
the least number of rows, as the row number decides the cost of storing and
running the ensemble model. The three types of codes are denoted as
BDC$(k,r,n)$, BCC$(k,r,n)$, and BTC$(k,r,n)$ respectively.
## 3 A Toy Example: Defending a Single Attacker
### Binary Classification
We start with the simple case of defending a single attacker. Given a code
matrix $H$, we describe the attacker as a one-hot vector $\mathbf{x}$ and the
models’ outputs as a vector $\mathbf{y}$ (all vectors are by default column
vectors). The attack vector has $x_{j}=1$ if user $j$ is the attacker. Here we
introduce our first independence assumption, stating that the injection of a
backdoor should be independent to data from those users besides the attacker.
If the backdoor can be successfully injected when a model is trained on all
users, the same backdoor should be injected when the model is trained on a
subset of users (as long as the subset includes the attacker). Under an
independent attack, $\mathbf{y}$ has two possible outcomes: If the true label
is $0$ and the attack target is $1$, then $y_{i}=1$ if $\exists
j,H_{ij}=1\land x_{j}=1$, or in short, $\mathbf{y}=H\mathbf{x}$. If the true
label is $1$ and the attack target is $0$, then $\mathbf{y}=1-H\mathbf{x}$.
For BDC codes to detect the attack, $H$ must not contain any column that
equals to $\mathbf{1}$ (a vector of all 1’s) so that at least one model is not
backdoored. Meanwhile, $H$ should not include any zero columns or zero rows,
otherwise a user’s data are not used or a model is not trained. For BCC codes
to find the true label, the code additionally cannot include two complementary
columns, or $\forall i\forall j,H_{\cdot i}\oplus H_{\cdot j}\neq\mathbf{1}$,
in which $\oplus$ represents the XOR operation. For BTC codes to track the
location of the attacker (i.e., reconstructing $\mathbf{x}$ from $H$ and
$\mathbf{y}$), we need to further ensure that $H$ does not have two identical
columns. When an attack happens, the attacker can be uniquely identified by
matching $\mathbf{y}$ or $1-\mathbf{y}$ to one of the columns of $H$. Examples
of the BCC and BTC codes are given in Figure 2 (c) and (d). Note that although
codes (a) and (c) are both valid BCC codes with the same number of rows, code
(c) has a higher row weight $r$, meaning that each model is trained on more
data. However, code (c) cannot be decoded by majority vote and requires more
complicated methods. The random matrix given in code (b) is BCC. In general, a
random matrix is BCC by a probability approaches 1 when $m$ or $n/r$ is large
enough.
### Multi-Class Classification
In a multi-class classification scenario, the class prediction $\mathbf{y}$ is
no longer a binary vector. To make the above codes available, we make a
further assumption of the determinism of the attack. An attack is
deterministic if the same input image with the same backdoor trigger is always
classified to the same class. In other words, the attack target is
deterministically decided by the attacker and will not be altered by the
randomness during training. When clean models (non-backdoored models) are
accurate enough, the class predictions should contain only two labels: one
being the attack target and the other being the true label. As such, we can
map one label to 1 and the other label to 0 so that the problem is equivalent
to the binary case. Problems with inaccurate clean models with be discuss
later.
### The Number of Attackers
In all the above discussions, $k$ refers to the number of cooperative
attackers that use the same backdoor trigger. For the case that many
independent groups of attackers attack the model using different triggers, $k$
is decided by the size of the largest group. For example, the above codes (a)
to (d) are effective codes when all users are attackers, as long as the
attackers are independent and use different triggers. If any two users share
the same trigger, we then need more powerful codes with $k=2$, which are
introduced in the next section.
Figure 2: Subset selection in matrix form.
## 4 Defending Multiple Attackers
In this section, we extend the defense to multiple attackers $k>1$ and derive
the construction of optimal BDC, BCC, and BTC codes. Here we begin with the
formal definition of codes with arbitrary $k$. For a code matrix $H$, we
denote the Boolean sum of any $k$ columns of $H$ as a $k$-sum, and the Boolean
sum of any less than or equal to $k$ columns as a $\bar{k}$-sum. Comparing to
the $k=1$ case in Section 3, the following definition replaces the conditions
on single columns to conditions on $\bar{k}$-sums. The reason of using
$\bar{k}$-sums is due to the independency assumption: a model is backdoored if
it is trained on any of the $1\sim k$ attackers.
###### Definition 1.
(1) $H$ is BDC$(k,r,n)$ if $H$ contains no zero column in its $n$ columns, has
at least $r$ 1’s in each row, and has no $\bar{k}$-sum equals to $\mathbf{1}$.
(2) $H$ is BCC$(k,r,n)$ if $H$ is BDC$(k,r,n)$ and has no XOR of any two
$\bar{k}$-sums equals to $\mathbf{1}$. (3) $H$ is BTC$(k,r,n)$ if $H$ is
BCC$(k,r,n)$ and has no two $\bar{k}$-sums equal to each other.
### Minimal Backdoor Correction Codes
Direct construction of BDC and BCC codes of any size can be difficult so we
start by looking at a special case with minimal $n$. The corresponding codes
are denoted as minimal codes.
###### Lemma 2.
For $k\geq 1$, $r\geq 1$, the minimal $n$ for BDC$(k,r,n)$ and BCC$(k,r,n)$ to
exist is $n=k+r$.
Proof. If $n<k+r$ then all rows have $<k$ 0’s and all $k$-sums are
$\mathbf{1}$. ∎
When $n=k+r$, we can give the tight lower bound of $m$ for all BDC$(k,r,k+r)$
codes:
###### Theorem 3.
For $k\geq 1$, $r\geq 1$, if $H$ is BDC$(k,r,k+r)$, then its number of rows
$m\geq\binom{k+r}{k}$. The solution for $m=\binom{k+r}{k}$, denoted as
$\mathcal{H}^{(k,r)}$, is unique up to row and column permutations and can be
constructed by:
$\displaystyle\mathcal{H}^{(k,1)}$ $\displaystyle=I_{k+1},$ (1)
$\displaystyle\mathcal{H}^{(1,r>1)}$
$\displaystyle=\begin{bmatrix}\mathbf{1}&\mathcal{H}^{(1,r-1)}\\\
0&\mathbf{1}^{T}\end{bmatrix},$ (2) $\displaystyle\mathcal{H}^{(k>1,r>1)}$
$\displaystyle=\begin{bmatrix}\mathbf{1}&\mathcal{H}^{(k,r-1)}\\\
\mathbf{0}&\mathcal{H}^{(k-1,r)}\end{bmatrix}.$ (3)
Sketch of proof. For the lower bound of $m$, first assume each row to have
exactly $k$ 0’s. For each row, there is only one $k$-sum to produce 0 in this
row. When rows are unique, each $k$-sum can have only one 0, and the location
of this 0 is unique. Since there are $\binom{k+r}{k}$ different $k$-sums, $m$
cannot be smaller than $\binom{k+r}{k}$. Cases with non-unique rows and rows
with $<k$ 0’s can be reduced to the above case. The construction can be
verified by induction and the uniqueness is obvious. ∎
Next we show that BCC and BDC codes are closely connected. In fact, a BCC code
can always be constructed from a BDC code by adding one extra row of 1’s,
meaning the cost of backdoor correction is almost negligible if the attack can
be detected:
###### Lemma 4.
If $H$ is BDC$(k,r,n)$, then $\begin{bmatrix}\mathbf{1}^{T}\\\ H\end{bmatrix}$
is BCC$(k,r,n)$.
Proof. All $\bar{k}$-sums have 1 in the first row and $1\oplus 1=0$. ∎
This leads the main result about minimal BCC codes. In fact, minimal BCC codes
and minimal BDC codes are the same when $r>1$.
###### Theorem 5.
For $k\geq 1$, $r>1$, the minimal BCC$(k,r,k+r)$ has $m=\binom{k+r}{k}$. For
$r=1$, the minimal BCC$(k,1,k+1)$ has $m=k+2$.
Sketch of proof. Prove by induction: $\mathcal{H}^{(1,r>1)}$ has the XOR of
any two $\bar{k}$-sums $\neq\mathbf{1}$ since each column has more 1’s than
0’s. For $\mathcal{H}^{(k>1,r>1)}$, if one $\bar{k}$-sum includes the first
column then its first $\binom{k+r-1}{k}$ rows are all 1’s. The XOR has at
least one 0 in these rows since a $\bar{k}$-sum in $\mathcal{P}^{(k,r-1)}$ has
at least one 1. If no $\bar{k}$-sum includes the first column then the problem
is reduced to the $\bar{k}$-sums of $\mathcal{H}^{(k-1,r)}$. For $r=1$,
obviously $I_{k+1}$ is BDC but not BCC. Lemma 4 gives $m=k+2$. ∎
Figure 3(a) shows some examples of minimal BCC codes to visualize the pattern
of recursive construction. At this point we obtain BCC codes that can defend
any number of attackers $k$ with any large data utilization ratio $r/n$ for
the special case $n=k+r$. Next we look at how to construct general BCC codes
for larger $n$.
Figure 3: Codes for multi-attacker defense (black represents one and white
represents zero).
### General Backdoor Correction Codes
The above result can be generalized to BCC codes with any large $n$ by column
duplication. For given $k$, $r$, and $n$, we calculate the relative row weight
$r/n$, select a proper BCC$(k,r_{0},k+r_{0})$ with $r_{0}/(k+r_{0})\approx
r/n$ as a starting point, and duplicate its columns until $r$ and $n$ are
satisfied:
###### Corollary 5.1.
For $k\geq 1$, $r\geq 1$, $n\geq k+r$, $p=\lfloor\frac{n-r}{k}\rfloor$, and
$r_{0}=\lceil\frac{r}{p}\rceil$, let $H$ with columns
$\left[\mathbf{h}_{1},\mathbf{h}_{2},\dots,\mathbf{h}_{k+r_{0}}\right]$ be
BCC$(k,r_{0},k+r_{0})$, then the following code is BCC$(k,r,n)$:
$\bigg{[}\mathbf{h}_{1},\dots,\mathbf{h}_{n-p(k+r_{0})},\underbrace{H,\dots,H}_{\text{repeat
p times}}\bigg{]}.$
Proof. For any $\bar{k}$-sum in the above code there exists an equal
$\bar{k}$-sum in $H$. Therefore $H$ being BCC implies the above code being
BCC. The row weight $\geq pr_{0}\geq r$. ∎
Figure 3(b) gives an example BCC$(2,4,8)$ generated from BCC$(2,2,4)$. The
result obtained by column duplication might not be optimal when $n$ is not
divided by $r$. However, since finding the optimal solution is NP-hard in
general, we consider the construction good enough. The result gives several
important implications:
* •
For fixed $k$ and $r/n$, the cost of backdoor correction is independent of the
absolute value of $n$.
* •
For fixed $k/n$ and $r$, the cost grows linearly.
* •
For fixed $k/n$ and $r/n$, the cost grows exponentially.
Such a result suggests a fundamental tradeoff between data utilization and
backdoor robustness. When either of the data utilization or backdoor
robustness is measured by absolute values, the total cost is manageable.
However, if the goal is to achieve both high data utilization ratio and
robustness ratio, there is no scalable solution.
### Backdoor Tracking Codes
Finally, we show that BCC codes can be concatenated with other familiar codes
in coding theory to achieve additional properties and create novel defenses.
For backdoor tracking, BTC codes further require the matrix to have no two
$\bar{k}$-sums equal to each other. In literature about superimposed codes and
group testing, a matrix with unique $\bar{k}$-sums is called a
$\bar{k}$-separable matrix. Here we directly construct BTC codes using the
existing results of $\bar{k}$-separable matrices:
###### Corollary 5.2.
If $H_{1}$ is BCC$(k,r,n)$, $H_{2}$ is $\bar{k}$-separable, and the row weight
of $H_{2}$ is at least $r$, then $H=\begin{bmatrix}H_{1}\\\
H_{2}\end{bmatrix}$ is BTC$(k,r,n)$.
Proof. The code $H$ is BCC$(k,r,n)$ since $H_{1}$ is BCC$(k,r,n)$ and $H_{2}$
has row weight $r$. The code $H$ is $\bar{k}$-separable since $H_{2}$ is
$\bar{k}$-separable. Therefore $H$ is BTC$(k,r,n)$. ∎
Previous study shows that $\bar{k}$-separable matrices with $n$ columns have a
lower bound of $m>O(k\log n)$ and a typical construction of $m=O(k^{2}\log n)$
(Du and Hwang 2000). Since BCC codes typically have a much smaller size,
concatenating a BCC code with a separable matrix gives a good enough BTC code.
Figure 3(c) shows an example of a BTC$(2,2,8)$ code, with its top 4 rows a
BCC$(2,2,8)$ code and bottom 7 rows a $\bar{2}$-separable matrix.
## 5 Decoding Algorithm
In Section 4, we study theoretical properties of the codes in an idealized
setting and assume that the class prediction vector $\mathbf{y}$ contains only
two groups: the true label and the attack target. This section fills the gap
between the idealized code design and practical defenses, where the noise in
$\mathbf{y}$ (i.e. inaccurate classification of clean models) is considered.
Here we propose a probabilistic solution. We model the noise with a
probability distribution and decode the noisy output by probability
maximization. Given a set of validation data, we can statistically estimate
the distribution of clean data classification by confusion matrices. For model
$i$, $C^{(i)}_{jk}$ represents the probability of classifying clean data in
class $j$ as class $k$. The attack can be specified by a prior probability of
the attack $A$, a success rate of the attack $S$, and a distribution of the
number of attackers $Q(\|\mathbf{x}\|_{1})$. Given the class predictions
$\mathbf{y}$, the number of users $n$, and the number of classes $c$, the
probability of the models being attacked is given by:
$\displaystyle Pr(\text{attack}=\text{True}|\text{pred}=\mathbf{y})=$
$\displaystyle\frac{A\sum_{\mathbf{x},t,l}p(\mathbf{x},\mathbf{y},t,l)}{A\sum_{\mathbf{x},t,l}p(\mathbf{x},\mathbf{y},t,l)+(1-A)c\sum_{l}\prod_{i}C^{(i)}_{ly_{i}}},$
$\displaystyle\text{ in which
}p(\mathbf{x},\mathbf{y},t,l)=\frac{Q(\|\mathbf{x}\|_{1})}{\binom{n}{\|\mathbf{x}\|_{1}}}\times$
$\displaystyle\prod_{i}\begin{cases}S\delta_{ty_{i}}+(1-S)C^{(i)}_{ly_{i}}&\text{if
}\sum_{j}H_{ij}x_{j}>0\\\ C^{(i)}_{ly_{i}}&\text{otherwise}\end{cases}.$
Here $\delta_{ij}=1$ if $i=j$ or $0$ if $i\neq j$. The true label $l$ and the
attackers $\mathbf{x}$ can be obtained by maximizing over the distributions
$Pr(\text{label}=l|\text{pred}=\mathbf{y})\sim
A\sum_{\mathbf{x},t}p(\mathbf{x},\mathbf{y},t,l)+(1-A)c\prod_{i}C^{(i)}_{ly_{i}}$
and
$Pr(\text{attackers}=\mathbf{x}|\text{attack}=\text{True},\text{pred}=\mathbf{y})\sim\sum_{t,l}p(\mathbf{x},\mathbf{y},t,l).$
See Appendix for detailed derivations.
In the noisy setting, codes with a high $k$ can defend a reduced number of
attackers that is less than $k$. For example, the code BCC$(2,4,8)$ with $k=2$
in Figure 3(b) can reliably defend one attacker but not two. If the six models
have five outputs of class A and one output of class B, then there could be
two equally likely cases: 1) two attackers have attack target A and the true
class is B, and 2) the true class is A and one model misclassifies it as B.
For a single attacker, the attack will results in three outputs of class A and
three outputs of class B. The chance for three models to make the same
misclassification is small.
## 6 Experiments
### Experiment Setup
We evaluate the proposed defense on MNIST (LeCun et al. 1998) and CIFAR-10
(Krizhevsky, Hinton et al. 2009). The models we tested include a 4-layer CNN
for MNIST and ResNet18 (He et al. 2016) for CIFAR-10. The attack method is
based on direct data poisoning with random 3px$\times$3px backdoor triggers
and random target classes (Qiao, Yang, and Li 2019). We use 12 users for
backdoor correction and 16 users for backdoor detection (for the ease of code
construction), with 1$\sim$3 attackers. Each attacker poisons 5% of their data
to perform the attack.
To synthesize the non-IID data, we follow the method from (Hsu, Qi, and Brown
2019) and use Dirichlet distribution to decide the number of data per class
(a.k.a. class distribution) for each user. A hyper-parameter $\alpha$
(concentration parameter of Dirichlet distribution) controls the degree of
unevenness of the class distribution. In the experiments we use $\alpha=10$,
1, and 0.1. Examples of per-user class distributions are given Figure 4.
Figure 4: Examples of class distributions controlled by $\alpha$. Each color
represents a class and each row represents a user.
We investigate eight different codes, plotted in Figure 5, including six codes
for backdoor correction and two codes for backdoor tracking. Codes (a) and (b)
represent non-overlapping partition, following (Levine and Feizi 2020). Codes
(c) and (d) are the proposed optimal BCC$(4,4,12)$ and BCC$(2,6,12)$. Codes
(e) and (f) are random matrices with constant row weight, following (Jia et
al. 2020). The row weights are set to 4 and 6 to match with code (c) and (d).
Codes (g) and (h) are backdoor tracking codes in which (g) is BTC$(1,11,16)$
and (h) is BTC$(2,4,16)$.
Figure 5: Six BCC codes and two BTC codes used in experiments. Figure 6:
Classification accuracy of BCC codes under 0$\sim$3 attackers.
### Backdoor Correction
We use classification accuracy under attack to evaluate the backdoor
correction performance. Figure 6 summarizes the results from the two datasets
with four different IID/non-IID settings. Each line represents the
classification accuracy (averaged from 5 repeated runs) of a code under
0$\sim$3 attackers (0 means no attack). Code (a) and (b) are decoded by
majority vote. Code (c) and (d) are decoded by the proposed decoder. Code (e)
and (f) are decoded using both approaches, denoted as (e/f)-maj and (e/f)-dec
respectively. We have the following observations from the results:
* •
Codes decoded by the proposed decoder have a slight drop in accuracy on clean
data, because some clean data are mistakenly considered as attacks. Despite
the above effect, code (a) and (b) have the lowest clean data accuracy in
general due to the low data utilization ratio. The accuracy drop is more
prominent for smaller $\alpha$.
* •
Codes decoded by majority vote perform poorly when $r>1/(2k+1)$ due to the
majority vote bound. When using the propose decoder, random codes (e) and (f)
have higher robustness but are weaker than the proposed codes. The gap is
especially large when data utilization ratio is high (code (d) versus
code(f)-dec).
* •
The proposed codes have good performance when the number of attackers is below
the codes’ designed capability. As code (c) with $k=4$ is effective against
1$\sim$3 attackers, code (d) with $k=2$ can only reliably defend one attacker.
### Backdoor Tracking
To evaluate the backdoor tracking capability, we use the decoder to predict
the binary vector $\mathbf{x}$ that corresponds to the attackers’ locations,
and compare the predicted attackers with the true attackers. The metric is
based on true positives (TP) and false positives (FP), where TP should be as
close to $k$ as possible and FP should be near to zero. Our decoder is allowed
to predict any number of attackers, but it will prioritize on finding the
minimum number of attackers when $Q(\|\mathbf{x}\|_{1})$ is a uniform
distribution.
We perform the experiments using IID data and summarize the results in Table
1. Means and standard deviations are gathered from 10 repeated runs. As we can
observe from the results, code (g) can reliably track one attacker, while code
(h) can track up to two attackers with a lower accuracy. Such result is
consistent with the codes’ theoretical properties. For codes (a)$\sim$(f), we
obtain an average TP of 0.2, 0.5, 0.5 for 1$\sim$3 attackers and an average FP
of 0.8, 1.1, 1.2. The backdoor tracking capability is close to random guess.
Table 1: True-positives and false-positives of attacker tracking. Dataset | $k$ | Code (g) | Code (h)
---|---|---|---
TP | FP | TP | FP
MNIST IID | 1 | 1.0$\pm$0.0 | 0.0$\pm$0.0 | 1.0$\pm$0.0 | 0.01$\pm$0.0
2 | 0.29$\pm$0.42 | 0.71$\pm$0.42 | 1.99$\pm$0.0 | 0.01$\pm$0.0
3 | 0.27$\pm$0.39 | 0.73$\pm$0.39 | 2.68$\pm$0.94 | 0.31$\pm$0.94
CIFAR IID | 1 | 0.97$\pm$0.02 | 0.03$\pm$0.02 | 0.83$\pm$0.21 | 0.38$\pm$0.36
2 | 0.18$\pm$0.24 | 0.82$\pm$0.24 | 1.72$\pm$0.36 | 0.26$\pm$0.27
3 | 0.2$\pm$0.4 | 0.8$\pm$0.4 | 2.36$\pm$0.56 | 0.36$\pm$0.34
### Discussion
The above experiments show that our codes achieve their design goal, which is
to obtain a high data utilization ratio while maintaining the robustness. A
significant gain in classification accuracy can be observed on non-IID
datasets. Comparison with random matrices indicates the importance of careful
code design, especially when data utilization ratio is high.
Improving the data utilization or tracking the attackers comes with the cost
of using more models. This limits our method to be mostly effective against a
small number of attackers. However, the scalability issue is a fundamental
limitation of all subset aggregation-based method and cannot be addressed by
designing better codes. To reduce the overall cost, potential solutions could
be using a large number of cheaper models or sharing parts of computation
across different models.
## 7 Related Works
### Backdoor Attacks and Defenses
Backdoor attacks were originated as a data-poisoning attack, where attackers
inject backdoors by poisoning a portion of training data (Gu, Dolan-Gavitt,
and Garg 2017; Turner, Tsipras, and Madry 2018). In Federated learning,
attackers could directly poison the gradients which further increase the
flexibility of the attack (Bagdasaryan et al. 2018). Most backdoor defenses
rely on extracting abnormal statistical features of backdoored data, which are
mostly heuristic based (Tran, Li, and Madry 2018; Chen et al. 2018a). When
attackers have knowledge of the features used by the defender. they can
adversarially create backdoor samples that bypass the defense (Tan and Shokri
2019).
Provable defenses use ensemble models to guarantee a certain degree of
backdoor robustness. Although ensemble models are costly to train and
inference, this is the only know type of defense that can provide robustness
regardless of the exact attack method. In randomized smoothing-based methods
(Weber et al. 2020; Wang et al. 2020), the defense assumes that attackers only
apply small perturbations (with bounded $L_{2}$ norm) to the training image.
In subset aggregation-based method (Levine and Feizi 2020; Jia et al. 2020),
the defense assumes that attackers only change a small number of training
data. Both types of methods are not developed in the context of collaborative
learning and assume that each training sample can be independently poisoned.
In such settings where $n\sim 10^{4}$ (each training sample is considered as a
user) and $k\sim 10^{2}$ (hundreds of poisoned samples), our method will have
similar solutions as the previous works.
### Coding Theory and ML
A major junction between coding theory and ML is in the area of error
correction codes (ECCs). ECCs are applied to ML most notably as error
correcting output codes (Dietterich and Bakiri 1994) for improving model
robustness (Verma and Swami 2019). In Federated learning, DRACO (Chen et al.
2018b) uses ECCs for gradient encoding and provide Byzantine resilience by
introducing redundant gradients.
Superimposed codes emerge from information retrieval (Kautz and Singleton
1964) and have applications in non-adaptive group testing (Du and Hwang 2000).
Several variants of the code such as disjunct matrices and separable matrices
are discussed in (Chen and Hwang 2007).
## 8 Conclusion
We propose a new subset aggregation-based defense against backdoor attacks in
collaborative learning and overcome the data utilization issue in previous
defenses. A coding-based framework is introduced to analyze the optimal subset
selection. Theoretically, we prove the fundamental tradeoff between data
utilization ratio and backdoor robustness, which indicates the limitation of
all subset aggregation methods. In practice, we show that by combining optimal
codes with a probabilistic decoder, our defense can achieve a high
classification accuracy while maintaining backdoor robustness.
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## Proofs and Derivations
Theorem 3. For $k\geq 1$, $r\geq 1$, if $H$ is BDC$(k,r,k+r)$, then its number
of rows $m\geq\binom{k+r}{k}$. The solution for $m=\binom{k+r}{k}$, denoted as
$\mathcal{H}^{(k,r)}$, is unique up to row and column permutations and can be
constructed by:
$\displaystyle\mathcal{H}^{(k,1)}=I_{k+1},$
$\displaystyle\mathcal{H}^{(1,r>1)}=\begin{bmatrix}\mathbf{1}&\mathcal{H}^{(1,r-1)}\\\
0&\mathbf{1}^{T}\end{bmatrix},$
$\displaystyle\mathcal{H}^{(k>1,r>1)}=\begin{bmatrix}\mathbf{1}&\mathcal{H}^{(k,r-1)}\\\
\mathbf{0}&\mathcal{H}^{(k-1,r)}\end{bmatrix}.$
Proof. We first prove the lower bound of $m$, which includes three cases:
Case 1: rows are unique and all rows have exactly $k$ 0’s. Assume $H$ is
BDC$(k,r,k+r)$ with $m<\binom{k+r}{k}$. Since rows of $H$ are unique and there
exists $\binom{k+r}{k}$ different binary sequences with $k$ 0’s and $r$ 1’s,
at least one binary sequence with $k$ 0’s and $r$ 1’s does not appear in any
row of $H$. Denote this sequence as $a$ and the indices of 0’s in $a$ as
$S=\\{i_{1},i_{2},\dots,i_{k}\\}$. Select elements of $a$ by the index set $S$
and denote the result as $a_{S}=\\{a_{i_{1}},a_{i_{2}},\dots,a_{i_{k}}\\}$,
then all elements of $a_{S}$ are 0’s. For any other binary sequence $b$ with
$k$ 0’s and $r$ 1’s such that $b\neq a$, select elements of $b$ using the same
index set $S$. The result $b_{S}$ has at least one 1 and the Boolean sum of
$b_{S}$ is 1. Let $H_{S}$ be the $k$ columns of $H$ selected by $S$, and $s$
be the Boolean sum of $H_{S}$, then $s$ is a $k$-sum of $H$. Since $a$ is not
in $H$, we have $s=\mathbf{1}$, which contradicts the assumption of $H$ being
BDC$(k,r,k+r)$.
Case 2: rows are unique and some rows have less than $k$ 0’s. If any row has
less than $k$ 0’s, then any $k$-sum of $H$ equals 1 in this row. If $H$ is
BDC$(k,r,k+r)$ then the code obtained by removing this row from $H$ is also
BDC$(k,r,k+r)$. Repeat the process until all rows with less than $k$ 0’s are
removed, then the problem is reduced to Case 1.
Case 3: rows are not unique. Similarly, if $H$ is BDC$(k,r,k+r)$ then the code
obtained by removing a non-unique row from $H$ is BDC$(k,r,k+r)$. Repeat the
process until all non-unique rows are removed, then the problem is reduced to
Case 2. Since there are no other possible cases, $H$ always has
$m\geq\binom{k+r}{k}$.
Next we verify that $\mathcal{H}^{(k,r)}$ has $m=\binom{k+r}{k}$ and satisfies
the requirements of BDC$(k,r,k+r)$.
When $r=1$, obviously $m=k+1$ and $I_{k+1}$ is BDC$(k,1,k+1)$.
When $k=1$ and $r>1$, assume that $\mathcal{H}^{(1,r-1)}$ is BDC$(1,r-1,r)$
and has $m=r$. Then $\mathcal{H}^{(1,r)}$ has $m=r+1$. Consider the row weight
of the first $r-1$ rows, the result is 1 plus the row weight of
$\mathcal{H}^{(1,r-1)}$, which gives $1+(r-1)=r$. Since the last row of
$\mathcal{H}^{(1,r)}$ has $r$ 1’s, $\mathcal{H}^{(1,r)}$ has a row weight of
$\min\\{r,r\\}=r$. The last $r-1$ columns are not $\mathbf{1}$ since their
first $r-1$ element are not all 1’s (based on the assumption of
$\mathcal{H}^{(1,r-1)}$). The first column is not $\mathbf{1}$ by
construction. Since $\mathcal{H}^{(1,r)}$ has a row weight of $r$ and no
column equals $\mathbf{1}$, $\mathcal{H}^{(1,r)}$ is BDC$(1,r,r+1)$.
When $k>1$ and $r>1$, assume that $\mathcal{H}^{(k,r-1)}$ has
$m=\binom{k+r-1}{k}$ and $\mathcal{H}^{(k-1,r)}$ has $m=\binom{k+r-1}{k-1}$.
Then $\mathcal{H}^{(k,r)}$ has
$m=\binom{k+r-1}{k}+\binom{k+r-1}{k-1}=\binom{k+r}{k}$. Further assume that
$\mathcal{H}^{(k,r-1)}$ is BDC$(k,r-1,k+r-1)$ and $\mathcal{H}^{(k-1,r)}$ is
BDC$(k-1,r,k+r-1)$. Then for $\mathcal{H}^{(k,r)}$, the row weight is given by
the minimum of the row weights of the first $\binom{k+r-1}{k}$ rows and the
last $\binom{k+r-1}{k-1}$ rows, which is $\min\\{1+(r-1),r\\}=r$. For any
$\bar{k}$-sum in $\mathcal{H}^{(k,r)}$, if the addends of the $\bar{k}$-sum
includes the first column, then the $\bar{k}$-sum is not $\mathbf{1}$ since
the last $\binom{k+r-1}{k-1}$ elements of the $\bar{k}$-sum is equal to a
$\overline{k-1}$-sum of $\mathcal{H}^{(k-1,r)}$, which is not $\mathbf{1}$. If
the addends of the $\bar{k}$-sum does not include the first column, then the
$\bar{k}$-sum is not $\mathbf{1}$ since the first $\binom{k+r-1}{k}$ elements
of the $\bar{k}$-sum is equal to a $\bar{k}$-sum of $\mathcal{H}^{(k,r-1)}$,
which is not $\mathbf{1}$. Since $\mathcal{H}^{(k,r)}$ has a row weight of $r$
and no $\bar{k}$-sum equals $\mathbf{1}$, $\mathcal{H}^{(k,r)}$ is
BDC$(k,r,k+r)$.
By induction, $\mathcal{H}^{(k,r)}$ is BDC$(k,r,k+r)$ and has
$m=\binom{k+r}{k}$ for all positive integers $k$ and $r$.
At last we show that the solution for $m=\binom{k+r}{k}$ is unique up to row
and column permutations. As proved earlier, if $m=\binom{k+r}{k}$, then all
rows are unique and each row includes exactly $k$ 0’s. Since there are only
$\binom{k+r}{k}$ possible binary sequences with $k$ 0’s and $r$ 1’s, all of
these sequences must be selected exactly once to construct the code, which
implies uniqueness of the selection. ∎
Theorem 5. For $k\geq 1$, $r>1$, the minimal BCC$(k,r,k+r)$ has
$m=\binom{k+r}{k}$. For $r=1$, the minimal BCC$(k,1,k+1)$ has $m=k+2$.
Proof. For $k\geq 1$, $r>1$, Theorem 3 states that $\mathcal{H}^{(k,r)}$ with
$m=\binom{k+r}{k}$ is BDC$(k,r,k+r)$. We need to verify that
$\mathcal{H}^{(k,r)}$ is also BCC$(k,r,k+r)$, meaning that the XOR of any two
$\bar{k}$-sums is not equal to $\mathbf{1}$.
When $k=1$, $r>1$, every columns of $\mathcal{H}^{(1,r)}$ have more 1’s than
0’s. By monotonicity of Boolean sums, every $\bar{k}$-sums of
$\mathcal{H}^{(1,r)}$ have more 1’s than 0’s. If the XOR of two $\bar{k}$-sums
equals $\mathbf{1}$, their total number of 1’s must be equal to their total
number of 0’s, which is impossible.
When $k>1$, $r>1$, assume that the XOR of any two $\bar{k}$-sums is not equal
to $\mathbf{1}$ for $\mathcal{H}^{(k-1,r)}$, then consider
$\mathcal{H}^{(k,r)}$ in three cases:
Case 1: no $\bar{k}$-sum of $\mathcal{H}^{(k,r)}$ includes the first column.
The XOR of any two $\bar{k}$-sums is not equal to $\mathbf{1}$ since the last
$\binom{k+r-1}{k-1}$ elements are not all 1’s (guaranteed by
$\mathcal{H}^{(k-1,r)}$).
Case 2: one and only one $\bar{k}$-sum of $\mathcal{H}^{(k,r)}$ includes the
first column. The first $\binom{k+r-1}{k}$ elements of one $\bar{k}$-sum are
all 1’s. The first $\binom{k+r-1}{k}$ elements of the other $\bar{k}$-sum have
at least one 1 since $\mathcal{H}^{(k,r-1)}$ contains no zero column. The XOR
of the two $\bar{k}$-sums has at least one 0.
Case 3: both $\bar{k}$-sums of $\mathcal{H}^{(k,r)}$ include the first column.
Obviously the first $\binom{k+r-1}{k}$ elements of the XOR of the two
$\bar{k}$-sums are equal to 0.
By induction, $\mathcal{H}^{(k,r)}$ is BCC$(k,r,k+r)$ for all positive
integers $k$ and $r>1$.
For $r=1$, obviously $I_{k+1}$ is BDC but not BCC. With Lemma 4, adding an
additional row of 1’s to $I_{k+1}$ turns the code to be BCC$(k,1,k+1)$, which
gives $m=k+2$. ∎
### Probabilistic Decoding
For model $i$, we use confusion matrices $C^{(i)}_{jk}$ to estimate the
probability of classifying clean data in class $j$ as class $k$, which gives:
$Pr(\text{pred}_{i}=y_{i}|\text{attack}=\text{False},\text{label}=l)=C_{ly_{i}}^{(i)},$
in which $\text{pred}_{i}=y_{i}$ is the class prediction of model $i$,
$\text{attack}=\text{False}$ refers to the cases that no attack happens, and
$\text{label}=l$ represents the true label of the data.
Assuming that the attack has a fixed target $\text{targ}=t$, then when an
attack succeeds, denoted as $\text{atkSucc}=\text{True}$ (do not confuse with
$\text{attack}=\text{True}$, which means an attack happens but not necessarily
succeeds), we have:
$\displaystyle
Pr(\text{pred}_{i}=y_{i}|\text{atkSucc}=\text{True},\text{targ}=t)=\delta_{ty_{i}}$
$\displaystyle\text{in which}\quad\delta_{ij}=\begin{cases}1\text{ if }i=j\\\
0\text{ otherwise}\end{cases}.$
Further assuming that a failed attack does not affect the model’s prediction,
which gives:
$Pr(\text{pred}_{i}=y_{i}|\text{atkSucc}=\text{False},\text{label}=l)=C_{ly_{i}}^{(i)}.$
The rate of attack $A$ specifies our prior assumption of the likelihood of an
attack to happen:
$Pr(\text{attack}=\text{True})=A.$
The success rate of attack $S$ connects the events of an attack to happen and
to succeed as:
$\displaystyle
Pr(\text{atkSucc}=\text{True}|\text{attack}=\text{True},\text{attackers}=\mathbf{x})$
$\displaystyle=\begin{cases}S&\text{if }\sum_{j}H_{ij}x_{j}>0\\\
0&\text{otherwise}\end{cases}.$
Here $H$ is the code matrix, $\mathbf{x}$ is the binary vector representing
all attackers, and $H_{ij}x_{j}>0$ means that model $i$ is backdoored.
Connecting the above equations gives:
$\displaystyle
Pr(\text{pred}_{i}=y_{i}|\text{attack}=\text{True},\text{attackers}=\mathbf{x},\text{targ}=t,$
$\displaystyle\text{label}=l)$
$\displaystyle=\begin{cases}S\delta_{ty_{i}}+(1-S)C^{(i)}_{ly_{i}}&\text{if
}\sum_{j}H_{ij}x_{j}>0\\\ C^{(i)}_{ly_{i}}&\text{otherwise}\end{cases}.$
To further specify the distribution of the locations of the attackers, we use
$Q(k)$ to represent the probability of having $k$ attackers out of $n$ users.
The permutation symmetry of the locations gives:
$Pr(\text{attackers}=\mathbf{x})=\frac{Q(\|\mathbf{x}\|_{1})}{\binom{n}{\|\mathbf{x}\|_{1}}}.$
Assuming independent model predictions, we denote $\text{pred}=\mathbf{y}$ as
the class predictions of all models and define $p(\mathbf{x},\mathbf{y},t,l)$
as:
$\displaystyle
p(\mathbf{x},\mathbf{y},t,l)\vcentcolon=Pr(\text{pred}=\mathbf{y},\text{attackers}=\mathbf{x}|\text{attack}=\text{True},$
$\displaystyle\text{targ}=t,\text{label}=l)$
$\displaystyle=\frac{Q(\|\mathbf{x}\|_{1})}{\binom{n}{\|\mathbf{x}\|_{1}}}\prod_{i}\begin{cases}S\delta_{ty_{i}}+(1-S)C^{(i)}_{ly_{i}}&\text{if
}\sum_{j}H_{ij}x_{j}>0\\\ C^{(i)}_{ly_{i}}&\text{otherwise}\end{cases}.$
Then assuming attack targets and true labels are independent and uniformly
distributed in $c$ classes, i.e., $Pr(\text{targ}=t,\text{label}=l)=1/c^{2}$,
we have:
$\displaystyle
Pr(\text{pred}=\mathbf{y}|\text{attack}=\text{True})=\sum_{\mathbf{x},t,l}p(\mathbf{x},\mathbf{y},t,l)/c^{2}\text{,
and}$ $\displaystyle
Pr(\text{pred}=\mathbf{y}|\text{attack}=\text{False})=\sum_{l}\prod_{i}C^{(i)}_{ly_{i}}/c.$
Using Bayes’s rule, we finally arrive at:
$\displaystyle Pr(\text{attack}=\text{True}|\text{pred}=\mathbf{y})=$
$\displaystyle\frac{A\sum_{\mathbf{x},t,l}p(\mathbf{x},\mathbf{y},t,l)}{A\sum_{\mathbf{x},t,l}p(\mathbf{x},\mathbf{y},t,l)+(1-A)c\sum_{l}\prod_{i}C^{(i)}_{ly_{i}}}.$
Let $m$ be the number of models, the time complexity of the decoder is
$O(n^{k}mc^{2})$.
## Experimental Details
### Data Preparation
MNIST. The dataset contains 60,000 images in the training set and 10,000
images in the testing set. All images in the dataset are normalized before
training. The dataset can be downloaded from
http://yann.lecun.com/exdb/mnist/.
CIFAR-10. The dataset contains 50,000 images in the training set and 10,000
images in the testing set. Each image is randomly cropped into 32px by 32px,
horizontally flipped and normalized before training. The dataset can be
downloaded from https://www.cs.toronto.edu/~kriz/cifar.html.
All experiments are performed on 12 or 16 users. In each experiment, we
randomly permute the training set and split the training dataset according to
the Dirichlet distribution. The testing set is split to 2 parts. Among them,
5,000 images are used for confusion matrix estimation. The other 5,000 images
are used for testing the decoder’s defense capability.
### Model Design
CNN for MNIST. The network consists of 2 convolutional layers and 2 fully
connected layers. The first convolutional layer has 32 3$\times$3 convolution
kernels with a stride of 1 and ReLU activation. The second convolutional layer
has 64 3$\times$3 convolution kernels with a stride of 1, followed by a
2$\times$2 max-pooling layer and a dropout layer. Followed by the
convolutional layers are two fully connected layers with output size 128 and
10.
ResNet18 for CIFAR. The ResNet18 is adopted from an open source implementation
https://github.com/kuangliu/pytorch-cifar.
### Training
We use Adam optimizer of model training with a batchsize of 128. When a model
is trained on data from multiple users, we iterate over the selected users
with the granularity of a batch. Data in a single batch always come from a
single user.
For hyper-parameter tuning, we manually explore the poison ratio from 1e-2 to
1e-1, the learning rate from 1e-4 to 1e-2, and the number of training epochs
from 10 to 30. The final results are obtained using the following settings:
The attack uses randomly generated 3px$\times$3px triggers with 5% poison
ratio. The triggers are generated by a random permutation of five 1’s and four
0’s on MNIST and by 27 i.i.d. random RGB values following a clipped Gaussian
distribution on CIFAR. We choose Adam as the optimizer, with learning rate
1e-3 and weight decay 1e-4. Models are trained for 10 epochs on MNIST and 20
epochs on CIFAR. The experiment are done on a server with two Intel Xeon
E5-2687W CPUs, 755 GiB of memory, and four Nvidia TITAN RTX GPUs.
### Testing
In the testing stage, we fix the attack rate $A$ to 0.5 and the attack success
rate to 0.99. The attacker distribution $Q$ is a uniform distribution covering
0$\sim$3 attackers.
|
# Relic gravitational waves from the chiral magnetic effect
Axel Brandenburg1,2,3,4 Yutong He1,2 Tina Kahniashvili3,4,5 Matthias
Rheinhardt6 Jennifer Schober7 1Nordita, KTH Royal Institute of Technology and
Stockholm University, Hannes Alfvéns väg 12, SE-10691 Stockholm, Sweden
2Department of Astronomy, AlbaNova University Center, Stockholm University,
SE-10691 Stockholm, Sweden
3McWilliams Center for Cosmology and Department of Physics, Carnegie Mellon
University, 5000 Forbes Ave, Pittsburgh, PA 15213, USA
4Faculty of Natural Sciences and Medicine, Ilia State University, 3-5
Cholokashvili Avenue, 0194 Tbilisi, Georgia
5Department of Physics, Laurentian University, Ramsey Lake Road, Sudbury, ON
P3E 2C, Canada
6Department of Computer Science, Aalto University, PO Box 15400, FI-00076
Aalto, Finland
7Laboratoire d’Astrophysique, EPFL, CH-1290 Sauverny, Switzerland
(Astrophys. J. 911, 110 (2021), http://doi.org/10.3847/1538-4357/abe4d7;
Received 2021 January 18; accepted 2021 February 7; published 2021 April 22)
###### Abstract
Relic gravitational waves (GWs) can be produced by primordial magnetic fields.
However, not much is known about the resulting GW amplitudes and their
dependence on the details of the generation mechanism. Here we treat magnetic
field generation through the chiral magnetic effect (CME) as a generic
mechanism and explore its dependence on the speed of generation (the product
of magnetic diffusivity and characteristic wavenumber) and the speed
characterizing the maximum magnetic field strength expected from the CME. When
the latter exceeds the former (regime I), the regime applicable to the early
universe, we obtain an inverse cascade with moderate GW energy that scales
with the third power of the magnetic energy. When the generation speed exceeds
the CME limit (regime II), the GW energy continues to increase without a
corresponding increase of magnetic energy. In the early kinematic phase, the
GW energy spectrum (per linear wavenumber interval) has opposite slopes in
both regimes and is characterized by an inertial range spectrum in regime I
and a white noise spectrum in regime II. The occurrence of these two slopes is
shown to be a generic consequence of a nearly monochromatic exponential growth
of the magnetic field. The resulting GW energy is found to be proportional to
the fifth power of the limiting CME speed and the first power of the
generation speed.
###### Subject headings:
gravitational waves—early universe—turbulence—magnetic fields—MHD
## 1\. Introduction
The chiral magnetic effect (CME) describes an electric current along a
magnetic field carried by electrically charged chiral fermions (Vilenkin,
1980). This effect has been discussed as one of several possible mechanisms
for significantly amplifying primordial magnetic fields in the early universe
(Boyarsky et al., 2012, 2015). It works as a dynamo effect that destabilizes
the state of vanishing magnetic field and causes an arbitrarily weak seed
field to grow exponentially for a limited time (Joyce & Shaposhnikov, 1997).
Excitation sets in when the fermion chiral asymmetry is large enough. However,
owing to the existence of a conservation law for the sum of magnetic helicity
and chiral asymmetry, the CME becomes continuously depleted until nearly all
the initial chiral asymmetry is turned into magnetic helicity (Boyarsky et
al., 2012, 2015). Thus, the initial chiral asymmetry determines the final
value of the product of the mean squared magnetic field $B_{\rm rms}^{2}$ and
the magnetic correlation length $\xi_{\rm M}$, forming a proxy for magnetic
helicity in case of a fully helical field. For realistic parameters describing
our universe, $B_{\rm rms}^{2}\xi_{\rm M}$ is expected to be of the order of
or below $(10^{-18}\,{\rm G})^{2}\,{\rm Mpc}$ (Brandenburg et al., 2017b).
This value is below the lower limit of $B_{\rm rms}^{2}\xi_{\rm
M}>(10^{-16}\,{\rm G})^{2}\,{\rm Mpc}$ that is inferred from the non-
observations of GeV-energy halos around TeV blazars (Aharonian et al., 2006;
Neronov & Vovk, 2010; Taylor et al., 2011). Yet the question can be raised,
whether the resulting magnetic stress could still be large enough to produce
measurable gravitational waves (GWs).
Another severe problem are the very small length scales associated with the
CME. An upper bound for the wavenumber associated with the chiral asymmetry in
comoving units is $k_{*}\equiv k_{\rm B}T/\hbar c=12\,{\rm cm}^{-1}$, where
$k_{\rm B}$ is the Boltzmann constant, $\hbar$ is the reduced Planck constant,
$c$ is the speed of light, and $T=2.7\,{\rm K}$ is the present day
temperature. Assuming a field strength of $1\,\mu{\rm G}$, the value of
$k_{*}$ is compatible with the upper bound on the magnetic helicity of
$(10^{-18}\,{\rm G})^{2}\,{\rm Mpc}$ (Brandenburg et al., 2017b). This value
of $k_{*}$ corresponds to very small length scales, because the CME is a
microphysical effect involving just $\hbar$, $c$, and $k_{\rm B}$ as relevant
natural constants, but not Newton’s constant or the Planck mass; see also
Brandenburg et al. (2017a). The Hubble radius, by contrast, does involve
Newton’s constant and is much bigger ($1.8\times 10^{15}\,{\rm cm}$). In units
of the inverse Hubble radius, the characteristic scale of the CME corresponds
to a wavenumber of about $2\times 10^{16}$; see Equation (1) of Kahniashvili
et al. (2013), and is associated with a very high GW frequency of $4\times
10^{11}\,{\rm Hz}$; see Equation (51) of Kosowsky et al. (2002). On the other
hand, at the time of the electroweak phase transition, the Hubble scale
corresponds to a frequency in the mHz range, which is the range accessible to
the Laser Interferometer Space Antenna. Larger length scales have been argued
to be possible by invoking strongly out-of-equilibrium magnetic field
generation during preheating (Díaz-Gil, 2008a, b), or during inflation (Sharma
et al., 2020; Okano & Fujita, 2021). In addition, the actual GW frequency
could be several orders of magnitude smaller owing to the inverse cascade
associated with the CME. By the time the magnetic field has reached its
maximum, its typical length scale can therefore be significantly larger than
the scale at which the field was originally produced. After that time, the
magnetic length scales continue to increase as the magnetic energy decreases.
However, Roper Pol et al. (2020b) found that the resulting GW energy is
determined just by the maximum field strength. It is therefore unclear whether
the late phase of magnetic decay is still relevant to GW production.
Although the CME may not open a viable pathway for explaining the primordial
magnetic field, it has the advantage of providing a self-consistent mechanism
for explaining not just a certain field strength and length scale, but also a
certain time dependence of its generation, independent of any extra
assumptions. Thus, it may serve as a proxy for other generation mechanisms. It
is then interesting to investigate GWs produced by the CME as a mechanism that
is likely to contain qualitatively valid aspects of primordial magnetic field
generation; see the recent work by Anand et al. (2019) for analytic approaches
addressing GW production from the CME at energies much above the electroweak
scale, or the approaches of Sharma et al. (2020) and Okano & Fujita (2021)
addressing GW production from helical magnetogenesis during inflation. These
works give more optimistic prospects about the resulting magnetic field
generation than Brandenburg et al. (2017b). Therefore, in the present study
our aim is to understand the detailed relationship between the strengths of
magnetic field and GWs, as well as their typical time and length scales.
In the past, theoretical GW energy spectra have been calculated mostly using
analytical approaches; see Deryagin et al. (1987) for an early pioneering
investigation and Caprini et al. (2019) for a recent review. Numerical
approaches have recently been applied to GWs, driven by acoustic turbulence
from first order phase transitions (Hindmarsh et al., 2015). A general
uncertainty in simulating relic GWs from primordial turbulent sources is due
to our ignorance about suitable initial conditions or generation mechanisms.
When a turbulent state is invoked as initial condition, the GW amplitude is
determined almost entirely by the fact that then the GW source, i.e., the
turbulent stress, jumps instantaneously from zero to a finite value (Roper Pol
et al., 2020a). By contrast, when driving turbulence gradually by applying
some forcing in the magnetohydrodynamic (MHD) equations, the resulting GW
amplitude depends on the details of how the turbulence develops and later
declines; see Kahniashvili et al. (2021) for a more systematic investigation.
These problems motivate our present study of GWs from the CME, too.
A number of interesting aspects of turbulence from the CME are already known.
In particular, depending on the relative rates of magnetic field generation,
on the one hand, and depletion of the CME, on the other, different regimes of
turbulence can be distinguished (Brandenburg et al., 2017b). If the depletion
is low, the maximum magnetic field strength is high and a turbulent spectrum
with an inertial range emerges before the turbulence starts to decay in a
self-similar fashion. If the depletion is high, on the other hand, no
turbulent inertial range develops. How the resulting GW amplitude depends on
the governing parameters of the CME-driven field generation is unclear and
illuminating this is the main purpose of this paper. Although the process is
physically motivated, we choose parameters that are motivated by our attempt
to understand the relationship between magnetic field generation and the
resulting GWs in any conceivable regime. Our parameters are therefore not
those relevant to the early universe, nor are they necessarily physically
realizable. Nevertheless, the present work may prove to be important for
guiding our intuition about GW production from primordial turbulent sources.
## 2\. The model
### 2.1. Basic equations
The MHD equations for an ultrarelativistic quark-gluon plasma in a flat
expanding universe in the radiation-dominated era after the electroweak phase
transition can be written in terms of conformal time and comoving coordinates
such that the expansion no longer appears explicitly (Brandenburg et al.,
1996, 2017a; Durrer & Neronov, 2013), except for the GW equation; see below.
The bulk motions are assumed to be subrelativistic.
We quantify the chiral asymmetry through the imbalance between the number
densities $n_{\rm L}$ and $n_{\rm R}$ of left- and right-handed fermions,
respectively, as
${\mu}_{5}=24\,\alpha_{\rm em}\,(n_{\rm L}-n_{\rm R})\,(\hbar c/k_{\rm
B}T)^{2},$ (1)
employing the normalization used by Rogachevskii et al. (2017). Here,
$\alpha_{\rm em}$ is the fine structure constant. The index 5 is commonly
chosen in this context and reminiscent of the fifth Dirac matrix $\gamma_{5}$,
central in defining particle chirality. We should point out that our
${\mu}_{5}$ has the unit of inverse length and is related to the chiral
chemical potential (with units of energy) through an extra $\hbar
c/4\alpha_{\rm em}$ factor; see Schober et al. (2020).
We follow here the normalization of Roper Pol et al. (2020a, b), where the
Heaviside-Lorentz system of units is used for the magnetic field and the scale
factor $a(t)$ is set to unity at the time $t_{*}$ of the electroweak phase
transition (denoted by an asterisk). The Hubble parameter $H$ at $t_{*}$ is
$H_{*}=t_{*}^{-1}$. All quantities are made nondimensional by normalizing time
by $t_{*}$, velocities by the speed of light $c$, and the density $\rho$ by
the critical density $\rho_{\rm crit}$ for a flat universe. Spatial
coordinates are then normalized by the Hubble scale $c/H_{*}$. Consequently,
${\mu}_{5}$ is normalized by $H_{*}/c$. To obtain the comoving magnetic field
in gauss, one has to multiply it by $\sqrt{4\pi\rho_{\rm crit}}c$.
The governing equations for the magnetic field $\bm{B}$ and ${\mu}_{5}$ can
then be written as (Rogachevskii et al., 2017; Schober et al., 2018)
$\displaystyle{\partial\bm{B}\over\partial t}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\nabla$}{}\times[\bm{u}\times\bm{B}+\eta({\mu}_{5}\bm{B}-\mbox{\boldmath$J$}{})],\;\;\;\mbox{\boldmath$J$}{}=\mbox{\boldmath$\nabla$}{}\times\bm{B},\;$
(2) $\displaystyle{{\rm D}{}{\mu}_{5}\over{\rm D}{}t}$ $\displaystyle=$
$\displaystyle-\lambda\,\eta\left({\mu}_{5}\bm{B}-\mbox{\boldmath$J$}{}\right)\cdot\bm{B}+D_{5}\nabla^{2}{\mu}_{5}-\Gamma_{\rm\\!f}{\mu}_{5},$
(3)
where ${\rm D}{}/{\rm D}{}t\equiv\partial/\partial
t+\bm{u}\cdot\mbox{\boldmath$\nabla$}{}$ is the advective derivative, $\eta$
is the magnetic diffusivity, $\lambda$ characterizes the depletion of
${\mu}_{5}$ as the magnetic field increases, $D_{5}$ is a chiral diffusion
coefficient, and $\Gamma_{\rm\\!f}$ is the flipping rate (see Boyarsky et al.,
2021, for a recent calculation). These equations have been derived under the
assumption $\eta\to 0$; see Rogachevskii et al. (2017) for details.
Brandenburg et al. (2017b) found that for $k_{\rm B}T=100\,{\rm GeV}$ and if
$\mu_{50}$ is produced thermally, relevant to the time of the electroweak
phase transition, $\Gamma_{\rm\\!f}/\eta{\mu}_{5}^{2}\approx 10^{-7}$, that
is, the time $1/\Gamma_{\rm\\!f}$ is much longer than the e-folding time of
the fastest growing magnetic mode; see Section 2.2. Hence, we put
$\Gamma_{\rm\\!f}=0$ from now on. The plasma velocity $\bm{u}$ and the density
$\rho$ (which includes the rest mass density) obey the momentum and energy
equations
$\displaystyle{{\rm D}{}\bm{u}\over{\rm D}{}t}$ $\displaystyle=$
$\displaystyle{2\over\rho}\mbox{\boldmath$\nabla$}{}\cdot\left(\rho\nu\mbox{\boldmath${\sf
S}$}{}\right)-{1\over 4}\mbox{\boldmath$\nabla$}{}\ln\rho+{\bm{u}\over
3}\left(\mbox{\boldmath$\nabla$}{}\cdot\bm{u}+\bm{u}\cdot\mbox{\boldmath$\nabla$}{}\ln\rho\right)$
(4) $\displaystyle-$
$\displaystyle{\bm{u}\over\rho}\left[\bm{u}\cdot(\mbox{\boldmath$J$}{}\times\bm{B})+\eta\mbox{\boldmath$J$}{}^{2}\right]+{3\over
4\rho}\mbox{\boldmath$J$}{}\times\bm{B},$
$\displaystyle{\partial\ln\rho\over\partial t}$ $\displaystyle=$
$\displaystyle-\frac{4}{3}\left(\mbox{\boldmath$\nabla$}{}\cdot\bm{u}+\bm{u}\cdot\mbox{\boldmath$\nabla$}{}\ln\rho\right)+{1\over\rho}\left[\bm{u}\cdot(\mbox{\boldmath$J$}{}\times\bm{B})+\eta\mbox{\boldmath$J$}{}^{2}\right]\\!,$
where ${\sf
S}_{ij}=(u_{i,j}+u_{j,i})/2-\delta_{ij}\mbox{\boldmath$\nabla$}{}\cdot\bm{u}/3$
are the components of the rate-of-strain tensor with commas denoting partial
derivatives, $\nu$ is the kinematic viscosity, and the ultrarelativistic
equation of state $p=\rho/3$ has been employed. In the following, we assume
uniform $\nu$, $\eta$, and $D_{5}$ and vary them such that $\nu=\eta=D_{5}$.
The GW equation in the radiation era for the scaled strain tensor ${\sf h}$
with ${\sf h}_{ij}=a{\sf h}_{ij}^{\rm phys}$ is written in Fourier space as
(Roper Pol et al., 2020a, b)
$\frac{\partial^{2}}{\partial
t^{2}}\tilde{h}_{+/\times}(\bm{k},t)+k^{2}\tilde{h}_{+/\times}(\bm{k},t)={6\over
t}\tilde{T}_{+/\times}(\bm{k},t),$ (5)
where $\tilde{h}_{+/\times}={\sf e}_{ij}^{+/\times}({\sf P}_{il}{\sf
P}_{jm}-{\textstyle{1\over 2}}{\sf P}_{ij}{\sf P}_{lm})\,\tilde{{\sf
h}}_{lm}(\bm{k},t)$ are the Fourier-transformed $+$ and $\times$ modes of
${\sf h}$, with ${\sf
e}^{+}_{ij}(\bm{k})\,\,=\,e_{i}^{1}e_{j}^{1}-e_{i}^{2}e_{j}^{2}$ and ${\sf
e}^{\times}_{ij}(\bm{k})\,=\,e_{i}^{1}e_{j}^{2}+e_{i}^{2}e_{j}^{1}$ being the
linear polarization basis, $\mbox{\boldmath$e$}{}^{1}$ and
$\mbox{\boldmath$e$}{}^{2}$ are unit vectors perpendicular to $\bm{k}$ and
perpendicular to each other, and ${\sf P}_{ij}(\bm{k})=\delta_{ij}-k_{i}k_{j}$
is the projection operator. $\tilde{T}_{+/\times}$ are defined analogously and
normalized by the critical density. The stress is composed of magnetic and
kinetic contributions, ${\sf T}_{ij}=\frac{4}{3}\gamma_{\rm Lor}^{2}\rho
u_{i}u_{j}-B_{i}B_{j}+...$, where $\gamma_{\rm Lor}=(1-\bm{u}^{2})^{-1/2}$ is
the Lorentz factor, and the ellipsis denotes terms proportional to
$\delta_{ij}$, not contributing to $\tilde{T}_{+/\times}$. Since we use the
nonrelativistic equations, we put $\gamma_{\rm Lor}=1$, except for one case
shown in Appendix A, where $\gamma_{\rm Lor}\neq 1$. Our equations apply to
the time after the electroweak phase transition $t_{\ast}$, so our normalized
time obeys $t\geq 1$. Furthermore, to compute the relic observable GW energy
at the present time, we have to multiply ${\cal E}_{\rm GW}^{\rm sat}$ by the
square of the ratio of the Hubble parameters and the fourth power of the ratio
of scale factors between the moment of the electroweak phase transition and
today, which is $1.64\times 10^{-5}$; see Roper Pol et al. (2020a, b) for
details.
As already alluded to above, the system of equations (2), (3) describing the
CME must be regarded as partly phenomenological and subject to extensions and
modifications. A purely helical magnetic field with wavenumber
$k={\mu}_{5}={\rm const}{}$, for example, can never decay if
$\Gamma_{\rm\\!f}=0$, and yet it would lead to Ohmic heating. However, those
effects are not critical to the dynamics that we are concerned with in this
paper and will therefore be ignored. Likewise, an extra
$-{\mu}_{5}\mbox{\boldmath$\nabla$}{}\cdot\bm{u}$ term on the right-hand side
of Equation (3) is necessary for a proper conservation equation. However, this
would not make a noticeable difference because
$\mbox{\boldmath$\nabla$}{}\cdot\bm{u}$ is always small; see Appendix A for a
demonstration. It should also be noted that, in comparison with earlier work,
this is the first time that the CME has been solved together with Equations
(4), which contain additional 4/3 factors. We refer to Appendix A of
Brandenburg et al. (2017a) for the differences to standard MHD.
### 2.2. Basic phenomenology of the chiral magnetic effect
The CME introduces two important characteristic quantities into the system:
$\lambda$ and the initial value of ${\mu}_{5}$, ${\mu}_{50}={\mu}_{5}(t=1)$,
both assumed uniform. Different evolutionary scenarios can be envisaged
depending on their values. Following Brandenburg et al. (2017b), we use the
fact that $\lambda^{-1}$ has the dimension of energy per unit length and
${\mu}_{50}$ has the dimension of inverse length, and identify two
characteristic velocities:
$v_{\lambda}={\mu}_{50}/\lambda^{1/2},\quad\quad v_{\mu}={\mu}_{50}\eta.$ (6)
We recall that we have used here dimensionless quantities. We can identify two
regimes of interest:
$\eta k_{1}<v_{\mu}<v_{\lambda}\quad\mbox{(regime I)},$ (7) $\eta
k_{1}<v_{\lambda}<v_{\mu}\quad\mbox{(regime II)},$ (8)
where $k_{1}$ is the smallest wavenumber in the domain and $\eta
k_{1}<v_{\mu}$ is necessary for magnetic field excitation. The case
$v_{\lambda}<\eta k_{1}$ is highly diffusive and was not considered. In regime
I, if the ratio $v_{\lambda}/v_{\mu}=\left[\eta\,\lambda^{1/2}\right]^{-1}$ is
large, the $\lambda$ term is unimportant and ${\mu}_{5}$ will only change
slowly as the magnetic field grows. Once the magnetic field exceeds a critical
value of around $v_{\mu}$, it becomes turbulent; see Brandenburg et al.
(2017b). In that paper, both $v_{\mu}$ and $v_{\lambda}$ were assumed to be
less than the speed of sound, but this is not a physically imposed constraint
and will be relaxed in the present work. Brandenburg et al. (2017b) also found
that the crossover between the regimes occurs when $v_{\lambda}/v_{\mu}\approx
8$. Regarding the resulting GW production, however, we shall find evidence for
a crossover at $v_{\lambda}/v_{\mu}\approx 1$. One should also remember that
$v_{\mu}$ and $v_{\lambda}$ do not correspond to physically realizable speeds
and are therefore not constrained to be below unity. Let us mention at this
point that, using the calculation of Arnold et al. (2000) for the value of
$\eta$ and the expression $\lambda=3\hbar c\,(8\alpha_{\rm em}/k_{\rm
B}T)^{2}$ from Rogachevskii et al. (2017), Brandenburg et al. (2017b)
estimated that $v_{\mu}\approx 2\times 10^{-5}$ and $v_{\lambda}\approx 0.05$
for ${\mu}_{50}=2\times 10^{16}$.
If ${\mu}_{50}\neq 0$, the CME determines primarily the magnetic helicity that
can subsequently be generated. This is a direct consequence of the
conservation law for the (weighted) sum of mean magnetic helicity density and
mean ${\mu}_{5}$, i.e., the total mean chirality (Rogachevskii et al., 2017),
${\textstyle{1\over
2}}\lambda\,\langle\mbox{\boldmath$A$}{}\cdot\bm{B}\rangle+\langle{\mu}_{5}\rangle={\rm
const}{},$ (9)
where $A$ with $\bm{B}=\mbox{\boldmath$\nabla$}{}\times\mbox{\boldmath$A$}{}$
is the magnetic vector potential, and the brackets denote averaging over a
closed or periodic volume; see Appendix A for a discussion of the accuracy of
Equation (9). If the initial magnetic helicity is arbitrarily small, the
constant in Equation (9) can be set to ${\mu}_{50}$. Neglecting the influence
of the turbulent flow $\bm{u}$ and inhomogeneities of ${\mu}_{5}$, the
generated magnetic field is fully helical (Beltrami), and its helicity can be
characterized by its wavenumber $k_{\rm M}$ and the mean magnetic energy
density $\langle\bm{B}^{2}\rangle/2$ through
$\langle\mbox{\boldmath$A$}{}\cdot\bm{B}\rangle\approx\langle\bm{B}^{2}\rangle/k_{\rm
M}$. Therefore, once all the initial ${\mu}_{5}$ is used up, we have
$\langle\bm{B}^{2}\rangle/k_{\rm M}\approx 2{\mu}_{50}/\lambda.$ (10)
Interestingly, the value of $\eta$ does not enter this estimate. It does,
however, determine the initial growth rate $\gamma(k)$ of the magnetic field,
which adopts its maximum, $\gamma_{0}\equiv\eta{\mu}_{50}^{2}/4$, at the
wavenumber $k_{\mu}\equiv{\mu}_{50}/2$. Using $k_{\rm M}\approx k_{\mu}$, we
expect
$\langle\bm{B}^{2}\rangle\lesssim{\mu}_{50}^{2}/\lambda\equiv
v_{\lambda}^{2},$ (11)
so large magnetic fields are expected for large values of ${\mu}_{50}$ and
small values of $\lambda$. The fact that $v_{\lambda}$ characterizes the
maximum magnetic field strength justifies the name “limiting CME speed”. On
the other hand, as one can express $v_{\mu}$ by the maximum growth rate and
the corresponding wavenumber as $2\gamma_{0}/k_{\mu}$, we may call it
“generation speed” in analogy to “phase speed” for a wave.
### 2.3. Magnetic energy spectrum from the CME
To estimate the amount of magnetic energy production from the CME, we adopt
the semi-empirical model of Brandenburg et al. (2017b), who proposed to
construct the magnetic energy spectrum such that it had the $k^{-2}$ slope
that is characteristic of magnetically dominated turbulence, with energy
injection predominantly at the wavenumber $k_{\mu}$. For an intermediate time
interval around the magnetic energy maximum, they then proposed the following
form for the magnetic energy spectrum $E_{\rm M}(k)$ (with normalization $\int
E_{\rm M}(k)dk=\langle\bm{B}^{2}\rangle/2\equiv{\cal E}_{\rm M}$) as a
function of wavenumber $k$ and the parameters $\eta$, ${\mu}_{50}$, and
$\lambda$ that govern the CME:
$E_{\rm M}(k)=C_{5}\,{\mu}_{50}^{3}\eta^{2}k^{-2}\quad\mbox{($k_{\lambda}\leq
k\leq k_{\mu}$)}$ (12)
where $C_{5}\approx 16$ is a Kolmogorov-type constant,
$k_{\lambda}=\sqrt{\lambda C_{5}/C_{\lambda}}\,{\mu}_{50}\eta\approx
4{\mu}_{50}\eta\lambda^{1/2}$ (13)
is the wavenumber corresponding to the outer scale of the $k^{-2}$ subrange,
and $C_{\lambda}\approx 1$ is another empirical constant (Brandenburg et al.,
2017b). Of course, Equation (12) can only hold if $k_{\lambda}\leq k_{\mu}$.
In regime I, $k_{\lambda}$ is the typical wavenumber of the magnetic field
when it has reached maximum strength.
A detailed sketch illustrating the different spectral subranges is Figure 1 of
Brandenburg et al. (2017b), who also confirmed the form of Equation (12)
through simulations. The present simulations also support the existence of the
different subranges.
### 2.4. GW energy scaling
The work of Roper Pol et al. (2020b) has shown that the GW energy is not just
proportional to the square of the magnetic energy, but also proportional to
the square of the dominating length scale (or inverse wavenumber) of $\bm{B}$.
For example, their Runs ini2 and ini3 have the same magnetic energy, but in
ini3, the spectral peak was at a ten times smaller wavenumber, corresponding
to just ten turbulent eddies per Hubble horizon. The resulting GW energy was
then about a hundred times larger. To leading order, the GW energy, normalized
by the critical energy of the universe, is given by ${\cal E}_{\rm
GW}=\langle\dot{h}_{+}^{2}+\dot{h}_{\times}^{2}\rangle/6$; see Roper Pol et
al. (2020a) for details regarding the 1/6 factor and additional correction
terms. Roper Pol et al. (2020b) studied different types of turbulence and
confirmed the quadratic relationship between the maximum magnetic energy,
${\cal E}_{\rm M}^{\max}$ and the saturation value of the GW energy, ${\cal
E}_{\rm GW}^{\rm sat}$ in the form
${\cal E}_{\rm GW}^{\rm sat}\approx(q{\cal E}_{\rm M}^{\max}/k_{\rm
peak})^{2},$ (14)
where $k_{\rm peak}$ is the wavenumber of the peak of the spectrum ($k_{\rm
peak}=600$ in most of their cases, and $60$ in the case where a hundred times
larger ${\cal E}_{\rm GW}^{\rm sat}$ was found, suggesting an inverse
quadratic relationship), and $q$ is an empirical efficiency parameter that is
about $0.9$ for their cases with a turbulent initial MHD state (but no
forcing), $1.8$ in their simulations with forced MHD turbulence, and $11$ in
their simulations of forced acoustic turbulence, where ${\cal E}_{\rm
M}^{\max}$ has to be replaced by the maximum kinetic energy. Larger values of
$q$ correspond to more efficient conversion of magnetic or kinetic energy into
GW energy. The reason why acoustic turbulence is more efficient is unclear,
but may be speculated to lie in its more vigorous time dependence.
### 2.5. Numerical aspects
We solve Equations (2)–(5) using the Pencil Code (Pencil Code Collaboration,
2021), which is a finite difference code that is third order in time and sixth
order in space, except that Equation (5) is solved exactly between subsequent
time instants; see Roper Pol et al. (2020a) for details. For most of our
simulations, we use $512^{3}$ meshpoints, which turned out to be sufficient
for the present investigations. The lowest wavenumber in our computational
domain, $k_{1}$, is chosen to be $100$ for many of our runs. The side length
of the cubical computational domain is then $2\pi/k_{1}$, which is chosen to
be large enough so that the governing dynamics is well captured by the
simulations, but small enough to resolve the smallest length scales. In many
cases, we verified that the results are independent of the choice of $k_{1}$.
Throughout his work, we present spectra of various quantities. We denote this
operation as $\mbox{\rm Sp}(\cdot)$, which is performed as integration over
concentric shells in wavenumber space. For a scalar quantity $f$ it reads
$\mbox{\rm Sp}(f(\bm{x}))=k^{2}\int|\tilde{f}(\bm{k})|^{2}d\Omega_{k}$, where
$\Omega_{k}$ is the solid angle in $\bm{k}$ space, while for the tensor ${\sf
h}$ we put $\mbox{\rm Sp}(\mbox{\boldmath${\sf h}$}{})=\mbox{\rm
Sp}(h_{+})+\mbox{\rm Sp}(h_{\times})$, and likewise for
$\dot{\mbox{\boldmath${\sf h}$}{}}$ and ${\sf T}$. Thus, the GW energy
spectrum is given by $E_{\rm GW}(k)\equiv\mbox{\rm
Sp}(\dot{\mbox{\boldmath${\sf h}$}{}})/6$ and the magnetic one by $E_{\rm
M}(k)\equiv[\mbox{\rm Sp}(B_{x})+\mbox{\rm Sp}(B_{y})+\mbox{\rm
Sp}(B_{z})]/2$.111 Let us note in this connection that one commonly denotes
the GW energy spectrum per logarithmic wavenumber interval by ${\cal E}_{\rm
GW}(k)$, which is distinguished from ${\cal E}_{\rm GW}$ by the argument $k$.
It is related to $E_{\rm GW}(k)$ through ${\cal E}_{\rm GW}(k)=kE_{\rm
GW}(k)$.
## 3\. Results
We have performed a range of simulations where we vary $\eta$, $\lambda$, and
${\mu}_{50}$, studying the influence of these parameters in turn.
### 3.1. Comparison with earlier GW energy scaling
To put our new simulations into context, it is convenient to compare our
values of ${\cal E}_{\rm GW}^{\rm sat}$ for given ${\cal E}_{\rm M}^{\max}$
with those obtained by Roper Pol et al. (2020b). We show in Figure 1 a plot
similar to their Figure 7, depicting our simulations with ${\mu}_{50}=10^{4}$,
grouped into four series with $\lambda^{1/2}$ in the range from $5\times
10^{4}$ to $5\times 10^{3}$. In each of those series, we vary $\eta$. The
resulting values of ${\cal E}_{\rm M}^{\max}$ and ${\cal E}_{\rm GW}^{\rm
sat}$ are summarized in Table LABEL:Tsummary, along with the four input
parameters $\eta$, $\lambda^{1/2}$, ${\mu}_{50}$, and $k_{1}$, as well as
several derived quantities: $\eta k_{1}$, $v_{\mu}$, $v_{\lambda}$,
$\eta{\mu}_{50}^{2}$, and $k_{\lambda}$ (provided
$k_{\lambda}\leq{\mu}_{50}/2$). In the last column, we also give according to
Equation (14)
$q=k_{\rm peak}\sqrt{{\cal E}_{\rm GW}^{\rm sat}}/{\cal E}_{\rm M}^{\max},$
(15)
where we estimate $k_{\rm peak}=k_{\mu}\min(1,v_{\mu}/v_{\lambda})$. This
means that $k_{\rm peak}=k_{\mu}$ when $v_{\mu}>v_{\lambda}$ (regime II) and
$k_{\rm peak}=k_{\lambda}/4$ when $v_{\mu}<v_{\lambda}$ (regime I); see also
Equation (13).
Figure 1.— ${\cal E}_{\rm GW}^{\rm sat}$ versus ${\cal E}_{\rm M}^{\max}$ for
runs with ${\mu}_{50}=10^{4}$, grouped into four series with
$\lambda^{1/2}=5\times 10^{4}$ (series A), $2\times 10^{4}$ (series B),
$10^{4}$ (series C), and $5\times 10^{3}$ (series D). In each series we vary
$\eta$. Closed (open) circles refer to cases where $v_{\mu}/v_{\lambda}<1$
($>1$), corresponding to regime I (II). For orientation, the data of Roper Pol
et al. (2020b) are shown in gray; ‘ac’ – acoustic, ‘hel’ – helically forced
MHD, ‘ini’ – turbulent initial MHD state (no forcing). Table 1 Summary of
Runs from series A–G.
Run $\eta$ $\lambda^{1/2}$ ${\mu}_{50}$ $\eta k_{1}$ $v_{\mu}$ $v_{\lambda}$
$\eta{\mu}_{50}^{2}$ $k_{1}$ $k_{\lambda}$ ${\cal E}_{\rm M}^{\max}$ ${\cal
E}_{\rm GW}^{\rm sat}$ $q$ A1 $1\times 10^{-6}$ $5\times 10^{4}$ $10^{4}$
$1\times 10^{-4}$ $0.01$ $0.2$ $100$ $100$ $2000$ $4.6\times 10^{-3}$
$8.9\times 10^{-14}$ $0.032$ A2 $2\times 10^{-6}$ $5\times 10^{4}$ $10^{4}$
$2\times 10^{-4}$ $0.02$ $0.2$ $200$ $100$ $4000$ $6.4\times 10^{-3}$
$4.3\times 10^{-13}$ $0.10$ A3 $5\times 10^{-6}$ $5\times 10^{4}$ $10^{4}$
$5\times 10^{-4}$ $0.05$ $0.2$ $500$ $100$ $(10000)$ $8.5\times 10^{-3}$
$1.1\times 10^{-12}$ $0.31$ A4 $1\times 10^{-5}$ $5\times 10^{4}$ $10^{4}$
$1\times 10^{-3}$ $0.1$ $0.2$ $1000$ $100$ — $9.2\times 10^{-3}$ $1.7\times
10^{-12}$ $0.71$ 167mm.5pt.7pt 2pt A5 $2\times 10^{-5}$ $5\times 10^{4}$
$10^{4}$ $2\times 10^{-3}$ $0.2$ $0.2$ $2000$ $100$ — $9.5\times 10^{-3}$
$2.5\times 10^{-12}$ $1.7$ A6 $5\times 10^{-5}$ $5\times 10^{4}$ $10^{4}$
$5\times 10^{-3}$ $0.5$ $0.2$ $5000$ $100$ — $9.6\times 10^{-3}$ $4.9\times
10^{-12}$ $2.3$ A7 $1\times 10^{-4}$ $5\times 10^{4}$ $10^{4}$ $1\times
10^{-2}$ $1$ $0.2$ $10000$ $100$ — $9.7\times 10^{-3}$ $1.0\times 10^{-11}$
$3.3$ A8 $2\times 10^{-4}$ $5\times 10^{4}$ $10^{4}$ $2\times 10^{-2}$ $2$
$0.2$ $20000$ $100$ — $9.7\times 10^{-3}$ $2.4\times 10^{-11}$ $5.1$ A9
$5\times 10^{-4}$ $5\times 10^{4}$ $10^{4}$ $5\times 10^{-2}$ $5$ $0.2$
$50000$ $100$ — $9.7\times 10^{-3}$ $6.6\times 10^{-11}$ $8.4$ A10 $1\times
10^{-3}$ $5\times 10^{4}$ $10^{4}$ $5\times 10^{-2}$ $10$ $0.2$ $1\times
10^{5}$ $50$ — $9.2\times 10^{-3}$ $1.4\times 10^{-10}$ $12$ A11 $2\times
10^{-3}$ $5\times 10^{4}$ $10^{4}$ $1\times 10^{-1}$ $20$ $0.2$ $2\times
10^{5}$ $50$ — $9.2\times 10^{-3}$ $2.2\times 10^{-10}$ $15$ A12 $5\times
10^{-3}$ $5\times 10^{4}$ $10^{4}$ $2\times 10^{-1}$ $50$ $0.2$ $5\times
10^{5}$ $50$ — $9.2\times 10^{-3}$ $3.0\times 10^{-10}$ $18$ B1 $1\times
10^{-6}$ $2\times 10^{4}$ $10^{4}$ $1\times 10^{-4}$ $0.01$ $0.5$ $100$ $100$
$800$ $1.6\times 10^{-2}$ $4.7\times 10^{-12}$ $0.027$ B2 $2\times 10^{-6}$
$2\times 10^{4}$ $10^{4}$ $2\times 10^{-4}$ $0.02$ $0.5$ $200$ $100$ $1600$
$2.5\times 10^{-2}$ $3.0\times 10^{-11}$ $0.087$ B3 $5\times 10^{-6}$ $2\times
10^{4}$ $10^{4}$ $5\times 10^{-4}$ $0.05$ $0.5$ $500$ $100$ $4000$ $4.0\times
10^{-2}$ $1.6\times 10^{-10}$ $0.31$ B4 $1\times 10^{-5}$ $2\times 10^{4}$
$10^{4}$ $1\times 10^{-3}$ $0.1$ $0.5$ $1000$ $100$ $(8000)$ $5.1\times
10^{-2}$ $3.0\times 10^{-10}$ $0.68$ B5 $2\times 10^{-5}$ $2\times 10^{4}$
$10^{4}$ $2\times 10^{-3}$ $0.2$ $0.5$ $2000$ $100$ — $5.7\times 10^{-2}$
$4.1\times 10^{-10}$ $1.4$ 167mm.5pt.7pt 2pt B6 $5\times 10^{-5}$ $2\times
10^{4}$ $10^{4}$ $5\times 10^{-3}$ $0.5$ $0.5$ $5000$ $100$ — $6.0\times
10^{-2}$ $4.8\times 10^{-10}$ $3.7$ B7 $1\times 10^{-4}$ $2\times 10^{4}$
$10^{4}$ $1\times 10^{-2}$ $1$ $0.5$ $10000$ $100$ — $6.0\times 10^{-2}$
$5.6\times 10^{-10}$ $3.9$ B8 $2\times 10^{-4}$ $2\times 10^{4}$ $10^{4}$
$2\times 10^{-2}$ $2$ $0.5$ $20000$ $100$ — $6.0\times 10^{-2}$ $9.4\times
10^{-10}$ $5.1$ B9 $5\times 10^{-4}$ $2\times 10^{4}$ $10^{4}$ $5\times
10^{-2}$ $5$ $0.5$ $50000$ $100$ — $6.0\times 10^{-2}$ $2.6\times 10^{-9}$
$8.4$ B10 $1\times 10^{-3}$ $2\times 10^{4}$ $10^{4}$ $1\times 10^{-1}$ $10$
$0.5$ $1\times 10^{5}$ $100$ — $6.0\times 10^{-2}$ $6.0\times 10^{-9}$ $12$ C1
$5\times 10^{-6}$ $10^{4}$ $10^{4}$ $5\times 10^{-4}$ $0.05$ $1$ $500$ $100$
$2000$ $1.1\times 10^{-1}$ $5.6\times 10^{-9}$ $0.33$ C2 $1\times 10^{-5}$
$10^{4}$ $10^{4}$ $1\times 10^{-3}$ $0.1$ $1$ $1000$ $100$ $4000$ $1.6\times
10^{-1}$ $9.9\times 10^{-9}$ $0.64$ C3 $2\times 10^{-5}$ $10^{4}$ $10^{4}$
$2\times 10^{-3}$ $0.2$ $1$ $2000$ $100$ $(8000)$ $2.0\times 10^{-1}$
$1.6\times 10^{-8}$ $1.3$ C4 $5\times 10^{-5}$ $10^{4}$ $10^{4}$ $5\times
10^{-3}$ $0.5$ $1$ $5000$ $100$ — $2.3\times 10^{-1}$ $1.8\times 10^{-8}$
$3.0$ 167mm.5pt.7pt 2pt C5 $1\times 10^{-4}$ $10^{4}$ $10^{4}$ $1\times
10^{-2}$ $1$ $1$ $10000$ $100$ — $2.3\times 10^{-1}$ $2.1\times 10^{-8}$ $6.2$
C6 $2\times 10^{-4}$ $10^{4}$ $10^{4}$ $2\times 10^{-2}$ $2$ $1$ $20000$ $100$
— $2.4\times 10^{-1}$ $2.4\times 10^{-8}$ $6.6$ C7 $5\times 10^{-4}$ $10^{4}$
$10^{4}$ $5\times 10^{-2}$ $5$ $1$ $50000$ $100$ — $2.4\times 10^{-1}$
$4.8\times 10^{-8}$ $9.1$ C8 $1\times 10^{-3}$ $10^{4}$ $10^{4}$ $5\times
10^{-2}$ $10$ $1$ $1\times 10^{5}$ $50$ — $2.3\times 10^{-1}$ $9.0\times
10^{-8}$ $13$ C9 $2\times 10^{-3}$ $10^{4}$ $10^{4}$ $1\times 10^{-1}$ $20$
$1$ $2\times 10^{5}$ $50$ — $2.3\times 10^{-1}$ $1.4\times 10^{-7}$ $16$ C10
$5\times 10^{-3}$ $10^{4}$ $10^{4}$ $2\times 10^{-1}$ $50$ $1$ $5\times
10^{5}$ $50$ — $2.3\times 10^{-1}$ $1.8\times 10^{-7}$ $18$ D1 $1\times
10^{-5}$ $5\times 10^{3}$ $10^{4}$ $2\times 10^{-3}$ $0.1$ $2$ $1000$ $200$
$2000$ $4.5\times 10^{-1}$ $2.3\times 10^{-7}$ $0.54$ D2 $2\times 10^{-5}$
$5\times 10^{3}$ $10^{4}$ $2\times 10^{-3}$ $0.2$ $2$ $2000$ $100$ $4000$
$6.3\times 10^{-1}$ $5.5\times 10^{-7}$ $1.2$ D3 $5\times 10^{-5}$ $5\times
10^{3}$ $10^{4}$ $5\times 10^{-3}$ $0.5$ $2$ $5000$ $100$ $(10000)$ $8.5\times
10^{-1}$ $7.2\times 10^{-7}$ $2.5$ D4 $1\times 10^{-4}$ $5\times 10^{3}$
$10^{4}$ $1\times 10^{-2}$ $1$ $2$ $10000$ $100$ — $9.3\times 10^{-1}$
$7.7\times 10^{-7}$ $4.7$ 167mm.5pt.7pt 2pt D5 $2\times 10^{-4}$ $5\times
10^{3}$ $10^{4}$ $4\times 10^{-2}$ $2$ $2$ $20000$ $200$ — $9.6\times 10^{-1}$
$7.0\times 10^{-7}$ $8.7$ D6 $5\times 10^{-4}$ $5\times 10^{3}$ $10^{4}$
$1\times 10^{-1}$ $5$ $2$ $50000$ $200$ — $9.7\times 10^{-1}$ $1.0\times
10^{-6}$ $10$ D7 $1\times 10^{-3}$ $5\times 10^{3}$ $10^{4}$ $2\times 10^{-1}$
$10$ $2$ $1\times 10^{5}$ $200$ — $9.7\times 10^{-1}$ $1.6\times 10^{-6}$ $12$
D8 $2\times 10^{-3}$ $5\times 10^{3}$ $10^{4}$ $4\times 10^{-1}$ $20$ $2$
$2\times 10^{5}$ $200$ — $9.7\times 10^{-1}$ $1.9\times 10^{-6}$ $14$ E1
$5\times 10^{-6}$ $10^{4}$ $2\times 10^{4}$ $1\times 10^{-3}$ $0.1$ $2$ $2000$
$200$ $4000$ $4.4\times 10^{-1}$ $8.6\times 10^{-8}$ $0.67$ E2 $1\times
10^{-5}$ $10^{4}$ $2\times 10^{4}$ $2\times 10^{-3}$ $0.2$ $2$ $4000$ $200$
$8000$ $6.3\times 10^{-1}$ $1.5\times 10^{-7}$ $1.2$ E3 $2\times 10^{-5}$
$10^{4}$ $2\times 10^{4}$ $4\times 10^{-3}$ $0.4$ $2$ $8000$ $200$ $(16000)$
$8.1\times 10^{-1}$ $1.9\times 10^{-7}$ $2.2$ E4 $5\times 10^{-5}$ $10^{4}$
$2\times 10^{4}$ $1\times 10^{-2}$ $1$ $2$ $20000$ $200$ — $9.3\times 10^{-1}$
$2.0\times 10^{-7}$ $4.9$ 167mm.5pt.7pt 2pt E5 $1\times 10^{-4}$ $10^{4}$
$2\times 10^{4}$ $2\times 10^{-2}$ $2$ $2$ $40000$ $200$ — $9.6\times 10^{-1}$
$2.1\times 10^{-7}$ $9.6$ E6 $2\times 10^{-4}$ $10^{4}$ $2\times 10^{4}$
$4\times 10^{-2}$ $4$ $2$ $80000$ $200$ — $9.7\times 10^{-1}$ $2.6\times
10^{-7}$ $10$ E7 $5\times 10^{-4}$ $10^{4}$ $2\times 10^{4}$ $1\times 10^{-1}$
$10$ $2$ $2\times 10^{5}$ $200$ — $9.7\times 10^{-1}$ $4.6\times 10^{-7}$ $13$
E8 $1\times 10^{-3}$ $10^{4}$ $2\times 10^{4}$ $2\times 10^{-1}$ $20$ $2$
$4\times 10^{5}$ $200$ — $9.7\times 10^{-1}$ $6.2\times 10^{-7}$ $16$ F1
$5\times 10^{-6}$ $10^{3}$ $2\times 10^{3}$ $2\times 10^{-4}$ $0.01$ $2$ $20$
$50$ $40$ $9.4\times 10^{-2}$ $7.2\times 10^{-11}$ $0.00091$ F2 $1\times
10^{-5}$ $10^{3}$ $2\times 10^{3}$ $5\times 10^{-4}$ $0.02$ $2$ $40$ $50$ $80$
$1.4\times 10^{-1}$ $6.7\times 10^{-9}$ $0.012$ F3 $2\times 10^{-5}$ $10^{3}$
$2\times 10^{3}$ $1\times 10^{-3}$ $0.04$ $2$ $80$ $50$ $160$ $2.5\times
10^{-1}$ $1.7\times 10^{-7}$ $0.067$ F4 $5\times 10^{-5}$ $10^{3}$ $2\times
10^{3}$ $1\times 10^{-3}$ $0.1$ $2$ $200$ $25$ $400$ $4.5\times 10^{-1}$
$3.3\times 10^{-6}$ $0.41$ F5 $1\times 10^{-4}$ $10^{3}$ $2\times 10^{3}$
$2\times 10^{-3}$ $0.2$ $2$ $400$ $25$ $800$ $6.3\times 10^{-1}$ $8.2\times
10^{-6}$ $0.91$ F6 $2\times 10^{-4}$ $10^{3}$ $2\times 10^{3}$ $5\times
10^{-3}$ $0.4$ $2$ $800$ $25$ $(1600)$ $8.1\times 10^{-1}$ $1.3\times 10^{-5}$
$1.8$ F7 $5\times 10^{-4}$ $10^{3}$ $2\times 10^{3}$ $1\times 10^{-2}$ $1$ $2$
$2000$ $25$ — $9.3\times 10^{-1}$ $1.6\times 10^{-5}$ $4.3$ 167mm.5pt.7pt 2pt
F8 $1\times 10^{-3}$ $10^{3}$ $2\times 10^{3}$ $2\times 10^{-2}$ $2$ $2$
$4000$ $25$ — $9.6\times 10^{-1}$ $1.8\times 10^{-5}$ $8.9$ G1 $1\times
10^{-5}$ $5\times 10^{2}$ $10^{3}$ $2\times 10^{-4}$ $0.01$ $2$ $10$ $25$ $20$
$9.8\times 10^{-2}$ $1.1\times 10^{-10}$ $0.00054$ G2 $2\times 10^{-5}$
$5\times 10^{2}$ $10^{3}$ $5\times 10^{-4}$ $0.02$ $2$ $20$ $25$ $40$
$1.5\times 10^{-1}$ $7.5\times 10^{-9}$ $0.0060$ G3 $5\times 10^{-5}$ $5\times
10^{2}$ $10^{3}$ $1\times 10^{-3}$ $0.05$ $2$ $50$ $25$ $100$ $2.9\times
10^{-1}$ $6.0\times 10^{-7}$ $0.066$ G4 $1\times 10^{-4}$ $5\times 10^{2}$
$10^{3}$ $2\times 10^{-3}$ $0.1$ $2$ $100$ $25$ $200$ $4.5\times 10^{-1}$
$4.8\times 10^{-6}$ $0.24$ G5 $2\times 10^{-4}$ $5\times 10^{2}$ $10^{3}$
$5\times 10^{-3}$ $0.2$ $2$ $200$ $25$ $400$ $6.3\times 10^{-1}$ $1.6\times
10^{-5}$ $0.63$ G6 $5\times 10^{-4}$ $5\times 10^{2}$ $10^{3}$ $1\times
10^{-2}$ $0.5$ $2$ $500$ $25$ $(1000)$ $8.6\times 10^{-1}$ $3.9\times 10^{-5}$
$1.8$ G7 $1\times 10^{-3}$ $5\times 10^{2}$ $10^{3}$ $1\times 10^{-2}$ $1$ $2$
$1000$ $10$ — $9.3\times 10^{-1}$ $5.4\times 10^{-5}$ $4.0$ 167mm.5pt.7pt 2pt
G8 $2\times 10^{-3}$ $5\times 10^{2}$ $10^{3}$ $2\times 10^{-2}$ $2$ $2$
$2000$ $10$ — $9.6\times 10^{-1}$ $6.2\times 10^{-5}$ $8.2$
Note: Dotted lines separate regime I from regime II runs. Bracketed
$k_{\lambda}$ values and hyphens mean that $k_{\lambda}$ exceeds $k_{\mu}$.
In view of any type of driven or decaying MHD turbulence, the dependence of
${\cal E}_{\rm GW}^{\rm sat}$ on $\eta$ seems not very intuitive as we find it
to increase with increasing $\eta$ although one would have expected that
smaller $\eta$ would cause a more vigorous time dependence. However, the
increase of ${\cal E}_{\rm GW}^{\rm sat}$ with $\eta$ is plausible due to the
fact that the maximum growth rate of $\bm{B}$ is proportional to $\eta$ – a
specific of the CME.
In all of our simulations of series A–D, the parameter $q$ is even lower than
in the least efficient simulations of Roper Pol et al. (2020b). This is rather
surprising and might indicate that the turbulence from the CME has a much less
vigorous time dependence than the cases considered there. For understanding
the reason behind this, it is necessary to study the present results in more
detail by inspecting the magnetic and GW energy spectra. We begin by analyzing
their mutual relation at late times when the magnetic energy has already
reached a maximum and the GW energy has achieved a steady state.
### 3.2. Late time GW spectra from the CME
We consider the case $\eta=10^{-6}$, $\lambda=4\times 10^{8}$, and
${\mu}_{50}=10^{4}$, which corresponds to Run B1. This means that
$v_{\mu}=0.01$ and $v_{\lambda}=0.5$, so $v_{\lambda}/v_{\mu}=50$, and we are
clearly in regime I.
The CME leads to exponential magnetic field generation, followed by subsequent
turbulent decay. At the time of the magnetic maximum, an approximate $k^{-2}$
magnetic energy spectrum with a short inertial range develops (Brandenburg et
al., 2017b). We then expect a $k^{-4}$ spectrum for the GW energy and a
$k^{-6}$ spectrum for ${\sf h}$; see Roper Pol et al. (2020b). There is a
trend for this to happen also in the present case, although ${\cal E}_{\rm
GW}(k)$ does not have clear power law subranges; see Figure 2. This is because
the turbulence is not steady and both energy spectra look very different even
just shortly before the magnetic field saturates, as will be shown below.
Figure 2.— Magnetic and GW energy spectra for Run B1 with
${\mu}_{50}=10^{4}$, $\lambda=4\times 10^{8}$, and $\eta=10^{-6}$, which is in
regime I with $(v_{\mu}=0.01)<(v_{\lambda}=0.5)$. $E_{\rm M}(k)$ (red) is
shown at the time of magnetic maximum (solid, $t=1.92$), the time when the
$k^{-2}$ spectrum is most clear ($t=3$, dashed), and at selected other times
(dotted, $t=1.71$, $1.77$, $1.83$, $1.89$, $1.94$, $2.00$, $2.15$, $2.32$,
$2.52$, $2.74$, and $3.00$, while $E_{\rm GW}(k)$ (solid blue) is from the
simulation’s end time ($t=14$), when it can be approximated by $k^{2}\mbox{\rm
Sp}(\mbox{\boldmath${\sf h}$}{})/6$ (dashed-dotted blue). The black horizontal
dashed-dotted line marks the saturation limit of Equation (10),
${\mu}_{50}/\lambda$, and the vertical dashed line marks the position of
$k_{\mu}$. Figure 3.— Similar to Figure 2, but for Run B10 with
$\eta=10^{-3}$, which is in regime II with $(v_{\mu}=10)>(v_{\lambda}=0.5)$.
$E_{\rm M}(k)$ (solid red) is at the time when the magnetic energy has
attained its maximum ($t=1.001$), dotted red lines show $E_{\rm M}(k)$ at
$t=1.0008$, $1.003$, $1.008$, $1.024$, and $1.075$, while $E_{\rm GW}(k)$ is
from the simulation’s end time ($t=1.075$), when $E_{\rm GW}(k)\approx
k^{2}\mbox{\rm Sp}(\mbox{\boldmath${\sf h}$}{})/6$.
For runs in regime II, however, we find an approximate $k^{-0.5}$ profile for
$E_{\rm GW}(k)$; see Figure 3. This is closer to the case of stationary
turbulence; see Table LABEL:Tregimes for a comparison of some characteristic
properties. $E_{\rm M}(k)$ shows an approximate $k^{5}$ subinertial range.
This is steeper than the $k^{4}$ spectrum expected based on causality
arguments (Durrer & Caprini, 2003). However, as we will see later more
clearly, at early times and close to $k=k_{\mu}$ the magnetic energy spectra
show a dent, explaining therefore the apparent steeper spectrum at early
times; a $k^{4}$ subinertial range can still be identified at other times. In
particular, for fully helical magnetic fields, a $k^{4}$ spectrum spectrum
always emerges, regardless of the initial slope; see Figure 3(a) of
Brandenburg & Kahniashvili (2017).
Table 2 Spectral properties of GWs in regimes I, II, and in stationary
turbulence.
| Run B1 | Run B10 | Stationary
---|---|---|---
| Regime I | Regime II | turbulence∗
$\mbox{\rm Sp}(\dot{\mbox{\boldmath${\sf h}$}{}})/\mbox{\rm Sp}(\mbox{\boldmath${\sf h}$}{})$ | $0.89\,(2\gamma_{0})^{2}$ | $0.96\,(2\gamma_{0})^{2}$ | $0.30\,k_{\rm f}^{2}$
$\mbox{\rm Sp}(\mbox{\boldmath${\sf T}$}{})/\mbox{\rm Sp}(\dot{\mbox{\boldmath${\sf h}$}{}})$ | $1.1\,(2\gamma_{0}{\mu}_{50})^{2}$ | $0.98\,(2\gamma_{0})^{2}$ | $0.10\,k_{\rm f}^{2}$
$\mbox{\rm Sp}(\mbox{\boldmath${\sf h}$}{})$, kinematic | $k^{-2}$ | $k^{2}$ | —
$\mbox{\rm Sp}(\mbox{\boldmath${\sf h}$}{})$, saturated | $k^{-2}$ | $k^{-0.5}$ | $k^{0}$
${}^{*}$Run K0 of \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Kahn21}{\@@citephrase{(}}{\@@citephrase{)}}},
$k_{\rmf}=600$; the ellipsis means no growth. ${}^{*}$Run K0 of
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Kahn21}{\@@citephrase{(}}{\@@citephrase{)}}},
$k_{\rmf}=600$; the ellipsis means no growth.footnotemark:
Figure 4.— Evolution of ${\cal E}_{\rm M}$, ${\cal E}_{\rm GW}$, and
$\langle{\mu}_{5}\rangle$ for Runs B1–B10. The light and dark gray bars on the
right of each panel indicate regimes I and II, respectively. Note the
occurrence of a slow final saturation phase of ${\cal E}_{\rm GW}$ for all
runs in regime II (Runs B7–B10). Run 8 (red dashed-dotted line), Run 9 (red
dashed line), and Run 10 (upper black dotted line) overlap in ${\cal E}_{\rm
M}$ and $\langle{\mu}_{5}\rangle$, but are well separated in ${\cal E}_{\rm
GW}$.
### 3.3. GW spectra during the early growth phase
At early times, as discussed above, ${\cal E}_{\rm M}(t)$ grows exponentially
at a rate $2\gamma_{0}=\eta{\mu}_{50}^{2}/2$ and ${\cal E}_{\rm GW}(t)$ grows
at a rate $4\gamma_{0}$. Across the different runs, this rate varies by three
orders of magnitude. To compare the evolution of GW and magnetic energies for
the different runs, it is thus convenient to plot both quantities versus
$4\gamma_{0}t$. The result is shown in Figure 4 for the runs of series B. One
clearly sees a slow final saturation phase of ${\cal E}_{\rm GW}(t)$ for all
runs in regime II (Runs B7–B10), while ${\cal E}_{\rm M}(t)$ and
$\langle{\mu}_{5}\rangle(t)$ are almost unchanged across different runs.
During the exponential growth phase, ${\mu}_{5}$ is close to its initial
value, ${\mu}_{50}=10^{4}$. It drops fastest in regime I, where $\eta$ is
small (Runs B1–B5). However, in contrast to Figure 1, where we saw a marked
qualitative change as we move from regime I to regime II, no such change is
seen in Figure 4 between regime I (Runs B1–B5) and II (Runs B7–B10).
In the case of stationary GW spectra (see, e.g., Kahniashvili et al., 2021),
and also in the previous section, we always have $\mbox{\rm
Sp}(\dot{\mbox{\boldmath${\sf h}$}{}})\approx k^{2}\mbox{\rm
Sp}(\mbox{\boldmath${\sf h}$}{})\approx k^{-2}\mbox{\rm
Sp}(\mbox{\boldmath${\sf T}$}{})$, but this is not so in the early exponential
growth phase. Nevertheless, in both these regimes, we find $\mbox{\rm
Sp}(\mbox{\boldmath${\sf T}$}{})\propto k^{2}$. This is a consequence of the
almost monochromatic magnetic field generation in a narrow range around
$k=k_{\mu}$, which implies that the spectral slope of $E_{\rm M}(k)$ for
$k<k_{\mu}$ is always steeper than that of white noise ($\propto k^{2}$), so
we call it “blue noise”. However, the square of a field with a blue noise
spectrum always has a white noise spectrum (Brandenburg & Boldyrev, 2020).
This explains why $\mbox{\rm Sp}(\mbox{\boldmath${\sf T}$}{})\propto k^{2}$.
Figure 5.— Comparison of GW energy spectra during the kinematic growth stage
for runs in regime I (B1–B5) and regime II (B7–B10). Note the change of slope
at a certain wavenumber that increases as we go from regime I to regime II.
Figure 6.— Time-evolving magnetic and GW energy spectra along with spectra of
stress ${\sf T}$ and strain ${\sf h}$ for Run B1 (regime I) at $t-1=0.2$,
$0.25$, $0.3$, $0.35$, $0.4$ in black, $0.45$, $0.5$, $0.55$, $0.6$, $0.65$ in
blue, $0.7$, $0.75$, $0.8$, $0.9$, $1.4$ in red, and the time of maximum
${\cal E}_{\rm M}$ at $1.9$, again in black. In panel (a), the dotted
horizontal line marks the level of $C_{5}{\mu}_{50}\eta^{2}$, and the
horizontal dashed-dotted line the level of $C_{\lambda}{\mu}_{50}/\lambda$.
Vertical dotted and dashed lines mark the positions of $2k_{\mu}={\mu}_{50}$
and $k_{\mu}$, respectively. The red filled symbol denotes the peak of $E_{\rm
M}(k)$ at the time of the magnetic maximum. Figure 7.— Similar to Figure 6,
but for Run B10 with $\eta=10^{-3}$, which is in regime II with
$(v_{\mu}=10)>(v_{\lambda}=0.5)$, at $t-1=2$, $3$, $4$, $5$, $6\times 10^{-4}$
in black, $6.5$, $7$, $7.5$, $8$, $9\times 10^{-4}$ in blue, and $0.001$
(maximum ${\cal E}_{\rm M}$), $0.002$, $0.007$, and $0.0075$ in red. ${\cal
E}_{\rm M}(t)$ reaches a maximum at $t-1=1.1\times 10^{-3}$ and ${\cal E}_{\rm
GW}(t)$ at $t-1=0.02$. The upward arrow in panel (c) emphasizes the change in
slope.
To see how the transition from a $k^{-2}$ profile for small $k$ toward a
$k^{2}$ profile for large $k$ occurs in $E_{\rm GW}(k)$, we plot it in Figure
5 during the kinematic growth phase. The times have been arranged such that
all spectra coincide at $k=k_{1}\equiv 100$. We clearly see the emergence of a
breakpoint from a $k^{2}$ spectrum at low $k$ toward a $k^{-2}$ spectrum at
larger $k$. The breakpoint shifts toward larger wavenumbers as we go from
regime I to regime II, although it can no longer be identified for Runs
B7–B10.
Furthermore, in both regimes I and II, we find $\mbox{\rm
Sp}(\dot{\mbox{\boldmath${\sf h}$}{}})\propto\mbox{\rm
Sp}(\mbox{\boldmath${\sf h}$}{})$ during the early growth phase, but their
slopes are different in the two regimes. In Figures 6 and 7, we compare the
spectra for Runs B1 (regime I) and B10 (regime II), including magnetic and GW
energy spectra along with the spectra of stress and strain. We clearly see
that at early times, $\mbox{\rm Sp}(\mbox{\boldmath${\sf h}$}{})$ and
$\mbox{\rm Sp}(\dot{\mbox{\boldmath${\sf h}$}{}})=6{\cal E}_{\rm GW}(k)$ all
have the same slope proportional to $k^{-2}$ and $k^{2}$ in regimes I and II,
respectively. Specifically, at $k=k_{\mu}$, we find for the ratio $\mbox{\rm
Sp}(\dot{\mbox{\boldmath${\sf h}$}{}})/\mbox{\rm Sp}(\mbox{\boldmath${\sf
h}$}{})\approx(2\gamma_{0})^{2}$ in both regimes. It is important to emphasize
that, even though $\gamma(k)$ depends on $k$, the stress spectrum grows at the
maximum rate $\gamma_{0}$ at all $k$. For $k\leq k_{\mu}$, this can simply be
understood as a consequence of the result of Brandenburg & Boldyrev (2020)
that the square of a field with a blue noise spectrum always has a white noise
spectrum.
For $k>{\mu}_{50}$, the magnetic energy spectrum always drops rapidly. Based
again on the results of Brandenburg & Boldyrev (2020), since the spectrum is
here a red one, the magnetic stress spectrum also drops rapidly with the same
slope. Following Brandenburg & Boldyrev (2020), in the range
$k_{\mu}<k<{\mu}_{50}$ the spectrum is slightly shallower than $k^{2}$ and it
peaks approximately at $k={\mu}_{50}$.
In Table LABEL:Tregimes, we summarize the spectral properties during the early
kinematic growth phase and contrast it with the saturated phase. In regime I,
we also find $\mbox{\rm Sp}(\mbox{\boldmath${\sf T}$}{})/\mbox{\rm
Sp}(\dot{\mbox{\boldmath${\sf h}$}{}})\approx(2\gamma_{0})^{2}$, but in regime
II, there is an extra ${\mu}_{50}^{2}$ factor (see Table LABEL:Tregimes),
which is a consequence of the different slopes of both curves. The reason for
the change of slopes in regimes I and II is explained in the next section.
### 3.4. Difference in the slopes between regimes I and II
To understand the change in the spectral slopes between regimes I and II
during the kinematic growth stage it is convenient to restrict our attention
to the case of a purely monochromatic exponential growth of $\bm{B}$ at the
wavenumber $k_{\mu}$ with the rate $\gamma_{0}=\eta{\mu}_{50}^{2}/4$. As
explained in Section 3.3, the magnetic stress increases then at all $k$ at the
rate $2\gamma_{0}$; see also Figures 6 and 7 for a direct confirmation of this
property.
Let us now assume that $\tilde{T}(\bm{k},t)$, representing the Fourier
transform of one of the two polarization modes of the stress, $T_{+}$ and
$T_{\times}$, is given by
$\tilde{T}(\bm{k},t)=\theta(t-1)\,\tilde{T}_{0}(k)\,e^{2\gamma_{0}(t-1)},$
(16)
where $\theta(t)$ is the Heaviside step function, and $\tilde{T}_{0}(k)$ is
assumed to depend just on $k=|\bm{k}|$.
Using $\tilde{h}(k,1)=\dot{\tilde{h}}(k,1)=0$ as initial conditions, we can
solve Equation (5) during the early growth phase in closed form as
$\displaystyle\tilde{h}(k,t)=\frac{6\tilde{T}_{0}(k)}{4\gamma_{0}^{2}+k^{2}}\left[e^{2\gamma_{0}\tau}-\cos
k\tau-\frac{2\gamma_{0}}{k}\sin k\tau\right]_{\tau=t-1},$ (17)
where $\tilde{h}$ stands for either $\tilde{h}_{+}$ or $\tilde{h}_{\times}$.
In practice, we are always interested in the case where the exponential term
dominates over the cosine and sine terms. When $k\ll 2\gamma_{0}$, $\mbox{\rm
Sp}(h)$ and $\mbox{\rm Sp}(\dot{h})$ are proportional to $\mbox{\rm
Sp}(T_{0})$. In particular, when $\tilde{T}_{0}(k)$ is a white noise spectrum,
we have $\mbox{\rm Sp}(h)\propto k^{2}|\tilde{T}_{0}(k)|^{2}\propto k^{2}$.
However, when $k\gg 2\gamma_{0}$, we find $\mbox{\rm Sp}(h)\propto\mbox{\rm
Sp}(\dot{h})\propto k^{2}|\tilde{T}_{0}(k)/k^{2}|^{2}\propto k^{-2}$, with the
breakpoint being at $k_{0}=2\gamma_{0}$.
To compare with the results of our simulations, let us try to numerically
determine the breakpoint $k=k_{\rm GW}$ as
$k_{\rm GW}^{-1}=\int k^{-1}E_{\rm GW}(k)\,{\rm d}{}k\,\left/\int E_{\rm
GW}(k)\,{\rm d}{}k\right..$ (18)
We have calculated it for the models of series B and D and find that our
analytic prediction $k_{0}=2\gamma_{0}$ matches the numerical results rather
well; see Figure 8. Representing $\mbox{\rm Sp}(\dot{h})$ according to
Equation (17) by
$E_{\rm GW}^{\rm
model}\propto\left[\frac{k}{k_{0}^{2}+k^{2}}\,e^{-(k/{\mu}_{50})^{4}}\right]^{2},$
(19)
where the exponential factor is intended to model the cutoff near
$k={\mu}_{50}$, we find $k_{\rm GW}=(\pi/2)\,k_{0}$, which is why we have
compensated $k_{\rm GW}$ in Figure 8 by this value. The reason why there are
departures for small and large values of $\eta{\mu}_{50}^{2}$ is that the
wavenumber range used for the integration is limited. In addition to
estimating $k_{0}$ as $k_{\rm GW}$ from Equation (18), we compute a fit to the
model spectrum of Equation (19). We do this by minimizing the mean squared
difference between the actual spectrum and the model spectrum. Those results
are also shown in Figure 8 (open symbols).
Figure 8.— Dependence of $k_{\rm GW}$ from Equation (18), normalized by
$k_{0}\pi/2$, on $\eta{\mu}_{50}^{2}$ for runs of series B (red filled
symbols) and D (blue filled symbols). Run B5 is highlighted in boldface (cf.
Figure 9). The dashed-dotted line gives an approximate fit through the data
points near their plateau, and the solid line goes through unity, the
theoretically expected value. The red open symbols denote the values of
$k_{0}$ obtained by fitting the spectra of Figure 5 to the model spectrum of
Equation (19), similar to what is done in Figure 9. Figure 9.— Comparison of
the GW energy spectrum for Run B5 and the model spectrum of Equation (19).
Figure 10.— Dependence of ${\cal E}_{\rm M}^{\rm max}$ and ${\cal E}_{\rm
GW}^{\rm sat}$ on $\eta$, and their mutual parametric dependence for runs of
series A–D with ${\mu}_{50}=10^{4}$ and $\lambda^{1/2}=5\times 10^{4}$,
$2\times 10^{4}$, $10^{4}$, $5\times 10^{3}$, respectively, series E with
${\mu}_{50}=2\times 10^{4}$, $\lambda^{1/2}=10^{4}$, series F with
${\mu}_{50}=2000$, $\lambda^{1/2}=1000$, and series G with ${\mu}_{50}=1000$,
$\lambda^{1/2}=100$. Filled (open) symbols denote runs in regime I (II). The
dotted line in panel (c) is for $q=13\,({\cal E}_{\rm M}^{\rm max})^{1/2}$.
In Figure 9, we show a comparison of one of the GW energy spectra of Figure 5
with Equation (19). While it provides an excellent description of $E_{\rm
GW}(k)$ in the bulk of the $k$ range, the exponential factor is not sharp
enough to model the simulation data near the cutoff.
### 3.5. Change of slope toward late times
We see in Figure 4(b) that for all runs in regime I, ${\cal E}_{\rm GW}$
saturates quickly after ${\cal E}_{\rm M}$ reaches its maximum, while for runs
in regime II, ${\cal E}_{\rm GW}$ continues to display a slow saturation
behavior. To understand this unusual behavior, we must look again at Figure 7,
showing the evolution of the spectra in Run B10, which is in regime II. We see
that, at the time when ${\cal E}_{\rm M}$ reaches its maximum, the peak of
$E_{\rm M}(k)$ is still at $k\approx{\mu}_{5}$. After that, ${\cal E}_{\rm M}$
decays such that ${\cal E}_{\rm M}/k_{\rm M}={\rm const}{}$, so based on the
earlier results of Roper Pol et al. (2020b), we would expect ${\cal E}_{\rm
GW}$ to stay constant. Looking at the evolution of $E_{\rm GW}(k)$ for Run B10
near equilibration in Figure 7(c), we observe a change in slope. This could be
responsible for the occurrence of a slow final saturation phase of ${\cal
E}_{\rm GW}$ for the runs in regime II, and especially for Run B10, seen in
Figure 4.
To discuss this possibility quantitatively, let us assume a simplified
spectrum of the form
$E_{\rm GW}(k,t_{\rm bef})=3{\cal E}_{0}\,k^{2}/{\mu}_{50}^{3}\quad\mbox{if
$k<2k_{\mu}={\mu}_{50}$}$ (20)
for the time $t_{\rm bef}$ before the slope changes. For $k>2k_{\mu}$ we
assume a sharp fall-off and therefore ignore that contribution. This $k^{2}$
spectrum is normalized such that $\int E_{\rm GW}(k,t_{\rm bef})\,{\rm
d}{}k={\cal E}_{0}$. The spectrum is then assumed to change to a new power law
$\propto k^{s}$, with an exponent $s$, of the form
$E_{\rm GW}(k,t_{\rm aft})=3{\cal E}_{0}\,k^{s}/{\mu}_{50}^{s+1}$ (21)
for the time $t_{\rm aft}$ after the slope has changed. Employing the same
${\cal E}_{0}$ in Equations (20) and (21), accounts for the fact that $E_{\rm
GW}({\mu}_{50})=3{\cal E}_{0}/{\mu}_{50}$ is no longer changing in time; see
Figure 7(c). For $s>-1$, the resulting GW energy is $3{\cal E}_{0}/(s+1)$. In
Figure 3, we found $s=-0.5$, so the resulting GW energy is then $\approx
6{\cal E}_{0}$, which is compatible with the late-time excess of ${\cal
E}_{\rm GW}$ in Run B10 relative to Run B6. It should be noted, however, that
the change of slope occurs at a time when the magnetic field is about to reach
the scale of the domain. It is therefore conceivable that $s=-0.5$ could be an
artifact of the finite domain size. In particular, $s=0$ is what has
previously been found based on numerical simulations (Roper Pol et al., 2020b)
including larger domains.
Figure 11.— Dependence of ${\cal E}_{\rm M}^{\rm max}$ and ${\cal E}_{\rm
GW}^{\rm sat}$ on $\lambda$, and their mutual parametric dependence for runs
of series K–N. Filled (open) symbols denote runs in regime I (II). The dotted
line in panel (c) is for $q=7\,{\cal E}_{\rm M}^{\max}$. Figure 12.—
Dependence of ${\cal E}_{\rm M}^{\rm max}$ and ${\cal E}_{\rm GW}^{\rm sat}$
on ${\mu}_{50}$, and their mutual parametric dependence for runs of series
U–X. Filled (open) symbols denote runs in regime I (II). The dashed line in
panel (c) is for $q=10$. Figure 13.— Dependence of ${\cal E}_{\rm GW}^{\rm
sat}$ on $v_{\lambda}^{5}v_{\mu}$ for all runs of series A–G with the same
colors as in Figure 10 (dashed lines with open circles), series K–N with the
same colors as in Figure 11 (solid lines with filled symbols), and series U–X
with the same colors as in Figure 12 (dotted lines with diamonds). The dashed-
dotted line has slope unity. The inset shows the same plot, extended down to
$v_{\lambda}^{5}v_{\mu}=6\times 10^{-12}$, corresponding to the CME estimate
for the early universe, and denoted by a big red triangle in the lower left.
Table 3 Summary of Runs from series K–N.
Run $\eta$ $\lambda^{1/2}$ ${\mu}_{50}$ $\eta k_{1}$ $v_{\mu}$ $v_{\lambda}$
$\eta{\mu}_{50}^{2}$ $k_{1}$ $k_{\lambda}$ ${\cal E}_{\rm M}^{\max}$ ${\cal
E}_{\rm GW}^{\rm sat}$ $q$ K1 = A5 $2\times 10^{-5}$ $5\times 10^{4}$ $10^{4}$
$2\times 10^{-3}$ $0.2$ $0.2$ $2000$ $100$ — $9.5\times 10^{-3}$ $2.5\times
10^{-12}$ $1.7$ K2 = B5 $2\times 10^{-5}$ $2\times 10^{4}$ $10^{4}$ $2\times
10^{-3}$ $0.2$ $0.5$ $2000$ $100$ — $5.7\times 10^{-2}$ $4.1\times 10^{-10}$
$1.4$ K3 = C3 $2\times 10^{-5}$ $1\times 10^{4}$ $10^{4}$ $2\times 10^{-3}$
$0.2$ $1$ $2000$ $100$ $8000$ $2.0\times 10^{-1}$ $1.6\times 10^{-8}$ $1.3$ K4
= D2 $2\times 10^{-5}$ $5\times 10^{3}$ $10^{4}$ $2\times 10^{-3}$ $0.2$ $2$
$2000$ $100$ $4000$ $6.3\times 10^{-1}$ $5.5\times 10^{-7}$ $1.2$ L1 = A6
$5\times 10^{-5}$ $5\times 10^{4}$ $10^{4}$ $5\times 10^{-3}$ $0.5$ $0.2$
$5000$ $100$ — $9.6\times 10^{-3}$ $4.9\times 10^{-12}$ $2.3$ L2 = B6 $5\times
10^{-5}$ $2\times 10^{4}$ $10^{4}$ $5\times 10^{-3}$ $0.5$ $0.5$ $5000$ $100$
— $6.0\times 10^{-2}$ $4.8\times 10^{-10}$ $3.7$ L3 = C4 $5\times 10^{-5}$
$1\times 10^{4}$ $10^{4}$ $5\times 10^{-3}$ $0.5$ $1$ $5000$ $100$ —
$2.3\times 10^{-1}$ $1.8\times 10^{-8}$ $3.0$ L4 = D3 $5\times 10^{-5}$
$5\times 10^{3}$ $10^{4}$ $5\times 10^{-3}$ $0.5$ $2$ $5000$ $100$ $10000$
$8.5\times 10^{-1}$ $7.2\times 10^{-7}$ $2.5$ M1 = A1 $1\times 10^{-6}$
$5\times 10^{4}$ $10^{4}$ $1\times 10^{-4}$ $0.01$ $0.2$ $100$ $100$ $2000$
$4.6\times 10^{-3}$ $8.9\times 10^{-14}$ $0.032$ M2 = B1 $1\times 10^{-6}$
$2\times 10^{4}$ $10^{4}$ $1\times 10^{-4}$ $0.01$ $0.5$ $100$ $100$ $800$
$1.6\times 10^{-2}$ $4.7\times 10^{-12}$ $0.027$ N1 = A3 $5\times 10^{-6}$
$5\times 10^{4}$ $10^{4}$ $5\times 10^{-4}$ $0.05$ $0.2$ $500$ $100$ $10000$
$8.5\times 10^{-3}$ $1.1\times 10^{-12}$ $0.31$ N2 = B3 $5\times 10^{-6}$
$2\times 10^{4}$ $10^{4}$ $5\times 10^{-4}$ $0.05$ $0.5$ $500$ $100$ $4000$
$4.0\times 10^{-2}$ $1.6\times 10^{-10}$ $0.31$ N3 = C1 $5\times 10^{-6}$
$1\times 10^{4}$ $10^{4}$ $5\times 10^{-4}$ $0.05$ $1$ $500$ $100$ $2000$
$1.1\times 10^{-1}$ $5.6\times 10^{-9}$ $0.33$
Table 4 Summary of Runs from series U–X.
Run $\eta$ $\lambda^{1/2}$ ${\mu}_{50}$ $\eta k_{1}$ $v_{\mu}$ $v_{\lambda}$
$\eta{\mu}_{50}^{2}$ $k_{1}$ $k_{\lambda}$ ${\cal E}_{\rm M}^{\max}$ ${\cal
E}_{\rm GW}^{\rm sat}$ $q$ U1 $1\times 10^{-4}$ $5\times 10^{2}$ $2\times
10^{2}$ $1\times 10^{-3}$ $0.020$ $0.4$ $4$ $10$ $40$ $1.8\times 10^{-2}$
$1.5\times 10^{-12}$ $0.00068$ U2 $1\times 10^{-4}$ $5\times 10^{2}$ $3\times
10^{2}$ $1\times 10^{-3}$ $0.030$ $0.6$ $9$ $10$ $60$ $4.2\times 10^{-2}$
$3.0\times 10^{-10}$ $0.0062$ U3 $1\times 10^{-4}$ $5\times 10^{2}$ $4\times
10^{2}$ $1\times 10^{-3}$ $0.040$ $0.8$ $16$ $10$ $80$ $7.4\times 10^{-2}$
$9.2\times 10^{-9}$ $0.026$ U4 $1\times 10^{-4}$ $5\times 10^{2}$ $5\times
10^{2}$ $1\times 10^{-3}$ $0.050$ $1$ $25$ $10$ $100$ $1.1\times 10^{-1}$
$7.1\times 10^{-8}$ $0.058$ U5 $1\times 10^{-4}$ $5\times 10^{2}$ $7\times
10^{2}$ $1\times 10^{-3}$ $0.070$ $1$ $49$ $10$ $140$ $2.2\times 10^{-1}$
$8.0\times 10^{-7}$ $0.14$ U6$\,=$G4 $1\times 10^{-4}$ $5\times 10^{2}$
$1\times 10^{3}$ $1\times 10^{-3}$ $0.10$ $2$ $100$ $10$ $200$ $4.4\times
10^{-1}$ $7.9\times 10^{-6}$ $0.32$ V1 $5\times 10^{-6}$ $10^{4}$ $2\times
10^{3}$ $5\times 10^{-4}$ $0.010$ $0.2$ $20$ $100$ $400$ $4.5\times 10^{-3}$
$3.2\times 10^{-15}$ $0.0012$ V2 $5\times 10^{-6}$ $10^{4}$ $3\times 10^{3}$
$5\times 10^{-4}$ $0.015$ $0.3$ $45$ $100$ $600$ $1.0\times 10^{-2}$
$4.6\times 10^{-13}$ $0.010$ V3 $5\times 10^{-6}$ $10^{4}$ $4\times 10^{3}$
$5\times 10^{-4}$ $0.020$ $0.4$ $80$ $100$ $800$ $1.8\times 10^{-2}$
$1.4\times 10^{-11}$ $0.041$ V4 $5\times 10^{-6}$ $10^{4}$ $5\times 10^{3}$
$5\times 10^{-4}$ $0.025$ $0.5$ $125$ $100$ $1000$ $2.9\times 10^{-2}$
$7.1\times 10^{-11}$ $0.074$ V5 $5\times 10^{-6}$ $10^{4}$ $7\times 10^{3}$
$5\times 10^{-4}$ $0.035$ $0.7$ $245$ $100$ $1400$ $5.6\times 10^{-2}$
$8.4\times 10^{-10}$ $0.18$ V6$\,=$C1 $5\times 10^{-6}$ $10^{4}$ $1\times
10^{4}$ $5\times 10^{-4}$ $0.050$ $1$ $500$ $100$ $2000$ $1.1\times 10^{-1}$
$5.6\times 10^{-9}$ $0.33$ V7 $5\times 10^{-6}$ $10^{4}$ $2\times 10^{4}$
$1\times 10^{-3}$ $0.10$ $2$ $2000$ $200$ $4000$ $4.4\times 10^{-1}$
$8.6\times 10^{-8}$ $0.67$ W1 $2\times 10^{-5}$ $10^{4}$ $2\times 10^{3}$
$2\times 10^{-3}$ $0.040$ $0.2$ $80$ $100$ $1600$ $8.0\times 10^{-3}$
$1.2\times 10^{-13}$ $0.017$ W2 $2\times 10^{-5}$ $10^{4}$ $3\times 10^{3}$
$2\times 10^{-3}$ $0.060$ $0.3$ $180$ $100$ $2400$ $1.8\times 10^{-2}$
$2.2\times 10^{-11}$ $0.16$ W3 $2\times 10^{-5}$ $10^{4}$ $4\times 10^{3}$
$2\times 10^{-3}$ $0.080$ $0.4$ $320$ $100$ $3200$ $3.2\times 10^{-2}$
$1.8\times 10^{-10}$ $0.34$ W4 $2\times 10^{-5}$ $10^{4}$ $5\times 10^{3}$
$2\times 10^{-3}$ $0.10$ $0.5$ $500$ $100$ $4000$ $5.0\times 10^{-2}$
$8.0\times 10^{-10}$ $0.57$ W5 $2\times 10^{-5}$ $10^{4}$ $7\times 10^{3}$
$2\times 10^{-3}$ $0.14$ $0.7$ $980$ $100$ $5600$ $9.7\times 10^{-2}$
$3.4\times 10^{-9}$ $0.84$ W6$\,=$C3 $2\times 10^{-5}$ $10^{4}$ $1\times
10^{4}$ $2\times 10^{-3}$ $0.20$ $1$ $2000$ $100$ $8000$ $2.0\times 10^{-1}$
$1.6\times 10^{-8}$ $1.3$ W7 $2\times 10^{-5}$ $10^{4}$ $2\times 10^{4}$
$2\times 10^{-3}$ $0.40$ $2$ $8000$ $100$ $16000$ $7.8\times 10^{-1}$
$1.9\times 10^{-7}$ $2.2$ X1 $1\times 10^{-3}$ $2\times 10^{4}$ $2\times
10^{3}$ $1\times 10^{-1}$ $2.0$ $0.1$ $4000$ $100$ — $2.4\times 10^{-3}$
$2.1\times 10^{-11}$ $3.8$ X2 $1\times 10^{-3}$ $2\times 10^{4}$ $5\times
10^{3}$ $1\times 10^{-1}$ $5.0$ $0.2$ $25000$ $100$ — $1.5\times 10^{-2}$
$5.5\times 10^{-10}$ $7.7$ X3$\,=$B10 $1\times 10^{-3}$ $2\times 10^{4}$
$1\times 10^{4}$ $1\times 10^{-1}$ $10$ $0.5$ $1\times 10^{5}$ $100$ —
$6.0\times 10^{-2}$ $6.0\times 10^{-9}$ $12$ X4 $1\times 10^{-3}$ $2\times
10^{4}$ $2\times 10^{4}$ $1\times 10^{-1}$ $20$ $1$ $4\times 10^{5}$ $100$ —
$2.3\times 10^{-1}$ $3.5\times 10^{-8}$ $16$ X5 $1\times 10^{-3}$ $2\times
10^{4}$ $5\times 10^{4}$ $5\times 10^{-1}$ $50$ $2$ $2\times 10^{6}$ $500$ —
$1.5\times 10^{0}$ $3.1\times 10^{-7}$ $18$
### 3.6. Dependence on $\eta$ for given $\lambda$ and ${\mu}_{50}$
We have already seen that, as we increase $\eta$, we gradually move from
regime I to regime II. Let us now also determine the functional dependence of
both ${\cal E}_{\rm M}^{\max}$ and ${\cal E}_{\rm GW}^{\rm sat}$ on $\eta$.
This is shown in Figure 10 for the runs of series A–G.
In Figure 10(c), we see the dotted line describing a cubic dependence, ${\cal
E}_{\rm GW}^{\rm sat}\approx 1.7\times 10^{-6}\,({\cal E}_{\rm
M}^{\max})^{3}$. A similar scaling has also been suggested by Neronov et al.
(2021) based on the consideration of characteristic time and length scales.
Using Equation (15), this implies a square root dependence of the efficiency
parameter $q$, $q\approx 13\,({\cal E}_{\rm M}^{\max})^{1/2}$.
As expected from Equation (11), smaller values of $\lambda$ lead to an
increase of ${\cal E}_{\rm M}^{\max}$. Values close to unity become not only
more unrealistic because of Big Bang nucleosynthesis constraints (Grasso &
Rubinstein, 2001), but they also can more easily lead to numerical problems.
In all cases, we see that there is a change in slope and that ${\cal E}_{\rm
M}^{\max}$ reaches a plateau when $v_{\mu}/v_{\lambda}=\eta\lambda^{1/2}$
approaches a critical value of around one half. Interestingly, ${\cal E}_{\rm
GW}^{\rm sat}$ still continues to increase approximately linearly with $\eta$,
so this cannot be explained by an increase of ${\cal E}_{\rm M}^{\max}$.
However, we have seen in Section 3.5 that there is a change in the slope of
$E_{\rm GW}(k)$, which results in larger GW energy when the slope changes from
$k^{2}$ to $k^{0}$ or even $k^{-0.5}$; see Equation (21).
### 3.7. Dependence on $\lambda$ for given $\eta$ and ${\mu}_{50}$
As expected, ${\cal E}_{\rm M}^{\max}$ scales inversely proportional to
$\lambda$. This can be seen in the first panel of Figure 11, where we plot the
runs of series K–N; see also Table LABEL:Tsummary_lam for a summary. In the
other panels, we also show the dependence of ${\cal E}_{\rm GW}^{\rm sat}$ on
$\lambda$ and the mutual parametric dependence of ${\cal E}_{\rm GW}^{\rm
sat}$ on ${\cal E}_{\rm M}^{\max}$. We see that the dependence of ${\cal
E}_{\rm GW}^{\rm sat}$ on $\lambda$ is steeper than $\lambda^{-2}$ –
approximately like $\propto\lambda^{-5/2}$, according to Figure 11(b). The
dependence of ${\cal E}_{\rm GW}^{\rm sat}$ on ${\cal E}_{\rm M}^{\max}$ is
therefore also steeper than quadratic, namely approximately cubic; see Figure
11(c).
It is instructive to see how well the dependence of ${\cal E}_{\rm GW}^{\rm
sat}$ on $\eta$, $\lambda$, and ${\mu}_{50}$ can be expressed just in terms of
$v_{\lambda}$ and $v_{\mu}$. The approximately linear dependence of ${\cal
E}_{\rm GW}^{\rm sat}$ on $\eta$ seen in Figure 10(b) for regime II would then
also suggest its linear dependence on $v_{\mu}$. Furthermore, the approximate
scaling ${\cal E}_{\rm GW}^{\rm sat}\propto\lambda^{-5/2}$ seen in Figure
11(b) would suggest ${\cal E}_{\rm GW}^{\rm sat}\propto v_{\lambda}^{5}$. The
combined dependence would then be
${\cal E}_{\rm GW}^{\rm sat}\propto v_{\lambda}^{5}v_{\mu},$ (22)
implying ${\cal E}_{\rm GW}^{\rm sat}\propto{\mu}_{50}^{6}$. In the next
section we see that this suggestion agrees reasonably well with our data.
### 3.8. Dependence on ${\mu}_{50}$ for given $\eta$ and $\lambda$
Let us finally determine the dependence of ${\cal E}_{\rm M}^{\max}$ and
${\cal E}_{\rm GW}^{\rm sat}$ on ${\mu}_{50}$, keeping $\eta$ and $\lambda$
unchanged. The results are shown in Figure 12. We clearly see the expected
quadratic dependence of ${\cal E}_{\rm M}^{\max}$ on ${\mu}_{50}$. The
dependence of ${\cal E}_{\rm GW}^{\rm sat}$ on ${\mu}_{50}$ is much steeper
and shows a break at ${\mu}_{50}\approx 500$ for runs of series U and $5000$
for runs of series V and W. However, all those runs are in regime I; see Table
LABEL:Tsummary_mu. We have therefore added the runs of series X, which are in
regime II. Nevertheless, the basic slopes are unchanged.
In Figure 12(c), we have plotted ${\cal E}_{\rm GW}^{\rm sat}$ versus ${\cal
E}_{\rm M}^{\max}/k_{\mu}$. This allows us to estimate an upper bound for the
empirical parameter $q$ in Equation (14) if $k_{\rm peak}$ is replaced by
$k_{\mu}$. We find $q<10$.
In view of Equation (14), using the dependence of $k_{\rm peak}$ on $v_{\mu}$
and $v_{\lambda}$, as given below Equation (15), we have
$q\propto(v_{\mu}v_{\lambda})^{1/2}$ in regime II and
$q\propto(v_{\mu}^{3}/v_{\lambda})^{1/2}$ in regime I. This has also been
verified using our numerical data.
### 3.9. Combined dependence
Equation (22) has the advantage that one can now summarize all of the
numerical data in one plot. The result is shown in Figure 13. In its inset, we
also show ${\cal E}_{\rm GW}^{\rm sat}$ for the set of parameters given by
Brandenburg et al. (2017b) for the early universe, $v_{\mu}=2\times 10^{-5}$
and $v_{\lambda}=0.05$, corresponding to $v_{\lambda}^{5}v_{\mu}=6\times
10^{-12}$. It should be noted, however, that those values are rather
uncertain, because both are proportional to ${\mu}_{50}$, for which only
uncertain upper bounds can be proposed.
Looking at Figure 13, we see that a few runs fall outside the linear trend.
This applies especially to the runs of series F and G (red dotted and blue
dotted lines, respectively). Also the runs of series U and X (black dashed and
orange dashed lines, respectively) show major departures. However, it is not
immediately obvious what is special about them.
Looking at Figure 13, we see that data points from one series are identical
with data points from another. This is because those data points are from the
same runs, but have alternative names, see the indications in Tables
LABEL:Tsummary_lam and LABEL:Tsummary_mu.
### 3.10. Numerical limitations
Because of certain numerical constraints, the parameters of our simulations
have to stay within specific empirical limits. The purpose of this section is
to discuss the nature of those constraints and to see how they depend on the
choice of the parameters. Let us begin with $\eta$, which we were able to vary
by more than four orders of magnitude. For smaller values of $\eta$, we go
deeper into regime I, provided $\eta\lambda^{1/2}<1$. The main limitation here
is the large separation of dynamical and diffusive time scales. These time
scales are proportional to ${\mu}_{50}^{-1}$ and $(\eta{\mu}_{50}^{2})^{-1}$,
respectively. This separation of time scales results in long run times that
make the simulations more computationally costly. In addition, there is a
large separation in spatial scales between ${\mu}_{50}^{-1}$ and $\eta$, which
corresponds to large magnetic Reynolds numbers, requiring a large number of
mesh points. And, as we have now seen, for decreasing $\eta$, the magnetic and
GW energies become very small. For larger $\eta$, on the other hand, we go
deeper into regime II, provided $\eta\lambda^{1/2}>1$. The main limitation
here is the shortness of the numerical time step, which depends on the mesh
spacing $\delta$ as $\sim\delta^{2}/\eta$.
Next, let us discuss the value of ${\mu}_{50}$, which we have been able to
vary by a little over two orders of magnitude. Clearly, for the dynamo
instability to exist, the mesh spacing cannot be too coarse, and ${\mu}_{50}$
must not exceed the largest resolved wavenumber in the domain
$\pi/\delta=k_{1}N/2$. Therefore, for a given number of mesh points $N$,
$k_{1}$ cannot be too small. It cannot be too large either, because then we
would no longer be able to capture the largest length scales in the system. In
particular, if $k_{1}$ is too large, it could lead to artifacts resulting from
the finiteness of the domain, as already discussed in Section 3.5. As we see
from Table LABEL:Tsummary, we have varied $k_{1}$ by a factor of 20. It should
be noted that it is not a physical parameter, since the intention is to
simulate an infinitely extended domain. Therefore, the final results should be
independent of $k_{1}$. An example is seen by inspecting Figure 1 for the runs
of series A, where the three uppermost open black symbols show a small shift
to the left. This is because here $k_{1}$ has been decreased from 100 to 50.
In Figure 2, for example, $k_{1}$ is not small enough to capture the maximum
GW energy properly.
Finally, the parameter $\lambda$ determines the limiting CME speed
$v_{\lambda}$. We have varied $\lambda^{1/2}$ by over two orders of magnitude.
For the smallest values in Table LABEL:Tsummary, we also needed to decrease
the value of ${\mu}_{50}$ to prevent the magnetic energy from exceeding the
critical density, which corresponds to a value of unity. This could lead to
the production of shocks which, in turn, requires more mesh points, larger
viscosity, or both. Furthermore, the neglect of special relativistic effects
could no longer be justified.
## 4\. Conclusions
The present work has revealed a scaling relation for the GW energy from the
CME: ${\cal E}_{\rm GW}^{\rm sat}\propto v_{\lambda}^{5}v_{\mu}$. Based on
earlier dimensional arguments and numerical findings for the resulting
magnetic field energy (Brandenburg et al., 2017b), it was already anticipated
that, within the framework of the standard description of the CME including
its dependence on temperature and the effective number of degrees of freedom,
the resulting GWs would be too weak to be detectable. This is indeed confirmed
by our present work. Furthermore, we have also shown that the conversion from
magnetic to GW energy is generally less efficient than for forced and decaying
turbulence; see Figure 1. Here, we have been able to estimate the efficiency
parameter $q$ in Equation (14) as being roughly
$\propto(v_{\mu}v_{\lambda})^{1/2}$ in regime II, but
$\propto(v_{\mu}^{3}/v_{\lambda})^{1/2}$ in regime I. It should also be
emphasized that, even though $q$ can reach values of the order of ten (see
Tables LABEL:Tsummary, LABEL:Tsummary_lam, and LABEL:Tsummary_mu), which is
similar to the value for acoustic turbulence, the final GW energy production
is still poor owing to the small length scales associated with the CME.
Magnetic field generation by the CME can occur in two different regimes;
regimes I and II, depending on the relation of magnetic field generation and
limiting CME speeds, $v_{\mu}$ and $v_{\lambda}$, respectively. In the present
work, we have regarded the CME as a generic mechanism that allows us to study
how GW energy production can be related to the strengths of generation and the
limiting CME speed. Whether or not other magnetogenesis mechanisms can really
be described in similar ways needs to be seen. It is interesting to note,
however, that our finding regarding the proportionality of the GW energy to
the fifth power of $v_{\lambda}$ is reminiscent of the earlier results of
Gogoberidze et al. (2007) who found the GW energy to be proportional to the
fifth power of the turbulent velocity; see their Equation (40). It should be
noted, however, that the additional dependence on $v_{\mu}$ cannot be
neglected and results in the increase of ${\cal E}_{\rm GW}$ with increasing
values of $\eta$; see Figure 10(b).
Our work has also revealed new unexpected GW energy spectra. In regime I, the
spectra were not of clean power law form, and the spectral energy was falling
off with wavenumber faster than in any earlier simulations. This means that
the GW energy ${\cal E}_{\rm GW}=\int E_{\rm GW}(k)\,{\rm d}{}k$ depends
significantly on its lower integration bound $k_{1}$ so that it will be
important to include even smaller wavenumbers in future simulations. This
could restore a quadratic scaling for Runs A1–A4 and Runs B1–B5 in Figure 1
and Figure 10(c). In regime II, on the other hand, we have seen that large GW
energies can be generated. This was rather surprising and counterintuitive,
because this regime implies a lack of a turbulent cascade in $E_{\rm M}(k)$
with just a spectral bump traveling toward lower wavenumbers. This traveling,
on the other hand, happened rather rapidly, which contributed to the large GW
energies in that case. The physical reality of this regime is however
questionable.
Software and Data Availability. The source code used for the simulations of
this study, the Pencil Code (Pencil Code Collaboration, 2021), is freely
available on https://github.com/pencil-code/. The DOI of the code is
https://doi.org/10.5281/zenodo.2315093 (Brandenburg, 2018). The simulation
setup and the corresponding data are freely available from
https://doi.org/10.5281/zenodo.4448211; see also
http://www.nordita.org/~brandenb/projects/GWfromCME/ for easier access.
Support through grants from the Swedish Research Council (2019-04234), the
Shota Rustaveli National Science Foundation of Georgia (FR18-1462), and the
European Research Council (694896) are gratefully acknowledged. We acknowledge
the allocation of computing resources provided by the Swedish National
Allocations Committee at the Center for Parallel Computers at the Royal
Institute of Technology in Stockholm. J.S. acknowledges the funding from the
Swiss National Science Foundation under Grant No. 185863. The computations and
data handling were enabled by resources provided by the Swedish National
Infrastructure for Computing (SNIC) at the Center for Parallel Computers at
the Royal Institute of Technology in Stockholm, partially funded by the
Swedish Research Council through grant agreement no. 2018-05973.
## Appendix A The compression term in Equation (2)
At the end of Section 2.1, we noted that for
$\mbox{\boldmath$\nabla$}{}\cdot\bm{u}\neq 0$, the conservation of the total
chirality requires an extra term,
$-{\mu}_{5}\mbox{\boldmath$\nabla$}{}\cdot\bm{u}$, on the right-hand side of
Equation (3). For $\Gamma_{\rm\\!f}=0$, this equation can then also be written
as
${\partial{\mu}_{5}\over\partial
t}=-\mbox{\boldmath$\nabla$}{}\cdot({\mu}_{5}\bm{u})-\lambda\,\eta\left({\mu}_{5}\bm{B}-\mbox{\boldmath$J$}{}\right)\cdot\bm{B}+D_{5}\nabla^{2}{\mu}_{5},$
(A1)
expressing the conservation of ${\mu}_{5}$ for $\bm{B}=\boldsymbol{0}$. To
illustrate the effect of the ${\mu}_{5}\mbox{\boldmath$\nabla$}{}\cdot\bm{u}$
term, we consider here a simple one-dimensional example with a prescribed
(kinematic) velocity field $\bm{u}=(u_{0}\sin kx,0,0)$ and periodic boundary
conditions. This is obviously an artificial way of demonstrating the
consequences for the generation of $\bm{B}$. To have an effect on the
conservation of ${\mu}_{5}$, we also consider an initial profile of the form
${\mu}_{5}(x,0)={\mu}_{50}\cos kx$, so $\langle{\mu}_{5}(x,0)\rangle=0$. In
Figure 14, we show $B_{y}(x)$ and ${\mu}_{5}(x)$ at $t=10$ for $k=1$,
${\mu}_{50}=100$, $\lambda=100$, $\eta=10^{-3}$, and $u_{0}=10^{-2}$. We used
a weak seed magnetic field with zero helicity as the initial condition.
Figure 14.— Comparison of the profiles of $B_{y}(x)$ and ${\mu}_{5}(x)$ for
$k=1$, ${\mu}_{50}=100$, $\lambda=100$, $\eta=10^{-3}$, and $u_{0}=10^{-2}$
with (black) and without (red) the
${\mu}_{5}\mbox{\boldmath$\nabla$}{}\cdot\bm{u}$ term included. The initial
profile of ${\mu}_{5}$ is also shown (blue dashed). Figure 15.— Comparison
of ${\cal E}_{\rm GW}(t)$ for the cases where $\gamma_{\rm Lor}\neq 1$ and the
${\mu}_{5}\mbox{\boldmath$\nabla$}{}\cdot\bm{u}$ term is included (black
solid) and where it is omitted (red dashed) with a case where $\gamma_{\rm
Lor}=1$ and the ${\mu}_{5}\mbox{\boldmath$\nabla$}{}\cdot\bm{u}$ term is
included (blue dotted).
When conservation of the total chirality is invoked by including the
${\mu}_{5}\mbox{\boldmath$\nabla$}{}\cdot\bm{u}$ term, there is a small
enhancement of $B_{y}$ around $kx=\pm 0.7$ and a small decrease at $\pm 2.3$.
This is caused by compression at $kx=0$ and expansion at $kx=\pm\pi$. In this
example, when the ${\mu}_{5}\mbox{\boldmath$\nabla$}{}\cdot\bm{u}$ term is
absent, the total chirality becomes negative and reaches about 6% of its
initial rms value. Finally, we show in Figure 15 the evolution of ${\cal
E}_{\rm GW}(t)$ for Run D8, where the magnetic field is one of the largest and
the effect of the ${\mu}_{5}\mbox{\boldmath$\nabla$}{}\cdot\bm{u}$ term is
expected to be strong. We compare the case where $\gamma_{\rm Lor}\neq 1$ and
the ${\mu}_{5}\mbox{\boldmath$\nabla$}{}\cdot\bm{u}$ term is included with a
case where it is omitted, and a case where $\gamma_{\rm Lor}=1$ and the
${\mu}_{5}\mbox{\boldmath$\nabla$}{}\cdot\bm{u}$ term is included. The effect
of the latter is here extremely small. We also see that the inclusion of the
$\gamma_{\rm Lor}$ term affects the detailed time evolution of ${\cal E}_{\rm
GW}(t)$, but not the final overall saturation level.
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|
# Reaching the sensitivity limit of a Sagnac gyroscope through linear
regression analysis
Angela D.V. Di Virgilio, Umberto Giacomelli, Andrea Simonelli and Giuseppe
Terreni
INFN Sez. di Pisa,
Polo Fibonacci, Largo B Pontecorvo 3, I-56127 Pisa, Italy
<EMAIL_ADDRESS>
&Andrea Basti, Nicolò Beverini, Giorgio Carelli, Donatella Ciampini, Francesco
Fuso, Enrico Maccioni and Paolo Marsili
Università di Pisa and INFN,
Dipartimento di Fisica "E. Fermi", Largo B Pontecorvo 3, I-56127 Pisa, Italy
&Carlo Altucci, Francesco Bajardi, Salvatore Capozziello, and Raffaele Velotta
Universitá di Napoli and INFN,
Dipartimento di Fisica, Complesso Univ. Monte Sant’Angelo, via Cintia, Napoli,
Italy &Alberto Porzio
CNR-SPIN and INFN, Napoli, Complesso Univ. Monte Sant’Angelo, via Cintia,
Napoli, Italy &Antonello Ortolan
INFN-National Lab. of Legnaro, viale dell’Università 2, I-35020, Legnaro (PD),
Italy
###### Abstract
The sensitivity to angular rotation of the top class Sagnac gyroscope
GINGERINO is carefully investigated with standard statistical means, using 103
days of continuous operation and the available geodesic measurements of the
Earth angular rotation rate. All features of the Earth rotation rate are
correctly reproduced. The sensitivity of fractions of frad/s is attained for
long term runs. This excellent sensitivity and stability put Sagnac gyroscopes
at the forefront for fundamental physics, in particular for tests of general
relativity and Lorentz violation, where the sensitivity plays the key role to
provide reliable data for deeper theoretical investigations. The achieved
sensitivity overcomes the conventionally expected one for Sagnac ring laser
gyroscopes.
## 1 Introduction
Ring Laser Gyroscopes (RLGs) exploit the Sagnac effect – i.e. the interference
of two counter-propagating photon beams in a closed optical path – to measure
absolute rotation of the apparatus with respect to the local inertial frame
[1]. For the last 40 years large-scale versions of RLGs have been regarded as
a promising tool for geodesy and fundamental physics researches [2, 3]. A
relevant contribution to geodesy is a more accurate estimate of Earth Rotation
Parameters (ERP) [4], i.e. polar motion and sub-daily variations of universal
time, that correspond to direction and modulus of the Earth angular velocity
vector, respectively. RLGs eventually should provide to the International
Earth Rotation System (IERS) [5] a continuous, high–resolution measurement of
the ERP, complementary to the well-established methods based on VLBI and GNSS
(e.g. GPS, LSR, DORIS, etc.) data. Further to ERP measurements with ground-
based instrumentation, challenging tests of fundamental physics can be carried
out with RLGs by studying the tiny residuals of the proper time difference
between the counter-propagating photon beams, which last once any known
rotation contribution (geophysical, geodetic or local) has been taken into
account and subtracted. These residuals are directly connected to metric
description of local space-time geometry [6] or searches for Lorentz
violations [7].
Unfortunately, the non–linearities induced by laser dynamics have made less
attractive the RLG applications, and prevented the full comprehension of long
term stability and sensitivity of RLGs. The laser dynamics corrections [8, 9,
10, 11, 12, 13] are unavoidable if the RLG top sensitivity must be pushed
towards testing space-time structures and symmetries, beyond the experimental
results in gravitational or particle physics already available in the
literature [14, 15]. In this regard, we have demonstrated that effects of non
linear dynamics can be cancelled at a level of one part in $10^{3}$ and this
was enough to push the sensitivity of our RLG to sub-prad/s rotation rates
[16], paving the way to explore fundamental physics thanks to such an
unprecedented sensitivity for a RLG. It is remarkable that also cold atoms in
a Sagnac interferometer achieved a pretty good sensitivity. However, there is
a $\sim 5$ orders of magnitude gap in rotation sensitivity between current
devices and large-scale RLGs [19, 17, 18]. Another interesting gyroscope
design is based on passive optical cavities which have been studied for
gravitational waves detectors [20, 21], and are foreseen for space based
gravitational waves antennas [22, 23, 24], but they have not yet demonstrated
sufficient sensitivity.
Notable General Relativity (GR) tests with gyroscopes are the measurement of
the De Sitter effect due to the curvature of space-time around the Earth and
of the Lense–Thirring effect due to the Earth rotation (dragging of the local
inertial frames) [25]. Such tests are based on the comparison between the
Earth angular velocity vector as estimated by IERS and the corresponding
measurements obtained by an array of RLGs. Moreover, a RLG array can
reconstruct the “local geometry” of null geodetics of space-time and test
whether it fully corresponds to the GR description or it requires GR
extensions or modifications [26, 27]. Though at the level of the solar system
GR well fits experimental observations, it suffers several shortcomings from
the very small up to the cosmological scales. For example, it cannot predict
the right correlation between mass and radius of some neutron stars [28, 29],
the galaxy rotation curve without introducing Dark Matter [30, 31], or the
accelerated expansion of the Universe in the late time without introducing
Dark Energy [32]. Dark Matter and Dark Energy are supposed to represent the
$26.8\%$ and 68$\%$ of the Universe content, but have never been detected
directly.
At the small scales, while Strong and Electroweak interactions can be dealt
with under the standard of quantum field theory, many issues arise in the
attempt to merge the formalism of GR with that of quantum mechanics [33, 34,
35]. Indeed, in view of a possible quantum scheme, the spacetime metric should
represent both a dynamical field and the background. This is not the case of
other interactions, whose treatment is simplified by the assumption that the
spacetime is supposed to be flat. Also, from quantum field theory in curved
spacetime, a discrepancy of 120 orders of magnitude occurs between the
theoretically predicted value of the cosmological constant and the
experimentally observed one.
Modified theories of gravity arose with the purpose of solving such
shortcomings, by considering alternatives to the Einstein–Hilbert action [36,
27]. GR can be modified in several ways, such as introducing the coupling
between geometry and scalar fields [37, 38], higher-order curvature invariants
[39, 40, 41], torsion and non-metricity [26, 42], or by not requiring the
equivalence principle to hold _a priori_ [43].
In this context, experimental observations play a fundamental role. One the
one hand, they can be used to constrain dynamical degrees of freedom occurring
in modified theories of gravity [44], selecting physically relevant theories.
One the other hand, observations may address the research for GR extensions
towards viable models, also suggesting the scales in which such extensions are
needed.
For instance, in [45], post-newtonian approximation is used to put upper
limits to the functional form of a higher-order scalar-tensor action; in [46]
$f(R)$ gravity is constrained by solar system tests; in [47] the fundamental
plane of galaxies is addressed to geometric contributions; in [48] non-local
theories of gravity are selected by S2 star orbit.
Other interesting tests searching for new physics involve the local Lorentz
invariance, since well motivated extensions of the Standard Model for particle
physics predict Lorentz violation terms, that can be checked by the
interference of the two counter-propagating beams of light [7].
All such unique features of RLGs motivated us to propose the GINGER experiment
[49, 50] and to build its test bed GINGERINO [51, 52] at the underground Gran
Sasso laboratory. GINGERINO is a 3.6 m side RLG that has been taking data in
an almost continuous basis since 2017; it runs unattended and free running
ensuring a duty–cycle around $80\%$ even in the absence of an active geometry
control [53]. This large amount of data gives us the possibility to improve
the comprehension of the instrument, understand the sensitivity limits, and
assess the feasibility of long term operation. Our activity is aimed at
improving the experimental set-up of the future GINGER array taking full
advantage of the very careful analysis of GINGERINO data, where weak points of
the system design can be identified, and the origin of disturbances or
limitations in the sensitivity can be investigated. In general terms, the
study of very high sensitivity apparatus is rather difficult, since it deals
with noise. GINGERINO has the advantage to be sensitive to the global geodetic
signals of the Earth, as Chandler and Annual wobbles, polar motions and
variations of the universal time. These are rather small signals independently
and constantly measured by the international system IERS with very high
accuracy. Therefore, the analysis can effectively look for those signals in
the GINGERINO data.
It has been recently demonstrated that GINGERINO reaches the sensitivity limit
of 40 frad/s in 3.5 days of integration [16]. The analysis has shown the
dominant role of the tilt measurements in the identification and subtraction
of the local disturbances. Aim of the present analysis is to improve the
identification and subtraction of the local and instrumental disturbances by
developing a more effective treatment of the signals produced by the
tiltmeters implemented in the RLG set-up. To fully exploit the GINGERINO
sensitivity we have also improved the cross calibration procedure with the
IERS data. The approach enables attaining an extremely high sensitivity, even
better than one hundred times the Lense-Thirring effect with a bandwidth
corresponding to a 600 s integration time and long term operation. Plan of the
paper is as follows. In Sect. 2 the general scheme of the analysis is
outlined. Sect. 3 reports the results of the analysis applied to 103 days of
data, illustrating the calibration procedure and the relative sensitivity, and
indicating the main instrumental limits of the apparatus. In Sect. 4 we
investigate the occurrence of signals due to deformation of the Earth crust.
Sect. 5 reports a general discussion about the analysis and the GR tests.
Conclusions are eventually drawn in Sect. 6.
## 2 Purpose and general scheme of the analysis
Purpose of the analysis is to reconstruct the Earth angular rotation rate and
the instrumental disturbances using the RLG data and the environmental signals
with the final goal of improving sensitivity and investigating its effective
limit. The procedure uses linear regression (LR) model [54, 55, 56, 57]
minimising the square of the difference between the evaluated and
independently measured rotation rates. To this end it is mandatory to subtract
the contributions induced by the non linear laser dynamics [12, 13]. In the
following the main properties of the Sagnac frequency, the Earth rotation
rate, the experimental set-up, and the analysis components are summarised.
### 2.1 The Sagnac frequency
The Sagnac gyroscope is identified by the oriented area $\mathbf{A}$ enclosed
by the optical path and the perimeter $P$ corresponding to its length. The
Sagnac beating signal $\omega_{s}$ 111The conventional symbol to denote the
Sagnac signal is $f_{s}$, as it is practical to measure beating frequencies in
Hz. We use instead $\omega_{s}=2\pi f_{s}$ because the angular frequency is
used in back scattering calculations and its introduction leads to an
adimensional scale factor in Eq. 1. is proportional to the scalar product
between $\mathbf{A}$ and the total angular velocity $\mathbf{\Omega}_{T}$ of
the RLG optical cavity. Without loss of generality, we can write
$\displaystyle\omega_{s}=$ $\displaystyle
2\pi\frac{\mathbf{A}}{\lambda\,P}\cdot\mathbf{\Omega}_{T}$
$\displaystyle\mathbf{\Omega}_{T}=$
$\displaystyle\mathbf{\Omega}_{\oplus}+\mathbf{\Omega}_{loc}$ (1)
$\displaystyle\textrm{SF}=$ $\displaystyle 2\pi\frac{A}{\lambda\,P}\
\cos\gamma\,$
where $\lambda$ is the laser wavelength, and $\mathbf{\Omega}_{loc}$ is the
sum of all possible angular velocities associated with Earth crust
deformations and instrument infinitesimal rotations. In general, local
rotations are unknown in amplitude and direction but very small compared to
the Earth rotation rate $\Omega_{\oplus}$, and so they contribute to the
Sagnac signal as an additive perturbation. Here, $\mathbf{\Omega}_{\oplus}$
describes the Earth rotation rate and its orientation, $\gamma$ is the angle
between the area vector and the rotation axis, corresponding to the laboratory
co-latitude for horizontal RLGs, and SF is the scale factor. In our analysis
the $\cos\gamma$ is associated with the scale factor to simplify the
discussion, since effects of geometrical scale factor changes cannot be
distinguished from orientation changes. The modulus and direction of
$\mathbf{\Omega}_{\oplus}$ changes in time, however it is continuously
monitored by IERS. In the following we will use uppercase $\Omega$ and
lowercase $\omega$ for angular velocity (in units of rad/s) and the
corresponding Sagnac angular frequency, respectively222Note, however, that for
the sake of clarity we will use frequency units, in Hz, when numerical
evaluations of quantities and uncertainties related to angular frequencies
$\omega$ will be given all through the text and figures. In addition, their
time dependence will not be explicitly indicated.
### 2.2 Geodesic signals
The international system IERS provides the data to describe the Earth motion
on a daily basis, from which it is possible to reconstruct the effect on
GINGERINO, called geodesic signal $\Omega_{{}_{IERS}}$, as the sum of the
average Earth rotation rate $\Omega_{\oplus}$, the Length of Day (LoD)
changes, celestial pole offsets, UT1 - UTC, polar motion and diurnal and semi-
diurnal variations produced by ocean tides [5]. In the present analysis the
$\Omega_{{}_{IERS}}$ time series is downloaded from Earth orientation center
of the Paris observatory [58].
### 2.3 GINGERINO experimental set up and data analysis
GINGERINO is a RLG with a square laser cavity, 3.6m in side. It is installed
horizontally with its area vector aligned with the local vertical. Its design
is based on a hetero-lithic (HL) mechanical structure, the 4 mirrors at the
corners of the cavity are contained inside vacuum tight boxes and connected
together by vacuum tubes. Interested readers can find more details in the
literature [51, 52]. The mechanical structure is attached to a cross shaped
monument made of granite, connected in the center to the underneath bedrock
through a reinforced concrete block. The set-up is located underground, where
typical day-and-night temperature variations are strongly suppressed, and far
from anthropic disturbances. The apparatus is protected by a cabinet, far from
the large experimental halls of the Gran Sasso laboratory, moreover the
electronics is contained in a separated room. The laser optical cavity is
aligned at the beginning of the run, and after that it operates continuously
and unattended. The geometry is not electronically controlled, this implies
that mode jumps and split lasing mode occur. Routinely more than $90\%$ of the
data are of good quality [52]; however, for the present analysis data around
laser mode jumps are discarded, leading us to keep no more than $80\%$ of the
complete data set.
### 2.4 Model and parameters of the analysis
The Sagnac frequency $\omega_{s}$ has to be evaluated by taking into account
laser dynamics contribution $\omega_{LD}$, (i.e.
$\omega_{s}=\omega_{s0}-\omega_{LD}$, where $\omega_{s0}$ is a first
estimation of the Sagnac signal, as described in details in Refs. [12, 13, 16,
59]), and then it can be expressed as
$\omega_{s}=K_{cal}\,\omega_{{}_{IERS}}+\omega_{loc}\,,\\\ $ (2)
where $K_{cal}$ is a cross calibration constant, very close to 1, found by
comparing $\omega_{s}$ and $\omega_{{}_{IERS}}$, the angular frequency related
to the $\Omega_{{}_{IERS}}$ data, and the term representing local rotations
$\omega_{loc}=\omega_{env}+\omega_{ins}$, which takes into account signals of
environmental and geophysical origin $\omega_{env}$, and rotations of
instrumental origin $\omega_{ins}$. The present analysis does not distinguish
between $\omega_{ins}$ and $\omega_{env}$, but they are kept separated as, in
principle, we could get rid of $\omega_{ins}$ with an improved design of RLG.
To this aim, we are investigating the causes of the disturbances of an
instrumental origin with a dedicated analysis [61].
The $\omega_{LD}$ contribution is evaluated including in the LR the 6
explanatory variables that correspond to the signals collected at the output
ports of the square cavity, i.e. the beat note of the two counter propagating
beams, the DC amplitudes of the two mono beams $IS_{1,2}$, their AC amplitudes
$PH_{1,2}$, and their relative phase $\epsilon$. Other explanatory variables
come from the available environmental signals: temperature, pressure, air flow
speed, the two channels ($\zeta_{1,2}$) of the tiltmeter located on top of the
granite table. The procedure is iterative as some explanatory variables have
to be elaborated using a rough estimate of $\omega_{s}$, $\omega_{loc}$ and
some environmental signals.
### 2.5 Explanatory variables of the linear regression (LR)
The present analysis extends and refines the multiple linear regression
procedure outlined in our recent work [16]. It is worth noticing that the
environmental infinitesimal rotations affect at the same time the Sagnac
angular frequency $\omega_{s}$ and the laser dynamics contribution
$\omega_{LD}$. For this reason $\omega_{LD}$ is evaluated at the beginning
with the complete set of available explanatory variables related to laser
dynamics and environmental sources. In any case, the final result remains
unchanged if all the terms are kept in the LR procedure until its end. Outputs
of the analysis are an estimate of the geodesic signal reconstructed from the
RLG data $\omega_{geo}$, the local angular velocity $\omega_{loc}$, and the
residual of the model $\Delta M_{LR}$, which provides an insight into physical
phenomena not included in the proposed model. The term accounting for local
rotations and the residuals read, respectively,
$\displaystyle\omega_{loc}=\omega_{s0}-\sum_{i=1}^{N}a_{i}\,\omega_{LDi}-K_{cal}\
\omega_{{}_{IERS}}$ (3) $\displaystyle\Delta
M_{LR}=\omega_{s0}-K_{cal}\omega_{{}_{IERS}}-\sum_{i=1}^{N}a_{i}\omega_{LDi}$
(4)
$\displaystyle-\sum_{j=1}^{M}b_{j}\,env_{j}-\sum_{k=1}^{K}c_{k}\,\tilde{F}_{k},$
where $env_{j}$ are temperature, pressure, air flow speed, and the two
tiltmeter signals $\zeta_{1,2}$, the time series $\tilde{F}_{k}$ are the
product of the $\zeta_{1,2}$, or the DC monobeam signals $PH_{1,2}$, mainly
with $\omega_{loc}$, but also with residuals of an intermediate stage, and
$a_{i}$, $b_{j}$, $c_{k}$ are weights of the LR procedure. The use of the
explanatory variables $\tilde{F}_{k}$ has been suggested by our recent work
[16]. However, $\zeta_{1,2}$ and the relative products with $\omega_{s}$ were
used in Ref. [16] as separated explanatory variables. The present analysis
indicated that the effective variable was the combination of the two, in the
form of $\zeta_{1,2}\omega_{loc}$. To obtain $K_{cal}$ and $\omega_{s}$, the
LR is iterated a few times using at the first step only $\omega_{LD}$ to
provide a first evaluation of $\omega_{s}$. The calibration value $K_{cal}$ is
determined by imposing that at a fixed time T0, arbitrarily chosen, the
intercept $I_{T_{0}}$ of the LR is close to zero, i.e.
$K_{cal}=(\Omega_{geo}(T_{0})+I_{T_{0}})/\Omega_{{}_{IERS}}(T_{0})$; in the
iterative estimation of $\omega_{s}$ this procedure is repeated. 333To
correctly evaluate $\Omega_{{}_{IERS}}$ it is required to measure both the
angle $\theta$ of the area vector of the RLG with respect to the rotational
axis and the geometrical scale factor GSF $=2\pi A/(\lambda P)$. With a single
RLG, as in the case of GINGERINO, we can estimate only a combination of the
two quantities. Due to local rotations, the orientation of our apparatus is
not fixed in time at the level of its sensitivity. Therefore, the model cross
calibrates at the time $T_{0}$ and any change recovered by first and second
order expansions is reconstructed by the LR analysis using the information of
the explanatory variables. Assuming FS the scale factor for horizontal
orientation, the effective scale factor at the cross calibration point is
$K_{cal}\,\textrm{FS}$, and so
$\omega_{s}(T_{0})-\omega_{loc}(T_{0})=K_{cal}\,\omega_{{}_{IERS}}(T_{0})$.
The estimation of the geodesic angular frequency $\omega_{geo}$ is given by
$\omega_{geo}=\omega_{s0}-\sum_{i=1}^{N}a_{i}\,\omega_{LDi}-\sum_{j=1}^{M}b_{j}\,env_{j}-\sum_{n=1}^{K1}c_{n}\,\tilde{F}_{n}\;.$
(5)
We noticed that $K_{cal}$ depends mostly on the absolute orientation of the
device. Being $\theta$ the colatitude, and assuming that the geometry of the
laser cavity is stable in time because the temperature is rather stable and
geometrical variations are consequently small, it is possible to estimate the
inclination of the RLG at the cross calibration time,
$\theta_{css}=\arcsin({\sin{\theta}\cdot K_{cal}})$.
Although the complete set of explanatory variables is used in the initial
analysis, we eventually keep only the statistical independent ones that affect
the residuals. Accordingly, we neglect in our final elaboration pressure and
air flow speed signals, since they demonstrated a negligible effect.
## 3 Analysis results
The LR method has been applied to a set of data from day 1 to 103 of year
2020. The variables indicated in Eqs. 3-6 are time series with
$\frac{1}{600}$Hz sampling rate. Typical results changing the cross
calibration point reconstruct correctly $\Omega_{geo}$, the angular velocity
variations associated with $\omega_{geo}$. For example, at cross calibration
$T_{0}$, modified Julian date 58886.23611, $K_{cal}=1.000120452$,
corresponding to a misalignment with respect to the RLG horizontal orientation
of about 0.1 mrad (at the cross calibration time). We emphasize that the
relevant signals are reproduced with a standard deviation of the residuals
$\Delta M_{LR}$ at the nHz level. Figure 1 shows the evaluated true Sagnac
signal $\omega_{s}$, and Fig. 2 compares the effects on the RLG of
$\Omega_{geo}$ and $\Omega_{IERS}$, expressed in units of frequency, Hz
(corresponding in our notation to $\omega_{geo}/2\pi$ and
$\omega_{{}_{IERS}}/2\pi$, respectively). The agreement between the IERS data
and the evaluated $\Omega_{geo}$ is so good that the two curves are almost
perfectly over imposed each other in Fig. 2.
Figure 1: The evaluated Sagnac frequency $\omega_{s}/(2\pi)$. Figure 2:
$\Omega_{geo}$ evaluated from the LR analysis applied to GINGERINO data
compared with $K_{cal}\cdot\Omega_{IERS}$. The detail of 20 days is shown in
the inset. The data removed from the analysis are evident.
Remarkably, the output of the LR analysis for the laser dynamics and local
disturbances leads to non negligible contributions, as shown in Fig. 3,
reporting the evaluated $\omega_{LD}$ and $\omega_{loc}$, i.e. the sum of the
corresponding explanatory variables multiplied by the LR coefficients
corresponding to laser dynamics and environmental monitor signals. Their
standard deviations, in units of frequency, are as large as 6 mHz and 4.5 mHz,
respectively, strongly suggesting that the related effects must be carefully
accounted for in order to improve the instrumental sensitivity. Note that
$\omega_{LD}$ has a bias close to 0.5 in units of frequency, remarking the
importance of the laser dynamics correction not only for the sensitivity, but
as well for the accuracy, which is the key issue of GR tests.[12, 13]
Figure 3: The total laser dynamics contribution $\omega_{LD}$ (orange) and
local disturbances $\omega_{loc}$ (green) evaluated with the LR analysis
applied to the complete time span.
The residual $\Delta M_{LR}$ is associated with portions of the data which
cannot be explained by the model. Inclusion in the set of explanatory
variables of those created by multiplying $\omega_{loc}$ by $\zeta_{1,2}$ and
the DC mono beam signals has impact in the results: we denote such variables
as projectors, since they are scalar products. In particular, for GINGERINO,
mainly $\zeta_{1,2}$ signals are used in the projectors, being the use of the
$PH_{1,2}$ less relevant. Typical $\Delta M_{LR}$ using a set of 4 or 5
projectors leads to a standard deviation as small as 4 nHz in units of
frequency (corresponding to $0.7$ frad/s in angular velocity).
The Overlapping Allan Deviation (OAD) of the residuals $\Delta M_{LR}$, which
provides information on noise and measurement sensitivity as a function of the
integration time $\tau$, is shown in Fig. 4. Remarkably, OAD is always below 6
parts in $10^{12}$ of the Earth rotation rate, well below the target
meaningful for fundamental physics and geodesy, reported to be 1 part in
$10^{9}$ [50, 49, 16, 7]. Moreover, the analysis has been repeated using 30
days of June 16-July 15 2018, obtaining a very similar behaviour of the OAD.
Figure 4: Typical Overlapping Allan Deviation of residuals $\Delta M_{LR}$,
relative to mean $\omega_{s}$.
### 3.1 Calibration by sinusoidal signal injection
We implemented a supplementary method to estimate GINGERINO sensitivity by
adding to $\omega_{s}$ two sinusoidal signals. Then we run the LR procedure,
including the explanatory variables that correspond to these probe signals. By
looking at the results, in particular at the precision of the estimated probe
parameters, we are able to calculate the signal to noise ratio and the
sensitivity of the method at the frequency of the sinusoidal signals. In
particular, the amplitude of the probes is $2.6\cdot 10^{-7}$ Hz
(corresponding to the magnitude of the Lense-Thirring effect at the GINGERINO
site), whereas their periods are 40 days and 0.5 days, respectively. They are
recovered by the LR analysis with a signal to noise ratio of $\sim 500$ and
$\sim 1000$, respectively, corresponding approximately to noise floors of 0.5
nHz and 0.3 nHz in units of frequency (i.e. 0.1 frad/s and 0.05 frad/sec in
angular velocity), with a bandwidth corresponding to a 600 s integration time.
### 3.2 Correlation between $\omega_{loc}$ and $\zeta_{1,2}$
The analysis shows a strong correlation between RLG and tiltmeter signals.
Note that GINGERINO is a HL mechanical device, and its mirrors are not rigidly
connected to the monument because there are mechanical levers used to align
the square cavity. Those levers are not fixed and, in case of monument tilts,
the mechanical components will have different equilibrium configurations to
compensate gravitational force. Since the mechanical parts are connected to
each other, the whole effect is an effective rotation of the laser cavity.
From $\omega_{loc}$, it is possible to estimate the effective phase $\phi$ of
the rotation. It has been straightforward to see a linear relation between the
effective inclination of the monument and the reconstructed $\phi$ obtained by
time integration of $\omega_{loc}$, the effective rotation of the device with
respect to the ground. This is a clear indication that the GINGERINO cavity
rotates when the granite table changes its orientation [61]. The correlation
is not always linear, indicating that the HL mechanical cavity has a complex
behaviour.
## 4 Close look to the main features of the Earth rotation rate
Known geophysical signals provide a real and effective playground for the
analysis of GINGERINO data. The Earth rotation rate contains several important
features, as LoD effects, Earth normal modes and deformations induced by
tides. Accordingly $\Omega_{IERS}$ is the sum of different contributions,
which can or not be taken into account in the analysis. Comparing the results
obtained with different contributions to $\Omega_{{}_{IERS}}$ it is possible
to isolate each contribution by subtracting the different $\Omega_{geo}$
elaborated by the different input models. In this way the analysis has been
already able to reproduce LoD effects by providing the term $\Delta\omega_{3}$
[16]. This procedure has been repeated to evaluate LoD and the variations
produced by ocean tides contributions, obtaining always results in agreement
with the expectations, with a discrepancy even smaller than 1 part in
$10^{3}$, consistent with the sensitivity of the apparatus, as shown for
example in Fig. 4.
### 4.1 Effects of tides and Earth normal modes
Since Earth is not a rigid body, tides and normal modes of the Earth induce
crust deformations. As a consequence, angular rotations of the GINGERINO site
occur; in the following we focus on the effects due to tides and normal modes
of the Earth. The angular rotation signal provided by IERS contains global
signals, as polar motions, but also local signals caused by the deformation of
the crust induced by tides or ocean loading. Figure 5 compares the Amplitude
Spectral Density (ASD) of $\omega_{IERS}$ and $\omega_{geo}$; the semi diurnal
peak, at a frequency slightly above $2\times 10^{-5}$ Hz, is caused by the
solid Earth tide, $\omega_{geo}$ and $\omega_{IERS}$ are in good agreement
each other in that frequency region.
Figure 5: Comparison of the ASD of $\omega_{IERS}$ and $\omega_{geo}$: the
main features due to polar motion and to solid Earth tide are clearly visible
and well in agreement in both frequency and amplitude. Smaller and very tiny
resonances at higher frequency are well visible.
Deformations associated with Earth normal modes have a typical frequency above
$0.3$ mHz (corresponding to a period below 53.9 prime minutes). Figure 6
reports an expanded view of the ASD in the relevant frequency range. As
demonstrated by the comparison of predictions and analysis results, very small
and tiny peaks due to deformations are well reproduced, although peak
amplitude is systematically smaller by more than $10\%$ with respect to the
predictions based on IERS data.
Figure 6: Detail of the rotation associated with Earth normal modes. The
agreement with IERS expectation shows equal frequency, while the amplitude of
the different components is systematically smaller by more than $10\%$. In the
plot the positions of several well known resonances are qualitatively
indicated for comparison.
Owing to the small amplitude of the signals relating to normal modes, much
smaller than typical disturbances of the apparatus, at the present stage of
the analysis we cannot definitely conclude that the instrumental sensitivity
is large enough to truly reconstruct their occurrence in the considered
frequency range. In general the agreement is very good in the semi-diurnal
signal, corresponding to a larger amplitude, while normal modes are found
systematically smaller in amplitude.
### 4.2 Attempts to evaluate the rotation caused by deformation
Further analysis has been done on this important subject, in order to recover
the contribution of the diurnal and semidiurnal variations produced by ocean
tides, which will be called $\Omega_{def}$ in angular velocity and as
$\omega_{def}$ in the relative effect on the Sagnac signal. The analysis
pipeline has been repeated using $\Omega^{\prime}_{IERS}$, i.e. the expected
global signal without the tide effects $\Omega_{def}$, as it can be obtained
with the available online tools. For this analysis the June-July 2018 data
set, 30 days, has been used, since at that time the temperature variations
were a factor 5 smaller than for the 2020 data set, and the corresponding
standard deviation of $\omega_{loc}$ was 4 times smaller. The estimation of
$\omega_{def}$ is done subtracting the global and local signals estimated in
the two different analysis. In general, $\Omega_{geo}-\Omega^{\prime}_{geo}$
is equal to $\Omega_{def}$ at the level of tens of frad/s, while the
difference $\omega^{\prime}_{loc}-\omega_{loc}$ is close to $\omega_{def}$
with a noise of fractions of $\mu$Hz. Figure 7 shows the amplitude spectral
density of $\omega_{def}$ and $\omega^{\prime}_{loc}-\omega_{loc}$. In this
case, the Earth normal modes are below the noise spectral density.
Figure 7: ASD of $\Omega_{def}$ as expected by IERS and evaluated with the
present analysis.
## 5 Discussions and findings
The analysis is based on the hypothesis that the Sagnac frequency $\omega_{s}$
is the sum of different components: laser disturbances $\omega_{LD}$, geodesic
global signals $\omega_{geo}$ (to be compared to the IERS measurement
$\omega_{{}_{IERS}}$, which is used in the model) and local disturbances
$\omega_{loc}$. The model attempts to estimate $\omega_{loc}$ as the sum of
terms independently evaluated using the environmental signals as explanatory
variables, in particular temperature, tilts and mono beam amplitudes.
GINGERINO is affected by many disturbances, but since their amplitudes are
very small, it is taken for granted that they can be eliminated from
$\omega_{s}$ with the linear regression, assuming linear (or second order)
expansion of the transfer functions of the environmental signals. It is worth
noticing that the $\omega_{LD}$ contribution is comparable to $\omega_{loc}$
and cannot be neglected, or in other words the LR with environmental signals
is not able to identify the local disturbances. It is crucial to identify and
subtract $\omega_{LD}$, since it prevents the estimation of the angular
velocity of the apparatus by means of a linear analysis approach, owing to the
non linearities of the laser dynamics and the relevance of their effects. The
described interpretation of GINGERINO data is confirmed by the comparison with
the monolithic RLG of the Wettzell Observatory G, which is not affected by
large disturbances of instrumental origin, and low frequency angular rotations
around the vertical axis of geophysical origin[60].
The general outcome of our analysis is that $\omega_{s}$ is dominated by local
disturbances, which can be eliminated below the 10 nHz noise level in units of
frequency by using the tiltmeter data. A sensitivity of the order of the nHz,
or even higher, can be reported. However, the method relies on the accuracy of
the explanatory variables and it cannot be considered predictive. In fact, we
have the convergence of LR analysis (with larger residuals), even if we
perturb some explanatory variables (for instance by slightly changing the
Earth rotation model).
The analysis shows that the sensitivity of the apparatus is orders of
magnitude better than expected [1]. Sensitivity is the key point for any
fundamental physics application of RLGs, and this discrepancy with the
expected noise level will deserve further investigation from theoretical side.
On the one hand, it will be necessary to more carefully check the analysis
procedure, in particular with more extended data sets, in order to verify
whether part of the rotation signal is cancelled out. On the other hand, the
development of a full Monte Carlo simulation of GINGERINO would create
complete data sets, containing the laser dynamics and known angular rotation
signals, to be subsequently reconstructed using the reported LR analysis.
### 5.1 Lesson learned on GR and Lorentz violation tests
So far the measurements of the Lense–Thirring effect have been done by space
experiments, using the gravity map of the Earth independently measured by the
GRACE mission, and providing latitude averaged measurements [62, 63, 64, 65].
The promise of the GINGER project is the direct and local measurement of the
main GR features of the rotating Earth, namely the de Sitter and Lense-
Thirring effects. The sensitivity study indicates the feasibility of Lense-
Thirring tests at the $0.1\%$ level, a factor 10 improvement with respect to
the first GINGER proposal. Such a sensitivity level is also an important goal
to discriminate among different theories of gravitation [66]. For the GR
tests, high accuracy is necessary, and the cross calibration has to be
replaced by independent measurements of the scale factor. This is feasible
with accurate measurements of the geometrical scale factor and its electronic
control [53]. At the same time, the inclination angle $\theta$ has to be
evaluated independently. The scheme proposed in the GINGER project is to
evaluate the relative angle using the RLG of the array oriented at the maximum
Sagnac signal (area vector parallel to the north pole direction), which, being
sensitive only at second order to local tilts, can be used as reference to
evaluate the inclination angles with respect to the rotation axis of the other
RLGs of the array. This is a crucial issue, since in this way the measurement
of the inclination angle is limited only by the RLG noise. We note that the
use of different schemes based on external metrology systems is in principle
feasible. However, those external systems have to ensure continuous operation
and angle accuracy at the prad level at least, otherwise the final sensitivity
of the RLG array will be limited by the relative angle measurements.
Improvements to the HL mechanical scheme are suggested by our analysis, since
it shows that local disturbances are mostly of an instrumental origin. This is
also an important point for the capability of the analysis to determine
differences with $\Omega_{{}_{IERS}}$. At present, local disturbances are
hundreds of times larger than the signals we are looking for. Mechanical
deformations and uncontrolled rocking of the HL set-up have to be reduced by
improving the mechanical scheme. To this aim, specific tests can be carried
out to measure the positions of the mirror holders with respect to the support
structure as a function of changes of inclination and temperature. Long term
thermal stability is also of paramount importance, see [61] for details.
Certainly the optimal experimental setup for testing gravitational theories is
to operate two or more RLG arrays in separate sites. This setup is also
advantageous for the other applications of RLGs in geophysics and geodesy, as
enabled by the cross-disciplinary nature of the experiment.
The same remarks hold for high-sensitivity Lorentz violation tests with the
advantage that the expected effect is modulated in time. Also in this case,
the study of residuals $\Delta M_{LR}$ will provide hints for any new physics
not modeled in the LR analysis [7].
## 6 Conclusion
GINGERINO is a top sensitivity RLG, running far from external disturbances and
protected from large thermal excursions in the deep underground environment.
Its data are compared with the global signal of the Earth rotation provided by
IERS. The laser dynamics induces fluctuations of the Sagnac signals, which
pose severe limitations to rotation sensitivity, due to their non linear
nature. The analysis takes into account and eliminates the nonlinear laser
disturbances and recovers the global IERS signal with all its features. At the
same time disturbances of an instrumental and local origin are estimated. The
obtained residuals, i.e. the unmodeled part of the Sagnac signal, with a
standard deviation of the order of a few nHz, indicate a rotational
sensitivity below the frad/s level. Disturbances are mainly of an instrumental
origin, suggesting the need for improvements in the mechanical design. By
injecting probe signals in the GINGERINO data, we conclude that the
sensitivity is $0.1$ frad/s with 600 s integration time, more than a factor
one hundred below the expected noise for this class of instruments. In the
near future it will be necessary to investigate this aspect from a theoretical
side, and it will be necessary to develop a detailed Monte Carlo study,
including the non linear dynamics of the laser, in order to carefully
investigate the analysis procedure. Once again we remark that GR tests require
the RLG array of the GINGER project which, in principle, can provide
independent measurements of the scale factors and relative angles[49].
Work is in progress to apply this analysis scheme to the RLG of the geodesic
observatory of Wettzell, to verify whether it is possible to remove laser
dynamics fluctuations and improve its sensitivity.
## Acknowledgments
We thank the Gran Sasso staff in support of the experiments, particularly
Stefano Gazzana, Nazzareno Taborgna and Stefano Stalio . We are thankful for
technical assistance to Alessio Sardelli and Alessandro Soldani of INFN
Sezione di Pisa and Francesco Francesconi of Dipartimento di Fisica. A special
thank to Gaetano De Luca of Istituto Nazionale di Geofisica e Vulcanologia for
regularly checking Gingerino operation status.
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|
11institutetext: University of Bologna / INRIA FoCUS Team, Bologna, Italy
22institutetext: Royal Holloway, University of London, Egham, UK
# Fair Refinement for Asynchronous Session Types††thanks: Research partly
supported by the H2020-MSCA-RISE project ID 778233 “Behavioural Application
Program Interfaces (BEHAPI)”.
(extended version)
Mario Bravetti 11 Julien Lange 22 Gianluigi Zavattaro 11
###### Abstract
Session types are widely used as abstractions of asynchronous message passing
systems. Refinement for such abstractions is crucial as it allows improvements
of a given component without compromising its compatibility with the rest of
the system. In the context of session types, the most general notion of
refinement is the asynchronous session subtyping, which allows to anticipate
message emissions but only under certain conditions. In particular,
asynchronous session subtyping rules out candidates subtypes that occur
naturally in communication protocols where, e.g., two parties simultaneously
send each other a finite but unspecified amount of messages before removing
them from their respective buffers.
To address this shortcoming, we study fair compliance over asynchronous
session types and fair refinement as the relation that preserves it. This
allows us to propose a novel variant of session subtyping that leverages the
notion of controllability from service contract theory and that is a sound
characterisation of fair refinement. In addition, we show that both fair
refinement and our novel subtyping are undecidable. We also present a sound
algorithm, and its implementation, which deals with examples that feature
potentially unbounded buffering.
###### Keywords:
Session types Asynchronous communication Subtyping.
## 1 Introduction
The coordination of software components via message-passing techniques is
becoming increasingly popular in modern programming languages and development
methodologies based on actors and microservices, e.g., Rust, Go, and the
Twelve-Factor App methodology [twelvefactor]. Often the communication between
two concurrent or distributed components takes place over point-to-point fifo
channels.
Abstract models such as communicating finite-state machines [BZ83] and
asynchronous session types [HYC16] are essential to reason about the
correctness of such systems in a rigorous way. In particular these models are
important to reason about mathematically grounded techniques to improve
concurrent and distributed systems in a compositional way. The key question is
whether a component can be _refined_ independently of the others, without
compromising the correctness of the whole system. In the theory of session
types, the most general notion of refinement is the asynchronous session
subtyping [ESOP09, CDY2014, MariangiolaPreciness], which leverages asynchrony
by allowing the refined component to anticipate message emissions, but only
under certain conditions. Notably asynchronous session subtyping rules out
candidate subtypes that occur naturally in communication protocols where,
e.g., two parties simultaneously send each other a finite but unspecified
amount of messages before removing them from their buffers.
We illustrate this key limitation of asynchronous session subtyping with
Figure 1, which depicts possible communication protocols between a spacecraft
and a ground station. For convenience, the protocols are represented as
session types (bottom) and equivalent communicating finite-state machines
(top). Consider $T_{S}$ and $T_{G}$ first. Session type $T_{S}$ is the
abstraction of the spacecraft. It may send a finite but unspecified number of
telemetries ($\mathit{tm}$), followed by a message $\mathit{over}$ — this
phase of the protocol typically models a for loop and its exit. In the second
phase, the spacecraft receives a number of telecommands ($\mathit{tc}$),
followed by a message $\mathit{done}$. Session type $T_{G}$ is the abstraction
of the ground station. It is the _dual_ of $T_{S}$, written
$\overline{T_{S}}$, as required in standard binary session types without
subtyping. Since $T_{G}$ and $T_{S}$ are dual of each other, the theory of
session types guarantees that they form a _correct composition_ , namely both
parties terminate successfully, with empty queues.
However, it is clear that this protocol is not efficient: the communication is
half-duplex, i.e., it is never the case that more than one party is sending at
any given time. Using full-duplex communication is crucial in distributed
systems with intermittent connectivity, e.g., in this case ground stations are
not always visible from low orbit satellites.
The abstraction of a more efficient ground station is given by type
$T^{\prime}_{G}$, which sends telecommands before receiving telemetries. It is
clear that $T^{\prime}_{G}$ and $T_{S}$ forms a correct composition.
Unfortunately $T^{\prime}_{G}$ is not an asynchronous subtype of $T_{G}$
according to earlier definitions of session subtyping [ESOP09,
MariangiolaPreciness, CDY2014]. Hence they cannot formally guarantee that
$T^{\prime}_{G}$ is a safe replacement for $T_{G}$. Concretely, these
subtyping relations allow for anticipation of emissions (output) only when
they are preceded by a _bounded_ number of receptions (input), but this does
not hold between $T^{\prime}_{G}$ and $T_{G}$ because the latter starts with a
loop of inputs. Note that the composition of $T^{\prime}_{G}$ and $T_{S}$ is
not existentially bounded, hence it cannot be verified by related
communicating finite-state machines techniques [LangeY19, BouajjaniEJQ18,
GenestKM06, GenestKM07].
$0$$1$$2$$!{\mathit{\mathit{tc}}}$$!{\mathit{\mathit{done}}}$$?\mathit{\mathit{tm}}$$?\mathit{\mathit{over}}$ | $0$$1$$2$$?\mathit{\mathit{tm}}$$?\mathit{\mathit{over}}$$!{\mathit{\mathit{tc}}}$$!{\mathit{\mathit{done}}}$ | $0$$1$$2$$!{\mathit{\mathit{tm}}}$$!{\mathit{\mathit{over}}}$$?\mathit{\mathit{tc}}$$?\mathit{\mathit{done}}$
---|---|---
$T^{\prime}_{G}$ | $T_{G}=\overline{T_{S}}$ | $T_{S}$
$T^{\prime}_{G}$ | = | $\mu\mathbf{t}.\oplus\\{\mathit{tc}:\mathbf{t},\mathit{done}:\mu\mathbf{t^{\prime}}.~{}\&\\{\mathit{tm}:\mathbf{t^{\prime}},\mathit{over}:\mathbf{end}\\}\\}$
---|---|---
$T_{G}$ | = | $\mu\mathbf{t}.~{}\&\\{\mathit{tm}:\mathbf{t},\mathit{over}:\mu\mathbf{t^{\prime}}.\oplus\\{\mathit{tc}:\mathbf{t^{\prime}},\mathit{done}:\mathbf{end}\\}\\}$
$T_{S}$ | = | $\mu\mathbf{t}.\oplus\\{\mathit{tm}:\mathbf{t},\mathit{over}:\mu\mathbf{t^{\prime}}.~{}\&\\{\mathit{tc}:\mathbf{t^{\prime}},\mathit{done}:\mathbf{end}\\}\\}$
Figure 1: Satellite protocols. $T^{\prime}_{G}$ is the refined session type of
the ground station, $T_{G}$ is the session type of ground station, and $T_{S}$
is the session type of the spacecraft.
In this paper we address this limitation of previous asynchronous session
subtyping relations. To do this, we move to an alternative notion of correct
composition. In [MariangiolaPreciness] the authors show that their subtyping
relation is fully abstract w.r.t. the notion of _orphan-message-free_
composition. More precisely, it captures exactly a notion of refinement that
preserves the possibility for all sent messages to be consumed along _all_
possible computations of the receiver. In the spacecraft example, given the
initial loop of outputs in $T^{\prime}_{G}$, there is an extreme case in which
it performs infinitely many outputs without consuming any incoming messages.
Nevertheless, this limit case cannot occur under the natural assumption that
the loop of outputs eventually terminates, i.e., only a finite (but
unspecified) amount of messages can be emitted.
The notion of correct composition that we use is based on _fair_ compliance,
which requires each component to always be able to eventually reach a
successful final state. This is a liveness property, holding under _full
fairness_ [GlabbeekH19], used also in the theory of should testing [RV07]
where “every reachable state is required to be on a path to success”. This is
a natural constraint since even programs that conceptually run indefinitely
must account for graceful termination (e.g., to release acquired resources).
Previously, fair compliance has been considered to reason formally about
component/service composition with _synchronous_ session types [Padovani16]
and _synchronous_ behavioural contracts [BravettiZ08]. A preliminary
formalisation of fair compliance for _asynchronous_ behavioural contracts was
presented in [wsfm08], but considering an operational model very different
from session types.
Given a notion of fair compliance defined on an operational model for
asynchronous session types, we define _fair refinement_ as the relation that
preserves it. Then, we propose a novel variant of session subtyping called
_fair asynchronous session subtyping_ , that leverages the notion of
controllability from service contract theory, and which is a sound
characterisation of fair refinement. We show that both fair refinement and
fair asynchronous session subtyping are undecidable, but give a sound
algorithm for the latter. Our algorithm covers session types that exhibit
complex behaviours (including the spacecraft example and variants). Our
algorithm has been implemented in a tool available online [tool].
#### Structure of the paper
The rest of this paper is structured as follows. In § 2 we recall syntax and
semantics of asynchronous session types, we define _fair compliance_ and the
corresponding _fair refinement_. In § 3 we introduce _fair asynchronous
subtyping_ , the first relation of its kind to deal with examples such as
those in Figure 1. In § 4 we propose a sound algorithm for subtyping that
supports examples with unbounded accumulations, including the ones discussed
in this paper. In § 5 we discuss the implementation of this algorithm.
Finally, in § 6 we discuss related works and future work. We give proofs for
all our results and examples of output from our tool in the appendix.
## 2 Refinement for Asynchronous Session Types
In this section we first recall the syntax of two-party session types, their
reduction semantics, and a notion of compliance centered on the successful
termination of interactions. We define our notion of refinement based on this
compliance and show that it is generally undecidable whether a type is a
refinement of another.
### 2.1 Preliminaries: Asynchronous Session Types
#### Syntax
The formal syntax of two-party session types is given below. We follow the
simplified notation used in, e.g., [BravettiCZ17, BCZ18], without dedicated
constructs for sending an output/receiving an input. Additionally we abstract
away from message payloads since they are orthogonal to the results of this
paper.
###### Definition 1 (Session Types).
Given a set of labels $\mathcal{L}$, ranged over by $l$, the syntax of two-
party session types is given by the following grammar:
$\begin{array}[]{lrl}T&::=&\ \ \oplus\\{{l}_{i}:{T}_{i}\\}_{i\in
I}\quad\mid\quad\&\\{{l}_{i}:{T}_{i}\\}_{i\in
I}\quad\mid\quad\mu\mathbf{t}.T\quad\mid\quad\mathbf{t}\quad\mid\quad\mathbf{end}\end{array}$
Output selection $\oplus\\{{l}_{i}:{T}_{i}\\}_{i\in I}$ represents a guarded
internal choice, specifying that a label $l_{i}$ is sent over a channel, then
continuation $T_{i}$ is executed. Input branching
$\&\\{{l}_{i}:{T}_{i}\\}_{i\in I}$ represents a guarded external choice,
specifying a protocol that waits for messages. If message $l_{i}$ is received,
continuation $T_{i}$ takes place. In selections and branchings each branch is
tagged by a label $l_{i}$, taken from a global set of labels $\mathcal{L}$. In
each selection/branching, these labels are assumed to be pairwise distinct. In
the sequel, we leave implicit the index set $i\in I$ in input branchings and
output selections when it is clear from the context. Types $\mu\mathbf{t}.T$
and $\mathbf{t}$ denote standard recursion constructs. We assume recursion to
be guarded in session types, i.e., in $\mu\mathbf{t}.T$, the recursion
variable $\mathbf{t}$ occurs within the scope of a selection or branching.
Session types are closed, i.e., all recursion variables $\mathbf{t}$ occur
under the scope of a corresponding binder $\mu\mathbf{t}.T$. Terms of the
session syntax that are not closed are dubbed (session) terms. Type
$\mathbf{end}$ denotes the end of the interactions.
The dual of session type $T$, written $\overline{T}$, is inductively defined
as follows: $\overline{\oplus\\{{l}_{i}:{T}_{i}\\}_{i\in
I}}=\&\\{{l}_{i}:{\overline{T}}_{i}\\}_{i\in I}$,
$\overline{\&\\{{l}_{i}:{T}_{i}\\}_{i\in
I}}=\oplus\\{{l}_{i}:{\overline{T}}_{i}\\}_{i\in I}$,
$\overline{\mathbf{end}}=\mathbf{end}$, $\overline{\mathbf{t}}=\mathbf{t}$,
and $\overline{\mu\mathbf{t}.T}=\mu\mathbf{t}.\overline{T}$.
#### Operational characterisation
Hereafter, we let $\omega$ range over words in $\mathcal{L}^{\ast}$, write
$\epsilon$ for the empty word, and write $\omega_{1}\\!\cdot\\!\omega_{2}$ for
the concatenation of words $\omega_{1}$ and $\omega_{2}$, where each word may
contain zero or more labels. Also, we write
$T\\{\nicefrac{{T^{\prime}}}{{\mathbf{t}}}\\}$ for $T$ where every free
occurrence of $\mathbf{t}$ is replaced by $T^{\prime}$.
We give an asynchronous semantics of session types via transition systems
whose states are configurations of the form:
$[T_{1},\omega_{1}]|[T_{2},\omega_{2}]$ where $T_{1}$ and $T_{2}$ are session
types equipped with two sequences $\omega_{1}$ and $\omega_{2}$ of incoming
messages (representing unbounded buffers). We use $s$, $s^{\prime}$, etc. to
range over configurations.
In this paper, we use explicit unfoldings of session types, as defined below.
###### Definition 2 (Unfolding).
Given session type $T$, we define $\mathsf{unfold}(T)$:
$\mathsf{unfold}(T)=\begin{cases}\mathsf{unfold}(T^{\prime}\\{\nicefrac{{T}}{{\mathbf{t}}}\\})&\text{if
$T=\mu\mathbf{t}.T^{\prime}$}\\\ T&\text{otherwise}\end{cases}$
Definition 2 is standard, e.g., an equivalent function is used in the first
session subtyping [GH05]. Notice that $\mathsf{unfold}(T)$ unfolds all the
recursive definitions in front of $T$, and it is well defined for session
types with guarded recursion.
###### Definition 3 (Transition Relation).
The transition relation $\rightarrow$ over configurations is the minimal
relation satisfying the rules below (plus symmetric ones):
1. 1.
if $j\in I$ then $[\oplus\\{{l}_{i}:{T}_{i}\\}_{i\in
I},\omega_{1}]|[T_{2},\omega_{2}]\rightarrow[T_{j},\omega_{1}]|[T_{2},\omega_{2}\\!\cdot\\!l_{j}]$;
2. 2.
if $j\in I$ then $[\&\\{{l}_{i}:{T}_{i}\\}_{i\in
I},l_{j}\\!\cdot\\!\omega_{1}]|[T_{2},\omega_{2}]\rightarrow[T_{j},\omega_{1}]|[T_{2},\omega_{2}]$;
3. 3.
if $[\mathsf{unfold}(T_{1}),\omega_{1}]|[T_{2},\omega_{2}]\rightarrow s$ then
$[T_{1},\omega_{1}]|[T_{2},\omega_{2}]\rightarrow s$.
We write $\rightarrow^{*}$ for the reflexive and transitive closure of the
$\rightarrow$ relation.
Intuitively a configuration $s$ reduces to configuration $s^{\prime}$ when
either (1) a type outputs a message $l_{j}$, which is added at the end of its
partner’s queue; (2) a type consumes an expected message $l_{j}$ from the head
of its queue; or (3) the unfolding of a type can execute one of the
transitions above.
Next, we define successful configurations as those configurations where both
types have terminated (reaching $\mathbf{end}$) and both queues are empty. We
use this to give our definition of compliance which holds when it is possible
to reach a successful configuration from all reachable configurations.
###### Definition 4 (Successful Configuration).
The notion of successful configuration is formalised by a predicate $s\surd$
defined as follows:
$[T,\omega_{T}]|[S,\omega_{S}]\surd\;\;\mbox{iff}\;\;\mathsf{unfold}(T)\\!=\\!\mathsf{unfold}(S)\\!=\\!\mathbf{end}\
\text{ and }\ \omega_{T}\\!=\\!\omega_{S}\\!=\\!\epsilon$
###### Definition 5 (Compliance).
Given a configuration $s$ we say that it is a correct composition if, whenever
$s\rightarrow^{*}s^{\prime}$, there exists a configuration $s^{\prime\prime}$
such that $s^{\prime}\rightarrow^{\ast}s^{\prime\prime}$ and
$s^{\prime\prime}\surd$.
Two session types $T$ and $S$ are _compliant_ if $[T,\epsilon]|[S,\epsilon]$
is a correct composition.
Observe that our definition of compliance is stronger than what is generally
considered in the literature on session types, e.g., [LangeY19, LY17, DY13],
where two types are deemed compliant if all messages that are sent are
eventually received, and each non-terminated type can always eventually make a
move. Compliance is analogous to the notion of _correct session_ in
[Padovani16] but in an asynchronous setting.
A consequence of Definition 5 is that it is generally _not_ the case that a
session type $T$ is compliant with its dual $\overline{T}$, as we show in the
example below.
###### Example 1.
The session type $T=\&\\{l_{1}:\mathbf{end},\
l_{2}:\mu\mathbf{t}.\oplus\\{l_{3}:\mathbf{t}\\}\\}$ and its dual
$\overline{T}=\oplus\\{l_{1}:\mathbf{end},\
l_{2}:\mu\mathbf{t}.\&\\{l_{3}:\mathbf{t}\\}\\}$ are not compliant. Indeed,
when $\overline{T}$ sends label $l_{2}$, the configuration
$[\mathbf{end},\epsilon]|[\mathbf{end},\epsilon]$ is no longer reachable.
### 2.2 Fair Refinement for Asynchronous Session Types
We introduce a notion of refinement that preserves compliance. This follows
previous work done in the context of behavioural contracts [BravettiZ08] and
_synchronous_ multi-party session types [Padovani16]. The key difference with
these works is that we are considering asynchronous communication based on
(unbounded) fifo queues. Asynchrony makes fair refinement undecidable, as we
show below.
###### Definition 6 (Refinement).
A session type $T$ refines $S$, written $T\sqsubseteq S$, if for every
$S^{\prime}$ s.t. $S$ and $S^{\prime}$ are compliant then $T$ and $S^{\prime}$
are also compliant.
In contrast to traditional (synchronous and asynchronous) subtyping for
session types [GH05, MariangiolaPreciness, ESOP09], this refinement is not
covariant on outputs, i.e., it does not always allow a refined type to have
output selections with less labels.111The synchronous subtyping in [GH05]
follows a channel-oriented approach; hence it has the opposite direction and
is contravariant on outputs.
###### Example 2.
Let $T=\mu\mathbf{t}.\oplus\\{l_{1}:\mathbf{t}\\}$ and
$S=\mu\mathbf{t}.\oplus\\{l_{1}:\mathbf{t},\ l_{2}:\mathbf{end}\\}$. We have
that $T$ is a synchronous (and asynchronous) subtype of $S$. However $T$ is
_not_ a refinement of $S$. In particular, the type
$\overline{S}=\mu\mathbf{t}.~{}\&\\{l_{1}:\mathbf{t},\ l_{2}:\mathbf{end}\\}$
is compliant with $S$ but not with $T$, since $T$ does not terminate.
Next, we show that the refinement relation $\sqsubseteq$ is generally
undecidable. The proof of undecidability exploits results from the tradition
of computability theory, i.e., Turing completeness of queue machines. The crux
of the proof is to reduce the problem of checking the reachability of a given
state in a queue machine to the problem of checking the refinement between two
session types.
#### Preliminaries
Below we consider only state reachability in queue machines, and not the
typical notion of the language recognised by a queue machine (see, e.g.,
[BravettiCZ17] for a formalisation of queue machines). Hence, we use a
simplified formalisation, where no input string is considered.
###### Definition 7 (Queue Machine).
A queue machine $M$ is defined by a six-tuple $(Q,\Sigma,\Gamma,\$,s,\delta)$
where:
* •
$Q$ is a finite set of states;
* •
$\Sigma\subset\Gamma$ is a finite set denoting the input alphabet;
* •
$\Gamma$ is a finite set denoting the queue alphabet (ranged over by
$A,B,C,X$);
* •
$\$\in\Gamma-\Sigma$ is the initial queue symbol;
* •
$s\in Q$ is the start state;
* •
$\delta:Q\times\Gamma\rightarrow Q\times\Gamma^{*}$ is the transition function
($\Gamma^{*}$ is the set of sequences of symbols in $\Gamma$).
Considering a queue machine $M=(Q,\Sigma,\Gamma,\$,s,\delta)$, a configuration
of $M$ is an ordered pair $(q,\gamma)$ where $q\in Q$ is its current state and
$\gamma\in\Gamma^{*}$ is the queue. The starting configuration is $(s,\$)$,
composed of the start state $s$ and the initial queue symbol $\$$.
Next, we define the transition relation ($\rightarrow_{M}$), leading a
configuration to another, and the related notion of state reachability.
###### Definition 8 (State Reachability).
Given a queue machine $M\\!\\!=\\!\\!(Q,\Sigma,\Gamma,\$,s,\delta)$, the
transition relation $\rightarrow_{M}$ over configurations $Q\times\Gamma^{*}$
is defined as follows. For $p,q\in Q$, $A\in\Gamma$, and
$\alpha,\gamma\in\Gamma^{*}$, we have
$(p,A\alpha)\rightarrow_{M}(q,\alpha\gamma)$ whenever
$\delta(p,A)=(q,\gamma)$. Let $\rightarrow_{M}^{*}$ be the reflexive and
transitive closure of $\rightarrow_{M}$.
A target state $q_{f}\in Q$ is _reachable_ in $M$ if there is
$\gamma\in\Gamma^{*}$ s.t. $(s,\$)\rightarrow_{M}^{*}(q_{f},\gamma)$.
Since queue machines can deterministically encode Turing machines (see, e.g.,
[BravettiCZ17]), checking state reachability for queue machines is
undecidable.
###### Theorem 2.1 ().
Given a queue machine $M$ and a target state $q_{f}$ it is possible to reduce
the problem of checking the reachability of $q_{f}$ in $M$ to the problem of
checking refinement between two session types.
In the light of the undecidability of reachability in queue machines, we can
conclude that refinement (Definition 6) is also undecidable.
### 2.3 Controllability for Asynchronous Session Types
Given a notion of compliance, controllability amounts to checking the
existence of a compliant partner (see, e.g., [Loh08, Wei08, BZ09a]). In our
setting, a session type is _controllable_ if there exists another session type
with which it is compliant.
Checking for controllability algorithmically is not trivial as it requires to
consider infinitely many potential partners. For the synchronous case, an
algorithmic characterisation was studied in [Padovani16]. In the asynchronous
case, the problem is even harder because each of the infinitely many potential
partners may generate an infinite state computation (due to unbounded
buffers). The main contribution of this subsection is to give an algorithmic
characterisation of controllability in the asynchronous setting. Doing this is
important because controllability is an essential ingredient for defining fair
asynchronous subtyping, see Section 3.
$0$$1$$2$$3$$4$$?\mathit{l_{1}}$$?\mathit{l_{3}}$$?\mathit{l_{2}}$$!{\mathit{l_{4}}}$$!{\mathit{l_{5}}}$$!{\mathit{l_{6}}}$
Figure 2: Example of an uncontrollable session type, see Example 3.
###### Definition 9 (Characterisation of Controllability,
$T\,\mathsf{ctrl}$).
Given a session type $T$, we define the judgement $T\,\mathsf{ok}$ inductively
as follows: end ok
end∈T T{$\nicefrac{{\mathbf{end}}}{{\mathbf{t}}}$} ok μt.T ok
T ok &{l:T} ok
∀i ∈I . T_i ok ⊕{l_i:T_i}_i∈I ok where $\mathbf{end}\in T$ holds if
$\mathbf{end}$ occurs in $T$.
We write $T\,\mathsf{ctrl}$ if there exists $T^{\prime}$ such that ($i$)
$T^{\prime}$ is obtained from $T$ by syntactically replacing every input
prefix $\&\\{{l}_{i}:{T}_{i}\\}_{i\in I}$ occurring in $T$ with a term
$\&\\{{l_{j}}:{T_{j}}\\}$ (with $j\in I$) and ($ii$) $T^{\prime}\,\mathsf{ok}$
holds.
Notice that a type $T$ such that $T\,\mathsf{ctrl}$ is indeed controllable, in
that $\overline{T^{\prime}}$, the dual of type $T^{\prime}$ considered above,
is compliant with $T$ (the predicate $\mathbf{end}\\!\in\\!T$ in the premise
of the rule for recursion guarantees that a successful configuration is always
reachable while looping). Moreover the above definition naturally yields a
simple algorithm that decides whether or not $T\,\mathsf{ctrl}$ holds for a
type $T$, i.e., we first pick a single branch for each input prefix
syntactically occurring in $T$ (there are finitely many of them) and then we
inductively check if $T^{\prime}\,\mathsf{ok}$ holds.
The following theorem shows that the judgement $T\,\mathsf{ctrl}$, as defined
above, precisely characterises controllability (i.e., the existence of a
compliant type).
###### Theorem 2.2 ().
$T\,\mathsf{ctrl}$ holds if and only if there exists a session type $S$ such
that $T$ and $S$ are compliant.
###### Example 3.
Consider the session type
$T=\mu\mathbf{t}.~{}\&\\{l_{1}:\&\\{l_{2}:\oplus\\{l_{4}:\mathbf{end},\
l_{5}:\mu\mathbf{t^{\prime}}.\oplus\\{l_{6}:\mathbf{t^{\prime}}\\}\\},\
l_{3}:\mathbf{t}\\}\\}$. See Figure 2 for a graphical representation of $T$.
$T\,\mathsf{ctrl}$ does _not_ hold because it is not possible to construct a
$T^{\prime}$ as specified in Definition 9 for which $T^{\prime}\,\mathsf{ok}$
holds. By Theorem 2.2, there is no session type $S$ that is compliant with
$T$. Hence $T$ is not controllable.
## 3 Fair Asynchronous Session Subtyping
In this section, we present our novel variant of asynchronous subtyping which
we dub _fair asynchronous subtyping_.
First, we need to define a distinctive notion of unfolding. Function
$\mathsf{selUnfold}(T)$ unfolds type $T$ by replacing recursion variables with
their corresponding definitions only if they are guarded by an output
selection. In the definition, we use the predicate
$\oplus\mathit{g}(\mathbf{t},T)$ which holds if all instances of variable
$\mathbf{t}$ are output selection guarded, i.e., $\mathbf{t}$ occurs free in
$T$ only inside subterms ${\oplus\\{{l}_{i}:{T}_{i}\\}_{i\in I}}$.
###### Definition 10 (Selective Unfolding).
Given a term $T$, we define $\mathsf{selUnfold}(T)=$
$\begin{cases}\oplus\\{{l}_{i}:{T}_{i}\\}_{i\in I}&\text{if
}T={\oplus\\{{l}_{i}:{T}_{i}\\}_{i\in I}}\\\
\&\\{l_{i}:\mathsf{selUnfold}(T_{i})\\}_{i\in I}&\text{if
}T={\&\\{{l}_{i}:{T}_{i}\\}_{i\in I}}\\\
T^{\prime}\\{\nicefrac{{\mu\mathbf{t}.T^{\prime}}}{{\mathbf{t}}}\\}&\text{if
}T={\mu\mathbf{t}.T^{\prime}}\text{,
$\oplus\mathit{g}(\mathbf{t},T^{\prime})$}\\\
\mu\mathbf{t}.\mathsf{selUnfold}(\mathsf{selRepl}(\mathbf{t},\mathbf{\hat{t}},T^{\prime})\\{\nicefrac{{\mu\mathbf{t}.T^{\prime}}}{{\mathbf{\hat{t}}}}\\})\
\mathit{with}\ \mathbf{\hat{t}}\ \mathit{fresh}&\text{if
}T={\mu\mathbf{t}.T^{\prime}}\text{,
$\lnot\oplus\mathit{g}(\mathbf{t},T^{\prime})$}\\\ \mathbf{t}&\text{if
}T={\mathbf{t}}\\\ \mathbf{end}&\text{if }T={\mathbf{end}}\end{cases}$
where, $\mathsf{selRepl}(\mathbf{t},\mathbf{\hat{t}},T^{\prime})$ is obtained
from $T^{\prime}$ by replacing the free occurrences of $\mathbf{t}$ that are
inside a subterm $\oplus\\{{l}_{i}:{S}_{i}\\}_{i\in I}$ of $T^{\prime}$ by
$\mathbf{\hat{t}}$.
###### Example 4.
Consider the type
$T=\mu\mathbf{t}.\&\\{l_{1}:\mathbf{t},\,l_{2}:\oplus\\{l_{3}:\mathbf{t}\\}\\}$,
then we have
$\mathsf{selUnfold}(T)=\mu\mathbf{t}.\&\\{l_{1}:\mathbf{t},\,l_{2}:\oplus\\{l_{3}:\mu\mathbf{t}.~{}\&\\{l_{1}:\mathbf{t},\,l_{2}:\oplus\\{l_{3}:\mathbf{t}\\}\\}\\}\\}$
i.e., the type is only unfolded within output selection sub-terms. Note that
$\mathbf{\hat{t}}$ is used to identify where unfolding must take place, e.g.,
$\mathsf{selRepl}(\mathbf{t},\mathbf{\hat{t}},\&\\{l_{1}:\mathbf{t},\,l_{2}:\oplus\\{l_{3}:\mathbf{t}\\}\\})={\&\\{l_{1}:\mathbf{t},\,l_{2}:\oplus\\{l_{3}:\mathbf{\hat{t}}\\}\\}}$.
The last auxiliary notation required to define our notion of subtyping is that
of _input contexts_ , which are used to record inputs that may be delayed in a
candidate super-type.
###### Definition 11 (Input Context).
An input context $\mathcal{A}$ is a session type with several holes defined by
the syntax:
$\mathcal{A}\ ::=\
\quad[\,]^{k}\quad\mid\qquad\&\\{{l}_{i}:{\mathcal{A}}_{i}\\}_{i\in
I}\quad\mid\qquad\mu\mathbf{t}.{\mathcal{A}}\quad\mid\qquad\mathbf{t}$
where the holes $[\,]^{k}$, with $k\in K$, of an input context $\mathcal{A}$
are assumed to be pairwise distinct. We assume that recursion is guarded,
i.e., in an input context $\mu\mathbf{t}.{\mathcal{A}}$, the recursion
variable $\mathbf{t}$ must occur within a subterm
$\&\\{{l}_{i}:{\mathcal{A}}_{i}\\}_{i\in I}$.
We write $\mathit{holes}(\mathcal{A})$ for the set of hole indices in
$\mathcal{A}$. Given a type $T_{k}$ for each $k\in K$, we write
$\mathcal{A}[{T_{k}}]^{k\in K}$ for the type obtained by filling each hole $k$
in $\mathcal{A}$ with the corresponding $T_{k}$.
In contrast to previous work [MariangiolaPreciness, ESOP09, CDY2014,
BravettiCZ17, sefm19, BCLYZ19], these input contexts may contain recursive
constructs. This is crucial to deal with examples such as Figure 1.
We are now ready to define the fair asynchronous subtyping relation, written
$\operatorname{\leq}$. The rationale behind asynchronous session subtyping is
that under asynchronous communication it is unobservable whether or not an
output is anticipated before an input, as long as this output is executed
along all branches of the candidate super-type. Besides the usage of our new
recursive input contexts the definition of fair asynchronous subtyping differs
from those in [MariangiolaPreciness, ESOP09, CDY2014, BravettiCZ17, sefm19,
BCLYZ19] in that controllability plays a fundamental role: the subtype is not
required to mimic supertype inputs leading to uncontrollable behaviours.
###### Definition 12 (Fair Asynchronous Subtyping, $\operatorname{\leq}$).
A relation $\,\mathcal{R}\\!\\!\;$ on session types is a controllable
subtyping relation whenever
$(T,S)\in\mathcal{R}$ implies:
1. 1.
if $T=\mathbf{end}$ then $\mathsf{unfold}(S)=\mathbf{end}$;
2. 2.
if $T=\mu\mathbf{t}.{T^{\prime}}$ then
$(T^{\prime}\\{\nicefrac{{T}}{{\mathbf{t}}}\\},S)\in\mathcal{R}$;
3. 3.
if $T=\&\\{{l}_{i}:{T}_{i}\\}_{i\in I}$ then
$\mathsf{unfold}(S)=\&\\{{l}_{j}:{S}_{j}\\}_{j\in J}$, $I\supseteq K$, and
$\forall k\in K\ldotp(T_{k},S_{k})\in\mathcal{R}$, where $K=\\{k\in
J\;|\;S_{k}\text{ is controllable}\\}$;
4. 4.
if $T=\oplus\\{{l}_{i}:{T}_{i}\\}_{i\in I}$ then
$\mathsf{selUnfold}(S)=\mathcal{A}[{\oplus\\{{l}_{i}:{S_{k}}_{i}\\}_{i\in
I}}]^{k\in K}$ and $\forall i\in I.\,(T_{i},\mathcal{A}[{{S_{ki}}}]^{k\in
K})\in\mathcal{R}$.
$T$ is a controllable subtype of $S$ if there is a controllable subtyping
relation $\mathcal{R}$ s.t. $(T,S)\,\in\,\mathcal{R}$.
$T$ is a _fair asynchronous subtype_ of $S$, written
$T\,\operatorname{\leq}\,S$, whenever: $S$ controllable implies that $T$ is a
controllable subtype of $S$.
Notice that the top-level check for controllability in the above definition is
consistent with the inner controllability checks performed in Case $(3)$.
#### Subtyping simulation game
Session type $T$ is a fair asynchronous subtype of $S$ if $S$ is not
controllable or if $T$ is a controllable subtype of $S$. Intuitively, the
above co-inductive definition says that it is possible to play a simulation
game between a subtype $T$ and its supertype $S$ as follows. Case (1) says
that if $T$ is the $\mathbf{end}$ type, then $S$ must also be $\mathbf{end}$.
Case (2) says that if $T$ is a recursive definition, then it simply unfolds
this definition while $S$ does not need to reply. Case (3) says that if $T$ is
an input branching, then the sub-terms in $S$ that are controllable can reply
by inputting at most some of the labels $l_{i}$ in the branching
(contravariance of inputs), and the simulation game continues (see Example 5).
Case (4) says that if $T$ is an output selection, then $S$ can reply by
outputting _all_ the labels $l_{i}$ in the selection, possibly after executing
some inputs, after which the simulation game continues. We comment further on
Case (4) with Example 6.
###### Example 5.
Consider $T=\&\\{l_{1}:\mathbf{end},\ l_{2}:\mathbf{end}\\}$ and
$S=\&\\{l_{1}:\mathbf{end},\
l_{3}:\mu\mathbf{t}.\oplus\\{l_{4}:\mathbf{t}\\}\\}$. We have
$T\operatorname{\leq}S$. Once branch $l_{3}$, that is uncontrollable, is
removed from $S$, we can apply contravariance for input branching. We have
$I=\\{1,2\\}\supseteq\\{1\\}=K$ in Definition 12.
###### Example 6.
Consider $T_{G}$ and $T^{\prime}_{G}$ from Figure 1. For the pair
$(T^{\prime}_{G},T_{G})$, we apply Case (4) of Definition 12 for which we
compute
$\mathsf{selUnfold}(T_{G})=\mathcal{A}[\oplus\\{\mathit{tc}:\mu\mathbf{t^{\prime}}.\oplus\\{\mathit{tc}:\mathbf{t^{\prime}},\mathit{done}:\mathbf{end}\\},\mathit{done}:\mathbf{end}\\}]$
with
$\mathcal{A}=\mu\mathbf{t}.\&\\{\mathit{tm}:\mathbf{t},\mathit{over}:[\,]^{1}\\}$.
Observe that $\mathcal{A}$ contains a recursive sub-term, such contexts are
not allowed in previous works [MariangiolaPreciness, ESOP09, CDY2014].
The use of selective unfolding makes it possible to express $T_{G}$ in terms
of a _recursive_ input context $\mathcal{A}$ with holes filled by types (i.e.,
closed terms) that start with an output prefix. Indeed selective unfolding
does not unfold the recursion variable $\mathbf{t}$ (_not_ guarded by an
output selection), which becomes part of the input context $\mathcal{A}$.
Instead it unfolds the recursion variable $\mathbf{t}^{\prime}$ (which is
guarded by an output selection) so that the term that fills the hole, which is
required to start with an output prefix, is a closed term.
Case (4) of Definition 12 requires us to check that the following pairs are in
the relation: ($i$)
$(T^{\prime}_{G},\mathcal{A}[\mu\mathbf{t^{\prime}}.\oplus\\{\mathit{tc}:\mathbf{t^{\prime}},\mathit{done}:\mathbf{end}\\}])$
and ($ii$)
$(\mu\mathbf{t^{\prime}}.~{}\&\\{\mathit{tm}:\mathbf{t^{\prime}},\mathit{over}:\mathbf{end}\\},\mathcal{A}[\mathbf{end}])$.
Observe that
$T_{G}=\mathcal{A}[\mu\mathbf{t^{\prime}}.\oplus\\{\mathit{tc}:\mathbf{t^{\prime}},\mathit{done}:\mathbf{end}\\}]$.
Hence, we have $T^{\prime}_{G}\leq T_{G}$ with
$\mathcal{R}\\!=\\!\left\\{(T^{\prime}_{G},T_{G}),(\mathbf{end},\mathbf{end}),(\mu\mathbf{t^{\prime}}.\&\\{\mathit{tm}\\!:\mathbf{t^{\prime}},\mathit{over}\\!:\mathbf{end}\\},\mu\mathbf{t}.\&\\{\mathit{tm}\\!:\mathbf{t},\mathit{over}\\!:\mathbf{end}\\})\right\\}$
and $\mathcal{R}$ is a controllable subtyping relation.
We show that fair asynchronous subtyping is sound w.r.t. fair refinement. In
fact, fair asynchronous subtyping can be seen as a sound coinductive
characterisation of fair refinement. Namely this result gives an operational
justification to the syntactical definition of fair asynchronous session
subtyping. Note that $\operatorname{\leq}$ is not complete w.r.t.
$\sqsubseteq$, see Example 7.
###### Theorem 3.1 ().
Given two session types $T$ and $S$, if $T\operatorname{\leq}S$ then
$T\sqsubseteq S$.
###### Example 7.
Let $T=\oplus\\{l_{1}:\&\\{l_{3}:\mathbf{end}\\}\\}$ and
$S=\&\\{l_{3}:\\!\oplus\\{l_{1}:\mathbf{end},\ l_{2}:\mathbf{end}\\}\\}$. We
have $T\sqsubseteq S$, but $T$ is not a fair asynchronous subtype of $S$ since
$\\{l_{1}\\}\neq\\{l_{1},l_{2}\\}$, i.e., covariance of outputs is not
allowed.
Unfortunately, fair asynchronous session subtyping is also undecidable. The
proof is similar to the one of undecidability of fair refinement, in
particular we proceed by reduction from the termination problem in queue
machines.
###### Theorem 3.2 ().
Given two session types $T$ and $S$, it is in general undecidable to check
whether $T\operatorname{\leq}S$.
## 4 A Sound Algorithm for Fair Asynchronous Subtyping
We propose an algorithm which soundly verifies whether a session type is a
fair asynchronous subtype of another. The algorithm relies on building a tree
whose nodes are labelled by configurations of the simulation game induced by
Definition 12. The algorithm analyses the tree to identify _witness_ subtrees
which contain input contexts that are growing following a recognisable
pattern.
###### Example 8.
Recall the satellite communication example (Figure 1). The spacecraft with
protocol $T_{S}$ may be a replacement for an older generation of spacecraft
which follows the more complicated protocol $T^{\prime}_{S}$, see Figure 3.
Type $T^{\prime}_{S}$ notably allows the reception of telecommands to be
interleaved with the emission of telemetries. The new spacecraft may safely
replace the old one because $T_{S}\operatorname{\leq}T^{\prime}_{S}$.
However, checking $T_{S}\operatorname{\leq}T^{\prime}_{S}$ leads to an
infinite accumulation of input contexts, hence it requires to consider
infinitely many pairs of session types. E.g., after $T_{S}$ selects the output
label $\mathit{tm}$ twice, the subtyping simulation game considers the pair
$(T_{S},T^{\prime\prime}_{S})$, where also $T^{\prime\prime}_{S}$ is in Figure
3. The pairs generated for this example illustrate a common recognisable
pattern where some branches grow infinitely (the $\mathit{tc}$-branch), while
others stay stable throughout the derivation (the $\mathit{done}$-branch). The
crux of our algorithm is to use a finite parametric characterisation of the
infinitely many pairs occurring in the check of
$T_{S}\operatorname{\leq}T^{\prime}_{S}$.
The _simulation tree_ for $T\operatorname{\leq}S$, written
$\mathit{simtree}(T,S)$, is the labelled tree representing the simulation game
for $T\operatorname{\leq}S$, i.e., $\mathit{simtree}(T,S)$ is a tuple
$(N,n_{0},\twoheadrightarrow,\lambda)$ where $N$ is its set of nodes,
$n_{0}\in N$ is its root, $\twoheadrightarrow$ is its transition function, and
$\lambda$ is its labelling function, such that $\lambda(n_{0})=(S,T)$. We omit
the formal definition of $\twoheadrightarrow$, as it is straightforward from
Definition 12 following the subtyping simulation game discussed after that
definition. We give an example below.
Notice that the simulation tree $\mathit{simtree}(T,S)$ is defined only when
$S$ is controllable, since $T\operatorname{\leq}S$ holds without needing to
play the subtyping simulation game if $S$ is not controllable. We say that a
branch of $\mathit{simtree}(T,S)$ is _successful_ if it is infinite or if it
finishes in a leaf labelled by $(\mathbf{end},\mathbf{end})$. All other
branches are _unsuccessful_. Under the assumption that $S$ is controllable, we
have that all branches of $\mathit{simtree}(T,S)$ are successful if and only
if $T\operatorname{\leq}S$. As a consequence checking whether all branches of
$\mathit{simtree}(T,S)$ are successful is generally undecidable. It is
possible to identify a branch as successful if it visits finitely many pairs
(or node labels), see Example 6; but in general a branch may generate
infinitely many pairs, see Examples 8 and 12.
$0$$1$$2$$3$$4$$5$$?\mathit{\mathit{tc}}$$?\mathit{\mathit{done}}$$!{\mathit{\mathit{tm}}}$$!{\mathit{\mathit{over}}}$$?\mathit{\mathit{tc}}$$?\mathit{\mathit{done}}$$!{\mathit{\mathit{tm}}}$$!{\mathit{\mathit{over}}}$
---
| | $T^{\prime}_{S}$ | = | $\mu\mathbf{t}$ | $.\&\big{\\{}$ | $\mathit{tc}:$ | $\oplus\\{\mathit{tm}:\mathbf{t},\mathit{over}:\mu\mathbf{t^{\prime}}.~{}\&\\{\mathit{tc}:\mathbf{t^{\prime}},\mathit{done}:\mathbf{end}\\}\\},$
---|---|---|---|---|---
| | | | $\mathit{done}:$ | $\mu\mathbf{t^{\prime\prime}}.\oplus\\{\mathit{tm}:\mathbf{t^{\prime\prime}},\mathit{over}:\mathbf{end}\\}\big{\\}}$
| $T^{\prime\prime}_{S}$ | = | | $\phantom{.}\&\big{\\{}$ | $\mathit{tc}:$ | $\&\\{$ | $\mathit{tc}:$ | $T^{\prime}_{S}$, |
---|---|---|---|---|---|---|---|---
| | | | | | $\mathit{done}:$ | $\mu\mathbf{t^{\prime\prime}}.\oplus\\{\mathit{tm}:\mathbf{t^{\prime\prime}},\mathit{over}:\mathbf{end}\\}$ | $\\}$,
| | | | $\mathit{done}:$ | $\mu\mathbf{t^{\prime\prime}}.\oplus\\{\mathit{tm}:\mathbf{t^{\prime\prime}},\mathit{over}:\mathbf{end}\\}$ | $\big{\\}}$
Figure 3: $T^{\prime}_{S}$ is an alternative session type for $T_{S}$, see
Example 8.
In order to support types that generate unbounded accumulation, we
characterise finite subtrees — called witness subtrees, see Definition 13 —
such that all the branches that traverse these finite subtrees are guaranteed
to be successful.
#### Notation
We give a few auxiliary definitions and notations. Hereafter $\mathcal{A}$ and
$\mathcal{A}^{\prime}$ range over _extended_ input contexts, i.e., input
contexts that may contain distinct holes with the same index. These are needed
to deal with unfoldings of input contexts, see Example 9.
The set of _reductions_ of an input context $\mathcal{A}$ is the minimal set
$\mathcal{S}$ s.t. ($i$) $\mathcal{A}\in\mathcal{S}$; ($ii$) if
$\&\\{l_{i}:\mathcal{A}_{i}\\}_{i\in I}\in\mathcal{S}$ then $\forall i\in
I.\mathcal{A}_{i}\in\mathcal{S}$ and ($iii$) if
$\mu\mathbf{t}.\mathcal{A}^{\prime}\in\mathcal{S}$ then
$\mathcal{A}^{\prime}\\{\nicefrac{{\mu\mathbf{t}.\mathcal{A}^{\prime}}}{{\mathbf{t}}}\\}\in\mathcal{S}$.
Notice that due to unfolding (item ($iii$)), the reductions of an input
context may contain extended input contexts. Moreover, given a reduction
$\mathcal{A}^{\prime}$ of $\mathcal{A}$, we have that
$\mathit{holes}(\mathcal{A}^{\prime})\subseteq\mathit{holes}(\mathcal{A})$.
###### Example 9.
Consider the following extended input contexts: A_1 = μt . &{ l_1 : [ ]^1, l_2
: &{ l_3 : t } }
A_2 = &{ l_3 : μt . &{ l_1 : [ ]^1, l_2 : &{ l_3 : t } } }
unfold(A_1) = &{ l_1 : [ ]^1, l_2 : &{ l_3 : μt . &{ l_1 : [ ]^1, l_2 : &{ l_3
: t } } } } Context $\mathcal{A}_{2}$ is a reduction of $\mathcal{A}_{1}$,
i.e., one can reach $\mathcal{A}_{2}$ from $\mathcal{A}_{1}$, by unfolding
$\mathcal{A}_{1}$ and executing the input $l_{2}$. Context
$\mathsf{unfold}(\mathcal{A}_{1})$ is also a reduction of $\mathcal{A}_{1}$.
Observe that $\mathsf{unfold}(\mathcal{A}_{1})$ contains two distinct holes
indexed by $1$.
Given an extended context $\mathcal{A}$ and a set of hole indices $K$ such
that $K\subseteq\mathit{holes}(\mathcal{A})$, we use the following shorthands.
Given a type $T_{k}$ for each $k\in K$, we write $\mathcal{A}\lfloor
T_{k}\rfloor^{k\in K}$ for the extended context obtained by replacing each
hole $k\in K$ in $\mathcal{A}$ by $T_{k}$. Also, given an extended context
$\mathcal{A}^{\prime}$ we write
$\mathcal{A}\langle\mathcal{A}^{\prime}\rangle^{K}$ for the extended context
obtained by replacing each hole $k\in K$ in $\mathcal{A}$ by
$\mathcal{A}^{\prime}$. When $K=\\{k\\}$, we often omit $K$ and write, e.g.,
$\mathcal{A}\langle\mathcal{A}^{\prime}\rangle^{k}$ and $\mathcal{A}\lfloor
T_{k}\rfloor^{k}$.
$(T_{S},\;\mathcal{A}\lfloor T^{\prime}_{S},T^{\prime}_{1}\rfloor^{\\{1,2\\}})$$(\mu\mathbf{t^{\prime}}.\&\\{\mathit{tc}:\mathbf{t^{\prime}},\mathit{done}:\mathbf{end}\\},\;\mathcal{A}\langle\mathcal{A}\lfloor T^{\prime\prime}_{1}\rfloor^{1}\rangle^{1}\lfloor\mathbf{end}\rfloor^{2})$$(\mu\mathbf{t^{\prime}}.\&\\{\mathit{tc}:\mathbf{t^{\prime}},\mathit{done}:\mathbf{end}\\},\;\mathcal{A}\lfloor T^{\prime\prime}_{1}\rfloor^{1}\lfloor\mathbf{end}\rfloor^{2})$$(\mathbf{end},\;\mathbf{end})$$(T_{S},\;\mathcal{A}\langle\mathcal{A}\lfloor T^{\prime}_{S}\rfloor^{1}\rangle^{1}\lfloor T^{\prime}_{1}\rfloor^{2})$$!{\mathit{\mathit{over}}}$$?\mathit{\mathit{tc}}$$?\mathit{\mathit{done}}$$!{\mathit{\mathit{tm}}}$$(T_{S},\;T^{\prime}_{S})$$!{\mathit{\mathit{tm}}}$$(\mu\mathbf{t^{\prime}}.\&\\{\mathit{tc}:\mathbf{t^{\prime}},\mathit{done}:\mathbf{end}\\},\;\mathcal{A}\lfloor T^{\prime\prime}_{1}\rfloor^{1}\lfloor\mathbf{end}\rfloor^{2})$$(\mathbf{end},\;\mathbf{end})$$(\mu\mathbf{t^{\prime}}.\&\\{\mathit{tc}:\mathbf{t^{\prime}},\mathit{done}:\mathbf{end}\\},\;T^{\prime\prime}_{1})$$(\mathbf{end},\;\mathbf{end})$$(\mu\mathbf{t^{\prime}}.\&\\{\mathit{tc}:\mathbf{t^{\prime}},\mathit{done}:\mathbf{end}\\},\;T^{\prime\prime}_{1})$$!{\mathit{\mathit{over}}}$$?\mathit{\mathit{done}}$$?\mathit{\mathit{tc}}$$?\mathit{\mathit{done}}$$?\mathit{\mathit{tc}}$= $\mathcal{A}$ $=\;$ | $\&\\{\mathit{tc}:[\,]^{1},\mathit{done}:[\,]^{2}\\}$
---|---
$T^{\prime}_{1}$ $=\;$ | $\mu\mathbf{t^{\prime\prime}}.\oplus\\{\mathit{tm}:\mathbf{t^{\prime\prime}},\mathit{over}:\mathbf{end}\\}$
$T^{\prime\prime}_{1}$ $=\;$ | $\mu\mathbf{t^{\prime}}.~{}\&\\{\mathit{tc}:\mathbf{t^{\prime}},\mathit{done}:\mathbf{end}\\}$
Figure 4: Simulation tree for $T_{S}\leq T^{\prime}_{S}$ (Figures 1 and 3),
the root of the tree is in bold.
###### Example 10.
Using the above notation and posing
$\mathcal{A}=\&\\{\mathit{tc}:[\,]^{1},\mathit{done}:[\,]^{2}\\}$, we can
rewrite $T^{\prime\prime}_{S}$ (Figure 3) as
$\mathcal{A}\langle\mathcal{A}\lfloor
T^{\prime}_{S}\rfloor^{1}\rangle^{1}\lfloor\mu\mathbf{t^{\prime\prime}}.\oplus\\{\mathit{tm}:\mathbf{t^{\prime\prime}},\mathit{over}:\mathbf{end}\\}\rfloor^{2}$.
###### Example 11.
Consider the session type below
$S=\&\\{l_{1}:\&\\{l_{1}:T_{1},\ l_{2}:T_{2},\
l_{3}:T_{3}\\},\;l_{2}:\&\\{l_{1}:T_{1},\ l_{2}:T_{2},\
l_{3}:T_{3}\\},\;l_{3}:T_{3}\\}.$
Posing $\mathcal{A}=\&\\{l_{1}:[\,]^{1},l_{2}:[\,]^{2},l_{3}:[\,]^{3}\\}$ we
have $\mathit{holes}(\mathcal{A})=\\{1,2,3\\}$. Assuming $J=\\{1,2\\}$ and
$K=\\{3\\}$, we can rewrite $S$ as $\mathcal{A}\langle\mathcal{A}\lfloor
T_{j}\rfloor^{j\in J}\rangle^{J}\lfloor T_{k}\rfloor^{k\in K}$.
###### Example 12.
Figure 4 shows the partial simulation tree for $T_{S}\leq T^{\prime}_{S}$,
from Figures 1 and 3 (ignore the dashed edges for now). Notice how the branch
leading to the top part of the tree visits only finitely many node labels (see
dotted box), however the bottom part of the tree generates infinitely many
labels, see the path along the $!{\mathit{\mathit{tm}}}$ transitions in the
dashed box.
#### Witness subtrees
Next, we define witness trees which are finite subtrees of a simulation tree
which we prove to be successful. The role of the witness subtree is to
identify branches that satisfy a certain accumulation pattern. It detects an
input context $\mathcal{A}$ whose holes fall in two categories: ($i$) growing
holes (indexed by indices in $J$ below) which lead to an infinite growth and
($ii$) constant holes (indexed by indices in $K$ below) which stay stable
throughout the simulation game. The definition of witness trees relies on the
notion of _ancestor_ of a node $n$, which is a node $n^{\prime}$ (different
from $n$) on the path from the root $n_{0}$ to $n$. We illustrate witness
trees with Figure 4 and Example 13.
###### Definition 13 (Witness Tree).
A tree $(N,n_{0},\twoheadrightarrow,\lambda)$ is a _witness tree_ for
$\mathcal{A}$, such that $\mathit{holes}(\mathcal{A})=I$, with
$\emptyset\subseteq K\subset I$ and $J=I\setminus K$, if all the following
conditions are satisfied:
1. 1.
for all $n\in N$ either
$\lambda(n)=(T,\mathcal{A}^{\prime}\langle\mathcal{A}\lfloor
S_{j}\rfloor^{j\in J}\rangle^{J}\lfloor S_{k}\rfloor^{k\in K})$ or
$\lambda(n)=(T,\mathcal{A}^{\prime}\langle\mathcal{A}\langle\mathcal{A}\lfloor
S_{j}\rfloor^{j\in J}\rangle^{J}\rangle^{J}\lfloor S_{k}\rfloor^{k\in K})$,
where $\mathcal{A}^{\prime}$ is a reduction of $\mathcal{A}$, and it holds
that
* •
$\mathit{holes}(\mathcal{A}^{\prime})\subseteq K$ implies that $n$ is a leaf
and
* •
if $\lambda(n)=(T,\mathcal{A}[S_{i}]^{i\in I})$ and $n$ is not a leaf then
$\mathsf{unfold}(T)$ starts with an output selection;
2. 2.
each leaf $n$ of the tree satisfies one of the following conditions:
1. (a)
$\lambda(n)=(T,S)$ and $n$ has an ancestor $n^{\prime}$ s.t.
$\lambda(n^{\prime})=(T,S)$
2. (b)
$\lambda(n)=(T,\mathcal{A}\langle\mathcal{A}\lfloor S_{j}\rfloor^{j\in
J}\rangle^{J}\lfloor S_{k}\rfloor^{k\in K})$ and $n$ has an ancestor
$n^{\prime}$ s.t. $\lambda(n^{\prime})=(T,\mathcal{A}[S_{i}]^{i\in I})$
3. (c)
$\lambda(n)=(T,\mathcal{A}[S_{i}]^{i\in I})$ and
$n$ has an ancestor $n^{\prime}$ s.t.
$\lambda(n^{\prime})=(T,\mathcal{A}\langle\mathcal{A}\lfloor
S_{j}\rfloor^{j\in J}\rangle^{J}\lfloor S_{k}\rfloor^{k\in K})$
4. (d)
$\lambda(n)=(T,\mathcal{A}^{\prime}[S_{k}]^{k\in K^{\prime}})$ where
$K^{\prime}\subseteq K$
and for all leaves $(T,S)$ of type (2c) or (2d) $T\operatorname{\leq}S$ holds.
Intuitively Condition (1) says that a witness subtree consists of nodes that
are labelled by pairs $(T,S)$ where $S$ contains a fixed context $\mathcal{A}$
(or a reduction/repetition thereof) whose holes are partitioned in growing
holes ($J$) and constant holes ($K$). Whenever all growing holes have been
removed from a pair (by reduction of the context) then this means that the
pair is labelling a leaf of the tree. In addition, if the initial input is
limited to only one instance of $\mathcal{A}$, the l.h.s. type starts with an
output selection so that this input cannot be consumed in the subtyping
simulation game.
Condition 2 says that all leaves of the tree must validate certain conditions
from which we can infer that their continuations in the full simulation tree
lead to successful branches. Leaves satisfying Condition (2a)
straightforwardly lead to successful branches as the subtyping simulation
game, starting from the corresponding pair, has been already checked starting
from its ancestor having the same label. Leaves satisfying Condition (2b) lead
to an infinite but regular “increase” of the types in $J$-indexed holes —
following the same pattern of accumulation from their ancestor. The next two
kinds of leaves must additionally satisfy the subtyping relation — using
witness trees inductively or based on the fact they generate finitely many
labels. Leaves satisfying Condition (2c) lead to regular “decrease” of the
types in $J$-indexed holes — following the same pattern of reduction from
their ancestor. Leaves satisfying Condition (2d) use only constant $K$-indexed
holes because, by reduction of the context $\mathcal{A}^{\prime}$, the growing
holes containing the accumulation $\mathcal{A}$ have been removed.
###### Remark 1.
Definition 13 is parameterised by an input context $\mathcal{A}$. We explain
how such contexts can be identified while building a simulation tree in
Section 5.
###### Example 13.
In the tree of Figure 4 we highlight two subtrees. The subtree in the dotted
box is not a witness subtree because it does not validate Condition (1) of
Definition 13, i.e., there is an intermediary node with a label in which the
r.h.s type does not contain $\mathcal{A}$.
The subtree in the dashed box is a witness subtree with 3 leaves, where the
dashed edges represent the ancestor relation,
$\mathcal{A}=\&\\{\mathit{tc}:[\,]^{1},\mathit{done}:[\,]^{2}\\}$, $J=\\{1\\}$
and $K=\\{2\\}$. We comment on the leaves clockwise, starting from
$(\mathbf{end},\mathbf{end})$, which satisfies Condition (2d). The next leaf
satisfies condition (2c), while the final leaf satisfies Condition (2b).
#### Algorithm
Given two session types $T$ and $S$ we first check whether $S$ is
uncontrollable. If this is the case we immediately conclude that
$T\operatorname{\leq}S$. Otherwise, we proceed in four steps.
1. S1
We compute a finite fragment of $\mathit{simtree}(T,S)$, stopping whenever
($i$) we encounter a leaf (successful or not), ($ii$) we encounter a node that
has an ancestor as defined in Definition 13 (Conditions (2a), (2b), and (2c)),
($iii$) or the length of the path from the root of $\mathit{simtree}(T,S)$ to
the current node exceeds a bound set to two times the depth of the AST of $S$.
This bound allows the algorithm to explore paths that will traverse the super-
type at least twice. We have empirically confirmed that it is sufficient for
all examples mentioned in Section 5.
2. S2
We remove subtrees from the tree produced in S1 corresponding to successful
branches of the simulation game which contain finitely many labels.
Concretely, we remove each subtree whose each leaf $n$ is either successful or
has an ancestor $n^{\prime}$ such that $n^{\prime}$ is in the same subtree and
$\lambda(n)=\lambda(n^{\prime})$.
3. S3
We extract subtrees from the tree produced in S2 that are potential
_candidates_ to be subsequently checked. The extraction of these finite
candidate subtrees is done by identifying the forest of subtrees rooted in
ancestor nodes which do not have ancestors themselves.
4. S4
We check that each of the candidate subtrees from S3 is a witness tree.
If an unsuccessful leaf is found in S1, then the considered session types are
not related. In S1, if the generation of the subtree reached the bound before
reaching an ancestor or a leaf, then the algorithm is unable to give a
decisive verdict, i.e., the result is _unknown_. Otherwise, if all checks in
S4 succeed then the session types are in the fair asynchronous subtyping
relation. In all other cases, the result is _unknown_ because a candidate
subtree is not a witness.
###### Example 14.
We illustrate the algorithm above with the tree in Figure 4. After S1, we
obtain the whole tree in the figure (11 nodes). After S2, all nodes in the
dotted boxed are removed. After S3 we obtain the (unique) candidate subtree
contained in the dashed box. This subtree is identified as a witness subtree
in S4, hence we have $T_{S}\operatorname{\leq}T^{\prime}_{S}$.
We state the main theorem that establishes the soundness of our algorithm,
where $\twoheadrightarrow\\!\\!{}^{*}$ is the reflexive and transitive closure
of $\twoheadrightarrow$.
###### Theorem 4.1 ().
Let $T$ and $S$ be two session types with
$\mathit{simtree}(T,S)=(N,n_{0},\twoheadrightarrow,\lambda)$. If
$\mathit{simtree}(T,S)$ contains a witness subtree with root $n$ then for
every node $n^{\prime}\in N$ s.t.
$n\twoheadrightarrow\\!\\!{}^{*}\,n^{\prime}$, either $n^{\prime}$ is a
successful leaf, or there exists $n^{\prime\prime}$ s.t.
$n^{\prime}\twoheadrightarrow{}n^{\prime\prime}$.
We can conclude that if the candidate subtrees of $\mathit{simtree}(T,S)$
identified with the strategy explained above are also witness subtrees, then
we have $T\operatorname{\leq}S$.
## 5 Implementation
To evaluate our algorithm, we have produced a Haskell implementation of it,
which is available on GitHub [tool]. Our tool takes two session types $T$ and
$S$ as input then applies Steps S1 to S4 to check whether
$T\operatorname{\leq}S$. A user-provided bound can be given as an optional
argument. We have run our tool on a dozen of examples handcrafted to test the
limits of our algorithm (inc. the examples discussed in this paper), as well
as on the 174 tests taken from [BCLYZ19]. All of these tests terminate under a
second.
For debugging and illustration purposes, the tool can optionally generate
graphical representations of the simulation and witness trees, and check
whether the given types are controllable. We give examples of these in the
appendix.
Our tool internally uses automata to represent session types and uses strong
bisimilarity instead of syntactic equality between session types. Using
automata internally helps us identify candidate input contexts as we can keep
track of states that correspond to the input context computed when applying
Case (4) of Definition 12. In particular, we augment each local state in the
automata representation of the candidate supertype with two counters: the
$c$-counter keeps track of how many times a state has been used in an input
context; the $h$-counter keeps track of how many times a state has occurred
within a hole of an input context. We illustrate this with Figure 5 which
illustrates the internal data structures our tool manipulates when checking
$T_{S}\leq T^{\prime}_{S}$ from Figures 1 and 3. The state indices of the
automata in Figure 5 correspond to the ones in Figure 1 (2nd column) and
Figure 3 (3rd column).
The first row of Figure 5 represents the root of the simulation tree, where
both session types are in their respective initial state and no transition has
been executed. We use state labels of the form $n_{c,h}$ where $n$ is the
original identity of the state, $c$ is the value of the $c$-counter, and $h$
is the value of the $h$-counter. The second row depicts the configuration
after firing transition $!{\mathit{\mathit{tm}}}$, via Case (4) of Definition
12. While the candidate subtype remains in state $0$ (due to a self-loop) the
candidate supertype is unfolded with $\mathsf{selUnfold}(T^{\prime}_{S})$
(Definition 10). The resulting automaton contains an additional state and two
transitions. All previously existing states have their $h$-counter
incremented, while the new state has its $c$-counter incremented. The third
row of the figure shows the configuration after firing transition
$!{\mathit{\mathit{over}}}$, using Case (4) of Definition 12 again. In this
step, another copy of state $0$ is added. Its $c$-counter is set to $2$ since
this state has been used in a context twice; and the $h$-counters of all other
states are incremented.
Using this representation, we construct a candidate input context by building
a tree whose root is a state $q_{c,h}$ such that $c>1$. The nodes of the tree
are taken from the states reachable from $q_{c,h}$, stopping when a state
$q^{\prime}_{c^{\prime},h^{\prime}}$ such that $c^{\prime}<c$ is found. A leaf
$q^{\prime}_{c^{\prime},h^{\prime}}$ becomes a hole of the input context. The
hole is a constant ($K$) hole when $h^{\prime}=c$, and growing ($J$)
otherwise. Given this strategy and the configurations in Figure 5, we
successfully identify the context
$\mathcal{A}=\&\\{\mathit{tc}:[\,]^{1},\mathit{done}:[\,]^{2}\\}$ with
$J=\\{1\\}$ and $K=\\{2\\}$.
Last transition | State of $T_{S}$ | Representation of $T^{\prime}_{S}$
---|---|---
$\epsilon$ | $0$ | $0_{0,0}$$1_{0,0}$$2_{0,0}$$3_{0,0}$$4_{0,0}$$5_{0,0}$$?\mathit{\mathit{tc}}$$?\mathit{\mathit{done}}$$!{\mathit{\mathit{tm}}}$$!{\mathit{\mathit{over}}}$$?\mathit{\mathit{tc}}$$?\mathit{\mathit{done}}$$!{\mathit{\mathit{tm}}}$$!{\mathit{\mathit{over}}}$
$!{\mathit{\mathit{tm}}}$ | $0$ | $0_{0,1}$$1_{0,1}$$2_{0,1}$$3_{0,1}$$4_{0,1}$$5_{0,1}$$0_{1,0}$$?\mathit{\mathit{tc}}$$?\mathit{\mathit{done}}$$?\mathit{\mathit{tc}}$$?\mathit{\mathit{done}}$$!{\mathit{\mathit{tm}}}$$!{\mathit{\mathit{over}}}$$?\mathit{\mathit{tc}}$$?\mathit{\mathit{done}}$$!{\mathit{\mathit{tm}}}$$!{\mathit{\mathit{over}}}$
$!{\mathit{\mathit{over}}}$ | $1$ | $0_{0,2}$$1_{0,2}$$2_{0,2}$$3_{0,2}$$4_{0,2}$$5_{0,2}$$0_{1,1}$$0_{2,0}$$?\mathit{\mathit{tc}}$$?\mathit{\mathit{done}}$$?\mathit{\mathit{tc}}$$?\mathit{\mathit{done}}$$?\mathit{\mathit{tc}}$$?\mathit{\mathit{done}}$
Figure 5: Internal representation of the simulation tree for $T_{S}\leq
T^{\prime}_{S}$ (fragment).
## 6 Related and Future Work
#### Related work
We first compare with previous work on refinement for asynchronous
communication by some of the authors of this paper. The work in [wsfm08] also
considers fair compliance, however here we consider binary (instead of
multiparty) communication and we use a unique input queue for all incoming
messages instead of distinct named input channels. Moreover, here we provide a
sound characterisation of fair refinement using coinductive subtyping and
provide a sound algorithm and its implementation. In [sefm19] the asynchronous
subtyping of [MariangiolaPreciness, ESOP09, CDY2014, BravettiCZ17] is used to
characterise refinement for a notion of correct composition based on the
impossibility to reach a deadlock, instead of the possibility to reach a final
successful configuration as done in the present paper. The refinement from
[sefm19] does not support examples such as those in Figure 1.
Concerning previous notions of synchronous subtyping, Gay and Hole [GH05,
GH99] first introduced the notion of subtyping for _synchronous_ session
types, which is decidable in quadractic time [LangeY16]. This subtyping only
supports covariance of outputs and contravariance of inputs, but does not
address anticipation of outputs. Padovani studied a notion of fair subtyping
for _synchronous_ multi-party session types in [Padovani16]. This work notably
considers the notion of _viability_ which corresponds, in the synchronous
multiparty setting, to our notion of controllability. We use the term
controllability instead of viability following the tradition of service
contract theories like those based on Petri nets [Loh08, Wei08] or process
calculi [BZ09a]. In contrast to [Padovani16], asynchronous communication makes
it much more involved to characterise controllability in a decidable way, as
we do in this paper. Fair refinement in [Padovani16] is characterised by
defining a coinductive relation on normal form of types, obtained by removing
inputs leading to uncontrollable continuations. Instead of using normal forms,
we remove these inputs during the asynchronous subtyping check. A limited form
of variance on output is also admitted in [Padovani16]. Covariance between the
outputs of a subtype and those of a supertype is possible when the additional
branches in the supertype are not needed to have compliance with potential
partners. In [Padovani16] this check is made possible by exploiting a
_difference_ operation [Padovani16, Definition 3.15] on types, which
synthesises a new type representing branches of one type that are absent in
the other. We observe that the same approach cannot work to introduce variance
on outputs in an asynchronous setting. Indeed the interplay between output
anticipation and recursion could generate differences in the branches of a
subtype and a supertype that cannot be statically represented by a (finite)
session type.
Padovani also studied an alternative notion of fair _synchronous_ subtyping in
[Padovani13]. Although the contribution of that paper refers to session types,
the formal framework therein seems to deviate from the usual session type
approach. In particular, it considers shared channel communication instead of
binary channels: when a partner emits a message, it is possible to have a race
among several potential receivers for consuming it. As a consequence of this
alternative semantics, the subtyping in [Padovani13] does not admit variance
on input. Another difference with respect to session type literature is the
notion of _success_ among interacting sessions: a composition of session is
successful if at least one participant reaches an internal successful state.
This approach has commonalities with testing [DH84], where only the test
composed with the system under test is expected to succeed, but differs from
the typical notion of success considered for session types. In [Barbanerad10,
BernardiH16] (resp. [MariangiolaPreciness]) it was proved that the Gay-Hole
synchronous session subtyping (resp. orphan message free asynchronous
subtyping) coincides with refinement induced by a successful termination
notion requiring interacting processes to be both in the $\mathbf{end}$ state
(with empty buffers, in the asynchronous case).
Several variants of asynchronous session subtyping have been proposed in
[ESOP09, MariangiolaPreciness, CDY2014] and further studied in our earlier
work [BravettiCZ17, sefm19, BCLYZ19]. All these variants have been shown to be
undecidable [BCZ18, LY17, BravettiCZ17]. Moreover, all these subtyping
relations are (implicitly) based on an unfair notion of compliance.
Concretely, the definition of asynchronous subtyping introduced in this paper
differs from the one in [MariangiolaPreciness, CDY2014] since no additional
constraint guaranteeing absence of orphan-messages is considered. Such a
constraint requires the subtype not to have output loops whenever an output
anticipation is performed, thus guaranteeing that at least one input is
performed in all possible paths. In this paper, absence of orphan messages is
guaranteed by enforcing types to (fairly) reach a successful termination.
Moreover, our novel subtyping differs from those in [MariangiolaPreciness,
CDY2014, ESOP09] since we use recursive input contexts (and not just finite
ones) for the first time — this is necessary to obtain
$T^{\prime}_{G}\operatorname{\leq}T_{G}$ and
$T_{S}\operatorname{\leq}T^{\prime}_{S}$ (see Figures 1 and 3). Notice that
not imposing the above mentioned orphan-message-free constraint of
[MariangiolaPreciness, CDY2014] is consistent with recursive input contexts
that allows for input loops in the supertype whenever an output anticipation
is performed. In [BCLYZ19], we proposed a sound algorithm for the asynchronous
subtyping in [MariangiolaPreciness]. The sound algorithm that we present in
this paper substantially differs from that of [BCLYZ19]. Here we use witness
trees that take under consideration both increasing and decreasing of
accumulated input. In [BCLYZ19], instead, only regular growing accumulation is
considered.
#### Future work
In future work, we will investigate how to support output variance in fair
asynchronous subtyping. We also plan to study fairness in the context of
asynchronous multiparty session types, as fair compliance and refinement
extend naturally to several partners. Finally, we will investigate a more
refined termination condition for our algorithm using ideas from [BCLYZ19,
Definition 11].
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## Appendix 0.A Proofs
### 0.A.1 Undecidability of Fair Refinement
The proof of undecidability of fair refinement is as follows: given a queue
machine $M$, and a target state $q_{f}$, we define two session types $T$ and
$S$ such that $T\sqsubseteq S$ if and only if state $q_{f}$ is reachable in
$M$. Hereafter, we use convenient notations for denoting output selections and
input branchings. Instead of using labels indexed on an indexing set $I$, as
in the input branching syntax $\&\\{{l}_{i}:{T}_{i}\\}_{i\in I}$, we also use
explicitly distinct labels, as in $\&\\{l:T_{l},m:T_{m}\\}$ (we use the same
notation for output selections). We also use the union operator to combine
disjoint sets of labels, for instance, instead of writing
$\oplus\\{l_{k}:T_{k}\\}_{k\in I\cup J}$, we use the notation
$\oplus\\{l_{i}:T_{i}\\}_{i\in I}\cup\\{l_{j}:T_{j}\\}_{j\in J}$ (we use the
same notation for input branchings).
We also make use of the notion of bisimilarity between session types.
###### Definition 14 (Type bisimilarity).
A relation $\,\mathcal{R}\\!\\!\;$ on session types is a bisimulation whenever
$(T,S)\in\mathcal{R}$ implies:
1. 1.
if $T=\mathbf{end}$ then $\mathsf{unfold}(S)=\mathbf{end}$;
2. 2.
if $T=\oplus\\{{l}_{i}:{T}_{i}\\}_{i\in I}$ then
${\mathsf{unfold}(S)}=\oplus\\{{l}_{i}:{S}_{i}\\}_{i\in I}$ with $\forall i\in
I.\,(T_{i},S_{i})\in\mathcal{R}$;
3. 3.
if $T=\&\\{{l}_{i}:{T}_{i}\\}_{i\in I}$ then
${\mathsf{unfold}(S)}=\&\\{{l}_{i}:{S}_{i}\\}_{i\in I}$ with $\forall i\in
I.\,(T_{i},S_{i})\in\mathcal{R}$;
4. 4.
if $T=\mu\mathbf{t}.{T^{\prime}}$ then
$(T^{\prime}\\{T/\mathbf{t}\\},S)\in\mathcal{R}$.
$T$ is bisimilar to $S$, written $T\sim S$, if there is a bisimulation
$\mathcal{R}$ such that $(T,S)\in\mathcal{R}$.
In the following we need a result about bisimilar session types: if $T$ is
compliant with $S$, then it is also compliant with all of its bisimilar types,
namely $T$ is compliant with $S^{\prime}$ for every $S^{\prime}\sim S$. This
is an immediate corollary of the following (trivial to prove) Lemma.
###### Lemma 1.
Consider the configuration $[T,\omega_{T}]|[S,\omega_{S}]$ and the session
type $R$ s.t. $T\sim R$. We have that:
* •
$[T,\omega_{T}]|[S,\omega_{S}]\surd$ if and only if
$[R,\omega_{T}]|[S,\omega_{S}]\surd$;
* •
$[T,\omega_{T}]|[S,\omega_{S}]\stackrel{{\scriptstyle}}{{\rightarrow}}[T^{\prime},\omega_{T}^{\prime}]|[S^{\prime},\omega_{S}^{\prime}]$
if and only if there exists $R^{\prime}\sim T^{\prime}$ s.t.
$[R,\omega_{T}]|[S,\omega_{S}]\stackrel{{\scriptstyle}}{{\rightarrow}}[R^{\prime},\omega_{T}^{\prime}]|[S^{\prime},\omega_{S}^{\prime}]$.
###### Corollary 1.
Consider three session types $T$, $T^{\prime}$, and $S$, with $T\sim
T^{\prime}$. We have that $T$ is compliant with $S$ if and only if
$T^{\prime}$ is compliant with $S$.
We start by defining the type $T=[\\![M,q_{f},E]\\!]$. This type reproduces
the finite control of the queue machine $M$, with a couple of differences:
($i$) it initialises the queue with symbol $\$$, and ($ii$) the state $q_{f}$
produces the additional ending symbol $E$ to communicate the end of the
computation, then it consumes all symbols in the queue and successfully
terminates when $E$ is read from the queue. In this way, the queue is empty
when it successfully terminates.
###### Definition 15 (Finite Control Encoding).
Let $M=(Q,\Sigma,\Gamma,\$,s,\delta)$ be a queue machine, $q_{f}\in Q$,
$E\not\in\Gamma$ the additional ending symbol, and $x\in\Sigma^{*}$. Let
$x=A_{1}\cdots A_{n}$; we define $[\\![M,q_{f},E]\\!]$ as follows:
$[\\![M,q_{f},E]\\!]\ =\oplus\\{\$:[\\![{s}]\\!]^{\emptyset}\\}$
where, given $q\in Q\setminus\\{q_{f}\\}$ and $\mathcal{S}\subseteq Q$,
$[\\![{q}]\\!]^{\mathcal{S}}$ is defined as follows:
$\begin{array}[]{l}[\\![{q}]\\!]^{\mathcal{S}}=\left\\{\begin{array}[]{l}\mu\mathbf{q}.\&\\{{A}\\!:\\!{\oplus\\{{B^{A}_{1}}:{\cdots\oplus\\{{B^{A}_{n_{A}}}:{[\\![{q^{\prime}}]\\!]^{\mathcal{S}\cup
q}}\\}}\\}}\\}_{A\in\Gamma}\\\\[2.84526pt] \hskip 25.6073pt\text{if
}q\not\in{\mathcal{S}}\text{ and }\delta(q,A)=(q^{\prime},B^{A}_{1}\cdots
B^{A}_{n_{A}})\\\ \\\ \mathbf{q}\qquad\mbox{if
$q\in{\mathcal{S}}$}\end{array}\right.\end{array}$
while
$[\\![{q_{f}}]\\!]^{\mathcal{S}}=\oplus\big{\\{}E:\big{(}\mu\mathbf{\mathbf{t}}.\&\\{{A}\\!:\\!{\mathbf{t}}\\}_{A\in\Gamma}\cup\\{E:\mathbf{end}\\}\big{)}\
\big{\\}}$
We now define the type $S=[\\![M,E]\\!]$, that repeatedly behaves like a
producer/consumer for all the symbols of the queue alphabet plus the ending
symbol $E$, with the difference that after producing and consuming the ending
symbol $E$, the type becomes $\mathbf{end}$.
###### Definition 16 (Producer/consumer).
Let $M=(Q,\Sigma,\Gamma,\$,s,\delta)$ be a queue machine and $E\not\in\Gamma$
be the ending symbol. We define $[\\![M,E]\\!]$ as
$[\\![M,E]\\!]=\mu\mathbf{\mathbf{t}}.\oplus\\{{A}:{\&\\{A:\mathbf{t}\\}}\\}_{A\in\Gamma}\cup\\{E:\&\\{E:\mathbf{end}\\}\\}$
Even though $T=[\\![M,q_{f},E]\\!]$ and $S=[\\![M,E]\\!]$ are rather different
and appear unrelated, we have that under some conditions $T\sqsubseteq S$
holds. Namely, $T\sqsubseteq S$ if and only if $q_{f}$ is reachable in $M$. To
prove this, we first characterize the set of types that are compliant with
$S=[\\![M,E]\\!]$.
###### Lemma 2.
Given a queue machine $M$ and the ending symbol $E$, consider
$S=[\\![M,E]\\!]$. We have that a session type $S^{\prime}$ is compliant with
$S$ if and only if $S^{\prime}\sim\overline{S}$.
###### Proof.
Let $S=[\\![M,E]\\!]$.
We first prove the only-if part. It is trivial to see that $\overline{S}$ is
compliant with $S$; this holds because in the configuration
$[S,\epsilon]|[\overline{S},\epsilon]$ the two parties alternate inputs and
outputs in such a way that their buffers have maximal length 1, and moreover
the possibility to successfully terminate by selecting the ending label $E$ is
never disallowed. By Corollary 1 we have that also all types
$S^{\prime}\sim\overline{S}$ are compliant with $S$.
We now move to the if part. Consider $R$ such that $[S,\epsilon]|[R,\epsilon]$
is a correct composition. We have that $\mathsf{unfold}(R)$ cannot start with
an output selection; in fact, if, for instance, it starts with an output
selection and it selects any label $A$, the type $S$ can select a branch with
a different label $A^{\prime}$, thus blocking. The initial input branching of
$\mathsf{unfold}(R)$ must have branchings labeled with all the symbols in
$\Gamma$ plus the ending symbol $E$, in that these are the labels that can be
initially selected by $S$. In each continuation of $R$, the unfolding of the
type should start with an output selection, otherwise the entire system is
blocked in that the continuation of $S$ after the initial output selection
starts with an input branching. Moreover, given that these input branchings of
the continuation of $S$ have only the initially selected label, the output
selection in the continuation of $R$ can have only such label. After each of
these output selections of the continuation of $R$, the same reasoning can be
applied, excluding the case in which the label $E$ was initially selected. In
this case, the continuation of $R$ should be such that its unfolding is
$\mathbf{end}$. This because, the continuation of $S$ becomes $\mathbf{end}$
after executing the input branching labeled with $E$. These constraints that
we have just proved holding for the type $R$ guarantee that
$R\sim\overline{S}$. ∎∎
###### Theorem 0.A.1.
Given a queue machine $M$, the target state $q_{f}$, and the ending symbol
$E$, consider $T=[\\![M,q_{f},E]\\!]$ and $S=[\\![M,E]\\!]$. We have that $T$
is compliant with $\overline{S}$ if and only if $q_{f}$ is reachable in $M$.
###### Proof.
Consider the queue machine $M$, the types $T=[\\![M,q_{f},E]\\!]$ and
$S=[\\![M,E]\\!]$ and the initial configuration
$[[\\![{s}]\\!]^{\emptyset},\epsilon]|[\overline{S},\epsilon]$. The first
transition is
$[T,\epsilon]|[\overline{S},\epsilon]\stackrel{{\scriptstyle}}{{\rightarrow}}[[\\![{s}]\\!]^{\emptyset},\epsilon]|[\overline{S},\$]$.
We now define a partial mapping function $\\{\\!\\!\\{{\,}\\}\\!\\!\\}$ from
configurations (reachable from the initial configuration
$[[\\![{s}]\\!]^{\emptyset},\epsilon]|[\overline{S},\$]$) to configurations in
the queue machine computation:
* •
$\\{\\!\\!\\{{[[\\![{q}]\\!]^{\emptyset},\omega_{T}]|[S^{\prime},\omega_{S}^{\prime}]}\\}\\!\\!\\}=(q,\omega_{T}\cdot\omega\cdot(\omega_{S}^{\prime})^{R})$
where
* –
$\omega=\epsilon$ if $S^{\prime}$ starts with an input branching, or
$\omega=A$ if $S^{\prime}$ starts with an output selection with unique label
$A$,
* –
the operator $\cdot$ stands for concatenation, and
* –
and $\beta^{R}$ is the reverse of $\beta$.
Notice that
$\\{\\!\\!\\{{[[\\![{s}]\\!]^{\emptyset},\epsilon]|[\overline{S},\$]}\\}\\!\\!\\}$
is defined and it coincides with the initial configuration of the queue
computation $(s,\$)$.
In the following we use the following notation:
* •
$[[\\![{q}]\\!]^{\emptyset},\omega_{T}]|[S^{\prime},\omega_{S}^{\prime}]\Rightarrow[[\\![{q^{\prime}}]\\!]^{\emptyset},\omega_{T}^{\prime}]|[S^{\prime\prime},\omega_{S}^{\prime\prime}]$
if
* –
$[[\\![{q}]\\!]^{\emptyset},\omega_{T}]|[S^{\prime},\omega_{S}^{\prime}]\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}[[\\![{q^{\prime}}]\\!]^{\emptyset},\omega_{T}^{\prime}]|[S^{\prime\prime},\omega_{S}^{\prime\prime}]$
and
* –
all intermediary traversed configurations are not in the domain of the partial
mapping function $\\{\\!\\!\\{{\,}\\}\\!\\!\\}$.
Given that, excluding the final state $q_{f}$, for each state $q$ of the queue
machine $[\\![{q}]\\!]^{\emptyset}$ reproduces the dequeue/enqueue actions of
state $q$ and $\overline{S}$ is a simple forwarder that repeatedly produces
and consumes the same labels, we have that given $q\neq q_{f}$ we have
$(q,\gamma)\rightarrow_{M}(q^{\prime},\gamma^{\prime})$ if and only if
$[[\\![{q}]\\!]^{\emptyset},\omega_{T}]|[S^{\prime},\omega_{S}^{\prime}]\Rightarrow[[\\![{q^{\prime}}]\\!]^{\emptyset},\omega_{T}^{\prime}]|[S^{\prime\prime},\omega_{S}^{\prime\prime}]$
with
$\\{\\!\\!\\{{[[\\![{q}]\\!]^{\emptyset},\omega_{T}]|[S^{\prime},\omega_{S}^{\prime}]}\\}\\!\\!\\}=(q,\gamma)$
and
$\\{\\!\\!\\{{[[\\![{q^{\prime}}]\\!]^{\emptyset},\omega_{T}^{\prime}]|[S^{\prime\prime},\omega_{S}^{\prime\prime}]}\\}\\!\\!\\}=(q^{\prime},\gamma^{\prime})$.
We now prove the only-if part of the theorem. Assume that $T$ is compliant
with $\overline{S}$. This means that there exists a computation leading to the
final successful configuration. The unique occurrence of $\mathbf{end}$ is
inside the type $[\\![{q_{f}}]\\!]^{\mathcal{S}}$, hence we have
$[[\\![{s}]\\!]^{\emptyset},\epsilon]|[\overline{S},\$]\Rightarrow\ldots\Rightarrow[[\\![{q_{f}}]\\!]^{\emptyset},\omega_{T}]|[S^{\prime},\omega_{S}^{\prime}]$
thus implying that state $q_{f}$ is reachable in $M$.
We now prove the if part. Assume that $q_{f}$ is reachable in $M$. Consider
$[[\\![{s}]\\!]^{\emptyset},\epsilon]|[\overline{S},\$]\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}[T^{\prime},\omega_{T}^{\prime}]|[S^{\prime},\omega_{S}^{\prime}]$.
There are two possible cases: either (i) it is possible to extend the sequence
of transitions as follows
$[T^{\prime},\omega_{T}^{\prime}]|[S^{\prime},\omega_{S}^{\prime}]\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}[[\\![{q}]\\!]^{\emptyset},\omega_{T}^{\prime\prime}]|[S^{\prime\prime},\omega_{S}^{\prime\prime}]$,
for some state $q$, (ii) or during the sequence of transitions
$[[\\![{s}]\\!]^{\emptyset},\epsilon]|[\overline{S},\$]\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}[T^{\prime},\omega_{T}^{\prime}]|[S^{\prime},\omega_{S}^{\prime}]$
a configuration is traversed in which the l.h.s. type is
$[\\![{q_{f}}]\\!]^{\emptyset}$.
In the first case (i), we have that
$(s,\$)\rightarrow_{M}^{*}\\{\\!\\!\\{{[[\\![{q}]\\!]^{\emptyset},\omega_{T}^{\prime\prime}]|[S^{\prime\prime},\omega_{S}^{\prime\prime}]}\\}\\!\\!\\}$;
moreover, in this computation of the queue machine the state $q_{f}$ is not
traversed. This means that such a queue machine computation can be extended to
reach $q_{f}$, hence the sequence of transitions
$[[\\![{s}]\\!]^{\emptyset},\epsilon]|[\overline{S},\$]\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}[[\\![{q}]\\!]^{\emptyset},\omega_{T}^{\prime\prime}]|[S^{\prime\prime},\omega_{S}^{\prime\prime}]$
can be additionally extended to reach a configuration where the l.h.s. type is
$[\\![{q_{f}}]\\!]^{\emptyset}$. From such a configuration, we have that there
are only finitely many transitions leading to the final successful
configuration (in this final transitions both the queues are emptied and both
types become $\mathbf{end}$).
In the second case (ii), we have that a configuration whose l.h.s. type is
$[\\![{q_{f}}]\\!]^{\emptyset}$. As just observed, this means that the
configuration
$[T^{\prime},\omega_{T}^{\prime}]|[S^{\prime},\omega_{S}^{\prime}]$ is an
intermediary configuration in the final sequence of transitions leading to the
final successful configuration (in which both the queues are emptied and both
types are $\mathbf{end}$). ∎∎
We can finally conclude that our encoding of queue machines into session types
correctly reduces state reachability into refinement.
See 2.1
###### Proof.
We show that given a queue machine $M=(Q,\Sigma,\Gamma,\$,s,\delta)$ and the
target state $q_{f}$ it is possible to compute two types $T$ and $S$ such that
$T\sqsubseteq S$ _if and only if_ $q_{f}$ is reachable in $M$. It is
sufficiente to take an additional ending symbol $E\not\in\Gamma$ and consider
$T=[\\![M,q_{f},E]\\!]$ and $S=[\\![M,E]\\!]$.
We first prove the only-if part. Let $T\sqsubseteq S$. By Lemma 2 we have that
$S$ is compliant with $\overline{S}$. Given that $T\sqsubseteq S$, also $T$ is
compliant with $\overline{S}$. By Theorem 0.A.1 this implies that $q_{f}$ is
reachable in $M$.
We now prove the if part. Assume that $q_{f}$ is reachable in $M$. Then by
Theorem 0.A.1 we have that $T$ is compliant with $\overline{S}$. By Corollary
1 we have that $T$ is compliant with all $S^{\prime}$ such that
$S^{\prime}\sim\overline{S}$, but by Lemma 2 this means that $T\sqsubseteq S$
because the set of the types compliant with $S$ coincides with the set of
types bisimilar to $\overline{S}$. ∎∎
### 0.A.2 Controllability Characterisation
In this section we will prove the following theorem about controllability
characterisation.
See 2.2
We start by introducing some notions and definitions that will be needed in
the proof.
First of all we present an equivalent definition, based on purely structural
induction, of the $\,\mathsf{ok}$ predicate introduced in Definition 9
characterizing session type controllability.
###### Definition 17.
Given a session type $T$, we define the judgment $T\,\mathsf{ok}$ inductively
as follows: t ok
end ok
end∈ T ∨∃t’ : t’ ≠ t ∧t’ ∈ free(T) T ok μt.T ok
T ok &{l:T} ok
∀i ∈I . T_i ok ⊕{l_i:T_i}_i∈I ok where $\mathsf{free}(T)$ is the set of
variables $t$ occurring free in $T$.
In the following we will use a reformulation of session types in terms of
equation sets. In equation set notations we will use terms $T$ that have the
same syntax as those used to denote session types, excluding the
$\mu\mathbf{t}.\\_$ recursion operator. Notice that in such notations we
consider possibly open terms $T$ (i.e. such that $\mathsf{free}(T)$ is not
empty). Session types are, thus, denoted by
$T\\{\mathbf{t}=T_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}\\}$, with
$\mathsf{Vars}$ being a set of variables $\mathbf{t}$ that includes all
variables in $\mathsf{free}(T)$ and also in $\mathsf{free}(T_{\mathbf{t}})$
for all $\mathbf{t}\in\mathsf{Vars}$.
Formally, given a session type $T$ (we assume with loss of generality that
each of its recursions uses a variable with a different name) we consider its
equivalent equation set notation
$\mathsf{esn}(T)=T_{\mathsf{init}}\\{\mathbf{t}=T_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}\\}$,
defined as follows:
* •
$\mathsf{Vars}$ is the set of variable names used in the recursions of $T$
* •
$T_{\mathsf{init}}$ is the only term without recursion operators satisfying:
there exists a set of terms $T^{\prime}_{\mathbf{t}}$, one for each variable
$\mathbf{t}\in\mathsf{free}(T_{\mathsf{init}})$, such that
$T_{\mathsf{init}}\\{T^{\prime}_{\mathbf{t}}/\mathbf{t}\mid\mathbf{t}\in\mathsf{free}(T_{\mathsf{init}})\\}=T$
* •
each $T_{\mathbf{t}}$, with $\mathbf{t}\in\mathsf{Vars}$, is the only term
without recursion operators satisfying: there exists a set of variables
$\mathsf{Vars}_{\mathbf{t}}\subseteq\mathsf{free}(T_{\mathbf{t}})$ and a set
of terms $T^{\prime}_{\mathbf{t}^{\prime}}$, one for each variable
$\mathbf{t}^{\prime}\in\mathsf{Vars}_{\mathbf{t}}$, such that
$T_{\mathbf{t}}\\{T_{\mathbf{t}^{\prime}}/\mathbf{t}^{\prime}\mid\mathbf{t}^{\prime}\in\mathsf{Vars}_{\mathbf{t}}\\}=T^{\prime\prime}$
with $\mu\mathbf{t}.T^{\prime\prime}$ occurring in $T$.
###### Definition 18 (Unfolding).
Given session type in equation set notation we define its unfolding
$\mathsf{unfold}(T\\{\mathbf{t}=T_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}\\})$
as follows:
$\mathsf{unfold}(T\\{\mathbf{t}=T_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}\\})=\begin{cases}\mathsf{unfold}(T_{\mathbf{t}^{\prime}}\\{\mathbf{t}=T_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}\\})&\text{if
$T=\mathbf{t}^{\prime}$}\\\
T\\{\mathbf{t}=T_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}\\}&\text{otherwise}\end{cases}$
Notice that unfolding is well defined because we consider session types with
guarded recursion in equation set notation.
The transition relation for configurations
$[T_{1}\\{\mathbf{t}=T_{1,\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}_{1}\\},\omega_{1}]|[T_{2}\\{\mathbf{t}=T_{2,\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}_{2}\\},\omega_{2}]$,
with
$T_{i}\\{\mathbf{t}=T_{i,\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}_{i}\\}$,
for $i\in\\{1,2\\}$, being session types in equation set notation, is defined
as in Definition 3 by using the above definition of unfolding (and by assuming
that the $\\{\mathbf{t}=T_{i,\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}_{i}\\}$
equational part is copied, for both $T_{1}$ and $T_{2}$, after every
transition).
Given $T_{1}$ and $T_{2}$ session types, it obviously holds (by standard
arguments) that the transition system of $[T,\epsilon]|[S,\epsilon]$ is
bisimilar to that of $[\mathsf{esn}(T),\epsilon]|[\mathsf{esn}(S),\epsilon]$,
hence that: $T$ and $S$ are compliant if and only if $\mathsf{esn}(T)$ and
$\mathsf{esn}(S)$ are compliant.
We now define predicate $\,\mathsf{ctrl}$ for session types in equation set
notation. $\,\mathsf{ctrl}$ is defined as in Definition 9, by assuming that
predicate $\,\mathsf{ok}$ is, instead, defined as follows.
$T\\{\mathbf{t}=T_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}\\}\,\mathsf{ok}$
if there exists an indexing (total order) ${\mathbf{t}}_{i}$ on the variables
of $\mathsf{Vars}$ such that $\\{\mathbf{t}_{i}\mid 1\leq i\leq
n\\}=\mathsf{Vars}$ and, for all $i$, with $1\leq i\leq n$, it, holds:
$\mathbf{end}\\!\in\\!T_{i}\vee\exists\mathbf{t}_{j}\\!:\\!j<i\wedge\mathbf{t}_{j}\\!\in\\!\mathsf{free}(T_{i})$
Moreover, as in Definition 9, in order to establish $\,\mathsf{ctrl}$ of a
session type $T\\{\mathbf{t}=T_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}\\}$
input prefix replacement must preliminarily be performed, so to obtain session
types
$T^{\prime}\\{\mathbf{t}=T^{\prime}_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}^{\prime}\\}$
where $\mathsf{Vars}^{\prime}\subseteq\mathsf{Vars}$ and in both term
$T^{\prime}$ and all terms $T^{\prime}_{\mathbf{t}}$, with
$\mathbf{t}\in\mathsf{Vars}^{\prime}$, all input prefixes have a single label.
###### Proposition 1.
$T$ $\,\mathsf{ctrl}$ if and only if $\mathsf{esn}(T)$ $\,\mathsf{ctrl}$.
###### Proof.
We first show that $T$ $\,\mathsf{ctrl}$ implies $\mathsf{esn}(T)$
$\,\mathsf{ctrl}$. Given $T^{\prime}$ obtained by input prefix replacement
from $T$ (so to have input prefixes with single choices) that satisfies the
$\,\mathsf{ok}$ predicate, we correspondingly consider
$\mathsf{esn}(T^{\prime})$, which is an input prefix replacement of
$\mathsf{esn}(T)$. $\mathsf{esn}(T^{\prime})\,\mathsf{ok}$ is an immediate
consequence of $T^{\prime}\,\mathsf{ok}$ by considering the indexing
${\mathbf{t}}_{i}$ of variable names used in the recursions of $T$ obtained as
follows. We incrementally assign indexes to variables (starting from $1$)
according to a depth-first visit of the syntax tree of $T$ as follows. When we
are at a $\mu\mathbf{t}.T^{\prime\prime}$ node, we have two cases. Either
$\mathbf{t}$ has already an assigned index (not possibile at the beginning) or
not. In the latter case: we consider all $\mu\mathbf{t}^{\prime}.\\_$
operators occurring in $T^{\prime\prime}$, if any, that syntactically include
$\mathbf{end}$ or variable $\mathbf{t}^{\prime\prime}$ such that
$\mathbf{t}^{\prime\prime}\\!\neq\\!\mathbf{t}\wedge\mathbf{t}^{\prime\prime}\\!\in\\!\mathsf{free}(T^{\prime\prime})$
and we assign an index to all such $\mathbf{t}^{\prime}$ (incrementing the
last assigned index) in increasing order from the innermost to the outermost;
then we assign an index to $\mathbf{t}$ (incrementing the last assigned
index). Finally, in both cases, we visit all the $\mu\mathbf{t}^{\prime}.\\_$
descendants (with no other recursion node in-between) of the
$\mu\mathbf{t}.\\_$ node, if any.
We now show that $\mathsf{esn}(T)$ $\,\mathsf{ctrl}$ implies $T$
$\,\mathsf{ctrl}$. Given
$T_{\mathsf{init}}\\{\mathbf{t}=T_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}\\}$
obtained by input prefix replacement from $\mathsf{esn}(T)$ that satisfies the
$\,\mathsf{ok}$ predicate, we correspondingly consider the only term
$T^{\prime}$ which is an input prefix replacement of $T$ such that
$\mathsf{esn}(T^{\prime})=T_{\mathsf{init}}\\{\mathbf{t}=T_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}\\}$.
We show that $T^{\prime}\,\mathsf{ok}$ (Definition 17 above) by structural
induction:
* •
For the base cases ${\mathbf{t}\,\mathsf{ok}}$ and
${\mathbf{end}\,\mathsf{ok}}$ we have nothing to show.
* •
$\&\\{{l}:{T^{\prime\prime}}\\}\,\mathsf{ok}$ and
$\oplus\\{{l}_{i}:{T^{\prime\prime}}_{i}\\}_{i\in I}\,\mathsf{ok}$ are a
direct consequence of the induction hypothesis, i.e.
$T^{\prime\prime}\,\mathsf{ok}$ and $\forall i\in I.\
T^{\prime\prime}_{i}\,\mathsf{ok}$, respectively.
* •
$\mu\mathbf{t}.T^{\prime\prime}\,\mathsf{ok}$ is a direct consequence of the
induction hypothesis $T^{\prime\prime}\,\mathsf{ok}$ and of the fact that:
$\mathbf{end}\\!\in\\!T^{\prime\prime}\vee\exists\mathbf{t}^{\prime}\\!:\\!\mathbf{t}^{\prime}\\!\neq\\!\mathbf{t}\wedge\mathbf{t}^{\prime}\\!\in\\!\mathsf{free}(T^{\prime\prime})$.
The latter is shown as follows. From
$T_{\mathsf{init}}\\{\mathbf{t}=T_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}\\}\,\mathsf{ok}$
we know that there exists a variable indexing ${\mathbf{t}}_{i}$ such that,
for all $i\in I$ it, holds:
$\mathbf{end}\\!\in\\!T_{i}\vee\exists\mathbf{t}_{j}\\!:\\!j<i\wedge\mathbf{t}_{j}\\!\in\\!\mathsf{free}(T_{i})$.
So, given index $i$ such that $\mathbf{t}_{i}=\mathbf{t}$, we have to show:
$\mathbf{end}\\!\in\\!T^{\prime\prime}\vee\exists
z\\!:\\!z\\!\neq\\!i\wedge\mathbf{\mathbf{t}}_{z}\\!\in\\!\mathsf{free}(T^{\prime\prime})$.
What we know is that
$\mathbf{end}\\!\in\\!T_{i}\vee\exists\mathbf{t}_{j}\\!:\\!j<i\wedge\mathbf{t}_{j}\\!\in\\!\mathsf{free}(T_{i})$,
so there are two cases:
1. 1.
Either it holds
$\mathbf{end}\\!\in\\!T^{\prime\prime}\vee\mathbf{t}_{j}\\!\in\\!\mathsf{free}(T^{\prime\prime})$
and we are done (with $z=j$).
2. 2.
Or $\mu\mathbf{t}_{j}.T^{\prime\prime\prime}$, for some
$T^{\prime\prime\prime}$, is a subterm of $T^{\prime\prime}$. In this case we
show that: $\mathbf{end}\\!\in\\!T^{\prime\prime\prime}\vee\exists
z\\!:\\!z\\!\neq\\!i\wedge\mathbf{\mathbf{t}}_{z}\\!\in\\!\mathsf{free}(T^{\prime\prime\prime})$.
To do this we consider index $j$ and the defining term $T_{j}$ in its
equation: we know that
$\mathbf{end}\\!\in\\!T_{j}\vee\exists\mathbf{t}_{k}\\!:\\!k<j\wedge\mathbf{t}_{k}\\!\in\\!\mathsf{free}(T_{j})$.
Now again we have the same two cases, considering index $k$ instead of $j$ and
term $T^{\prime\prime\prime}$ instead of term $T^{\prime\prime}$. Notice that
we cannot proceed like this forever because the syntax of $T^{\prime\prime}$
is finite, hence case $1.$ must eventually apply. Moreover when this happens,
we are sure that the variable $\mathbf{t}_{z}$ that we detect is different
from $\mathbf{t}=\mathbf{t}_{i}$ (i.e. $z\neq i$) because the indexing of the
variables that we consider are always strictly smaller than $i$. ∎
∎
We are now in a position to prove the desired theorem. We prove implications
in the two opposite directions one at a time.
###### Theorem 0.A.2.
If there exists a session type $S$ such that $T$ and $S$ are compliant then
$T$ $\,\mathsf{ctrl}$.
###### Proof.
Since $T$ and $S$ are compliant, as observed above, we have also that
$\mathsf{esn}(T)$ and $\mathsf{esn}(S)$ are compliant. Therefore (the
transition system of) configuration
$[\mathsf{esn}(T),\epsilon]|[\mathsf{esn}(S),\epsilon]$ is a correct
composition according to Definition 5.
We now show that $\mathsf{esn}(T)$ $\,\mathsf{ctrl}$: by Proposition 1 this
implies that $T$ $\,\mathsf{ctrl}$. In order to do this we need to enrich the
transition system representation of the behaviour of configurations
$[T_{1}\\{\mathbf{t}=T_{1,\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars_{1}}\\},\omega_{1}]|[T_{2}\\{\mathbf{t}=T_{2,\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars_{2}}\\},\omega_{2}]$.
We assume the transition relation $\rightarrow$ defined in Definition 3 to be
enriched as follows: $\rightarrow$ transitions originated from outputs of
$T_{1}$ (rule $1.$ of Definition 3) are assumed to be decorated with the label
$l_{j}$ of the performed output (denoted by
$\,\mathop{\longrightarrow}\limits^{\overline{l_{j}}}\,$), while $\rightarrow$
transitions originated from inputs of $T_{1}$ (rule $2.$ of Definition 3) are
assumed to be decorated with the label $l_{j}$ of the performed input (denoted
by $\,\mathop{\longrightarrow}\limits^{l_{j}}\,$). Notice that, in case of
transitions originated from inputs or outputs of $T_{2}$ no decoration is
added to transitions $\rightarrow$. Moreover, rule $3.$ (about recursion
unfolding) of Definition 3 is assumed to just copy the decoration labeling the
transition (if there is any).
We now consider such an enriched transition system over configurations
$[T_{1}\\{\mathbf{t}=T_{1,\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars_{1}}\\},\omega_{1}]|[T_{2}\\{\mathbf{t}=T_{2,\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars_{2}}\\},\omega_{2}]$.
We use $s$ to range over these configurations. We say that a configuration
$s=[T_{1}\\{\mathbf{t}=T_{1,\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars_{1}}\\},\omega_{1}]|[T_{2}\\{\mathbf{t}=T_{2,\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars_{2}}\\},\omega_{2}]$
exposes variable $\mathbf{t}^{\prime}\in\mathsf{Vars_{1}}$ if
$T_{1}=\mathbf{t}^{\prime}$. Moreover, we denote transition systems paths
starting from a given configuration $s$, i.e. finite sequences of transitions
$s\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,s_{1}\,\mathop{\longrightarrow}\limits^{\alpha_{2}}\,s_{2}\dots\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,s_{n}$
(where $\alpha_{i}$ decorations can be $\varepsilon$ in case of non decorated
$\rightarrow$ transitions), by means of strings
$\langle\alpha_{1},s_{1}\rangle\langle\alpha_{2},s_{2}\rangle\dots\langle\alpha_{n},s_{n}\rangle$
(strings over pairs $\langle\alpha^{\prime},s^{\prime}\rangle$ with
$\alpha^{\prime}$ being a decoration or $\varepsilon$ and $s^{\prime}$ a
configuration).
Assuming
$\mathsf{esn}(T)=T_{\mathsf{init}}\\{\mathbf{t}=T_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}\\}$,
we now detect an indexing on the variables in the subset
$\mathsf{Vars^{\prime}}$ of $\mathsf{Vars}$, which includes variables
$\mathbf{t}$ such that: a configuration $s$ that exposes $\mathbf{t}$ is
reachable from the initial configuration
$[\mathsf{esn}(T),\epsilon]|[\mathsf{esn}(S),\epsilon]$. We proceed as
follows. If $\mathsf{Vars^{\prime}}\neq\emptyset$, then we consider any
reachable configuration $s$ that exposes some variable
$\mathbf{t}\in\mathsf{Vars}$. Since
$[\mathsf{esn}(T),\epsilon]|[\mathsf{esn}(S),\epsilon]$ is a correct
composition, the configuration $s$ must reach a configuration $s^{\prime}$
such that $s^{\prime}\surd$. We consider the path from $s$ to $s^{\prime}$ and
the last configuration $s^{\prime\prime}$ of such a path that exposes a
variable. We denote such a variable with $\mathbf{t}_{1}$, the configuration
$s^{\prime\prime}$ that exposes it with $s_{1}$, and the path (string) from
$s_{1}$ that leads to $s^{\prime}$ (part of the path from $s$ to $s^{\prime}$
considered above) with $\mathsf{path}_{1}$. In any subsequent $k$-th step,
with $k\geq 2$, we consider the set
$\mathsf{Vars}_{k}=\mathsf{Vars^{\prime}}-\\{\mathbf{t}_{h}\mid h<k\\}$. If
$\mathsf{Vars}_{k}\neq\emptyset$, then we consider any reachable configuration
$s$ that exposes some variable $\mathbf{t}\in\mathsf{Vars}_{k}$. Since
$[\mathsf{esn}(T),\epsilon]|[\mathsf{esn}(S),\epsilon]$ is a correct
composition, the configuration $s$ must reach a configuration $s^{\prime}$
such that $s^{\prime}\surd$. We consider the path from $s$ to $s^{\prime}$ and
the first configuration $s^{\prime\prime}$ of such a path that either exposes
a variable in $\\{\mathbf{t}_{h}\mid h<k\\}$ or is such that
$s^{\prime\prime}\surd$. Again we consider the path from $s$ to
$s^{\prime\prime}$ and the last configuration $s^{\prime\prime\prime}$ of such
a path that: is different from $s^{\prime\prime}$ and exposes a variable (such
a variable must exist, because $s$ exposes a variable, and belong to
$\mathsf{Vars}_{k}$ because of the way we have selected $s^{\prime\prime}$).
We denote such a variable with $\mathbf{t}_{k}$, the configuration
$s^{\prime\prime\prime}$ that exposes it with $s_{k}$, and the path (string)
from $s_{k}$ that leads to $s^{\prime\prime}$ (part of the path from $s$ to
$s^{\prime\prime}$ considered above) with $\mathsf{path}_{k}$.
We now consider terms $T^{\prime}_{k}$ for each variable
$\mathbf{t}_{k}\in\mathsf{Vars^{\prime}}$. We build $T^{\prime}_{k}$ terms
inductively by taking
$T^{\prime}_{k}=\mathsf{term}(T_{\mathbf{t}_{k}},s_{k},\mathsf{path}_{k})$,
where $\mathsf{term}(T^{\prime},s,\mathsf{optpath})$, with $\mathsf{optpath}$
being either a $\mathsf{path}$ or $*$ (that represents being outside the
path), is defined as follows.
* •
$\mathsf{term}(\mathbf{t},s,\varepsilon)=\mathbf{t}$
* •
$\mathsf{term}(\mathbf{end},s,\varepsilon)=\mathbf{end}$
* •
$\mathsf{term}(\&\\{{l}_{i}:{T}_{i}\\}_{i\in I},s,\langle
l_{j},s^{\prime}\rangle\mathsf{path})=\&\\{{l_{j}}:{\mathsf{term}(T_{j},s^{\prime},\mathsf{path})}\\}$
* •
$\mathsf{term}(\oplus\\{{l}_{i}:{T}_{i}\\}_{i\in
I},s,\langle\overline{l_{j}},s^{\prime}\rangle\mathsf{path})=\oplus\\{{l}_{i}:{T^{\prime}}_{i}\\}_{i\in
I}$
where $T^{\prime}_{j}\\!=\\!\mathsf{term}(T_{j},s^{\prime},\mathsf{path})$
and, for all $i\\!\in\\!I$, $i\\!\neq\\!j$:
$T^{\prime}_{i}\\!=\\!\mathsf{term}(T_{i},s_{i},*)$ with
$s\,\mathop{\longrightarrow}\limits^{\overline{l_{i}}}\,s_{i}$
* •
$\mathsf{term}(T^{\prime},s,\langle\varepsilon,s^{\prime}\rangle\mathsf{path})=\mathsf{term}(T^{\prime},s^{\prime},\mathsf{path})$
* •
$\mathsf{term}(\mathbf{t},s,*)=\mathbf{t}$
* •
$\mathsf{term}(\mathbf{end},s,*)=\mathbf{end}$
* •
$\mathsf{term}(\&\\{{l}_{i}:{T}_{i}\\}_{i\in
I},s,*)=\&\\{{l_{j}}:{\mathsf{term}(T_{j},s_{j},*)}\\}$ if $s$ has some
$\,\mathop{\longrightarrow}\limits^{l}\,$ transition
where $j$ is any $i\in I$ such that
$s\,\mathop{\longrightarrow}\limits^{l_{j}}\,s_{j}$
* •
$\mathsf{term}(\oplus\\{{l}_{i}:{T}_{i}\\}_{i\in
I},s,*)=\oplus\\{{l}_{i}:{\mathsf{term}(T_{i},s_{i},*)}_{i}\\}_{i\in I}$ if
$s$ has some $\,\mathop{\longrightarrow}\limits^{\overline{l}}\,$ transition
where, for all $i\\!\in\\!I$,
$s\,\mathop{\longrightarrow}\limits^{l_{i}}\,s_{i}$
* •
$\mathsf{term}(T^{\prime},s,*)=\mathsf{term}(T^{\prime},s^{\prime},*)$ if
$T^{\prime}\notin\\{\mathbf{t},\mathbf{end}\\}$ and $s$ has neither
$\,\mathop{\longrightarrow}\limits^{l}\,$ nor
$\,\mathop{\longrightarrow}\limits^{\overline{l}}\,$ transitions
where $s^{\prime}$ is the first configuration having some
$\,\mathop{\longrightarrow}\limits^{l}\,$ transition or some
$\,\mathop{\longrightarrow}\limits^{\overline{l}}\,$ transition in the path
from $s$ to a configuration $s^{\prime\prime}$ such that
$s^{\prime\prime}\surd$ (such a path must exist because
$[\mathsf{esn}(T),\epsilon]|[\mathsf{esn}(S),\epsilon]$ is a correct
composition)
where we use $\varepsilon$ to represent the empty string.
We also take
$T^{\prime}_{\mathsf{init}}=\mathsf{term}(T_{\mathsf{init}},[\mathsf{esn}(T),\epsilon]|[\mathsf{esn}(S),\epsilon],*)$.
We now have that
$T^{\prime}_{\mathsf{init}}\\{\mathbf{t}_{k}=T^{\prime}_{k}\mid\mathbf{t}_{k}\in\mathsf{Vars}^{\prime}\\}$
is a session type in equation notation: $\mathsf{Vars}^{\prime}$ must include
all variables in $\mathsf{free}(T^{\prime}_{\mathsf{init}})$ and also in
$\mathsf{free}(T^{\prime}_{k})$ for all
$\mathbf{t}_{k}\in\mathsf{Vars}^{\prime}$ because, otherwise, a configuration
$s$ exposing the variable that is not included in $\mathsf{Vars}^{\prime}$
would have been reachable from the initial configuration
$[\mathsf{esn}(T),\epsilon]|[\mathsf{esn}(S),\epsilon]$ (which contradicts the
definition of $\mathsf{Vars}^{\prime}$). Moreover, due to the way
$\mathsf{term}$ is defined,
$T^{\prime}_{\mathsf{init}}\\{\mathbf{t}_{k}=T^{\prime}_{k}\mid\mathbf{t}_{k}\in\mathsf{Vars}^{\prime}\\}$
is obtained from
$T_{\mathsf{init}}\\{\mathbf{t}=T_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}\\}$
by performing input replacement that yields input prefixes with single inputs.
Finally, being $s_{k}$ the last configuration exposing a variable inside a
path ending with a configuration $s$ that either exposes a variable in
$\\{\mathbf{t}_{h}\mid h<k\\}$ (and not having previous configurations
exposing such variables) or is such that $s\surd$, each of the
$T^{\prime}_{k}$ satisfies the constraint
$\mathbf{end}\\!\in\\!T^{\prime}_{k}\vee\exists\mathbf{t}_{h}\\!:\\!h<k\wedge\mathbf{t}_{h}\\!\in\\!\mathsf{free}(T^{\prime}_{k})$.
∎∎
###### Theorem 0.A.3.
If $T$ $\,\mathsf{ctrl}$ then there exists a session type $S$ such that $T$
and $S$ are compliant.
###### Proof.
If $T$ $\,\mathsf{ctrl}$ then
$\mathsf{esn}(T)=T_{\mathsf{init}}\\{\mathbf{t}=T_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}\\}$
$\,\mathsf{ctrl}$. That is, there exists an input prefix replacement that
yields a session type
$T^{\prime}_{\mathsf{init}}\\{\mathbf{t}=T^{\prime}_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}^{\prime}\\}$
such that $\mathsf{Vars}^{\prime}\subseteq\mathsf{Vars}$ (and in both term
$T^{\prime}_{\mathsf{init}}$ and all terms $T^{\prime}_{\mathbf{t}}$, with
$\mathbf{t}\in\mathsf{Vars}^{\prime}$, all input prefixes have a single label)
and that satisfies the $\,\mathsf{ok}$ predicate, i.e. there exists an
indexing $\mathbf{t}_{i}$ of the $\mathsf{Vars}^{\prime}$ variables, such
that:
$\mathbf{end}\\!\in\\!T^{\prime}_{\mathbf{t}_{i}}\vee\exists\mathbf{t}_{j}\\!:\\!j<i\wedge\mathbf{t}_{j}\\!\in\\!\mathsf{free}(T^{\prime}_{\mathbf{t}_{i}})$.
We take $S$ to be the unique session type such that
$\mathsf{esn}(S)=\overline{T^{\prime}_{\mathsf{init}}}\\{\mathbf{t}=\overline{T^{\prime}_{\mathbf{t}}}\mid\mathbf{t}\in\mathsf{Vars}^{\prime\prime}\\}$
and $\mathsf{Vars}^{\prime\prime}\subseteq\mathsf{Vars}^{\prime}$.
Considered the transition system over configurations
$[T_{1}\\{\mathbf{t}=T_{1,\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars_{1}}\\},\omega_{1}]|[T_{2}\\{\mathbf{t}=T_{2,\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars_{2}}\\},\omega_{2}]$,
ranged over by $s$, we say that $s$ exposes variable
$\mathbf{t}^{\prime}\in\mathsf{Vars_{1}}$ if $T_{1}=\mathbf{t}^{\prime}$. Now,
given any configuration $s$ reachable from the initial configuration
$[\mathsf{esn}(T),\epsilon]|[\mathsf{esn}(S),\epsilon]$ we consider the last
configuration, if any, that exposes a variable. If there is such a
configuration, then the exposed variable must be an indexed variable
$\mathbf{t}_{i}\in\mathsf{Vars}^{\prime\prime}$: denoted with
$T^{\prime\prime}\\{\mathbf{t}=T_{\mathbf{t}}\mid\mathbf{t}\in\mathsf{Vars}\\}$
the left-hand session type in $s$, we have that $T^{\prime\prime}$ is either
$\mathbf{t}_{i}$ or a subterm of $T_{\mathbf{t}_{i}}$, so it is possible to
either reach a configuration $s^{\prime}$ such that $s^{\prime}\surd$ (in case
$\mathbf{end}\in T^{\prime\prime}$), or to reach a configuration exposing a
variable $\mathbf{t}_{j}\in\mathsf{Vars}^{\prime\prime}$. Moreover, in the
case $T^{\prime\prime}$ is $\mathbf{t}_{i}$, we also have that, for such a
reachable configuration, it holds $j<i$. If, instead, there is no such a
configuration, then, we have that $T^{\prime\prime}$ is a subterm of
$T_{\mathsf{init}}$, so it is possible to either reach a configuration
$s^{\prime}$ such that $s^{\prime}\surd$ (in case $\mathbf{end}\in
T^{\prime\prime}$), or to reach a configuration exposing a variable
$\mathbf{t}_{i}\in\mathsf{Vars}^{\prime\prime}$. We thus have that
$\mathsf{esn}(T)$ and $\mathsf{esn}(S)$ are compliant, hence $T$ and $S$ are
compliant. ∎∎
### 0.A.3 Soundness of Fair Asynchronous Subtyping w.r.t. Fair Refinement
###### Lemma 3.
Consider the session type
$T=\mathcal{A}[{\oplus\\{{l}_{j}:{T_{k}}_{j}\\}_{j\in J}}]^{k\in K}$. Let
$P_{2}=[T,\omega_{T}]|[S,\omega_{S}]$ and
$P_{1}^{i}=[\mathcal{A}[{T_{ki}}]^{k\in
K},\omega_{T}]|[S,\omega_{S}\\!\cdot\\!l_{i}]$, for every $i\in J$. If $P_{2}$
is a correct composition then one of the following holds:
* •
$\mathcal{A}$ does not contain any input branching and
$P_{2}\stackrel{{\scriptstyle}}{{\rightarrow}}P_{1}^{i}$, for every $i\in J$;
* •
$\mathcal{A}$ contains an input branching and $P_{1}^{i}$ (for every $i\in J$)
and $P_{2}$ have at least one outgoing transition.
For every possible transition
$P_{1}^{i}\stackrel{{\scriptstyle}}{{\rightarrow}}P_{1}^{\prime}$ we have that
one of the following holds:
1. 1.
$P_{1}^{i}$ does not consume the label $l_{i}$ and there exist
$\mathcal{A}^{\prime}$, $W$, $T^{\prime}_{wj}$ (for every $w\in W$, $j\in J$),
$S^{\prime}$, $\omega_{T}^{\prime}$ and $\omega_{S}^{\prime}$ s.t.
$P_{1}^{\prime}=[\mathcal{A^{\prime}}[{T^{\prime}_{wi}}]^{w\in
W},\omega_{T}^{\prime}]{|}[S^{\prime},\omega_{S}^{\prime}\\!\cdot\\!l_{i}]$
and
$P_{2}\stackrel{{\scriptstyle}}{{\rightarrow}}[\mathcal{A^{\prime}}[{\oplus\\{{l}_{j}:{T^{\prime}_{w}}_{j}\\}_{j\in
J}}]^{w\in W},\omega_{T}^{\prime}]{|}[S^{\prime},\omega_{S}^{\prime}]$;
2. 2.
$P_{1}^{i}$ consumes the label $l_{i}$, hence
$P_{1}^{\prime}=[\mathcal{A}[{T_{ki}}]^{k\in
K},\omega_{T}]{|}[S^{\prime},\omega_{S}]$, and $\exists j\in\\{1,\ldots,m\\}$
s.t.
$P_{2}\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}[T_{ji},\omega_{T}^{\prime}]{|}[S^{\prime},\omega_{S}]$
and
$\omega_{T}=a_{1}\\!\cdot\\!\dots\\!\cdot\\!a_{w}\\!\cdot\\!\omega_{T}^{\prime}$,
where $a_{1},\dots,a_{w}$ are the labels in one of the paths to $[\,]^{j}$ in
$\mathcal{A}$.
For every possible transition
$P_{2}\stackrel{{\scriptstyle}}{{\rightarrow}}P_{2}^{\prime}$ we have that
there exist $\mathcal{A}^{\prime}$, $W$, $T^{\prime}_{wj}$ (for every $w\in
W$, $j\in J$), $S^{\prime}$, $\omega_{T}^{\prime}$ and $\omega_{S}^{\prime}$
s.t.
$P_{2}^{\prime}=[\mathcal{A^{\prime}}[{\oplus\\{{l}_{j}:{T^{\prime}_{w}}_{j}\\}_{j\in
J}}]^{w\in W},\omega_{T}^{\prime}]{|}[S^{\prime},\omega_{S}^{\prime}]$ and
$P_{1}^{i}\stackrel{{\scriptstyle}}{{\rightarrow}}[\mathcal{A^{\prime}}[{T^{\prime}_{wi}}]^{w\in
W},\omega_{T}^{\prime}]{|}[S^{\prime},\omega_{S}^{\prime}\\!\cdot\\!l_{i}]$.
###### Lemma 4.
Consider $P_{1}=[\mathcal{A}[{T_{k}}]^{k\in K},\omega_{T}]{|}[S,\omega_{S}]$
and $P_{2}=[T_{j},\omega_{T}^{\prime}]{|}[S,\omega_{S}]$ with
$\omega_{T}=a_{1}\\!\cdot\\!\dots\\!\cdot\\!a_{w}\\!\cdot\\!\omega_{T}^{\prime}$,
where $a_{1},\dots,a_{w}$ are the labels in one of the paths to $[\,]^{j}$ in
$\mathcal{A}$. We have that if $P_{2}$ is a correct composition, then also
$P_{1}$ is a correct composition.
###### Proof.
By contraposition, assume $P_{1}$ is not a correct composition. This implies
the existence of $P_{1}^{\prime}$, from which it is not possible to reach a
successful configuration, such that
$P_{1}\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}P_{1}^{\prime}$. If the
labels $a_{1},\dots,a_{w}$ were not consumed, we extend
$P_{1}\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}P_{1}^{\prime}$ to
$P_{1}\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}P_{1}^{\prime\prime}$ by
allowing the l.h.s. type to consume all the labels $a_{1},\dots,a_{w}$. We
have that also from $P_{1}^{\prime\prime}$ is not possible to reach a
successful configuration. We now reorder the transitions in
$P_{1}\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}P_{1}^{\prime\prime}$ such
that in the initial $w$ steps the l.h.s. type consumes the labels
$a_{1},\dots,a_{w}$. After these transitions the configuration $P_{2}$ is
reached. This implies that also
$P_{2}\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}P_{1}^{\prime\prime}$, but
this is not possible because $P_{2}$ is a correct composition and from
$P_{1}^{\prime\prime}$ no successful configuration can be reached. ∎∎
###### Lemma 5.
Consider the session type
$T=\mathcal{A}[{\oplus\\{{l}_{j}:{T_{k}}_{j}\\}_{j\in J}}]^{k\in K}$. Let
$P_{2}=[T,\omega_{T}]|[S,\omega_{S}]$ and
$P_{1}^{i}=[\mathcal{A}[{T_{ki}}]^{k\in
K},\omega_{T}]|[S,\omega_{S}\\!\cdot\\!l_{i}]$, for every $i\in J$. If $P_{2}$
is a correct composition then, for every $i\in J$, there exists
$[T^{\prime},\omega_{T}^{\prime}]|[S^{\prime},\omega_{S}^{\prime}]$ such that
$P_{1}^{i}\rightarrow^{*}[T^{\prime},\omega_{T}^{\prime}]|[S^{\prime},\omega_{S}^{\prime}]$
and $[T^{\prime},\omega_{T}^{\prime}]|[S^{\prime},\omega_{S}^{\prime}]\surd$.
###### Proof.
Given that $P_{2}$ is a correct composition, then there exists
$[T^{\prime},\omega_{T}^{\prime}]|[S^{\prime},\omega_{S}^{\prime}]$ s.t.
$[\mathcal{A}[{\oplus\\{{l}_{j}:{T_{k}}_{j}\\}_{j\in J}}]^{k\in
K},\omega_{T}]|[S,\omega_{S}]\rightarrow^{*}[T^{\prime},\omega_{T}^{\prime}]|[S^{\prime},\omega_{S}^{\prime}]$
and $[T^{\prime},\omega_{T}^{\prime}]|[S^{\prime},\omega_{S}^{\prime}]\surd$.
During this sequence of transitions, the input context $\mathcal{A}$ will
become without input branchings, because a configuration that contains one
type with an input branching is not successful. In other terms there exist a
prefix of the sequence of transitions, at the end of which the input context
becomes without input branchings. We proceed by induction on the length of
such a prefix. If the length is zero, we can apply the first item of Lemma 3
to conclude that $P_{2}\stackrel{{\scriptstyle}}{{\rightarrow}}P_{1}^{i}$, for
every $i\in J$, hence also $P_{1}^{i}$ can reach a successful configuration.
In the inductive step, we consider the first transition of $P_{2}$, we apply
the last item of Lemma 3 to show that also $P_{1}^{i}$, for every $i\in J$,
can perform a transition such that it is possible to apply again the
hypothesis on the reached configurations. This is possible because if $P_{2}$
is correct, also the configurations it can reach are correct. ∎∎
###### Proposition 2.
Consider the session type
$T=\mathcal{A}[{\oplus\\{{l}_{j}:{T_{k}}_{j}\\}_{j\in J}}]^{k\in K}$. We have
that if $[T,\omega_{T}]|[S,\omega_{S}]$ is a correct composition then, for
every $i\in J$, we have that also $[\mathcal{A}[{T_{ki}}]^{k\in
K},\omega_{T}]|[S,\omega_{S}\\!\cdot\\!l_{i}]$ is a correct composition.
###### Proof.
By contraposition, assume $i\in J$ s.t.
$P_{1}^{i}=[\mathcal{A}[{T_{ki}}]^{k\in
K},\omega_{T}]|[S,\omega_{S}\\!\cdot\\!l_{i}]$ is not a correct composition.
This means the existence of
$P_{1}^{i}\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}P^{\prime}$ such that
$P^{\prime}$ cannot reach a successful configuration. By induction on the
length of this sequence of transition we show that, differently from what
assumed, $P^{\prime}$ can reach a successful configuration. If the length is
0, we simply apply Lemma 5 to show that $P_{1}^{i}=P^{\prime}$ can reach a
successful configuration. If the length is not 0, we consider two possible
cases: (i) the initial transition of
$P_{1}^{i}\stackrel{{\scriptstyle}}{{\rightarrow}}P^{\prime\prime}$ of
$P_{1}^{i}\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}P^{\prime}$ consumes the
label $l_{i}$ from the the queue of the r.h.s. type or (ii) it does not. In
case (i) we use the corresponding item 2 in Lemma 3 to see that we can apply
Lemma 4 on $P_{2}$ and $P^{\prime\prime}$, in order to conclude that
$P^{\prime\prime}$ is a correct composition. Given that
$P^{\prime\prime}\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}P^{\prime}$ we
can conclude that $P^{\prime\prime}$ can reach a successful configuration. In
case (ii) we use the corresponding item 1 in Lemma 3 to conclude that we can
apply again the inductive hypothesis on the shortest sequence of transitions
$P^{\prime\prime}\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}P^{\prime}$. This
is possible because $P_{2}$ has a corresponding transition to
$P_{2}\stackrel{{\scriptstyle}}{{\rightarrow}}P_{2}^{\prime}$, such that
$P^{\prime\prime}$ and $P_{2}^{\prime}$ still satisfies the assumption in the
statement of the Lemma. In particular $P_{2}^{\prime}$ is a correct
composition because also $P_{2}$ is a correct composition. ∎∎
###### Lemma 6.
If $[S,\omega_{S}]|[R,\omega_{R}]$ is a correct composition then $S$ is
controllable.
###### Proof.
We show the existence of a type $T$ such that $[S,\epsilon]|[T,\epsilon]$ is a
correct composition.
Consider a type $T$ defined as follows. Assume $\omega_{S}=l_{1}^{S}\cdots
l_{k}^{S}$ and $\omega_{R}=l_{1}\cdots l_{w}^{R}$. The type $T$ initially
performs $k$ outputs with single output labels $l_{1}$, $\cdots$, $l_{k}$,
respectively. After such outputs, it becomes like $R$, with the difference
that along all of its paths, the initial $w$ input branchings are replaced by
one of its continuation as follows: the $i$-th input branching is replaced by
its continuation in the branch labeled with $l_{i}^{R}$.
We now show by contraposition that $[S,\epsilon]|[T,\epsilon]$ is a correct
composition. If $[S,\epsilon]|[T,\epsilon]$ is not correct, then there exists
$[S,\epsilon]|[T,\epsilon]\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}[S^{\prime},\omega_{S}^{\prime}]|[T^{\prime},\omega_{T}^{\prime}]$
such that from
$[S^{\prime},\omega_{S}^{\prime}]|[T^{\prime},\omega_{T}^{\prime}]$ it is not
possible to reach a successful configuration. It is not restrictive to assume
that during
$[S,\epsilon]|[T,\epsilon]\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}[S^{\prime},\omega_{S}^{\prime}]|[T^{\prime},\omega_{T}^{\prime}]$
the r.h.s. type has produced the queue $\omega_{S}$ (in fact, if it has not
produced them, we continue the computation performing them). We can also
assume that outputs in $T$, corresponding to outputs in $R$ along an initial
path with less than $w$ inputs have been all performed (also in this case, if
these outputs were not performed, we continue the computation executing them).
We have that also $[S,\omega_{S}]|[R,\omega_{R}]$ can perform a computation
$[S,\omega_{S}]|[R,\omega_{R}]\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}[S^{\prime},\omega_{S}^{\prime}]|[T^{\prime},\omega_{T}^{\prime}]$.
Given that $[S,\omega_{S}]|[R,\omega_{R}]$ is a correct composition, we have
that from $[S^{\prime},\omega_{S}^{\prime}]|[T^{\prime},\omega_{T}^{\prime}]$
will be possible to reach a successful configuration, thus contradicting the
above assumption. ∎∎
###### Proposition 3.
Given two session types $T$ and $S$, if $T\operatorname{\leq}S$ then, for
every $\omega$, $R$, and $\omega_{R}$ such that $[S,\omega]|[R,\omega_{R}]$ is
a correct composition, there exist $T^{\prime}$, $\omega^{\prime}$,
$R^{\prime}$, and $\omega_{R}^{\prime}$ such that
$[T,\omega]|[R,\omega_{R}]\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}[T^{\prime},\omega^{\prime}]|[R^{\prime},\omega_{R}^{\prime}]$
and $[T^{\prime},\omega^{\prime}]|[R^{\prime},\omega_{R}^{\prime}]\surd$.
###### Proof.
Given that $[S,\omega]|[R,\omega_{R}]$ is a correct composition, there exist
$S^{\prime}$, $\omega^{\prime\prime}$, $R^{\prime\prime}$, and
$\omega_{R}^{\prime\prime}$ such that
$[S,\omega]|[R,\omega_{R}]\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}[S^{\prime},\omega^{\prime\prime}]|[R^{\prime\prime},\omega_{R}^{\prime\prime}]$
and
$[S^{\prime},\omega^{\prime}]|[R^{\prime\prime},\omega_{R}^{\prime\prime}]\surd$.
We proceed by induction on the length of this sequence of transition.
If the length is 0, then $[S,\omega]|[R,\omega_{R}]\surd$, that implies
$\mathsf{unfold}(S)=\mathbf{end}$, that also implies
$\mathsf{unfold}(T)=\mathbf{end}$ (because $T\operatorname{\leq}S$), from
which we have $[T,\omega]|[R,\omega_{R}]\surd$.
If the length is greater than 0, we proceed by case analysis on the possible
first transition
$[S,\omega]|[R,\omega_{R}]\stackrel{{\scriptstyle}}{{\rightarrow}}[S^{\prime\prime},\omega^{\prime\prime\prime}]|[R^{\prime\prime\prime},\omega_{R}^{\prime\prime\prime}]$.
If the transition is inferred by $R$ it is sufficient to observe that
$S^{\prime\prime}=S$ and
$[T,\omega]|[R,\omega_{R}]\stackrel{{\scriptstyle}}{{\rightarrow}}[T,\omega^{\prime\prime\prime}]|[R^{\prime\prime\prime},\omega_{R}^{\prime\prime\prime}]$,
and then apply the inductive hypothesis because
$[S^{\prime\prime},\omega^{\prime\prime\prime}]|[R^{\prime\prime\prime},\omega_{R}^{\prime\prime\prime}]$
is a correct composition in that it is reachable from a correct composition.
We now consider that the transition is inferred by $S$.
We first discuss the case in which
$\mathsf{unfold}(S)=\oplus\\{{l}_{i}:{S}_{i}\\}_{i\in I}$. In this case, the
above transition is
$[S,\omega]|[R,\omega_{R}]\stackrel{{\scriptstyle}}{{\rightarrow}}[S_{i},\omega^{\prime\prime\prime}]|[R^{\prime\prime\prime},\omega_{R}^{\prime\prime\prime}]$,
for some $i\in I$. Given that $T\operatorname{\leq}S$, and $S$ is controllable
by Lemma 6, we have $\mathsf{unfold}(T)=\oplus\\{{l}_{i}:{T}_{i}\\}_{i\in I}$
with $T_{i}\operatorname{\leq}S_{i}$, for every $i\in I$. This ensures that
$[T,\omega]|[R,\omega_{R}]\stackrel{{\scriptstyle}}{{\rightarrow}}[T_{i},\omega^{\prime\prime\prime}]|[R^{\prime\prime\prime},\omega_{R}^{\prime\prime\prime}]$.
Then we can apply the inductive hypothesis because
$T_{i}\operatorname{\leq}S_{i}$ and
$[S_{i},\omega^{\prime\prime\prime}]|[R^{\prime\prime\prime},\omega_{R}^{\prime\prime\prime}]$
is a correct composition.
We now discuss the case in which
$\mathsf{unfold}(S)=\&\\{{l}_{i}:{S}_{i}\\}_{i\in I}$. There are two possible
subcases: (i) also $T$ starts with an input branching, i.e.,
$\mathsf{unfold}(T)=\&\\{l_{j}:T_{j}\\}_{j\in J}$, or (ii) $T$ starts with an
output selection, i.e., $\mathsf{unfold}(T)=\oplus\\{l_{j}:T_{j}\\}_{j\in J}$.
In case (i), the above transition is
$[S,\omega]|[R,\omega_{R}]\stackrel{{\scriptstyle}}{{\rightarrow}}[S_{i},\omega^{\prime\prime\prime}]|[R^{\prime\prime\prime},\omega_{R}^{\prime\prime\prime}]$,
for some $i\in I$. Given that $T\operatorname{\leq}S$, and $S$ is controllable
by Lemma 6, we have $\mathsf{unfold}(T)=\&\\{{l}_{j}:{T}_{j}\\}_{j\in J}$,
$J\supseteq K$, and $\forall k\in K\ldotp T_{k}\operatorname{\leq}S_{k}$,
where $K=\\{k\in I\;|\;S_{k}\text{ is controllable}\\}$. Given that
$[S,\omega]|[R,\omega_{R}]$ is a correct composition and
$[S,\omega]|[R,\omega_{R}]\stackrel{{\scriptstyle}}{{\rightarrow}}[S_{i},\omega^{\prime\prime\prime}]|[R^{\prime\prime\prime},\omega_{R}^{\prime\prime\prime}]$,
also the latter configuration is a correct composition. By Lemma 6 we have
that $S_{i}$ is controllable. This implies that $i\in K$, hence also $i\in J$.
This ensures that
$[T,\omega]|[R,\omega_{R}]\stackrel{{\scriptstyle}}{{\rightarrow}}[T_{i},\omega^{\prime\prime\prime}]|[R^{\prime\prime\prime},\omega_{R}^{\prime\prime\prime}]$.
Then we can apply the inductive hypothesis because
$T_{i}\operatorname{\leq}S_{i}$ and
$[S_{i},\omega^{\prime\prime\prime}]|[R^{\prime\prime\prime},\omega_{R}^{\prime\prime\prime}]$
is a correct composition.
In case (ii), given that $T\operatorname{\leq}S$, and $S$ is controllable, we
have $\mathsf{selUnfold}(S)=\mathcal{A}[{\oplus\\{{l}_{i}:{S_{k}}_{i}\\}_{i\in
J}}]^{k\in K}$, and $\mathsf{unfold}(T)=\oplus\\{l_{j}:T_{j}\\}_{j\in J}$ with
$T_{j}\operatorname{\leq}\mathcal{A}[{S_{kj}}]^{k\in K}$, for every $j\in J$.
We first observe that the sequence of transitions
$[S,\omega]|[R,\omega_{R}]\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}[S^{\prime},\omega^{\prime\prime}]|[R^{\prime\prime},\omega_{R}^{\prime\prime}]$,
with
$[S^{\prime},\omega^{\prime\prime}]|[R^{\prime\prime},\omega_{R}^{\prime\prime}]\surd$,
includes at least one output selection $l_{j}$ executed by one of the output
selections filling the holes in $\mathcal{A}$. This label $l_{j}$ is the first
one emitted by the l.h.s. type after it has executed input branchings in
$\mathcal{A}$. We have that the same sequence of transitions, excluding the
output of $l_{j}$, can be executed from the configuration
$[\mathcal{A}[{S_{kj}}]^{k\in K},\omega]|[R,\omega_{R}\\!\cdot\\!l_{j}]$. Such
a sequence is $[\mathcal{A}[{S_{kj}}]^{k\in
K},\omega]|[R,\omega_{R}\\!\cdot\\!l_{j}]\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}[S^{\prime},\omega^{\prime\prime}]|[R^{\prime\prime},\omega_{R}^{\prime\prime}]$,
with
$[S^{\prime},\omega^{\prime\prime}]|[R^{\prime\prime},\omega_{R}^{\prime\prime}]\surd$;
notice that it is shorter than the above one. We now consider
$[T,\omega]|[R,\omega_{R}]\stackrel{{\scriptstyle}}{{\rightarrow}}[T_{i},\omega]|[R,\omega_{R}\\!\cdot\\!{l_{j}}]$.
We can now apply the inductive hypothesis on the shorter sequence
$[\mathcal{A}[{S_{kj}}]^{k\in
K},\omega]|[R,\omega_{R}\\!\cdot\\!l_{j}]\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}[S^{\prime},\omega^{\prime\prime}]|[R^{\prime\prime},\omega_{R}^{\prime\prime}]$,
because $T_{j}\operatorname{\leq}\mathcal{A}[{S_{kj}}]^{k\in K}$ and by
Proposition 2 also $[\mathcal{A}[{S_{kj}}]^{k\in
K},\omega]|[R,\omega_{R}\\!\cdot\\!l_{j}]$ is a correct composition. ∎∎
See 3.1
###### Proof.
If $S$ is not controllable, then the thesis trivially holds because
$T\sqsubseteq S$ for every $T$.
We now consider $S$ controllable, and we prove the thesis by showing that if
$T\operatorname{\leq}S$ then, for every $\omega$, $R$, and $\omega_{R}$ such
that $[S,\omega]|[R,\omega_{R}]$ is a correct composition, we have that the
following holds:
* •
if
$[T,\omega]|[R,\omega_{R}]\rightarrow[T^{\prime},\omega^{\prime}]|[R^{\prime},\omega_{R}^{\prime}]$
then there exists $S^{\prime}$ such that
$T^{\prime}\operatorname{\leq}S^{\prime}$ and
$[S^{\prime},\omega^{\prime}]|[R^{\prime},\omega_{R}^{\prime}]$ is a correct
composition.
The above implies the thesis because, given $T\operatorname{\leq}S$ and the
correct composition $[S,\epsilon]|[R,\epsilon]$, if there exists a computation
$[T,\epsilon]|[R,\epsilon]\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}[T^{\prime},\omega^{\prime}]|[R^{\prime},\omega_{R}^{\prime}]$,
we can apply the above result on each step of the computation to prove that
there exists $S^{\prime}$ such that $T^{\prime}\operatorname{\leq}S^{\prime}$
and $[S^{\prime},\omega^{\prime}]|[R^{\prime},\omega_{R}^{\prime}]$ is a
correct composition. Then, by Proposition 3, we have that there exist
$T^{\prime\prime}$, $\omega^{\prime\prime}$, $R^{\prime\prime}$, and
$\omega_{R}^{\prime\prime}$ such that
$[T^{\prime},\omega^{\prime}]|[R^{\prime},\omega_{R}^{\prime}]\stackrel{{\scriptstyle}}{{\rightarrow}}^{*}[T^{\prime\prime},\omega^{\prime\prime}]|[R^{\prime\prime},\omega_{R}^{\prime\prime}]$
and
$[T^{\prime\prime},\omega^{\prime\prime}]|[R^{\prime\prime},\omega_{R}^{\prime\prime}]\surd$.
We now prove the above result. The transition
$[T,\omega]|[R,\omega_{R}]\stackrel{{\scriptstyle}}{{\rightarrow}}[T^{\prime},\omega^{\prime}]|[R^{\prime},\omega_{R}^{\prime}]$
can be of four possible kinds:
1. 1.
the consumption of a message from the r.h.s. queue, i.e.
$[T,\omega]|[R,l\\!\cdot\\!\omega_{R}^{\prime}]\rightarrow[T,\omega]|[R^{\prime},\omega_{R}^{\prime}]$;
2. 2.
the insertion of a new message in the l.h.s. queue, i.e.
$[T,\omega]|[R,\omega_{R}]\rightarrow[T,\omega\\!\cdot\\!l]|[R^{\prime},\omega_{R}]$;
3. 3.
the consumption of a message from the l.h.s. queue, i.e.
$[T,l\\!\cdot\\!q^{\prime}]|[R,\omega_{R}]\rightarrow[T^{\prime},\omega^{\prime}]|[R,\omega_{R}]$;
4. 4.
the insertion of a new message in the r.h.s. queue, i.e.
$[T,\omega]|[R,\omega_{R}]\rightarrow[T^{\prime},\omega]|[R,\omega_{R}\\!\cdot\\!l]$.
In the first two cases, we simply observe that there exists also
$[S,\omega]|[R,l\\!\cdot\\!\omega_{R}^{\prime}]\rightarrow[S,\omega]|[R^{\prime},\omega_{R}^{\prime}]$
(resp.
$[S,\omega]|[R,\omega_{R}]\rightarrow[S,\omega\\!\cdot\\!l]|[R^{\prime},\omega_{R}]$),
that $T\operatorname{\leq}S$, and also
$[S,\omega]|[R^{\prime},\omega_{R}^{\prime}]$ (resp.
$[S,\omega\\!\cdot\\!l]|[R^{\prime},\omega_{R}]$) is a correct composition
because reachable from the correct composition
$[S,\omega]|[R,l\\!\cdot\\!\omega_{R}^{\prime}]$ (resp.
$[S,\omega]|[R,\omega_{R}]$).
In the third case we have that $\mathsf{unfold}(T)$ starts with an input
branching. Given that $T\operatorname{\leq}S$, and $S$ is controllable, also
$\mathsf{unfold}(S)$ must start with an input branching, i.e.
$\mathsf{unfold}(S)=\&\\{{l}_{i}:{S}_{i}\\}_{i\in I}$. By definition of
$\operatorname{\leq}$ we have that
$\mathsf{unfold}(T)=\&\\{{l}_{j}:{T}_{j}\\}_{j\in J}$, $J\supseteq K$, and
$\forall k\in K\ldotp T_{k}\operatorname{\leq}S_{k}$, where $K=\\{k\in
I\;|\;S_{k}\text{ is controllable}\\}$. Given that
$[S,l\\!\cdot\\!q^{\prime}]|[R,\omega_{R}]$ is a correct composition, there
exists $i\in I$ s.t. $l=l_{i}$ and
$[S,l\\!\cdot\\!q^{\prime}]|[R,\omega_{R}]\stackrel{{\scriptstyle}}{{\rightarrow}}[S_{i},\omega^{\prime}]|[R,\omega_{R}]$.
The former configuration is a correct composition, hence also the latter is
such. This implies, by Lemma 6, that $S_{i}$ is controllable, hence $i\in K$
and also $i\in J$. Thus, we have
$[T,l\\!\cdot\\!q^{\prime}]|[R,\omega_{R}]\stackrel{{\scriptstyle}}{{\rightarrow}}[T_{i},\omega^{\prime}]|[R,\omega_{R}]$,
with $T_{i}\operatorname{\leq}S_{i}$. We conclude this case by observing again
that $[S_{i},\omega^{\prime}]|[R,\omega_{R}]$ is a correct composition in that
reachable from the correct composition
$[S,l\\!\cdot\\!q^{\prime}]|[R,\omega_{R}]$.
In the fourth and last case, we have that $\mathsf{unfold}(T)$ starts with an
output selection, and $T^{\prime}$ is the continuation in the branch with
label $l$. Given that $T\operatorname{\leq}S$, and $S$ is controllable, we
have $\mathsf{selUnfold}(S)=\mathcal{A}[{\oplus\\{{l}_{j}:{S_{k}}_{j}\\}_{j\in
I}}]^{k\in K}$, and $T^{\prime}\operatorname{\leq}S_{km}$, for every $k\in K$
and some $m\in I$ such that $l_{m}=l$. It remains to show that
$[\mathcal{A}[{{S_{km}}}]^{k\in K},\omega]|[R,\omega_{R}\\!\cdot\\!l]$ is a
correct composition, but this follows from Proposition 2 and the fact that
$[\mathcal{A}[{\oplus\\{{l}_{j}:{S_{k}}_{j}\\}_{j\in I}}]^{k\in
K},\omega]|[R,\omega_{R}]$, with $l=l_{m}$ for some $m\in I$, is a correct
composition. In fact
$\mathsf{selUnfold}(S)=\mathcal{A}[{\oplus\\{{l}_{j}:{S_{k}}_{j}\\}_{j\in
I}}]^{k\in K}$ and $[S,\omega]|[R,\omega_{R}]$ is a correct composition. ∎∎
### 0.A.4 Undecidability of Fair Asynchronous Subtyping
Also the proof of undecidability of fair asynchronous subtyping exploits queue
machines. In this case we reduce the problem of checking the nontermination of
a queue machine to the problem of checking subtyping between two session
types. In Definition 8 we have defined
$(q,\gamma)\rightarrow_{M}(q^{\prime},\gamma^{\prime})$ denoting computation
steps of a queue machine. We have that one queue machine $M$ terminates if and
only if there exists a configuration with empty queue that is reachable from
the initial configuration, i.e.,
$(s,\$)\rightarrow_{M}^{*}(q^{\prime},\epsilon)$. This holds because the
transition function is total in queue machines, hence if the queue is not
empty there is always a possible transition. In case the queue machine does
not terminate, we have that
$(q,\$)\rightarrow_{M}^{*}(q^{\prime},\gamma^{\prime})$ implies the existence
of an additional computation step
$(q^{\prime},\gamma^{\prime})\rightarrow_{M}(q^{\prime\prime},\gamma^{\prime\prime})$.
Given a queue machine $M=(Q,\Sigma,\Gamma,\$,s,\delta)$ and an additional
ending symbol $E\not\in\Gamma$, we now define the types
$T=[\\![\\![{M,\\_,E}]\\!]\\!]$ and $S=[\\![\\![{M,E}]\\!]\\!]$ in such a way
that $M$ does not terminate if and only if $T\operatorname{\leq}S$. The
encodings $[\\![\\![{M,\\_,E}]\\!]\\!]$ and $[\\![\\![{M,E}]\\!]\\!]$ are
similar to the corresponding encodings $[\\![M,q_{f},E]\\!]$ and
$[\\![M,E]\\!]$ defined in Definitions 15 and 16, but with the following
differences:
* •
there is no specific target state $q_{f}$;
* •
the encoding $[\\![\\![{M,E}]\\!]\\!]$ starts with an input branching with
only one branch labeled with the initial queue symbol $\$$ and continuation
corresponding to the producer/consumer $[\\![M,E]\\!]$ as defined in
Definition 16;
* •
in order to be a potential subtype of $S=[\\![\\![{M,E}]\\!]\\!]$, all of the
output selections in $T=[\\![\\![{M,\\_,E}]\\!]\\!]$ must have branchings for
all of the symbols in $\Gamma\cup\\{E\\}$ (because these are the labels in the
output selection in the potential supertype); among all of these branchings
only one will be consistent with the encoding of the finite control, while the
continuations in the other branchings are guaranteed to be always good
subtypes (this is guaranteed by a type that nondeterministically produces
symbols, and that after producing the ending symbol $E$ it is able to
recursively consume all possible symbols in $\Gamma$, and than become
$\mathbf{end}$ after consuming the ending symbol $E$).
###### Definition 19 (New Finite Control Encoding).
Let $M=(Q,\Sigma,\Gamma,\$,s,\delta)$ be a queue machine and let
$E\not\in\Gamma$ be the additional ending symbol. We define
$[\\![\\![{M,\\_,E}]\\!]\\!]$ as follows:
$[\\![\\![{M,\\_,E}]\\!]\\!]\ =[\\![\\![{s}]\\!]\\!]^{\emptyset}$
with, given $q\in Q$ and $\mathcal{S}\subseteq Q$,
$[\\![\\![{q}]\\!]\\!]^{\mathcal{S}}$ is defined as follows:
$\begin{array}[]{l}[\\![\\![{q}]\\!]\\!]^{\mathcal{S}}=\left\\{\begin{array}[]{l}\mu\mathbf{q}.\&\\{{A}\\!:\\!{{\\{\\!\\!\\{{B^{A}_{1}\cdots
B^{A}_{n_{A}}}\\}\\!\\!\\}}_{q^{\prime}}^{\mathcal{S}\cup\\{q\\}}}\\}_{A\in\Gamma}\\\\[2.84526pt]
\hskip 25.6073pt\text{if }q\not\in{\mathcal{S}}\text{ and
}\delta(q,A)=(q^{\prime},B^{A}_{1}\cdots B^{A}_{n_{A}})\\\ \\\
\mathbf{q}\qquad\mbox{if $q\in{\mathcal{S}}$}\end{array}\right.\end{array}$
where
$\begin{array}[]{l}{\\{\\!\\!\\{{B_{1}\cdots
B_{m}}\\}\\!\\!\\}}_{r}^{\mathcal{T}}\\!=\\!\left\\{\\!\\!\begin{array}[]{ll}\\!{[\\![\\![{r}]\\!]\\!]}^{\mathcal{T}}&\text{if
}m=0\\\
\begin{array}[]{ll}\\!\\!\\!\\!\oplus&\\!\\!\\!\\!\big{(}\big{\\{}B_{1}:{\\{\\!\\!\\{{B_{2}\ldots
B_{m}}\\}\\!\\!\\}}_{r}^{\mathcal{T}}\big{\\}}\cup\\\
&\\!\big{\\{}{A:T}\big{\\}}_{A\in\Gamma\setminus\\{B_{1}\\}}\cup\\{E:T^{\prime}\\}\big{)}\end{array}&\text{otherwise}\end{array}\right.\end{array}$
with
$T=\mu\mathbf{\mathbf{t}}.\big{(}\oplus\\{{A}:{\mathbf{t}}\\}_{A\in\Gamma}\cup\\{E:T^{\prime}\\}\big{)}$
and
$T^{\prime}=\mu\mathbf{\mathbf{t}}.\big{(}\&\\{{A}\\!:\\!{\mathbf{t}}\\}_{A\in\Gamma}\cup\\{E:\mathbf{end}\\}\big{)}$.
###### Definition 20 (New Producer/consumer).
Let $M=(Q,\Sigma,\Gamma,\$,s,\delta)$ be a queue machine and $E\not\in\Gamma$
be the ending symbol. We define $[\\![\\![{M,E}]\\!]\\!]$ as
$[\\![\\![{M,E}]\\!]\\!]=\&\\{\$:[\\![M,E]\\!]\\}$
with $[\\![M,E]\\!]$ as defined in Definition 16.
We now prove that the two above types $T=[\\![\\![{M,\\_,E}]\\!]\\!]$ and
$S=[\\![\\![{M,E}]\\!]\\!]$ are such that $T\operatorname{\leq}S$ if and only
if the machine $M$ does not terminate.
###### Theorem 0.A.4.
Given a queue machine $M$ and the ending symbol $E$, consider
$T=[\\![\\![{M,\\_,E}]\\!]\\!]$ and $S=[\\![M,E]\\!]$. We have that
$T\operatorname{\leq}S$ if and only if $M$ does not terminate.
###### Proof.
We first consider the only-if part, proving the contrapositive statement, that
is, if the queue machine $M$ terminates then
$T\not\\!\\!\operatorname{\leq}S$. If the queue machine terminates, we have
that $(s,\$)\rightarrow_{M}^{*}(q^{\prime},\epsilon)$. Consider now the pair
of types $(T,S)$ with $T=[\\![\\![{M,\\_,E}]\\!]\\!]$ and $S=[\\![M,E]\\!]$.
If, by contradiction, $T\operatorname{\leq}S$, we have that by definition
Definition 12 there exists a fair asynchronous subtyping relation
$\mathcal{R}$ such that $(T,S)\in\mathcal{R}$. We now show that, by definition
of fair asynchronous subtyping relation, $\mathcal{R}$ will have to include
other pairs of types $(T^{\prime\prime},S^{\prime\prime})$ corresponding with
configurations $(q^{\prime\prime},\gamma^{\prime\prime})$ reachable in the
queue machine $M$. Consider the type $T$. It starts with an input branching,
including the initial queue symbol $\$$. Then it has a sequence of output
selections, including the sequence of symbols to be emitted by the queue
machine after having consumed $\$$. Consider now the type $S$. It starts with
an input branching with only label $\$$, followed by an output selection on
all symbols, including label $E$ having continuation $\&\\{E:\mathbf{end}\\}$.
The latter ensures that $S$ is controllable. If we consider the the
constraints imposed by the Definition 12 on fair asynchronous subtyping
relations, we can conclude that $\mathcal{R}$ should contain a pair of types
$(T^{\prime},S^{\prime})$ where $T^{\prime}$ is the type corresponding to the
new state of the queue machine (reached after the above sequence of output
selections, including the sequence of symbols to be emitted by the queue
machine after having consumed $\$$) and $S^{\prime}$ is like $S$, with the
difference that before the output selection there is a sequence of input
branchings, each one with only one label, corresponding with the symbols in
the queue after the first computation step. This reasoning can be repeatedly
applied to prove that $\mathcal{R}$ should also contain other pairs of types
$(T^{\prime\prime},S^{\prime\prime})$, one for each configuration
$(q^{\prime\prime},\gamma^{\prime\prime})$ reachable in the queue machine $M$.
Consider the pair $(T_{f},S_{f})\in\mathcal{R}$ corresponding to the
terminating configuration $(q^{\prime},\epsilon)$. The type $T_{f}$, as all
the types representing states in the queue machine, starts with an input
branching. The type $S_{f}$, on the other hand, represents the empty queue, so
it is like $[\\![M,E]\\!]$, i.e., there are no input branchings before the
output selection. This means that $(T_{f},S_{f})$ does not satisfy the item
for input selection in Definition 12. Hence $\mathcal{R}$ cannot be a fair
asynchronous subtyping, but this contradicts the above initial assumption.
We now move to the if part. Assume that the queue machine $M$ does not
terminate. We show that there exists a fair asynchronous subtyping relation
$\mathcal{R}$ that contains the pair $(T,S)$, hence $T\operatorname{\leq}S$.
There are two kinds of pairs in $\mathcal{R}$: (i) the pairs discussed in the
above only-if part of the proof that corresponds to branches of the types
involved in reproducing the computation of the queue machine $M$, and (ii)
other pairs corresponding to alternative branches. The pairs of type (i)
satisfy the constraints imposed by Definition 12 because output selections of
the l.h.s. type can always be mimicked by the r.h.s. type (that always include
an output selection after a sequence of input branchings with only one label),
and input branchings can always be mimicked by the r.h.s. type because the
queue is always non-empty in the queue machine computation. Also the pairs of
type (ii) satisfy the constraints imposed by Definition 12. In fact, these
pairs are generated considering the alternative branches in
${\\{\\!\\!\\{{B_{1}\cdots B_{m}}\\}\\!\\!\\}}_{r}^{\mathcal{T}}$ in
Definition 19. The l.h.s. type in the pairs $(T^{\prime},S^{\prime})$
associated with these branches, are of two kinds: (a) they are able to
recursively perform all possible outputs until the label $E$ is selected, or
(b) they are able to recursively perform all possible inputs until the label
$E$ is selected. In the first case (a), the constraints in Definition 12 are
satisfied because the r.h.s. type is always able to mimick output selections
(see the above observation). In the second case (b), we have that the output
$E$ has been previously selected by the last pair of kind (a) considered.
Hence, the r.h.s. type is a sequence of input branchings, with only one label,
where all inputs excluding the last one are different from $E$, and the last
one having label $E$ has continuation $\mathbf{end}$. This guarantees that all
these pairs satisfy the constraints in Definition 12, under the assumption
that also a final pair $(\mathbf{end},\mathbf{end})$ belongs to $\mathcal{R}$.
We have that $(T,S)\in\mathcal{R}$ in that this is the first pair of the kind
(i) above. Hence we can conclude that $T\operatorname{\leq}S$. ∎∎
See 3.2
###### Proof.
Direct consequence of Theorem 0.A.4. ∎∎
### 0.A.5 Soundness of the Algorithm w.r.t. Fair Asynchronous Subtyping
The soundness of the algorithm reported in Section 4 w.r.t. fair asynchronous
session subtyping relies on Theorem 4.1 that guarantees that given a _witness
tree_ $(N,n_{0},\twoheadrightarrow,\lambda)$ such that $\lambda(n_{0})=(T,S)$,
then $T\operatorname{\leq}S$.
The definition of witness tree consider nestings of input contexts
$\mathcal{A}$. In the proof of Theorem 4.1 we need the notation
$\mathcal{A}^{h}\lfloor S_{j}\rfloor^{j\in J}$, to generalize to nestings of
input contexts with parametric depth, defined as follows:
* •
$\mathcal{A}^{1}\lfloor S_{j}\rfloor^{j\in J}$ is $\mathcal{A}\lfloor
S_{j}\rfloor^{j\in J}$
* •
$\mathcal{A}^{h}\lfloor S_{j}\rfloor^{j\in J}$ is
$\mathcal{A}\langle\mathcal{A}^{h-1}\lfloor S_{j}\rfloor^{j\in J}\rangle^{J}$,
when $h>1$.
Given a witness tree for $\mathcal{A}$, we define a family of isomorphic trees
with labels in which the r.h.s. type has incrementally increased nestings of
the input context $\mathcal{A}$ in the growing holes.
###### Definition 21 ($h$-th Witness Tree).
Given a witness tree $\mathcal{T}=(N,n_{0},\twoheadrightarrow,\lambda)$ for
$\mathcal{A}$, and $h\geq 1$, we inductively define $\mathcal{T}^{h}$ as
follows:
* •
$\mathcal{T}^{1}=\mathcal{T}$;
* •
for $h>1$, given
$\mathcal{T}^{h-1}=(N^{h-1},n_{0}^{h-1},\twoheadrightarrow^{h-1},\lambda^{h-1})$
we define
$\mathcal{T}^{h}=(N^{h},n_{0}^{h},\twoheadrightarrow^{h},\lambda^{h})$ with
$N^{h}=N^{h-1}$, $n_{0}^{h}=n_{0}^{h-1}$,
$\twoheadrightarrow^{h}=\twoheadrightarrow^{h-1}$, and
$\lambda^{h}(n)=\mathcal{A}^{\prime}\langle\mathcal{A}^{h}\lfloor
S_{j}\rfloor^{j\in J}\rangle^{J}\lfloor S_{k}\rfloor^{k\in K}$ if
$\lambda^{h-1}(n)=\mathcal{A}^{\prime}\langle\mathcal{A}^{h-1}\lfloor
S_{j}\rfloor^{j\in J}\rangle^{J}\lfloor S_{k}\rfloor^{k\in K}$.
We now prove that given a witness subtree $\mathcal{T}$ of a simulation tree,
all the trees in the family $\mathcal{T}^{h}$ faithfully represent the
subtyping simulation game.
###### Lemma 7.
Consider a witness tree
$\mathcal{T}^{1}=(N^{1},n_{0}^{1},\twoheadrightarrow^{1},\lambda^{1})$
contained in a simulation tree. For every $h\geq 1$, we have that
$\twoheadrightarrow^{h}$ in
$\mathcal{T}^{h}=(N^{h},n_{0}^{h},\twoheadrightarrow^{h},\lambda^{h})$ is
compatible with the subtyping simulation game, i.e.,
$n\twoheadrightarrow^{h}n^{\prime}$ is present in $T^{h}$ if and only if there
exists a simulation tree $(M,m_{0},\twoheadrightarrow,\lambda)$ including
$m\twoheadrightarrow^{h}m^{\prime}$ with $\lambda(m)=\lambda^{h}(n)$ and
$\lambda(m^{\prime})=\lambda^{h}(n^{\prime})$.
###### Proof.
We proceed by induction. If $h=1$, the thesis directly follows from the fact
that $\mathcal{T}^{1}$ is contained in a simulation tree.
If $h>1$, by inductive hypothesis we have that the thesis holds for
$\mathcal{T}^{h-1}$. We prove that the thesis holds also for $\mathcal{T}^{h}$
showing that there exists a simulation tree including
$m\twoheadrightarrow{}m^{\prime}$ with $m^{\prime}$ labeled with
$(T^{\prime},\mathcal{A}^{\prime\prime}\langle\mathcal{A}^{v^{\prime}}\lfloor
S^{\prime}_{j}\rfloor^{j\in J}\rangle^{J}\lfloor S^{\prime}_{k}\rfloor^{k\in
K})$ if and only if there exists a simulation tree including
$t\twoheadrightarrow{}t^{\prime}$ with $t^{\prime}$ labeled with
$(T^{\prime},\mathcal{A}^{\prime\prime}\langle\mathcal{A}^{v^{\prime}+1}\lfloor
S^{\prime}_{j}\rfloor^{j\in J}\rangle^{J}\lfloor S^{\prime}_{k}\rfloor^{k\in
K})$. The proof is by case analysis, considering the three possible steps in
the subtyping simulation game at the basis of the definition of
$\twoheadrightarrow{}$.
If $T$ starts with a recursive definition, the thesis trivially holds because
$\twoheadrightarrow{}$ simply modify the l.h.s. type by unfolding its initial
recursion and leaves the r.h.s. type unchanged.
If $T$ starts with an input branching, by Definition 12 we have that the
r.h.s. type contains an entire context $\mathcal{A}$ in its growing holes. We
initially consider $m\twoheadrightarrow{}m^{\prime}$ with $m^{\prime}$ labeled
with
$(T^{\prime},\mathcal{A}^{\prime\prime}\langle\mathcal{A}^{v^{\prime}}\lfloor
S^{\prime}_{j}\rfloor^{j\in J}\rangle^{J}\lfloor S^{\prime}_{k}\rfloor^{k\in
K})$. This means that by applying $\mathsf{unfold}($) to the r.h.s. type we
obtain an input context starting with an input branching satisfying the
constraints imposed by Definition 12. The step of the subtyping simulation
game corresponding to $m\twoheadrightarrow{}m^{\prime}$ selects a branch of
the input branching such that its continuation
$\mathcal{A}^{\prime\prime}\langle\mathcal{A}^{v^{\prime}}\lfloor
S^{\prime}_{j}\rfloor^{j\in J}\rangle^{J}\lfloor S^{\prime}_{k}\rfloor^{k\in
K}$ is controllable. Now consider $t$ with label
$(T,\mathcal{A}^{\prime}\langle\mathcal{A}^{v+1}\lfloor S_{j}\rfloor^{j\in
J}\rangle^{J}\lfloor S_{k}\rfloor^{k\in K})$. The application of
$\mathsf{unfold}($) modifies the outer context in the same way thus obtaining
a type starting with the same input branching, simply with an additional
nesting of $\mathcal{A}$ in the holes in $J$. The continuation
$\mathcal{A}^{\prime\prime}\langle\mathcal{A}^{v^{\prime}+1}\lfloor
S^{\prime}_{j}\rfloor^{j\in J}\rangle^{J}\lfloor S^{\prime}_{k}\rfloor^{k\in
K}$ is also controllable because it is an input contexts with the set of
indexed holes, hence the same set of types $S^{\prime}_{j}$ and
$S^{\prime}_{k}$. Hence it is possible to apply a corresponding step in the
subtyping simulation game $t\twoheadrightarrow{}t^{\prime}$ with $t^{\prime}$
labeled with
$(T^{\prime},\mathcal{A}^{\prime\prime}\langle\mathcal{A}^{v^{\prime}+1}\lfloor
S^{\prime}_{j}\rfloor^{j\in J}\rangle^{J}\lfloor S^{\prime}_{k}\rfloor^{k\in
K})$. Notice that the same reasoning can be applied assuming that
$t\twoheadrightarrow{}t^{\prime}$ with $t^{\prime}$ labeled with
$(T^{\prime},\mathcal{A}^{\prime\prime}\langle\mathcal{A}^{v^{\prime}+1}\lfloor
S^{\prime}_{j}\rfloor^{j\in J}\rangle^{J}\lfloor S^{\prime}_{k}\rfloor^{k\in
K})$ to prove that there exists also the corresponding step in the subtyping
simulation game $m\twoheadrightarrow{}m^{\prime}$. In this case we use the
assumption that in the growing holes of the r.h.s. type of the label of $m$ we
have an entire context $\mathcal{A}$, thus guaranteeing the presence of the
same $S^{\prime}_{j}$ in all the continuations of the initial input branching
present in the outer context.
If $T$ starts with an output selection, we initially consider
$m\twoheadrightarrow{}m^{\prime}$ with $m^{\prime}$ labeled with
$(T^{\prime},\mathcal{A}^{\prime\prime}\langle\mathcal{A}^{v^{\prime}}\lfloor
S^{\prime}_{j}\rfloor^{j\in J}\rangle^{J}\lfloor S^{\prime}_{k}\rfloor^{k\in
K})$. This means that by applying $\mathsf{selUnfold}()$ to the r.h.s. type we
obtain an input context filled with types starting with output selections
satisfying the constraints imposed by Definition 12. Notice that the
application of $\mathsf{selUnfold}()$ to the outer input context does not
remove holes, but at most replicates some of them. Moreover, the application
of $\mathsf{selUnfold}()$ applies to the innermost types $S_{j}$ and $S_{k}$
by unfolding the variables inside outputs replacing them with their
definitions (already present in $S_{j}$ and $S_{k}$ given that these are
closed terms). The considered step in the subtyping simulation game modifies
(the unfoldings of) $S_{j}$ and $S_{k}$ by resolving initial output
selections, thus obtaining $S_{j}^{\prime}$ and $S_{k}^{\prime}$. Now consider
$t$ with label $(T,\mathcal{A}^{\prime}\langle\mathcal{A}^{v+1}\lfloor
S_{j}\rfloor^{j\in J}\rangle^{J}\lfloor S_{k}\rfloor^{k\in K})$. What we have
just observed about the step $m\twoheadrightarrow{}m^{\prime}$ of subtyping
simulation game, holds also for this new pair of types. The application of
$\mathsf{selUnfold}()$ respectively modifies the outer input context and the
inner types $S_{j}$ and $S_{k}$ in the same way, and also the same resolution
of the initial output selections in $S_{j}$ and $S_{k}$ is possible. Hence
there exists $t\twoheadrightarrow{}t^{\prime}$ with $t^{\prime}$ labeled with
$(T^{\prime},\mathcal{A}^{\prime\prime}\langle\mathcal{A}^{v^{\prime}+1}\lfloor
S^{\prime}_{j}\rfloor^{j\in J}\rangle^{J}\lfloor S^{\prime}_{k}\rfloor^{k\in
K})$. Notice that the same reasoning can be applied assuming that
$t\twoheadrightarrow{}t^{\prime}$ with $t^{\prime}$ labeled with
$(T^{\prime},\mathcal{A}^{\prime\prime}\langle\mathcal{A}^{v^{\prime}+1}\lfloor
S^{\prime}_{j}\rfloor^{j\in J}\rangle^{J}\lfloor S^{\prime}_{k}\rfloor^{k\in
K})$ to prove that there exists also the corresponding step in the subtyping
simulation game $m\twoheadrightarrow{}m^{\prime}$. ∎∎
We now prove a key result, namely, that given a witness subtree $\mathcal{T}$
of a simulation tree, we have that all branches in the simulation tree that
traverse $\mathcal{T}$ follows paths also present in the family of trees
$\mathcal{T}^{h}$ or in simulation trees
$\mathit{simtree}(T^{\prime},S^{\prime})$ where $(T^{\prime},S^{\prime})$ is a
leaf of $\mathcal{T}$ for which we know that
$T^{\prime}\operatorname{\leq}S^{\prime}$.
###### Proposition 4.
Let $T$ and $S$ be two session types with
$\mathit{simtree}(T,S)=(N,n_{0},\twoheadrightarrow,\lambda)$. If
$\mathit{simtree}(T,S)$ contains a witness tree $\mathcal{T}$ with root $n$,
then for every node $n^{\prime}\in N$ such that
$n\twoheadrightarrow\\!\\!{}^{*}\,n^{\prime}$ we have that
$\lambda(n^{\prime})$ is a label present either in $\mathcal{T}^{h}$, for some
$h$, or in
$\mathit{simtree}(T^{\prime},S^{\prime})=(N^{\prime},n_{0}^{\prime},\twoheadrightarrow,\lambda^{\prime})$
with $T^{\prime}\operatorname{\leq}S^{\prime}$.
###### Proof.
We proceed by induction on the length of
$n\twoheadrightarrow\\!\\!{}^{*}\,n^{\prime}$.
If the length is 0, then $n^{\prime}$ is the root of $\mathcal{T}$ hence its
label is obviously in $\mathcal{T}^{1}$.
If the length is greater than 1, consider
$n\twoheadrightarrow\\!\\!{}^{*}\,n^{\prime\prime}\twoheadrightarrow{}n^{\prime}$.
By inductive hypothesis we have that $\lambda(n^{\prime\prime})$ is a label
present either in $\mathcal{T}^{h}$, for some $h$, or in
$\mathit{simtree}(T^{\prime},S^{\prime})=(N^{\prime},n_{0}^{\prime},\twoheadrightarrow,\lambda^{\prime})$
with $T^{\prime}\operatorname{\leq}S^{\prime}$.
We start from the latter case, i.e., there exists $m^{\prime\prime}$ in
$\mathit{simtree}(T^{\prime},S^{\prime})=(N^{\prime},n_{0}^{\prime},\twoheadrightarrow,\lambda^{\prime})$
such that $\lambda^{\prime}(m^{\prime\prime})=\lambda(n^{\prime\prime})$. We
have that there exists $m^{\prime\prime}\twoheadrightarrow{}m^{\prime}$ in
$\mathit{simtree}(T^{\prime},S^{\prime})$ s.t.
$\lambda^{\prime}(m^{\prime})=\lambda(n^{\prime})$.
We now consider the former case, i.e., there exists one node in
$\mathcal{T}^{h}$, for some $h$, labeled with $\lambda(n^{\prime\prime})$. Let
$m^{\prime\prime}$ be such node. There are two possibilities, either (i) the
node $m^{\prime\prime}$ is a leaf in $\mathcal{T}^{h}$, or (ii) it is not a
leaf. In the case (ii) we have that $\mathcal{T}^{h}$ contains
$m^{\prime\prime}\twoheadrightarrow{}m^{\prime}$, with $m^{\prime}$ labeled
with $\lambda(n^{\prime})$. If $m^{\prime\prime}$ is a leaf, we consider the
four kinds of leaves separately.
If $m^{\prime\prime}$ is a leaf of type 2a, then there exists an ancestor
$m^{\prime\prime\prime}$ of $m^{\prime\prime}$ in $\mathcal{T}^{h}$ with the
same label $\lambda(n^{\prime\prime})$. Given that the ancestor is not a leaf,
$\mathcal{T}^{h}$ contains
$m^{\prime\prime\prime}\twoheadrightarrow{}m^{\prime}$, with $m^{\prime}$
labeled with $\lambda(n^{\prime})$.
If $m^{\prime\prime}$ is a leaf of type 2b in $\mathcal{T}$, we have
$\lambda(n^{\prime\prime})=$ $(T^{\prime},\mathcal{A}^{h+1}\lfloor
S_{j}\rfloor^{j\in J}\lfloor S_{k}\rfloor^{k\in K})$. The node
$n^{\prime\prime}$ has an ancestor $n^{\prime\prime\prime}$ in
$\mathcal{T}^{h}$ s.t.
$\lambda(n^{\prime\prime\prime})=(T^{\prime},\mathcal{A}^{h}\lfloor
S_{j}\rfloor^{j\in J}\lfloor S_{k}\rfloor^{k\in K})$. Consider now the
corresponding node $m^{\prime\prime\prime}$ in $\mathcal{T}^{h+1}$. We have
that $m^{\prime\prime\prime}$ is labeled with
$(T^{\prime},\mathcal{A}^{h+1}\lfloor S_{j}\rfloor^{j\in J}\lfloor
S_{k}\rfloor^{k\in K})=\lambda(n^{\prime\prime})$. Given that
$m^{\prime\prime\prime}$ is not a leaf, $\mathcal{T}^{h+1}$ contains
$m^{\prime\prime\prime}\twoheadrightarrow{}m^{\prime}$, with $m^{\prime}$
labeled with $\lambda(n^{\prime})$.
If $m^{\prime\prime}$ is a leaf of type 2c in $\mathcal{T}$, we have
$\lambda(n^{\prime\prime})=(T^{\prime},\mathcal{A}^{h}\lfloor
S_{j}\rfloor^{j\in J}\lfloor S_{k}\rfloor^{k\in K})$. We have two cases. If
$h=1$, by definition of witness tree,
$T^{\prime}\operatorname{\leq}\mathcal{A}^{h}\lfloor S_{j}\rfloor^{j\in
J}\lfloor S_{k}\rfloor^{k\in K}$. The node $n^{\prime\prime}$ has the same
label as the root of $\mathit{simtree}(T^{\prime},\mathcal{A}^{h}\lfloor
S_{j}\rfloor^{j\in J}\lfloor S_{k}\rfloor^{k\in K})$. Hence such a simulation
tree includes a transition from its root to a node labeled with
$\lambda(n^{\prime})$. If $h>1$ the node $n^{\prime\prime}$ has an ancestor
$n^{\prime\prime\prime}$ in $\mathcal{T}^{h}$ such that
$\lambda(n^{\prime\prime\prime})=(T^{\prime},\mathcal{A}^{h+1}\lfloor
S_{j}\rfloor^{j\in J}\lfloor S_{k}\rfloor^{k\in K})$. Consider now the
corresponding node $m^{\prime\prime\prime}$ in $\mathcal{T}^{h-1}$. We have
that $m^{\prime\prime\prime}$ is labeled with
$(T^{\prime},\mathcal{A}^{h}\lfloor S_{j}\rfloor^{j\in J}\lfloor
S_{k}\rfloor^{k\in K})=\lambda(n^{\prime\prime})$. Given that
$m^{\prime\prime\prime}$ is not a leaf, $\mathcal{T}^{h-1}$ contains
$m^{\prime\prime\prime}\twoheadrightarrow{}m^{\prime}$, with $m^{\prime}$
labeled with $\lambda(n^{\prime})$.
If $m^{\prime\prime}$ corresponds to leaf of type 2d in $\mathcal{T}$, we have
that the label $\lambda(n^{\prime\prime})$ of $m^{\prime\prime}$ is the same
as the label in the corresponding node in $\mathcal{T}$, i.e.
$(T^{\prime},\mathcal{A}^{\prime}[S_{k}]^{k\in K^{\prime}})$. In fact labels
of the leaves of type 2d in $\mathcal{T}$ do not change when moving to
$\mathcal{T}^{h}$. This because the input context $\mathcal{A}^{\prime}$ does
not include growing holes. By definition of witness tree we have that
$T^{\prime}\operatorname{\leq}\mathcal{A}^{\prime}[S_{k}]^{k\in K^{\prime}}$.
The node $n^{\prime\prime}$ has the same label as the root of
$\mathit{simtree}(T^{\prime},\mathcal{A}^{\prime}[S_{k}]^{k\in K^{\prime}})$.
Hence such a simulation tree includes a transition from its root to a node
labeled with $\lambda(n^{\prime})$. ∎∎
We can now conclude with the main result needed to prove the soundness of the
algorithm proposed in Section 4. See 4.1
###### Proof.
Let $\mathcal{T}$ be the witness subtree with root in $n$. By Proposition 4 we
have that $\lambda(n^{\prime})$ is a label present either in
$\mathcal{T}^{h}$, for some $h$, or in
$\mathit{simtree}(T^{\prime},S^{\prime})=(N^{\prime},n_{0}^{\prime},\twoheadrightarrow,\lambda^{\prime})$
with $T^{\prime}\operatorname{\leq}S^{\prime}$. In the latter case the thesis
trivially holds because all nodes $m^{\prime}$ in
$\mathit{simtree}(T^{\prime},S^{\prime})$ are either successful or there
exists $m^{\prime}\twoheadrightarrow{}m^{\prime\prime}$. In the former case
there are two cases: either there exists an intermediary node (non-leaf) in
one $\mathcal{T}^{h}$, for some $h$, labeled with $\lambda(n^{\prime})$ is an
intermediary, or such a node can be only in leaf positions. In the first case
the thesis trivially holds because all intermediary nodes have successors. The
second case can occur only for leaves of type 2c in $\mathcal{T}$, or
corresponding to leaves of type 2d in $\mathcal{T}$. Both cases imply that
$\lambda(n^{\prime})=(T^{\prime},S^{\prime})$ with
$T^{\prime}\operatorname{\leq}S^{\prime}$. Hence $n^{\prime}$ has the same
label as the root of $\mathit{simtree}(T^{\prime},S^{\prime})$ and, as above,
the thesis trivially holds because all nodes $m^{\prime}$ in
$\mathit{simtree}(T^{\prime},S^{\prime})$ are either successful or there
exists $m^{\prime}\twoheadrightarrow{}m^{\prime\prime}$. ∎∎
## Appendix 0.B Example of tool outputs
Below we give examples of diagrams generate by our tool [tool]. We note that
the tool internally uses automata to represent session types. In addition, we
use strong bisimilarity of automata instead of equality between session types,
see the two right-most nodes in Figure 9 for an example of two bisimilar
configurations.
In the simulation and witness trees each configuration (box) contains the
following information. The label (top) of the box is node-id: state-l.h.s
(tag) where node-id is the unique identifier of the configuration in the tree,
state-l.h.s refers to the state of the candidate subtype (e.g., $T$ in
Definition 13). The tag is used internally to identify marked nodes, see Step
S2, where K means “keep” and R means (potentially) “remove”. The automaton
under the box label represents the candidate super-type, i.e. $S$ in
Definition 13, where the diamond-shaped node is the starting state.
### 0.B.1 Ground stations example — Figure 1
This example generates a simulation tree that can be completely pruned in Step
S2, hence there is no witness tree to check.
|
---|---
Refined ground station ($T^{\prime}_{G}$) | Ground station ($T_{G}$)
Figure 6: Input session types (as communicating finite-state machines). Figure
7: Simulation tree for Figure 6.
### 0.B.2 Spacecraft example — Figures 1 and 3
|
---|---
Spacecraft ($T_{S}$) | Older spacecraft ($T^{\prime}_{S}$)
Figure 8: Input session types (as communicating finite-state machines). Figure
9: Simulation tree for Figure 8. Figure 10: Witness tree for Figure 8, subtree
of Figure 9.
### 0.B.3 Additional example
|
---|---
Subtype | Supertype
Figure 11: Input session types. Figure 12: Simulation tree for Figure 11.
Figure 13: Witness tree for Figure 11, subtree of Figure 12.
|
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